Geometry of Membrane Sigma Models by Jan Vysok´y

A thesis submitted in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY IN MATHEMATICS (Double Doctoral Agreement between Jacobs University and Czech Technical University)

Date of Defense: 15th July, 2015 Approved Dissertation Committee: Prof. Dr. Peter Schupp (supervisor, chair) Dr. Branislav Jurˇco(supervisor) Prof. Dr. Dr. h.c. mult. Alan T. Huckleberry Prof. Dr. Olaf Lechtenfeld

Bibliographic entry

Title: Geometry of membrane sigma models Author: Ing. Jan Vysok´y Czech Technical University in Prague (CTU) Faculty of Nuclear Sciences and Physical Engineering Department of Physics Jacobs University Bremen (JUB) Department of Physics and Earth Sciences Degree Programme: Co-directed double Ph.D. (CTU - JUB) Application of natural sciences (CTU) Mathematical sciences (JUB) Field of Study: Methematical engineering (CTU) Mathematical sciences (JUB) Supervisors: Ing. Branislav Jurˇco,CSc., DSc. Charles University Faculty of Mathematics and Physics Mathematical Institute of Charles University Prof. Dr. Peter Schupp Jacobs University Bremen Department of Physics and Earth Sciences Academic Year: 2014/2015 Number of Pages: 209 Keywords: generalized geometry, string theory, sigma models, membranes.

Abstract

String theory still remains one of the promising candidates for a unification of the theory of gravity and quantum field theory. One of its essential parts is relativistic description of moving multi-dimensional objects called membranes (or p-branes) in a curved spacetime. On the classical field theory level, they are described by an action functional extremalising the volume of a manifold swept by a propagating membrane. This and related field theories are collectively called membrane sigma models. Differential geometry is an important mathematical tool in the study of string theory. It turns out that string and membrane backgrounds can be conveniently described using objects defined on a direct sum of tangent and cotangent bundles of the spacetime manifold. Mathe- matical field studying such object is called generalized geometry. Its integral part is the theory of Leibniz algebroids, vector bundles with a Leibniz algebra bracket on its module of smooth sections. Special cases of Leibniz algebroids are better known Lie and Courant algebroids. This thesis is divided into two main parts. In the first one, we review the foundations of the theory of Leibniz algebroids, generalized geometry, extended generalized geometry, and Nambu- Poisson structures. The main aim is to provide the reader with a consistent introduction to the mathematics used in the published papers. The text is a combination both of well known results and new ones. We emphasize the notion of a generalized metric and of corresponding orthogonal transformations, which laid the groundwork of our research. The second main part consists of four attached papers using generalized geometry to treat selected topics in string and membrane theory. The articles are presented in the same form as they were published.

Bibliografick´yz´aznam

N´azevpr´ace: Geometrie membr´anov´ych sigma model˚u Autor: Ing. Jan Vysok´y Cesk´evysok´euˇcen´ıtechnick´evˇ Praze (CVUT)ˇ Fakulta jadern´aa fyzik´alnˇeinˇzen´yrsk´a Katedra fyziky Jacobs University Bremen (JUB) Department of Physics and Earth Sciences Studijn´ıprogram: Doktor´atpod dvoj´ımveden´ım(CVUTˇ - JUB) Aplikace pˇr´ırodn´ıch vˇed(CVUT)ˇ Matematick´evˇedy(JUB) Studijn´ıobor: Matematick´einˇzen´yrstv´ı(CVUT)ˇ Matematick´evˇedy(JUB) Skolitel´e:ˇ Ing. Branislav Jurˇco,CSc., DSc. Univerzita Karlova Matematicko-fyzik´aln´ıfakulta Matematick´y´ustav UK Prof. Dr. Peter Schupp Jacobs University Bremen Department of Physics and Earth Sciences Akademick´yrok: 2014/2015 Poˇcetstran: 209 Kl´ıˇcov´aslova: zobecnˇen´ageometrie, teorie strun, sigma modely, membr´any.

Abstrakt

Strunov´ateorie st´alez˚ust´av´anadˇejn´ymkandid´atemna sjednocen´ı teorie gravitace a kvan- tov´eteorie pole. Jej´ıd˚uleˇzitousouˇc´ast´ıje relativistick´ypopis pohybu v´ıcerozmˇern´ych objekt˚u naz´yvan´ych membr´any (pˇr´ıpadnˇe p-br´any) v zakˇriven´emprostoroˇcase. Ten je na ´urovni kla- sick´eteorie pole pops´anfunkcion´alemakce extremalizuj´ıc´ımobjem variety vytnut´eˇs´ıˇr´ıc´ıse membr´anou.Tuto a souvisej´ıc´ıpoln´ıteorie souhrnnˇenaz´yv´amemembr´anov´esigma modely. Diferenci´aln´ıgeometrie pˇredstavuje d˚uleˇzit´ymatematick´yn´astroj pˇristudiu strunov´eteorie. Ukazuje se, ˇzestrunov´aa membr´anov´apozad´ı lze v´yhodnˇepopsat pomoc´ı objekt˚udefino- van´ych na direktn´ımsouˇctuteˇcn´ehoa koteˇcn´ehofibrovan´ehoprostoru k prostoroˇcasov´evarietˇe. Studiem tohoto objektu se zab´yv´atzv. zobecnˇen´ageometrie. Jej´ıned´ılnou souˇc´ast´ıje teorie Leibnizov´ych algebroid˚u,vektorov´ych fibrovan´ych prostor˚u,jejichˇzmoduly hladk´ych ˇrez˚ujsou vybaveny Leibnizovou algebrou. Speci´aln´ımipˇr´ıpadyLeibnizov´ych algebroid˚ujsou zn´amˇejˇs´ı Lieovy a Courantovy algebroidy. Tato pr´aceje rozdˇelenado dvou hlavn´ıch ˇc´ast´ı.V prvn´ırekapitulujeme z´akladyteorie Leib- nizov´ych algebroid˚u, zobecnˇen´egeometrie, rozˇs´ıˇren´ezobecnˇen´egeomerie a Nambu-Poissonov´ych struktur. C´ılemje poskytnout ˇcten´aˇriucelen´yz´akladmatematick´eteorie pouˇzit´ev publiko- van´ych pracech, text je kombinac´ızn´am´ych i nov´ych v´ysledk˚u.D˚urazje kladen pˇredevˇs´ımna pojem zobecnˇen´emetriky a pˇr´ısluˇsn´ych ortogon´aln´ıch transformac´ı,jenˇzposlouˇzilyjako z´aklad naˇsehov´yzkumu. Druhou hlavn´ıˇc´ast´ıje pˇr´ılohatvoˇren´aˇctyˇrmiˇcl´ankyuˇz´ıvaj´ıc´ımizobecnˇenou geometrii na vybran´epartie teorie strun a membr´an.Pr´acejsou otiˇstˇeny ve stejn´epodobˇe,v jak´ebyly publikov´any.

Acknowledgements

I would like to thank my supervisors, Branislav Jurˇcoand Peter Schupp, for their guidance and inspiration throughout my entire doctorate, unhesitatingly encouraging me in my scientific career. I would like to thank my wonderful wife, Kamila, for her endless support and patience with the weird and sometimes rather unpractical mathematical physicist. Last but not least, I would like to thank my family and all my friends for all the joy and fun they incessantly bring into my life.

Contents

1 Introduction 13 1.1 Organization ...... 14 1.2 Overview of the theoretical introduction ...... 14 1.3 Guide to attached papers ...... 17 1.4 Conventions ...... 18

2 Leibniz algebroids and their special cases 21 2.1 Leibniz algebroids ...... 21 2.2 Lie algebroids ...... 24 2.3 Courant algebroids ...... 27 2.4 Algebroid connections, local Leibniz algebroids ...... 30

3 Excerpts from the standard generalized geometry 39 3.1 Orthogonal group ...... 39 3.2 Maximally isotropic subspaces ...... 43 3.3 Vector bundle, extended group and Lie algebra ...... 44 3.4 Derivations algebra of the Dorfman bracket ...... 46 3.5 Automorphism group of the Dorfman bracket ...... 47 3.6 Twisting of the Dorfman bracket ...... 50 3.7 Dirac structures ...... 50 3.8 Generalized metric ...... 51 3.9 Orthogonal transformations of the generalized metric ...... 55 3.10 Killing sections and corresponding isometries ...... 59 3.11 Indefinite case ...... 60 3.12 Generalized Bismut connection ...... 63

9 4 Extended generalized geometry 67 4.1 Pairing, Orthogonal group ...... 67 4.2 Higher Dorfman bracket and its symmetries ...... 69 4.3 Induced metric ...... 70 4.4 Generalized metric ...... 73 4.5 Doubled formalism ...... 76 4.6 Leibniz algebroid for doubled formalism ...... 78 4.7 Killing sections of generalized metric ...... 82 4.8 Generalized Bismut connection II ...... 83

5 Nambu-Poisson structures 87 5.1 Nambu-Poisson manifolds ...... 87 5.2 Leibniz algebroids picture ...... 92 5.3 Seiberg-Witten map ...... 94

6 Conclusions and outlooks 97

Appendices 105

A Proofs of technical Lemmas 107

B Paper 1: p-Brane Actions and Higher Roytenberg Brackets 111

C Paper 2: On the Generalized Geometry Origin of Noncommutative Gauge Theory 135

D Paper 3: Extended generalized geometry and a DBI-type effective action for branes ending on branes 155

E Paper 4: Nambu-Poisson Gauge Theory 203

10 List of used symbols

R ...... field of real numbers M ...... smooth finite-dimensional manifold C∞(M) ...... module of smooth real-valued functions on M Xp(M) ...... module of smooth p-vector fields on M Ωp(M) ...... module of smooth differential p-forms on M Ω•(M) ...... whole exterior algebra of M q p (M) ...... module of tensors of type (p, q) T (M) ...... whole tensor algebra of M T, ...... canonical pairing of two objects Γ(h· E·i) ...... module of smooth global sections of vector bundle E over M ΓU (E) ...... module of smooth local sections of E defined on U M p p p ⊆q X (E) ...... module of global smooth sections of Λ E, similarly for Ω (E), p (E) A := B ...... expression B is used to define the expressionT A BDiag(g, h) ...... block diagonal matrix with g and h on the diagonal ...... Lie derivative along a vector field X X(M) LX ∈ iX ...... insertion operator (or inner product) of a vector field X X(M) [ , ] ...... Schouten-Nijenhuis bracket of multivector∈ fields · · S Hom(E,E0) ..... smooth vector bundle morphisms from E to E0 over the identity on the base End(E) ...... shorthand notation for Hom(E,E) Aut(E) ...... subset of End(E) consisting of fiber-wise bijective maps

11

Chapter 1

Introduction

After more then forty years of history, string theory still stands as one of the most promising attempts to unify the gravitational and quantum physics. Originating as a quantum theory of one-dimensional strings moving in the space-time, it evolved throughout the years to include fermionic particles (superstring theory) and extended objects such as D-branes, membranes or p-branes. It also grew into a challenging and sophisticated theory. An effort of a single person to review the string theory is as difficult as to ask a crab living on a Normandy beach to describe the Atlantic Ocean. As a welcome side-effect, string theory fueled the development in various, old and new, areas of mathematics. In particular, attempting to be a theory of gravity, it pushed forward many areas of differential geometry. Quite recently, one such mathematical theory rose to prominence as a useful tool in string theory. It was pioneered in early 2000s, as it appeared in three subsequent papers [43–45] of Nigel Hitchin and in the Ph.D. thesis [39] of his student Marco Gualtieri. In a nutshell, generalized geometry is a detailed study of the geometry of the generalized tangent bundle TM T ∗M. Such a vector bundle has a surprisingly rich structure, in particular it possesses a canonical⊕ indefinite metric and a bracket operation. This allows one to describe various geometrical objects in a new intrinsic way. Note that Hitchin already recognized the possible applications of generalized geometry in string theory and commented on this several times in the above cited papers. It turned out that certain symmetries of string theory can be naturally explained in terms of generalized geometry. For example, string T -duality can be viewed as an orthogonal trans- formation, see [22,38,86], following the work of P. Bouwknegt et al. [14,16,17]. Recently there has been found a way to describe D-branes as Dirac structures, isotropic and involutive sub- bundles of E, see [4–6]. For a nice overview of further interesting applications in physics and string theory see [64]. There are also ways to proceed in the opposite direction, constructing field theories out of generalized geometry mathematical objects. Besides the very well-known Poisson sigma models [54,83] , there exist Courant sigma models [55,82] using the AKSZ mech- anism to construct actions from a given Courant algebroid, Dirac sigma models [68, 69] and Nambu sigma models [18,59,84]. Suitable modifications of generalized geometry can be useful in string related physics. For applications in M-theory, see the work of Hull [52] and Berman et al. in [8–10]. A nice modification of generalized geometry was used to describe supergravity effective actions in [24,25]. Finally, note that generalized geometry is closely related to recently very popular modification of field theory, called double field theory. See the works of C.M. Hull, B. Zwiebach and O. Hohm [47, 50, 51], and especially the recent review paper [48].

13 By membrane sigma models we mean various field actions emanating from the bosonic part of Polyakov-like action for membrane as introduced by Howe-Tucker in [49] and for string case independently in [28] and [19], and named after Polyakov who used it to quantize strings in [79]. We focus in particular on its gauge-fixed version, which can be related to its dual version, non- topological Nambu sigma model as defined in [59]. A need to find a suitable mathematics underlying these field actions leads to a necessity to extend the tools of standard generalized geometry to more general vector bundles. Driven by this desire we refer to results of our work and consequently also of this thesis as to geometry of membrane sigma models.

1.1 Organization

This thesis is divided in two major parts. The first half is an attempt to bring up a consistent introduction to the mathematics used in the papers. The intention is to give definitions and derive important properties in detail as well as enough examples to illustrate sometimes quite abstract theory. The main idea is to provide a self-contained text with a minimal necessity to refer to external literature, which inevitably leads to a rephrasing of old and well known results. We will stress new contributions where needed. The second half consists of four published papers which use these mathematical tools to describe and develop several aspects of membrane sigma model theory. These papers are attached in the exactly same form as they were published in the journals. All of them were published as joint work with both my supervisors.

1.2 Overview of the theoretical introduction

Let us briefly summarize the content of the following chapters. This section is intended to be main navigation guide for the reader. Chapter 2 brings up a quick review of elementary properties of Leibniz algebroids. We could take the liberty to disregard the chronological order in which this theory appeared in mathematical world. Generalizations usually develop from more elementary objects, but it is sometimes simpler to provide the original structures as special examples of the more general ones. This is why we start with a general definition of Leibniz algebroids in Section 2.1. The basic and the most important example is the (higher) Dorfman bracket. Further, an induced Lie derivative on the tensor algebra of Leibniz algebroid is introduced. A known subclass of Leibniz algebroids are those with a skew-symmetric bracket, denoted as Lie algebroids. They are described in Section 2.2. Most importantly, a Lie algebroid induces an analogue of the exterior differential on the module of its differential forms, allowing for a Cartan calculus on the exterior algebra. In fact, this exterior differential contains the same data as the original Lie algebroid, and can be used as its equivalent definition. The exterior algebra of multivectors on the Lie algebroid bundle can be equipped with an analogue of Schouten- Nijenhuis bracket, turning it into a Gerstenhaber algebra. This observation is essential for the definition of Lie bialgebroid. A need of understanding Lie bialgebroids underpins the subject of Section 2.3, Leibniz algebroids with an invariant fiber-wise metric called Courant algebroids. They first appeared

14 in the form of their most important example, the Dorfman bracket with an appropriate anchor and pairing. Finding a suitable set of axioms for this new type of algebroid proved necessary to define a double of Lie bialgebroid. We present this as one of the examples of Courant algebroids. The second chapter is concluded by Section 2.4 dealing with linear connections on Leibniz algebroids, and related notions of torsion and curvature. For Lie algebroids, this is pretty straightforward and essentially it can be defined just by mimicking the ordinary theory of linear connections. For Courant and general Leibniz algebroids, this is more involved task. We present a new way how to approach this using a concept of local Leibniz algebroids. For Courant algebroids, one recovers the known definitions of torsion operator. To the best of our knowledge a working way how to define a curvature operator is presented, as well as a Ricci tensor and scalar. We explain this on the example which uses the Dorfman bracket. Chapter 3 contains an introduction to the geometry of the generalized tangent bundle, usually called a generalized geometry. This thesis covers only a little portion of this wide branch of mathematics suitable for our needs. These are the reasons why we have completely omitted the original backbone of generalized geometry - generalized complex structures. In Section 3.1 we study an orthogonal group of the direct sum of a vector space and its dual equipped with a natural pairing. Its Lie algebra can conveniently be described in terms of tensors. This also allows one to construct important special examples of orthogonal maps. We present a very useful simple ways of block matrix decompositions. Although almost trivial at first glance, this observation allows one to prove some quite complicated relations later. Since the natural pairing does not have a definite quadratic form, it allows for isotropic subspaces. This is a subject of Section 3.2. Examples of maximally isotropic subspaces are presented. Everything can be without any issues generalized from vector spaces to vector bundles, replacing linear maps with vector bundle morphism, et cetera. One can consider a more general group of transformations preserving the fiber-wise pairing on generalized tangent bundle. We call this group the extended orthogonal group. Its corresponding Lie algebra can be shown to be a semi-direct product of the ordinary (fiber-wise) orthogonal Lie algebra and the Lie algebra of vector fields. This is what Section 3.3 is about. The generalized tangent bundle has a canonical Courant algebroid structure, defined by the Dorfman bracket. It is natural to examine the Lie algebra of its derivations and the group of its automorphisms. This was one of the main reasons why generalized geometry and the Dorfman bracket came to prominence in string theory. We study these structures in Section 3.4 and Section 3.5. One can also find an explicit formula integrating the Dorfman bracket derivation to obtain a Dorfman bracket automorphism. To our belief this was not worked out in such detail before. It happens that some orthogonal maps do not preserve the bracket, and can be used to ”twist” it, to define different (yet isomorphic) Courant algebroids. This defines a class of twisted Dorfman brackets, introduced in Section 3.6. Dirac structures are maximally isotropic subbundles of the generalized tangent bundle which are involutive under Dorfman bracket. They constitute a way how to describe presymplectic and Poisson manifolds in terms of generalized geometry. They are described in Section 3.7. For the purposes of applications of generalized geometry in string theory, the most important concept is the one of a generalized metric. It can be defined in several ways, where some of them make sense in a more general setting then others. In Section 3.8 we discuss all these possibilities and show when they are equivalent. The orthogonal group acts naturally on the space of generalized metrics. We examine the consequences and properties of this action in Section 3.9. In particular, we show that Seiberg-Witten open-closed relations can be interpreted in this way.

15 Having a fiber-wise metric, we may study its algebra of Killing sections. This is done in Section 3.10. We give an explicit way to integrate these infinitesimal symmetries to actual generalized metric isometries. A generalized metric is by definition positive definite. We can modify some of its definitions to include also the indefinite case. This up brings certain issues discussed in Section 3.11. Finally, to any generalized metric we may naturally assign a Courant algebroid metric compatible connection, called a generalized Bismut connection. Its various forms are shown in Section 3.12, and its scalar curvature is calculated. Chapter 4 is to be considered as the most important of this thesis. Its main idea is to extend the concepts of generalized geometry to a higher generalized tangent bundle suitable for applications in membrane theory. This effort poses interesting problems to deal with. First note that there is no natural fiber-wise metric present. Instead, one can use a pairing with values in differential forms. However, its orthogonal group structure becomes quite complicated and depends on the rank of involved vector bundles. We examine this in Section 4.1. In particular, a set of examples of Dirac structures with respect to this pairing is very limited. On the other hand, the Dorfman bracket generalizes into a bracket of very similar properties. We investigate its algebra of derivations and its group of automorphisms in detail in Section 4.2. A next important step is to define an extended version of the generalized metric. For these reasons, we start with a description of a way how to use a manifold metric to induce a fiberwise metric on higher wedge powers of the tangent and cotangent bundle. There are several consequences following from the construction. In particular, one can calculate its signature out of the signature of the original metric. Moreover, for Killing equations and metric compatible connections, it is important to examine the way how Lie and covariant derivatives of the induced metric can be calculated using Lie and covariant derivatives of the original manifold metric. Finally, we derive a very important formula proving the relation of their determinants. All of this can be found in Section 4.3. We proceed to the actual definition of a generalized metric in Section 4.4. Similarly to the ordinary generalized metric, one can express it either in terms of a metric and a differential form, or in terms of dual fields. One can show that in this extended case, one of the dual fields is not automatically a multivector, and the two dual fiberwise metrics are not induced from each other. Finally, we generalize the open-closed relations as a particular transformation of the generalized metric. As we have already noted, there is no useful orthogonal group to encode the open-closed relations as an example of an orthogonal transformation. However, there is a natural pairing on the ”doubled” vector bundle, produced from the original one by adding (in the sense of a direct sum) its dual bundle. In this ”doubled formalism”, as we call it, one can define a generalized metric in the usual sense. The membrane open-closed relations can be now easily encoded as an orthogonal transformation of a relevant generalized metric. The doubled formalism is a subject of Section 4.5. Having now a larger vector bundle with working orthogonal structure, we would like to find a suitable integrability conditions for its subbundles. One such Leibniz algebroid is examined in Section 4.6. We show how closed differential forms and Nambu-Poisson structures can be realized as Dirac structures of this Leibniz algebroid. We conclude by showing how its bracket can be twisted by a certain orthogonal map, obtaining a twist similar to the one of Dorfman bracket. One of the reasons to use the induced metric in the definition of generalized metric is the fact that Killing equations are easy to solve in this case, as we show in Section 4.7. It also

16 allows us to find a simple example of a generalized metric compatible connection. We show how to interpret this connection in the doubled formalism and we calculate its scalar curvature in Section 4.8. The final Chapter 5 of this thesis is devoted to a natural generalization of Poisson manifolds called Nambu-Poisson structures. In Section 5.1 we present and prove various equivalent ways how to express the fundamental identity for a Nambu-Poisson tensor. Interestingly, it can be shown that an algebraic part of the fundamental identity not only forces its decomposability, but also its complete skew-symmetricity of this tensor. Similarly to the ordinary Poisson case, Nambu-Poisson structures can be realized as certain involutive subbundles with respect to the Dorfman bracket. This interpretation allows one to easily define a twisted Nambu-Poisson structure. This and one quite interesting related observation are contained in Section 5.2. Finally, in Section 5.3, we examine in detail the construction of the Nambu-Poisson structure induced diffeomorphism called a Seiberg-Witten map. It involves the flows of time-dependent vector fields which are for the sake of clarity recalled there.

1.3 Guide to attached papers

The second part of this thesis consists of four attached papers. All of them are available for download at arXiv.org archive, three of them can be openly accessed directly through the respective journals. Papers are presented here in their original form, exactly as they were published. We sometimes refer to equations in the theoretical introduction of this thesis to link the introduced mathematical theory with its applications in the papers. The first attached paper is p-Brane Actions and Higher Roytenberg Brackets [61]. A main subject of this paper is a study of Nambu sigma model proposed in [59, 84]. We modify its action slightly to include a twist with a B-field. Using relations (4.49 - 4.51), its equivalence to a p-brane sigma model action is shown. We introduce a slightly modified higher analogue of Courant algebroid bracket (2.36) which we call in accordance with [42] a higher Roytenberg bracket. We use the results of [32] to show that the Poisson algebra of generalized charges of the Nambu sigma model closes and it can be described by the higher Roytenberg bracket. This generalizes the results [2, 42]. Next, we turned our attention to the topological Nambu sigma model. It can be viewed as a system with constrains. We have proved that a consistency of the constrains with time evolution forces the tensor Π defining the sigma model to be a Nambu-Poisson tensor. See Chapter 5 of this thesis for details. Using the Darboux coordinates of Theorem 5.1.2 we were able to explicitly solve the equations of motion. We have concluded the paper by showing that coefficients of the generalized Wess-Zumino term produced out of topological Nambu sigma model are exactly the structure functions of the higher Roytenberg bracket. On the Generalized Geometry Origin of Noncommutative Gauge Theory [60]. Idea of this paper was to use the tools of generalized geometry to explain and simplify certain steps derived originally in [58], [62] and [63]. A main observation was that different factoriza- tions of the generalized metric correspond to Seiberg-Witten open closed relations (3.119). For the first time we interpret this as an orthogonal transformation (3.17) of a generalized metric, see Section 3.9 of this thesis. Moreover, also adding a fluctuation F to the B-field background can be interpreted as an orthogonal transformation (3.16). These two transformation do not commute (this is in fact a direct consequence of Lemma 3.1.2). This immediately leads to the correct definition of non-commutative versions of the respective fields. We were able to use

17 this formalism to quickly re-derive the identities essential for the equivalence of classical DBI action and its semi-classically noncommutative counterpart. The third of the appended papers is Extended generalized geometry and a DBI-type effective action for branes ending on branes [56]. In this article, our intentions were to generalize the approach taken in the previous paper [60] to simplify the calculations in [59] leading to the proposal of a p-brane generalization of DBI effective action. To reach this goal, we were able to explain the p-brane open-closed relations (4.56 - 4.59) in terms of the factorization of the generalized metric (4.47). Higher generalized tangent bundle does not possess a canonical orthogonal group structure (it does for p = 1). This was the reason why we have approached this through the addition of its dual. See Section 4.5 of this thesis for details. Using the extended generalized geometry, we were able to show the equivalence of respective DBI actions very quickly. Moreover, using the doubled formalism, we could define an analogue of a so called back- ground independent gauge. Reason for this name comes from the famous paper [85], and it is related to the actual background independence of the corresponding non-commutative Yang- 1 Mills action. For p = 1, this corresponds to the choice θ = B− in (3.119). We generalize this idea to p 1 including also the case of a degenerate 2-form B for p = 1. A choice of suit- ≥ able Nambu-Poisson tensor Π singles out particular directions on the p0-brane, which we call noncommutative directions. This allows us to derive a generalization of a double scaling limit, see [85], introducing an infinitesimal parameter  into DBI action. Calculating an expansion of DBI in the first order of  yields a possible generalization of a matrix model. This lead us to the writing of the short paper Nambu-Poisson Gauge Theory [57]. The letter follows the ideas for p = 1 published in [76]. Nambu-Poisson theory gauge theory was originally invented in [46] as en effective theory on a M5-brane for a large longitudinal C-field in M-theory. Our idea was to start from scratch without any reference to M-theory and branes. We define covariant coordinates to be functions of space-time coordinates transforming in a prescribed way under gauge transformations parametrized by a (p 1)-form, using a pre- scribed (p+1)-ary Nambu-Poisson bracket. Following [62,63], we use the− covariant coordinates to define a Nambu-Poisson gauge field and a corresponding Nambu-Poisson field strength. The second part of this paper is devoted to an explicit construction of covariant coordinates using the Seiberg-Witten map described in this thesis in Section 5.3. We propose a simple Yang-Mills action for this gauge theory and relate it to the DBI action expansion obtained in the above described paper [56].

1.4 Conventions

The main aim of this section is to introduce a notation used throughout this entire work. We always work with a finite-dimensional smooth second-countable Hausforff manifold, which we usually denote as M and its dimension as n. For implications of this definition, I would recommend an excellent book [71]. In particular, one can use the existence of a partition of unity and its implications. We consider all objects to be real, in particular all vector spaces, vector bundles, bilinear forms, etc. Now, let us clarify our index notations. We reserve the small Latin letters (i, j, k etc.) to label the components corresponding to a set of local coordinates (y1, . . . , yn) on M, or sometimes to some local frame field on M. We reserve small Greek letters (α, β, γ, etc.) to label the components with respect to some local basis of the module of smooth sections of a

18 vector bundle E. We use the capital Latin letters (I, J, K etc.) to denote the strictly ordered multi-indices, that is I = (i1, . . . , ip) for some p N, where 1 i1 < < ip n. Particular value of p should be clear from the context.∈ We always hold≤ to the··· Einstein≤ summation convention, where repeated indices (one upper, one lower) are assumed to be summed over I (i1...ip) their respective ranges. For example, v wI stands for the sum 1 i <

dyI = dyi1 ... dyip , ∂ = ∂ . . . ∂ . (1.1) ∧ ∧ I i1 ∧ ip I p p By definition, dy and ∂I form a local basis of Ω (M) and X (M) respectively. We will often use a generalized Kronecker symbol. We define it as follows

+1 both p-indices are strictly ordered and one is an even permutation of the other, δj1...jp = 1 both p-indices are strictly ordered and one is an odd permutaion of the other, i1...ip −  0 in all other cases.  J J It is defined so that (dy )I = δI . A Levi-Civita symbol i1...ip can be then defined as i1...ip = δ1...p . i1...ip

Let E,E0 be two vector bundles over M. By Hom(E,E0) we mean a module of smooth vector bundle morphisms from E to E0 over the identity map IdM . Under our assumptions on M, Hom(E,E0) coincides with the module of C∞(M)-linear maps from Γ(E) to Γ(E0), and we will thus never distinguish between the vector bundle morphisms and the induced maps of sections. We define End(E) := Hom(E,E), and Aut(E) := End(E), is fiber-wise bijective . {F ∈ F } Now, let p 0 be a fixed integer, and C Ωp+1(M) be a differential (p + 1)-form on M. This induces a≥ vector bundle morphism C : X∈p(M) Ω1(M) defined as [ → J J i C[(Q ∂J ) := Q CiJ dy , (1.2) for all Q = QJ ∂ Xp(M), and C = C(∂ , ∂ . . . , ∂ ). It is straightforward to check that C J ∈ iJ i j1 jp [ is a well-defined C∞(M)-linear map of sections (all indices are properly contracted). At each p point m M, C[ thus defines a linear map from Λ TmM to Tm∗ M, with the corresponding matrix (C∈ ) [C ] . Collecting those matrices, we get a matrix of functions (C ) = [|m i,J ≡ |m iJ |[ i,J CiJ . Here comes our convention. In the whole thesis, we will denote objects C, C[ and (C[)i,J with the single letter C, and the particular interpretation will always be clear from the context. T 1 p T By C we mean the transpose map from X (M) to Ω (M). Note that C (X) = iX C, for all X X(M). ∈ Similarly, for Π Xp+1(M), we define the vector bundle morphism Π] :Ωp(M) X(M) as ∈ → ] J iJ Π (ξJ dy ) = ξJ Π ∂i, (1.3) for all ξ Ωp(M), and ΠiJ = Π(dyi, dyj1 , . . . , dyjp ). We again do not distinguish Π, Π] and the matrix∈ (Π])i,J = ΠiJ . The transpose map ΠT then maps from Ω1(M) to Xp(M). These conventions may seem to be quite unusual when compared to the standard generalized geometry papers. They are however well suited for matrix multiplications, and proved to be the best choice to get rid of unnecessary ( 1)p factors in all formulas. −

19 20 Chapter 2

Leibniz algebroids and their special cases

In this chapter, we will introduce a framework useful to describe various algebraical and geo- metrical aspects of the objects living on vector bundles. Fields arising in string and membrane sigma models theory can be viewed as vector bundle morphisms of various powers of tangent and cotangent bundles, which justifies the efforts to understand the canonical structures com- ing with those vector bundles. We will proceed in a rather unhistorical direction, starting from the most recent definitions, and arriving to the oldest.

2.1 Leibniz algebroids

The basic idea goes as follows. Let E π M be a vector bundle. By definition, E is a collection of isomorphic vector spaces E at each→ point m M. Let us say that every E can be m ∈ m equipped with a Leibniz algebra bracket [ , ]m, that is an R-bilinear map from Em Em to · · × Em, satisfying the Leibniz identity

[v, [v0, v00]m]m = [[v, v0]m, v00]m + [v0, [v, v00]m]m, (2.1) for all v, v0, v00 Em. This bracket needs not to be skew-symmetric. Leibniz algebras were first introduced by Jean-Luis∈ Loday in [73]. Now, suppose that this bracket changes smoothly from point to point. More precisely, if e, e0 Γ(E) are smooth sections, then formula ∈ [e, e0](m) := [e(m), e0(m)]m, (2.2) defines a smooth section [e, e0] Γ(E). We have just constructed a simplest example of Leibniz 1 ∈ algebroid . Note that the bracket satisfies [e, fe0] = f[e, e0] for all e, e0 Γ(E) and f C∞(M), and also the Leibniz identity ∈ ∈

[e, [e0, e00]] = [[e, e0], e00] + [e0, [e, e00]]. (2.3) This special case is too simple in a sense that the bracket depends only on the point-wise values of the incoming sections. In fact, all examples which we will encounter in this thesis are not of this type. Let us now give a formal definition of a general Leibniz algebroid. 1Consider that E has its typical fiber equipped with a Leibniz algebra bracket. If E can be locally trivialized by Leibniz algebra isomorphisms, one calls this example a Leibniz algebroid bundle.

21 Definition 2.1.1. Let E π M be a vector bundle, and ρ Hom(E,TM). Let [ , ] : → ∈ · · E Γ(E) Γ(E) Γ(E) be an R-bilinear map. We say that (E, ρ, [ , ]E) is a Leibniz algebroid, if × → · ·

For all e, e0 Γ(E), and f C∞(M), there holds the Leibniz rule: • ∈ ∈

[e, fe0]E = f[e, e0]E + (ρ(e).f)e0. (2.4)

The bracket [ , ] defines a Leibniz algebra on Γ(E), it satisfies the Leibniz identity • · · E

[e, [e0, e00]E]E = [[e, e0]E, e00]E + [e0, [e, e00]E]E, (2.5)

for all e, e0, e00 Γ(E). ∈ The map ρ is called the anchor of Leibniz algebroid, since it allows the sections of E to act on the module of smooth functions on M.

Let us make a few remarks to this definition.

In the math literature, Leibniz algebroids are often called Loday algebroids. There is • an extensive work on this topic by Y. Kosmann-Schwarzbach in [66] and especially by J. Grabowski an his collaborators [36]. However, we stick to the name Leibniz alge- broid which was used in relation to Nambu-Poisson structures [41, 53], or in relation to generalized geometry, as in [7]. In some literature, there is an another axiom present, namely that ρ is a bracket homo- • morphism: ρ([e, e0]E) = [ρ(e), ρ(e0)], (2.6)

for all e, e0 Γ(E). However, it follows directly from the compatibility of Leibniz rule and Leibniz∈ identity. Our motivating example thus corresponds to the case ρ = 0, so called totally intransitive • Leibniz algebroid. Note that in the general case, there is no way how to consistently induce a Leibniz algebra bracket on the single fibers of E. There is an interesting subtlety in the presented definition of the Leibniz algebroid. The • bracket [ , ] has two inputs, but Leibniz rule controls only the right one. One can prove · · E the following. Let e0 Γ(E) be a section such that e0 = 0 for some open subset U M. ∈ |U ⊆ Then ([e, e0] ) = 0. This can be shown by choosing f = 1 χ, where χ is a bump E |U − function supported inside U, and χ(m) = 1 for chosen m M. Then e0 = (1 χ)e0 and consequently ∈ −

[e, e0] = [e, (1 χ)e0] = (1 χ)[e, e0] + (ρ(e).(1 χ))e0. E − E − E −

Evaluating this at m, we get [e, e0] (m) = 0. We can repeat this for any m U, and E ∈ we get ([e, e0] ) = 0. This property allows one to restrict the second input of [ , ] to E |U · · E ΓU (E). For a general Leibniz algebroid, there is however no way to do this in the first input.

Let us now examine some structures induced by Leibniz algebroid bracket on E. First note that it induces an analogue E of Lie derivative on the tensor algebra (E). It is defined as L T follows. Assume e, e0 Γ(E), α Γ(E∗) and f C∞(M). ∈ ∈ ∈ 22 0 E 1. On (E) = C∞(M), we define it as f = ρ(e).f. T0 ∼ Le 1 E 2. For (E) = Γ(E), we set e0 = [e, e0] , T0 ∼ Le E 0 3. For (E) = Γ(E∗), we define it by contraction T1 ∼ E α, e0 := ρ(e). α, e0 α, [e, e0] . (2.7) hLe i h i − h Ei It follows from the Leibniz rule that the right-hand side is C∞(M)-linear in e0, and thus defines an element α Γ(E∗). Le ∈ 4. For general τ p(E), E is defined as ∈ Tq Le [ Eτ](e , . . . , e , α , . . . , α ) = ρ(e).τ(e , . . . , e , α , . . . , α ) Le 1 q 1 p 1 q 1 p τ( Ee , . . . , e , α , . . . , α ) ... (2.8) − Le 1 q 1 p − ... τ(e , . . . , e , α ,..., Eα ), − 1 q 1 Le p for all e , . . . , e Γ(E) and α , . . . , α Γ(E∗). 1 q ∈ 1 p ∈ We can easily prove the following properties of the Lie derivative. Lemma 2.1.2. Lie derivative E satisfies the Leibniz rule: Le E(τ σ) = E(τ) σ + τ E(σ), (2.9) Le ⊗ Le ⊗ ⊗ Le for all τ, σ (E). Moreover, it can be restricted to the exterior algebra Ω•(E), and forms its degree 0 derivation:∈ T E E E (ω ω0) = (ω) ω0 + ω (ω0), (2.10) Le ∧ Le ∧ ∧ Le for all ω, ω0 Ω•(E). Next, the commutator of Lie derivatives is again a Lie derivative: ∈ E E E E E 0 0 = 0 , (2.11) Le Le − Le Le L[e,e ]E for all e, e Γ(E). Note that this also implies E = 0, although in general [e, e] = 0. 0 [e,e]E E Finally, there∈ holds also the identity L 6 E E i 0 i 0 = i 0 , (2.12) Le ◦ e − e ◦ Le [e,e ]E where both sides are now considered as operators only on the submodule Ω•(E).

Proof. Leibniz rule (2.9) follows from the Leibniz rule (2.4) for the bracket [ , ]E and the definition formula (2.8). When τ = ω Ωq(E), the expression on the right-hand· side· of (2.8) ∈ can be seen to be skew-symmetric in (e1, . . . , eq), and the derivation property (2.10) then follows from the Leibniz rule (2.9). Finally, the left-hand side of (2.11) is a commutator and thus obeys 0 (2.9). It thus suffices to prove this on tensors of type (0, 0), (1, 0) and (0, 1). For f 0 (E), the condition (2.11) is equivalent to (2.6), for (1, 0) it reduces to the Leibniz identity (2.5),∈ T and for (0, 1) it follows from the relation E E E E [ , 0 ]α, e00 = [ρ(e), ρ(e0)]. α, e00 α, [ , 0 ]e00 . (2.13) h Le Le i h i − h Le Le i To finish the proof we have to show (2.12). Because both sides are derivations of degree 1 of − the exterior algebra Ω•(E), we have to prove the result on forms of degree 0 and 1 only. The first case is trivial, the second gives E ρ(e). α, e0 α, e0 = α, [e, e0] , h i − hLe i h Ei which is precisely the definition of the Lie derivative on Ω1(E). Note that Leibniz identity for [ , ] was not used to prove (2.12). · · E  23 We see that there is still one piece missing in the Cartan puzzle, namely the analogue of +1 the differential: dE :Ω•(E) Ω• (E). There is however no way around this for general Leibniz algebroid. There are two→ reasons - usual Cartan’s formula for differential not only fails to define a form of a higher degree, it does not give a tensorial object at all. To conclude the subsection on Leibniz algebroids, we now bring up an example, which will play a significant role hereafter.

p Example 2.1.3. Let E = TM Λ T ∗M, where p 0. We will denote the sections of E as formal sums X + ξ, where X ⊕X(M) and ξ Ωp(M≥). We define the anchor ρ simply as the projection onto TM: ρ(X + ξ)∈ = X. The bracket,∈ which we will denote as [ , ] is defined as · · D [X + ξ, Y + η] := [X,Y ] + η i dξ, (2.14) D LX − y for all X + ξ, Y + η Γ(E). It is a straightforward check to see that (E, ρ, [ , ] ) forms a ∈ · · D Leibniz algebroid. The bracket [ , ]D is called the Dorfman bracket. For p = 1, it first appeared in [29], for p > 1 it appeared· · in [39, 45]. It proved to be a useful tool to describe Nambu-Poisson manifolds [41]. To illustrate the previous definitions, note that on Ω1(E), the induced Lie derivative E has the form L E (α + Q) = α + (dξ)(Q) + Q, (2.15) LX+ξ LX LX for all X + ξ Γ(E) and all α + Q Ω1(E). ∈ ∈

2.2 Lie algebroids

Having the concept of Leibniz algebroid defined, it is easy to define a Lie algebroid. This structure is much older, it first appeared in [80]. Lie algebroids play the role of an ”infinitesimal” object corresponding to Lie groupoids. While a Lie algebra is the tangent space at the group unit with the extra structure coming from the group multiplication, Lie algebroid is a vector bundle over a set of units of a Lie groupoid. However; not every Lie algebroid corresponds to a Lie groupoid, see [27]. For an extensive study of Lie groupoids, Lie algebroids and related subjects, see the book [74]. Definition 2.2.1. Let (L, l, [ , ] ) be a Leibniz algebroid. We say that (L, l, [ , ] ) is a Lie · · L · · L algebroid, if [ , ]L is skew-symmetric, and hence a Lie bracket. Leibniz identity (2.5) is now called the Jacobi· · identity (note that it can be reordered using the skew-symmetry of the bracket). Example 2.2.2. There are several standard examples of Lie algebroids

1. Consider L = TM, l = Id and let [ , ] = [ , ] be a vector field commutator. TM · · L · · 2. A generalization of the example in the previous section, where each fiber of Em is equipped with a Lie algebra bracket, with a smooth dependence on m. In particular, for M = m , we see that every Lie algebra is an example of Lie algebroid. { } 3. This is a classical example, which probably first appeared in [35]. According to [67], it was discovered independently by several authors during 1980s. Look there for a complete list of references. It is sometimes called Koszul or Magri bracket, or simply a cotangent Lie algebroid.

24 2 Let Π X (M) be a bivector on M. Choose L = T ∗M, and define the anchor ρ as ρ(α) =∈ Π(α) for all α Ω1(M). Finally, define the bracket [ , ] as ∈ · · Π [α, β] = β i dα. (2.16) Π LΠ(α) − Π(β) This bracket is skew-symmetric when Π is, and satisfies the Jacobi identity if and only if Π is a Poisson bivector, that is f, g = Π(df, dg) defines a Poisson bracket on M. We will investigate this in more detail{ in} Section 5.

π 4. Consider a principal G-bundle P M. There is a C∞(M)-module ΓG(TP ) of G- invariant vector fields on P , which→ turns out to be isomorphic to the module Γ(T P/G) of sections of the quotient bundle T P/G (quotient with respect to the right translation of vector fields induced by the principal bundle group action). This isomorphism induces a bracket on Γ(T P/G) using the vector field commutator on Γ(TP ). Finally, the tangent map T (π): TP TM descends to the quotient T (π): T P/G TM, defining an anchor. For details→ of this construction, see the sections 3.1, 3.2 of→ [74]. The resulting § § Lie algebroid is called the Atiyah-Lie algebroid. b

The newly imposed skew-symmetry of the bracket [ , ] of a Lie algebroid allows for new · · L structures on the exterior algebra Ω•(L) and multivector field algebra X•(L). First, see that we finally have a differential (absent in general Leibniz algebroid case). We state this as a proposition.

Proposition 2.2.3. Let (L, l, [ , ]L) be a Lie algebroid. We define an R-linear map dL : +1 · · p Ω•(L) Ω• (L) as follows. Let ω Ω (L), and e , . . . , e Γ(L). Set → ∈ 0 p ∈ p (d ω)(e , . . . , e ) = ( 1)il(e ).ω(e ,..., e , . . . , e ) L 0 p − i 1 i p i=0 X (2.17) + ( 1)i+jω([e , e ] , e b,..., e ,..., e , . . . , e ), − i j L 0 i j p i

ω d (ω ω0) = d ω ω0 + ( 1)| |ω d ω0, (2.18) L ∧ L ∧ − ∧ L 2 for all ω, ω0 Ω•(L), such that ω is defined. Moreover, the map dL squares to zero: dL = 0. Finally, let ∈L be a Lie derivative| defined| by Leibniz algebroid structure on L. Then the Cartan magic formulaL holds: Lω = d i ω + i d ω, (2.19) Le L e e L for all e Γ(L) and ω Ω•(L). ∈ ∈ Proof. First, one can check the first assertion, which is quite straightforward. Next, one has to prove that the right-hand side is C∞(M)-linear in e0, . . . , ep and thus dL is a well-defined operator on tensors of L. This follows from the Leibniz rule (2.4). The most difficult step is to show (2.18), which is a quite tedious but straightforward proof by induction and we skip it here. To show d2 = 0, one notes that d2 = 1 d , d , where , is the graded commutator, L L 2 { L L} {· ·} 25 2 2 and thus dL is a derivation of Ω•(M) of degree 2. It thus suffices to check dL = 0 on degrees 0 and 1. One obtains 2 (d f)(e, e0) = ρ(e).ρ(e0).f ρ(e0).ρ(e).f ρ([e, e0] ).f, L − − L which is precisely the homomorphism property (2.6). For α Ω1(E), we get ∈ 2 (d α)(e, e0, e00) = ([ρ(e), ρ(e0)] ρ([e, e0]L).α(e00) + cyclic e, e0, e00 L − { } (2.20) + α, [[e, e0] , e00] + [e00, [e, e0] ] + [e0, [e00, e] ] . h L L L L L Li The first line again vanishes due to (2.6), and the second line due to Jacobi identity (2.5). The last assertion is an equality of two degree 0 derivations of Ω•(L), and it thus has to be verified for degree 0 and 1 forms, which is easy. 

Interestingly, dL is not only an induced structure, it contains all the information about the original Lie algebroid. More precisely, having any vector bundle L with a degree 1 derivation dL of the exterior algebra Ω•(L) squaring to 0, we can define the anchor l as l(e).f := d f, e , (2.21) h L i for all f C∞(M) and e Γ(L), and then a bracket [ , ] by ∈ ∈ · · L α, [e, e0] = l(e). α, e0 l(e0). α, e (d α)(e, e0), (2.22) h Li h i − h i − L 1 for all e, e0 Γ(L) and α Ω (L). It then follows by simple calculations using just (2.18) that (L, l, [ , ] )∈ is a Lie algebroid.∈ · · L The second object induced by a Lie algebroid is an analogue of Schouten-Nijenhuis bracket, we again define it using a proposition, this time without any proof. For a detailed discussion on this topic, see [67]. Proposition 2.2.4. Let (L, l, [ , ] ) be a Lie algebroid. Then there is a unique bracket [ , ] · · L · · L defined on the multivector field algebra X•(L), which has the following properties:

1 0 For e X (L) Γ(L), and f X (L) C∞(M), we have [e, f] = ρ(e).f. • ∈ ≡ ∈ ≡ L 1 For e, e0 X (L) Γ(L), [ , ] coincides with the Lie algebroid bracket. • ∈ ≡ · · L (X•(L), [ , ] ) forms a Gerstenhaber algebra, which amounts to the following: • · · L 1. [ , ] is a degree 1 map, that is [P,Q] = P + Q 1 for P,Q X•(L). · · L − | L| | | | | − ∈ 2. For each P X•(L), [P, ] is a derivation of the exterior algebra X•(L) of degree P 1, that∈ is there holds· | | − ( P 1) Q [P,Q R] = [P,Q] R + ( 1) | |− | |Q [P,R] . (2.23) ∧ L L ∧ − ∧ L 3. It is graded skew-symmetric, that is ( P 1)( Q 1) [P,Q] = ( 1) | |− | |− [Q, P ] . (2.24) L − − L 4. It satisfies the graded Jacobi identity ( P 1)( Q 1) [P, [Q, R] ] = [[P,Q] ,R] + ( 1) | |− | |− [Q, [P,R] ] . (2.25) L L L L − L L All properties are assumed to hold for all P, Q, R X•(L) with a well-defined degree. ∈ The bracket [ , ] is called a Schouten-Nijenhuis bracket corresponding to the Lie algebroid · · L (L, l, [ , ]L). For L = TM, it reduces to the original well-known Schouten-Nijenhuis bracket of multivector· · fields.

26 2.3 Courant algebroids

Let us now consider a second special case of Leibniz algebroids, the one most relevant for the generalized geometry. We will stick to the modern and more used definition, which views Courant algebroid as a Leibniz algebroid with an additional structure. Historically, though, it appeared to be a much more complicated object. It appeared in [72] as a double corresponding to a pair of compatible Lie algebroids (we will show this as an example) in an attempt to generalize the concept of Manin triple to the Lie algebroid setting. The modern definition accounts to the thesis [81] of Roytenberg, who proved that the original skew-symmetric bracket can be replaced by a Leibniz algebroid bracket together with a set of simpler axioms. We present this form here. Definition 2.3.1. Let E π M be a vector bundle. By fiber-wise metric on E, we mean a → symmetric C∞(M)-billinear non-degenerate form , : Γ(E) Γ(E) C∞(M). It follows h· ·iE × → from the C∞(M)-bilinearity that for every m M it defines a non-degenerate symmetric ∈ bilinear form (metric) on the fiber Em.

Definition 2.3.2. Let (E, ρ, [ , ]E) be a Leibniz algebroid. Let , E be a fiber-wise metric on E. We say that (E, ρ, , ·, [· , ] ) forms a Courant algebroidh· ,·i if h· ·iE · · E 1. The form , is invariant with respect to the bracket: h· ·iE ρ(e). e0, e00 = [e, e0] , e00 + e0, [e, e00] , (2.26) h iE h E iE h EiE 0 for all e, e0, e00 Γ(E). Equivalently, if gE 2 (E) is a tensor corresponding to , E, we require Eg∈ = 0 for all e Γ(E). ∈ T h· ·i Le E ∈ 2. For all e, e0 Γ(E), the symmetric part of the bracket is governed by ρ and , E in the sense: ∈ h· ·i 1 [e, e] , e00 = ρ(e00). e, e , (2.27) h E iE 2 h iE for all e, e0 Γ(E). Equivalently, let gE : E E∗ be the induced vector bundle iso- ∈ → 1 T morphism. Define an R-linear map : C∞(M) Γ(E) as := gE− ρ d, where T D → D ◦ ◦ ρ Hom(T ∗M,E∗) is the transpose of the anchor. We can then rewrite the axiom simply∈ as 1 [e, e] = e, e , (2.28) E 2Dh iE and this can be polarized to

[e, e0] = [e0, e] + e, e0 . (2.29) E − E Dh iE We see that [ , ] is skew-symmetric up to the of the function e, e0 . · · D D h iE We will call , or g the Courant metric on E. h· ·iE E Courant algebroid can be viewed as a generalization of the quadratic Lie algebra, since for M = m , it reduces to a Lie algebra equipped with a non-degenerate ad-invariant symmetric { } bilinear form. It was noted in [69] that the invariance of , E cannot be achieved without the sacrifice of the skew-symmetry, i.e. there is no Lie algebroidh· ·i with an invariant fiber-wise metric , E, unless it is totally intransitive, that is ρ = 0. Note that the control over the symmetric parth· ·i of the bracket allows one to derive the Leibniz rule in its first input. We get

[fe, e0] = f[e, e0] (ρ(e0).f)e + e, e0 f. (2.30) E D − h iED 27 Recall the remark under the Definition 2.1.1, where we have noted that general Leibniz bracket [e, e0]E depends only on the values of section e0 in an arbitrarily small neighborhood, but nothing can be said about e. This is not true for Courant algebroids, where we can use (2.30) to prove that e = 0 implies ([e, e0] ) = 0. Altogether, we can restrict [ , ] to the module |U E |U · · E of the local sections ΓU (E), which proves useful when working with a local basis for Γ(E). Remark 2.3.3. Interestingly, Kosmann-Schwarzbach has shown in [65] that the axiom of Leibniz rule (2.4) is superfluous in the definition of Courant algebroid, and can be derived from (2.26, 2.27).

Example 2.3.4. Let us now give a few examples of Courant algebroids.

1. Consider the Leibniz algebroid from Example 2.1.3 for p = 1. Then there is a canonical 1 pairing , E of vector fields and 1-forms on Γ(E) = X(M) Ω (M). Explicitly, the map is thenh· ·i(f) = 0 + df Γ(E). We have ⊕ D D ∈ 1 [X + ξ, X + ξ] = d(i ξ) = X + ξ, X + ξ . (2.31) D X 2Dh iE The invariance of the pairing reduces to showing that X.( η, Z + ζ, Y ) = [X,Y ], ζ + η i dξ, Z + η, [X,Z] + Y, ζ i dξ . h i h i h i hLX − Y i h i h LX − Z i This is easy to show after one uses the definitions of and the exterior differential d. L 3 Now, see that [ , ]D can be modified in the following way. Let H Ω (M) be a 3-form on M, and define· · a new bracket [ , ]H as ∈ · · D [X + ξ, Y + η]H = [X + ξ, Y + η] H(X,Y, ), (2.32) D D − · for all X +ξ, Y +η Γ(E). Because H is completely skew-symmetric and C∞(M)-linear, ∈ H both axioms (2.26, 2.27) remain valid also for [ , ]D . Plugging into Leibniz identity (2.5) · · 3 H shows that it holds if and only if dH = 0, that is H Ωclosed(M). The bracket [ , ]D is called the H-twisted Dorfman bracket. ∈ · ·

2. Let (L, l, [ , ] ) and (L∗, l∗, [ , ] ∗ ) be a pair of Lie algebroids, where L∗ is the dual vector · · L · · L bundle to L. One says that (L, L∗) forms a Lie bialgebroid, if dL is a derivation of the 2 Schouten-Nijenhuis bracket [ , ] ∗ , that is · · L ω 1 d [ω, ω0] ∗ = [d ω, ω0] ∗ + ( 1)| |− [ω, d ω0] ∗ , (2.33) L L L L − L L for all ω, ω0 Ω•(L) = X•(L∗). Define E = L L∗. Denote the sections of E as ∈ ∼ ⊕ ordered pairs (e, α), where e Γ(L) and α Γ(L∗). The anchor ρ is defined as ρ(e, α) = ∈ ∈ l(e) + l∗(α). The bracket [ , ] is defined as · · E L∗ L [(e, α), (e0, α0)] = [e, e0] + e0 i (d ∗ e), [α, α0] ∗ + α0 i 0 (d α) , (2.34) E L Lα − α L L Le − e L for all (e, α), (e0, α0) Γ( E). Finally, let , be a fiber-wise metric on E induced
by the ∈ h· ·iE canonical pairing of L and L∗. Then (E, ρ, , , [ , ] ) is a Courant algebroid, called the h· ·iE · · E double of the Lie bialgebroid (L, L∗). The actual proof of this statement is straightforward but takes quite some time to go through. See [72] for details. Conversely, let (E, ρ, , , [ , ] ) be a Courant algebroid, and L and L be two com- h· ·iE · · E 1 2 plementary Dirac structures, that is E = L L . Then L = L∗ and (L ,L ) can be 1 ⊕ 2 2 ∼ 1 1 2 equipped with a structure of Lie bialgebroid. (E,L1,L2) is called a Manin triple.

2 This condition is in fact equivalent to the one with L and L∗ interchanged [75].

28 3. Let us show an example of a double of Lie bialgebroid. Let (L, l, [ , ] ) be any Lie · · L algebroid. Choose L∗ to be a trivial Lie algebroid (L∗, 0, 0). Since [ , ] ∗ = 0, the Lie · · L bialgebroid condition (2.33) holds trivially. The resulting bracket on E = L L∗ is then ⊕ L [(e, α), (e0, α0)] = ([e, e0], α0 i 0 d α), (2.35) E Le − e L

for all (e, α), (e0, α0) Γ(E). We will call it the Dorfman bracket corresponding to Lie algebroid (L, l, [ , ] ).∈ For L = (T M, Id , [ , ]), one obtains part 1. of this example. · · L TM · · 4. Let L = (T M, Id , [ , ]) be the canonical tangent bundle Lie algebroid, and set L∗ = TM · · (T ∗M, Π, [ , ] ) to be the cotangent Lie algebroid from Example 2.2.2, 3. One can find · · Π in [75] that d ∗ = [Π, ] , where [ , ] is the ordinary Schouten-Nijenhuis bracket on L − · S · · S X•(M). By definition of Schouten-Nijenhuis bracket, d ∗ is thus a derivation of [ , ] , L · · S which is exactly the Lie bialgebroid condition (2.33). The pair (L, L∗) therefore forms a Lie bialgebroid, called a triangular Lie bialgebroid. The resulting double bracket (2.34) on E can be after some effort written as

[X + ξ, Y + η]E = [X + Π(ξ),Y + Π(η)] Π η i dξ − L(X+Π(ξ)) − (Y +Π(η)) (2.36) + η i dξ, L(X+Π(ξ)) − (Y +Π(η))
for all X +ξ, Y +η Γ(E). Although complicated at first glance, it can be rewritten using the Dorfman bracket∈ (2.14). Indeed, define a bundle map eΠ : E E as eΠ(X + ξ) = X + Π(ξ) + ξ for all X + ξ Γ(E). We can then write → ∈ Π Π Π [X + ξ, Y + η]E = e− [e (X + ξ), e (Y + η)]D. (2.37)

Π Π Π Moreover, ρ = prTM e , and e (X +ξ), e (Y +η) E = X +ξ, Y +η E. This shows that bracket (2.36) is in fact◦ just ah ”twist” of the Dorfmani bracket.h Therei is one important remark to be said. The bracket written in the form (2.36) in fact does not require Π to be a Poisson bivector in order to define a Courant algebroid. However; for general Π, the bracket (2.34) is not the same as (2.36). H If one uses [ , ]D in the formula (2.37) instead of [ , ]D, one obtains a bracket which is in [42] called· the· Roytenberg bracket. We also use this· · name for its higher version.

There is a famous classification of Severaˇ of a particular class of Courant algebroids. Let 1 T (E, ρ, , E, [ , ]E) be any Courant algebroid. Define a map j : T ∗M E as j = gE− ρ . One saysh· ·i that· a· Courant algebroid is exact, if there is a short exact sequence→ ◦

j ρ 0 T ∗M E TM 0. (2.38)

In particular, ρ has to be surjective and j injective, and Im j = ker ρ. Note that inclusion Im j ker ρ holds for any Courant algebroid, because ρ = 0. Severaˇ proved in [89] that up to⊆ an isomorphism, exact Courant algebroids are uniquely◦ D determined by a class [H] ∈ H3(M, R). In particular, there is an isotropic splitting s : TM E, s(X), s(Y ) E = 0 for all X,Y X(M), such that one can write → h i ∈ [s(X) + j(ξ), s(Y ) + j(η)] = s([X,Y ]) + j( η i dξ H(X,Y, )), (2.39) E LX − Y − · 3 where H Ωclosed(M). For different splitting s0 of the sequence, H changes to H0 = H + dB ∈ 2 for some 2-form B Ω (M), but [H0] = [H]. Map Ψ : TM T ∗M E defined as Ψ(X +ξ) = ∈ ⊕ → 29 H s(X) + j(ξ) then defines a Courant algebroid isomorphism from (TM T ∗M, pr , [ , ] ) to ⊕ TM · · D (E, ρ, [ , ]E). Every exact Courant algebroid is thus isomorphic to a one equipped with an H-twisted· · Dorfman bracket. Dorfman and H-twisted Dorfman brackets of Example 2.3.4, 1. are exact, whereas a Manin triple of Lie bialgebroid in general is not. This can be seen on example of the Dorfman bracket of a Lie algebroid, 2.3.4, 3. where any Lie algebroid L with a non-surjective anchor will give a non-exact Courant algebroid. On the other hand, the example 2.3.4, 4. is an example of exact Manin triple, in particular [H] = [0] in this case. To conclude, let us briefly note on the older, skew-symmetric version of Courant alge- broid brackets. Let (E, ρ, , E, [ , ]E) be a Courant algebroid. Define [ , ]E0 to be its skew- symmetrization: h· ·i · · · · 1 1 [e, e0]0 := ([e, e0] [e0, e] ) = [e, e0] e, e0 . (2.40) E 2 E − E E − 2Dh iE Let us examine what happened to the Leibniz rule. Plugging into (2.40), we obtain

1 [e, fe0]0 = f[e, e0]0 + (ρ(e).f)e0 e, e0 f. (2.41) E E − 2h iD

Invariance of , with respect to the bracket [ , ]0 becomes h· ·iE · · E 1 1 ρ(e). e0, e00 = [e, e0]0 + e, e0 , e00 + e0, [e, e00]0 + e, e00 , (2.42) h iE h E 2Dh iE iE h E 2Dh iEiE for all e, e0, e00 Γ(E). Note that Leibniz rule for [ , ] implies ρ = 0. This also shows that ∈ · · E ◦ D [ , ]0 also satisfies the homomorphism property (2.6): · · E

ρ([e, e0]E0 ) = [ρ(e), ρ(e0)], (2.43) for all e, e0 Γ(E). The most complicated calculation is to see that Leibniz identity for [ , ]0 ∈ · · E fails in the following sense. Define a map T : Γ(E) Γ(E) Γ(E) C∞(M) as × × → 1 T (e, e0, e00) := [e, e0]0 , e00 + cyclic e, e0, e00 . (2.44) 6h E iE { } Then there holds the following identity:

[[e, e0]0 , e00]0 + [[e00, e]]0 , e0]0 + [[e0, e00]0 , e]0 = T (e, e0, e00). (2.45) E E E E E E D For the proof of this statement see [81]. Now let us just say that equations (2.41, 2.42, 2.43, 2.45) form a set of axioms of the original definition of Courant algebroid, as proposed in [72]. The skew-symmetric version of the bracket has its advantages, in particular in relation to strongly homotopy Lie algebras.

2.4 Algebroid connections, local Leibniz algebroids

For Lie algebroids there is a straightforward way to define linear connections [34]. For Courant algebroids, or even Leibniz algebroids, matters become more complicated, see [1]. Let us recall the basic definitions first.

30 Definition 2.4.1. Let E be a vector bundle. Map : X(M) Γ(E) Γ(E) is called a linear connection on vector bundle E, if ∇ × →

(fX, e) = f (X, e), (X, fe) = f (X, e) + (X.f)e, (2.46) ∇ ∇ ∇ ∇ for all X X(M) and e Γ(E). We write e := (X, e). ∈ ∈ ∇X ∇ Remark 2.4.2. Equivalently, we can view as follows. Let (E) be a vector bundle over M such that its space of sections Γ( (E)) has∇ the form D D Γ( (E)) = : Γ(E) Γ(E) (fe) = f (e) + (X.f)e, e Γ(E), for X X(M) . (2.47) D {F → | F F ∀ ∈ ∈ } Define a : Γ( (E)) X(M) as a( ) = X, and let [ , ] = . Then ( (E), a, [ , ]) is a Lie algebroid.D → We can then viewF linear connectionF G Fas ◦ a G vector − G ◦ F bundle morphismD · · Hom(TM, (E)) defined as (X) = fitting in the commutative∇ diagram ∇ ∈ D ∇ ∇X

TM ∇ (E) D a . (2.48) 1TM TM

Note that both TM and (E) are Lie algebroids. One can easily extend this definition to any D Lie algebroid (L, l, [ , ]L). For more details concerning the construction of vector bundle (E), see [74]. · · D

Every linear connection on E induces an analogue of the curvature operator. For X,Y X(M) and e Γ(E) it is defined using the standard formula: ∈ ∈ R(X,Y )e = e e e. (2.49) ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] 2 1 It is C∞(M)-linear in all inputs, hence R Ω (M) (E). In view of Remark (2.4.2), we ∈ ⊗ T1 may view R as a failure of to be a Lie algebroid morphism. If gE is any fiber-wise metric on E, we can say that is metric∇ compatible with g if ∇ E

X.g (e, e0) = g ( e, e0) + g (e, e0), (2.50) E E ∇X E ∇X for all X X(M) and e, e0 Γ(E). Obviously, there is no analogue of torsion for connections ∈ ∈ on a vector bundle. Now, let (E, ρ, , E, [ , ]E) be a Courant algebroid. One can define the Courant algebroid connection accordingh· ·i to [1]· · as follows:

Definition 2.4.3. Let (E, ρ, , E, [ , ]E) be a Courant algebroid. A map : Γ(E) Γ(E) Γ(E) is a Courant algebroidh· connection·i · · , if ∇ × →

(fe, e0) = f (e, e0), (e, fe0) = f (e, e0) + (ρ(e).f)e0, (2.51) ∇ ∇ ∇ ∇ for all e, e0 Γ(E), and is metric compatible with Courant metric , in the sense that ∈ ∇ h· ·iE

ρ(e). e0, e00 = e0, e00 + e0, e00 , (2.52) h iE h∇e iE h ∇e iE for all e, e0, e00 Γ(E). As usual, we will write e0 := (e, e0). ∈ ∇e ∇ 31 As before, we can naively define a curvature operator R corresponding to as ∇

R(e, e0)e00 = 0 e00 0 e00 0 e00, (2.53) ∇e∇e − ∇e ∇e − ∇[e,e ]E for all e, e0, e00 Γ(E). This is C∞(M)-linear in e0 and e00, but not in e. Instead, we get ∈

R(fe, e0)e00 = fR(e, e0)e00 e, e0 E f e00. (2.54) − h i ∇D Let us remark that there is a class of connections which define a tensorial curvature operator. We say that is an induced Courant algebroid connection, if = 0 for some vector ∇ ∇e ∇ρ(e) bundle connection 0. Because ρ D = 0, the anomalous term in (2.54) disappears, and R is a well defined tensor∇ on E. Second◦ possibility is to restrict R to sections of some isotropic involutive subbundle D E, where e, e0 = 0. ⊆ h iE Unlike for vector bundle connections, there is a well defined analogue of the torsion operator. There are two independent, but essentially equivalent definitions. In [40], a torsion tensor T (E) is given as ∈ T3 1 T (e, e0, e00) = e0 0 e [e, e0]0 , e00 + ( 00 e, e0 00 e0, e , (2.55) h∇e − ∇e − E iE 2 h∇e iE − h∇e iE for all e, e0, e00 Γ(E). By definition, it is skew-symmetric in (e, e0). In fact, the Courant metric compatibility condition∈ (2.52) can be used to show that T Ω3(E). In [1], a Courant algebroid torsion was defined as C Ω3(E) in the form ∈ ∈ 1 1 C(e, e0, e00) = [e, e0]0 , e00 e0 0 e, e00 + cyclic(e, e0, e00). (2.56) 3h E iE − 2h∇e − ∇e iE This expression in manifestly completely skew-symmetric in all inputs, but at first glance it does not resemble the conventional definition of torsion operator. Interestingly, these two definitions coincide.

Lemma 2.4.4. Let (E, ρ, , E, [ , ]E) be a Courant algebroid, and a Courant algebroid connection. Then h· ·i · · ∇ T (e, e0, e00) = C(e, e0, e00). (2.57) − Proof. This can be done by using (2.52) and Courant algebroid axioms (2.26, 2.28). Note that both T and C are well defined based on the Leibniz rule (2.1.1), but equivalent only due to the other Courant algebroid axioms. 

For general Leibniz algebroids, there is no metric , E present and definitions (2.55, 2.56) make no sense anymore. There is however a way to defineh· ·i a connection, a torsion and even a curvature operator for a special (and quite wide) class of Leibniz algebroids.

Definition 2.4.5. Let (E, ρ, [ , ] ) be a Leibniz algebroid. If there exists a C∞(M)-trilinear · · E map L : Γ(E∗) Γ(E) Γ(E) Γ(E), such that × × → [fe, e0] = f[e, e0] (ρ(e0).f)e + L(df, e, e0), (2.58) E E − for all e, e0 Γ(E) and f C∞(M), we call (E, ρ, [ , ] , L) a local Leibniz algebroid. Here ∈ ∈ · · E d : C∞(M) Γ(E∗) is an R-linear map defined by → df, e = ρ(e).f, (2.59) h i for all e Γ(E) and f C∞(M). ∈ ∈ 32 Note that L is not uniquely determined by equation (2.58), and has to be a part of the defini- tion of a local Leibniz algebroid. Moreover, the compatibility of (2.58) with the homomorphism property (2.6) implies ρ(L(df, e, e0)) = 0. (2.60) For a given Leibniz algebroid (E, ρ, [ , ] ) with a well-defined subbundle ker ρ, one can always · · E find L such that this property can be extended to ρ(L(β, e, e0)) = 0 for all e, e0 Γ(E) and ∈ β Γ(E∗). To achieve this, choose some fiber-wise metric on E∗ and define L(β, e, e0) := 0 for ∈ all β Γ(Ann ker(ρ)⊥). ∈ Example 2.4.6. In fact, all examples in this thesis can be equipped with the structure of a local Leibniz algebroid.

Let (L, l, [ , ]L) be a Lie algebroid. The choice of L = 0 shows that (L, l, [ , ]L, L) is a • local Leibniz· · algebroid. · · Let (E, ρ, , , [ , ] ) be a Courant algebroid. We see that from (2.30) that • h· ·iE · · E 1 L(df, e, e0) = e, e0 g− (df). (2.61) h iE E There is one canonical way to extend L. Define

1 L(β, e, e0) = e, e0 g− (β), (2.62) h iE E

for all e, e0 Γ(E) and β Γ(E∗). However, note that this choice does not satisfy ∈ ∈ ρ(L(β, e, e0)) = 0. For E with well-defined subbundle ker ρ, one can extend L trivially to some complement of Ann(ker ρ). Note that different choices of this complement will lead to different extensions.

Let E = TM T ∗M be equipped with the Dorfman bracket (2.14), and ρ(X + ξ) = X. • ⊕ The kernel of ρ is the subbundle T ∗M E. We have a short exact sequence ⊆ j ρ 0 T ∗M E TM 0, (2.63)

where j is an inclusion. Choosing a complement of ker ρ corresponds to the choice of a splitting s Hom(TM,E) of this sequence. We can restrict ourselves to isotropic ∈ splittings, that is s(X), s(Y ) E = 0 for all X,Y X(M). The set of such splittings is in 2 h i 2 ∈ fact Ω (M), and for any B Ω (M) the complement to T ∗M is exactly the subbundle ∈ G = X + B(X) X TM E. (2.64) B { | ∈ } ⊆

Note that G0 = TM. This gives us also a splitting of E∗, in particular

E∗ = Ann(ker ρ) G . (2.65) ⊕ B

We can now define L to be trivial on GB, let us write it with subscript B. We get

L (α + V, e, e0) = L (α B(V ) + (V + B(V )), e, e0) B B − 1 (2.66) = e, e0 g− (α B(V )) = e, e0 (α B(V )). h iE E − h iE − We see how L can explicitly depend on the choice of the complement.

33 For an arbitrary Leibniz algebroid (E, ρ, [ , ]E) we can define a Leibniz algebroid connection in the same way as in Definition 2.4.3, except· · that we do not require the metric compatibility (2.52).∇ Assume that this Leibniz algebroid is local. Then we can in fact define a torsion operator!

Proposition 2.4.7. Let (E, ρ, [ , ]E, L) be a local Leibniz algebroid. Define an R-bilinear map T : Γ(E) Γ(E) Γ(E) as · · × → λ T (e, e0) = e0 0 e [e, e0] + L(e , e, e0), (2.67) ∇e − ∇e − E ∇eλ k λ k for all e, e0 Γ(E). Here (eλ)λ=1 is an arbitrary local frame of E, and (e )λ=1 the corresponding ∈ 1 dual one. Then T is C∞(M)-linear in e and e0 and consequently T 2 (E). We call T a torsion operator corresponding to . ∈ T ∇

Proof. Direct calculation. 

Let us emphasize that T is not in general skew-symmetric in (e, e0). This is not a problem since we can always take its skew-symmetric part. Moreover, its definition certainly depends on the choice of the map L.

Interestingly, for induced connections e = ρ0 (e), this is not the case. To see this, choose k ∇ ∇ the local frame (e ) adapted to the splitting E = ker(ρ) ker(ρ)⊥ with respect to some λ λ=1 ⊕ fiber-wise metric gE on E. Because is induced, only those eλ in Γ(ker(ρ)⊥) contribute to the sum in (2.67). But in this case eλ ∇Ann(ker ρ), where the map L is determined uniquely3 ∈ For Courant algebroid (E, ρ, , E, [ , ]E) and L in the form (2.62), the torsion operator (2.67) can be simply related to theh· ·i one defined· · by (2.55).

Proposition 2.4.8. Let (E, ρ, , , [ , ] ) be a Courant algebroid, and L be a map defined h· ·iE · · E by (2.62). Denote the torsion operator (2.55) as TG, and let T be the torsion operator (2.67). Let be a Courant algebroid connection. Then ∇

T (e, e0, e00) = T (e, e0), e00 . (2.68) G h iE

Proof. This can be verified by a direct calculation. Use the fact that

λ L(e , e, e0), e00 = 00 e, e0 . (2.69) h ∇eλ iE h∇e iE

This shows that for Courant algebroid connections, the symmetric part of the map K(e, e0) = λ L(e , e, e0) does not depend on at all. Indeed, we have ∇eλ ∇

K(e, e0) + K(e0, e) = 00 e, e0 + e, 00 e0 = ρ(e00). e, e0 . h∇e iE h ∇e iE h iE

This in fact proves that T (e, e0) is skew-symmetric in (e, e0), because

e00,T (e, e0) + T (e0, e) = e00, [e, e0] + [e0, e] + ρ(e00). e, e0 = 0. h iE h − E EiE h iE We have used the axiom (2.28) in the last step. 

3Note that sections of the form df for some f C (M) locally generate Γ(Ann(ker ρ)). ∈ ∞

34 For more general (local) Leibniz algebroids, there also exists a notion of a generalized torsion introduced for special examples in [24, 25]. They proceed as follows. Consider a local Leibniz E algebroid (E, ρ, [ , ]E, L). Let be the Lie derivative induced by [ , ]E. In their paper this is · · L · k· called the Dorfman derivative. Consider a local holonomic frame (eα)α=1, that is [eα, eβ]E = 0. α β We will use the shorthand notation f,α := ρ(eα).f. Let e = v eα and e0 = w eβ. We have

E α β α β λ βµ e0 = v w w (v v L ) e . (2.70) Le { ,α − ,α − ,µ λα } β

Their idea is to define a ”covariantized” Dorfman derivative e∇ by replacing commas with semicolons: L α β α β λ βµ ∇e0 = v w w (v v L ) e . (2.71) Le { ;α − ;α − ;µ λα } β Here e = vβ e . This can be rewritten in terms of and L as ∇eα ;α β ∇ µ ∇e0 = e0 0 e + L(e , e, e0). (2.72) Le ∇e − ∇e ∇eµ Torsion operator T is in [24, 25] defined as difference of these two Lie derivatives:

E T (e, e0) = ( ∇ )e0. (2.73) Le − Le Comparing this with (2.67) we see from (2.72) that the two definitions coincide. Note the importance of the local frame holonomicity for a validity of this assertion. We have shown that any local Leibniz algebroid allows one to define a tensorial torsion operator. We can use a very similar approach to get a well-defined curvature operator.

Proposition 2.4.9. Let (E, ρ, [ , ] , L) be a local Leibniz algebroid, such that ρ(L(β, e, e0)) = 0 · · E for all β Γ(E∗) and e, e0 Γ(E). Let be a Leibniz algebroid connection on E. Then the ∈ ∈ ∇ map R defined for all e, e0, e00 Γ(E) as ∈

0 0 0 λ 0 R(e, e0)e00 = e e e00 e ee00 [e,e ]E e00 + L(e , e,e )e00, (2.74) ∇ ∇ − ∇ ∇ − ∇ ∇ ∇eλ 1 is C∞(M)-linear in all inputs, and thus R (E). We call R a generalized Riemann tensor. ∈ T3

Proof. The statement can be directly verified. One has to use (2.6) to show the C∞(M)- linearity in e00. The additional correction term containing L cancels the wrong term coming from the bracket term and its first input. The condition ρ(L(β, e, e0)) = 0 is necessary to keep the C∞(M)-linearity in e00. 

First, note that because of the condition ρ(L(β, e, e0)) = 0, the additional term vanishes for induced connections, which in fact shows that in this case the usual curvature operator formula works and defines a tensorial R. Next, see that in general R is not skew-symmetric in (e, e0), which can be fixed by a skew-symmetrization if necessary. Having a curvature operator R, we can define the corresponding Ricci tensor Ric as a contraction of R in two indices. Namely set

λ Ric(e, e0) = e ,R(e , e0)e , (2.75) h λ i k λ k for all e, e0 Γ(E), where (e ) is some local frame of E, and (e ) the corresponding ∈ λ λ=1 λ=1 dual one of E∗. For Courant algebroid connections, R has some remarkable properties.

35 Proposition 2.4.10. Let (E, ρ, , E, [ , ]E) be a Courant algebroid with ker ρ E being a well defined subbundle. Let L be definedh· ·i trivially· · on some complement to Ann(ker⊆ρ). Let be ∇ a Courant algebroid connection. Then R(e, e0) is skew-symmetric in (e, e0), and

R(e, e0)f, f 0 + R(e, e0)f 0, f = 0, (2.76) h iE h iE for all e, e0, f, f 0 Γ(E). ∈

Proof. Let E∗ = Ann(ker ρ) V for some subbundle V , such that L V = 0. Choose a local λ j k j⊕m | k k m frame e = (g , f ), where (g )j=1 is a local frame of Ann(ker ρ), and (f )k=1− a local frame of k m m V . Let eλ = (gj, fk) be a corresponding dual basis. Then (fk)k=1− generates ker ρ, and (gi)i=1 its complement. We have

λ k 1 k L(e , e, e), e0 = L(g , e, e), e0 = e, e g− (g ), e0 h ∇eλ iE h ∇gk iE hh∇gk iE E iE 1 k 1 k k = [ ρ(g ). e, e ] g , e0 = ρ( g , e0 g + f , e0 f ). e, e 2 k h iE h i 2 h i k h i k h iE 1 = ρ(e0). e, e . 2 h iE

λ This proves that for Courant algebroid connections, we have L(e , eλ e, e) = [e, e], and the two non-trivial contributions in R(e, e) cancel. Note that this shows that∇ also T (e, e) = 0 for such an L. The proof of (2.76) is analogous to the one for ordinary connections, using the metric compatibility (2.52). Note that in this process one has to use ρ(L(eλ, e, e)) = 0. ∇eλ 

Example 2.4.11. Consider E = TM T ∗M and the usual Dorfman bracket (2.14). Extend ⊕ L to all γ + Z Γ(E∗) as ∈ L(γ + Z,X + ξ, Y + η) = X + ξ, Y + η (0 + γ). (2.77) h iE This corresponds to the choice B = 0 in (2.66). Now consider a Courant algebroid connection on E. We have ∇ (Y + η) = 0 (Y + η) + 00(Y + η), (2.78) ∇X+ξ ∇X ∇ξ for all X + ξ, Y + η Γ(E), for some vector bundle connection 0 on E, and a map 00 : 1 ∈ ∇ ∇ Ω (M) Γ(E) Γ(E). Note that 00 must be C∞(M)-linear in the second input, and thus × 1→ 1 ∇ 1 in fact 00 X (M) 1 (E). We can view 00 as C∞(M)-linear map 00 :Ω (M) End(E). What are∇ the∈ implications⊗ T of the Courant metric∇ compatibility (2.52)?∇ We get →

X. e, e0 = 0 e, e0 + e, 0 e0 , (2.79) h iE h∇X iE h ∇X iE 0 = 00e, e0 + e, 00e0 . (2.80) h∇ξ iE h ∇ξ iE

This implies that 0 and 00 have to be of the block form ∇ ∇ M X ΠX Aξ θξ X0 = ∇ M , ξ00 = T , (2.81) ∇ BX ∇ Cξ A  ∇X   − ξ  2 2 M where ΠX , θξ X (M), BX ,Cξ Ω (M), Aξ End(TM), and is an ordinary connection M ∈ ∈ ∈ ∇ M on M. in the bottom-right corner of 0 is the usual extension of on 1-forms. All ∇X ∇X ∇ 36 objects are assumed to be C∞(M)-linear in X and ξ. For the curvature tensor, we get

pr (R(X,Y )(Z + ζ)) = RM (X,Y )Z + ( M Π) (ζ) ( M Π) (ζ) (2.82) 1 ∇X Y − ∇Y X + Π M (ζ) + Π (B (Z)) Π (B (Z)) T (X,Y ) X Y − Y X B (X,Y )(Ak(Z) + θk(ζ)), − k pr (R(X,Y )(Z + ζ)) = RM (X,Y )ζ + ( M B) (Z) ( M B) (Z) (2.83) 2 ∇X Y − ∇Y X + B M (Z) + B (Π (ζ)) B (Π (ζ)) T (X,Y ) X Y − Y X B (X,Y )(Ck(Z) (Ak)T (ζ)), − k − pr (R(ξ, η)(Z + ζ)) = (A A + θ C )(Z) + (A θ θ AT )(ζ) (ξ η) (2.84) 1 ξ η ξ η ξ η − ξ η − ↔ Π (ξ, η)(Ak(Z) + θk(ζ)), − k pr (R(ξ, η)(Z + ζ)) = (C A AT C )(Z) + (C θ + AT AT )(ζ) (ξ η) (2.85) 2 ξ η − ξ η ξ η ξ η − ↔ Π (ξ, η)(Ck(Z) (Ak)T (ζ)), − k − M pr1(R(X, η)(Z + ζ)) = ( X A)η(Z) + A η,T M ( ,X) (Z) (2.86) ∇ h · i M + ( X θ)η(ζ) + θ η,T M ( ,X) (ζ) ∇ h · i + (Π C θ B )(Z) (Π AT + A Π )(ζ), X η − η X − X η η X M pr2(R(X, η)(Z + ζ)) = ( X C)η(Z) + C η,T ( ,X) (Z) (2.87) ∇ h · i M T T ( X A)η (ζ) A η,T ( ,X) (ζ) − ∇ − h · i + (B A + AT B )(Z) + (B θ C Π )(ζ). (2.88) X η η X X η − η X M By ( X Π)Y we mean the following. Bivector ΠY depends C∞(M)-linearly on Y , and thus ∇ 2 M 2 defines a tensor Π 1 (M). One can then calculate its covariant derivative ( X Π) 1 (M), ∈ T 2 ∇ ∈ T and finally for each Y X(M) the tensor ( X Π)Y 0 (M). Similarly for the other objects. T M denotes the torsion∈ operator of the connection∇ ∈M T. ∇

37 38 Chapter 3

Excerpts from the standard generalized geometry

In this chapter, we will recall some basic facts about the standard generalized geometry, that is the geometry of the vector bundle E = TM T ∗M. We will focus only on the topics relevant for the following chapters (including the papers).⊕ In the previous chapter, we have shown that E is equipped with the Courant algebroid bracket (2.14), and the natural pairing , E. Note that E is sometimes called the generalized tangent bundle. Pioneering works inh· generalized·i geometry are those of Hitchin [43,45] and especially the Ph.D. thesis of Gualtieri [39]. We will focus on a very detailed explicit analysis of the involved objects, which eventually will prove to be useful for physical applications. In the section describing the generalized metric, we have used the approach taken in [69].

3.1 Orthogonal group

First assume that V is an n-dimensional real vector space. The direct sum W = V V ∗ is ⊕ equipped with the canonical pairing , W , which defines a symmetric non-degenerate bilinear n h· ·i 1 n form on W . If = (ei)i=1 is any basis of V , there is a canonical basis (e1, . . . , en, e , . . . , e ) of i nE W , where (e ) is the basis of V ∗ dual to . In this basis, the pairing , has the matrix i=1 E h· ·iW 0 1 g = , (3.1) W 1 0   where 1 denotes the n n unit matrix. This proves that , has the signature (n, n). The × h· ·iW group of operators on W preserving , W is thus O(n, n). Let O(n, n). We will often use the formal block decomposition of linearh· ·i maps on W , that is weO will ∈ write O O = 1 2 , (3.2) O O3 O4   where O End(V ), O End(V ∗) and O Hom(V ∗,V ), O Hom(V,V ∗). The matrix g 1 ∈ 4 ∈ 2 ∈ 3 ∈ W can be thus also viewed as a formal block decomposition of the isomorphism g : V V ∗ W ⊕ → V ∗ V induced by , . The orthogonality condition ⊕ h· ·iW (v + α), (v0 + α0) = v + α, v0 + α0 (3.3) hO O iW h i 39 can be now rewritten in terms of O by expanding the 2 2 block matrix equation T g = i × O W O gW . One obtains a set of three relations:

T T O3 O1 + O1 O3 = 0 (3.4) T T O4 O2 + O2 O4 = 0 (3.5) T T O3 O2 + O1 O4 = 1 (3.6)

1 We can get more equations. First note that − = g g , which explicitly gives O W O W T T 1 O4 O2 − = . (3.7) O OT OT  3 1  1 T 1 The map − is again orthogonal, there holds − gW − = gW . We have three more (of course notO independent) equations: O O

T T O4O3 + O3O4 = 0, (3.8) T T O2O1 + O1O2 = 0, (3.9) T T O2O3 + O1O4 = 1. (3.10)

We will now focus on the maps of the form = exp , where o(n, n). Lie algebra o(n, n) is defined as the space of linear endomorphismsO of WA, which areA ∈skew-symmetric with respect to , . Thus, every o(n, n) thus has to satisfy the condition h· ·iW A ∈

(v + α), v0 + α0 + v + α, (v0 + α0) = 0, (3.11) hA iW h A iW for all v + α, v0 + α0 W . Let us write in a formal block matrix form ∈ A N Π = , (3.12) A BN 0   where N End(V ), N 0 End(V ∗), Π Hom(V ∗,V ) and B Hom(V,V ∗). Plugging into ∈ ∈ T ∈ ∈ (3.11) gives a block matrix equation gW +gW = 0, expansion of which yields a set of three equations A A T T T B + B = 0, Π + Π = 0,N 0 + N = 0. (3.13) This gives an easy way to interpret the conditions for the respective blocks. We see that map 2 2 T B has to be induced by a 2-form B Λ V ∗, Π by a bivector Π Λ V , and N 0 = N . The conclusion is that (as a vector space)∈ the Lie algebra o(n, n) can∈ be decomposed as−

2 2 o(n, n) = End(V ) Λ V ∗ Λ V, (3.14) ∼ ⊕ ⊕ where each o(n, n) has a block form A ∈ N Π = (3.15) A B N T  −  2 2 for N End(V ), B Λ V ∗ and Π Λ V . Note that this simply reflects the fact that for any ∈ ∈ ∈ 2 2 i 2 i two finite-dimensional vector spaces V,W , one has Λ (V W ) ∼= i=0 Λ V Λ − W . We can now proceed with the examples of O(n, n) transformations.⊕ ⊗ L Example 3.1.1. Let us now show three main classes of O(n, n) transformations.

40 1. B-transform: Choose N = Π = 0 in of the form (3.15). Denote by eB its exponential, that is eB = exp . Explicitly, A A 1 0 eB = . (3.16) B 1   As a map, it has the form eB(v + α) = (v, α + B(v)), for all v + α W . By construction, eB O(n, n). Below, it will play an important role in relation to∈ the Dorfman bracket. ∈ 2.Π -transform: Now, choose so that N = B = 0. Denote by eΠ its exponential, that is eΠ = exp . Explicitly, A A 1 Π eΠ = . (3.17) 0 1   As a map, it has the form eΠ(v + α) = (v + Π(α), α), for all v + α W . It will play an important role in the description of (Nambu-)Poisson structures. ∈

3. Group Aut(V ): Every invertible map A Aut(V ) defines an O(n, n) transformation A in the form ∈ O A 0 A = T . (3.18) O 0 A−   T As a map, it works as (v + α) = A(v) + A− (α). OA

O(n, n) as a Lie group has four connected components, and the above three examples generate its identity component. We will often make use of the following simple observation:

Lemma 3.1.2. Let be a block 2 2 matrix in the form M × AB = . (3.19) M CD   The matrices A and D have to be square, but can be of different dimensions. Then

If A is invertible, there exists a unique decomposition • 1 1 0 A 0 1 A− B = 1 1 (3.20) M CA− 1 0 D CA− B 0 1    −    of into a product of block lower unitriangular, block diagonal, and block upper unitri- angularM matrices.

If D is invertible, there exists a unique decomposition • 1 1 1 BD− A BD− C 0 1 0 = − 1 (3.21) M 0 1 0 D D− C 1       of into a product of block upper unitriangular, block diagonal, and block lower unitri- angularM matrices.

If is invertible, and there exists some decomposition of into a product of block • lowerM unitriangular, block diagonal, and block upper unitriangularM matrices, then A is invertible, and the decomposition is precisely of the form (3.20).

41 If is invertible, and there exists some decomposition of onto a product of block • upperM unitriangular, block diagonal, and block lower unitriangularM matrices, then D is invertible, and the decomposition is precisely of the form (3.21).

Proof. If A is invertible, we can construct the right-hand side of (3.20) and verify by direct calculation that the product gives . Now assume that M 1 0 S 0 1 V = (3.22) M U 1 0 T 0 1       for some matrices U, V, S, T . Expanding the right-hand side gives AB SSV = . CD UST + USV     1 1 This shows that S = A, and since A is invertible, we get U = CA− , V = A− B and T = 1 D CA− B, which are exactly the blocks in (3.20). This proves the uniqueness assertion. − The proof of the invertible D case is analogous. Now assume that is invertible and there exists some its decomposition of the form (3.22). We see that det =M det S det T , which forces both S and T to be invertible. But we have shown that S hasM to be A, and· thus A is invertible, and we have shown that the blocks U, V, T are uniquely determined by . The proof for the other decomposition is analogous. M  Consider now a general orthogonal transformation , parametrized as in (3.2). If one O assumes that either O1 or O4 is invertible, there always exists one of the decompositions in Lemma 3.1.2. Are the maps in the decomposition orthogonal? The answer is given by the following proposition. Proposition 3.1.3. Let O(n, n) be parametrized as in (3.2). O ∈ 2 2 Let O1 End(TM) be an invertible map. Then there exist B Λ V ∗ and Π Λ V , • such that∈ ∈ ∈ 1 0 O1 0 1 Π = T . (3.23) O B 1 0 O− 0 1    1    Moreover, any such B and Π are unique.

2 2 Let O4 End(T ∗M) be an invertible map. Then there exist B0 Λ V ∗ and Π0 Λ V , • such that∈ ∈ ∈ T 1 Π O− 0 1 0 = 0 4 . (3.24) O 0 1 0 O4 B0 1       Moreover, any such B0 and Π0 are unique.

Proof. Let O1 be an invertible map. By Lemma 3.1.2, there exists a decomposition (3.20), and 1 1 we get B = O3O1− , A = O1, and Π = O1− O2. We have to show that B is induced by a 2-form T 1 T T on V , that is B +B = 0. This reduces to O3O1− +O1− O3 = 0. Multiply this equation by O1 T T T from the right, and by O1 from the left. This gives O1 O3 + O3 O1 = 0. But this is exactly the 2 1 T T equation (3.4). To show that Π Λ V , we are required to prove that O1− O2 + O2 O1− = 0. This reduces precisely to (3.9). At∈ this point we know that

B O1 0 Π = e 1 e . O 0 O4 O3O− O2  − 1  42 B Π 1 Because e and e are in O(n, n), so has to be the middle block. This requires O4 O3O1− O2 = T − O1− . The proof of the second part is analogous. 

Note that there are elements of O(n, n) which can be decomposed in both ways. However, not every orthogonal map, not even from the identity component of O(n, n), can be written in B Π B 2Π this form. Consider for example n = 2 and = e e e− e , where O 0 1 0 1 B = , Π = . (3.25) 1 0 1− 0 −    The resulting orthogonal map has the form 0 0 0 1 0 0 1− 0 =   . (3.26) O 0 1 0 0 − 1 0 0 0      It is a product of exponentials, hence it lies in the identity component of O(n, n). On the other hand, it clearly cannot be decomposed in any way of Proposition 3.1.3.

3.2 Maximally isotropic subspaces

Having the pairing , W with the signature (n, n), it is natural to study its isotropic subspaces. We say that subspaceh· ·iP W is isotropic, iff p, q = 0 for all p, q P . In particular, we will be interested in maximally isotropic⊆ subspaces.h Leti us recall a well-known∈ fact from the theory of quadratic forms. For the proof, see for example [70]. Note that maximally isotropic subspaces are sometimes called Lagrangian subspaces. Lemma 3.2.1. All maximally isotropic subspaces of (W, , ) are n-dimensional. h· ·iW Because , is induced by the canonical pairing, there are two obvious maximally isotropic h· ·iW subspaces, namely V and V ∗, viewed as subspaces of W . By definition, every orthogonal trans- formation O(n, n) applied on V or V ∗ induces an isotropic subspace. O ∈ Example 3.2.2. Let us recall some standard examples of maximally isotropic subspaces of (W, , ). h· ·iW 2 B Let B Λ V ∗, and define the subspace G := e (V ). Explicitly, • ∈ B G = v + B(v) v V V V ∗. (3.27) B { | ∈ } ⊆ ⊕ We can thus view G as a graph of the linear map B Hom(V,V ∗). Conversely, let B ∈ B Hom(V,V ∗) be any linear map. One can always construct the subspace (3.27). It is ∈ always an n-dimensional subspace of W . One readily checks that GB is isotropic if and 2 only if B Λ V ∗. ∈ 2 Π Let Π Λ V , and define the subspace G := e (V ∗). Explicitly, • ∈ Π G = α + Π(α) α V ∗ V V ∗. (3.28) Π { | ∈ } ⊆ ⊕ We can thus view G as a graph of the map Π Hom(V ∗,V ). Conversely, let Π Π ∈ ∈ Hom(V ∗,V ) be any linear map. Onc can always construct the subspace (3.28). It is always an n-dimensional subspace of W . One readily checks that GΠ is isotropic if and only if Π Λ2V . ∈ 43 Let ∆ V be any subspace of V . Let Ann(∆) V ∗ be the annihilator subspace of V ∗, • that is⊆ the vector space defined as ⊆

Ann(∆) = α V ∗ v ∆, α(v) = 0 . (3.29) { ∈ | ∀ ∈ } Then ∆ Ann(∆) W forms a maximally isotropic subspace. ⊕ ⊆ 2 Let E V be a vector subspace of V , and let θ Λ E∗. Define the vector space • ⊆ ∈ L(E, θ) = v + α V V ∗ v E, and α = θ(v) (3.30) { ∈ ⊕ | ∈ } Then L(E, θ) is a maximally isotropic subspace. Moreover, every maximally isotropic subspace is of this form for some E and θ, see [39].

3.3 Vector bundle, extended group and Lie algebra

We can generalize everything from the previous two sections to the vector bundle E = TM ⊕ T ∗M. Subspaces will be replaced by subbundles, and linear maps are promoted to vector bundle morphisms over the identity map on M. For example, we define

O(n, n) = Aut(E) (e), (e0) = e, e0 for all e Γ(E) . (3.31) {F ∈ | hF F iE h iE ∈ } Similarly for the orthogonal Lie algebra:

o(n, n) = End(E) (e), e0 + e, (e0) = 0 for all e Γ(E) . (3.32) {F ∈ | hF iE h F iE ∈ } By a direct generalization of (3.14) we would arrive to

o(n, n) = End(TM) Ω2(M) X2(M). (3.33) ∼ ⊕ ⊕ We now have O(n, n) transformations of the form of B-transforms, Π-transforms and Aut(TM) at our disposal, for B Ω2(M) and Π X2(M). Finally, instead of maximally isotropic subspaces, we will talk∈ about maximally∈ isotropic subbundles of E. All examples from the previous subsection generalize naturally. Of course, we can study also slightly more general objects. In particular, define extended au- tomorphism group EAut(E) of E to be a group of fiber-wise bijective vector bundle morphisms over diffeomorphisms. Note that any ( , ϕ), where ϕ Diff(M), induces an automorphism (denoted by the same letter) of Γ(EF). Indeed, let∈e Γ(E). Define (e) Γ(E) as (F (e))(ϕ(m)) = (e(m)) for all m M. ∈ F ∈ F F ∈ Using this notation, we can define the extended orthogonal group EO(n, n) as

EO(n, n) = ( , ϕ) EAut(E) (e), (e0) ϕ = e, e0 . (3.34) { F ∈ | hF F iE ◦ h iE} Its structure is in fact very simple, as the following lemma proves. Lemma 3.3.1. Let Diff(M) be the group of diffeomorphisms of M. Then

EO(n, n) = O(n, n) o Diff(M), (3.35)

1 where Diff(M) acts on O(n, n) by conjugation: ϕ I 0 := T (ϕ) 0 T (ϕ)− , for all ϕ F ◦ F ◦ 1 ∈ Diff(M). Define the map (T (ϕ), ϕ) EAut(E) by putting T (ϕ)(X + ξ) = ϕ (X) + (ϕ− )∗(ξ), for all X + ξ Γ(E). ∈ ∗ ∈ 44 Proof. Let (F, ϕ) EO(n, n). It is not difficult to show that (T (ϕ), ϕ) EO(n, n). Define ∈ ∈ 0 Aut(A) as = 0 T (ϕ). Then, by definition, 0 O(n, n). Moreover, O(n, n) forms aF normal∈ subgroupF ofFEO◦ (n, n). It only remains to determineF ∈ the multiplication rule. Let ( , ψ) EO(n, n) and = T (ψ). We get G ∈ G G0 ◦ 1 = [T (ϕ) T (ϕ)− ] T (ϕ ψ). F ◦ G F0 ◦ ◦ G0 ◦ ◦ ◦ This proves the semi-direct structure assertion (3.35). 

What is the Lie algebra Eo(n, n) corresponding to the group EO(n, n)? First, recall the vector bundle (E) defined in Remark 2.4.2. For any Lie algebroid (L, l, [ , ]L), any vector bundle morphismD R : L (E) preserving the brackets is called a representation· · of Lie algebroid L on the vector bundle→ D E. See [74] for details. We claim that Γ( (E)) is exactly the Lie algebra corresponding to EAut(E). To see this, assume that ( , ϕ )D is a 1-parameter subgroup of automorphisms in EAut(E). In particular, Ft t ϕt is a 1-parameter subgroup of Diff(M), hence a flow of some vector field X X(M). Define d ∈ : Γ(E) Γ(E) as (e) := t(e) for all e Γ(E). Note that for f C∞(M), we F → F dt t=0 F− ∈ ∈ have t(fe) = (f ϕt) t(e). Differentiating this condition with respect to t at t = 0 gives F− ◦ F−

d (fe) = f (e) + [ f(ϕ )]e = f (e) + (X.f)e. (3.36) F F dt t F t=0

This proves that Γ( (E)), and moreover a( ) = d ϕ . We can also write the relation F ∈ D F dt t=0 t of and t as exp (t ) := t. F F F F−

Let us now return to the Lie algebra Eo(n, n). Let ( t, ϕt) be a 1-parameter subgroup of F d EO(n, n). The corresponding element of Eo(n, n) will be = dt t=0 t. Since t EO(n, n), we have F F− F ∈

t(e), t(e) E = e, e0 E ϕt. (3.37) hF− F− i h i ◦ Differentiating this with respect to t at t = 0 leads us to the definition

Eo(n, n) := Γ( (E)) a( ). e, e0 = (e), e0 + e, (e0) , e, e0 Γ(E) . (3.38) {F ∈ D | F h iE hF iE h F iE ∀ ∈ } Recall that ( (E), a, [ , ]) is the Lie algebroid defined in Remark 2.4.2. Lemma 3.3.1 suggests that Eo(n, n)D can be also· · written as a semi-direct product, this time of Lie algebras. Before proceeding to the lemma, note that there is a Lie algebroid representation R : TM (E) of the Lie algebroid (T M, Id , [ , ]) which takes values in Eo(n, n). Indeed, let X →X D(M). M · · ∈ Define R(X) Γ( (E)) as R(X)(e) = [X +0, e]D. It follows from (2.26) that R(X) Eo(n, n). We can now state∈ D ∈ Lemma 3.3.2. Lie algebra Eo(n, n) can be decomposed as

Eo(n, n) = X(M) n o(n, n), (3.39) where X(M) acts on o(n, n) by Lie derivatives.

Proof. Let Eo(n, n). Define 0 o(n, n) as = R(a( )) + 0. This proves the as- sertion on theF ∈ level of vector spaces.F ∈ First note thatF [R(X),RF (Y )]F = R([X,Y ]) because R is a Lie algebroid representation. From (3.33) we see that every 0 o(n, n) corresponds to a triplet (N,B, Π) End(TM) Ω2(M) X2(M). Going throughF ∈ the construction of ∈ ⊕ ⊕ this correspondence above (3.14) it is straightforward to show that if 0 (N,B, Π), then [R(X), ] ( N, B, Π) where N is viewed as (1, 1)-tensor onFM≈. This proves the F0 ≈ LX LX LX assertion (3.39) on the level of Lie algebras. 

45 3.4 Derivations algebra of the Dorfman bracket

Let us now focus on the Dorfman bracket (2.14). It satisfies the Leibniz rule (2.4) in the right input, and Courant algebroid induced Leibniz rule (2.30) in the left input. We will now examine the Lie algebra Der(E) of its derivations defined as

Der(E) = Γ( (E)) ([e, e0] ) = [ (e), e0] + [e, (e0)] , e, e0 Γ(E) . (3.40) {F ∈ D | F D F D F D ∀ ∈ } Let us emphasize that Der(E) is not a C∞(M)-module. Recall the map R defined just before Lemma 3.3.2. It follows from the Leibniz identity (2.5) that for every X X(M), we have R(X) Der(E). Now observe that any Der(E) can be decomposed as ∈ ∈ F ∈ = R(a( )) + . (3.41) F F F0 Note that is now C∞(M)-linear, or equivalently End(E). Moreover, it is a difference F0 F0 ∈ of two derivations, hence itself a derivation. We can now focus on finding all 0 Der(E) End(E). First, there are now certain restrictions forced by the compatibilityF of∈ the Leibniz∩ rule (2.4) and the derivation property (3.40). Indeed, evaluating the derivation 0 on [e, fe0] in two ways, we obtain F ρ( (e)) = 0, (3.42) F0 for all e Γ(E). This shows that must have a formal block form ∈ F0 0 0 0 = , (3.43) F F21 F22   where F21 Hom(TM,T ∗M) and F22 End(T ∗M). Next, there comes the compatibility with the left Leibniz∈ rule (2.30). We obtain∈ the condition

e, e0 ( f) = (e), e0 + e, (e0) f, (3.44) h iEF0 D {hF0 iE h F0 iE}D which has to hold for all e, e0 Γ(E) and f C∞(M). In particular, for e, e0 = 0 this implies ∈ ∈ h iE 0(e), e0 E + e, 0(e0) E = 0. This immediately implies that F21(X),Y + X,F21(Y ) = 0, hF i h F i 2 h i h i and thus F21(X) = B(X) for B Ω (M). Choosing e = X X(M) and e0 = ξ Ω(M), we get ∈ ∈ ∈ ξ, X F (df) = F (ξ),X df. (3.45) h i 22 h 22 i This has to hold for any (f, X, ξ), which is possible only if F22(ξ) = λξ for some λ C∞(M). We see that has to have the form ∈ F0 0 0 = . (3.46) F0 B λ 1  ·  It remains to plug this into condition (3.40) to find the conditions on B and λ. We have [X + ξ, Y + η] = B([X,Y ]) + λ( η i dξ), (3.47) F0 D LX − Y [ (X + ξ),Y + η] = i d(B(X) + λξ), (3.48) F0 D − Y [X + ξ, (Y + η)] = (B(Y ) + λη). (3.49) F0 D LX Inserting this into (3.40) yields two independent equations B([X,Y ]) = (B(Y )) i d(B(X)), (3.50) LX − Y λ η = (λη). (3.51) LX LX Recall that B(X) = iX B. We can use the usual Cartan formulas to rewrite (3.50) as a − 2 condition dB = 0, that is B Ωclosed(M). Second equation forces λ to be locally constant, that is λ Ω0 (M). We have∈ just proved the following proposition. ∈ closed 46 Proposition 3.4.1. Let Der(E) be the space of derivations of the Dorfman bracket [ , ]D, that is (3.40) holds. Then as a vector space, it decomposes as · · . Der(E) = X(M) Ω0 (M) Ω2 (M). (3.52) ⊕ closed ⊕ closed

Every Der(E) decomposes uniquely as = R(X) + λ + B, where R(X)(Y + η) = F ∈ 1F F F ([X,Y ], X η) for all X,Y X(M) and η Ω (M). Vector bundle endomorphisms λ and B are definedL as ∈ ∈ F F 0 0 0 0 = , = , (3.53) Fλ 0 λ 1 FB B 0  ·    where λ Ω0 (M) and B Ω2 (M). Nontrivial commutation relations are ∈ closed ∈ closed [R(X),R(Y )] = R([X,Y ]), (3.54)

[R(X),FB] = F B, (3.55) LX [ , ] = . (3.56) Fλ FB FλB On the Lie algebra level, we thus have

0 2 Der(E) = X(M) n (Ωclosed(M) n Ωclosed(M)), (3.57)

0 2 0 where Ωclosed(M), Ωclosed(M) are viewed as Abelian Lie algebras, Ωclosed(M) acts on 2-forms 2 0 2 in Ωclosed(M) by multiplication, and X(M) acts on Ωclosed n Ωclosed(M) by Lie derivatives. Finally, when we restrict to the subalgebra Eo(n, n), we have

2 Der(E) Eo(n, n) = X(M) n Ω (M). (3.58) ∩ closed Proof. We have proved the first part in the text above. The commutation relations can be directly calculated. 

3.5 Automorphism group of the Dorfman bracket

Let us now examine the group of Dorfman bracket automorphisms. Its subgroup of orthogonal automorphisms is well-known for a long time and it is in fact one of the main reasons why the Dorfman bracket and generalized geometry play such an important role in string theory. We roughly follow the proof of Gualtieri in [39]. From the Courant algebroid perspective is makes sense to restrict to EO(n, n), because in this case one obtains an automorphism of the whole Courant algebroid structure. However, for the sake of generalization to Leibniz algebroids where there is no pairing anymore, we will discuss the whole automorphism group. We define the Dorfman bracket automorphism group AutD(E) as

Aut (E) := ( , ϕ) EAut(E) [ (e), (e0)] = [e, e0] , e, e0 Γ(E) . (3.59) D { F ∈ | F F D F D ∀ ∈ }

This group decomposes similarly as EO(n, n) in Lemma 3.3.1. Indeed, let ( , ϕ) in AutD(E). 1 F Then recall the map (T (ϕ), ϕ), defined as T (ϕ)(X+ξ) = ϕ (X)+(ϕ− )∗(ξ) for all X+ξ Γ(E). It follows from the usual properties of the Lie derivative∗ and the exterior differential∈ that (T (ϕ), ϕ) Aut (E). Define Aut(E) as = T (ϕ). It follows that Aut (E), ∈ D F0 ∈ F F0 ◦ F0 ∈ D and we can thus focus on finding all vector bundle morphisms 0 over the identity preserving the bracket. F

47 First note that it follows from the compatibility of (3.59) and Leibniz rule (2.4) that its projection using ρ satisfies ρ( 0(e)) = ρ(e) for all e Γ(E). This shows that 0 has to be of the block form F ∈ F 1 0 0 = . (3.60) F F21 F22   The Leibniz rule compatibility in the left input (2.30) gives the condition

e, e0 ( f) = (e), (e0) f, (3.61) h iEF0 D hF0 F0 iED for all e, e0 Γ(E). Very similarly to the equation (3.44), this proves that F21(X) = B(X) for 2 ∈ B Ω (M), and F22(ξ) = λ ξ for some λ C∞(M). We require 0 to be fiber-wise bijective, and∈ thus λ(m) = 0 for all m· M. This restricts∈ to have the blockF form 6 ∈ F0 1 0 = . (3.62) F0 B λ 1  ·  Finally, we have to plug into the condition (3.59). We have F0 [X + ξ, Y + η] = [X,Y ] + B([X,Y ]) + λ η i dξ , (3.63) F0 D {LX − Y } [ (X + ξ), (Y + η)] = [X,Y ] + (B(Y ) + λη) i d(B(X) + λξ). (3.64) F0 F0 D LX − Y Combining these two expressions gives the same conditions on B and λ as (3.40). We thus 2 0 0 get B Ωclosed(M), and λ Ωclosed(M). Let G(Ωclosed(M)) denote the Abelian group of ∈ ∈ 0 everywhere non-zero closed 0-forms on M. Note that G(Ωclosed(M)) is in fact isomorphic to a direct product of k copies of (R 0 , ), where k is a number of connected components of M. We have just proved the following\{ proposition:} ·

Proposition 3.5.1. Let AutD(E) be the group of automorphisms (3.59) of the Dorfman bracket (2.14). Then it has the following group structure:

2 0 AutD(E) = (Ωclosed(M) o G(Ωclosed(M))) o Diff(M), (3.65)

2 0 where Ωclosed(M) is viewed as an Abelian group with respect to addition, G(Ωclosed(M)) acts 2 2 0 on Ωclosed(M) by multiplication, and Diff(M) acts on Ωclosed(M) o G(Ωclosed(M)) by inverse pullbacks. Every ( , ϕ) Aut (E) can be uniquely decomposed as F ∈ D = eB T (ϕ), (3.66) F ◦ Sλ ◦ B 2 where e = exp B, and Sλ(X + ξ) = X + λξ, for unique B Ωclosed(M) and locally constant F 0 ∈ everywhere non-zero function λ G(Ωclosed(M)). Finally, the subgroup of AutD(E) consisting of (extended) orthogonal transformations∈ is

2 AutD(E) EO(n, n) = Ω o Diff(M). (3.67) ∩ closed B0 Proof. Only the multiplication rules remain to be proved. Let = e Sλ0 T (ϕ0). By the direct calculation, one obtains G ◦ ◦

B+λ(ϕ−1)∗B0 = e S −1 ∗ 0 T (ϕ ϕ0). (3.68) F ◦ G ◦ λ(ϕ ) λ ◦ ◦ Symbolically, this yields the multiplication rule

1 1 (B, λ, ϕ) (B0, λ0, ϕ0) = (B + λ(ϕ− )∗B0, λ(ϕ− )∗λ0, ϕ ϕ0), (3.69) ∗ ◦ which is exactly the double semi-direct product (3.65). 

48 Finally, let us show that every Dorfman bracket derivation Der(E) can explicitly be F ∈ integrated to a 1-parameter subgroup exp (t ) AutD(E) of the group of Dorfman bracket automorphisms. Note that t ( , ) for someF ⊆ > 0, and in general, t cannot be expanded to ∈ − R. exp (t ) is thus a 1-parameter subgroup with ”certain conditions on parameters”. F We have shown in Proposition 3.4.1 that every Der E can uniquely be written as 0 F ∈ 2 X = R(X) + Fλ + B for X X(M), λ Ωclosed(M), and B Ωclosed(M). Let φt be the flowF correspondingF to X, for t∈ ( , ). There∈ lies the reason of∈t limitations: X may not be a complete vector field. We will∈ now− look for exp (t ) in the form F B(t) X exp (t ) = e Sµ(t) T (φ t), (3.70) F ◦ ◦ − where B(t) Ω2 (M), and µ(t) G(Ω0 (M)) for every t ( , ). We have ∈ closed ∈ closed ∈ − X X X exp (t )(Y + η) = φ t (Y ) + B(t)(φ t (Y )) + µ(t)φt ∗(η), (3.71) F − ∗ − ∗ for all Y + η Γ(E). Differentiating with respect to t at t = 0 gives the condition ∈ d d (Y + η) = (1 + B(0))[X,Y ] + [ B(t)](Y ) + [ µ(t)]η + µ(0) η. (3.72) F dt dt Lx t=0 t=0

Comparing this with our parametrization of gives the conditions on B(t), and µ(t): F d d µ(t) = λ, µ(0) = 1, B(t) = B,B(0) = 0. (3.73) dt dt t=0 t=0

First two conditions give µ (t) = exp tλ. To find the solution for B, we will use the 1-parameter X subgroup property of exp (t ). Note that we have φt ∗(λ) = λ. This follows from the fact that flows cannot flow outside ofF the single connected component. Using the multiplication rule (3.68), we get

X∗ B(t)+exp tλ φt (B(s)) X exp (t ) exp (s ) = e · Se(t+s)λ T (φ (t+s)). (3.74) F ◦ F ◦ ◦ − Comparing this to exp ((t + s) ) gives the condition F X B(t + s) = B(t) + exp tλ φ ∗(B(s)). (3.75) · t Differentiate both sides with respect to s at s = 0. This yields X B˙ (t) = exp tλ φ ∗B, (3.76) · t and consequently t X B(t) = exp kλ φ ∗B dk. (3.77) { · k } Z0 It is straightforward to check that such B(t) indeed satisfies (3.75) and the two initial conditions (3.73). We have thus made our way to the following proposition: Proposition 3.5.2. Let Der(E) be a derivation of the Dorfman bracket. Then there F ∈ is an  > 0 and a 1-parameter subgroup exp (t ) AutD(E), where t ( , ), such that d F ⊆ ∈ − 0 = dt t=0 exp (t ). Explicitly, if = R(X) + λ + B for X X(M), λ Ωclosed(M), and F 2 F F F F ∈ ∈ B Ωclosed(M), we have ∈ B(t) X exp (t ) = e Sµ(t) T (φ t), (3.78) F ◦ ◦ − X where µ(t) = exp tλ, φt is the flow of X, and t X B(t) = exp kλ φ ∗B dk. (3.79) { · k } Z0

49 Proof. We have shown above that exp (t ) integrates . We just have to show that for each F F t, exp (t ) AutD(E). According to Proposition 3.5.1, this happens if and only if B(t) Ω2 (MF ),∈ and µ(t) G(Ω0 (M)). But this clearly holds because d commutes with the∈ closed ∈ closed integration and pullbacks. 

3.6 Twisting of the Dorfman bracket

We have shown in Proposition 3.5.1 that Aut(E) preserving the bracket must be of 2 F ∈0 the form (3.60) for B Ωclosed(M) and λ Ωclosed(M). Let us now focus only on O(n, n) transformations, and thus∈ set λ 1. In this∈ case simply = eB. What happens with the ≡ F bracket for dB = 0? This is what we will examine in this section. Define a new bracket [ , ]D0 as 6 · · B B B [e, e0]D0 = e− [e (e), e (e0)]D, (3.80) for all e, e0 Γ(E). Rewriting this bracket explicitly gives ∈ [X + ξ, Y + η]0 = [X,Y ] + (η + B(Y )) i d(ξ + B(X)) B([X,Y ]) D LX − Y − = [X + ξ, Y + η] + (B(Y )) i d(B(X)) B([X,Y ]) (3.81) D LX − Y − = [X + ξ, Y + η] dB(X,Y, ). D − ·

This proves that [ , ]D0 is precisely the H-twisted Dorfman bracket (2.32), where H = dB. One can in fact show· · something more general: Twisted Dorfman brackets corresponding to the different representatives of the same cohomology class [H] H3(M, R) are related precisely by a B-transform. ∈ Proposition 3.6.1. Let H Ω3 (M), and B Ω2(M). Then ∈ closed ∈ B B H B H+dB [e (e), e (e0)]D = e ([e, e0]D ). (3.82)

Proof. Just repeat the calculation (3.81). 

3.7 Dirac structures

In Section 3.2, we have introduced maximally isotropic subspaces and their examples. Gen- eralizing this to the vector bundle E = TM T ∗M, we have an additional structure at our disposal, namely the Dorfman bracket. It is a⊕ well known fact that subbundles of TM which are involutive with respect to the vector field commutator bracket are of an additional geo- metrical significance - they are tangent bundles to integral submanifolds of distributions. This justifies why it is interesting to study the subbundles of E involutive with respect to the Dorfman bracket [ , ]D. In particular, one is interested in involutive subbundles, where the skew-symmetry ”anomaly”· · (2.28) disappears. This is precisely the main idea leading to the definition of Dirac structures. Let us remark that study of Dirac structures was in fact the origin of all Courant algebroid brackets, see [26]. Definition 3.7.1. Let (E, ρ, , , [ , ] ) be a Courant algebroid. A subbundle L E is h· ·iE · · E ⊆ called an almost Dirac structure, if for all e, e0 Γ(L), we have e, e0 = 0. ∈ h iE An almost Dirac structure L is called a Dirac structure, if L is involutive with respect to [ , ] , that is [e, e0] Γ(L) for all e, e0 Γ(L). · · E E ∈ ∈ Note that [ , ]E Γ(L) Γ(L) together with ρ Γ(L) forms a Lie algebroid structure on L. · · | × | 50 We can identify several examples of almost Dirac structures of E = TM T ∗M in Example 3.2.2, if we just replace subspaces with subbundles, and elements of exterior⊕ powers of vector spaces by sections of corresponding vector bundles. We will now show under which conditions these become Dirac structures.

Example 3.7.2. Let E = TM T ∗M be equipped with the Dorfman bracket (2.14). ⊕ 2 Let GB be the graph (3.27) of a 2-form B Ω (M). We can examine the involutivity. • Let X + B(X), Y + B(Y ) Γ(G ). Then ∈ ∈ B [X + B(X),Y + B(Y )] = [X,Y ] + (B(Y )) i d(B(X)). D LX − Y We see that [X + B(X),Y + B(Y )] Γ(G ) iff D ∈ B (B(Y )) i d(B(X)) = B([X,Y ]), (3.83) LX − Y for all X,Y X(M). This is once more the condition (3.50), equivalent to dB = 0. We conclude that∈ G is a Dirac structure iff B Ω2 (M). B ∈ closed 2 Let GΠ be a graph (3.28) of a bivector Π X (M). Involutivity condition implies the • equation ∈ [Π(ξ), Π(η)] = Π( η i dξ), (3.84) LΠ(ξ) − Π(η) for all ξ, η Ω1(M). We will show in Chapter 5 that this is equivalent to the Jacobi ∈ identity for f, g := Π(df, dg). We conclude that GΠ is a Dirac structure if and only if Π X2(M){ is a Poisson} bivector. ∈ Let ∆ TM be a subbundle (that is in fact a smooth distribution on M). Let L = • ∆ Ann(∆)⊆ E. Examining the involutivity condition shows that L is a Dirac structure, iff⊕ [∆, ∆] ∆,⊆ that is ∆ is an integrable distribution. ⊆

3.8 Generalized metric

We will now introduce a key concept for the applications of generalized geometry in string theory. In the context of generalized geometry, it appeared first in [39]. The name generalized metric was probably used for the first time by Hitchin in [43]. Generalized metric has several equivalent formulations, which we all present here.

Definition 3.8.1. Let E be a vector bundle with a fiber-wise metric , E. Let τ End(E) be an involution of E, that is τ 2 = 1. We say that τ is a generalizedh· metric·i , if the∈ formula

G (e, e0) := e, τ(e0) , (3.85) τ h iE for all e, e0 Γ(E), defines a positive definite fiber-wise metric G on E. When talking about ∈ τ generalized metric, we will not distinguish between τ and Gτ .

There are some remarks to be made about τ. It follows from the definition of a generalized metric, that τ must be symmetric with respect to , , and consequently also orthogonal. h· ·iE There is one nice property of involutive maps.

51 Lemma 3.8.2. Let V be a finite-dimensional real vector space, and A End(V ) satisfies 2 ∈ A = 1. Then A has eigenvalues 1 and it is diagonalizable, that is V = V+ V , and ± ⊕ − A(v+ + v ) = v+ v . − − − 2 Proof. Let p(z) = z 1. Then p(A) = 0, and a minimal polynomial mA of A thus must divide p. For m (z) = z 1,− this would imply A = 1. In all other cases m (z) = (z +1)(z 1). This A ± ± A − shows that all roots of mA have multiplicity 1, which is equivalent to A being diagonalizable. Moreover, its roots are precisely the eigenvalues of A. 

We can now use this result to reformulate the definition of generalized metric in case when , has a constant signature (the same at each fiber). h· ·iE Proposition 3.8.3. Let E be a vector bundle with a fiber-wise metric , E of constant sig- nature (p, q). Definition 3.8.1 of generalized metric τ is then equivalenth· to·i a definition of a positive subbundle V E of maximal possible rank p. + ⊆ Proof. First, let τ End(E) be a generalized metric. It induces an involution τ in each ∈ m fiber Em. By previous Lemma, there exist its 1 eigenspaces Vm+ and Vm , such that Em = ± − Vm+ Vm . By definition of generalized metric τ, Vm+ and Vm are positive definite and ⊕ − − negative definite subspaces respectively. By our assumption, this implies dim Vm+ = p, and dim Vm = q. Now define − V := ker (τ 1). (3.86) ± ∓ We have just proved that vector bundle morphisms τ 1 have both constant rank, and V ∓ ± are thus well-defined subbundles of E. Moreover, rank V+ = p, and V+ is a positive definite subbundle. Note that E = V+ V , and V = (V+)⊥, where the orthogonal complement is taken with respect to , . ⊕ − − ⊥ h· ·iE Conversely, let V E be a positive-definite subbundle or rank p. + ⊆ First, having a vector space W with positive definite subspace W W of dimension p with + ⊆ respect to signature (p, q) metric , W , define W := (W+)⊥. Clearly W = W+ W . Is W a negative definite subspace? If thereh· ·i would be a− non-zero strictly positive vector⊕w −W , we− ∈ − could define W+0 = W R w , which would be a positive definite subspace of W of dimension p + 1. This cannot happen.⊕ { If} w W would be a non-zero isotropic vector, we can take any ∈ − nonzero v W+, and define w0 = v+w. Then w0, w0 W = v, v W > 0, and W+0 = W+ R w0 would be a∈ positive definite subspace of W ofh dimensioni ph+ 1,i which is again impossible.⊕ { We} conclude that necessarily w, w < 0. h iW To a positive definite subbundle V+ of rank p, we can define V = (V+)⊥, where is taken with respect to , . This is a well-defined rank q subbundle,− which is by the previous⊥ h· ·iE discussion negative definite, and V = V+ V . We can now define τ End(E) as τ(e+ +e ) := ⊕ − ∈ − e+ e , for e V . One checks easily that τ satisfies all properties required by Definition − − ± ∈ ± 3.8.1. 

To get back to E = TM T ∗M, we now bring an interpretation of the generalized metric most useful for actual calculations.⊕ Proposition 3.8.4. Let E be a vector bundle with a fiber-wise metric , of signature (n, n), h· ·iE and let E = L L∗, where L and L∗ are isotropic subbundles with respect to , . Note that ⊕ h· ·iE L∗ can be identified with the vector bundle dual to L. 2 Generalized metric τ on E is then equivalent to a unique pair (g, B), where g Γ(S L∗) is a positive definite fiber-wise metric on L, and B Ω2(L) is a 2-form on L. ∈ ∈ 52 Proof. Let τ be a generalized metric. We meet the requirements of Proposition 3.8.3, and thus E = V+ V , where rank V = n, and V+ and V form the positive and negative definite ⊕ − ± − subbundles with respect to , . Now, because V L = V L∗ = 0 , we see that V must h· ·iE + ∩ + ∩ { } + be a graph of some vector bundle morphism A Hom(L, L∗). A can uniquely be decomposed 2 ∈ 2 as A = g + B, where g Γ(S L∗), and B Ω (L). Every section e Γ(V+) can be thus written as e = X + (g + B∈)(X), where X Γ(∈L). We obtain ∈ ∈ e, e = X + (g + B)(X),X + (g + B)(X) = 2 g(X),X = 2g(X,X). (3.87) h iE h iE h iE

Note that the canonical pairing between L and L∗ is provided by , E. Because V+ is the positive definite subbundle, we see that g(X,X) > 0 for all nonzeroh· ·iX Γ(L), proving the positivity of the metric g. See that A is always a vector bundle isomorphism.∈ Also note that V must be for the same reasons the graph of some vector bundle morphism A Hom(L, L). − T ∈ From V+,V E = 0 it follows that A = g + B = A . h −i − − 2 e Conversely, let (g, B) be a pair, where g Γ(S L∗) is a positive definite metric and B 2 e ∈ ∈ Ω (L). We can define V+ to be the graph of A = g + B. Repeating the calculation (3.87) shows that V+ is a positive definite subbundle of rank n. This by Proposition 3.8.3 defines a generalized metric on the vector bundle E. 

In the rest of this section, we will assume E = TM T ∗M, and L = TM, L∗ = T ∗M. These satisfy the requirements of the previous proposition.⊕ However, keep in mind that everything works also for general L and L∗.

We will now rewrite the map τ and the corresponding fiber-wise metric Gτ in terms of g and B. First, note that we can explicitly construct the projectors P : E V . Define two isomorphisms Ψ : TM V as ± → ± ± → ± Ψ (X) = X + ( g + B)(X), (3.88) ± ± for all X X(M). Next, note that we can rewrite X + ξ Γ(E) as ∈ ∈ 1 1 X + ξ = (X + (g + B)(X)) + (X + ( g + B)(X)) 2 2 − 1 1 1 1 1 1 + (g− (ξ) + (g + B)(g− (ξ))) (g− (ξ) + ( g + B)(g− (ξ))) 2 − 2 − (3.89) 1 1 1 1 1 1 (g− B(X) + (g + B)(g− B(X))) + (g− B(X) + ( g + B)(g− B(X))) − 2 2 − 1 1 1 1 1 1 = Ψ+ X + g− (ξ) g− B(X) + Ψ X g− (ξ) + g− B(X) . 2 − 2 − − We thus obtain

1 1 1 P (X + ξ) = Ψ X g− (ξ) g− B(X) . (3.90) ± 2 ± ± ∓ We have defined V as 1 eigenbundles of τ. Hence
± ± 1 1 1 1 1 1 τ(X + ξ) = Ψ+(X + g− (ξ) g− B(X)) Ψ (X g− (ξ) + g− B(X)) 2 − − 2 − − (3.91) 1 1 1 1 = g− (ξ) g− B(X) + (g Bg− B)(X) + Bg− (ξ). − − This proves that τ has a formal block form

1 1 g− B g− τ = − 1 1 . (3.92) g Bg− B Bg−  −  53 Fiber-wise metric Gτ in the block form is obtained from τ by multiplying it by matrix gE, which is the same as (3.1). Thus

1 1 g Bg− B Bg− Gτ = − 1 1 . (3.93) g− B g−  −  Now recall Lemma 3.1.2. We see that the top-left and bottom-right blocks are invertible, and thus both decompositions of Gτ exist. We find

1 B g 0 1 0 Gτ = 1 . (3.94) 0 1 0 g− B 1     −  B T B This proves that G = (e− ) e− , where is the block diagonal metric τ GE GE 1 = BDiag(g, g− ). (3.95) GE This observation gives us two interesting facts. First, note that the blocks in the decompo- sition are unique, which re-proves the uniqueness assertions of the preceding proposition. Next, this helps us to prove that not every positive definite fiber-wise metric on E is a generalized metric. We see that det (Gτ ) = 1. Define G := λGτ , where λ = 1 is a positive real constant. G is clearly a positive definite fiber-wise metric on E, but det6 (G) = λ2n = 1. We can now give the last equivalent definition of the generalized metric. 6 Proposition 3.8.5. Let E be a vector bundle, and , be a fiber-wise metric on E. We say h· ·iE that fiber-wise metric G is a generalized metric, if G is positive definite and G Hom(E,E∗) ∈ defines an orthogonal map. We use the fact that the dual vector bundle E∗ is naturally equipped 1 with an induced fiber-wise metric , ∗ = g− . h· ·iE E We claim that this definition coincides with Definition 3.8.1.

Proof. Denote by g Hom(E,E∗) the vector bundle isomorphism induced by a fiber-wise E ∈ metric , E. Let τ End(E) be a generalized metric according to Definition 3.8.1. The propertiesh· ·i of τ can be∈ now written as

T T 2 gEτ = τ gE, τ gEτ = gE, τ = 1. (3.96)

The metric G and τ are related simply as G = g τ. We have to show that e, e0 = τ τ E h iE G (e), G (e0) ∗ . This can be rewritten as the condition h τ τ iE 1 Gτ gE− Gτ = gE. (3.97)

T Plugging in for Gτ translates into τ gEτ = gE. Conversely, let G be a generalized metric 1 according to the definition in 3.8.5. This implies that G satisfies (3.97). Define τ := gE− G. We have ◦ T 1 g τ τ g = g (g G) (Gg− )g = G G = 0. E − E E E − E E − This proves that τ is symmetric with respect to , . Then h· ·iE T 1 1 1 τ gEτ = (GgE− )gE(gE− G) = GgE− G = gE, where we have used (3.97) in the last step. This proves that τ is orthogonal with respect to , . The property τ 2 = 1 follows automatically (any map which is both symmetric and h· ·iE orthogonal is an involution). 

54 To conclude this section, note that there is no actual reason to choose the map A ∈ Hom(TM,T ∗M) in order to describe V+ in the proof of Proposition 3.8.4. One can as well 1 1 1 describe V+ using the map A− Hom(T ∗M,TM), which decomposes as A− = G− + Π, where G is positive definite metric∈ on M, and Π X2(M). The two descriptions are related as ∈ 1 1 (g + B)− = G− + Π. (3.98)

One can find G and Π explicitly in terms of (g, B) as

1 G = g Bg− B, (3.99) − 1 1 1 Π = g− B(g Bg− B)− . (3.100) − −

To obtain this use the decomposition (3.21) for Gτ in the form (3.93), and then note that Gτ can be also decomposed as

1 0 G 0 1 Π Gτ = 1 − . (3.101) Π 1 0 G− 0 1       A comparison of the blocks gives exactly the relations (3.99, 3.100).

3.9 Orthogonal transformations of the generalized metric

There is a natural action of the orthogonal group on the space of generalized metrics. We will analyze this action mainly in terms of the corresponding fields (g, B). We consider E = TM T ∗M. Let τ be a generalized metric, and O(n, n). Define τ 0 End(E) as ⊕ O ∈ ∈ 1 τ 0 := − τ . (3.102) O O T T 2 Then Gτ 0 = gEτ 0 = O gEτ = Gτ . Clearly τ 0 = 1. This means that τ 0 is also a generalized metric. CorrespondingO eigenbundlesO O are related as

τ 0 1 τ V = − V . (3.103) ± O ±

We have also proved that there is always a unique pair (g, B) corresponding to τ. Let (g0,B0) be a pair corresponding to τ 0. How are (g0,B0) and (g, B) related? Let A = g + B, and τ τ 0 A0 = g0 + B0. By definition, V+ = GA, and V+ = GA0 , where GA and GA0 are the graphs of the respective vector bundle morphisms. We will use the notation introduced in Section 3.1. Let X X(M). We have ∈ 1 T T T T − (X + A(X)) = (O + O A)(X) + (O + O A)(X). O 4 2 3 1 T T Define Y = (O4 + O2 A)(X). Then

1 T T T T 1 − (X + A(X)) = Y + (O + O A)(O + O A)− (Y ). O 3 1 4 2 T T If the inverse of O4 + O2 A exists, we get the following formula for A0:

T T T T 1 A0 = (O3 + O1 A)(O4 + O2 A)− . (3.104)

Recall the isomorphisms Ψ Hom(TM,V ) defined by (3.88), and let Ψ Hom(T ∗M,V ) ± ∈ ± 1 ± ∈ 1 ± be similarly induced isomorphisms: Ψ (ξ) = ξ + ( G− + Π)(ξ), for all ξ Ω (M). Define ± ± e ∈ e 55 new vector bundle morphisms Φ , Υ by the following commutative diagram (in fact there are two independent diagrams, one± for± +, one for ): − Υ± T ∗M T ∗M 0 τ τ Ψe ± Ψe ± −1 τ τ 0 (3.105) A V O V A0 ± ± Ψτ τ0 ± Ψ± Φ TM ± TM All involved maps are vector bundle isomorphisms, and so have to be Φ , Υ . We can find explicit formulas: ± ± T T T T T Φ+ = O4 + O2 A, Φ = O4 O2 A , (3.106) − − T T 1 T T T Υ+ = O1 + O3 A− , Υ = O1 O3 A− . (3.107) − − This proves that the inverse in (3.104) exists. These maps in fact transform between the involved Riemannian metrics. Proposition 3.9.1. There hold the following conjugation relations: 1 1 T g0− = Φ g− Φ , (3.108) ± ± 1 1 T g0 B0g0− B0 = Υ (g Bg− B)Υ . (3.109) − ± − ± Proof. We will prove only one of the four equations, because the other ones follow in the same τ τ way. First note that Ψ (X), Ψ (Y ) = 2g(X,Y ), and similarly for τ 0. Hence h + + iE τ τ τ 0 τ 0 2g(X,Y ) = Ψ (X), Ψ (Y ) = Ψ (Φ (X)), Ψ (Φ (Y )) = 2g0(Φ (X), Φ (Y )). h + + iE hO + + O + + iE + + T Thus g = Φ+g0Φ+, which is precisely (3.108) with the + sign. 

Now, note that formula (3.104) can be rewritten as 1 T T A0 = Υ+AΦ+− = Φ− AΥ . (3.110) − − The latter expression can be found as the analogue of (3.104) derived using the subbundle V . − Together with the previous proposition, we can find transformation rules for B0. T 1 T 1 B0 = ((Υ+ Φ+− )g + Υ+B)Φ+− = ( (Υ Φ− )g + Υ B)Φ− . (3.111) − − − − − − − Using the orthogonal group, we can describe the set of all generalized metrics in a more intrinsic way. To do so, we first need to prove two following lemmas. Lemma 3.9.2. The action of O(n, n) on the set of generalized metrics is transitive.

B T B Proof. Let G and G0 be two generalized metrics on E. Then G = [e− ] Ee− , and G0 = B0 T B0 B B0 G [e− ] E0 e− . Because e− and e− are O(n, n) transformations, it suffices to show that G T there exists O(n, n), such that E0 = E . For any two Riemannian metrics g and g0 O ∈ G O G O T on M, there is a vector bundle isomorphism N Aut(TM), such that g0 = N gN. Define := , where is a block diagonal map ∈ O ON ON N 0 N = T . (3.112) O 0 N −   T Obviously O(n, n) and 0 = . This finishes the proof. ON ∈ GE ON GEON  56 On the other hand, the action of O(n, n) is not free, as the next lemma shows.

Lemma 3.9.3. Let G be a generalized metric. Let O(n, n)G O(n, n) be its stabilizer sub- group. Then ⊆ O(n, n)G = O(n) O(n). (3.113) ∼ ×

Proof. Any morphism stabilizing G must preserve the subbundles V+ and V , it is thus O − block diagonal with respect to decomposition E = V+ V . Moreover, , E has the form ⊕ − h· ·i

e+ + e , e+0 + e0 E = e+, e+0 + e , e0 , (3.114) h − −i h i − h − −i− for all e , e0 V , and , are positive definite fiber-wise metrics on V . This proves ± ± ∈ ± h· ·i± ± that O(n, n) iff both its diagonal blocks are in O(n). We conclude that O(n, n)G = O(n)OO ∈(n). ×  These two observations are sufficient to describe the set of all generalized metrics on E. Proposition 3.9.4. The set of all generalized metrics is the coset space O(n, n)/(O(n) O(n)). ×

Proof. There always exists at least one generalized metric, for example E for some Riemannian metric g. On every manifold M, there exists some g (it is constructedG using the partition of unity). The space of all generalized metrics is its orbit by Lemma 3.9.2. But every orbit is isomorphic to O(n, n)/O(n, n)G, and thus by Lemma 3.9.3 to O(n, n)/(O(n) O(n)). ×  Example 3.9.5. Let us conclude this section with a few examples of the O(n, n) actions on the generalized metric G described by a pair of fields (g, B).

2 Z Let Z Ω (M) be a 2-form. Set = e− . In particular, we have • ∈ O O = 1,O = 0,O = Z,O = 1. (3.115) 1 2 3 − 4 1 Hence Φ = 1, Υ = 1 + Z( g + B)− . We get g0 = g, and ± ± ± B0 = ( (Υ 1)g + Υ B) = B + Z. (3.116) ∓ ± − ± Z T B T B Z Of course, we could have seen this directly from G0 = (e− ) [(e− ) Ee− ]e− . In G 1 particular, if B and B0 are related by a gauge transformation, B0 = B+da for a Ω (M), we can interpret the gauge transformation as the O(n, n) transformation of the generalized∈ metric (g, B) (g, B + da). 7→ Let θ X2(M) be a 2-vector. Set = eθ. In particular, we have • ∈ O

O1 = 1,O2 = θ, O3 = 0,O4 = 1. (3.117) This implies Φ = 1 + θ( g + B), Υ = 1. The resulting relations are more clear in ± ± ± terms of dual fields (G, Π) and (G0, Π0) defined by (3.98). We get

G0 = G, Π0 = Π θ. (3.118) − One can thus write the relations as 1 1 = + θ. (3.119) g + B g0 + B0 These are precisely the open-closed relations of Seiberg-Witten as they appeared in [85].

57 Let N Aut(E), and let = . Then • ∈ O ON T O1 = N,O2 = 0,O3 = 0,O4 = N − , (3.120) 1 T T T and consequently Φ = N − , and Υ+ = N . We get g0 = N gN, and B0 = N BN. This proves that a± change of frame in TM and its consequences for g and B can be incorporated as a special case of O(n, n) transformation.

d+1 µ Consider M = R for d > 0, and coordinates (x , x•), µ 1, . . . , d . Define the vector • bundle morphism T Aut(E) as ∈ { } ∈ µ µ T (∂µ) = ∂µ,T (dx ) = dx ,T (∂ ) = dx•,T (dx•) = ∂ . (3.121) • • It is easy to see that T O(n, n), where now n = d + 1. We can write the matrix of T in block form as ∈ 1d 0 0 0 0 0 0 1 T =   , (3.122) 0 0 1d 0  0 1 0 0   where 1d is a d d identity matrix. We can write the matrix of metric g, and matrix of B Ω2(M) in block× forms as ∈ gˆ g BBˆ g = • ,B = , (3.123) gT g BT 0•  • •• − •  whereg ˆ is a d d matrix (ˆg)µν = gµν , and similarly with other components. We thus have × 1 0 0 0 0 0 1 0 O = d ,O = ,O = ,O = d . (3.124) 1 0 0 2 0 1 3 0 1 4 0 0         1d 0 1d 0 1d 0 Φ = T T = T T . (3.125) ± g B g g B 1 0 g ± • − • ± •• ± • − •   ± •• These maps are invertible, because g > 0. We get ••

1 1d 0 1d 0 1d 0 Φ− = 1 T T = 1 T T 1 (3.126) ± 0 g g + B 1 g ( g + B ) g  ± ••  ∓ • •  ± •• ∓ • • ± ••  The simplest way to (g0,B0) is now through (3.104), because we have already calculated 1 T T 1 Φ− (O + O A)− . We have + ≡ 4 2 gˆ + Bˆ g + B OT + OT A = . (3.127) 3 1 0• 1 •   Plugging into (3.104) now gives gˆ + Bˆ + 1 (g + B )( gT + BT ) 1 (g + B ) g•• • • g•• • • A0 = − • • (3.128) 1 T T 1 g ( g + B ) g ! •• − • • •• Reading off the symmetric and skew-symmetric part, we obtain

1 T T 1 1 gˆ0 =g ˆ + (B B g g ), g0 = B , g0 = , (3.129) g • • − • • • g • •• g •• •• •• 1 T T 1 Bˆ0 = Bˆ + (g B B g ),B0 = g . (3.130) g • • − • • • g • •• ••

58 But these are exactly the well-known Buscher rules [20] emerging from string theory’s T -duality. The generalized geometry thus allows to describe T -duality as an orthogonal transformation of the generalized metric.

3.10 Killing sections and corresponding isometries

Let G be a given generalized metric on E. Up to now, we have discussed only the isometries formed from O(n, n) maps, concluding that O(n) O(n) is the subgroup preserving G. We can generalize the notion of isometry as follows. × Definition 3.10.1. We define the group EIsom(G) of extended isometries as

EIsom(G) = ( , ϕ) EAut(E) G( (e), (e0)) ϕ = G(e, e0) . (3.131) { F ∈ | F F ◦ }

We have shown that EIsom(G) O(n, n) ∼= O(n) O(n). We do not intend to find the whole group EIsom(G). Instead, we∩ will find an important× class of examples - solutions to the Killing equation. To find it, let us assume that ( t, ϕt) EIsom(G) is a one-parameter d F ⊆ subgroup and define Γ( (E)) as = dt t=0 t. By differentiating (3.131) with respect to t at t = 0, we obtainF ∈ D F F−

a( ).G(e, e0) = G( (e), e0) + G(e, (e0)), (3.132) F F F for all e, e0 Γ(E). This is still a way too complicated equation to solve, and we therefore ∈ restrict to in the form (e0) = [e, e0]D for fixed e Γ(E), and all e0 Γ(E). Note that Der(E),F and a( ) = ρF(e). Requiring to satisfy (3.132)∈ leads to the∈ definition of Killing Fequation. ∈ F F Definition 3.10.2. Let G be a generalized metric. We say that e Γ(E) is a Killing section of G, if it satisfies the Killing equation ∈

ρ(e).G(e0, e00) = G([e, e0]D, e00) + G(e0, [e, e00]D), (3.133) for all e0, e00 Γ(E). ∈ Now assume that G (g, B). We will examine the condition (3.133) in terms of the fields ≈ B T B 1 g and B. Recall that G = (e− ) Ee− , where E = BDiag(g, g− ). Moreover, we can B B G B dB G use the fact that e− [e, e0]D = [e− (e), e− (e0)]D , following from (3.82). Finally, note that B ρ(e− (e)) = ρ(e). These observations allow us to rewrite (3.133) as

B B dB B dB ρ(e− (e)). (f 0, f 00) = ([e− (e), f 0] , f 00) + (f 0, [e− (e), f 00] ), (3.134) GE GE D GE D B for all f 0, f 00 Γ(E). This proves that e is a Killing section of G, iff f := e− (e) is a Killing ∈ section of E. We will thus find all Killing sections of simpler generalized metric E, but using G G 1 dB-twisted Dorfman bracket instead. Write f = X0 + ξ0 for X0 X(M) and ξ0 Ω (M). ∈ ∈ Plugging into (3.134) for f, and writing f 0 = Y + η, f 00 = Z + ζ, we obtain

1 X0. g(Y,Z) + g− (η, ζ) = g([X0,Y ],Z) + g(Y, [X0,Z]) { } 1 1 + g− ( 0 η i dξ0, ζ) + g− (η, 0 ζ i dξ0) (3.135) LX − Y LX − Z 1 1 g− (dB(X0,Y, ), ζ) g− (η, dB(X0,Z, )). − · − · 59 This gives three equations

X0.g(Y,Z) = g([X0,Y ],Z) + g(Y, [X0,Z]), (3.136) 1 1 1 X0.g− (η, ζ) = g− ( 0 η, ζ) + g− (η, 0 ζ), (3.137) LX LX 1 1 0 = g− (i dξ0, ζ) + g− (dB(X0,Y, ), ζ) (3.138) Y · all valid for all Y,Z X(M) and η, ζ Ω1(M). They are equivalent to ∈ ∈

0 g = 0, dξ0 = i 0 dB (3.139) LX − X B Now let X0 + ξ0 = e− (X + ξ) = X + ξ B(X). We arrive to the following proposition: − Proposition 3.10.3. Let G be a generalized metric corresponding to fields (g, B). A section e = X + ξ is a Killing section of G, iff

g = 0, dξ = B. (3.140) LX −LX Since = [X + ξ, ] Der(E), we can use the result of Proposition (3.5.2) to find the correspondingF automorphism· ∈ of Dorfman bracket. By construction, we expect exp (t ) to be an element of EIsom(G). Also note that Eo(n, n), and thus exp (t ) EO(n, nF). Note F ∈ F ∈ that = R(X) + dξ. Using the Killing equation, we evaluate the integral defining the 2-form B(t):F F t t X X X B(t) = φ ∗(dξ) dk = φ ∗( B) dk = B φ ∗B. { k } − { k LX } − t Z0 Z0 The corresponding 1-parameter subgroup of AutD(E) then has the form

B(t) X exp (t ) = e T (φ t). (3.141) F ◦ − Finally, we are able to prove the following statement:

Proposition 3.10.4. Let = [e, ]D, where e Γ(E) is a Killing section of G. Then exp (t ) Aut (E) is an extendedF isometry· of G,∈exp (t ) EIsom(G). F ∈ D F ∈ B T B Proof. This is a direct calculation. Note that for G = (e− ) e− , we get GE X∗ X∗ φt B X φt B X G(exp (t )(e), exp (t )(e0)) = E e (T (φ t)(e)), e (T (φ t)(e0)) . F F G − − X∗ φt B X X B
Now note that e T (φ t) = T (φ t) e . Condition (3.131) then becomes ◦ − − ◦ X X X E(T (φ t)(f),T (φ t)(f 0)) = E(f, f 0) φt . (3.142) G − − G ◦ X Rewriting this using the definition of E, we obtain the condition φ t∗g = g. Here we use the second of the conditions (3.140) andG the fact that Killing vector field− X generates a flow preserving the metric g. 

3.11 Indefinite case

The concept of generalized metric has proved to be a useful tool to encode a Riemannian metric g and a 2-form B into a single object G, or its equivalents τ and V+. For applications of this

60 tool in physics, we should discuss also the case when g is an indefinite metric. We clearly have to abandon the interpretation using the definite subbundles V . The obvious candidate for indefinite generalized metric is G in the block form (3.93), since± all expressions make sense. For given metric g and 2-form B, we define generalized metric G to be a fiber-wise metric 1 1 1 B g 0 1 0 g Bg− B Bg− G = 1 = − 1 1 . (3.143) 0 1 0 g− B 1 g− B g−     −   −  1 A first question comes with the invertibility of the map g Bg− B. To answer it, we prove the following lemma: − 2 Lemma 3.11.1. Let V be a finite-dimensional vector space, g S V ∗ be a non-degenerate 2 ∈ bilinear form on V , and B Λ V ∗. Let A Hom(V,V ∗) be a linear map defined as A = g +B. ∈ ∈ 1 Then the map A is invertible if and only if the bilinear form g Bg− B is non-degenerate. − Proof. First, assume that A is invertible. Then so is AT = g B. Next, note that we can write 1 1 −1 g Bg− B = (g + B)g− (g B). This proves that g Bg− B is non-degenerate. In fact, we can− take the determinant of− this formula to get − 2 1 [det (g + B)] = det (g) det (g Bg− B). (3.144) − This proves the converse statement. 

For a positive definite g, the map g + B is always invertible. However, for indefinite g, this is no more true. Consider for example 1 0 0 1 g = ,B = . (3.145) 0 1 1 0  −  −  The map A = g + B is then certainly singular. 1 Then comes a question of the signature of G = g Bg− B. But this in fact follows from T − 1 the observation in the proof above, because G = A g− A, and the signature of G thus must be same as the one of g. If the signature of g is (p, q), we can take the square root of (3.144) to obtain q 1 q 1 1 det (g + B) = [( 1) det (g)] 2 [( 1) det (g Bg− B)] 2 . (3.146) ± − − − Note the sign in the formula. For g > 0, determinant on the left-hand side is always positive. Indeed, define± f(t) = det (g + tB). This is a continuous nonzero function of t, and f(0) > 0. This proves that f(1) > 0. For indefinite g, the signs for det (g) and det (g + B) can be different. Consider for example 1 0 0 λ g = ,B = . (3.147) 0 1 λ 0  −  −  Then det (g + B) = 1 + λ2, and det (g) = 1. We can thus choose λ > 1 and the two signs differ (the reason is of− course the singularity− at λ = 1). We can still consider an orthogonal transformation of generalized metric G. Consider T O(n, n) and simply define G0 := G . Is there always metric g0 and 2-form B0 such Othat ∈ O O 1 B0 g0 0 1 0 G0 = 1 (3.148) 0 1 0 g0− B0 1     −  holds true? A partial answer is given by the following proposition. We use the notation of sections 3.1 and 3.9.

61 Proposition 3.11.2. Let G be a generalized metric in a sense of (3.143), and define G0 = T G for O(n, n). O O O ∈ There exist metric g0 and 2-form B0 such that G0 has the form (3.143), if and only if the map Φ OT + OT (g + B) is invertible. + ≡ 4 2

Proof. First note that bottom-right corner of G0 defines a fiber-wise bilinear form h0 on T ∗M given by formula

T 1 T 1 T 1 T 1 h0 = O g− O + O Bg− O O g− BO + O (g Bg− B)O . (3.149) 4 4 2 4 − 4 2 4 − 4

Next see that h0 can be written as 1 T h0 = Φ+g− Φ+. (3.150)

T T This can be verified directly, using the orthogonality property O4 O2 +O2 O4 = 0 for . Taking the determinant of this relation gives O

2 det (h0) = [det (Φ+)] / det (g). (3.151)

This proves that h0 is invertible if and only if Φ+ is.

2 Now assume that G0 has the form (3.148) for metric g0 and B0 Ω (M). In this case 1 ∈ h0 = g0− , and (3.151) proves that Φ+ is invertible.

Conversely, let Φ+ be an invertible map. Formula (3.151) proves that h0 is invertible (and 1 thus a fiber-wise metric on T ∗M). Define g0 := h0− . Now recall Proposition 3.8.5. Even in T indefinite case, G Hom(E,E∗) is an orthogonal map, and so is G0 = G . Moreover, this ∈ O O map has invertible bottom-right corner h0. We can now use an analogue of Proposition 3.1.3 2 to show that G0 can be decomposed as (3.148), proving that B0 Ω (M). ∈ 

Note that in indefinite case, Φ+ is not always invertible, not even in the case when g + B Θ is. Consider = e− of Example 3.9.5 for n = 2. Define O 1 0 0 b 0 t g = ,B = , Θ = . (3.152) 0 1 b 0 t 0  −  −  − 

2 Then det (g + B) = b 1, and for b = 1, g + B is invertible. The map Φ+ = 1 Θ(g + B) has the explicit form − 6 ± − 1 + tb t Φ = . (3.153) + t 1 + tb   Now det (Φ ) = (1 + tb)2 t2. Equation det (Φ ) = 0 now has two roots: t = 1/(b + − + − ± 1). For every b, we can thus find Θ making Φ+ a singular matrix, and consequently G0 is indecomposable in the sense of (3.148). We see that everything in principle carries out to the indefinite case, but in every step one has to make assumptions concerning the invertibility of involved maps. In order to avoid unnecessary discussions here and there, we thus usually stick to the positive definite case.

62 3.12 Generalized Bismut connection

Let G be a generalized metric on the Courant algebroid E = TM T ∗M. We may look for Courant algebroid connections compatible with G. Recall that by⊕ Definition 2.4.3, is a ∇ Courant algebroid connection if it preserves the Courant metric , E. There exists a simple example of such connection. It first appeared in [33] and was studiedh· ·i from the perspective of Courant algebroids in [40]. Before we introduce its definition, recall that G induces an operator C : V V defined as ± → ∓ C(Ψ (X)) = Ψ (X), (3.154) ± ∓ for all X X(M). By direct calculation using the projectors (3.90), one arrives to an explicit formula ∈ C(X + ξ) = X ξ + 2B(X), (3.155) − for all X + ξ Γ(E). Note that by definition C2 = 1, and C is an anti-orthogonal map with ∈ respect to , E. For any e Γ(E), denote e P (e) to simplify the notation. h· ·i ∈ ± ≡ ± Definition 3.12.1. The generalized Bismut connection is for e, e0 Γ(E) defined as ∇ ∈

ee0 = ([e+, e0 ]D) + ([e , e+0 ]D)+ + ([C(e+), e+0 ]D)+ + ([C(e ), e0 ]D) . (3.156) ∇ − − − − − − We have used a more abstract definition as it appeared in [40]. It is easy to see that fee0 = f ee0. This follows from the fact that V+,V E = 0, and (e+) = (e )+ = 0 for all ∇e Γ(E).∇ To prove the second property note thath ρ −Ci= ρ, and we get− − ∈ ◦

e(fe0) = f ee0 + (ρ(e+).f)e0 + (ρ(e ).f)e+0 + (ρ(e+).f)e+0 + (ρ(e ).f)e0 ∇ ∇ − − − − = f e0 + (ρ(e).f)e0. ∇e We will prove the metric compatibility with , later. It is easier to determine the action of h· ·iE the connection on the sections of the special form. Let e = Ψ+(X), and e0 = Ψ (Y ). Only one of the four terms contributes, namely the first one. Next note that Ψ = eB−Ψ0 , where Ψ0 (X) = X g(X). Hence ± ± ± ∓ B 0 0 H [Ψ+(X), Ψ (Y )]D = e [Ψ+(X), Ψ (Y )]D . − − B 1 1 Here H = dB. This simplifies the calculation, because P (e (Y + η)) = 2 Ψ (Y g− (η)). Then − − − 0 0 H [Ψ+(X), Ψ (Y )]D = [X,Y ] X (g(Y )) iY d(g(X)) H(X,Y, ). − − L − − · Combining all the observations gives

1 1 1 1 ee0 = Ψ [X,Y ] + g− ( X (g(Y )) + iY d(g(X))) + g− H(X,Y, )) . (3.157) ∇ − 2{ L } 2 · The terms not containing B form a well known object. Indeed, we have 

LC 1 1 Y = [X,Y ] + g− ( (g(Y )) + i d(g(X))) , (3.158) ∇X 2{ LX y } where LC is the Levi-Civita connection on M corresponding to the metric g. We can thus write ∇ LC 1 1 ee0 = Ψ ( X Y + g− H(X,Y, )). (3.159) ∇ − ∇ 2 · Repeating the same procedure for all the possible combination of signs, we would prove the following lemma. ±

63 Lemma 3.12.2. Let be the generalized Bismut connection defined by (3.156), and H = dB. Then ∇

(Ψ (Y )) = Ψ ( + Y ), (3.160) ∇Ψ±(X) + + ∇X Ψ (X)(Ψ (Y )) = Ψ ( − Y ), (3.161) ∇ ± − − ∇X for all X,Y X(M). ± is a couple of connections on M defined as ∈ ∇

LC 1 1 ± Y = Y g− H(X,Y, ). (3.162) ∇X ∇X ∓ 2 ·

Connections ± are metric compatible with g, and equations (3.160 - 3.162) can be considered as an equivalent∇ definition of the generalized Bismut connection.

This lemma has one immediate consequence. We can now prove that is in fact induced by an ordinary vector bundle connection. To show this, let e = 0+α for α ∇Ω1(M). It suffices 1 1 1 1 ∈ to show that e = 0. But in this case e = Ψ+( 2 g− (α)) Ψ ( 2 g− (α)), and the statement follows from (3.160,∇ 3.161). − − Our aim now is to find an expression for the generalized Bismut connection acting on a section in a general form e = Y + η. To do so, introduce an auxiliary connection e = B B ∇ e− B e . We have ∇e (e) b B 1 1 1 1 e (Y + η) = Ψ+(Y + g− (η)) + Ψ (Y g− (η)). 2 2 − −

B 1 1 In particular e (X) = Ψ+(X) + Ψ (X). This shows that 2 2 −

B 1 1 1 1 e X (Y + η) = Ψ (X)Ψ+(Y + g− (η)) + Ψ (X)Ψ (Y g− (η)) ∇ 2∇ + 2∇ + − − 1 + 1 1 1 b = Ψ+( (Y + g− (η))) + Ψ ( − (Y g− (η))) 2 ∇X 2 − ∇X − 1 B 0 + 1 1 B 0 1 = e Ψ+( X (Y + g− (η))) + e Ψ ( X− (Y g− (η))) 2 ∇ 2 − ∇ − Hence 1 0 + 1 1 0 1 X (Y + η) = Ψ+( X (Y + g− (η))) + Ψ ( X− (Y g− (η))). (3.163) ∇ 2 ∇ 2 − ∇ −

It is now simpleb to plug in from (3.162) for ±. This results in the formal block form of : ∇ ∇ LC 1 1 1 X 2 g− H(X, g− (?), ) b X = ∇ − · . (3.164) ∇ 1 H(X, ?, ) LC − 2 · ∇X  The ? symbol indicatesb where the input enters. In other words, we have

LC 1 1 1 LC 1 (Y + η) = ( Y g− H(X, g− (η), )) + ( η H(X,Y, )). (3.165) ∇X ∇X − 2 · ∇X − 2 · Now it isb easy to write the original G-compatible connection . One obtains ∇ LC 1 1 1 1 0 X 2 g− H(X, g− (?), ) 1 0 X = ∇ − · . (3.166) ∇ B 1 1 H(X, ?, ) LC B 1   − 2 · ∇X  −  This is the form of generalized Bismut connection as it appeared in [33].

64 Proposition 3.12.3. Generalized Bismut connection is compatible with the pairing , E, and with the generalized metric G. ∇ h· ·i

Proof. Directly from the definition of , it is easy to see that is compatible with , and ∇ ∇ h· ·iE G, if and only if is compatible with , and . The latter property can be easily checked ∇ h· ·iE GE explicitly using the block form (3.164)b of .  b ∇ Now recall the definition of the torsionb operator (2.67). We will use L suitable for Courant algebroids, namely 1 L(β, e, e0) = e, e0 g− (β). (3.167) h iE E We will make advantage of the simpler connection to calculate T . Indeed, we have ∇ B B B H λ T (e (e), e (e0)) = e e0 0 e [e, e0] + L(e , e, e0) . ∇e − ∇e −b D ∇eλ B B B Thus T (e (e), e (e0)) = e T (e, e0), where T is the torsion of connection
with H-twisted b b b ∇ Dorfman bracket. Let us now calculate T explicitly. One gets b b b 1 1 1 1 1 1 T (X + ξ, Y + η) = g−bH(X, g− (η), ) + g− H(Y, g− (ξ), ) − 2 · 2 · (3.168) 1 1 1 1 b H(X,Y, ) H(g− (ξ), g− (η), ). − 2 · − 2 · This proves that and consequently is torsion-free if and only if H = 0. Torsion T can be now calculated in∇ a straightforward manner∇ using eB and T . According to theb remark under (2.74), we may define the curvature operator of using the usual formula: b ∇ R(e, e0)e00 = 0 e00 0 e00 0 e00, (3.169) ∇e∇e − ∇e ∇e − ∇[e,e ]D for all e, e0, e00 Γ(E). Using the relation of to , we get the expression ∈ ∇ ∇ B B B B R(e (e), e (e0))e (e00) = e e e0 e00 e0 ee00 [e,e0]H e00 . (3.170) ∇ ∇ b − ∇ ∇ − ∇ D B B B B
Hence R(e (e), e (e0))e (e00) = e (R(e, e0)e00b),b where Rbis theb curvatureb operator of using H ∇ the H-twisted Dorfman bracket [ , ]D . It is not difficult to calculate R explicitly. We get · · b b b LC 1 1 LC 1 LC 1 R1(X,Y )(Z + ζ) = R (X,Y )Z g− (( H)(Y, g− (ζ), ) (b H)(X, g− (ζ), )) − 2 ∇X · − ∇Y · 1 1 1 1 1 1 b + g− H(X, g− H(Y,Z, ), ) g− H(Y, g− H(X,Z, ), ) (3.171) 4 · · − 4 · · 1 1 R (X,Y )(Z + ζ) = RLC (X,Y )ζ ( LC H)(Y,Z, ) + ( LC H)(X,Z, ) (3.172) 2 − 2 ∇X · 2 ∇Y · 1 1 1 1 1 1 b + H(X, g− H(Y, g− (ζ), ), ) H(Y, g− H(X, g− (ζ), ), ). 4 · · − 4 · ·

We have used R1 and R2 to denote the TM and T ∗M components of R respectively. One can λ now calculate the corresponding Ricci tensor Ric, defined as Ric(e, e0) = e , R(e , e0)e . Note h λ i that it has onlyb two non-trivialb components (with respect to the blockb decomposition). One obtains b b b

LC 1 1 1 k Ric(X,Y ) = Ric (X,Y ) H(Y, g− H(∂ ,X, ), g− (dy )), (3.173) − 4 k · 1 LC 1 1 k bRic(ξ, Y ) = ( H)(Y, g− (ξ), g− (dy )). (3.174) −2 ∇∂k b 65 Finally, we may use the generalized metric E to calculate the trace of Ric and obtain the corresponding scalar curvature . One gets G R b 1 λ 1 ijk = Ric(b − (e ), e ) = (g) H H . (3.175) R GE λ R − 4 ijk To conclude this section,b noteb that we can use this result to calculate the scalar curvature of the generalized Bismut connection. R Proposition 3.12.4. Let be the generalized Bismut connection corresponding to the gener- alized metric G. Let Ric be∇ its Ricci tensor, and let be the scalar function on M defined as R 1 λ = Ric(G− (e ), e ), (3.176) R λ where (e )2n is some local frame on E. Then = , that is λ λ=1 R R 1 = (g) H bHijk. (3.177) R R − 4 ijk

Proof. The result follows from the definition of the connection , the relation (3.170), and the B T B ∇ fact that G = (e− ) e− . GE  b We see that the scalar curvature of the generalized Bismut connection does not depend on B, but only on a cohomology class [H] of its exterior derivative H = dB.

66 Chapter 4

Extended generalized geometry

The main aim of this chapter is to generalize the objects of the standard generalized geometry p in order to work also on the vector bundle E = TM Λ T ∗M. The main issue is that the most straightforward generalization of the orthogonal group⊕ O(n, n) suitable for E does not seem to be useful for a description of the generalized metric. This lead us to the idea of embedding the generalized geometry of E into the larger vector bundle E E∗, already equipped with the canonical O(d, d) structure. ⊕

4.1 Pairing, Orthogonal group

p p Let E = TM Λ T ∗M. We have Γ(E) = X(M) Ω (M). Define a non-degenerate C∞(M)- ⊕ ⊕p 1 bilinear symmetric form , : Γ(E) Γ(E) Ω − (M) as h· ·iE → → X + ξ, Y + η = i η + i ξ, (4.1) h iE X Y for all X + ξ, Y + η Γ(E). Although it is not an ordinary C∞(M)-valued pairing, one can still define its orthogonal∈ group O(E) as usual, that is

O(E) = Aut(E) (e), (e0) = e, e0 , e, e0 Γ(E) . (4.2) {F ∈ | hF F iE h iE ∀ ∈ } We will now examine its Lie algebra o(E), defined as

o(E) = End(E) (e), e0 + e, (e0) = 0, e, e0 Γ(E) . (4.3) {F ∈ | hF iE h F iE ∀ ∈ } In fact, the structure of this algebra greatly depends on p and the dimension n of the manifold M. Write in the formal block form as F A Π = T . (4.4) F C A0 −  Plugging into the condition (4.3) gives the following set of equations: F T T iY C (X) + iX C (Y ) = 0, (4.5)

iY A0(ξ) + iA(Y )ξ = 0, (4.6)

iΠ(ξ)η + iΠ(η)ξ = 0. (4.7) One can now discuss the consequences of these equations. This is a straightforward but a little bit technical linear algebra. We present only the results in the form of a proposition.

67 Proposition 4.1.1. Let End(E) have a formal block form (4.4). Then, depending on p, we have the following conditionsF ∈ for o(E). F ∈ 1. p = 0: All fields are arbitrary, that is

1 o(E) = End(TM) X(M) Ω (M) C∞(M). (4.8) ⊕ ⊕ ⊕

T 2 2 2. p = 1: In this case o(E) = o(n, n), and thus A0 = A , Π X (M), C Ω (M), and − ∈ ∈ o(E) = End(TM) X2(M) Ω2(M). (4.9) ⊕ ⊕

3. 1 < p < n 1: In this case A = λ 1, A0 = λ 1, where λ C∞(M), Π = 0, and C Ωp+1(M−). Hence, · − · ∈ ∈ p+1 o(E) = X (M) C∞(M). (4.10) ⊕ n n 4. p = n 1: A = λ 1, A0 = λ 1, for λ C∞(M). Π X (M), and C Ω (M). Thus, − · − · ∈ ∈ ∈ n n o(E) = X (M) Ω (M) C∞(M). (4.11) ⊕ ⊕

5. p = n: In this case A = λ 1, A0 = λ 1, for any λ C∞(M), C = Π = 0, and therefore · − · ∈

o(E) = C∞(M). (4.12)

We see that possible choices for o(E) are very different for different values of p. In par- ticular note that for 1 < p < n 1, there is no (p + 1)-vector Π defining a skew-symmetric transformation, and thus no eΠ −defining an orthogonal transformation. This proves that for general p, Nambu-Poisson manifolds cannot be realized as Dirac structures. This was proved by Zambon in [91], and it has in fact lead to the theory of Nambu-Dirac manifolds examined by Hagiwara in [41].

Example 4.1.2. For a general p, there are thus fewer generic examples of orthogonal trans- formations, let us recall them here

p+1 p C-transform: Let C Ω (M). It defines a map C Hom(Λ TM,T ∗M), and we will • define eC Aut(E) by∈ its formal block form ∈ ∈ 1 0 eC = . (4.13) CT 1 −  Note that for p = 1, we have C = CT . It follows from Proposition 4.1.1 that eC O(E). − ∈

Let λ C∞(M) be everywhere non-zero smooth function. Define the map as • ∈ Oλ 1 (X + ξ) = λX + ξ, (4.14) Oλ λ for all X + ξ Γ(E). Obviously O O(E). ∈ λ ∈

68 4.2 Higher Dorfman bracket and its symmetries

Let us now examine the Dorfman bracket from Example 2.1.3. Recall that it is defined as

[X + ξ, Y + η] = [X,Y ] + η i dξ, (4.15) D LX − Y for all X + ξ, Y + η Γ(E). To distinguish it from its p = 1 version, we sometimes refer to ∈ it as the higher Dorfman bracket. We have shown that for ρ = prTM , the triplet (E, ρ, [ , ]E) forms a Leibniz algebroid. Due to its structure similar to p = 1 Dorfman bracket, it also· · has some properties similar to Courant algebroid axioms:

Lemma 4.2.1. Let [ , ] be the Dorfman bracket (4.15), and , be the pairing (4.1). Then · · D h· ·iE 1 [X + ξ, X + ξ] = d X + ξ, X + ξ , (4.16) D 2 h iE for all X + ξ Γ(E), and the pairing , is invariant with respect to [ , ] in the sense that ∈ h· ·iE · · D Y + η, Z + ζ = [X + ξ, Y + η] ,Z + ζ + Y + η, [X + ξ, Z + ζ] , (4.17) Lρ(X+ξ)h iE h D iE h DiE for all X,Y,Z X(M), and ξ, η, ζ Ωp(M). ∈ ∈

Proof. Direct calculation and definitions. 

We can directly generalize the derivations algebra and the automorphism group of the Dorfman bracket. The derivation of the results is completely analogous to the p = 1 case provided in Section 3.4 and Section 3.5. We thus omit proofs of following propositions.

Proposition 4.2.2. Define the Lie algebra Der E of derivations of the Dorfman bracket (4.15) as in (3.40). Then as a vector space, it decomposes as . Der(E) = X(M) Ω0 (M) Ωp+1 (M). (4.18) ⊕ closed ⊕ closed

Every Der(E) decomposes uniquely as = R(X) + λ + C , where R(X)(Y + η) = ([X,Y ]F, ∈η), for all X,Y X(M) and η FΩp(M). VectorF bundleF endomorphisms and LX ∈ ∈ Fλ C are defined as F 0 0 0 0 = , = , (4.19) Fλ 0 λ 1 FC CT 0  ·  −  where λ Ω0 (M) and C Ωp+1 (M). Nontrivial commutation relations are ∈ closed ∈ closed [R(X),R(Y )] = R([X,Y ]), (4.20)

[R(X),FC ] = F B, (4.21) LX [ , ] = . (4.22) Fλ FB FλC On the Lie algebra level, we thus have

0 p+1 Der(E) = X(M) n (Ωclosed(M) n Ωclosed(M)), (4.23)

0 p+1 0 where Ωclosed(M), Ωclosed(M) are viewed as Abelian Lie algebras, Ωclosed(M) acts on forms in p+1 0 p+1 Ωclosed(M) by multiplication, and X(M) acts on Ωclosed n Ωclosed(M) by Lie derivatives.

69 Proposition 4.2.3. Let AutD(E) be the group of automorphisms (3.59) of the Dorfman bracket (4.15). Then it has the following group structure:

p+1 0 AutD(E) = (Ωclosed(M) o G(Ωclosed(M))) o Diff(M), (4.24)

p+1 0 where Ωclosed(M) is viewed as an Abelian group with respect to addition, G(Ωclosed(M)) acts p+1 p+1 0 on Ωclosed(M) by multiplication, and Diff(M) acts on Ωclosed(M) o G(Ωclosed(M)) by inverse pullbacks. Every ( , ϕ) Aut (E) can uniquely be decomposed as F ∈ D = eC T (ϕ), (4.25) F ◦ Sλ ◦ C p+1 where e = exp C , and Sλ(X + ξ) = X + λξ for the unique C Ωclosed(M) and λ 0 F 0 ∈ ∈ G(Ωclosed(M)). By G(Ωclosed(M)) we mean the multiplicative group of everywhere non-zero closed 0-forms (locally constant functions).

Similarly to the p = 1 Dorfman bracket, we expect something interesting to happen when we twist it with a non-trivial C-transformation. Let C Ωp+1(M), and in general dC = 0. ∈ 6 Define a new bracket [ , ]0 as · · D C C C [e, e0]D0 = e− [e (e), e (e0)]D, (4.26)

dC for all e, e0 Γ(E). It turns out, due calculations similar to Section 3.6, that [ , ]0 = [ , ] , ∈ · · D · · D where the H-twisted higher Dorfman bracket [ , ]H is for given H Ωp+2 (M) defined as · · D ∈ closed [X + ξ, Y + η]H = [X + ξ, Y + η] H(X,Y, ), (4.27) D D − · for all X + ξ, Y + η Γ(E). There holds also a complete analogue of Proposition 3.6.1, where all objects can straightforwardly∈ be replaced by their p > 1 generalizations.

4.3 Induced metric

p Before proceeding to an analogue of the generalized metric suitable for E = TM Λ T ∗M, let 2 ⊕ us examine in detail the following construction. Let g Γ(S T ∗M) be an arbitrary metric on M. Our intention is to define an induced fiber-wise metric∈ g on ΛpTM. First, let us define a type (0, 2p) tensor g on M as

σ e g(V ,...,V ,W ,...,W ) = ( 1)| |g(V ,W ) g(V ,W ), (4.28) 1 pe 1 p − σ(1) 1 × · · · × σ(p) p σ S X∈ p for all Ve,...,V ,W ,...,W X(M). First note that g is skew-symmetric in first and last p 1 p 1 p ∈ inputs. Moreover, one can interchange (V1,...,Vp) and (W1,...,Wp): e g(V1,...,Vp,W1,...,Wp) = g(W1,...,Wp,V1,...,Vp). (4.29) This proves that g defines a fiber-wise symmetric bilinear form on ΛpT M. In local coordinates, e e ∗ it has the form g = δk1...kp g . . . g . (4.30) e i1...ipj1...jp i1...ip k1j1 kpjp 1 To prove that g is a fiber-wise metric, define a type (2p, 0) tensor g− on M as e 1 1 1 g− (α , . . . , α , β , . . . , β ) = g− (α , β ) g− (α , β ), (4.31) e 1 p 1 p σ(1) 1 × · · · ×e σ(p) p σ S X∈ p e 70 1 for all α1, . . . , αp, β1, . . . , βp Ω (M). Now note that we can view g as a vector bundle p ∈p 1 p morphism g from Λ TM to Λ T ∗M, and g− as a vector bundle morphism from Λ T ∗M to p 1 Λ TM. It is straightforward to check that g− g = 1. This proves thate g is non-degenerate p ◦ and thus ae fiber-wise metric on Λ TM. e There is now one interesting question to pose.e Whate is the signature of gefor given signature (r, s) of g? The answer is given by the following lemma e Lemma 4.3.1. Let g be a metric of signature (r, s), and let g be the fiber-wise metric on ΛpTM defined by (4.28). Then g has the signature ( n N(r, s, p),N(r, s, p)), where the p − number N(r, s, p) is given by a formula
e e p/2 d e s r N(r, s, p) := . (4.32) 2k 1 p 2k + 1 kX=1  −  − 

n Proof. Choose an orthonormal frame (Ei)i=1 = (e1, . . . , er, f1, . . . , fs) for g:

g(e , e ) = δ , g(f , f ) = δ , g(e , f ) = 0. (4.33) i j ij i j − ij i j We can calculate g in the basis E = E ... E . One gets g(E ,E ) = δJ , proving I i1 ∧ ∧ ip I J ± I that EI form an orthonormal basis for g. It thus remains to track the sign. For given s r ± odd j 1, . . . , p ethere is Nj(r, s, p) := j p j different strictlye ordered p-indices I, such that exactly∈ { j indices} in I correspond to negative− norm orthonormal basis vectors. These are e

precisely the p-indices I where g(EI ,EI ) = 1. Resulting N(r, s, p) is just a sum of Nj(r, s, p) over all odd j: − s r N(er, s, p) = . j p j j odd, 1 j p    X≤ ≤ − This is exactly the formula (4.32). 

We can now calculate some relevant examples. For a positive definite g, we have (r, s) = (n, 0), and thus N(n, 0, p) = 0. This means that also g is positive definite. For Lorentzian g, we have two possibilities: (r, s) = (1, d) or (r, s) = (d, 1). Note that for p even, one gets N(r, s, p) = N(s, r, p). e

(r, s) = (d, 1): We get • p/2 d e 1 d d N(d, 1, p) = = . (4.34) 2k 1 p 2k + 1 p 1 kX=1  −  −   −  (r, s) = (1, d): For even p, we get N(d, 1, p) = N(1, d, p). For odd p, we obtain • p/2 d e d 1 d N(1, d, p) = = . (4.35) 2k 1 p 2k + 1 p kX=1  −  −    By construction of g, it is clear that geometrical properties of g will follow from those of g. As an example, we can calculate its Lie derivative. e e 71 Lemma 4.3.2. Let g be the fiber-wise metric (4.28). Then we have

( g)(P,Q) = X.g(P,Q) g( P,Q) g(P, Q), (4.36) eLX − LX − LX p 0 for all P,Q X (M), wheree on the left-hande side ge is viewed ase a tensor g 2p(M). Moreover, this Lie derivative∈ can be calculated as ∈ T ( g)(V ,...,V ,W ,...,W ) = e e LX 1 p 1 p p σ (4.37) ( 1)| | g(V ,W ) ... ( g)(V ,W ) g(V ,W ). −e σ(1) 1 × × LX σ(k) k × · · · × σ(p) p σ S k=1 X∈ p X In particular, g = 0 implies g = 0. LX LX Proof. Equation (4.36) follows frome the definition of Lie derivative and the fact that it com- mutes with contractions. Equation (4.37) follows from the fact that

X σ X X (φ ∗g)(V ,...,V ,W ,...,W ) = ( 1)| |(φ ∗g)(V ,W ) (φ ∗g)(V ,W ). t 1 p 1 p − t σ1 1 × · · · × t σ(p) p σ S X∈ p e (4.38) Now differentiate this at t = 0 to obtain (4.37). 

Remark 4.3.3. The converse statement is not true. Consider M = R2, and the Minkowski metric 1 0 g = . (4.39) −0 1   Its algebra of Killing vectors is spanned by generators of two translations and a Lorentz boost. 1 2 For p = 2, the metric g is given by single component g(12)(12) = 1. If X = X ∂1 + X ∂2, the Killing equation for g gives − 1 2 e X ,1 + X ,2 =e 0. (4.40) To be in Killing algebrae of g, X1 and X2 have to have the form

X1 = cx2 + a, X2 = cx1 + b, for a, b, c R. Note that such X1, X2 indeed solve (4.40). On the other hand, (4.40) has many more solutions,∈ for example X1 = f(x2), X2 = 0 for an arbitrary smooth function f. This shows that the Killing algebra of g can be strictly larger than the one of g. Also, note that Killing algebra for g does not have to be finite-dimensional. e n n Now, see that ine chosen coordinates, one can view gIJ as a square p p matrix, and gij as a square n n matrix. Are determinants of these matrices related? × ×

e i Lemma 4.3.4. Let A be a square n n matrix, denote its components as A j. Define an n n matrix B labeled by strictly ordered× p-indices I and J as p × p

I I k1 kp B J = δk1...kp A j1 ...A jp . (4.41) Then A is invertible if and only if B is invertible. Moreover, there holds a determinant formula:

n−1 det (B) = [det (A)](p−1). (4.42)

72 1 Proof. Let A be invertible. Define B− as

1 I I 1 k 1 k (B− ) = δ (A− ) 1 ... (A− ) p . (4.43) J k1...kp j1 jp

I 1 J I It is straightforward to check that B J (B− ) K = δK . The opposite statement follows from the formula (4.42). Its proof is more complicated and can be found in the Appendix A. 

To conclude this section, we shall examine how a covariant derivative acts on g. This will be important for the generalized Bismut connection in one of the following sections. e Lemma 4.3.5. Let be any connection on M. Let g be the metric (4.28) viewed as (0, 2p)- tensor. Then we can∇ write its covariant derivative as e ( g)(P,Q) = X.g(P,Q) g( P,Q) g(P, Q), (4.44) ∇X − ∇X − ∇X for all P,Q Xp(M). Moreover, one has ∈ e e e e ( g)(V ,...,V ,W ,...,W ) = ∇X 1 p 1 p p σ (4.45) ( 1)| | g(V ,W ) ( g)(V ,W ) g(V ,W ). − e σ(1) 1 × · · · × ∇X σ(k) k × · · · × σ(p) p σ S k=1 X∈ p X In particular, if g = 0, then g = 0. ∇X ∇X Proof. Equation (4.44) follows frome the definition of covariant derivative and the fact that it commutes with tensor contractions. Next, let m M, and γ be the integral curve of X X(M) γ ∈ ∈ starting at m. Let τt be the parallel transport from m γ(0) to γ(t) induced by connection . Then ≡ ∇ p γ σ γ γ (τ tg)(V1,...,Vp,W1,...,Wp) = ( 1)| | (τ tg)(Vσ(1),W1) (τ tg)(Vσ(p),Wp). − − − × · · · × − σ S k=1 X∈ p X e Differentiation of this equation with respect to t at t = 0 gives the assertion of the lemma. 

4.4 Generalized metric

We would like to define a positive definite fiber-wise metric G on E of similar properties as the generalized metric on TM T ∗M defined in Section 3.8. There is no canonical fiber-wise ⊕p 1 metric on E, except for the Ω − (M)-valued pairing , E. The definition suitable for the h· ·i C T C generalization to E is the formal block form (3.93), in particular its form G = (e− ) Ee− . For any metric g, define G g 0 E = 1 , (4.46) G 0 g−   p p+1 C where g is the induced metric on Λ TM defined by (4.28). Let C Ω (M), and let e− be the O(E) map defined by (4.13). Then set e ∈

e 1 T 1 1 C g 0 1 0 g + Cg− C Cg− G := 1 T = 1 T 1 . (4.47) 0 1 0 g− C 1 g− C g−         e e e 73 e e If g is positive definite, then so is G. For g of a general signature (r, s), one can determine the signature of G using Lemma 4.3.1. The reason why we first consider only g induced by g and C Ωp+1(M) follows from the physics - the inverse of G naturally appears in the Hamiltonian ∈ density of gauge-fixed Polyakov-like action for a p-brane. It appeared in exactlye this form in the paper of Duff and Lu [30]. See also related concepts in [52] suitable for various M-geometries. There is still one characterization, which could possibly survive the generalization, because p 1 the dual bundle E∗ can be equipped with a X − (M)-valued pairing , ∗ , defined similarly h· ·iE to (4.1). We can then ask if G viewed as G Hom(E,E∗) defines an orthogonal map with ∈ respect to , and , ∗ . But this is not true for p > 1. This can be most easily seen from h· ·iE h· ·iE the fact that E itself is not orthogonal for p > 1, and thus even in the C = 0 case, G is not orthogonal. G Note that in order to introduce the orthogonal transformations, we have to allow for more general fields in generalized metric. In particular, for p > 1 we would not assume that g is p induced from g via (4.28). Moreover, the vector bundle morphism C Hom(Λ TM,T ∗M) ∈ does not have to be induced by a (p + 1)-form C. Note that every positive definite fiber-wisee metric G on E can then be decomposed as (4.47). It will turn out that a generalized metric G0 related to G by open-closed relations will not have G and G related by (4.28). 1 T For positive definite g, the symmetric bilinear form g + Cg− C is invertible, and thus defines a metric on M. Using the decomposition Lemmae (3.1.2), we see that there exists a unique decomposition e

1 0 GN 0 1 ΠN G = T 1 − , (4.48) Π 1 0 G− 0 1 − N   N    where the fields GN , GN and ΠN have the form e G = g + Cg 1CT , (4.49) N e − T 1 GN = g + C g− C, (4.50) e 1 T 1 1 1 T 1 1 ΠN = (g + Cg− C )− Cg− = g− C(g + C g− C)− . (4.51) e −e − There is a historical reason behind the N subscript. Fields (G , G , Π ) correspond to the e e e N N N Nambu sigma model dual to the membrane sigma model described by fields (g, g, C). Note p that in general, GN is not induced from GN via (4.28), and ΠN eHom(Λ T ∗M,TM) is not p+1 ∈ induced by ΠN X (M). e ∈ e Example 4.4.1. Let us show an example proving the preceding assertion. Consider M = R3, 3 and let g be the Euclidean metric on R . Consider p = 2. Let (∂(12), ∂(13), ∂(23)) be a local basis 2 2 of X (M). The induced metric g has the unit matrix in this basis. Any C Hom(Λ TM,T ∗M) induced by a (p + 1)-form C has the matrix ∈ e 0 0 c C = 0 c 0 , c −0 0   where c := C123. We have 1 + c2 0 0 1 + c2 0 0 G = 0 1 + c2 0 , G = 0 1 + c2 0 .  0 0 1 + c2  0 0 1 + c2   e  

74 This shows that G is not of the form (4.28) because such G must be quadratic in the elements of G, for example G = δkl G G = G2 = (1 + c2)2 = 1 + c2. The vector (12)(12) 12 k1 l2 12 6 bundle morphism ΠN ehas the matrix e e 2 1 0 0 (1 + c )− c 2 1 − ΠN = 0 (1 + c )− c 0 ,  2 1  (1 + c )− c 0 0 −   which shows that Π in this case is induced by a 3-vector Π X3(M). N N ∈ Example 4.4.2. Finding a case when ΠN is not induced by (p + 1)-vector is also not difficult, but one has to go to higher dimensions. Consider n = 5, p = 2, M = R5 and g the Euclidean metric. There are 10 strictly ordered 2-indices, let us order them lexicographically. Define a 3-form C as C = dx1 dx2 dx3 + dx3 dx4 dx5. We get G = 2g + dx3 dx3, and thus ∧ ∧ ∧ ∧ N ⊗ 1 1 1 1 G− = g− ∂ ∂ . N 2 − 6 3 ⊗ 3

iJ ik J Then (ΠN ) = GN CkJ , because gIJ = δI . We can now simply calculate ΠN explicitly. One obtains − 1(23) 1 2(13) 1 3(12) 1 (ΠN ) = e , (ΠN ) = , (ΠN ) = . (4.52) −2 2 −3

This proves that ΠN is not induced by a 3-vector.

To conclude this section, we can briefly discuss the p 1 analogue of the open-closed p ≥ relations (3.119). Let Π Hom(Λ T ∗M,TM) be any vector bundle morphism. Define new ∈ generalized metric G0 as Π T Π G0 = (e ) Ge . (4.53) Recall that eΠ Aut(E) is defined as ∈ 1 Π eΠ = . (4.54) 0 1  

1 ΠN 1 ΠN T We immediately see that this has a solution by rewriting (4.48) as G− = e N− (e ) , 1 1 (Π Π) 1 (Π Π) T G where = BDiag(G, G− ). This proves that G0− = e N − − (e N − ) . We thus have GN GN

e G0 = GN , G0 = GN , Π0 = ΠN Π. (4.55) N N N − We see that everything works as in the casee ofe ordinary open-closed relations. We can also use (4.55) to write down the explicit relations between (g, g, C) and new fields, usually denoted as (G, G, Φ). We get e 1 T 1 T e g + Cg− C = G + ΦG− Φ , (4.56) T 1 T 1 g + C g− C = G + Φ g− Φ, (4.57) e e 1 1 1 T Cg− = ΦG− (G + ΦG− Φ )Π, (4.58) e − e 1 1 T 1 g− C = G− Φ Π(G + Φ G− Φ). (4.59) e e − e Similarly to the above examples, G is in general not ine the form (4.28), and Φ is not necessarily induced by a (p + 1)-form Φ Ωp+1(M). ∈ e 75 4.5 Doubled formalism

This section will provide a more rigid framework for the generalized metric G defined by (4.47). The main clue leading to this approach was the fact that open-closed relations (4.56 - 4.59) can be rewritten in the formal block matrix form

1 1 g C − G Φ − 0 Π = + . (4.60) CT g ΦT G ΠT 0 −  −  −  Let W be a vector bundle W = TM ΛpTM. Define the following vector bundle morphisms: e ⊕ e g 0 0 C G 0 0 Φ 0 Π = , = , = , Ξ = , Θ = . G 0 g B CT 0 H 0 G ΦT 0 ΠT 0   −    −  −  (4.61) We can now rewritee (4.60) in the form resemblinge the original open-closed relations (3.119):

1 1 ( + )− = ( + Ξ)− + Θ. (4.62) G B H See that and are positive definite fiber-wise metrics on W , , Ξ Ω2(W ), and Θ X2(W ). This suggestsG thatH we should focus on the generalized geometryB of W∈. In particular, to∈ consider the vector bundle V = W W ∗. This vector bundle is equipped with a natural pairing ⊕ n , V , and thus also with a natural orthogonal group O(d, d), where d = n + p . This configurationh· ·i allows one to define a generalized metric on V using the formalism of Section 3.8.
By the generalized metric we mean all forms equivalent to Definition 3.8.1. Let us see how this allows one to describe the generalized metric of Section 4.4. Note that we do not assume that C Ωp+1(M), and g is in general ont of the form (4.28). ∈ p Definition 4.5.1. Let GV be a generalized metric on V = W W ∗, where W = TM Λ TM. e ⊕ ⊕ We can view GV as an element of Hom(V,V ∗). Note that E and E∗ are subbundles of both V and V ∗.

We say that G is a relevant generalized metric, if G (E) E∗. V V ⊆

Let us now show that the restriction of a relevant generalized metric GV to the subbundle E is exactly the generalized metric G defined by (4.47). First, note that every GV is uniquely determined by a positive definite metric on W , Ω2(W ), and has the formal block form G B ∈ 1 1 − − GV = G − BG1 BBG 1 . (4.63) − −  −G BG 

It is straightforward to show that GV is a relevant generalized metric, if and only if and have the form G B g 0 0 C = , = , (4.64) G 0 g B CT 0   −  where g is a Riemannian metric on M, g is a positive definite fiber-wise metric on ΛpTM, and p C Hom(Λ TM,T ∗M). We have showne that G is uniquely determined by a map = + . ∈ V A G B This map now reads e g C = . (4.65) A CT g − 

To see how G and G fit together, note that V can also be written as V = E E∗. Moreover, V e ⊕ E and E∗ are complementary maximally isotropic subbundles of V , and , coincides with h· ·iV 76 the canonical pairing of E and E∗. Because GV is a relevant generalized metric, we see that the involution corresponding to G satisfies (E) E∗. We can write as a block TV V TV ⊆ TV matrix with respect to the splitting V = E E∗ as ⊕ 0 H = . (4.66) TV GN   2 Now V has to be symmetric with respect to , V , and V = 1. These two properties give T 1 h· ·i T N = 0, and H = G− . An examination of G shows that it is exactly the generalized metric (4.47). Moreover, the corresponding eigenbundles V have the form ± V = e G(e) e E . (4.67) ± { ± | ∈ }

In the isotropic splitting V = E E∗, a relevant generalized metric GV is thus described by a pair (G, 0), where G is a positive⊕ definite fiber-wise metric in E, and 0 Ω2(E). Again, let us emphasize that this description does not single out G where g is an induced∈ metric (4.28), and C Ωp+1(M). ∈ We can consider O(d, d) transformations an their action one the generalized metric GV , T similarly to Section 3.9. Clearly, not for any V O(d, d), the new metric GV0 := V GV V is again a relevant one. O ∈ O O Example 4.5.2. Let us show some examples of O(d, d)-transformations. We will follow the structure of Example 3.9.5. We only have to discuss the conditions under which the new generalized metric GV0 becomes relevant. We assume that GV is of the form (4.63), where and are parametrized by (g, g, C) as in (4.64). G B 2 T Let Ω (V ), and choose = e−Z . The new generalized metric G0 = G is • Z ∈ e OV V OV V OV described by a pair ( 0, 0), and we get G B

0 = , 0 = + . (4.68) G G B B Z

Clearly ( 0, 0) describes a relevant metric, if and only if 0 is again block off-diagonal. This happensG B if and only if is block off-diagonal: B Z 0 Z = , (4.69) Z ZT 0 −  p where Z Hom(Λ TM,T ∗M). The fiber-wise metric G0 corresponding to GV0 is then described∈ by a triplet (g, g, C + Z).

2 Θ T Let Θ X (V ). Define V = e . The new generalized metric GV0 = V GV V is • described∈ by a pair ( , Ξ),eO and we get the relation O O H 1 1 ( + )− = ( + Ξ)− + Θ. (4.70) G B H

To see which Θ give a relevant GV0 is similar to the previous example - one just has to 1 1 switch to the dual fields describing GV . In particular, if = + , let − = N− + ΘN for a positive definite fiber-wise metric on V , and ΘA GX2(VB). OneA can showG that GN N ∈ for a relevant GV they have the form

G 0 0 Π = N , Θ = N , (4.71) GN 0 G N ΠT 0  N  − N  e 77 where (GN , GN , ΠN ) are the fields (4.49 - 4.51). Similarly to the p = 1 case, we have Θ0 = Θ Θ. This proves that G0 is a relevant generalized metric, if and only if Θ is N N − V block off-diagonal:e 0 Π Θ = , (4.72) ΠT 0 −  p where Π Hom(Λ T ∗M,TM). To conclude, if G0 corresponding to G0 is described by ∈ V a triplet (G, G, Φ), we get exactly the equation (4.60). This is in turn equivalent to the set of equations (4.56 - 4.59). e Let End V , and choose V = , where is defined as in (3.120). The new • N ∈ T O ON ON generalized metric G0 = G is described by a pair ( 0, 0), where V OV V OV G B T T 0 = , 0 = . (4.73) G N GN B N BN

Criteria for to give a relevant generalized metric GV0 are in this case more intricate. Indeed, let Nhave the block form N N N = 1 2 . (4.74) N N3 N4  

Then GV0 defines a relevant generalized metric, if and only if

T T N1 gN2 + N3 gN4 = 0, (4.75) T T N2 gN1 + N4 gN3 = 0, (4.76) T T T N2 CN4 N4 C N2 = 0, (4.77) e − e N T CN N T CT N = 0. (4.78) 1 3 − 3 1

Let (g0, g0,C0) be the fields corresponding to the fiber-wise metric G0. There are two simple solutions to this set of equations. First consider N2 = N3 = 0. This gives

e T T T g0 = N1 gN1, g0 = N4 gN4,C0 = N1 CN4. (4.79)

The second example is N1 = N4 = 0.e In this case,e one obtains T T T T g0 = N gN , g0 = N gN ,C0 = N C N . (4.80) 3 3 2 2 − 3 2

This example shows that criteriae toe give a relevant generalized metric are not universal - they can depend on the original generalized metric GV .

4.6 Leibniz algebroid for doubled formalism

We have just shown that the vector bundle V = W W ∗ proves to be a natural way to describe the generalized geometry on the vector bundle⊕ E. To complete this discussion, we have to introduce a suitable bracket structure on V . In particular, we would like to define a bracket [ , ]V , which will restrict to the higher Dorfman bracket on (4.15) on E. To do so, we will use the· · following lemma:

78 Lemma 4.6.1. Let (E, ρ, [ , ] ) be a Leibniz algebroid. Then there is always a Leibniz algebroid · · E structure on V = E E∗, restricting to [ , ] on E. In particular, define ⊕ · · E E [e + α, e0 + α0] = [e, e0] + α0, (4.81) V E Le for all e, e0 Γ(E), and α, α0 Γ(E∗). The anchor ρV Hom(V,TM) is defined as ρV = ∈ ∈ E ∈ ρ pr1. Then (V, ρV , [ . ]V ) is a Leibniz algebroid. By we mean the induced Lie derivative (2.7).◦ · · L

Proof. The Leibniz rule for [ , ] follows from (2.10). For the Leibniz identity, we have · · V E E [e + α, [e0 + α0, e00 + α00] ] = [e, [e0, e00] ] + 0 e00, (4.82) V V E E Le Le E [[e + α, e0 + α0]V , e00 + α00]V = [[e, e0]E, e00]E + 0 e00, (4.83) L[e,e ]E E E [e0 + α0, [e + α, e00 + α00] ] = [e0, [e, e00] ] + 0 e00. (4.84) V V E E Le Le

One now sees that the Leibniz identity for [ , ]V follows from the one of [ , ]E and its property (2.11). The bracket [ , ] clearly restricts to· · [ , ] on E. · · · · V · · E  p Now consider E = TM Λ T ∗M with the higher Dorfman bracket (4.15). We have already calculated E for this bracket,⊕ see (2.15). We will denote the sections of V as (X, P, α, ξ) for X X(M),LP Xp(M), α Ω1(M), and ξ Ωp(M). Then, ∈ ∈ ∈ ∈ [(X, P, α, ξ), (Y, Q, β, η)] = [X,Y ], Q, β + (dξ)(Q), η i dξ , (4.85) V LX LX LX − Y for all (X, P, α, ξ), (Y, Q, β, η) Γ(V ). We claim that this is the bracket most
suitable to describe the generalized geometry∈ on E V . The bracket (4.85) was briefly mentioned by Hagiwara in [41] to describe Nambu-Dirac⊆ structures.

Remark 4.6.2. Note that with respect to the splitting V = W W ∗, this bracket strongly resembles the usual Dorfman bracket. Indeed, see that there is a⊕ Leibniz algebroid bracket on W defined as [(X,P ), (Y,Q)] = ([X,Y ], Q), (4.86) W LX for all (X,P ), (Y,Q) Γ(W ). Then W (β, η) = ( β, η), and we have ∈ L(X,P ) LX LX [(X, P, α, ξ), (Y, Q, β, η)] = [(X,P ), (Y,Q)] , W (β, η) i d (α, ξ) , (4.87) V W L(X,P ) − (Y,Q) W where i d (α, ξ) is a formal operation defined as i d (α, ξ) := ( (dξ)(Q), i
dξ). There (Y,Q) W (Y,Q) W − Y is no actual definition of the differential dW .

We have introduced the bracket (4.85) in order to be able to define Dirac structures of the vector bundle V . Since we have the fiber-wise metric , V , we can study involutive maximally isotropic subbundles of V . One can roughly followh· Section·i 3.7. We will make use of the following simple technical Lemma

p Lemma 4.6.3. Let T q (M) be a tensor field on M, completely skew-symmetric in all upper ∈ T p q indices, and in all lower indices (equivalently T Hom(Λ TM, Λ TM)), such that X T = 0 for all X X(M). Then ∈ L ∈ 1. For p = q, T = λ 1, where λ Ω0 (M). · ∈ closed 2. For p = q, T = 0. 6

79 Proof. See Appendix A.  Example 4.6.4. Let us bring up some important examples of Dirac structures of V .

By definition (Lemma 4.6.1), E and E∗ define Dirac structures of V . • Subbundles W and W ∗ define Dirac structures of V . • 2 Let Ω (W ) be an arbitrary 2-form on W . We know that eB O(d, d). This implies • B ∈ ∈ that eB(W ) is a maximally isotropic subbundle of V . Let be of the block form B BC = , (4.88) B CT B −  2 2 p p B where B Ω (M), B Ω (Λ TM), and C Hom(Λe TM,T ∗M). The subbundle e (W ) ∈ ∈ ∈ is a graph of Hom(W, W ∗), that is B ∈ e G = (w, (w)) w W V. (4.89) B { B | ∈ } ⊆

The involutivity condition [G ,G ]V G gives two conditions B B ⊆ B (B(Y ) + C(Q)) + d( CT (X) + B(P ))(Q) = B([X,Y ]) + C( Q), (4.90) LX − LX ( CT (Y ) + B(Q)) i d( CT (X) + B(P )) = CT ([X,Y ]) + B( Q)). (4.91) LX − − Y − e − LX p These two equationse have to hold for all X,Ye X(M) and for all P,Qe X (M). First note that there is only one term containing the∈ pair (P,Q) in the first∈ condition. This implies d(B(P )) = 0 for all P Xp(M). Hence df B(P ) = 0 for all P Xp(M). For ∈ ∧ ∈ p < n, this gives B = 0. For p = n, there is no non-trivial B, because ΛpTM has rank 1. e e The part of (4.90) containing the pair (X,Y ) gives X (B(Y )) = B([X,Y ]). This is e L e equivalent to X B = 0. Using Lemma 4.6.3, we get B = 0. There remain only two non-trivial equations.L First, consider the part of (4.91) containing the pair (X,Y ). One obtains (CT (Y )) i d(CT (X)) = CT ([X,Y ]). (4.92) LX − Y This condition is C∞(M)-linear in Y , but in general not in X. This forces the following condition to hold for every f C∞(M): ∈ df [i CT (Y ) + i CT (X)] = 0. (4.93) ∧ X Y T T We see that necessarily iX C (Y ) + iY C (X) = 0. This proves that C has to be induced p+1 T by C Ω (M), C (X) = iX C. Using this in (4.92) gives iY iX dC = 0, that is ∈p+1 C Ωclosed(M). There still remains one condition for the terms containing the pair (X,Q∈ ) in (4.90). One can show that it already follows from dC = 0. We have just proved that G is a Dirac structure of V , if and only if is of the form B B 0 C = ,C Ωp+1 (M). (4.94) B CT 0 ∈ closed − 

Now note that we have three more isotropic subbundles which we can produce using eB. In particular eB(W ∗), eB(E), and eB(E∗). None of these other choices give something interesting.

80 2 Θ Let Θ X (W ). Then e (W ∗) is a maximally isotropic subbundle of V . Let Θ have the • block form∈ π Π Θ = . (4.95) ΠT π −  Θ The subbundle e (W ∗) is in fact the graph of the map Θ: e G = (Θ(µ), µ) µ Γ(W ∗) W W ∗. (4.96) Θ { | ∈ } ⊆ ⊕ Let us examine the consequences of the involutivity condition [GΘ,GΘ]V GΘ. We get the set of two equations ⊆ [π(α) + Π(ξ), π(β) + Π(η)] = π β + (dξ)( ΠT (β) + π(η)) (4.97) Lπ(α)+Π(ξ) − + Π π(α)+Π(ξ)η iπ(β)+Π(η)dξ ,
L − e ( ΠT (β) + π(η)) = ΠT β + (dξ)( ΠT (β) + π(η)) (4.98) Lπ(α)+Π(ξ) − − Lπ(α)+Π(ξ) −
+ π π(α)+Π(ξ)η iπ(β)+Π(η)dξ .
e L − e Since these two equations have to hold for all α, β Ω1(M) and
ξ, η Ωp(M), we can find these equivalent to the set of moree simpler equations:∈ ∈ [π(α), π(β)] = π( β), (4.99) Lπ(α) [π(α), Π(η)] = Π( η), (4.100) Lπ(α) [Π(ξ), π(β)] = π β (dξ)(ΠT (β)) Π(i dξ), (4.101) LΠ(ξ) − − π(β) [Π(ξ), Π(η)] = π (dξ)(π(η)) + Π( η i dξ), (4.102) LΠ(ξ)
− Π(η) (ΠT (β)) = ΠT ( β), (4.103) Lπ(α) Lπ(α)
(π(η)) = π( eη), (4.104) Lπ(α) Lπ(α) T T T Π(ξ)(Π (β)) = Π Π(ξ)β (dξ)(Π (β)) + π(iπ(β)dξ). (4.105) L e e L − Let us focus on the very first equation. It can be rewritten
as π(α)π = 0. It is not linear in α, which forces π(α) π(β) = 0, for all α, β Ω1(M). ThisLe means that vector fields π(α) and π(β) are linearly∧ dependent for all α, β∈. This would mean that the map π has rank at most 1 everywhere. But π is skew-symmetric, and thus π = 0. This radically simplifies the rest of the equations. In particular, we are left with the two of them: [Π(ξ), Π(η)] = Π( η i dξ), (4.106) LΠ(ξ) − Π(η) (ΠT (β)) = ΠT β (dξ)(ΠT (β)) . (4.107) LΠ(ξ) LΠ(ξ) − First note that the first equation can be rewritten as
( Π)(η) = Π(i dξ), (4.108) LΠ(ξ) − Π(η) where Π on the left-hand side is viewed as a type (0, p + 1) tensor. We will show in the next chapter that this condition forces Π to be induced by a (p + 1)-vector Π Xp+1(M), which is moreover a Nambu-Poisson tensor. Compare with (5.13) of Lemma∈ 5.1.4 and use Lemma 5.1.6 to prove the skew-symmetry of Π. The second condition equation can then easily be seen to be the transpose to the first Θ one. Note that there is no condition on π whatsoever. We conclude that e (W ∗) = GΘ is a Dirac structure, if and only if 0 Θ e Θ = , Θ Xp+1(M), π X2(ΛpTM), (4.109) ΘT π ∈ ∈ −  p and Θ is a Nambu-Poisson tensor. The bivector π on Λe TM can be completely arbitrary. e 81 e Let L E be an arbitrary subbundle of E. We can define a subbundle ∆ of V as • ⊆

∆ = L L⊥, (4.110) ⊕

where L⊥ E∗ is the annihilator subbundle of L. By definition, ∆ is a maximally isotropic subbundle⊆ of V . Then ∆ is a Dirac structure of V iff L is involutive under higher Dorfman bracket (4.15).

To conclude this section, let us consider the twisting of the bracket [ , ]V . We follow the idea of (4.26) and below. Let Ω2(W ) be of the same block form as in· (4.88).· We define a B ∈ new bracket [ , ]0 as · · V [v, v0]V0 = e−B[eB(v), eB(v0)]V , (4.111) for all v, v0 Γ(V ). We expect to obtain the bracket in the form ∈

[v, v0]0 = [v, v0] d (v, v0), (4.112) V V − B where d : Γ(V ) Γ(V ) Γ(V ) is to be determined now. The calculation shows that for B × → v = (X, P, α, ξ), v0 = (Y, Q, β, η), we have pr d (v, v0) = 0, and W B T pr ∗ (d (v, v0)) = (C(Q) + B(Y )) + d(C (X) B(P ))(Q) (4.113) T M B −LX − + B[X,Y ] + C( Q), LX T T e pr p ∗ (d (v, v0)) = (C (Y ) B(Q)) i d(C (X) B(P )) (4.114) Λ T M B LX − − Y − CT ([X,Y ]) + B( Q). − e LX e

Now, note that d (v, v0) is C∞(M)-linear in v0, but for a general it is not C∞(M)-linear in B e B v. Let us now require this property. This implies that for any f C∞(M) there must hold ∈ df i B(Y ) + (Y.f)B(X) = 0, (4.115) ∧ X (df B(P ))(Q) = 0. (4.116) ∧ The first condition can be rewritten as i (df B(X)) = 0. It then follows that these two Y ∧e conditions imply B = B = 0. Next, from (4.114) we obtain the condition

df (i CT (Y ) + i CT (X)) = 0. (4.117) e ∧ X Y This proves that C has to be induced by C Ωp+1(M). Let us see how d looks now. We obtain ∈ B d (v, v0) = 0, 0, (i dC)(Q), i i dC . (4.118) B − X Y X One can conclude that the twist by eB defines a C∞(M)-bilinear
map d , if and only if p Bp+1 B = B = 0, and C Hom(Λ TM,T ∗M) is induced by a (p + 1)-form C Ω (M). ∈ ∈ e 4.7 Killing sections of generalized metric

One can naturally generalize Section 3.10 to the p 1 setting. In particular, we can define Killing sections as in (3.133). Also the derivation of the≥ explicit form of Killing equations is very

82 similar. Assume that g is of the form (4.28), and C Ωp+1(M). One can see that e Γ(E) is C T C ∈ ∈ a Killing section of G = (e− ) e− , if and only if there holds GE C e C dC C dC ρ(e− (e)). (f 0, f 00) = ([e− (e), f 0] , f 00) + (f 0, [e− (e), f 00] ), (4.119) GE GE D GE D C for all f 0, f 00 Γ(E). Let f := e− (e). We can thus study the Killing equation for the section ∈ dC f and the metric E, but now using the twisted bracket [ , ]D . Let f = X0 +ξ0 for X0 X(M) p G · · ∈ and ξ0 Ω (M), and write f 0 = Y + η, f 00 = Z + ζ. One gets ∈ 1 X0. g(Y,Z) + g− (η, ζ) = g([X0,Y ],Z) + g(Y, [X0,Z]) { } 1 + g− 0 η i dξ0 dC(X0,Y, ), ζ (4.120) LX − Y − · e 1 + g− η, X0 ζ iZ dξ0 dC(X0,Z, )
. e L − − · This yields a set of four separate equations
e

X0.g(Y,Z) = g([X0,Y ],Z) + g(Y, [X0,Z]), (4.121) 1 1 1 X0.g− (η, ζ) = g− ( 0 η, ζ) + g− (η, 0 ζ), (4.122) LX LX 1 0 = g− ( i dξ0 dC(X0,Y, ), ζ), (4.123) − Y − · e e 1 e 0 = g− (η, iZ dξ0 dC(X0,Z, )). (4.124) e − − · The first equation is an ordinary Killing equation for the vector field X0, that is X0 g = 0. The 1 e L second equation yields 0 g− = 0. This in turn yields 0 g = 0. For g of the form (4.28), LX LX Lemma 4.3.2 shows that this already follows from 0 g = 0. Last two equations force LX e e e dξ0 = i 0 dC. (4.125) − X

Now let e = X + ξ. Then X0 = X, and ξ0 = ξ + iX C. Plugging into the above conditions gives: Proposition 4.7.1. Let e = X + ξ. Then e satisfies a generalized Killing equation (3.133) for C T C G = (e− ) e− , if and only if the following conditions hold: GE g = 0, dξ = C. (4.126) LX −LX Moreover, one can also restate Proposition 3.10.4 for the p > 1 case. It has the completely same form and there is no reason to repeat it here explicitly.

4.8 Generalized Bismut connection II

We will now generalize the notion of generalized Bismut connection defined in Section 3.12. p Let G be a generalized metric (4.47) on E = TM Λ T ∗M. We assume that g is of the form (4.28). Let H = dC. We are looking for an example⊕ of the vector bundle connection on E compatible with G. The most straightforward way is to use the form (3.166)e and replace g with g where necessary to make it work. We define the connection as ∇ LC 1 1 1 1 0 X 2 g− H(X, , g− (?)) 1 0 e X = ∇ · . (4.127) ∇ CT 1 1 H(X, ?, ) LC CT 1 −  − 2 · ∇X    e By LC we mean the Levi-Civita connection of g acting on vector fields and on p-forms. Using Lemma∇ 4.3.5 it is easy to check that is metric compatible with G. ∇X 83 We can now examine this connection from a more conceptual viewpoint, using the doubled formalism on V = W W ∗. Let G be the generalized metric on V = W W ∗ corresponding ⊕ V ⊕ to G, and let Ψ : W V be the two isomorphisms induced by GV . Let and be the ± → ± G B fields corresponding to the generalized metric G . Let ∗ be the dual connection induced by V ∇ on the vector bundle E∗. Explicitly ∇

LC 1 1 C X 2 H(X, ,?) 1 C X∗ = 1 1 ∇ 1 LC· − . (4.128) ∇ 0 1 g− H(X, g− (?), ) 0 1   − 2 · ∇X   

V Define a new connection on V e= E E∗ as a block diagonal combination of and ∗: ∇ ⊕ ∇ ∇

V (e + α) = e + ∗ α, (4.129) ∇X ∇X ∇X

V for all e Γ(E) and α Γ(E∗). By construction, is compatible with the generalized ∈ ∈ ∇ metric GV . Moreover, it can be written in a way resembling the original generalized Bismut connection. Define a bilinear map : Γ(W ) Γ(W ) Γ(W ∗) using the twisting map d H × → B defined by (4.118). Let w, w0 Γ(W ). We can view them as elements of Γ(V ). Define ∈

(w, w0) = pr ∗ d (w, w0). (4.130) H W B

V With respect to the splitting V = W W ∗, the connection can be written in the block form ⊕ ∇ LC 1 1 1 1 0 − (X, − (?)) 1 0 V = ∇X − 2 G H G , (4.131) ∇X 1 1 (X,?) LC 1 B  − 2 H ∇X  −B 

LC p p where acts diagonally on W = TM Λ TM and on W ∗ = T ∗M Λ T ∗M. Note that this connection∇ is also compatible with the⊕ natural pairing , on V ,⊕ that is h· ·iV

V V X. v, v0 = v, v00 + v, v00 , (4.132) h iV h∇X iV h ∇X iV

V for all v, v0 Γ(V ). By definition, we obtain by restriction of onto the subbundle E V . In accordance∈ with Lemma 3.12.2, one can write∇ V using the isomorphisms∇ Ψ and a pair⊆ of ∇ ± connections ± on the vector bundle W . One gets ∇

+ (Ψ (w0)) = Ψ ( w0), (4.133) ∇Ψ±(w) + + ∇w 0 Ψ±(w )(Ψ (w0)) = Ψ ( w−w0), (4.134) ∇ − − ∇ for all w, w0 Γ(W ), and ± is a pair of connections on W defined as ∈ ∇

LC 1 1 ±w0 = w0 − (w, w0). (4.135) ∇w ∇pr1w ∓ 2G H

Returning to the original connection , we may calculate its torsion and curvature opera- ∇ C C tors. Define a simpler connection using the formula X = e X e− . This is a connection compatible with , having the form∇ of the middle block∇ as in∇ (4.127). By definition, it will GE b b 84 have the same scalar curvature as . Let X,Y X(M) and Z + ζ Γ(E). We get ∇ ∈ ∈ LC R1(X,Y )(Z + ζ) = R (X,Y )Z (4.136)

1 1 LC 1 1 1 LC 1 + g− ( H)(Y, , g− (ζ)) g− ( H)(X, , g− (ζ)) b 2 ∇X · − 2 ∇Y · 1 1 1 1 1 1 g− H(X, , g− H(Y,Z, )) + g− H(Y, , g− H(X,Z, )), − 4 · e · 4 · e · R (X,Y )(Z + ζ) = RLC (X,Y )ζ (4.137) 2 e e 1 1 ( LC H)(Y,Z, ) + ( LC H)(X,Z, ) b − 2 ∇X · 2 ∇Y · 1 1 1 1 1 1 + H(X, g− H(Y, , g− (ζ)), ) H(Y, g− H(X, , g− (ζ)), ) 4 · · − 4 · ·

The Ricci tensor Ric has only two non-triviale components. Namely, we get e

LC 1 1 k 1 Ric(b X,Y ) = Ric (X,Y ) + H(Y, g− (dy ), g− H(∂ ,X, )), (4.138) 4 k · 1 LC 1 k 1 bRic(ξ, Y ) = ( H)(Y, g− (dy ), g− (ξ)). (4.139) 2 ∇∂k e

1 λ Finally, define a scalarb curvature = Ric( − (e ), ee). One obtains R GE λ 1 b =b (g) H HijK . (4.140) R R − 4 ijK The indices of H are raised by metricb g, and (g) is the scalar curvature of the Levi-Civita connection of g. Let be the scalar curvatureR of the original connection defined using the R C T C 1 λ ∇ generalized metric G = (e− ) e− , that is = Ric(G− (e ), e ). By construction, the GE R λ two connections have the same scalar curvature, that is we get = . R R b

85 86 Chapter 5

Nambu-Poisson structures

In this chapter, we will discuss in detail the definitions and structures induced by a Nambu bracket on a manifold. This (p + 1)-ary bracket was for p = 2 introduced in 1972 by Y. Nambu in [78] as an attempt to generalize the classical Hamiltonian mechanics. Nambu defines a trinary bracket f, g, h for a triplet of functions of three variables (x, y, z) as { } ∂(f, g, h) f, g, h = . (5.1) { } ∂(x, y, z)

He notes that such a bracket has some remarkable properties, in particular it is completely skew-symmetric and it satisfies the Leibniz rule.

ff 0, g, h = f f 0, g, h + f, g, h f 0. (5.2) { } { } { } Interestingly, he does not attempt to generalize the third usual property of Poisson bracket, the Jacobi identity. An axiomatic definition of Nambu brackets was introduced more than twenty years later in [87], where the term Nambu-Poisson manifold appears for the first time. The author already suspects and emphasizes throughout his paper that Nambu-Poisson structures are much more rigid objects than Poisson structures. This was finally proved by several different people in 1996, for example in [3]. For the complete list of references and many more interesting remarks, see the survey [88] of I. Vaisman.

5.1 Nambu-Poisson manifolds

Definition 5.1.1. Let p 1 be a fixed integer, and let ,..., : C∞(M) C∞(M) ≥ {· ·} × · · · × → C∞(M) be an R-(p+1)-linear map. We say that ,..., is a Nambu bracket, if the following properties hold: {· ·}

1. The map ,..., is completely skew-symmetric. {· ·} 2. It satisfies the Leibniz rule:

f , . . . , f g = f , . . . , f g + f f , . . . , g . (5.3) { 1 p+1 · p+1} { 1 p+1} p+1 p+1{ 1 p+1}

87 3. It satisfies the fundamental identity:

f1, . . . , fp, g1, . . . , gp+1 = f1, . . . , fp, g1 , . . . , gp+1 { { }} {{ } } (5.4) + + g ,..., f , . . . , f , g . ··· { 1 { 1 p p+1}} Both the Leibniz rule and the fundamental identity are assumed to hold for all involved smooth functions.

We see that for p = 1, the definition reduces to the ordinary Poisson bracket on M. Next, note that for any f C∞(M), the bracket g , . . . , g 0 := f, g , . . . , g defines again a ∈ { 1 p} { 1 p} Nambu-Poisson bracket. Both axioms can be read as follows. To any p-tuple (f1, . . . , fp) of smooth functions, we may assign an operator X := f , . . . , f , . (5.5) (f1,...,fp) { 1 p ·}

Leibniz rule proves that X(f1,...,fp) is a vector field on M, X(f1,...,fp) X(M). The fundamental identity then requires X to be a derivation of the bracket ∈,..., . (f1,...,fp) {· ·} It follows from the Leibniz rule (5.3) that ,..., in fact depends only on differentials of incoming functions. It thus makes sense to define{· a Nambu-Poisson·} tensor Π by Π(df , . . . , df ) := f , . . . , f , (5.6) 1 p+1 { 1 p+1} for all f1, . . . , fp+1 C∞(M). The complete skew-symmetry of the bracket ,..., is clearly equivalent to Π being∈ a (p + 1)-vector, Π Xp+1(M). The fundamental identity{· can·} be then rewritten simply as ∈ X Π = 0. (5.7) L (f1,...,fp) Now, let us recall the fundamental theorem for the theory of Nambu-Poisson manifolds. The proof can be found for example in [3, 31, 88], and we thus omit it here. Theorem 5.1.2. Let p 2, and Π Xp+1(M). Then Π is a Nambu-Poisson tensor, if and only if for every x M,≥ such that Π(∈x) = 0, there is a neighborhood U x, and a set of local coordinates (x1, . . .∈ , xn) on U, such that6 locally 3 ∂ ∂ Π = ... . (5.8) ∂x1 ∧ ∧ ∂xp+1 The components of Π in this coordinates are thus Πi1...ip+1 = i1...ip+1 , and the corresponding bracket ,..., has the local form {· ·} ∂(f , . . . , f ) f , . . . , f = 1 p+1 (5.9) { 1 p+1} ∂(x1, . . . , xp+1) We will call (x1, . . . , xn) the Darboux coordinates for Π.

Note that the only if part of this theorem is not true for p = 1. The simple counter-example is the canonical Poisson structure Π on R2n, n > 1: n ∂ ∂ Π = . (5.10) ∂qj ∧ ∂p j=1 j X In these coordinates, the component matrix Πij is invertible. On the other hand, the matrix in coordinates (5.8) has always the rank 2, hence it cannot be invertible. This proves that Π defined by (5.10) is not decomposable. For Poisson manifolds, there holds a more subtle statement, called Darboux-Weinstein theorem [90]. We will now use Theorem 5.1.2 to reformulate the fundamental identity in several ways more useful for our purposes.

88 Lemma 5.1.3. Let Π Xp+1(M), and p > 1. Then Π satisfies the fundamental identity (5.7), if and only if there holds∈ Π = Π, dξ Π, (5.11) LΠ(ξ) −h i for all ξ Ωp(M). ∈

Proof. If part is simple, for any p-tuple (f1, . . . , fp) choose ξ = df1 ... dfp. Then Π(ξ) = p ∧ ∧ ( 1) X(f ,...,f ), and since dξ = 0, we get X Π = 0. − 1 p L (f1,...,fp) Conversely, assume that Π is a Nambu-Poisson tensor. It suffices to prove (5.11) for ξ in the form ξ = g df1 ... dfp, where g, f1, . . . , fp C∞(M). At points x M where Π(x) = 0 the statement holds· ∧ trivially.∧ We can thus assume∈ that we can work locally∈ with Π in the form (5.8). We have

p p+1 Π(ξ)Π = ( 1) gX Π = ( 1) X(f ,...,f ) idgΠ. L − L (f1,...,fp) − 1 p ∧ We have used the fundamental identity (5.7) to get rid of one term. We can write

p ∂g ∂ ∂ ∂ ∂ i Π = ( 1)r+1 ...... , dg − ∂xr ∂x1 ∧ ∧ ∂xr 1 ∧ ∂xr+1 ∧ ∧ ∂xp r=1 − X and consequently, we obtain

p p+1 p+1 ∂ r ∂g ∂ ∂ ( 1) X idgΠ = ( 1) ... X ... − (f1,...,fp) ∧ − ∂x1 ∧ ∧ (f1,...,fp) ∂xr ∂xr ∧ ∧ ∂xp r=1 Xp ∂g = ( 1)p+1( Xr )Π = ( 1)p+1(X .g)Π − (f1,...,fp) ∂xr − (f1,...,fp) r=1 X = ( 1)p+1 Π, df ... df dg Π = Π, dξ Π. − h 1 ∧ ∧ p ∧ i −h i This proves the assertion of the Lemma. 

Note that (5.11) does not hold for p = 1. To include this case, one must modify it as 1 Π = ( Π, dξ Π i (Π Π)). (5.12) LΠ(ξ) − h i − p + 1 dξ ∧ For p > 1, the decomposability implies Π Π = 0, and the proof is then just an easy alteration. For p = 1, this is not necessarily true, as illustrates∧ the example (5.10). The proof of (5.12) is in this case left for an interested reader. We will now derive a more conceptual reformulation of the fundamental identity, which will give an immediate geometrical description of Nambu-Poisson structures. Lemma 5.1.4. Let p 1, and Π Xp+1(M). Then Π is a Nambu-Poisson tensor iff ≥ ∈ [ Π](η) = Π(i dξ), (5.13) LΠ(ξ) − Π(η) for all ξ, η Ωp(M). ∈ Proof. For p = 1, this follows from 1 ( Π)(η) + Π(i dξ) = i i [Π, Π] , (5.14) LΠ(ξ) Π(η) 2 η ξ S

89 for all ξ, η Ω1(M). This can be verified directly in coordinates, or using several explicit forms of the Schouten-Nijenhuis∈ bracket [ , ] . See for example [67]. · · S We will focus on the p > 1 case here. Assume that Π is a Nambu-Poisson tensor. At points where Π(x) = 0, the equation (5.13) holds trivially. We can thus again assume that Π is of the form (5.8). Note that (5.13) is C∞(M)-linear in η. Moreover, Π and Π(ξ) only have components with indices ranging only in 1, . . . , p + 1 . We thus have to check (5.13) { } only for η in the form η = dx[r] := dx1 ... dxr ... dxp+1, where r 1, . . . , p + 1 and ∧ ∧ ∧ ∧ ∈ { } dxr denotes the omitted term. Choose one such η. Both sides of (5.13) are vector fields. If we examine the k-th component of the both sidesd for k = r, the left hand side vanishes. The r+1 kJ 6 right-handd side gives ( 1)  (dξ)rJ . The only non-trivial contribution to the sum can come − ˆ from J = [k] := (1,..., k, . . . , p + 1), but then (dξ)r[k] = 0 because r [k]. Thus also the right-hand side vanishes. For k = r, the left-hand side gives ∈

p+1 ( Π)(dx[r])r = ( Π)r[r] = ( 1)r+1( Π)1...p+1 = ( 1)r ξ qJ = ( 1)r(dξ) . LΠ(ξ) LΠ(ξ) − LΠ(ξ) − J,q − 1...p q=1 X The right-hand side can be rewritten as

r r rJ r [Π(i [r] dξ)] = ( 1) Π (dξ) = ( 1) (dξ) . − Π(dx ) − rJ − 1...p A comparison of both sides gives the result. Conversely, if (5.13) holds, we can plug in ξ = df ... df to obtain the fundamental identity (5.7). 1 ∧ ∧ p  We can immediately use this lemma to prove some important observations. First note that the identity (5.13) has two parts, differential and algebraical. To see this, rewrite it in some 1 n I J local coordinates (y , . . . , y ) . We write ξ = ξI dy , η = dy , and take the k-th component of the result. We get

p [( Π)(η)]k = ( Π)kJ = ξ ΠnI ΠkJ ΠkI ΠnJ Πjq I Πkj1...n...jp LΠ(ξ) LΠ(ξ) I ,n − ,n − ,n q=1 X p
ξ ΠkI ΠnJ + Πjq I Πkj1...n...jp . − I,n q=1 X
The right-hand side gives Π(i dξ)k = ΠkL(dξ) ΠmJ = ξ δnI ΠkLΠmJ . − Π(η) − mL − I,n mL The terms proportional to ξI give the differential part of the fundamental identity: p ΠnI ΠkJ ΠkI ΠnJ Πjq I Πkj1...n...jp = 0. (5.15) ,n − ,n − ,n q=1 X The terms proportional to ξI,n give the quadratic equation for Π, the algebraical part of the fundamental identity:

p kI nJ jq I kj1...n...jp nI kL mJ Π Π + Π Π = δmLΠ Π . (5.16) q=1 X This one is trivially satisfied for p = 1. For p > 1 it is in fact this part which forces Π to be decomposable. We can use this to immediately prove the following:

90 Lemma 5.1.5. Let p > 1. Let Π be a Nambu-Poisson tensor, and f C∞(M). Then n ∈ Π0 := fΠ is also a Nambu-Poisson tensor. In particular, every Π X (M), n = dim M, is a Nambu-Poisson tensor. ∈

Proof. Multiplication of Π by f does not change the algebraic part (5.16). Because Π is Nambu-Poisson, we can choose Darboux coordinates where ΠiJ = iJ . It suffices to prove the differential identity, choose ξ = dxI . We thus have to show that

I (fΠ) = 0. (5.17) L(fΠ)(dx ) We have

I I (fΠ) = ((fΠ)(dx ).f)Π + f I Π L(fΠ)(dx ) L(fΠ)(dy ) I 2 I = f(Π(dx ).f)Π + f I Π f(Π(dy ) i Π) LΠ(dy ) − ∧ df = f (Π(dxI ).f)Π Π(dyI ) i Π . − ∧ df We have used the fundamental identity for Π in the last step. It
thus suffices to show that

(Π(dxI ).f)Π = Π(dyI ) i Π. (5.18) ∧ df This is equation which has a single non-trivial component, that is (1, . . . , p). We get

nI kI nJ  ∂nf =  ∂nf kJ .

The only non-trivial contribution is possible for I = [r] for r 1, . . . , p + 1 . We then get ∈ { } ( 1)r+1∂ f = ( 1)r+1∂ fnJ  = ( 1)r+1∂ f. − r − n rJ − r This finishes the proof of the first part. The second part follows easily. Every Π Xn(M) can ∈ be locally written as Π = f∂ ... ∂ . The n-vector Π = ∂ ... ∂ is (at least locally 1 ∧ ∧ n 1 ∧ ∧ n well defined) Nambu-Poisson tensor, and we can use the preceding proof to show that Π = fΠ satisfies the fundamental identity. e  e There is one very interesting observation, noted in [37]. If one assumes that Leibniz rule (5.3) holds in every input (instead of in just one as we did), and that fundamental identity (5.4) holds, then the complete skew-symmetry of the bracket already follows. We reformulate this in the language of the corresponding tensors and vector bundle morphisms:

p Lemma 5.1.6. Let Π Hom(Λ T ∗M,TM) be an arbitrary vector bundle morphism satisfying (5.16), where now ΠiJ ∈:= dyi, Π(dyJ ) . Then Π is induced by a (p + 1)-vector on M, that is Π(ξ) = Π( , ξ) for Π Xp+1h (M). i · ∈ Proof. We will show that the algebraical identity (5.16) implies that ΠiJ = 0, whenever i J, ∈ in arbitrary coordinates. In particular, for any non-zero α Tx∗M, we can choose the local 1 ∈ i2 ip coordinates in M, such that dy x = α. This will prove that α, Π(α dy ... dy = 0, which implies the complete skew-symmetry| of Π. Let us thush prove this∧ assertion,∧ ∧ let i J. Choose k = n = i and J = I in (5.16). We obtain ∈

p iJ 2 jq J ij1...i...jp iJ iL mI (Π ) + Π Π = δmLΠ Π . q=1 X 91 By assumption, we have i = jr for some r 1, . . . , p . Thus only the q = r term on the left- hand side contributes to the sum. Moreover,∈ { the right-hand} side vanishes identically, because iJ δmL is completely skew-symmetric in top indices. We thus get 2(ΠiJ )2 = 0, which proves the assertion. 

5.2 Leibniz algebroids picture

Recall now Example 2.2.2 and the Koszul bracket induced on cotangent bundle by a Poisson tensor. Can one construct such a bracket also for a Nambu-Poisson structure? p Proposition 5.2.1. Let Π Hom(Λ T ∗M,TM) be a vector bundle morphism. Let L = p ∈ Λ T ∗M, and define the R-bilinear bracket [ , ]Π : Γ(L) Γ(L) Γ(L) as · · × → [ξ, η] := η i dξ, (5.19) Π LΠ(ξ) − Π(η) p for all ξ, η Ω (M). Then (L, Π, [ , ]Π) is a Leibniz algebroid, if and only if Π is a Nambu- Poisson tensor.∈ · ·

Proof. The Leibniz rule (2.4) holds for any Π. We will show that the Leibniz identity (2.5) for [ , ]Π is equivalent to the fundamental identity (5.7) for Π. In particular, we will use its version (5.13).· · For ξ, η Ωp(M), define the vector field V (ξ, η) as ∈ V (ξ, η) := [Π(ξ), Π(η)] Π( η i dξ). (5.20) − LΠ(ξ) − Π(η) We claim that V (ξ, η) = 0 if and only if Π is a Nambu-Poisson tensor. This follows from the fact that commutes with contractions, and thus L [Π(ξ), Π(η)] = (Π(η)) = ( Π)(η) + Π( η) LΠ(ξ) LΠ(ξ) LΠ(ξ) = Π( η i dξ) + ( Π)(η) + Π(i dξ) . LΠ(ξ) − Π(η) { LΠ(ξ) Π(η) } We see that Π satisfies (5.13) if and only if V (ξ, η) = 0. Combining this with Lemma 5.1.6 proves that V (ξ, η) = 0 if and only if Π is a Nambu-Poisson tensor. Moreover, note that we have V (ξ, η) = [Π(ξ), Π(η)] Π([ξ, η] ). (5.21) − Π Let us now examine the Leibniz identity for [ , ] : · · Π [ξ, [η, ζ]Π]Π = Π(ξ)( Π(η)ζ iΠ(ζ)dη) iΠ( ζ i dη)dξ, (5.22) L L − − LΠ(η) − Π(ζ) [[ξ, η]Π, ζ]Π = Π( i dξ)ζ iΠ(ζ)d( Π(ξ)η iΠ(η)dξ), (5.23) L LΠ(ξ)− Π(η) − L − [η, [ξ, ζ]Π]Π = Π(η)( Π(ξ)ζ iΠ(ζ)dξ) iΠ( ζ i dξ)dη. (5.24) L L − − LΠ(ξ) − Π(ζ) Arranging this into the Leibniz identity for [ , ]Π and using the Cartan formulas to rewrite several terms, one arrives to the condition · · ζ i dη + i dξ = 0. (5.25) LV (ξ,η) − V (ξ,ζ) V (η,ζ) We are now ready to finish the proof. First, when Π is a Nambu-Poisson tensor, then V (ξ, η) = 0, hence (5.25) holds. This proves that (L, Π, [ , ] ) is a Leibniz algebroid. Con- · · Π versely, if (L, Π, [ , ]Π) is a Leibniz algebroid, we know that the anchor Π is a bracket homo- morphism (2.6).· Glancing· at (5.21), we see that this is equivalent to V (ξ, η) = 0. Hence Π is a Nambu-Poisson tensor. 

92 There is a natural way how to explain the origin of the bracket (5.19). Consider now p arbitrary Π Hom(Λ T ∗M,TM). We view its graph G as a subbunle of E: ∈ Π p G = Π(ξ) + ξ ξ Λ T ∗M . (5.26) Π { | ∈ }

We can now study its involutivity. Note that GΠ is not an isotropic subbundle with respect to the pairing (4.1). Instead, it forms an example of an almost Nambu-Dirac structure, defined and studied in detail in [41].

Proposition 5.2.2. The subbundle GΠ is involutive under the higher Dorfman bracket (4.15), if and only if Π is a Nambu-Poisson tensor.

Proof. Let Π(ξ) + ξ and Π(η) + η be sections of GΠ. Then [Π(ξ) + ξ, Π(η) + η] = [Π(ξ), Π(η)] + η i dξ. (5.27) D LΠ(ξ) − Π(η)

The right-hand side is again a section of GΠ iff [Π(ξ), Π(η)] = Π η i dξ . (5.28) LΠ(ξ) − Π(η) Using the properties of Lie derivative, this is equivalent to

( Π)(η) = Π(i dξ). (5.29) LΠ(ξ) − Π(η) This is exactly the fundamental identity for Π written in the form (5.13). 

We can use this to clarify the structure of the bracket (5.19). Indeed, define a vector bundle p isomorphism Ψ Hom(Λ T ∗M,GΠ) as Ψ(ξ) = Π(ξ) + ξ. Assume that Π is a Nambu-Poisson tensor. The relation∈ is then Ψ([ξ, η]Π) = [Ψ(ξ), Ψ(η)]D. (5.30) The anchor for [ , ] is then in fact a composition ρ Ψ. Indeed, we have · · Π ◦ (ρ Ψ)(ξ) = Π(ξ). (5.31) ◦ This gives an alternative proof of the ”if part” of Proposition 5.2.1. To conclude this section, see that Proposition 5.2.2 allows one to easily define the twisted version of Nambu-Poisson structure.

p+2 H Definition 5.2.3. Let H Ω (M) be a closed (p + 2)-form, and let [ , ]D be a twisted ∈ p · · Dorfman bracket (4.27). Let Π Hom(Λ T ∗M,TM), and let G E be its graph (5.26). We ∈ Π ⊆ say that Π is an H-twisted Nambu-Poisson tensor if GΠ defines a subbundle involutive under [ , ]H . · · D This definition by itself does not point at all to the statement of the following proposition, which may seem somewhat surprising. It was first observed and proved in [18]. Proposition 5.2.4. For p = 1, Definition 5.2.3 gives a usual H-twisted Poisson manifold, where Π X2(M) satisfies the condition ∈ 1 [Π, Π] (ξ, η, ζ) = H(Π(ξ), Π(η), Π(ζ)), (5.32) 2 S 1 for all ξ, η, ζ Ω (M). Recall that [ , ]S is the Schouten-Nijenhuis bracket of multivector fields. For p > 1, Definition∈ 5.2.3 gives no· new· structure at all.

93 Proof. For p = 1, the statement follows from the fact that 1 [Π, Π] (ξ, η, ) = [Π(ξ), Π(η)] Π( η i dξ). (5.33) 2 S · − LΠ(ξ) − Π(η) The involutivity of G under [ , ]H gives the condition Π · · Π [Π(ξ), Π(η)] = Π( η i dξ H(Π(ξ), Π(η), )) (5.34) LΠ(ξ) − Π(η) − · A combination of these two relations gives the assertion of the proposition. For p > 1, we can rewrite the involutivity of G under [ , ]H as Π · · D ( Π)(η) = Π(i dξ + H(Π(ξ), Π(η), )). (5.35) LΠ(ξ) − Π(η) · Now recall that the algebraic part (5.16) of the fundamental identity comes from the failure of (5.13) to be C∞(M)-linear in ξ. But the addition of H does not change this part! Hence Π n satisfies (5.16). It was shown in [3] that this in fact proves that there is a local frame (eλ)λ=1, such that Π = e1 ... ep+1. Thus the only components of H contributing to (5.35) are those corresponding∧ to the∧ first p + 1 vectors of the frame. But H is a (p + 2)-form, and those components vanish due to skew-symmetry. Hence H in no way contributes to (5.35) and Π satisfies the untwisted fundamental identity (5.13). 

5.3 Seiberg-Witten map

We will now show that given a p-form A, one can use a Nambu-Poisson tensor Π to define a diffeomorphism of M. It is a direct generalization of the Seiberg-Witten map [85]. In the presented form, it was introduced for p = 1 in [63] as a dual analogue of Moser’s lemma in symplectic geometry [77]. Its generalization to p > 1 were presented in [59] and [23]. Let us first recall a few facts about time-dependent vector fields and their flows. Definition 5.3.1. Let I R be an open interval, and let V : I X(M) be a map. A value ⊆ • → of this map at given t I is a vector field denoted as Vt. We say that V is a time-dependent ∈ 1 n • vector field, if in every local coordinate set (y , . . . , y ) the components of Vt depend smoothly on t. General time-dependent tensor fields are defined analogously. Remark 5.3.2. Equivalently, we may view V as a vector field on the extended manifold M I in the form • × V (x, t) = Vt(x) + ∂t. (5.36) • This interpretation is however not useful for higher tensor fields.

For a time-dependent vector field, a notion of integral curve still makes sense, except that one has to specify its starting time. For s I, γ : J M is an integral curve starting at ∈ s → m M at the time s, ifγ ˙s(t) = Vt(γs(t)) for all t J, and γs(s) = m. Here J I is an open interval.∈ The vector field local flow theorem generalizes∈ as follows: ⊆ Proposition 5.3.3. Let V be a time-dependent vector field defined on an open interval I R. Then there exists an open• subset I I M and a smooth map ψ : M called⊆ a time-dependent local flow of VE, such⊆ × that× E → • 1. For each s I and m M, the set = t I (t, s, m) is an open subinterval ∈ ∈ Es,m { ∈ | ∈ E} of I containing s. The map ψs,m := ψ( , s, m): s,m M is a maximal integral curve of V starting at m at the time s. · E → • 94 2. For any t , and q = ψ (t), there holds = , and ψ = ψ . ∈ Es,m s,m Et,q Es,m t,q s,p 3. For any (t, s) I I, the set Mt,s = m M (t, s, m) is an open subset of M. ∈ × { ∈ | ∈ E} 1 The map ψ := ψ(t, s, ): M M is a diffeomorphism, and ψ− = ψ . t,s · t,s → s,t t,s s,t 4. Let m M , ant ψ (m) M . Then p M and ∈ t,s t,s ∈ v,t ∈ v,s ψ ψ = ψ . (5.37) v,t ◦ t,s v,s Proof. The proof is in fact a careful application of the ordinary local flow theorem for a vector field V mentioned in Remark 5.3.2. For details see [71].  • Having a local flow, there is a well defined generalization of Lie derivative. Let T be a • time-dependent tensor field, and ψt,s be a local flow of a time-dependent vector field V . Define a new time-dependent tensor field as •

τ d ( V T )s = ψt,s∗ (Tt). (5.38) L • • dt t=s

A direct calculation similar to the one for ordinary tensor fields shows that τ ( V T )s = ∂sTs + Vs Ts. (5.39) L • • L We have used the superscript τ to distinguish the generalization from the ordinary Lie derivative standing on the right-hand side (which is assumed to be given by usual algebraic formula). Lie derivative is a tool useful to describe the invariance of tensor fields with respect to flows. Let us show that for time-dependent tensor fields, τ plays the same role. L Lemma 5.3.4. Let V be a time-dependent vector field. Let T be a time-dependent tensor τ • • field satisfying V T = 0. Then for any (t, s) I I, one has ψt,s∗ (Tt) = Ts on Mt,s. L • • ∈ ×

d Proof. First let us show that the assumption in fact implies that dt ψt,s∗ (Tt) = 0, for all t I. We will now use the composition rule (5.37). Indeed, one has ∈

d d d τ ψt,s∗ (Tt) = ψt∗+a,s(Tt+a) = ψt,s∗ ψt∗+a,t(Tt+a) = ψt,s∗ ( V (T )t) = 0. (5.40) dt da da L • • a=0 a=0

This proves that ψt,s∗ (T t) = Ts0 for some T 0 and all t I. Setting t = s shows that T 0 = T .  • ∈ • • We now have all ingredients prepared to introduce the Seiberg-Witten map. Let Π Xp+1(M) be a Nambu-Poisson tensor, and let A Ωp(M). Denote F = dA. We can use F to∈ ∈ F define a new Nambu-Poisson tensor Π0 as follows. Let e Aut(E) be the map (4.13) induced ∈ by F . Let GΠ E be a graph of Π which is by definition involutive under the Dorfman bracket. We have shown⊆ in Proposition 4.2.3 that eF is an automorphism of the Dorfman bracket. This F proves that the subbundle e (GΠ) is also involutive under the Dorfman bracket. If there is p F Π0 Hom(Λ T ∗M,TM) such that GΠ0 = e (GΠ), we know that Π0 is again a Nambu-Poisson tensor.∈ Let Π(ξ) + ξ Γ(G ). We have ∈ Π eF (Π(ξ) + ξ) = Π(ξ) + (1 F T Π)(ξ). (5.41) − T T If Π0 exists, there must hold Π(ξ) = Π0(1 F Π)(ξ). Let us assume that 1 F Π is an invertible map. Hence − − T 1 Π0 = Π(1 F Π)− . (5.42) − 95 Now define a time-dependent tensor field Π as • T 1 Π = Π(1 tF Π)− . (5.43) t −

Assume that it is well defined for I = ( , 1 + ) for some  > 0. Clearly Π0 = Π, Π1 = Π0. Using the same argument as above, Π is− a Nambu-Poisson tensor for every t I. In particular, t ∈ it satisfies the fundamental identity (5.13). Plug ξ = A into this condition. It gives Πt(A)Πt = Π F T Π . Next, examine the time derivative of Π . One obtains L − t t t T 1 T T 1 T ∂ Π = Π(1 tF Π)− F Π(1 F Π)− = Π F Π . (5.44) t t − − t t # # If we define a time-dependent vector field A as At = Πt(A), we have just proved that τ • # Π = 0. Using Lemma 5.3.4 we see that ψt,s∗ Πt = Πs. In particular, define a diffeomorphism LA• • ρ := ψ . The map ρ Diff(M) is called the Seiberg-Witten map. By construction, A 1,0 A ∈ ρ∗ (Π0) = Π. Note that it is essential that Π is a Nambu-Poisson tensor for every t I. A t ∈ To conclude this section, note that for p > 1, the form F Ωp+1(M) used to define a time-dependent tensor field Π does not have to be closed to define∈ a set of Nambu-Poisson tensors. This is true because in• fact for any F Ωp+1(M) there holds ∈

t 1 Π = (1 Π,F )− Π. (5.45) t − p + 1h i

This can be proved easily in Darboux coordinates (5.8) for Π. This shows that Πt is just a scalar multiple of Π and the assertion follows from Lemma 5.1.5.

96 Chapter 6

Conclusions and outlooks

We have introduced an extension of the generalized geometry suitable for a description of membrane sigma models. Our intention was to follow the outline of the standard generalized geometry, in particular in the case of generalized metric. This was not possible until we have in- troduced a doubled formalism. To our delight, we were able to use it to significantly simplify the calculations required in particular to relate commutative and semi-classically non-commutative p-DBI actions. Moreover, it proved useful to discover the membrane analogue of background- independent gauge and the double scaling limit. Of course, this formalism is mathematically interesting in its own right. We should focus on the future prospects of the ideas presented in this thesis. We will now point out the sections which require further investigation. Let us start with the mathematical side of things. We have defined connections on local Leibniz algebroids in Section 2.4. This direction is definitely worth of pursuing. For example, one can study a class of Courant algebroid connections which are compatible with the general- ized metric and their torsion operator (2.55) vanishes. A set of such connections is larger then in the case of the ordinary Riemannian geometry. However, it turns out that they such connec- tions have a quite nice form allowing for the calculation of their scalar curvature. Interestingly, this scalar function is exactly the one multiplying the integral density in string effective actions. One should relate this viewpoint to the generalized geometry treatment of string effective ac- tions in [11, 12]. It would be also necessary to generalize such Courant algebroid connections to the Leibniz algebroid setting, in particular using the doubled formalism of Sections 4.5 and 4.6. Killing sections of Section 3.10 have an interesting role in string theory, since one can construct generalized charges using such sections. Those charges are conserved in time evolution if and only if the respective sections satisfy the generalized Killing equations. Moreover, such sections are closely related to T-duality, see [38]. There has to exist some link between these observations. It would be also important to find a geometrical explanation to membrane duality rotations od Duff and Lu in [30]. Understanding T-duality analogues for membranes could possibly give an equivalent derivation od p-DBI actions. There are several directions where to proceed with the physics presented in the attached papers. A Nambu sigma model proposed using AKSZ construction in [18] is a little bit different from the one defined in [59]. It would be interesting to analyze this disparity. In particular, the latter version used also in our paper [61] is not invariant with respect to worldvolume reparametrizations. Is there a way to define an invariant and possibly more general Nambu

97 sigma model action? An important feature of topological Poisson sigma models is the existence of the gauge trans- formations, see for example [13]. It is in fact a direct consequence of the fact that topological Poisson sigma models are a theory with constraints, and constraints themselves are integrals of motion. Since the topological Nambu sigma model is also a theory with constraints, there should be a similar process leading to its gauge symmetries. Nambu-Poisson structures can be possibly an interesting object on its own. Usual Poisson structures and Poisson sigma models turned out to be a crucial element in the integration of Lie algebroids. Is there a similar use for Nambu-Poisson structures and Nambu sigma models? By construction, standard generalized geometry (and its extended variant presented here) is not suitable to describe supersymmetric theories due to its lack of Grassmanian variables. There are its extensions used in supergravity [24, 25] and M-theory [52]. It would be inter- esting to modify generalized geometry to work in a world of graded geometry, in particular supermanifolds in the sense of [21]. The proposed p-DBI action in [59] is obviously only the bosonic part of a (yet unknown) full supersymmetric action. An understanding of generalized (super)geometry could give us answers necessary to derive it. The guiding principle of ”doubling” and related construction of generalized metric can be easily generalized to more general vector bundles. There is an intriguing relation between Leibniz algebroids and Lie algebra representations, see [7]. This reference provides a huge class of interesting Leibniz algebroid examples, which can treated similarly as we did within our extended generalized geometry. This can be useful to understand better the spherical T-duality, [15]. The author hopes that this thesis and his research proved once more the importance of understanding the geometry underlying the theoretical physics. Not only it brings new ways how to understand and verify known things, but it could also provide the missing tools to push the knowledge of our world further.

98 Bibliography

[1] A. Alekseev and P. Xu, “Derived Brackets and Courant Algebroids.” http://www.math.psu.edu/ping/anton-final.pdf. [2] A. Alekseev and T. Strobl, Current algebras and differential geometry, JHEP 0503 (2005) 035, [hep-th/0410183]. [3] D. Alekseevsky and P. Guha, On decomposability of Nambu-Poisson tensor, Acta Math. Univ. Comenianae LXV (1996), no. 1 1. [4] T. Asakawa, H. Muraki, and S. Watamura, D-brane on Poisson manifold and Generalized Geometry, arXiv:1402.0942. [5] T. Asakawa, H. Muraki, S. Sasa, and S. Watamura, Generalized geometry and nonlinear realization of generalized diffeomorphism on D-brane effective action, Fortsch.Phys. 62 (2014) 749–753, [arXiv:1405.4999]. [6] T. Asakawa, S. Sasa, and S. Watamura, D-branes in Generalized Geometry and Dirac-Born-Infeld Action, JHEP 1210 (2012) 064, [arXiv:1206.6964]. [7] D. Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry, Journal of Geometry and Physics 62 (May, 2012) 903–934, [arXiv:1101.0856]. [8] D. S. Berman, H. Godazgar, M. Godazgar, and M. J. Perry, The Local symmetries of M-theory and their formulation in generalised geometry, JHEP 1201 (2012) 012, [arXiv:1110.3930]. [9] D. S. Berman, H. Godazgar, and M. J. Perry, SO(5,5) duality in M-theory and generalized geometry, Phys.Lett. B700 (2011) 65–67, [arXiv:1103.5733]. [10] D. S. Berman and M. J. Perry, Generalized Geometry and M theory, JHEP 1106 (2011) 074, [arXiv:1008.1763]. [11] R. Blumenhagen, A. Deser, E. Plauschinn, and F. Rennecke, Non-geometric strings, symplectic gravity and differential geometry of Lie algebroids, JHEP 1302 (2013) 122, [arXiv:1211.0030]. [12] R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, and C. Schmid, The Intriguing Structure of Non-geometric Frames in String Theory, Fortsch.Phys. 61 (2013) 893–925, [arXiv:1304.2784]. [13] M. Bojowald and T. Strobl, Classical solutions for Poisson sigma models on a Riemann surface, JHEP 0307 (2003) 002, [hep-th/0304252].

99 [14] P. Bouwknegt, Lectures on cohomology, T-duality, and generalized geometry, Lect.Notes Phys. 807 (2010) 261–311. [15] P. Bouwknegt, J. Evslin, and V. Mathai, Spherical T-Duality, arXiv:1405.5844. [16] P. Bouwknegt, J. Evslin, and V. Mathai, T duality: Topology change from H flux, Commun.Math.Phys. 249 (2004) 383–415, [hep-th/0306062]. [17] P. Bouwknegt, K. Hannabuss, and V. Mathai, T duality for principal torus bundles, JHEP 0403 (2004) 018, [hep-th/0312284]. [18] P. Bouwknegt and B. Jurco, AKSZ construction of topological open p-brane action and Nambu brackets, Rev.Math.Phys. 25 (2013) 1330004, [arXiv:1110.0134]. [19] L. Brink, P. Di Vecchia, and P. S. Howe, A Locally Supersymmetric and Reparametrization Invariant Action for the Spinning String, Phys.Lett. B65 (1976) 471–474. [20] T. Buscher, A Symmetry of the String Background Field Equations, Phys.Lett. B194 (1987) 59. [21] A. S. Cattaneo and F. Sch¨atz, Introduction to Supergeometry, Reviews in Mathematical Physics 23 (2011) 669–690, [arXiv:1011.3401]. [22] G. R. Cavalcanti and M. Gualtieri, Generalized complex geometry and T-duality, ArXiv e-prints (June, 2011) [arXiv:1106.1747]. [23] C.-H. Chen, K. Furuuchi, P.-M. Ho, and T. Takimi, More on the Nambu-Poisson M5-brane Theory: Scaling limit, background independence and an all order solution to the Seiberg-Witten map, JHEP 1010 (2010) 100, [arXiv:1006.5291]. [24] A. Coimbra, C. Strickland-Constable, and D. Waldram, Supergravity as Generalised Geometry I: Type II Theories, JHEP 1111 (2011) 091, [arXiv:1107.1733]. [25] A. Coimbra, C. Strickland-Constable, and D. Waldram, Supergravity as Generalised + Geometry II: Ed(d) R and M theory, arXiv:1212.1586. × [26] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990) 631–661. [27] M. Crainic and R. L. Fernandes, Integrability of Lie brackets, ArXiv Mathematics e-prints (May, 2001) [math/0105033]. [28] S. Deser and B. Zumino, A Complete Action for the Spinning String, Phys.Lett. B65 (1976) 369–373. [29] I. Y. Dorfman, Dirac structures of integrable evolution equations, Physics Letters A 125 (1987), no. 5 240–246. [30] M. J. Duff and J. X. Lu, Duality Rotations in Membrane Theory, Nuclear Physics B 347 (1990) 394–419. [31] J. Dufour and N. Zung, Poisson Structures and Their Normal Forms. Progress in Mathematics. Birkh¨auserBasel, 2005. [32] J. Ekstrand and M. Zabzine, Courant-like brackets and loop spaces, Journal of High Energy Physics 3 (Mar., 2011) 74, [arXiv:0903.3215].

100 [33] I. T. Ellwood, NS-NS fluxes in Hitchin’s generalized geometry, JHEP 0712 (2007) 084, [hep-th/0612100]. [34] R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Advances in Mathematics 170 (2002), no. 1 119–179. [35] B. Fuchssteiner, The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems, Progress of Theoretical Physics 68 (1982), no. 4 1082–1104. [36] J. Grabowski, D. Khudaverdyan, and N. Poncin, The Supergeometry of Loday Algebroids, ArXiv e-prints (Mar., 2011) [arXiv:1103.5852]. [37] J. Grabowski and G. Marmo, Non-antisymmetric versions of Nambu-Poisson and algebroid brackets, Journal of Physics A Mathematical General 34 (May, 2001) 3803–3809, [math/0104122]. [38] M. Grana, R. Minasian, M. Petrini, and D. Waldram, T-duality, Generalized Geometry and Non-Geometric Backgrounds, JHEP 0904 (2009) 075, [arXiv:0807.4527]. [39] M. Gualtieri, Generalized complex geometry, ArXiv Mathematics e-prints (Jan., 2004) [math/0401221]. [40] M. Gualtieri, Branes on Poisson varieties, ArXiv e-prints (Oct., 2007) [arXiv:0710.2719]. [41] Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A 35 (2002) 1263. [42] N. Halmagyi, Non-geometric String Backgrounds and Worldsheet Algebras, JHEP 0807 (2008) 137, [arXiv:0805.4571]. [43] N. Hitchin, Brackets, forms and invariant functionals, ArXiv Mathematics e-prints (Aug., 2005) [math/0508]. [44] N. Hitchin, Instantons, Poisson Structures and Generalized K¨ahlerGeometry, Communications in Mathematical Physics 265 (July, 2006) 131–164, [math/0503]. [45] N. Hitchin, Generalized Calabi-Yau manifolds, Quart.J.Math.Oxford Ser. 54 (2003) 281–308, [math/0209099]. [46] P.-M. Ho, Y. Imamura, Y. Matsuo, and S. Shiba, M5-brane in three-form flux and multiple M2-branes, JHEP 0808 (2008) 014, [arXiv:0805.2898]. [47] O. Hohm, C. Hull, and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 1008 (2010) 008, [arXiv:1006.4823]. [48] O. Hohm, D. Lust, and B. Zwiebach, The Spacetime of Double Field Theory: Review, Remarks, and Outlook, arXiv:1309.2977. [49] P. S. Howe and R. Tucker, A Locally Supersymmetric and Reparametrization Invariant Action for a Spinning Membrane, J.Phys. A10 (1977) L155–L158. [50] C. Hull and B. Zwiebach, Double Field Theory, JHEP 0909 (2009) 099, [arXiv:0904.4664]. [51] C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 0909 (2009) 090, [arXiv:0908.1792].

101 [52] C. Hull, Generalised Geometry for M-Theory, JHEP 0707 (2007) 079, [hep-th/0701203].

[53] R. Ibanez, M. de Leon, J. C. Marrero, and E. Padron, Leibniz algebroid associated with a Nambu-Poisson structure, Journal of Physics A Mathematical General 32 (Nov., 1999) 8129–8144, [math-ph/9906027].

[54] N. Ikeda, Two-dimensional gravity and nonlinear gauge theory, Annals Phys. 235 (1994) 435–464, [hep-th/9312059].

[55] N. Ikeda, Chern-Simons gauge theory coupled with BF theory, Int.J.Mod.Phys. A18 (2003) 2689–2702, [hep-th/0203043].

[56] B. Jurˇco,P. Schupp, and J. Vysok´y, Extended generalized geometry and a DBI-type effective action for branes ending on branes, JHEP 1408 (2014) 170, [arXiv:1404.2795].

[57] B. Jurˇco,P. Schupp, and J. Vysok´y, Nambu-Poisson Gauge Theory, Phys.Lett. B733 (2014) 221–225, [arXiv:1403.6121].

[58] B. Jurˇcoand P. Schupp, Noncommutative Yang-Mills from equivalence of star products, Eur.Phys.J. C14 (2000) 367–370, [hep-th/0001032].

[59] B. Jurˇcoand P. Schupp, Nambu-Sigma model and effective membrane actions, Phys.Lett. B713 (2012) 313–316, [arXiv:1203.2910].

[60] B. Jurˇco,P. Schupp, and J. Vysok´y, On the Generalized Geometry Origin of Noncommutative Gauge Theory, JHEP 1307 (2013) 126, [arXiv:1303.6096].

[61] B. Jurˇco,P. Schupp, and J. Vysok´y, p-Brane Actions and Higher Roytenberg Brackets, JHEP 1302 (2013) 042, [arXiv:1211.0814].

[62] B. Jurˇco,P. Schupp, and J. Wess, Noncommutative gauge theory for Poisson manifolds, Nucl.Phys. B584 (2000) 784–794, [hep-th/0005005].

[63] B. Jurˇco,P. Schupp, and J. Wess, NonAbelian noncommutative gauge theory via noncommutative extra dimensions, Nucl.Phys. B604 (2001) 148–180, [hep-th/0102129].

[64] P. Koerber, Lectures on Generalized Complex Geometry for Physicists, Fortsch.Phys. 59 (2011) 169–242, [arXiv:1006.1536].

[65] Y. Kosmann-Schwarzbach, Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory, ArXiv Mathematics e-prints (Oct., 2003) [math/0310359].

[66] Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Annales de l’institut Fourier 46 (1996), no. 5 1243–1274.

[67] Y. Kosmann-Schwarzbach and F. Magri, Poisson-nijenhuis structure, in Annales de l’IHP Physique th´eorique, vol. 53, pp. 35–81, Elsevier, 1990.

[68] A. Kotov, P. Schaller, and T. Strobl, Dirac sigma models, Commun.Math.Phys. 260 (2005) 455–480, [hep-th/0411112].

[69] A. Kotov and T. Strobl, Generalizing Geometry - Algebroids and Sigma Models, arXiv:1004.0632.

[70] T. Lam, Introduction to Quadratic Forms over Fields. American Mathematical Soc.

102 [71] J. Lee, Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Springer, 2012.

[72] Z.-J. Liu, A. Weinstein, P. Xu, et al., Manin triples for Lie bialgebroids, J. Differential Geom 45 (1997), no. 3 547–574.

[73] J. Loday, Une version non commutative des alg`ebres de Lie: les alg´ebres de Leibniz. Institut de Recherche Math´ematiqueAvanc´eeStrasbourg: Prepublication de l’Institut de Recherche Math´ematiqueAvanc´ee.IRMA, 1993.

[74] K. C. Mackenzie, General theory of Lie groupoids and Lie algebroids, vol. 213 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005.

[75] K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (02, 1994) 415–452.

[76] J. Madore, S. Schraml, P. Schupp, and J. Wess, Gauge theory on noncommutative spaces, Eur.Phys.J. C16 (2000) 161–167, [hep-th/0001203].

[77] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294.

[78] Y. Nambu, Generalized Hamiltonian Dynamics, Phys. Rev. D 7 (Apr., 1973) 2405–2412.

[79] A. M. Polyakov, Quantum geometry of bosonic strings, Physics Letters B 103 (July, 1981) 207–210.

[80] J. Pradines, Th´eoriede Lie pour les groupo¨ıdesdiff´erentiables. Calcul diff´erenetiel dans la cat´egorie des groupo¨ıdesinfinit´esimaux, C. R. Acad. Sci. Paris S´er.A-B 264 (1967) A245–A248.

[81] D. Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds, ArXiv Mathematics e-prints (Oct., 1999) [math/9910078].

[82] D. Roytenberg, AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories, Lett.Math.Phys. 79 (2007) 143–159, [hep-th/0608150].

[83] P. Schaller and T. Strobl, Poisson structure induced (topological) field theories, Mod.Phys.Lett. A9 (1994) 3129–3136, [hep-th/9405110].

[84] P. Schupp and B. Jurˇco, Nambu Sigma Model and Branes, PoS CORFU2011 (2011) 045, [arXiv:1205.2595].

[85] N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 9909 (1999) 032, [hep-th/9908142].

[86] J. Streets, Generalized geometry, T-duality, and renormalization group flow, ArXiv e-prints (Oct., 2013) [arXiv:1310.5121].

[87] L. Takhtajan, On Foundation of the generalized Nambu mechanics, Commun.Math.Phys. 160 (1994) 295–316, [hep-th/9301111].

[88] I. Vaisman, A survey on Nambu-Poisson brackets, Acta Math. Univ. Comenianae 68 (1999), no. 2 213–241.

103 [89] P. Severa,ˇ Definition and classification of exact Courant algebroids , their origin in 2dim variational problems, suggested a connection with gerbes., Letters to Alan Weinstein (1998-2000). [90] A. Weinstein, The local structure of poisson manifolds, J. Differential Geom. 18 (1983), no. 3 523–557.

[91] M. Zambon, L-infinity algebras and higher analogues of Dirac structures and Courant algebroids, ArXiv e-prints (Mar., 2010) [arXiv:1003.1004].

104 Appendices

105

Appendix A

Proofs of technical Lemmas

Lemma 4.3.4 •

Proof. Let us prove the formula (4.42). Both sides can be viewed as smooth functions of i n,n 1 matrix elements A j. We will restrict to the open subset GL(n, R) = R det− ( 0 ). \ { } It is dense in Rn,n, and the general result will follow by the continuity of both functions. Recall that there holds a formula

∂ det (A) 1 j i = det (A)(A− ) i, (A.1) ∂A j

Hence we get

∂ (n−1) n 1 (n−1) 1 j [det A] p−1 = − [det A] p−1 (A− ) . ∂Ai p 1 i j  −  n−1 This proves that for F = [det A](p−1), we have

∂ n 1 1 j ln F = − (A− ) . (A.2) ∂Ai | | p 1 i j  −  We will now show that the same equation holds for B, that is

∂ n 1 1 j ln det B = − (A− ) . (A.3) ∂Ai | | p 1 i j  −  Let us calculate this explicitly. We get

I ∂ 1 J ∂B J 1 J ∂ I k k det (B) = det (B)(B− ) = det (B)(B− ) [δ A 1 ...A p ] i I i I i k1...kp j1 jp ∂A j ∂A j ∂A j p 1 J I k1 kr jr kp = det (B)(B− ) I δk1...kp A j1 . . . δi δj ...A jp . r=1 X 107 One can now insert a unit matrix to get

p ∂ 1 J I k k k jr 1 m det (B) = det (B)(B− ) δ A 1 ...A r ...A p δ (A− ) ∂Ai I k1...kp j1 m jp j i j r=1 Xp 1 J I jr 1 m = det (B)(B− ) I B j1...m...jp δj (A− ) i = r=1 X 1 j n 1 1 j = det (B) (A− ) = det (B) − (A− ) . i p 1 i J,j J   X∈ − This proves the equation (A.3). We thus have

n−1 det (B) = K[det (A)](p−1). (A.4)

for some K which is locally constant on GL(n, R). To finish the proof, we have to prove that K = 1 on both components of GL(n, R). For the group unit component, we can I I choose A = 1. In this case B J = δJ , and thus det (B) = 1. For the second component, choose A to be Minkowskian metric of signature (n 1, 1). From the proof of Lemma 4.3.1, we see that B is diagonal metric with 1 on the− diagonal, where the number of n 1 ± negative ones is N(n 1, 1, p) = p−1 . We have − −
n−1 det (A) = 1, det (B) = ( 1)(p−1). − − We see that again K = 1. This finishes the proof for A GL(n, R), and the assertion of ∈ the lemma follows by the continuity. 

Lemma 4.6.3 • 1 n Proof. Let us work in fixed local coordinate system (y , . . . , y ) on M. Let I = (i1 < < ip) be a strictly ordered p-index, and J = (j1 < < jq) a strictly ordered q-index. By··· assumption, we have ···

q p i ...m...i 0 = ( T )I = XmT I + Xm T I Xil T 1 p , (A.5) LX J J,m ,jq j1...m...jq − ,m J r=1 X Xl=1

for all X X(M). In particular, it must hold also for fX, where f C∞(M). This gives a necessary∈ condition ∈

q p m I il i1...m...ip f,jr X Tj1...m...jq = X f,mTJ . (A.6) r=1 X Xl=1 First assume that I = J. In particular, this includes the case p = q. With no loss of generality, we may assume6 that there is i I, such that i = J.6 Choose X = ∂ , and a ∈ a 6 ia f = yia . This gives q δia T I = T I . (A.7) jr j1...ia...jq J r=1 X But because ia = J, the Kronecker symbol in the left-hand side sum is always zero. I 6 Hence TJ = 0. We can now assume that p = q. Moreover, we have already proved that

108 J J TI = λI δI , where λI C∞(M). We want to show that λI = λJ for all (I,J). First consider (I,J), such that∈ both p-indices differ in exactly one index. There is thus i I, a ∈ and j J, such that I i = J j . Choose X = ∂ and f = yjb in (A.6). This b ∈ \{ a} \{ b} ia gives λI = λJ .

Now let I and J be general p-indices. There is always a chain [K0,...,Km] of p-indices, where I = K0, J = Km, and (Ki,Ki+1) differ in exactly one index. This proves λI = λJ . I I Hence TJ = λ δJ for λ C∞(M). As a map T thus has a form T = λ 1. By definition of Lie derivative,· we have∈ ·

(T (Q)) = ( T )(Q) + T ( Q), (A.8) LX LX LX for all Q Xp(M). Using the assumption and the above form of T , we get ∈ (λQ) = λ Q, (A.9) LX LX for all Q X(M). This is possible, if and only if λ Ω0 (M). ∈ ∈ closed 

109 110 Appendix B

Paper 1: p-Brane Actions and Higher Roytenberg Brackets

111

JHEP02(2013)042 Springer tric flux com- , February 7, 2013 January 14, 2013 November 5, 2012 : : : 10.1007/JHEP02(2013)042 Published Received Accepted doi: -Poisson structures. -brane models. We find that ensional analogs of generalized p orfman, Roytenberg and Courant iences and Physical Engineering, s, Charles University, Published for SISSA by c [email protected] , and Jan Vysok´y b Peter Schupp a Flux compactifications, p-branes, Sigma Models Motivated by the quest to understand the analog of non-geome [email protected] SISSA 2013 -brane actions and higher Roytenberg brackets E-mail: Mathematical Institute, Faculty of MathematicsPrague 186 and 75, Physic Czech Republic Jacobs University Bremen, 28759 Bremen, Germany Czech Technical University in Prague,Prague Faculty 115 of 19, Nuclear Czech Sc Republic [email protected] b c a c p Poisson sigma models and corresponding dual string and Abstract: pactification in the context of M-theory, we study higher dim Branislav Jurˇco,

higher generalizations of the algebraic structuresplay due an to important D role and establish their relation toKeywords: Nambu JHEP02(2013)042 ] - s 1 2 5 7 9 9 11 13 15 17 18 ]. Recently, 17 ]. See also [ -fluxes, where 8 R es the structures of gma models are inti- -duals of geometric flux re standard Calabi-Yau T try transformations. The we refer to [ tric structures become more perfluous dimensions. Intro- elaxed, allowing patching not hysics and cosmology in four- ry. The fields of Poisson sigma n the ] and can be further generalized 10 ]. It was observed by Alekseev and 12 , ] and Bonelli and Zabzine in [ 11 16 – 1 – ] are also at the heart of Kontsevich’s approach to 6 , 5 -brane action ], whose quantization is related to twisted Poisson sigma p 3 ], such as Dorfman bracket and Dirac structures. This was fur ]. For a recent review with a comprehensive list of reference 7 15 , ]. An example are toroidal compactifications with 14 2 , 1 ], that the current algebra of sigma models naturally involv 13 ]. Poisson sigma models [ 4 From a mathematical point of view, it is known that Poisson si mately connected to amodels lot of can be interesting differential interpreted geomet as Lie algebroid morphisms [ ther developed by Ekstrand and Zabzine in [ in terms ofStrobl generalized in (complex) [ geometrygeneralized [ geometry [ models [ deformation quantization [ in the more generalfor context an of interesting AKSZ conception topological of field membrane theory, symmetries. non-associative structures arise [ compactifications, but at the samegeneral: time the the notion underlying of geome aonly compactifying by manifold diffeomorphisms needs but toresulting also be non-geometric by r flux more compactifications general can string appear symme i 1 Introduction To relate ten-dimensional superstringdimensional theory spacetime, to it particle isducing p necessary fluxes to in compactify this the context su helps to overcome problems of mo A Higher Roytenberg bracket, structure functions B Nambu-Poisson structures 6 Equations of motion, solution 7 Generalized Wess-Zumino terms 8 Conclusion 3 Higher Roytenberg bracket 4 Charge algebra 5 Topological model, consistency of constraints Contents 1 Introduction 2 Nambu sigma model and compactifications [ JHEP02(2013)042 1, the ] for a 23 p > ]. ], Halmagyi dentity of a 28 20 ], see [ + 1)-dimensional and a bivector Π, 22 p B relevant models and ], compute the corre- sigma-model, which is hing Schouten bracket, 29 lly embedded in eleven- heory [ nstruction [ , he structure functions of cobi identity that governs plicit non-trivial solution. Finally, in [ ], Halmagyi observed that eory and their relation to n three-dimensional mem- sistency of the topological ze relevant facts about the and five-branes. This mo- their algebra to close under e model and its relation to 2-form 27 ch we call a higher Royten- + 2)-dimensional world vol- r action to a three-manifold ion for differential equations 19 relevant higher algebraic and acobi identity. For tures that we have described p l as a differential part and it is part). In section 6 we derive n a target manifold. The topo- ]. Evidence for the latter choice is (twisted) Poisson bi-vector to a Poisson sigma models has recently ]. In [ 21 = 1, the consistency of the equations 18 p +1)-vector Π to twist a higher Dorfman p -brane action. p ]. For 26 . Another possibility is to impose the so-called – 2 – – p 24 -brane action p ]: this Nambu-sigma model features a ( 27 -brane action. We find that Π should satisfy the fundamental i p = 1 case. One possibility is to impose the condition of a vanis The known string theories as well as supergravity are natura This paper is organized as follows: In section 2 we review the p +1)-vector field we face the question how to generalize the Ja p ( dimensional M-theory, whose buildingtivates blocks the are study of membranes above. higher dimensional In analogs this ofalready article, the a higher struc we version wouldbrane, of but like the still to Poisson features go sigma-model a bi-vector on beyond field. an the Generalizing ope th Courant the but that will be non-trivial only for even world volume and corresponding higher-orderlogical tensor fields version o of the model can also be obtained by an AKSZ co sponding Hamiltonian and remark on the dual recent review and manygeneralized references. geometry were discussed Local in symmetries [ in M-th comes from the study of actions for multiple membranes in M-t in the Hamiltonian of the Polyakov model, characterized by a shows that the sameusing bracket appears Stokes theorem. if one lifts the first orde fundamental identity of a Nambu-Poisson structure [ of motion. In this articlegeometric we structures. solve A this suitable problem higher andbeen generalization study of proposed the by two of us [ Nambu-Poisson fundamental identity has an algebraicthus as not wel clear how it could be related to a consistency condit appears a more general forma of more world sheet general currents bracket, and found which he calls a Roytenberg bracket. D-branes have been identified with Dirac structures [ In this section we review the Nambu sigma model following [ higher Roytenberg bracket and Nambu Poisson structures. 2 Nambu sigma model and ume and derivethe generalized higher Wess-Zumino Roytenberg terms bracket. that In involve the t appendices we summari Nambu-Poisson structure (differential asthe well equations as of motion algebraic In of the section topological 7 model we and lift find the an topological ex part of the action to a ( of motion of the topological sigma model action implies the J compute the Hamiltonian. In section 3 we use a ( part of the bracket and obtain aberg new bracket. Courant bracket In like sectionthe structure, 4 higher whi we Roytenberg discuss bracket. the In charge algebra section of 5 th we verify the con JHEP02(2013)042 e es M are with (2.5) (2.3) (2.4) (2.1) (2.2) in the e ) and G h e η , can  esponding J and H rectangular X,P, G (
, and a fiber- g ∂X 1 p i n ij where the latter X G oordinates on − L 0 × . We will assume i ∂ p ...p n iJ 1 X , ) 0 B < i I  in a local section basis ∂ i i J − . Likewise, we will use the following X e te strictly ordered multi- dX IJ ··· K J ], is 0 σP B I e e η p ∂ G . i i . M e 29 < d form an -dimensional target manifold . We also choose some local . K = ( η η , are I n m 1 R means the integration over the B I iJ IJ i iJ i → + 27 but it still drops out of K e σ Π G dX J X p i g ∂X e η mI d − P ] = + , I :Σ . e η e I Π η j J K sense that first using the equations of motion X K IJ − i ) g ∂X with local components I , and ), where 1 g ∂X g I ∂X K ; similarly for p + 1)-form p X,P, , we obtain the Hamiltonian e − η i [ iJ G ij J J p iK e ) g ∂X G e η + B 1 ( B can dX + 1)-vector Π with components − 2 1 = − p H + – 3 – , . . . , i ∧ G i J 1 i + ( η e i η  P j TM ... σ η IJ = p i p = e i = ( ∧ η G d i P 1 ij I η − i ) Z 1 ) of Σ. The Hamiltonian can be conveniently written in are Cartesian coordinates for a Lorentzian metric p 2 1 − := := dX µ G i σ I . Lower case Latin characters will always correspond to thes ( = e K ] = and column index 1 2 K M I i , . . . , σ − + 1)-dimensional world volume Σ with a set of local coordinat 1 +1)-vector Π and a ( dX  σ X,P p [ ) on p σ n ), H +) on Σ. Furthermore, we consider an +1 . Metric matrices with upper indices denote as usual the corr X p component of the world volume form p ( cannot be directly expressed in terms of i d i -indices), that is ∂ i y p ∂y ,..., Z , . . . , y on the vector bundle Λ X 1 0 = . . . p + ∂ y e are auxiliary fields, which transform under change of local c , i G − ] := ). We assume that J X and then constructing the Hamiltonian yields the same result p e η ∧· · ·∧ , η is equipped with a metric tensor field 1 i i ∂ η e η, X ∂y M and The “Nambu-sigma model” action, as introduced in [ Let us consider a ( The canonical momenta corresponding to the fields η, [ Note that , . . . , σ e η ≡ 1 , equipped with a ( 0 S I σ M indices (mostly coordinates. We will use upper case Latin characters to deno that coordinates ( computation, as it should.for The construction is robust in the matrix Π with row index Here and in the rest of the paper, the integration over where matrix notation: the components of the ( space-like coordinates ( ∂ denotes the 1 wise metric signature ( notation: substituting the Euler-Lagrange equation for ( inverses. For the components of the smooth map where Starting with the canonical Hamiltonian according to their index structure. JHEP02(2013)042 . g ∂X (2.7) (2.8) (2.6) (2.9) TM p (2.12) (2.10) (2.11) (2.13) (2.14) Λ and ⊕ P using their , troduced by M , s ∗ J   e η : T ) g ∂X ) g e g, C ∂X C ) as , respectively. Next, B ] with properly fixed g, e . + G − 32 2.3 B ! η ( ( dinate expressions of sec- B T ) can then be rewritten as T and . T X . X Π ! G 0 2.5 0 ] B ∂ I ∂ V − i ]. ) − P + 1 , , g ∂X 33 ) and get e 1 1 T K e g . , GA

− − ) and ( g ∂X e 2.8 1 Π ) G T Π T C, e g = T T − Π) 1 G A 2.4 C T ( e Π − 1 in (

V K G and view it as the matrix of the fiberwise − T g e − G T 1 2 η ] and Howe-Tucker [ g =  ∂X e g T V = [ = 1 generalized geometry. 1 2 e 0 − G C C 31 σ ! e G ), and integrate out the fields + Π + Π p p 1 η − – 4 – + + B d and 1 − 0 Π 1 1 = 1 case, see [ g e g g − G 2.1 − − 1 0 1 X p Z e 0 −  G has a natural interpretation as a (twisted) higher G G

= = T · ! = = η g ∂ e ] = i G G I = ( = ( ! 1 2 T C e 1 G GA K e g g ]. For K are column vectors defined in the obvious way. We next Π can also be expressed in terms of X as the matrix of a linear endomorphism of − T 0 T

27 X,P  ∂ e [ A G, A 1 0 Π σ 2 1 g ∂X = H −  G, +1

σ K p . Then, we can rewrite the Hamiltonian ( d and +1 = . The defining equations ( p Z d X matrices corresponding to the metrics A TM 0 p
∂ TM Z ] = p n Λ )-row column vectors p , ], Brink-Di Vecchia-Howe [
Λ e p n ⊕ ) is just the Polyakov-style Howe-Tucker membrane action in K ] = 30 . We can view × X, η , ⊕ can always be inverted, i.e. we can uniquely express the field e [ X
K M + 2.9 [ p n K S ∗ S M A n , T ∗ e η and T , where , V η and K 1) analog of the generalized metric of A n The background fields If we start again with the action ( = × p > The action ( tions of Note that these vectors have the same index structure as coor gauge (coordinates on Σ), see [ K use the Euler-Lagrange equations to eliminate equations of motion, we get the action and where n Note that where we define ( using Let us note that the matrix metric on Finally, we can define the matrix ( Deser-Zumino [ JHEP02(2013)042 . ]. g˜a ga T 36 T (3.3) (3.4) (3.1) (3.2) ˜a (2.15) a ]. This , where = d 27 ] or [ + 1)-form ˜ ◦ A p 35 an bracket: j G T . = ˜ A )-bilinear pairing ! , D ] of the well-known C M D , and denote by the ( ] 27 + E ∞ D ] , + 1)-vector Π as well as ], since the bracket was ) as B ) 3 C , p E 0 1 1 2 19 . The Dorfman bracket is , e dξ e actions for multiple branes 1 (

es needed in the following. ) ), defined as V Γ( e -brane relations of [ [ , W p E .f . , = i ) and equating ) by a ( ). We define the anchor map 2 1 ξ → 1 Γ( a e ) = (
− − e M ξ sed ) ( e ) g C ( W ρ → + [ η i p 1 0 1 C + ( M -brane version [ ) and 1 Ω ( D V p ] 1 − + ( V E ) + 3 ( L − ! ∈ = G η D ρ p 0 g Γ( ) features the inverse of the matrix ] , e ( − 2 ˜a V 1 D ] + × i ] 2.9 :Ω 0 ˜ − , e ξ, η = 2 ) 1 g = 1 e E , e ]. We present a higher analog of this bracket [

D − i 1 V,W – 5 – f ). are higher e η e 34 G = : Γ( = = [ + C M g -forms ( 1 ). This bracket is a particular example of a Leibniz = [[ D ]. We can write these relations in terms of the higher p ] D D e g, C − · ] ] ∞ 9 D g 2 , M η ] · g, ( C . Redefining ξ, W − p D
+ ] ∈ , fe 1 where Ω + 3 1 ) and ) as f , T e ∈ , e V [ . We define a non-degenerate and Π = 1 0 Π 2 h M M Π and ξ, W e − ). ( ga ( as [ M ]. If we define 1 , T ξ, η e X ∗ E + G, 1 · − a ) and T 35 e p [ Γ( V
∈ p [ E G, Ω GA = Λ -field, it is sometimes more convenient to introduce a ( ∈ T Γ( . For further reading on higher Dorfman bracket see e.g. [ 1 Φ 0 1 ) and 3 B → ⊕ A ∈ H ) ) is the inclusion, we have the following properties of Dorfm GA , e M V,W 2 2 ( T E = E as the projection onto the first direct summand of X , e , e TM A Γ( Γ( ˜ 1 1 A ∈ e e × = → ֒ ) TM ) E E + 1)-form V,W → M p for all for all ( -bilinear bracket on sections [ : Γ( p Instead of the Let R E 1. Derivation property: ·i : :Ω , by a ( provides an alternative derivation of the general open-clo j The Hamiltonian corresponding to the action ( Φ and write In this sectionThe we name will “Roytenberg recall bracket”originally was some introduced introduced of by by Halmagyi the Roytenberghere, [ algebraic in which is [ structur essentially a higher Dorfman bracket twisted new approach should alsoending be on useful branes. in the context of effective 3 Higher Roytenberg bracket open-closed string relations, cf. alsogeneralized [ metric and algebroid bracket, see [ the for vector fields h· for all The relations between ρ same character also the induced map of sections JHEP02(2013)042 : ). # 3.4 (3.9) (3.8) (3.6) (3.7) (3.5) ). All (3.12) (3.11) (3.10) , 3.11
) and ( H ) 3.3 : 1 we present the e ( ρ M A ). All the other i ) 2 , H. 3.3 e i ( V ): ρ i D i ] E 3 W at is , -twisted higher Dorfman i , e )) H 1 1 ))) + e ) + -twisted case. [ e 1 ( , . Using this notation, one can e , 2 H 2 dξ ( i )-linear map of sections Π ( E e 2 ]. In appendix ∂ = 1 we get exactly the structure h pr as , W ( M , pr i p 28 ) iK + 1)-vector Π on ( i + ( ) 2 d e ξ p i Π ∞ − . ( ( ( 3 ρ ) ) e, e ) K TM C i 2 # ξ ξ η , e e ( ( ( Π Dh ) with respect to the pairing ( D → ρ # ] V i 1 2 2 − ) as )) + L ), we can define E 3.6 Π = 2 − E , e : ξ = e V i + Π 1 M ( )) p ρ ] + e ( 2 D ξ [ 2 – 6 – ] h e 1) ) = pr +2 ( : ( p ξ − 2 e, e ) ) and ( R ) = [ V,W Ω 1 as higher Roytenberg bracket. This bracket together ξ ) = + ·i pr e ( i ) into Γ( ( ( , 3.5 R ∈ 3 j ρ ) ] = [ V i h· 1 ) = ( · ( M e ) , e ξ , ( ( H ρ · ( 2 = ρ p H e ( D # ] L h R ( i η Π ) j 2 1 ). e + ( , e ρ E 1 ) defines again a Leibniz algebroid, i.e., it satisfies ( e )] + L h 2 Γ( + 2)-form e ξ, W 3.9 ( ∈ p ). + 3 , ρ ). Finally, we define the following bracket on Γ( E ) ). We refer to [ has to be closed, in order to keep the property ( V , e 1 [ E E ). Define new anchor map 2 e Γ( ( ), for example the map induced by a ( ]. H -invariant in the following sense: Γ( , e the projection onto the second summand of ρ Γ( M ∈ as 1 ( 20 M E 2 ∈ e e p ∈ ( = [ E 2 Ω is pr 2 X R , e ] , e ∈ ·i 1 2 → 1 , e ξ e ) for all h· for all , e 1 Secondly, assume that we have an arbitrary Firstly, given a ( The form M e 3. Dorfman bracket is skew-symmetric up to “coboundary”, th 2. [ ( p with the anchor map ( for all for all define new non-degenerate pairing and the “twisted” inclusion of Ω Denote as for all Ω This bracket can be easily modified in two ways: More interestingly, it alsoof satisfies the ( properties are straightforwardcoordinate to form check; of see the also higher [ Roytenberg bracket. For properties of higher Dorfman bracket are also valid for the bracket on functions of [ JHEP02(2013)042 (4.5) (4.3) (4.4) (4.2) (4.1) ). The , , ] for the rated the ...p ...p 13 3.11 1 1 )) )) ). The key is i ). We can use R η i 4.1 η 3.1 + + .
ξ, W ding charge algebra is ) ξ, W + , B , is the pairing ( (  ar in the Hamiltonian as- + ] ) V J ξ J ssed in the previous section. h R ( V ( eev and Strobl in [ e h ·i # K ∗ t it here. ( , g , ∂X Π as defined in ( J ∗ more general charges, involving ) is the pairing ( X J ξ ′ h· − ξ X σ Q V ∧ ·i + i ]. Here we will calculate the charge , + ∧ i − i h· df + 37 K σ P df i ξ )( ) and i ( δ σ )( V V i j ( σ  )[ δ ) + : ( ) defined as ) 3.12 ) and σ ξ σg ξ ˜ σ ( Q ( = p ( σg d 3.2 + # p } σf ) = 1. More general observations from the super- d σf ′ Π – 7 – p that appear explicitly in the Hamiltonian. Here V Z p ). The canonical equal-time Poisson brackets are and p σ d ( d Z ( J − f − j Q e 2.3 e Z K Q ) − V Z ,P ) D ) ] f R η σ ] ) = e Q and ) = ( ξ η i ξ i + -tuples of world volume coordinates. We consider the gen- + + X + K p ) = ]: With { ξ V (Σ) is a test function. The appearance of Courant algebroid V ( = = 16 . This can be done in a straightforward manner; however it is ξ, W ( f + ∞ f ξ, W } } B ) ) + Q e C V Q η η + ( V f ∈ + + V ([ Q f ([ fg W W fg e ( ( Q 1, following the approach of Ekstrand and Zabzine, who integ g g Q e − Q ≥ ,Q , − ) and ) ) is the higher Dorfman bracket ( is the higher Roytenberg bracket ( p are the space-like ξ ξ E ′ R D ] ] + + Γ( · · , , · · V V ∈ σ, σ ( ( f f ξ Let us return to the Hamiltonian ( e Q Q + { { where [ where [ The resulting Poisson bracket of the charges is structures in the current algebra was first observed by Aleks the Poisson bracket is V corresponding to the currents geometry point of view were done by Guttenberg in [ eralized charges calculation is straightforward but quite lengthy and we omi the following relation between charges this result to find the Poisson brackets for the charges algebra for Poisson-sigma model, i.e. the special case where In this section wesociated study to the the algebra Nambu-sigmagoverned of by model. the the higher currents Roytenberg We that bracket find that appe that we have the discu correspon 4 Charge algebra currents to generalized charges.background In fields fact, Π we and shall consider easier to use the results of [ JHEP02(2013)042 ), = E (4.6) (4.8) (4.7) (4.9) W Γ( (4.14) (4.13) (4.12) (4.10) (4.11) Π and . ∈ e } G, hat leads ξ ) I + G, . dy V ( ) is conserved:
J ξ } e . For the special ) nJ K i ) implies that the + ) ∂ IJ ( e V dξ G j ( ( ∂ 4.11 Q ij ,Q , − ) ) ) 1 )–( , onditions ξ − J ) rvation of such charges. To G j + 4.9 ( mnJ .) Let us observe that there ↔ . V = 1, one finds that the charge ↔ ,Q ( I dB M ) i g m Q ξ osing . , { + ( = ) + ( W +
)) ]. , + ξ f

V (Π) = 0 B } nJ 19 ( + ( ) jL ) , kn W ) J = 0 Q ) ) W V dξ L 1 i { ( } dξ ( dy − 1 1 2 ( dB − ( − G IJ Q ,H − ) = ξ + ( ) e ( W G e ) preserves the background fields ξ } i G ( d IJ – 8 – ) ( I ) mnJ j ) := + e 1 = mjL 3.9 K W ∂ ξ − ) = V ij L dB e dB dξ ( ) + G ,Q B m 1 ( ) m ( Q ξ − V { ) = − ( W up to an exact term. W W G + ), we shall consider only the charges G Q L ). The left hand side of this condition can be conveniently ( be the Lie derivative with respect to the vector field nI (( B i V nL nL 4.5 Π ( W K = 1. We are interested to obtain conditions on 2.3 Π W ) to the more general conditions ( Π L Q kJ L f { IL in ,Q -times covariant tensor field on Π ) 2 1 e G G p ξ 4.13 ) can be rewritten as + = = = + ) to carry out the straightforward but tedious calculation t ij V kI 4.12 IJ ) ( ) 4.5 G Q e = G ( { (Π) under the anchor map ( + 1)-form field ( } 1 2 W p W ξ W L L ,H L + ) closes and it is described by the higher Roytenberg bracket ) ξ ). The following set of conditions ensure that the charge V is the Hamiltonian ( is viewed as a 2 4.5 ξ + ( e = 1, this was already observed by Halmagyi [ H G # V ( p Π Using this result, we can determine conditions for the conse Let us note that choosing constant test functions Q − { preserves the ( which would guarantee that image of to the following result. Let rewritten using the Leibniz rule for Poisson bracket: case avoid the anomalous term in ( The particular solution ( The special choice ( for a constant test function Now we can use ( algebra ( where all terms on the right-hand side vanish and we get a new set of c V exists a particular simplification of these conditions: cho (Here JHEP02(2013)042 ) by (5.3) (5.4) (5.5) (5.6) (5.1) (5.2) (4.16) (4.15) 2.1 nd that ) be the · , ), we have · . ) for special 2.5  ) is conserved, J 4.1 ξ appearing in the , g + ∂X i G ) and ( ) are equivalent to
 V X ) with respect to the ) ( 0 . J 2.4 ∂ Q 4.11 , ! iJ 4.15 g ∂X ) h is obtain from ( B u-Poisson structure. )–( η R W )( kJ ] · sistency of the constraints is − 2 , B 4.9 · he higher Roytenberg bracket, J . e η + ! e, e i [ I 1 η k , being the corresponding Lagrange , e 1 − P K 0 iJ ) I e ( . I e I G e η Π e ) η kI I σ dy 0 G − p Π ( ) + ( dy ) d I , 2 (

, the inverse of matrix − I −· 1 e T η I Z , e σ − g ( ∂X R δ ! Q I = 0 − ] G e η 1 g ∂X Q − I ξ , we obtain ). Using the notation of ( V – 9 – I e + I K e ] = η =

e, e e i η 2.2  ). The conditions ( ) = X σ H σ 0 e η, P p ( M ) := ∂ d I ∗ i ) = ([ given by η η e 2 X, T K [  Z p ). In the other words, the charge + , e M σ Λ can be expressed in terms of the charges ( H ) have an interesting geometrical meaning. Let ( 1 − ∗ M e +1 T ∗ ( ⊕ p . as given in ( H p ) T ξ, W d ] = 4.11 Λ k p e ( + P Λ Z ⊕ ρ )–( TM e η, P V ⊕ ( Γ( 4.9 ] := X, = 0. We will show that algebra of constraints closes on shell a TM [ ∈ 1 TM H as well as ): − e η, X ξ e Γ( I G 2.7 η, e is a “Killing section” of the fiberwise metric ( + K [ ∈ = ξ S 2 1 V − + , e 1 G = e V e The conditions ( The action has the form = e Hamiltonian ( fiberwise metric on multipliers. Looking at Lagrange-Euler equation for which should be viewed as a set of constraints, with choices of test functions: In this section wesetting examine the topological sigma model, whic higher Roytenberg bracket. 5 Topological model, consistency of constraints the equation for all Let if The canonical Hamiltonian of this model can be written as the constraints are compatibleensured with by time the evolution. vanishing The ofwhich in certain con turn structure is functions related of to t the fundamental identity of a Namb with canonical momenta JHEP02(2013)042 ) , )- ) 3.1 σ ) for dB (5.7) (5.8) ( (5.9) ( (5.10) (5.11) ) − ·i . ) are B.6 , ′ ) − · r ′ σ h· ′ ), since in ( σ σ ( ∂σ M ( 5.7 e ...p K ∂δ 1 clearly vanishes )
nishes when the ) σ ( R i ce it just leads again to J ˆ q...p ], such (nontrivial) Π ˆ , dy r... 36 aints and consistency of t term in ( I ... , 1 1, we are forced to add the 0 dy
ental identity for a ( n therefore be expressed in h ) . . = 0 is thus equivalent to the ( i − ≈ rt of the fundamental identity ng current algebra ∗ R ) R ′ i ) implies that the graph of Π, X M J = 0 σ ˆ q...p ) = 0 . ∧ ) and the constraints ”). To see this, it is sufficient to 5.8 − ′ J ˆ r... dim , dy ≈ σ )) ˆ q...p σ I ... IJ q = 0 dy ( ... )( 1 ( χ δ 1
− · ) } dy ) K ) ’s and thus they Poisson commute with ). In terms of the structure functions of ) I ) h ′ ) σ σ R e m R K ( dy ( i σ i < p < ( M 4.5 P
( δ J J ( # IJ K dy d ( Π S IK q KL r i # , dy , dy χ I + – 10 – Π I , χ − i K ) k ( ) + dy dy ∗ σ I h 1. For 1 h K ( MJ ∗ ) X dy − S IJ q M ( X IJk doesn’t vanish. To see it let us note that vanishing of ) ) ( χ dy + J R { ( M R . Imposing the condition to vanish. This leads precisely to the condition ( ≡ r, q i # dy and we shall henceforth assume that this is the case. Even )( ( ′ I Π KJ q , is isotropic with respect to the canonical pairing 2 i R # σ dB q IJ } χ ( Π ) ∗ χ i sgn( − , dy K − = dim q J X M σ p = p 6= =1 ) do not contain any ( ( MI r r X p dy δ h S H Ω 5.9 + − ). However, as was noticed by Zambon in [ ∈ = = 1 the twisting of Nambu-Poisson structures is redundant sin ξ M and } } | ∗ ) = 1 and ) ) I ′ ′ T p > ξ p σ σ p ( ) is just a sign, irrelevant for the discussion. The first term ( dy ( Λ # J is equivalent to M = e r, q K ⊕ e R K , η i + Π ) , = 0. The second term, in fact, also weakly vanishes (i.e. it va , I ) ξ σ J ( σ { TM ( I IJ q , dy dy = e χ Note that for IJ q J K 2 { χ = Π dy { It is hence natural to ask for h The Poisson brackets between the new constraints show that these constraints turns out tofor be ensured a by Nambu-Poisson the algebraic tensor. pa Indeed, geometrically ( The anomalous terms can be dealt with using secondary constr each other, i.e. constraints equations are used; this is denoted by “ an ordinary Nambu-Poisson structure. for following set of constraints to the system: G on Γ( exists only for general the expression ξ then, there still seems to be a problem with the anomalous las the Roytenberg bracket (cf. appendix) we obtain the followi where sgn( assumption that Π fulfills the differential part of the fundam terms of the Roytenberg bracket by equation ( The constraint algebra and time evolution of constraints ca The new constraints ( twisted Nambu-Poisson tensor JHEP02(2013)042 . ) ) ). m X 5.6 = 0 M 5.12 ∗ repeat (5.13) (5.12) ) with T ) using ˆ q...p p 4.6 . By ( 5.1 ˆ V r... ) ( ... ∗ 1 sth (
are consistent ) X # R Π i i . The reasoning IJ q J Π χ G , dy I and dy ’s. Hence, we just have ). h ) I s of motion for the IJ q )) = 0, clearly gives zero. M M e χ tion of charges ( ( K p ssume that Π is a Nambu- stly just state the results. p dy ( e Hamiltonian. This follows ( opological action ( = 0. The most difficult part atural coordinates associated Ω n bracket. Of course, among # all X gebra. Π , I bracket via the charge algebra. i ∈ ) 3 ( e , ∗ K ξ ). This is again simple using ( 1)-contractions with X dB } . − 5.3 V = 0 i } p ) ( ) of the pullback bundle = 1). ,H := e η ) Π ( Σ with Π( p y constraints follow already from the first ones, J ,H σ G # ) ( ( Π ρ ∈ σ MIJ rq dy i m 1 ( ( 1), we will find an explicit solution of these χ J p i − ), the constraints η p # X P { Λ { | 5.3 p > – 11 – := R i ’s. Doing this, new anomalous terms proportional to I ) = e J η ) = ) = Π e σ K σ ( V ( ) can be shown to be a consequence of the following ( i , dy m ) ˙ I e P η ˙ ) to be conserved, one gets the condition ( X ξ 5.11 # dy h Π + L ) is conserved for arbitrary V ξ ( ) will appear. We can treat these again as new constraints and R Q i J ] (0 + , dy Q 38 I dy h ) M dy ( # = 0 by the above discussion. 6= 0, the validity of ( Π i ) IJ q )) denotes the projection onto the first summand of the graph N ). For the charge χ p ( dy ρ ( 4.5 # X Given the results of this section, we will shall henceforth a To conclude this section, we shall investigate the conserva Straight from the definition, one can calculate the equation Since the Hamiltonian is of the form ( Π Alternatively, one can introduce new constraints i ( 3 ∗ Note that the charge to check their Poisson brackets with the the procedure until we arrive at anomalous terms containing is just an easythe application equations of of the motion Leibniz we rule will for find the also Poisso the constrains with the dynamics, i.e., theyimmediately weakly from Poisson the commute above with discussion th of the constraints al X where we have introduced the section this is identically equal toi.e., zero. from Note that all these auxiliar the Hamiltonian formalism and previouswith results. every Nambu-Poisson Using structure the (for n equations. The calculations involve theThey higher are Roytenberg again quite long, but straightforward and we will mo and ( respect to the dynamics governed by the Hamiltonian ( observation made in [ comes with the calculation of fields. Indeed, the calculation of If Π( In this section we will derive the equations of motion of the t 6 Equations of motion, solution Evaluating the left hand side expression at a These will obviously Poisson commute with each other and with Poisson tensor (which may be twisted in the case where itself is not very illuminating and we skip the details here. JHEP02(2013)042 )– el: (6.9) (6.5) (6.6) (6.2) (6.7) (6.8) (6.3) (6.1) (6.4) 6.2 ) and . σ L ( m +1). (The g ∂X ,P  ) σ ( ): kmL m ) are arbitrary real ˆ q, . . . , p X 6.4 R dB ( ∈ )–( ...p ]). In these coordinates 1 kJ ted with every Nambu- ) ˆ m . p, . . . , 6.2 Π L 39 . ta symbol: C ...p
− 1 dX p ) B ,..., . i n of ( N ion of the field equations ( ∧ ∂
motion considerably. We define i J L ...k...l e η , − dX e ) impose 1 K d ] = (1 ) ( ∧ . ml +1 + B 6.3 p + 1, where i ∂ B +1 p, q ∂ 6= 0 (see e.g. [ p m n i kmL mN ∂x ,l ( ) , e η K ...i B , ) evolve in accordance with the following x 1 # d ∧ kJ i ( ǫ kJ Π mL = 0 Π m > p 5.1 = 0 − Π ... = − m k p k =1 i m + Π ∧ for X n 1)-index [ – 12 – P ∂ ˙ iL +1 X ∂σ 1 p − ∂X + ) m B ,m ...p , ∂ , p 1 ...i C ∂x −· , where Π( ) 1 − kJ m J kL i σ i ( e N η P Π = M δ Π B ), such that J K ). First, note that . This is not straightforward to see, one has to use the Q e n η m ∈ mJ mI , ,m Π = dX } ) = − x X 4.5 kJ ∧ i K ) = P = Π = Π = +1) and ( σ ( I mN , . . . , x + Π g ∂X i ), one gets the following result: The fields , . . . , p m m e 1 η ˙ 1 ˙ P P d x ) are in these coordinates equivalent to X ( g ∂X 4.5 mK ˆ r, . . . p mL,k ∈ { B 5.9 − ). Furthermore, the equations ( B = 0, we get the equations of motion for the untwisted sigma mod k k + P kJ , B 5.12 m ,..., Π ,m J ) and ( P e η (  kJ J 5.6 e η Π mI mJ ] = (1 + 1 and − − r +1 and we thus get = Π = Π = I m m ˙ ) of the sigma model defined by action ( ˙ The constraints ( P ). We will use the natural local coordinates that are associa m > p m > p X σ -index [ g ∂X ( p J 6.3 e hats denote omitted indices.) This choice of local coordinatesa simplifies the equations of constants. One can then easily deduce the following solutio Now we will show that there always exists a non-trivial solut for which exist around every point the components of Π can be expressed in terms of the Levi-Civi ( Poisson tensor, namely ( set of equations: Hence Now, using ( This can be done again using ( for η consequences of ( In particular, for JHEP02(2013)042 . # + 1)- p ...p 1 ) ] r,q [ n. dX ∧ k sentially the classic ˙ X : Π defines a ( d ( X ] r,q cal terms of the model to ). [ oytenberg bracket appears ; for ] r [ km ts in the resulting generalized 6.4 rg relations are satisfied. B tion to verify that this solution g ∂X ) =1 +1 , p k X  r +1 ] q ˙ r =1 [ X , +1 6= ...p , r p X 1 , r,q m m +1 I B g r ∂X f C + ], adapted to the twisted Poisson sigma E k 1) − ˙ = = X 41 = − ) ) are coordinates adapted to this foliation. (1 , I m m ,k n e ] = 1 case, where the generalized Wess-Zumino η r = ( +1 X 40 X [ – 13 – r p ] m r [ 1) e B η − , . . . , x − 1 ] x r = ( [ r km P dB ( , and (  ) indeed also solves the equation ( ]. M =1 6.3 20 +1 p X k,r (Σ) are arbitrary smooth functions on Σ; are arbitrary real constants; " ∞ are constant in the directions transversal to this foliatio 0 R ], C r ) and ( dσ X + 1, ∈ ∈ + 1, + 1, are arbitrary constants in space-like variables on Σ. + 1, + 1, p m 6= [ I Z 6.2 m p p ≤ f C E I = ≤ ≤ m m m > p r m > p r P where where and if where We shall use the Lagrangian formalism this time and follow es There is a nice geometrical interpretation of the solutions +2) dimensions, the structure functions appear as coefficien (i) For (v) for (ii) for (iv) for p (iii) for approach of Wess, Zumino, and Witten [ terms are topological if and only if the associated Roytenbe 7 Generalized Wess-Zumino terms In this section, we encounterin yet the another context way of how( the the higher Nambu R sigma model: Lifting the topologi Hence the fields Wess-Zumino terms. This resembles the dimensional foliation in model by Halmagyi in [ Although straightforward, it is actuallyto a equations lengthy ( computa JHEP02(2013)042 (7.1) (7.6) (7.5) (7.2) (7.4) (7.3) , ) ) B K . ( e ) ∗ A , hence it is in J N X JK , ↔ − on and where M ) i I e 4 ( A B A − ( i ∗ − ∧ A ) J X p i e j iJ A − A Π IJ iJ dX . ∧ IJ Π − J ∧ dM K J . M − e J e A A ∧ e J J A ∧ +2)-dimensional manifold. Using ... iJ dX , J ∧ p ( J p e ∧ Π A dX e dX ∧ ∗ A r ∧
dσ JK j − I ∧ ∧ ∧ ) J ∧ ) = +1 and introducing an auxiliary e l , which is zero on Σ I J A I ∧ p e J M I A [ e dX e A IJ e d IJ A dy A ) can be rewritten as − dX ) , 1 2 ( dX ... ∧ 1 2 ) 1 i M + + J ∧ J ∧ . ∧ − 2.1 i + i e A 0 A ∧ : + k ... e , . . . , σ 1 i dy J G A as ( iJ 1 is a smooth ( ( J iJ e N dA A ∧ – 14 – A dσ # e Π dσ 1 2 J ∧ A dX Π 1 i J , σ Π N ∧ ∧ e i j i + A e 0 η η − ∧ ∧ ) − − i J σ J − I J i j ( e = = A dX A dX e e o ) A ( A A klJ i J J ∧ I ∧ 1 2 + dX dX iJ r . e A i j A k dB , where dy ∧ ∗ as Π ( = = Π i top ) 1 J dX − i ) I dX A J ; however, we assume an arbitrary extension of e ∂N − ! i ψ Σ ψ L e ,k r dy ij ψ p + 2)! ( ( Z ) and 1-forms ∂N iJ d 1) 1 # p dX i ( − Π ( Π − and N i A ] = ( G − − Z − i e ( ∗ A p ψ =1 2 1 r X X . ] = − ) = 1 1 2 2 e N A Σ -forms X, A, top = = Z [ p L ( IJ top d ] = S X, A, [ e M A top S Topological part of this action has the form Let us suppose that Σ = Define the X, A, [ This term is zero on Σ = 4 S Stoke’s theorem, we can lift the action to We define new fields where we have added a new term general non-zero on (The hat denotes a factor that is omitted.) Choosing the orientation on Σ as Minkowski world volume metric, the action ( JHEP02(2013)042 ] 1 ]) 27 35 gher (7.8) (7.7) p > , . l ) NL s out that e obtained ), one finds ψ J M ∧ top ↔ i ∧ L racket (see [ A I ( N ( d e ∧ A -brane action. The − k p ∧
ψ ion, we have obtained i K  = 1 case. ′ oytenberg bracket. Its kl e J A p F e e the appropriate A i ∧ 1 2 had been introduced in [ ctures related to this new ∧ A ]. We define a higher analog e-Tucker of generalized Poisson sigma + ,K I k ∧ . p i e arameterized by sections of the xt, we have defined generalized 34 A ψ al analog of a generalized metric J A L for the e s, which resemble that of higher ization of the Roytenberg bracket ∧ A jlK ∧ ) IJ ...k...j J ′ ∧ 1 J e j S ψ e L dB A 2 1 ( e ∧ M A ∧ + -brane models. In this context, we have M + k l LJi p e i ′ ψ A + 2) ψ k J A R p p ∧ Ji 1 2 ) 1 ′ I ′ kM J e Q – 15 – − A D I ...k...j + 1)( J + ↔ and redefinition of the fields into 1 + e ′ lL p ψ ij J I ( J ( e D IJ ) corresponding to a re-scaled 3-form flux ψ ∧ e ψ = A − + ∧ + Π M ∧ l p dM e J r l A ψ e jlK A ψ ∧ d ∧ H ∧ . We have found that we can use previous results of Ekstrand ...k...j L k ...k...j 1 + k e 1 j are structure functions of skew-symmetric version of the hi A ψ j i M ψ ′ J e ∗ ψ ′ klL dX T dA ,R p ′ LM ,k ,k ′ klJ H ′ ∧ I I Λ 1 2 r r H S i ,S j j ′ 1 2 2 1 ⊕  ψ Π Π − − − − ] to calculate the world sheet algebra of the charges. It turn ,D + 1)-vector Π, we have further twisted this structure and hav ′ p p =1 =1 r r TM X X 16 p ) = 1 1 2 2 ,H ′ ]). Starting with the definition of a twisted higher Dorfman b top = = 42 ,F ′ L ( IJ Q d We observe that Let us summarize the main results: By a Legendre transformat dM = 1 version was originally introduced by Roytenberg in [ Roytenberg bracket (see appendix where a new Courantp bracket like structure, which we call a higher R and Zabzine [ Dorfman brackets, canbracket be play easily a fundamental verified. rolecharges throughout for this The the article. model, algebraic with Ne a stru vector complicated bundle structure that is p and using a ( in coordinate-free intrinsic form, such that its propertie and identified as a dual to the gauge-fixed Polyakov-style How Putting the above expression for that 8 Conclusion In this article, wemodels have and studied the higher corresponding dimensional dual analogs string and play an important role and that Nambu-Poisson structures ar found that higher algebraic structures related to a general resulting quadratic form can be viewed(see as e.g. higher-dimension [ generalization of the Poisson structures that are relevant the Hamiltonian corresponding to the Nambu sigma model that JHEP02(2013)042 ]. M ∗ 19 T p ]. The Λ 19 ⊕ = 1 in [ p TM + 2)-dimensional p + 1)-vector field Π -brane action turns p p and eventually imply = 1 case: a factorized X have introduced in this t to a set of constraints observation, but we can alog of the calculation of p 1 and confirms the results sing secondary constraints ytenberg bracket up to an s are compatible with time reted as constraints on the . It therefore factorizes and r. However, an anomalous cture functions of the skew- ly if the higher Roytenberg ave introduce in this paper. tion. We have been let to a arried out using the higher ressions for the equations of yzed the consistency of these raints on the auxiliary fields the charge algebra. This has ed gebra itself. The constraints Weinstein coordinates in the ent to the differential part of fields ground ( es certain structure functions ed methods. The observation p > ropic subbundle part of the tal identity of a Nambu-Poisson ranteed locally for any Nambu- ess-Zumino terms) incorporates ns of motion and find an explicit pace for the other fields of the es found by Halmagyi in [ : they constitute Killing equations can be rewritten in the terms of generalized m P – 16 – . One can further find the parameterizing sections ). R , using Stoke’s theorem. After some redefinitions of A ·i , h· ∂N . This was already observed by Halmagyi in the case e η and , such that Σ = ]: we have lifted the topological part of the action to a ( η N 20 Studying the consistency of the topological model, one is le the fields, the resulting Lagrangianthe density fields (generalized coupled W to newsymmetric background version fields, of which the are higherThe the Roytenberg stru generalized bracket Wess-Zumino that terms werelations are h are topological satisfied if (see and appendix on of the charges, such thatset of they partial are differential conserved equations under that generalize time the evolu on world volume effectively reduce the availablemodel. dimensionality The of multi-vector Π target isis s of thus maximal forced rank to in bethat this of we subspace Nambu-Poisson type. have obtained Thisand in is true this the for article conclusion using is, more sophisticat however, also valid in the well-studi that are usually understood as constraints on the embedding Halmagyi does not furtherin fact comment now on understand the this implication in the of present this context: the const with respect to a certain fiber-wise metric. The topological equations have an interesting geometrical interpretation Poisson structure, we have been ablenon-trivial to simplify solution. the equatio Thiscase is of similar Poisson to sigmaHalmagyi the models. in use [ of Finally, Darboux- we have present the an charges. Using special coordinates, whose existence is gua to be a Nambu-Poisson structure.motion We of have the derive topological explicitbeen model, exp possible, using since once the more canonical momenta results for anomaly. This anomaly vanishes if one restricts to some isot the Poisson bracket of the charges closes under the higher Ro structure also holds. It is thus natural to consider the back of the higher Roytenbergthe bracket to fundamental vanish, identity which satisfied isterm by remains equival in a the Nambu-Poisson Poissonfor tenso bracket, the which model. can We beevolution, have dealt provided shown that with that the u algebraic these part secondary of constraint the fundamen out to be a system withconstraints constraints, under as expected. time We evolution havecan anal and be with written the inarticle constraint the and al terms the calculation ofRoytenberg bracket. of the their generalized Consistency Poisson charges under bracket that time can evolution we forc be c with respect to a twisted pairing conditions on the multi-vectorauxiliary Π, fields but that can also be interp JHEP02(2013)042 = K (A.7) (A.8) (A.4) (A.1) (A.3) (A.5) (A.6) (A.2) dy and k ) twisted by a ∂ ∂y MEYS (Mobility) 3.12 = , k ∂ klL , ) will indeed ensure the cket ( ity in Prague, grant No. 2 J H V etrized version of the higher . . mL , dX , er Bouwknegt for discussions. L ful comments on a preliminary g relations): cial example of a more general or hospitality at IMPA. . Π L klL L ∧ and
dy l lJ . Denote H L dy 1 dy L so ensures consistency. Π V lJ L JI M dX IJ J klL k kI Π S S ∧ H D ) denotes an ordered multi-index with kI − , k p + + Π + + L ˇ CR P201/12/G028. The research of J.V. p m ,n mL + Π m m dX IJ ∂ ∂ ∂ p Π mI S J klJ m , . . . , j lJ Π 1 ...k...j m k IJm kl 1 2 H 1 j Π – 17 – F R Q nJ ...k...j 1 mj = j Π L lkL = ( = = = δ Π L + 2) H − J ,k ,k R R R p , IJ I I ] ] ] ′ , + l ,n r r 1 J J j j S lJ mJ ,k , ∂ Π Π and mJ k Π Π , dy , dy + 1)( p ∂ Π mJ p p I k j [ =1 =1 p r r X X ∂ klJ lkL ( Π nI [ with suitable vector fields dy [ − − − H H dX 2 = ∧ V = = = = = Π H ∧ J L L m ... 1 J k m k kl IJ IJm V ∧ ) be a set of local coordinates on D Q F S 1 R n j . Then, one has p k dX . dy ≡ , . . . , y p 1 ∧ J y ... dX ∧ Let ( 1 < . . . < j + 2)-form flux k 1 p We denote by a prime theRoytenberg structure functions bracket. of For the example skew-symm dy The structure functions have the following form (Roytenber A Higher Roytenberg bracket,Here structure we functions summarize the local form of the higher Roytenberg bra was supported by GrantSGS10/295/OHK4/3T/14. Agency We of also thank the to DAAD Czechfor (PPP) Technical supporting and Univers ASCR our & collaboration. B.J. thanks H. Bursztyn f ( j where version of the manuscript. Also,The it research is of a B.J. pleasure was to supported thank by Pet grant GA consistency of the equations ofPoisson motion, bi-vector but satisfying this the is Jacobi just identity, a which spe al Acknowledgments We would like to thank Noriaki Ikeda for discussions and help bi-vector (i.e. Π = JHEP02(2013)042 ) M ( (B.4) (B.3) (B.6) (B.1) (B.5) X -twisted → H ) M ] as needed in ( p 21 :Ω on structure: Let Π . # A ,  )) -twisted higher Dorfman Π) H 1 there occurs an interest- )) ; (B.2) , H ∧ ) ξ ) , ( dξ n structures [ ) = 0. We call Π an ( # (Π dξ p > ) π ( Π ] η dξ ) i ( i η dH ( # ctor, the other part of the identity − , ξ, η Π # i 1 ([ + 1 Π ( i dξ is closed under the higher Dorfman # p ( # ) − } η Π ) ( − ) ) can be stated in the following equivalent − # η (Π) = 0 M ( Π ) ) ( i )] = Π p B.1 ξ p ( ( η ) = df ), such that ( (Π)Π Ω # # η be defined as ∧ of Π is closed under ( # Π ]. Π dξ ... – 18 – M ) i ) splits into two parts – one part is a differential ∈ # . We call Π a Nambu-Poisson structure, if ( L we define the induced map Π Π ∧ − 35 1  , M ξ G ) +2 M ( df − B.1 := ξ ( p M p | ( (Π)) ) = # π Ω ) Ω ] η # ] or [ ξ ξ Π on ( ( ∈ L -twisted Nambu-Poisson structure can be defined using # [Π 38 # . → i ξ, η A Π the condition ( (Π) = [ H ) H ) + ∂ ) L ξ ξ 1 ( ( ( M iK (Π)) ( # ) it holds that # ≥ p A ξ Π ( ) ). Π ; . Then it holds that p it holds that Ω . Let K L # ) ) { ξ ) Π M M × M ( ( M M L = = ) p ( ( M ( ∞ p p ( Ω A p M Π C Ω Ω ). This definition is correct, however, for ξ + 1)-vector field on ( i ∈ Ω ); p G p p ∈ ∈ ∈ M ∈ ( + 1)-vector field p 1) 3.2 p :Ω ξ ξ, η p − Ω ξ, η ξ, η π ). Equivalently, a For arbitrary ] · ∈ , · 3.7 [ , . . . , f ) = ( 1 + 1)-vector on ξ f ξ, η ( p for all for all bracket ( # Let Π be a ( For any ( There seems to be a natural way to define a twisted Nambu-Poiss A 4. for any 3. let 1. The graph 2. for any Lemma 1. for all ing thing: The fundamental identity ( for all the condition identity similar to the Jacobi identity for the Poisson bive ways: this paper. For details see, e.g., [ as Here we recall some fundamental properties of Nambu-Poisso B Nambu-Poisson structures Nambu-Poisson structure, if the graph bracket ( be a ( JHEP02(2013)042 ). ]. ans 1, as B.2 (B.7) SPIRE ]. p > ing ( (1990) 394 (1994) 435 IN (2005) 455 (2005) 035 , ][ Handbook of , 03 235 260 SPIRE , in B 347 IN -models and σ , ][ JHEP Non-geometric fluxes, ) comes from the fact , arXiv:1204.3714 -models , σ B.1 math/0406445 , [ (2011) 385401 ]. Annals Phys. Nucl. Phys. ]. , 6= 0, such that , ), we see that if we add a part ennecke, ) ]. +1)-vector whose support is an x A 44 art of ( p SPIRE B.6 . SPIRE IN [ arXiv:1207.0926 [ IN (2005) 400 Commun. Math. Phys. +1 SPIRE -models and quantization of p ][ , ∂ σ IN 54 J. Phys. , V. Cortes ed., European Mathematical ]. ]. ∂x ][ where Π( , ∧ x -models ... σ (2012) 012 SPIRE SPIRE IN IN ∧ – 19 – Membrane 09 1, the fundamental identity ensures the existence 1 Lie algebroid morphisms, Poisson ][ ][ ∂ Nongeometric flux compactifications Dirac arXiv:1004.0632 ∂x hep-th/9405110 p > [ J. Geom. Phys. . If we now consider ( q-alg/9709040 [ , JHEP ξ , Π = ]. ]. ]. ]. ]. Current algebras and differential geometry Generalised T-duality and non-geometric backgrounds 28 Poisson structure induced (topological) field theories Generalizing geometry — Algebroids and Duality rotations in membrane theory , the algebraic part of identity will stay untouched. This me ) around every point SPIRE (1994) 3129 hep-th/0508133 hep-th/0512005 ξ [ [ (2003) 157 n SPIRE SPIRE SPIRE )-linear in IN ) is in fact still an ordinary Nambu-Poisson tensor, satisfy IN IN IN ]. Conversely, every decomposable ( ][ -twisted Nambu-Poisson tensor is therefore redundant for M 66 ][ ][ ][ A 9 ( 39 B.6 H Deformation quantization of Poisson manifolds. . ∞ , . . . , x 1 C x )-linear in (2005) 085 (2006) 009 Lectures on AKSZ topological field theories for physicists Two-dimensional gravity and nonlinear gauge theory ]. ]. M 10 05 ( ) is not ∞ C SPIRE SPIRE B.2 hep-th/0411112 hep-th/0410183 IN IN hep-th/9312059 arXiv:1106.0316 Society, Switzerland Z¨urich, (2012), pseudo-Riemannian geometry and supersymmetry off-shell closed gauge symmetries asymmetric strings and nonassociative geometry non-geometric flux backgrounds [ [ [ Lett. Math. Phys. [ [ Mod. Phys. Lett. JHEP [ JHEP [1] J. Shelton, W. Taylor and B. Wecht, [9] M.J. Duff and J.X. Lu, [7] M. Kontsevich, [8] N. Ikeda, [5] N. Ikeda, [6] P. Schaller and T. Strobl, [3] R. Blumenhagen, A. Deser, D. E. L¨ust, Plauschinn and F. R [4] D. Mylonas, P. Schupp and R.J. Szabo, [2] A. Dabholkar and C. Hull, [10] M. Bojowald, A. Kotov and T. Strobl, [12] A. Kotov, P. Schaller and T. Strobl, [13] A. Alekseev and T. Strobl, [11] A. Kotov and T. Strobl, has already been noticed in [ integrable distribution is Nambu-Poisson. The algebraic p that a Π satisfying ( For details, see e.g. [ References that ( that is The concept of an of coordinates ( is purely algebraic. Interestingly for JHEP02(2013)042 , ] , action ]. (2003) 281 ]. (2011) 074 (2011) 074 ]. 54 , 03 06 SPIRE , SPIRE , IN (2007) 045020 SPIRE IN [ ][ arXiv:1110.3930 -brane action and ]. -model [ IN p JHEP The gauge structure of σ ]. JHEP , , ][ , , D 75 ]. SPIRE IN SPIRE [ mpson, ]. IN The local symmetries of M-theory (1976) 471 (2012) 012 Multiple membranes in M-theory ][ SPIRE -branes to topological M-theory IN [ p 01 PoS(CORFU2011)045 Phys. Rev. SPIRE , B 65 ]. , arXiv:1208.5884 IN [ ]. ’s ]. ][ 2 arXiv:1206.6964 [ Quart. J. Math. Oxford Ser. JHEP M , , SPIRE math/0401221 ]. Target space symmetries in topological theories. SPIRE IN , SPIRE IN ][ (1977) L155 IN – 20 – Phys. Lett. ][ hep-th/9301111 A locally supersymmetric and reparametrization D-branes in generalized geometry and (2013) 064 [ , ][ SPIRE ]. (2012) 064 IN 01 [ A 10 -model and effective membrane actions -model and branes 10 σ σ SPIRE arXiv:1203.2910 [ Generalized geometry and M theory IN Courant-like brackets and loop spaces JHEP [ AKSZ construction of topological open ]. ]. ]. (1994) 295 From current algebras for Modeling multiple , ]. ]. ]. A locally supersymmetric and reparametrization invariant J. Phys. JHEP A complete action for the spinning string ]. Nambu- Nambu , hep-th/0106042 , [ 160 SPIRE SPIRE SPIRE hep-th/0507051 arXiv:0805.4571 [ [ SPIRE SPIRE SPIRE IN IN IN IN IN IN (2012) 313 SPIRE [ [ ][ ][ ][ (1976) 369 ][ arXiv:1110.0134 IN , ][ On foundation of the generalized Nambu mechanics Non-geometric string backgrounds and worldsheet algebras Non-geometric backgrounds and the first order string Generalized complex geometry (2002) 021 Generalized Calabi-Yau manifolds B 65 B 713 (2005) 015 (2008) 137 02 ]. 09 07 JHEP SPIRE , arXiv:1205.2595 arXiv:1008.1763 IN hep-th/0611108 math/0209099 arXiv:0903.3215 invariant action for the spinning string for a spinning membrane 1 [ Phys. Lett. generalised diffeomorphisms Phys. Lett. [ and their formulation in[ generalised geometry [ arXiv:1203.3546 arXiv:0906.2891 Commun. Math. Phys. JHEP Dirac-Born-Infeld action JHEP [ [ Nambu brackets [26] D.S. Berman, M. Cederwall, A. Kleinschmidt and D.C. Tho [32] P.S. Howe and R. Tucker, [33] L. Baulieu, A.S. Losev, and N.A. Nekrasov, [29] P. Schupp and B. Jurˇco, [30] S. Deser and B. Zumino, [31] L. Brink, P. Di Vecchia and P.S. Howe, [27] B. and Jurˇco P. Schupp, [28] P. Bouwknegt and B. Jurˇco, [25] D.S. Berman, H. Godazgar, M. Godazgar and M.J. Perry, [22] J. Bagger and N. Lambert, [23] J. Bagger, N. Lambert, S. Mukhi and C. Papageorgakis, [24] D.S. Berman and M.J. Perry, [20] N. Halmagyi, [21] L. Takhtajan, [18] T. Asakawa, S. Sasa and S. Watamura, [19] N. Halmagyi, [15] M. Gualtieri, [16] J. Ekstrand and M. Zabzine, [17] G. Bonelli and M. Zabzine, [14] N. Hitchin, JHEP02(2013)042 , es (1984) 455. (2011) 437. , 92 ds ]. Acta Math. A 54 Archivum Math. , , y s and Courant SPIRE IN , ][ . Sci. China , ]. Commun. Math. Phys. (2002) 1263 , SPIRE IN ]. arXiv:1003.1004 ][ A 35 SPIRE ]. IN – 21 – (2012) 1 [ ][ J. Phys. , ]. SPIRE IN math/0112152 [ 10N4 ][ SPIRE On decomposability of Nambu-Poisson tensor IN [ -models and Integrability of Generalized Complex Structur (1996) 1. σ Consequences of anomalous Ward identities hep-th/0609015 [ (2002) 123 On higher analogues of Courant algebroids LXV 61 (1971) 95 A note on quasi Lie bialgebroids and twisted Poisson manifol Brackets, hep-th/0605148 Nambu-Dirac manifolds L-infinity algebras and higher analogues of Dirac structure Lectures on generalized complex geometry and supersymmetr B 37 Nonabelian bosonization in two-dimensions J. Symplectic Geom. (2007) 004 , 06 (2006) 119 [ Phys. Lett. JHEP Lett. Math. Phys. Univ. Comenianae 42 algebroids [41] E. Witten, [42] M. Zabzine, [38] Y. Hagiwara, [39] D. Alekseevsky and P. Guha, [35] Y. Bi and Y. Sheng, [37] S. Guttenberg, [34] D. Roytenberg, [40] J. Wess and B. Zumino, [36] M. Zambon, Appendix C

Paper 2: On the Generalized Geometry Origin of Noncommutative Gauge Theory

135

JHEP07(2013)126 y Springer July 4, 2013 July 22, 2013 : : March 25, 2013 , : braic Geometry, d geometry point Accepted Published ve and semiclassically Received 10.1007/JHEP07(2013)126 doi: ralized geometry of D-branes. s, Charles University, c,d Published for SISSA by [email protected] , and Jan Vysok´y c Dedicated to Bruno Zumino on the occasion of his 90th birthda Peter Schupp a,b D-branes, Non-Commutative Geometry, Differential and Alge We discuss noncommutative gauge theory from the generalize [email protected] Czech Technical University in Prague, Faculty of Nuclear Sciences andPrague Physical 115 Engineering, 19, Czech Republic E-mail: Mathematical Institute, Faculty of MathematicsPrague 186 and 75, Physic Czech Republic CERN, Theory Division, CH-1211 Geneva 23, Switzerland Jacobs University Bremen, 28759 Bremen, Germany [email protected] b c d a Open Access Sigma Models Keywords: of view.noncommutative We DBI argue actions is that naturally encoded the in the equivalence gene between the commutati On the generalized geometrynoncommutative origin gauge of theory Abstract: Branislav Jurˇco, JHEP07(2013)126 ] 3 6 7 ]. ], 1 3 8 5 7 10 11 12 12 , 6 , a more ]. In [ θ 11 sted) Poisson ]. Field equations 4 ], it has been shown 17 ac structures. This was and a bivector B naturally involves structures Lie algebroid morphisms [ to be intimately related to a ]. In [ d (complex) geometry [ ar if one lifts the topological ations ] and also generalized to Dirac es. Here we mention only few urn, at least in some instances, tion of Poisson manifolds (and 16 8 a has been shown to close under ated to the present paper. Topo- , geometry. For instance, the topo- -field [ H ], or coisotropic submanifolds [ 10 , 9 – 1 – ], it was observed that in the first order (non- 15 ]. In [ 14 ] recently appeared to be a powerful mathematical tool for 2 , ] and [ 1 13 ] and are at the heart of deformation quantization [ 3 ], where the graph defined by the corresponding (possibly twi 7 2.1 Fiberwise metric,2.2 generalized metric Factorizations of2.3 generalized metric, open-closed Dorfman rel bracket, Dirac structures, D-branes general form of world-sheet currentsa appears. more Their general algebr bracket, the so called Roytenberg bracket [ further developed in [ topological) Poisson sigma model characterized by a 2-form that the structure constants of the Roytenberg bracket appe Poisson sigma models can besigma twisted models by [ a 3-form it has been observed thatof the current generalized algebra of geometry, sigma such models as the Dorfman bracket and Dir lot of interesting differential,logical in Poisson particular sigma generalized modelsLie are algebroids) of [ interest forof the (topological) integra Poisson sigmaand models as can such can be further interpreted be as generalized in terms of generalize the description of variousinstances aspects of of its relevance string that andlogical are field and more non-topological theori or less Poisson directly sigma rel models are known Generalized geometry [ 1 Introduction 4 Non-topological Poisson-sigma model and5 Polyakov action Seiberg-Witten map 6 Noncommutative gauge theory and DBI action 3 Gauge field as an orthogonal transformation of the generalized metric Contents 1 Introduction 2 Generalized geometry structure is replaced by aD-branes more can general be Dirac related structure. to Dirac In structures t [ JHEP07(2013)126 . ], M 10 ∗ . As T F ⊕ ns of an d metric to a three- and TM θ θ and a possibly ty of D-branes ′ ], where, among F 21 d geometry origin of ]. We will discuss this berg-Witten map. In on D-brane volume cor- quivalence of commuta- , the closed background 25 ndpoints, in the presence θ F ivalence between the com- , natural generalized metric mmutative gauge theories ized geometry. We empha- etric defined by the closed and a bivector ual sections, let us note that y is relevant for discussions tring endpoints in the open ty in string theory we refer, m, in our case or le for a direct generalization 24 Dirac structures. d Witten [ oded in two different ways of d naturally interpreted within ertain orthogonal transforma- e gauge field and closely related issues) and ctions to the usual DBI action, B n the noncommutative version ransformations of er we follow the proposal of [ Poisson sigma models play a role. and Φ. It comes as a relation between two G ]. A thorough discussion of noncommutativity – 2 – 20 ] and [ ], which fits naturally to their role in both the integration 19 3 , . Finally, we recall the definition of the Dorfman bracket, 18 θ . The natural question is whether the so obtained generalize ). The positive answer is given by two different factorizatio ′ B θ and ]. Let us also note that the (semiclassical) noncommutativi g 23 and the open string backgrounds , 22 B , -field was recognized in [ g Before going into a more detailed description of the individ The paper is organizedIn as the follows. second section, we review basic definitions of general The purpose of the present paper is to unravel the generalize In the third section, we observe that adding the gauge field Noncommutativity of open strings, more precisely of their e B modified bivector can again be rewritten in the open string variables (with som expressing a generalized metric on a D-brane. mutative and semiclassically noncommutative DBIargue actions that ( the necessity of suchgeneralized an geometry. equivalence can In be the seen discussion,Roughly an non-topological speaking, wetive intend and to semiclassically convince noncommutative the DBI actions reader is that enc the e noncommutative gauge theory. We will mainly focus on the equ on the D-brane, thebackgrounds pullback of the generalized space-time m where D-branes correspond to leaves of foliations defined by responds to an action of an orthogonal transformation on the in detail in a forthcoming paper. can be seen astopological the Poisson sigma (semiclassical) model [ noncommutativityas of well the as s deformation quantization of Poisson structures. other things, alsowas the discussed equivalence via ofparticular, a commutative it field and was nonco argued redefinitionof that known the the Dirac-Born-Infeld under higher (DBI) action derivative the canthe terms be name effective i viewed D-brane as Sei corre action.e.g., For to reviews [ on noncommutativi in string theory followed in the famous article of Seiberg an part of first order Poisson sigma characterized by a 2-form of a orthogonal transformations defined by a bivector and a 2-for This allows us to recover the formulas relating, via a bivect almost everything in thisto paper Nambu-Poisson is structures presented and in M-theory a membranes, form cf. suitab [ generalized metrics, which are connectedtion by induced the action by of the a bivector c dimensional WZW term.of It non-geometric this backgrounds. respect, generalized geometr fields Dirac structures and their relation to D-branes. In the latt size the behavior of a generalized metric under orthogonal t JHEP07(2013)126 . ′ ]. is × θ ) 26 n (2.1) (2.2) f the , E 2 and θ ) : Γ( . · } , , defined as ) ), where · i E 2 n, n , e on 1 e ivalence is a direct h ·i is a symmetric non- , me relating the non- eneral framework, we = ve and semiclassically h· tween open and closed R anifestly invariant un- i -branes, i.e. those which 2 he semiclassical Seiberg- fined by graphs of Pois- D er, what we believe is new → be a smooth manifold and ical Poisson-sigma model, y correspond to the same sidering D-branes which are p , Oe ions of Seiberg and Witten, )-bilinear map ( ] should replace the Poisson , 1 E M D-brane which is a symplectic 7 ) tions combined with a Seiberg- ized geometry, see, e.g., [ vity. e metric M ξ wo (Poisson) bivectors r discussion of DBI actions, also Witten map, once one recalls its × ( ( Oe its main ingredients as developed h . This interpretation is the most p ∞ W ′ i E θ C )) ( : E ) + p ) η and is a · Γ( ( , ) the set of vector bundle automorphisms · θ V ∈ E i 2 ,( = , the latter one closely related to the nocom- n, n ′ ( , e i M – 3 – 1 F ) on η O · e ∈ , ∀ · + ( p | and )) F ξ, W ). This fiberwise metric has signature ( E ( + E V Γ( h Aut ∈ Γ( ) η ∈ , the Seiberg-Witten map is naturally interpreted in terms o + O θ { W ( . Hence, we denote by . A fiberwise metric ( , ) = ) ), such that for each M ξ M ∗ M + ( n, n T ( ∞ V O ⊕ C → TM We believe that analogous results hold also for more general In the final section, we discuss the equivalence of commutati In the fifth section, we briefly recall the interpretation of t In the fourth section, we use the above mentioned relation be ) = E leaf of the Poisson structure, describing the noncommutati noncommutative DBI actionconsequence of of a the (gauge D-brane. fieldWitten dependent) map. open-closed We The rela show discussionand here that interesting is is this not the completely clear equ new.in generalized geometry previous Howev origin sections. of Everything works very naturally for a for every ( relevant one for our discussionsymplectic in leaves the of final section. When con corresponding Dirac structure. commutativity parameters (Poisson bivectors) Witten map as a local diffeomorphism on the D-brane world volu mutative gauge field strength. Thishints equality, towards crucial the for appearance ou ofinterpretation the as semiclassical the Seiberg- local coordinate change between the t variables (including gaugeits fields) Hamiltonian to andder show the the that corresponding open-closedgeneralized non-topolog Polyakov metric. field action redefinitions are as m they geometricall focus on a particularE choice of vector bundle. Namely, let 2 Generalized geometry 2.1 Fiberwise metric,In generalized this metric section we recallAlthough some most basic of facts the regarding general involved objects can be defined in a more g sigma models. a dimension of are related toson more tensors. general Dirac For structures such than D-branes, the Dirac ones sigma de models of [ Γ( degenerate bilinear form. There exists a canonical fiberwis which includes the field strengths a consequence, we find a generalization of open-closed relat preserving this fiberwise metric. That is JHEP07(2013)126 = 0 (2.3) (2.4) (2.7) (2.6) (2.9) (2.8) (2.5) (2.12) (2.11) (2.10) 12 A = 1. For T 22 2 . A τ 1 + − 11 A 22 21 A e uniquely decom- A T 12 − A = B = 0, . In the block matrix form 1 21 . − , and A ) ! , N . ξ , T 11 ) . ( 12 , ξ V ) ! ) A T ! V . A ξ , ( i − ξ 1 ( ) V ξ + 2 V θ ,V − 11 B · N , ξ ! ( · , e 11 A be any invertible smooth vector bundle + ( ) + T . B A 1 ! − θ ξ θ ! is given as ξ − ) + 0 e as 1 T 21 − ( ) be a bivector. The induced vector bundle θ = = = + V N e B + ) transformations, which we will use in the θ τ TM A N ( 1 0 e h 1 0 1 e B θ θ M N B V O ξ 0 ( V by the same letter, i.e., we define , N N i V →

O

)-linear map of sections, such that n, n i X – 4 – := 11 ( − B 2

. In this paper we will always denote the induced − M = τ ) = = M A − ) = O Λ ) ∗ ( ξ ), i.e., ) := e ξ = M 2 TM ! ! T ξ = ∞ ∈ + : ) := ) = + , e ! ξ ξ ξ V θ C V + n, n 1 to ( N V V ( ξ e N V V ( θ (

(

( V θ O ( θ B

B e B e e N e TM in N ) be a O O  denotes the map transpose to E 12 22 Γ( A A M ∗ 11 21 ) be a 2-form on = 1, we find → T A A ). Finally, let ) ). Similarly, let M 22  → ( E E E A 2 ). Correspondingly, we have Γ( ). Correspondingly, the map Ω Γ( T 11 M : Γ( ∗ A ∈ ∈ M ∈ M T ( τ ( ) ) 1 + : ), we put ξ B ) transformation with the invertible upper-left block can b ξ X Ω E T 12 + + ∈ − ∈ A Γ( n, n V V ( V ξ N T 21 ∈ O A Let now 2 , e 1 e for all and More explicitly, for posed as a product of the form morphism is again denoted by the same letter, that is Any In the block matrix form In the block matrix form for all vector bundle morphism from where morphism over identity. We define the map for all ( for all ( sequel. Let There are three important examples of JHEP07(2013)126 , , − is − V M V τ , we ) · = ·i (2.13) (2.14) (2.15) (2.20) (2.16) (2.18) (2.17) (2.19) , , · ⊥ and + h· , we can V e ( is positive + B generalized V ·i ). Moreover, = 0, and the 1 eigenvalues , − h· and , M ) n, n , ) can be written ∗ . ξ g ( , } a skew-symmetric . ( T } 1 O ! M ! − B ∩ 2.19 ∗ M ∈ η g 1 W ∗ + T τ is positive definite, set a ( T V B ∈ and 1 0 ) + . ·i ∈ − ξ ! ξ , is a Riemannian metric on ( A ! ξ 1 | 1 h· 1 g | − ξ − ξ V − ! . ξ 1 Bg E = 0 and as covariant 2-tensor field on 0 − ) + , ! g ) + ξ i B Bg A ( ) 1 ξ 1 ) + 1 0 g ( 1 2 B g − − TM 1 e 1 − V ′− ( . − ( − ∩ , on which A − Bg g A B , τ ! . Using the orthogonality of we get that n , corresponding to +1 and ). Finally, we can observe that V + 1 1 − − B Bg V e − − − B M + 1 ⊕ h V g V g B g V − 1 0 1 } ≡ { 1 + } ≡ { can be viewed as graphs of invertible smooth is positive definite, we get that =

− V −

– 5 – i τ can then be written in the block matrix form ) Bg T g − and 2 ) · TM = V of rank = τ V − is a symmetric part of ! TM − , ) , e + · · ∈ )( ) E ξ g g V ∈ ! = 1, it is orthogonal and thus V , E 1 , we have · B 1 e 2 V and

+ 1

1 ( V − τ of | V − τ − | + = h = ) ) V V τ Bg V ! ) V ( , where ′ ( B Bg η ξ − V 1 B A A B g g + − , respectively. We can view

1 . From this equivalent formulation, i.e. using + + + 1 to get a generalized metric on − τ g B M Bg g V V − ∗ . This will give ) = ( { { = − + ξ, W − T τ ). In the block matrix form, ξ and vice versa, and using the knowledge of the signature of = g g A + E = = + ·i ⊥ → ). Also, because −

, V V Γ( − + ) and negative definite on Γ( E | V ( h· V V ( = can be an arbitrary 2-form on τ + ∈ τ Γ( ′ defines a positive definite fiberwise metric, we refer to it as a is a symmetric map, that is, V TM ) B . From now on, we will always assume that this is the case. Sinc A : . This means that ξ ∈ τ τ ) ′ − E 2 + V . From the positive definiteness of ., . , e = 1, we get two eigenbundles A V 1 A, A 2 e τ From positive definiteness on The important observation is that the block matrix in formul := +1 and , respectively. Using the fact that ( τ , whereas V | with respect to of definite on Γ( from for all get the direct sum decomposition Conversely, for any subbundle symmetric, same for τ vector bundle morphisms: as a product of simpler matrices. Namely, and write uniquely metric on If such ( where The corresponding fiberwise metric ( uniquely reconstruct for all ( part of we see that M JHEP07(2013)126 , G B . (2.22) (2.21) (2.23) (2.26) (2.27) (2.25) (2.24) τ ) · , · ( -flux. The is globally O H is changed by B = ′ H τ and a 2-form τ ) · on the generalized . g , ] is trivial. · θ . ! H τ − θ e )) = 1 and − rphism 2 sume that 2 e ′ dles: 0 1 1 ( , τ . . } n metric ,O ! ! ) map ) θ, M ! 1 ∗ 1 have to be closed, and this will , e Φ) T ( Φ Φ 0 1 n, n B 1 0 Φ 1 1 ( 1 0 O ∈ 1 0 − , we get the matrix equations − ohomology class [ − − − O . Clearly ) on the space of generalized metrics. ξ G is invertible, we can proceed the other G G . , described by a Riemannian metric = ( | Φ θ . θ Φ τO i ! ) − H ! n, n − 1 ) ! H − does not ξ + 1 e ( e 2 ) of − 1 ( − 1 V − e 0 1 O G 0 − H − θ ( , which has the form B 0 O G ( τ − g G e ) θ G G 2.23 = e − 0 g ,O = 0 B = G – 6 – ′ 0 1 G )) τ = B 1 − + G + 1 e ! g G G ( V − ! τ ! ) and ( O B ( + ( Bg and Φ. Since 1 0 1 τ = Φ define 1 Φ 0 1 ξ h 1 Φ 0 1 − 1 2.22 { G

τ

. The action of the − g We thus consider only the models with trivial = = = i = M 1 ! 2 Bg G + 1 G H , e V ) θ 1 0 1 -flux will be discussed elsewhere. e

( H ′ globally. τ = ) and given h G dB n, n ( = O is again a generalized metric. We may use the notation ( H ′ ∈ τ ) gives us a new generalized metric · O , · H be a 2-vector field on θ There exists a natural action of the group Note the important fact that the 2-form More precisely, we assume that the corresponding integral c . Namely, we have 1 θ − By the previous discussion, there exists a unique Riemannia metric and a 2-form Φ. Hence such that Let Comparing the two expressions ( Let us start with a (different) generalized metric 2.2 Factorizations of generalized metric, open-closed relations From that, we can easily find the relation between +1 eigenbun Since e remain true throughoutdefined, the i.e. whole paper. Nevertheless, we as Hence ( For each which can be uniquely solved for case of the non-trivial way around as well. We also know how the corresponding endomo JHEP07(2013)126 , . . is g ). G M θ 2.1 (2.29) (2.28) (2.30) (2.31) (2.32) . Define ic ( ]. ). If nd only if M 27 as its graph, , there exists 2.28 θ E of θ G , , Φ and can be equipped with e geometric interpreta- , D G is a closed 2-form on ) )] M η ). dξ B nt bracket is the antisym- , . In particular, for given ∗ ( } θ . If we exchange the tangent T + W y fman bracket and Dirac struc- G i M . 2.29 ⊕ has the same properties as ∗ } W . . − 1-eigenbundles. The open-closed ( 1 } T 1 xt of duality rotations see [ ) − B ± − η ∈ , which is maximally isotropic with M . As before, define a vector bundle TM TM ( , e ∗ G ξ ) E V M ∈ T ξ | = + Φ) L of ) ∈ V + ξ G E | ξ ( L ] + V 1 | ) relating closed and open string backgrounds ( − + ( ) B 2 V ). Define a subbundle θ ξ ( e ), one has ( V,W – 7 – , ξ B θ · = M ( + Φ) = [ be any rank-2 covariant tensor field on ( 1 θ + + = [ 2 , respectively, − G D ξ ] ) B Ω V D { η ] M { B + ( η ∗ of the generalized metric ∈ ) = = + as ξ ξ = 1 T + and Φ, and conversely, for given ( + θ B { − θ B g B G ( G G = G G is a Dirac structure, if and only if ξ, W and by ξ, W H B + + ) are the symmetric and antisymmetric parts of ( V ]. Our vector bundle G + V TM [ TM 28 have identical graphs as well as V [ 2.25 B . , e ). The corresponding pairing is the canonical fiberwise metr G → B is a Dirac structure with respect to the Dorfman bracket, if a 26 E θ , M Γ( 2 G and ∗ and and involutive under the Dorfman bracket ( ) and ( ∈ 1 T ) and its graph g · ) is the Seiberg-Witten formula ) − ·i : ξ , V, G 2.24 θ h· be a rank-2 contravariant tensor field on ( + 2.28 B θ , we can find a unique V ]. Consider the inverse θ 10 ) = Let A Dirac structure is a (smooth) subbundle For Φ = 0 the open-closed relations can be given a slightly mor Furthermore, for any closed For an earlier appearance of this type of formulae in the conte V 2 and is a Poisson bivector. Similarly, let ( Obviously, respect to for all ( in the presence of a noncommutative structure represented b B a structure of ametrization Courant of the algebroid. Dorfman bracket: The corresponding Coura that is and cotangent bundles and morphism tion [ Formulae ( we get using the above formula that Poisson, ( It is known that 2.3 Dorfman bracket, DiracHere we structures, briefly D-branes recall sometures, relevant facts see, concerning the e.g., Dor [ relations, for Φ = 0, is a simple consequence of that. θ a unique pair B Again, one can show that JHEP07(2013)126 . ∈ is F θ H (3.2) (3.1) + (2.33) B . supporting = ′ ! M , where 1 B , where zed metric θ e be a 2-form (at H F , but by gauged 1 0 by the action of − F H ′ , , leaves of foliations G ! ! as ∼ ! ′ 1 by presympletic leaves, θ θ G will be given as a graph , − F − 1 0 ). The generalized metric i M L − η 1 0 1 . 0 1 1 is an automorphism of the + ! 2.23 er, . Hence, in this case we will will be the foliation generated 1 B θ . Further, let ! ! ! e ′ 1 B T M Φ . As shown before, we can always 1 0 ξ, W − 0 , see ( B F Φ 1 − 1 0 1 0 + N − − G − e ) is no longer true for V 0 N h ! , such that we get 1 !

! on the generalized metric = ′ 2.32 M − 1 0 1 ) action. In particular, there exists a vector i θ 0 and 2-form − ) − G H 0 − g η will make sense only when considered on the D-brane. g on e G ! 0 n, n G 0 g ′ – 8 – F + ( . This can be achieved under some assumptions, 1 0 ′ θ G − ) transform O 0 F W ) has the following block matrix form: ! N ! ( ). ′ F ) of the Dirac structure under the anchor map. We ! B n, n T B ( 0 will be closed and defined only on a submanifold of L + , e ( provides a singular foliation of N 1 0 1 , such that 1 Φ 0 1 O ) 2.21 , which is described by the same F ). In the other words, ρ ′ 1 Φ 0 1 ξ L

E H g, B + TM ! ! = Γ( ! 1 ′ V ). The gauge transformation defines new 2-form F ( → 3 ′ by the 1 ) transformation ∈ H 1 0 θ B 1 0 1 θ 1 0 e ) G h

η n, n TM ( and Φ, see ( and the corresponding foliation of = : = + ! O ′ θ ′ for some gauge field G ) holds. ′ N F G G W ) corresponds the generalized metric F ( 1 0 1 , 2.33 )

ξ g, B = ] for arguments in favor of the identification “D-branes is related to + ′ = Φ+ ′ 10 ′ V G G ), but ( by action of M on the generalized metric One may ask, if there is a bivector Generally, a Dirac structure ( G Later, when discussing DBI action, corresponding to the pair ( ′ 3 X θ ′ 2 − a D-brane. In which case, all expression involving described by fields bundle morphism however, only up to a certain additional get where that is, 2-form Φ Indeed, examine the block matrix decomposition: G e defined by Dirac structures”. In the case we will consider lat and Λ corresponding Courant algebroid. Note that ( by Hamiltonian vector fields, i.e., by symplectic leaves of identify the symplectic leaves and D-branes. 3 Gauge field asLet an us orthogonal start with transformation a of given Riemannian the metric generali of a Poisson tensor To the pair ( this point an arbitrary one for all ( refer to [ which is generated by its image JHEP07(2013)126 . is 2.3 θ (3.4) (3.5) (3.6) (3.8) (3.9) (3.7) (3.3) . (3.10) (3.12) (3.11) ! 1 . . ′ F 1 0 )) ′ − Nθ we have F )) ′ N ′ ) we see that the θ ! )) F ′ θ (Φ + F 2.28 − 1 ices, can be uniquely and ar block matrix — of . − 0 1 1 1 θ (Φ + o ( : G 1 − ′ ) ) ′ (Φ + − tion, we assume that F 1 of the DBI action and non- ! e F N G ′ − ve expression. Since we want ) ) 1 , ′ ′ F, F 1 G ′ 1 − F ) F − ng the DBI action. 1 0 ′ − e F ) ) (Φ + F , ′ has to be again a Dirac structure of )). What we find are the following F θ θ − F θ (Φ + ′ ! − θ G + Φ + e − (Φ + . . G TM N ′ ′ ) 1 (1 + G ′ θ F ( G − O θ using the tools described in subsection det( ′ ( F T G G ′ · = (1 + F . For the graphs of T G = θ ( 2 N 1 − 1 0 = = ) θ N T e M − 1 Γ(Aut( θ ) – 9 – F N + ( (Φ + − − N = and G ) ′ ∈ G ′ − T F θF θ ) is invertible. For this, use the decomposition of θ F e − e F θF −

) = N = θF. e θF N ) as F θ 1 = 1 (1 + − − − + e ) )) = det( F ! G 2.11 F F 1 ) and B ) ), we find that ′ = = 1 + = (1 + ( 1 + + ′ ′ F M F 1 0 − ( 3.7 θ is a bivector on N : F − g B B X ′ ) ( ), we get the equalities 2 N θ 1 + F ) can equivalently be written as Λ (Φ + − g ! 3.2 , we may proceed by inserting 1 = g T ( + ′ ) ∈ θ 3.8 and N F ′ F B ′ − ( , θ = F + 0 1 1 ) ) according to ( − , 1 ′ ) and ( is closed. B g − θ M ( ) and ( ( g is a Poisson bivector. Similarly, one can see that ! n, n ) 3.1 F 2 ′ ( − 3.7 Ω F θ g O is an automorphism of Dorfman bracket, Φ 1 + ∈ 1 0 ∈ ′ − F B det( e F ( F

− Furthermore, following the same type of arguments leading t Finally, let us examine the objects e θ . Hence, − expression for Comparing ( Taking the determinant of ( This equality will play the central role when later discussi and with decomposed into acourse, product only of if a wee assume diagonal that and (1 + an upper triangul Now it is enough to note that the product of the last three matr E We will concentrate oncommutative the case gauge important theory. for the Therefore, discussion in the rest of this sec equations ( Poisson and to modify Φ to Φ+ It suffices to consider the three rightmost matrices in the abo Since JHEP07(2013)126 i ) ). η n 1 r in (4.5) (4.2) (4.3) (4.4) (4.1) , σ 0 σ , . . . , y 1 y , equipped ). We also ˜ η M is therefore a η, ′ with signature owing notation: F h ). Integrating out := ( e coordinates. For erves the (maximal) and 3.5 T V. ). Finally, we define a 1 E cal coordinates ( − according to their index X . ˜ 1 G M ! T ′ ) and ( 0 V. V, θ X, ∂ a model from the generalized T T ′ σV 0 3.8 2 θ ∂ 0 and introduce auxiliary fields 1 d − is closed. − ), (

Z N ! F := ( ′ we will use the following notation: Ψ + Ψ Ψ + Ψ 2 1 3.7 + F F T ¯ ˜ G G 1 T − M − V T T − N by ( = Ψ Ψ ! B ′ ′ → 1 2 2 1 -dimensional target manifold − ¯ V coupling to the boundary (and extending to F := F n σ σ 1 2 2 ′ still makes perfect sense as a Dirac structure. − F g − ¯ A d d ) can equivalently be written as G ¯ g F – 10 – :Σ ¯ θ doesn’t need to be closed. The last remark: In G ¯ + Φ − ′ Z Z T G 4.2 − B X F F -dimensional column vector Ψ ′ is an isotropic subbundle of and is Poisson nor that e

n σV , Φ and ¯ ′ F ). We also choose some local coordinates ( θ 2 ] := ] := ¯ G G F d N = − G dA Φ ˜ Z ¯ G e e Φ + η, X η, X T = 1 2

N η, η, [ [ − F we obtain the Polyakov action expressed equivalently eithe and a 2-form Φ. We can assume Σ with a non-empty boundary = S S θ η ), the action ( ¯ G . Hence ] := ¯ Φ := F 3.8 , X -dimensional column vector and ˜ are Cartesian coordinates for a Lorentzian metric [ G n ), ( S η µ ) is not invertible, GN being related to σ 3.7 T will still define an almost Dirac structure. , 2-vector 4 F θF N F G . Let us also note, that G assume an abelian gauge field ). In this section it will be convenient to introduce the foll θ := M and e X ¯ M ( matrix G i B y , which transform under change of local coordinates on , n . Lower case Latin characters will always correspond to thes j 2 g e η Let us consider a 2-dimensional world-sheet Σ with a set of lo = +) on Σ. Furthermore, we consider an M Here, we neither need to assume that × , 4 i Σ. On n − the components of the smooth map on and ∂ Σ, the field strength being with a metric open or closed variables We put 2 ( Similarly, 4 Non-topological Poisson-sigma modelIn and this Polyakov section action wegeometry review point of the view non-topological developed Poisson-sigm in the previous sections. X structure. We combine them in a 2 case that (1 + We assume that where Using relations ( the auxiliary fields introduce another 2 2-form on isotropy property of This is no more an automorphism of Dorfman bracket but it pres with Our (non-topological) Poisson-sigma model action is JHEP07(2013)126 i - ) – η 34 4.2 ned. onal = 0) (4.6) (5.1) holds . The F θ ˜ G ). G = tF tF θ e leads to the expressed as ). Hence, we ¯ G ′ t ]. = ) can be found, 3.2 G 37 t is applied to the θ 3.2
G s, are used. As can 1 η − ) and ( ) with 1 1 1 ) appeared (with r this, note that rela- ). The auxiliary fields 3.1 4.5 on from the commutative oordinates given by a map variables. Let us note that, 4.5 given by ( X, η l methods akin to the ones of elds, the ˜ ′ 1 atrix ∂ tical. To write down the result G , ) with respect to Υ ′ := ( 5.1 ]-flux. The interested reader may find some G 1 T , t T H . ) with − ]. Here we follow the approach of [ θ t ) ) Υ 1 A 33 4.6 , simply observe that ( F θ t t t , tF θ ) and the action ( θ dσ θ 32 − 4.2 Z −L 2 1 (1 + = – 11 – θ = t θ t t This map can be derived using a generalization of = ], see [ θ ] = t ∂ and the related non-commutative gerbe in [ t ∂ 31 θ 5 . H θ X, η [ ) or in the Hamiltonian corresponding to the action ( H can explicitly be seen either in the Hamiltonian correspond ) = ′ ′ is a bona-fide Dirac structure even for non-invertible (1 + 4.5 Partial differentiation of ( θ ]. The Hamiltonian ( G θ ( ∗ 6 G ) and their transposes are obtained from the nonzero off-diag 29 ρ 1]. Of course, we have to presume that the formula is well-defi tF , e -dimensional column vector Υ ]. Polyakov actions like the first one in ( 3.10 [0 n are indeed Poisson for all 17 t ∈ θ t . ) can alternatively be expressed as the equality of matrices ˜ G 3.8 , such that = 0, in [ , this can be rewritten as = M the matrix given by the two equivalent decompositions ( F ), ( ¯ dA G ′ → 3.7 = ) can be found in [ G = 0, the relation between the action ( ] in the context of Poisson-Lie T-duality. The generalized metric F M 3.1 F As said before, here we assume only topologically trivial [ Let us note again that 30 5 6 ], where it was shown that the Seiberg-Witten field redefiniti : ing to the Polyakov action ( equality where straightforwardly be checked, thesewe Hamiltonians introduce are a iden new 2 for the respective graphs. To see that these relations in the form ( become the canonical momenta and the Hamiltonian is in ( For ρ Moser’s lemma: Consider the family of Poisson bivectors to the non-commutative setting has its origin in a change of c again for have the same Hamiltonian usingfor either the closed or the open after the equations of motions for one half of the auxiliary fi blocks after the similarity transformation with the block m differential equation relevant discussion concerning nontrivial in [ parameterized by For an approach to theZumino’s non-abelian famous case, decent using equations cohomologica 36 [ 5 Seiberg-Witten map tions ( Actually, all this can be seen rather straightforwardly. Fo JHEP07(2013)126 . D . (5.3) (5.2) } . We ] that ′ . Also, θ M 36 ∗ ij (i.e., the ∗ A explicitly θ ρ ), and the ]. T ) transfor- . To avoid θ ρ A A ∈ 38 ( := ρ t . Actually, all n, n θ t ], to define the ij ( , η ) ) O in order to carry 10 η θ ( T structure − N y the . Obviously, 1 that we can safely restrict φ ) + . This differential equation ertheless, according to our ralized geometry. Namely, n map is the map of graphs θ eeded for our discussion of η he matrix ( ( = spective D-branes. In such a ) we see that so is T = ρ w for quite some time [ suggestion of [ − nguish between the tensor itself oncommutative DBI actions fol- 0 3.4 θ N . ′ . ) . 1 θ , are Morita equivalent [ ij ′ ) still makes sense and we can argue − Nθ θ ) θ . Thus { ′ j θ det l ) and the (semiclassical) Seiberg-Witten 5.3 θ ) and has understood the (semiclassical) = 2 J ∗ A i J } and k ρ 3.7 only changes coordinates on the symplectic ) = 3.7 ( t J ), if one considers D-branes which are sym- θ M θ θ A ( ∗ ), ( ) = ρ , hence we shall use the notation ), ( ∗ t ′ T – 12 – A ) = θ φ 2.11 ( ∈ kl 2.25 ′ ∗ A 2.25 θ ρ = det ( ( is invertible. From ( ), ( is a diffeomorphism of the D-brane world-volume 2 the formula ( ∗ A η, η ), ( − θ ′ ρ det A θ ), with initial condition J ρ 2.24 ) + 2.24 η ,A · ( , such that is infinitesimally generated by the vector field ( t θ t φ θ A ρ have in fact the same symplectic foliations for all t θ } 7→ { ) := A , including in particular M ( t ∗ t θ θ T and hence also . We have i k ∈ θ ’s are the same, we see that t ∂ρ ∂x θ . More explicitly, entering the decomposition ( θ η, η ). We can thus restrict ourselves to the non-degenerate case = G θ N k i ) + O 7→ J η ( θ ′ ′ In the next section, we will discuss the case when the Poisson Finally, on the level of Dirac structures, the Seiberg Witte θ G { : ∗ For degenerate immediately have that denote corresponding Dirac structure) will be used, following the leaves (of we have seen that thehave relations their ( root in generalizedthe geometry. equivalence of Actually, the it commutative is andlows kno (semiclassically) once n one hasSeiberg-Witten established map ( as a (local) D-brane diffeomorphism. Nev plectic leaves. 6 Noncommutative gauge theoryIn and DBI the action previousnoncommutativity sections of we D-branes have as described a all consequence ingredients of n their gene our discussion without thecase, loss the of generality (Seiberg-Witten) to map any of the re ρ D-branes as its symplectic leaves. The above argument shows as follows: Since the map We thus get that can be integrated to a flow possible confusion, we will forand a its moment components notationally in disti coordinates. Therefore we introduce t Let us assume for a moment that depends on the choice of gauge potential out the computation of the Jacobian. Hence, the Seiberg-Witten map can be seen as the map induced b with a vector field mation kernels of all Poisson structures The Poisson structures JHEP07(2013)126 . (6.4) (6.2) (6.1) (6.3) , one nc DBI S and an S , ) =: ) to rewrite ) ′ ′ ˆ F F . ction, we have Seiberg-Witten 2.24 ) ′ ˆ F Φ + ), again using the ic matrix + + Φ + Φ). ]. utative and noncom- 3.7 ˆ etric and an antisym- G G 39 G, ( ( cture having constant rank. 2 + Φ + 2 / / 1 1 semiclassically noncommu- G ism. ( n between the D-brane and 2 As argued before, under this ound fields (open and closed det he equivalence of the commu- / ffective closed and open string 1 es, see [ ) det 8  y is new. Moreover, the discus- or the sake of completeness and . and corresponding field strength . ˆ θ θ θ 2 θF , and the relation ( the D-brane is a submanifold, such that /  A ) det 2 1 ] and will be elaborated in detail in . , i.e, a submanifold of target space- 2 / closed relations, here we start from a given 4 / d 1  / | 1 25 θF 1 ) (1 + A , 2  1 B det / ) and the relation ( 24 1 − g G s + Φ) + (1 + 1 ˆ 2 G 6.1 ]. These are related as g G det / AS det x 1 det s 21 d − 1  d G – 13 – det( s det( det S | x g s of dimension 2 Z  d 1 / s d G = 1 g | D s S ) = Z = | G F ) = s -theory branes [ = F G ) = + | M F + A and determine uniquely the open variables ( B , respectively [ s θ + + B + G S g B | + ( , g 2 Namely, assume that our D-brane is of a particular kind, i.e. + ( / A and 1 2 g 7 / ( . s ), pick a 1 2 θ g det / 1 s g, B det 1 to it defines a regular Poisson structure, i.e. a Poisson stru g s θ det 1 x g d s ), and performing the change of coordinates according to the 1 d g . While describing the Seiberg-Witten map in the previous se x 5.3 Z d D equipped with a line bundle with a connection d := Z M Before we turn to the discussion of the DBI action and its comm Assume that we have a D-brane It is straight-forward to modify everything to the case where Recall, in accordance with our above discussion of the open- c 8 7 DBI . Also, consider the restrictions (pullbacks) of the backgr S Integrating over the D-brane world-volume the restriction of A most intriguing relation is obtained from ( We can use the formula for the determinant of a sum of a symmetr map, we finally obtain atative relation DBI between the actions commutative and above mentioned formula formetric the matrix: determinant of a sum of a symm this as assumption, the Seiberg-Witten map is a D-brane diffeomorph mutative description, we discuss thecoupling relation constants between the e which comes as symplectic leaf of the Poisson structure a forthcoming paper. Heretative we and will include (semiclassically) the noncommutative DBI derivationthe of actions reader’s t f convenience. For related work based on dualiti closed background ( antisymmetric matrix time sion generalizes to the case of best knowledge, the direct relation to generalized geometr recalling ( seen that it isthe quite Poisson natural tensor to assume that there is a relatio ones) to F JHEP07(2013)126 ˜ λ is ′ ˆ (6.5) F and . This ′ θ ommons G nition, (2003) 281 . This makes ¯ G 54 ) is the effective it is defined as = θ 6.4 ˇ ˜ CR P201/12/G028. G AD (PPP) and ASCR ts on an earlier version of Gravity”. h Technical University in t GA tative and semiclassically efully acknowledge support t the necessity of a relation ccasion of his 90th birthday. ve the details to the reader. . nder the gauge transformation g to the Poisson tensor d reproduction in any medium, =0 t | ). Expressed directly in terms of ) λ on the l.h.s. in ( ( 4.5 n ) t c DBI ∂ . Hence, an alternative — but completely s S Quart. J. Math. Oxford Ser. ˜ + 1)! g G , 4 ) + n / ( 1 , A ( } t – 14 – ˜ λ θ det , ( ′ ij ) decompositions of the generalized metric ˆ F X { 3.2 = = ) is to start from the matrix equality ) can equivalently be expressed using either the “commu- ′ ˜ λ ij is the integral of 6.4 ˆ 4.6 F δ ]. c DBI S SPIRE IN ][ This article is distributed under the terms of the Creative C Generalized Calabi-Yau manifolds , the action ), and similarly for the other objects. As a result of this defi ˜ ij G ) or “noncommutative” ( θ ( transforms semiclassically noncommutatively, i.e., 3.1 A ∗ ). ρ dλ math/0209099 6.4 [ Finally, the Hamiltonian ( Let us note: The commutative DBI action := = [1] N. Hitchin, ij ˆ equivalent — way of obtainingnoncommutative DBI the actions relation ( between the commu up to the inverse of the closed coupling constant is maybe the most directlike ( hint from generalized geometry abou the relation to the Polyakov action more transparent. We lea tative” ( D-brane action obtained fromthe the matrix Polyakov action ( We would like to dedicate thisWe article would to Bruno like Zumino to on the thankof o Satoshi the Watamura manuscript. for important The commen research of B.J. was supported by gran Acknowledgments The hat “ˆ” has the following meaning: On matrix elements of Here, the curly bracket is the Poisson bracket correspondin & MEYS (Mobility) forfrom supporting the our DFG collaboration within and the grat Research Training GroupOpen 1620 Access. “Models The research of J.V. wasPrague, supported grant by No. Grant SGS13/217/OHK4/3T/14. Agency of We the also Czec thank to DA θ the semiclassically noncommutative fieldδA strength, which u provided the original author(s) and source are credited. References Attribution License which permits any use, distribution an is the (semiclassical) noncommutative gauge parameter. JHEP07(2013)126 . ]. , SPIRE IN (2005) 455 (2005) 035 ][ , (2011) 074 03 ds 260 ]. severa/letters/ (2002) 341 , coisotropic , 03 -models and ∼ ]. ]. σ 43 , , JHEP SPIRE , ]. -model IN JHEP -models σ SPIRE , , σ ][ (1999) 030 math/0406445 IN [ ][ , SPIRE 06 ]. IN [ math/0309180 ]. [ ]. ]. ]. J. Geom. Phys. JHEP , , (2005) 400 Commun. Math. Phys. SPIRE SPIRE , SPIRE IN SPIRE ]. IN 54 IN ]. ]. ][ math/9902090 IN [ arXiv:1206.6964 ][ ][ [ (2004) 157 ][ math/0308180 Noncommutative geometry from strings and [ math/0401221 -models 69 SPIRE , σ SPIRE SPIRE IN IN IN – 15 – http://sophia.dtp.fmph.uniba.sk/ ][ D-branes in generalized geometry and , Lie algebroid morphisms, Poisson ][ ][ (2000) 591 Dirac (2012) 064 (2004) 33 Noncommutative field theory J. Geom. Phys. math/0112152 10 [ , 212 hep-th/0106048 67 [ hep-th/9812219 [ Coisotropic submanifolds in Poisson geometry and branes in A path integral approach to the Kontsevich quantization hep-th/9810072 Courant-like brackets and loop spaces [ ]. From current algebras for p-branes to topological M-theory ]. ]. ]. ]. ]. Current algebras and differential geometry String theory and noncommutative geometry WZW — Poisson manifolds JHEP Lett. Math. Phys. Noncommutative open string and D-brane ]. Generalizing geometry — Algebroids and , , SPIRE hep-th/9908142 hep-th/0507051 arXiv:0805.4571 [ [ [ (2002) 123 SPIRE SPIRE SPIRE SPIRE SPIRE IN (2001) 977 (1999) 151 IN IN IN IN IN SPIRE [ [ ][ 61 (1999) 016 ][ ][ ][ 73 IN A Note on quasi Lie bialgebroids and twisted Poisson manifol On the integration of poisson manifolds, Lie algebroids and D-branes and deformation quantization ][ 02 Lett. Math. Phys. Non-geometric string backgrounds and worldsheet algebras Non-geometric backgrounds and the first order string Generalized complex geometry , B 550 Letters to Alan Weinstein. 2 (1999) 032 (2005) 015 (2008) 137 Commun. Math. Phys. JHEP , 09 09 07 , ˇ Severa, hep-th/9903205 arXiv:0903.3215 hep-th/0410183 math/0104189 hep-th/0411112 off-shell closed gauge symmetries branes JHEP Rev. Mod. Phys. Nucl. Phys. [ JHEP Lett. Math. Phys. arXiv:0906.2891 [ JHEP the Poisson sigma model [ [ Dirac-Born-Infeld action arXiv:1004.0632 [ formula submanifolds [8] and C. T. Klimˇc´ık Strobl, [9] P. [6] A. Kotov and T. Strobl, [7] A. Kotov, P. Schaller and T. Strobl, [3] A.S. Cattaneo, [4] A.S. Cattaneo and G. Felder, [5] M. Bojowald, A. Kotov and T. Strobl, [2] M. Gualtieri, [21] N. Seiberg and E. Witten, [22] M.R. Douglas and N.A. Nekrasov, [18] C.-S. Chu and P.-M. Ho, [19] V. Schomerus, [20] F. Ardalan, H. Arfaei and M. Sheikh-Jabbari, [16] D. Roytenberg, [17] N. Halmagyi, [13] J. Ekstrand and M. Zabzine, [14] G. Bonelli and M. Zabzine, [15] N. Halmagyi, [11] A.S. Cattaneo and G. Felder, [12] A. Alekseev and T. Strobl, [10] T. Asakawa, S. Sasa and S. Watamura, JHEP07(2013)126 , ]. oids , , ts , ] (1990) 394 SPIRE IN (2003) 207 ][ hep-th/9905218 , B 347 , 378 , May 17–23, ry , , A cohomological approach hep-th/0102129 ]. [ Nucl. Phys. Phys. Rept. hep-th/0105192 , [ , SPIRE (1990) 631. umino, ]. ]. ]. IN [ ]. 319 (2001) 148 PoS(CORFU2011)045 SPIRE SPIRE , SPIRE ]. IN IN SPIRE IN (2001) 047 ][ ][ Noncommutative gerbes and deformation IN ][ B 604 ][ 06 SPIRE Target space symmetries in topological theories. IN ][ – 16 – hep-th/0206231 The Seiberg-Witten map for noncommutative gauge ]. ]. JHEP Duality invariant Born-Infeld theory , Nucl. Phys. -model and branes -model and effective membrane actions Noncommutative gauge theory for Poisson manifolds Non-abelian noncommutative gauge theory via , SPIRE SPIRE . σ σ hep-th/0001032 IN hep-th/0005005 IN [ [ [ [ arXiv:1203.2910 [ Trans. Amer. Math. Soc. hep-th/9512040 [ ]. , Gauge equivalence of Dirac structures and symplectic group ]. Continuous Advances in QCD 2002/Arkadyfest Poisson-Lie T duality and loop groups of Drinfeld doubles Noncommutative Yang-Mills from equivalence of star produc Nambu Nambu- hep-th/0106042 [ Duality rotations in membrane theory (2010) 261 SPIRE SPIRE IN (2000) 367 (2000) 784 (1985) 477 IN (1996) 65 (2012) 313 ˇ Severa, ][ 807 ][ Lectures on cohomology, T-duality, and generalized geomet . C 14 hep-th/0206101 (2002) 021 Dirac manifolds B 584 B 253 , Quantum field theory on noncommutative spaces Cohomology of gauge groups: cocycles and Schwinger terms B 372 B 713 02 ]. ]. ]. , talk presented at JHEP SPIRE SPIRE SPIRE , IN IN IN hep-th/0109162 arXiv:1205.2595 quantization 1 to the nonAbelian Seiberg-Witten map math/0202099 [ noncommutative extra dimensions [ Eur. Phys. J. Nucl. Phys. Nucl. Phys. theories Phys. Lett. Lect. Notes Phys. [ [ Phys. Lett. [ University of Minnesota, U.S.A. (2002), [37] P. Aschieri, I. Bakovic, B. and Jurˇco P. Schupp, [28] T. Courant, [29] L. Baulieu, A.S. Losev and N.A. Nekrasov, [38] H. Bursztyn and O. Radko, [39] D. Brace, B. Morariu and B. Zumino, [36] B. P. Jurˇco, Schupp and J. Wess, [34] B. and Jurˇco P. Schupp, [35] B. P. Jurˇco, Schupp and J. Wess, [32] D. Brace, B. Cerchiai, A. Pasqua, U. Varadarajan[33] and B. B. Z Cerchiai, A. Pasqua and B. Zumino, [30] and C. P. Klimˇc´ık [31] B. Zumino, [26] P. Bouwknegt, [27] M.J. Duff and J.X. Lu, [24] B. and Jurˇco P. Schupp, [25] P. Schupp and B. Jurˇco, [23] R.J. Szabo, 154 Appendix D

Paper 3: Extended generalized geometry and a DBI-type effective action for branes ending on branes

155

JHEP08(2014)170 -brane p Springer velop a geo- April 23, 2014 : August 12, 2014 August 29, 2014 , : : Received 10.1007/JHEP08(2014)170 Accepted Published uge. Leading terms in the doi: eneralize the concept of semi- losed relations to the -classical) matrix model. mutative and noncommutative ysics, Mathematical Institute, -brane. We calculate the power ′ e, p Published for SISSA by b,c [email protected] , -branes ending on a and Jan Vysok´y p Dedicated to the memory of Julius Wess and Bruno Zumino. b -branes governed by a Poisson tensor. We find a natural -brane backgrounds. With tools of generalized geometry p D . 3 Peter Schupp 1404.2795 a The Authors. c p-branes, Differential and Algebraic Geometry, M-Theory

Starting from the Nambu-Goto bosonic membrane action, we de , [email protected] E-mail: Jacobs University Bremen, Campus Ring 1, 28759 Bremen,Department Germany of Physics, CzechFaculty Technical University of in Nuclear Pragu Sciences Prague andBˇrehov´a7, 115 Physical 19, Engineering, Czech Republic Charles University in Prague,Sokolovsk´a83, Faculty Prague of 186 Mathematics 75, and Czech Ph Republic [email protected] b c a Open Access Article funded by SCOAP double scaling limit are given by a generalization ofKeywords: a (semi ArXiv ePrint: series expansion of the action in background independent ga Abstract: metric description suitable for Branislav Jurˇco, Extended generalized geometry andeffective a action DBI-type for branes ending on branes description of the correspondenceversions of recently of proposed an com effective action for we derive the pertinent generalization of the string open-c case. Nambu-Poisson structures are usedclassical in noncommutativity this of context to g JHEP08(2014)170 1 4 5 9 36 37 38 27 29 32 34 37 10 14 17 21 23 ally commutative geometry. Low of extended objects are novel g theory. These symmetries ing theory and the equivalence geometry. Particularly interest- otentially observable phenomena nts of the stringy symmetries can – 1 – as transformation of the fibrewise metric F B.1 Pseudoinverse ofB.2 a 2-form Integrable forms A.1 Scalar density noncommutative DBI action ing examples of these symmetriesof are T-duality commutative/noncommutative in descriptions closed in str have their open natural strin settingsenergy in effective theories generalized link geometry the fundamental andin theories (target) non to spacetime. p Interestingly, the spacetime remna 1 Introduction Among the most intriguing featurestypes of of symmetries fundamental and theories concomitant generalized notions of B Background independent gauge 11 Matrix model 12 Conclusions and discussion A Nambu-Poisson structures 9 Background independent gauge 10 Non-commutative directions, double scaling limit 6 Gauge field 7 Seiberg-Witten map 8 Nambu gauge theory; equivalence of commutative and semiclassic 3 Membrane actions 4 Nambu sigma model 5 Geometry of the open-closed brane relations Contents 1 Introduction 2 Conventions JHEP08(2014)170 ] ], ], ]. ], 1 8 19 42 39 , ] for , , 6 ], and ]. 37 18 -brane. 9 38 ′ ], where , 20 p 5 ]. 36 ]. The way 44 4 mbu-Poisson – , 2 ]. The axioms [ -symmetry and elf-duality con- 43 ′ κ rtance for string 21 α The main idea of [ ] that tackles the quest ]. This “commutative- ]. It was further elab- lready in 1973 [ o a new Nambu gauge 1 7 ane is well known: the symmetry and geometry 47 , been proposed in [ are a current focus of re- abelian DBI action [ khtajan [ – 6 ut that the requirement of ed on T-duality arguments. ct generalization of Poisson 45 stency with onjectured in the pioneering -branes ending on a utative descriptions [ . More precisely, we intended p onic part of the effective action heet quantization of the funda- ] with its covariance problems est to ours is the one of [ rst principles in string theory is ] and K. Furuuchi et al. [ uely [ ring that its equations of motion lined in [ e need of actual string computa- ] and others. See also [ 40 ermine, in the case of a M5-brane, ing theory and supergravity. Here 17 n to all orders in – 35 – 14 31 ]. For a modern treatment of Nambu- 23 , 22 – 2 – ], this direction is not pursued there. Recently, a ]. The role of generalized geometries in M-theory 45 -symmetry, we refer to the reviews [ 50 κ , 49 ]. ]. For these and alternative formulations, e.g., those of [ 26 41 – 24 ]) motivated by the works of [ 30 – 27 ], was to introduce open-closed membrane relations, and a Na ]. See also the work of P.-M. Ho et al. [ ]. Although Hitchin certainly recognized the possible impo -symmetry as a guiding principle and features a non-linear s 9 κ 13 48 – ] and proved three years later in [ 10 21 Nambu-Poisson structures were first considered by Y. Nambu a The main objective of this paper is to study this interplay of Generalized geometry was introduced by N. Hitchin in [ Matrix-model like actions using Nambu-Poisson structures In this paper, we focus only on the bosonic part of the action. The result forDirac-Born-Infeld the action case provides an of effective descriptio open strings ending on a single D-br dition. It avoids the use of an auxiliary chiral scalar [ (nonlinear) selfduality. in the case ofto higher extend, dimensional clarify and extended further objects develop (branes) the construction out orated in [ backgrounds, and commented on itfocus in of [ applications of generalizedwe geometry, mention is closely superstr related work [ search (see e.g. [ further reference. Among the earlywhich approaches, uses the one clos following a suggestion ofbased [ on superspace embedding and of Nambu-Poisson structures, although theystructures, seem are to in bepaper a fact dire [ very restrictive.Poisson structures This see was [ already c essentially uniquely. Interesting open problemsthe are to form det of the full supersymmetric action and to check consi generalized and axiomatized more then 20 years later by L. Ta inspired by [ map which can betheory [ used to relateon ordinary relation of higher M2/M5 gauge“commutative-noncommutative to duality” theory Nambu-Poisson determines the structures. t bos It turns o Moreover, there is are equivalent commutative andthe non-comm equivalency condition fixesnoncommutative the duality” action has essentially been uniq In used the also context to of study the M2/M5 the brane non- system a generalization has that this effective action hasrather originally indirect: been derived the from effective action fi double is as determined consistency conditions by for requi anmental anomaly string. free A world more s direct target space approach can be bas to find an all-order effective action for a system of multiple tions: “string theory with no strings attached.” fix these effective theories essentially uniquely without th JHEP08(2014)170 ] n 53 s a , s) was 52 = 1 case. p of the previ- has not been has its origin in the equivalence of TM p 8 Λ TM correct mathematical p n-closed relations, can ⊕ hogonal transformation tion ground field redefinitions, ng Hamiltonian density is o describe open-closed re- ambu-Poisson structures. of open variables and this e directions. To the best of TM ]. An outline of the relation and Λ nds. On the other hand, the metric on the vector bundle on. We present various forms ric backgrounds. See, for ex- as a fluctuation of the original aic manipulations are used to requires the introduction of a rices. Finally, we compare our uge fixing can be used to find s is mainly pursued in [ = ]. Finally, there is an interest- 62 i-classical) Seiberg-Witten map uctures of generalized geometry F 63 erg and Witten. TM W = 1 string case, we find the need isson sigma model of the , we review basic facts concerning p 3 with using a generalization of Moser’s lemma. ∗ 7 W ]. A further focus is the construction of the + 1-dimensional brane world volume and the ⊕ 51 p – 3 – W +1)-form gauge field ], or the lecture notes [ p 61 . Compared to the M ∗ T ], we have used generalized geometry to describe the relatio 1 brane case. p ]. One should also mention the use of generalized geometry in Λ 66 60 ]. Generalized geometry (mostly Courant algebroid bracket , ⊕ p > 55 59 M ∗ T , we describe the sigma model dual to the membrane action. It i , we introduce the ( 4 sets up the geometrical framework for the field redefinitions ⊕ 6 ] and [ 5 58 – ], see also [ TM ]. Finally, in [ p 56 54 65 Λ , ⊕ Based on the results of the previous sections, we prove in sec In section This map is explicitly constructed in section This paper is organized as follows: in section Section In section 64 considered in the context of M-theory before. target manifold diffeomorphism, which generalizesfrom the (sem the string to the membrane background. We show thatof this the can generalized be metric viewedoriginal as describing an the background ort membrane can backgrou description equivalently can be be described extendedidentify to in the include terms pertinent fluctuations. background Algebr fields. The construction two parts are relatedour to knowledge, the this temporal particular and structure spatial worldvolum a commutative and semiclassically noncommutativeof DBI the acti same action usingaction determinant to identities existing of proposals block mat for the M5-brane action. The key ingredient isbe the chosen fact to be thatformulation of Π, a the Nambu-Poisson which analogue tensor. of appears a in symplectic Attention volume the is form ope paid for N to a a convenient form ofa the fiberwise action. metric We on showgeneralizing a that the certain the well-known vector correspondi open-closed bundle. relations of We Seib present back straightforward generalization of the non-topological Po of T-duality with generalizeding geometry interpretation can of be D-branes found inin in string [ [ theory as Diracbetween str string theory and non-commutative geometry. classical membrane actions. In particular, we recall how ga field theories based onand objects in of [ generalized geometry. Thi the description of T-duality, see [ was previously examined by C.M. Hull in [ gauge fixing of the auxiliary metric on the for a second “doubling” of the geometry. The split in ous sections. Anlations extension as of an generalized orthogonalTM geometry transformation is of used the t generalized also used in relationample, to [ worldsheet algebras and non-geomet JHEP08(2014)170 i M dy ] to ∗ ) be , we 5 e the T ij M g ) coor- ( y → = 1 case. +1 ) and is used to nd rewrite p ) is indicted p 5 Ω M = 1 case, we p TM ( p al coordinates p p ∈ X :Λ ), respectively. B ral interpretation ♭ denoted by Greek of , . . . , σ M -dimensional target B ); the induced map 1 ( n J q σ 1 we have to consider ∂ M . Let ( ). We denote the corre- n M +1 cussion of the underlying p > . We use the letter Π also p J ay. We identify a possible out that this double scaling itting using a block matrix or any metric tensor he much easier oped in section X ive directions to expand the ) ) and Ω -parts of the exterior algebras for dy q ∈ J he double scaling limit of [ , . . . , y M ξ 1 ( q . We use the shorthand notation y , etc. Upper case Latin characters = n X ξ . In analogy with the ≤ C p for +1)-tensors on in the local basis i ∂ p < i in order to obtain a well defined Nambu- ♭ i, j, k, . . . J . The degree . We assume that the -tuples of indices corresponding to ( B ξ p j C p ]. α ··· iJ ∂ – 4 – dy ∂σ 11 < . We do not distinguish between vector bundle ∧ J . Similarly, let Π 1 i ≡ ∂ i ) ) = Π J ) are denoted by ... J α ξ ≤ Q ∧ ( ∂ ∂ )-linear maps of smooth sections. We will usually use M ♯ ( matrix of 1 as a time parameter. Integration over all coordinates is ♭ j is introduced, we assume that it is positive definite, i.e., = M 0 n ( dy ,B σ i Q M ∞ × ∂ = h C
of the paper, we use background independent gauge, double ) with 1 J p on n 0 is a fixed positive integer. Furthermore, we assume that we p = the components of the inverse contravariant tensor. g dy 11 + 1)-form background field ij i,J , where g p > p ) i and , . . . , i is defined as Π ) and forms Ω( , whereas the integration over space coordinates ( . We define the corresponding vector bundle map B 1 dy , etc. As usual, σ p i j J M M ∂ , we introduce a convenient splitting of the tangent bundle a ( +1 ∂y Q , we show that the Nambu-Poisson structure Π can be chosen to b p TM = ( X 9 10 d iJ ∧ + 1)-dimensional compact orientable worldvolume Σ with loc I also for the → is equipped with a set of local coordinates ( p B R ... α, β, . . . ). We may interpret B , etc. will denote strictly ordered ∧ p M M ), that is ( ∗ . Indices corresponding to the worldvolume coordinates are 1 ) = j σ ∂ T Q M p p ∂y ( ( ) is a Riemannian manifold. With this choice we will find a natu d ♭ 1 + 1)-form on We use the following convention to handle ( In the final section In section In section Where-ever a metric , . . . , σ :Λ R ≡ B p 0 ♯ J M, g σ Π of Ω a ( morphisms and the induced the letter denote, as usually, by as dinates, e.g., ∂ of membrane backgrounds in terms of generalized geometry. F sponding indices by lower case Latin characters 2 Conventions Thorough the paper, scaling limit, and coordinatesDBI adapted action up to to the firstlimit non-commutat order cuts in the off scaling thecandidate parameter. infinite for It turns series the in generalizationNambu-Poisson gauge a of theory physically a we refer meaningful matrix to w model. [ For a dis (This is a nice example of the power of generalized geometry. cut off the series expansion of the effective action. derive the results in a way that looks formally identical to t characters as manifold are given a ( ( of vector fields I, J, K, . . . indicated by all membrane backgrounds informalism. coordinates We adapted introduce an to appropriate this generalization spl of t pseudoinverse of the ( ( call this choice “background independent gauge”. However, Poisson tensor Π. The generalized geometry formalism devel both algebraic and geometric properties of JHEP08(2014)170 , . i p ab M X ...i h ...p = 0, 1 (3.2) (3.1) (3.4) (3.3) i 1 → δ a 0 h ≡ :Σ , p , and corre- X ab ...i ...p h 1 X 1 i ], the subgroup of δ . We define the ◦ ns of motion of the , . det 67 i ). In the rest of ...p ≡ 

, and y 1 1 ) al derivatives of ge, the first term in λ g ) − p 3.1 ij p J = 1) g . Using the equations ...i λ j p 1 i i − , momentum tensor into one T dX ǫ ) X = X 1 p b ). We reserve the symbol -index constitutes an even 2 ij ( − ∂ J ≡ p p g i = ( 00 j − on Σ and find the classically λ p h . dy X J , X ( p a ij h ...i = ∗ β g ∂ 1 αβ . Clearly, with these conventions ...l i ′ j ∂ ic string action to higher dimen- p g ∂X g i 1 i X ǫ ). As discussed in [ l ) T X s a symmetry of the gauge-fixed p-brane ǫ J X = p β = 3.2 ts can thus be consistently imposed at the i α ∂
dy i ∂ J X λ ( + det ( ♯ p X l 1) ij Π , one gets back to ( α ∂ dX ) such that g , ) and ( ∂ det ( , that is, p j P i − J S ··· p X q αβ -brane. In a similar manner as for the string 3.1 dy ( 0 1 . Using now the equations of motion for p h σ h i ∂ -brane action, simply measuring the volume of dX h  – 5 – i − p X +1 = h , in , . . . , σ 1 p 1 imply an energy-momentum tensor with vanishing compo- X l 0 γδ ij d √ 0 ∂ g g σ i,J ∂ 00 σ j h -form Z  γδ = p X +1 h ). Similarly, p σ -brane: to be +1 whenever the top I p i 1 if for the odd permutation, and 0 otherwise. In other β p T 1. Using reparametrization invariance, one can always d p ∂ and p +1 − i dy p αβ b g ∂X ( ...j ...i d 0 Z ≡ h . We use the convention X ∗ ] = . 1 1 ] ] h j ′ i 1 2 p α p p p , 2 δ iJ X j X Z i ∂ T [ 0 T , that is (Π) a 2 1 ♯ h ≡ NG = Π . . . δ ) = ] = i S 1 be a smooth map. We use the notation 1 ] = j i αβ [ [ i,J X . These constraints must be considered along with the equatio δ X ( )] [ X, h · M d 00 [ g ( ! T gf P ’s: P ∗ p = S S → αβ i X matrix of Π = and h and (Π) a n p p 0 dX :Σ = [ 0 can be chosen arbitrarily (but fixed), and iJ ) splits into two parts, one of them containing only the spati T × denotes the space-like components of the metric. In this gau are components of the positive definite target space metric ...j ...i ) and also transforms the pertinent components of the energy-
1 ), to ensure equivalence with the actions ( B 1 X = j i p n αβ ab ij 3.2 -tuple of scalar fields describing the δ 0 3.4 g g λ > 3.4 = h n a for spatial components of the T -brane: Let p i,J The gauge constraints on J ) 1 B g one gets the gauge fixed Polyakov action where words, is the where Thus, in this notation we have permutation of the bottom one, action, one can introduce an auxiliary Riemannian metric ( equivalent Polyakov action of the action ( and the spatial components of the metric The most straightforward generalizationsional of world the volumes relativist is the Nambu-Goto 3 Membrane actions generalized Kronecker delta where the diffeomorphism symmetries thataction remains ( after gaugeanother fixing (even i if theylevel are of not states. set equal to zero). The constrain the paper, we will choose the spondingly action ( (at least locally) choose coordinates ( of motion for for the nents ∂X where JHEP08(2014)170 ] M 68 m play (3.7) (3.8) (3.5) (3.9) (3.6) T (3.10) T p . Any is non- C g 1 − . TM is a positive e g p

) of Λ I e g C iJ E . C + J . g ! is not necessarily )-row vector ). Since p
j e g p n . p g ∂X iP . 3.5 k g ∂X i

] has the form iJ . Note that from the X + IJ X ork of Duff and Lu [ 0 C [ e g . e we define ! . . . g n J M J C i∂ 1 C ) j 2 S been discussed around that 1 1 an formalism and vice versa. Ψ g ∂X k g ∂X − C − i g I g 1 ] + ! p X T − ecial case ( IJ e g , which shows that 0 X g e g ...k [ g C ∂X as a fibrewise bilinear form on the 1 J − T I,J σ∂ k I gf P + + e g δ δ g C C S e g +1 g ij ∂X . − p ≡ I g 1 d j ) =

! p − ] = i j J † 1 ) g X J , one can define the basis ( g ∂X Z p X 0 X − ( T Ψ [ i ,E 0 ∂ π g + M i i I C ( − g ∂X tot P i∂ ij E ∈ − σ X S – 6 – g 0

)( j

. . . g ∂ ) = +1 m ) is the orthonormal basis for the quadratic form p 1 T m  X n . We can view j C d ( 0 σ ( ! e g Ψ = ∗ ∂ JI (1) i Z π e g +1 i X p iP . g X g ∂X 1 2 d = ) 0 Σ , . . . , e

∂ 1 π Z is a positive definite metric on ( Z e i  IJ σ TM ] = p e g σ p 1 2 g − d in the Hamiltonian and a similar expression X [ sgn +1 p can be any positive definite fibrewise metric on Λ p C Z ), we can add the following coupling term to the action: ] = ] = tot P 1 Σ d 1 2 e g ∈ S − X X X M π [ [ g Z ( − , where ( C T . Moreover, at any p gf 1 2 P i = S +1 e C S p ] = IJ ∧ Ω + TM ), i.e., ] = e g p e g . In this basis one has ∈ it follows that X ... [ 3.5 X,P g [ M C ∧ tot P ∈ tot P 1 S i e H m = I For any From now on, unless explicitly mentioned, we may assume that ) at E m ( definite fibrewise metric on Λ degenerate, we can pass fromThe the corresponding Lagrangian to Hamiltonian the has Hamiltoni the form The resulting gauge fixed Polyakov action g the role of “open membrane metrics” and first appeared in the w already in 1990. Hamilton densities for membranes have also This can be written in the compact matrix form by defining an ( as The action then has the block matrix form further discussions will, of course, be valid also for the sp Using this notation, one can write The second term can be rewritten in a more convenient form onc The expression vector bundle Λ From now on, assume that symmetry of of the form ( JHEP08(2014)170 . C (3.15) (3.16) (3.12) (3.11) (3.14) (3.13) denote definite A , . ! and a 2-form ) and consider a ! ! . In analogy with g 1 M β ( R C + Π − e g 1 1 +1 1 , one gets exactly the − p C − g ! ambu sigma models, let − ]. We thank D. Berman for 1 e and X ) ! G C − 70 Φ) 0 g Φ ) C = 1 1 ∈ 1 1 . T − − C T e g − − g 1 ds C ) ! G g nian metric 0 Π Π T − 1 ] for the string case. More recently, T T T C g − − C Φ 69 e g C − − ]. Again, in analogy with the case

e G + 1 C d, e.g., in [ + Φ + 0 1 1 cf. [ + e g . e g e + G 1 ( 1 = 1 and + Φ ( g − ! − ! ( p 1 g e 0 g e g G ! + 1)-vector Π ( − − T e 1 Φ G p g T − 1 can be written as a product of block lower C 0 − − g C T C g − T ) Φ G 1 − G T Π

– 7 – ). For − −

as to the closed background fields. Let C ! ) T − 1

1 T M ! = 1 ! − C ( 1 defined on sections as C e T g − p = 0 α Q − 1 ) A ) 1 0 X ) has been known, in this or a similar form, to experts for a long 1 C C −

T T T − e g and − Φ + ∈ 0 Π TM Π = 1 case, Φ ! 1 C 3.10 g

p , i.e., 1 g − ) = − e g p ( − Λ R B + e = T

G R , e G ), that is, T g ⊕ = C ( Q + G + g C C 1 1 3.9 1 + Φ and Φ as to the open backgrounds. More explicitly, we have the − M − + Φ − − ∗ e g , β

G A G T ! G

( Q ( e T Φ G ) and = + The block matrix in the Hamiltonian can be viewed as positive on Φ T 1 1 M α 2 ( ( − − Φ G G ]. 1 e G − A G Ω 67

∈ = = of the form B B α, β = 1 case, we shall refer to Before we proceed with our discussion of the corresponding N p We believe that the Hamiltonian ( 2 = 1, we will refer to such that the equality holds. This generalization was introduced and used in [ inverse of the generalized metric corresponding to a Rieman us introduce another parametrization of the background fiel triangular, diagonal and upper triangular matrices: for all matrix the matrix in the action ( the Note that, analogously to the Further, let us assume an arbitrary but fixed ( p time but we werethe not Hamiltonian able as to wellbringing trace as this it the paper in open to even our membrane older attention. metrics literature, appeare fibrewise metric time, see e.g. [ This matrix is always invertible, explicitly: JHEP08(2014)170 e (3.23) (3.25) (3.17) (3.18) (3.24) (3.19) (3.20) (3.21) (3.22) Π. Using ). ↔ − , we may prove . Note, that the c 3.25 o . In particular, A . G , . B ! ! ! = 1 1 . Φ, and Π C c ) , , 1 T , T 1 0 T G T − Π ↔ Π . C g C Π Φ) G 1 tive expressions of above in- em yields ( 1 or 1 1 0 e C )Π 1 0 1 G T − − − , 1 ! T e g e g f the matrix directly seen from the equality e matrices on the diagonal. Note omposability of a 2x2 block ma- − G G Φ C − r triangular, diagonal, and upper ]) are usually written simply as C ! T 1 B 5 − ) − ) + Π ↔ + Φ) such that the above relations are . C e g 0 1 1 T ≡ ! Π) + Π e g g , respectively. Hence the open-closed 1 + Φ , , e g e C o T o , G, ), the above open-closed relations give − T Φ e ) 0 + Π g G ΦΠ Φ G A G 1 ! + Φ T T G, 0 det ( e − 1 0 g G − − Π( 3.10 = C ↔ C − e g g G 1 1 1 c e + Φ and (1 (1 − g G T − + ( + − 0 1 1 o A e g 0 1 G Φ det det e g − − G ) equivalently in terms of the open backgrounds 1 − A C – 8 – e . This time we have + Φ + Φ ≡ G G e − g = 0 ( g T T + G e 3.9 G G G C ) = ! ) A g C 1 T Π) C ( = = = = 1 ! 1 T + T 1 1 − T ΦΠ Φ C C 1 0 g Φ can be found straightforwardly. Obviously, the inverse g − + 1)-vector Π. 1 1 1 ! C T e − 1 p − − − − − G 1 can be decomposed in two different ways, either C g − g g C − Φ c e g 1 0 T e T g ) there exist unique ( ! + A C C C C in the closed and open variables. For the former we we shall us = (1 = (1 e g − 0 1 1 1 1 + + e G g, C 1 Π 0 1 − −

, the open-closed relations (see [ e g g e g g

g g, det ( = = and = = c c e g A A A G and for the latter we introduce = 1 and c , again using the formula for the inverse of the block matrix p 1 G − B In terms of the corresponding Hamiltonian ( In the sequel it will be convenient to distinguish the respec For fixed Π, given ( and = c , Φ and the (so far auxiliary) ( troduced matrices A triangular block matrices, thethat triangular for ones having unit latter form is just equivalent totrix the with statement the about invertible the upper dec left block as a product of lowe the useful identity: just another factorization of the matrix G To show this, just note that relations can be expressed either way: or as Taking the determinant of both expressions and comparing th following set of open-closed relations: To conclude this section, note that taking the determinant o fulfilled, and vice versa.A The explicit expressions are most and the explicit expression for relations are obtained simply by interchanging these relations, we can write the action ( JHEP08(2014)170 . , c a N A M and Π (4.8) (4.6) (4.1) (4.2) (4.7) (4.5) (4.4) (4.3) , i = is not is not ). For η N o e N G 3.9 A e , G TM . N 1 ), G − → , or vice versa. ) 3.5 C N . M 1 ∗ Π ’s ( is a bivector. − and T N g ) p , e N g G C ,  1 g T N :Λ . onians. First, find the − lyakov action ( ] Ψ e † g η N T + Π change of coordinates on C 1 − N 1, Π losed ones, we fix Φ to be zero and + X,P, e [ Υ + Υ G e g ( 1 ( − N . p > ’s. C ]. − L is a fibrewise positive definite metric as functions of A Π 1 i N ! † − 71 1 N N N G X Υ g N − N Π e G 0 G 1 2 , e Π − G ∂ − N i — can be introduced  e = G 1 T N σ 3 = , and . This also means that it is not in general a σP 1 − N Π N p +1 − N G d . The corresponding (non-topological) Nambu p — using new background fields M − G M e g e – 9 – G d !

C 1 i Z N J 1 − expressed with help of Nambu background fields Z = 1, it is easy to show that Π e η = ). To see this, introduce new auxiliary fields iη − Π ) 1 , supposing that the open-closed relations A p 1 −

] = , , c A T − 3.9 1 1 − e η ) , A C − − A T T N 1 ) ) ] = C, e 1 η − Π C T N N e g − 1 N Π Π X,P, and ), that is g − [ C e is a skew-symmetrized tensor product of N N G e T g o + 1)-vector on e X, η, e g + 4.1 N G G C C [ A p ), one gets the correspondence between closed and Nambu sigm g N T N c NSM the matrix ( + + = 1 case, we may ask whether there is a Nambu sigma model H + Π e − g g p N NSM 3.14 1 + Π + Π S = = = − N A 1 1 )-row vector Υ = − N − N G N N N
( e p n is a Riemannian metric on e G G Π G G − N + = = ( = ( G . Using ( n e g g N C . It is important to note that in general, for can be either of Π , A N TM e p G Also note that even if Yet another parametrization of Again, it is instructive to pass to the corresponding Hamilt The converse relations are: Clearly, , Here, instead of fixing Π and finding open variables in terms of c 3 , which transform according to their index structure under a N J e Nambu-Poisson tensor. However; for η Define an ( sigma model then has the form: which we refer to as Nambu background fields necessarily induced by a ( the detailed treatment of Nambu sigma models see [ hold. Using the equations of motion for Υ, one gets back the Po canonical Hamiltonian to ( find, again using the open-closed relations, unique in general the skew-symmetrized tensor product of In analogy with the 4 Nambu sigma model where G We will denote as classically equivalent to the action ( on Λ background fields: JHEP08(2014)170 M ∗ . (5.2) (5.1) (4.9) T (4.11) (4.10) p Λ TM . p ⊕ Λ G . ⊕ = 1 case, one ! ). The matrix and vice versa, TM p additional labor N iP TM ng Π — in open g ∂X 3.10 , Π , = ]. One expects that ! N ! 66 P V e introduce the notation W G N , e N G . N e ! ∗ N G ), that is G 0 w in [ C Π ons of the matrix ! W 1 T e vector bundle as T N 3.10 ⊕ , 0 . C T N Π ] 1 0 ). Denote ! − W Π W N α Σ ) are usually written simply as . e

T N G → N 3.16 X,P N Π ! . In analogy with the [ = 4.5 ∗ e ! η N N 0 tot ! P 0 e W e )–( + Π G G + Π P V H T 1 1 N 4.3 1

− N − N 0 Π Π 0 G B ] = G G − coincides with ( – 10 – ) can naturally be explained using the language =

, one can uniquely find T ! C ! = X,P 1 case. In the previous section we have already men- [ ! using block matrices as 3.24 NSM N 1 P V + ! ∗ e g, C H g iP α Σ g ∂X , relations ( p > g, 1 Π 0 1 W NSM ! ] to the vector bundle g

). Here we discuss an another approach to a generalization

0 e g H σ Θ 66 → ), one obtains exactly the Hamiltonian ( = 0 g p = e g d 3.12

4.9 W G : Z = 1 2 B ! , ) to ( − P V G ). Next, define the map Θ : = 1 and

4.4 ] = p W G )–( Φ) as to augmented Poisson sigma model. Γ( X,P ∈ e 4.3 G, [ P G, + NSM V H = 1, the open-closed relations ( for it. This shows that to any Define the maps Note that for We will refer to the Poisson sigma model, when expressed — usi The main goal of this section is to show that we can without any p can be thus written as N for all since they both come from the respective unique decompositi If one plugs ( when using the NambuG background fields, in which case we shall expects that resulting Hamiltonian of generalized geometry. Wesimilar have observations developed apply also this for pointtioned of the vie possibility to define the generalized metric onof th the generalized geometry starting from equation ( G by the inverse of the matrix ( Indeed, we get Second, use the equations of motion to get rid of variables ( 5 Geometry of theFor open-closed brane relations adapt the whole formalism of [ JHEP08(2014)170 : . , t ∗ is ∗ Φ ·i , A W . G e W G, h· (5.3) ⊕ ! . f ⊕ ∗ G, Q W W we finally W + + Ψ ⊕ − , β ) V W ) on W Q , if the fibrewise M ∗ ( ! ). Note that this and 1 , corresponding to 1 ∞ W − . + M ∗ , it follows that , that they are both − . has zero intersection ( C ) + Σ( V ⊕ + } p ! T W P V + 1 Ω → t W W V ⊕ BBG ) using the fields ) ∈ 1 1 0 ∈ ∗ = 1. The purpose of this BG −B − W 1 5.1 Ψ P p W vector bundle ) + Ψ( , − ⊂ + ⊕ ! W 1 ( −G , V ± − 0 | α i W ) for G − BG as in ( V ) . G . can be reconstructed using the data ) 2 Γ(

∗ − 0 P G E ) + T ), and Σ V 3.24 T ( ) × + Θ W · it follows that V + ! ) , T ⊕ ( ) can be then written as simply as M · ∗ , ( ! + β P V → 1 p + V B 1 + Ξ V W )( X E = + + Σ , that is: 3.16 1 0 1 B ∗ W i ⊕ H = ∈ α V

≡ h + ). ∗ be a vector bundle endomorphism squaring to W – 11 –

= W Q = Ξ: . From the orthogonality of T p ∗ n + Ψ is a generalized metric on , W ) , = B 2 ±G ! β W P W and H T ⊕ 1 : Γ( T 1 + 1 + ), is a positive definite fibrewise metric on − + ,E ), ⊕ − ·i ∗ 1 W G W is orthogonal and symmetric with respect to the inner , ) + ( , n Q E M
W h· ( . From the positive definiteness of ( P W + Ψ) p n T 1 + B BBG ⊕ β + Ω 1 → + BG + − + V 1 ∈ ,W W ∗ ( n G − { Q Γ( W = + Σ can be written as a sum of a symmetric and a skew-symmetric par = + −G , and is thus a graph of a unique vector bundle isomorphism form the positive definite and negative definite subbundles o α, β ∈ ). Then define ⊕ ∗ A α ± ∗ G − BG A 2 − ), . It follows immediately from the properties of = 1. We say that : V W ,W +

V W W ·i 2 T ,E M , : ( 1 . The open-closed relations ( P T Γ( h· , orthogonal to each other, and thus + Σ E X and and +
T ∈ 1 of p α n . Moreover, it defines two eigenbundles e g, C ∈ + V ± ·i , or equivalently the fibrewise metric ( h W + V g, , + T . The map to get + Σ h· P ∗ n α V,W B + W We define an inner product Now, let V → ( and The map G Note that the above block matrix can be decomposed as a produc obtain that: product It follows from definition that respectively. From the positive definiteness of instead of eigenvalues bilinear form section is to obtain these relations from the geometry of the both with of rank Moreover, a positive definite fibrewise metric on defined for all We see that they have exactly the same form as ( for all with respect to identity, that is, W to be the natural pairing between pairing has the signature ( for all JHEP08(2014)170 G ), − e g and 3.9 (5.4) (5.5) T ) · , · has to be a G . , orthogonal with are vector bundle ∗ have to be positive ! ! e g 1 M W . We will show that closed relations. For D ∗ , that is 1 g, T ⊕ T − W , TO D g 1 0 i 1 1 W → 0 1 1 − → − , we can define a new map , e g ∗ O → + Ψ T ! ∗ ! TM , W = , p ence. Also, note that ( Q ! , β , i D W ′ ! 0 e g 2 1 : the fibrewise bilinear form . + + Σ ! :Λ corresponds to two unique fields T V − Q ⊕ T T D α g ,E V V Q . + Σ) 1 )) D T T 0 ) · W 1 , namely: ( are not arbitrary, since E α ! T : D + − h C,D e ! ! O O is again a generalized metric. Obviously, B e g 0 V , , Θ( − ( O e g = ′ ) D 1 T · e g i M e g, D ( T 1 Θ 0 1 − −h ) T C BC − = 0 is relevant for our purpose. 0 g g D 2 O

O g, D − = g E , and an orthogonal vector bundle isomorphism – 12 – D ). It can easily be seen that the vector bundle ( i =

= B = ( ! ′ can be seen from the equalities O = M ′ 1 , defined as ! = = ! , . One immediately gets that T ( ∗ + 1 are symmetric and skew-symmetric fibrewise bilinear T ) p e 1 D ) Q B 1 V + Ψ) − ! ! and · W − e Ω W B e g , E have to be positive definite. Inspecting the action ( g · β + + Σ V V = Q Q 1 0 ( G ( D T ⊕ , that is, ∈ , Ξ corresponding to T

α V B Θ( are related using ·i hO 1 0 1 D and H , D G , B Ψ

W 1

+ , e g h· ). Given a generalized metric Θ − V ∗ e → e g = = + Σ ∗ D W α ! h W − e g ), and ⊕ g ⊕ ), and Σ T M g D W ( D 2 M W ( Γ( Ω , respectively. The fields

: 1 can be parametrized as ∈ ∈ Ω Θ . It can be easily checked that e B TM 2 , B ∈ p G TO ,E 1 , or 2-tensor 1 is a symmetric covariant 2-tensor on − E α, β are related as D g . This means that to given 1 O ′ − B Now, let us turn our attention to the explanation of the open- T g = ) · T , there exists a unique pair ′ , · isomorphism for all open-closed relations are a special case of this correspond We have also proved that every generalized metric ( and Now, consider an arbitrary skew-symmetric morphism Θ : O the respective eigenbundles T for all forms on Λ respect to the inner product morphisms, definite. The conditions imposed on we see that only the case when D where The maps We see that there are two equivalent conditions on positive definite fibrewise metric on this, consider the vector bundle automorphism JHEP08(2014)170 . ·i , (5.8) (5.6) (5.7) h· . Note . B . ! . + } ! ∗ 1 G ) we get the . 1 Θ 0 1 0 − W Θ. Note that, e g ! 5.5 1 ∈ ! 0 − 1 . 1 Θ 0 1 g ↔ − } satisfies ∗

B 1 0 ′ + Σ) . ! − W = T 1 α Θ 1 ( 1 − . From ( ∈ G ′ − | − ! τ Θ ) G 1 -closed relations: so as · − , t us turn our attention to the 0 , Ξ and Θ − · 1 BBG G + Σ) . 1 − + Σ) BG α Θ 0 − Θ) G ( 1 α BG | − − B ↔ and ( 1 − B , , ! , τ − and Ξ can be found uniquely. Inverse −G ) ) B ! ! · (1 )Θ + , G − BG B 1 e + Σ) π H Π 1 0 1 C · , B − 1 + Σ) + ( α B + 1 T G − 1 − ) α ! g G Π π G ↔ H ! − ( + Ξ corresponding to B T 0
− C – 13 – = ( Θ H , Θ 1

Θ 1 1 0 1 0 1 − + ! − − − + Σ) + ( 1 G − BG e g

1 ). We also know how to handle this relation on the

G − BG − and any Θ, α Θ = − 0 e g T ( ) = ), is orthogonal with respect to the inner product = = (1 + Θ = ( 1 5.3 ∗ B C C B + Ξ) be the generalized metric corresponding to − 1 1 , ! 1 Ξ = ! ) 1 1 − − W + 1 G − H 1 B − T − ( e g 0 − − ). One has H H G g ⊕ Ξ 0 ( C + Ξ H 1 0 0 T H − BH 5.1 Ξ { W G C + ( Ξ . Let − g { Γ( = 1 ! Θ

), we obtain that Ξ − 1 ∈ H − = T − 1 + − e 0 H = = V − T 5.4 + Ξ H 1 Θ B V + Σ − − 1 0 −H e H − α BG H − in the form ( = + ′

! B T Q + + V can be expressed as + 0 1 1 Ξ G − BG G

T V + V We have proved that for given relations can be obtained by interchanging case of actually, the last equation follows from the first two. Now le Comparing both expressions, we get the explicit form of open Using the decomposition of the matrices, we can write this al Using the relation ( that level of the positiverelation definite fibrewise metrics ( Parametrize Θ as But this is precisely the relation ( Its inverse is simply for all We see that the vector bundle morphism JHEP08(2014)170 , M (5.9) , can to the W F to ]. ∗ . 66 W ! = 1 − ! 1 e g rom uctuation − ). C e π 0 e g ) C 4.5 C 1 ) gives exactly the open- 1 , ially uniquely fixed, there not be related simply by − )Π + )–( − g T ! g 5.8 -branes can end. T 0 e expects that it is also true T p N 4.3 is a skew-symmetric fibrewise C 0 ired in the discussion of DBI d and open membrane actions , 1 C )–( C N following paragraphs show that − ! − + e g T N 5.6 0 , e g

C is block diagonal, and Ξ is block 0 Π Π ( N , where T + p + − H 0 Φ Φ g is defined only on a some submanifold , and Θ

≥ ! ( + Θ − ′ e π Π p W = F 1 1

− N − T N g H Π is skew-symmetric fibrewise bilinear form on π ). T Θ Ξ = − e π = C to be defined globally on the entire manifold -brane, , , ′ π – 14 – 1 3.20 p ) − F − ! ! = 0. This means that we choose Θ to be of the ! T ) T e 0 N )–( G C C 0 B , and e π e 1 1 G )Π 0 G − − + = C g e g 3.17 N

TM 1 0 G T 0 ( C π G − ) is then = C g →

+ T 5.7 + H = 1, we have already used this observation in [ g ). The relations between the open membrane variables and = C ( M can be explained in a similar fashion. Indeed, note that the e g p ∗ N + T T N 3.20 p H e g C Π ( 1 , as transformation of the fibrewise metric )–( − − N . Parametrizing them as e g 0

∗ ), we obtain exactly the set of equations ( e F G C , 3.17 = W ), Π : Λ 5.9 N + G M g . At this point we consider ( 2

C is invertible, and its inverse, the vector bundle morphism f X is a fibrewise positive definite metric on B ). Defining ∈ N . Right-hand side of ( + π H 5.2 4 M G . The idea is the following: suppose that we would like to add a fl We also show that corresponding open backgrounds are essent ∗ Later, this submanifold will correspond to a 4 M T +1)-form p p Λ where map is no ambiguity at all. For 6 Gauge field be split into symmetric and skew-symmetric part: closed relations ( Nambu fields bilinear form on off-diagonal, we have to choose for their versions taking intoit account the is fluctuations. true The “upopen-closed to relations in an the isomorphism”, form fluctuated ( backgrounds can In this section, weactions. would In like the to previous sections developare we the have related shown equalities using how the requ the close generalized geometry point of view. On it is now straightforward to see that the set of equations ( and expanding ( We see that to obtain a generalized metric where where form ( ( although everything works also inof the case when JHEP08(2014)170 d has as . (6.3) (6.4) (6.5) (6.1) (6.2) , this F ) with M ) there ∗ ) can be TM F TM p T . 3.10 p p Λ + 6.1 ! Λ Λ ) ⊕ , which is an ⊕ ⊕ F F e g, C − M . + ′ ∗ M e , which we denote g, ∗ Π T TM C e T ≡ ( Φ), corresponding to of ! es ( TM − T ) p ′ e ) sigma model, described G, Π 1 0 1 , Λ e F F Π − G, ⊕ e e ! . ! 0 e g hisms of M F (Φ + . This pairing is defined as ∗ , Φ 1 , − in the Hamiltonian ( Ξ − is an orthogonal transformation T , e − 0 − M , defined as: Q c c g 1 0 1 W β 0 1 1 Π ! i i G G e α Q T , TM + ) ! ′ p ! is not orthogonal with respect to the ! 1 F Π Σ + e 0 Λ e R 0 G e G − ! ′ F V α T e i 1 i 1 ) ⊕ 1 F − ( − e − 0 − F 0 T = = 1 0 e ≡ G M G i Π i in the form 1 0 + ∗ + 1)-form in ? F =

R T p F o − C – 15 – ). It can be shown that any orthogonal in the open variables ( ! F ( ! e + Ξ = + describes a fluctuation of the background Φ. More G 1 c 1 − o − M ′ ). In this notation, the transformation ( e ( ! G T T G

is a ( , β F p Π ) ,W )) on F ′ α = 1 case, M e Q 1 0 Φ X Q ( − ≡ F F M p

− p e ( ∈ + + Σ 1 0 Ω ! 1 Π

iff = , is orthogonal with respect to the canonical pairing (value e α − F , for non-zero Π, is never orthogonal on V Π ∈ h p R h F c corresponds to replacing e (Φ + Π , M − M ), where X Ξ e ∗ ′ ∗ − G , Q = 0 1 1 F T F T

o p

p ′ + c Λ Λ G Π Φ+ e C ⊕ G corresponds to an endomorphism of ⊕ ) and ? e = ! ) and Σ G, as ! to 1 M . Dually to the previous discussion, F o TM o can be rewritten as F ; although ( M G, TM T C T 1 ( − ) G G G 1 0 Ω F X Π 0 1 1 e TM − ∈ p ∈

and ( )) on

Λ ′ = ( = M ⊕ Π ( α, β . Note that unlike in the V,W e 1 F c F M − G p Now, it is natural to ask whether to the gauged closed variabl Going from ∗ − , defined as e T c F for all in Ω endomorphism of precisely, we ask whether one can write where we can express to be identically 0. On the other hand, its transpose map, ( Translated into the language of the corresponding automorp correspond some open variablesby and some hence Π an augmented Nambu of boils down to the question for all canonical pairing (valued in written as G as augmented Nambu sigma model. If we define the automorphism The matrix We know that JHEP08(2014)170 ): is F ′ T F 6.6 . To (6.6) (6.7) (6.9) (6.8) Π sition − (6.10) (6.11) in the F . e − . . T 1 − Π ! ! ) reads ) 1 F T − 1 M,N 6.5 T Π − ) Π F 1 T = 1 − − Π ) N F F 1 0 T − = (1 Π and − M 1 1 (1 0 1 − . (1 r triangular matrix, it has to ! Π) onal term. Note that . ′ ′ ! T F F 1 ! , F Π and F ′ e , − 1 1 F Π are (necessarily) invertible vector − ′ . Hence, we assume that 1 − ! T − e ! ) ) 1 is invertible. The decomposition of Π F 1 Π 0 T F ′ T ! − , 0 T T T ′ Π F F 0 − T F, TM Π N N Π Π Π 1 p 1 F Π Π − T 1 − − F Λ = Π(1 − 0 0 ) ) − F M 1 − − Π T , examine the inverse of the equation ( 0 ′ T T −

Π − − M → (1 M ′ 1 Π F , and vector bundle morphisms ′ ? ,Π 1 T

F = − F Π 1 = (1 ) ′ – 16 – −

− ′ Π e ! F − − ,Π ) and ′ TM − = T F ′ p F e = F ,Π T Π Π T 0 1 F F F = Π − ,Π T :Λ = (1 Π = (1 1 ′ − − e − Π 1 Π − 1 e − F F − ) (1 − − − N Π e F 0 1 + Π 1 ) T e e as is lower triangular. For a matrix to have a decomposition F (1 F Π F Π) − T ′ 1 F e F is invertible iff 1 T T 1 Π F Π and − e 1 + Π F e =

F − matrix. We are now looking for a solution of the equation Π ′ T − − M ! e
∗ F 1 ! p Π n (1 1 T 1 = (1 − − F − ′ ) × = 0, which, of course, in general is not satisfied. The decompo . In general, this is not possible. Explicitly the equation ( ) have one of the following equivalent forms: ) ′ → F
F = = Π(1 T F p n F ′ ′ T 6.6 Π T Π M F Π F ∗ 1 0 − and T − ′ : (1 (1 F T reads M − Π Π We can decompose We can conclude that the fields 1 0 1 − − e

F bundle morphisms. where is an invertible From this we see that find an alternative description of into a product of ahave block an upper invertible triangular, bottom diagonal right and block, lowe that is 1 upper triangular, whereas decomposition ( We thus get that on the right-hand side therefore has to contain a block-diag This implies Π e for some Π The left hand side of this equation is which shows that 1 JHEP08(2014)170 := 1 (6.15) (6.16) (6.14) (6.21) (6.17) (6.18) (6.12) (6.19) (6.20) (6.13) − ¯ G s, we get , } Ψ . † ′ . . Π N. T 1 e
− − ) ′ ) ! Υ+Υ , F M F 0 1 1 , N T T − − ′
) ! ′ F c ′ 0 T M , ! 0 F (Φ + A } Π lowing notation:

† 1 Ψ ) T iP − F ′ ′ Υ g ∂X ce between closed and open . As in the previous sections, F o { F 0 Π G Π . . If we now put

F o − (Φ + σ A T hey are not related simply by = (1 + Π 1 − e gets † F c ) A N ! ′ − (Φ+ ′ 1 +1 )

Ψ G p e − − F G F = { F = (1 e d
) 1 T + ′ σ F c e g − F Z T + 1 ! ! F 1 . +1 − − A e 0 Π G p M ′ − C ) ( ! ! d = iP g 1 ∂X F ) T ′ } + (Φ + in the form − := T 0

, i.e., set Z ¯ Π ) iP ′ F Ψ e c g ∂X G G F o † σ ¯ 1 2 e F F ¯ + (Φ + F p G

A

G T d ) g − + F o = 1 + = G ′ ¯ Φ + – 17 – N ( F , one can write down the corresponding Polyakov } Z T G 1 C and (1 Υ+Υ F o ( − Ψ 1 2 1 T Π T T ) 1 (Φ+ − − F c ) = ′ N G ! − M − − ¯ F c

Φ F F ) A e 1 † = A ¯ iP F G g ∂X − + = † ] = ! 1 + Π and Ψ T T { T

Υ ¯ ) F c Φ + C M F c 0 Π σ { ( ( σ N = F A 1 σ p G − X,P − +1 − [ d T , + F p

g +1 0 F o d (1 ¯ T p Φ := T Z M C d ) = , F Π ( A F 1 2 Z

NSM 1 F Z , in particular comparing the respective bottom right block , F o ′ − − 2 1 − − H F c F c e Π e g GN + A e ) A G T = C F = 1 + = 1 ] = ] = ] = = ′ N + X F o N + ( [ M = C e g G X,P e ¯ [ G tot,F , P X, η, η + ( [ S g M tot,F P 1 H − F NSM Equivalently, one can gauge the matrix Thus, we have found a factorization of G S T Similarly, comparing the top left blocks of the inverses, on the important identity Comparing this to To express this matrix in open variables we introduce the fol 7 Seiberg-Witten map In the previous section,fields, we including have their developed the respective corresponden fluctuations. However, t M the (gauged) open-closed relations are equivalent to or (augmented) Nambu sigma models, i.e., using the matrices JHEP08(2014)170 1 ” is M F (7.3) (7.1) (7.2) . We p > M as a formal is again a ) seems to ′ ′ ) on , respectively A 6.16 ♯ ′ , whenever D ] coordinates on the η = 1 it is a Poisson he other hand, note p and Π + ) holds, and hence, for ♯ . A.7 . , we see that Π + 1)-vector. ξ, W and side of ( ) factor coming under the +1 Π p p 1 ( N mmutative” counterpart. + A.2 ( −  ) is related to the determinant 2 , i ions. We may therefore use any vector bundle isomorphisms V  [ i Π We will now show that for ) holds. The rest of the proof is F 1 7.1 ,F 7 closed, we can use the fact that 5 − − ,F Π e A. h  ). It can be seen that the Dorfman Π. Π F ) in which ( i 1, any Nambu-Poisson tensor and any h 1 n = − +1)-vector (see appendix 1 + 1 ) ,F D 6.11 p 1 + 1 T p p > = 1 case. We use a direct generalization of Π )] ) and lemma h p F η p ]. The most intriguing part will be to define − , . . . , x 1, and ). The resulting diffeomorphism will be called Π 7 7.1 − 1 1 + 1 + 1 + 1)-vector, whereas for x . For 1 −  – 18 – ≥ p p T W 6.16  + 1)-form. ( ) F p − p are graphs of the maps Π F T ) = Π = (1 1 − ′ T F ′  Π − F , e Π ). For this, one has to prove that ) is closed under the Dorfman bracket, which is according G Π = T ξ − ′ ), respectively, into the picture. The determinant of the ′ F Π − + Π . Now, using ( G and (Π V 6.13 iJ ( ǫ Π is invertible. In a more formal approach we also could treat Π T r F Π = (1 det (1 ), ( − = G T . Again, use those in which ( ≡ e = 1 case: for F ) seems to be a likely candidate to appear in the “commutative iJ Π M iJ p 6.12 F − -gauged tensor Π 6.16 is not necessarily a ( 1, one can see that iJ F ′ ). This is easily verified using ( , where F , its holds Π p > = Π is closed. F 1, ) satisfies [ A.1 , the Jacobian of which could cancel the det G i F F M − ,F A.1 p > e Π equivalent to the Nambu-Poisson fundamental identity. On t , defined by ( h = N To include the Next, see that the scalar function in front of Π in ( In the following, let Π be a Nambu-Poisson ( This observation suggests that we should look for a change of First, for We assume that 1 ′ A.1 5 + 1)-form Π p G (see lemma to closed. But this implies that To prove this identity, note thatlocal both coordinates sides are on scalar funct the semi-classical construction used first in [ Nambu-Poisson tensor. straightforward. of the vector bundle isomorphism 1 bracket ( that for Π with components Π a Seiberg-Witten map in analogy to the string carefully a substitute for a determinant of a Nambu-Poisson left-hand side of ( membrane DBI action, whereas thecontain determinant as on a the factor right-h a likely candidate to appear in its “nonco can examine the ( and open-closed relations. Instead, the discussion brings new power series in Π. where This can easily be checked in coordinates ( this tensor is always a Nambu-Poisson ( bivector if determinant from the right-hand side of ( manifold JHEP08(2014)170 . ) ) n A )). e x x 7.7 ( (7.7) (7.5) (7.4) (7.8) (7.6) gives: by an A t ρ ,... to be the ( A i | -form ) e . This is x ϕ p x 5 1 analogue Π( ≥ | ) = -form to Π, that is, p x for a p , we then have . t | )( J ϕ +1 | r. Then recall the p ( dA ∗ A ). To see this, note ...i ρ = 1 ) states precisely that A with respect to i ( ǫ ′ t ♯ t ≡ F ′ p on the 7.5 ) +1 +1 p . p x i Π, cf. Footnote e ( J sly, it preserves the singular 1 ). Indeed, the equation ( +1 b = Π = Π, maps Π ϕ − p ) ′ 0 ♯ t coordinates. We have c distribution) define ... 7.7 ally only in the directions of the ...i T 1 1 A 1 for the determinant of the matrix i i )=Π. This is the F ′ ) coordinates is thus a block upper e J | By definition of Π e x Π e J t (Π +1 | 6 dinates. This gives a good definition even if +1 . +1 p , . p ∗ A 1 1 . Equation ( | ′ t +1 +1 ) and ( ′ t p j in ( − i ′ t p p ) ρ . ...j | | Π Π . Contracting the resulting vector field x in some (arbitrarily) chosen local coordi- 1 Π A ) ) ♯ t (e.g., a matrix component, determinant, 7.1 covariant j T J ρ T i x x A ǫ k Π( F ...J ϕ J | F ′ t dy of b ′ t 1 = . Differentiation of Π Π( Π( 1 := (1 ) = Π ′ j | | i −L e ′ t | Π is defined as J = ′ t J e +1 J = = Π – 19 – − p ♯ t | (Π η ) being ′ t ′ t = Π ∗ t i A ...j ) = = = φ Π ′ 1 x Π 1 t t ( j | ′ t , which gives +1 ǫ ∂ ∂ ∗ A 1 J and p 1 Π | ρ . φ ♯ t − A A ,...j A  1 ≡ is closed, that is at least locally := L i j , together with condition Π ′ i ♯ t = b A F ) of the density F b ρ , A X (Π ξ = Π and Π b Π ∗ A the determinant of A.8 h ′ 0 ρ | J . One can choose, for instance, the special coordinates ( | 1 + 1 , with M i p k b ) holds. We will use the notation X will always denote the function defined as ∂x − ∂ gives exactly can easily be calculated using ( b ϕ 1 ) on i | A.7 = n  e . From now, for any function J k dy | ), and choose i )) ) coordinates, rewritten as x J ( e corresponding to x k A A.3 e x t ρ , . . . , x ∂ ( φ is, using similar arguments as above, a Nambu-Poisson tenso = 1, one can (around every regular point of the characteristi i 1 e x x ′ t p +1)-coordinates. The Jacobi matrix in which ( ∂ p Denote Further, denote by This equation can be rewritten as Further on, assume that For = M 6 i k e where the time-dependent vector field nates ( J property ( can be, in ( the flow obviously chosen so that Π on The Jacobian equality with of the well knownfoliation semiclassical defined Seiberg-Witten by map. Π. Obviou We emphasize the dependence of this ma To justify this, note thatfirst Seiberg-Witten ( map acts nontrivi We have thus found the map that Π Jacobian of the transformationΠ to is the degenerate. Darboux-Weinstein coor Define a 1-parametric family of tensors Π explicit addition of the subscript etc.), the symbol Recall now the definition ( JHEP08(2014)170 )). un- x ( (7.9) A (7.10) (7.11) (7.13) (7.12) A ompo- -brane ρ p )( ncommu- M . Finally, in 1 − 1)-form gauge , as expected. ) | ) ∗ A does not depend − ρ x t p ( ) = (det Π( dλ . | ). Since these are the x ivide both sides with | + / e | J )( ∗ A Darboux coordinates in +1 | ) p ρ nsor Π , x M e x | | rmation parameter. It is = ) ) b Π( oreover, the determinant of x x | +1 det . p ,... la to | | n — with a ( b Π( Π( 1 ) ) | | , ...i e x x x 1 ative”. i , ). Fortunately, the determinant of dλ ǫ =0 +1) b Π( Π( t p | | +1

( , +1 in Darboux-Weinstein coordinates directly p +1 = ′ 1 ) p p − i | ˆ − A.10 λ e F e J J ) (  i | ,...,i i k 1 ∂ T δA ′ i ) b J b t F ... F F , ∂ Λ 1 1 b i ∗ A Π b + 1)! d Π e ). ρ J + h 1)-form gauge parameter. This is the k − iJ ♯ t ( 7.8 = = A – 20 – − 1 + 1 1 L . As already mentioned, the Nambu-Poisson map p +1 = Π ( − p p A ′  b i − F =0 ∞ ,...,i k X δ b = det (1 F 1 1 ′ i , b  F b Π +1 h p Λ = = | , where i — induces a change in the flow, which is generated by the itself transforms as in ( J , and then use ( | ′ ∂ -potential | 1 +1 + 1 ) J p λ p evaluated in the covariant coordinates. Infinitesimally, c are calculated. We discuss this subtlety in the appendix ), we obtain the useful relation | p x ′ Λ | J ) d | 7.8 F − Π( x | iJ 1 Π(  | ) does not depend on the coordinates ( = 1 case, we define the (components of the) semiclassically no = Π p ] A.9 λ,A [ 7 in ( ) X transform as M 7.3 ˆ = 1, one can derive this relation by calculating F ). We see that ) and the definition of Π . We thus remain with the equation p +1 7.7 p A.9 Note that this expression does not depend on the choice of the The following observation is in order: the Nambu-Poisson te does: an infinitesimal gauge transformation For 7 ...j is then equal to the determinant of the top left block. We can d 1 A j e only coordinates changed by the Seiberg-Witten map, we get ( In other words, these determinants cancel out in the fractio the block der ( which the densities from ( or using ( is the semiclassically noncommutative ( triangular with identity matrix in the bottom right block. M Putting this back into ( J analogy with the analog of the exactstraightforwardly obtained Seiberg-Witten by application map of for the BCH the formu gauge transfo on the choice of the gauge which justifies the adjectives “semiclassically noncommut ρ i.e., the components of nents of tative field strength to be ǫ vector field transformation parameter JHEP08(2014)170 (8.1) ), we . The , -brane. ′ ] M 8.1 = 5 and p T assically ′ ) , using the ′ p . Our goal ′ F map. Hence, N p -branes ending and secondly, it p ≤ 8 + 1)-form defined (Φ + 1 p ). p an M5 brane. We − = 2 and ˜ G . ) and notice the factor p ) 6.19 ′ N t hand side of ( F only on the bosonic part n strings after we restrict the other agrangian density must be ], where the role of generalized etric. These considerations 7.10 ues is exact, for slowly varying old 71 olume of the larger iece of the action by a proper N such that some Nambu-Poisson tensor in ′ p + (Φ + Φ, calculated using the membrane ve dual descriptions similar to the work [ -brane with open ′ G . Hereafter, we denote by the same p e G, ) is the most natural choice to enter and -brane action ( M det[ p p G, is assumed to be a ( 8.1 · ] were assumed to be the closed membrane F T ) immediately imply Π . This makes sense since all of them are co- F . We shall now clarify the geometry underly- e to it. It would be tempting to take the square g, C to be a Nambu-Poisson submanifold of N 6.16 – 21 – N − g, ∗ A -background. The construction is based on two N ρ [1 2 +1 p C -brane . All the discussion related to Seiberg-Witten map in ′ p ), are assumed to be calculated entirely on N +1)-th root of ( p ] = det T 3.20 ) . Obviously, in order get a sensible action we have to form F the Nambu-Poisson tensor Π on )–( . Little subtlety comes with the Nambu-Poisson tensor Π. We F 1 + − M ) to construct the action. But, recall ( ) 3.17 C T ( 8.1 1 Π choose − ] appearing in it upon the application of the Seiberg-Witten F -brane submanifold as . The open membrane variables g ′ )˜ T p N − Π F I F + − -brane. This in fact requires ′ C = ( p [1 ′ + ( F +1) g p ( Denote the The open-closed membrane relations ( noncommutative DBI action − Actually, our exposition so far closely followed our previous to the 8 det[ on it while being submerged in a guiding principles: firstly, this effectivecommutative and action non-commutative should ones ha of the D-brane and ope is thus the construction of an effective action for a an integral density, which can be integrated over the world v our construction is valid for arbitrary values of And, in order to obtain a noncommutative action from the righ and having components only in the effective action thatan we integral look density, for. and therefore Aspower we we of need already the to said, determinant multiply the of that L the p pullback of the target spacegeometry m was emphasized. have to apply the Seiberg-Witten map det should feature expressions that also appear in the the previous section is assumed to take place on the submanif where pullbacks of closed variables. Finally, the field 8 Nambu gauge theory; equivalence of commutative and semicl ing the following discussion. Originally, of this action, we do not need to restrict ourselves to the val Here we considerwould a like system to of describefields, this multiple to system open by all M2 an orders effective branes in action ending the that on coupling constant. Since we focus root of the identity ( open-closed relations ( not the square root but the 2( backgrounds in the ambient background manifold variant tensor fields on latter option is to characters their pullbacks to the have basically two options. First,M we would like to restrict backgrounds to JHEP08(2014)170 . e g ack- (8.3) (8.5) (8.4) (8.6) (8.7) (8.8) (8.2) -brane , ′ ,  p must be  ), which T T , but it is ) , . . ) ′ | m ′ 8.2    . b Π F G ) F | T is the Nambu T ) ! ) ′ F ) b b F Φ+ F F F ( + and 1 + + (Φ + s over the e g + C − 1 ( m C ˜ and C − b G 1 ( g C ( ) ˜ 9 1 ( − 1 G ′ ) − g b − ′ F T eneralizations. The first g T g ) r the action ( )˜ F )˜ ) ) followed by the change of F F b F F Φ+ g + 8.3 + + + , in the case of factorizable +( . e Seiberg-Witten map. g C b C C C n of a tensor. We just evaluate the G ( ) ( (  + (Φ + 1 ′ 1 1 +1) − p − p p + ( vanishes for constant − − p G +1)

1 2( (  e g g | g p  )  2( Π T g ) | +1) +1) ) 1 1 p p 1 + 1 + +1) 1 F p   2( 2( det det = det · 2( 3.25 + and b g +1) +1) 1 1 G p p | det det G/ C · · 2( 2( det ( Π c +1) | ] 1 p · – 22 – g p T − ) ) in all other cases. ) 2( (det 1 g g det det ′ F )˜ ( − · · p m +1) p 8.5 Π p ) ) F g ( det g g +1) 2( -brane. Asking for − p ′ − p ( ( p + = p 2 2 1 1 1 1 2( +1 +1 [1 p p C m | | det ) is exactly equal to its “noncommutative” dual G 1 Π Π det det +1) det c | | + ( p m ( 1 8.2 m m g g m 1 1 1 m in this expression are the pull-backs of the corresponding b g g  g 1 x b G ). This follows from integrating of ( x x x C +1) +1 1 ) det x ′ p +1 +1 p +1 ′ ′ G ′ 2( d p p 7.12 +1 ( p ′ ) that the closed and open coupling constants d d d p Z , and denotes objects evaluated at covariant coordinates d +1) det p 8.1 g p Z Z Z · b − , ˜ 2( Z g − − − g = − = = = +1) det p p = is a “closed membrane” coupling constant. The integration i 2( m -DBI 1 p m -DBI -DBI G -DBI p p g S p det S S S = The factor involving the quotient of As desired, the action ( Let us give two alternative, but equivalent, expressions fo For the second one, let us note that det˜ Let us emphasize that this is not a coordinate transformatio -NCDBI m 1 9 p g S essential for the gauge invariance of ( it follows from ( where as before (NC) field strength ( one is obvious: integration variables on its right hand side according to th might turn out to be useful when looking for supersymmetric g component functions in different coordinates. related as A very similar expression can be found using ( fix the action essentially uniquely and we postulate where Hence, in this case: and the fields ground target space fields to this JHEP08(2014)170 - ty κ N tiv- (8.9) (8.10) -brane ′ p = 5 we have in the action ], for an early ′ ane action in p k 72 . to the , f noncommutativity ! 2 ) ) C .... k F = 2 and = 2 case relevant for the + e g linear self-duality and + p (tr p ate that our proposal can 2 he context of the M5 brane ynomial in s explained in detail above, ) 1 C hoose the bivector Π to be 18 ( k ]. However, the type of non- ty) give rise to the same action + T 74 (tr l in the action. At the same time ) -known deformation of the com- 2 sson tensor that already appeared F k 1 -symmetry and the requirement of 18 tr κ g + = 1, we get indeed the DBI D-brane + 1 6 C ] and [ 2 p g consistently decoupled from the back- ). Indeed,for ( − k 73 − k tr 10 8.2 can be interpreted as a low-energy (second

1 6 tr ′ 1 3 S − +1) 1 p k 2( – 23 – 1 + tr , we get ′ 1 3 r p -symmetry gives its explicit form, which can be ob- det g 1 and in particular for the κ = m 1 + 1 g -DBI action ( p det is the modified field strength. (See also [ p r p > x ], it is shown that consistency of the non-linear self-duali p jkl +1 x ) 39 p -symmetry and the form of the polynomial are determined. 6 ) = , d C d κ k 38 + Z Z and ]. Their action is, up to conventions, − − dA 6 1 (1 + ( 1 6 39 = = = ikl , ′ ) M det 1, we do not deform the commutative product — our “noncommuta will be justified later. S 38 C S +1) 1 M p + S p > 2( , dA x 6 ) approximation of our d = ( k = 1 string theory case. This is very different from the notion o i j = = 1, assuming that the pullback of the background 2-form k p x p To our surprise, we found that this action Finally, let us mention that noncommutative structures in t Let us note that in the case of a D-brane, i.e., Now we can compare our action, e.g, to the DBI part of the M5-br The notation +1 ′ 10 p order in M5 brane: for that we argue to be pertinent for d have previously been discussed, for example, in [ non-linear self-duality, the self-duality relations bein ground. More precisely, in [ has been determined by lengthy computation based on action. In the other extreme case, the projector specifying the tained without a priori specifying the form of the polynomia in the commutativity discussed in these earliermutative papers point-wise multiplication is along the a well (constant) Poi is restrictive enough that demanding proposal with a similar index structure.) The form of the pol indeed be extended to a full supersymmetric action. symmetry, the other on commutative/non-commutative duali in the low energy limit is very encouraging and seems to indic The fact that two very different approaches (one based on non- where 9 Background independent gauge For ity” has rather to becf. understood the in remark the at Nambu-Poisson the sense end a of the previous section. equation (2.9) of [ is non-degenerate and closed (that is symplectic), one can c JHEP08(2014)170 . , ). ∗ V V C 2 W (9.5) (9.4) (9.2) (9.3) (9.1) is an → Λ 2 Λ g ∈ V rt to get . : . See that B B ∈ P , the metric C structure W by V is a finite-dimensional of appendix = 1. is a linear map, lace V p ∗ a 2-form. Let B.1 V ∗ . for ) V , C. 2 C → C C, Λ ! − ]. Our aim is to generalize this + . The key is to keep in mind the − V 0 5 C ∈ p g . C. V ( . 1 case. Our goal is to find a suitable C T p 1 − 1 , Φ = C :Λ ) 0 − C ≥ − + Π P − p C C, Cg )Π) Φ =

1 , such that Π = Π to be an orthogonal projector with respect , and C 1 g − + Φ − = ′ C, P V + Φ is thus invertible. Note that we are now + G ), not assuming a non-degenerate 1 B Cg = 1 have the form g − ) = ker( , and a bilinear skew-symmetric form G gP – 24 – P,P , ( . Then there exists a unique bivector Π = p − 9.1 T ∗ C V ′ ′ Cg = − p ! V C P , not discussing any global properties of Π yet. 0 e g Λ − gP C 1 ker( + V ⊕ 0 g → T = Π ′ g

V P . Assume that V G : = ≡ C + Φ = = . If we define + Φ = (1 − G B C G W G is an inner product on Λ G by e g on ). We see that by defining C G 5.3 1, even giving milder assumptions on be an inner product on , and ≥ g V . From this we can read of the symmetric and skew-symmetric pa (that is a Poisson bivector corresponding to the symplectic p P C − ), one gets = 1 is viewed as a map ′ 9.2 C , and the 2-form P G We would like to generalize this procedure to Recall that open-closed relations for The situation is thus analogous to the previous one, if we rep Let us start on the level of linear algebra first. Assume that is again a positive definite metric, and by g We can view this as a generalization of ( construction for where G we get an inner product satisfying the inverse of The reader can find the proof of this statement in proposition where This equality can be rewritten as denote a projector orthogonal with respect to on the level of a single vector space choice for Π, such that Φ = vector space. Let open-closed relations ( Solving the open-closed relations then gives Using ( This is known as the background independent gauge [ inner product on JHEP08(2014)170 (9.6) (9.8) (9.9) (9.7) is an V is block- p ly proved Λ H → ). The equality V p 9.6 :Λ Θ = Θ gives . This proves that in P e P ) e g ( , as intended. However, ) imply that ⊥ to see that there exists a C ) 9.7 − C , and , , , g , B.1 V. ! ! ! p e (ker 0 0 0 Λ C P −B e ⊕ T . ⊕ T P. 0 P ) T 0 0 Π C Π − . g

V ( Ξ = ). The equality , C − − ⊥ T , = , = ⊆ ) !

) for this choice of Θ, using the same = C T B 0 ! C ! C 1 = = e 0 P C T P 5.3 Θ = Θ 0 − T C T Π ! ! P 0 Π 0 Π ker P T , ). One gets e 0 0 P −

0 C ⊕ P – 25 – P, ) = ker( − BG T

0 9.5 P − T = (ker = C,C ′ − P 0 Π Π = C ⊥ . This and the relation ( and Ξ are, leads to Φ = P = − = B ) then proves that Θ is block-off-diagonal, that is e g GP B ! ! Θ = e B T Θ T P 0 0 ′ C 9.6 ! C C P = ker T T e 0 P Π , we get 0 Π 0 Π C = B B 0 . Clearly, one has ) = ker P − − ), and ker( and Θ obtained by this procedure are of the suitable form, H P = P

C ker H ), we may again apply proposition B BP . We can now simply extract all the relations from ( ) = ker( is block-diagonal and Θ block-off-diagonal. This can be easi V is an orthogonal projector with respect to e . Exploring what P , such that H → ) = ker( V W ∗ P − P 2 → V Λ p V ∈ = 1 : gives ′ P P P with ker( = G B by examining the projector a block form, we have that is whether Therefore we have that Im( we do not know whether which translates into Rewriting the equation where diagonal. The second equality in ( Also see that ker( orthogonal projector with respect to Θ unique Θ calculation as we did in order to obtain ( where Π : Λ which translates into to Now we can solve the open-closed relations ( where JHEP08(2014)170 ) ∧ ). ≡ T ) 9.8 T C ... , and (9.12) (9.11) (9.13) (9.10) in the C . Note ∧ N , where ∗ and the e g 1 ∈ e N V e P 6= 0 on the x +1 ∈ ) p . It is thus a x ∗ ). x ) of ker( Λ ( and . We see that p , such that ( D , the map Π is a egration in DBI thus means that ∈ C − ), forming a local D C. ′ e g V P 9.11 + 1, and p +1 − p C C +1 oints p ). p to ensure that Π is a → Λ hat ) of 6= 0 for all ∗ neral ∈ C 3.5 ) +1 Φ = V , . . . , f ), and since such a Π is p ) = x p jectors 1 C ( dd the second assumption: or bundle morphism ) gives ( C induced (at a chosen point f , . . . , e m ( 1 C, C orthogonal (with respect to 9.10 e 1 V 5.3 t + 1)-form − . . + 1. T , . . . , e g p 1 → C D p C T e 6= 0. If not, we cannot go to the . +1. Since Π always has the same ∗ , +1 C ≥ . p ) p V +1 ) and ( T ) x p p +1 + Λ ( e p C ′ e 9.8 C ∈ e ∧ P = Π ∧ e g , we have that T T ′ ... C Π ... e P ∧ e P ∧ 1 = , where e , 1 ′ – 26 – e · e is induced by a ( G · N at every point, defining a smooth subbundle (it is . Assumption on the rank of 1 λ , λ C ⊥ ) − D T = T Π = Π C is spanned by orthonormal basis of the form 1 C P C − Π = D e g p C ) to find + = ker( + 1. ′ 9.7 , there exists a unique linear map Π : Λ p ∗ D gP V T ) = ′ + 1)-vector, more precisely Π → C P . Choose now an orthonormal basis ( p D V = denote the non-degenerate subspace of p to an open submanifold G V . This allows us to write Π explicitly as :Λ + 1. From the skew-symmetry of 1, first observe that the linear map Π : Λ N +1 ⊆ C p p , we get the first assumption on the linear map e ) hold. Plugging this Π into open-closed relations ( ), Π is a ( D 6= 0. Now, choosing an arbitrary complementary basis ( p > C ∧ ) = λ 3.5 ) by a Nambu-Poisson tensor has rank either 0 or 9.10 D ... Let Now assume that the linear map There will always arise problems with the smoothness of Π at p For In this case we find that Λ To use this for our purposes, we have to impose conditions on We now turn our attention to global properties. If we assume t ) = 0. If this set has measure zero, we can change the area of int , one can find counterexamples to the assumption that, for a ge M x ∧ +1)-vector (although it has a correct rank). We thus have to a ⊥ has to be of the special skew-symmetrized tensor product for ( -brane, we can define the subspace ) to its kernel, that is ′ r p ˆ e g ( D where top-level form on g therefore that rank( dim( that in this case, we always have the estimate rank( background-independent gauge. Let us hereafter assume tha action from rank as C on linear map form ( We have thus shown that, corresponding to the orthogonal pro Nambu-Poisson tensor. unique, this is the one. We can thus conclude that for rank( and ( It is easy to show that such a Π indeed satisfies ( Finally, we may examine ( and thus Around any point, we can choose a local orthonormal frame ( e an orthogonal complement to the kernel of constant rank vect p JHEP08(2014)170 . D D D for the that B + 1)-vector + 1)-vector . Note that + 1)-form of p p N p + 1, satisfying p + 1)-form of rank ) we get can be any nonzero p mbu-Poisson tensor. . In particular, the C 9.12 . dC )-form on + 1 of these coordinates. ) is a positively oriented p N is a ( p p = 0. Interestingly, both . mbu-Poisson ( − − p ′ , ′ point in “non-commutative” ompositions C p − ⊥ p ′ = 0. rst dC p under vector field commutator: e D ) D e forms, see appendix f . Using ( is D D . ⊕ ∧ , . . . f C f p . But forms with integrable kernel Γ( ⊥ + 1)-form of rank 1 | e − to define an integrable distribution we denote the codifferential defined D p ′ ) is a local orthonormal frame for D N ... , f p ∗ . This distribution has to be regular, δ dC C f = +1 ∧ T +1 N p 1 p ∧ 1 is already a non-vanishing ( is a ( − ) proves that Π is a smooth ( f p TM C − ... C ∧ p ′ p ∧ , . . . , e Λ 9.13 , . . . , e Λ 1 +1 1 1 , p e – 27 – e f e ⊥ → · to define a Poisson tensor Π is the integrability of = 0 are the non-homogeneous charge free Maxwell ∧ D . Since λ ⊥ C ⊕ ... = TN δC D = 0. : is integrable. By D , and a condition ∧ C . Also, note that in the whole discussion, we do not need T 1 ∗ D C δC ) = e C ∗ , such that ( C . This adds another condition on = ∗ is then by definition n N g TM = 0, g ,( , we see that the sufficient and necessary condition is that the . The expression ( Ω T is a smooth function. ) D dC is integrable, around each point there are coordinates such A.3 C 1 λ ∗ = 0, then D is an integrable everywhere non-vanishing ( δC . We say that tangent vectors contained in = ker( C D defines an integrable distribution in ∗ p D . D = 1, the discussion is very similar, except that the rank of D = Λ -brane, since are integrable regular distributions if p ′ p e ⊆ +1, the sufficient condition for integrability of D ⊥ ] p D One can find a simple equivalent criterion for Finally, we have to decide under which conditions Π forms a Na There exists a nice sufficient integrability condition: if For . In order to do so, assume now that ( + 1, such that D,D distribution have their own name, they are called integrabl We see that [ and thus, this condition is equivalent to the involutivity o definition and basic properties. We can conclude that Π is a Na subbundle D In the view of lemma on the basis for the sections of if and only if orthonormal local frame for the Hodge star is defined with respect to the induced metric on The metric volume form Ω Having a volume form, one can form the Hodge dual of rank p the Maxwell equations 10 Non-commutative directions, doubleBy scaling the limit construction of the preceding section, we have the dec and using the Hodge duality. Noteequations for that the field strength where the integrability of the distribution even integer not exceeding necessary and sufficient condition on is spanned by coordinate tangent vectors corresponding to fi the regular smooth distribution directions. Because JHEP08(2014)170 g N ] as and 5 g (10.1) . and Π 1 −
N C G 1 . Scaling the − g into description, , respectively. In ′ Cg . The right-hand ⊥ 2 α ij e ) is explicitly . Also, note that in D ′ • . . form corresponding to N Π πα 1 = Π ! and ! (2 − I ◦ ,Π 0 } ) eiberg and Witten in [ e F F j D N T − 1 . This gives non-vanishing • − • G g F , x II D ) tative” coordinates. This ter- i • F • F , x ubspaces, the matrices of . Π C . . { Π ng of the metric

1 ! 1 ! T • − ◦ − ! 0 1 0 = 0 in a way such that − e g 0 0 g is a positive definite fibrewise metric F ) • ′ • ◦ 1 0 0 0 • = 0 − Π g → g e ,F

• Π = (1 , T πα ′ •

(1 will not be a metric. The correct answer T C I ′ • α F • D = (2 ! T • F = 0 N Π − e − Π P − e g correspond to • G

1 0 0 , , = , Π j = 1, the background independent gauge could = 1, we have

and Nambu fields 1 = x – 28 –

N ! ! p 1 − p ◦ C 0 Π g − ′ , looks like in this block form. We have = 1 0 0 0 Π = • g and ′ 0 ). Reintroducing the Regge slope Π) Π = 1 T g

T πα • i T , 2 F

C x = F • , 4.11 ! − = Π 0 P T − . We also have Π 1 g 1 • C 0 0 1 − C ) − Π(1 •

Π ≡ are simply given as Cg = T ′ 2 • e ) P ′ are positive definite fibrewise metrics on Π F C ◦ -gauged tensor Π − πα e g F and (2 (1 • P − and g • e g = Π = • ′ is a positive definite fibrewise metric on N • and G g ⊥ Having this in hand, recall that for We can thus write all involved quantities in the block matrix D will be block diagonal: e be obtained in a completely different way. It was obtained by S Hence, the defining equations of Π can be written as this formalism the same fashion we obtain Examine how the Hence Denote Π the relation between closed variables on a following limit of the relation ( this decomposition. From the orthogonality of respective s g minology comes from the fact that for These local coordinates are accordingly called “non-commu side is non-vanishing when both where Now one would like to do the zero slope limit quantum-mechanical commutator of these coordinates. remain finite. This clearly requires the simultaneous scali as a whole will not work, since the resulting JHEP08(2014)170 1 2 g ’s is ǫ • ers ◦ g ) by e g (10.2) 3.2 B ∝ finite (we • , we have to N g 1 and G of the metric We immediately ∝ ◦ g ◦ 11 f the vector bundle . G . This suggests that 1 , • , − ◦ contains as many ling just ǫ g ) g rections differently was • IJ ∝ C = . 1 case. ) , 1 s is e Polyakov action ( 2 1 • ◦ ◦ ◦ − • ǫ e g g G und independent gauge. We N g e kept in the expansion of the ontext of the M2/M5 system that pansion of the DBI action. It p > • ∝ = C • in order to keep ◦ 2 . ) N ′ 2 ǫ p . ǫ ,C T • 1 C. πα ,G 1 C ∝ T 1 ∝ (2 • − ) is. This means that every component • ers of • − • g ◦ C J and commutative part − C e g 1 e g T • • • • − • , C g g C g (or 1 ( • also split accordingly as • + -fold tensor product of 0 gives in this limit + I C ∝ p • C 2 • N – 29 – 1 e g ◦ ) g 7→ ′ − • ǫ = g . The biggest issue comes with the fact that = ,G ) πα ′ ǫ, g e g • T • • as N (2 and Π N C ∝ πα 1 2 e 1 G 0. This is clearly not very plausible. However, this can G − • ǫ N − and • (2 is exactly the background independent gauge. This double e g • g . G ∝ − g g • → 1 G ′ ǫ, − C • ǫ = = α − by C ∝ • • ’s only. In fact, every component ( 1, • ◦ N N N = = ]. It would be interesting to compare the scaling in these pap g g 0 as Π G G • • ∝ . This would imply that ′ ). But the second relation is 76 p N N ◦ → ǫ g C again. We immediately see that scaling 1 Π G πα , 2 N ǫ ∝ 0 the background independent gauge. This corresponds to e G • into ∝ ] and [ TM e g → p • ǫ by Π and ǫ 75 is again a skew-symmetrized g Λ N • ⊕ e g , then ] for a previous discussion of the double scaling limit in the c ǫ independently! The correct answer is given by the geometry o 9 TM e g should scale differently. We must thus abandon the idea of sca ∝ ◦ See [ = • e g g 11 of if as the number of “commutative” indices in Note that not a tensor product of W still be fixed by using the remaining gauge fixing freedom of th scale This shows that have included Let us note thatdescribed in in the [ case of an M5 brane a scaling treating di Now, scaling came to different conclusions regarding the appropriate pow gives in limit scaling also the ratio between see that first naive answer would be wrong. One of the relation is given by scaling the non-commutative part Replacing Π scaling limit was then used toDBI determine action. which terms We should would b like to find an analogue of this in our differently. The resulting maps 11 Matrix model Now we will apply thewill use previous the generalization double of scaling the limit backgro to cut off the power series ex with the one introduced here. JHEP08(2014)170 . → T . ! ◦ ⊥ T T F ◦ ◦ 1 T (11.1) ◦ F F D − ◦ 1 1 F e g ⊕ 1 − − ◦ ◦ I e g e g − . ◦ son tensor F D I I e g lock of the : T ◦ ! F F ′ • 1 . . It will be of T T F ◦ ] ′ • Π 1 − X • F 1 T F Π g − ◦ ) ) 1 − 1 • g T − ◦ ◦ F + − Y • ) by multiplying it e g F + . The term of order ◦ T Y II + II 1 F • T F F F II − , F ◦ e g 1 C 1 1 e g + ′ • F ( ‘ − ◦ ! − ◦ 1 − • I 1 Π g -DBI action has the form 0 is g e C g − • − F . p T ( ) I e g ) e ctor bundle endomorphism g • 1 + T T ⊥ + ′ • ) F ) e gauge field II − II F 1 T • T D II Π F ◦ e g F Y F − ◦ • ) F 1 F + 1 F 1 (semi-classical) analogue of a e g e g ⊕ 1 + 1 − ◦ − F • ◦ • + − • g − 0 e 0 1 − F . In the block form C D Y C e g • e g + 1 ( I ( 1 analogue of the semiclassical pure T p > ◦ T 1 II C II 1 ) − Y ◦ ( → C F 1 + − • g F − F . , 1 F 1 ( g 1 − g ⊥ − 1 • p > − ◦ − ! − ! ◦ + ◦ + ( e g g T − I ◦ • I ◦ I D g Y T e g II g C Y F Y F

F + ( [1 + ⊕ F 1 1 T • 1 • Y I II II T − ◦ II e − Y − D • F e g Y +1) F F 1 e g F ! • I p e g : 1 ) det ( ) 1

e g 1 – 30 –

) 1 + • 2( F − , and at most quadratic in ◦ ′ • − • • F − 1 Y ) 0 Y a e g • e g e g = = • ◦ − • ) b Y ′ • + )Π X g F 1 det F • Y F T Π 1 · I e g C − + • ( ′ • + T − ◦ I ) F 1 gives a possible Y T g 1 g • 0 1 Π F ) − ( − 1 ◦ • T C − F I g 1 2 ) = det ( + e g ( − ◦ F I F 1 e g Y T (1 1 + F ◦ − ) • − ◦ det • • II g F e g + F F C = 1+ ◦

m det ( 1 ( 1 • − g F 1 g + − ◦ = 1 X ( II − • g • − − ◦ T e g F ′ • C ) g X ( = ( • 1 satisfies all the conditions required for Π to be a Nambu-Pois 1 + F − ◦ − • C g + e g = 1 + ), we have that Lagrangian of the commutative DBI 1 + Π , where • II − II

p C ’s and constant in is an invertible matrix. This is true because it is a top left b F 8.6 ( Y a 1 L 1 1 • = TM b − ◦ in block form as X − • − Y • coming from positive definite matrix g + 1) in the matrix variables g Y Y → Y , we have p I Y ⊥ Y 1+ . From ( D

− Assume that TM + 1) in N ◦ : = ⊕ p Y X Here we have used the following notations for the blocks of 2( on This can be decomposed as a product The second matrix has the form by invertible block-diagonal matrices. Hence, we can write order 2( D where the vector bundle endomorphism matrix model with higher brackets and an interacting with th turns out that we find an action describing a natural matrix model. Note that the secondX determinant is the determinant of the ve note that matrix Writing JHEP08(2014)170 . . ). ǫ ) ± . • 2 ) ) ǫ g ∝ 2 ( ( T II ǫ p o (11.2) ( F 1) o + ′ • −  + Π
• •  , such that g . T Y T )
T . The sign .
N ′ • 2 . ) = det ◦ ) ◦ ) ǫ • 2 D , it is clear that F Π ( g ′ F ǫ Y . o ( ( II + ( ) ) + is also integrable, o I 1 + F ) on b I Y p 1 − F  ( +1) ⊥ + 1 F p − ◦ +1 p T ( T ′ ′ • T
g D ′ • 2(
p umptions, the integral . • +1) 1 Π ◦ p e g Π tr( . Note that ( F ′ • II 2( on II 1 +1) F ǫ + p F ) det ( )Π I we get the noncommutative 1 uced above. Namely, in the g , . . . , x − T det ) = det 1 +1) I F ( | 1 ) − ◦ • ) ) p . − 1 2 ◦ F T T e g +2 e g • ′ II ) • x 1 e g p
◦ . All quantities with indices in 2( b g
F nding to Π and Π Π − ◦ ◦ ( g ◦ ′ • ⊥ e g Π( , x ( det 2 1 II F + I F Π 2 1 )
) D · | F . Therefore, we have ) • + F +1 • • b g 1 + ) g p I T ( e g g ǫ 1 I det T + ( induced by the basis of ′ • T 2 1 F det ′ • − ◦ F F 1 2 • · T F e g 1 ′ • +1 T Π Π g )Π ′ • D
p ) Π ( p Π ◦ | det • II T I Π ) span − ) T det g , . . . , x F ′ • F II − F ( x 1 II 1 +1 1 F Π 1 2 + (1 ′ x 1 +1 F − ◦ ∂ p − ◦ I p Π( g 1 | | e g 1 – 31 – +1 F is of order ) ∂x I tr p − ◦ + T det(1 m x g • ′ tr( F 1 1 T II b  g − 0 Π ± + F Π( x det g tr | • ) = det ′ • II + 1) 1 ,..., g g +1 F Π + 1) ( ( 1 p m ′ 1 • ) = +2 p 2 g 1 T p +1) • 1 ′ • g p ∂ 1 2( d x ). Using this, we can write p − F T ◦ Π 2( ′ • 2 ∂x g ǫ Z det 2( +1  Π + ( ′ p o tr • tr II ∓ + ), we have d F C =  ) = 1 + Z )=1 + 1 • +1) , and ( 1) + 1 11.1 +1) 1 − ◦ Y II p ∓ det( p D ( g − Y 2( 1 2( = • − +1) 1 Y -NCDBI • p + ( p Y 2( =1+ I S -DBI − Y )=1+ p II ) span S Y − Y det 1 +1 ( ), we wish to keep only the terms scaling at most as ◦ p = 1 ∂ − Y • Y 1 ( ∂x ( +1) Y 1 p − I • 2( Y +1) 1 Y +1) 1 p p ,... In the last part of the discussion assume that the distributi are now assumed to be in this coordinate basis. Under this ass − 2( 2( 1 ◦ det ∂ ⊥ Y ∂x det depends on the ordering of that basis. Next, see that det( Here we assume that one chooses the basis of Λ For the first factor in ( DBI action in the form: Changing the coordinates according to Seiberg-Witten map, The whole term in parentheses after This shows that Also, det At this point, we will employ the double scaling limit introd Putting all together, we obtain D Now, comparing the definitions of scalar densities correspo so we can use the set of local coordinates ( ( density in the action can be written as JHEP08(2014)170 , ǫ = } ble } ≡ B B A b ) and X (11.3) b . , X +1 a , p mutative b X {· ··· , { ), and } -fold tensor ion appears. a + B p , we have x k +1 ( p  A b onventions, for , . . . , a b F ) ∗ A 1 ρ b ρ a X kA kJ b b F = F } } a A + A nd notation ,..., b ) for labeling the coor- b X b 1 J X X b b , ctive action for a system , n body of the paper, the e coordinates, we reserve b b a F a X } b b can be rewritten as i, j, k X for multiindices. Note that X valent descriptions: a com- isson tensor is replaced by a ed the construction outlined B }{ { } his automatically. This term }{ b +1)-indices ( X +1 ), ( A kB A p , p , b b a b F X b } +1 X + 1) , b 1 b p p , X X a b p AB a { b b X e g is a skew-symmetrized b X b B 2( X k • { }{ = ,..., b e g + . This gives an independent justification F A , . . . , x 1 1 a } A F b )( ,..., X ). x k 1 A b + 1) , 1 X b b kJ F b a -indices containing at least one commutative { X p b b F p X , b X 10.2 b , +1 2( p { + a – 32 – b b , or X -indices containing only noncommutative indices b X { p in ( ) leads to a series in positive integer powers of +1 aJ lA ) for { 1 + p b b bI F ◦ F a b  e g } F b g = kl -brane. The leading principle was to have an action I ) ′ 11.2 A b g 1 a ◦ +1 ) for p b p b ... b g ). Also, note that from the definition of aB X F | ′ ( I, J, K and = , 1 } ⊥ ) 2 1 b b a Π x • A 1 e D b a e g X b nA b g X { Π( b det ! , F | A, B, C 1 p a ) and ( kA kn m b 1 b X e b F g b g = D ), ( { ( x 1 generalization of the matrix model. Note that using the dou } +1 +1 =1 ′ ′ A +1 n p p ′ b X p + 1) + 1) 1 1 are block diagonal, no confusion concerning range of summat , p > d P p p . Finally, we also introduce usual index raising/lowering c ) for labeling the coordinates ( a e g is the Nambu-Poisson bracket corresponding to Π, } b 2( 2( Z = p X a . We can even drop the restriction of the summations to noncom , . . . , x ·} -indices labeling -branes ending on a ∓ • A }{ + + b and ). Here, we have used the fact that p p X a, b, c k g A +2 g = b p ). To simplify the expressions, we shall also use the shortha +1 F b X p p x ,..., , -indices labeling a ,..., {· p 1 b X a ]. We discussed in detail the bosonic part of an all-order effe Finally, to distinguish the noncommutative and commutativ { b 1 -NCDBI , . . . , b , . . . , b X p 1 1 , S b b directions, since the Nambu-Poissoncorresponds bracket to takes a care of t product of scaling limit for the expansion of ( Implementing this notation, we can write ( since both dinates ( {· of the independent scaling of automatically truncating higher-order powers in (thus allowing, similarly to themutative and DBI a action, “noncommutative” for one.noncommutativity means two As a mutually explained semicalssical equi in one, the in mai which the Po Note that the first non-cosmological term example, in [ of multiple 12 Conclusions and discussion In this paper we have extended, clarified and further develop the letters ( index (thus where summation now goes over all (not strictly ordered) ( where ( JHEP08(2014)170 : , , v ), ∗ M is. B ∗ e W T TM -field art of 3.10 M ∗ to B 13 → . . There- T defines a he gener- p ∗ W B i M Λ is therefore ∗ ,. W + ρ T ⊕ g ) and the ma- p ⊕ hT Λ W 3.16 ⊕ ituation, we found TM : (and similarly of ion, the DBI action, ρ =1 string case and allows p TM TM p . The map lized tangent bundle, we ite intriguing. Extending urant algebroid structure. generalized tangent space , such that d an essential role. All key wever, we can still consider ∗ led form ( ndle for a meaningful interpreta- ent bundle there is a natural TM llow for a natural “sum” of a and Λ W , etc. hor map of maximal rank, on which the d/doubled generalized tangent e a generalized metric, is to give ⊕ =1, the sum of the background defined by the sum brane, the ordinary point-wise product ve definite. Such a subbundle can . It already has been appreciated TM , viewed as a map from p M ∗ oduced for the W W M T 2 ∗ +1)-form. This observation is further is thus an analogue of the usual T on ⊕ spatial world-sheet directions. p B p e T → TM , we can mimic the standard constructions i ). From here it is just a small step to recognize TM ., . – 33 – h . There are several possibilities to do this. To 3.9 ). Finally, the corresponding Hamiltonian ( ∗ . , the appearance of p W 4.1 TM ⊕ p Λ W ⊕ is a general section of Λ ) by zeros. The map ∗ B -brane backgrounds, one can consider the action of the map TM +1)-form. What this sum should be is indicated by the Polyako p W p = ⊕ It turned out that this requirement determines the bosonic p W W 12 , where ∗ . For instance, one can speak of the orthogonal group, define t W M , with ⊕ . Although, this observation perfectly applies also in our s ∗ ∗ T ), tells us what the relation to the generalized metric on W M plays a prominent role. It enters naturally the Polyakov act W ∗ ) is related to the split into one temporal and ⊕ T ⊕ B 4.9 → M p ∗ ∗ + Λ W T g TM p W ⊕ Nevertheless, we found the doubled generalized geometry qu However, we are still facing a problem; we lack a canonical Co In our derivation of the action, generalized geometry playe Let us comment on this more: in the string case, The doubled generalized geometry formalism can also be intr Let us notice, that in our approach to noncommutativity of five ⊕ 13 12 W and extended to End( and Λ choose the one suitable for type membrane action in its matrix form ( “too simple” to control theLeibniz symmetric algebroid part structures of on any bracket. Ho an elegant formulation of the theory. For any Hence, at the end,use we it do only for not a really nice use embedding the of full the generalized doubled tangent genera bu natural (+) pairing on generalized tangentbe bundle characterized is as positi a graph of the map from on the above comments: sincefunction-valued on non-degenerated the doubled pairing generalized tang the doubled generalized tangenttion bundle of as the a “sum” right of framework the metric and a highertrix rank form ( of the Nambucf. sigma also model ( ( supported by the form of the open closed relations in the doub alized metric using an involutive endomorphism remains undeformed. with fore, it is quite naturalmetric to and a look higher for rank ( a formalism which would a which leave us only with a projection onto the tangent bundle Nambu-Poisson one. Buscher’s rules, etc. In generalizeda geometry, one subbundle way of defin the generalized tangent bundle fibre-wise metric on the doubled generalized tangent bundle The reason lies basically in very limited choices for the anc ingredients, have their origin inin the the generalized geometry literature thatTM the presence of a (p+1)-form leads to a it very useful tospace double it, i.e., to consider the the extende fields the effective action essentially uniquely. JHEP08(2014)170 ) ]. ⊕ ⋉ M ) 25 ( ], we W (A.1) M X 70 , → manifolds . Notably, ) 51 , defined as – M ] — based on M ] as needed in ( M 49 p 39 ∗ 21 to take the block , T y) for supporting :Ω p B 38 ˇ Snobl, and Satoshi ♯ Λ A ⊕ 0 “Models of Gravity” the vector bundle TM ntity, we need to recall the s which helped to improve ). We find worth to pursue dξ, , Chong-Sun Chu, Pei-Ming se of interest in this paper. . SGS13/217/OHK4/3T/14 ized geometry is much bigger m and use the advantages of n structures [ W ato, Libor e result of [ Dorfman bracket, see e.g. [ i ubject to the field content one ERN for hospitality. We grate- 8.10 the hospitality of the Center for − ngly, its full group of orthogonal η V 1) a semi-direct product Diff( L ˇ CR P201/12/G028 (B.J.), by the Grant ] + -theory and supergravity [ p > M . It turns our that there is a Leibniz algebroid, V,W ). This bracket coincides with the one defined M ]. – 34 – ∗ we define the induced map = [ M ( 25 T D ] ⊕ M ). η +1 p closed M ] or [ + on ( TM Ω p 24 A . Ω ∈ i ∂ ξ, W ∈ to be an isomorphism of the bracket forces C -bilinear bracket on the sections of iK B + e R A ξ, η V [ K ξ is the group of locally constant non-zero functions on ), with = G ) and 5.1 A ] to study Nambu-Dirac manifolds. Moreover, Nambu-Poisson ξ M i ( 24 p + 1)-vector field X 1) p ∈ − ), where G ⋊ ) = ( V,W ξ ( ♯ Finally, let us again notice the striking similarity with th Also, for an alternative formulation of the fundamental ide For any ( Relating our approach, based on the generalized geometry on , with the usual generalized geometries in +1 A closed p ∗ Watamura for helpful discussions. B.J.Theoretical and P.S. Sciences, appreciate Taipei, Taiwan, R.O.C.fully B.J. acknowledge thanks financial C support by the grant GA Here we recall some fundamental properties of Nambu-Poisso (J.V.), and by the(J.V., DFG P.S.). within We the thankour the Research collaboration. DAAD Training (PPP) Group We and 162 the also ASCR manuscript. thank & the MEYS (Mobilit referee for his comment A Nambu-Poisson structures Agency of the Czech Technical University in Prague, grant No such that the condition for Dorfman bracket, i.e., the this paper. For details see, e.g., [ Ho, Petr Dalibor Hoˇrava, Noriaki Kar´asek, Ikeda, Matsuo S a deeper understanding of this similarity in the future. Acknowledgments It is a pleasure to thank Tsuguhiko Asakawa, Peter Bouwknegt having a natural function-valuedHowever, pairing the field as content we coming with didthen such for a we doubled started our general with ca and we have to reducea it very accordingly. different approach — and discussed after equation ( notice the following. Awants choice to of describe. a generalized In geometry principle, is one s can double each of of the W as by Hagiwara in [ transform of generalized geometry for all off-diagonal form ( appear naturally as its Nambu-Diracautomorphisms structures. can Interesti be calculated, giving (for (Ω this coincides with the group of all automorphisms of higher JHEP08(2014)170 ); A.1 (A.5) (A.6) (A.4) (A.2) (A.7) ction, and be a smooth ) M ( this is not true in ∞ . C = 1  ∈ . We may ask whether p f quivalent reformulations Π) ∧ TM , )) ; (A.3) ) under the Dorfman bracket. . The answer is given by the , → dξ (Π . Let ) dξ Π ( M 1 ) can be stated in the following ( π ) is a Nambu-Poisson tensor. dξ ] . ) G i η M η ( ) ( ♯ 6= 0, there exist local coordinates ∗ , ♯ +1 A.2 Π p > ξ, η T p ) 1 Π i ∂ + 1 i p ( ([ x TM ♯ ♯ p ∂x − Π :Λ +1 ) − p ♯ − η ( (Π) = 0 is closed under the Dorfman bracket ( ) ) and the involutivity condition have a good ξ )] = Π p } Γ(Λ ( ) = ) η ♯ df ∧ · · · ∧ (Π)Π ( η Π + 1)-vector on be defined as ∧ Π ∈ ♯ ( 1 M dξ – 35 – ♯ p , where Π( ) G ... L ( i Π . We call Π a Nambu-Poisson structure if ∂ Π ∧ p , ∂x 1  M M ) Ω . M := ( , the condition ( ξ df − (Π)) p ( ( 1 ∈ iJ ) ♯ π ∈ ♯ ] ξ ǫ Ω ) = Π ( ξ ≥ x ♯ x [Π L | = Π → p ξ, η [ ξ Π( L ) (Π) = ( ) +1 ξ ( M . Then any )+ ♯ ...p ( it holds that ξ iJ 1 p Π is again a Nambu-Poisson tensor. For ( δ ) ). ; . Then it holds that ♯ it holds that Ω L + 1 ) ) Π Π ) = p M M × { f ( ( M M vector bundle morphism Π ) p ( ( be a Nambu-Poisson tensor, and M iJ = ∞ p p ( = Ω p M n Π C Ω Ω + 1)-vector field on ( Π ∈ Ω any p p ∈ ∈ ∈ G ∈ For an arbitrary Let Let p . Then , which is not induced by ( :Ω ♯ ξ ξ, η 1, a Nambu-Poisson tensor can be multiplied by any smooth fun 1, around any point ξ, η ξ, η π M ] · ), such that , n · [ , . . . , f p > p > 1 f for all for all This lemma has a simple useful consequence There is an interesting little technical detail. One of the e For Let Π be a ( For , . . . , x 4. for any 1. The graph 3. let 2. for any 1 x there exists Π meaning also for ( following lemma: Lemma A.1. general. Lemma A.3. function on equivalent ways: for all But see that the both, the definition of In this coordinates Π one gets again a Nambu-Poisson tensor: Lemma A.2. of fundamental identity was the closedness of the graph JHEP08(2014)170 , so η (A.8) in the -tensor M . This will + 1) =  p p M J ( ∈ . =
k ...m...j ) be arbitrary local ), allow us to define 1 K n ij mK lM δ . Π A.7 lJ . ,m Π , and mK lM  K , . . . , y δ k p r 1 j iM y lJ ). Then, the only non-trivial = Π Π , p Π } i ) p =1 ...m...j iM r 1 X , = be a contravariant = 0, whenever M K,m ij . . . m ( , ξ − +1 -vector, and hence a Nambu-Poisson p i 1 Π Π m p = Π ) − Ω K kM ξ m  p . r mJ ( + 1). Let ( i = j j ♯ ∈ +1) Π p p Π Π ξ = ( ( ∂y ( ∂x = 0 lM ,m | + 1)-vector. − ) p ). Let ...m...j 2 α, =1  1 +1)-vector. This relation is tensorial in M ) r h X ξ iK p dξ ij is a p ( Π + A.1 kM Π – 36 – lJ ) + Π ) = − K ξ as be a vector bundle morphism, such that its graph = det Π r ), in which Π has the form ( mJ ( j | | ♯ 2(Π n ,m ) ) Π α, ξ Π iM . Right-hand side vanishes due to skew-symmetry of , where Π of weight x x iJ q k { iK | TM Π( Π p ) =1 Π( Π( . Then -th component of the identity. The left-hand side is Π r = X = | = | x , . . . x = Π i ) → 1  i r q mK Π + ) form the differential part of the identity, whereas the terms x under the Dorfman bracket can immediately be rewritten Π( M | Π m ( G M p dξ K,m K Π  ∗ mJ ) is ) ξ ξ Ω J G T K Π p ξ − ∈ dy A.3 ( iK ♯ ξ = =0, and Π is thus a ( :Λ Π Π form the algebraic part: i + 1)-vector. To do this, choose ♯ iJ ( ♯ p Π kM and Π Π) K,m ) ) ξ ξ − ( , and look at the Let ♯ M J ( Π . Hence, we obtain 1 δ L dy Ω ( = ∈ η The closedness of ), where Π is now not necessarily a ( α A.3 This proves that Π as ( the symbol term in the sum is the one for tensor. Proof. coordinates. Define the function We will use this algebraic identity to show that Π choose proportional to above identity. Assume that for all is closed under higher Dorfman bracket ( The terms proportional to defined by a well-behaved scalar density prove that Π is a ( The right-hand side of ( Interestingly, the coordinates ( A.1 Scalar density Lemma A.4. JHEP08(2014)170 2, + 1 +1, p p (A.9) (A.10) n > = or -th power n n = 0. We thus . For L . It is of course ♯ iJ . be an inner product +1 on the choice of the . p ∂ g | ) +1 ∂y ). Now, see that transforms, with respect p x . ). This is indeed a scalar ′ ∂ = det Π ). We can split it as ) = 1 and | k ǫ k | ) M M Π( x ∂x ) is up to sign the ( J | x x ( p ( i x i y iJ e Π( x ∧ · · · ∧ | Π( , | ′ = + 1), as can easily be seen using | 1 = i ) , such that an orthogonal projector, such that ∂ p i y ∧ · · · ∧ x ( ∂y ) is another set of such coordinates, Π . e x V 1 n 1 n − ′ ′ Π( | ) coordinates. We can ask, whether , | ∂ ) y → ∂x x ) forces det( ! M, +1 p Π = Π V ) = K Π( -vector | , . . . , x A.9 : ) coordinates on 2 ǫ ′ 1 M ′ LM J = +1 x P x = det +1) submatrix corresponding to the first p ∂

– 37 – P,P p J ) of weight ( +1 ∂x ), as p = e y ∂ = x × = det ( = 2 is a special case contained in the next question. J det ). This implies that . Let ∂x C . | 7→ | ). Let ( n ) V ) ). The determinant of Π Π +1 n y x x x + 1) ( be the matrix of the vector bundle map Π ∧ · · · ∧ p j>p Π( Π( | 1 = 1, x iJ ...n ( ∧ · · · ∧ ∂ 1 p ±| ∂x , . . . , x 1 M is a ( 1 be a finite-dimensional vector space. Let = ∂ x ǫ -form on = Π ∂x be the matrix of Π in ( 2 J ) = iJ V 1 n x matrix. We can thus ask whether ij a | ) = det ) = ∗ n Π( x +1. Let Π Let x V p × M 2 Π( whenever Π is decomposable. The answer is clearly negative f Π( | . Then there exists a unique = Λ n ) ij n = 0. The case is calculated with respect to ( ∈ C the Jacobi matrix of the transformation ′ | ij C ) = 1, let Π J x p 1 and , and thus detΠ Π( = det Π | ≥ ) = ker( | ...n , and ) 1 Further, we have to be careful with the dependence of For p P V x Π( of both sets of coordinates. The condition in ( Proposition B.1. B Background independent gauge B.1 Pseudoinverse of a 2-form where the top-left block of Π on that is ker( get the important observation that This means that and moreover det Denote by special local coordinates ( that is, the Jacobian of the coordinate transformation the chain rule. to the change the special coordinates ( | where density (with respect to a change where det Π Let this is a square modulo the sign, depending on the ordering of the basis of Ω JHEP08(2014)170 = can (B.1) (B.2) k ) is in P 2 ∆ B.1 . Finally, , such that 1 is called an is, and that Σ + C = Σ gA T . uniquely. It coincides A is not unique, and the ! . Define Π 0 1 O − . The form and a matrix A . M ). The condition ( 1 T ,..., ock-diagonal form of the ma- O − 0 P , that is , gebraic part of Nambu-Poisson A g and thus . The (unique) matrix . T ! , k k 1 + 0 C O λ − 1 , and it is given, in the block form, as . k , . This map is skew-symmetric with Σ g 2 − k , , gAOΣO Σ Σ 1 V ! 2 blocks. g λ 0 − ) denotes the value of the induced vector 0 , respectively, in an arbitrary fixed basis − − × P = 0 = 0 A 1 + 1)-form on → P be the matrix of a bivector we are looking (

= T − = AO∆ p , C 0 0 C 0 V g O dC Π g = ∧ , Σ : . This means that + T block of ∧ ) ) 1 C k ,..., C

– 38 – 2 . This fixes ) − k C P C be a ( ! 1 + 0. Note that the matrix ( 2 1 P = ∆ A . Let 1 when evaluated on ( ( − 0 − 1 C Σ C T λ AOΣ × g > + g C = − ( 1 k k M = Σ = 1 ≡ A ∗ λ Σ 0 − + ) T T − Π e g . This matrix is skew-symmetric (in the ordinary sense). C Σ O k → 2k 2 , . . . , λ

e be the matrix diagonalizing CA 1 translates into 1 ∆ λ ΠgAOΣO ΠgAO − A 1 TM P p A − be the matrices of AO∆ has the form = = diag A = = :Λ P Σ T ), and ′ C Σ ′ 0), where the number of 1’s is 2 . Let . Now it is easy to see that Π is a bivector, if and only if P O e C C e k C C 2 ), where on the left-hand side and 1 − ∆ ,..., g M g 0 ( , , where p , rank( = . Indeed, we have T 1 2 1 C = X g is the invertible top left 2 is unique up to the reordering of the 2 Σ ∈ = 0 + e C Σ be a smooth manifold, and let Let . In more detail, one can find an orthogonal matrix k Σ Σ ′ ,..., P OΣO e . First construct the map C This shows that we can write M = V ′ Π = Πholdsifandonlyif e bundle morphism where matrix where fact a very restrictive one.fundamental identity: Also, it is very similar to the al Denote with the Moore-Penrose pseudoinverse of the matrix of B.2 Integrable forms Let Proof. P respect to We thus get that ( where Standard linear algebra saystrix that there exists a standard bl define the matrix for all integrable form if it holds diag(1 C be now written as for. The equation Π JHEP08(2014)170 as ′ M , where ∈ M x , there have ), such that x -forms, and let M ( p X ). One gets ∈ M ( 6= 0, at least at some : V X )) ∈ + 1)-form at grability od forms. . Second, take the original P ) if and only if it is decom- , defined at every p ( V ′ . . Q } C B.1 M ( ∧ V ) leads to the desired decomposi- i . That means that there exists a ) = 0 V ), such that , the open submanifold of . p P . ′ C . . ( )) = 0 ) ≡ ( = p -forms , such that locally x 6= 0 = 0 C +1 C M V α ( 1 i λ ( p p ) P
) = 0 C ∧ α V x 1) satisfies ( ∧ λ ( i ), and thus, by induction hypothesis, is C C ( , . . . , α − = 0 case is a trivial statement, any 1 form ) is a nonzero ( ∧ V 1 ( V ∧ i ∧ C ... C i p P α ( ) B.1 | ) C ( ). We have to show that = ∧ ... P ∧ Q C 1 ( ∧ M M ) , taking – 39 – α +1 ∧ ( x − C 1 p 1 C Q . The T 1 α λ = α is a distribution on V − C p )( ( p ), such that ∈ · = C = C C X C )) V M V 6= 0. First, see that for any V ). We have to show that it is decomposable. , such that i C ( 0. Assume that statement holds for all i C ∈ of { ) satisfies ( V P ( p -form. Then i ) ( M 6= 0. We can now apply the induction hypothesis to this C X x Q = B.1 ( K C ) ( +1)-form. Denote by p > ∈ ( x V to both sides with an arbitrary x ∈ i + 1) C p V of linearly independent K x i V p ))( i ( ) P C ( is a ( +1 p -form V i be a C p ) and C . Thus, locally we can write Let M x , . . . , α , such that ( linearly independent 1-forms ( 1 Let X ) and apply α M p ( )) is a scalar function, and since ∈ 6= 0, the 6= 0, also ( ∈ B.1 P + 1)-form satisfying ( ) ) ( V x x p x C ( Let us proceed by induction on -tuple ( λ ))( V i We can now relate integrability of distributions to the inte Let us now clarify where integrable forms got their name from Let C ( be a ( + 1) 6= 0. The kernel distribution V -form to get p i Proof. is decomposable. Now choose C neighborhood of to exist ( p ( This finishes the proof, because taking tion. Since Definition B.3. condition ( decomposable. Let us take any posable around every point But Note that this distribution is not necessarily a smooth one. Lemma B.2. C But this can be rewritten as which follows from the assumptions on JHEP08(2014)170 , . \ ]. j K 77 M dα (B.5) (B.4) (B.3) ∈ -form. . The . How- x spanned V M f x 6= 0. We ’s. +1) j n ). It gives p U ( α λ , so it has to + 1)-forms to B.4 is an integrable K p rank, and hence a K . For each are on . M }} K , we have grable ( ′ + 1 by equation ) into the second defining M is an integrable be the corresponding volume Π ) hold trivially and we can . B.3 Ω C mma. ′ , and a set of local coordinates , . . . , p B.2 x valent to the integrability of 1 M , . ∋ . ), ( ′ -vector +1 . ∈ { ), and plug it into ( x = 0 p i U M +1))-dimensional regular smooth dis- +1 K B.1 ∀ dy p p +1 , + 1) ( p α Γ( ∧ p α integrable if and only if + 1)-form. Then by the previous lemma, − ∧ ( to the whole manifold + 1. The second condition for integrable + 1)-tuple of linearly independent 1-forms, C. ∈ ∧ n p ... C p under the commutator of vector fields, which K ... )) = 0 V ∧ x ... K ∧ 1 Ω = ( is not integrable in general. For details see [ ). Since we are on – 40 – i ∧ Π 1 , . . . , p i α dy 1 α . But this is, using the Cartan formula for -vector if and only if K ( · B.1 } α V = λ i . Define a = 2 ∧ | + 1)-form. i + 1 = C j +1)) has to be annihilated by all vectors of M p p ( -form. Then M C dα C is integrable (( ). Obviously, this holds for the above defined x for ), such that sections of the subbundle − +1). To see that this is a smooth regular distribution, note T i , there exists a ( n p K ′ is an integrable ( +1 + 1) ( ( B.4 , . . . , p ∈ p + 1)-forms, dy , there is a neighborhood p 1 ′ M . Now take any C − ( p V } -form on = n ∈ { M ∈ { ). Then i x the integrability conditions ( = ∈ be an orientable smooth manifold. Let + 1 be a clearly satisfies ( , . . . , y α j ′ + 1) 1 . By this extension one gets a smooth singular distribution o x x +1)) linearly independent linear equations for the components o } C p M C p . Hence, a smooth distribution in K , y can be determined easily as M 0 ( ( ′ ∂ is thus is an integrable ( k { − x \ n x ( Let , and Let , . . . , p C +1)) = K 1 ). It turns out that it is equivalent to TM One can extend the distribution p 1 K be a ∂x M -dimensional regular smooth distribution on ( x . − K C n ∈ B.2 λdy ( ∈ { B.5 is a Nambu-Poisson x is the kernel of a smooth vector bundle morphism of a constant ,..., + 1)) j = First assume that ) = 0 for all 1 p j Π ∂ 1 ( K ∂x Conversely, assume that At Let us conclude this section by relating the concepts of inte To see that it is also integrable, plug the expression ( α , . . . , x dα , define − + 1)-forms translates as ( ′ ( 1 x p n V ever, even for integrable ( ( tribution. At every by ( M have the local form We see that this equivalent to involutivity of the subbundle conclude that i equation ( is in turn, using the Frobenius integrability theorem, equi Nambu-Poisson structures. This is given byLemma the B.6. following le for all Remark Lemma B.4. set form. Let This is a set of that ( Then dimension of Proof. The subspace ( such that locally around every point of subbundle of JHEP08(2014)170 . M ommons ). amental identity . In the view of ′ , B.2 , N ), ( hep-th/9908105 , , +1 fold B.1 p he previous lemma. There, omponents of Π, so in the redited. 6= 0. Then there exist local , . . n ]. ) . . . dy tions ( . x ansversal directions. We can now dx +1) ∧ , p ( n +1) 1 ∧ -model ∂ p − ( σ SPIRE n dy dx ∂ + 1)-form, written (around any point − ... IN n ∧ p ∧ ∂x ][ ∧ ∂x vanishes. The conditions on integrability ∧ ... +1)) ∧ p C ∧ ]. ( ... ) = 0. Let Π be a Nambu-Poisson tensor. By 1 +1)+1 − p ... x ∧ n ( – 41 – ( ( dx ∧ 1 − · C n ]. ∂ 1 SPIRE ω ∂x ). Writing the volume form in these local coordinates ∂ IN dx · [ ∂x -model and effective membrane actions · . . . dx σ Quantum string theory effective action g B.5 λ has the explicit form ω ∧ , such that SPIRE Ω = arXiv:1203.2910 [ to conclude that Π is a Nambu-Poisson tensor on x 1 ), which permits any use, distribution and reproduction in C IN = Π = [ dx Π = C · A.3 Nambu- g (1989) 2767 can be, for an integrable ( ) around (1985) 1 Ω = (2012) 313 n CC-BY 4.0 C A 4 ) = 0 if and only if Born-Infeld action, supersymmetry and string theory x This article is distributed under the terms of the Creative C Dirac-Born-Infeld action from Dirichlet B 261 , . . . , x B 713 1 x ]. 6= 0) in the local form ( , one would expect that this is enough. Inspection of the fund ) x 6= 0. We thus see that ( A.3 Clearly, Π( SPIRE ω C IN Nucl. Phys. Mod. Phys. Lett. [ Phys. Lett. The converse statement follows basically from the proof of t [1] B. and Jurˇco P. Schupp, [3] R.G. Leigh, [4] A.A. Tseytlin, [2] E.S. Fradkin and A.A. Tseytlin, coordinates ( are, at these points, satisfied trivially. Assume that Π( It is easy to check that it satisfies both integrability condi where one finds the local expression for Π as previous comment, at singular points of Π, References apply (the proof of) lemma any medium, provided the original author(s) and source are c shows that allfundamental partial identity there derivatives are are no partial contracted derivatives in with tr the c as we have shown that lemma Note that this is a top-level multivector field on the submani In these coordinates, the volume form Ω is where Open Access. Proof. Attribution License ( JHEP08(2014)170 , . ] , ]. ]. ]. SPIRE 48 , SPIRE -brane ] ] 5 IN IN , -form flux (2011) 437 (1996) 103 3 ][ M , SPIRE 37 Adv. Theor. IN rsion) [ A 54 , ge theory , -form background arXiv:0906.3172 [ ion (2013) 46 [ 1) ]. , Bracket description of 1 − Chin. J. Phys. , p ( ]. hep-th/0102129 arXiv:1001.2300 [ [ . , SPIRE Sci. China (1973) 2405 , arXiv:0804.3629 IN [ ]. (2009) 050 ][ SPIRE Universe Lett. Math. Phys. , D 7 rder solution to the ]. ]. , IN 08 ][ SPIRE ]. (2001) 148 (2010) 127 IN (2002) 1263 -brane worldvolume action with -brane in three-form flux and multiple 5 SPIRE SPIRE ][ More on the Nambu-Poisson 5 (2008) 105 JHEP 03 IN IN ]. M , M ]. ]. ][ ][ SPIRE 06 -brane in large RR A 35 Phys. Rev. B 604 IN , . Dp ][ JHEP SPIRE arXiv:1006.5291 SPIRE SPIRE , [ IN -brane in NS-NS and RR backgrounds IN IN JHEP – 42 – ][ 3 hep-th/9301111 [ ][ ][ J. Phys. 2, D (1996) 1 ]. , Nambu-Poisson gauge theory M Nucl. Phys. -brane in large C-field background arXiv:0805.2898 Non-Abelian noncommutative gauge theory via , [ 5 hep-th/0701130 LXV SPIRE (2010) 100 [ M arXiv:1403.6121 from [ On decomposability of Nambu-Poisson tensor IN ]. 5 String solitons in the (1994) 295 hep-th/0006018 10 ][ ]. String theory and noncommutative geometry M A toy model of open membrane field theory in constant S-duality for Effective action for 160 (2008) 014 SPIRE arXiv:1302.6919 hep-th/0006058 hep-th/9908142 [ [ [ SPIRE On higher analogues of Courant algebroids JHEP IN (2007) 913 , IN 08 (2014) 221 [ ][ 39 (2002) 1259 [ Some remarks concerning Nambu mechanics On foundation of the generalized Nambu mechanics (second ve The non-Abelian Born-Infeld action and noncommutative gau Nambu-Dirac manifolds Non-linearly extended self-dual relations from the Nambu- On the general structure of the non-Abelian Born-Infeld act 4 JHEP Generalized Hamiltonian dynamics B 733 , A concise review on Gauge symmetries from Nambu-Poisson brackets (2013) 056 (2000) 033 (1999) 032 arXiv:0912.0445 ]. ]. ]. 05 07 09 -brane in a constant C-field background -branes SPIRE SPIRE SPIRE 5 2 arXiv:1003.1350 IN IN IN Nambu-Poisson structure and Seiberg-Witten map M Math. Phys. M theory: scaling limit, backgroundSeiberg-Witten independence map and an all o Acta Math. Univ. Comenianae [ [ Commun. Math. Phys. [ arXiv:1311.3393 JHEP Gen. Rel. Grav. Phys. Lett. JHEP JHEP noncommutative extra dimensions [ (2010) 1 [ [8] S. Terashima, [9] C.-H. Chen, K. Furuuchi, P.-M. Ho and T. Takimi, [6] L. Cornalba, [7] B. P. Jurˇco, Schupp and J. Wess, [5] N. Seiberg and E. Witten, [20] Y. Nambu, [19] K. Furuuchi, [23] P. Guatheron, [24] Y. Hagiwara, [25] Y. Bi and Y. Sheng, [21] L. Takhtajan, [22] D. Alekseevsky and P. Guha, [18] K. Furuuchi and T. Takimi, [15] P.-M. Ho and Y. Matsuo, [16] P.-M. Ho, Y. Imamura, Y. Matsuo and S. Shiba, [17] P.-M. Ho, [13] P.-M. Ho and C.-T. Ma, [14] P.-M. Ho and Y. Matsuo, [10] P.-M. Ho, [11] B. P. Jurˇco, Schupp and J. Vysok´y, [12] P.-M. Ho and C.-T. Ma, JHEP08(2014)170 ] ] J. ] , , ] , = 11 ]. heory, a (2012) 3 -Lie D 3 15 -branes (2009) 66 (2003) 281 2 SPIRE nt algebroids IN M 54 hep-th/9906142 ][ [ arXiv:0806.0335 ]. ]. ]. [ arXiv:1003.4694 B 811 [ hep-th/9611008 (1997) 103 [ (2007) 045020 five-brane with the -branes and ]. 22 5 SPIRE SPIRE SPIRE (2000) 1 IN M IN Transverse five-branes in IN Living Rev. Rel. BPS solutions of the tiny D 75 ]. = 11 ][ ][ ]. , ][ ]. 2 SPIRE (2009) 161 / ]. nk, 329 D ]. (1997) 62 (2010) 026 IN 1 Nucl. Phys. ]. ][ , SPIRE arXiv:1110.2687 07 SPIRE [ SPIRE IN C 64 IN SPIRE IN SPIRE -form flux, quantum Nambu geometry ][ B 394 IN Phys. Rev. ][ 3 IN SPIRE ][ J. Geom. Phys. , -branes ][ IN ][ , JHEP 2 ’s Phys. Rept. 2 , RR ][ M , Quart. J. Math. Oxford Ser. An action for the superfive-brane in hep-th/0603120 M hep-th/0501001 [ , [ arXiv:1012.2236 [ A note on topological (2012) 055401 hep-th/9701037 Phys. Lett. Eur. Phys. J. [ – 43 – Quantized Nambu-Poisson manifolds in a Classification of all brane system and a generalized Nahm’s equation , 5 Covariant action for a hep-th/0211139 = 5, [ A 45 -symmetry and applications M hep-th/9712059 arXiv:1003.1004 [ - p (2008) 100 κ (2005) 001 hep-th/0406214 2 (2011) 075 , [ arXiv:0711.0955 [ -strings in large hep-th/0412310 06 M [ (1997) 41 1 04 04 Taking off the square root of Nambu-Goto action and obtaining ]. ]. D = 11 Modeling multiple Gauge symmetry and supersymmetry of multiple ]. ]. J. Phys. The D , (2012) 1 [ ]. (2003) 038 JHEP JHEP (1998) 007 B 398 JHEP , Tiny graviton matrix theory: DLCQ of IIB plane-wave string t SPIRE SPIRE , 01 (2004) 017 , SPIRE SPIRE IN IN 04 (2005) 136 IN IN SPIRE ][ ][ 10N4 09 (2008) 065008 ][ ][ algebras and higher analogues of Dirac structures and Coura IN Algebraic structures on parallel ][ JHEP ∞ Superbranes and superembeddings , JHEP L B 713 Generalized Calabi-Yau manifolds Phys. Lett. , D 77 Five-brane effective action in M-theory JHEP , Brane effective actions, , Model of M-theory with eleven matrices ]. ]. ]. ]. -branes in C-field 5 M SPIRE SPIRE SPIRE SPIRE IN arXiv:1110.2422 math/0209099 hep-th/9610234 IN hep-th/0611108 arXiv:0709.1260 IN IN algebra reduced model [ [ [ [ matrix theory conjecture [ Phys. Rev. [ Nucl. Phys. [ Filippov-Lie algebra gauge theory action [ supergravity chiral field string-fivebrane duality and Symplectic Geom. graviton matrix theory [ [44] J. Simon, [43] D.P. Sorokin, [29] J. DeBellis, C. and S¨amann R.J. Szabo, [28] M. Sato, [45] N. Hitchin, [41] E. Witten, [42] P.S. Howe and E. Sezgin, [39] L. Bao, M. Cederwall and B.E.W. Nilsson, [40] P. Pasti, D.P. Sorokin and M. Tonin, [36] J.M. Maldacena, M.M. Sheikh-Jabbari and M. Van Raamsdo [37] M.M. Sheikh-Jabbari, [38] M. Cederwall, B.E.W. Nilsson and P. Sundell, [34] J. Bagger and N. Lambert, [35] A. Gustavsson, [31] A. Basu and J.A. Harvey, [32] M.M. Sheikh-Jabbari and M. Torabian, [33] J. Bagger and N. Lambert, [30] C.-S. Chu and G.S. Sehmbi, [27] J.-H. Park and C. Sochichiu, [26] M. Zambon, JHEP08(2014)170 , ] , ] ]. (2005) 455 (2005) 035 -model ]. ]. σ , ]. (2011) 074 03 260 ]. SPIRE , , , 03 -models and , IN SPIRE SPIRE ry [ σ itchin ]. SPIRE IN IN , JHEP IN SPIRE ][ ][ hep-th/0701203 , ]. [ -model ][ IN ometry ]. JHEP σ -models , ][ SPIRE σ math.DG/0406445 [ IN ]. [ SPIRE SPIRE IN IN ][ ][ SPIRE (2007) 079 IN Supergravity as generalised geometry Supergravity as generalised geometry math.DG/0508618 07 ][ T-duality, generalized geometry and (2005) 400 , Commun. Math. Phys. arXiv:1212.1586 arXiv:1402.0942 , [ [ ]. arXiv:0807.4527 54 [ ]. ]. arXiv:1206.6964 [ JHEP , math.DG/0401221 -models SPIRE , σ D-brane on Poisson manifold and generalized ]. SPIRE SPIRE IN arXiv:1107.1733 IN IN – 44 – [ (2014) 019 ][ math.DG/0503432 D-branes in generalized geometry and [ Lie algebroid morphisms, Poisson ][ ][ (2009) 075 03 Dirac SPIRE (2012) 064 On the generalized geometry origin of noncommutative arXiv:1303.6096 (2014) 1450089 [ 04 Generalized complex geometry and T-duality IN J. Geom. Phys. [ 10 , JHEP Courant-like brackets and loop spaces A 29 (2011) 091 ]. , JHEP (2006) 131 From current algebras for p-branes to topological M-theory ]. ]. ]. ]. ]. Current algebras and differential geometry , JHEP 11 Generalizing geometry — algebroids and , (2013) 126 265 (2010) 261 SPIRE arXiv:0805.4571 hep-th/0501062 hep-th/0507051 [ [ [ SPIRE SPIRE SPIRE SPIRE SPIRE 07 IN IN IN IN IN IN JHEP [ [ [ ][ 807 , ][ ][ Lectures on cohomology, T-duality, and generalized geomet and M-theory Non-geometric backgrounds and the first order string Non-geometric string backgrounds and worldsheet algebras JHEP Generalized complex geometry Generalized complex geometry, generalized branes and the H + , Brackets, forms and invariant functionals Instantons, Poisson structures and generalized ge K¨ahler Generalised geometry for M-theory R (2008) 137 (2005) 022 (2005) 015 Int. J. Mod. Phys. × , ]. ]. ) 07 03 09 d ( d E SPIRE SPIRE arXiv:0903.3215 hep-th/0410183 hep-th/0411112 IN IN off-shell closed gauge symmetries Dirac-Born-Infeld action geometry gauge theory Lect. Notes Phys. non-geometric backgrounds arXiv:0906.2891 JHEP arXiv:1106.1747 JHEP [ arXiv:1004.0632 JHEP [ [ [ II: [ Commun. Math. Phys. I: type II theories [54] A. Kotov and T. Strobl, [52] A. Kotov, P. Schaller and T. Strobl, [65] T. Asakawa, H. Muraki and S. Watamura, [66] B. P. Jurˇco, Schupp and J. Vysok´y, [62] P. Bouwknegt, [63] M. R. Gra˜na, Minasian, M. Petrini and D. Waldram, [64] T. Asakawa, S. Sasa and S. Watamura, [60] N. Halmagyi, [61] G.R. Cavalcanti and M. Gualtieri, [57] G. Bonelli and M. Zabzine, [58] J. Ekstrand and M. Zabzine, [59] N. Halmagyi, [55] R. Zucchini, [56] A. Alekseev and T. Strobl, [53] M. Bojowald, A. Kotov and T. Strobl, [50] A. Coimbra, C. Strickland-Constable and D. Waldram, [51] C.M. Hull, [47] N. Hitchin, [48] M. Gualtieri, [49] A. Coimbra, C. Strickland-Constable and D. Waldram, [46] N. Hitchin, JHEP08(2014)170 ]. ]. , , (1990) 394 SPIRE d theta ]. SPIRE IN [ (2011) 074 IN [ B 347 , 06 SPIRE IN ][ JHEP (1990) 398 , Progress in , (1990) 610 Critical fields on the A noncommutative (2000) 193 Nucl. Phys. ]. , dell, dell, ]. B 343 B 335 B 492 ]. SPIRE SPIRE IN hep-th/0005026 IN [ ][ ]. SPIRE ][ IN Nucl. Phys. ][ , Phys. Lett. Nucl. Phys. , SPIRE , IN – 45 – (2000) 173 The eleven-dimensional five-brane ][ -brane actions and higher Roytenberg brackets p hep-th/0109107 [ B 590 hep-th/0404049 [ Generalized geometry and M-theory hep-th/9606118 Open membranes, ribbons and deformed Schild strings [ ]. ]. Poisson structures and their normal forms Duality rotations in membrane theory (2002) 012 SPIRE arXiv:1211.0814 [ SPIRE Nucl. Phys. IN Deformation independent open brane metrics and generalize IN , 02 (1996) 85 ][ (2004) 045007 ][ JHEP B 386 D 70 Duality rotations in string theory , (2013) 042 ]. Membrane symmetries and anomalies 02 -brane and noncommutative open strings SPIRE 5 hep-th/0006112 IN arXiv:1008.1763 M-theory five-brane M [ Mathematics. Basel Birkh¨auser, Switzerland (2005). Phys. Rev. JHEP Phys. Lett. [ [ parameters [69] M.J. Duff, [76] E. Bergshoeff, D.S. Berman, J.P. van der Schaar and[77] P. Sun J. Dufour and N. Zung, [74] D.S. Berman and B. Pioline, [75] E. Bergshoeff, D.S. Berman, J.P. van der Schaar and P. Sun [71] B. P. Jurˇco, Schupp and J. Vysok´y, [72] E. Bergshoeff, M. de Roo and T. Ort´ın, [73] D.S. Berman et al., [68] M.J. Duff and J.X. Lu, [70] D.S. Berman and M.J. Perry, [67] I. Bars, Appendix E

Paper 4: Nambu-Poisson Gauge Theory

203

Physics Letters B 733 (2014) 221–225

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Nambu–Poisson gauge theory ∗ Branislav Jurcoˇ a, Peter Schupp b, Jan Vysoký b,c, a Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute, Prague 186 75, Czech Republic b Jacobs University Bremen, 28759 Bremen, Germany c Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Prague 115 19, Czech Republic article info abstract

Article history: We generalize noncommutative gauge theory using Nambu–Poisson structures to obtain a new type of Received 2 April 2014 gauge theory with higher brackets and gauge fields. The approach is based on covariant coordinates Received in revised form 15 April 2014 and higher versions of the Seiberg–Witten map. We construct a covariant Nambu–Poisson gauge theory Accepted 24 April 2014 action, give its first order expansion in the Nambu–Poisson tensor and relate it to a Nambu–Poisson Available online 29 April 2014 matrix model. Editor: M. Cveticˇ © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 3 Keywords: (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP . Nambu–Poisson structures Noncommutative gauge theory Matrix models M-theory

1. Introduction Nambu–Poisson (p + 1)-tensor. Based on these covariant coordi- nates, we introduce Nambu–Poisson gauge fields in Section 4.In In this letter, we introduce a higher analogue of noncommuta- Section 5, we construct Nambu–Poisson gauge fields explicitly, us- tive (abelian) pure gauge theory. What we consider here is a de- ing a suitable generalization [6–8] of the Seiberg–Witten map [9], formation, in the presence of a background (p + 1)-rank Nambu– starting from an ordinary (p −1)-form gauge potential. We give ex- Poisson tensor, of an abelian gauge theory with a p-form gauge plicit expressions for all components of the Nambu–Poisson field potential, i.e., a (p − 1)-gerbe connection. Our approach, for p > 1, strength. In Section 6, we give the corresponding (semiclassi- is similar to that of [1] which deals with the more familiar case of cally) “noncommutative” action and its first order expansion in the p = 1. A Nambu–Poisson gauge theory was pioneered by P.-M. Ho Nambu–Poisson tensor. Up to this order the result is unambiguous, et al. in [2] as the effective theory of M5-brane for a large longitu- because quantum corrections from any type of quantization of the dinal C-field background in M-theory. Related work can be found Nambu–Poisson structure will only affect higher orders. We con- in their papers [3–5]. clude the letter by relating the action to (the semiclassical version We formulate the theory independently of string/M-theory. of) a Nambu–Poisson matrix model. Nevertheless, the motivation comes from M-theory branes; more We only briefly comment on deformation quantization of explicitly from an effective DBI-type theory proposed for the de- Nambu–Poisson structures in this letter. A satisfactory descrip- scription of multiple M2-branes ending on an M5-brane, where the tion of Nambu–Poisson noncommutative gauge theory beyond the Nambu–Poisson 3-tensor enters as a pseudoinverse of the 3-form semiclassical level will require a suitable analogue of Kontsevich’s field C [6,7]. We develop the theory at a semiclassical level, briefly formality, solving in particular the deformation quantization prob- commenting on the issue of quantization at the end. lem for an arbitrary Nambu–Poisson structure. The paper is organized as follows: After discussing conven- tions in Section 2, we introduce in Section 3 covariant coordinates, 2. Conventions which transform nontrivially with respect to gauge transformations parametrized by a (p − 1)-form, the gauge transformation being We assume that n-dimensional space–time M is equipped with described in terms of a (p + 1)-bracket arising from a background a rank p + 1 Nambu–Poisson structure Π,with1< p < n.1 The corresponding Nambu–Poisson bracket is denoted by {·,...,·}.In

* Corresponding author. E-mail addresses: [email protected] (B. Jurco),ˇ 1 The discussion could be extended to include also the well known case p = 1, [email protected] (P. Schupp), vysokjan@fjfi.cvut.cz (J. Vysoký). but for clarity and brevity we concentrate here on p > 1andreferto[7] for p = 1. http://dx.doi.org/10.1016/j.physletb.2014.04.043 0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 222 B. Jurˇco et al. / Physics Letters B 733 (2014) 221–225 order to keep notation close to the familiar p = 1 case, we write Covariantizing the individual components of this tensor using the 1 ij1... jp { f ,λ}:=Π(df , dλ) = ! Π ∂i f (dλ) j ... j for a (p − 1)-form λ diffeomorphism ρ, we obtain the Nambu–Poisson (“noncommuta- p 1 p ˆ  and a function f . In the special case, where dλ factorizes as tive”) field strength F with components = ∧···∧ { }≡{ }   aproductdλ dλ1 dλp ,wehave f ,λ f ,λ1,...,λp .We  ∗  ˆ i1...i p+1 := i1...i p+1 consider a set of local coordinates (x1,...,xn) on M and denote F ρ F . (6) ∗ the corresponding indices by lower case Latin characters i, j, k, Using (5) and a hat to denote the application of ρ , etc. Upper case Latin characters I, J , K , etc. denote strictly ordered = ··· ˆ  i1...i p+1 ˆ  i1...i p+1 ˆ i1...i p+1 p-tuples of indices, i.e. J ( j1,..., j p) with 1  j1 < < j p  n. F = Π − Π iJ     With this notation, Π(df , dλ) = Π ∂ f (dλ) .Often,wewillomit ∗  ∗ i J = Π i1...i p+1 − Π i1...i p+1 . (7) indices altogether, implicitly implying matrix multiplication of the ρ ρ T i = iK underlying rectangular matrices as in (Π F ) j Π F Kj.Weuse Rewriting this with the help of (2) as Roman characters a, B, etc. for indices and multi-indices taking     ˆ  i ...i + i i + i i + values only in the “noncommutative” directions 1,...,p + 1. F 1 p 1 = xˆ 1 ,...,xˆ p 1 − x 1 ,...,x p 1 (xˆ), (8) ˆ  3. Covariant coordinates the gauge transformation of F can be easily determined:     δ Fˆ i1...i p+1 = Fˆ i,Λ . (9) Before we introduce in the next section the Nambu–Poisson Λ gauge potential2 Aˆ and field strength Fˆ ,letusdefine“covari- From now on we will assume without loss of generality that ant coordinates”3 as functions xˆi (x), i = 1,...,n of the space–time the local coordinates xi are adapted to the Nambu–Poisson struc- { i }n { i } coordinates x i=1, which transform under gauge transformations ture Π, i.e., x are local coordinates around some point M, where parametrized by a (p − 1)-form Λ as Π is non-zero, such that5   i i δΛxˆ = xˆ ,Λ , (1) Π = ∂1 ∧···∧∂p+1. (10) where the bracket is a p + 1 Nambu–Poisson bracket (cf. Section 2 With this choice of coordinates, we find for notation). We assume our fixed (but arbitrarily chosen) coor-     ˆ  i ...i + i i + i i + dinates xi to be invariant under gauge transformations. We also F 1 p 1 = xˆ 1 ,...,xˆ p 1 − x 1 ,...,x p 1 , (11) ∈ assume that they can be expanded around any point x M,at where the second bracket is in fact either zero or equal to the p +1 ˆi = i +··· least in the sense of formal power series, as x x .Hence,at epsilon symbol in the noncommutative directions 1,...,p + 1. Ro- least formally, we can always solve for xi as functions of covariant man indices a ,...,a + shall henceforth denote these directions. ˆi i = ˆi +··· 1 p 1 coordinates x ,i.e.x x .Wedenotebyρ the (formal) dif- Furthermore, we will focus on the case where for the covariantiz- feomorphism on M corresponding to this change of local variables ∗ ˆi = i i ∗ ing map ρ acts trivially (i.e. x x ) on coordinates x with indices on M and write xˆi = ρ (xi ) for the respective local coordinate + ∗ in the commutative directions p 2,...,n. It follows from (1) that functions. The change of coordinates defined by ρ is also called only the covariant coordinates in the noncommutative directions “covariantizing map”. The diffeomorphism ρ can be used to define transform non-trivially under gauge transformations and that the  {· ·} a new Nambu–Poisson structure Π with bracket ,..., : gauge fields Aˆ i are trivial for i = p + 2,...,n. Also, all the field      ∗  ∗ ∗ strengths, except those indexed solely by noncommutative indices ρ xi1 ,...,xi p+1 := ρ xi1 ,...,ρ xi p+1   i = 1,...,p + 1, will automatically be zero. With these conven- ˆ ≡ xˆi1 ,...,xˆi p+1 . (2) tions, we can use the p +1 epsilon tensor to lower the index on Aa and introduce another kind of gauge potential uniquely determined ˆ 4. Nambu–Poisson gauge fields by complete antisymmetrization of its non-zero components A B labeled by strictly ordered p-tuples of indices, with individual in- Here and in the subsequent sections, we follow closely the dices taking values in the labels of the noncommutative directions semiclassical parts of [10,11], where the p = 1caseisdescribed. ˆ := ˆ a Using covariant coordinates xˆi ,wedefinetheNambu–Poisson A B aB A . (12) (“noncommutative”) gauge potential with components labeled by The components Aˆ transform in a more familiar looking manner = 4 B upper indices i 1,...,n as (but recall that we are still dealing with a p + 1 Nambu–Poisson   ∗ bracket and a (p − 1)-form gauge parameter Λ): Aˆ i = xˆi − xi = ρ xi − xi. (3) ˆ = +{ˆ } Its gauge transformation follows from (1) δΛ A B (dΛ)B A B ,Λ . (13)     ˆ i ˆ i i Similarly, we define the corresponding field strength with compo- δΛ A = A ,Λ + x ,Λ . (4) ˆ  nents FaB by    Next, we introduce the contravariant tensor F with components ˆ  ˆ  bC bC  +  F = − (14) F i1...i p 1 as the difference of the Nambu–Poisson structures Π , aB aC Π Π bB. see Eq. (2), and Π: ˆ  The components FaB transform as expected  i ...i +  i ...i + i ...i +   F 1 p 1 = Π 1 p 1 − Π 1 p 1 . (5) ˆ  = ˆ  δΛ FaB FaB,Λ . (15)  A straightforward check reveals that Fˆ can be consistently ex- 2 aB This is the higher analog of the p = 1 noncommutative gauge potential. ˆ  3 Covariant with respect to the gauge transformation (4).Forp = 1theycor- tended to be antisymmetric in all of its indices. Finally, FaB can be respond to background independent operators of [9]; they are actually dynamical fields. 4 See [12–14] for an alternative approach related to area-preserving diffeomor- 5 Here we ignore, for simplicity, points where Π could possibly be zero and focus phisms. on globally non-degenerate Nambu–Poisson structures. B. Jurˇco et al. / Physics Letters B 733 (2014) 221–225 223

ˆ = expressed in terms of the gauge potential A B .Forthis,weneed where the time-dependent vector field At is defined as At a (p + 1 − q)-ary Nambu bracket defined as6 = iJ Πt (a) Πt a J ∂i and is the corresponding Lie derivative. LAt   i1...iq i1 iq {·,...,·} := x ,...,x , ·,...,· . Eq. (21) implies that the flow φt corresponding to At ,together with the initial condition Π0 = Π,mapsΠt to Π, that is, Now, using (3), (11), (12) and (14) we obtain ∗ = p−1  φt (Πt ) Π. (22)   p+1 ˆ  ˆ = + (σ (k)−1) F + = (dA)1...p+1 + (−1) k r 1 := ∗  = 1...p 1 We have thus found the map ρa φ1, such that ρa (Π ) Π. r=0 σ ∈S(r,n−r) This is the higher form gauge field (p > 1) analogue of the well ˆ ˆ σ (1)...σ (r) known semiclassical Seiberg–Witten map. We emphasize the de- × sgn(σ ){A[σ (r+1)],...,A[σ (p+1)]} , (16) pendence of this map on the p-form a by an explicit addition ∈ − − [ ] where σ S(r,n r) is an (r,n r) shuffle, and a is the multi- of the subscript a. The following observation is important: The ··· − + ··· + index 1 (a 1)(a 1) (p 1). This formula is a generalization Nambu–Poisson tensor Π is gauge invariant (because it depends = t to p > 1 of the well-known p 1 formula for the (noncommuta- on the p-potential a only via the gauge invariant p + 1formfield tive) field strength that involves the 2-bracket (“commutator”) of strength f = da), but the Nambu–Poisson map ρa is not: The in- gauge fields. finitesimal gauge transformation δλa = dλ,witha(p − 1)-form In the next section we will use a higher analog of the Seiberg– gauge transformation parameter λ, induces a change in the flow, Witten map in order to construct explicit expressions for the co- iJ which is generated by the vector field X[λ,a] = Π dΛ J ∂i , where variant coordinates and noncommutative gauge fields. This will the (p − 1)-form Λ, explicitly given by allow us to also supplement the remaining components of the ∞ k Nambu–Poisson gauge field strength (14), i.e., the ones with at  ( + ∂t ) (λ) At least one index in a commutative direction. Λ = L , (23) (k + 1)! = k=0 t 0 5. Nambu–Poisson gauge fields via Seiberg–Witten map is the semiclassically noncommutative (p − 1)-form gauge param- eter. This leads to the following rule for the gauge transformation We start with a brief summary of the relevant facts concerning ∗ of coordinates xˆi := xi ,cf.(1): the Seiberg–Witten map as it applies in the present context. We a ρa ( ) refer the reader to a detailed exposition in [7]. All order solution   δ xˆi = xˆi ,Λ . (24) to the Seiberg–Witten map related to Nambu–Poisson M5-brane λ a a theory can be found in [8]. Hence, the generalized Seiberg–Witten map provides us with an Let us consider a p-form gauge potential a on M with cor- explicit construction, based on ordinary higher gauge fields, of responding field strength F = da. Infinitesimally, under a gauge the covariant coordinates xˆi that we introduced in Section 3.As transformation given by a (p − 1)-form λ, ˆi ≡ ˆi  ≡ a consequence, we can identify x xa and Π ΠF .Moreover, xˆi = xˆi = xi , for the “commutative” directions i = p + 2,...,n.All = = a δλa dλ, δλ F 0. (17) discussion of the previous Sections 3 and 4 applies directly. Using the (p + 1)-form F we construct from a given Nambu– Having the ordinary p-form gauge field a at our disposal we ˆ  Poisson tensor Π the F -gauged tensor which we denote for now can now define the full Nambu–Poisson field strength F with all 7 by ΠF , components (in noncommutative as well as in commutative direc-  −  − tions), such that its components in the noncommutative directions := − T 1 = − T 1 1 p+1 ˆ  ΠF 1 Π F Π Π 1 F Π . (18) x ,...,x coincide with those of FaB (14). For this let These expressions are to be interpreted as matrix equations for     the corresponding maps sending p-forms to 1-forms, cf. Section 2.  −1 −1 F := F 1 − Π T F = 1 − F Π T F (25) The superscript T stands for the transposed map. For p > 1, the 8 (p + 1)-tensor ΠF is always a Nambu–Poisson one, furthermore, and define we also have due to factorizability of Π, ˆ  := ∗   − FiJ ρA FiJ, (26) 1 1 ΠF = 1 − Π, F  Π, (19)  p + 1 i.e., the components of F evaluated in the covariant coordinates. It is a rather straightforward check to see that for all indices  = iJ ≡ T where Π, F Π FiJ Tr(Π F ). i1,...,i p+1 taking values only in the set {1,...,p + 1} we get ex-  Now we define a 1-parametric family of Nambu–Poisson ten- actly the Fˆ of (14). T −1 aB sors Π := (1 − tΠ F ) Π,cf.Footnote7, interpolating between  t Now we turn our attention to the remaining components of Fˆ Π and Π . Differentiation of Π with respect to t gives: F t (including commutative directions). Starting from (25) and (26), T we can with the help of Footnote 7 and the explicit expression ∂t Πt = Πt F Πt . (20) for Π in coordinates (10) use a construction very similar to the This equation can be rewritten as one leading to (16). We find that the resulting expressions in- volve a covariant scalar function that depends on Aˆ (and hence via =− ∂t Πt Πt , (21) the generalized Seiberg–Witten map also on the ordinary p-form LAt gauge potential a):

6 p  With some abuse of notation we allow also for the case p = q, i.e., the “1-ary”   p+1 ˆ (σ (k)−1) bracket, which will become useful later. f [A]:=1 + (−1) k=r+1 7 − T We assume that 1 Π F is invertible. In a more formal approach we also could r=0 σ ∈S(r,n−r) treat ΠF as a formal power series in Π. 8 ˆ ˆ σ (1)...σ (r) Even for a non-closed F . × sgn(σ ){A[σ (r+1)],...,A[σ (p+1)]} . 224 B. Jurˇco et al. / Physics Letters B 733 (2014) 221–225

ˆ  ˆ  Firstly, let us consider FaK with the index a taking on values in Shifting the components F1...p+1 of the Nambu–Poisson field { + } 1,...,p 1 , and K containing at least one index in one of the strength by the constants 1...p+1, will not affect the gauge in- commutative directions p + 2,...,n.Wefind variance of the action (31).Using(11) and (14) we see that the ˆ   action (31) with shifted F takes the form of a semiclassical ver- Fˆ = f [Aˆ]Fˆ , (27) aK aK sion of a Nambu–Poisson matrix model:   ˆ = ∗ n a A where FaK ρ FaK is the component FaK of the ordinary (com- S M = d x xˆ , xˆ {xˆa, xˆ A } mutative) field strength evaluated at the covariant coordinates xˆi .  Secondly, for the components of Fˆ with index k taking value in 1   n ˆa1 ˆap+1 ˆ ˆ = d x x ,...,x {xa ,...,xa + }, (34) {p + 2,...,n}, and A containing only the indices lying in the set p! 1 p 1 {1,...,p + 1}, where the summation in the second expression runs over all (not  strictly ordered) (p + 1)-indices (a ,...,a + ) and (b ,...,b + ), Fˆ = f [Aˆ]Fˆ . (28) 1 p 1 1 p 1 kA kA with all of them in the noncommutative direction. We could actu- ˆ  ally drop the a priori restriction of the summation to noncommu- Finally, for the components FkL, where k takes value in the set {p + 2,...,n} and L contains at least one index of the same set, tative directions, since the Nambu–Poisson bracket automatically we have takes care of this. For a more detailed discussion of the (semiclas- sical) matrix model we refer to [7]. + p 1 Given an appropriate quantization [·,...,·] of the Nambu– ˆ  = ˆ + [ ˆ] − a+1 ˆ ˆ FkL FkL f A ( 1) Fk[a] FaL. (29) Poisson bracket and trace of the quantized Nambu–Poisson struc- a=1 ture, the Nambu–Poisson matrix model takes the form Under (ordinary) infinitesimal gauge transformations δ , all com- 1   λ ˜ ˆa1 ˆap+1 ˆ ˆ ˆ  S M = Tr x ,...,x [xa ,...,xa + ]. (35) ponents of F transform as p! 1 p 1     There have been several attempts to find a consistent quantization δ Fˆ = Fˆ ,Λ , (30) λ of Nambu–Poisson structures. One of these [15] is in fact suitable justifying calling it “Nambu–Poisson” or “(semiclassically) noncom- for our purposes (at least in the case p = 2): It is an approach mutative” field strength. based on nonassociative star product algebras on phase space, Note that unlike for the noncommutative components, the full whose Jacobiator defines a quantized Nambu–Poisson bracket on  tensor Fˆ cannot be extended to be a totally antisymmetric one. configuration space. Let us mention without going into details that this approach can be adapted to provide a consistent quantization 6. Action of the Nambu–Poisson gauge theory described in this letter, includ- ing a quantization of the generalized Seiberg–Witten maps. Details For simplicity, we assume Euclidean space–time signature.9 The of this construction are beyond the scope of the present letter and action will be reported elsewhere.

1   dnxFˆ Fˆ iJ (31) Acknowledgements g iJ M It is a pleasure to thank Tsuguhiko Asakawa, Peter Bouwknegt, is by construction invariant under ordinary commutative as well Chong-Sun Chu, Pei-Ming Ho, Petr Horava,ˇ Dalibor Karásek, No- as under Nambu–Poisson (semiclassically noncommutative) gauge riaki Ikeda, Matsuo Sato, and Satoshi Watamura for helpful dis- transformations. This can easily be verified using partial integra- cussions. B.J. and P.S. appreciate the hospitality of the Center for tion. The coupling constant g is dimensionless in n = 2(p + 1) Theoretical Sciences, Taipei, Taiwan, ROC. B.J. thanks CERN for spacetime dimensions, i.e. for example for p = 1, n = 4(NC hospitality. We gratefully acknowledge support by grant GACR Maxwell) and for p = 2, n = 6 (M2–M5 system). In the following P201/12/G028 (B.J.), by the DFG (GRK 1620) within the Re- we will set g = 1. search Training Group 1620 “Models of Gravity” (J.V., P.S.), and  We expand Fˆ in a power series in Π by the Grant Agency of the Czech Technical University in Prague,   grant No. SGS13/217/OHK4/3T/14 (J.V.). We thank the DAAD (PPP) ˆ  = + kL + kL + 2 (54446342) and ASCR & MEYS (Mobility) (54446342) for support- FiJ FiJ ALΠ FiJ,k FiLΠ FkJ o Π . (32) ing our collaboration. The corresponding expansion of the action (31) is References n ˆ  ˆ  iJ n iJ 1 iJ kL d xF F = d x FiJF − FiJF FkLΠ iJ p + 1 [1] J. Madore, S. Schraml, P. Schupp, J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J. C 16 (2000) 161–167, arXiv:hep-th/0001203. M M   [2] P.-M. Ho, Y. Imamura, Y. Matsuo, S. Shiba, M5-brane in three-form flux and + 2F iJF ΠkL F + o Π 2 . (33) multiple M2-branes, J. High Energy Phys. 0808 (2008) 014, arXiv:0805.2898. iL kJ [3] P.-M. Ho, Y. Matsuo, A toy model of open membrane field theory in constant 3-form flux, Gen. Relativ. Gravit. 39 (2007) 913–944, arXiv:hep-th/0701130. A quantization of the underlying Nambu–Poisson structure will not [4] P.-M. Ho, Y. Matsuo, M5 from M2, J. High Energy Phys. 0806 (2008) 105, arXiv: add quantum corrections to the action at the given order of expan- 0804.3629. [5] P.-M. Ho, A concise review on M5-brane in large C-field background, Chin. sion. J. Phys. 48 (2010) 1, arXiv:0912.0445. [6] B. Jurco,ˇ P. Schupp, Nambu-sigma model and effective membrane actions, Phys. Lett. B 713 (2012) 313–316, arXiv:1203.2910. 9 Another simple possibility would be consider the Minkowskian space–time, [7] B. Jurco,ˇ P. Schupp, J. Vysoký, Extended generalized geometry and a DBI-type with Π extending in the spatial directions only. In case of a general metric g we effective action for branes ending on branes, in preparation, http://arxiv.org/ would have to use the inverse metric matrix elements evaluated in the covariant abs/1404.2795.  coordinates to rise the indices of Fˆ and the density defined by the metric also [8] C.-H. Chen, K. Furuuchi, P.-M. Ho, T. Takimi, More on the Nambu–Poisson evaluates in the covariant coordinates. M5-brane theory: scaling limit, background independence and an all order so- B. Jurˇco et al. / Physics Letters B 733 (2014) 221–225 225

lution to the Seiberg–Witten map, J. High Energy Phys. 1010 (2010) 100, arXiv: [12] P.-M. Ho, Gauge symmetries from Nambu–Poisson brackets, Universe 1 (4) 1006.5291. (2013) 46. [9] N. Seiberg, E. Witten, String theory and noncommutative geometry, J. High En- [13] P.-M. Ho, C.-T. Ma, S-duality for D3-brane in NS–NS and R–R backgrounds, ergy Phys. 9909 (1999) 032, arXiv:hep-th/9908142. arXiv:1311.3393. [10] B. Jurco,ˇ P. Schupp, J. Wess, Noncommutative gauge theory for Poisson mani- [14] P.-M. Ho, C.-T. Ma, Effective action for Dp-brane in large RR (p − 1)-form back- folds, Nucl. Phys. B 584 (2000) 784–794, arXiv:hep-th/0005005. ground, J. High Energy Phys. 1305 (2013) 056, arXiv:1302.6919. [11] B. Jurco,ˇ P. Schupp, J. Wess, Nonabelian noncommutative gauge theory via [15] D. Mylonas, P. Schupp, R.J. Szabo, Membrane sigma-models and quantization of noncommutative extra dimensions, Nucl. Phys. B 604 (2001) 148–180, arXiv: non-geometric flux backgrounds, J. High Energy Phys. 1209 (2012) 012, arXiv: hep-th/0102129. 1207.0926.

Statutory Declaration (on Authorship of a Dissertation)

I, ______hereby declare that I have written this PhD thesis independently, unless where clearly stated otherwise. I have used only the sources, the data and the support that I have clearly mentioned. This PhD thesis has not been submitted for conferral of degree elsewhere.

I confirm that no rights of third parties will be infringed by the publication of this thesis.

Bremen, February 12, 2015

Signature ______