Aspects of Generalized Geometry: Branes with Boundary, Blow-ups, Brackets and Bundles
Charlotte Sophie Kirchhoff-Lukat
Department of Applied Mathematics and Theoretical Physics University of Cambridge
This dissertation is submitted for the degree of Doctor of Philosophy
Trinity College September 2018
Declaration
I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other university. This dissertation is my own work and details on all collaborations are declared in Chapter 1.
Charlotte Sophie Kirchhoff-Lukat September 2018
Acknowledgements
I would like to extend sincere thanks to the many people who helped and supported me throughout my PhD research: My supervisor Professor Malcolm Perry showed great patience and encouraged me to find and pursue my own research interests. Professor Marco Gualtieri proposed the project that became Part I of this thesis, and supervised and guided me throughout my research. I further thank him and the Department of Mathematics at the University of Toronto for hosting me during a productive and intellectually stimulating long-term research visit. I learned mathematics at a faster rate during this time than at any other time in my mathematical education. Professor Madeleine Jotz-Lean advised me throughout the course of my PhD, and we had a productive and close collaboration which resulted in the publication on which Part II is based. Professor Gil Cavalcanti and the Mathematical Institute at Utrecht University hosted me during a short research visit. I value our discussions on toric and stable generalized complex Lefschetz fibrations. Dr. David Skinner, my departmental adviser at DAMTP, supported and guided me regarding the progression of my PhD, and gave helpful advice concerning my future career. Trinity College, Cambridge, has not only supported me financially with an Internal Grad- uate Studentship, but also provided an excellent academic and living environment. I further thank the college for awarding me the Rouse-Ball Travelling Scholarship in Mathematics, which enabled my research visit to the University of Toronto. Special thanks to Dr. Jean Khalfa and the whole tutorial team, who helped me both in academic and in personal matters. The STFC covered my university and college fees for the duration of my PhD degree, and the Cambridge Philosophical Society awarded me a Research Studentship to fund the continuation of my research in my final year. Professor Nigel Hitchin and Professor Ivan Smith each took the time to discuss my research project in-depth across several meetings, taught me about Hyperkähler geometry and the symplectic flux respectively, and advised me regarding the next steps. Sincere thanks to the examiners of my thesis, Dr. Ailsa Keating and Professor Gil Cavalcanti for their careful scrutiny of this text, for the mathematical discussion of the content, and for their helpful critique of my mathematical writing. Throughout my PhD I have had the opportunity to discuss various aspects of Poisson and symplectic geometry in particular, and geometry more broadly, with a number of inspiring mathematical colleagues. These discussions have helped to broaden my mathematical knowledge and interests. Dr. Ana Rita Pires, Professor Eva Miranda and Dr. Geoffrey Scott have helped me deepen my knowledge of log symplectic geometry, Francis Bischoff and Dr. Nikita Nikolaev answered many questions on various aspects of geometry, Davide Alboresi and I discussed pseudoholomorphic curves and Floer theory, and Alec Barns-Graham and I exchanged ideas on quantum field theory, flag manifolds, quivers and other areas where mathematics and physics intersect. Finally, I wish to thank my parents Christiane Kirchhoff and Rüdiger Lukat for their unwavering both practical and emotional support and encouragement in all matters during the entirety of my studies, my husband Hans Schmidt, and my close friends in the UK and in Germany, and in particular my fellow DAMTP PhD students Chandrima Ganguly and Markus Kunesch for making thesis-writing much more enjoyable and less solitary.
vi Abstract
This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form TM ⊕ E∗,E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side.
Contents
Notation xiii
1 Introduction 1
2 Generalized Geometry 7 2.1 Courant algebroids ...... 7 2.2 Leibniz algebroids ...... 10 2.3 Generalized geometric structures ...... 11 2.3.1 Generalized complex geometry ...... 13 2.3.2 Generalized complex branes ...... 17 2.4 Generalized geometry in theoretical high-energy physics ...... 18 2.4.1 Double and exceptional field theory ...... 19 2.4.2 String compactifications ...... 21
3 Double vector bundles 23 3.1 Double vector bundles, duals and linear splittings ...... 23 3.2 The tangent double and the cotangent double of a vector bundle ...... 27 3.3 The first jet bundle of a vector bundle...... 29
I Lagrangian branes and symplectic methods for stable generalized complex manifolds 31
4 Logarithmic and elliptic symplectic geometry 33 4.1 Logarithmic symplectic geometry ...... 34 4.2 Complex log symplectic and elliptic symplectic geometry ...... 36 4.3 Stable generalized complex structures ...... 41 4.3.1 Generalized complex branes in stable generalized complex manifolds . 42
5 Log and elliptic symplectomorphisms and flux 43 5.1 The groupoid exponential map ...... 44 Contents
5.2 The Flux homomorphism for log and elliptic symplectic manifolds ...... 47 5.3 Lagrangian neighbourhood theorem for log symplectic manifolds ...... 52 5.4 Small deformations of Lagrangians in log symplectic manifolds ...... 55 5.4.1 Log and elliptic Lagrangians as coisotropic submanifolds in a Poisson manifold ...... 57
6 Lagrangian branes with boundary in stable generalized complex manifolds 61 6.1 Definition and basic properties ...... 61 6.2 Real oriented blow-up of the anticanonical divisor ...... 65 6.2.1 Lagrangian branes under blow-up ...... 74 6.2.2 Neighbourhoods of Lagrangian branes with boundary ...... 77 6.3 Small deformations of Lagrangian branes with boundary ...... 80
7 Lefschetz thimbles in stable generalized complex Lefschetz fibrations 85 7.1 Ehresmann connections for log symplectic Lefschetz fibrations ...... 87 7.2 Stable generalized complex Lefschetz fibrations under blow-up ...... 90 7.3 Lefschetz thimbles in stable generalized complex Lefschetz fibrations and examples of Lagrangian branes with boundary ...... 92 7.3.1 Example: The Hopf surface ...... 94 7.3.2 Examples of genus-one boundary Lefschetz fibrations over the disk .. 97
8 Lagrangian branes with boundary and complex branes in holomorphic log symplectic manifolds 101 8.1 Example: C × T 2 ...... 102 8.2 Example: Hopf Surface ...... 103 8.3 Example: CP 2 ...... 106
9 Conclusions and Outlook: Stable Hamiltonian systems from stable gener- alized complex manifolds 109
II Dorfman brackets and double vector bundles 113
10 Brackets on TM ⊕ E∗ and the standard VB-Courant algebroid 115 10.1 Dorfman brackets and dull brackets ...... 115 10.2 The E∗-valued Courant algebroid structure on the fat bundle Eb ...... 118 10.3 Linear sections of TE ⊕ T ∗E → E ...... 120 10.3.1 Linear closed 3-forms ...... 123
11 Dorfman brackets and natural lifts 127 11.1 The equivalence of Dorfman brackets and natural lifts ...... 127
x Contents
11.2 Links to known results on Omni-Lie algebroids, on Dorfman connections and on the standard VB-Courant algebroid ...... 130 11.3 Standard examples ...... 131 11.3.1 Lift of the Courant-Dorfman bracket ...... 131 11.3.2 Another lift to TTM ⊕ T ∗TM ...... 133 11.3.3 More general examples ...... 134 11.4 Twisted Courant-Dorfman bracket over vector bundles ...... 135 11.5 Symmetries of Dorfman brackets ...... 137
Appendix A Details of proofs 141 A.1 On the proofs of Theorems 10.10 and 10.11 ...... 141 A.2 On the proof of Theorem 11.4 ...... 143
Appendix B A non-local Leibniz algebroid 145
Bibliography 147
xi
Notation
Unless otherwise specified, manifolds in this text are smooth manifolds, vector bundles are real and smooth vector bundles and submanifolds are embedded. M,N Smooth manifolds. N is also used to denote a submanifold.
X(M) The sheaf of smooth vector fields on M. Similarly, the sheaves of degree-k multi- vector fields will be denoted by Xk(M).
X(M, − log Z) The sheaf of sections of TM(− log Z), i.e. the sheaf of logarithmic vector fields on M with respect to the codimension-1 submanifold Z. Similarly the elliptic vector fields on M with respect to the elliptic divisor D will be denoted by X(M, − log |D|). iX α Insertion of a vector (field) X ∈ X(M) into the first component of a differential form α ∈ Ωk(M).
£X · Lie derivative with respect to the vector field X ∈ X(M).
E Vector bundle E → M.
Γ(E) The sheaf of smooth sections of E.
Der(E) The space of derivations of Γ(E), i.e. the space of R-linear maps δ : Γ(E) → Γ(E) for which there exists a vector field X ∈ X(M) (referred to as the symbol of δ) such that δ(fe) = fδ(e) + X(f)e ∀e ∈ Γ(E), f ∈ C∞(M). tot(E) The total space of a vector bundle E (as a manifold). prA Projection onto the named summand A of a direct sum A ⊕ B: prA : A ⊕ B → A.
·, · This notation is both used for the Courant-Dorfman bracket (see equation (2.2)), as J K 3 well as for more general Dorfman brackets (see Definition 10.1). ·, · H ,H ∈ Ωcl(M) J K is the H-twisted Courant-Dorfman bracket.
2 2 π, ω π ∈ X (M), ω ∈ Ωcl(M) are used to denote Poisson structures and (log/elliptic) symplectic forms respectively. The same notation is used for the associated vector bundle morphisms π : T ∗M → TM, ω : TM → T ∗M.
xiii Contents e↑ The canonical vertical lift of a vector bundle section e ∈ Γ(E) to a vertical ↑ ′ vector field (i.e. one tangent to the fibres of E) in X(tot E), given by e (em) = d ′ (em + te(m)). dt t=0
Db If D : Γ(E) → Γ(E) is a derivation on a vector bundle E, Db is the corresponding linear vector field on E given by equation (3.5).
ℓε The linear function on a vector bundle E given by ℓε(em) = ⟨ε(m), em⟩, where ε ∈ Γ(E∗).
ψe If (D; A, B; M) is a double vector bundle (see Definition 3.1) with core C → M, we can associate to each section ψ ∈ Γ(B∗ ⊗ C) a core linear section ψe. See Section 3.1.
xiv Chapter 1
Introduction
Generalized geometry is a framework introduced by Hitchin and subsequently studied by his students in the early 2000s, although the study of some of its basic underlying structures, such as Courant algebroids, is far older. Its basic general principle is: Instead of considering geometric structures on the tangent bundle TM of a smooth manifold M, often with suitable integrability conditions, one considers geometric structures on the direct sum of the tangent and cotangent bundles TM ⊕ T ∗M, again frequently including an integrability condition. The Lie bracket on vector fields X(M) is replaced with the so-called Courant-Dorfman bracket on Γ(TM ⊕ T ∗M). While at first this may seem arbitrary, several of these structures, Dirac structures and generalized complex structures among them, have garnered wide-spread interest because they provide a unified description of a priori distinct conventional geometric structures. For example, generalized complex manifolds include both complex and symplectic manifolds as examples. The development of aspects of the theory has furthermore been guided and inspired by phenomena in theoretical physics, and generalized geometry has proved very adept at providing new descriptions and insights into phenomena that arise in string theory. The double tangent bundle TM ⊕ T ∗M is naturally equipped with a number of additional structures: a non-degenerate symmetric bilinear form of split signature, the projection to TM, and the Courant-Dorfman bracket on Γ(TM ⊕ T ∗M), a generalisation of the Lie bracket on vector fields. Together these make up the so-called standard Courant algebroid. Courant algebroids themselves are a fertile field of study, and various generalisations and relations to higher geometric structures have appeared in the literature in recent years, for example the integration of so-called exact Courant algebroids to symplectic 2-groupoids [LBŠ12]. Defining geometric structures on other Courant algebroids constitutes a further generalisation of generalized geometry. Several such generalisations can be found in the mathematical and theoretical physics literature. This thesis consists of two parts of rather different content and flavour, concerning those two different branches of generalized geometry research: Part I focuses entirely on a subclass
1 Introduction of generalized complex manifolds, so-called stable generalized complex manifolds, and their submanifolds. Stable generalized complex manifolds are very similar to symplectic manifolds: They are equipped with a generically non-degenerate Poisson structure that only degenerates on a codimension-2 submanifold. This allows us to extend a number of techniques from symplectic to stable generalized complex manifolds. The new type of submanifold introduced in this text is called Lagrangian branes with boundary. They are Lagrangian in the symplectic bulk of the manifold and intersect the degeneracy locus in their boundary. Even though Lagrangian branes with boundary are not generalized complex branes, which is the usual notion of submanifold without boundary in generalized complex geometry, they arise naturally in examples, notably as Lefschetz thimbles in the stable generalized complex analogue of Lefschetz fibrations:
Example. (See Example 7.21.) CP 2 admits a stable generalized complex structure and a compatible boundary Lefschetz fibration (which is a regular genus-2 Lefschetz fibration over the open disk for the symplectic bulk of the manifold) with 3 Lefschetz singularities. The associated Lefschetz thimbles are Lagrangian branes with boundary.
We study these submanifolds as a first step towards the aim of extending Lagrangian intersection Floer homology to (stable) generalized complex geometry and ultimately con- structing a Fukaya category, which we hope to achieve in future work. As Lefschetz thimbles, Lagrangian branes with boundary should be considered as objects in a construction involving Fukaya-Lefschetz theory. The homological mirror symmetry conjecture [Kon94] relates the derived Fukaya category associated to a symplectic manifold to the derived category of coherent sheaves of a complex manifold. Since both complex and symplectic manifolds are examples of generalized complex manifolds, a natural question is whether there is a (derived) category associated to each generalized complex manifold, of which the Fukaya category and the category of coherent sheaves are examples. Currently, neither mirror symmetry nor generalized complex geometry are sufficiently well understood to make this question tractable. But understanding stable generalized complex manifolds and their Lagrangian submanifolds could serve as a stepping stone towards a more general approach. Stable generalized complex manifolds turn out to be in correspondence with a particular subclass of logarithmic symplectic manifolds, generically symplectic Poisson manifolds whose Poisson structures degenerate on a codimension-1 submanifold:
Theorem. (See Theorems 6.13 and 6.15, Lemma 6.14.) Under real oriented blow-up of the degeneracy locus, every stable stable generalized complex manifold is related to a logarithmic symplectic manifold with boundary. Conversely, every logarithmic symplectic manifold with boundary whose boundary is a principal U(1)-bundle and whose logarithmic symplectic form satisfies a set of compatibility conditions involving the U(1)-action vector field can be blown down to a stable generalized complex manifold. These processes are inverse and
2 the blow-down map defines an isomorphism of Lie algebroid cohomology associated to the respective manifolds.
Crucially, this process can also be applied to stable generalized complex Lefschetz fibrations to construct Lefschetz thimbles like in the example above (see Proposition 7.17). Lagrangian branes lift under this correspondence and conversely blow down to submani- folds with and without boundary depending on their intersection with the boundary of the blow-up. This allows us to apply the Lagrangian neighbourhood theorem for logarithmic symplectic geometry, Theorem 5.18, to find a standard local normal form around a Lagrangian brane with boundary, and to determine small deformations of Lagrangian branes with and without boundary:
Theorem. (See Theorem 6.22 and Remark 5.26) Let M be a stable generalized complex manifold with degeneracy locus D.
(i) Let L be a Lagrangian generalized complex brane which intersects D transversely. Small deformations of L up to Hamiltonian isotopy are given by H1(L, log |L ∩ D|), the natural first Lie algebroid cohomology associated to L.
(ii) Let (L, ∂L) be a Lagrangian brane with boundary, in particular L ∩ D = ∂L. Small deformations of L up to Hamiltonian isotopy are given by H1(L, log ∂L), the first logarithmic cohomology, the natural Lie algebroid cohomology associated to a manifold with a chosen codimension-1 submanifold.
Part II studies particular generalisations of the Courant-Dorfman bracket, which we simply refer to as Dorfman brackets, in the context of double vector bundles, a certain type of vector bundle whose base space is itself a vector bundle. The double vector bundle in question is the standard VB-Courant algebroid TE ⊕ T ∗E → E (where E → M is a smooth vector bundle). VB-Courant algebroids are studied in [LB12] and constitute a natural class of Courant algebroids on double vector bundles. It turns out that Dorfman brackets can be viewed as restrictions of the Courant-Dorfman bracket, which is thus in a certain sense universal. This relation fits into a wider pattern of geometric structures obtained as substructures or restrictions of the standard VB-Courant algebroid. Based on the bijection between the linear sections of the standard VB-Courant algebroid, sections that are also vector bundle morphisms, and sections of the Omni-Lie algebroid of E∗ [CL10, CLS11], which we prove, we obtain standard form for linear sections:
Theorem. (See Theorem 10.8.) Linear sections χ of TE ⊕ T ∗E → E are in one-to-one- ∗ ∗ correspondence with pairs consisting of a derivation dχ of E and a tensor φχ ∈ Γ(E ⊗ T M) ∗ which together define a derivation Dχ of TM ⊕ E .
This result can be applied to find a characterisation of Dorfman brackets on TM ⊕ E∗ in terms of the standard VB-Courant algebroid:
3 Introduction
Theorem. (See Theorem 11.2.) Dorfman brackets ·, · on TM ⊕ E∗ are in one-to-one J K correspondence with R-linear maps Ξ from Γ(TM ⊕ E∗) to the linearsections of TE ⊕ T ∗E which are natural with respect to the Courant-Dorfman bracket on TE ⊕ T ∗E, also denoted by ·, · : J K Ξ(ν1), Ξ(ν2) = Ξ ν1, ν2 J K J K This is just one of a number of cases where a geometric structure on TM ⊕ E∗ can be characterised in terms of a subspace of sections of TE ⊕ T ∗E, which we summarize in Section 11.2. These connections hint at a more general relationship that could be explored in future work.
In Chapters 2 and 3 of this thesis we provide a general review of background material on generalized geometry and double vector bundles: We introduce Courant algebroids and generalized geometric structures, and provide an overview of the applications of generalized geometry to theoretical physics. We then review the definition and basic properties of double vector bundles and the most important examples, the tangent and cotangent bundle of a smooth vector bundle. Chapter 3 and parts of Section 2.1 are based on the introductory section in [JK16], and in part follow the exposition in [JL18]. Part I arose out of a collaboration with Professor Marco Gualtieri at the University of Toronto, who proposed the project and supervised me during the process of research. Chapter 4 contains background material on logarithmic and elliptic symplectic geometry, particular generalisations of symplectic geometry which involve generically non-degenerate Poisson structures that only drop in rank on lower-dimensional submanifolds. Section 5.1 reviews groupoid bisections and the exponential map, while sections 5.2 to 5.4 contain original results: We generalise the concept of symplectic flux to logarithmic and elliptic symplectic geometry, prove a Lagrangian neighbourhood theorem for Lagrangian submanifolds in logarithmic symplectic manifolds, and employ these results to find the space of small deformations for such logarithmic Lagrangians. In Chapter 6, which is entirely original, we introduce the main object of study in this part of the thesis: Lagrangian branes with boundary in stable generalized manifolds. The main result in Section 6.2 is a theorem relating stable generalized complex structures to particular logarithmic symplectic structures via a real oriented blow-up, which allows us to also define a standard local neighbourhood for branes with boundary. Finally, we apply the result on small deformations of Lagrangians in logarithmic symplectic manifolds to find the small deformations of Lagrangian branes with boundary. Chapter 7 describes the natural generalisation of the concept of Lefschetz fibration to logarithmic and elliptic symplectic geometry as introduced by [CK16, CK17] (background and derivative material are clearly marked with citations), as well as our own result on the connections between the two. We also for the first time investigate Lefschetz thimbles, natural Lagrangian submanifolds in Lefschetz fibrations, in this context and show that Lagrangian branes with boundary arise as such. Finally, in Chapter 8 we present a range of examples of
4 Lagrangian branes with boundary in holomorphic logarithmic symplectic geometry, which can be related to complex submanifolds by local Hamiltonian isotopies, and in Chapter 9 provide an outlook on future work: With what is now known on Lagrangian-type submanifolds in stable generalized complex manifolds, we aim to generalize Lagrangian intersection Floer homology to this context, and eventually construct an analogue of the Fukaya category. Part II is based on an equal-parts collaboration with Dr. Madeleine Jotz-Lean (Georg- August-Universität Göttingen) initiated by the author. The text has in a similar form been submitted for publication and is available in pre-print form: [JK16] Section 10.1 contains a short review of prerequisites, whereas the rest of Chapter 10 and all of Chapter 11 are largely original work (citations of previously known results are clearly marked). Chapter 10 studies the standard VB-Courant algebroid TE ⊕ T ∗E over a smooth vector bundle E → M. We prove an original theorem characterising its so-called linear sections, which can also be viewed as the sections of the Omni-Lie algebroid, a vector bundle associated to each smooth vector bundle E → M introduced in [CL10]. We provide a proof of this relationship. Chapter 11 employs our characterisation of linear sections to prove the main result of this part: An equivalence that allows us to view Dorfman brackets on vector bundles of the form TM ⊕ E∗,E → M a smooth vector bundle, as restrictions of the standard Courant-Dorfman Dorfman bracket on TE ⊕ T ∗E via a natural lift. We discuss our result in the context of other relations between brackets and lifts to the standard VB-Courant algebroids and present a range of examples.
5
Chapter 2
Generalized Geometry
2.1 Courant algebroids
The central structure in generalized geometry is the vector bundle TM ⊕T ∗M → M equipped ∗ with the natural projection prTM : TM ⊕ T M → TM, the canonical non-degenerate symmetric bilinear form ⟨·, ·⟩ and the so-called Courant-Dorfman bracket, a generalisation of the Lie bracket on X(M). This bracket was first studied by Courant [Cou90] in the antisymmetric form
∗ ∗ ∗ ·, · C : Γ(TM ⊕ T M) × Γ(TM ⊕ T M) → Γ(TM ⊕ T M), J K 1 (X, θ), (Y, η) C = [X,Y ] + £X η − £Y θ − d (iX η − iY θ) (2.1) J K 2 and by Dorfman [Dor87], who proposed the alternative form1
(X, θ), (Y, η) = [X,Y ] + £X η − iY dθ (2.2) J K These brackets are equivalent and related by
A, B C = A, B − B,A (2.3) J K J K J K A, B = A, B C + d⟨A, B⟩ (2.4) J K J K ∗ (TM ⊕ T M, ·, · , ⟨·, ·⟩, prTM ) is today called the standard Courant algebroid and usually J K considered with the bracket in Dorfman form:
Definition 2.1. [Cou90] The direct sum of the tangent and cotangent bundle TM ⊕ T ∗M
together with the projection ρ = prTM as an anchor map, the symmetric split-signature
1According to [KS13], Dorfman’s development of the bracket was independent of Weinstein and Courant, whose bracket does not yet appear in their first article on Dirac structures, [CW88].
7 Generalized Geometry pairing ⟨· , ·⟩ defined by
⟨(Xm, θm), (Ym, ηm)⟩ = θm(Ym) + ηm(Xm) (2.5)
∗ for all m ∈ M, vm, wm ∈ TmM and αm, βm ∈ TmM and the Courant-Dorfman bracket ∗ (X, θ), (Y, η) = ([X,Y ], £X η − iY dθ) for all (X, θ), (Y, η) ∈ Γ(TM ⊕ T M), form the J K standard example of a Courant algebroid, called the standard Courant algebroid over M.
The general concept of Courant algebroid was first introduced by Liu, Weinstein and Xu [LWX97]. In the modern literature, there are several further generalisations. We now define the notion of Courant algebroid with pairing in a vector bundle V → M, of which ordinary Courant algebroids are a subclass:
Definition 2.2. ([PP01a], Section 1., [PP01b], Section 1.) An anchored vector bundle is a vector bundle Q → M endowed with a vector bundle morphism ρQ : Q → TM over the identity.
Definition 2.3. (Defined in [CLS10], Definition 2.1. Note that there are multiple notions of Courant algebroids whose pairing takes values in a real vector bundle, however, these are not all obviously equivalent, or equivalent at all. The particular notion used in this text appears first in [JL15], Section 5.1.) Let V → M be a vector bundle. A Courant algebroid with pairing in V is an anchored vector bundle (E, ρ) over the same base M together with a morphism ρe: E → Der(V ), a non-degenerate pairing ⟨· , ·⟩: E ×M E → V with values in V , and an R-bilinear bracket · , · on the smooth sections Γ(E), which satisfy the following J K conditions:
(i) The symbol of ρe(e) is ρ(e) ∈ X(M) for all e ∈ Γ(E).
(ii) e1, e2, e3 = e1, e2 , e3 + e2, e1, e3 , J J KK JJ K K J J KK (iii) ρe(e1)⟨e2, e3⟩ = ⟨ e1, e2 , e3⟩ + ⟨e2, e1, e3 ⟩, J K J K (iv) e1, e2 + e2, e1 = D⟨e1, e2⟩, J K J K (v) ρe e1, e2 = [ρe(e1), ρe(e2)] J K ∞ for all e1, e2, e3 ∈ Γ(E) and f ∈ C (M). D : Γ(V ) → Γ(E) is given by ⟨Dv, e⟩ = ρe(e)(v) for all v ∈ Γ(V ).
Note that
(vi) e1, fe2 = f e1, e2 + (ρ(e1)f)e2 J K J K ∞ for e1, e2 ∈ Γ(E) and f ∈ C (M) follows immediately from (iii).
8 2.1 Courant algebroids
A Courant algebroid is a Courant algebroid with pairing in the trivial bundle V = R×M → M [LWX97, Roy99]. In this case D = ρ∗ ◦d: C∞(M) → Γ(E), where E is identified ∗ with E via the pairing, ρe = ρ, and (v) follows from (iii) and the non-degeneracy of the pairing (see [Uch02], Remark 2.3). Example 2.1 does indeed constitute a Courant algebroid in the sense of this definition.
Most relevant for generalized geometry are exact Courant algebroids:
Definition 2.4. (See [Š98], “Some trivialities on C. a.’s (classification of exact ones)”.) A Courant algebroid E → M is called exact if the sequence
ρ∗ ρ 0 → T ∗M → E → TM → 0 (2.6) is exact.
Note that the inclusion ρ∗ : T ∗M → E is isotropic. It is always possible to choose an ∼ ∗ isotropic splitting s : TM → E of the sequence (2.6), so (E, ρ) = (TM ⊕ T M, prTM ) as an anchored vector bundle. As first described by Ševera in letters to Weinstein [Š98], this immediately allows us to classify exact Courant algebroids on a smooth manifold M by classes [H] ∈ H3(M): The splitting s allows us to transport the Courant bracket ·, · on E J K to the H-twisted Courant-Dorfman bracket
∗ (X, θ), (Y, η) H = [X,Y ] + £X η − iY dθ + iY iX H, (X, θ), (Y, η) ∈ Γ(TM ⊕ T M), J K 3 where H ∈ Ωcl is a closed 3-form. The cohomology class [H] determines the exact Courant algebroid E up to isomorphism, and picking a different splitting for E also only changes H by an exact form.
Definition 2.5. (See [ŠW01], §3. The exposition in this text follows [Gua11].) A B-transform of TM ⊕ T ∗M is an isomorphism
B ∗ ∗ 2 e : TM ⊕ T M → TM ⊕ T M, (Xm, θm) 7→ (Xm, θm + iXm Bm), for some B ∈ Ω (M)
Proposition 2.6. (Symmetries of exact Courant algebroids)(See [ŠW01], §3. This text follows Proposition 2.2 in [Gua11].)
(i) The B-transform eB naturally maps the exact Courant algebroid (TM ⊕ T ∗M,H) to ∗ (TM ⊕ T M,H + dB), in particular it leaves the bracket ·, · H invariant iff dB = 0 J K and otherwise constitutes an isomorphism of exact Courant algebroids.
(ii) The full group of symmetries of an exact Courant algebroid E = (TM ⊕ T ∗M,H) is therefore 2 Symm(E) = Ωcl(M) ⋊ Diff[H](M),
9 Generalized Geometry
where Diff[H](M) is the subgroup of diffeomorphisms of M that preserve the cohomology class [H]. In particular, the symmetries of the standard Courant algebroid (TM ⊕ T ∗M, 0) are
∗ 2 Symm(TM ⊕ T M) = Ωcl(M) ⋊ Diff(M).
The only symmetries of the Lie bracket [·, ·] on X(M) are the diffeomorphisms of M. So when passing from ordinary geometry, which considers the tangent bundle with the Lie bracket, to generalized geometry, which considers exact Courant algebroids with a Courant bracket, we consider an enlarged group of symmetries in which diffeomorphisms are a proper subgroup.
2.2 Leibniz algebroids
Leibniz algebroids, also referred to as Loday algebroids in parts of the literature, constitute an approach to generalising Lie algebroids, taking the Courant-Dorfman bracket as a model. In this particular form of generalisation, we drop the requirement for the bracket to be antisymmetric and only demand that it satisfies the Leibniz identity for functions in its second entry, while still satisfying the Jacobi identity in Leibniz form. Clearly Courant algebroids constitute a particular subclass of Leibniz algebroid. While exact Courant algebroids with H-twisted Courant-Dorfman bracket are the model for the so-called C-bracket in double field theory, its generalisations in exceptional field theory are modelled on more general Leibniz algebroid brackets. In fact, all of these particular Leibniz algebroids are examples of Dorfman brackets, which we study in detail in Part II of this thesis. In Section 10.1, we present the most important examples, and in Appendix B we give an example of a non-local Leibniz algebroid. Certain concepts from generalized geometry (based on exact Courant algebroids) generalize to particular Leibniz algebroids.
Definition 2.7. (i) (Definition 3.1 in [IdLMP99]. See [Lod93], Section 1., for the earlier notion of Leibniz algebra.) A Leibniz algebroid is an anchored vector bundle (Q → M, ρ)
endowed with a bracket · , · on Γ(Q) satisfying q1, fq2 = f q1, q2 + ρ(q1)(f)q2 ∀f ∈ ∞ J K J K J K C (M), q1, q2 ∈ Γ(Q) and the Jacobi identity in Leibniz form
q1, q2, q3 = q1, q2 , q3 + q2, q1, q3 J J KK JJ K K J J KK
for all q1, q2, q3 ∈ Γ(Q).
(ii) A Leibniz algebroid Q is transitive if the anchor ρ : Q → TM is surjective.
(iii) ([Bar12], Definition 3.5.) A transitive Leibniz algebroid is split if there is a section σ : TM → Q of the anchor map such that σ(X(M)) is closed under the Leibniz bracket.
10 2.3 Generalized geometric structures
It follows from the definition of Leibniz algebroid that
ρ q1, q2 = [ρ(q1), ρ(q2)] for all q1, q2 ∈ Γ(Q). J K [Bar12] Now consider a split transitive Leibniz algebroid Q: It forms a split short exact sequence of vector bundles: ρ 0 → E∗ ,→ Q → TM → 0 (2.7) with E∗ = ker ρ, which we view as the dual to a vector bundle E → M. The splitting map σ : TM → Q induces an isomorphism Q =∼ TM ⊕ E∗. Since σ(X(M)) is closed under the
Leibniz bracket and ρ ◦ σ = idTM , we have σ(X), σ(Y ) = σ[X,Y ], so this is the requirement J K that the splitting be compatible with the Leibniz algebroid bracket. In [Bar12], Baraglia studies a particular subclass of Leibniz algebroids, called closed-form algebroids, whose Dorfman bracket can be expressed as the derived bracket of an L∞-algebroid. All commonly studied Leibniz algebroids, including those relevant for the theoretical physics applications, are such closed-form algebroids.
2.3 Generalized geometric structures
Generalized geometry studies geometric structures on exact Courant algebroids, and as the name of this framework suggests, ordinary differential geometric structures can be viewed as examples of their generalized counterparts – one of the most appealing aspects of generalized geometry is that in several instances different ordinary geometric structures are examples of the same generalized structure. Generalized geometric structures can usually be phrased in terms of specific subbundles of TM ⊕T ∗M or the exact Courant algebroid under consideration. In this section, we will introduce the most important examples of generalized geometric structures, including generalized complex structures, the subject of Part I of this text.
Definition 2.8. [Cou90] Let E → M be an exact Courant algebroid.
(i) An almost Dirac structure is is a subbundle L ⊂ E of rank n = dim M, which is isotropic with respect to ⟨·, ·⟩ (i.e. a maximal isotropic subbundle).
(ii) A Dirac structure is an almost Dirac structure which is also integrable w.r.t. to the (twisted) Courant-Dorfman bracket:
∀A, B ∈ Γ(L): A, B H ∈ Γ(L). (2.8) J K
Lemma 2.9. (Proposition 2.3.3 in [Cou90]) An almost Dirac structure L is Dirac if and only if ⟨ A, B ,C⟩ = 0 ∀A, B, C ∈ Γ(E). (2.9) J K 11 Generalized Geometry
Remark 2.10. (See [LWX97]. We follow [Gua03] for the explicit exposition.) The Jacobiator of the Courant-Dorfman bracket is given by Jac(A, B, C) = d(Nij(A, B, C)), where
1 Nij(A, B, C) = (⟨ A, B ,C⟩ + ⟨ B,C ,A⟩ + ⟨ C,A ,B⟩) , A, B, C ∈ Γ(E) (2.10) 3 J K J K J K
L is a Dirac structure if and only if Nij |L vanishes. Note that the restriction of Nij to an isotropic subbundle is a tensor.
Dirac structures probably constitute the most important example of a generalized geo- metric structure, and Dirac geometry has bloomed into an active field of research. Examples of Dirac structures include:
Example 2.11. (Poisson structures) Let π ∈ X2(M) be a Poisson bivector. Then
∗ ∗ Lπ = Graph(π : T M → TM) = {ξ + iξπ|ξ ∈ T M}
∗ is a Dirac structure in the standard Courant algebroid TM ⊕ T M. The integrability of Lπ is precisely equivalent to the Jacobi identity for π: Nij |L is tensorial, so it is sufficient to consider it on exact one-forms: 1 Nij |L(idf π + df, idgπ + dg, idhπ + dh) = (⟨ idf π + df, idgπ + dg , idhπ + dh⟩ + cyclic perm.) 3 J K 1 = ⟨−i π + d{f, g}, i π + dh⟩ + cyclic perm. 3 d{f,g} dh 2 = ({h, {f, g}} + cyclic perm.) 3
2 Example 2.12. (Presymplectic structures) Similarly, if ω ∈ Ωcl(M) is a presymplectic structure, ∗ Lω = Graph(ω : TM → T M) = {X + iX ω|X ∈ TM}
∗ is a Dirac structure in the standard Courant algebroid TM ⊕ T M; the integrability of Lω is equivalent to the closedness of ω. ∗ Clearly Lω ∩ T M = {0}. If ω is symplectic, i.e. non-degenerate, Lω satisfies Lω ∩ TM = {0} ∗ and Lω ∩ T M = {0}. In fact, symplectic structures on M are in one-to-one correspondence with Dirac structures in the standard Courant algebroid that satisfy these two properties.
Example 2.13. (Foliations) Let F ⊂ TM be a distribution of constant rank on M. Then
∗ LF = F ⊕ Ann(F) ⊂ TM ⊕ T M is a Dirac structure if and only if the distribution F is integrable, i.e. tangent to the leaves of a regular foliation on M.
12 2.3 Generalized geometric structures
Definition 2.14. (See [Gua03], Proposition 6.1; the exposition in this text follows [CG11], Section 1.1.) A generalized (Riemannian) metric is an endomorphism G ∈ Γ(End(TM⊕T ∗M)) such that
∗ (i) G is orthogonal w.r.t. ⟨·, ·⟩, i.e. ⟨GA, GB⟩ = ⟨A, B⟩ ∀A, B ∈ (TM ⊕ T M)p, p ∈ M.
∗ (ii) G is symmetric, i.e. ⟨GA, B⟩ = ⟨A, GB⟩ ∀A, B ∈ (TM ⊕ T M)p, p ∈ M.
(iii) G is positive definite, i.e ⟨GA, A⟩ > 0 ∀A ∈ TM ⊕ T ∗M.
Obviously every generalized metric G is self-inverse, G2 = Id. Thus TM ⊕ T ∗M is the orthogonal sum of the +1- and −1-eigenbundles C± of G. ⟨·, ·⟩|C± is positive definite/ negative definite respectively, since G is positive definite. Clearly generalized metrics can equivalently be defined by a choice W ⊂ TM ⊕ T ∗M of maximal subbundle on which ⟨·, ·⟩ is positive definite: ⟨·, ·⟩ is then negative definite on W ⊥, and TM ⊕ T ∗M = W ⊥W ⊥ (⊥ the orthogonal direct sum of subbundles), and G|W = + Id, G|W ⊥ = − Id defines a generalized metric in the sense of Definition 2.14. ∗ ∗ C+ ⊂ TM ⊕ T M can be written as the graph of a linear map A : TM → T M. Write A as the sum of its symmetric and its antisymmetric part A = g + b. Then
C± = {X ± iX g ± iX b|X ∈ TM}, so the choice of a generalized metric is also equivalent to the choice of a Riemannian metric g and smooth two-form b. By virtue of unifying metrics and 2-forms into one geometric object, generalized Rie- mannian metrics are at the heart of double field theory, a reformulation of supergravity, the low-energy limit of closed string theory. We will describe the basics of this theory in Section 2.4.
2.3.1 Generalized complex geometry
Generalized complex structures are the most important generalized geometric structures for the purpose of this text: Part I studies stable generalized complex manifolds, a particular subclass. Again let E =∼ (TM ⊕ T ∗M,H) be an exact Courant algebroid.
Definition 2.15. ([Hit03], Definition 1 ,[Gua03], Definitions 4.1, 4.18) A generalized complex structure J on the exact Courant algebroid E is J ∈ Γ(End E) s.t. J 2 = − Id, J is orthogonal with respect to ⟨·, ·⟩, and the +i-eigenbundle of J , L ⊂ E ⊗ C, is a complex Dirac structure, i.e. integrable with respect to the twisted Courant-Dorfman bracket ·, · H . J K Equivalently, generalized complex structures can be characterised solely by their +i-eigenbundle • ∗ L, or by their canonical bundle K ⊂ ∧ TCM:
13 Generalized Geometry
Note that TM ⊕ T ∗M (and thus any exact Courant algebroid E) always admits a Spin structure, and there is a non-canonical isomorphism of the spin bundle S to the exterior algebra ∧•T ∗M. The natural action of TM ⊕ T ∗M on ∧•T ∗M is
∗ • ∗ (X, ξ) · φ = iX φ + ξ ∧ φ ∀(X, ξ) ∈ TM ⊕ T M, φ ∈ ∧ T M
Definition 2.16. (See [Che96], 3.1; the exposition in this text follows [Gua03], Chapter 2.5.)
• ∗ ∗ (i) Let φ ∈ ∧ Tp M, p ∈ M be a spinor. Its null space Lφ ⊂ TpM ⊕ Tp M is
∗ Lφ = {A ∈ TpM ⊕ Tp M|A · φ = 0}.
Similarly, if K ⊂ ∧•T ∗M is a line subbundle, its bundle of null spaces is
[ ∗ LK = {A ∈ (TM ⊕ T M)p|A · φ = 0 ∀φ ∈ Kp} p∈M
• ∗ (ii) A spinor φ ∈ ∧ T M is called pure when Lφ is maximally isotropic, i.e. is of rank dim M. Similarly a line subbundle K is a pure spinor subbundle when all φ ∈ K are pure spinors in the above sense.
Obviously these concepts extend to (TM ⊕ T ∗M) ⊗ C and the complex spin bundle • ∗ isomorphic to ∧ TCM.
Proposition 2.17. (See [Gua03], Definition 4.14. The exact formulation presented here is based on [CG15].)
(i) If L ⊂ E ⊗ C is a complex Dirac structure and L ∩ L¯ = {0} (L¯ the complex conjugate), L is the +i-eigenbundle of a generalized complex structure.
• ∗ (ii) Let K ⊂ ∧ TCM be a sub-line bundle. The bundle of null spaces of K, LK ⊂ E, is
the +i-eigenbundle of a generalized complex structure JK if and only if K satisfies the following properties:
(a) At every point, a local generating section ρ ∈ Γ(K) can be written as
ρ = eB+iω ∧ Ω,
where B, ω are smooth real two-forms and Ω is the wedge product of k linearly independent complex one-forms. (b) ωn−k ∧ Ω ∧ Ω¯ ̸= 0.
(c) There exists X + ξ ∈ E ⊗ C such that dρ + H ∧ ρ = iX ρ + ξ ∧ ρ.
14 2.3 Generalized geometric structures
Condition (a) and (b) ensure that the null space of K is a maximal isotropic subbundle (i.e. K is a pure spinor line bundle); condition (c) is an integrability condition;
equivalent to L being closed under the Courant-Dorfman bracket ·, · H . K is called J K the canonical bundle of JK .
For details and proofs see Section 4 in [Gua03].
Remark 2.18. Note that a generalized complex manifold is always even-dimensional. At ∗ every p ∈ M we can construct a maximal isotropic subspace of (TM ⊕T M)p spanned by pairs ∗ of the form (Ai, J Ai), where all Ai are linearly independent: Pick some A1 ∈ (TM ⊕ T M)p ∗ with ⟨A1,A1⟩ = 0, so ⟨J A1, J A1⟩ = 0 as well. Now pick A2 ∈ (TM ⊕ T M)p orthogonal to the span of {A1, J A1} and such that ⟨A2,A2⟩ = 0. Again this implies ⟨J A2, J A2⟩ = 0.
Iterate this process until no further such Ai can be found, i.e. a maximal isotropic subspace has been reached. Since such a maximal coisotropic subspace has dimension equal to dim M, the dimension of M must be even. (See Proposition 4.5 in [Gua03].)
Example 2.19. (Complex structures) Let I ∈ Γ(End(TM)) be a complex structure on
M. Then JI given by −I 0 ! JI = 0 I∗ is a generalized complex structure for the standard Courant algebroid TM ⊕ T ∗M.
2 Example 2.20. (Symplectic structures) Let ω ∈ Ωcl(M) be a symplectic form on M. Then Jω given by 0 −ω−1! Jω = ω 0 is a generalized complex structure for the standard Courant algebroid TM ⊕ T ∗M.
Thus generalized complex structures encompass both complex and symplectic structures; i.e. generalized complex geometry provides a unified description of two of the most important types of structure in modern geometry.
2 Example 2.21. Let I be the standard complex structure on C = {(z1, z2)|z1 = x1+iy1, z2 = x2 + iy2} and ω = dx1 ∧ dy2 + dy1 ∧ dx2 a symplectic form. Then
(t2 − 1)I −tω−1 ! Jt = , t ∈ [0, 1] (2t − t3)ω (1 − t2)I∗ is a family of generalized complex structures which interpolate the complex and the symplectic structure. (A similar smooth interpolation can be done on any holomorphic symplectic manifold.)
15 Generalized Geometry
Definition 2.22. The type of a generalized complex structure J at a point p ∈ M is
C Type(J ) = codim (prTpM⊗C L)
Remark 2.23. The integer k in Proposition 2.17 is the type of J at p:
∗ C C codim prTpM⊗C L = dim (Lp ∩ TC,pM)
Now, if K at p ∈ M is spanned by ρ = eB+iω ∧ Ω as in Proposition 2.17,
∗ ξ ∈ L ∩ TCM ⇔ ξ ∧ Ω = 0.
Ω is the wedge product of k linearly independent complex one-forms, so ξ ∧ Ω = 0 if and only if ξ is a linear combination of these one-forms. So k = Type(J ).
Generalized complex structures of type 0 are locally equivalent to symplectic structures via B-transform, while generalized complex structures of maximum type n locally correspond to complex structures. In general, at every point a generalized complex structure of type k is equivalent to the direct sum of a complex structure of complex dimension k and a transverse symplectic structure of real dimension 2n − 2k (see [Gua03]). Any generalized complex structure J ∈ Γ(End(TM ⊕ T ∗M)) induces a Poisson structure on M as follows:
Proposition 2.24. (See [Cra11], Proposition 2.2.) The map π = prTM ◦J |T ∗M defines a bivector (i.e. is an anti-selfdual map) which is Poisson.
We frequently only consider exact Courant algebroids up to (closed) B-transform, so it is useful to look at how a generalized complex structure behaves under such a transformation:
Corollary 2.25. (Generalized complex structures under B-transform) If J is a generalized complex structure with +i-eigenbundle L and canonical bundle K for the exact Courant algebroid (TM ⊕ T ∗M,H) and B ∈ Ω2(M). Under the B-transform eB, these structures transform as
J 7→ eBJ e−B L 7→ eBL −B 1 2 1 3 K 7→ e ∧ K = 1 − B + B − B + ... ∧ φ φ ∈ K 2 3 and define a generalized complex structure J ′ for the exact Courant algebroid (TM ⊕ T ∗M,H + dB).
16 2.3 Generalized geometric structures
2.3.2 Generalized complex branes
Generalized complex branes are a class of natural submanifolds for generalized complex manifolds. There are several similar, but non-equivalent definitions for branes carrying complex line bundles, compare for example [Gua11] and [Col14]. These two definitions both involve a complex line bundle supported on the submanifold. The definition used in this text does not include a complex line bundle, but the submanifolds underlying the branes in both definitions named above will always be branes in the sense of our definition. The definition we use has previously been used by [CG15, CG09] and others.
Definition 2.26. A generalized complex brane in a generalized complex manifold (M,H, J ) is a pair (Y,F ) of a submanifold ι : Y,→ M and a two-form F ∈ Ω2(Y ) such that
(i) dF = ι∗H
∗ ∗ ∗ (ii) τF = {(X, ξ) ∈ TY ⊕ T M|Y s.t. ι ξ = iX F } ⊂ (TM ⊕ T M)|Y is preserved by J :
J (τF ) = τF
Remark 2.27. This definition is invariant under B-transforms in the following sense: If B ∈ Ω2(M), we have B τF 7→ e τF . (2.11)
B So e τF is invariant under the new generalized complex structure and the brane (Y,F ) gets mapped to a new brane (Y,F − ι∗B), ι : Y,→ M.
Remark 2.28. Generalized complex branes should not be confused with generalized complex submanifolds in the sense of [BBB04], Definition 5.9, a different concept of natural submanifold in a generalized complex manifold: These submanifolds carry a generalized complex structure themselves, which is equal to the restriction of the generalized complex structure on the larger manifold in a well-defined sense. Generalized complex branes do in general not carry a generalized complex structure. Generalized complex submanifolds are comparable to symplectic submanifolds in sym- plectic manifolds, whereas generalized complex branes are comparable to Lagrangian or coisotropic submanifolds. (In fact, not only comparable: Symplectic and Lagrangian sub- manifolds are examples of generalized complex submanifolds and generalized complex branes respectively.)
Example 2.29. (Complex branes, Example 7.7 in [Gua03]) Let JI be a generalized complex structure like in Example 2.19. A pair (Y,F ) as in 2.26 defines a generalized complex brane if and only if
• I(TY ) = TY , i.e. Y ⊂ M is a complex submanifold,
17 Generalized Geometry
∗ • J (iX F ) + iJX F ∈ Ann TY ∀X ∈ TY , which means that F is a (1, 1)-form on Y .
If F is an integral form, it is the curvature of a unitary holomorphic line bundle.
Example 2.30. (Lagrangian and coisotropic branes, Example 7.8 in [Gua03]) Now let Jω be a generalized complex structure like in Example 2.20. Consider a pair (Y,F ) as in
Definition 2.26. Choose a two-form B on a neighbourhood of Y that extends F , i.e. B|Y = F . (Y,F ) is a generalized complex brane if and only if the following are satisfied:
• ω−1(Ann TY ) ⊂ TY , or equivalently TY ω-orth ⊂ TY , i.e. Y is a coisotropic submanifold of (M, ω).
−1 ω-orth • ω (iX B) ∈ TY ∀X ∈ TY , which implies iX F = 0 ∀X ∈ TY . Thus F descends ω-orth to a two-form on T Y/T Y . Note that F + iω|Y describes a non-degenerate form of type (2, 0) on T Y/T Y ω-orth, so the T Y/T Y ω-orth is complex even-dimensional.
• (ω + BωB−1)(TY ) ⊂ Ann(TY ), or equivalently ω(X,X′) = ω−1(BX,BY ) ∀X,X′ ∈ −1 ω-orth TY , which imply that (ω|Y ) F is an almost complex structure on T Y/T Y .
If dim M = 2n, these conditions imply dim Y = n + 2k, k ∈ N. F is a closed form, so T Y/T Y ω-orth carries a holomorphic symplectic structure, which can be viewed as a holomor- phic symplectic structure transverse to the Lagrangian foliation of the coisotropic submanifold Y . In the case k = 0, Y is Lagrangian and F = 0. So Lagrangian branes are the lowest- dimensional generalized complex branes in symplectic manifolds and carry a flat line bundle.
The precise nature of generalized complex branes in these two examples motivated their name: String theorists are particularly interested in the so-called string A- and B-model. These are topological quantum field theories obtained from particular supersymmetric string theories, which are related by mirror symmetry. The A-model is defined by a symplectic structure, the B-model by its complex structure. Branes in these theories are precisely complex submanifolds with holomorphic line bundles on the B-side, and Lagrangian and coisotropic branes on the A-side – so these submanifolds are exactly the generalized complex branes in each model.
2.4 Generalized geometry in theoretical high-energy physics
The theory of generalized geometry has both been inspired by and inspired developments in theoretical high-energy physics, in particular string theory. In this section, we will give a brief overview of some of the different ways in which generalized geometry intersects with theoretical physics.
18 2.4 Generalized geometry in theoretical high-energy physics
2.4.1 Double and exceptional field theory
Double field theory is a new formulation for the low-energy limit of superstring theory, 10-dimensional supergravity, in which the O(n, n, R)-symmetry of the theory induced by T-duality (T-duality transformations themselves are in O(n, n, Z)) is made manifest. The spacetime dimension n is taken to be 10 in this case. (See [HZ09, HHZ10].) T-duality relates the two closed superstring theories in 10 dimensions, type IIA and type IIB. As a theory that is invariant under the induced O(n, n, R) transformations, double field theory then contains both type IIA and type IIB supergravity, unifying both theories into one. To this effect, an additional set of coordinates is introduced and the theory is developed on a doubled space, a space of twice the original spacetime dimension. This is equipped with a flat split-signature metric and a so-called generalized Lie derivative – these structures are similar to those on an exact Courant algebroid E =∼ (TM ⊕ T ∗M,H); the dual coordinates locally essentially correspond to fibre coordinates for T ∗M, and the generalized Lie derivative reduces to the twisted Courant-Dorfman bracket when certain constraints on the coordinate dependance of fields on the doubled space are introduced. These constraints on what constitutes a “physical field” serve to reduce the degrees of freedom of the theory to their original number – after applying them, the theory reduces to generalized geometry. An obvious appeal of this approach is the following: In the type IIA 10-dimensional supergravity theory (the low-energy limit of type IIA superstring theory) the bosonic (as opposed to fermionic, i.e. spinorial) fields are
g : a (pseudo-)Riemannian metric, the spacetime metric describing gravity B : a two-form field called the Kalb-Ramond field which enters the theory solely in terms of its derivative H = dB φ : a scalar field or function called the Dilaton
As we have seen in the previous section, a Riemannian metric g and a two-form B can be combined into a generalized metric G. It turns out that after the introduction of the doubled space the entire action of type A 10-dimensional supergravity can be rewritten in terms of the generalized metric G (and the dilaton φ). [HHZ10] This action is then manifestly O(n, n, R)-invariant, where O(n, n, R) is the symmetry group preserving the canonical symmetric bilinear form ⟨·, ·⟩. The O(n, n, R)-symmetry is induced by T-duality (O(n, n, Z)) from the full string theory and is present without the double field theory description, but not obviously visible. Since the O(n, n, R)-transformations mix original spacetime coordinates and momentum coordinates from the doubled space, this formulation also works for so-called non-geometric backgrounds, which include T-folds.A T-fold is a spacetime whose local charts are T n-fibrations given as subspaces of a T 2n-fibration.
19 Generalized Geometry
They are patched by local T-duality transformations in the larger T 2n-fibration that in general do not descend to charts of the T n-subspaces defining the spacetime. [Hul05] As described above, the so-called section conditions (there is a weak and a strong section condition) reduce the now 2n-dimensional theory back to its original number of degrees of freedom, n. In the case of 10-dimensional double field theory, there are two ways of solving the section conditions: One solution is type IIA supergravity, the other type IIB, so double field theory encompasses both closed string theories in 10 dimensions. Double field theory was long mostly treated as a local theory, and the global nature of the doubled space was not specified. In recent years different ways of patching charts on double spaces have been suggested [Hul15, RS15, Pap15]. However, there is no real consensus on what kind of mathematical objects the doubled space and the physical fields are, and no consistent mathematical description of the physical fields. The framework was later extended to the lower-dimensional supergravity theories (which can all be obtained by compactification from a unique 11-dimensional supergravity theory) (see e.g. [BP11, HS14a, HS14b]). This analogous approach is usually referred to as extended or exceptional field theory, and after applying its respective versions of the section conditions, the theories reduce to Leibniz algebroids of the form TM ⊕ E, where E → M is, depending on the dimension of the theory, one of the following vector bundles:
TM ⊕ E = TM⊕ ... R ∧2 T ∗M ∧2 T ∗M ⊕ ∧5T ∗M ∧2 T ∗M ⊕ ∧5T ∗M ⊕ (T ∗M ⊗ ∧7T ∗M)
(See [Bar12, BP11, PPW08].) The brackets on these vector bundles are all examples of Dorfman brackets which we study in Part II of this thesis. (The explicit Dorfman brackets used in exceptional generalized geometry are also briefly discussed there.) These lower-dimensional supergravity theories also have an induced symmetry, U-duality, with symmetry groups SL(2),SL(5),SO(5, 5),E(6,6),E(7,7),E(8,8) respectively, according to dimension. The exceptional field theory formulation using extended spaces and Dorfman brackets again allows for the formulation of an action which is invariant under U-duality and thus makes the symmetry manifest. Furthermore, [GF14] employs the theory of transitive (but not exact) Courant algebroids, which are in this case obtained via the generalized reduction of Courant algebroids (see [BCG07]) from an exact Courant algebroid, to describe heterotic string theory in a similar way:
20 2.4 Generalized geometry in theoretical high-energy physics
Heterotic string theories [Gro87] are built on the principle that along a closed string, left- and right-moving excitations are completely independent of each other. It is thus possible to consider hybrid theories of a left-moving bosonic string (no supersymmetry; needs a 26-dimensional spacetime to be consistent) and a right-moving superstring (as above; consistent in 10 dimensions). The left-over 16 dimensions of the bosonic string are compactified on an even, self-dual lattice. There are two inequivalent such lattices which result in two 16-dimensional tori: The maximal tori in E8 × E8 and Spin(32)/Z2. Thus there are two different heterotic string theories with gauge groups E8 × E8 and Spin(32)/Z2.
In [GF14], Garcia-Fernandez considers a principal bundle P = PYM ×M PM , where PYM has a compact structure group that is either contained in E8 × E8 or in SO(32), and PM is the SL(10, R)-bundle of oriented frames of TM. The transitive Courant algebroid that describes the heterotic theory is a reduction of TP ⊕ T ∗P twisted by some canonical 3-form, and is itself (non-canonically) isomorphic to TM ⊕ ad P ⊕ T ∗M. The fields of heterotic string theory are described in terms of an admissible generalized metric.
2.4.2 String compactifications
Generalized complex geometry also arises very naturally in string theory in the context of compactification: As already mentioned in this section, superstring theory is only consistent in 10 spacetime dimensions. In order to use it to describe a 4-dimensional universe similar to the physical universe, we consider string theory on a spacetime manifold
M 10 = R1,3 × M¯ 6, where M¯ is a compact 6-dimensional manifold. The theory on M¯ 6 can then be analysed separately, and its effect on the 4-dimensional effective theory calculated. However, in order for the theory to keep its supersymmetry and conformal symmetry, the manifold M¯ has to to satisfy a range of conditions: For example, a necessary condition for the spacetime supersymmetry to be unbroken is that M¯ with (non-zero) closed three-form H be generalized Kähler:
Definition 2.31. [Gua03]A generalized Kähler structure on (M,H) is a pair of generalized ∗ complex structures J1, J2 with respect to the exact Courant algebroid (TM ⊕ T M,H) such that
[J1, J2] = 0 and −J1J2 is a generalized Riemannian metric, i.e. positive definite.
In string theory, this condition was originally formulated in [GHR84] in terms of M¯ having a bihermitian geometry, which means that M¯ carries a Riemannian metric g, a closed three-form H, and two complex structures J± with respect to which the metric is hermitian.
21 Generalized Geometry
± ± Furthermore, they must satisfy ∇ J± = 0, where ∇ are the so-called Bismut connections ± 1 −1 ∇ = ∇LC ± 2 g H, ∇LC being the Levi-Civita connection of g.
Theorem 2.32. (See [Gua03], Section 6.4, in particular Remark 6.27.) Bihermitian structures and generalized Kähler structures are in one-to-one correspondence.
In addition to the spacetime symmetry, the compactification also needs to preserve the conformal invariance of the theory, which corresponds, when calculating string interactions up to one loop, to the additional requirement that the generalized Kähler structure on M¯ be generalized Calabi-Yau metric [HT09]:
Definition 2.33. (i) ([Hit03], Definition 2) A generalized complex manifold (M, J ,H) is generalized Calabi-Yau if its canonical bundle K admits a global, nowhere-zero section ρ ∈ Γ(K), dρ = 0.
(ii) ([Gua03], Definition 6.40) A generalized Kähler structure (J1, J2) is generalized Calabi-
Yau metric if both J1 and J2 are generalized Calabi-Yau in the sense above, with
closed sections ρ1 ∈ Γ(K1), ρ2 ∈ Γ(K2) such that
(ρ1, ρ1) = c(ρ2, ρ2), c ∈ R,
• ∗ where the bilinear form on complex spinors α, β ∈ ∧ TCM is given by
(α, β) := (∗(α) ∧ β)top.
• ∗ (·)top picks out the top-degree term of an element in ∧ TCM and ∗ : φ1 ∧ · · · ∧ φk 7→
φk ∧ · · · ∧ φ1 for one-forms φi.
Thus we see that the consistency condition for string compactifications can be phrased in terms of the existence of certain generalized complex structures on the internal spacetime manifold.
22 Chapter 3
Double vector bundles
Double vector bundles appear implicitly in a number of texts on connections of vector bundles. The first detailed account of the theorey is given in [Pra74a]. Further general treatments of the theory can be found in [Pra77, Mac05, GSM10]. Double vector bundles are manifolds with two compatible vector bundle structures over base spaces which are themselves vector bundles over a common base M. In this chapter we recall basic definitions and investigate the two most important examples of double vector bundles in the context of this thesis: the tangent and cotangent double of a smooth vector bundle E → M, which both naturally carry the structure of a double vector bundle. Lastly, we also revise some basic results on the first jet bundle of a vector bundle.
3.1 Double vector bundles, duals and linear splittings
In this section, we follow the notation and exposition in [Mac05], Chapter 9.
Definition 3.1. (see e.g. [Mac05], Definition 9.1.1.) A double vector bundle (D; A, B; M) is a commutative square πB D / B
πA qB A / M qA of vector bundles D → B,D → A, A → M,B → M, which satisfy a number of further conditions: Both (πB, qA) and (πA, qB) have to be vector bundle morphisms. The two additions +A and +B for the vector bundles D → A and D → B respectively (when it is obvious from the context which addition operation is used, we will often just write +) need to define vector bundle morphisms, i.e. they satisfy
(d1 +A d2) +B (d3 +A d4) = (d1 +B d3) +A (d2 +B d4) (3.1)
23 Double vector bundles
for all d1, d2, d3, d4 ∈ D with πA(d1) = πA(d2), πA(d3) = πA(d4) and πB(d1) = πB(d3),
πB(d2) = πB(d4). A → M,B → M are the side bundles of D.
The core C of (D; A, B; M) is the intersection of the kernels of the projections πA and of A B πB. The zero section of A → M is 0 : M → A; 0 is the zero section of B → M. Write 0˜A, 0˜B for the zero sections of D → A, D → B respectively. C has a natural vector bundle ′ ′ A ′ B structure over M: Let c, c ∈ Cm, i.e. πA(c) = πA(c ) = 0m, πB(c) = πB(c ) = 0m for some ˜A ˜B ′ B ′ A m ∈ M. Note that for all m ∈ M, 0 A = 0 B , and that πB(c +A c ) = 0 , πA(c +B c ) = 0 , 0m 0m m m ′ ′ so both c +A c , c +B c are again in Cm.
′ ′ ′ ′ ′ A ′ (c +A c ) −B (c +B c ) = ((c +A c ) −B c) −B c = ((c +A c ) −B (c +A 0˜ A )) −B c 0m ′ A ′ ′ ′ B = ((c −B c) +A (c −B 0˜ A )) −B c = c −B c = 0˜ B , 0m 0m so addition in C → M is well-defined. The inclusion C,→ D is denoted by
−1 A −1 B Cm ∋ c 7−→ c ∈ πA (0m) ∩ πB (0m).
∞ The space of sections ΓB(D) of D → B can be generated as a C (B)-module by two particular classes of sections, namely linear and core sections (see [Mac11]):
(i) Each section c: M → C has a corresponding core section c† : B → D:
c†(b ) = 0 + c(m), m ∈ M, b ∈ B . m ebm A m m
Similarly, there is a core section A → D, which we also denote byl c†. The argument distinguishes them where necessary. The space of core sections of D over B is written c as ΓB(D).
(ii) A section ξ ∈ ΓB(D) is called linear if ξ : B → D is a vector bundle morphism from B → M to D → A over a ∈ Γ(A):
ξ B / D
qB πA M a / A
ℓ ∗ We write the space of linear sections of D over B as ΓB(D). ψ ∈ Γ(B ⊗ C) has an associated linear section ψe: B → D over the zero section 0A : M → A, given by
ψe(bm) = e0bm +A ψ(bm).
ψe takes values inside the core, so we refer to it as a core-linear section.
24 3.1 Double vector bundles, duals and linear splittings
Example 3.2. (Standard examples of double vector bundles)
(i) Let A, B → M be vector bundles over M. We can always form the double vector
bundle (A ×M B; A, B; M) with trivial core C = 0, where the maps A ×M B → A, B are simply the canonical projections.
(ii) (Example 9.1.4 in [Mac05].) If C → M is a third vector bundle over M, we can form
the double vector bundle (A ×M B ×M C; A, B; M). D = A ×M B ×M C can be viewed ∗ ∗ ∗ ∗ as qAB ⊕ qAC over A, and qBA ⊕ qBC over B. C is the core of this double vector bundle, which we call the trivial double vector bundle over A and B with core C. Up to isomorphism of double vector bundles (see below) all double vector bundles are of this form.
Definition 3.3. (See [Mac05], Definition 9.1.2.) A morphism of double vector bundles ′ ′ ′ ′ ′ (φ; φA, φB; f):(D; A, B; M) → (D ; A ,B ; M ) is a tuple of four maps φ : D → D , φA : A → ′ ′ ′ A , φB : B → B , f : M → M where each of the pairs (φ, φB), (φ, φA), (φA, f), (φB, f) forms ′ a morphism of vector bundles. If M = M and f = IdM , we say the morphism is over M.
We then also say that φ is a morphism over (φA, φB).
Definition 3.4. A linear splitting of the double vector bundle (D; A, B; M) is an injective
morphism of double vector bundles Σ: A ×M B,→ D over the identity on the sides A and B.
Local linear splittings are simply double vector bundle charts. The original definition of double vector bundles in [Pra74b] uses double vector bundle charts. Any double vector bundle admits a global linear splitting, which follows from the existence of a global double splitting proved in [Pra74a]. In [JL18], Jotz-Lean shows that a linear splitting Σ in the sense of the above definition is ℓ ∞ equivalent to a splitting σA : Γ(A) → ΓB(D) of the short exact sequence of C (M)-modules
∗ ℓ 0 → Γ(B ⊗ C) → ΓB(D) → Γ(A) → 0, (3.2)
where C is the core of (D; A, B; M) and the third map sends a linear section (ξ, a) to the base
section a ∈ Γ(A). We call σA a horizontal lift. Given a linear splitting Σ, the corresponding
horizontal lift σA is
σA(a)(bm) = Σ(a(m), bm).
Conversely, a horizontal lift σA defines
Σ(am, bm) = σA(a)(bm), where a ∈ Γ(A) s.t. a(m) = am.
This can easily be checked to be well-defined and a linear splitting. ℓ The existence of local linear splittings implies furthermore that the linear sections ΓB(D) ∞ ℓ form a locally free C (M)-module, so ΓB(D) is in fact the space of sections of a vector
25 Double vector bundles bundle Ab over M. This bundle is called the fat vector bundle of (D; A, B; M) (see [JL15], 6.2). The short exact sequence (3.2) of C∞(M)-modules induces a short exact sequence of vector bundles 0 → B∗ ⊗ C → Ab → A → 0. (3.3)
A double vector bundle (D; A, B; M) has two natural duals – one with respect to each vector bundle structure – given as follows:
Definition 3.5. (i) The vertical dual of (D; A, B; M) is
A πC∗ D∗A / C∗
∗ q ∗ πA C A / M qa
∗ ∗ ∗ qC∗ : C → M is the ordinary vector bundle dual of the core C. πA : D A → A is the A ∗ ∗ dual vector bundle to qA : D → A. The projection πC∗ : D A → C is given by
D A E D ˜A E −1 πC∗ (Φ), c = Φ, 0a +B c¯ , where c ∈ Cm, Φ: πA (a) → D
∗ ∗ The addition +C∗ in D A → C is given by
′ ′ ′ ′ Φ +C∗ Φ , d +B d = ⟨Φ, d⟩ + Φ , d .
See [Mac05], 9.2, for a the simple proof that this is well-defined.
(ii) The horizontal dual of (D; A, B; M) is
∗ πB D∗B / B
B q ∗ πC∗ C C∗ / M qC∗
The unknown projection and addition are defined analogously:
D B E D ˜B E πC∗ (Ψ), c = Ψ, 0b +A c¯ ′ ′ ′ ′ Φ +C∗ Φ , d +A d = ⟨Φ, d⟩ + Φ , d .
The core vector bundles of the vertical dual (D∗A; A, C∗; M) and the horizontal dual (D∗B; B,C∗,M) are B∗ → M and A∗ → M respectively.
26 3.2 The tangent double and the cotangent double of a vector bundle
3.2 The tangent double and the cotangent double of a vector bundle
Let qE : E → M be a vector bundle. The tangent bundle TE has two natural vector bundle structures: Firstly, by definition as the tangent bundle of the manifold tot(E), secondly as a vector bundle over TM. The projection of TE → TM is the derivative of the projection of E → M. Addition and scalar multiplication in TE are similarly given as derivatives of the operations on E: They are defined by the addition and scalar multiplication of paths in E to which elements of TE are tangent.
pE TE / E
T qE qE TM / M pM
The diagram above forms a double vector bundle (TE; TM,E; M) with core E → M (see [Mac05], Example 9.1.3 and §3.4). The inclusion of the core E,→ TE, ¯· : E → −1 E −1 TM d p (0 ) ∩ (T qE) (0 ) sends em ∈ Em to e¯m = tem ∈ T0E E. Thus the core vector E dt t=0 m field obtained from e ∈ Γ(E) is the canonical vertical lift e↑ : E → TE, which has flow e↑ e↑ ′ ′ ℓ ℓ φ : E × R → E, φt (em) = em + te(m). Elements of ΓE(TE) = X (E) are called linear vector fields. There is a well-known (see e.g. [Mac05]) correspondence of linear vector fields ξ ∈ Xℓ(E) covering X ∈ X(M) and derivations D : Γ(E) → Γ(E) over X ∈ X(M), namely
∗ ∗ ξ(ℓε) = ℓD∗(ε) and ξ(qEf) = qE(X(f)) (3.4)
∗ for all ε ∈ Γ(E ), where ℓε is the linear function on tot(E) naturally defined by ε, and f ∈ C∞(M). D∗ : Γ(E∗) → Γ(E∗) denotes the dual derivation to D on Γ(E). Write Db for the linear vector field in Xl(E) corresponding to a derivation D of Γ(E) under 3.4. If D is a derivation over X ∈ X(M), Db is explicitly given by d Db(em) = TmeX(m) +E (em − tD(e)(m)) (3.5) dtt=0
for em ∈ E and any e ∈ Γ(E) such that e(m) = em.
The horizontal dual of (TE; TM,E; M) is the double vector bundle
cE T ∗E / E
rE qE E∗ / M qE∗
which we call the cotangent double of E → M (again following [Mac05], 9.4).
27 Double vector bundles
∗ We define the projection rE as follows: for θem , rE(θem ) ∈ Em, ′ d ′ ⟨rE(θem ), em⟩ = θem , em + tem dtt=0
′ for all em ∈ Em. ∗ ∗ The sum θe +r ωe′ ∈ T ′ E of θe and ωe′ such that rE(θe ) = rE(ωe′ ) = εm ∈ E m E m em+em m m m m m is uniquely determined by
∗ ′ ′ ′ ′ ⟨θem +E ωem , vem +TM vem ⟩ = ⟨θem , vem ⟩ + ⟨ωem , vem ⟩
′ ′ ′ for all vem ∈ Tem E, vem ∈ Tem E s.t. (qE)∗(vem ) = (qE)∗(vem ).[Mac05] ∗ Let ε ∈ Γ(E ). The one-form dℓε is linear over ε: rE(dem ℓε) = ε(m) holds for all m ∈ M and the sum
′ ′ dem ℓε +rE dem ℓε = dem+em ℓε.
∗ 1 ∗ E∗ The one-form qEθ, θ ∈ Ω (M) is a core section of TE → E, since rE((qEθ)(em)) = 0m . ∗ l ∗ For φ ∈ Γ(Hom(E,T M)) the associated core-linear section φe ∈ ΓE(T E) is φe(em) = ∗ ∗ ∗ ∗ (Tem qE) φ(em) for all em ∈ E. dem ℓε and dem (qEf) span Tem for all ε ∈ Γ(E ) and f ∈ C∞(M). Finally, ∗ ∼ d ℓfε = qE d ℓε + ε ⊗ d f holds for all ε ∈ Γ(E∗) and f ∈ C∞(M). Now we take the direct sum of TE and T ∗E over E as a double vector bundle, namely
πE TE ⊕ T ∗E / E
ΦE qE TM ⊕ E∗ / M qTM⊕E∗
∗ ∗ with side projection ΦE = (qE)∗ ⊕ rE and core E ⊕ T M. For all sections (e, θ) of E ⊕ T M, ↑ q q ◦ we define the vertical section (e, θ) ∈ ΓE(T E E ⊕ (T E E) ) as the pair
↑ ′ d ′ t ′ (e, θ) (em) = em + te(m), (Tem qE) θ(m) (3.6) dtt=0
′ ↑ ∗ for all em ∈ E. By construction the vertical sections (e, θ) are core sections of TE⊕T E → E. As a vector bundle, TE ⊕ T ∗E → E is naturally equipped with the standard Courant algebroid structure, which is compatible with the double vector bundle structure in the sense that it is linear over E → M, i.e. in particular the pairing and the Courant-Dorfman bracket
28 3.3 The first jet bundle of a vector bundle.
of two linear sections are again linear sections [LB12]. The double vector bundle
ΦE :=(qE ∗,rE ) TE ⊕ T ∗E / TM ⊕ E∗
πE E / M qE
turns out to be the standard example of the general concept of VB-Courant algebroid ([LB12], see also [Jot17]). ∗ ∗ The anchor Θ = prTE : TE ⊕ T E → TE restricts to ∂E = prE : E ⊕ T M → E on the ∗ core, and defines an anchor ρTM⊕E∗ = prTM : TM ⊕ E → TM on the side.
3.3 The first jet bundle of a vector bundle.
We also recall some basic results about the first jet bundle of a vector bundle in this chapter, which will be used in Part II.
Definition 3.6. The first jet bundle J 1E of a vector bundle E over M is the space
1 J E := {ηm ∈ Hom(TmM,Tem E) | m ∈ M, em ∈ Em}.
1 It has a projection to prE : J E → E to E, ηm ∈ Hom(TmM,Tem E) 7→ em and a projection 1 to pr: J E → M to M, ηm 7→ m. This second projection is the projection of a vector
′ bundle structure over M; for ηm ∈ Hom(TmM,Tem E) and µm ∈ Hom(TmM,Tem E), we have
′ αηm + βµm ∈ Hom(TmM,Tαem+βem E),
(αηm + βµm)(vm) = αηm(vm) +TM βµm(vm),
where +TM is the addition in the tangent prolongation TE → TM of the vector bundle E → M.
1 E For each φm ∈ Hom(TmM,Em) we get an element ι(φm) ∈ J Em with prE(ι(φm)) = 0m, E d ι(φm)(vm) = Tm0 (vm) + tφm(vm). dt t=0 Two elements ηm ∈ Hom(TmM,Tem E) and µm ∈ Hom(TmM,Tem E) differ by such an element φm ∈ Hom(TmM,Em) and we have a short exact sequence
pr 0 −→ Hom(TM,E) −→ι J 1E −→E E → 0
of vector bundles over M. The corresponding sequence
pr 0 −→ Γ(Hom(TM,E)) −→ι Γ(J 1E) −→E Γ(E) → 0
29 Double vector bundles is canonically split by the map
1 1 1 1 j : Γ(E) → Γ(J E), (j e)m ∈ Hom(TmM,Tem E), (j e)m(vm) = Tme(vm).
In particular, given m ∈ M and two sections e, e′ ∈ Γ(E) with e(m) = e′(m), we find 1 1 ′ (j e)m = (j e )m + ι(φm) for a φm ∈ Hom(TmM,Em). In other words, there is a canonical isomorphism
1 ∼ ∗ 1 Γ(J E) = Γ(E) ⊕ Γ(T M ⊗ E), µ 7→ (prE µ, µ − j (prE µ)). (3.7)
1 1 1 1 1 Furthermore, we have j (e1 + e2) = j e1 + j e2 and J (fe) = fj e + ι(d f ⊗ e) for all ∞ e, e1, e2 ∈ Γ(E) and f ∈ C (M).
1 1 Note finally that every element µ ∈ Jm(E) can be written µ = (j e)m with a local section e ∈ Γ(E). Furthermore, two local sections e, e′ ∈ Γ(E) define the same element 1 1 ′ 1 ′ (j e)m = (j e )m =: µ ∈ Jm(E) if and only if Tme = Tme as vector space morphisms ′ ′ TmM → Te(m)E. That is, e(m) = e (m) and Tme(vm) = Tme (vm) for all m ∈ TmM. The ′ ′ later is equivalent to vm⟨ϵ, e⟩ = (Tmevm)(ℓϵ) = (Tme vm)(ℓϵ) = vm⟨ϵ, e ⟩ for all vm ∈ TmM and all ϵ ∈ Γ(E∗), and so to
⟨dϵ(m) ℓe,Tmϵvm⟩ = ⟨dϵ(m) ℓe′ ,Tmϵvm⟩
∗ 1 1 ′ for all vm ∈ TmM and all ϵ ∈ Γ(E ). Hence, (j e)m = (j e )m if and only if dϵ ℓe = dϵ ℓe′ for ∗ ∗ all ϵ ̸= 0 ∈ Em; by continuity then dϵ ℓe = dϵ ℓe′ for all ϵ ∈ Em.
30 Part I
Lagrangian branes and symplectic methods for stable generalized complex manifolds
31
Chapter 4
Logarithmic and elliptic symplectic geometry
Leading into our work on stable generalized complex structures, in this chapter we review results on two particular classes of Poisson manifolds: In both logarithmic symplectic (log symplectic for short; also called b-symplectic or b-Poisson) and elliptic symplectic geometry, we consider Poisson structures that are generically non-degenerate, but degenerate on a lower-dimensional submanifold. Such Poisson structures correspond to symplectic forms outside their degeneracy locus, which can be viewed as singular on the degeneracy locus. This symplectic description allows for the generalisation of a number of results from symplectic geometry to these contexts, notably the existence of Darboux coordinates and neighbourhood theorems for Lagrangian submanifolds (see also Section 5.3, where we prove the Lagrangian neighbourhood theorem for Lagrangian submanifolds in log symplectic manifolds). Logarithmic and elliptic symplectic geometry fit into the more general framework of so-called Lie algebroid symplectic forms. More specifically, they are examples of E-symplectic structures, a concept widely used in the context of symplectic forms defined on locally free subsheaves of X(M) in recent years, and explicitly defined in [MS18] to extend techniques from symplectic geometry to a wider class of Poisson structures.
Definition 4.1. Let ρA : A → M be a Lie algebroid with bracket [·, ·]A.
(i) (Lie algebroid Poisson structures in this sense seem to first appear explicitly in [Pop09], Definition 1. The exposition here follows [CK17].) An A-Poisson structure is π ∈ Γ(∧2A)
such that [π, π]A = 0.
(ii) A Poisson structure π′ on M is said to be of A-type if there is an A-Poisson structure π such that ′ ρA(π) = π .
33 Logarithmic and elliptic symplectic geometry
π′ is of non-degenerate A-type if there is a corresponding A-Poisson structure π which is non-degenerate.
(iii) ([NT01], Definition 2.11) An A-symplectic structure is a non-degenerate Lie algebroid 2 ∗ 2-form ω ∈ Γ(∧ A ) s.t. dA ω = 0.
Obviously A-symplectic structures are in one-to-one correspondence with non-degenerate A-Poisson structures.
Definition 4.2. Let E ⊂ X(M) be a locally free C∞(M)-submodule which satisfies the involutivity condition [E, E] ⊂ E. There is a vector bundle E TM whose sections are precisely E, which we call E-tangent bundle. The natural inclusion E ⊂ X(M) induces a vector bundle E morphism ρE : TM → TM over the identity and the restriction of the ordinary vector field Lie bracket [·, ·] makes the E-tangent bundle into a Lie algebroid. Its dual bundle is denoted by E T ∗M and called the E-cotangent bundle. The sections Γ(∧k(E T ∗M)) =: E Ωp(M) are called E-forms of degree k. All E-forms form a differential complex induced by the Lie algebroid structure on E TM. A non-degenerate ω ∈ E Ω2(M) such that dω = 0 is called an E-symplectic form, and the triple (M, E, ω) is an E-symplectic manifold.
E-symplectic forms are examples of Lie algebroid symplectic forms whose Lie algebroid has a smaller or equal rank to dim M.
4.1 Logarithmic symplectic geometry
The most well-studied and in several ways the simplest generically non-degenerate Poisson structures are logarithmic or b-symplectic structures, where the degree of degeneracy is the smallest possible:
Definition 4.3. ([GMP14], Definition 14; this notion is referred to as b-Poisson there.) A Poisson structure π ∈ X2(M) on a manifold M is called logarithmic if πn ∈ X2n(M) vanishes transversely, i.e. its graph intersects the zero section of ∧2nTM transversely.
The Poisson structure π is then everywhere non-degenerate, except along a smooth codimension-1 submanifolds Z ⊂ M, on which it is of rank 2n − 2.
Remark 4.4. In [GMP14] M is assumed to be oriented, in which case Z, as the pre-image of a regular value of a function M → R (obtained from picking a trivialisation of ∧2nTM), is also oriented. However, the results below do not require this assumption.
The logarithmic vector fields on (M,Z) are X(M, − log Z) := {X ∈ X(M) s.t. X|Z ∈ X(Z)}. These form a locally free sheaf, so there is a smooth vector bundle TM(− log Z) of rank = dim(M), which we call the logarithmic tangent bundle. (M, X(M, − log Z)) is an
34 4.1 Logarithmic symplectic geometry example of an E-manifold as defined above: The inclusion a : X(M, − log Z) → X(M) induces a vector bundle morphism, and the Lie bracket on X(M) restricts to X(M, − log Z). Note that π ∈ X2(M, − log Z), and as such, π is non-degenerate! In particular, it has a well-defined inverse ω = π−1 ∈ Γ(∧2T ∗M(log Z)), where T ∗M(log Z) is the dual bundle to TM(− log Z). We call Ω•(M, log Z) := Γ(∧•T ∗M(log Z)) the complex of logarithmic differential forms. The bivector π being Poisson is precisely equivalent to dω = 0, where d also denotes the differential for logarithmic forms. Thus, ω is a symplectic structure for the Lie algebroid TM(− log Z); this is called a logarithmic symplectic or, for short, log symplectic structure. Note that the anchor a : TM(− log Z) → TM is an isomorphism on M \ Z, but not globally. When restricted to Z, we instead obtain a short exact sequence
a 0 → ker a|Z → TM(− log Z)|Z → TZ → 0.
Choose a tubular neighbourhood U of Z, such that U is diffeomorphic to a neighbourhood of the zero section in the normal bundle NZ. The normal bundle carries a canonical Euler ∂ 1 vector field given by e = x ∂x , x any fibre coordinate . As a section of TM(− log Z), e is non-zero on the entire tubular neighbourhood of Z. Of course e ∈ X(M, − log Z) depends on the identification U =∼ NZ, but its restriction (as a section of the log tangent bundle) to Z does not: If f is a second vanishing function for Z, we have
−1! ∂ f ∂f ∂ ∂ f = x = x ∂f x ∂x ∂x ∂x Z x=0 x=0 Z
∂ ∂ Clearly we have a x ∂x |Z = 0, so x ∂x |Z ∈ ker a at every point of Z. ker a|Z has rank 1, ∂ so x ∂x |Z spans ker a|Z at every point of Z and is uniquely defined. In particular ker a|Z is a trivial line bundle with canonical non-vanishing section. Using e ∈ X(M, − log Z), we can define the so-called residue of a logarithmic form, namely
∗ • res α = ι (ieα) ∈ Ω (Z), ι : Z,→ M, which is well-defined by the above observations, and also descends to a map of the cohomology groups. In particular, we obtain a short exact sequence of differential complexes
0 → Ωk(M) → Ωk(M, log Z) →res Ωk−1(Z) → 0.
1By fibre coordinate we mean the following: Choose a local basis b for NZ. Then any point p ∈ NZ over the corresponding local open neighbourhood can be written as p = x(p)b; x is the fibre coordinate, and at the same time a local defining function for Z. We complete x to a full set of coordinates for NZ by picking some set of coordinates on NZ.
35 Logarithmic and elliptic symplectic geometry
We now present a number of known results without proof which will be used later in the text:
Theorem 4.5. (Mazzeo-Melrose theorem, [MM98]) The logarithmic cohomology, the cohomology of the complex Ω•(M, log Z), is given by
Hk(M, log Z) = Hk(M) ⊕ Hk−1(Z).
If x1 is a local vanishing function for Z and x1, x2, . . . , xn, y1 . . . yn are coordinates for M, TM(− log Z) and T ∗M(log Z) are locally spanned by
∂ ∂ ∂ ∂ ∂ x1 , ,..., , ,... ∂x1 ∂x2 ∂xn ∂y1 ∂yn and dx1 , dx2,..., dxn, dy1,... dyn x1 respectively.
Theorem 4.6. (Logarithmic Darboux coordinates, Theorem 37 in [GMP14]) Let (M 2n,Z2n−1, ω) be a log symplectic manifold. On an open neighbourhood around every p ∈ Z, there are coordinates x1, x2, . . . , xn, y1, . . . , yn such that x2, . . . , xn, y1, . . . , yn restrict to coordinates on Z, and the log symplectic form ω has the form
dx1 ω = ∧ dy1 + dx2 ∧ dy2 + ··· + dxn ∧ dyn. x1
4.2 Complex log symplectic and elliptic symplectic geometry
There is an analogous complex notion of log symplectic forms, which uses the concept of complex divisor, an extension of the concept of divisor from complex to smooth geometry:
Definition 4.7. (See [CG15], Section 1.)
(i) A complex divisor on a smooth manifold M is a pair D = (U, s) of a complex line bundle U → M and a section s ∈ Γ(U) which intersects the zero section transversely. We also write D = {s = 0} and also call it the complex divisor. (D ⊂ M is a smooth codimension-2 submanifold.)
(ii) The vanishing ideal associated to D is
∗ ∞ Is = Im (s : Γ(U ) → CC (M))
36 4.2 Complex log symplectic and elliptic symplectic geometry
(iii) The complex logarithmic tangent bundle TCM(− log D) associated to D is the smooth vector bundle whose sections are
Z ∈ XC(M) s.t. Z(Is) ⊂ Is
(These form a locally free sheaf and are thus indeed the sections of a smooth vector bundle.) ∼= Remark 4.8. s vanishes transversely along D, so dν s : ND → U|D, the normal derivative ∼ ∗ of s, defines an isomorphism of vector bundles over D. Furthermore, clearly U|W = p ND, where W is a tubular neighbourhood of D and p : W → D the projection map induced by the vector bundle structure ND → D.
In fact, TCM(− log D) inherits the Lie bracket from TCM, as well as an anchor a : ∗ TCM(− log D) → TCM and is thus a (complex) Lie algebroid. Its dual bundle is TCM(log D), and there is a differential complex of complex logarithmic forms
k k+1 d: ΩC(M, log D) → ΩC (M, log D)
∂ We can define z ∂z ∈ XC(M, − log D) on a neighbourhood of D (where z is now a complex local vanishing function for D, e.g. given by a local trivialisation of U), and the complex residue of a complex log form ∗ res β = ι i ∂ β . z ∂ z k k−1 This is also a morphism of differential complexes res : ΩC(M, log D) → ΩC (D).
Definition 4.9. ([CG15], Definition 1.6) An elliptic divisor is a pair (R, q) of a real line bundle R → M with section q ∈ Γ(R) which vanishes critically on a smooth codimension-2 submanifold D ⊂ M s.t. the normal Hessian of q along D is positive definite. The vector ∗ ∞ fields preserving the associated ideal Iq = Im(q : R → C (M)) form a locally free sheaf and are thus the sections of a smooth real vector bundle TM(− log |D|), the elliptic tangent bundle associated to (R, q). Since q ∈ Γ(R) vanishes critically along D, the normal Hessian
Hess(q) ∈ Γ(Sym2 N ∗D ⊗ R) is well-defined. Remark 4.10. The section q is a trivialisation of the real line bundle R away from D, so R is in fact an oriented (and thus trivial) real line bundle. Like the real and complex logarithmic tangent bundle, the elliptic vector bundle is a Lie algebroid. Just like for real logarithmic forms, it makes sense to consider the symplectic forms for the elliptic Lie algebroid TM(− log |D|):
37 Logarithmic and elliptic symplectic geometry
Definition 4.11. An elliptic symplectic form ω ∈ Ω2(M, log |D|) := Γ(∧2T ∗M(log |D|)) is s.t. ω is non-degenerate as a two-form on TM(− log |D|) and dω = 0 ∈ Ω2(M, log |D|).
Clearly, π = ω−1 defines a Poisson structure on M that is non-degenerate on M \ D and degenerates to lower rank on D.
Complex and elliptic divisors are related as folllows: Any complex divisor (U, s) defines an elliptic divisor (U ⊗ U,¯ s ⊗ s¯). In this case D is always co-oriented. Conversely, any elliptic divisor whose vanishing locus D is co-oriented is in fact of the form (U ⊗ U,¯ s ⊗ s¯) with (U, s) a complex divisor, and (U, s) is unique up to isomorphism once a co-orientation is chosen. From now on, we focus on this case only. If z a local complex vanishing function for D and r = |z|, θ = arg(z):
∂ ∂ R = r , t = ∂r ∂θ
The elliptic tangent bundle can be viewed as
∼ TM(− log |D|) = TCM(− log D) ×M TCM(− log D), since they are vector bundles of the same rank, and given a pair of complex log vector fields
(X,Y ) ∈ XC(M, log D) and an element f ⊗ g¯ ∈ Is⊗s¯, we can define
(X, Y¯ )(f ⊗ g¯) := X(f) ⊗ Y¯ (¯g) = X(f) ⊗ Y (g) ∈ Is⊗s¯.
The theory of residues for elliptic differential forms (as described in [CG15]) is more complicated than in the logarithmic case: Upon restriction to D, we obtain a short exact sequence
0 → R ⊕ t → TM(− log |D|)|D → TD → 0 (4.1)
∂ The line bundle R is the trivial line bundle spanned by the Euler vector field r ∂r (r a radial coordinate around D s.t. {r = 0} = D). t =∼ ∧2N ∗D ⊗ R is the adjoint bundle of infinitesimal rotations preserving the normal Hessian. When D is co-oriented, t inherits an orientation and is thus also trivial.
Let ιD : D,→ M be the inclusion. By dualising (4.1), we can define two different residues for elliptic forms:
Definition 4.12. (i) The elliptic residue is
k k−2 ∗ rese :Ω (M, log |D|) → Ω (D), rese(ρ) = ιD(i ∂ i ∂ ρ) r ∂ r ∂ θ
38 4.2 Complex log symplectic and elliptic symplectic geometry
• Denote the kernel of rese by Ω0(M, log |D|). It is a natural subcomplex of the elliptic de Rham complex.
(ii) Consider the the short exact sequence for the Atiyah algebroid of the circle bundle S1(ND) (Since we have assumed that D is co-oriented, this is in fact a principal U(1)-bundle) associated to ND:
0 → t → At(S1(ND)) → TD → 0
1 It is a quotient of (4.1) by R, i.e. TM(− log |D|)|D is an extension of At(S (ND)) by a trivial bundle. Thus the elliptic residue filters through the radial residue
k k−1 1 ∗ resr :Ω (log |D|) →Γ(∧ At(S (ND)) ), resr(ρ) = ir ∂ ρ (4.2) ∂ r D k−1 i ∂ resr(ρ) = rese(ρ), and if rese(ρ) = 0, resr(ρ) ∈ Ω (D). (4.3) ∂ θ
There is a Lie algebroid morphism
ι : TM(− log |D|) ⊗ C → TCM(− log D) from the definition of TM(− log |D|) as the fibre product of the complex logarithmic tangent bundle with its complex conjugate.
Proposition 4.13. (Relation between complex log forms and elliptic forms, Propo- ∗ ∗ • • sition 1.11 in [CG15]) The composition Im of ι :ΩC(M, log D) → Ω (M, log |D|) ⊗ C with the projection to the imaginary part surjects onto the elliptic differential forms with vanishing elliptic residue, denoted by Ω0(M, log |D|). The kernel of this map is given by the • real smooth forms, injected into ΩC(M, log D) as the real part. We obtain the short exact sequence of complexes
• • Im∗ • 0 → ΩR(M) → ΩC(M, log D) → Ω0(M, log |D|) → 0
Proof. If w is a complex coordinate such that D = {w = 0} and (x3, . . . , x2n) are coordinates for D, a general complex log form can be written as
ρ = dlog w ∧ α + β,
∗ where both α, β are generated by (dw, ¯ dx3,..., dx2n). If this is to satisfy Im (ρ) = 0, we have 0 = dlog w ∧ α − dlogw ¯ ∧ α¯ + β − β,¯
39 Logarithmic and elliptic symplectic geometry
with α = dlogw ¯ ∧ α1 + α2, β = dlogw ¯ ∧ β1 + β2, αi, βi generated by (dx3,..., dx2n):
0 = dlog w ∧ dlogw ¯ ∧ (wα ¯ 1 + wα¯1)
+ dlog w ∧ (α2 − wβ¯1) + dlogw ¯ ∧ (¯α2 − wβ¯ 1)
+ (β2 − β¯2)
Each of the summands vanishes independently, i.e. α1, α2, and thus α are divisible by w and thus ρ = dlog w ∧ α + β must have been smooth. This shows that a complex log form is determined by its imaginary part, an elliptic form, up to adding a real smooth form (to its real part.)
3 Definition 4.14. Let H ∈ Ωcl(M) and D a complex divisor on M.A complex log symplectic 2 form is a complex logarithmic two-form σ ∈ ΩC(M, log D) with non-degenerate imaginary part Im(σ) =: ω such that dσ = a∗H, i.e. in particular dω = 0, so ω is an elliptic symplectic form.
2 ′ 2 ′ We consider two log symplectic forms σ ∈ ΩC(M, log D), σ ∈ ΩC(M, log D ) for com- plex divisors D,D′ and closed 3-forms H,H′ respectively to be equivalent if there is a diffeomorphism of divisors ψ :(M,D) → (M,D′) such that
ψ∗σ′ = σ + b, b ∈ Ω2(M) a real two-form which satisfies ψ∗H′ = H + db.
Theorem 4.15. (Darboux coordinates for complex log symplectic forms, Theorem 3.31 in [CG15]) Let (M, D, σ) be a complex log symplectic manifold and p ∈ D. Then i there are coordinates (w, z, x , yj) on a local neighbourhood around p (w a complex vanishing i function for D, z a second complex coordinate, x , yj real coordinates) such that
X j σ = dlog w ∧ dz + i dx ∧ dyj. j
Theorem 4.16. Let D be an elliptic divisor on M.
(i) (Theorem 1.8 in [CG15]) The restriction of forms to M \ D and the radial residue
map resr together define an isomorphism
∼ Hk(M, log |D|) →= Hk(M \ D) ⊕ Hk−1(S1ND).
(ii) (Theorem1.12 in [CG15]) When applied to elliptic forms with vanishing elliptic residue, the same map defines an isomorphism
k =∼ k k−1 H0 (M, log |D|) → H (M \ D) ⊕ H (D).
40 4.3 Stable generalized complex structures
4.3 Stable generalized complex structures
We now finally return to generalized complex geometry and consider a class of generalized complex structures which can be described in terms of complex log symplectic, and ultimately a subclass of elliptic symplectic forms: Recall that a generalized complex structure is uniquely • ∗ ∗ defined by its canonical bundle K ⊂ ∧ TCM. Consider the section s ∈ Γ(K ) which projects
any ρ ∈ Kp, p ∈ M to its degree-zero-component:
⟨ρ, sp⟩ := ρ0 ∈ R
Definition 4.17. A stable generalized complex structure is one where D = (K∗, s) is a complex divisor, which we then call the anticanonical divisor. By abuse of notation, we also write D = {s = 0} and call it the anticanonical divisor.
Theorem 4.18. (Theorem 3.2 in [CG15]) Any stable generalized complex strucutre J on (M,H) defines a complex log symplectic form σ = B + iω for the anticanonical divisor D = (K∗, s). Conversely, given a complex divisor D and a log symplectic form σ for a particular pair (M,H), we can construct a stable generalized complex structure. These two assigments are inverse to each other. In this correspondence, any local trivialisation of the canonical line bundle K satisfies ∗ σ a ρ = ρ0e .
∗ Let Γσ ⊂ TM(− log D) ⊕ T M(log D) denote the graph of σ. Then the correspondence is
∗ LJ = a∗Γσ = {a(X) + η ∈ TCM|X + a η ∈ Γσ},
B′ −B′ ′ 2 LJ the +i-eigenbundle of J . Under B-transforms J 7→ e J e ,B ∈ Ωcl(M), σ 7→ σ + B′ = (B + B′) + iω – the equivalence relations for stable generalized cmplex structures and complex log symplectic forms are indeed the same.
Theorem 4.19. (Theorem 3.7 in [CG15]) Let M be a smooth manifold. The forgetful map taking a pair (J ,H) of a closed 3-form H and stable generalized complex structure J ,
which is integrable w.r.t. H, to the pair (Q, o) of the real Poisson structure Q = prTM ◦J |T ∗M and the co-orientation o of the anticanonical divisor D defines a bijection between gauge equivalence classes of stable generalized complex structures (up to B-transforms) and elliptic symplectic structures ω = Q−1 with vanishing elliptic residue and co-oriented degeneracy locus.
J corresponds to a complex log symplectic form σ whose imaginary part is ω. The imaginary part of a complex log form ω has vanishing elliptic residue, and as we can see above, is invariant under B-transforms. We know that a complex log form like σ is determined by its imaginary part up to adding smooth 2-forms.
41 Logarithmic and elliptic symplectic geometry
4.3.1 Generalized complex branes in stable generalized complex manifolds
We have already defined generalized complex branes as a natural class of submanifolds in gen- eralized complex manifolds. We will now summarise results from [CG15] on half-dimensional branes in stable generalized complex manifolds – since these are very close to symplectic manifolds, their half-dimensional branes behave similarly to Lagrangian submanifolds: Stable generalized complex manifolds are generically symplectic, so their half-dimensional branes will be generically Lagrangian w.r.t. the elliptic symplectic form ω. The aspect that sets them apart from branes in pure symplectic manifolds is their intersection with the anticanonical divisor D.
Proposition 4.20. (Proposition 3.42 in [CG15]) Any submanifold L ⊂ M in a stable generalized complex manifold (M, J ) which is transverse to the anticanonical divisor D and Lagrangian for the elliptic symplectic structure underlying J inherits a smooth 2-form F = ι∗B (ι : L,→ M inclusion), making it into a generalized complex brane.
Note that because L ⋔ D, the elliptic divisor on M pulls back to form an elliptic divisor on L, so ω pulls back to L as an elliptic form, and it makes sense to demand that this pullback be zero. For these Lagrangian branes intersecting D transversely there is the following natural Lagrangian neighbourhood theorem:
Theorem 4.21. (Elliptic Lagrangian neighbourhood theorem, Theorem 3.38 in [CG15]) If (M, D, ω) is an elliptic symplectic manifold and L a compact Lagrangian sub- manifold transverse to D, there exists a tubular neighbourhood of L which is elliptic sym- plectomorphic to a tubular neighbourhood of the zero section in T ∗L(log |L ∩ D|) equipped with the natural elliptic symplectic form on the elliptic cotangent bundle.
The elliptic cotangent bundle of any manifold equipped with an elliptic divisor carries a natural elliptic symplectic structure, defined in the same way as for the ordinary cotangent ∗ ∗ bundle: T M(log |D|) has a pullback elliptic divisor with singular locus T M(log |D|)|D, and the natural elliptic symplectic form is the derivative of the tautological elliptic one-form.
42 Chapter 5
Log and elliptic symplectomorphisms and flux
First consider a compact log symplectic manifold (M, Z, ω).A diffeomorphism of the pair (M,Z) is simply a diffeomorphism that preserves Z. These diffeomorphisms clearly form a subgroup of the diffeomorphism group of M, whose Lie algebra is precisely given by the logarithmic vector fields X(M, − log Z). As a consequence, the pullback and push-forward of logarithmic forms and vector fields with respect to such logarithmic diffeomorphisms are well-defined in a natural way, and a logarithmic symplectomorphism is simply such a diffeomorphism φ which satisfies φ∗ω = ω.
Now, for a compact elliptic symplectic manifold (M, D, ω), this is less obvious: For a general diffeomorphism of φ : M → M, and the chosen elliptic divisor D = (R, s) on M, we can always consider the pullback divisor φ∗D = (φ∗D, φ∗s). This is isomorphic to the original divisor, and, up to isomorphism, gives rise to the same elliptic tangent and cotangent bundle. But the space of elliptic vector fields inside the smooth vector fields will in general be different, even if φ preserves D. In order to compare the symplectic form before and after pullback with a diffeomorphism, we need the notion of elliptic vector field to stay the same, i.e. if X ∈ X(M − log |D|), we
need φ∗(X) ∈ X(M, − log |D|) with respect to the original elliptic divisor.
Proposition 5.1. The flow φt of a time-dependent elliptic vector field Xt ∈ X(M, − log |D|) preserves the space of elliptic vector fields under push-forward, i.e.
Y ∈ X(M, − log |D|) ⇒ (φt)∗Y ∈ X(M, − log |D|)
The diffeomorphisms obtained in this manner form a subgroup of the identity component of the diffeomorphism group which we call the elliptic diffeomorphism group.
43 Log and elliptic symplectomorphisms and flux
This result follows from general Lie groupoid and Lie algebroid theory, which we outline in the following section. Within the subgroup of elliptic diffeomorphisms, an elliptic symplectomorphism is one that satisfies φ∗ω = ω, and we call this subgroup of the identity component of all diffeomorphisms of M the (identity component of the) elliptic symplectomorphism group. After outlining the background theory on Lie groupoids and Lie algebroids which proves Proposition 5.1, we define the log and elliptic symplectic flux and generalise a number of results on the ordinary symplectic flux to this setting. We further prove a Lagrangian neighbourhood theorem for compact Lagrangian submanifolds in a logarithmic symplectic manifold (M,Z) which intersect the degeneracy locus transversely. Finally, we apply these results to find the small deformations of such log Lagrangians, as well as for Lagrangian branes in stable generalized complex manifolds which intersect the anticanonical divisor transversely.
5.1 The groupoid exponential map
This section is a review of material covered in [Mac05], Chapter 1 and 3, and follows the notation of that text. Let G ⇒ M be a Lie groupoid and AG its associated Lie algebroid. Denote the source and target map by (α, β) respectively, and the tangent bundle along the fibres of α by T αG.
Definition 5.2. (i) A bisection of G is a smooth map σ : M → G so that α ◦ σ = IdM and β ◦ σ : M → M is a diffeomorphism.
(ii) The left-translation with respect to a bisection σ is
Lσ : G → G, g 7→ σ(βg)g
The corresponding right-translation is
−1 Rσ : G → G, g 7→ gσ((β ◦ σ) (αg)
The inner automorphism associated to σ is
−1 Iσ : G → G, g 7→ σ(βg)gσ(αg)
(iii) The group multiplication for bisections σ, τ is given by
(σ ∗ τ)(x) = σ((β ◦ τ)(x))τ(x) ∀x ∈ M
(iv) A vertical vector field on G is vertical with respect to α, i.e. X ∈ Γ(T αG).
44 5.1 The groupoid exponential map
(v) A vector field X on G is right-invariant if it is vertical, and X(hg) = (Rg)h,∗(X(h)) for all (g, h) ∈ G ∗ G.
(vi) The vector bundle AG of G is the pullback vector bundle of T αG → G along the unit embedding 1 : M → G. The anchor a : AG → TM, a vector bundle morphism over the identity on M, is given by the following composition: ⊂ β AG T αG TG ∗ TM (∗) β M 1 G G M This defines a vector bundle morphism over the identity on M, since β ◦ 1 = IdM .
Remark 5.3. A vertical vector field on G is right-invariant if and only if
X(g) = (Rg)∗(X(1β(g))) ∀g ∈ G, (5.1) so it is fully determined by its values on the submanifold of identity elements. Conversely, any section of AG extends to a smooth right-invariant vector field on G via X˜(g) = (Rg)∗(X(βg)). This implies that sections of AG and right-invariant vector fields on G are in one-to-one correspondence. The Lie bracket on AG is induced by the Lie bracket on the right-invariant vector fields.
Any morphism F : G → G′ of Lie groupoids over f : M → M induces a morphism ′ A(F ) = F∗ : AG → AG of the corresponding Lie algebroids.
Proposition 5.4. (Flow and exponential map) Let W ⊂ M be open, take Xt ∈ ΓW (AG) to be a time-dependent local section of the Lie algebroid. Then for each x0 ∈ W there is an open neighbourhood U ⊂ W containing x0, called a flow neighbourhood for Xt, an ε > 0, and a unique family of local bisections Φt ∈ ΓU , |t| < ε such that
d ′ ′ Φt (x) = Xt(ψt(x)) ∀|t| < ε, x ∈ U and Φ0 = 1 dt t′=t
Furthermore, the family of local diffeomorphisms ψt = β ◦ Φt is the flow of a(X).
Proof. This is constructed from the flow on G for the right-invariant vertical vector field X˜t corresponding to Xt ∈ Γ(A): The local flow ϕt : U → Ut of X˜t satisfies α ◦ ϕt = α, since X˜t is α-vertical. Define U = β(U),Ut = β(Ut), ψt : U → Ut such that ϕt U Ut β β ψt U Ut y ˜ commutes. Since ϕt(hξ) = ϕt(h)ξ ∀ξ ∈ Gx (because X is right-invariant), ψt is well-defined.
45 Log and elliptic symplectomorphisms and flux
β is a submersion, so ψt is smooth. The inverse of ϕt has the same properties as ϕt, so −1 (ψt) ◦ ϕt ◦ β = β. Thus ψt is a local diffeomorphism. For x ∈ U, we have:
d d ′ ′ ˜ ′ ψt (x) = ′ β(ϕt (h)) = β∗(X(ϕt(h))) for h ∈ U with β(h) = x dt t′=t dt t′=t
This shows that ψt : U → Ut is a local flow for a(X) = β∗(X˜).
The bisections Φt are now defined as follows: The above properties of ϕt imply (see
Proposition 1.4.12 in [Mac05]) that each ϕt is the restriction of a unique local left-translation
U U LΦt : G → Gt , where the bisection Φt ∈ ΓU G is defined by
−1 x Φt(x) = ϕt(h)h , h any element of U ∩ G
Restricting to just time-independent sections X ∈ Γ(AG) allows us to make a stronger statement: Φt then additionally satisfies
(i) Φt+s = Φt ∗ Φs, when |s|, |t|, |s + t| < ε,
−1 (ii) Φ−t = (Φt)
(iii) {β ◦ Φt : U → Ut} is a local 1-parameter group of diffeomorphisms for a(X) ∈ ΓW (TM).
Because of these properties, we write also: For Φt the bisection associated to X ∈ Γ(AG):
Φt = exp(tX)
This allows us to define an exponential map X 7→ exp(X) on the sheaf of germs of local sections of AG with values in the sheaf of germs of local bisections of G.
Proposition 5.5. (Proposition 3.6.3 in [Mac05]) The set of values exp(tX(m)) =
Φt(m),X ∈ Γ(AG), m ∈ m, and t ∈ R such that exp is defined, is the identity-subgroupoid of G.
Theorem 5.6. (Theorem 3.6.4 in [Mac05]) Let X ∈ Γ(AG) and g0 ∈ G with β(g0) = y0 ∈ M. Then the integral curve for X˜ through g0 is infinitely extendable in both directions if and only if the integral curve for a(X) is infinitely extendable in both directions. In particular, X˜ is complete if and only if a(X) is.
Finally, just like there is a canonical adjoint action of a Lie group on its Lie algebra, we can define an adjoint action of Lie groupoid bisections of G on the Lie algebroid AG:
46 5.2 The Flux homomorphism for log and elliptic symplectic manifolds
Definition 5.7. Let σ be a bisection of G over U ⊂ M, with (β ◦ σ)(U) = V . Then U V Iσ : GU → GV is a morphism of Lie algebroids over β ◦ σ : U → V . We define the adjoint action of σ on AG by
Ad(σ) = (Iσ)∗ : AG|U → AG|V
Proposition 5.8. (Proposition 3.7.1 in [Mac05]) With this notation:
(i) For X,Y ∈ ΓU (AG), Ad(σ)[X,Y ] = [Ad(σ)X, Ad(σ)Y ].
(ii) If X ∈ ΓU (AG) with U a flow neighbourhood for X, then V = (β ◦ σ)(U) is a flow neighbourhood for Ad(σ)X and, for |t| sufficiently small,
−1 exp(t Ad(σ)X) = I˜σ(exp(tX)), where I˜σ(exp(tX)) : y 7→ Iσ(exp(tX)((β ◦ σ) (y)))
(iii) If X,Y ∈ ΓU (AG) and U is a flow neighbourhood for X, then
d [X,Y ] = − Ad(exp(tX))(Y ) dt t=0
These general results can now be applied to the elliptic tangent bundle TM(− log |D|) and its frame groupoid to prove Proposition 5.1: Proposition 5.4 and Proposition 5.8 allow us to construct a bisection Φt of the groupoid corresponding to the flow of each elliptic vector field X. This bisection acts adjointly on any other elliptic vector field Y to produce a new elliptic vector field, and the diagram (∗) defining the anchor a : AG → TM allows us to conclude that
a(Ad(Φt)X) = ψ∗(a(X)).
5.2 The Flux homomorphism for log and elliptic symplectic manifolds
Analogously to ordinary symplectic manifolds, we can define flux homomorphisms for log and elliptic symplectic manifolds to pick out Hamiltonian diffeomorphisms (i.e. the endpoints of Hamiltonian isotopies) in the identity component of the log or elliptic symplectomorphism group respectively. To simplify the notation, in this section only, we will denote the elliptic and logarithmic objects in the same manner: The singularity locus is D, the elliptic or log symplectic form is ω, the log or elliptic tangent bundle is TM(− log D), the elliptic or log cohomology is H•(M, log D), and so on. Where there is a difference between the logarithmic and the elliptic case, it will be specifically indicated. Denote by Symp0(M, ω) the identity component of the group of (log or elliptic) symplectomorphisms of a log or elliptic symplectic manifold (M,D) with (log or elliptic) symplectic form ω, and by Symp^0(M, ω) its universal cover.
47 Log and elliptic symplectomorphisms and flux
The results and proofs in this section closely follow [MS98] and [Oh15], which describe the theory for ordinary symplectic manifolds.
Definition 5.9. The flux homomorphism is
Z 1 1 Flux : Symp^0(M, ω) → H (M, log D), Flux({ψt}) = [iXt ω] dt, 0 where {ψt}, t ∈ [0, 1] is a representative of a homotopy class of paths in Symp0(M, ω) with ψ0 = Id and endpoint ψ1, and Xt its associated time-dependent log or elliptic vector field.
Theorem 5.10. The flux homomorphism, as above, is well-defined, and a group homomor- phism.
First of all, note that the group structure on Symp^0(M, ω) can be defined in two different, but equivalent ways:
1. The product of two homotopy classes of smooth paths [φt], [ψt] is [φt ◦ ψt], or
2. the concatenation of the paths [χt] with
( φ2t 0 ≤ t ≤ 1/2 χt = (5.2) ψ2t−1 ◦ φ1 1/2 < t ≤ 1
It is easy to check that the paths in these two definitions are homotopic with fixed endpoints and thus define the same element in the universal cover.
Lemma 5.11. (See Lemma 2.3.4 in [Oh15]) Let X,Y be log/elliptic symplectic vector fields. Then their Lie bracket is the Hamiltonian vector field associated to the smooth function
f = −ω(X,Y )
The proof of this proceeds exactly as in the ordinary symplectic case.
s Lemma 5.12. (Banyaga’s Lemma, [Ban78]) Let {φt } be a smooth two-parameter family of diffeomorphisms on M. Denote
∂φs ∂φs X = Xs = t ◦ (φs)−1,Y = Y s = t ◦ (φs)−1. t ∂t t t ∂s t
Then ∂Y ∂X = + [Y,X] ∂t ∂s This formulation of Banyaga’s Lemma and a proof can be found in [Oh15], Lemma 2.4.2. The following lemma generalises Lemma 2.4.3 in [Oh15] to log and elliptic forms:
48 5.2 The Flux homomorphism for log and elliptic symplectic manifolds
′ Lemma 5.13. Let {ψt}, {ψt} ∈ Symp^0(M, ω) two paths from the identity to the same ′ ′ endpoint ψ = ψ1 = ψ1. Let Xt,Xt be the log/elliptic vector fields associated to these paths. ′ If {ψt}, {ψt} are homotopic relative to their ends, then the log/elliptic one-form
Z 1 ′ iXt−X ω dt 0 t is exact.
s ′ Proof. Let ψt be a homotopy between ψt, ψt relative to {0, 1} i.e.
s s ψ0 = IdM , ψ1 = ψ, 0 ≤ s ≤ 1
s s Denote by Xt the vector field associated to the path {ψt , 0 ≤ t ≤ 1}. If we can prove that the log/elliptic one-form
d Z 1 i s ω dt Xt ds 0 is exact for all s, the result follows. s s s We have already defined Xt as in Lemma 5.12; define Yt accordingly as well. All ψt s s are log/elliptic symplectomorphisms, so the vector fields Xt ,Yt are symplectic vector fields. Compute: d Z 1 Z 1 ∂ Z 1 i s ω dt = (i s ω) dt = i ∂ Xs ω dt, Xt Xt t ds 0 0 ∂s 0 ∂ s which we can rewrite using Lemma 5.12:
d Z 1 Z 1 i s ω dt = i ∂ Y s ω dt. Xt t s s ds 0 0 ∂ t +[Xt ,Yt ]
Now, the first part of the integral is simply
i s s ω = 0 ∀s, Y1 −Y0
s s since ψ1 = ψ ∀s, ψ0 = IdM ∀s. As the Lie bracket of two log/elliptic symplectic vector fields, s s [Xt ,Yt ] is a log/elliptic Hamiltonian vector field (proof proceeds exactly as in the ordinary symplectic case) and thus the second term is exact for all s.
1 This shows that Flux : Symp^0(M, ω) → H (M, log D) is well-defined. The group homomorphism property follows easily from the fact that if the isotopy of symplectomorphisms {φt} is generated by Xt, and {ψt} by Yt, ψt ◦ φt is generated by
Yt + (ψt)∗Xt. Furthermore, φ∗X − X is Hamiltonian whenever X is a symplectic vector field and φ a symplectomorphism.
49 Log and elliptic symplectomorphisms and flux
Theorem 5.14. (Compare Theorem 10.12 in [MS98]) Let ψ ∈ Symp0(M, ω). ψ is a log/elliptic Hamiltonian symplectomorphism if there exists a symplectic isotopy ψt with
ψ0 = IdM , ψ1 = ψ such that
Flux({ψt}) = 0.
Conversely, given a symplectic isotopy {ψt} with Flux({ψt}) = 0, it is homotopic (with fixed endpoints) to a Hamiltonian isotopy.
The proof again proceeds exactly as in the ordinary symplectic case; with the vector fields and forms being log or elliptic:
Proof. If ψ is Hamiltonian, there is a Hamiltonian isotopy ψt with ψ1 = ψ. This is associated to a smooth family of Hamiltonian functions Ht : M → R. Let Xt be the family of log/elliptic vector fields generating ψt. So:
Z 1 Z 1 Flux({ψt}) = [iXt ω] dt = [dHt] dt = 0 0 0
Conversely, let ψt be a symplectic isotopy with ψ as its endpoint, and assume that Flux({ψt}) = R 1 0. This implies that 0 iXt ω dt is exact. To prove the statement, we have to show that the R T integrand is exact for all t, which is equivalent to showing that 0 iXt ω dt is exact for each T ∈ [0, 1].
R 1 • Step 1: ψt can be modified by a Hamiltonian isotopy such that 0 iXt ω dt = 0 R 1 s (not just exact). Write 0 iXt ω dt = dF . Let φF be the flow of the Hamiltonian vector field associated to F . This is Hamiltonian for each s, so we can prove the theorem for 1 −1 1 −1 the composition (φF ) ◦ ψ instead of for the original ψ. (φF ) ◦ ψ is the endpoint of t −1 the concatenation of (φF ) and ψt, as defined in (5.2), which is generated by a family ′ R 1 X i ′ ω dt = 0 of log/elliptic vector fields t such that 0 Xt . From now on, we assume that ψ = ψ1 is the endpoint of a symplectic isotopy with
Z 1 iXt ω dt = 0 0
• Step 2: Constructing the Hamiltonian isotopy. Now, for every t (separately), s R consider θt ∈ Symp0(M, ω), s ∈ , the flow of the log/elliptic symplectic vector field R t Yt = − 0 Xλ dλ, i.e. d θs = Y ◦ θs ∀s, θ0 = Id ds t t t t s s We have Y0 = Y1 = 0, so θ0 = θ1 = Id for all s. Claim: 1 φt = θt ◦ ψt
50 5.2 The Flux homomorphism for log and elliptic symplectic manifolds
is the Hamiltonian isotopy from the identity to ψ1 = ψ. In fact, since Flux is a group homomorphism:
1 Flux({φt}0≤t≤T ) = Flux({θt }0≤t≤T ) + Flux({ψt}0≤t≤T ) Z T (1) s = Flux({θT }0≤s≤1) + [iXt ω] dt 0 (2) Z T = [iYT ω] + [iXt ω] dt 0 = 0
1 s (1): There is a homotopy connecting {θt }0≤t≤T and {θT }0≤s≤1: In the following, τ is the parameter for each isotopy, and σ the parameter of the homotopy. We take 0 ≤ τ, σ ≤ 1 and consider
( σ ( 1 σ θ2σT τ 0 ≤ τ ≤ 1/2 0 0 1 θ2τT 0 ≤ τ ≤ 1/2 Hτ = σ(−2τ+2) ; Ht = θ0 = Id,Ht = −2τ+2 θsT 1/2 < τ ≤ 1 θT 1/2 < τ ≤ 1
0 Clearly, this is a homotopy between the constant isotopy at the identity (Ht = Id) and 1 1 1−s Ht , which is in fact the concatenation of {θt }0≤t≤T and θT . Since this concatenation is null-homotopic, the two original paths are homotopic. s (2): θT is the flow of YT .
∗ Lemma 5.15. (Compare Lemma 10.14 in [MS98]) Now assume that M = T L(log DL) for some compact (L, DL) with either a logarithmic or elliptic structure. The log or elliptic structure om (L, DL) pulls back to the log/elliptic cotangent bundle, and there is a canonical log/elliptic symplectic form ω = dλ, λ the tautological log/elliptic one-form on the log/elliptic cotangent bundle. Let ψt a log/elliptic symplectic isotopy on M, then
∗ Flux({ψt}) = [ψ1λ − λ]
Proof. Let Xt be the family of log/elliptic symplectic vector fields which generates ψt.
(1) (2) d [i ω] = [i ψ∗ω] = [ψ∗(i ω) = [ψ∗(i ω)] = [ψ∗(£ λ)] = [ψ∗λ] (5.3) Xt Xt t t (ψt)∗Xt t Xt t Xt dt t
For (1), we use that for any (log/elliptic) symplectic vector field X and any log/elliptic symplectomorphism φ, φ∗X − X is Hamiltonian. (2) is the application of the general identity for the Lie derivative of forms with respect to a time-dependent vector field:
d ∗ ∗ ′ ψt′ α = ψt £Xt α dt t′=t
51 Log and elliptic symplectomorphisms and flux
We obtain the result by integrating the right-hand side of (5.3) from t = 0 to 1.
∗ ∗ Corollary 5.16. If M = T L(log DL),D = T L(log DL)|DL with exact symplectic form, as above,
Flux(π1(Symp0(M, ω))) = 0.
1 Thus we obtain a morphism Flux : Symp0(M, ω) → H (M, log D) whose kernel is precisely given by Hamiltonian diffeomorphisms.
Proof. Lemma 5.15 immediately proves the first sentence. This establishes that the flux only depends on the endpoint of a symplectic isotopy, and thus the isotopies with Hamiltonian endpoints are precisely those with zero flux.
5.3 Lagrangian neighbourhood theorem for log symplectic mani- folds
Let (M, Z, ω) be a real logarithmic symplectic manifold. We can prove a Lagrangian neigh- bourhood theorem for compact Lagrangians that intersect the singularity locus transversely, employing the same techniques as in the proof of Weinstein’s original Lagrangian neighbour- hood theorem and the version for stable generalized complex manifolds (see Theorem 4.21 and [CG15]). As far as we are aware, this theorem and proof have not previously appeared in the literature.
Proposition 5.17. If ιN : N,→ M is a submanifold which intersects Z transversely, there are induced morphisms
ιN,∗ : TN(− log N ∩ Z) → TM(− log Z) ∗ • • ιN :ΩM (log Z) → ΩN (log N ∩ Z)
Proof. We only need to consider the pushforward of x ∂ , where x is a vanishing function ∂x N∩Z for N ∩ Z. Obviously, since N ⋔ Z, any vanishing function for Z on M will provide one for N ∩ Z in N via pullback. Assume x = ι∗x˜. Then we can define the pushforward ∂ ∂ ιN,∗ =x ˜ , ∂x N∩Z ∂x˜ N∩Z
∂ which is well-defined and smooth: x˜ can be chosen in such a way that x˜ ∂x˜ is tangent to N. The definition of the pullback for logarithmic forms is then obvious.
Theorem 5.18. (Lagrangian neighbourhood theorem for log symplectic mani- folds) Let (M, Z, ω) be a real log symplectic manifold with degeneracy locus Z and log
52 5.3 Lagrangian neighbourhood theorem for log symplectic manifolds
symplectic form ω, and ιL : L,→ M a compact Lagrangian submanifold which intersects the degeneracy locus Z transversely. Then there is a neighbourhood (U, U∩Z) of L in M which is isomorphic to a neighbourhood of the zero section in T ∗L(log L ∩ Z), i.e. there exists a diffeomorphism onto its image
φ :(U, U ∩ Z) → T ∗L(log L ∩ Z)
∗ ∗ such that φ (ω0) = ω, where ω0 is the standard log symplectic form on T L(log L ∩ Z), and (φ(U), φ(U ∩ Z)) is a tubular neighbourhood of (L, L ∩ Z).
Remark 5.19. Throughout this text, we will in particular be interested in applying to the case where Z = ∂M and Z ∩ L = ∂L.
For the proof, we will need the Whitney extension theorem and a Lemma, both of which can be found in [Can01]:
Theorem 5.20. (Whitney extension theorem) Let M be an n-dimensional manifold and X a k-dimensional submanifold (k < n). Suppose that at each p ∈ X we are given a
linear isomorphism Lp : TpM → TpM s.t. Lp|TpX = IdTpX and Lp depends smoothly on p. Then there exists an embedding h : U → M of some open neighbourhood U of X in M such
that h|X = IdX and and dhp = Lp ∀p ∈ X.
Lemma 5.21. (Proposition 8.3 in [Can01]) Let V be a 2n-dimensional vector space
with two syplectic forms Ω0, Ω1. Let U ⊂ V be a Langrangian subspace for both Ω0 and Ω1. Let W ⊂ V be any complement to U. Then from W we can canonically construct a linear ∗ isomorphism L : V → V s.t. L|U = IdU and L Ω1 = Ω0.
Proof of Theorem 5.18. The proof proceeds like that of the original Weinstein Lagrangian neighbourhood theorem, see for example [Can01]. In this case, we start by choosing a tubular neighbourhood for (L, L ∩ Z) whose intersection with Z is a tubular neighbourhood for L ∩ Z. ∼ Claim: The cokernel of ιL,∗ is TM(− log Z)|L/ Im(ιL,∗) = NL.
Proof. We consider the sheaves of sections and show that they are isomorphic as locally
free sheaves. Γ (TM(− log Z)|L/T L(− log L ∩ Z)) includes into Γ(NL) = Γ(TM|L/T L) via
the anchor. The inverse map is as follows: Let X + X(L) ∈ Γ(TM|L/T L). We have ′ assumed L ⋔ Z, so (TL + TZ)|L∩Z = TM|L∩Z , i.e. we can write X|L∩Z = X + Y , where ′ ′ ′ X ∈ Γ(TL|L∩Z ),Y ∈ Γ(TZ|L∩Z ). Extend X to a section on all of L. ⇒ X − X + X(L) = ′ X + X(L), and X − X ∈ Γ(TM(− log Z)|L). ′ Lastly, check that the class of X − X in Γ(TM(− log Z)|L/T L(− log L ∩ Z)) does not depend on the choice of X′ and its extension. Let X′′ ∈ X(L) be another section s.t. ′′ X − X |L∩Z ∈ X(Z). Then
′ ′′ ′′ ′ (X − X )|L∩Z − (X − X )|L∩Z = (X − X )|L∩Z ∈ Γ(TL ∩ TZ)|L∩Z = Γ(T (L ∩ Z))
53 Log and elliptic symplectomorphisms and flux
So X + X(L) 7→ X − X′ + X(L, − log L ∩ Z) is well-defined.
Since L ⋔ Z and Z ⊂ M a codimension-1 submanifold, there is a tubular neighbourhood U of L in M such that U ∩ Z is a tubular neighbourhood of L ∩ Z in Z. Now, since ∼ ∗ TM(− log Z)|L/ Im(ιL,∗) = NL and ιLω = 0, we obtain an isomorphism
∗ ω : TM(− log Z)|L/ Im(ιL,∗) → T L(log L ∩ Z), which maps the tubular neighbourhood (U, U ∩ Z) of (L, L ∩ Z) to a tubular neighbourhood of the zero section in T ∗L(L ∩ Z) in such a way that U ∩ Z gets mapped to the fibre over L ∩ Z. ∗ Thus we can now view both ω and the natural log symplectic form on T L(log L ∩ Z), ω0, ∗ ∗ as log symplectic forms on (U, U ∩ Z), both of which satisfy ιLω = 0 = ιLω0. The projection p : TL(− log L ∩ Z) → L induces an isomorphism on log cohomology p∗ : H•(L, log L ∩ Z) → H•(U, log U ∩Z). p is homotopic to the identity on U, so p∗ is an isomorphism on cohomology: ∗ • • ∗ Since p ◦ ι = IdL, ι ◦ p ∼ IdU , ι : H (U, log U ∩ Z) → H (L, L ∩ Z) is the inverse of p on ∗ ∗ cohomology. Thus, since ιLω = ιLω0 = 0, ω, ω0 are in the same log cohomology class on U (in fact, both are trivial in cohomology on U).
1 ⇒ ω − ω0 = dα for some α ∈ Ω (U, log U ∩ Z).
We now consider the family of cohomologous closed log forms ωt = tω + (1 − t)ω0, which is in fact a family of log symplectic forms close to L, since ωt is non-degenerate on a small tubular neighbourhood of L for all t ∈ [0, 1]: Note that every fibre of TU(− log U ∩ Z)|L has a Lagrangian subspace with respect to both ω and ω0 given by TL(− log L ∩ Z). If we pick a metric on TU(− log U ∩ Z), it defines smoothly varying orthogonal complements Wp to TpL(− log L ∩ Z) at every point p ∈ L. Using Lemma 5.21, we obtain smoothly varying isomorphisms hp : TU(− log U ∩ Z)|p → TU(− log U ∩ Z)|p for each p ∈ L. Then with the Whitney extension theorem, we obtain an embedding h′ :(U ′,U ′ ∩ Z) → (U, U ∩ Z) of some ′ ′ ′ neighbourhood (U ,U ∩ Z) of (L, L ∩ Z) such that h |L = IdL and (dh)p = hp ∀p ∈ L. Thus: ′∗ ∗ h (ω)p = hp(ω|p) = ω0|p ∀p ∈ L
′ ′ ∗ So replacing (U, U ∩ Z) by (U ,U ∩ Z) and ω by h ω means that ω|L = ω0|L, so ωt = ′′ ′′ ′ ′ tω + (1 − t)ω0 is non-degenerate on some neighbourhood (U ,U ∩ Z) ⊂ (U ,U ∩ Z) of (L, L ∩ Z). Assume that we had chosen this neighbourhood to begin with. Now we can apply the Moser argument:
−1 Xt := −ωt (α), where dα = ω − ω0
54 5.4 Small deformations of Lagrangians in log symplectic manifolds
∗ is a well-defined logarithmic vector field, in particular it is smooth. We assume that ιLα = 0, which is clearly always possible. L was assumed to be compact, so this time-dependent
log vector field can be integrated to a family of diffeomorphisms ψt, t ∈ (0, 1) on a small ∗ neighbourhood of L in U, which preserve U ∩ Z. We have ψt|L : L → L, since ιLα = 0. Furthermore ψ0 = Id, so:
∗ ψt (ωt) = ω0 ∀t ∈ [0, 1]
∗ Thus there is a neighbourhood of (L, L ∩ Z) in (U, U ∩ Z) with diffeomorphism ψ1(ω) = ω0, which proves the theorem.
Corollary 5.22. If L ⊂ M a compact Lagrangian such that L ⋔ Z, each connected component of L ∩ Z lies inside a single symplectic leaf of ω−1 in Z and is Lagrangian inside this leaf.
Proof. From the neighbourhood theorem, we obtain a tubular neighbourhood of L with 1 2 n coordinates (x , x , . . . x , y1 . . . , yn) around a point of L ∩ Z such that xi are coordinates for ∗ L, x1 a vanishing function for Z, yi are fibre coordinates for T L(− log L ∩ Z), and where the log symplectic form is given by
dx1 ω = ∧ dy + X dxi ∧ dy x1 1 i i>1
Clearly, the symplectic leaves of ω−1 are given by the integrable distribution ker(res ω).
The intersection L ∩ Z in these coordinates is given by yi = 0, x1 = 0 and thus clearly
T (L ∩ Z) ⊂ ker(res ω) = ker dy1, so each connected component of L ∩ Z will lie inside a single symplectic leaf. The symplectic form on the symplectic leaves will clearly be given by
′ X i ω = dx ∧ dyi, i>1
so L ∩ Z will be Lagrangian inside the symplectic leaf.
5.4 Small deformations of Lagrangians in log symplectic mani- folds
In this section, we consider compact Lagrangians (L, L ∩ Z) inside a log symplectic manifold (M, Z, ω), and we assume that L ⋔ Z. We have just proved a Lagrangian neighbourhood theorem for this scenario, Theorem 5.18. We consider a neighbourhood U =∼ T ∗L(log L ∩ Z) of L as in the theorem, equipped with the canonical log symplectic form ω = dλ.
55 Log and elliptic symplectomorphisms and flux
Definition 5.23. A strong map of pairs f :(A, B) → (M,N) (where B ⊂ A, N ⊂ M are smooth submanifolds) is a smooth map with f −1(N) = B.
Definition 5.24. A small deformation of L is a second Lagrangian L′ connected to L by a smooth family of Lagrangians
φ : L × [0, 1] → M, φ(L, 0) = 0, φ(L, 1) = L′, where each φt := φ(·, t):(L, L ∩ Z) → (M,Z) is a smooth embedding and a strong map of 1 pairs, and all φt are C -close to φ0.
Note that if we have any smooth family of log Lagrangian embeddings φ : L × [0, 1] → 1 M, φ(L, 0) = L, for sufficiently small t, φt will always be C -close to φ0. Furthermore, the images φ(L, t) will intersect the fibres in the tubular neighbourhood U =∼ T ∗L(log L ∩ Z) of L transversely. Any such small deformation of L can then be written as the graph of a logarithmic one-form on L in U – and it is easy to check that with the canonical log symplectic form on T ∗L(log L∩Z), such a graph will be Lagrangian if and only if the log one-form is closed.
Proposition 5.25. Small deformations of L up to local Hamiltonian isotopy (i.e. such that the image of L never leaves U) are given by
H1(L, log L ∩ Z).
Proof. Given the results on the log flux homomorphism, this proof proceeds exactly as for ordinary Lagrangians in an ordinary symplectic manifold. If we consider the graph of a small one-form α on L in U, this is connected to L by the smooth family of Lagrangians
φ : L × [0, 1] → U, φ(l, t) = Graph(tα)|l, l ∈ L
If α = df is exact, the Hamiltonian isotopy ψt given by the flow of the Hamiltonian vector
field Xf maps ψ1 : L 7→ Graph(df). What is left to show: If the graph of a closed log one-form α ∈ Γ(T ∗L(log L ∩ Z)) is the image of L under a Hamiltonian isotopy, α must have been exact.
We assume that there is a Hamiltonian isotopy {ψt} such that ψ1(L) = Graph(α). We know that Flux({ψt}) = 0. According to Lemma 5.15, we have
∗ ∗ ∞ Flux({ψt}) = [ψt λ − λ] ⇒ ψ1λ = λ + dg, g ∈ C (U)
56 5.4 Small deformations of Lagrangians in log symplectic manifolds
ι : L → T ∗L(log L ∩ Z) inclusion of L as the zero section α : L → T ∗L(log L ∩ Z)
ψ1(L) = Graph(α) ⇒ α = ψ1 ◦ ι
Now, the canonical one-form λ on T ∗L(log L∩Z) has the property σ∗λ = σ ∀σ ∈ Ω1(L, log L∩ ∗ ∗ ∗ ∗ ∗ Z). So: α = α λ = i (ψ1(λ)) = i (λ + dg) = di (g).
Remark 5.26. We can immediately make an analogous statement for compact Lagrangian branes in a stable generalized complex manifold which intersect the degeneracy locus D transversely: For such branes, [CG15] prove an analogous Lagrangian neighbourhood theorem, and we can then apply the same reasoning as above to conclude that small deformations of such a brane (L, L ∩ D) up to Hamiltonian isotopy are given by
H1(L, log |L ∩ D|).
5.4.1 Log and elliptic Lagrangians as coisotropic submanifolds in a Poisson manifold
Lagrangians in logarithmic symplectic manifolds and generalized complex branes in stable generalized complex manifolds which intersect the degeneracy locus transversely are both examples of coisotropic submanifolds with respect to the associated Poisson structures.
[CF07] define an L∞-algebroid structure on the normal bundle NC of any coisotropic submanifold C, and [SZ13] show that if the Poisson structure is fibrewise entire on a tubular neighbourhood of C, small deformations of C as a coisotropic submanifold are given by the degree-1 Maurer-Cartan elements of this L∞-algebroid. In both cases discussed here, the established Lagrangian neighbourhood theorems show that ω−1 is indeed fibrewise entire, and we can apply the standard form of the log or elliptic symplectic form respectively to explicitly compute the Maurer-Cartan elements. This provides an alternative route to obtain small deformations of log and elliptic Lagrangians that intersect the singularity locus transversely, which is to be understood as a consistency check.
Definition & Proposition 5.27. Let C ⊂ (M, π) be a coisotropic submanifold in a Poisson manifold, i.e. π(N ∗C) ⊂ TC. Some basic definitions and results on coisotropic submanifolds:
∞ (i) The vanishing ideal of C is I(C) = {f ∈ C (M)|f|C = 0}. The condition {I(C),I(C)} ⊂ I(C),where {·, ·} is the Poisson bracket, equivalently defines coisotropic submanifolds.
(ii) The distribution π(N ∗C) ⊂ TC defines the characteristic foliation of C. It is involutive,
since it is spanned by Hamiltonian vector fields XH = π(dh), h ∈ I(C), which commute on the coisotropic submanifold C.
57 Log and elliptic symplectomorphisms and flux
(iii) Each coisotropic submanifold C has an associated Lie algebroid structure on N ∗C uniquely defined by the conditions
[df, dg] = d{f, g}, f, g ∈ I(C) (5.4) [hα, β] = h[α, β] − π(β)(h)α, α, β ∈ Γ(N ∗C), h ∈ C∞(C) (5.5)
Thus Γ(∧•NC) carries the structure of a differential complex.
(iv) NC carries the structure of an L∞-algebroid, or, more precisely, an L∞[1]-algebroid: • Γ(∧ NC)[1] is equipped with graded symmetric brackets λk, k ≥ 1 of degree 1:
• ⊗k • λk : Γ(∧ NC)[1] → Γ(∧ NC)[1]
λ1 = d is the differential of the differential complex on NC. λ2 is a chain map which obeys the Jacobi identity up to exact terms, so it induces another Lie bracket on
λ1-cohomology. In particular, Maurer-Cartan elements of Γ(NC[1]) satisfy
1 MC(α) := X λ α⊗k = 0, λ α⊗k = P ([[... [π, α], α] ... ], α]), k! k k k≥1
where P : X(NC) → Γ(NC) is restriction to C composed with projection X(NC) →
Γ(NC) induced by the canonical splitting T (NC)|C = NC ⊕ TC. In this expression, α ∈ Γ (∧•NC) is viewed as a vertical multivector field on the total space of NC in the canonical way, and [·, ·] is the Schouten bracket of multivector fields.
Definition 5.28. ([SZ13], Section 1.1) A fibrewise entire function on a vector bundle E → C is a function f : U → R defined on some tubular neighbourhood of the zero section such that the restriction f|Ux to each Ex ∩ U, x ∈ C can be written as a convergent power series. A multivector field X on a tubular neighbourhood U =∼ NC of a coisotropic submanifold C in (M, π) is fibrewise entire if there are local coordinates xi on C and corresponding fibre coordinates yj for NC such that
∂ ∂ X = h (xi, yj) ∧ , IJ ∂xI ∂yJ where I,J are multi-indices so that |I| + |J| = deg(X) and all hIJ are fibrewise entire functions on U.
Remark 5.29. Note that the notion of fibrewise entire functions and multivector fields on a tubular neighbourhood near a submanifold C depends on the choice of a tubular neighbourhood and its particular identification with NC.
Theorem 5.30. (Theorem 1.12 in [SZ13]) If M = NC, π fibrewise entire on NC, the space of C1-small deformations of a coisotropic submanifold C ⊂ (M, π) is locally given by
58 5.4 Small deformations of Lagrangians in log symplectic manifolds
the Maurer-Cartan elements in Γ(NC) (in the sense that the Maurer-Cartan series of these
elements converges to zero) with respect to the L∞-structure on NC[1] established above.
We now apply these results to log Lagragians which intersect the degeneracy locus transversely:
Proposition 5.31. Let (L, L ∩Z) ⊂ (M, Z, ω) be a Lagrangian submanifold in a logarithmic symplectic manifold which intersects the singularity locus Z transversely.
The space of Maurer-Cartan elements of the L∞-algebroid structure on NL is isomorphic to 1 Ωcl(L, log L ∩ Z). Proof. Pick a log Lagrangian neighbourhood (U, U ∩ Z) of (L, L ∩ Z) according to Theorem
5.18, equipped with the standard log symplectic form ω = ω0 and the inverse Poisson structure −1 i π = ω . , which in adapted local coordinates (x, q , y, pi) around each point in U ∩ Z has the form ∂ ∂ ∂ ∂ π = −x ∧ + X ∧ . ∂x ∂y ∂p ∂qi i i As a first step, we verify that the Lie algebroid structure induced on N ∗L by π is indeed isomorphic to TL(− log L ∩ Z) under the identification NL =∼ T ∗L(log L ∩ Z),N ∗L =∼ TL(− log L ∩ Z) induced by ω: Since the Lie algebroid structure on N ∗L is uniquely defined by the conditions (5.4) and (5.5), it suffices to check that ω([ω−1(·), ω−1(·)]) (5.6)
on N ∗L satisfies these. ([·, ·] the Lie bracket on TL(− log L ∩ Z).) Let f, g ∈ I(L). We know that for any elliptic Hamiltonian vector fields on M
[ω−1(df), ω−1(dg)] = ω−1(d{f, g}).
−1 −1 −1 −1 Since f, g ∈ I(L), ω (df)|L, ω (dg)|L ∈ Γ(TL(− log ∂L)). We have [ω (df), ω (dg)]|L = −1 −1 [ω (df)|L, ω (dg)|L], which shows that (5.4) is satisfied. Now let h ∈ C∞(L), α, β ∈ Γ(N ∗L).
ω [ω−1(hα), ω−1(β)], · = ω [hω−1(α), ω−1(β)], · = ω h[ω−1(α), ω−1(β)], · − ω ω−1(β)(h)ω−1(α), · = hω [ω−1(α), ω−1(β)], · − ω−1(β)(h)α
This shows that the bracket in (5.6) satisfies (5.5) as well. Finally, we need to establish that the higher terms in the Maurer-Cartan equation vanish. Recall that for a ∈ Γ(NL):
⊗k λk a = P ([[... [π, a], a] ... ], a]),
59 Log and elliptic symplectomorphisms and flux where we view a as a vertical vector field on the total space of NL via the canonical vertical lift a↑. [π, a] is of course in general non-zero (as established above, it is da, viewing a as a log form). But since the component functions of both π and a have no dependence on the fibre coordinates, [π, a] is an entirely vertical bivector field. Thus [[π, a], a] vanishes identically: It only contains partial derivatives in the fibre directions, but neither π nor a depend on the fibre coordinates. All higher terms then also vanish automatically.
Remark 5.32. We can make an exactly analogous argument for generalized complex branes
(L, DL) in a stable generalized complex manifold (M, D, ω), which intersect the anticanonical divisor transversely in DL, a codimension-2 submanifold inside L. (L, DL) carries an elliptic divisor, which is the pullback of the elliptic divisor on (M,D). The differential complex −1 • structure on NL induced by π = ω is isomorphic to the differential complex Ω (L, log |DL|), i.e. λ1 = d, the differential for the elliptic differential forms on (L, DL). Furthermore, the ⊗k higher terms in the Maurer-Cartan equation λk α , k > 1 vanish, so the Maurer-Cartan elements are simply closed elliptic one-forms. Thus, up to elliptic Hamiltonian diffeomorphism, deformations are simply given by the first elliptic cohomology of (L, DL).
60 Chapter 6
Lagrangian branes with boundary in stable generalized complex manifolds
In this chapter we introduce and investigate the principal objects in the focus of Part I of this thesis: Lagrangian branes with boundary. In Section 4.3.1, we have presented results on generically Lagrangian submanifolds L of stable generalized complex manifolds (M 2n,D2n−2, σ = B + iω) which intersect the anticanonical divisor D transversely. These are all generalized complex branes in the sense of Definition 2.26. Now we instead consider generically Lagrangian submanifolds with boundary (Ln, (∂L)n−1) of (M 2n,D2n−2, σ) which intersect D cleanly in their boundary. This implies that the intersection is not transverse, and in fact these submanifolds are not generalized complex branes. We make sense of the pullback of elliptic differential forms to logarithmic differential forms on Lagrangian branes with boundary and subsequently study the so-called real oriented blow-up of the anticanonical divisor D in a stable generalized complex manifold: This allows us to relate stable generalized complex structures to a subclass of log symplectic structures, and Lagrangian branes (with and without boundary) to logarithmic Lagrangians.
6.1 Definition and basic properties
Definition 6.1. An n-dimensional submanifold with boundary ιL :(L, ∂L) ,→ (M, D, ω) is a Lagrangian brane with boundary if
L ∩ D = ∂L and T (L ∩ D) = TL ∩ TD|L∩D (clean intersection) (6.1)
61 Lagrangian branes with boundary in stable generalized complex manifolds and ∗ ιLω = 0
∗ Of course a priori ιLω is only defined outside D as the pullback of an ordinary differential form, but Proposition 6.4 illustrates how to make sense of this expression on the entirety of M. In order to give a simple proof, we consider natural local neighbourhoods for branes with boundary. Let (Y, ∂Y ) ⊂ (M,D) be any submanifold with boundary in a manifold equipped with a complex (and thus an induced elliptic) divisor, which intersects D cleanly in its boundary. For such manifolds, which include Lagrangian branes with boundary, there is a natural notion of local neighbourhood, similar to tubular neighbourhoods of submanifolds without boundary, or submanifolds of a manifold with boundary, although these neighbourhoods are not open submanifolds of M: We can choose a tubular neighbourhood of D in M in such a way that a collar neigh- bourhood of ∂Y in Y defines a rank-1 subbundle in ND.
Definition 6.2. Let V ⊂ D be a tubular neighbourhood of ∂Y in D, isomorphic to ND.
Consider the restriction ND|V , a trivial rank-2 bundle. In every fibre over ∂Y , pick a wedge
2 W := R>0 × R>0 ∪ {(0, 0)} around the 1-dimensional subspace defined by Y , where the tip of of wedge is the base point. Since ND|∂Y is trivial, such a choice can be consistently made across ∂Y , and extend 2 to ND|V , to glue together to a smooth W -bundle over V . Let Vˆ be the image of this W 2-bundle inside the tubular neighbourhood of D in M. A wedge neighbourhood of (Y, ∂Y ) ⊂ (M,D) consists of the smooth gluing of such a W 2-neighbourhood Vˆ with a tubular neighbourhood of Y \ (∂Y × [0, 1)) inside M \ D.
Note that such a space is not a smooth manifold, but instead has the following local type near ∂N: Open neighbourhoods of points in ∂N inside the wedge neighbourhood are of the form W 2 × Rdim M−2. We call such spaces wedge manifolds and equip them with a smooth structure: A map is smooth on W 2 × Rk−2 if it is smooth away from {0} × Rk−2 and can be extended to a smooth map on some proper tubular neighbourhood of {0} × Rk−2 in Rk. In the case where the wedge neighbourhood is embedded in M as above, it inherits its smooth structure from M.
Lemma 6.3. If ιN : N,→ M is a submanifold with (smooth) boundary in a manifold with a complex and induced elliptic divisor, such that ∂N = N ∩ D and
T (N ∩ D) = TN ∩ TD|N∩D, (6.2)
62 6.1 Definition and basic properties we can, inside a wedge neighbourhood of an open neighbourhood of ∂N in N, choose the ∂ polar coordinates (r, θ) in such a way that r ∂r is tangent to N in an open neighbourhood of ∂N.
Proof. Since a complex divisor is given by a transversely vanishing section of a complex line bundle, we can locally describe it by a complex function z = reiθ = a + ib, which is however only defined up to multiplication by a nowhere vanishing complex function g = |g|eiσ, where |g| is a smooth map from an open neighbourhood of D to the positive real numbers and σ a smooth map to S1. In addition to the polar coordinates (r, θ) we choose coordinates σ y3, . . . , y2n to describe a full tubular neighbourhood of D. By multiplying z by e , where i ∂ σ only depends on the y , we can always rotate z so that ∂a |∂N is tangent to N and inward-pointing. Then in a small neighbourhood of D, one of the equations determining N is
θ = λ(a, yi),
i π where λ is a smooth function with lima→0 λ(a, y ) = 0, so |λ| < 4 in some neighbourhood of q D. (If we choose the yi correctly, the other equations determining N are r = a2 + β2(a, yi), β a smooth function, of the form yj = 0 for j > k, and λ only depends on yi, i ≤ k.) Clearly λ can be extended to a small neighbourhood of N simply by as a constant function in b. On the intersection of the tubular neighbourhood of D and some wedge neighbourhood of {b = 0} the transformation
i z 7→ e−iλ(a,y )z, i.e θ 7→ θ′ = θ − λ(a, yi)
′ ∂ ′ ′ is a diffeomorphism which takes N to {θ = 0}, so ∂a′ , a = r cos(θ ) is tangent to N in that ∂ ′ ∂ neighbourhood, as is r = a ′ . ∂r N ∂a N
Proposition 6.4. If ιN : N,→ M is a submanifold with (smooth) boundary in manifold with a complex and induced elliptic divisor, such that ∂N = N ∩ D and
T (N ∩ D) = TN ∩ TD|N∩D, (6.3) the elliptic divisor (R, q) on M induces morphisms
ιN,∗ : TN(− log N ∩ D) → TM(− log |D|)|N (6.4) ∗ k k ιN :Ω (M, log |D|) → Ω (N, log N ∩ D), (6.5) where TN(− log N ∩ D) is the real logarithmic tangent bundle for N ∩ D inside N, and Ωk(N, log N ∩ D) the real logarithmic differential forms.
∗ This ensures that that the pullback ιLω in Definition 6.1 makes sense.
63 Lagrangian branes with boundary in stable generalized complex manifolds
Proof. It is sufficient to construct ιN,∗ : TN(− log N ∩ D) → TM(− log |D|) on ∂N = N ∩ D and to show that this extends the ordinary pushforward map on the interior smoothly. Locally, the elliptic divisor is given by a function r2 with D = {r = 0} (As seen above, r2 is only determined up to multiplication with a positive real function). √ x = ι∗r2 is a smooth function on (N, ∂N), more particularly, it is a vanishing function
∂ for ∂N. It is a well-known fact from log geometry that x ∂x ∈ Γ(TN(− log ∂N)|∂N ) is ∂N independent of the choice of vanishing function. Similarly, in elliptic geometry r ∂ only ∂r D depends on the elliptic divisor, not the function r2. So we set:
! ∂ ∂ ι∗ x = r ∂x ∂N ∂r N∩D
Then ι∗ : TN(− log ∂N) → TM(− log |D|) is smooth: If we choose (r, θ) as in Lemma 6.3, ∂ ∂ x ∂x extends to r ∂r on a wedge neighbourhood of a neighbourhood of ∂N in D. This allows us to see that ι∗ clearly pushes log vector fields on N forward to restrictions of elliptic vector fields to N in a smooth manner.
Example 6.5. (Standard local example) According to Theorem 3.21 in [CG15], if M is a stable generalized complex manifold with anticanonical divisor D, the associated complex logarithmic symplectic form can be written in local coordinates (w, z, q3, . . . , qn, p3, . . . , pn)
(w, z complex coordinates, qi, pi real) around any point in D as
dw Ω = ∧ dz + i X dp ∧ dq w j j j
If we write w = reiθ, z = x + iy, we obtain
dr ω = Im(Ω) = ∧ dy + dθ ∧ dx + X dp ∧ dq . r j j j
Then the following planes in R2n define Lagrangian branes with boundary in the sense of Definition 6.1:
∗ {y = const, θ = const, i (dpj ∧ dqj) = 0 (e.g. pj = const)}
Proposition 6.6. If the stable generalized complex structure is given by the complex log symplectic form σ = B + iω, a Lagrangian brane with boundary carries a natural logarithmic two-form F = i∗B with non-vanishing residue.
This proposition will be proved at the end of section 6.2.2. Hence, Lagrangian branes with boundary are not generalized complex branes. This text will argue that they should nonetheless be considered when studying submanifolds
64 6.2 Real oriented blow-up of the anticanonical divisor of stable generalized complex manifolds, and show how they fit into a general framework of branes in stable generalized complex manifolds. Towards this goal, we will establish a correspondence of stable generalized complex manifolds and log symplectic manifolds, as well as their Lagrangian submanifolds, in the following section.
6.2 Real oriented blow-up of the anticanonical divisor
In this section we establish a connection between elliptic symplectic geometry and log symplectic geometry via the so-called real oriented blow-up. In particular this allows us to relate every stable generalized complex manifold to a log symplectic manifold, and the Lagrangian branes to Lagrangian submanifolds with respect to the log symplectic structure. The real oriented blow-up of a compact submanifold Y involves replacing this submanifold with its normal sphere bundle S1(NY ), thus producing a manifold with boundary [M; Y ], a procedure that is well-defined up to diffeomorphism.1 There is always a natural lift of vector fields on M tangent to Y to vector fields on [M; Y ] tangent to ∂[M; Y ] = S1(NY ).
Definition 6.7. The real oriented blow-up of the origin in Rn is [Rn; {0}] := [0, ∞) × Sn−1 with the associated blow-down map β :[Rn; {0}] → Rn, (r, Ω) 7→ rΩ, where Ω ∈ Sn−1 and Sn−1 ,→ Rn the usual embedding of the unit sphere.
Obviously, the blow-down map β is a diffeomorphism away from ∂[Rn; {0}] =∼ {0} × Sn−1; in fact, the blow-down map is the projection map for the collapse of the submanifold ∂[Rn; {0}]. Note also: [Rn; {0}] =∼ Rn \{0} ⊔ Sn−1.
Lemma 6.8. (see [Mel96]) The GL(n, R)-action on Rn lifts to a GL(n, R)-action on [Rn; {0}].
Proof. If A ∈ GL(n, R), AΩ A · (r, Ω) = |AΩ|r, , |AΩ| defines a smooth GL(n, R)-action.
Definition 6.9. If E → M is a rank-k vector bundle over the smooth manifold M, we can identify M with the zero section of E and define the real oriented blow-up of E along M by
G [E; M] := [Em; {0}]. m∈M
This is naturally a smooth manifold with boundary; in fact a fibre bundle over M with fibres diffeomorphic to [Rk; {0}]. The blow-down map is also defined fibre-wise.
1Recall that in a stable generalized complex manifold (M,D) the radial residue of the elliptic symplectic form naturally lives on S1(ND), the normal sphere bundle to to D.
65 Lagrangian branes with boundary in stable generalized complex manifolds
Note that now [E; M] =∼ E \ M ⊔ Sk−1M, where Sk−1M is the unit sphere bundle of the normal bundle NM of M inside E.
Lemma 6.10. (see [Mel96]) Any smooth vector field X ∈ X(E) which is tangent to the zero section lifts to a smooth vector field X˜ on [E; M] which is tangent to ∂[E; M]. ˜ “Lifts to” means that conversely, β∗(Xe) = Xβ(e) ∀e ∈ [E; M].
Definition 6.11. If M is a smooth manifold and Y ⊂ M an embedded submanifold (both without boundary), we can define the real oriented blow-up of M along Y as follows: Choose a tubular neighbourhood U of Y , and view it as an open neighbourhood of the zero section in the normal bundle NY . Then the real-oriented blow-up [U; Y ] is defined as above. Since [U; Y ] \ ∂[U; Y ] can be identified with U \ Y , we obtain [M; Y ] by gluing M \ Y and [U; Y ] with this identification map.
In [Mel96], Melrose shows that this is, up to diffeomorphism, independent of the particular tubular neighbourhood of Y ⊂ M that is chosen. Consider now a manifold M with submanifold Y , and the real oriented blow-up [M; Y ]. Let Z ⊂ M be another submanifold. There are two obvious cases in which the lift of Z to the blow-up [M; Y ] is defined:
If Z ⊂ Y, β∗(Z) := β−1(Z) ⊂ ∂[M; Y ]. (6.6) If Z = cl(Z \ Y ), β∗(Z) := cl β−1(Z \ Y ) ⊂ [M; Y ] (6.7)
Clearly, we can blow these lifted submanifolds back down and obtain β(β∗(Z)) = Z.
Proposition 6.12. (i) If E → Y is a complex line bundle, its complex vector bundle structure induces a smooth free U(1)-action on the real oriented blow-up E˜ = [E; Y ], ˜ ˜ ˜ which restricts to Y := ∂E s.t. β|Y˜ : Y → Y is a principal U(1)-bundle.
(ii) If M˜ is a manifold with boundary D˜ s.t. there is a U(1)-principal bundle structure β : D˜ → D, there exists a smooth structure on
M := (M˜ \ D˜) ⊔ D,
which is canonical up to diffeomorphism, makes M into a manifold without boundary, and [M; D] =∼ M˜ . β extends to a β : M˜ → M and D has a complex normal bundle in M.
(iii) Let (M,˜ D˜) be a manifold with boundary ∂M˜ = D˜ such that β : D˜ → D is a principal U(1)-bundle. Then there is a canonical induced complex divisor on (M,D) which is determined up to isomorphism.
66 6.2 Real oriented blow-up of the anticanonical divisor
Proof. (i) Since E → Y is a complex line bundle, its structure group can be reduced to U(1). Its sphere bundle S1(E) is in fact the associated principal U(1)-bundle associated ∼ 1 to E, whose U(1)-action naturally extends to [E; Y ] = S (E) × R+ as a free action.
(ii) Consider the open cover of M˜ given by M˜ \ D˜ and a collar neighbourhood of D˜, U˜ =∼ D˜ × [0, 1). Since D˜ has the structure of a U(1)-principal bundle, it comes with a free U(1)-action that we can extend to the collar neighbourhood (this depends on the choice of collar neighbourhood, of course). Collapse the U(1)-fibres in U˜ to obtain the quotient space U, which still carries a U(1)-action, now with fixed-point set D. U can be viewed as a neighbourhood of the zero section of the complex vector bundle associated to β : D˜ → D. Glue U to M˜ \ D˜ using the same gluing functions as for U˜. This is M. From the description of the real oriented blow-up, it is clear that M˜ =∼ [M; D]. Lastly, we show explicitly that the diffeomorphism type of M does not depend on the choice of collar neighbourhood U˜: Consider two different collar neighbourhood
embeddings λi : D˜ × [0, 1) → M,˜ i = 1, 2. Write B for the unique disk bundle obtained from collapsing the U(1)-fibres in D˜ inside D˜ × [0, 1]. Denote:
U˜i : = λ(D˜ × [0, 1))
Ui : = U˜i/ ∼, where ∼ denotes the collapse of the U(1)-fibres in D˜ ˜ 1/2 ˜ h 1 i Ui : = λi D × 0, 2 −1 ˜ ˜ φ12 = λ2 ◦ λ1 : U1 → U2 ˜ 1/2 Mi : = M \ Ui ∪λi B
′ ˜ ˜ 1/2 ˜ ˜ 1/2 Note that we can find a diffeomorphism ψ12 : M \ U1 → M \ U2 which restricts to ˜ 1 φ12 on λ1 D × 2 , 1 and is the identity outside some neighbourhood of U1, since ˜ 1/2 ˜ ˜ M \ Ui are both naturally diffeomorphic to M \ D. A natural diffeomorphism ψ12 : M1 → M2 is then given by
′ ˜ 1/2 ψ (m) if m ∈ M1 \ U 12 1 ψ12(m) := φ12(m) if m ∈ U1 \ D , m if m ∈ D
′ which is well-defined by the choice of ψ12.
(iii) We have already established that the U(1)-action on D˜ induces a complex structure on ND, the normal bundle to D in M. Pick a hermitian metric h for this complex line bundle. Then the function ND → R, v 7→ h(v, v)
67 Lagrangian branes with boundary in stable generalized complex manifolds
clearly defines an elliptic divisor for D in tot(ND), and since we have already picked an orientation of ND, this induces a complex divisor on tot(ND) (the complex line bundle being p∗ND → tot(ND), where p : ND → D is the vector bundle projection). h is unique up to multiplication with a positive real function on D, but such a rescaling is merely an isomorphism of elliptic divisors. Thus the induced complex divisor is also unique up to isomorphism. We can embed a neighbourhood of the zero section in ND into M as a tubular neighbourhood of D, and extend the elliptic divisor defined by h to the entirety of M in some positive smooth way. Since the real line bundle associated to an elliptic divisor is trivial, this is again unique up to isomorphism.
Note that the choice of a vanishing function for D˜ in M˜ and an extension of the U(1)-action to a collar neighbourhood of D˜ fix the choice of elliptic divisor on a tubular neighbourhood of D, if these are to be compatible with the blow-down map β : M˜ → M.
~ M M θ r r θ ~ D D
Figure 6.1 Real oriented blow-up of the anticanonical divisor
Theorem 6.13. (Relation between stable generalized complex and log symplectic manifolds via real oriented blow-up I) Every stable generalized complex manifold (M, D, ω) can be related to a log symplectic manifold with boundary (M,˜ D,˜ ω˜) via the real oriented blow-up of the anticanonical divisor. With β : M˜ → M the blow-down map,
ω˜ = β∗(ω), which is a non-degenerate log two-form.
This result is a consequence of the following lemma:
Lemma 6.14. Let D = (U, s) be a complex divisor on a smooth manifold M; also denote its zero locus by D. Consider the real oriented blow-up M˜ := [M; D], D˜ := ∂M˜ .
(i) The lift of vector fields associated with the real oriented blow-up induces a map
β∗ : X(M, − log |D|) → X(M,˜ − log D˜)
68 6.2 Real oriented blow-up of the anticanonical divisor
which maps a local C∞(M)-basis to a local C∞(M˜ )-basis.
(ii) There is a well-defined vector bundle morphism
β∗ : T M˜ (− log D˜) → TM(− log |D|)
(iii) This β∗ : T M˜ (− log D˜) → TM(− log |D|) further induces a pullback
β∗ :Ω•(M, log |D|) → Ω•(M,˜ log D˜),
which maps local bases to local bases, and a vector bundle morphism
k ∗ k ∗ β∗ : ∧ T M˜ (log D˜) → ∧ T M(log |D|)
(iv) The pullback β∗ induces an isomorphism on cohomology:
β∗ : H•(M, log |D|) → H•(M,˜ log D˜)
Proof. (i) All vector fields which are tangent to D lift to logarithmic vector fields w.r.t D˜ on the real oriented blow-up M˜ , and elliptic vector fields are certainly tangent to D. On M˜ \ D˜, β is a diffeomorphism, so local bases of vector fields get mapped to each other. Coordinates (yi, θ, r) on a neighbourhood of D pull back to coordinates on a neighbourhood of D˜. The associated local basis of elliptic vector fields is
∂ ∂ ∂ r , , , ∂r ∂θ ∂yi
which clearly lift to a local basis of log vector fields around D˜. ˜ ˜ (ii) Since β|M˜ \D˜ is a diffeomorphism, and T M(− log D),TM(− log |D|) are isomorphic to T M,TM˜ outside D,D˜ respectively, it suffices to define β∗ : T M˜ (− log D˜) → TM(− log |D|) on D˜ and ensure that it forms a smooth vector bundle isomorphism together with the standard definition away from D˜. Choose a tubular neighbourhood of D; write M = ND, and p : ND → D for the projection of the normal bundle. Fix a hermitian metric for ND. Also choose a ˜ U(1)-principal connection α for β|D˜ : D → D. This immediately defines a complex linear connection on the associated complex line bundle ND, and splittings of the short
69 Lagrangian branes with boundary in stable generalized complex manifolds
exact sequences of vector bundles
0 → t′ →T D˜ → β∗(TD) → 0 (6.8) 0 → VD →TM → p∗(TD) → 0 (6.9)
0 → R ⊕ t →TM(− log |D|)|D → TD → 0 (6.10)
′ ˜ ∂ t is the trivial line bundle spanned by the U(1)-action vector field on D, ∂θ . t is the ∂ trivial line bundle spanned by the elliptic vector field ∂θ |D on D. R is the trivial line ∂ bundle given by the elliptic vector field r ∂r restricted to D. The splitting of the third sequence (6.10) is induced by the splitting of the second: The connection is unitary, so the associated horizontal lift will lift any section of p∗(TD) to
an elliptic vector field. For any X ∈ TdD we can choose an extension to a section in
X(D), which in turn lifts to X¯ ∈ X(M, − log |D|)|D). The splitting of the sequence is defined by
X 7→ X¯|d,
which is independent of the chosen extension. Thus we obtain an isomorphism
∼ TM(− log |D|)|D = R ⊕ t ⊕ TD
Upon choice of hermitian metric on ND, the real oriented blow-up of ND along the zero ˜ ˜ 1 ˜ ˜ section [ND; D] is canonically identified with D×[0, ∞) (D = S ND). T M(− log D)|D˜ ˜ ˜ ˜ ∂ is canonically isomorphic to R ⊕ T D, with R the trivial line bundle spanned by r ∂r |D˜ . The U(1)-principal connection for D˜ with its associated splitting (6.8) then defines an isomorphism ˜ ˜ ∼ ˜ ′ ∗ T M(− log D)|D˜ = R ⊕ t ⊕ β (TD)
There is now clearly a vector bundle morphism R˜ ⊕ t′ ⊕ β∗(TD) → R ⊕ t ⊕ TD over
β|D˜ given by
∂ ∂ r 7→ r ∂r d˜ ∂r β(d˜)
∂ ∂ 7→ ∂θ d˜ ∂θ β(d˜) ˜ ∗ (X, d) ∈ β (TD)d˜ 7→ X ∈ Tβ(d˜)D
which in turn allows us to define
70 6.2 Real oriented blow-up of the anticanonical divisor
∼ ˜ ˜ = ˜ ′ ∗ T M(− log D)|D˜ R ⊕ t ⊕ β (TD)
β∗ =∼ TM(− log D)|D R ⊕ t ⊕ TD ˜ ˜ This joins smoothly with the standard definition of (β|M˜ \D˜ )∗ : M \ D → M \ D to give
β∗ : T M˜ (− log D˜) → TM(− log |D|)
The way we have defined this map, the definition of β a priori depends on the choice of U(1)-principal connection. But since we have used the same connection to define the
isomorphism in both the upper and the lower line of the above diagram, β∗ is invariant under a change of this connection.
k (iii) These morphisms can be defined using the ones in 1. and 2.: If α ∈ ΩM (− log |D|) and X˜1,..., X˜k ∈ T M˜ (− log D˜),
∗ β (α)(X˜1,..., X˜k) := α(β∗(X˜1), . . . , β∗(X˜k))
is a well-defined map of differential complexes. It is easy to check that local bases get mapped to local bases using the usual coordinates around D, (yi, θ, r). Similarly if α˜ ∈ ∧kT ∗M(log D˜), we can define a pointwise pushforward using the lift of
elliptic to log vector fields: Let X1,...,Xk ∈ X(M, − log |D|).
∗ ∗ β∗(α)(X1,...,Xk) := α(β (X1), . . . , β (Xk)).
(iv) Recall the results for the logarithmic and elliptic de Rham cohomology:
k ∼ k k−1 1 H (M, log |D|) = H (M \ D) ⊕ H (S ND), [α] 7→ ([α|M\D], [resr α]) Hk(M,˜ log D˜) =∼ Hk(M˜ ) ⊕ Hk−1(D˜), [˜α] 7→ ([˜α − s(resα ˜)], [resα ˜]), where s : Hk−1(D˜) → Hk(M,˜ log D˜) some section of res
M˜ is a manifold with boundary ∂M˜ = D˜, and so Hk(M˜ ) =∼ Hk(M˜ \ D˜) =∼ Hk(M \ D).
k ˜ ˜ k ˜ ˜ k−1 ˜ H (M, log D) → H (M \ D) ⊕ H (D), [˜α] 7→ ([˜α|M˜ \D˜ ], [resα ˜])
71 Lagrangian branes with boundary in stable generalized complex manifolds
is an isomorphism, since we can map
[˜α] 7→ ([˜α − s(resα ˜)], [resα ˜]) 7→ [˜α − s(resα ˜)]|M˜ \D˜ , [resα ˜] 7→ [˜α]|M˜ \D˜ , [resα ˜]
Each of these maps is an isomorphism on cohomology, thus the composition is. Now, we obtain a commutative diagram β∗ Hk(M, log |D|) Hk(M,˜ log D˜)
=∼ =∼ Id Hk(M \ D) ⊕ Hk−1(S1ND) Hk(M˜ \ D˜) ⊕ Hk−1(D˜) Three of the morphisms in this diagram are isomorphisms, so β∗ is one as well.
Theorem 6.13 follows immediately from this result. The following theorem illustrates the converse relationship:
Theorem 6.15. (Relation between stable generalized complex and log symplectic manifolds via real oriented blow-up II) Let (M,˜ D˜ = ∂M,˜ ω˜) be a real logarithmic symplectic manifold with a U(1)-principal bundle structure β : D˜ → D and associated blow-down β :(M,˜ D˜) → (M,D) (see Proposition 6.12, (ii)). Assume that
∂ ˜ 1. i ∂ (res ω˜) = 0, where is the action vector field of the U(1)-action on D. ∂ θ ∂θ
2. d i ∂ ω˜ D˜ = 0. Note that the first assumption implies that i ∂ ω˜|D˜ is actually a smooth ∂ θ ∂ θ one-form on D˜, so its exterior derivative on D˜ is defined.
Then (M,D) carries and induced elliptic divisor and ω˜ induces a gauge equivalence class of stable generalized complex strcutures ω with anticanonical divisor D.
Proof. Since β : D˜ → D carries the structure of a U(1)-principal bundle, ND is a complex vector bundle and thus in particular oriented. Let π˜ = ω˜−1 be the Poisson structure associated to the real logarithmic symplectic structure on M˜ . π˜n ∈ X2n(M˜ ) is then a section that vanishes transversely on D˜. n 2n Claim: β∗π˜ ∈ X (M) defines an elliptic divisor with vanishing locus D ⊂ M. Clearly, the pointwise pushforward of π˜n is defined, we only need to ensure that this actually gives a smooth section, and that it has a positive definite normal Hessian on D.
72 6.2 Real oriented blow-up of the anticanonical divisor
According to the normal form theorem for log symplectic forms, we can choose a collar neighbourhood around D˜ with coordinate r such that the log symplectic form ω˜ takes the form dr ω˜ = ∧ Ω˜ + Σ, r I
where Σ is a closed two-form on D˜ and Ω˜ I = resω ˜ is a closed one-form on D˜ (which are pulled back to the collar neighbourhood).
By assumption, i ∂ (resω ˜) = 0, so ∂ θ ˜ £ ∂ ΩI = 0 ∂ θ ˜ Furthermore, i ∂ (resω ˜) = 0 implies that i ∂ ω˜|D˜ = i ∂ Σ =: ΩR. By assumption, this form is ∂ θ ∂ θ ∂ θ ˜ closed, and £ ∂ ΩR = 0. ∂ θ Taken together, we obtain that Ω˜ I , Ω˜ R are horizontal one-forms on D˜, which are invariant under the U(1)-action. This implies that they are pulled back from smooth (closed) one-forms
ΩI , ΩR on D. The next step is to show that given the above, the U(1)-action on D˜ can always be ˜ ˜ ˜ extended to a collar neighbourhood U = D × [0, 1) in such a way that £ ∂ ω˜ = 0 on all of U. ∂ θ We begin by extending the U(1)-action to U˜ as a free action in some way. (This amounts ˜ ∼ ˜ to fixing the diffeomorphism D × [0, 1) = U.) Assume £ ∂ ω˜ ̸= 0. Consider the family of ∂ θ logarithmic forms it ∗ ω˜t := (e ) ω,˜ t ∈ [0, 2π).
(eit)∗ is the pullback with respect to the U(1)-action diffeomorphism. We can average over t to obtain the U(1)-invariant log form
1 Z ω¯ = ω˜t dt 2π S1
We have d ω˜t = £ ∂ ω˜ = d i ∂ ω˜ dt t=0 ∂ θ ∂ θ
Since we assumed i ∂ ω˜|D˜ = 0, we obtain ω˜t|D˜ = ω˜|D˜ = ω¯|D˜ . In particular, ω¯ is also ∂ θ non-degenerate, i.e. a log symplectic form, at least upon restriction to a smaller collar neighbourhood of D˜. ω˜ − ω¯ is a smooth two-form which vanishes on D˜. D˜ is a deformation retract of its collar neighbourhood, so ω˜ − ω¯ = dα, with α a smooth one-form on the collar neighbourhood of D˜. ˜ Since D was assumed to be compact and ω˜|D˜ = ω¯|D˜ , we can apply the Moser argument to the family of non-degnerate (on a small neighbourhood of D˜) log forms
′ ωs := sω˜ + (1 − s)¯ω, s ∈ [0, 1]
73 Lagrangian branes with boundary in stable generalized complex manifolds
′ −1 Xs := (ωs) (α) is a logarithmic vector field which integrates to an isotopy φs with
∗ ′ ∗ φs(ωs) =ω, ¯ φ1(˜ω) =ω ¯
Write φ := φ1 for this diffeomorphism. We have:
∗ ∗ 0 = £ ∂ φ (˜ω) = φ £ ∂ ω˜ ⇒ £ ∂ ω˜ = 0 ∂ θ φ∗ ∂ θ φ∗ ∂ θ
∂ Since φ is a diffeomorphism, φ∗ ∂θ is again the action vector field of a U(1)-action on a collar neighbourhood of D˜, and this is the extension of the U(1)-action we have been looking for. Locally on an open set in D˜, we can now write
˜ Σ = dθ ∧ ΩR +σ, ˜ where i ∂ σ˜ = 0, ∂ θ so on a neighbourhood near the boundary:
dr ω˜ = ∧ Ω˜ + dθ ∧ Ω˜ +σ, ˜ r I R where θ is chosen such that £ ∂ ω˜ = 0 on the entire collar neighbourhood. σ˜ is a horizontal, ∂ θ closed two-form on D˜, i.e. it is also the pullback of a closed two-form σ on D. n n In particular, this implies that β∗π˜ is smooth on D. The normal Hessian of β∗π˜ as a section of ∧2nTM is clearly positive definite, since π˜n itself vanishes transversely, and we obtain an n 2n elliptic divisor (β∗π˜ , ∧ TM) with vanishing locus D, which is co-oriented. Using the local expression for ω˜ on a collar neighbourhood established above, and the ∗ ∗ ∗ established fact that Ω˜ I = β ΩI , Ω˜ R = β (ΩR), σ˜ = β (σ) for smooth forms on D, it is clear that ω˜ is the pullback of dr ω = ∧ Ω + dθ ∧ Ω + σ. r I R Such an expression exists for each open set of a covering of D, and because the coordinate transformations for a tubular neighbourhood U =∼ ND are compatible with those for U˜ =∼ D˜ × [0, 1), these patch to a well-defined elliptic symplectic form ω ∈ Ω2(M, log |D|). Lastly, we have
rese(ω) = 0, so ω and the already established co-orientation of D together define the gauge-equivalence class of a stable generalized complex structure on (M,D).
6.2.1 Lagrangian branes under blow-up
Let (M,˜ D,˜ ω˜) and (M, D, ω) be a logarithmic symplectic manifold and stable generalized complex manifold related by real oriented blow-up as above. We begin by showing that
74 6.2 Real oriented blow-up of the anticanonical divisor
Lagrangian branes with and without boundary in (M,D) lift to logarithmic Lagrangians in (M,˜ D˜):
Proposition 6.16. (Lift of Lagrangian branes to the real oriented blow-up)
(i) Let L ⊂ M be a Lagrangian brane which intersects D transversely. Then L˜ = [L; L ∩ D] ⊂ M˜ is a Lagrangian submanifold with boundary which intersects D˜ transversely.
(ii) Let L ⊂ M be a Lagrangian brane with boundary. The lift of L, L˜ := β∗(L) = cl β−1(L \ (L ∩ D)), is a Lagrangian submanifold in M˜ which intersects the singular ˜ ∂ ˜ ˜ ˜ locus D transversely, and such that ∂θ is nowhere tangent to L ∩ D. L intersects each U(1)-fibre in at most one point.
Proof. (i) [L; L ∩ D] naturally embeds into (M,˜ D˜) as L˜ = cl β−1(L \ (L ∩ D)): Since L ⋔ D, we can always choose a small tubular neighbourhood of D in which L is a fibre ˜ ∗ ˜ ˜ ˜ of ND. L is clearly Lagrangian with respect to ω˜ = β (ω), since β(L) = L. L ⋔ D is obvious.
(ii) Again, L˜ is clearly Lagrangian with respect to ω˜ = β∗(ω), since β(L˜) = L. It intersects D˜ transversely: D˜ is codimension-1 in M˜ . If L˜ did not intersect D˜ transversely, we ˜ ˜ would have T L|L˜∩D˜ ⊂ T D|L˜∩D˜ , which would also imply TL|L∩D ⊂ TD|L∩D, which is a contradiction – we assumed L ∩ D to be a clean intersection. Because of the clean intersection, we can choose a tubular neighbourhood of D such that L is a rank-1 ∂ subbundle of ND over ∂L. This means in particular that ∂θ is not tangent to L in some neighbourhood of D, so it will not be tangent to L˜ ∩ D˜ either. If L˜ intersected any U(1)-fibre in more than one point, L ∩ D would not be the boundary of L.
β
D ~ D
Figure 6.2 Log Lagrangians under blow-down: Depending on the intersection with D˜, the result can be a brane with or without boundary. Not all branes without boundary obtained by blow-down are smooth.
Conversely, let L˜ ⊂ M˜ be a compact Lagrangian submanifold with boundary, s.t. ∂L˜ ⊂ D˜ ˜ ˜ ˜ and L ⋔ D. According to Theorem 5.18 every connected component of ∂L will lie inside
75 Lagrangian branes with boundary in stable generalized complex manifolds a symplectic leaf of D˜ and be Lagrangian inside this leaf. We know that the symplectic foliation of D˜ is precisely given by the distribution ker(resω ˜) = ker(Ω˜ I ). There are two cases of interest, in which β(L˜) is a smooth submanifold in M:
Theorem 6.17. (Blow-down of logarithmic Lagrangians) Let (M, D, ω) be a stable generalized complex manifold and (M,˜ D,˜ ω˜) its blow-up. Let L˜ ⊂ M˜ be a Lagrangian submanifold with boundary that intersects D˜ transversely, and D˜ ∩ L˜ = ∂L. ∂ ˜ ˜ ˜ ˜ (i) If ∂θ ∈ X(L∩D), the U(1)-action restricts to ∂L. Any such L is log Hamiltonian isotopic to a L˜′ whose image under the blow-down map L′ = β(L˜′) is a smooth Lagrangian brane without boundary.
∂ ˜ ˜ ˜ (ii) If ∂θ is nowhere tangent to T (D ∩ D) and β|L˜ is injective, β(L) =: L is a Lagrangian brane with boundary in D. 2
Proof. (i) In this case, the free U(1) action restricts to ∂L˜, which itself becomes a
U(1)-principal bundle over β(∂L˜) = DL. Pick a collar neighbourhood of D˜, U˜ = D˜ × [0, 1), in such a way that
dr ω = ∧ Ω˜ + dθ ∧ Ω˜ +σ, ˜ r I R
and £ ∂ ω = 0. We know that ∂ θ
∗ ˜ ∗ ˜ ι∂L˜ΩI = 0, ι∂L˜(dθ ∧ ΩR +σ ˜) = 0,
so L˜′ := ∂L˜ × [0, 1) is a log Lagrangian. Obviously the U(1)-action also restricts to its boundary. Clearly, β(L˜′) =: L′ is a smooth elliptic Lagrangian without boundary ′ in M, equipped with a pullback elliptic divisor DL′ , and L ⋔ D. Pick an elliptic Lagrangian neighbourhood for L′ according to Theorem 4.21. This corresponds to a log Lagrangian neighbourhood of L˜′ in M˜ (via pullback of coordinates). At least in some neighbourhood of D˜, L˜ is contained in the thus obtained log neighbourhood of L˜′ and can be written as the graph of a closed log one-form α˜ ∈ Ω1(L˜′, log ∂L˜′). Since the 1 ′ 1 ˜′ ˜′ pullback β∗ :Ω (L , log |DL′ |) → Ω (L , log ∂L ) is an isomorphism on cohomology, α˜ has to be in the same cohomology class as a form α˜′ which is the pullback of a smooth elliptic form α′ on L′. Thus the graph of α˜, L˜, is locally log Hamiltonian isotopic to a Lagrangian which blows down to a smooth Lagrangian brane without boundary in M. This argument takes place inside a tubular neighbourhood of D˜, it is however possible to cut off any Hamiltonian function with a bump function, so the Hamiltonian isotopy above can be extended by the identity outside a neighbourhood of D˜.
2 ˜ We will later consider a similar example where β|L˜ is not injective, but β(L) is nonetheless smooth and intersects D in a smooth codimension-one submanifold, just not its boundary. But since branes with boundary the focus of this text, this case is excluded for now.
76 6.2 Real oriented blow-up of the anticanonical divisor
∂ ˜ ˜ (ii) Since ∂θ is nowhere tangent to L, ∂L intersects each U(1)-fibre transversely. Since β|∂L˜ is injective, the intersection with each fibre is either empty or in exactly one point. ˜ Thus β|L˜ is a diffeomorphism onto its image, and L := β(L) is a smooth submanifold of M, with boundary in D.
Example 6.18. Consider M = T ∗L(− log |Y |), where L is a compact n-dimensional manifold and Y ⊂ L a codimension-2 submanifold given as the zero-locus of a complex divisor. Equip
M with the canonical elliptic symplectic form ω0. ∗ Then the real oriented blow-up M˜ of T L(− log |Y |)|Y inside M with the pullback-form ∗ ∗ β ω0 is isomorphic to T L˜(− log ∂L˜), where L˜ := [L; Y ] = cl(L \ Y ) is the lift under the real
oriented blow-up, equipped with the canonical logarithmic symplectic form ω˜0. Conversely, if L˜ is an n-dimensional manifold with boundary ∂L˜, s.t. ∂L˜ is a U(1)-principal ∗ bundle, we can consider the blow-down of (T L˜(− log ∂L˜), ω˜0), and the result will be isomor- ∗ phic to (T L(− log |Y |), ω0), where L is the blow-down L˜ → L and Y = ∂L/U˜ (1).
6.2.2 Neighbourhoods of Lagrangian branes with boundary
We have now established that every Lagrangian brane with boundary (L, ∂L) ⊂ (M,D) is the blow-down of a log Lagrangian submanifold with boundary L˜ which intersects D˜ transversely. For such Lagrangians L˜, we have proved a Lagrangian neighbourhood theorem, Theorem 5.18. Choose a neighbourhood U˜ of L˜ according to this theorem. Its image under the blow-down U := β(U˜) is a wedge neighbourhood in the sense of Definition 6.2.
β ~ L D D
Figure 6.3 Lagrangian wedge neighbourhood as a blow-down
Proposition 6.19. (Normal form for wedge neighbourhoods) Let (L, ∂L) be a La- grangian brane with boundary in a stable generalized complex manifold (M, D, ω), and (L,˜ ∂L˜) the corresponding log Lagrangian in the real oriented blow-up (M,˜ D,˜ ω˜). Let (U,˜ ∂U˜) ⊂ T ∗L˜(log ∂L˜) be a Lagrangian neighbourhood of (L,˜ ∂L˜) in the sense of Theorem 5.18. Identify U˜ with the tubular neighbourhood of the zero section in T ∗L˜(log ∂L˜) and write U˜ for either. Write ψ˜ : U˜ → U˜ for the diffeomorphism such that
˜∗ ψ ω˜|U˜ =ω ˜0,
77 Lagrangian branes with boundary in stable generalized complex manifolds
∗ where ω˜0 is the standard log symplectic form on the log cotangent bundle T L˜(log ∂L˜).
Then ω˜0 is the pullback of an elliptic symplectic form ω0 on the wedge neighbourhood
(U, DU ) := (β(U˜), β(∂U˜)) which locally around each point in DU can be expressed in local coordinates as
dr ω = ∧ dy + dθ ∧ dx + X dqi ∧ dp , 0 r i i
(r, x, qi) coordinates on L. Furthermore, the diffeomorphism ψ˜ of U˜ descends to a diffeomor- ∗ phism of the wedge neighbourhood ψ : U → U such that ψ ω = ω0.
Proof. First we need to show that ω˜0 can indeed be written as the pullback of an elliptic form on U that extends smoothly around the wedge of U. According to Lemma 6.3, we ∂ can choose (r, θ) in such a way that r ∂r is tangent to L in a small neighbourhood of D, a property that persists after blow-up. We have already established that the brane in the blow-up, L˜, intersects each U(1)-fibre in at most one point, so we can choose the tubular ˜ ˜ ∂ neighbourhood of L in such a way that near D, ∂θ is tangent to the fibres. We can view ∂ ˜ ∂θ |L˜ as spanning a sub-line bundle of NL|∂L˜. We identify NL˜ with T ∗L˜(log L˜ ∩ D˜) using ω˜. On an open set near D˜, we can write
dr ω˜ = ∧ γ + dθ ∧ γ +ϵ. ˜ r r θ
˜ ∂ ∗ ˜ ˜ The subbundle of NL spanned by ∂θ gets mapped to the subbundle of T L(log L ∩ D) ∗ spanned by ι γθ, ι : L,˜ → M˜ .
dr Let ξr be the fibre coordinate associated to r , and χ the fibre coordinate associated to ∗ ι γθ. Then ω˜0 has the form
dr ω˜ = − ∧ dξ − ι∗γ ∧ dχ +ρ ˜ 0 r r θ
∗ ˜ ˜ ˜ with ρ˜ a two-form on T L(log L ∩ D) s.t. i ∂ ρ˜ = 0, i ∂ ρ˜ = 0. r ∂ r ∂ χ This implies ∗ i ∂ (resω ˜0) = 0, d i ∂ ω˜0 D˜ = d ι γθ|D˜∩L˜ = 0. ∂ χ ∂ χ We can choose the tubular neighbourhood embedding for L˜ such that θ = ω∗(χ) (viewing ω ∗ as a map NL˜ → T L˜(log L˜ ∩ D˜)). According to Theorem 6.15 this means that ω˜0 is indeed the pullback of a locally defined elliptic form ω0 on (U, U ∩ D), with respect to the same
78 6.2 Real oriented blow-up of the anticanonical divisor
−1 n −1 n elliptic divisor as ω: (˜ω ) and (˜ω0 ) are both of the form
∂ ∂ ∂ ∂ fr ∧ ∧ ∧ · · · ∧ , f ̸= 0 ∂r ∂θ ∂y3 ∂y2n
with respect to the same (r, θ).
The diffeormorphism ψ˜ : U˜ → U˜ relating ω˜ and ω˜0 is the time-1 flow of the time-dependent log vector field ˜ −1 Xt = −ω˜t (˜α), ω˜t = tω˜ + (1 − t)˜ω0, ω˜ − ω˜0 = d˜α
We have: ∗ ∗ ω˜ = β (ω), ω˜0 = β (ω0),
∗ L is Lagrangian with respect to both ω and ω0, so ω − ω0 = dα, where β (α) =α ˜. Clearly,
˜ −1 β∗(Xt) = −ωt (α) =: Xt,
∗ which is an elliptic vector field whose time-1 flow takes ψ (ω) = ω0.
We thus obtain a standard local neighbourhood of branes with boundary in stable generalized complex manifolds, which is a wedge neighbourhood in the sense of Definition 6.2. With the results from this section, it is easy to prove Proposition 6.6:
Proof. of Proposition 6.6 According to Proposition 6.19, we can pick coordinates on a wedge neighbourhood (U, U ∩ D) of a brane with boundary so that
dr ω = ∧ dy + dθ ∧ dx + σ r
Then, up to addition of a smooth closed two-form, the real part of a corresponding log symplectic form σ = B + iω is
dr B = ∧ dx + dθ ∧ dy, r
∗ ∗ with ι : L → M, res(ι B) = ι dx|∂L. We have already established that after real oriented blow-up
=∼ ∗ ∂ ω˜ : NL˜ → T L˜(log L˜ ∩ D˜), 7→ dx ∂θ L is an isomorphism. Thus ι∗ dx ̸= 0 everywhere.
79 Lagrangian branes with boundary in stable generalized complex manifolds
6.3 Small deformations of Lagrangian branes with boundary
Based on our results so far, the tangent space to the deformation space of a Lagrangian brane with boundary (L, ∂L) ⊂ (M, D, ω), while keeping the elliptic symplectic structure constant, is given by elliptic symplectic vector field modulo elliptic symplectic vector fields tangent to L, all restricted to L. Clearly, elliptic vector fields on (M,D) which are also tangent to L precisely restrict to logarithmic vector fields on L. Since L is Lagrangian with respect to the elliptic symplectic form ω, ω defines an isomorphism X(M, − log |D|)| ω : L → Ω1(L, log ∂L) X(L, − log ∂L) The elliptic vector fields that define deformations of L while preserving the elliptic symplectic 1 form are symplectic, so their image under this isomorphism will be precisely Ωcl(L, log ∂L) – i.e. infinitesimal deformations of a Lagrangian brane with boundary preserving the elliptic symplectic structure are in one-to-one correspondence with closed logarithmic one-forms. Similarly, infinitesimal Hamiltonian deformations of (L, ∂L) are given by elliptic Hamilto- nian vector fields restricted to L, modulo elliptic Hamiltonian vector fields which are tangent to L. Under the above isomorphism, this subspace of vector fields will precisely map to the exact log one-forms on L.
Corollary 6.20. Thus, infinitesimal deformations of L modulo infinitesimal Hamiltonian isotopies are given by H1(L, log ∂L).
If there is a smooth moduli space of deformations of branes with boundary (up to Hamiltonian isotopy), it will thus locally look like H1(L, log ∂L). This is the space we will now investigate:
Definition 6.21. A small deformation of the Lagrangian brane with boundary L is a
Lagrangian brane with boundary L1 such that there exists a smooth family of embeddings
φ :(L, ∂L) × [0, 1] → (M,D) which are strong maps of pairs, such that φ(L, 0) = L, φ(L, 1) = L1, and such that all φ(L, t) are Lagrangian branes with boundary in the sense of Definition 6.1 which are C1-close to L.
When we lift such a family φ(L, t) of Lagrangian branes with boundary to the real oriented blow-up of (M,D), (M,˜ D˜), we obtain a family of log Lagrangians which define a C1-small deformation of L˜ = β∗(L) in the sense of Definition 5.24. Conversely, the image of any sufficiently C1-small deformation of L˜ under the blow-down map β is a small deformation of Lagrangian branes with boundary as above. If we pick a log Lagrangian neighbourhood for L˜ and a β-related wedge neighbourhood of L, the lifts of sufficiently small deformations φ(L, t) to the blow-up will intersect the fibres
80 6.3 Small deformations of Lagrangian branes with boundary
of T ∗L(log ∂L) transversely and can thus be written as graphs of closed log one-forms on L. Conversely, under the identification of the tubular neighbourhood of L˜ with a neighbourhood of the zero section of T ∗L(log ∂L). followed by the blow-down, every sufficiently small closed log one-form defines a small deformation of the Lagrangian brane with boundary L.
Theorem 6.22. (Small deformations of Lagrangian branes with boundary) Up to lo- cal Hamiltonian isotopy (i.e. Hamiltonian isotopy that stays within the wedge neighbourhood) the small deformations of a brane with boundary L ⊂ (M,D) are given by H1(L, log L ∩ D), i.e. the first cohomology of logarithmic forms on L with respect to ∂L = L ∩ D.
Remark 6.23. Although wedge neighbourhoods are not tubular neighbourhoods in the usual sense, and drop in dimension by one over ∂L, these are the natural neighbourhoods containing small deformations of branes with boundary: According to the definition, we only consider deformations whose boundary stays inside D. All such small deformations precisely sweep out a wedge neighbourhood of L.
In order to prove this theorem, we need the following lemma:
Lemma 6.24. If a brane with boundary and a small deformation of it are related by a local log Hamiltonian isotopy in a tubular neighbourhood in (M,˜ D˜) and by a local elliptic symplectic isotopy in the corresponding wedge neighbourhood in (M,D), there is also a local elliptic Hamiltonian isotopy between them in (M,D).
Proof. Let L, L′ ⊂ (M,D) be branes with boundary that are related by an elliptic symplectic ′ ′ ∗ isotopy φt : φ1(L) = L contained in a wedge neighbourhood U of L, and such that L˜ = β (L) ∗ is given as a section of T L(log ∂L) in the Lagrangian neighbourhood of L. Now, φt lifts ∗ ′ ∗ ′ to a log symplectic isotopy φ˜t of L˜ = β (L), L˜ = β (L ) by lifting the generating elliptic symplectic vector field. Denote
1 Flux({φ˜t}) = [α], α ∈ Ω (L,˜ log L˜ ∩ D˜)
– since U˜ = β−1(U) is homotopy equivalent to L˜, we can choose a representative α ∈ Ω1(L,˜ log L˜ ∩ D˜), viewed as a log form on U˜ via pullback. We further assumed that L,˜ L˜′ are
related by a Hamiltonian isotopy ψt in (M,˜ D˜). In particular, this will have vanishing flow, and we know that L˜′ is the graph of an exact log one-form on L˜.
−1 ′ (ψ1) (L˜ ) = L˜
−α Further, there is the standard symplectic isotopy Λt between the zero section and the graph of −α with flux −α given by the symplectic vector field associated to −α. The composition
81 Lagrangian branes with boundary in stable generalized complex manifolds
−α −1 ˜ Λt ◦ ψt ◦ φt has the properties
−α −1 ˜ (Λ1 ◦ (ψ1) ◦ φ1)(L) = Graph(−α) −α −1 ˜ Flux(Λt ◦ (ψt) ◦ φt) = [α − α] = 0
Thus, L and the graph of −α are related by a local Hamiltonian isotopy in the tubular neighbourhood isomorphic to T ∗L(log ∂L), and so according to Proposition 5.25 α must have ∗ been an exact form. Thus φ˜t was a Hamiltonian isotopy, and since β is an isomorphism ′ of elliptic and log cohomology, so was φt. Thus L, L ⊂ (M,D) are related by Hamiltonian isotopy.
Proof of Theorem 6.22. In Section 5.4 we have already shown that small deformations of L˜ = β∗(L) ⊂ M˜ up to local Hamiltonian isotopy correspond to H1(L,˜ L˜ ∩ D˜). By the definition above, small deformations of L˜ are clearly in one-to-one correspondence with small deformations of L. Since any elliptic Hamiltonian flow on (M,D) will lift to a log Hamiltonian flow on (M,˜ D˜) via the lift of elliptic to log vector fields under real oriented blow-up, it is clear that if two Lagrangian branes with boundary in (M,D) are Hamiltonian isotopic, their preimages in (M,˜ D˜) are, too. And thus, if one is a local deformation of the other, it corresponds to an exact log one-form, as long as the Hamiltonian isotopy is local. Conversely, we need to show that if a deformation is given as the graph of an exact log form on L˜ in (M,˜ D˜), its image in (M,D) is related to the original brane by a smooth elliptic Hamiltonian isotopy. Although this is the simpler direction in the context of ordinary Lagrangians in symplectic manifolds, it turns out to be less intuitive here: Obviously, the graph of an exact log one-form df on L as a Lagrangian in T ∗(log ∂L) is log Hamiltonian isotopic to the zero section via the flow of the log Hamiltonian vector field associated to f, but this Hamiltonian vector field does not descend as a smooth elliptic vector field to (M,D). Instead, we begin by constructing a smooth symplectic isotopy between the original brane ′ 1 and the image of the graph of a sufficiently small closed log one-form: Let α ∈ Ωcl(L, log ∂L). Note that there is always an extension α ∈ Ω1(U, log |D ∩ U|) to some wedge neighbourhood ′ ∗ of L such that α|L = α and such that β α defines a map
∗ ∗ ∗ ∗ β α : U → T L(log ∂L), (β α)(ξp) ∈ Tp L(log ∂L).
′ dr Namely, if α = fr(r, x, qi) r + fx(r, x, qi) dx + fqi (r, x, qi) dqi in local coordinates (r, x, qi) on L, we pick the extension
dr α = f (r cos θ, x, q ) + f (r cos θ, x, q ) dx + f (r cos θ, x, q ) dq , r i r x i qi i i
82 6.3 Small deformations of Lagrangian branes with boundary which is a smooth elliptic one-form on (U, U ∩ D), although it is of course not closed. This extension exists across all of L. Now consider the following isotopy of diffeomorphisms on a neighbourhood of the zero-section of T ∗L(log ∂L), which descends to (M,D) under the blow-down map:
ψt : ξp 7→ ξp + tα(ξp).
′ One can check that ψt are indeed diffeomorphisms, as long as we pick (U, U ∩ D) and α ′ to be sufficiently small. We have: ψ1(L) = Graph(α ). Of course, the ψt do not preserve the elliptic symplectic form on (U, U ∩ D). Instead (for ω = ω0 the standard local elliptic ∗ symplectic form on a wedge neighbourhood): ψt ω = ω + t dα 1 Since dα|L = 0, there is a choice of smooth α¯ ∈ Ω (U, log |U ∩ D|) such that α¯|L = 0 and d¯α = dα. Now we apply the relative Moser theorem using the flow φs of the elliptic vector −1 ∗ ∗ field ω (tα¯) (for each t), which preserves L, and satisfies φt (ψt ω) = ω. Thus, we have ′ 1 defined an elliptic symplectic isotopy between L and the graph of α ∈ Ωcl(L, log ∂L). If α′ = df is an exact log one-form on L, the existence of a symplectic isotopy between the two resulting branes with boundary in (M,D) implies the existence of a Hamiltonian isotopy: See Lemma 6.24.
Remark 6.25. Lagrangian branes with boundary are coisotropic submanifolds with respect to the Poisson structure ω−1. But in contrast to the Lagrangians which intersect the degeneracy locus transversely and whose deformations we discussed in Section 5.4, we do not have a standard local form for the Poisson structure on a full tubular neighbourhood of the brane, only on a wedge neighbourhood. However, both the explicit computation of the L∞-structure, as well as the result on deformations of coisotropic submanifolds with respect to a fibrewise entire Poisson structures require the Poisson structure to be known on a full tubular neighbourhood. Thus these results are at present not applicable to general Lagrangian branes with boundary. We can find examples where ω−1 is fibrewise entire on a full neighbourhood of a brane with boundary (L, ∂L), and where the Maurer-Cartan elements of the L∞-structure on NL 1 do indeed again reduce to Ωcl(L, log ∂L).
83
Chapter 7
Lefschetz thimbles in stable generalized complex Lefschetz fibrations
Lefschetz fibrations are an important tool in symplectic geometry, and in particular in the study of symplectic 4-manifolds. In this chapter, we are going to review some recent results on the extension of the concept of Lefschetz fibrations to the logarithmic and elliptic symplectic setting from [CK16, CK17], and then define Lefschetz thimbles in these settings. In stable generalized complex Lefschetz fibrations, Lagrangian branes with boundary occur as Lefschetz thimbles.
Definition 7.1. Let M 2n be a compact connected oriented manifold. A Lefschetz fibration on M is a map f : M → Σ, to some compact connected oriented surface Σ2 which only has isolated critical points, and around each critical point pi ∈ M there are local complex coordinates (z1, . . . , zn), as well as a local complex coordinate around f(pi), compatible with the orientation of M and Σ such that f takes the form
2 2 f(z1, . . . , zn) = z1 + ··· + zn.
Remark 7.2. If n = 2, the condition above is equivalent to the existence of a local complex coordinates (w1, w2) around each critical point such that f(w1, w2) = w1w2:
w1 = z1 + iz2
w2 = z1 − iz2
A Lefschetz fibration on M can always be slightly changed to make the regular fibres F ⊂ M connected and each fibre contain at most one critical point.
85 Lefschetz thimbles in stable generalized complex Lefschetz fibrations
For 4-manifolds, we have the following relations between the existence of Lefschetz fibrations and symplectic structures:
Theorem 7.3. (See [GS99], Theorem 10.2.18.) If a closed 4-manifold M 4 admits a Lefschetz fibration f : M → Σ such that the homology class of the fibre F is non-zero in H2(M, R), M admits a symplectic structure s.t. the fibration is symplectic, i.e. the regular fibres are symplectic submanifolds.
Since the fibration is symplectic, the symplectic form ω induces a unique Ehresmann connection Hω : T Σ → TM whose image is precisely the symplectic orthogonal to the regular fibre (TF )ω−orth. Recall that an Ehresmann connection on a fibration allows for the horizontal lifting of paths: If γ : [0, 1] → Σ is a path in the base, there is a unique path γ˜ : [0, 1] → M in M such that f ◦ γ˜ = γ for each point in f −1(γ(0)), the fibre over the endpoint of the path, and such that γ˜′(t) ∈ H(γ′(t)) ∀t ∈ [0, 1]. There is a converse statement to Theorem 7.3 in a certain sense, which in its original formulation by Donaldson involves the concept of Lefschetz pencils. We state a less precise variation of the theorem here:
Theorem 7.4. (Based on [Don99]) Every symplectic 4-manifold is related to one admitting a Lefschetz fibration by the symplectic blow-up of a finite number of points.
There are a number of variations of the definition of Lefschetz fibration for non-compact symplectic manifolds over non-compact base surfaces. In what follows, we use the following notion of Lefschetz fibration with compact fibres for open symplectic manifolds:
Definition 7.5. Let Σ be an oriented punctured surface, i.e. obtained by removing a finite number of discs from a closed oriented surface. Let M 2n be an even-dimensional manifold, not necessarily compact. A Lefschetz fibration f : M → Σ is a proper map with only finitely many critical values and such that each critical fibre contains only one critical point. Furthermore, there exist complex charts with coordinates (z1, . . . , zn) around each critical point pi and w around f(pi) which are compatible with the orientation of M and Σ, and such that f has the form 2 2 f(z1, . . . , zn) = z1 + . . . zn.
A Lefschetz fibration is called symplectic if M is equipped with a symplectic form whose pullback to each regular fibre is non-degenerate.
Remark 7.6. Since we assumed f to be proper, the fibres of such a Lefschetz fibration will in particular be compact.
Lefschetz thimbles are a natural class of Lagrangian submanifolds in symplectic Lefschetz fibrations:
86 7.1 Ehresmann connections for log symplectic Lefschetz fibrations
Definition 7.7. Let f :(M, ω) → Σ be a Lefschetz fibration with critical points p1, . . . , pk ∈
M. Choose a regular base point x ∈ Σ and a collection of paths γi : [0, 1] → Σ from x to f(pi) such that the only critical value in Im(γi) is f(pi). The Lefschetz thimble Lγi (or just Li) over γi is the union of all horizontal lifts of γi which end in the critical point pi.
Proposition 7.8. (See e.g. [Sei01].) The Lefschetz thimble Lγi is a smooth embedded n n-dimensional disk D , in particular it is smooth at pi. Lγi intersects each regular fibre in n−1 the pre-image of γi in a Lagrangian sphere S (also called the vanishing cycle associated to pi), and is overall Lagrangian with respect to ω.
Lefschetz thimbles play an important role in the definition of Fukaya category for Lefschetz fibrations. See [Sei08, Sei12].
7.1 Ehresmann connections for log symplectic Lefschetz fibra- tions
Just like for ordinary symplectic structures, it makes sense to consider Lefschetz fibrations which admit a log symplectic structure. They will be Lefschetz fibrations over a surface with a marked hypersurface, and the logarithmic structure is such that the singular locus fibres over that hypersurface in the base. These fibrations have been defined and studied in detail in [CK16] (using slightly different terminology than in this text – to avoid any confusion, we explicitly introduce this terminology here):
Definition 7.9. A b-manifold is a pair (M,Z) of a manifold M and a hypersurface Z ⊂ M, equipped with the logarithmic tangent bundle TM(− log Z). A b-manifold (M,Z) is called b-oriented if TM(− log Z) is oriented. A b-map between two b-manifolds (M,ZM ), (N,ZN ) −1 is a map f : M → M such that f (ZN ) = ZM and f is transverse to ZN . Write f :(M,ZM ) → (N,ZN ) . 2n 2 A b-Lefschetz fibration or logarithmic Lefschetz fibration is a b-map f :(X ,ZX ) → (Σ ,ZΣ) between compact connected b-oriented b-manifolds such that for each critical point x in the set ∆ of all critical points there exist complex coordinate charts compatible with the orientations induced by the b-orientations, centred at x and f(x) in which f takes the form
Cn C 2 2 f : → , (z1, . . . , zn) 7→ z1 + ··· + zn
Remark 7.10. (i) Since a b-map f is transverse to ZN , i.e. Im(f) ⋔ ZN , f :(M,ZM ) → (N,ZN ) induces a morphism
f∗ : TM(− log ZM ) → TN(− log ZN )
87 Lefschetz thimbles in stable generalized complex Lefschetz fibrations
which maps f∗ : RM ↠ RN , where RM = ker aM |ZM ⊂ TM(− log ZM )|ZM ,RN =
ker aN |ZN ⊂ TN(− log ZN )|ZN . The respective canonical sections will be mapped to each other at every point. The reason that this is well-defined: Any vanishing function −1 for ZN pulls back to a vanishing function for ZM , because f (ZN ) = ZM .
(ii) From the local model around a Lefschetz singularity x ∈ M we can see that df|x = 0,
so since f is transverse on ZM , the set of critical points ∆ and ZM are disjoint.
Let f :(X,ZX ) → (Σ,ZΣ) be a logarithmic Lefschetz fibration. There is a commutative diagram of vector bundles over X (V the vertical distribution of f):
f∗ ∗ 0 V TX(− log ZX ) f (T Σ(− log ZΣ)) 0 aX aΣ f∗ 0 V TX f ∗(T Σ) 0 See Proposition 2.14 in [CK16] for a proof that ker(f∗ : TX(− log ZX ) → T Σ(− log ZΣ)) and ker(f∗ : TX → T Σ) can indeed be identified via the anchor aX : TX(− log ZX ) → TX.
When restricted to ZX , we obtain a diagram with exact rows and columns: 0 0
f∗ RX RΣ
f∗ ∗ 0 V TX(− log ZX )|ZX f (T Σ(− log ZΣ)|ZΣ ) 0
f∗ ∗ 0 V TZX f (TZΣ) 0
0 0
Proposition 7.11. Assume that H˜ : T Σ(− log ZΣ) → TX(− log ZX ) is a section of f∗ : ˜ ˜ TX(− log ZX ) → T Σ(− log ZΣ). If this is such that H|ZX (RΣ) = RX , H induces an Ehresmann connection H : T Σ → TX (defined on all of Σ!). ∼ Proof. First note that TX|X\ZX = TX(− log ZX )|X\ZX via the anchor map a : TX(− log ZX ) → ∼ TX induced by the inclusion of log vector fields, and similarly T Σ|Σ\ZΣ = T Σ(− log ZΣ)|Σ\ZΣ . ˜ Thus H induces an Ehresmann connection H : T Σ|Σ\ZΣ → TX|X\ZX . So it suffices to show that this extends in a well-defined manner and smoothly to H : T Σ → TX.
Any splitting s : TZΣ → T Σ(− log ZΣ)|ZΣ of the short exact sequence
0 → RΣ → T Σ(− log ZΣ)|ZΣ → TZΣ → 0
88 7.1 Ehresmann connections for log symplectic Lefschetz fibrations
induces the same map H : TZΣ → TZX that is compatible with the splitting outside ZΣ:
The difference between the two splittings is in RΣ, so by the assumption H˜ (RΣ) = RX ,
H = a ◦ H˜ ◦ s : TZΣ → TZX does not depend on the choice of s. ′ Consider a tubular neighbourhood of ZΣ, with x a local vanishing function for ZΣ = {x′ = 0}. Since f defines a logarithmic Lefschetz fibration, x = x′ ◦ f defines a local ∂ vanishing function for ZX on a tubular neighbourhood of ZX . The vector field x ∂x is defined everywhere on the tubular neighbourhood and, as a section of TX(− log ZX ) its restriction ′ ∂ to ZX generates RX (and similarly for x ∂x′ on Σ). ˜ ˜ ′ ∂ ∂ According to the assumption H(RΣ) = RX , we have H(x ∂x′ ) = x ∂x + xv, where v ∈ Γ(V ) ∂ on the tubular neighbourhood. Locally, ∂x′ is a normal vector field to ZΣ, which extends to the tubular neighbourhood, and the obvious induced Ehresmann connection outside ZΣ
∂ ∂ H : 7→ + v ∂x′ ∂x extends to ZΣ itself. In the case where NZΣ is trivial (which automatically means NZX ′ is trivial, too), x , x are coordinates on the entire tubular neighbourhoods of ZΣ or ZX ∂ ∂ respectively, and so ∂x′ , ∂x are well defined as normal vector fields everywhere. In this case it is obvious that H is well-defined.
If NZΣ is not orientable: With a chosen tubular neighbourhood embedding, the normal coordinates (= fibre coordinates for the normal bundle) on different patches around ZΣ are related by multiplication with a non-zero function on ZΣ:
∂ 1 ∂ x′ 7→ gx′, g ̸= 0 ⇒ 7→ ∂x′ g ∂x′
′ ∂ ′ ∂ ∂ ∗ ∂ ∗ 1 Since gx ∂gx′ = x ∂x′ , we must have x ∂x + xv = (f g)x ∂(f ∗g)x + (f g)xv¯, i.e. v¯ = g v, on the new coordinate neighbourhood – this makes H as above consistent.
Now assume that X is equipped with a log symplectic form ω s.t. the pullback of ω to
V = ker f∗ is non-degenerate, in particular the fibres of f in X \ ZX are symplectic. We call such b-Lefschetz fibrations log symplectic. Consider the unique splitting H˜ : T Σ(− log ZΣ) →
TX(− log ZX ) s.t. the image of H˜ is the symplectic orthogonal of the vertical distribution V . ˜ Proposition 7.12. On ZX : H(RΣ) = RX ⇔ The fibres of f|ZX are are made up of leaves of the symplectic foliation of ω in ZX . ˜ Proof. ⇒: Since the image of H is the symplectic orthogonal of V , we obtain iRX ω|V = 0, i.e. V ⊂ ker(res ω). But ker(res ω) precisely defines the symplectic foliation of ω in ZX , and since both it and V have dimension 2n − 2, the fibres of f|ZX must be made up of symplectic
89 Lefschetz thimbles in stable generalized complex Lefschetz fibrations leaves of ω. ω−orth ⇐: Now by assumption V ⊂ ker(res ω) ⇒ RX ⊂ V . Since the image of H˜ is the log ˜ symplectic orthogonal of V , there has to be a Y ∈ T Σ(− log ZΣ)|ZΣ s.t. H(Y ) spans RX . ⇒ Y ∈ RΣ, and H˜ (RΣ) = RX .
Remark 7.13. All logarithmic Lefschetz fibrations we want to consider satisfy this: In 4 2 2 1 [CK16], Theorem 3.4 and 3.7, a log symplectic structure ωX for f :(X ,ZX ) → (Y ,ZY ) (with orientable, compact, homologically essential fibres F and compact base Y 2) is constructed 2 1 from a log symplectic structure ωY for (Y ,ZY ). This uses a closed, and non-degenerate fibrewise smooth form, as well as the pullback of ωY . Since the logarithmic (singular) term is pulled back from the base, ker(res ωX ) will contain the tangent spaces to the fibres of the fibration, i.e. the tangent spaces to the symplectic leaves. Corollary 7.14. Given a fibration as in Proposition 7.12, a path γ : [0, 1] → Y with
γ(1) ∈ YZ s.t. γ intersects Z transversely, and a Lagrangian submanifold l ⊂ F of a (regular)
fibre F of f, the Lagrangian L swept out by parallel transport of l along γ will intersect ZX transversely. Proof. This parallel transport is given by the lift of γ to X using the Ehresmann connection H defined by ω, with starting points in l. The lift with any Ehresmann connection is such that tangent vectors to the lifted paths project to the tangent vector of the path, thus any path intersecting the boundary transversely will have a lift that does, too.
7.2 Stable generalized complex Lefschetz fibrations under blow- up
Let (M, D, ω) be a manifold with elliptic divisor D. Let (M,˜ D,˜ ω˜) be the real oriented ˜ ˜ blow-up of D. Let β : M → M be the blow-down map s.t. β|D˜ : D → D is a principal U(1)-bundle. Analogously to logarithmic Lefschetz fibrations, there is a notion of Lefschetz fibration for manifolds equipped with an elliptic divisor. [CK17] define and study these so-called boundary Lefschetz fibrations in detail as a specific case of Lie algebroid Lefschetz fibrations. Let (Σ,Z) be a surface, with Z a separating hypersurface (i.e. a line).
Definition 7.15. Let f :(M,D) → (Σ,Z) be a map of pairs such that Im(f∗) ⊂ TZ. The normal Hessian of f along D is the map
Hν(f) : Sym2(ND) → f ∗(NZ) which associates to f its normal Hessian at each point: Since Im(f) ⊂ TZ, the map
ν(df): ND → NZ
90 7.2 Stable generalized complex Lefschetz fibrations under blow-up
is the zero map. We can consider a local defining function z for Z and set h := f ∗z, which ν satisfies dh|D = 0. H (f) is defined as the Hessian of this function at each point; it is easy to check that this is independent of the chosen vanishing function z. A strong map of pairs f :(M,D) → (Σ,Z) is called a boundary map if its normal Hessian ν H (f) is definite along D. A boundary map f is called fibrating if f|D : D → Z is a submersion.
A boundary Lefschetz fibration is a fibrating boundary map f such that f|X\D : X \D → Σ\Z is a Lefschetz fibration (see Definition 5.21 in [CK17]).
Remark 7.16. (i) Any fibrating boundary map f :(M,D) → (Σ,Z) is submersive in a punctured neighbourhood around D. (Of course, f is not submersive on a full open neighbourhood of D.)
(ii) By passing to a cover of Σ, we can always assume that the generic fibres of a boundary Lefschetz fibration are connected. If the generic fibres near D are connected, the fibres
of f|D : D → Z are also.
First note the following (see [CK17]): If f :(M,D) → (Σ,Z) is a boundary map to the surface Σ such that Z is a separating submanifold, NZ is in particular orientable and there
is a global defining function z for Z s.t. f(X) ⊂ Σ+, the locus where z ≥ 0. Then f defines ′ ′ ′ ′ a boundary map f :(M,D) → (Σ ,Z ), where Σ = Σ+ ∩ f(X),Z = Z ∩ f(D). So if Z is separating, we can always assume that it is in fact the boundary of Σ. Furthermore, whenever (Σ,Z) is any manifold admitting a log symplectic structure which is also oriented, Z is separating.
Proposition 7.17. Assume that f :(M,D) → (Σ,Z) is a boundary Lefschetz fibration over a surface with boundary (Σ, ∂Σ = Z). Then there exists a real branched double cover of (Σ,Z) over Z˜ =∼ Z, ρ :(Σ˜, Z˜) → (Σ,Z) (in which Z˜ is separating) s.t. f factors through (Σ˜,Z): f = ρ ◦ f ′, and f ′ ◦ β :(M,˜ D˜) → (Σ˜, Z˜)
is a logarithmic Lefschetz fibration.
Branched cover over a surface with boundary Around every boundary component of Σ (either R or S1), there is a collar neighbourhood s.t. the boundary component is given by the vanishing of a positive coordinate. Denote this coordinate by x. For simplicity, write Z for a single boundary component. Now, a trivial branched cover of Σ over the boundary Z can be constructed as follows: Consider Z × [0, 1) × R (where the first two factors are a collar neighbourhood of Z in Σ). There is a smooth surface defined by the graph of y2 = x inside Z × [0, 1) × R ∋ (z, x, y). Its closure has boundary Z × {±1}, so we have doubled the previous boundary of the
91 Lefschetz thimbles in stable generalized complex Lefschetz fibrations collar neighbourhood. To obtain the full branched double cover Σ˜, glue one copy each of Σ \ (Z × [0, 1)) to each “arm” of the new branched surface. If there are multiple boundary components, glue one to all the +1-boundaries, the other to the −1-boundaries. Up to diffeomorphism, this procedure defines a unique new surface Σ˜ without boundary, in which Z is a separating hypersurface. ρ : Σ˜ → Σ, locally around Z defined by
(z, y2, y) 7→ (z, y2)(z ∈ Z) and away from the boundary components by the identity on each leaf, defines a branched covering.
Proof of Proposition 7.17. Let ρ :(Σ˜, Z˜) → (Z, Σ) be a branched double cover as just constructed. [CK17] prove a standard local form for boundary Lefschetz fibrations: There are coordinates (r, θ, x3, . . . , x2n) around D in (M,D) and (x, z) around Z in (Σ,Z) (x a vanishing function for Z) such that the boundary Lefschetz fibration f :(M,D) → (Σ,Z) near D takes the form 2 f(r, θ, x3, . . . , x2n) = (r , x2n)
This factors through the +1-arm of (Σ˜, Z˜) with coordinates (y, z) around Z˜ (y a vanishing function for Z˜) as ′ f ρ 2 f :(r, θ, x3, . . . , x2n) → (r, x2n) → (r , x2n)
If we compose this f ′ with the blow-down map β, we obtain a logarithmic Lefschetz fibration
f˜ = f ′ ◦ β :(M,˜ D˜) → (Σ˜, Z˜),
˜−1 ˜ ˜ ′ ∂ ∂ ˜ ˜ since f (Z) = D. β is a submersion, and f∗( ∂r ) = ∂x , so f is transverse to Z.
7.3 Lefschetz thimbles in stable generalized complex Lefschetz fibrations and examples of Lagrangian branes with bound- ary
From the previous section we know that a boundary Lefschetz fibration f :(M,D) → (Σ,Z) blows up to a log Lefschetz fibration f˜ :(M,˜ D˜) → (Σ,Z). Assume that (M,D) is equipped with a stable generalized complex structure given by the elliptic symplectic form ω in such a way that ω is non-degenerate on ker(f∗ : TM(− log |D|) → T Σ(− log Z)), in particular the fibres of f in M \ D are symplectic. (Similarly to log Lefschetz fibrations, the fibres of a boundary Lefschetz fibration are either entirely in D or entirely in M \ D.) We call this a stable generalized complex Lefschetz fibration.
92 7.3 Lefschetz thimbles in stable generalized complex Lefschetz fibrations and examples of Lagrangian branes with boundary
Proposition 7.18. (i) A stable generalized complex Lefschetz fibration f :(M,D) → (Σ,Z) induces the structure of a log symplectic Lefschetz fibration on f˜ :(M,˜ D˜) → (Σ,Z).
(ii) If H˜ : f ∗(T Σ(− log Z)) → TM(− log |D|) is the elliptic Ehresmann connection associ- ated to ω, we have:
H˜ (RΣ) = RM ⇔ The fibres of f|D are given by ker(resr ω),
where RM ⊂ ker a, a : TM(− log |D|)|D → TD is the subbundle spanned by the Euler vector fieldon ND.
Proof. (i) From the real oriented blow-up of D inside M, we obtain the following commu- tative diagram with exact rows: 0 V˜ T M˜ (− log D˜) f˜∗(T Σ(− log Z)) 0
β∗ ρ∗ =∼
0 V TM(− log |D|) f ∗(T Σ(− log Z)) 0
Note that β∗ is fibrewise an isomorphism, and it induces a fibrewise isomorphism on V˜ . Thus if ω is non-degenerate on V , ω˜ = β∗ω is non-degenerate on V˜ .
(ii) “⇒”: By definition of H˜ , Im(H˜ ) is the symplectic orthogonal of V in TM(− log |D|)), so ′ ∗ ′ iRM ω|V = 0. Consider V = a(V ) ⊂ TD. We have res ω(v) = ιD(iRM ω)(v) = 0 ∀v ∈ V , so V ′ ⊂ ker(res ω). But both ker(res ω) and V ′ have rank 2n − 3, so they are equal. ′ “⇐”: Now we assume that the fibres of f|D have the tangent distribution V = ker(res ω). ω−orth Since the elliptic residue of ω is zero, this implies RM ⊂ V . Thus there exists
Y ∈ T Σ(− log Z) such that H˜ (Y ) spans RM , and since f∗(RM ) = RΣ,Y spans RΣ, so
H˜ (RΣ) = RM .
Lefschetz Thimbles Let f :(M, D, ω) → (Σ,Z) be a stable generalized complex Lefschetz
fibration whose fibres in D correspond to ker(resr ω). Consider the associated log symplectic Lefschetz fibration f˜ :(M,˜ D,˜ ω˜) → (Σ˜, Z˜). According to what we have just shown, this admits an Ehresmann connection H induced by its log symplectic form. Like for ordinary symplectic Lefschetz fibrations, we can consider Lefschetz thimbles with respect to H over a path in the base surface which ends at the image of one of the Lefschetz singularities in the interior (i.e. a critical value of f˜). If the path is chosen such that it hits Z transversely, the associated Lefschetz thimble will be a logarithmic Lagrangian that intersects D˜ transversely in its (spherical) boundary. Now, when looking at the image of such a thimble under the blow-down map β, there are two main cases of interest according to Theorem 6.17:
93 Lefschetz thimbles in stable generalized complex Lefschetz fibrations
D M
f
Σ
Figure 7.1 A Lefschetz thimble which is a brane with boundary
∂ 1. If ∂θ is not tangent to and β is injective on the boundary of the Lefschetz thimble, the image in (M,D) will be a Lagrangian brane with boundary, a Lefschetz thimble whose boundary lies in the anticanonical divisor.
2. If the U(1)-action of D˜ restricts to the boundary sphere of the thimble, the thimble blows down to a Lagrangian brane without boundary (not always smooth, but always Hamiltonian isotopic to a Lagrangian that does blow down smoothly). In the case where the total space of the fibration is a 4-manifold, [BCK17] define the boundary vanishing cycle associated to the singular locus of a boundary Lefschetz fibration. This case then precisely occurs when the boundary vanishing cycle is the same as the vanishing cycle of the Lefschetz singularity at the other end of the thimble. The result is a Lagrangian brane which is topologically an S2. (By choosing the correct base path, we can always obtain a smooth Lagrangian S2.)
In the following, we examine some examples of Lagrangian branes produced by the parallel transport of Lagrangian spheres in the fibres of stable generalized complex Lefschetz fibrations:
7.3.1 Example: The Hopf surface
Consider the complex manifold X = (C \{0}) /(z ∼ 2z). This is clearly diffeomorphic to S3 × S1, viewing S3 as
3 n 2 2 2 o S = (z0, z1) ∈ C ||z0| + |z1| = 1
We can make X into a boundary Lefschetz fibration without any singular fibres as follows 3 2 (see [CK17]): Compose the Hopf fibration p : S → S , (z0, z1) 7→ [z0 : z1] with the standard height function h : S2 → I = [0, 1]. Furthermore consider the S1-coordinate given by
1 η = log |z |2 + |z |2 . (7.1) 2 0 1
94 7.3 Lefschetz thimbles in stable generalized complex Lefschetz fibrations and examples of Lagrangian branes with boundary
z1
D
z0
Figure 7.2 Hopf surface in C2 \{0} with anticanonical divisor
Since |z0| ∼ 2|z0|, |z1| ∼ 2|z1|, we obtain η ∼ η + log 2. Then
f(z0, z1) = (h([z0 : z1]), η)
defines a boundary Lefschetz fibration F : M → I × S1. z = r eiθ0 , z = r eiθ1 t = r0 (t, η, θ , θ ) M z = 0 If 0 0 1 1 and r1 , 0 1 are coordinates for away from 1 . (1/t, η, θ0, θ1) are coordinates away from z0 = 0. ′ ′ 2 We can pick coordinates t , 1/t on I s.t. the height function maps (t, θ0 − θ1) ∈ S to t2 ∈ I, and similarly on the other coordinate patch. In these coordinates, f becomes 2 f(t, η, θ0, θ1) = (t , η), i.e. the fibres of f are precisely the (θ0, θ1)-tori.
1 dt′ 1 d(1/t′) ∧ dη = − ∧ dη 2 t′ 2 1/t′ is a well-defined logarithmic symplectic form on I × S1. We can pull it back to M via f to obtain dt d(1/t) ∧ dη = − ∧ dη t 1/t This can be completed to the following elliptic symplectic form on M:
dt d(1/t) ω = ∧ dη − dθ ∧ dθ = − ∧ dη − dθ ∧ dθ (7.2) t 0 1 1/t 0 1
This is actually the imaginary part of the holomorphic log symplectic form dz0 ∧ dz1 dr1 dr0 dr0 dr1 Ω = i = ∧ dθ0 − ∧ dθ1 + i ∧ − dθ0 ∧ dθ1 (7.3) z0z1 r1 r0 r0 r1
The anticanonical divisor of Ω is D = {z0 = 0} ∪ {z1 = 0}. The symplectic orthogonal distribution to the fibres is spanned by
∂ ∂ ∂ t = −1/t , ∂t ∂(1/t) ∂η
95 Lefschetz thimbles in stable generalized complex Lefschetz fibrations and the corresponding Ehresmann connection
H˜ : T (I × S1)(− log({0} × S1 ∪ {1} × S1)) → TM(− log |D|) lifts ∂ ∂ ∂ ∂ t′ 7→ t , 7→ , ∂t′ ∂t ∂η ∂η so induces an Ehresmann connection H : T (I × S1) → TM. 1 Now consider any path γ in I × S from (0, η0) to (1, η0) with η(s) = η0 constant. Under
H, this lifts to a corresponding path with t(s), η = η0, θ0 = const., θ1 = const. We can for example obtain the following Lagrangian branes with boundary from parallel transporting Lagrangian circles in the fibres along such a path in the base:
Example 7.19. (i) Circle with θ0 = const., θ1 ∈ [0, 2π): When parallel-transported to
the component of D with z1 = 0, this circle closes up. Parallel-transporting all along the path γ yields a Lagrangian brane with boundary that is diffeomorphic to D2 and
intersects {z0 = 0} in its circular boundary, {z1 = 0} in a point. Evidently we can 2 exchange z0 and z1 to obtain a similar D -brane with boundary in {z1 = 0}.
(ii) Circle with θ0 = θ1 = θ ∈ [0, 2π): When parallel-transported all along the path γ, we
obtain a cylindrical Lagrangian brane with boundary, which intersects both {z0 = 0}
and {z1 = 0} in a circle.
Figure 7.3 Lagrangian (ii) in a neighbourhood of either component of the anticanonical divisor.
(iii) Circle with 2θ0 = θ1 = θ ∈ [0, 2π): When parallel-transported all along the path γ, we obtain a smooth Lagrangian brane with boundary that is topologically a Möbius
band. It intersects the {z1 = 0}-locus in a circle which is its boundary, and {z0 = 0} also in a circle, the zero section of the Möbius band as a subset of the Möbius line bundle. Note that when lifted to the real oriented blow-up, this Lagrangian intersects
96 7.3 Lefschetz thimbles in stable generalized complex Lefschetz fibrations and examples of Lagrangian branes with boundary
both components of the singular locus in a circle, but on the blow-up of {z0 = 0} the blow-down map is not injective.
Figure 7.4 Lagrangian (iii) in an open neighbourhood of both components of the anticanonical divisor: On the left, we see the neighbourhood of {z0 = 0} and on the right the neighbourhood of {z1 = 0}.
(iv) Circle with 3θ0 = θ1 = θ ∈ [0, 2π): When parallel-transported all along γ, this does
not result in a smooth submanifold: Away from {z0 = 0}, this is an open Lagrangian
cylinder which will intersect {z1 = 0} in its circular boundary, but at {z0 = 0} there is a triple intersection of leaves of the cylinder. As in the previous case, the blow-down map is not injective on the boundary of the lift of this Lagrangian to the real oriented blow-up.
Figure 7.5 Lagrangian (iv) in an open neighbourhood of both components of the anticanonical divisor: On the left, we see the neighbourhood of {z0 = 0} and on the right the neighbourhood of {z1 = 0}.
7.3.2 Examples of genus-one boundary Lefschetz fibrations over the disk
Example 7.20. Boundary Lefschetz fibration with one Lefschetz singularity. [CK17] study this example (see Example 8.4 in that text): If we consider the 4-dimensional genus-1
97 Lefschetz thimbles in stable generalized complex Lefschetz fibrations
Lefschetz fibration over D2 with one singular fibre with vanishing cycle b ∈ H1(T 2), a genera- tor, the monodromy around ∂D2 is the Dehn twist with b. Thus this Lefschetz fibration can be completed to a boundary Lefschetz fibration with a stable generalized complex structure whose anticanonical divisor fibres over ∂D2. From Proposition 6.2 and 6.5 in [CK17] we obtain coordinates (s, x, y, z) for a neighbour- hood of the anticanonical divisor where s is a radial coordinate for the distance from the anticanonical divisor, and (x, y, z) angular coordinates such that
(x, y, z) ∼ (x, y + 1, z) (x, y, z) ∼ (x, y, z + 1) (x, y, z) ∼ (x + 1, y, z − y)
The projection to a tubular neighbourhood of ∂D2 is (s, x, y, z) 7→ (s2, x); (y, z) are angular coordinates for the torus fibres. (These coordinates are for what is referred to as the standard 1-model). Note that the z-coordinate encodes the vanishing cycle b. On the other hand, z is is the angular coordinate in the fibre of the complex line bundle over D that defines the standard 1-model. Thus as r → 0, the z-circle shrinks to zero. So any Lefschetz thimble for the single Lefschetz singularity in this example over a path in the base from the singularity to the boundary will be topologically an S2: The vanishing cycle sweeps out a disk when moving along a path away from the singularity, which closes up to a sphere as the vanishing cycle shrinks back to a point when approaching the anti-canonical divisor. As described in [CG09] and [CK17], such spheres can be blown down in a way that is compatible with the stable generalized complex structure. After the blow-down, we obtain another boundary Lefschetz fibration for the Hopf surface.
a
a+3b a-3b
b
Figure 7.6 Lefschetz singularities in Example 7.21. The total space of this boundary Lefschetz fibration is CP 2.
Example 7.21. There are other genus-1 Lefschetz fibrations over the open disk with multiple Lefschetz singularities, but whose monodromy is still the power of a Dehn twist – and whenever this is the case, they can be completed to a closed boundary Lefschetz fibration (see Proposition 6.5 in [CK17]). For example (Example 8.5 in [CK17]), if a, b are generators
98 7.3 Lefschetz thimbles in stable generalized complex Lefschetz fibrations and examples of Lagrangian branes with boundary
2 of H1(T ), there is a Lefschetz fibration with three singularities in the interior of the disk with associated vanishing cycles (in counter-clockwise order)
a − 3b, a, a + 3b.
The global monodromy around all three singularities is 9b, so we can complete this to a
boundary Lefschetz fibration by gluing in the standard 9-model tot(L9) (see Proposition 6.5 in [CK17]), such that the total space admits a stable generalized complex structure. Topologically, the resulting closed total space of this boundary Lefschetz fibration is CP 2 (see also Example 5.3 in [CG09]). In order to extend the Lefschetz thimbles associated to the three Lefschetz singularities into the anticanonical divisor, we follow the C∞-log surgery as described in Section 4 of [CG09]: We consider the honest Lefschetz fibration of CP 2#9CP 2 over S2, with paths from all 3 + 9 Lefschetz singularities to a regular reference fibre. We trivialise this Lefschetz fibration around the reference fibre and perform a C∞-log transform to obtain a generalized complex Lefschetz fibration over the disk. If we identify the regular fibre with the standard
a a a-3b a+3b a+3b a-3b
b b S² b D²
Figure 7.7 Performing a C∞-log transform on the Lefschetz fibration of CP 2#9CP 2 over S2 to obtain a generalized complex Lefschetz fibration over the disk.
torus in such a way that the homology base of cycles a, b corresponds to the canonical circles in the standard torus, with b the boundary vanishing cycle, the Lefschetz thimbles associated to the 9 Lefschetz singularities with cycle b are again 2-spheres, just like in the previous example. As described in [CG09], these spheres can be blown down. Using this surgery, the Lefschetz thimbles associated to the remaining three Lefschetz singularities are branes with boundary.
99
Chapter 8
Lagrangian branes with boundary and complex branes in holomorphic log symplectic manifolds
As previously established, Lagrangian branes with boundary as studied in this paper are not generalized complex branes in the usual sense. We will now consider a stable generalized complex structure on a complex surface X which is given by a holomorphic log Poisson structure π, or equivalently a holomorphic complex log symplectic form Ω = B + iω. In this setting, all one-dimensional complex submanifolds are generalized complex branes: The pullback of any holomorphic two-form to such a submanifold is zero for dimensional reasons and all complex submanifolds will intersect the degeneracy locus (itself a complex submanifold) in a set of points. Of course the converse is not true even for half-dimensional branes: There are generalized complex branes ι : L,→ X in X with ι∗ω = 0, but ι∗B ̸= 0. We have already established that for a Lagrangian brane with boundary, ι∗B is a log two-form with non-vanishing residue with respect to the boundary. However, there are instances where such branes inside holomorphic Poisson manifolds are related to complex submanifolds via Hamiltonian isotopies outside the anticanonical divisor, as illustrated by the following examples. Most of the complex submanifolds obtained this way in the known examples are non- algebraic submanifolds, it is however not clear how general this feature is. Furthermore, given a particular complex structure with respect to which a chosen stable generalized complex structure is holomorphic Poisson, no existence result for the deformation of a brane with boundary into a complex submanifold is currently known. The construction is ad hoc in each example. Note that a particular stable generalized complex structure can be holomorphic Poisson with respect to several different complex structures.
101 Lagrangian branes with boundary and complex branes in holomorphic log symplectic manifolds 8.1 Example: C × T 2
Consider M = C × T 2 with complex coordinates (w, z) and stable GC structure given by
dw dr dr Ω = B + iω = ∧ dz = ∧ dx − dθ ∧ dy + i ∧ dy + dθ ∧ dx w r r
Write w = reiθ, z = x + iy, (w, z) = (r, θ, x, y). Consider a submanifold L given as follows:
L = {(r, θ, θ, f(r))}, f(r) smooth.
All such submanifolds are Lagrangian branes:
dr ∂f ι∗ ω = ∧ dr + dθ ∧ dθ = 0 L r ∂r
Proposition 8.1. If Lt = {(r, θ, θ, ft(r))} is a smooth family of such branes outside D = −1 {r = 0}, we can find a time-dependent Hamiltonian vector field Xt = ω (dgt) with gt a smooth family of smooth maps whose flow φt reproduces the family Lt:
φt(L0) = Lt
dft dr dft dr Proof. Since ft = ft(r), − dt r is an exact one-form away from r = 0, so − dt r = dgt. Now we consider the time-dependent Hamiltonian vector field
df ∂ X = ω−1(dg ) = t t t dt ∂y
Its flow satisfies: dφy df t = t , dt dt so φt(r, θ, x, y) = (r, θ, x, y + ft(r) − f0(r)).
Thus φt(L0) = {φt(r, θ, θ, f0(r))} = {(r, θ, θ, ft(r))} = Lt. Note that this flow is everywhere well-defined for all t ∈ [0, 1] and r > 0.
L = {z = −i log w} = {(r, θ, θ, − log r)} defines a cylindrical complex brane in M which does not intersect D = {w = 0}, instead it wraps around the y-direction faster and faster as r → 0.
Consider the family of branes Lt = {(r, θ, θ, (t − 1) log(r + t))}. For t > 0 this is a family of Lagrangian branes with boundary, and we have L0 = L, L1 = {(r, θ, θ, 0)}.
φt(r, θ, x, y) = (r, θ, x, y + (t − 1) log(r + t) + log(r)) is the Hamiltonian flow that maps these branes into each other. It is well-defined and smooth away from the anticanonical divisor. The closer one approaches the anticanonical divisor, the
102 8.2 Example: Hopf Surface
more the flow has to move the brane with boundary {(r, θ, θ, 0)} in order to make it complex (or vice versa).
8.2 Example: Hopf Surface
This example follows exactly the same pattern as the first: We consider the Hopf surface X with the same coordinates and stable generalized complex structure as in Section 7.3.1 and show that outside the anticanonical divisor the branes with boundary in Example 7.19 can be deformed into complex submanifolds:
iσ Example 8.2. (i) A complex submanifold of X is given by L = {z1 = const. = ae }.
Now, this obviously intersects the anticanonical divisor at {z0 = 0} in a point, but
does not intersect {z1 = 0}, instead wrapping infinitely often around the η-direction as r0 t = a → ∞. L lies over the path
1 r 7→ r2, log(r2 + a2) 0 0 2 0
in the base. In terms of the coordinates (t, η, θ0, θ1):
1 L = t, log(a) + log t2 + 1 , 2
or in terms of t′ = 1/t:
1 1 L = t′, log(a) + log + 1 . 2 t′2
We can interpolate between L = L0 and the brane with boundary
L1 = {(t, 0, θ, σ)}
with the family of branes
Ls = {(t, fs(r), θ, σ)} , !! 1 t2 f (t) = (1 − s) log(a) + log + 1 , s 2 (1 + st)2
and this interpolation can again be realised in terms of a time-dependent Hamiltonian vector field, namely ∂fs(t) ∂ ∂s ∂η
103 Lagrangian branes with boundary and complex branes in holomorphic log symplectic manifolds
(ii) Next, consider the following cylindrical Lagrangian, which is parametrised by one complex coordinate z = reiθ, r ̸= 0:
1 1 L := 2r2, log r2 + , θ, −θ 2 2r2
Under the map f, L projects to the path in [0, 1] × S1
1 1 r 7→ 4r4, log r2 + 2 2r2
n 1 o In terms of the complex coordinates (z1, z2), this is L = z, 2z . Clearly, this is a complex submanifold that does not intersect the anticanonical divisor {z0 = 0} ∪ {z1 = 0}. Now consider the following family of Lagrangians:
1 t 1 L := {(t, f (t), θ, −θ)} , f (r) = (1 − s) log(1/2) + log + s s s 2 1 + ts t + s
This family of Lagrangians interpolates between L0 = L and the brane with boundary (two S1 boundary components, one each in the two connected components of the anticanonical divisor)
L1 = {(t, 0, θ, −θ)}
Whenever s > 0, Ls extends into the the anticanonical divisor at either end. The closer 2 s is to zero, the stronger the brane Ls and its corresponding base path (t , fs(t)) wrap in the η-direction.
∂fs ∂ The Hamiltonian vector field which flows L0 into Ls is given by ∂s ∂η .
Example 8.3. The Hopf surface with a different generalized complex structure. 3 1 2 Consider the Hopf surface M = S × S = C \{0}/((z0, z1) ∼ 2(z0, z1)) with complex y = z , z = z1 coordinates 0 z0 and stable generalized complex structure given by the holomorphic log symplectic form dz dz dy dz Ω = B + iω = 0 ∧ 1 = ∧ z0 z1 y z
This coordinate chart is only defined where z0 ̸= 0. If we write z = reiθ, y = seiλ:
ds dr dr ds Ω = ∧ − dλ ∧ dθ + i dλ ∧ + ∧ dθ s r r s
Now consider a submanifold
L = {(r, θ, f1(θ), f2(r))} with points in M written as (r, θ, s, λ).
104 8.2 Example: Hopf Surface
Note that in order for this to be well-defined, we need to have
n f1(θ + 2π) = 2 f1(θ) for some n ∈ N0.
L is clearly a Lagrangian submanifold for ω. t t Let Lt = {(r, θ, f1(θ), f2(r))} be a family of Lagrangian branes such that f1(θ) ̸= 0 and n log(2)θ/2π f1(θ) = e with n ∈ N0 fixed. Now
! ∂ dr ∂f t dr ∂f t F = ι∗ B = log(f t) dθ ∧ − 2 dr ∧ dθ = − ∧ dθ c + r 2 t Lt ∂θ 1 r ∂r r ∂r
So: ! d ∂ df t dr F = −r 2 ∧ dθ dt t ∂r dt r ! df t dr db = dI∗ 2 + t dθ dt r dt
t df2 dr Clearly, when r ̸= 0, dt r = dgt for some family of functions gt. Set
df t ∂ X = ω−1(dg ) = 2 t t dt ∂λ
The flow φt of the time-dependent vector field Xt preserves ω and φt(L0) = Lt:
t 0 φt(r, θ, s, λ) = (r, θ, s, λ + f2(r) − f2 (r))
Now choose the particular case
t log 2 θ t log(2r + t) f (θ) = e 2π , f (r) = (t − 1) , t ∈ [0, 1] 1 2 2π
n θ/2π log(2r) o a log 2 In particular, L0 = r, θ, 2 , − 2π = {(y, z)|y = z } with a = −i 2π , so L0 is a complex brane which intersects neither {z0 = 0} nor {z1 = 0}. For t > 1, Lt is a Lagrangian brane with boundary at {z1 = 0} that does not intersect {z0 = 0}. The full family of
Lagrangian branes is obtained from L0 via
log(2r) log(2r + t) φ (r, θ, s, λ) = r, θ, s, λ − + (t − 1) t 2π 2π
105 Lagrangian branes with boundary and complex branes in holomorphic log symplectic manifolds 8.3 Example: CP 2
2 Now consider C with an affine coordinate chart (z0, z1) and generalised complex structure
dz ∧ dz Ω = i 0 1 z0z1 − 1
Then we have a disk-shaped brane with boundary
n o L = teiθ, te−iθ , t ∈ [0, 1]
It is possible to deform this brane with boundary (minus its boundary) into a subset of the complex line {z0 = 0} using a Hamiltonian isotopy, the image being a complex disk (without boundary). The family of branes is
−iθ Ls = {(sz,¯ z), z = re , r ∈ [0, 1)}, and the corresponding Hamiltonian isotopy
2 φs(z0, z1) = (1 − sr1)z0 + sz¯1
Note in particular that this is smooth at r1 = 0. We now need to show that this is indeed a Hamiltonian isotopy:
Proof. Away from z0 = z1 = 0, we can define the following coordinate transformation:
iσ w = z0z1 − 1 =: re , u = i log(z1) =: x + iy
dw Note that by definition x is actually a circular coordinate. In these coordinates, Ω = w ∧ du, and 2 2 L = {(t − 1, θ + i log(t))},Ls = {(st − 1, θ + i log(t))}
2y set fs(y) = 1 − se , s ∈ [0, 1] and consider the time-dependent Hamiltonian vector field
∂ ∂ X = log(f (y))r , s ∂s s ∂r which integrates to 2y φs(r, θ, x, y) = (r 1 − se , σ, x, y)
(only defined where y < 0, i.e. r1 < 1), so
2 φs(L0) = φs({(1, π, θ, log(t))}) = {(1 − st , π, θ, log(t))} = Ls.
106 8.3 Example: CP 2
Now, transforming back into the original coordinates, we find
(1 − sr2)w + 1 (1 − sr2)(z z − 1) + 1 (1 − sr2)z z + sr2 z0 1 1 0 1 1 0 1 1 2 φs (z0, z1) = = = = (1−sr1)z0+sz¯1 z1 z1 z1
In particular, this extends to z1 = 0, so is well-defined in the entire affine neighbourhood we are considering. By cutting off the Hamiltonian function with a bump function, we can perform this deformation in a an open neighbourhood of the origin in the coordinate neighbourhood. q 2 2 (As chosen, the entire family of branes sits in an open ball with r1 + r2 < 2.) The original disk-shaped brane has t ∈ [0, 1], so L0, t ∈ [0, 1) is also only a small part of
the full complex submanifold z0 = 0. In contrast to the previously examined examples, this is not a complex brane with wraps infinitely often around the anticanonical divisor. If we instead consider the compatible stable generalised complex Lefschetz fibration away from
z1 = 0 given by (r, σ, x, y) 7→ (r2, y),
the complex brane {z0 = 0} sits entirely over r = 1 and thus over a path at fixed distance to the anticanonical divisor. Furthermore, instead of the interior of a brane with boundary being mapped subjectively to an open complex submanifold, in this example the image is only a small open set in the
full complex curve z0 = 0.
107
Chapter 9
Conclusions and Outlook: Stable Hamiltonian systems from stable generalized complex manifolds
In Part I we have studied stable generalized complex manifolds through the lens of their associated elliptic symplectic form. We have shown that stable generalized complex structures are related to certain logarithmic symplectic structures via the real oriented blow-up of the anticanonical divisor. Stable generalized complex manifolds are in many ways the simplest class of examples of generalized complex manifolds that are neither symplectic nor complex (and include underlying manifolds that do not admit either a symplectic or a complex structure), and since the structure can be described in terms of an elliptic symplectic form, extending more techniques from symplectic geometry to stable generalized complex geometry is a natural continuation of the work presented here so far. The main focus of this part of the thesis has been on Lagrangian branes with boundary, a new class of submanifold with boundary for stable generalized complex manifolds that is not included in the previously studied class of generalized complex branes. And yet, since they are Lagrangian with respect to the elliptic symplectic form, and in particular the restricted symplectic form away from the anticanonical divisor, they can be expected to appear in the construction of a generalisation of the Fukaya category for stable generalized complex manifolds which considers non-compact Lagrangians. Fukaya categories are fundamental objects in the study of symplectic manifolds. In spite of their name, they are not categories in the usual sense, but A∞-categories (see [Sei08]), whose morphisms are not associative. Instead for every collection of objects (X0,...,Xd) there is a collection of composition maps
d µ : Hom(Xd−1,Xd) ⊗ · · · ⊗ Hom(X0,X1) → Hom(X0,Xd)[2 − d],
109 Conclusions and Outlook: Stable Hamiltonian systems from stable generalized complex manifolds which satisfy a particular collection of associativity relations each containing µd of several different degrees. In particular, the lowest degree relation is (µ1)2 = 0, and we can recover an 2 1 honest category from each A∞-category by taking µ up to µ -cohomology as the composition operation (which is associative), however, obviously information is lost in this process. The objects of a Fukaya category are (a particular subclass of) Lagrangian submanifolds, and morphisms are given by the Lagrangian intersection Floer (co-)homology of two La- grangians. This particular incarnation of Floer (co-)homology is defined on the intersection points of two transversely intersecting Lagrangian submanifolds. The definition of the chain map for this (co-)homology theory involves moduli spaces of pseudo-holomorphic discs, disks whose boundary lies inside the two Lagrangians and which are holomorphic with respect to a compatible almost complex structure. The construction of different Fukaya categories for symplectic manifolds is an active field of research. Fundamental works on this subject include [Sei08, FOOO09, FOOO10]. An attempt for the construction of a Fukaya category for stable generalized complex manifolds could be modelled on the following: For an exact symplectic manifold (M, ω = dλ) with a complete Liouville vector field X, we can define the so-called wrapped Fukaya category [AS10]: Outside a compact subset, M R ∂ has the form ∂M × with X being outward-pointing at ∂M. On ∂M, we can write X = t ∂t , where t is the coordinate for the factor R. ∂M inherits a contact form s.t. ω = d(tα). The objects of the wrapped Fukaya category are exact Lagrangians L ⊂ M which are invariant under the flow of the Liouville vector field outside a compact subset, and such that λ|L = 0 outside a compact set – in summary, non-compact L have to coincide with ∂L × (1, ∞) outside a compact subset, where ∂L is a Legendrian submanifold of ∂M. In order to be able to consider a finite set of intersection points between any two such Lagrangians, we “wrap” one of them by considering its image under the Hamiltonian flow associated to the Reeb vector field.
Now consider a stable generalized complex manifold (M, D, ω). If D is compact, ω|M\D is never exact; however, using the normal form of ω in a tubular neighbourhood of D, we can nonetheless make a similar construction: Let U be a tubular neighbourhood of D in M: U \ D =∼ D˜ × (0, 1/e), where D˜ = S1(ND). Recall that on D˜ × [0, 1/e), we have a standard local form for ω: dr ω = ∧ Ω˜ + dθ ∧ Ω˜ +σ, ˜ r I R where Ω˜ I , Ω˜ R, σ˜ are all closed forms on D˜ which are pullbacks of forms on D. Setting
t = − log(r), away from D we can write:
ω = − dt ∧ Ω˜ I + dθ ∧ Ω˜ R +σ ˜
110 If we set α = Ω˜ I , β = dθ ∧ Ω˜ R + σ˜, these forms are invariantly defined everywhere on U, and since ω is elliptic symplectic, α ∧ βn−1 ̸= 0. We can write: ω = d(tα) + β,
where both α, β can be viewed as closed forms on D˜. Of course, unlike in the exact/ contact case, α defines an integrable distribution in D˜. The pullback of β to each leaf is symplectic. Such a structure is an example of a so-called stable Hamiltonian structure [CV15], for which Floer theory results have already been established (e.g. [CM05]). Using the coordinate t instead of r, U \ D =∼ D˜ × (1, ∞). We now consider Lagrangians L = ∂L × R ⊂ D˜ × R, where ∂L ⊂ ker α (i.e. contained in one leaf of the corresponding foliation) and ∂L Lagrangian with respect to β. When these Lagrangians are extended into D, they can be branes with and without boundary. (As illustrated in previous examples, they can also end up not being smooth.) As the degeneracy locus of a log symplectic structure, D˜ fibres over S1. There is a vector ˜ ∂ field R on D such that α(R) = 1, iRβ = 0, which is a lift of the vector field ∂ϑ , ϑ the angle along the S1. We can use this as the analogue of the Reeb vector field to wrap Lagrangians. This construction is a candidate for a Fukaya category for stable generalized complex manifold. However, as described so far, this is really a construction for the log symplectic manifold obtained as the real oriented blow-up. How can we take the full generalized complex structure into account? For example, we know that the log symplectic real oriented blow-up of the anticanonical divisor inherits a U(1)-action from the stable generalized complex structure – how can this additional structure be used to impose further restrictions? In future work, we will attempt to use this approach to generalise Lagrangian intersection Floer homology to stable generalized complex structures, and to define a Fukaya category in this context. The way generalized complex geometry unifies the description of complex and symplectic manifolds and their submanifolds has raised questions about mirror symmetry in this more general context from its inception. The successful construction of a Fukaya category for (some) stable generalized complex manifolds could allow future investigations in this direction using the homological language for mirror symmetry.
111
Part II
Dorfman brackets and double vector bundles
113
Chapter 10
Brackets on TM ⊕ E∗ and the standard VB-Courant algebroid
In this chapter we introduce Dorfman brackets, a particular type of Leibniz algebroid, on vector bundles of the form TM ⊕ E∗, where E → M is a vector bundle. Such Leibniz algebroids fit into a particular double vector bundle, the associated standard VB-Courant algebroid (TE ⊕ T ∗E; TM ⊕ E∗,E; M) (see Chapter 3.2). The linear sections of this double vector bundle lie over sections of TM ⊕ E∗ → M, and there are different notions of lift, associating a linear section of TE ⊕ T ∗E → E to each section of TM ⊕ E∗. We prove a result ℓ ∗ on the form of these linear sections, and show that the fat vector bundle Eb of ΓE(TE ⊕ T E) is isomorphic to the so-called Omni-Lie algebroid studied in [CL10, CLS11]. We further prove a result on the precise form of the Courant-Dorfman bracket on linear sections which is the basis for the relationship between the Courant-Dorfman bracket on TE ⊕ T ∗E and Dorfman brackets on TM ⊕ E∗ investigated in the next chapter.
10.1 Dorfman brackets and dull brackets
In this section, we introduce Dorfman brackets, a term we use for the brackets of transitive Leibniz algebroids (see Section 2.2), and recall some results on dull algebroids, a generalisation of Lie algebroids introduced in [JL18].
Definition 10.1. A transitive Leibniz algebroid E′ is, as an anchored vector bundle, iso- ∗ morphic to TM ⊕ E with ρ = prTM and E → M. We call its bracket ·, · a Dorfman J K bracket1.
If E′ is a split transitive Leibniz algebroid (see Section 2.2) with splitting σ : TM → E′, we can use σ to define the isomorphism E′ → TM ⊕ E∗. We then obtain a Dorfman bracket
1Occasionally the term “Dorfman bracket” is used for the bracket of arbitrary Leibniz algebroids in the literature, but in this thesis it will exclusively refer to the case where the anchor is surjective and the underlying vector bundle is split.
115 Brackets on TM ⊕ E∗ and the standard VB-Courant algebroid with the property
def (X, 0), (Y, 0) = σ(X), σ(Y ) = σ[X,Y ] = ([X,Y ], 0) (10.1) J K J K Correspondingly, we call a Dorfman bracket split precisely if it has this property.
Consider a Dorfman bracket · , · : Γ(Q) × Γ(Q) → Γ(Q). Its dual map is J K D : Γ(Q) → Der(Q∗),
′ ′ ′ ′ ∗ defined by ρ(q)⟨q , τ⟩ = ⟨q , Dqτ⟩ + ⟨ q, q , τ⟩ for all q, q ∈ Γ(Q) and τ ∈ Γ(Q ). The Jacobi J K identity in Leibniz form for · , · is equivalent to J K
Dq1 ◦ Dq2 − Dq2 ◦ Dq1 = D q1,q2 (10.2) J K for all q1, q2 ∈ Γ(Q). We also refer to both (10.2) and the Jacoby identity in Leibniz form as the Dorfman condition. D allows the extension of the Dorfman bracket to all tensor bundles of Q via the Leibniz rule. In the theoretical physics applications, this operation is called the generalized Lie derivative due to its Lie algebra property.
Definition 10.2. [JL18]A dull algebroid is an anchored vector bundle (Q → M, ρ) endowed with a bracket · , · on Γ(Q) satisfying ρ q1, q2 = [ρ(q1), ρ(q2)], and the Leibniz identity in J K J K both terms
f1q1, f2q2 = f1f2 q1, q2 + f1ρ(q1)(f2)q2 − f2ρ(q2)(f1)q1 J K J K ∞ for all f1, f2 ∈ C (M), q1, q2 ∈ Γ(Q).
Consider a dull algebroid (Q, ρ, · , · ). Then the bracket can be dualised to a map J K ∗ ∗ ′ ′ ′ ∆: Γ(Q) × Γ(Q ) → Γ(Q ), ρ(q)⟨q , τ⟩ = ⟨ q, q , τ⟩ + ⟨q , ∆qτ⟩ J K for all q, q′ ∈ Γ(Q) and τ ∈ Γ(Q∗). The map ∆ is then a Dorfman (Q-)connection on Q∗ (see [JL18]), i.e. an R-bilinear map with
∗ (i) ∆fqτ = f∆qτ + ⟨q, τ⟩· ρ d f,
(ii) ∆q(fτ) = f∆qτ + ρ(q)(f)τ and
∗ ∗ (iii) ∆q(ρ d f) = ρ d(£ρ(q))
∞ ′ ∗ for all f ∈ C (M), q, q ∈ Γ(Q), τ ∈ Γ(Q ). The curvature of ∆ is the map R∆ : Γ(Q) × ∗ ∗ ′ ′ Γ(Q) → Γ(Q ⊗ Q ) defined on q, q ∈ Γ(Q) by R∆(q, q ) := ∆q∆q′ − ∆q′ ∆q − ∆ q,q′ . For all J K
116 10.1 Dorfman brackets and dull brackets
∞ ∗ f ∈ C (M) and q1, q2, q3 ∈ Γ(Q), τ ∈ Γ(Q ), we have
⟨R∆(q1, q2)τ, q3⟩ = ⟨ q1, q2 , q3 + q2, q1, q3 − q1, q2, q3 , τ⟩. JJ K K J J KK J J KK Note that if we apply the definition for the curvature of a Dorfman connection to the dual map D of a Dorfman bracket, the Dorfman condition is precisely the requirement that the curvature is zero.
Most commonly studied examples of Leibniz algebroid (e.g. in [Bar12]), including those describing gauge symmetries of supergravity in theoretical physics, are Dorfman brackets. We present the most important and well-known examples here.
Example 10.3. The bracket of a Courant algebroid E is a Dorfman bracket. Using the nondegenerate pairing to identify E with its dual, we find that D is in this case the “adjoint
action”: De = e, · for e ∈ Γ(E). J K Example 10.4. On any vector bundle of the form TM ⊕E∗ with E = ∧k1 TM ⊕· · ·⊕∧kl TM, there is a Dorfman bracket
∗ (X, α), (Y, β) = [X,Y ] + £X β − iY d α for (X, α), (Y, β) ∈ Γ(TM ⊕ E ) (10.3) J K For simplicity of notation, consider the special case TM ⊕ ∧kT ∗M for the rest of this example – the more general case works in the same way. Let (T, θ) ∈ Γ(∧kTM ⊕ T ∗M). Then we have
D E D(X,α)(T, θ), (Y, β) = X ⟨(T, θ), (Y, β)⟩ − ⟨ (X, α), (Y, β) , (T, θ)⟩ J K = ⟨£X θ, Y ⟩ + ⟨£X T, β⟩ + ⟨iY d α, T ⟩ D k E = (£X T, £X θ + (−1) d α(T, ·)),Y + β
k which shows D(X,α)(T, θ) = (£X T, £X θ + (−1) d α(T, ·)).
Example 10.5. [Bar12] extensively discusses a generalisation of example 10.4, so-called closed-form Leibniz algebroids. All commonly studied examples of Dorfman brackets belong to this class of Leibniz algebroids. In addition to the terms in (10.3), closed form algebroids can for example contain terms that mix different degrees of differential forms:
(k−1)j (0; αk, 0, 0), (0; 0, βj, 0) = (−1) (0; 0, 0, d αk ∧ βj) (10.4) J K for the Dorfman bracket on TM ⊕ ∧kT ∗M ⊕ ∧jT ∗M ⊕ ∧k+j+1T ∗M.
117 Brackets on TM ⊕ E∗ and the standard VB-Courant algebroid
Terms of this type correspond to terms of the following form in D:
D E D(0;αk,0,0)(Tk,Tj,Tk+j+1; θ), (Y ; βk, βj, βj+k+1)
= − ⟨ (0; αk, 0, 0), (Y ; βk, βj, βj+k+1) , (Tk,Tj,Tk+j+1; θ)⟩ J K D (k−1)j+1 E = (0; iY d αk, 0, (−1) d αk ∧ βj), (Tk, 0,Tk+j+1; 0) D (k−1)j+1 k E = (0, (−1) Tk+j+1¬ d αk, 0; (−1) iTk d α), (Y ; βk, βj, βk+j+1) and therefore
(k−1)j+1 D(0;αk,0,0)(0, 0,Tk+j+1; 0) = (−1) (0,Tk+j+1¬ d αk, 0; 0), (10.5) where ¬ denotes contraction over the first (in this case) (k + 1) indices.
Example 10.6. A more complex example of closed-form algebroid underlies the so-called
E7-exceptional generalised geometry (see [Bar12, Hul07]). The vector bundle
TM ⊕ ∧2T ∗M ⊕ ∧5T ∗M ⊕ (∧7T ∗M ⊗ T ∗M) (10.6)
∗ carries a natural E7 × R -structure and the Dorfman bracket (see [Bar12])
(X; α2, α5, u), (Y ; β2, β5, v) J K = ([X,Y ]; £X β2 − iY d α2, £X β5 − iY d α5 + d α2 ∧ β2, £X v − d α2 ⋄ β5 + d α5 ⋄ β2), where (d α ⋄ β)(X) = (iX d α) ∧ β for all X ∈ X(M). The dual map D is then given as follows:
D(X;α2,α5,u)(T2,T5,T7 ⊗ Z; θ) is
(£X T2 − T5¬ d α2 + T7¬iZ d α5, £X T5 − T7¬iZ d α2, 0; £X θ + d α2(T2, ·) − d α5(T5, ·))
Remark 10.7. Note that all examples for Dorfman brackets in this text are local, i.e. their brackets are given in terms of differential operators in both components. There are non-local Leibniz algebroids, for an example see Appendix B.
10.2 The E∗-valued Courant algebroid structure on the fat bundle Ec
l ∗ As previously established for general double vector bundles, the space ΓE(TE ⊕ T E) is ∞ ∞ l ∗ a C (M)-module: choose f ∈ C (M) and χ ∈ ΓE(TE ⊕ T E) a linear section over ∗ ∗ ∗ l ∗ ν ∈ Γ(TM ⊕ E ). Then qEf · χ is linear over fν ∈ Γ(TM ⊕ E ). The space ΓE(TE ⊕ T E) is a locally free and finitely generated C∞(M)-module (this follows from the existence of l ∗ local splittings). Hence, there is a vector bundle Eb over M such that ΓE(TE ⊕ T E) is
118 10.2 The E∗-valued Courant algebroid structure on the fat bundle Eb
∞ l ∗ isomorphic to Γ(Eb) as C (M)-modules, the fat vector bundle defined by ΓE(TE ⊕ T E). We prove below that it is isomorphic to Der(E∗) ⊕ J 1(E∗), where Der(E∗) is the bundle of derivations on E∗, and J 1(E∗) the first jet bundle.
First recall that (3.4) defines a bijection between the linear vector fields Xl(E) and Γ(Der(E∗)). It is easy to see from (3.4) that this bijection is a morphism of C∞(M)-modules. l l ∗ Hence, the fat bundle defined by X (E) = ΓE(TE) is the vector bundle Der(E ). l ∗ Next note that ΓE(T E) fits in the short exact sequence
∗ e· l ∗ rE ∗ 0 −→ Γ(Hom(E,T M)) −→ ΓE(T E) −→ Γ(E ) −→ 0,
of C∞(M)-modules, where the second map sends φ ∈ Γ(Hom(E,T ∗M)) to the core-linear l ∗ ∗ l ∗ section φe ∈ ΓE(T E), φe(e) = (TeqE) φ(e) for all e ∈ E, and the third map sends θ ∈ ΓE(T E) ∗ 1 ∗ l ∗ 1 to its base section rEθ in Γ(E ). We define Ψ: Γ(J E ) → ΓET E) by Ψ(j ϵ) = d ℓϵ for ϵ ∈ ∗ ∗ l ∗ ∗ ∞ Γ(E ) and Ψ(ιφ) = φf ∈ ΓE(T E) for φ ∈ Γ(Hom(TM,E )). The map Ψ is C (M)-linear and we get the following commutative diagram of morphisms of C∞(M)-modules
ι prE∗ 0 / Γ(Hom(TM,E∗)) / Γ(J 1E∗) / Γ(E∗) / 0
(·)∗ Ψ Id 0 / Γ(Hom(E,T ∗M)) / Γl (T ∗E) / Γ(E∗) / 0 E rE e·
with short exact sequences in the top and bottom rows. Since the left and right vertical arrows are isomorphisms, Ψ is an isomorphism by the five lemma. Since Ψ is an isomorphism V V of C∞(M)-modules, we obtain a vector bundle isomorphism ψ : J 1E∗ → T ∗E, where T ∗E is l ∗ the fat bundle defined by ΓE(T E). Finally we obtain a vector bundle isomorphism
∗ 1 ∗ 1 ∗ Θ: Der(E ) ⊕ J (E ) → E,b (Dm, (j ϵ)m) 7→ evm Dc , d ℓϵ . (10.7)
l ∗ ∗ Recall that for a linear section χ ∈ ΓE(TE ⊕ T E), there exists a section ν ∈ Γ(TM ⊕ E ) such that πTM⊕E∗ ◦ χ = ν ◦ qE. The map χ 7→ ν induces a short exact sequence of vector bundles 0 −→ E∗ ⊗ (E ⊕ T ∗M) ,→ Eb −→ TM ⊕ E∗ −→ 0.
Note that the restriction of the pairing on TE ⊕ T ∗E to linear sections of TE ⊕ T ∗E defines a nondegenerate pairing on Eb with values in E∗. Since the Courant bracket of linear sections is again linear, the vector bundle Eb inherits a Courant algebroid structure with pairing in E∗ (see [Jot17]). In particular, the Courant algebroid structure on TE ⊕ T ∗E defines a Leibniz bracket on sections of Der(E∗) ⊕ J 1(E∗) and a pairing with values in E∗ on
∗ 1 ∗ ∗ 1 ∗ Der(E ) ⊕ J (E ) ×M Der(E ) ⊕ J (E ) .
119 Brackets on TM ⊕ E∗ and the standard VB-Courant algebroid
This is called an Omni-Lie algebroid in [CL10], see also [CLS11]. The symmetric bilinear ∗ 1 nondegenerate pairing with values in E on Eb is given by ⟨Θ(D(m)), Θ((j ϵ)m + ιφm)⟩ = ∗ ∗ ⟨Dc , d ℓϵ + φe⟩(m) = D(ϵ)(m) + φ (X)(m) for D a derivation with symbol X ∈ X(M). Here, the second term is the evaluation at m ∈ M of the linear function ℓDϵ+φ∗X , when identified with Dϵ + φ∗X ∈ Γ(E∗). Hence, the corresponding symmetric bilinear nondegenerate pairing ∗ 1 ∗ ∗ 1 with values in E on J (E ) ⊕ Der(E ) is given by Dm, (j ϵ)m + ιφm = Dm(ϵ) + φ(X)(m) ∗ ∗ ∗ for ϵ ∈ Γ(E ), φ ∈ Γ(Hom(TM,E )) and Dm ∈ Dm(E ) with symbol X ∈ X(M).
10.3 Linear sections of TE ⊕ T ∗E → E
In this section we build on the techniques summarised in Section 3.2 and we prove original results on linear sections of TE ⊕ T ∗E → E. Those results will be the basis of our main theorem in Section 11. l ∗ ∗ We consider a linear section χ ∈ ΓE(TE ⊕ T E) over a pair (X, ε) ∈ Γ(TM ⊕ E ). Given a section e ∈ Γ(E), the difference χ(e(m)) − (TmeX(m), de(m) ℓε) projects to e(m) in E and ∗ ∗ to 0m ∈ TM ⊕ E and we can define a section Dχ(e, 0): M → E ⊕ T M by
↑ χ(e(m)) − (TmeX(m), de(m) ℓε) = −Dχ(e, 0) (e(m)) for all m ∈ M. By construction and the scalar multiplication in the fibers of TE ⊕ T ∗E → ∗ TM ⊕ E , we get Dχ(re, 0) = rDχ(e, 0) for a real number r ∈ R, and Dχ(e1 + e2, 0) = ∞ Dχ(e1, 0) + Dχ(e2, 0) for e1, e2 ∈ Γ(E). For a smooth function f ∈ C (M), we have χ((fe)(m)) = χ(f(m)e(m)) and
↑ Tm(fe)X(m), df(m)e(m) ℓε = Tm(f(m)e)X(m) + (X(f)e) (f(m)e(m)), df(m)e(m) ℓε .
Hence, we find that
Dχ(fe, 0) = fDχ(e, 0) + (X(f)e, 0). (10.8)
∗ ∗ Now we set Dχ : Γ(E ⊕ T M) → Γ(E ⊕ T M), Dχ(e, θ) = Dχ(e, 0) + (0, £X θ). (10.8) and
Theorem 10.10 below shows that Dχ is a smooth derivation. We have found the following result:
Theorem 10.8. Let χ be a linear section of TE ⊕T ∗E → E over a pair (X, ε) ∈ Γ(TM ⊕E∗). ∗ ∗ Then there exists a unique derivation Dχ : Γ(E⊕T M) → Γ(E⊕T M) with symbol X ∈ X(M) and which satisfies
(i) Dχ(e, θ) = Dχ(e, 0) + (0, £X θ) and
↑ (ii) χ(e(m)) = (TmeX(m), de(m) ℓε) − Dχ(e, 0) (e(m)), for all e ∈ Γ(E) and θ ∈ Ω1(M).
120 10.3 Linear sections of TE ⊕ T ∗E → E
Conversely, given a pair (X, ε) ∈ Γ(TM ⊕E∗) and a smooth derivation D : Γ(E ⊕T ∗M) → ∗ Γ(E ⊕ T M) over X ∈ X(M), we write χε,D for the linear section defined by
↑ χε,D(e(m)) = (TmeX(m), de(m) ℓε) − D(e, 0) (e(m))
l ∗ for all e ∈ Γ(E). Note that (1) in the last theorem shows that for each χ ∈ ΓE(TE ⊕ T E) ∗ ∗ there exist a derivation dχ ∈ Γ(Der(E)) and a tensor φχ ∈ Γ(E ⊗ T M) with Dχ(e, 0) =
(dχe, φχ(e)). More precisely, dχ = prE ◦Dχ ◦ ιE : Γ(E) → Γ(E) is a derivation of E with ∗ symbol X and the vector bundle morphism is φχ = prT ∗M ◦Dχ ◦ ιE : E → T M. The linear section χ can then be written χ = dcχ, d ℓε − φfχ
Remark 10.9. With the results in Section 10.2, we can phrase this correspondence in terms ∼ ∗ 1 ∗ of the bundle isomorphism Eb = Der(E ) ⊕ J (E ): χ = (dcχ, d ℓε − φfχ) ∈ Γ(Eb) corresponds 1 ∗ 1 ∗ to (dχ, j ε − ι(φχ)) in Γ(Der(E ) ⊕ J (E )).
We can use these results on linear sections to prove the following:
Theorem 10.10. Let χ be a linear section of TE ⊕ T ∗E → E over (X, ε) ∈ Γ(TM ⊕ E∗). The Courant-Dorfman bracket on sections of TE ⊕ T ∗E → E satisfies
↑ ↑ rχ, τ z = Dχτ
and the pairing ↑ ∗ ⟨χ, τ ⟩ = qE⟨(X, ε), τ⟩.
∗ for all τ ∈ Γ(E ⊕ T M). The anchor satisfies prTE(χ) = dcχ.
We prove the first identity in Appendix A.1. The second and third identities follow immediately from (10.3). We now state our first main theorem.
l ∗ Theorem 10.11. Choose two linear sections χ1, χ2 ∈ ΓE(TE⊕T E), over pairs (X1, ε1), (X2, ε2) ∈ Γ(TM ⊕ E∗). Then we have
V ∼ χ , χ = [d , d ], d ℓ ∗ − pr ∗ ◦[D ,D ] ◦ ι 1 2 χ1 χ2 prE∗ Dχ (X2,ε2) T M χ1 χ2 E J K 1 (10.9)
= χpr ∗ D∗ (X ,ε ),[D ,D ] E χ1 2 2 χ1 χ2
and ⟨χ1, χ2⟩ = ℓpr ∗ (D∗ (X ,ε )+D∗ (X ,ε )). E χ1 2 2 χ2 1 1 The Theorem is again proved in Appendix A.1 and gives us an expression for the induced E∗-valued Courant bracket on Der(E∗) ⊕ J 1(E∗):
121 Brackets on TM ⊕ E∗ and the standard VB-Courant algebroid
∗ Corollary 10.12. Choose d1, d2 ∈ Γ(Der(E )) with symbols X1,X2 ∈ X(M) and choose 1 ∗ ∗ ∗ ∗ µ1, µ2 ∈ Γ(J (E )) corresponding as in (3.7) to (ε1, φ1), (ε2, φ2) ∈ Γ(E ) ⊕ Γ(T M ⊗ E ). Then 1 (d1, µ1), (d2, µ2) = [d1, d2], £d1 µ2 − £d2 µ1 + j ⟨d2, µ1⟩ , (10.10) J K where the Der(E∗)-Lie derivative on J 1(E∗) is defined in equation (19) of [CL10]:
∗ ∗ ∗ ∗ £dµ = £d(ε, φ) = (dε, (£X ◦ φ − φ ◦ d ) ) where d is a derivation of E∗ with symbol X ∈ X(M) and µ = (ε, φ) ∈ Γ(J 1E∗) ≃ Γ(E∗ ⊕ Hom(TM,E∗)). Thus, our theorem proves that the E∗-valued Courant algebroid structure on Der(E∗) ⊕ J 1(E∗) given in [CL10] is precisely induced from TE ⊕ T ∗E via the isomorphism Ψ from 10.2.
Proof. With the correspondence between Γ(Eb) and Γ(Der(E∗) ⊕ Γ(J 1(E∗)) = Γ(Der(E∗) ⊕ ∗ ∗ ∗ E ⊕ Hom(TM,E )), (di, (εi, φi)) corresponds to χi = χεi,Di with Di(e, 0) = (di(e), −φi (e)). ∗ ∗ Then we have prE Dχ1 (X2, ε2) = d1(ε2) + φ1(X2) as well as
∗ ∗ ∗ ∗ ∗ ∗ prT ∗M ([Dχ1 ,Dχ2 ](e, 0)) = −φ1(d2e) + φ2(d1e) − £X1 (φ2(e)) + £X2 (φ1(e)).
By the considerations in Section 10.2, we have further ⟨d2, µ1⟩ = d2ε1 + φ1(X2). We 1 ∗ ∗ ∗ l ∗ get using the isomorphisms Γ(J E ) ≃ Γ(E ⊕ Hom(TM,E )) and ΓE(TE ⊕ T E) ≃ Γ(J 1E∗ ⊕ Der(E∗)):
r z (d1, µ1), (d2, µ2) = (d1, ε1, φ1), (d2, ε2, φ2) = dc1, d ℓε1 + φf1 , dc2, d ℓε2 + φf2 J K J K
= χϵ1,D1 , χϵ2,D2 = χ(d1(ε2)+φ1(X2)),[D1,D2] J K = ([d1, d2], d1(ε2) + φ1(X2), £d1 φ2 − £d2 φ1)
= ([d1, d2], 0, 0) + (0, d2ε1 + φ1(X2), 0) + (0, £d1 (ε2, φ2) − £d2 (ε1, φ1)) 1 = ([d1, d2], j ⟨d2, µ1⟩ + £d1 µ2 − £d2 µ1).
∗ Note that the derivation Dχ defines as follows a derivation of Hom(E,E ⊕ T M):
(Dχϕ)(e) = Dχ(ϕ(e)) − ϕ(dχ(e)) for all e ∈ Γ(E).
Corollary 10.13. In the situation of the preceding theorem, the Courant-Dorfman bracket ∗ satisfies χ, ϕe = D]χϕ for ϕ ∈ Γ(Hom(E,E ⊕ T M)). J K ∗ ∗ ∗ ∗ Proof. The section ϕ ∈ Γ(E ⊗(E⊕T M)) can be written as ϕ = (φ1, φ2), with φ1 ∈ Γ(E ⊗E) ∗ ∗ 1 ∗ ∗ and φ2 ∈ Γ(Hom(TM,E )). Furthermore, φ defines a section of Der(E ) ⊕ J (E ): φ1 is ∗ 1 ∗ a derivation of E with symbol 0 ∈ X(M) and φ2 ≃ ιφ2 is a section of J E . Therefore φe is simply the corresponding core-linear section under the correspondence outlined above.
122 10.3 Linear sections of TE ⊕ T ∗E → E
Choose χ = (d, µ) with d a derivation of E∗ over X ∈ X(M) and µ = j1ε + ιψ ∈ Γ(J 1E∗). Then the results above yield
∗ ∗ ∗ ∗ ∗ ∗ ∗ (d, µ), (φ1, φ2) = (d, ε, ψ), (φ1, 0, φ2) = ([d, φ1], (£X ◦ φ2 − φ2 ◦ d ) + φ1 ◦ ψ), J K J K
which is easily seen to be Dχϕ.
10.3.1 Linear closed 3-forms
ℓ ∗ Having already looked at linear one-forms as part of ΓE(TE ⊕ T E), we now also introduce higher-degree linear forms. Let E be as usual a vector bundle over M.A k-form H on E is linear if the induced vector bundle morphism H♯ : ⊕k−1 TE → T ∗E over the identity on E is also a vector bundle morphism over a map h: ⊕k−1 TM → E∗ on the other side of the double vector bundles [BC12]. According to Proposition 1 in [BC12], a linear k-form H ∈ Ωk(E) can be written
H = d Λµ + Λω
with µ ∈ Ωk−1(M,E∗) and ω ∈ Ωk(M,E∗). Here, given ω ∈ Ωk(M,E∗), the k-form k Λω ∈ Ω (E) is given by ∗ Λω(em) = (Tem qE) (⟨ω, e⟩(m)),
where ⟨ω, e⟩ ∈ Ωk(M) is the obtained k-form on M. Note that in the equation for H, we have µ = (−1)k−1h.
∗ Example 10.14. For instance, we have seen in §3.2 that for ε ∈ Γ(E ), the 1-form d ℓε ∈ Ω1(E) is linear. Since it projects to ε ∈ Γ(E∗), we know that any linear 1-form on E can ∗ 0 ∗ ∗ 1 ∗ be written d ℓε + φe for ε ∈ Γ(E ) = Ω (M,E ) and φ ∈ Γ(Hom(E,T M)) = Ω (M,E ). An 0 ∞ 1 easy computation shows Λε = ℓε ∈ Ω (E) = C (E) and Λφ = φe ∈ Ω (E).
k−1 ∗ Proposition 10.15. Consider a linear k-form H = d Λµ + Λω, with µ ∈ Ω (M,E ) and ω ∈ Ωk(M,E∗). Then H is closed, d H = 0, if and only if ω = 0.
Proof. H is closed if and only if Λω is closed. It is enough to evaluate d Λω on linear and l core vector fields on E. Take k linear vector fields Dci ∈ X (E) over Xi ∈ X(M), i = 1, . . . , k,
123 Brackets on TM ⊕ E∗ and the standard VB-Courant algebroid and one vertical vector field e↑ ∈ Xc(E). Then
k ↑ X i+1 ↑ (d Λω) Dc1,..., Dck, e = (−1) Dci Λω Dc1,...,ˆi . . . , Dck, e i=1 k ↑ + (−1) e Λω Dc1,..., Dck X i+j h i ↑ + (−1) Λω Dci, Dcj , Dc1,...,ˆi, . . . , ˆj, . . . , D[k−1, e 1≤i≤j≤k k X i+k h ↑i + (−1) Λω Dci, e , Dc1,...,ˆi, . . . , Dck . i=1
h ↑i Since Dci, e is again a vertical vector field and Λω vanishes on vertical vector fields, the first, third and fourth terms of this sum all vanish. The remaining term is
k ↑ k ∗ (−1) e Λω Dc1,..., Dck = (−1) qE⟨ω(X1,...,Xk), e⟩.
This is 0 for all X1,...,Xk ∈ X(M) and e ∈ Γ(E) if and only if ω = 0.
2 ∗ In what follows, we will consider closed linear 3-forms H = d Λµ with µ ∈ Ω (M,E ) the base map of H♯. Let us compute the inner product of such a 3-form with two linear vector fields on E. Recall that any linear vector field can be written Db ∈ Xl(E) with a derivation D : Γ(E) → Γ(E) over X ∈ X(M). The derivation D induces a derivation D : Ω1(M,E∗) → Ω1(M,E∗) by (Dω)(Y ) = D∗(ω(Y )) − ω[X,Y ] for all ω ∈ Ω1(M,E∗) and Y ∈ X(M). In particular, given a Dorfman bracket on sections of ∗ ∗ TM ⊕ E , the linear vector field prTE Ξ(ν) equals δcν, where ν is a section of TM ⊕ E and 1 ∗ δν is the derivation over prTM ν. We write δν for the induced derivation of Ω (M,E ).
2 ∗ l Proposition 10.16. Choose µ ∈ Ω (M,E ). Let Dc1, Dc2, Db ∈ X (E) be linear vector fields over X1,X2,X ∈ X(M) and let e be a section of E. Then
i i d Λµ = d ℓi i µ + D1^(iX2 µ) − D2^(iX1 µ) − i[X^,X ]µ (10.11) Dc2 Dc1 X2 X1 1 2 and ∗ i ↑ i d Λµ = −q ⟨iX µ, e⟩. (10.12) e Db E
124 10.3 Linear sections of TE ⊕ T ∗E → E
Proof. We have for e ∈ Γ(E):
↑ ↑ ↑ ie↑ i i d Λµ = Dc1 Λµ Dc2, e − Dc2 Λµ Dc1, e + e Λµ Dc1, Dc2 Dc2 Dc1 h i ↑ h ↑i h ↑i − Λµ Dc1, Dc2 , e + Λµ Dc1, e , Dc2 − Λµ Dc2, e , Dc1 ↑ ∗ = 0 − 0 + e (ℓµ(X1,X2)) − 0 + 0 − 0 = qE⟨µ(X1,X2), e⟩.
∗ ∗ This shows that i i d Λµ = d ℓi i µ + φe for a section φ ∈ Γ(E ⊗ T M). We have then Dc2 Dc1 X2 X1
ℓ⟨φ,X ⟩ = ⟨φ,e Dc3⟩ = i i i d Λµ − Dc3(ℓi i µ) 3 Dc3 Dc2 Dc1 X2 X1 = Dc1 Λµ Dc2, Dc3 − Dc2 Λµ Dc1, Dc3 + Dc3 Λµ Dc1, Dc2 h i h i h i − Λµ Dc1, Dc2 , Dc3 + Λµ Dc1, Dc3 , Dc2 − Λµ Dc2, Dc3 , Dc1 − Dc3(ℓ ) iX2 iX1 µ \ = Dc1(ℓµ(X2,X3)) − Dc2(ℓµ(X1,X3)) − Λµ [D1,D2], Dc3 + Λµ [D\1,D3], Dc2 − Λµ [D\2,D3], Dc1
= ℓ ∗ − ℓ ∗ − ℓ + ℓ − ℓ D1 (µ(X2,X3)) D2 (µ(X1,X3)) µ([X1,X2],X3) µ([X1,X3],X2) µ([X2,X3],X1)
= ℓ⟨D (i µ)−D (i µ)−i µ,X ⟩ 1 X2 2 X1 [X1,X2] 3 and we find (10.11). In order to prove (10.12), we use the equation
∗ ie↑ i i d Λµ = qE⟨µ(X1,X2), e⟩ Dc2 Dc1 above to find that ∗ i ie↑ i d Λµ = −i (qE⟨µ(X1), e⟩) Dc2 Dc1 Dc2 l ∗ ′ for all linear Dc2 ∈ X (E). Since ie′↑ ie↑ i d Λµ = 0 = −ie′↑ (qE⟨µ(X1), e⟩) for all e ∈ Γ(E), Dc1 ∗ we find that ie↑ i d Λµ = −qE⟨µ(X1), e⟩. Dc1 We will use the following lemma.
Lemma 10.17. Choose an element β ∈ Ω1(M,E∗) and a linear vector field Db ∈ Xl(E) over ∗ X ∈ X(M). Then i d Λβ is a linear 1-form over β(X) ∈ Γ(E ). More precisely, Db
i d Λβ = − d ℓβ + Dβ.g Db
Proof. We have
↑ ↑ ↑ ↑ ∗ i ↑ i d Λβ = Db⟨Λβ, e ⟩ − e ⟨Λβ, Db⟩ − ⟨Λβ, (De) ⟩ = −e (ℓ ) = −q ⟨e, β(X)⟩. e Db β(X) E
125 Brackets on TM ⊕ E∗ and the standard VB-Courant algebroid
∗ Therefore i d Λβ = − d ℓ + φe with a section φ ∈ Γ(Hom(E,T M)) to be determined. Xe β(X) We have
D ′E ′ φ,e Dc = ⟨i d Λβ + d ℓ , Dc⟩ Db β(X) ′ ′ = Db(ℓβ(Y )) − Dc(ℓβ(X)) − ℓβ[X,Y ] + Dc(ℓβ(X)) D ′E = ℓD∗(β(Y ))−β[X,Y ] = ℓ(Dβ)(Y ) = Dβ,g Dc for any linear vector field Dc′ ∈ Xl(E) over Y ∈ X(M). This shows that φ = Dβ.
126 Chapter 11
Dorfman brackets and natural lifts
Using our results on linear sections from Section 10.3, we now prove our main result of Part II: ∗ ∗ ℓ ∗ Every Dorfman bracket on TM ⊕ E corresponds to a lift Ξ : Γ(TM ⊕ E ) → ΓE(TE ⊕ T E), where the Dorfman condition is equivalent to the lift being natural, seding the Dorfman bracket in question to the Courant-Dorfman bracket on TE ⊕ T ∗E. In this way, every Dorfman bracket on TM ⊕ E∗ can be viewed as the restriction of the Courant-Dorfman bracket to a subspace of Γℓ(TE ⊕ T ∗E) – however, in general this subspace will not be one of sections of a subbundle! A number of similar results are already known for Dull and Lie brackets. In Section 11.2 we discuss how this new result and previously known results could fit into a general correspondence of brackets on TM ⊕ E∗ and litfts to the Omni-Lie algebroid Eb.
11.1 The equivalence of Dorfman brackets and natural lifts
Consider now an R-linear lift
∗ l ∗ Ξ: Γ(TM ⊕ E ) → ΓE(TE ⊕ T E), sending each section (X, ε) of TM ⊕ E∗ to a linear section over (X, ε). Then the lift defines an R-linear map
∗ ∗ D : Γ(TM ⊕ E ) → Der(E ⊕ T M), Ξ(X, ε) = χε,D(X,ε) .
Consider the dual
· , · : Γ(TM ⊕ E∗) × Γ(TM ⊕ E∗) → Γ(TM ⊕ E∗) J K
127 Dorfman brackets and natural lifts of D, written in bracket form and defined by
X⟨ν, τ⟩ = ⟨ν, D(X,ε)τ⟩ + ⟨ (X, ε), ν , τ⟩ J K for all (X, ε), ν ∈ Γ(TM ⊕ E∗) and τ ∈ Γ(E ⊕ T ∗M). Any bracket defined in this manner is
R-bilinear, anchored by prTM and satisfies a Leibniz identity in its second component. We easily get the following result.
Proposition 11.1. Lifts
∗ ℓ ∗ Ξ: Γ(TM ⊕ E ) → ΓE(TE ⊕ T E), sending each section (X, ε) of TM ⊕ E∗ to a linear section over (X, ε) are equivalent to ∗ ∗ ∗ R-bilinear brackets · , · : Γ(TM ⊕ E ) × Γ(TM ⊕ E ) → Γ(TM ⊕ E ), that are anchored J K by prTM and satisfy a Leibniz identity in the second component.
Define further the map δ : Γ(TM ⊕ E∗) → Der(E) by
δν = prE ◦Dν ◦ ιE.
∗ l ∗ As we have seen before, the lift Ξ: Γ(TM ⊕ E ) → ΓE(TE ⊕ T E) can be written
↑ Ξ(X, ε)(em) = (TmeX(m), dem ℓε) − (D(X,ε)(e, 0)) (em), (11.1) for any e ∈ Γ(E) with e(m) = em, or ∼ Ξ(X, ε) = (δ[(X,ε), d ℓε − prT ∗M D(X,ε) ◦ ιE).
In terms of sections of the Omni-Lie algebroid Der(E∗) ⊕ J 1(E∗), this says that anchored ∗ R-bilinear brackets on TM ⊕ E with Leibniz rule in the second component are in one-to-one correspondence with splittings
Ξ: Γ(TM ⊕ E∗) → Γ(Der(E∗) ⊕ J 1(E∗)), of the short exact sequence
0 → Γ(E∗ ⊗ (E ⊕ T ∗M)) → Γ(Der(E∗) ⊕ J 1(E∗)) → Γ(TM ⊕ E∗) → 0
Note that in either description the map Ξ is a map of sections only, so its image will in general not span a sub-vector bundle of Eb =∼ Der(E∗) ⊕ J 1(E∗). We prove the following theorem, which shows that a chosen lift as above is natural, if and only if the bracket · , · is a Dorfman bracket. J K
128 11.1 The equivalence of Dorfman brackets and natural lifts
Theorem 11.2. Let E be a smooth vector bundle over a manifold M. Consider an R-bilinear ∗ bracket · , · on sections of TM ⊕ E , that is anchored by prTM and satisfies the Leibniz J K identity in its second component. Then · , · is a Dorfman bracket if and only if the J K corresponding lift as in Proposition 11.1 or (11.1) is natural, i.e. if and only if
Ξ(ν1), Ξ(ν2) = Ξ ν1, ν2 J K J K ∗ for all ν1, ν2 ∈ Γ(TM ⊕ E ), where the bracket on the left-hand side is the Courant-Dorfman bracket.
The proof of this theorem follows from the general results in 10.3 and is given in Appendix A.2. Note that the proof of this theorem can also be adapted in a straightforward manner from the proof of the main theorem in [JL18] (see the following remark); the only difference being that D is not C∞(M)-linear in its lower entry. The proof in [JL18] is however independent of this property.
∗ l ∗ Remark 11.3. Note that horizontal lifts σ : Γ(TM ⊕ E ) → ΓE(TE ⊕ T E) satisfying ∗ σ(ν1 +ν2) = σ(ν1)+σ(ν2) and σ(f ·ν) = qEfσ(ν) are called linear. The horizontal lifts above are in general not linear; they are additive, but in general they are not C∞(M)-homogeneous. ∗ l ∗ Linear horizontal lifts σ : Γ(TM ⊕ E ) → ΓE(TE ⊕ T E) were proved in [JL18] to be equivalent to dull brackets on sections of TM ⊕ E∗, or equivalently to Dorfman connections Γ(TM ⊕ E∗) × Γ(E ⊕ T ∗M) → Γ(E ⊕ T ∗M). Let ∆: Γ(TM ⊕E∗)×Γ(E ⊕T ∗M) → Γ(E ⊕T ∗M) be a Dorfman connection and consider
the dual dull bracket · , · ∆. Note that the map J K ∗ ∇: Γ(TM ⊕ E ) × Γ(E) → Γ(E), ∇νe = prE(∆ν(e, 0))
∗ ∗ is a linear connection. Choose ν, ν1, ν2 ∈ Γ(TM ⊕ E ) and τ ∈ Γ(E ⊕ T M).[JL18] proves the following identities
D ∆ ∆ E 1. σ (ν1), σ (ν2) = ℓ ν1,ν2 ∆+ ν2,ν1 ∆ , J K J K D ∆ ↑E ∗ 2. σ (ν), τ = qE⟨ν, τ⟩,
∆ ↑ ↑ 3. prTE σ (ν) = ∇cν and prTE(τ ) = (prE τ) ,
∆ ↑ ↑ 4. qσ (ν), τ y = (∆ντ) ,
∆ ∆ ∆ 5. qσ (ν1), σ (ν2)y = σ ( ν1, ν2 ∆) − R∆(ν^1, ν2) ◦ ιE. J K Those results could now be easily deduced from Theorems 10.10 and 10.11, as we deduce our main result Theorem 11.2 from those.
129 Dorfman brackets and natural lifts
Here, we have the following result, a counterpart for Dorfman brackets of the results described in Remark 11.3. Note that since · , · is anchored by prTM , the sum ν1, ν2 + ν2, ν1 ∗ J ∗K J K J K is a section of E for all ν1, ν2 ∈ Γ(TM ⊕ E ).
∗ Theorem 11.4. Let · , · be a Dorfman bracket on sections of TM ⊕ E . For all ν, ν1, ν2 ∈ J K Γ(TM ⊕ E∗) and τ ∈ Γ(E ⊕ T ∗M), we have
D ↑E ∗ 1. ⟨Ξ(ν1), Ξ(ν2)⟩ = ℓ ν1,ν2 + ν2,ν1 and Ξ(ν), τ = qE⟨ν, τ⟩, J K J K ↑ ↑ 2. prTE (Ξ(ν)) = δcν and prTE(τ ) = (prE τ) ,
† † 3. qΞ(ν), τ y = Dντ . Proof. Those identities are all given by Theorems 10.10 and 10.11.
11.2 Links to known results on Omni-Lie algebroids, on Dorf- man connections and on the standard VB-Courant alge- broid
[CL10, CLS11] prove the following result on Lie algebroid structures on subbundles of TM ⊕E∗ versus Dirac structures inside the E∗-valued Courant-algebroid E := Der(E∗) ⊕ J 1(E∗). Note that such a Dirac structure is called reducible if its projection to TM ⊕ E∗ is surjective.
Theorem 11.5. (Theorem 3.7 in [CLS11]) There is a one-to-one correspondence between reducible Dirac structures L ⊂ E and projective Lie algebroids A ⊂ TM ⊕ E∗ such that A is the quotient Lie algebroid of L. (As a Dirac structure, L carries a Lie bracket induced by the Courant-Dorfman bracket.)
A projective Lie algebroid is a subbundle A ⊂ TM ⊕ E∗ with a Lie algebroid structure
(A, [·, ·]A, ρA), with anchor given by ρA = prTM |A.A reducible Dirac structure L ⊂ E is a Dirac structure the image of which in TM ⊕ E∗ under b: E → TM ⊕ E∗ is a regular subbundle. The correspondence in the theorem is such that A = b(L), and the Lie bracket is the quotient Lie bracket on A induced by the short exact sequence
0 → A0 → L →b A → 0
For details, see [CLS11]. This result, our results from Section 11 and the results from [JL18] as outlined in Remark 11.3, suggest the following relationships between subspaces of Γ(Eb) =∼ Γ(E) that are closed ∗ under ·, · and project to locally-free subsheaves of Γ(TM ⊕ E ), and R-bilinear brackets on J K subbundles of TM ⊕ E∗: Let V ⊂ Γ(Eb) be a subspace that is closed under ·, · and such that V maps to Γ(F ),F J K a subbundle of TM ⊕ E∗. Then, collectively, we have the following results:
130 11.3 Standard examples
Setting 1:
F = TM ⊕ E∗, V = ℑ(Ξ), where Ξ: Γ(TM ⊕ E∗) → Γ(Eb) is a splitting of p : Γ(Eb) → Γ(TM ⊕ E∗). This is just the setting of Proposition 11.1, i.e. such lifts precisely correspond to brackets on TM ⊕ E∗ that satisfy a Leibniz identity in the second component. Now, if V is additionally a sub-vector bundle of Eb and Ξ a morphism of vector bundles, we are in the setting of dull brackets and Dorfman connections, as studied in [JL18], i.e. the resulting bracket satisfies the Leibniz identity also in its first component. If instead (or additionally) the lift Ξ is natural, i.e. Ξ·, Ξ· = Ξ ·, · , the bracket on TM ⊕ E∗ J K J K satisfies the Dorfman condition (the Jacobi identity in Leibniz form). If V is such that ⟨ν, ν′⟩ = 0 for all ν, ν′ ∈ V, the bracket ·, · on TM ⊕ E∗ is antisymmetric J K (see Theorem 10.11).
Setting 2:
V = Γ(L),L ⊂ Eb a Dirac structure. This is the case studied by [CL10, CLS11] as described above. The parallels to the first setting are obvious: V is closed under ·, · , which is necessary ∗ J K to induce an R-bilinear bracket on its projection to TM ⊕ E at all, V is isotropic under ⟨·, ·⟩, so the resulting bracket is antisymmetric, and V is given by the sections of a vector bundle, i.e. the Leibniz rule in the first component is satisfied. However, in this case there is not necessarily a splitting Ξ: F → L.
Setting 3:
Of course the first two settings are not mutually exclusive: According to our results, Dirac structures L ⊂ Eb which project surjectively to TM ⊕ E∗ allow a lift Ξ: TM ⊕ E∗ → L, which is natural – a projective Lie bracket on TM ⊕ E∗ is in particular a Dorfman bracket.
11.3 Standard examples
We illustrate the result in Theorem 11.2 with the examples of standard Dorfman brackets on TM ⊕ E∗, by giving explicitly the lifts.
11.3.1 Lift of the Courant-Dorfman bracket
We consider here the case where E = TM and the Dorfman bracket on TM ⊕ T ∗M is the Courant-Dorfman bracket
(X1, θ1), (X2, θ2) = ([X1,X2], £X1 θ2 − iX2 d θ1) J K 1 ∗ for X1,X2 ∈ X(M) and θ1, θ2 ∈ Ω (M). First, recall that the derivation D : Γ(TM ⊕ T M) × ∗ ∗ Γ(TM ⊕ T M) → Γ(TM ⊕ T M) is just Dν1 ν2 = ν1, ν2 . Hence, by definition, the value J K 131 Dorfman brackets and natural lifts
Ξ(X, θ)(Y (m)) is
d ∗ TmYX(m) − (Y + t[X,Y ])(m), d ℓθ − (TY (m)pM ) (−iY d θ) . dtt=0
∗ l ∗ Using (3.5), we get Ξ: Γ(TM ⊕ T M) → ΓTM (TTM ⊕ T TM), Ξ(X, θ) = [[X, ·], d ℓθ − dfθ , (11.2)
∗ ∗ where dfθ is the one-form on TM defined by dfθ(v) = (TvpM ) (−iv d θ) ∈ Tv (TM) for all v ∈ TM. This choice of sign is for consistency with the notations in the next section for
the general case E, e.g. in the proof of (10.11). We have indeed ⟨dfθ, Db⟩ = ℓiX d θ for any derivation D of TM over X ∈ X(M), since evaluated at Y (m) ∈ TM, ⟨dfθ, Db⟩(Y (m)) is ∗ ⟨(TY (m)pM ) (−iY (m) d θ),TmY (X(m))⟩ = ⟨−iY (m) d θ, X(m)⟩ = ℓiX d θ(Y (m)). For the convenience of the reader, let us compute here explicitly the Courant-Dorfman
bracket Ξ(X1, θ1), Ξ(X2, θ2) of two images of Ξ. The Lie bracket of two linear vector fields V J l K Dc1, Dc2 ∈ X (E) is [Dc1, Dc2] = [D\1,D2] = D1 ◦ D2 − D2 ◦ D1. To see this, one only needs
to apply [Dc1, Dc2] on linear and pullback functions. Since [[X1, ·], [X2, ·]] is [[X1,X2], ·] by
the Jacobi identity, we find that the Lie bracket of the vector fields [\X1, ·] and [\X2, ·] is V
[[X1,X2], ·]. Let us compute £ (d ℓθ2 −dgθ2)−i d(d ℓθ1 −dgθ1). We have £ d ℓθ2 = [\X1,·] [\X2,·] [\X1,·] d [\X , ·](ℓ ) = d ℓ 1 θ2 £X1 θ2 and ∼ £ (dgθ2) = d(£X1 θ2). [\X1,·]
The second equation is more difficult to see and requires some explanations. Take Y ∈ X(M). Then
D E D E D VE £ d θ, [Y,d·] = [[X, ·] d θ, [Y,d·] − d θ, [[X,Y ], ·] [X,·] f f f [ = [X, ·]ℓiY d θ − ℓi[X,Y ] d θ = ℓ£X iY d θ−i[X,Y ] d θ D E = ℓiY d £X θ = d^£X θ, [Y,d·] .
D E D E £ d θ, Y ↑ d^£ θ, Y ↑ £ d θ = Since [X,·] f is easily seen to vanish, as X does, we find that [X,·] f
d^£X θ. Therefore we get
£ (d ℓθ2 − dgθ2) − i d(d ℓθ1 − dgθ1) [\X1,·] [\X2,·]
=£ (d ℓθ2 − dgθ2) + £ dgθ1 − d(i dgθ1) [\X1,·] [\X2,·] [\X2,·] = d ℓ − d(^£ θ ) + d(^£ θ ) − d⟨dgθ , [\X , ·]⟩ £X1 θ2 X1 2 X2 1 1 2 = d ℓ − d(^£ θ ) + d(^£ θ ) − d ℓ . £X1 θ2 X1 2 X2 1 iX2 d θ1
132 11.3 Standard examples
Since d(£X2 θ1) = d(iX2 d θ1), this shows
V ∼ Ξ(X1, θ1), Ξ(X2, θ2) = [[X1,X2], ·], d ℓ£X θ2−iX d θ1 − d(£X1 θ2 − iX2 d θ1) J K 1 2 = Ξ (X1, θ1), (X2, θ2) . J K Remark 11.6. There is a canonical isomorphism of double vector bundles
Σ: T (TM ⊕ T ∗M) → TTM ⊕ T ∗TM,
which maps the natural VB-Courant algebroid structure on T (TM ⊕ T ∗M), the tangent prolongation of the standard Courant algebroid on TM ⊕ T ∗M, to the standard VB-Courant algebroid structure on T (TM) ⊕ T ∗(TM). The lift Ξ is then precisely Ξ = Σ ◦ T , where T denotes the tangent prolongation of a section,
(s: M → TM ⊕ T ∗M) 7→ (T s : TM → T (TM ⊕ T ∗M)).
A precise description and proof can be found in [? ].
11.3.2 Another lift to TTM ⊕ T ∗TM
∗ l ∗ Consider this time the natural lift Ξ: Γ(TM ⊕ T M) → ΓTM (T (TM) ⊕ T (TM)), Ξ(X, θ) = ∗ [[X, ·], d ℓθ . This is equivalent to the Dorfman bracket · , · : Γ(TM ⊕ T M) × Γ(TM ⊕ J K T ∗M) → Γ(TM ⊕ T ∗M),
(X1, θ1), (X2, θ2) = ([X1,X2], £X1 θ2). J K
To see this, let us compute the Courant-Dorfman bracket of Ξ(X1, θ1) with Ξ(X2, θ2). We have
h i 2 Ξ(X1, θ1), Ξ(X2, θ2) = [\X1, ·], [\X2, ·] , £ d ℓθ2 − i d ℓθ1 . J K [\X1,·] [\X2,·] By the formulas found in the preceding example, we get
V Ξ(X1, θ1), Ξ(X2, θ2) = [[X1,X2], ·], d ℓ£X θ2 = Ξ([X1,X2], £X1 θ2). (11.3) J K 1 In fact, we call the lifts associated to Dorfman brackets “natural” because they generalise the properties of this lift.
133 Dorfman brackets and natural lifts
11.3.3 More general examples
More generally, according to (11.1) the lift corresponding to the Dorfman bracket in Example 10.4 has the same form:
↑ Ξ((X, α))(em) = (Tme)(X(m)) − (£X e) (em), d ℓα(em) − dgα = £dX ·, d ℓα − dgα (em) (11.4)
k for all em ∈ ∧ TM, where, in the second equality, we have used the definition of the derivation Db in (3.5). Here in order to be consistent with the next section, as well as the previous example, dgα is defined by:
∗ k ∗ dgα(em) = (Tem prTM ) ((−1) iem d α) = (Tem prTM ) (d α(·, em)).
∗ l ∗ In all examples so far, the lift Ξ: Γ(TM ⊕ E ) → ΓE(TE ⊕ T E) is really a direct sum of l ∗ l ∗ two lifts ΞTM : X(M) → ΓE(TE) and ΞE∗ : Γ(E ) → ΓE(T E). All the examples discussed so far are split Dorfman brackets. For these, we always have:
Proposition 11.7. For all split Dorfman brackets on TM ⊕ E∗, Ξ(X, 0) ∈ Xl(E).
Proof. We show that D(X,0)(e, 0) = (δ(X,0)e, 0):
D E D(X,0)(e, 0), (Y, 0) = X ⟨(Y, 0), (e, 0)⟩ − ⟨ (X, 0), (Y, 0) , (e, 0)⟩ J K = − ⟨([X,Y ], 0), (e, 0)⟩ = 0 for all Y ∈ X(M).
∗ l ∗ However, for general split Dorfman brackets ΞE∗ is a map Γ(E ) → Γe(TE ⊕ T E). For ↑ example the term in (10.5) gives rise to a term (e¬ d αk) (em) ∈ Γ(TE) in ΞE∗ (αk)(em). If the Dorfman bracket is not split, mixing can also occur in the TM-part of the lift: l ∗ ΞTM : TM → ΓE(TE ⊕ T E), as illustrated by the following example:
3 Example 11.8. Let H ∈ Ωcl(M) be a closed 3-form. Then (X1, θ1), (X2, θ2) H = (X1, θ1), (X2, θ2) + J K J K (0, iX2 iX1 H) (with ·, · the Courant-Dorfman bracket) is also a Dorfman bracket on TM ⊕ ∗ J K H T M. This Dorfman bracket is not split, and we have D(X,0)(Y, 0) = ([X,Y ], iY iX H) by Example 10.3, which shows
H ∗ Ξ (X, 0)(Y (m)) = (ΞTM (X), −pM (iY iX H))(Y (m)).
The following section studies in detail such twistings of Dorfman brackets in relation to their lifts.
134 11.4 Twisted Courant-Dorfman bracket over vector bundles
11.4 Twisted Courant-Dorfman bracket over vector bundles
Here we consider the standard Courant-Dorfman bracket on TE ⊕ T ∗E over a vector bundle E, twisted by a linear closed 3-form H ∈ Ω3(E). That is, we have
(X1, α1), (X2, α2) H = (X1, α1), (X2, α2) + (0, iX2 iX1 H). J K J K Given a form µ ∈ Ω2(M,E∗) and a Dorfman bracket · , · on sections of TM ⊕ E∗, we ∗ J K ∗ ∗ can define a twisted bracket · , · µ : Γ(TM ⊕ E ) × Γ(TM ⊕ E ) → Γ(TM ⊕ E ) by J K
(X1, ϵ1), (X2, ϵ2) µ = (X1, ϵ1), (X2, ϵ2) + (0, iX2 iX1 µ). J K J K
This satisfies a Leiniz equality in the second term (as always, with anchor prTM ) and is compatible with the anchor. We make the following definition.
Definition 11.9. Let · , · : Γ(TM ⊕ E∗) × Γ(TM ⊕ E∗) → Γ(TM ⊕ E∗) be a Dorfman 2 J ∗K bracket and µ ∈ Ω (M,E ) a form. Then we say that µ twists · , · if · , · µ satisfies the J K J K Jacobi identity in Leibniz form, i.e. if · , · µ is a new Dorfman bracket. J K In this section we will describe in terms of the lift associated to · , · a necessary and J K sufficient condition for µ to twist · , · . J K Example 11.10. The standard Dorfman bracket on TM ⊕ ∧kT ∗M (Example 10.4) is 2 k ∗ k+2 twisted by µ ∈ Ω (M, ∧ T M) if and only if µ ∈ Ωcl (M), i.e. actually antisymmetric in all components and closed.
µ ∗ ∗ ∗ We define the dual derivation D : Γ(TM ⊕ E ) × Γ(E ⊕ T M) → Γ(E ⊕ T M) to · , · µ J K and find µ D(X,ϵ)(e, θ) = D(X,ϵ)(e, θ) − (0, ⟨iX µ, e⟩). (11.5)
µ ∗ l ∗ The corresponding lift Ξ : Γ(TM ⊕ E ) → ΓE(TE ⊕ T E) as in (11.1) is then just
µ Ξ (X, ϵ) = Ξ(X, ϵ) + (0^, iX µ).
Recall that it is natural if and only if · , · µ satisfies the Jacobi identity. J K Theorem 11.11. With the notations above, we have
µ µ µ Ξ (ν1), Ξ (ν2) − d Λ = Ξ ν1, ν2 J K µ J K ∗ for all ν1, ν2 ∈ Γ(TM ⊕ E ).
135 Dorfman brackets and natural lifts
Proof. We just compute
Ξµ(ν ), Ξµ(ν ) = rΞ(ν ) + (0^, i µ), Ξ(ν ) + (0^, i µ)z 1 2 − d Λµ 1 X1 2 X2 J K − d Λµ (10.11) ^ ^ = Ξ(ν1), Ξ(ν2) − 0, d ℓiX iX µ + Dν1 (iX2 µ) − Dν2 (iX1 µ) − i[X^1,X2]µ J K 2 1 r z r z + Ξ(ν1), (0^, iX2 µ) + (0^, iX1 µ), Ξ(ν2) ^ ^ = Ξ ν1, ν2 − 0, d ℓiX iX µ + Dν1 (iX2 µ) − Dν2 (iX1 µ) − i[X^1,X2]µ J K 2 1 + 0, D ^(i µ) + 0, d ℓ − 0, D ^(i µ) ν1 X2 iX2 iX1 µ ν2 X1 µ = Ξ ν1, ν2 J K In the third equality, we have used Lemma 10.13. We are now ready to prove our main theorem.
Theorem 11.12. Consider a Dorfman bracket
· , · : Γ(TM ⊕ E∗) × Γ(TM ⊕ E∗) → Γ(TM ⊕ E∗) J K ∗ l ∗ and the corresponding lift Ξ: Γ(TM ⊕ T M) → ΓTM (TE ⊕ T E). Then a form µ ∈ Ω2(M,E∗) twists · , · if and only if J K
Ξ(ν1), Ξ(ν2) d Λ = Ξ ν1, ν2 µ J K µ J K ∗ for all ν1, ν2 ∈ Γ(TM ⊕ TM ).
In other words, µ twists a Dorfman bracket if and only its natural lift lifts the twisted bracket to the twist by d Λµ of the Courant-Dorfman bracket. Note that we also have the following result, which follows from (10.12) and (11.5).
Proposition 11.13. In the situation of the previous theorem, we have
↑ µ ↑ rΞ(ν), τ z = Dν τ , d Λµ for ν ∈ Γ(TM ⊕ E∗) and τ ∈ Γ(E ⊕ T ∗M), no matter if µ twists the Dorfman bracket or not.
Proof of Theorem 11.12. Assume that · , · µ is a Dorfman bracket. Then by Theorem 11.2, J K we have µ µ µ µ µ Ξ (ν1), Ξ (ν2) = Ξ ν1, ν2 µ = Ξ ν1, ν2 + Ξ (0, µ(X1,X2)). (11.6) J K J K J K µ Since Ξ (ν) = Ξ(ν) + (0^, iX µ), we find that
µ Ξ (0, µ(X1,X2)) = Ξ(0, µ(X1,X2)) (11.7)
136 11.5 Symmetries of Dorfman brackets
µ ∗ and also that prTE Ξ (ν) = prTE Ξ(ν) = δcν for all ν ∈ Γ(TM ⊕ E ). By Theorem 11.11, we
have Ξµ(ν ), Ξµ(ν ) = Ξµ ν , ν + 0, i i d Λ . 1 2 1 2 δ δ µ (11.8) J K J K cν2 cν1 (11.6), (11.7) and (11.8) yield together Ξ(0, µ(X1,X2)) = 0, i i d Λµ , and so δcν2 δcν1
Ξ(ν ), Ξ(ν ) = Ξ(ν ), Ξ(ν ) + 0, i i d Λ 1 2 d Λµ 1 2 δ δ µ J K J K cν2 cν1 = Ξ ν1, ν2 + Ξ(0, µ(X1,X2)) = Ξ ν1, ν2 µ. J K J K Example 11.14. Consider E = TM and choose the Courant-Dorfman bracket on TM ⊕T ∗M.
Recall from §11.3.1 the corresponding natural lift. Then if ν1 = (X1, θ1), we get δνX2 =
[X1,X2] and Dν1 (iX2 µ) = £X1 iX2 µ. As a consequence,
Dν1 (iX2 µ) − Dν2 (iX1 µ) − i[X ,X ]µ = iX2 £X1 µ − £X2 iX1 µ 1 2 (11.9) = iX2 iX1 d µ − d(iX2 iX1 µ)
and Dν1 (iX2 µ) − Dν2 (iX1 µ) − i[X1,X2]µ = − d(iX2 iX1 µ) if and only if µ is closed. We get then using (10.11)
Ξ(X , θ ), Ξ(X , θ ) 1 1 2 2 d Λµ J K = Ξ (X1, θ1), (X2, θ2) + 0, i i d Λµ [\X2,·] [\X1,·] J K
= Ξ (X1, θ1), (X2, θ2) + 0, d ℓiX iX µ − d(i^X2 iX1 µ) = Ξ (X1, θ1), (X2, θ2) µ. J K 2 1 J K 11.5 Symmetries of Dorfman brackets
In this section we use the known symmetries of the standard Courant algebroid over E to study a similar class of symmetries for Dorfman brackets on TM ⊕ E∗. 2 ∗ ∗ Consider B ∈ Ωcl(E). We denote by ΦB : TE ⊕ T E → TE ⊕ T E the vector bundle morphism over the identity on E that is defined by
ΦB(X, θ) = (X, θ + iX B)
1 for all X ∈ X(E) and θ ∈ Ω (E). Then ΦB is a symmetry of the Courant-Dorfman bracket on TE ⊕ T ∗E [BCG07]:
ΦB(χ1), ΦB(χ2) = ΦB χ1, χ2 J K J K ∗ for all χ1, χ2 ∈ Γ(TE ⊕ T E). According to [BC12] (see Section 11.4), given a form β ∈ Ω1(M,E∗), the closed form ∗ B = − d Λβ is linear. In particular, if · , · is a Dorfman bracket on TM ⊕E and Ξ: Γ(TM ⊕ J K
137 Dorfman brackets and natural lifts
∗ l ∗ E ) → ΓE(TE ⊕ T E) the associated lift, ΦB(Ξ(ν)) = Ξ(ν) + i B is a linear section of δbν ∗ ∗ ∗ TE ⊕ T E over Φβ(ν) = ν + (0, iX β) (see Lemma 10.17), where Φβ : TM ⊕ E → TM ⊕ E is the vector bundle morphism over the identity on M:
Φβ(X, ϵ) = (X, ϵ + iX β).
In this section we aim to understand when this map defines a symmetry of a Dorfman bracket on TM ⊕ E∗. We prove the following result.
Theorem 11.15. A form β ∈ Ω1(M,E∗) defines a symmetry of a Dorfman bracket ·, · via J K (X, ϵ) 7→ (X, ϵ + iX β) if and only if
Φ− d Λβ ◦ Ξ = Ξ ◦ Φβ
∗ l ∗ for the corresponding lift Ξ: Γ(TM ⊕ E ) → ΓE(TE ⊕ T E).
1 ∗ The proof relies on the following lemma. We set B := − d Λβ for β ∈ Ω (M,E ).
∗ Lemma 11.16. Choose φ ∈ Γ(Hom(E,E ⊕ T M)). Then d⟨ΦB(Ξ(ν)), φe⟩ is a core linear section of T ∗E → E for all ν ∈ Γ(TM ⊕ E∗) if and only if φ = 0. ˜ ˜ Proof. Since ⟨φ, ΦB(Ξ(ν))⟩ is linear, d⟨φ, ΦB(Ξ(ν))⟩ is a core linear section if and only if D ˜E ΦE d ΦB(Ξ(ν)), φ = 0. We have
D D ˜E E D D ˜E ↑E ΦE d ΦB(Ξ(ν)), φ , e = d ΦB(Ξ(ν)), φ , e ↑ D ˜E = e ΦB(Ξ(ν)), φ .
∗ Write ν = (X, ϵ) ∈ Γ(TM ⊕ E ). Since ΦB(Ξ(X, ϵ)) = Ξ(X, ϵ) + (0, d ℓβ(X) − δ^(X,ϵ)β), we find
D ˜E ΦB(Ξ(ν)), φ = ℓφ∗(X,ϵ+β(X))
↑ D ˜E ∗ ∗ and so e ΦB(Ξ(ν)), φ = qE⟨φ (X, ϵ + β(X)), e⟩. This vanishes for all e ∈ Γ(E) and all (X, ϵ) ∈ Γ(TM ⊕ E∗) if and only if φ∗(X, ϵ + β(X)) = 0 for all (X, ϵ) ∈ Γ(TM ⊕ E∗). In particular, φ∗(0, ϵ) must be 0 for all ϵ ∈ Γ(E∗) or, in other words, φ must have image in T ∗M. Using this, we find φ∗(X, 0) = φ∗(X, β(X)) for X ∈ X(M). Since this must vanish for all X ∈ X(M), we have shown that φ must be 0.
∗ Proof of Theorem 11.15. We define φ(X,ϵ) ∈ Γ(Hom(E,E ⊕ T M)) by
φ^(X,ϵ) = Ξ(0, iX β) − (0, i B) = Ξ(0, iX β) − 0, d ℓβ(X) − δ^(X,ϵ)β . (11.10) δ\(X,ϵ)
138 11.5 Symmetries of Dorfman brackets
We have used Lemma 10.17. Note that this difference is a core-linear section of TE ⊕ T ∗E because the linear sections Ξ(0, iX β) and 0, d ℓβ(X) − δ^(X,ϵ)β both project to (0, iX β) in Γ(TM ⊕ E∗). Consider
Φβ(X1, ϵ1), Φβ(X2, ϵ2) = (X1, ϵ1 + iX1 β), (X2, ϵ2 + iX2 β) J K J K ∗ in Γ(TM ⊕ E ). This lifts to Ξ (X1, ϵ1 + iX1 β), (X2, ϵ2 + iX2 β) , which equals J K
Ξ(X1, ϵ1) + Ξ(0, iX1 β), Ξ(X2, ϵ2) + Ξ(0, iX2 β) J K But this is
r z ΦB(Ξ(X1, ϵ1)) + φ^(X1,ϵ1), ΦB(Ξ(X2, ϵ2))) + φ^(X2,ϵ2) ,
which can be expanded to
r z ΦB(Ξ (X1, ϵ1), (X2, ϵ2) ) + ΦB(Ξ(X1, ϵ1)), φ^(X2,ϵ2) J K r z r z + φ^(X1,ϵ1), ΦB(Ξ(X2, ϵ2))) + φ^(X1,ϵ1), φ^(X2,ϵ2) (11.11)
The second and fourth terms are again core-linear (see Lemma 10.13 and Lemma 4.5 in [Jot17], respectively) so project to 0, but the third is
r z − ΦB(Ξ(X2, ϵ2)), φ^(X1,ϵ1) + 0, d⟨ΦB(Ξ(X2, ϵ2)), φ^(X1,ϵ1)⟩ .
The left-hand term is core-linear, so projects to 0. By Lemma 11.16, the right-hand term also ∗ has values in the core for arbitrary (X2, ϵ2) ∈ Γ(TM ⊕ E ) if and only if φ(X1,ϵ1) = 0. This ∗ happens exactly when (11.11) projects to (X1, ϵ1), (X2, ϵ2) + (0, i[X1,X2]β) on TM ⊕ T M, J K so when
(X1, ϵ1 + iX1 β), (X2, ϵ2 + iX2 β) = (X1, ϵ1), (X2, ϵ2) + (0, i[X1,X2]β). J K J K
Now φe(X,ϵ) = 0 is equivalent to (Ξ ◦ Φβ)(X, ϵ) = (ΦB ◦ Ξ)(X, ϵ) because
(Ξ ◦ Φβ)(X, ϵ) = Ξ(X, ϵ + β(X)) = Ξ(X, ϵ) + Ξ(0, iX β)
= Ξ(X, ϵ) + (0, i B) + φe(X,ϵ) = (ΦB ◦ Ξ)(X, ϵ) + φe(X,ϵ). δ\(X,ϵ)
Note that so far, we have not made any statement as to the existence of forms like in Theorem 11.15. The theorem rather provides a simple reformulation of the condition for being a symmetry.
139 Dorfman brackets and natural lifts
Example 11.17. Consider E = ∧kTM and the standard Dorfman bracket on TM ⊕∧kT ∗M k ∗ already studied earlier. Choose a morphism β : TM → ∧ T M and consider − d Λµ the associated linear 2-form on E = ∧kTM. For β to define a symmetry of the Dorfman bracket on TM ⊕ ∧kT ∗M, we need
Ξ(0, iX β)(em) = − d Λβ(δ\(X,αk))(em)
for all em ∈ E, which is equivalent to (0, d ℓiX β − d^iX β) = − d Λβ(£dX ). Both sides of this equation are sections of T ∗E, and they are equal if and only if they map all linear and all core vector fields in the same way. On core vector fields T ↑, for T ∈ Γ(∧kTM), we have
↑ ↑ ↑ ↑ d Λβ(£dX ,T ) = £dX (Λβ(T )) − T (Λβ(£dX )) − Λβ([£dX ,T ]) ∗ = 0 − qE⟨T, iX β⟩ − 0 = 0,
↑ ∗ ↑ l d ℓiX β(T ) = qE⟨T, iX β⟩ and d^iX β(T )(em) = 0. On a linear vector field Db ∈ X (E) over Y ∈ X(M), we have
d Λβ(£dX , Db) = £dX (Λβ(Db)) − Db(Λβ(£dX )) − Λβ([£dX , Db])
= ℓ ∗ , £X (iY β)−D (iX β)−i[X,Y ]β
∗ d^iX β(Db) = ℓiY d iX β and Db(ℓiX β) = ℓD (iX β). Thus we are left with the following condition on β:
∗ ∗ £X (iY β) − i[X,Y ]β − D (iX β) = iY d iX β − D (iX β) for all X,Y ∈ X(M), which is equivalent to β ∈ Ωk+1(M) and d β = 0.
140 Appendix A
Details of proofs
A.1 On the proofs of Theorems 10.10 and 10.11
Choose a linear section χ of TE ⊕ T ∗E → E over a pair (X, ε) ∈ Γ(TM ⊕ E∗). Then χ = dcχ, d ℓε − φfχ , following the notations set after Theorem 10.8. For simplicity, we write 1 θχ for d ℓε − φfχ ∈ Ω (E).
Lemma A.1. Choose linear sections χ, χ′ of TE ⊕ T ∗E → E over (X, ε), (X′, ε′) ∈ Γ(TM ⊕ E∗), a section e ∈ Γ(E) and a derivation D of E with symbol Y . Then
↑ ∗ 1. ⟨θχ, e ⟩ = qE⟨ε, e⟩,
⟨θ , D⟩ = ℓ ∗ ∗ 2. χ b D ε−φχ(Y ),
∗ 3. £e↑ θχ = qE (d⟨ε, e⟩ − φχ(e)),
∗ ∗ £ θ = d ℓ ∗ − (d^′ (φ )) 4. χ d ′ ε χ χ . dcχ′ χ ∗ 1 ∗ Note that in the last equation, φχ is an element of Ω (M,E ). For a derivation D of E over X ∈ X(M), the derivation D : Ω1(M,E∗) → Ω1(M,E∗) over X is defined by (Dω)(Y ) = D∗(ω(Y )) − ω[X,Y ] for all Y ∈ X(M).
Proof. The first identity is immediate. For the second, we recall (3.5). The pairing of Dˆ with
θχ at em is ∗ Y ⟨ε, e⟩ − ⟨φχ(e),Y ⟩ − ⟨ε, De⟩ = ⟨D ε, e⟩ − ⟨φχ(e),Y ⟩
at m. Hence we have found (2). Next we prove (3). We have
′↑ ↑ ′↑ h ↑ ′↑i ↑ ∗ ′ ⟨£e↑ θχ, e ⟩ = e ⟨θχ, e ⟩ − ⟨θχ, e , e ⟩ = e (qE⟨ε, e ⟩) = 0
141 Details of proofs for e′ ∈ Γ(E) and
↑ h ↑ i ∗ ∗ ∗ ↑ ⟨£e↑ θχ, Db⟩ = e ⟨θχ, Db⟩ − ⟨θχ, e , Db ⟩ = qE⟨D ε − φχ(Y ), e⟩ + ⟨θχ, (De) ⟩ ∗ ∗ ∗ ∗ ∗ = qE⟨D ε − φχ(Y ), e⟩ + qE⟨ε, De⟩ = qE(Y ⟨ε, e⟩ − ⟨Y, φχ(e)⟩)
∗ for a derivation D of E over Y ∈ X(M). Since qE (d⟨ε, e⟩ − φχ(e)) takes the same values on e′↑ and Dˆ, we are done. Finally, we compute using the first identity
↑ ↑ h ↑i ∗ ′ ⟨£ θχ, e ⟩ = dcχ′ ⟨θχ, e ⟩ − ⟨θχ, dcχ′ , e ⟩ = qE(X ⟨ε, e⟩ − ⟨ε, dχ′ e⟩) dcχ′ ∗ ∗ ↑ ∗ ∗ ↑ = q ⟨d ′ ε, e⟩ = ⟨d ℓd∗ ε, e ⟩ = ⟨d ℓd∗ ε − (dχ^′ (φ )) , e ⟩ E χ χ′ χ′ χ for e ∈ Γ(E). Similarly, using (2) above
h i V ⟨£ θ , D⟩ = d ′ ⟨θ , D⟩ − ⟨θ , d ′ , D ⟩ = ℓ ∗ ∗ ∗ − ⟨θ , [d ′ ,D]⟩ χ b cχ χ b χ cχ b d (D ε−φχ(Y )) χ χ dcχ′ χ′
∗ = ℓd (D∗ε−φ∗ (Y ))−[d ′ ,D]∗ε+φ∗ [X′,Y ] χ′ χ χ χ
∗ ∗ ∗ for a derivation D of E over Y ∈ X(M). An easy calculation shows [dχ′ ,D] = [dχ′ ,D ], which leads to
∗ ∗ ⟨£ θ , D⟩ = ℓ ∗ ∗ ∗ = ⟨d ℓ ∗ − (d^′ φ ) , Dˆ⟩. χ b D d ε−(d ′ φχ)(Y ) d ′ ε χ χ dcχ′ χ′ χ χ
∗ ∗ ′↑ Proof of Theorem 10.10. We write τ = (e, θ) ∈ Γ(E ⊕T M). First we find that ⟨£ qEθ, e ⟩ dbχ ∗ ′↑ ∗ ′↑ ∗ ∗ ∗ equals dcχ⟨qEθ, e ⟩−⟨qEθ, [dcχ, e ]⟩ = 0−0 = 0 and ⟨£ qEθ, Db⟩ = dcχ(qE⟨θ, Y ⟩)−⟨qEθ, [dcχ, Db]⟩ = dbχ ∗ ∗ ′ qE(X⟨θ, Y ⟩ − ⟨θ, [X,Y ]⟩) = qE⟨£X θ, Y ⟩ for all e ∈ Γ(E) and any derivation D of E over ∗ ∗ Y ∈ X(M). This shows £ qEθ = qE(£X θ). In the same manner, we have ie↑ d θχ = dbχ ↑ ∗ £e↑ θχ − d⟨θχ, e ⟩ = qE(−φχ(e)) by (1) and (3) in Lemma A.1. We get
↑ h ↑i ∗ ↑ ∗ rχ, τ z = dcχ, e , £ qEθ − ie↑ d θχ = (dχe) , qE(£X θ + prT ∗M Dχ(e, 0)) dbχ ↑ ∗ ↑ = (dχe) , qE(prT ∗M Dχ(e, θ)) = Dχτ , which proves Theorem 10.10.
Proof of Theorem 10.11. We simply compute
h i χ , χ = d , d , £ θ − i d θ . 1 2 dχ1 dχ2 d χ2 d χ1 (A.1) J K cχ1 cχ2
The TE-part is [d\χ1 , dχ2 ]. By definition of Dχ, we have prE ◦Dχ ◦ ιE ◦ prE = prE ◦Dχ and so [dχ1 , dχ2 ] = prE ◦[Dχ1 ,Dχ2 ] ◦ ιE.
142 A.2 On the proof of Theorem 11.4
The T ∗E-component of (A.1) is ∼ ∼ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ d ℓdχ ε2 − (dχ1 (φχ2 )) −d ℓdχ ε1 + (dχ2 (φχ1 )) + d ℓd ¨¨ε −φ (X ) 1 2 ¨χ2 1 χ1 2
∗ ∗ by Lemma A.1. First we find that ⟨dχ1 ε2 − φχ1 (X2), e⟩ equals
X1⟨ε2, e⟩ − ⟨ε2, dχ1 e⟩ − ⟨X2, φχ1 (e)⟩ ∗ = X1⟨ε2, e⟩ − ⟨(X2, ε2),Dχ1 (e, 0)⟩ = ⟨Dχ1 (X2, ε2), (e, 0)⟩
∗ ∗ ∗ for any e ∈ Γ(E). Then we find that ⟨(dχ1 φχ2 − dχ2 φχ1 ) (e),X⟩ equals
∗ ∗ ∗ ∗ ∗ ∗ dχ1 (φχ2 (X)) − φχ2 [X1,X] − dχ2 (φχ1 (X)) + φχ1 [X2,X] (e)
=X1⟨X, φχ2 (e)⟩ − ⟨X, φχ2 (dχ1 (e))⟩ − ⟨[X1,X], φχ2 (e)⟩
− X2⟨X, φχ1 (e)⟩ + ⟨X, φχ1 (dχ2 (e))⟩ + ⟨[X2,X], φχ1 (e)⟩
=⟨X, £X1 (prT ∗M Dχ2 (e, 0)) + prT ∗M ◦Dχ2 ◦ ιE ◦ prE ◦Dχ1 (e, 0)⟩
− ⟨X, £X2 (prT ∗M Dχ1 (e, 0)) + prT ∗M ◦Dχ1 ◦ ιE ◦ prE ◦Dχ2 (e, 0)⟩
for X ∈ X(M) and e ∈ Γ(E). Since £X1 (prT ∗M Dχ2 (e, 0)) equals
prT ∗M Dχ1 (0, prT ∗M Dχ2 (e, 0)) and prT ∗M ◦Dχ1 ◦ ιE ◦ prE ◦Dχ2 (e, 0) equals
prT ∗M Dχ1 (prE Dχ2 (e, 0), 0), we find that the first and fourth term add up to ⟨X, prT ∗M Dχ1 Dχ2 (e, 0)⟩. Similarly the second and third term add up to
−⟨X, prT ∗M Dχ2 Dχ1 (e, 0)⟩ and we get
∗ ∗ ∗ ∗ ⟨(dχ1 φχ2 − dχ2 φχ1 ) (e),X⟩ = ⟨prT M [Dχ1 ,Dχ2 ](e, 0),X⟩.
The proof of the second identity is left to the reader.
A.2 On the proof of Theorem 11.4
Recall that D has the following property:
D(X,ε)(e, θ) = D(X,ε)(e, 0) + (0, £X θ) (A.2)
for all (X, ε) ∈ Γ(TM ⊕ E∗) and (e, θ) ∈ Γ(E ⊕ T ∗M). (A.2) and the definition of δ yield together
δ ◦ prE = prE ◦D. (A.3)
We will use the following lemma.
Lemma A.2. · , · satisfies the Jacobi identity in Leibniz form if and only if J K
1. [δν1 , δν2 ] = δ ν1,ν2 and J K 143 Details of proofs
2. prT ∗M [Dν1 , Dν2 ] ◦ ιE = prT ∗M ◦D ν1,ν2 ◦ ιE J K ∗ for all ν1, ν2 ∈ Γ(TM ⊕ E ).
Proof. First note that by (A.3), we have
[δν1 , δν2 ] = prE ◦[Dν1 , Dν2 ] ◦ ιE. (A.4)
If · , · satisfies the Jacobi identity in Leibniz form, then (1) and (2) are immediate by (10.2). J K Conversely, (1) and (2) give using (A.4): [Dν1 , Dν2 ] ◦ ιE = D ν1,ν2 ◦ ιE. We have always J K 1 [Dν1 , Dν2 ](0, θ) = (0, £X1 £X2 θ−£X2 £X1 θ) = (0, £[X1,X2]θ) = D ν1,ν2 (0, θ) for all θ ∈ Ω (M). J K This shows that (1), (2) are equivalent to [Dν1 , Dν2 ] = D ν1,ν2 , which dualises to the Jacobi J K identity in Leibniz form for · , · . J K Now we can prove Theorem 11.2.
Proof of Theorem 11.2. We write τ = (e, θ), τi = (ei, θi) and ν = (X, ε), νi = (Xi, εi) for i = 1, 2. By (10.9), we have
V ∼ ∗ Ξ(ν1), Ξ(ν2) = [δν1 , δν2 ], d ℓpr ∗ D ν2 − prT ∗M ◦[Dν1 , Dν2 ] ◦ ιE . J K E ν1 By Lemma A.2, this is
V ∼ ∗ ∗ Ξ(ν1), Ξ(ν2) = δ ν1,ν2 , d ℓprE∗ Dν ν2 − prT M ◦D ν1,ν2 ◦ ιE J K J K 1 J K ∗ if and only if · , · satisfies the Jacobi identity in Leibniz form. Since Dν ν2 = ν1, ν2 , we J K 1 J K are done.
144 Appendix B
A non-local Leibniz algebroid
1 1 2 ∗ 2 ∗ Let M = S × S ≃ T and consider the vector bundle E¯ = T M ⊕ ∧ T M over M. Let 1 1 ∗ ∗ η ∈ Ω (S ) be the standard volume form on the circle and set ηx = pr1 η and ηy = pr2 η, 1 1 1 where pri : S × S → S are the projections, i = 1, 2. Then ηx ∧ ηy is a volume form on 1 M and ηx, ηy ∈ Ω (M) form a basis of one-forms such that the pullback of ηx along any 1 1 1 ιq : S ,→ S × {q} and the pullback of ηy to any {p} × S are the standard volume form on the circle. Define the following operations for integration along the first fibre. For f, g, h ∈ C∞(M) R fη + gη ∈ C∞(M) : S1 x y , Z Z ∗ fηx + gηy (p, q) := ιq(f)η S1 S1
R h η ∧ η ∈ Ω1(M) and S1 x y , Z Z ∗ h ηx ∧ ηy (p, q) := ιq(h)η ηy(p, q). S1 S1
R fη + gη ∈ C∞(M) 1 Clearly, the resulting function S1 x y is constant along the first S , i.e. only q R h η ∧ η a function of in the notation above. In the same manner, the one-form S1 x y is 1 constant along the first S and only has a ηy component. That is, the obtained functions and 1-forms are invariant along the fibers of pr2. Now we define a bracket on E¯ = T ∗M ⊕ ∧2T ∗M as follows: Z Z (α1, α2), (β1, β2) = 0, α1 β2 + α2 ∧ β1 J K S1 S1
and we prove that (E¯ = T ∗M ⊕ ∧2T ∗M, ·, · , 0: E¯ → TM is a Leibniz algebroid. Since the J K bracket is clearly C∞-linear in the second component and thus satisfies the Leibniz rule for functions with the zero-anchor, it suffices to check the Jacobi identity in Leibniz form. For R R 1 simplicity, we just write for S1 , and this is always the integration along the first S . We
145 A non-local Leibniz algebroid have
s Z Z { (α1, α2), (β1, β2), (γ1, γ2) = (α1, α2), 0, β1 γ2 + β2 ∧ γ1 J J KK Z Z Z Z = 0, α1 β1 γ2 + α1 β2 ∧ γ1 and in a similar manner
s Z Z { (α1, α2), (β1, β2) , (γ1, γ2) = 0, α1 β2 + α2 ∧ β1 , (γ1, γ2) JJ K K Z Z Z = 0, α1 β2 + α2 ∧ β1 ∧ γ1 Z Z Z Z = 0, α1 β2 ∧ γ1 − β1 α2 ∧ γ1 .
Therefore we get
(α1, α2), (β1, β2), (γ1, γ2) − (β1, β2), (α1, α2), (γ1, γ2) J J KK J J KK − (α1, α2), (β1, β2) , (γ1, γ2) Z Z Z Z Z JJ Z Z KZ K = 0, α1 β1γ2 + α1 β2 ∧ γ1 − β1 α1γ2 − β1 α2 ∧ γ1 Z Z Z Z − α1 β2 ∧ γ1 + β1 α2 ∧ γ1 = 0
This Leibniz algebroid is non-local, i.e. its bracket not given by a bilinear differential operator of any order.
146 Bibliography
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