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Geometry of Membrane Sigma Models by Jan Vysok´Y Geometry of Membrane Sigma Models by Jan Vysok´y A thesis submitted in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY IN MATHEMATICS (Double Doctoral Agreement between Jacobs University and Czech Technical University) Date of Defense: 15th July, 2015 Approved Dissertation Committee: Prof. Dr. Peter Schupp (supervisor, chair) Dr. Branislav Jurˇco(supervisor) Prof. Dr. Dr. h.c. mult. Alan T. Huckleberry Prof. Dr. Olaf Lechtenfeld Bibliographic entry Title: Geometry of membrane sigma models Author: Ing. Jan Vysok´y Czech Technical University in Prague (CTU) Faculty of Nuclear Sciences and Physical Engineering Department of Physics Jacobs University Bremen (JUB) Department of Physics and Earth Sciences Degree Programme: Co-directed double Ph.D. (CTU - JUB) Application of natural sciences (CTU) Mathematical sciences (JUB) Field of Study: Methematical engineering (CTU) Mathematical sciences (JUB) Supervisors: Ing. Branislav Jurˇco,CSc., DSc. Charles University Faculty of Mathematics and Physics Mathematical Institute of Charles University Prof. Dr. Peter Schupp Jacobs University Bremen Department of Physics and Earth Sciences Academic Year: 2014/2015 Number of Pages: 209 Keywords: generalized geometry, string theory, sigma models, membranes. Abstract String theory still remains one of the promising candidates for a unification of the theory of gravity and quantum field theory. One of its essential parts is relativistic description of moving multi-dimensional objects called membranes (or p-branes) in a curved spacetime. On the classical field theory level, they are described by an action functional extremalising the volume of a manifold swept by a propagating membrane. This and related field theories are collectively called membrane sigma models. Differential geometry is an important mathematical tool in the study of string theory. It turns out that string and membrane backgrounds can be conveniently described using objects defined on a direct sum of tangent and cotangent bundles of the spacetime manifold. Mathe- matical field studying such object is called generalized geometry. Its integral part is the theory of Leibniz algebroids, vector bundles with a Leibniz algebra bracket on its module of smooth sections. Special cases of Leibniz algebroids are better known Lie and Courant algebroids. This thesis is divided into two main parts. In the first one, we review the foundations of the theory of Leibniz algebroids, generalized geometry, extended generalized geometry, and Nambu- Poisson structures. The main aim is to provide the reader with a consistent introduction to the mathematics used in the published papers. The text is a combination both of well known results and new ones. We emphasize the notion of a generalized metric and of corresponding orthogonal transformations, which laid the groundwork of our research. The second main part consists of four attached papers using generalized geometry to treat selected topics in string and membrane theory. The articles are presented in the same form as they were published. Bibliografick´yz´aznam N´azevpr´ace: Geometrie membr´anov´ych sigma model˚u Autor: Ing. Jan Vysok´y Cesk´evysok´euˇcen´ıtechnick´evˇ Praze (CVUT)ˇ Fakulta jadern´aa fyzik´alnˇeinˇzen´yrsk´a Katedra fyziky Jacobs University Bremen (JUB) Department of Physics and Earth Sciences Studijn´ıprogram: Doktor´atpod dvoj´ımveden´ım(CVUTˇ - JUB) Aplikace pˇr´ırodn´ıch vˇed(CVUT)ˇ Matematick´evˇedy(JUB) Studijn´ıobor: Matematick´einˇzen´yrstv´ı(CVUT)ˇ Matematick´evˇedy(JUB) Skolitel´e:ˇ Ing. Branislav Jurˇco,CSc., DSc. Univerzita Karlova Matematicko-fyzik´aln´ıfakulta Matematick´y´ustav UK Prof. Dr. Peter Schupp Jacobs University Bremen Department of Physics and Earth Sciences Akademick´yrok: 2014/2015 Poˇcetstran: 209 Kl´ıˇcov´aslova: zobecnˇen´ageometrie, teorie strun, sigma modely, membr´any. Abstrakt Strunov´ateorie st´alez˚ust´av´anadˇejn´ymkandid´atemna sjednocen´ı teorie gravitace a kvan- tov´eteorie pole. Jej´ıd˚uleˇzitousouˇc´ast´ıje relativistick´ypopis pohybu v´ıcerozmˇern´ych objekt˚u naz´yvan´ych membr´any (pˇr´ıpadnˇe p-br´any) v zakˇriven´emprostoroˇcase. Ten je na ´urovni kla- sick´eteorie pole pops´anfunkcion´alemakce extremalizuj´ıc´ımobjem variety vytnut´eˇs´ıˇr´ıc´ıse membr´anou.Tuto a souvisej´ıc´ıpoln´ıteorie souhrnnˇenaz´yv´amemembr´anov´esigma modely. Diferenci´aln´ıgeometrie pˇredstavuje d˚uleˇzit´ymatematick´yn´astroj pˇristudiu strunov´eteorie. Ukazuje se, ˇzestrunov´aa membr´anov´apozad´ı lze v´yhodnˇepopsat pomoc´ı objekt˚udefino- van´ych na direktn´ımsouˇctuteˇcn´ehoa koteˇcn´ehofibrovan´ehoprostoru k prostoroˇcasov´evarietˇe. Studiem tohoto objektu se zab´yv´atzv. zobecnˇen´ageometrie. Jej´ıned´ılnou souˇc´ast´ıje teorie Leibnizov´ych algebroid˚u,vektorov´ych fibrovan´ych prostor˚u,jejichˇzmoduly hladk´ych ˇrez˚ujsou vybaveny Leibnizovou algebrou. Speci´aln´ımipˇr´ıpadyLeibnizov´ych algebroid˚ujsou zn´amˇejˇs´ı Lieovy a Courantovy algebroidy. Tato pr´aceje rozdˇelenado dvou hlavn´ıch ˇc´ast´ı.V prvn´ırekapitulujeme z´akladyteorie Leib- nizov´ych algebroid˚u, zobecnˇen´egeometrie, rozˇs´ıˇren´ezobecnˇen´egeomerie a Nambu-Poissonov´ych struktur. C´ılemje poskytnout ˇcten´aˇriucelen´yz´akladmatematick´eteorie pouˇzit´ev publiko- van´ych pracech, text je kombinac´ızn´am´ych i nov´ych v´ysledk˚u.D˚urazje kladen pˇredevˇs´ımna pojem zobecnˇen´emetriky a pˇr´ısluˇsn´ych ortogon´aln´ıch transformac´ı,jenˇzposlouˇzilyjako z´aklad naˇsehov´yzkumu. Druhou hlavn´ıˇc´ast´ıje pˇr´ılohatvoˇren´aˇctyˇrmiˇcl´ankyuˇz´ıvaj´ıc´ımizobecnˇenou geometrii na vybran´epartie teorie strun a membr´an.Pr´acejsou otiˇstˇeny ve stejn´epodobˇe,v jak´ebyly publikov´any. Acknowledgements I would like to thank my supervisors, Branislav Jurˇcoand Peter Schupp, for their guidance and inspiration throughout my entire doctorate, unhesitatingly encouraging me in my scientific career. I would like to thank my wonderful wife, Kamila, for her endless support and patience with the weird and sometimes rather unpractical mathematical physicist. Last but not least, I would like to thank my family and all my friends for all the joy and fun they incessantly bring into my life. Contents 1 Introduction 13 1.1 Organization . 14 1.2 Overview of the theoretical introduction . 14 1.3 Guide to attached papers . 17 1.4 Conventions . 18 2 Leibniz algebroids and their special cases 21 2.1 Leibniz algebroids . 21 2.2 Lie algebroids . 24 2.3 Courant algebroids . 27 2.4 Algebroid connections, local Leibniz algebroids . 30 3 Excerpts from the standard generalized geometry 39 3.1 Orthogonal group . 39 3.2 Maximally isotropic subspaces . 43 3.3 Vector bundle, extended group and Lie algebra . 44 3.4 Derivations algebra of the Dorfman bracket . 46 3.5 Automorphism group of the Dorfman bracket . 47 3.6 Twisting of the Dorfman bracket . 50 3.7 Dirac structures . 50 3.8 Generalized metric . 51 3.9 Orthogonal transformations of the generalized metric . 55 3.10 Killing sections and corresponding isometries . 59 3.11 Indefinite case . 60 3.12 Generalized Bismut connection . 63 9 4 Extended generalized geometry 67 4.1 Pairing, Orthogonal group . 67 4.2 Higher Dorfman bracket and its symmetries . 69 4.3 Induced metric . 70 4.4 Generalized metric . 73 4.5 Doubled formalism . 76 4.6 Leibniz algebroid for doubled formalism . 78 4.7 Killing sections of generalized metric . 82 4.8 Generalized Bismut connection II . 83 5 Nambu-Poisson structures 87 5.1 Nambu-Poisson manifolds . 87 5.2 Leibniz algebroids picture . 92 5.3 Seiberg-Witten map . 94 6 Conclusions and outlooks 97 Appendices 105 A Proofs of technical Lemmas 107 B Paper 1: p-Brane Actions and Higher Roytenberg Brackets 111 C Paper 2: On the Generalized Geometry Origin of Noncommutative Gauge Theory 135 D Paper 3: Extended generalized geometry and a DBI-type effective action for branes ending on branes 155 E Paper 4: Nambu-Poisson Gauge Theory 203 10 List of used symbols R ...................................................................................................... field of real numbers M ................................................................................ smooth finite-dimensional manifold C1(M) ........................................................ module of smooth real-valued functions on M Xp(M) ................................................................... module of smooth p-vector fields on M Ωp(M) ............................................................ module of smooth differential p-forms on M Ω•(M) ................................................................................... whole exterior algebra of M q p (M) ............................................................................... module of tensors of type (p; q) T (M) ....................................................................................... whole tensor algebra of M T; .................................................................................. canonical pairing of two objects Γ(h· E·i) ...................................... module of smooth global sections of vector bundle E over M ΓU (E) ........................................... module of smooth local sections of E defined on U M p p p ⊆q X (E) ........................ module of global smooth sections of Λ E, similarly for Ω (E), p (E) A := B ....................................................... expression B is used to define the expressionT A BDiag(g; h) .......................................... block diagonal matrix with g and h on the diagonal ................................................................ Lie derivative
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