Instituto Nacional de Matem´aticaPura e Aplicada Generalized reduction and pure spinors. Author: Thiago L. Drummond Rio de Janeiro 2010 2 Agradecimentos. Agrade¸co`aminha fam´ıliao apoio incondicional `aminha longa jornada longe de Bras´ılia,sem eles nada seria poss´ıvel. A` Francyne, por estar ao meu lado nos ´ultimos3 anos, per´ıodo mais cr´ıtico no doutorado. Ela foi meu porto seguro nas muitas idas e vindas ao mundo das id´eias. Aos meus amigos novos e antigos, sempre me proporcionando momentos de alegria. Principalmente, `aJu, Renatinha, Caio, Andr´e,Arthur, Marcus Vin´ıcius, Alvarenga e Bruno. Aos meus colegas de faculdade, Santiago, Lu- cas e Mauro. Eles tiveram uma influ^enciaenorme na minha forma¸c~aocomo matem´atico.Ao Professor Celius Magalh~aes,t~aodecisivo na minha entrada no mundo da pesquisa como tutor do PET. Aos meus amigos do IMPA, Thiago Fassarela, Luiz Gustavo, Rener, Andr´e Contiero. Aos professores do IMPA, Carlos Isnard, Marcelo Viana, Jorge Vit´orio. Agrade¸comuito aos meus colegas Cristi`anOrtiz, Alejandro Cabrera e Fer- nando del Carpio, pelas muitas conversas, palestras e troca de id´eiasque me ajudaram enormemente `aconcluir essa tese. A` meu orientador Henrique Bursztyn por sua confian¸cae enorme generosi- dade. Seu constante apoio e disponibilidade para conversar sobre minhas id´eias foram fundamentais para a conclus~aoda tese. Finalmente, ao CNPq e ao MCT pelo aux´ıliofinanceiro 1 2 Contents 1 Introduction. 5 1.1 Generalized geometry. 5 1.2 Generalized reduction. 8 1.3 The pure spinor point of view. 9 1.4 Contents. 11 2 Reduction on the linear algebra setting 15 2.1 Extensions of finite-dimensional vector spaces. 15 2.2 The split-quadratic category. 23 2.3 A reduction procedure. 26 2.4 Quotient morphism as a composition of pull-back and push-forward. 31 3 Spinors: Part I. 35 3.1 Clifford algebra . 35 3.1.1 Clifford modules for split-quadratic vector spaces. 39 3.1.2 Pure spinors. 43 3.2 Pull-back and push-forward of spinors. 49 3.2.1 Main theorem at the linear algebra level. 50 3.2.2 Dealing with non-transversality. 55 4 Reduction of generalized structures. 59 4.1 Generalized geometry. 59 4.1.1 Courant algebroids. 59 4.1.2 Symmetries of the Courant bracket. 64 4.2 Actions on Courant algebroids. 68 4.2.1 Extended actions. 68 4.2.2 Lifted actions. 72 4.2.3 Moment maps. 76 4.3 The reduced Courant algebroid. 78 4.4 Reduction of Dirac structures. 83 4.4.1 Reduction of generalized complex structures. 88 3 4 CONTENTS 5 Spinors: Part II 89 5.1 Preliminaries . 89 5.1.1 Clifford bundle . 89 5.1.2 Cartan calculus on Clifford modules. 92 5.2 Pure spinors. 96 5.3 Reduction of pure spinors. 102 5.3.1 Main theorem. 102 5.3.2 Integrability . 108 5.4 Applications. 116 5.4.1 Examples. 116 5.4.2 Reduction of generalized Calabi-Yau structures. 121 5.4.3 The T-duality map. 126 A More on pure spinors and the split-quadratic category. 133 A.1 The transform. 133 A.1.1 Pull-back and push-forward morphisms. 137 A.2 Zero set and pure spinors. 138 A.2.1 Polarization dependence. 139 A.2.2 Zero set. 142 B Push-forward on principal bundles. 149 C Some proofs. 155 Chapter 1 Introduction. The idea of \reducing" geometric structures is as old as the very notion of symmetry. The interplay between symmetries and reduction was significantly explored by the founders of classical mechanics (e.g. Poisson, Jacobi) who real- ized that integrals of motion of a mechanical system could be used to reduce its degrees of freedom (see e.g. [6]) and produce a \smaller" phase space. The inti- mate connection between symmetries and conservation laws is generally referred to as \Noether's principle" and is central in the study of differential equations and dynamics (see e.g. [37, 43]). The modern mathematical formulation of the theory of reduction takes place in the context of symplectic geometry. The usual set-up involves an action of a Lie group on a symplectic manifold equipped with a moment map (see e.g. [34, 46]); then the Marsden-Weinstein reduction theorem [39] produces a quotient symplectic manifold (physically representing the phase space with reduced degrees of freedom). This reduction procedure has played a key role in different areas of mathematics, including the study of moduli spaces in gauge theory and their applications to mathematical physics, see e.g. [5, 21]. In recent years, mathematical physics has motivated the study of a much broader class of geometrical structures beyond symplectic geometry; these in- clude e.g. Dirac structures [17, 45] and generalized complex structures [28, 24] and are commonly referred to as \generalized geometries". The main subject of this thesis is the study of symmetries and reduction of generalized geometries, extending symplectic reduction. 1.1 Generalized geometry. For a smooth manifold M, its generalized tangent bundle is TM = TM ⊕ T ∗M. In [17], T. Courant, following ideas of A. Weinstein (see also [18]), realized that by considering the generalized tangent bundle, one can unify different kinds of geometric structures (including e.g. pre-symplectic forms, Poisson structures and foliations). The motivation to treat Poisson and pre-symplectic geometry 5 6 CHAPTER 1. INTRODUCTION. on equal footing lies on the work of P. Dirac on constrained mechanics [20] where both geometries arise naturally. There is a natural symmetric non-degenerate bilinear form on TM given by ∗ gcan(X + ξ; Y + η) = iX η + iY ξ; X; Y 2 Γ(TM); ξ; η 2 Γ(T M): For a 2-form ! 2 Ω2(M) (resp. a bivector field π 2 X2(M)), there corresponds a subbundle of TM given by ∗ L! = f(X; iX !) j X 2 TMg (resp. Lπ = f(iξπ; ξ) j ξ 2 T Mg): Both L! and Lπ share the property of being Lagrangian (i.e., maximal isotropic) with respect to gcan. Following [17], we call Lagrangian subbundles of TM almost Dirac structures. For a distribution ∆ ⊂ TM, L∆ = ∆ ⊕ Ann (∆) ⊂ TM is also an almost Dirac structure. The main achievement of T. Courant [17] was the definition of a bracket - the Courant bracket - on the sections of TM which encompasses the well-known integrability conditions of Poisson and pre- symplectic structures as well as distributions. Its formula is [[X + ξ; Y + η]] = [X; Y ] + LX η − iY dξ; (1.1) for X + ξ; Y + η 2 Γ(TM ⊕ T ∗M). The Courant bracket, as written above, is not skew symmetric, hence it is not a Lie bracket. It does satisfy, however, a version of the Jacobi identity. Noticing that [[X + ξ; Y + η]] = −[[Y + η; X + ξ]]+ d gcan(X + ξ; Y + η); (1.2) one may verify that [[·; ·]] is a Lie bracket when restricted to isotropic subbundles of TM. A Dirac structure is an almost Dirac structure L ⊂ TM such that [[Γ(L); Γ(L)]] ⊂ Γ(L): For a distribution ∆ ⊂ TM, the corresponding subbundle L∆ is a Dirac struc- ture if and only if ∆ is involutive. Also, L! (resp. Lπ) is a Dirac structure if and only if d! = 0 (resp. [π; π] = 0, where [·; ·] is the Schouten bracket on multivector fields). It is also possible to incorporate a description of complex structures in this context, a fact which was realized by N. Hitchin [28]. Let J : TM ! TM be an almost complex structure on M and consider its −i-eigenbundle T0;1 ⊂ TM ⊗C. Then, L0;1 = T0;1 ⊕ Ann (T0;1) ⊂ TM ⊗ C is an almost Dirac structure relative to the C-bilinear extension of gcan. By complexifying the Courant bracket, one has that L0;1 is a Dirac structure if and only if J is integrable. This motivates the definition of generalized complex structures on M as being maximal isotropic subbundles L of TM ⊗ C satisfying 1.1. GENERALIZED GEOMETRY. 7 (i) L \ L¯ = 0; (ii) [[Γ(L); Γ(L)]] ⊂ Γ(L). It is also possible to see a symplectic structure on M as a generalized complex structure given by Li ! = f(X; i !(X; ·)) j X 2 TM ⊗ Cg ⊂ TM ⊗ C; (1.3) where ! 2 Ω2(M; R) is the symplectic 2-form. Generalized complex structures were intensively studied in [24]. Their importance lies on the fact that they provide a unified view of symplectic and complex geometries much desired by physicists who study mirror symmetry (for more on the relation between gen- eralized complex structures and physics, see [24] and references therein). Motivated also by some constructions from physics (e.g. [32]), P. Severaˇ and A. Weinstein [45] realized that the Courant bracket could be twisted by a closed 3 3-form H 2 Ω (M), defining the H-twisted Courant bracket [[·; ·]]H given, for X + ξ; Y + η 2 Γ(TM), by [[X + ξ; Y + η]]H = [X; Y ] + LX η − iY dξ + iY iX H: (1.4) The bundle TM endowed with gcan and [[·; ·]]H is the main example of an exact Courant algebroid [36, 45]. In general, an exact Courant algebroid consists of a vector bundle E over M endowed with a bracket [[·; ·]] on its sections, a non-degenerate symmetric bilinear form g and a map p : E ! TM, called the anchor, satisfying a set of axioms presented in x4.1. In the case of TM, the anchor is prTM , the projection on TM. One particularly important property of E is the existence of an exact sequence p∗ p 0 −! T ∗M −! E −! TM −! 0; (1.5) where we have identified E =∼ E∗ using g.