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Applications of Graded Manifolds to Poisson Geometry By Applications of graded manifolds to Poisson geometry by Miquel Cueca Ten Under the supervision of Prof. Dr. Henrique Bursztyn Instituto de Matematica Pura e Aplicada Rio de Janeiro, Brasil August 2019 Abstract In this thesis we use graded manifolds to study Poisson and related higher geome- tries shedding light on many new aspects of their connection. The relation between graded manifolds and classical geometric objects is based on a new geometrization functor that allows us to think of graded manifolds as collections of vector bun- dles carrying some additional structure. As a consequence of this geometrization of graded manifolds, we prove a graded version of the Frobenius theorem about in- tegrability of involutive distributions, classify odd degree symplectic Q-manifolds, relate the geometry of higher Courant algebroids to shifted cotangent bundles and semi-direct products of the adjoint representation (up to homotopy) of a Lie alge- broid, and give a new description of the infinitesimal data of a multiplicative Dirac structure. 1 Acknowledgements This thesis was possible thanks to my advisor Henrique Bursztyn. It is impossible to summarize his work and influence on me. During the last five years Henrique was really generous and spent a lot of his time teaching me, solving my doubts, listening to my silly mathematical ideas or just talking about life. He was always there to attract my attention to the right points or give his support in difficult days. I appreciate the freedom that he gave me as well as the way he explained to me the \big picture" whenever I was lost. Apart from his crazy idea that all sentences in English must have a subject, I have realized (sometimes many years after) that his advices always pointed in the right direction. I am grateful to him for these five years showing me the realm of Poisson geometry. There are also people involuntarily responsible for this story. The first one is Marco Zambon who, during a summer school at ICMAT, introduced me to the world of symplectic geometry and encouraged me to do my PhD at IMPA. The second one is Alexei Kotov, who gave a mini-course on Supergeometry at IMPA in January of 2015. After these interesting lectures, I start to learn more about supermanifolds and Henrique Bursztyn provided me with the unpublished manuscript [24]. This was the starting point for this work. I am also grateful to all the Poisson geometry community in Rio de Janeiro for many conversations, discussions and seminars. I am particularly grateful to Henrique's student's who overlapped with me at IMPA: Hudson Lima, Inocencio Ortiz, Mateus de Melo, Daniel Alvarez, Dan Ag¨ueroand Clarice Netto. Last but not least, I want to thank the \supermathematicians" Alejandro Cabr- era, Fernando del Carpio-Marek and Rajan Mehta for explaining to me their works on graded manifolds and spending their time with my troubles to understand graded geometry. Also, I thank the hospitality of Smith College during my two visits to Rajan Mehta. I thank CNPq for the financial support in the last 5 years. Finally, I should mention that the great mathematical environment at IMPA also contributed to my work. It is not easy to find an institution where interesting seminars, in many differ- ent areas, take place daily. I am very thankful to the Graduate Students Association (APG) for its concern and many initiatives to make this mathematically interesting environment more warm in contrast with the direction of IMPA who has forgotten the human aspects behind the students, and ignored our social environment. I hope this will change in the future. 2 Cos de podadora m`anegues d'astral l'encomana que m'has fet l'aigua li ha fet mal doncs tira-li farina i se t’endurir`a A la llum de la de mediterr`aniaque m'has vist cr´eixer,a la roja terra dels ametllers i tarongers dels quals s´ocfruit, a la t`ebiasorra de la platja i el suau vent del mar que sempre em portes esperan¸ca. Per`osobretot, a la gent, al poble que m'ha envoltat amb la seva estima, calor i for¸ca. Especialment agra¨ıt a aquelles quatre dones de la meva infantesa que, lligades per sempre mes a un temps passat, em van mostrar una forma diferent de veure les coses. Als meus pares i germ`a,pels bons moments, l’educaci´oi la paci`enciarebuda al llarg de molts anys. A la fam´ıliaen el sentit m´esample, per mostrar-me d'on vinc i qui s´ocaix´ıcom donar suport als moments dif´ıcilsmalgrat en certes ocasions convertir-se en una pesada llosa. Al Trompo, que en certa forma tamb´e´esfam´ılia, per oferir-me l'oportunitat de con`eixergent diversa, aleat`oriaaix´ıcom mostrar-me la cruesa de la paraula treball. Als amics que el temps m'ha deixat triar, n'he fet un grapat cadascun diferent de l'anterior i possiblement tots bojos com jo. Gr`aciesper somiar junts, per com- partir amb mi els moments f`acilso dif´ıcils,de riures o de lluita, de festes o estudis. Hem anat creixent junts amb els anys, fent-se majors que hi diuen, no s´ocmes que experi`encies amb tots vosaltres. A les parelles i als amants, per estimar-me i deixar-se estimar. Tristeza n~aotem fim felicidade sim A chuva do Rio que me viu chegando h´acinco anos com medo, a quente noite carioca do samba que se transforma em funk e viu o medo ir embora. A morna areia da praia e o vento do Atl^antico que continuam me trazendo esperan¸ca. A cerveja es- tupidamente gelada em torno da qual sempre apareceram amigos, conversas, choros e risadas. A Cata, Marcos, Mateus, Miguel e Yaya, que me acompanharam nessa luta matem´aticapelo doutorado. Por fazer do IMPA um lugar humano e me ensinar matem´aticaao mesmo tempo que compartilhamos as ilus~oesda vida. Aos amigos que fizeram do Rio minha casa e n~aos´oa cidade onde eu moro. Cinco anos que passaram mais r´apidodo que eu consegui perceber; grato pela amizade. A Felipe, que conheci na primeira semana e me acompanhou nesses cinco anos; obrigado por me guiar nos mil e um Rios que convivem, violenta e injustamente, nesta cidade. Por me acolher na sua casa quando precisei e me dar for¸canos mo- mentos de loucura matem´atica. 3 Contents 1 Introduction6 2 Graded manifolds 14 2.1 The category GMn ............................ 14 2.2 Vector fields and tangent vectors . 16 2.2.1 Vector fields . 17 2.2.2 Q-manifolds . 18 2.2.3 Tangent vectors at a point . 20 2.3 Submanifolds . 21 2.4 Graded Poisson and symplectic manifolds . 22 3 Geometrization of graded manifolds 25 3.1 The category CoBn ............................ 26 3.1.1 The geometrization functor . 27 3.1.2 Relation with other geometrizations . 32 3.2 Geometrization of submanifolds . 33 3.3 Geometrization of vector fields . 34 3.4 Geometrization of Q-manifolds . 36 3.4.1 Degree 1: Lie algebroids . 37 3.4.2 Degree 2 . 38 3.4.3 Degree n: Ln-algebroids . 40 3.5 Geometrization of graded Poisson manifolds . 42 3.5.1 Degree 1: Lie algebroids . 42 3.5.2 Degree 2 . 43 3.6 Geometrization of PQ-manifolds . 45 3.6.1 Degree 1: Lie bialgebroids . 45 3.6.2 Degree 2 . 46 4 The Frobenius theorem for graded manifolds 49 4.1 Distributions on graded manifolds . 49 4.2 The Frobenius theorem . 50 4.3 Integral submanifolds . 54 5 Lie algebroids through graded manifolds 56 5.1 Homology for Lie algebroids . 56 5.1.1 Left and right representations . 56 5.1.2 Divergence and generating operators . 58 4 5.1.3 Graded manifold approach . 60 5.1.4 The Lie bialgebroid case . 63 5.1.5 The derived category . 64 5.2 Semi-direct products with 2-term representations up to homotopy . 65 5.2.1 2-term representations up to homotopy . 66 5.2.2 VB-algebroids and Q-bundles . 67 5.2.3 L1-algebroids and Q-manifolds . 69 6 The geometry of graded cotangent bundles 72 6.1 Graded cotangent bundle of a graded manifold . 72 6.1.1 T ∗[n + 1]M and homotopy Poisson structures . 73 6.1.2 T ∗[2n + 1]M and odd degree symplectic manifolds . 75 6.2 Graded cotangent bundles of 1-manifolds . 77 6.2.1 Classical description of the manifolds T ∗[k]A[1] . 78 6.2.2 The symplectic structure . 80 6.3 The symplectic Q-structures of T ∗[k]A[1] . 85 6.3.1 Q-structures and twists . 85 6.3.2 Q-cohomology . 87 6.3.3 Brackets and higher Dirac structures . 88 6.3.4 Relation with the coadjoint representation up to homotopy . 90 6.4 AKSZ σ-models and integrating objects . 91 6.4.1 AKSZ σ-models . 91 6.4.2 Speculations on the integration of Q-manifolds . 91 7 IM-Dirac structures 93 7.1 Degenerate Courant algebroids and IM-Dirac structures . 93 7.1.1 LA-Courant algebroids and morphic Dirac structures . 97 7.2 Example: T [1]A∗[1]............................ 99 7.2.1 Integration to multiplicative Dirac structures . 104 7.3 Graded symplectic Q-groupoids . 105 A More on graded manifolds 108 A.1 Vector bundles . 108 A.2 Differential forms and Cartan calculus . 109 A.3 Graded symplectic manifolds (2-form definition) . 111 A.4 Graded groupoids and graded algebroids . 113 B Double vector bundles and 2-manifolds 117 B.1 Basic properties . 117 B.2 VB-groupoids and VB-algebroids .
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