There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

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Deformation Quantization of Poisson Manifolds

Quantum field theory on noncommutative spaces

General Topology-I

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. 
We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. 
Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.

Generalized complex geometry

Reduction of Courant algebroids and generalized complex structures

Given a Lie groupoid G over a manifold M, we show that multiplicative 2-forms on G relatively closed with respect to a closed 3-form phi on M correspond to maps from the Lie algebroid of G into T* M satisfying an algebraic condition and a differential condition with respect to the phi-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to phi-twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-Hamiltonian spaces and group-valued momentum maps.

/pdf/integration-of-twisted-dirac-brackets-25b5un3i04.pdf

Integration of twisted Dirac brackets

Nous etudions les transformations de jauge des structures de Dirac et la relation entre les equivalences de jauge et de Morita pour les varietes de Poisson. Nous decrivons comment la structure symplectique d'un groupoide symplectique est modifiee lors d'une transformation de jauge de la structure de Poisson de la section identite de ce groupoide et nous prouvons que des structures de Poisson integrables equivalentes sous une transformation de jauge sont equivalentes au sens de Morita. Comme exemple, nous etudions certains ensembles generiques de structures de Poisson sur les surfaces de Riemann : nous exhibons des invariants complets d'equivalence de jauge pour de telles structures qui, sur la sphere S 2 , donnent un invariant complet d'equivalence de Morita.

/pdf/gauge-equivalence-of-dirac-structures-and-symplectic-3q88i1p4nh.pdf

Gauge equivalence of Dirac structures and symplectic groupoids

The correspondence between Poisson structures and symplectic groupoids, analogous to the one of Lie algebras and Lie groups, plays an important role in Poisson geometry; it offers, in particular, a unifying framework for the study of hamiltonian and Poisson actions. In this paper, we extend this correspondence to the context of Dirac structures twisted by a closed 3-form. 
More generally, given a Lie groupoid $G$ over a manifold $M$, we show that multiplicative 2-forms on $G$ relatively closed with respect to a closed 3-form $\phi$ on $M$ correspond to maps from the Lie algebroid of $G$ into the cotangent bundle $T^*M$ of $M$, satisfying an algebraic condition and a differential condition with respect to the $\phi$-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-hamiltonian spaces and group-valued momentum maps.

In this paper we give a complete characterization of Morita equivalent star products on symplectic manifolds in terms of their characteristic classes: two star products ⋆ and ⋆' on (M,ω) are Morita equivalent if and only if there exists a symplectomorphism ψ\colon MM such that the relative class t(⋆, ψ⋆(⋆')) is 2 π i-integral. For star products on cotangent bundles, we show that this integrality condition is related to Dirac's quantization condition for magnetic charges.

Henrique Bursztyn

Papers

Reduction of Courant algebroids and generalized complex structures

Integration of twisted Dirac brackets

Gauge equivalence of Dirac structures and symplectic groupoids

Integration of twisted Dirac brackets

The Characteristic Classes of Morita Equivalent Star Products on Symplectic Manifolds