Higher Structures

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I'll report on a new integration theory in the context of cyclic cohomology, which is based on BV-type formalism, and which constructs, starting from A-infinity algebras, equivariantly closed differential forms on matrix spaces gl(n,A ).
In the case of the simplest associative algebra consisting of the identity element, my construction produces an equivariant extension of the matrix Airy integral.
In the general case, the asymptotic expansion of my matrix integrals are sums over stable ribbon graphs, giving cohomology classes of compactified moduli spaces of curves. My noncommutative BV integration theory is the "higher genus" counterpart of the theory of periods arising in the BV-formalism of polyvector fields on Calabi-Yau manifolds.

I will describe how to use the formality in th epresence of two branes to produce a quantization of polynomial Poisson structures via generators and relations.

Topological quantum field theories may be defined as appropriate functors from the cobordism category. The classical and perturbative versions thereof have started to be investigated only recently.
We discuss topological field theories in the AKSZ formalism (this includes Chern-Simons and BF theories as well as the Poisson sigma model). In this case, to boundary components (objects) one can associate a symplectic manifold and a coisotropic submanifold thereof, whereas to cobordisms one associates canonical relations. This very natural construction (a special case with more structure than the general one recently introduced by V. Fock), yields to new viewpoints on, say, the moduli space of flat connections or the symplectic groupoid of a Poisson manifold. This also constitutes the starting point for the perturbative quantization of these theories. The possibility of including boundaries of boundaries (and so on) naturally yields to a Lurie-type description. Eventually, one may hope to be able to reconstruct perturbative topological field theories by gluing simple pieces.
This is based on joint work with P. Mnev and N. Reshetikhin.

I will explain how one can get normal forms for interesting submanifolds of a generalized complex 4-manifold. These include neighborhood theorems for points in the type change locus, a whole compact component of the type change locus as well as Lagrangian submanifolds intersecting the type change locus. I will then use these results to produce examples of generalized complex 4-manifolds and to show that the problem of classifying them is related to the problem of classifying symplectic manifolds with (singular) symplectic tori.

We explain the homotpy notions of cyclic A-infinity algebras and and related developments. We explain a theorem of Kontsevich and Soibelman on its relationship with cyclic cohomology, and provide explicit correspondences in terms of strong homotopy inner products and negative cyclic cohomology.

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Inspired upon categories of sheaves and quasi-coherent sheaves in algebraic geometry, Grothendieck categories are often used as models for non-commutative spaces. Although these categories have a good intrinsic deformation theory, deformations do not necessarily preserve a concrete representation of the category as a linear site. In this talk, we will discuss homological conditions ensuring that the deformation theory is actually controlled by an underlying (non-linear) site. We will present two different applications to schemes, one to "local" representations using affine covers, and one to "global" representations of projective schemes using $\Z$-algebras.

The algebra of observables in classical (1+1)-dimensional field theory is a vertex Poisson algebra (VPA), naturally graded by conformal weight. Truncation at every value of the weight has a left adjoint, allowing for a successive approximation of the original VPA by structures with a finite number of operations. The first interesting truncation is at conformal weight 1, and is precisely a Courant-Dorfman algebra (CDA). VPA's freely generated by CDA's are special in that they come from vertex Lie algebroids via a "universal enveloping" construction, and admit a natural quasi-conformal structure. Examples include Heisenberg and Kac-Moody current algebras, as well as a Poisson sigma model current algebra studied by Alekseev and Strobl. This is an ongoing joint work with Paul Bressler.

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I will review the general construction of three dimensional TFT within BV framework. I will explain how the corresponding Feynman rules lead to the interesting characteristic classes and the knots invariant within this framework.

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