Talk

The goal of my talk will be to discuss the relation between two approaches to the geometric Langlands program. The first has been proposed by Beilinson and Drinfeld, using ideas and methods from conformal field theory (CFT). The second was initiated by Kapustin and Witten based on a topological version of four-dimensional maximally supersymmetric Yang--Mills theory and its reduction to a two-dimensional topological sigma model.

I will discuss a definition of bordism categories where bordisms are equipped with general geometric structures. This is motivated by applications such as the representation of cohomology theories through field theories. Then I will discuss the case of 1-dimensional TQFTs over a manifold X. It turns out, as one would hope, that these are nothing but vector bundles with connection over X. I'll explain how the difficulties entailed in this statement and their resolution are related to the problem of representing cohomology theories.

I will review the construction of ''higher operations'' on local and extended operators in topological field theory, and some applications of this construction in supersymmetric field theory. In particular, the higher operation on supersymmetric local operators in a 3d N=4 theory turns out to be induced by the holomorphic Poisson structure on the moduli space of the theory.

Bundle gerbes are the geometric objects which describe B-fields in string theory. Their sections, in turn, are the (twisted) Chan-Paton bundles that model the K-theory charges of D-branes. While this  describes the topological part of a spacetime geometry in string theory, the configuration space of strings consists of loop and path spaces. On these spaces, the same geometry takes a different form; we show that it translates to bundles of algebras and bimodules that generalise coloured, knowledgeable Frobenius algebras.

We present a version of higher Hochschild homology for spaces equipped with principal G-bundles. As coefficients we allow $E_\infty$-algebras with G-action. For this homology theory we establish an  equivariant version of excision to prove that it extends to an equivariant topological field theory with values in the $(\infty,1)$-category of co-spans in $E_\infty$-algebras. As an example we construct equivariant Dijkgraaf-Witten theories.  This is joint work in progress with Lukas Woike.

This is a join work with T. Pantev. In this talk, we will discuss moduli of flat bundles on smooth algebraic varieties, with possibly irregular singularities at infinity. For this, we use the notion of ''formal boundary'', previously studied by Ben Bassat-Temkin, Efimov and Hennion-Porta-Vezzosi, as well as the moduli of flat bundles at infinity. We prove that the fibers of the restriction map to infinity are representable. We also prove that this restriction map has a canonical Lagrangian structure in the sense of shifted symplectic geometry.

I will explain and state a conjecture of Kontsevich, that relates vertex models from statistical mechanics to $\mathbb{E}_n$-algebras.
I will also give the main ingredients of the proof of Kontsevich's conjecture, which is a joint work in progress with Damien Lejay.

According to Nekrasov and Shatashvili the Coulomb vacua of four-dimensional N=2 theories of "class S'', subjected to the Omega background in two of the four dimensions, correspond to the eigenstates of a quantisation of the Hitchin integrable system. The vacua may be found as the intersection between two Lagrangian branes in the Hitchin moduli space, one of which is the space of opers (or quantum Hamiltonians) and one is defined in terms of a system of Darboux coordinates on the corresponding moduli space of flat connections.

The Euler characteristic of a compact complex manifold M is a
classical cohomological invariant. Depending on the viewpoint, it is
most natural to interpret it as an index of an elliptic differential
operator on M, or as a supersymmetric index in superconformal field
theories "on M''. Refining the Euler characteristic but keeping with both
index theoretic interpretations, one arrives at the notion of complex
elliptic genera. We argue that superconformal field theory motivates
further refinements of these elliptic genera which result in a choice