Abstracts for Conference "Higher algebra and mathematical physics", August 13 - 17, 2018

The idea of topologically twisting a supersymmetric field theory was introduced in the physics literature in order to generate interesting new examples of topological field theories. The idea is very general, but systematically realising the examples it produces using mathematical models for topological quantum field theory (such as the functorial axioms of Atiyah-Segal or the theory of $E_n$-algebras) is not always possible.

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.

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.