Talks

I will explain and state a conjecture of Kontsevich, that relates vertex models from statistical mechanics to En-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.

The based loop group is an infinite-dimensional manifold equipped with a Hamiltonian action of a finite dimensional torus. This was studied by Atiyah and Pressley. We investigate the Duistermaat–Heckman distribution using the theory of hyperfunctions. In applications involving Hamiltonian actions on infinite-dimensional manifolds, this theory is necessary to accommodate the existence of the infinite order differential operators which aries from the isotropy representation on the tangent spaces to fixed points. (Joint work with James Mracek)

This is a joint 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.

It is conventional wisdom among physicists that anomalies of fermionic theories measure an obstruction to the existence of a well-defined (gauge-invariant) partition function. The aim of this talk is to use the formalism of Costello and Gwilliam to show how this wisdom is instantiated for perturbative anomalies of the massless free fermion. We will show how an action of a dg Lie algebra L on the massless free fermion theory gives rise to a line bundle over the formal moduli problem corresponding to L; the anomaly is precisely the failure of this line bundle to be trivial. Our running example will be the axial symmetry of the massless free fermion.

The generalized double semion model, introduced by Freedman and Hastings, is a lattice field theory similar to the toric code, with a gapped Hamiltonian whose space of ground states depends on the topology of the ambient manifold. In this talk, I’ll explain how to calculate its low-energy limit, which forms part of a topological field theory, in terms of characteristic classes of the ambient manifold.

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. After discussing some issues complicating a direct comparison we will formulate a proposal for a precise relation between two main ingredients in the two approaches.

I will describe examples of holomorphic N=1 super-symmetric vertex algebras with small (non-zero) values of the elliptic genus. I will speculate on a relation to certain patterns in the theory of topological modular forms.