Mathematical Physics

Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in single formal automorphism of the cluster variety, called the DT-transformation. An oriented surface S with punctures, and a finite number of special points on the boundary give rise to a moduli space, closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. We determine the DT-transformation of this space. This is a joint work with Alexander Goncharov.

I will present some results on three-dimensional gauge theory from the point of view of extended topological field theory. In this setting a theory is specified by describing its collection of boundary conditions - in our case, a collection of categories (standing in for 2d TFTs) with a prescribed symmetry group G. We will apply ideas from Seiberg-Witten geometry to construct a new commutative algebra of symmetries for categorical representations (or line operators in the gauge theory) -  a categorification of Kostant's description of the center of the enveloping algebra. (Joint with Sam Gunningham and David Nadler)

The notion of singular support for coherent sheaves was introduced by Arinkin and Gaitsgory in order to carefully state the geometric Langlands conjecture.  This is a conjectural equivalence of categories of sheaves on certain moduli spaces: in order to make the conjecture reasonable one needs to restrict to sheaves which satisfy a certain "singular support condition".  In this talk I'll explain how to think about this singular support condition from the point of view of boundary conditions in twisted N=4 gauge theory.  Specifically, Arinkin and Gaitsgory's singular support condition arises by considering only those boundary conditions which are compatible with a natural choice of vacuum state.  By allowing this vacuum state to move away from this natural choice we see aspects of a rich additional structure for the geometric Langlands correspondence.  This work is joint with Philsang Yoo.