Mathematical Physics

The Lagrangian Floer homology of a pair of holomorphic Lagrangian submanifolds of a hyperkahler manifold is expected to simplify, by work of Solomon-Verbitsky and others. This occurs in part because, in this setting, the symplectic action functional, the gradient flow of which computes Lagrangian Floer homology, is the real part of a holomorphic function. As noted by Haydys, thinking of this holomorphic function as a superpotential on an infinite-dimensional symplectic manifold gives rise to a quaternionic analog of Floer's equation for holomorphic strips: the Fueter equation. I will explain how this line of thought gives rise to a `complexification' of Floer's theorem identifying Fueter maps in cotangent bundles to Kahler manifolds with holomorphic planes in the base. This complexification has a conjectural categorical interpretation, giving a model for Fukaya-Seidel categories of Lefshetz fibrations, which should have algebraic implications for the study of Fukaya categories. This is a report on upcoming joint work with Aleksander Doan.

I will discuss a method to associate new types of quantum groups to certain holomorphic Lagrangian fibrations. This algebraic structure controls the enumerative geometry of the variety admitting the Lagrangian fibration in a way analogous to how a quantum group controls the enumerative geometry of a symplectic singularity. Various questions, including the monodromy of the quantum differential equation, can be answered using local-to-global techniques.

Zoom Link: https://pitp.zoom.us/j/99571261442?pwd=NDZTL1dIbXJPNHZwWVcvSEM4M2crZz09

 I will discuss a brane construction of rational gl(m|n) spin chains with spins valued in Verma modules and show that string dualities map the spin chain spectra to the vacua of certain 2d N=(2,2) gauge theories, thus realizing the 2d Bethe/Gauge correspondence for superspin chains. The talk is based on arXiv:2110.15112.

Zoom Link: https://pitp.zoom.us/j/98827189404?pwd=blNjdy9oYnc2cGUrQXdwV2xoOVpOdz09