# Holomorphic Floer Theory and the Fueter Equation

### APA

Rezchikov, S. (2022). Holomorphic Floer Theory and the Fueter Equation . Perimeter Institute. https://pirsa.org/22040124

### MLA

Rezchikov, Semon. Holomorphic Floer Theory and the Fueter Equation . Perimeter Institute, Apr. 22, 2022, https://pirsa.org/22040124

### BibTex

@misc{ pirsa_22040124, doi = {}, url = {https://pirsa.org/22040124}, author = {Rezchikov, Semon}, keywords = {Mathematical physics}, language = {en}, title = {Holomorphic Floer Theory and the Fueter Equation }, publisher = {Perimeter Institute}, year = {2022}, month = {apr}, note = {PIRSA:22040124 see, \url{https://pirsa.org}} }

## Abstract

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.