Holomorphic Floer Theory and the Fueter Equation


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


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


          @misc{ pirsa_PIRSA:22040124,
            doi = {10.48660/22040124},
            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}}

Semon Rezchikov Harvard University


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.