[Paper Review] Holomorphic Floer Theory and the Fueter Equation

Aleksander Doan, Semon Rezchikov|arXiv (Cornell University)|Oct 21, 2022

Geometric and Algebraic Topology4 citations

TL;DR

This paper proposes a 2-category $ \mathrm{Fuet}_{M}$ categorifying the Fukaya category of complex Lagrangians in a hyperkähler manifold $M$, using Fueter maps—solutions to a quaternionic generalization of the Cauchy-Riemann equation—on $[0,1]\times\mathbb{R}^2$ with boundary conditions on complex Lagrangians. The key result establishes a correspondence between Fueter maps and complex gradient trajectories when $M = T^*X$, linking the construction to the Fukaya–Seidel category of a holomorphic function $F: X \to \mathbb{C}$, thus generalizing Floer’s theorem to the holomorphic setting.

ABSTRACT

We outline a proposal for a $2$-category $\mathrm{Fuet}_M$ associated to a hyperkähler manifold $M$, which categorifies the subcategory of the Fukaya category of $M$ generated by complex Lagrangians. Morphisms in this $2$-category are formally the Fukaya--Seidel categories of holomorphic symplectic action functionals. As such, $\mathrm{Fuet}_M$ is based on counting maps to $M$ satisfying the Fueter equation with boundary values on holomorphic Lagrangians. We make the first step towards constructing this category by establishing some basic analytic results about Fueter maps, such as the energy bound and maximum principle. When $M=T^*X$ is the cotangent bundle of a Kähler manifold $X$ and $(L_0, L_1)$ are the zero section and the graph of the differential of a holomorphic function $F: X o \mathbb{C}$, we prove that all Fueter maps correspond to the complex gradient trajectories of $F$ in $X$, which relates our proposal to the Fukaya--Seidel category of $F$. This is a complexification of Floer's theorem on pseudo-holomorphic strips in cotangent bundles. Throughout the paper, we suggest problems and research directions for analysts and geometers that may be interested in the subject.

Motivation & Objective

• To develop a holomorphic categorification of Lagrangian Floer homology using Fueter maps on 3-manifolds with boundary conditions on complex Lagrangians.
• To establish analytic foundations for Fueter maps, including energy bounds and maximum principles, enabling future construction of the proposed 2-category.
• To demonstrate a correspondence between Fueter maps and complex gradient trajectories when $M = T^*X$, linking the theory to known results in symplectic topology.
• To suggest new research directions in analysis, geometry, and mathematical physics related to the Fueter 2-category and its connections to 3d mirror symmetry.

Proposed method

• Formal construction of a 2-category $\mathrm{Fuet}_M$ whose morphisms are defined via Fukaya–Seidel categories of holomorphic symplectic action functionals on $M$.
• Use of Fueter maps—solutions to the equation $I(U)\partial_\tau U + J(U)\partial_s U + K(U)\partial_t U = 0$—on $[0,1]\times\mathbb{R}^2$ with boundary values on complex Lagrangians.
• Adoption of a complexified version of Floer’s theorem, showing that Fueter maps correspond to complex gradient trajectories of a holomorphic function $F: X \to \mathbb{C}$ when $M = T^*X$.
• Introduction of taming triples to control behavior at infinity and ensure compactness in the analytic setup.
• Use of $\mathbb{C}^*$-symmetry to relate solutions under rescaling of the holomorphic function $F$, preserving the structure of the gradient flow equations.
• Perturbation of the holomorphic action functional $\mathscr{A}$ by $I$-holomorphic functions $F$, leading to a perturbed Fueter equation involving $\nabla f$ with $f = \mathrm{Re}(F)$.

Experimental results

Research questions

• RQ1 How can a 2-category be constructed that categorifies the subcategory of the Fukaya category generated by complex Lagrangians in a hyperkähler manifold?
• RQ2 What is the precise correspondence between Fueter maps and complex gradient trajectories in cotangent bundles of Kähler manifolds?
• RQ3 What analytic properties—such as energy bounds and maximum principles—hold for Fueter maps, and how do they support compactness and regularity?
• RQ4 How does the Fueter 2-category relate to known structures in 3d mirror symmetry and generalized Seiberg–Witten theory?
• RQ5 What are the implications of the $\mathbb{C}^*$-symmetry of the complex gradient flow equation for the moduli space of solutions?

Key findings

• Fueter maps from $[0,1]\times\mathbb{R}^2$ to a hyperkähler manifold $M$ with boundary values on complex Lagrangians are shown to satisfy an energy bound and a maximum principle, establishing foundational analytic control.
• When $M = T^*X$ for a Kähler manifold $X$, Fueter maps correspond exactly to complex gradient trajectories of a holomorphic function $F: X \to \mathbb{C}$, generalizing Floer’s theorem to the holomorphic setting.
• The Fueter equation for maps $U: [0,1]\times\mathbb{R}^2 \to M$ is equivalent to the anti-complex gradient flow equation $\partial_s U - I(U)(\partial_t U + \nabla \mathcal{A}_J(U)) = 0$, linking it to holomorphic Floer theory.
• The perturbed Fueter equation $\partial_s U - I(U)(\partial_t U - J(U)\partial_\tau U + \nabla f(U)) = 0$ arises from perturbing the holomorphic action functional by $\mathrm{Re}(F)$, preserving the structure of the theory.
• The $\mathrm{SO}(3)$ symmetry of the Fueter equation—rotating the complex structures—implies invariance under $\mathrm{U}(1)$-rotations of the $(s,t)$-plane, preserving the equation in rotated coordinates.
• The complex gradient flow equation $\partial_s u + I(u)\partial_t u + \nabla g(u) = 0$ (with $g = \mathrm{Im}(F)$) is shown to be equivalent to the Hamiltonian flow of $g$ under the symplectic form $\omega_J$, linking holomorphic Floer theory to symplectic geometry.