Holomorphic Floer Theory and the Fueter Equation

TL;DR

The paper develops a complexified, higher-categorical extension of Floer theory for hyperkähler manifolds by introducing the Fueter 2-category Fuet_M, whose objects are complex Lagrangians and whose morphisms arise from a holomorphic symplectic action functional and Fueter maps. It builds a bridge between infinite-dimensional complex Morse theory and holomorphic Floer theory, clarifying how Fukaya–Seidel data can be modeled in this setting and how decategorifications relate to classical Fukaya categories and loop-space invariants. A key analytic backbone is provided by energy identities, maximum principles, and a quaternionic convexity theory for Fueter maps, alongside a concrete cotangent-bundle setting (T^*X) where Fuet_{T^*X}(L0,L1) is conjectured to recover FS(X,F) for L0 as the zero section and L1 as the graph of dF. The work also situates these constructions within 3d mirror symmetry, predicting deep correspondences with Kapustin–Rozansky–Saulinas’s 2-categories and outlining a program to define and compute Fuet via complex Morse theory, with implications for toric and representation-theoretic contexts.

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 $F: X \to \mathbb{C}$, 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.

Figures (13)

• Figure 2: Vertical compostion in the Fueter $2$-category. Domain is $\mathbb{C} \times [0,1]$, boundary conditions as on the right. Grey denotes Lagrangian boundary conditions, blue denotes asymptotic pseudoholomorphic strips.
• Figure 3: Horizontal composition in the Fueter $2$-category. Domain is $\mathbb{R} \times (\mathbb{D}^2 \setminus \{3 \text{ points}\})$. As before, Lagrangians are boundary conditions while pseudo-holomorphic triangles and strips are asymptotic conditions.
• Figure 4: The Fukaya-Seidel category. Objects are closed Lagrangians and non-compact Lagrangians with prescribed asymptotic behavior. All objects can be computed from the distinguished objects corresponding to the Lefschetz thimbles.
• Figure 5: The complex Morse theory model $\mathrm{FS}(X,F)$ of the Fukaya-Seidel category does not involve Lagrangian submanifolds of $X$. The generating objects are taken to be critical points of $F$, and the morphisms are generated by gradient flows of $\mathrm{Re}(e^{-i \theta} F)$ between these critical points. In simple cases, these are naturally in bijection with intersection points between the thimbles, as shown above.
• Figure 6: Composition in the Fukaya--Seidel category is given by counts of pseudoholomorphic triangles.

Theorems & Definitions (81)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 2: Energy bound
• Remark 5
• Theorem 3: Maximum principle
• Remark 6
• Remark 7