Locality in the Fukaya category of a hyperk\"ahler manifold

TL;DR

The paper proves a locality property of the Fukaya category in hyperk"ahler manifolds, showing holomorphic Lagrangians are unobstructed and their Fukaya $A_inite$ algebra is formal, with implications for special Lagrangian conditions.

Contribution

It establishes the locality of the Fukaya category in hyperk"ahler manifolds and proves holomorphic Lagrangians are unobstructed and formal.

Findings

Holomorphic Lagrangians are unobstructed in the Fukaya category.

The Fukaya $A_inite$ algebra of a holomorphic Lagrangian is formal.

Holomorphic Lagrangians satisfy the special Lagrangian condition without instanton corrections.

Abstract

Let $(M,I,J,K,g)$ be a hyperk\"ahler manifold. Then the complex manifold $(M,I)$ is holomorphic symplectic. We prove that for all real $x, y,$ with $x^2 + y^2 = 1$ except countably many, any finite energy $(xJ+yK)$-holomorphic curve with boundary in a collection of $I$-holomorphic Lagrangians must be constant. By an argument based on the Lojasiewicz inequality, this result holds no matter how the Lagrangians intersect each other. It follows that one can choose perturbations such that the holomorphic polygons of the associated Fukaya category lie in an arbitrarily small neighborhood of the Lagrangians. That is, the Fukaya category is local. We show that holomorphic Lagrangians are tautologically unobstructed in the sense of Fukaya-Oh-Ohta-Ono. Moreover, the Fukaya $A_\infty$ algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.