Speculations on higher Fukaya categories

TL;DR

The article develops a framework for higher Fukaya categories ${n}\mathrm{Fuk}(X)$ associated to $(n-1)$-shifted symplectic stacks, positing an inductive definition via derived Lagrangian intersections that recovers the classical Fukaya category when $n=1$. By focusing on the 1-shifted cotangent stack $\mathrm{T}^{*}[1]M$ and the coadjoint stack $[\mathfrak{g}^{*}/G]$, the authors connect higher-Fukaya theory with Teleman’s gauge-theoretic results and 3D mirror symmetry, and they supply new results in $1$-shifted symplectic geometry (e.g., symplectic fibrations, Lefschetz fibrations) in support of the program. They develop a rich narrative around local systems of categories, schobers, and intrinsic mirror symmetry, suggesting a path to a 3D HMS framework via both torus and non-abelian groups. The work also explores how completed/embedded 2-categories (KRS, IndCohShvCat) may realize 3D A/B-model dualities and tests the proposal with explicit diagrams and test Lagrangians, hinting at a unifying picture that links Lefschetz fibrations, schobers, and gauge-theoretic phenomena through higher categorical Fukaya-type structures.

Abstract

We investigate a possible theory of higher Fukaya categories associated to $n$-shifted symplectic stacks, where $n \geq 0$. We consider two paradigmatic cases, the shifted cotangent stack of a smooth manifold and the coadjoint stack of a compact Lie group, drawing connections to the work of Teleman and 3D mirror symmetry. Our evidence includes some new results in $1$-shifted symplectic geometry.

Figures (1)

• Figure 1: Structures deriving from a Lefschetz fibration.

Theorems & Definitions (19)

• Remark 1.1
• Example 2.1
• Remark 4.1
• Theorem 4.2
• Remark 4.3
• Example 4.4
• Remark 4.5
• Conjecture 4.6
• Remark 4.7
• Theorem 4.8