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Fukaya-Seidel category and gauge theory

Andriy Haydys

TL;DR

The work develops a unified gauge-theoretic approach to Fukaya–Seidel-type structures by (i) formulating a finite-dimensional Fukaya–Seidel $A_ fty$-category for symplectic Lefschetz fibrations via antigradient flow lines of $ ext{Re}igl(e^{i heta}figr)$ and (ii) embedding this into an infinite-dimensional five-dimensional gauge theory whose reductions recover Floer-type, Hitchin, and Vafa–Witten-type frameworks. It establishes key analytic results—such as a priori $C^0$ and $C^ abla$-estimates, an action-energy identity, exponential decay, compactness up to breaking, and Fredholm properties—for the perturbed equations, ensuring a well-defined moduli theory. Dimensional reductions to four and three dimensions yield gradient-flow interpretations and Hitchin-type structures, linking the 5D theory to known invariants and suggesting conjectural invariants: integers for 5-manifolds, Floer-type homology for 4-manifolds, and Fukaya–Seidel-type categories for 3-manifolds. The paper thus offers a framework to relate symplectic and gauge-theoretic techniques in low-dimensional topology, with potential applications to extended quantum field theories and Calabi–Yau geometry as well as a conjectural pathway to Calabi–Yau threefold contexts via complex Chern–Simons functionals.

Abstract

Given a J-holomorphic Morse function on a symplectic manifold, a new construction of the Fukaya-Seidel category is outlined. Applying this construction in an infinite dimensional case, a Fukaya-Seidel-type category is associated to a smooth three-manifold. In this case the construction is based on a five-dimensional gauge theory.

Fukaya-Seidel category and gauge theory

TL;DR

The work develops a unified gauge-theoretic approach to Fukaya–Seidel-type structures by (i) formulating a finite-dimensional Fukaya–Seidel -category for symplectic Lefschetz fibrations via antigradient flow lines of and (ii) embedding this into an infinite-dimensional five-dimensional gauge theory whose reductions recover Floer-type, Hitchin, and Vafa–Witten-type frameworks. It establishes key analytic results—such as a priori and -estimates, an action-energy identity, exponential decay, compactness up to breaking, and Fredholm properties—for the perturbed equations, ensuring a well-defined moduli theory. Dimensional reductions to four and three dimensions yield gradient-flow interpretations and Hitchin-type structures, linking the 5D theory to known invariants and suggesting conjectural invariants: integers for 5-manifolds, Floer-type homology for 4-manifolds, and Fukaya–Seidel-type categories for 3-manifolds. The paper thus offers a framework to relate symplectic and gauge-theoretic techniques in low-dimensional topology, with potential applications to extended quantum field theories and Calabi–Yau geometry as well as a conjectural pathway to Calabi–Yau threefold contexts via complex Chern–Simons functionals.

Abstract

Given a J-holomorphic Morse function on a symplectic manifold, a new construction of the Fukaya-Seidel category is outlined. Applying this construction in an infinite dimensional case, a Fukaya-Seidel-type category is associated to a smooth three-manifold. In this case the construction is based on a five-dimensional gauge theory.

Paper Structure

This paper contains 17 sections, 29 theorems, 169 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fukaya--Seidel categories of symplectic Lef-schetz fibrations
  3. Symplectic Lefschetz fibrations
  4. Outline of the construction
  5. A priori $C^0$-estimates
  6. The action functional and the energy identity
  7. A priory $C^\infty$--estimates
  8. Asymptotic behaviour
  9. Compactness
  10. Fredholm property
  11. A gauge theory on 5-manifolds
  12. Dimensional reductions
  13. Dimension four
  14. Dimension three
  15. Invariants
  16. ...and 2 more sections

Key Result

Proposition 2.4

Suppose the closed domain $G$ bounded by the triangle $z_-z_0z_+$ contains no critical values of $f$ other than $z_\pm$. Let $\nu_j\in (0,1)$ and $\gamma_j\in\Gamma_{\nu_j}(m_-, m_+)$ be arbitrary sequences such that $\nu_j\to 0$. Then there exists a subsequence $j_k\to\infty$ such that $\gamma_{j_k

Figures (2)

  • Figure 1: The domain $\Omega$ with three long necks.
  • Figure 2: Graph of $\theta_\nu$.

Theorems & Definitions (65)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Theorem 2.7: Energy identity
  • proof
  • Proposition 2.8
  • ...and 55 more