Adiabatic Solutions of the Haydys-Witten Equations and Symplectic Khovanov Homology

TL;DR

This paper explores solutions to a simplified version of the Haydys-Witten equations, revealing connections to symplectic Khovanov homology and proposing a new approach to Witten's conjecture linking gauge theory and knot invariants.

Contribution

It introduces a decoupled version of the Haydys-Witten equations with Hermitian Yang-Mills structure and relates adiabatic solutions to extended Bogomolny equations, offering a novel perspective on Floer homology.

Findings

Decoupled Haydys-Witten equations exhibit Hermitian Yang-Mills structure.

Adiabatic solutions correspond to non-vertical paths in the extended Bogomolny equations moduli space.

Grothendieck-Springer resolution models the monopole solution space.

Abstract

An influential conjecture by Witten states that there is an instanton Floer homology of four-manifolds with corners that in certain situations is isomorphic to Khovanov homology of a given knot $K$. The Floer chain complex is generated by Nahm pole solutions of the Kapustin-Witten equations on $\mathbb{R}^3 \times \mathbb{R}^+_y$ with an additional monopole-like singular behaviour along the knot $K$ inside the three-dimensional boundary at $y=0$. The Floer differential is given by counting solutions of the Haydys-Witten equations that interpolate between Kapustin-Witten solutions along an additional flow direction $\mathbb{R}_s$. This article investigates solutions of a decoupled version of the Haydys-Witten equations on $\mathbb{R}_s \times \mathbb{R}^3 \times \mathbb{R}^+_y$, which in contrast to the full equations exhibit a Hermitian Yang-Mills structure and can be viewed as a lift of the extended Bogomolny equations (EBE) from three to five dimensions. Inspired by Gaiotto-Witten's approach of adiabatically braiding EBE-solutions to obtain generators of the Floer homology, we propose that there is an equivalence between adiabatic solutions of the decoupled Haydys-Witten equations and non-vertical paths in the moduli space of EBE-solutions fibered over the space of monopole positions. Moreover, we argue that the Grothendieck-Springer resolution of the Lie algebra of the gauge group provides a finite-dimensional model of this moduli space of monopole solutions. These considerations suggest an intriguing similarity between Haydys-Witten instanton Floer homology and symplectic Khovanov homology and provide a novel approach towards a proof of Witten's gauge-theoretic interpretations of Khovanov homology.