The Decoupled Haydys-Witten Equations and a Weitzenb\"ock Formula

TL;DR

This paper introduces a decoupled version of the Haydys-Witten equations, explores their relation to the full equations on manifolds with boundaries, and analyzes solution behaviors using a Weitzenb"ock formula and boundary conditions.

Contribution

It presents a decoupled form of the Haydys-Witten equations with a Hermitian Yang-Mills structure and analyzes their relation to the full equations using a Weitzenb"ock formula.

Findings

Decoupled equations exhibit Hermitian Yang-Mills structure.

Conditions identified for reduction from full to decoupled equations.

Detailed analysis of boundary behavior and solution expansions.

Abstract

The Haydys-Witten equations are partial differential equations on five-dimensional Riemannian manifolds that are equipped with a non-vanishing vector field $v$. Conjecturally, their solutions determine the Floer differential in a gauge-theoretic approach to Khovanov homology. This article introduces a certain decoupled version of the Haydys-Witten equations, a specialization of the Haydys-Witten equations that exhibits a Hermitian Yang-Mills structure. These equations exist whenever the vector bundle defined by the orthogonal complement of $v$ admits an almost Hermitian structure. We investigate the relation between the full Haydys-Witten equations and their decoupled version on manifolds with poly-cylindrical ends and boundaries, and find conditions under which the Haydys-Witten equations reduce to the decoupled equations. This relies on a Weitzenb\"ock-like formula that shows that the difference between the full Haydys-Witten equations and the decoupled equations is governed by the asymptotic behaviour of solutions near boundaries and non-compact ends. Regarding the analysis near boundaries, we provide a detailed analysis of the polyhomogeneous expansion of Haydys-Witten solutions with twisted Nahm pole boundary conditions, generalizing work of Siqi He in the untwisted case. The corresponding analysis at non-compact ends relies on a vanishing theorem by Nagy-Oliveira that was recently generalized by the author.