A family of instanton-invariants for four-manifolds and their relation to Khovanov homology
Michael Bleher
TL;DR
The paper advances a gauge-theoretic program linking four-manifold invariants to Khovanov homology by formulating a one-parameter family of Haydys-Witten Floer groups $HF_{\theta}(W^4)$ and detailing their origin from dimensional reductions of the Haydys-Witten equations. It develops the 5D/4D reduction framework to KW, VW, EBE, and Nahm-type equations, provides an elliptic-regularity analysis via indicial sets, and defines Nahm-pole boundary conditions with knot singularities within a Nahm-pole–TEBE boundary framework. A central contribution is the construction of Haydys-Witten homology as a Floer-type invariant, with a chain complex generated by KW solutions and differentials given by HW instantons, plus the demonstration that at $\theta=\pi/2$ the invariant recovers Khovanov homology for knots via the blow-up construction $W^4=[X^3\times\mathbb{R}^+,K]$. The work thus provides a concrete realization of Witten’s proposal that gauge-theoretic invariants of 4-manifolds capture knot homology data, and it establishes a robust analytic and topological framework for future rigorous treatment and applications. $HF_{\theta}(W^4)$ serves as a topological invariant of $W^4$ (and a functor on suitable 5D cobordisms), with the special case $W^4=[S^3\times \mathbb{R}^+,K]$ yielding a direct gauge-theoretic interpretation of Kh$^\bullet(K)$ when $\theta=\pi/2$.
Abstract
This article provides a review of the gauge-theoretic approach to Khovanov homology, framed in terms of a generalisation of Witten's original proposal. Concretely, the physical arguments underlying Witten's insights suggest that there is a one-parameter family of Haydys-Witten instanton Floer homology groups $HF_θ(W^4)$ for four-manifolds. At the heart of the proposal is a systematic investigation of the dimensional reductions of the Haydys-Witten equations. It is shown that on the five-dimensional cylinder $M^5=\mathbb{R}_s\times W^4$ with nowhere-vanishing vector field $v=\cosθ\, \partial_s+\sinθ\, w$, the Haydys-Witten equations provide flow equations for the $θ$-Kapustin-Witten equations on $W^4$. Similar reductions to lower dimensions include the twisted extended Bogomolny equations on three-manifolds and the twisted octonionic Nahm equations on one-manifolds, whose solutions provide natural boundary conditions along the boundary and corners of $W^4$. These reductions determine the indicial roots of the Haydys-Witten and $θ$-Kapustin-Witten equations with twisted Nahm-pole boundary conditions, which are required to establish elliptic regularity. Motivated by these insights, the groups $HF_θ(W^4)$ are defined in analogy with Yang-Mills instanton Floer theory: solutions of the $θ$-Kapustin-Witten equations on $W^4$ modulo Haydys-Witten instantons on the cylinder $\mathbb{R}_s\times W^4$ interpolating between them. The relation to knot invariants observed by Witten arises when the four-manifold is the geometric blow-up $W^4=[X^3\times\mathbb{R}^+,K]$ along a knot $K\subset X^3\times{0}$ in its three-dimensional boundary. This yields a precise restatement of Witten's conjecture as the equality between $HF^\bullet_{π/2}([S^3\times\mathbb{R}^+,K])$ and Khovanov homology $Kh^\bullet(K)$.