A Family of Instanton-Invariants for Four-Manifolds and Their Relation to Khovanov Homology

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 $H{F}_{\theta }\left(W^4\right)$ 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 $v=\cos \theta {\partial }_s+\sin \theta w$, the Haydys-Witten equations provide flow equations for the $\theta$-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 $\theta$-Kapustin-Witten equations with twisted Nahm-pole boundary conditions, which are required to establish elliptic regularity. Motivated by these insights, the groups $H{F}_{\theta }\left(W^4\right)$ are defined in analogy with Yang-Mills instanton Floer theory: solutions of the $\theta$-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 $H{F}_{\pi /2}^{\bullet }\left(\right[S^3\times {\mathbb{R}}^{+},K\left]\right)$ and Khovanov homology ${\mathrm{Kh}}^{\bullet }(K)$.

This work is part of the author's PhD thesis at Heidelberg University