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

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.

💡 Research Summary

The paper tackles Witten’s conjecture that a four‑dimensional instanton Floer homology defined on manifolds with corners coincides with Khovanov homology of a knot K. The conjectural Floer complex is generated by Nahm‑pole solutions of the Kapustin‑Witten (KW) equations on (\mathbb{R}^{3}\times\mathbb{R}^{+}{y}) with a monopole‑like singularity along K on the boundary (y=0). The differential counts solutions of the Haydys‑Witten (HW) equations on (\mathbb{R}{s}\times\mathbb{R}^{3}\times\mathbb{R}^{+}_{y}) that interpolate between two KW solutions as the flow parameter (s) goes from (-\infty) to (+\infty).

The author studies a decoupled version of the HW equations (dHW) on the same five‑dimensional space. Unlike the full HW system, dHW exhibits a Hermitian Yang‑Mills (HYM) structure: the equations split into a complex‑gauge‑invariant part (