This work proposes that adiabatic solutions of the decoupled Haydys-Witten equations on a five-dimensional product correspond to non-vertical paths in the moduli space of extended-Bogomolny-equation solutions fibred over monopole positions, giving a concrete handle on Witten's gauge-theoretic construction of Khovanov homology. It further argues that the Grothendieck-Springer resolution of the gauge Lie algebra provides a finite-dimensional model of this monopole moduli space. Students learn how the Hermitian Yang-Mills structure of the decoupled equations reduces a five-dimensional gauge-theory problem to path-following in a Higgs-bundle moduli space, and why this suggests a link between Haydys-Witten instanton Floer homology and symplectic Khovanov homology.
This mathematical-physics article contributes to Witten's gauge-theoretic program for Khovanov homology, in which a knot invariant is realized as the instanton Floer homology of a chain complex gener…