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Abstract

An influential conjecture by Witten states that there is a Floer theory based on Haydys-Witten instantons that provides a gauge theoretic approach to Khovanov homology. This thesis explores a novel approach towards a potential proof of this claim. One of the key insights is the existence of a Hermitian Yang-Mills structure for a ‘decoupled’ version of the Haydys-Witten and Kapustin-Witten equations. It is shown that, in favourable circumstances, any Haydys-Witten solution is already a solution of the decoupled equations. This utilizes a dichotomy that is proved to be satisfied by θ-Kapustin-Witten solutions on any ALE or ALF space, generalizing a corresponding result on ℝ^4. The Hermitian Yang-Mills structure gives rise to a Kobayashi-Hitchin-like correspondence. It is proposed that solutions are classified by intersections of Lagrangian submanifolds in the moduli space of solutions of the extended Bogomolny equations. In that interpretation, Haydys-Witten instantons are in correspondence with pseudo-holomorphic discs, leading to a conjectural equivalence with a Lagrangian intersection Floer homology. A physically motivated argument suggests that the latter is fully determined in a finite-dimensional model space, given by a Grothendieck-Springer resolution of the nilpotent cone inside the underlying Lie algebra. This provides a relation to symplectic Khovanov homology, which is known to be isomorphic to a grading-reduced version of Khovanov homology.