2024/12/17 by Bleher, Michael
#53C07 #57R58 #81T13 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2412.13285
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 HFθ(W4) 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 M5=ℝs× W4 with nowhere-vanishing vector field v=cosθ ∂s+sinθ w, the Haydys-Witten equations provide flow equations for the θ-Kapustin-Witten equations on W4. 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 W4. 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 HFθ(W4) are defined in analogy with Yang-Mills instanton Floer theory: solutions of the θ-Kapustin-Witten equations on W4 modulo Haydys-Witten instantons on the cylinder ℝs× W4 interpolating between them. The relation to knot invariants observed by Witten arises when the four-manifold is the geometric blow-up W4=[X3×ℝ+,K] along a knot K⊂ X3×0 in its three-dimensional boundary. This yields a precise restatement of Witten's conjecture as the equality between HF^\bulletπ/2([S3×ℝ+,K]) and Khovanov homology Kh^\bullet(K).