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

Abstract

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 Kapustin-Witten and 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.

Figures (7)

• Figure 1: A general knot $K$ in the boundary of $W^4 = S^1_t \times \Sigma \times \mathbb{R}^+_y$ varies with time. The adiabatic approach can be viewed as stretching the size of $S^1_t$, such that at any given time $t$, the fields $(A,\phi)$ are well-approximated by a solution of the extended Bogomolny equations.
• Figure 2: Isotopy $\beta_\bullet$ interpolating between the $S^1_t$-invariant strand $\beta_0$ centred at the origin of $\mathbb{C}$ and a non-trivial single stranded knot $\beta_1$. The isotopy describes a homotopy of trajectories $\zeta_q(t)$ that interpolate from the constant to the original trajectory $z_0(t)$.
• Figure 3: Illustration of the covering of $S^1_t \times \Sigma \times \mathbb{R}^+_y$ that is used in the iterative construction of approximate solutions of the decoupled Kapustin-Witten equations for a multi-stranded and time-dependent knot.
• Figure 4: The fixed points of rescaled parallel transport $h_\beta^{\mathrm{resc}}: \mathcal{Y}_{\pi,\rho,D} \rightarrow \mathcal{Y}_{\pi,\rho,D}$ along a pure braid $\beta$ determine inequivalent horizontal lifts of $\beta$. Crosses in the
• Figure 5: Left: Every knot $K$ can be viewed as closure of a bipartite braid $\beta \times \textrm{id}$ embedded in $[-L,L]_t \times \mathbb{C}$ by gluing in cups and caps. Right: Gluing instructions for cups and caps are captured by a crossingless matching $\mathfrak{m}$ consisting of $k$ disjoint arcs $\delta_i \subset \mathbb{C}$.

Theorems & Definitions (17)

• Theorem : He2019a
• Definition 1: Nahm Pole Boundary Condition
• Definition 2: Moduli Space of EBE Solutions
• Remark
• Definition 3: Higgs Bundle
• Definition 4: Effective Triples
• Theorem 1: He2019cHe2020b
• Definition 5: Moduli Space of decoupled Kapustin-Witten solutions
• Remark
• Remark