Adiabatic Solutions of the Haydys-Witten Equations and Symplectic Khovanov Homology
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| Format: | Preprint |
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2025
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| _version_ | 1866915765096022016 |
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| author | Bleher, Michael |
| author_facet | Bleher, Michael |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_01365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adiabatic Solutions of the Haydys-Witten Equations and Symplectic Khovanov Homology Bleher, Michael Mathematical Physics Geometric Topology 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. |
| title | Adiabatic Solutions of the Haydys-Witten Equations and Symplectic Khovanov Homology |
| topic | Mathematical Physics Geometric Topology |
| url | https://arxiv.org/abs/2501.01365 |