A Family of Instanton-Invariants for Four-Manifolds and Their Relation to Khovanov Homology
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908907477139456 |
|---|---|
| author | Bleher, Michael |
| author_facet | Bleher, Michael |
| contents | 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 $HF_θ\bigl(W^4\bigr)$ 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 $M^5=\mathbb{R}_s\times W^4$ with nowhere-vanishing vector field $v=\cosθ\partial_s+\sinθw$, the Haydys-Witten equations provide flow equations for the $θ$-Kapustin-Witten equations on $W^4$. 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 $W^4$. 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 $HF_θ\bigl(W^4\bigr)$ are defined in analogy with Yang-Mills instanton Floer theory: solutions of the $θ$-Kapustin-Witten equations on $W^4$ modulo Haydys-Witten instantons on the cylinder $\mathbb{R}_s\times W^4$ interpolating between them. The relation to knot invariants observed by Witten arises when the four-manifold is the geometric blow-up $W^4=\bigl[X^3\times\mathbb{R}^+,K\bigr]$ along a knot $K\subset X^3\times{0}$ in its three-dimensional boundary. This yields a precise restatement of Witten's conjecture as the equality between $HF^\bullet_{π/2}\bigl(\bigl[S^3\times\mathbb{R}^+,K\bigr]\bigr)$ and Khovanov homology $\mathrm{Kh}^\bullet(K)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13285 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Family of Instanton-Invariants for Four-Manifolds and Their Relation to Khovanov Homology Bleher, Michael Mathematical Physics Differential Geometry 57R58, 53C07, 81T13 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 $HF_θ\bigl(W^4\bigr)$ 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 $M^5=\mathbb{R}_s\times W^4$ with nowhere-vanishing vector field $v=\cosθ\partial_s+\sinθw$, the Haydys-Witten equations provide flow equations for the $θ$-Kapustin-Witten equations on $W^4$. 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 $W^4$. 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 $HF_θ\bigl(W^4\bigr)$ are defined in analogy with Yang-Mills instanton Floer theory: solutions of the $θ$-Kapustin-Witten equations on $W^4$ modulo Haydys-Witten instantons on the cylinder $\mathbb{R}_s\times W^4$ interpolating between them. The relation to knot invariants observed by Witten arises when the four-manifold is the geometric blow-up $W^4=\bigl[X^3\times\mathbb{R}^+,K\bigr]$ along a knot $K\subset X^3\times{0}$ in its three-dimensional boundary. This yields a precise restatement of Witten's conjecture as the equality between $HF^\bullet_{π/2}\bigl(\bigl[S^3\times\mathbb{R}^+,K\bigr]\bigr)$ and Khovanov homology $\mathrm{Kh}^\bullet(K)$. |
| title | A Family of Instanton-Invariants for Four-Manifolds and Their Relation to Khovanov Homology |
| topic | Mathematical Physics Differential Geometry 57R58, 53C07, 81T13 |
| url | https://arxiv.org/abs/2412.13285 |