Chern-Simons factorization algebras and knot polynomials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915802458882048 |
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| author | Costello, Kevin Francis, John Gwilliam, Owen |
| author_facet | Costello, Kevin Francis, John Gwilliam, Owen |
| contents | This work identifies the Reshetikhin-Turaev invariant of links in terms of a trace map on factorization homology. In particular, to recover the knot invariants associated to Chern-Simons theories, we construct a filtered $\mathcal{E}_3$-algebra $\mathcal{A}^λ$ by BV quantization of Chern-Simons theory for a semi-simple Lie algebra ${\frak g}$ with invariant pairing~$λ$, and we prove that a finite-dimensional representation $V$ of the Drinfeld-Jimbo quantum group $U_\hbar{\frak g}$ defines a perfect $\mathcal{A}^λ$ module~$\mathcal{V}$. For any framed link $K$ in $\mathbb{R}^3$, we then prove that there is an equality \[\int_{K\subset\mathbb{R}^3}{\rm tr}(V) = Z_V(K\subset\mathbb{R}^3) \] between the factorization homology trace for $V$ and the Reshetikhin-Turaev link invariant determined by~$V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12412 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Chern-Simons factorization algebras and knot polynomials Costello, Kevin Francis, John Gwilliam, Owen Quantum Algebra Mathematical Physics Algebraic Topology Geometric Topology This work identifies the Reshetikhin-Turaev invariant of links in terms of a trace map on factorization homology. In particular, to recover the knot invariants associated to Chern-Simons theories, we construct a filtered $\mathcal{E}_3$-algebra $\mathcal{A}^λ$ by BV quantization of Chern-Simons theory for a semi-simple Lie algebra ${\frak g}$ with invariant pairing~$λ$, and we prove that a finite-dimensional representation $V$ of the Drinfeld-Jimbo quantum group $U_\hbar{\frak g}$ defines a perfect $\mathcal{A}^λ$ module~$\mathcal{V}$. For any framed link $K$ in $\mathbb{R}^3$, we then prove that there is an equality \[\int_{K\subset\mathbb{R}^3}{\rm tr}(V) = Z_V(K\subset\mathbb{R}^3) \] between the factorization homology trace for $V$ and the Reshetikhin-Turaev link invariant determined by~$V$. |
| title | Chern-Simons factorization algebras and knot polynomials |
| topic | Quantum Algebra Mathematical Physics Algebraic Topology Geometric Topology |
| url | https://arxiv.org/abs/2602.12412 |