Chern-Simons factorization algebras and knot polynomials

Abstract

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 ${\mathcal{A}}^{\hat{I}}»$ by BV quantization of Chern-Simons theory for a semi-simple Lie algebra $\mathfrak{g}$ with invariant pairing~$\hat{I}»$, and we prove that a finite-dimensional representation $V$ of the Drinfeld-Jimbo quantum group $U_{\hslash }\mathfrak{g}$ defines a perfect ${\mathcal{A}}^{\hat{I}}»$ 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~$V$.

75 pages; preliminary version, comments welcome