Chern-Simons factorization algebras and knot polynomials

TL;DR

This paper connects Chern-Simons theory, factorization homology, and quantum group invariants to provide a new algebraic framework for understanding knot invariants, unifying topological quantum field theory and quantum algebra.

Contribution

It constructs a filtered $ ext{E}_3$-algebra from Chern-Simons theory and proves its modules recover Reshetikhin-Turaev knot invariants via factorization homology.

Findings

Established an equality between factorization homology trace and Reshetikhin-Turaev invariants.

Constructed a new algebraic model for Chern-Simons knot invariants.

Connected topological quantum field theory with quantum group representations.

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