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Chern-Simons factorization algebras and knot polynomials

Kevin Costello, John Francis, Owen Gwilliam

TL;DR

The paper provides a rigorous perturbative realization of Reshetikhin–Turaev knot invariants within the framework of factorization homology by quantizing Chern–Simons theory via the BV formalism. It constructs a filtered $\\mathcal{E}_3$-algebra $\\mathcal{A}^{\\lambda}$ of quantum observables and a perfect $\\mathcal{A}^{\\lambda}$-module $\\mathcal{V}$ so that the trace of $V$ on framed links coincides with RT invariants, i.e., $\\int_{K} {\\rm tr}(V)=Z_V(K)$. The authors develop a detailed correspondence between perturbative CS quantizations and braided monoidal deformations of $Rep_{fin}(\\mathfrak g)$, establish a bridge via Koszul duality and beta-factorization homology for line defects, and construct explicit fermionic defects that realize Wilson loops in representations. They further show that the perturbative invariants recover classical knot polynomials (Jones/HOMFLYPT) as $\\hbar$-expansions matching RT-type constructions with Drinfeld–Jimbo quantum groups $U_{\\hbar}(\\mathfrak g)$. The work thus provides a comprehensive, structurally rich pathway from perturbative CS theory to quantum-group knot invariants through higher algebra and constructible factorization algebras. The framework has potential to generalize to other TQFTs and defect setups, clarifying how higher-categorical structures govern topological observables.

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 $\mathcal{A}^λ$ 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~$V$.

Chern-Simons factorization algebras and knot polynomials

TL;DR

The paper provides a rigorous perturbative realization of Reshetikhin–Turaev knot invariants within the framework of factorization homology by quantizing Chern–Simons theory via the BV formalism. It constructs a filtered -algebra of quantum observables and a perfect -module so that the trace of on framed links coincides with RT invariants, i.e., . The authors develop a detailed correspondence between perturbative CS quantizations and braided monoidal deformations of , establish a bridge via Koszul duality and beta-factorization homology for line defects, and construct explicit fermionic defects that realize Wilson loops in representations. They further show that the perturbative invariants recover classical knot polynomials (Jones/HOMFLYPT) as -expansions matching RT-type constructions with Drinfeld–Jimbo quantum groups . The work thus provides a comprehensive, structurally rich pathway from perturbative CS theory to quantum-group knot invariants through higher algebra and constructible factorization algebras. The framework has potential to generalize to other TQFTs and defect setups, clarifying how higher-categorical structures govern topological observables.

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 -algebra by BV quantization of Chern-Simons theory for a semi-simple Lie algebra with invariant pairing~, and we prove that a finite-dimensional representation of the Drinfeld-Jimbo quantum group defines a perfect module~. For any framed link in , we then prove that there is an equality between the factorization homology trace for and the Reshetikhin-Turaev link invariant determined by~.
Paper Structure (60 sections, 49 theorems, 219 equations, 1 figure)

This paper contains 60 sections, 49 theorems, 219 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A guiding dictionary and our central result
  3. Sharpening the question
  4. The appearance of higher algebra
  5. Why are there local observables in Chern--Simons theory?
  6. Connecting with categories
  7. The main identification
  8. Wilson loops and modules for an $\mathcal{E}_3$-algebra
  9. Modules for an $\mathcal{E}_3$-algebra and knot invariants
  10. Back to knot polynomials
  11. Section by section outline
  12. Acknowledgements
  13. Links and factorization homology
  14. Recollections on filtered Koszul duality
  15. Recollections
  16. ...and 45 more sections

Key Result

Theorem 1.1

For each invariant pairing on a semisimple Lie algebra ${\mathfrak{g}}$, BV quantization of Chern--Simons theory gives a filtered $\mathcal{E}_3$-algebra $\mathcal{A}^\lambda$. A choice of finite-dimensional representation $V$ of the Drinfeld--Jimbo quantum group ${\rm U}_\hbar{\mathfrak{g}}$ define between the factorization homology trace and the Reshetikhin--Turaev invariant of the link.

Figures (1)

  • Figure 1: The factorization product map.

Theorems & Definitions (109)

  • Remark 1
  • Theorem 1.1
  • Theorem 1.2: Thm. 14.2.1, Chapter 5, CosBook
  • Theorem 1.3: Drinfeld
  • Proposition
  • Lemma
  • Theorem 1.4
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 99 more