From BV-BFV Quantization to Reshetikhin-Turaev Invariants

Abstract

We propose a program for bridging the gap between the perturbative BV-BFV quantization of Chern-Simons theory and the non-perturbative Reshetikhin-Turaev (RT) invariants of 3-manifolds, passing through factorization homology of $\mathbb{E}_n$-algebras and the derived algebraic geometry of character stacks. We conjecture that the modular tensor category underlying the RT construction arises as the $\mathbb{E}_2$-category from BV-BFV quantization of Chern-Simons theory on the disk, with the derived character stack $\mathrm{Loc}_G(Σ)$ and its shifted symplectic structure mediating the proposed identification. We formulate seven conjectures, including a main conjecture asserting natural equivalence of the BV-BFV and RT constructions as (3-2-1)-extended topological quantum field theories, develop a proof strategy via deformation quantization of shifted symplectic stacks, and clarify the role of $\mathbb{E}_n$-Koszul duality in translating between perturbative and non-perturbative data. Supporting evidence is examined in the abelian, low-genus, and Seifert fibered cases. Connections to resurgence, categorification, and the geometric Langlands program are discussed as further motivation, though significant technical gaps remain open.

Figures (10)

• Figure 1: Schematic overview of the program. The BV-BFV functor (top left) is an ordinary, perturbative TQFT valued in $\mathrm{\mathbf{Vect}}_\hbar$. The RT construction (top right) is a $(3\text{-}2\text{-}1)$-extended TQFT valued in $2\text{-}\mathrm{\mathbf{Vect}}$. The Main Conjecture asserts that the BV-BFV data can be promoted to a $(3\text{-}2\text{-}1)$-extended TQFT that agrees with RT. Solid arrows indicate connections supported by existing results; dashed arrows indicate conjectural identifications. The derived character stack $\mathrm{Loc}_G(\Sigma)$ sits at the center, mediating between all four frameworks.
• Figure 2: The BV-BFV framework, illustrated for a cobordism $M: \Sigma_{\mathrm{in}} \to \Sigma_1 \sqcup \Sigma_2$. The bulk 3-manifold $M$ (blue shading) carries BV data with a degree $(-1)$ symplectic structure; each boundary component (red circles) carries BFV data with a degree $0$ symplectic structure. The restriction map $\pi$ connects bulk and boundary fields, and the BV-BFV axiom states that the failure of the classical master equation in the bulk is precisely controlled by the BFV charge on the boundary.
• Figure 3: The RT invariant via surgery. (a) A closed 3-manifold $M$ is presented as surgery on a framed link $L = L_1 \cup L_2$ in $S^3$, with each component colored by a simple object $V_j$ of the MTC. (b) The RT invariant is computed by summing the colored link invariant $J_L(\vec{V})$ over all labelings by simple objects, weighted by quantum dimensions, and corrected by normalization factors depending on the signature $\sigma(L)$ and the total quantum dimension $\mathcal{D}$. Invariance under Kirby moves requires modularity of $\mathscr{C}$.
• Figure 4: Top: Factorization homology as a "continuous tensor product." The $\mathbb{E}_2$-algebra $\mathcal{A}$ is placed on each disk embedded in the surface $\Sigma$; the colimit over all disk embeddings produces the global invariant $\iint_\Sigma \mathcal{A}$. Bottom: The $\otimes$-excision property (Theorem \ref{['thm:AF-excision']}): cutting $\Sigma$ along a codimension-1 submanifold $N$ expresses the factorization homology as a relative tensor product over the data assigned to $N \times \mathbb{R}$.
• Figure 5: The PTVV hierarchy of shifted symplectic structures on character stacks (Theorem \ref{['thm:PTVV']}). A compact oriented $d$-manifold $\Sigma$ gives rise to $\mathrm{Loc}_G(\Sigma)$ with a $(2-d)$-shifted symplectic structure. The three cases $d = 1, 2, 3$ correspond precisely to the BFV, classical symplectic, and BV structures in the BV-BFV formalism. This dimensional pattern, produced by a single mechanism (AKSZ transgression of the Killing form via Poincaré duality), is the geometric foundation of the program.

Theorems & Definitions (86)

• Definition 2.1: BV manifold
• Definition 2.2: BV quantization
• Definition 2.3: BFV manifold
• Definition 2.4: BV-BFV data
• Proposition 2.5
• proof
• Proposition 2.7: Structure of the perturbative expansion
• Remark 2.8: Structure of the Feynman diagrams
• Proposition 2.9: CMR CMR2018
• Remark 2.10