From BV-BFV Quantization to Reshetikhin-Turaev Invariants

TL;DR

This paper aims to connect perturbative BV-BFV quantization of Chern-Simons theory with non-perturbative Reshetikhin-Turaev invariants of 3-manifolds through advanced geometric and algebraic frameworks.

Contribution

It formulates conjectures linking BV-BFV quantization to RT invariants, proposing a unified topological quantum field theory perspective and developing a proof strategy via deformation quantization.

Findings

Supporting evidence in abelian, low-genus, and Seifert fibered cases.

Formulation of seven conjectures relating BV-BFV and RT constructions.

Development of a proof strategy involving deformation quantization of shifted symplectic stacks.

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 $\mathrm{Loc}_G(\Sigma)$ 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.