From BV-BFV Quantization to Reshetikhin-Turaev Invariants

Source 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 [Formula: see text]-algebras and the derived algebraic geometry of character stacks. We conjecture that the modular tensor category underlying the RT construction arises as the [Formula: see text]-category from BV-BFV quantization of Chern–Simons theory on the disk, with the derived character stack [Formula: see text] of the complexified group and its shifted symplectic structure mediating the proposed identification. A recurring theme is the mismatch between the compact gauge group of Chern–Simons theory and the reductive algebraic group of derived algebraic geometry: we work throughout with [Formula: see text] and indicate where a Riemann–Hilbert type comparison is needed to return to the compact theory. We formulate a main conjecture asserting natural equivalence of the BV-BFV and RT constructions as [Formula: see text]-extended topological quantum field theories, together with several auxiliary conjectures and two questions, develop a proof strategy via deformation quantization of shifted symplectic stacks, and clarify the role of [Formula: see text]-Koszul duality, which relates the perturbative expansion around a flat connection to the local geometry of the character stack. We relate the perturbative expansions to the sheaf of vanishing cycles on the character stack of a [Formula: see text]-manifold, propose that a stratification of the character stack by stabilizer type organizes the trans-series decomposition, and propose that the Stokes data of the perturbative expansion should be read as a stability datum in the sense of Kontsevich–Soibelman, with the Chern–Simons functional playing the role of the central charge. Supporting evidence is examined in the abelian, low-genus, and Seifert fibered cases. Connections to resurgence, categorification, and the quantum geometric Langlands program are discussed as further motivation. Throughout, we distinguish between results in the literature, conjectures, and heuristics.