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 ${\mathrm{Loc}}_{G_{\mathbb{C}}}(\Sigma )$ 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 $G_{\mathbb{C}}$ 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 3-2-1-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 ${\mathbb{E}}_n$-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 $3$-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.

arXiv comment86 pages, 11 figures, to appear in Rev. Math. Phys