Quantization and Algebraic Index
Si Li
TL;DR
This work synthesizes BV quantization with algebraic index theory across 1d and 2d settings. It develops an effective BV framework, using (-1)-shifted symplectic structures, heat-kernel renormalization, and UV finiteness to derive algebraic index formulas via traces and Fedosov deformation quantization. In parallel, it extends to 2d chiral theories on elliptic curves, establishing elliptic traces through regularized integrals, chiral homology, and Beilinson–Drinfeld constructions, and showing a bridge to mirror symmetry via elliptic Gromov–Witten data. The results unify a BV-based quantization perspective with index theory, deformation quantization, and elliptic vertex algebras, offering a coherent path from local quantum field theory data to global index-type invariants with applications to mirror symmetry on elliptic curves.
Abstract
This article reviews the program on connecting Batalin-Vilkovisky (BV) quantization with index theories of algebraic type. We explain how the classical algebraic index theorem can be proved in terms of BV quantization of topological quantum mechanics. This is generalized to 2d chiral CFT in which we present an elliptic chiral analog of the algebraic index theory. As an application, we show how the generating function of all genus Gromov-Witten invariants on elliptic curves is mirror equivalent to an elliptic chiral index in the mirror BCOV theory.