Quantization and Algebraic Index

TL;DR

This paper explores the connection between BV quantization and algebraic index theory, extending classical results to 2d chiral CFT and linking Gromov-Witten invariants with elliptic chiral indices in mirror symmetry.

Contribution

It introduces a novel approach to relate BV quantization with algebraic index theorems and extends the theory to elliptic chiral CFT, connecting Gromov-Witten invariants with mirror symmetry.

Findings

Classical algebraic index theorem proved via BV quantization.

Extension to 2d chiral CFT with elliptic index analog.

Gromov-Witten invariants linked to elliptic chiral index in mirror symmetry.

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.