Batalin-Vilkovisky quantization and the algebraic index

TL;DR

This paper connects Fedosov's deformation quantization with BV quantization of a sigma model, using quantum field theory techniques to derive the algebraic index theorem through semi-classical analysis.

Contribution

It establishes a direct link between Fedosov's deformation quantization and BV quantization within a rigorous quantum field theory framework, enabling a new proof of the algebraic index theorem.

Findings

Fedosov's Abelian connections solve the quantum master equation

BV integration yields a natural trace on the deformation algebra

Semi-classical analysis reproduces the algebraic index theorem

Abstract

Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov's deformation quantization of a symplectic manifold X and the BV quantization of a one-dimensional sigma model with target X. This model is a quantum field theory of AKSZ type and is quantized rigorously using Costello's homotopic theory of effective renormalization. We show that Fedosov's Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration produces a natural trace map on the deformation quantized algebra. This formulation allows us to exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem via semi-classical analysis, i.e., one-loop Feynman diagram computations.