The Batalin-Vilkovisky Formalism and the Determinant Line Bundle

TL;DR

This paper demonstrates that the Batalin-Vilkovisky quantization of a family of free fermions naturally produces the determinant line bundle associated with the family of Dirac operators, linking quantum field theory and geometric analysis.

Contribution

It establishes a rigorous mathematical connection between BV quantization and the construction of the determinant line bundle for Dirac operators in families.

Findings

BV quantization yields the determinant line bundle for fermion families.

Provides a rigorous mathematical framework linking quantum field theory and geometry.

Shows the determinant line bundle arises naturally from BV formalism.

Abstract

Given a smooth family of massless free fermions parametrized by a base manifold $B$, we show that the (mathematically rigorous) Batalin-Vilkovisky quantization of the observables of this family gives rise to the determinant line bundle for the corresponding family of Dirac operators.