On Localization and Regularization

TL;DR

This paper investigates how different regularizations affect path integral localization, using specific examples like the harmonic oscillator and SU(2) Weyl character, and resolves the Weyl shift problem with advanced mathematical tools.

Contribution

It introduces a novel approach to the Weyl shift problem in path integrals by applying the Atiyah-Patodi-Singer eta-invariant and Borel-Weil theory.

Findings

Regularization choice significantly impacts path integral results.

Resolved the Weyl shift problem using eta-invariant and Borel-Weil theory.

Provided insights into localization techniques in quantum mechanics and representation theory.

Abstract

Different regularizations are studied in localization of path integrals. We discuss the effect of the choice of regularization by evaluating the partition functions for the harmonic oscillator and the Weyl character for SU(2). In particular, we solve the Weyl shift problem that arises in path integral evaluation of the Weyl character by using the Atiyah-Patodi-Singer $\eta$-invariant and the Borel-Weil theory.