On Path Integral Localization and the Laplacian

TL;DR

This paper introduces a new localization principle that unifies various existing localization methods and applies it to heat kernels on Lie groups, offering new formulas potentially useful for trace formulas.

Contribution

It presents a generalized localization principle unifying BRST, non-abelian, and conformal localization, with applications to heat kernels and homogeneous spaces.

Findings

Unified localization framework for different localization methods

Localized heat kernel calculations on compact Lie groups

Derived new formulas for homogeneous spaces

Abstract

We introduce a new localization principle which is a generalized canonical transformation. It unifies BRST localization, the non-abelian localization principle and a special case of the conformal Duistermaat-Heckman integration formula of Paniak, Semenoff and Szabo. The heat kernel on compact Lie groups is localized in two ways. First using a non-abelian generalization of the derivative expansion localization of Palo and Niemi and secondly using the BRST localization principle and a configuration space path integral. In addition we present some new formulas on homogeneous spaces which might be useful in a possible localization of Selberg's trace formula on locally homogeneous spaces.