On path integral localization and the Laplacian, the thesis

TL;DR

This thesis develops advanced path integral localization methods inspired by topological field theory, unifies several localization principles, explores their applications to trace formulas and integrable models, and provides educational resources for newcomers.

Contribution

It introduces a new unifying localization principle, applies localization techniques to trace formulas and integrable models, and offers pedagogical insights into topological field theory.

Findings

Unified localization principle encompassing BRST, non-Abelian, and conformal localization.

Derived a generalized Selberg trace formula for Lie groups.

Presented a new derivation of DeWitt's term and conjectured localization applicability to integrable models.

Abstract

In this thesis, we develop path integral localization methods that are familiar from topological field theory: the integral over the infinite dimensional integration domain depends only on local data around some finite dimensional subdomain. We introduce a new localization principle that unifies BRST localization, the non-Abelian localization principle and the conformal generalization of the Duistermaat-Heckman integration formula. In addition, it is studied if one can possibly derive a generalized Selberg's trace formula on locally homogeneous manifolds using localization techniques. However, a definite answer is obtained only in the Lie group case (we complete the work of R. Picken) in which it is an application of the Duistermaat-Heckman integration formula. Also a new derivation of DeWitt's term is reported. Furthermore, connections between evolution operators of integrable models and localization methods are studied. A derivative expansion localization is presented and it is conjectured to apply also to integrable models, for example the Toda lattice. Moreover, a pedagogical introduction to the localization techniques is given, as well as a list of selected references that might be useful for a beginning graduate student in mathematical physics or for a mathematician who would like to study the physical point of view to topological field theory and string theory.