On Quantum Integrability and the Lefschetz Number

TL;DR

This paper explores the connection between quantum integrability, equivariant cohomology, and the Lefschetz number, providing a geometric framework for evaluating certain phase space path integrals exactly.

Contribution

It extends the use of equivariant cohomology and localization to a broader class of models with arbitrary Hamiltonians related to Lie group Cartan subalgebras.

Findings

Path integrals are related to the Lefschetz number of a Dirac operator.

Equivariant characteristic classes offer a geometric understanding of quantum integrability.

The approach generalizes previous exact evaluations of phase space path integrals.

Abstract

Certain phase space path integrals can be evaluated exactly using equivariant cohomology and localization in the canonical loop space. Here we extend this to a general class of models. We consider hamiltonians which are {\it a priori} arbitrary functions of the Cartan subalgebra generators of a Lie group which is defined on the phase space. We evaluate the corresponding path integral and find that it is closely related to the infinitesimal Lefschetz number of a Dirac operator on the phase space. Our results indicate that equivariant characteristic classes could provide a natural geometric framework for understanding quantum integrability.