Coherent states for compact Lie groups and their large-N limits

TL;DR

This paper reviews heat-kernel coherent states for compact Lie groups, explores their mathematical properties and applications, and discusses recent large-N limit results for the Segal-Bargmann transform on unitary groups.

Contribution

It provides a comprehensive survey of heat-kernel coherent states and introduces new results on the large-N behavior of the Segal-Bargmann transform for U(N).

Findings

Connections to geometric quantization and Yang-Mills theory

Resolution of the identity for coherent states

Large-N asymptotic behavior of trace polynomial Laplacians

Abstract

The first two parts of this article surveys results related to the heat-kernel coherent states for a compact Lie group K. I begin by reviewing the definition of the coherent states, their resolution of the identity, and the associated Segal-Bargmann transform. I then describe related results including connections to geometric quantization and (1+1)-dimensional Yang--Mills theory, the associated coherent states on spheres, and applications to quantum gravity. The third part of this article summarizes recent work of mine with Driver and Kemp on the large-N limit of the Segal--Bargmann transform for the unitary group U(N). A key result is the identification of the leading-order large-N behavior of the Laplacian on "trace polynomials."