Notes on invariant measures for loop groups

TL;DR

This paper discusses invariant measures on loop groups and their generalizations, exploring their characterization, explicit representation, and relevance to quantum sigma models and three-dimensional Yang-Mills theory.

Contribution

It introduces conjectures and questions about the nature and explicit forms of invariant measures on loop groups and related spaces, connecting them to quantum field theories.

Findings

Existence of bi-invariant measures on hyperfunction completions of loop groups

Generalizations involving line bundle valued measures and symmetric spaces

Open questions on measure characterization and applications to quantum models

Abstract

Let $K$ denote a simply connected compact Lie group and let $G=K^{\mathbb C}$, the complexification. It is known that there exists an $LK$ bi-invariant probability measure on a natural hyperfunction completion of the complex loop group $LG$. There are various generalizations, involving positive line bundle valued measures on the hyperfunction completion, replacing $K$ with a symmetric space, replacing $LK$ (the configuration space of the principal chiral model) with (the homotopy equivalent space of ) gauge equivalence classes of $K$-connections on the 2-sphere (the configuration space of $YM_3$), and so on. The purpose of these notes is to publicize a number of conjectures and questions concerning how these measures are characterized, how they are explicitly represented, and how they are potentially relevant to quantum sigma models and $YM_3$.