Unitarising measures for Kac-Moody algebras

TL;DR

This paper introduces a probability measure related to Kac-Moody algebras and loop groups, providing a rigorous foundation for a formal path integral expression involving K"ahler geometry and representation theory.

Contribution

It constructs a probability measure on holomorphic forms linked to Kac-Moody algebras, clarifying the formal path integral in the context of loop group geometry.

Findings

Characterizes the measure via a covariance property.

Provides a rigorous interpretation of the formal path integral.

Connects the measure to Kac-Moody representation forms.

Abstract

Given a compact connected Lie group $G$ with dual Coxeter number $\check h$ and a level $\kappa<-2\check h$, we introduce a probability measure $\nu_\kappa$ on the space of holomorphic $\mathfrak g_{\mathbb C}$-valued $(1,0)$-forms in $\mathbb D$, in relation to the K\"ahler geometry of the loop group of $G$ and the action of a pair of Kac--Moody algebras at respective levels $\kappa$ and $-2\check h-\kappa>0$. We prove that $\nu_\kappa$ is characterised by a covariance property making rigorous sense of the formal path integral ``$\mathrm d\nu_\kappa(\gamma)=e^{-\check\kappa\mathscr{S}(\gamma)}D\gamma$", where $D\gamma$ is the non-existent Haar measure on the loop group and $\mathscr S$ is a K\"ahler potential for the right-invariant Kac--Moody metric. Infinitesimally, the covariance formula prescribes the Shapovalov forms of the Kac--Moody representations.