Unitarising measures for Kac-Moody algebras

TL;DR

This work constructs a rigorous unitarising measure $\nu_κ$ on the space $\mathcal A$ of holomorphic $\mathfrak g_\mathbb C$-valued $(1,0)$-forms by leveraging two commuting Kac–Moody realizations at levels $\kappa$ and $\check\kappa=-2\check h-\kappa$, and a universal KM action $\mathscr S$ that serves as a Kähler potential for the right-invariant KM metric. The measure is uniquely characterized by covariance identities under the left action of the analytic loop group and the right action of a holomorphic loop subgroup, with covariance moduli $\Omega$ and $\Lambda$ encoding the infinitesimal transformation rules; its moments reproduce a KM-compatible Shapovalov structure. Existence is established via a Wiener-measure approximation, followed by a tightness and moment-analysis argument, while uniqueness follows from a moment-determined covariance framework. The construction provides a probabilistic realization of an analytic Langlands–type program for loop groups, and suggests connections to coset WZW models, moduli-space measures, and Drinfeld–Sokolov reductions, paving the way for future couplings to WZW actions and Virasoro/$\mathcal W$-algebras in a rigorous probabilistic setting.

Abstract

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

Theorems & Definitions (45)

• Theorem 1.1
• Remark 1
• Example 1
• Remark 2
• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2