Unitarising measures for Kac-Moody algebras

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 Kähler 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 ``$\mathrm{d}{\nu }_{\kappa }(\gamma )=e^{-\check{\kappa }\mathcal{S}(\gamma )}D\gamma$", where $D\gamma$ is the non-existent Haar measure on the loop group and $\mathcal{S}$ is a Kähler potential for the right-invariant Kac–Moody metric. Infinitesimally, the covariance formula prescribes the Shapovalov forms of the Kac–Moody representations.