Unitarising measures for Kac-Moody algebras

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.

💡 Research Summary

The paper introduces a new family of probability measures νₖ on the space A of holomorphic 𝔤𝕔‑valued (1,0)‑forms on the unit disc 𝔻, where 𝔤 is the Lie algebra of a compact connected Lie group G and 𝔤𝕔 its complexification. The construction is carried out for levels κ < −2 ĥ (ĥ being the dual Coxeter number) and its dual level (\check κ = -2ĥ - κ > 0).

The authors first define a Kähler potential S on a dense subspace A_ω ⊂ A, which is a potential for the right‑invariant Kac–Moody metric on a certain subgroup of the loop group LG. Two natural infinite‑dimensional group actions are then introduced: a left action of the subgroup L_ω¹ G on A and a right action of the group (\widehat D_ω^∞ G_{\mathbb C}) of holomorphic maps on the punctured disc. For each action they construct continuous extensions Ω(χ,α) and Λ(α,h) of the differences S(χ·α) − S(α) and S(α·h) − S(α), respectively.

The main result (Theorem 1.1) states that for any (\check κ>0) there exists a unique Borel probability measure νₖ on A satisfying two covariance identities:

1. Left covariance – for every χ ∈ L_ω¹ G and bounded continuous F,
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