Unitarising measures for Kac-Moody algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910013373546496 |
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| author | Baverez, Guillaume |
| author_facet | Baverez, Guillaume |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06094 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unitarising measures for Kac-Moody algebras Baverez, Guillaume Representation Theory Mathematical Physics Group Theory Operator Algebras Probability 60G60, 17B67, 81T40 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. |
| title | Unitarising measures for Kac-Moody algebras |
| topic | Representation Theory Mathematical Physics Group Theory Operator Algebras Probability 60G60, 17B67, 81T40 |
| url | https://arxiv.org/abs/2602.06094 |