Wild orbits and generalised singularity modules: stratifications and quantisation

Abstract

We study isomorphism classes of untwisted irregular singular meromorphic connections on principal bundles over (wild) Riemann surfaces, for any complex reductive structure group $G$ and polar divisor. In particular we compute the stabilisers of suitable marked points on their principal part orbits, showing the stabilisers are connected and controlled by the corresponding filtration of (Levi factors of) nested parabolic subgroups of $G$; this uniquely determines the orbits as complex homogeneous manifolds for groups of jets of principal $G$-bundle automorphisms. Moreover, when the residue is semisimple we stratify the space of orbits by the stabilisers, relating this to local wild mapping class groups and generalising the Levi stratification of a Cartan subalgebra $\mathfrak{t} \subseteq \mathfrak{g} = \operatorname{Lie}(G)$: the dense stratum corresponds to the generic setting of irregular isomonodromic deformations à la Jimbo-Miwa-Ueno. Then we adapt a result of Alekseev-Lachowska to deformation-quantise nongeneric orbits: the $\ast$-product involves affine-Lie-algebra modules, extending the generalised Verma modules (in the case of regular singularities) and the `singularity' modules of F.-R. (in the case of generic irregular singularities). As in the generic case, the modules contain Whittaker vectors for the Gaiotto-Teschner Virasoro pairs from irregular Liouville conformal field theory; but they now provide all the quotients which are obtained when the corresponding parameters leave the aforementioned dense strata. We also construct Shapovalov forms for the corresponding representations of truncated (holomorphic) current Lie algebras, leading to a conjectural irreducibility criterion. Finally, we use these representations to construct new flat vector bundles of vacua/covacua à la Wess--Zumino-Novikov-Witten, equipped with connections à la Knizhnik-Zamolodchikov.

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Theorems & Definitions (58)

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