Spinor modules for Hamiltonian loop group spaces

Let $LG$ be the loop group of a compact, connected Lie group $G$. We show that the tangent bundle of any proper Hamiltonian $LG$-space $\mathcal{M}$ has a natural completion $\overline{T}\mathcal{M}$ to a strongly symplectic $LG$-equivariant vector bundle. This bundle admits an invariant compatible complex structure within a natural polarization class, defining an $LG$-equivariant spinor bundle $\mathsf{S}_{\overline{T}\mathcal{M}}$, which one may regard as the $\operatorname{Spin}_c$-structure of $\mathcal{M}$. We describe two procedures for obtaining a finite-dimensional version of this spinor module. In one approach, we construct from $\mathsf{S}_{\overline{T}\mathcal{M}}$ a twisted Spinc-structure for the quasi-Hamiltonian $G$-space associated to $\mathcal{M}$. In the second approach, we describe an ‘abelianization procedure’, passing to a finite-dimensional $T \subseteq LG$-invariant submanifold of $\mathcal{M}$, and we show how to construct an equivariant $\operatorname{Spin}_c$-structure on that submanifold.

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Received 22 June 2017

Accepted 16 July 2019

Published 30 July 2020