Spinor modules for Hamiltonian loop group spaces

TL;DR

This paper develops a framework for constructing spinor modules and Spin$_c$-structures on Hamiltonian loop group spaces, linking infinite-dimensional geometry with finite-dimensional models and quasi-Hamiltonian spaces.

Contribution

It introduces a natural completion of the tangent bundle for Hamiltonian $LG$-spaces, leading to an invariant spinor bundle and two methods for finite-dimensional approximation.

Findings

Constructed a strongly symplectic $LG$-equivariant vector bundle with a compatible complex structure.

Established procedures to derive finite-dimensional spinor modules from the infinite-dimensional setting.

Connected the infinite-dimensional structures to quasi-Hamiltonian $G$-spaces and abelianization techniques.

Abstract

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 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 Spin$_c$-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 $T\subset LG$-invariant submanifold of $\mathcal{M}$, and we show how to construct an equivariant Spin$_c$-structure on that submanifold.