Journal of Symplectic Geometry

Volume 18 (2020)

Number 3

Spinor modules for Hamiltonian loop group spaces

Pages: 889 – 937



Yiannis Loizides (Department of Mathematics, Pennsylvania State University, State College, Penn., U.S.A.)

Eckhard Meinrenken (Department of Mathematics, University of Toronto, Ontario, Canada)

Yanli Song (Department of Mathematics, Washington University, St. Louis, Missouri, U.S.A.)


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.

Received 22 June 2017

Accepted 16 July 2019

Published 30 July 2020