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.

中文翻译：

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哈密​​顿环群空间的旋量模

令$LG$成为紧致连通李群$G$的环群。我们证明了任何真哈密顿量 $LG$-空间 $\mathcal{M}$ 的切丛对于强辛 $LG$-等变向量丛具有自然完成 $\overline{T}\mathcal{M}$。这个丛在自然极化类中承认一个不变相容的复杂结构，定义了一个 $LG$-equivariant spinor bundle $\mathsf{S}_{\overline{T}\mathcal{M}}$，可以认为它是自旋$_c$-$\mathcal{M}$ 的结构。我们描述了获得该 Spinor 模块的有限维版本的两个过程。在一种方法中，我们从 $\mathsf{S}_{\overline{T}\mathcal{M}}$ 构造一个扭曲的 Spin$_c$-结构，用于与 $\mathcal 相关的准汉密尔顿 $G$-空间{M}$。在第二种方法中，我们描述了一个“阿贝尔化过程”，