This paper studies the spinor modules theory of loop groups.
Let \( G \) be a compact, connected Lie group and let the loop group \( LG \) be the Banach Lie group of \(G\)-valued loops of a fixed Sobolev class \( S > 1/2 \). The authors prove that the tangent bundle of any proper Hamiltonian loop group space \(M\) possesses a canonically defined \(LG-\)equivariant completion \(\overline{T}M\), such that any weakly symplectic 2-form \(\omega\) of any proper Hamiltonian loop group space extends to a strongly symplectic 2-form on \(\overline{T}M\).
Furthermore, it is proved that the bundle \(\overline{T}M\) possesses a distinguished \(LG-\)invariant polarization and a global \(LG-\)invariant \(\omega-\)compatible complex structure \(J\) within this polarization class, unique up to homotopy. This leads to the definition of \( LG-\)equivariant spinor bundle \( \mathrm{S}_{\overline{T}M} \), which is used to construct the twisted \( \mathrm{Spin}_c \)-structure for the associated quasi-Hamiltonian \(G\)-space \(M\). This is is a way to get a finite-dimensional version of the spinor module \( \mathrm{S}_{\overline{T}M} \).
The authors also discuss \textquoteleft abelianization procedure\textquoteright which is another way to get a finite-dimensional version of \( \mathrm{S}_{\overline{T}M} \). The idea is to shift to a finite-dimensional maximal torus \(T \subseteq LG-\)invariant submanifold of \(M,\) and construct an equivalent \(\mathrm{Spin}_c \)-structure on that submanifold. More precisely, if the moment map \(\Phi\) of a proper Hamiltonian \(LG\)-space is transverse to the Lie algebra \( \mathfrak{t}^* \) (as a space of constant connections valued in the Lie algebra of the maximal torus \( T \)), then the pre-image \(\Phi^{-1} (\mathfrak{t}^*)\) is a finite-dimensional pre-symplectic manifold that inherits a \(T\)-equivalent \(\mathrm{Spin}_c \)-structure.