Quantization of Hamiltonian loop group spaces

Abstract

We prove a Fredholm property for spin-c Dirac operators $\mathsf{D}$ on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group $K⋉\Gamma$, with $K$ compact and $\Gamma$ discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing $[\mathcal{X}]\cap [\mathsf{D}]$ yields an element of the formal completion $R^{-\infty }(T)$ of the representation ring of a maximal torus $T\subset G$; the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating $[\mathcal{X}]\cap [\mathsf{D}]$ corresponds to an element of the ring of projective positive energy representations of the loop group.

35 pages, title changed, small clarifications added

Cited by in corpus (1)

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