Quantization of Hamiltonian loop group spaces

Abstract

We prove a Fredholm property for spin-c Dirac operators D on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group K ⋉ Γ, with K compact and Γ 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 [X ] ∩ [D] yields an element of the formal completion R −∞ (T ) of the representation ring of a maximal torus T ⊂ H; the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating [X ] ∩ [D] corresponds to an element of the ring of projective positive energy representations of the loop group.

Quantization of Hamiltonian loop group spaces

“…In this article we prove analogous results for Hamiltonian loop group spaces. This work builds on earlier articles [31] (joint with E. Meinrenken) and [32]. We very briefly summarize some results from these papers here, and in somewhat greater detail in Section 3.…”

“…In [32] we proved that D has a well-defined index in R −∞ (T ). Moreover, if G is simple and simply connected, index(D) is the Weyl-Kac numerator (restricted to 1 ∈ S 1 rot ) of an element of the level k fusion ring R k (G), the analogue of the representation ring for level k positive energy representations of the loop group.…”

“…It is also closely related to a Fredholm criterion of Anghel [3], and to the property of being 'κ-coercive' for all κ > 0 in [8, Corollary 5.6]. We described the proof of a slightly more general result in [32,Appendix B].…”

“…In this section we briefly review the setup and results from [31,32], and begin to study the Witten deformation in our context.…”

“…Fix a maximal torus T ⊂ G. In earlier work [31] with E. Meinrenken, we constructed a finite-dimensional 'global transversal' Y ⊂ M, as well as a canonical spinor module S 0 → Y. The submanifold Y is a small 'thickening' of the possibly singular subset X = Φ −1 M (t * ) ⊂ M. In [32] we studied a Dirac-type operator D on Y acting on sections of ∧n − ⊗S, where S = S 0 ⊗ L and n − denotes the sum of the negative root spaces of G. The operator D was shown to represent an index pairing between a spin-c Dirac operator for S and the pull back of a Bott element for g/t, the latter formally playing the role of a Poincaré dual to X .…”