Spinor modules for Hamiltonian loop group spaces

Hamiltonian loop group spaces and a theorem of Teleman and Woodward

“…[55]). We briefly recall from [45] the construction of a finite dimensional smooth Spin c submanifold X ⊂ M that plays a fundamental role throughout the rest of the paper, analogous to the role played by the 'extended moduli spaces' of Jeffrey-Kirwan [37]; one advantage of our X is that it is smooth; a disadvantage is that it is less explicit. We describe the motivating examples of Hamiltonian loop group spaces coming from moduli spaces of flat connections on a compact surface with framing along the boundary.…”

“…Definition 2.7 ( [45]). Let (M, ω M , µ M ) be a proper Hamiltonian LG-space, and let R ⊂ Lg * be as constructed in Proposition 2.5.…”

“…There is a canonical level h ∨ spinor bundle S can for hol * M T M . The heuristic idea behind the construction in [45] is as follows. In finite dimensions, spinor modules satisfy 2-out-of-3: given a short exact sequence of even rank Euclidean vector bundles 0 → V 1 → V 2 → V 3 → 0 and given spinor modules for any two of the V i , one obtains a spinor module for the third.…”

“…Assuming the 2-out-of-3 property still holds in this infinite dimensional context, one obtains a spinor module for hol * M T M . With care this strategy can be carried out rigorously [45]. The dual Coxeter number appears because this is the level of the spin representation of the loop group.…”

“…Since in our case the manifold Σ is not closed, we impose a boundary condition (of generalized Atiyah-Patodi-Singer-type) to obtain a smooth G-equivariant Fredholm family (D Σ m ) m∈M , defining an element in the G-equivariant K-theory of M. In previous work with E. Meinrenken and Y. Song [45], we constructed a finite dimensional non-compact Spin c submanifold X ⊂ M, and studied the equivariant index of the Dirac operator [47]. The manifold X plays a similar role to the 'extended moduli spaces' used by Jeffrey-Kirwan [37] in their work on the cohomology ring of the moduli space of flat connections.…”