On the quantization of conjugacy classes

Abstract

Abstract. Let G be a compact, simple, simply connected Lie group. A theorem of FreedHopkins-Teleman identifies the level k ≥ 0 fusion ring R k (G) of G with the twisted equivariant K-homology at level k + h ∨ , where h ∨ is the dual Coxeter number of G. In this paper, we will review this result using the language of Dixmier-Douady bundles. We show that the additive generators of the group R k (G) are obtained as K-homology push-forwards of the fundamental classes of pre-quantized conjugacy classes in G.

Dirac Induction for Loop Groups

“…Together with the isomorphism R(H) ∼ = K 0 G (G/H), the result above shows that the induction map as defined in this paper, can be identified in K-theory as the push-forward along the map obtained by composing G/H → e → G. For this side of the story, see also [8,14].…”

“…The same homomorphism can be constructed topologically in equivariant (twisted) K-theory, cf. [4], see also [3,8,14].…”

Modular invariants and twisted equivariant K-theory

“…An elegant proof of this is given in [75]. The twist is crucial for finite dimensionality: e.g., [19] computes that the untwisted…”

“…More precisely, we get a natural embedding of the (finite-dimensional) Lie algebras of these stabilizers, into the (infinite-dimensional) loop algebra. The level 1 basic representation of that affine algebra then restricts to a coherent family of projective representations of those stabilizers, and from this the bundle is formed (see Equation (21) in [75]). By using the affine algebra representation, he obtains almost for free a global description of the bundle, avoiding our complicated explicit construction of unitaries and verification of their consistency conditions.…”

“…This same example was worked out in e.g., Example 1.7 of [41], using Mayer-Vietoris and K * ,cs , with the same result (the answers must agree since the space G is compact). A very explicit yet elegant calculation of K G * (G) for any compact simple G was done in [75] using the spectral sequence of [98].…”