Twisted equivariantK-theory with complex coefficients

Abstract

Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact group acting on itself by conjugation and relate the result to the Verlinde algebra and to the Kac numerator at q = 1. Verlinde's formula is also discussed in this context. IntroductionLet X be a locally compact topological space acted upon by a compact Lie group G. The equivariant K-theory K * G (X) was defined by Atiyah and Segal; some foundational papers are [3,30]. Twisted versions of K-theory, both equivariant and not equivariant, have recently attracted some attention. The equivariant twistings that we consider are classified up to isomorphism by an equivariant cohomology class(A twisting is a representative cocycle for such a class in some model of equivariant cohomology.) For torsion, non-equivariant twistings, the relevant K-theory was first introduced in [14]; subsequent treatments ([28] and, more recently, [1, 5, 8]) remove the torsion assumption. We recall for convenience the topologists' definition in the case ε = 0. Because the projective unitary group PU has classifying space K(Z; 3), a class [τ ] ∈ H 3 (X; Z) defines a principal PU-bundle over X up to isomorphism. Let F X be the associated bundle of Fredholm endomorphisms. The negative τ K(X) groups are the homotopy groups of the space of sections of F X ; the others are determined by Bott periodicity. In the presence of a group action, the equivariant τ K-groups arise from the space of invariant sections (a technical variation is needed in the equivariant case, pertaining to the topology on Fredholm operators [5, 31]). In this paper, we shall implicitly assume the basic topological properties of twisted K-theory; their justification is found in the papers mentioned earlier. However, since H 1 twistings get less coverage in the literature, we discuss them in more detail in § 4.One of the basic results [30] of the equivariant theory expresses, in terms of fixed-point data, the localisation of K * G (X) at prime ideals in the representation ring R G of G. The situation simplifies considerably after complexification, when the maximal ideals in R G correspond to (complex semi-simple) conjugacy classes. Recall that, in the non-equivariant case, the Chern character maps complex K-theory isomorphically onto complex cohomology. The localisation results can be assembled into a description of complex equivariant K-theory by a globalised Chern character [4,29,32] supported over the entire group. Part I of our paper generalises these results to the twisted case: the main result, Theorem 3.9, describes τ K * G (X), via the twisted Chern character ( § 2), in terms of (twisted) equivariant cohomology of fixed-point sets, with coefficients in certain equivariantly flat complex line bundles. For orbifolds, this is Vafa's discrete torsion [36,37]. We give a more detailed outline of Part I in § 1.In Part II, we apply our main result to the ...

Loop groups and twisted K-theory III

“…The complicating factors are: the presence of torsion in the group H 3 of twisting classes, an additional type of twistings classified by H 1 G (G; Z/2), related to gradings of the loop group, and finally, the fact that the two sides need not be quotients of R(G). A simple statement can still be given when G is connected and π 1 (G) is free, as in [FHT08,§6], precisely because both sides can be realised as quotients of R(G). A construction of the map via a correspondence induced by conjugacy classes was indicated in [Fre02]; we spell it out in Section 6.11 here.…”

“…A noteworthy complement to Theorem 5 is that K τ T (G G ) contains the Kac numerator formula for LG τ -representations (see §15.6). It would be helpful to understand this as a twisted Chern character, just as the the Kac numerator at q = 1 was discovered to describe the Chern character for K τ G (G) [FHT08].…”

“…These should be regarded as the LG s -equivariant K τ -groups of a point. The reader should note that defining K-theory for graded algebras is a more delicate matter in general [Bla86]; the shortcut above, also used in [FHT08,§4], relies on the semi-simplicity of the relevant categories of modules.…”

“…This can be further extended to an index theorem for generalised flag varieties of loop groups, in which twisted K-theory provides the topological side. We refer to [FHT08,§8] for a verification of this result in the special case of connected groups with free π 1 , and to [Tel04] for further developments concerning higher twistings of K-theory. There is in the transgressed case also a ring structure on R τ (BG), the fusion product, and R τ (LG) is called the Verlinde ring.…”

Sheaf theory for stacks in manifolds and twisted cohomology forS1–gerbes

“…One attempt is the approach of Freed, Hopkins and Teleman [11] in which the complex of smooth differential forms is replaced by similar objects in algebraic topology.…”