Loop groups and twisted K-theory I
Daniel S. Freed, Michael J. Hopkins, Constantin Teleman
TL;DR
This work develops twisted equivariant K-theory for compact Lie groups via groupoids, graded central extensions, and a Fredholm-model foundation, connecting twisted K-groups to loop-group representation theory. It introduces a robust framework of twistings and twist-induced K-theory on groupoids, proves core axioms (functoriality, excision, products, Thom isomorphism, pushforwards), and constructs a fundamental spectral sequence that computes twisted K-groups. The main result computes K_G^τ(G) for nondegenerate twists provided G is connected with torsion-free π1G, showing K_G^τ(G) ≅ Hom_{W_{aff}^{e}}(Λ^τ, H_c^n(𝔱)) and, in primitive cases, identifies K_G^τ(G) with R(G)/I^τ, the Verlinde algebra, thereby linking Verlinde fusion to Pontryagin product in twisted K-theory. These findings lay groundwork for relating twisted equivariant K-theory to the Verlinde ring of loop groups, with Part II and III extending the constructions to Dirac operators and general G.
Abstract
This is the first in a series of papers investigating the relationship between the twisted equivariant K-theory of a compact Lie group G and the "Verlinde ring" of its loop group. In this paper we set up the foundations of twisted equivariant K-groups, and more generally twisted K-theory of groupoids. We establish enough basic properties to make effective computations. Using the Mayer-Vietoris spectral sequence we compute the twisted equivariant K-groups of a compact connected Lie group G with torsion free fundamental group. We relate this computation to the representation theory of the loop group at a level related to the twisting.