Group dualities, T-dualities, and twisted K -theory

Abstract

This paper explores further the connection between Langlands duality and T-duality for compact simple Lie groups, which appeared in work of Daenzer-van Erp and Bunke-Nikolaus. We show that Langlands duality gives rise to isomorphisms of twisted K-groups, but that these K-groups are trivial except in the simplest case of SU (2) and SO (3). Along the way we compute explicitly the map on H3 induced by a covering of compact simple Lie groups, which is either 1 or 2 depending in a complicated way on the type of the groups involved. We also give a new method for computing twisted K-theory using the Segal spectral sequence, which is better adapted to many problems involving compact groups. Finally we study a duality for orientifolds based on complex Lie groups with an involution.