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

Abstract

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 H 3 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.

The multi-degree of coverings on Lie groups

“…Naturally, topological invariants of the coverings are useful to formulate, or to solve, the relevant extension problems. In this paper we introduce for each covering map of Lie groups an invariant, called the multi-degree of the covering; extend Schubert calculus to evaluate the invariant; and apply the results to two outstanding topological problems arising from the studies of the Wess-Zumino-Witten models and the topological Gauge theories [6,16,25]. The main tool in our approach is the Chow rings of Lie groups, introduced by Grothendieck [19] in 1958.…”

“…and investigated its extension problem [16,Appendix 1], where G is simply connected and simple. This problem was emphasized by Dijkgraaf and Witten in the work [6] on the topological Gauge theories, and by Mathai and Rosenberg [25] in the study on the relationship between Langlands duality and T-duality for compact Lie groups. Since the map c * in (1.6) is Kronecker dual to the map c * on H 3 (P G) we get from Corollary 1.2 the following result.…”

“…Remark 1.4. In the inspiring work [25] Mathai and Rosenberg have computed the leading terms a 1 of the sequences D(G, P G). In the cases of G = SU (n) with n divisible by 4 and G = Spin(2(2b + 1)), our results in Corollary 1.2 are different with the ones stated in [25, Theorem 1, (1), (3)].…”

A new approach to twisted K–theory of compact Lie groups

“…There are two of these, PSU(3) with fundamental group Z/3 and PSp(2) ∼ = SO(5) with fundamental group Z/2. The case of PSU(3) was studied in [32,Theorem 19 and Remark 20], so we consider here the case of PSp (2). Note first of all that the covering map Sp(2) π − → PSp(2) induces an isomorphism on H 3 by [32, Theorem 1], and that PSp(2) fits into a fibration (4) S 3 = Sp(1) → PSp(2) → RP 7 , which replaces the fibration Sp(1) → Sp(2) → S 7 used in the proof of Theorem 12.…”

“…This paper is an outgrowth of the paper [32] by Mathai and the author, where we started studying a new approach to the computation of the twisted K-theory of compact simple Lie groups. This problem was first studied by physicists (e.g., [36,31,19,9,11,12,22]) because of interest in the WZW (Wess-Zumino-Witten) model, which appears both in conformal field theory and as a string theory whose underlying spacetime manifold is a Lie group, usually compact and simple.…”