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

TL;DR

This paper investigates the relationship between Langlands duality and T-duality in the context of compact simple Lie groups, revealing that twisted K-theory is mostly trivial except for specific cases, and introduces new computational methods.

Contribution

It establishes isomorphisms of twisted K-groups via Langlands duality, computes the induced maps on cohomology, and develops a new spectral sequence approach for twisted K-theory calculations.

Findings

Twisted K-groups are trivial except for SU(2) and SO(3).

The induced map on $H^3$ can be 1 or 2, depending on the group type.

A new spectral sequence method simplifies twisted K-theory computations.

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, giving simpler computations of certain twisted K-theory groups of compact Lie groups relevant for D-brane charges in WZW theories and rank-level dualities. Finally we study a duality for orientifolds based on complex Lie groups with an involution.