Twisted crystallograpic T-duality via the Baum--Connes isomorphism

TL;DR

This paper introduces a new form of T-duality in crystallography, linking twisted equivariant K-theory groups of position and momentum tori through KK-theory, with applications in K-theory calculations.

Contribution

It establishes the twisted crystallographic T-duality as an isomorphism using KK-theory, connecting twisted K-groups of dual tori for crystallographic groups.

Findings

Proves the isomorphism between twisted K-groups of position and momentum tori.

Identifies the T-duality map with the Dirac homomorphism in KK-theory.

Demonstrates applications in K-theory computations.

Abstract

We establish the twisted crystallographic T-duality, which is an isomorphism between Freed-Moore twisted equivariant K-groups of the position and momentum tori associated to an extension of a crystallographic group. The proof is given by identifying the map with the Dirac homomorphism in twisted Chabert--Echterhoff KK-theory. We also illustrate how to exploit it in K-theory computations.