Crystallographic T-duality

TL;DR

This paper introduces crystallographic T-duality, a new mathematical framework inspired by topological crystalline materials, unifying and extending existing dualities, and aiding in topological phase classification and bulk-boundary analysis.

Contribution

It presents the concept of crystallographic T-duality, connecting K-theory with graded equivariant twists to topological crystalline materials, and extends dualities and conjectures in this context.

Findings

New topological T-dualities established

Unification of previously known dualities achieved

Enhanced tools for topological phase classification provided

Abstract

We introduce the notion of crystallographic T-duality, inspired by the appearance of $K$-theory with graded equivariant twists in the study of topological crystalline materials. Besides giving a range of new topological T-dualities, it also unifies many previously known dualities, motivates generalisations of the Baum-Connes conjecture to graded groups, provides a powerful tool for computing topological phase classification groups, and facilitates the understanding of crystallographic bulk-boundary correspondences in physics.