Crystallographic T-duality

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.

Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices

“…7.1) in the momentum space picture is T-dual to the geometrical restriction ι * in the position space picture, as is intuitively "obvious" and in the same vein as the results in [21]. In [18] we show further that "crystallographic T-dualities" exist for any crystallographic space group, and just as for pg, graded twists are essential in their formulation.…”

“…We have also given a glimpse of "crystallographic T-duality" in the case of G = pg, which was guided by the physical intuition of the bulk-boundary correspondence. In a subsequent work [18], we study such new dualities for general space groups, where further interesting phenomena arise.…”

“…Nowadays, topological phases subject to crystallographic symmetry constraints are also under intense study [46,4,30,53,18]. Most existing works address the classification problem -identifying and computing the invariants that classify bulk topological phases with given crystallographic symmetry.…”

“…The distinction of p11g from the group generated by an ordinary translation is achieved by the data of a nontrivial Z 2 -grading on p11g ∼ = Z recording the "above/below" exchange. The importance of such graded frieze/subperiodic groups in formulating crystallographic bulk-boundary correspondences is further explored in [18]. Our 2D pg example is a basic toy model in which a correspondence of crystallographic Z/2 bulk/edge phases can be formulated rigorously.…”

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

“…The basic T-duality isomorphisms K * (R/Z) ∼ = K * −1 ( Ẑ) can be formulated as a Fourier-Mukai transform, or as the Baum-Connes assembly map for Z composed with Poincaré duality. In [GT19], it was shown that every (ordinary) crystallographic space group G acting on d-dimensional Euclidean space V , induces crystallographic T-duality isomorphisms of twisted K-theories, (1.1) T G : K * +d,−v P (V /Π) → K * ,σ P ( Π).…”