Yosuke Kubota

Controlled Topological Phases and Bulk-edge Correspondence

In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev’s periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant K-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane–Mele

The joint spectral flow and localization of the indices of elliptic operators

{\mathbb{Z}_2}

\mathrm{KK}

Z2-invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse Mayer-Vietoris exact sequence. As a new example, we study the index of reflection-invariant systems.

-theory

In this paper, we study a generalization of twisted (groupoid) equivariant K-theory in the sense of Freed–Moore for ℤ2-graded C*-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele’s K-theory for ℤ2-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant K-group when the C*-algebra is trivially graded. It is applied for the bulk-edge correspondence of topological insulators with CT-type symmetries.

Notes on twisted equivariant

We introduce the notion of the joint spectral flow, which is a generalization of the spectral flow, by using Segals model of the connective

\mathrm{K}

K

-theory for

-theory spectrum. We apply it for some localization results of indices motivated by Wittens deformation of Dirac operators and rephrase some analytic techniques in terms of topology.