[Paper Review] Notes on twisted equivariant $\mathrm{K}$-theory for $\mathrm{C}^*$-algebras

Yosuke Kubota|arXiv (Cornell University)|Nov 17, 2015

Advanced Operator Algebra Research28 references3 citations

TL;DR

This paper introduces a generalized twisted equivariant K-theory for Z₂-graded C*-algebras using Fredholm operators on Hilbert modules with twisted representations, providing a simple presentation for trivially graded algebras. It establishes a connection to van Daele's K-theory and applies the framework to the bulk-edge correspondence in topological insulators with CT-type symmetries.

ABSTRACT

In this paper, we study a generalization of twisted (groupoid) equivariant $\mathrm{K}$-theory in the sense of Freed-Moore for $\mathbb{Z}_2$-graded $\mathrm{C}^*$-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 $\mathrm{K}$-theory for $\mathbb{Z}_2$-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant $\mathrm{K}$-group when the $\mathrm{C}^*$-algebra is trivially graded. It is applied for the bulk-edge correspondence of topological insulators with CT-type symmetries.

Motivation & Objective

• To generalize Freed-Moore's twisted equivariant K-theory for Z₂-graded C*-algebras using Fredholm operators on Hilbert modules with twisted representations.
• To compare this formulation with van Daele's K-theory for Z₂-graded Banach algebras, providing a unified framework.
• To offer a simplified presentation of twisted equivariant K-groups when the C*-algebra is trivially graded.
• To apply the theory to the bulk-edge correspondence in topological insulators with CT-type symmetries.
• To establish a connection between the boundary map in the Toeplitz exact sequence and the bulk-edge correspondence in quantum systems.

Proposed method

• Define the twisted equivariant KK-theory group $^\phi\mathrm{KK}^{\mathcal{G}}_{c,\tau}(A,B)$ as a bifunctor on $\phi$-twisted $\mathcal{G}$-C*-algebras.
• Construct the twisted equivariant K-theory group as $^\phi\mathrm{K}^{\mathcal{G}}_{c,\tau}(A) = {}^\phi\mathrm{KK}^{\mathcal{G}}_{c,\tau}(\mathbb{R}, A)$, generalizing Atiyah's Fredholm operator formulation.
• Establish an isomorphism between the twisted equivariant K-group and a van Daele-type K-group via Karoubi triples.
• Use the exponential map to describe the boundary map in the long exact sequence, simplifying its presentation.
• Apply the framework to the Toeplitz extension to derive the bulk-edge correspondence in topological insulators.
• Leverage the isomorphism between $^\phi\mathscr{K}_{0,c,\tau}^{\mathcal{G}}(A)$ and $\mathrm{KF}_n^{\mathcal{G}_0}(A)$ for CT-type symmetries.

Experimental results

Research questions

• RQ1 How can twisted equivariant K-theory be generalized for Z₂-graded C*-algebras using Fredholm operators with twisted representations?
• RQ2 What is the relationship between the Fredholm operator-based twisted K-theory and van Daele's K-theory for Z₂-graded Banach algebras?
• RQ3 Can a simplified presentation of twisted equivariant K-groups be obtained for trivially graded C*-algebras?
• RQ4 How does the boundary map in the long exact sequence relate to the bulk-edge correspondence in topological insulators?
• RQ5 To what extent does this framework classify gapped Hamiltonians in fermionic quantum systems with CT-type symmetries?

Key findings

• The twisted equivariant K-group $^\phi\mathrm{K}^{\mathcal{G}}_{c,\tau}(A)$ is isomorphic to the group of homotopy classes of twisted $\mathcal{G}$-equivariant families of Fredholm operators on $X$ when $A = C_0(X)$.
• A generalization of the Green-Julg theorem is established for the twisted equivariant KK-theory framework.
• For trivially graded C*-algebras, the twisted equivariant K-group admits a presentation similar to Karoubi's K-theory via Karoubi triples.
• The group $^\phi\mathscr{K}_{0,c,\tau}^{\mathcal{G}}(A)$ is isomorphic to $\mathrm{KF}_n^{\mathcal{G}_0}(A)$ for a CT-type symmetry with twist $(\phi, 0, \tau)$, linking the framework to Kitaev's periodic table.
• The boundary map $\partial$ in the Toeplitz exact sequence is given by $\partial[s] = [-\exp(\pi i \tilde{s})]$, providing a mathematical realization of the bulk-edge correspondence.
• The boundary map $\partial: {}^\phi\mathscr{K}_{0,c,\tau}^{\mathcal{G}}(A/I) \to {}^\phi\mathscr{K}_{-1,c,\tau}^{\mathcal{G}}(I)$ is realized as $0 \oplus \mathrm{id}_{\mathrm{KF}_{n-1}(A)}$ in the Toeplitz extension, confirming the edge state classification.