Finite volume calculation of $K$-theory invariants

Abstract

Odd index pairings of $K_1$-group elements with Fredholm modules are of relevance in index theory, differential geometry and applications such as to topological insulators. For the concrete setting of operators on a Hilbert space over a lattice, it is shown how to calculate the resulting index as the signature of a suitably constructed finite-dimensional matrix, more precisely the finite volume restriction of what we call the spectral localizer. In presence of real symmetries, secondary ${\mathbb{Z}}_2$-invariants can be obtained as the sign of the Pfaffian of the spectral localizer. These results reconcile two complementary approaches to invariants of topological insulators.

terminology changed (from Bott operator to spectral localizer), presentation of results improved, section of eta-invariant corrected, to appear in New York Journal of Mathematics

References in corpus (3)

• Disorder-induced Floquet Topological Insulators
• Topological Insulators in Random Lattices
• Effect of Strong Disorder on 3-Dimensional Chiral Topological Insulators: Phase Diagrams, Maps of the Bulk Invariant and Existence of Topological Extended Bulk States

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• The noncommutative geometry of the Landau Hamiltonian: Differential aspects
• A Guide to the Bott Index and Localizer Index
• Eigenvalue crossings in Floquet topological systems
• Skew localizer and ${\mathbb{Z}}_2$-flows for real index pairings
• The Bott index of two unitary operators and the integer quantum Hall effect