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Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices. (English) Zbl 1436.19011
The authors prove as the main result (Theorem 7.1 in this paper) that an index theorem for a twisted family of Toeplitz operators, as a generalization of the classical index theorem for Toeplitz operators, holds, in the sense that the topological push-forward (or bulk-edge) map of some \(\mathbb Z_2\)-equivariant (or twisted) \(K\)-theory groups for \(B_x \times B_y\) as a bundle over \(B_x=\mathbb R/ 2\pi \mathbb Z=B_y\) as a circle \(S^1\) with some sense, in the Gysin exact sequence in the topological \(K\)-theory is induced by the assignment to the input edge-bulk data \(U, V\) from the 1-dim Brillouin zone \(B_x\) and \(B_x \times B_y\) both to the unitary group \(U(2n)\), respectively, of the same mod \(2\) dimension as analytic invariant, of the kernels of the Toeplitz operators associated to the edge \(U\) as involving the multiplication operator for \(U\) on the Hilbert space \(L^2(S^1)\).
There are many more details, omitted here in the review.

MSC:
19L50 Twisted \(K\)-theory; differential \(K\)-theory
19L47 Equivariant \(K\)-theory
19K56 Index theory
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A53 (Semi-) Fredholm operators; index theories
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