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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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