an:07042683
Zbl 1436.19011
Gomi, Kiyonori; Thiang, Guo Chuan
Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices
EN
Lett. Math. Phys. 109, No. 4, 857-904 (2019).
00431375
2019
j
19L50 19L47 19K56 47L80 47B35 47A53
bulk-edge correspondence; topological crystalline insulators; twisted \(K\)-theory; equivariant \(K\)-theory; Toeplitz index theorem; Toeplitz operator; Fredholm index; circle bundle
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.
Takahiro Sudo (Nishihara)