Secondary Z_2 valued index invariantsOberseminar Global Analysis and Operator Algebras

by Maxim Braverman (Northeastern University)

Tuesday May 6, 2025, 2:15 PM → 3:15 PM Europe/Berlin

Seminar room 1.008 (Mathezentrum)

We investigate elliptic operators with a symmetry that forces their index to vanish. We study the secondary index, defined modulo 2. We examine Callias-type operators with this symmetry on non-compact manifolds and establish mod 2 versions of the Gromov-Lawson relative index theorem, the Callias index theorem, and the Boutet de Monvel’s index theorem for Toeplitz operators.

For paths of operators with similar symmetry, we study the secondary Z_2-valued spectral flow and prove an analog of the Atiyah-Patodi-Singer-Robbin-Salamon theorem, showing that this secondary spectral flow of the path A(t) is equal to the Z_2-valued index of the suspension operator d/dt+A(t). We compute the graded secondary spectral flow of a symmetric family of Toeplitz operators on a complete Riemannian manifold. In the case of a pseudo-convex domain, this leads to an odd version of the secondary Boutet de Monvel’s index theorem for Toeplitz operators. When this domain is simply a unit disc in the complex plane, we recover the bulk-edge correspondence for the Graf-Porta module for 2D topological insulators of type AII. (joint work with Ahmad Reza Haj Saeedi Sadegh)