Atiyah–Patodi–Singer Index Theorem―Domain-Wall Junction among Particle Theory, Condensed Matter Physics, and Mathematics

Abstract

The Atiyah–Patodi–Singer index theorem is related to the bulk-edge correspondence of symmetry protected topological insulators. The mathematical set-up for this theorem is, however, unnatural since it imposes on the fermion fields a non-local boundary condition known as the “APS boundary condition” by hand, which is unlikely to be realized in the condensed matter systems. In 2017, we showed that the same integer as the APS index can be given by the eta-invariant of the domain-wall fermion Dirac operator. Recently, we invited three mathematicians, Mikio Furuta, Shinichiroh Matsuo, and Mayuko Yama­shita to our group and proved that this correspondence is not a coincidence but generally true.