AtiyahPatodiSinger
The Atiyah-Patodi-Singer index theorem is a significant result in differential geometry and topology, establishing a deep connection between the analytic properties of differential operators and the topological invariants of manifolds. Formulated by Michael Atiyah, V. Patodi, and Isadore Singer, it generalizes the earlier Atiyah-Singer index theorem. The theorem relates the index of a generalized Dirac operator on a compact Riemannian manifold with boundary to certain topological invariants of the manifold and its boundary. The index of an operator is defined as the difference between the dimension of its kernel and the dimension of its cokernel.
The generalized Dirac operator considered in the theorem is typically an elliptic operator acting on sections
This theorem has had profound implications across various fields of mathematics and physics, including quantum field