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 of a vector bundle. The boundary condition plays a crucial role in the statement of the theorem. The Atiyah-Patodi-Singer index theorem provides a formula for this index in terms of integrals of curvature forms and the eta invariant associated with the boundary. The eta invariant is an analytic invariant that measures the spectral asymmetry of the boundary operator.

This theorem has had profound implications across various fields of mathematics and physics, including quantum field theory and string theory. It has been instrumental in developing new techniques for studying the topology of manifolds and understanding the behavior of quantum fields in curved spacetime. The interplay between analysis and topology highlighted by this theorem continues to inspire further research.