Atiyah–Patodi–Singer index theorem

Atiyah–Patodi–Singer index theorem
Name	Atiyah–Patodi–Singer index theorem
Field	Differential geometry; Michael Atiyah; Isadore Singer; Vikraman Balaji
Introduced	1970s
Contributors	Michael Atiyah, Isadore Singer, Laurent Schwartz

Atiyah–Patodi–Singer index theorem The Atiyah–Patodi–Singer index theorem is a foundational result in global analysis connecting elliptic operators on manifolds with boundary to topological invariants; it extends the Atiyah–Singer index theorem by incorporating spectral boundary conditions and the eta invariant. Developed in collaboration by Michael Atiyah and Isadore Singer with later refinements by others, the theorem bridges techniques from Bernhard Riemann-type geometry, Henri Poincaré duality, and spectral theory of David Hilbert-space operators.

Introduction

The theorem arose from work by Michael Atiyah and Isadore Singer in the context of the Atiyah–Singer index theorem program and interactions with researchers such as Raoul Bott, Hermann Weyl, and Atle Selberg. It addresses index problems for first-order elliptic operators on compact manifolds with boundary, complementing analytic frameworks used by John von Neumann and Marcel Riesz. The development involved collaboration and cross-fertilization with mathematicians like Alfred Tarski, Jean Leray, Shiing-Shen Chern, and Raoul Bott and influenced work by Edward Witten, Alain Connes, Daniel Quillen, and Mikhail Gromov.

Statement of the theorem

Let M be a compact Riemannian manifold with boundary ∂M and let D be a first-order elliptic differential operator, modeled on Dirac operators introduced by Paul Dirac and developed by Marcel Berger and Shiing-Shen Chern. Impose the Atiyah–Patodi–Singer spectral boundary condition defined in terms of the self-adjoint boundary operator B. The index of D equals the integral over M of a local characteristic form, constructed from characteristic classes appearing in work of Chern, Henri Cartan, and Hermann Weyl, plus a correction term given by the eta invariant of B. This formula parallels index formulae found by Atiyah–Singer, with secondary terms echoing notions from John Milnor and René Thom.

Eta invariant and spectral boundary conditions

The eta invariant, introduced in the theorem, refines spectral asymmetry concepts previously studied by Hermann Weyl and David Hilbert. For the boundary operator B, the eta invariant η(B) measures the signed asymmetry of eigenvalues, a spectral quantity related to studies by George Pólya and Emile Picard. The spectral boundary condition projects onto the nonnegative eigenspaces of B, an idea influenced by projection techniques of John von Neumann and Erwin Schrödinger. The eta invariant appears in subsequent connections with invariants studied by Andrew Casson, William Thurston, Simon Donaldson, and Edward Witten.

Proof outline and analytic techniques

The proof employs heat kernel methods and spectral theory developed by Israel Gelfand and André Weil, using asymptotic expansions akin to work by Hermann Weyl and Minakshisundaram-Pleijel analysis. Key analytic tools include parametrix constructions inspired by Laurent Schwartz and pseudodifferential operator calculus advanced by Joseph Kohn and Louis Nirenberg. The treatment of boundary contributions relies on Calderón projectors studied by Alessandro Calderón and boundary layer potentials explored by Franz Rellich. Gluing formulae and cobordism ideas draw on concepts from René Thom and Michel Kervaire.

Applications and examples

Applications span gauge theory and invariants in low-dimensional topology influenced by Simon Donaldson, Kronheimer, and Peter Ozsváth; anomalies in quantum field theory examined by Edward Witten and Alonzo Church-style formative studies; and index computations on manifolds with cylindrical ends related to work by Gerd Grubb and Richard Melrose. Concrete examples include Dirac operators on product manifolds related to Henri Poincaré-type constructions, spectral flow calculations linked to Ludwig Faddeev-Victor J. Emery methods, and relations to Reidemeister torsion pursued by John Milnor and Vladimir Turaev. The eta invariant also appears in investigations by Mikhail Gromov and Dennis Sullivan.

Generalizations and related results

Generalizations include extensions to families index theorems developed by Daniel Quillen and John B. Conway-style operator theory, equivariant versions influenced by Michael Atiyah and Daniel Quillen, and relations to noncommutative geometry pioneered by Alain Connes. Interactions with Seiberg–Witten theory advanced by Edward Witten and Nathan Seiberg echo the theorem's role in linking analysis and topology. Further developments by Richard Melrose, Paul Baum, Alan Carey, and Boris Tsygan expanded the analytical machinery to manifolds with more singular structures, feeding into contemporary research by Maxim Kontsevich, Dennis Sullivan, and Mikhael Gromov.

Category:Index theorems