Atiyah–Patodi–Singer

Atiyah–Patodi–Singer
Name	Atiyah–Patodi–Singer
Field	Differential geometry, Global analysis, Topology
Notable works	Index theorem, Eta invariant, Boundary value problems
Related people	Michael Atiyah, Vijay Patodi, Isadore Singer

Atiyah–Patodi–Singer

The Atiyah–Patodi–Singer framework unites ideas from Michael Atiyah, Vijay Patodi, and Isadore Singer to extend index theory for elliptic operators on manifolds with boundary, introducing the eta invariant and global boundary conditions. Developed in the early 1970s, this work influenced research across Donaldson theory, Seiberg–Witten theory, K-theory, and interactions with mathematical physics communities such as those around Edward Witten and Alain Connes. The trio's contributions impacted institutions including Princeton University, University of Cambridge, Harvard University, and conferences like the International Congress of Mathematicians.

Introduction

The Atiyah–Patodi–Singer program builds on prior advances by Atiyah–Singer index theorem, Atiyah–Bott fixed-point theorem, and analytic techniques used by Leray, Hodge, and Kodaira. It addresses elliptic operators originally studied by Gilkey, Seeley, and Boutet de Monvel on manifolds with boundary such as examples considered by Riemann, Gauss, Bernhard Riemann-related moduli problems, and later applied in settings explored by Thurston and Gromov. The formulation incorporates spectral data similar to considerations in Weyl law discussions and complements categorical approaches in Grothendieck-influenced Algebraic topology.

Atiyah–Patodi–Singer Index Theorem

The Atiyah–Patodi–Singer index theorem extends the Atiyah–Singer index theorem to compact manifolds with boundary by expressing the analytic index of a Dirac-type operator in terms of topological quantities such as characteristic classes from Chern–Weil theory, Pontryagin classes used by Hirzebruch, and the signature studied by Milnor and Hirzebruch signature theorem. The formula involves integrals over manifolds appearing in work by Chern, Simons, and Pontryagin, together with a correction term given by the eta invariant introduced by Atiyah, Patodi, and Singer and related to spectral flow concepts explored by Phillips and Robbin. This theorem influenced research by Freed, Uhlenbeck, Donaldson, Taubes, and Witten.

Eta Invariant and Spectral Asymmetry

The eta invariant measures spectral asymmetry of self-adjoint elliptic operators, connecting to zeta-regularization techniques used by Ray, Singer (Ray–Singer torsion), and Elizalde in quantum field contexts studied by Coleman and Hawking. It relates to analytic torsion from Ray–Singer analytic torsion and to anomalies considered by Adler and Bell in particle physics, while spectral flow relations tie into work by Phillips and Getzler. Connections also arise with Connes’s noncommutative geometry, Kasparov’s KK-theory, and index pairing results employed by Baum and Douglas.

Boundary Conditions and APS Boundary Problem

APS boundary conditions are global, nonlocal conditions defined using the spectral projection of the tangential operator, inspired by spectral decompositions studied by Weyl, Courant, and Hilbert. These boundary conditions differ from classical local conditions such as Dirichlet boundary condition and Neumann boundary condition used in problems addressed by Poisson and Laplace. The APS setup is relevant to manifold decompositions in the style of Milnor and to surgery techniques developed by Wall and Kervaire, and ties to gluing formulae examined by Bismut and Cheeger.

Proof Sketch and Analytical Techniques

The proof combines pseudodifferential operator calculus developed by Kohn, Nirenberg, Hörmander, and Seeley with heat kernel asymptotics refined by Minakshisundaram and Pleijel and later applied by Gilkey and Patodi. Key analytic inputs include the small-time expansion examined by McKean and Singer, boundary parametrix constructions appearing in Boutet de Monvel’s work, and spectral estimates influenced by Agmon and Lax–Milgram lemma-style functional analysis contributions traced through Riesz and Fredholm. Techniques parallel those in proofs by Atiyah–Singer index theorem collaborators and intersect with microlocal analysis advanced by Duistermaat and Guillemin.

Applications and Examples

The APS theory has been applied to signature theorems for manifolds with boundary relevant to Hirzebruch problems, to moduli spaces studied by Atiyah–Bott and Donaldson, and to invariants in low-dimensional topology examined by Witten, Floer, and Seiberg. In mathematical physics it appears in anomaly computations by Alvarez-Gaumé, Witten, and Friedan and in topological field theories influenced by Reshetikhin–Turaev and Chern–Simons theory developments. Concrete examples include metrics on manifolds considered by Yau and spectral computations on lens spaces studied by Reidemeister-type invariants and work by Milnor.

Generalizations and Related Theories

Generalizations include families index theorems studied by Bismut–Freed, equivariant versions influenced by Segal and Atiyah–Bott, and extensions to noncompact and singular spaces pursued by Melrose, Mazzeo, and Cheeger. Relationships connect to noncommutative index theory advanced by Connes and Kasparov, to K-homology frameworks developed by Higson and Roe, and to analytic torsion refinements by Bismut and Zhang. Further developments intersect with geometric flows investigated by Hamilton and Perelman and with categorical approaches in the spirit of Kontsevich and Deligne.

Category:Index theory