Atiyah–Singer

Contents

Atiyah–Singer is the collective name used to denote the family of results comprising the index theorem connecting analytical invariants of elliptic differential operators to topological invariants of manifolds. The theorem ties together notions from Michael Atiyah, Isadore Singer, Alexander Grothendieck, Hermann Weyl, and John von Neumann-influenced areas, linking analytic techniques of Paul Dirac, Eugene Wigner, Israel Gelfand and topological frameworks associated with Henri Poincaré, Luitzen Brouwer, Marston Morse and Raoul Bott.

Statement of the theorem

The central statement asserts that for an elliptic differential operator on a compact manifold the analytical index equals the topological index, relating the Fredholm operator-theoretic index of an operator introduced by Errett Bishop and Kiyoshi Oka to characteristic classes in K-theory as developed by Atiyah, Friedlander and Grothendieck. In modern form the theorem equates the analytical index of a Dirac operator or more general elliptic operator, defined using spectral theory from David Hilbert and John von Neumann, with a topological expression given by the pushforward in topological K-theory and characteristic classes such as the Chern character, Todd class, and A-roof genus associated to tangent bundles classified by Samuel Eilenberg and Norman Steenrod. The index theorem encompasses special cases such as the Gauss–Bonnet theorem, the Riemann–Roch theorem of Bernhard Riemann and Guido Castelnuovo, and the Hirzebruch–Riemann–Roch theorem of Friedrich Hirzebruch.

Origins trace to the interplay between spectral theory studied by David Hilbert and global analysis pursued by Atle Selberg, with earlier hints in the work of Carl Friedrich Gauss and Bernhard Riemann on curvature and topology. The collaborative formulation by Michael Atiyah and Isadore Singer in the 1960s synthesized inputs from Alfred Tarski-level abstraction in algebraic topology, the development of K-theory by Atiyah and Grothendieck, and advances in elliptic operator theory influenced by Lars Hörmander and Shmuel Agmon. Key milestones include applications to the Riemann–Roch program pursued by Jean-Pierre Serre and Alexander Grothendieck, the exposition by Raoul Bott and John Milnor, and subsequent expansions propelled by interactions with Edward Witten and Simon Donaldson in the 1980s, which connected to problems studied by Michael Freedman and Charles Fefferman.

Proofs use a blend of pseudodifferential operator theory from Lars Hörmander and Joseph J. Kohn, K-theory techniques from Michael Atiyah and Friedrich Hirzebruch, heat equation methods influenced by Mark Kac and Richard Feynman, and cobordism ideas cultivated by Lev Pontryagin and René Thom. The analytic approach employs heat kernel asymptotics building on work of Minakshisundaram and Pleijel, while topological proofs exploit the Thom isomorphism and pushforward maps formulated by Grothendieck and Jean-Louis Koszul. Later proofs introduced index formulas via equivariant methods of Bertram Kostant and localization techniques inspired by Nikita Nekrasov and Edward Witten, and K-homology reinterpretations attributable to Gennadi Kasparov and Paul Baum.

The theorem underpins major results in differential geometry and topology such as proofs of signature theorems by Hirzebruch, restrictions on manifold structures studied by John Milnor and Michael Freedman, and invariants used in Donaldson theory and Seiberg–Witten theory developed by Simon Donaldson and Edward Witten. In mathematical physics it informs anomalies in quantum field theory investigated by Alvaro Alvarez-Gaumé and Edward Witten, index calculations for Dirac operators in gauge theory as in work by Atiyah with N. Hitchin and I. Singer with M. F. Atiyah, and spectral flow results used by M. Atiyah and I. M. Singer in studies related to the Atiyah–Patodi–Singer eta invariant developed with V. K. Patodi. Consequences also include explicit computations in representation theory of Lie groups from the work of Bertram Kostant and Harish-Chandra, and rigidity theorems proven by M. F. Atiyah and F. Hirzebruch.

Generalizations extend to equivariant indices associated to actions of compact Lie groups like SU(2), SO(n), and U(1), family index theorems of Quillen and Bismut for parameterized families studied by Daniel Quillen and Jean-Michel Bismut, and index theory on manifolds with boundary via the Atiyah–Patodi–Singer framework involving V. K. Patodi. Noncommutative geometry generalizations by Alain Connes recast the theorem in the language of C*-algebras and cyclic cohomology developed by Alain Connes and Max Karoubi, while analytic torsion and zeta regularization techniques relate to work of Ray and Singer and later expansions by D. B. Ray and Isadore Singer. Further related results include heat kernel proofs by Patodi and applications to index problems in string theory pursued by Edward Witten and Cumrun Vafa.