McKean–Singer

McKean–Singer
Name	McKean–Singer
Field	Mathematics, Differential geometry, Global analysis
Statement	Relates the index of an elliptic operator to the supertrace of the heat kernel; shows analytic index equals topological index for Dirac-type operators
First proved	1967
Authors	Henry P. McKean, Isadore M. Singer

McKean–Singer McKean–Singer is a theorem in Mathematics connecting analytic and topological invariants via heat kernel methods. It played a seminal role in the development of the Atiyah–Singer index theorem, influenced work in Quantum field theory, and linked techniques from Partial differential equations, Spectral theory, and Differential geometry.

History and Background

The result emerged in the late 1960s amid interactions between researchers at Harvard University, Massachusetts Institute of Technology, and institutions where Henry P. McKean and Isadore M. Singer were active, building on earlier work by Atiyah, Michael Francis, Singer collaborators and influences from Israel Gelfand, Mikhail Gromov, and John Milnor. It was contemporaneous with breakthroughs by Michael Atiyah, Raoul Bott, Jean-Pierre Serre, Lars Hörmander, and Atle Selberg in spectral and topological analysis. The McKean–Singer approach integrated ideas from Richard Feynman's path integrals, Andrey Kolmogorov's heat kernel theory, and classical results of Élie Cartan and Hermann Weyl. Subsequent developments involved researchers such as Edward Witten, Daniel Quillen, Roger Penrose, Nigel Hitchin, and Shing-Tung Yau.

McKean–Singer Index Theorem

The theorem asserts that for a compact manifold equipped with a Dirac operator of Dirac-type, the analytic index equals the supertrace of the heat kernel as time tends to zero, thereby equating the analytic index to a local density whose integral is the topological index. Key figures linked to the conceptual framework include Michael Atiyah, Isadore M. Singer, Raoul Bott, Bertram Kostant, Alfred Tarski, and Henri Poincaré. The statement refines classical index formulas associated with the names of Atiyah–Singer, Gauss–Bonnet, Hirzebruch–Riemann–Roch, and the Lefschetz fixed-point theorem; antecedents and comparisons were explored by Hermann Weyl, Norbert Wiener, Marcel Berger, and Shiing-Shen Chern.

Proof Outline and Techniques

Proof methods use asymptotic expansion of the heat kernel, spectral decomposition, and supertrace cancellations on spaces of spinors; contributors and influences include Lars Hörmander, Peter Lax, Israel Gelfand, Louis Nirenberg, and Jean Leray. The argument invokes microlocal analysis developed by Joseph J. Kohn, J. J. Duistermaat, and Victor Guillemin, and heat equation techniques refined by E. B. Davies and Lawrence Evans. Index pairing and K-theory perspectives involve Alexander Grothendieck, Michael Atiyah, Daniel Quillen, Raoul Bott, and Jean-Louis Koszul. Functional-analytic input references work of John von Neumann, Marshall Stone, and Frigyes Riesz in operator theory.

Applications and Consequences

Applications span proofs and refinements of the Atiyah–Singer index theorem, geometry of spin manifolds, calculations in Seiberg–Witten theory, relations to anomalies in Quantum field theory, and links to the Index theorem in noncommutative geometry of Alain Connes. The McKean–Singer viewpoint influenced work by Edward Witten on supersymmetry, by Simon Donaldson on four-manifold invariants, and by Max Born-adjoining researchers in mathematical physics including Gerard 't Hooft, Alexander Polyakov, and Kenneth G. Wilson. It also informs computational techniques used in studies by Charles Fefferman, Dennis Sullivan, Jean Bourgain, and Terence Tao.

Examples and Computations

Canonical examples compute indices on spheres, tori, and complex projective spaces, with explicit cases studied by Shiing-Shen Chern, Hirzebruch, Jean-Pierre Serre, and Raoul Bott. Model computations for the de Rham complex reproduce the Gauss–Bonnet theorem as illustrated in work by Élie Cartan and Hermann Weyl; Dirac operator computations on Kähler manifolds reflect results of Shing-Tung Yau, Nigel Hitchin, and Raoul Bott. Heat kernel asymptotics and explicit coefficient calculations cite methods from Minakshisundaram, Srinivasa Varadhan, Lars Hörmander, and Peter Gilkey.

Extensions and Generalizations

Extensions include families index theorems by Michael Atiyah and Isadore M. Singer, equivariant generalizations linked to Bertram Kostant and Raoul Bott, and noncommutative versions developed by Alain Connes. Further generalizations involve connections to the Atiyah–Patodi–Singer index theorem of Michael F. Atiyah, Vladimir I. Arnold-inspired singularity theory, and advances by Eugene Wigner-related symmetry methods. Contemporary directions connect to research by Maxim Kontsevich, Edward Witten, Don Zagier, and Klaus Schulten bridging enumerative geometry, mathematical physics, and noncommutative topology.

Category:Index theorems