Atiyah–Patodi–Singer eta invariant

Atiyah–Patodi–Singer eta invariant
Name	Atiyah–Patodi–Singer eta invariant
Field	Differential geometry; Global analysis; Topology
Introduced	1970s
Founders	Michael Atiyah; Vijay Patodi; Isadore Singer

Atiyah–Patodi–Singer eta invariant The Atiyah–Patodi–Singer eta invariant is a spectral invariant arising in global analysis and differential topology associated to elliptic self-adjoint operators on manifolds with boundary. Developed in the collaborative work of Michael Atiyah, Vijay Patodi, and Isadore Singer, it plays a central role in formulations of the Atiyah–Singer index theorem, connections with Donaldson theory, ramifications for Chern–Simons theory, and interactions with the Pontryagin class and Riemannian metric choices on manifolds.

Definition and basic properties

The eta invariant is defined for a self-adjoint elliptic operator, notably the Dirac operator on a compact Riemannian manifold with boundary studied by Michael Atiyah, Vijay Patodi, and Isadore Singer in their series of papers; it measures spectral asymmetry of the operator by assigning an analytic value to the signed sum of eigenvalues reminiscent of constructions in Alexander Grothendieck-era index ideas and in the work of Atiyah–Bott. For closed manifolds the invariant ties to secondary characteristic classes such as the Pontryagin class and the Todd class via transgression; for manifolds with boundary it appears as a correction term in the Atiyah–Singer index theorem variant proved by the trio. Basic properties include dependence on the choice of Riemannian metric and connection up to spectral flow contributions studied by Daniel Freed and examined in relation to anomalies in Edward Witten's work on quantum field theory and Chern–Simons theory.

Eta function and analytic continuation

The construction uses the eta function η(s) = Σ sign(λ)|λ|^{-s} summed over nonzero eigenvalues λ of the operator; this analytic function is defined initially for Re(s) large and then analytically continued to s = 0, a technique that echoes methods developed by Bernhard Riemann for the Riemann zeta function and by Atiyah–Bott and Ray–Singer in analytic torsion. Analytic continuation employs spectral theory methods prominent in the work of Lars Hörmander and Mark Kac, and regularization ideas similar to those used by Paul Dirac and Richard Feynman in quantum field formulations. The value η(0) yields the eta invariant up to contributions from the kernel of the operator, paralleling constructions in the Selberg trace formula and aspects of the Hodge theorem.

Spectral asymmetry and index theorem applications

As a measure of spectral asymmetry, the eta invariant corrects index formulae in settings where classical index contributions from interior terms, like the Chern character paired with the Todd class, are insufficient; the Atiyah–Patodi–Singer index theorem equates an analytic index to topological data plus the eta invariant boundary correction. This correction has been invoked in the study of Donaldson theory, the Seiberg–Witten invariant, and invariants appearing in Floer homology developed by Andreas Floer and extended by Peter Kronheimer and Tomasz Mrowka. In mathematical physics, Edward Witten used eta invariants in anomaly cancellation and partition function phases, connecting to Chern–Simons theory on 3-manifolds studied by William Thurston and Dale Rolfsen.

Computation and examples

Explicit computations of eta invariants appear for standard geometric operators on familiar spaces: spherical space forms analyzed using methods of Heinz Hopf and Atle Selberg; lens spaces informed by computations in Reidemeister torsion and studies by John Milnor; and flat tori related to spectral lattice sums studied by Harold Davenport. Examples include Dirac operators on even and odd dimensional spheres, where symmetry simplifies η(0), and computations on lens space quotients and mapping tori important in 3-manifold topology and in work by Ciprian Manolescu. Computational techniques draw upon representation theory as in the work of Hermann Weyl and character formulae used by Bertram Kostant.

Relations to other invariants and extensions

The eta invariant relates to analytic torsion introduced by Daniel B. Ray and Isadore Singer and to the rho invariant of John Lott and Michel Hilsum in higher index theory contexts; it also interacts with the Atiyah–Patodi–Singer index theorem correction terms and with secondary invariants in K-theory as explored by Gennadi Kasparov and Alain Connes. Extensions include equivariant eta invariants considered by Segal-type theorists and families eta invariants in the work of Jeffrey Cheeger and Bismut whose families index theorem connects to Quillen's superconnection formalism. In noncommutative geometry, analogues appear in cyclic cohomology central to Alain Connes's program, and in higher rho invariants tied to the Novikov conjecture studied by Sergei Novikov and Gromov.

Analytical techniques and heat kernel methods

Analytic proofs and evaluations rely on heat kernel asymptotics developed by Raymond Seeley, Paul Gilkey, and Leonard Hörmander, using small-time expansions to extract local invariants like the A-hat genus and to control spectral regularization similar to techniques in index theory and in the analysis conducted by Nikolai Vilenkin. Heat kernel methods connect to short-time parametrix constructions in the pseudodifferential calculus elaborated by Kohn and Nirenberg and to the study of spectral flow pioneered by Booss-Bavnbek and Wojciech Ziemian?. The interplay of heat kernel coefficients with geometric curvature invariants underlies computational frameworks that support applications across differential topology, mathematical physics, and global analysis.

Category:Global analysis