Nonlocal Invariants in Index Theory

In its original form, the Atiyah-Singer Index Theorem equates two global quantities of a closed manifold, one analytic (the index of an elliptic operator) and one topological (a characteristic number). Because it relates invariants from different branches of mathematics, the Index Theorem has many applications and extensions to differential geometry, K-theory, mathematical physics, and other fields. This report focuses on advances in geometric aspects of index theory. For operators naturally associated to a Riemannian metric on a closed manifold, the topological side of the Index Theorem can often be expressed as the integral of local (i.e. pointwise) curvature expression. We will first discuss these local refinements in §1, which arise naturally in heat equation proofs of the Index Theorem. In §§2,3, we discuss further developments in index theory which lead to spectral invariants, the eta invariant and the determinant of an elliptic operator, that are definitely nonlocal. Finally, in §4 we point out some recent connections among these nonlocal invariants and classical index theory. 1. Local invariants in Atiyah-Singer index theory The Atiyah-Singer Index Theorem, first proved around 1963, equates the index, IndD = dim KerD − dim CokerD, of an elliptic operator D : Γ(E) → Γ(F ) taking sections of a bundle E over a closed manifold M to sections of a bundle F with 〈P (M,σtop(D)), [M ]〉, a characteristic number built from the topology of M and topological information contained in the top order symbol of D [5], [38]. For particular choices of D, this deep result encompasses Chern’s generalization of the Gauss-Bonnet theorem, the Hirzebruch signature theorem, and Hirzebruch’s generalization of the Riemann-Roch theorem, as well as giving many new results. Atiyah and Singer also proved the Families Index Theorem [6] for a family of elliptic operators {Dn} parametrized by n in a compact manifold N . This theorem identifies the Chern character of the index bundle Ind D in H∗(N ;Q) with a characteristic class on N built from the topology of N and the pushforward of the symbols of the Dn; the significance of this theorem is that the Chern character of a bundle determines the K-theory isomorphism class of the bundle up to torsion. Here Ind D = Ker D − Coker D is a virtual bundle whose fiber for generic n ∈ N is the formal difference Ker Dn −Coker Dn. For example, for a family of Dirac operators associated to a family of metrics gn on M parametrized by n ∈ N , we get ch(Ind D) = π∗Â(M/N), where Â(M/N) is the Â-polynomial of the bundle over N whose fiber at n is the bundle of spinors associated to gn, and π : M ×N → N is 1991 Mathematics Subject Classification. Primary 58G25; Secondary 58G10, 58G25, 58G26. Partially supported by the NSF. c ©1997 American Mathematical Society