International Congress of Mathematicians

Heat Kernels and the Index Theorems • • • 365By using the heat kernel method, one can show easily that ï](DQM,S) can be extendedto a meromorphic function on C, which is holomorphic at s = 0. Following[APS], we then define_ dim (ker £> eM ) + »} (-DOM, 0)n (D dM ) = 2 ( 3 - 5 )and call it the (reduced) eta invariant of DQM-The eta invariants of Dirac operators have played important roles in manyaspectsof topology, geometry and mathematical physics.In the next sections, we will discuss the role of eta invariants in the heat kernelapproaches to the index theorems on odd dimensional manifolds.4. Heat kernels and the index theorem on odd dimensionalmanifoldsLet M be now an odd dimensional smooth closed oriented spin manifold. Letg be a Riemannian metric on TM and S(TM) the associated Hermitian vectorbundle of (TAf,^)-spinors. 1 In this case, the associated Dirac operator D :Y(TM) —t Y(TM) is (formally) s elf-adjoint. 2 Thus, one can proceed as in Section3 to construct the Atiyah-Patodi-Singer projectionP>o • L 2 (S(TMj) -+ Ll 0 (S(TMj).Now consider the trivial vector bundle C* over M. We equip C* with thecanonical trivial metric and connection. Then F> 0 extends naturally to an orthogonalprojection from L 2 (S(TM) C^) to L% 0 (S(TM) C^) by acting as identityon C^. We still denote this extension by P>o-On the other hand, letg:M -• U(N)be a smooth map from M to the unitary group U(N). Then g can be interpretedas automorphism of the trivial complex vector bundle C N . Moreover g extendsnaturally to an action on L 2 (S(TM) ® C*) by acting as identity on L 2 (S(TMj).We still denote this extended action by g.With the above data given, one can define a Toeplitz operator T g as follows,T g = P>ogP>o • Lio (S(TM) C") —• L\ 0 (S(TM) ® C") . (4.1)The first important fact is that T g is a Fredholm operator. Moreover, it isequivalent to an elliptic pseudodifferential operator of order zero. Thus one cancompute its index by using the Atiyah-Singer index theorem [AS2], as was indicatedin the paper of Baum and Douglas [BD], and the result isindT, = - (A(TAf)ch(