International Congress of Mathematicians

366 Weiping Zhangwhere ch(g) is the odd Chern character associated to g.There is also an analytic proof of (4.2) by using heat kernels. For this onefirst applies a result of Booss and Wojciechowski (cf. [BW]) to show that thecomputation of ind T g is equivalent to the computation of the spectral flow of thelinear family of self-adjoint elliptic operators, acting of Y(S(TM) ® C N ), whichconnects D and gDg^1.The resulting spectral flow can then be computed byvariationsof ^-invariants, where the heat kernels are naturally involved.The above ideas have been extended in [DZ1] to give a heat kernel proof of afamily extension of (4.2).5. An index theorem for Toeplitz operators on odddimensional manifolds with boundaryIn this section, we describe an extension of (4.2) to the case of manifolds withboundary, which was proved recently in my paper with Xianzhe Dai [DZ2]. Thisresult can be thought of as an odd dimensional analogue of the Atiyah-Patodi-Singerindex theorem described in Section 3.This section is divided into three subsections. In Subsection 4.1, we extend thedefinition of Toeplitz operators to the case of manifolds with boundary. In Subsection4.2, we define an ^-invariant for cylinders which will appear in the statementof the main result to be described in Subsection 4.3.5.1. Toeplitz operators on manifolds with boundaryLet M be an odd dimensional oriented spin manifold with (nonempty) boundarydM. Then dM is also oriented and spin. Let g be a Riemannian metric onTM such that it is of product structure near the boundary dM. Yet S(TM) be theHermitian bundle of spinors associated to (M,g). Since dM ^ 0, the Dirac operatorD : Y(S(TMj) —t Y(S(TMj) is no longer elliptic. To get an elliptic operator,one needs to impose suitable boundary conditions, and it turns out that again wewill adopt the boundary conditions introduced by Atiyah, Patodi and Singer [APS].Let DQM '• Y(S(TM)\QM) —t Y(S(TM)\QM) be the canonically induced Diracoperator on the boundary dM. Then DQM is elliptic and (formally) self-adjoint.For simplicity, we assume here that DQM is invertible, that is, ker DOM = 0.Let PdM,>o denote the Atiyah-Patodi-Singer projection from L 2 (S(TM)\QM)to L 2 >Q(S(TM)\QM). Then (D, PQM;>Q) forms a self-adjoint elliptic boundary problem.We will also denote the corresponding elliptic self-adjoint operator by Dp 8M>0 .Yet L 2 P >Q(S(TMj) be the space of the direct sum of eigenspaces of nonnegativeeigenvalues of Dp aM >0 . Yet Pp BM >0 >o denote the orthogonal projectionfrom L 2 (S(TM)) to L 2 PgM^; 0 (S(TM)).Now let C^ be the trivial complex vector bundle over M of rank N, whichcarries the trivial Hermitian metric and the trivial Hermitian connection. We extendPp BMi>0 ,>o to act as identity on C*.Let g : M —¥ U(N) be a smooth unitary automorphism of C N . Then g extendsto an action on S(TM) ® C^ by acting as identity on S(TM).