Subelliptic boundary conditions for Spin ℂ –Dirac operators, gluing, relative indices, and tame Fredholm pairs

Let X be a Spin manifold with boundary, such that the Spin structure is defined near the boundary by an almost complex structure, which is either strictly pseudoconvex or pseudoconcave (and hence contact). Using generalized Szego projectors, we define modified partial differential-Neumann boundary conditions, Reo, for spinors, which lead to subelliptic Fredholm boundary value problems for the Spin-Dirac operator, eth(eo). To study the index of these boundary value problems we introduce a generalization of Fredholm pairs to the "tame" category. In this context, we show that the index of the graph closure of (eth(eo), Reo) equals the tame relative index, on the boundary, between Reo and the Calderon projector. Let X0 and X1 be strictly pseudoconvex, Spin manifolds, as above. Let phi : bX1 --> bX0, be a contact diffeomorphism, S0, S1 denote generalized Szego projectors on bX0, bX1, respectively, and R0(eo), R1(eo), the subelliptic boundary conditions they define. If X1 is the manifold X1 with its orientation reversed, then the glued manifold X = X0 coproduct operator(phi) X1 has a canonical Spin structure and Dirac operator, ethX(eo). Applying these results we obtain a formula for the relative index, R-Ind(S0, phi*S1), [formula: see text]. As a special case, this formula verifies a conjecture of Atiyah and Weinstein [(1997) RIMS Kokyuroku 1014:1-14] for the index of the quantization of a contact transformation between cosphere bundles.