Subelliptic boundary conditions for Spinℂ–Dirac operators gluing relative indices and tame Fredholm pairs

摘要

Let X be a Spinℂ manifold with boundary, such that the Spinℂ structure is defined near the boundary by an almost complex structure, which is either strictly pseudoconvex or pseudoconcave (and hence contact). Using generalized Szegő projectors, we define modified ∂̄-Neumann boundary conditions, ℛ^eo, for spinors, which lead to subelliptic Fredholm boundary value problems for the Spinℂ–Dirac operator, ð^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 (ð^eo, ℛ^eo) equals the tame relative index, on the boundary, between ℛ^eo and the Calderon projector. Let X0 and X1 be strictly pseudoconvex, Spinℂ manifolds, as above. Let φ : bX1 → bX0, be a contact diffeomorphism, S0, S1 denote generalized Szegő 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 ∐_φ X₁ has a canonical Spin_ℂ structure and Dirac operator, ð_X^eo. Applying these results we obtain a formula for the relative index, R-Ind(S₀, φ*S₁), 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.