Adiabatic Limit, Theta Function, and Geometric Quantization

Let 7r: (M, w) -> B be a non-singular Lagrangian torus fibration on a complete base B with prequantum line bundle (L, VL) -> (M, w). Compactness on M is not assumed. For a positive integer N and a compatible almost complex structure J on (M, w) invariant along the fiber of 7r, let D be the associated Spinc Dirac operator with coefficients in L (R) N. First, in the case where J is integrable, under certain technical condition on J, we give a complete orthogonal system {theta(b)}b is an element of BBS of the space of holomorphic L2-sections of L (R) N indexed by the Bohr-Sommerfeld points BBS such that each theta(b) converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber 7r-1(b) by the adiabatic(-type) limit. We also explain the relation of theta(b) with Jacobi's theta functions when (M, w) is T2n. Second, in the case where J is not integrable, we give an orthogonal family {(theta) over tilde (b)}b is an element of BBS of L2-sections of L (R) N indexed by BBS which has the same property as above, and show that each D (theta) over tilde (b) converges to 0 by the adiabatic(-type) limit with respect to the L2-norm.