Adiabatic Limit, Theta Function, and Geometric Quantization

Let (M, ) B be a non-singular Lagrangian torus fibration on a complete base B with prequantum line bundle (L, L) (M, ). Compactness on M is not assumed. For a positive integer N and a compatible almost complex structure J on (M, ) invariant along the fiber of, let D be the associated Spinᶜ Dirac operator with coefficients in L^ N. First, in the case where J is integrable, under certain technical condition on J, we give a complete orthogonal system \ b\₁ ₁_ ₁ₒ of the space of holomorphic L²-sections of L^ N indexed by the Bohr-Sommerfeld points B ₁ₒ such that each b converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber ^-1 (b) by the adiabatic (-type) limit. We also explain the relation of b with Jacobi's theta functions when (M, ) is T^2n. Second, in the case where J is not integrable, we give an orthogonal family \ { b\}₁ ₁_ ₁ₒ of L²-sections of L^ N indexed by B ₁ₒ which has the same property as above, and show that each D b converges to 0 by the adiabatic (-type) limit with respect to the L²-norm.