Adiabatic Limit, Theta Function, and Geometric Quantization

Let π (M,ω)→ B be a non-singular Lagrangian torus fibration on a complete base B with prequantum line bundle (L,∇L)→ (M,ω). Compactness on M is not assumed. For a positive integer N and a compatible almost complex structure J on (M,ω) invariant along the fiber of π, 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 \ ϑb_b∈ BBS of the space of holomorphic L²-sections of L⊗ N indexed by the Bohr-Sommerfeld points BBS such that each ϑb converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π⁻¹(b) by the adiabatic(-type) limit. We also explain the relation of ϑb with Jacobi's theta functions when (M,ω) is T²ⁿ. Second, in the case where J is not integrable, we give an orthogonal family \ ϑb_b∈ BBS of L²-sections of L⊗ N indexed by BBS which has the same property as above, and show that each D ϑb converges to $0$ by the adiabatic(-type) limit with respect to the L²-norm.