Adiabatic Limit, Theta Function, and Geometric Quantization

Abstract

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 conditions 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 𝛵²ⁿ. Second, in the case where 𝐽 is not integrable, we give an orthogonal family {ϑ~b}b ∈ 𝐵BS of 𝐿²-sections of 𝐿⊗ᴺ indexed by 𝐵BS which has the same property as above, and show that each 𝘋ϑ~b converges to 0 by the adiabatic(-type) limit with respect to the 𝐿²-norm.