Adiabatic Limit, Theta Function, and Geometric Quantization

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