Adiabatic Limit, Theta Function, and Geometric Quantization

Let $\pi \text{\hspace{0.17em}⁣}:(M,\omega )\to B$ be a non-singular Lagrangian torus fibration on a complete base $B$ with prequantum line bundle $\bigl(L,\nabla^L\bigr)\to (M,\omega)$\mathrm{.}Compactnesson$M$isnotassumed\mathrm{.}Forapositiveinteger$N$ 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 $\{{\vartheta }_b{\}}_{b\in B_{\mathrm{B}\mathrm{S}}}$ of the space of holomorphic $L^2$-sections of $L^{\otimes N}$ indexed by the Bohr-Sommerfeld points $B_{\mathrm{B}\mathrm{S}}$ 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$L^2$-sectionsof$L^{\otimes N}$indexedby$B_{\rm BS}$ 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.