Adiabatic limits, Theta functions, and geometric quantization

Abstract

Let $\pi\colon (M,\omega)\to B$ be a (non-singular) Lagrangian torus fibration on a compact, complete base $B$ with prequantum line bundle $(L,\nabla^L)\to (M,\omega)$. 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 $L^{\otimes N}$. Then, we give an orthogonal family $\{ {\tilde \vartheta}_b\}_{b\in B_{BS}}$ of sections of $L^{\otimes N}$ indexed by the Bohr-Sommerfeld points $B_{BS}$, and show that each ${\tilde \vartheta}_b$ converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber $\pi^{-1}(b)$ and the $L^2$-norm of $D{\tilde \vartheta}_b$ converges to $0$ by the adiabatic(-type) limit. Moreover, if $J$ is integrable, we also give an orthogonal basis $\{ \vartheta_b\}_{b\in B_{BS}}$ of the space of holomorphic sections of $L^{\otimes N}$ indexed by $B_{BS}$, and show 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 $T^{2n}$.Comment: 41 page