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Adiabatic Limit, Theta Function, and Geometric Quantization

Takahiko Yoshida

Abstract

Let $π\colon (M,ω)\to B$ be a non-singular Lagrangian torus fibration on a complete base $B$ with prequantum line bundle $\bigl(L,\nabla^L\bigr)\to (M,ω)$. 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 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 $π^{-1}(b)$ by the adiabatic(-type) limit. We also explain the relation of $\vartheta_b$ with Jacobi's theta functions when $(M,ω)$ 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.

Adiabatic Limit, Theta Function, and Geometric Quantization

Abstract

Let be a non-singular Lagrangian torus fibration on a complete base with prequantum line bundle . Compactness on is not assumed. For a positive integer and a compatible almost complex structure on invariant along the fiber of , let be the associated Spin Dirac operator with coefficients in . First, in the case where is integrable, under certain technical condition on , we give a complete orthogonal system of the space of holomorphic -sections of indexed by the Bohr-Sommerfeld points such that each converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber by the adiabatic(-type) limit. We also explain the relation of with Jacobi's theta functions when is . Second, in the case where is not integrable, we give an orthogonal family of -sections of indexed by which has the same property as above, and show that each converges to by the adiabatic(-type) limit with respect to the -norm.

Paper Structure

This paper contains 21 sections, 45 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main theorems
  4. Notations
  5. Unfolding Lagrangian fibrations
  6. Integral affine structures
  7. Lagrangian fibrations
  8. Lagrangian fibrations with complete bases
  9. The lifting problem of fiber-preserving symplectomorphisms to the prequantum line bundle
  10. Degree-zero harmonic spinors and integrability of almost complex structures
  11. Bohr--Sommerfeld points
  12. Almost complex structures
  13. A condition on the existence of nontrivial harmonic spinors of degree-zero
  14. The $\boldsymbol{\Gamma}$-equivariant case
  15. The integrable case
  16. ...and 6 more sections

Key Result

Theorem 1.1

Under the above setting, assume that $J$ is integrable and satisfies certain technical condition. Then, for each $t>0$, there exists a complete orthogonal system $\{\vartheta^t_b\}_{b\in B_{\rm BS}}$ of holomorphic $L^2$-sections of $L^{\otimes N}\to \bigl(M,N\omega ,J^t\bigr)$ indexed by the Bohr-- where $\langle \, ,\, \rangle_{L^{\otimes N}}$ is the Hermitian metric of $L^{\otimes N}$, $\delta_

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Proposition 2.8
  • ...and 63 more