Adiabatic Limit, Theta Function, and Geometric Quantization
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866914876594585600 |
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| author | Yoshida, Takahiko |
| author_facet | Yoshida, Takahiko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_04076 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Adiabatic Limit, Theta Function, and Geometric Quantization Yoshida, Takahiko Symplectic Geometry Differential Geometry 53D50 (Primary), 58H15, 58J05 (Secondary) 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. |
| title | Adiabatic Limit, Theta Function, and Geometric Quantization |
| topic | Symplectic Geometry Differential Geometry 53D50 (Primary), 58H15, 58J05 (Secondary) |
| url | https://arxiv.org/abs/1904.04076 |