绝热极限、Theta函数与几何量子化
Adiabatic Limit, Theta Function, and Geometric Quantization
AI总结：
本文研究非紧拉格朗日环面纤维化上的绝热极限，构造由Bohr-Sommerfeld点索引的L^2截面正交系，证明其收敛到delta函数截面，并推广到非可积近复结构情形。
摘要：
设π:(M,ω)→B是完备基B上的非奇异拉格朗日环面纤维化，带有预量子线丛(L,∇^L)→(M,ω)。不假设M紧致。对于正整数N和(M,ω)上沿π的纤维不变的相容近复结构J，设D为系数在L^⊗N中的相关Spin^c Dirac算子。首先，在J可积的情况下，在J的某个技术条件下，我们给出了全纯L^2-截面空间的一个完全正交系{ϑ_b}_{b∈B_BS}，由Bohr-Sommerfeld点B_BS索引，使得每个ϑ_b通过绝热（型）极限收敛到支撑在相应Bohr-Sommerfeld纤维π^{-1}(b)上的delta函数截面。我们还解释了当(M,ω)为T^{2n}时ϑ_b与Jacobi theta函数的关系。其次，在J不可积的情况下，我们给出了由B_BS索引的L^2-截面族{ϑ̃_b}_{b∈B_BS}，该族具有与上述相同的性质，并证明了每个Dϑ̃_b在L^2-范数意义下通过绝热（型）极限收敛到0。
英文摘要：
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.
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