圆2球面的线丛与Grauert管Hardy量子化的交织
Intertwining the line bundle and Grauert-tube Hardy quantizations of the round 2-sphere
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中文总结 AI 辅助
本文比较圆2球面单位余切丛上的两种Hardy量子化，计算其直线归一化重叠，推导相关迹级数与渐近，给出等变酉算子及Fredholm行列式等结果。
中文摘要 AI 辅助
我们比较圆2球面的单位余切丛上承载的两种自然Hardy量子化。通过$S^2\simeq CP^1$，该余切丛是典范丛$O(-2)$的单位圆丛，其Hardy空间组装了截面空间$H^0(CP^1,O(2\ell))$。通过实解析圆度量，虚时指数映射将同一余切丛与每个带有适配复结构的Grauert管边界等同，其Reeb流为测地流。两种Hardy空间均在$L^2(SO(3))$中实现，在左作用下无重数，且各选取一个Peter-Weyl重数空间中的直线。我们以闭式计算这些直线的归一化重叠为$\frac{\sqrt{\binom{2\ell}{\ell}}}{2^\ell}\frac{\sinh^\ell\tau}{\sqrt{P_\ell(\cosh 2\tau)}}$，其中$P_\ell$为勒让德多项式。归一化核向量定义了两种Hardy空间间的显式等变酉算子，且平方重叠是正迹类算子$\Pi_h\Pi_\tau\Pi_h$的特征值。我们推导了插入Hardy投影子的四个精确迹级数，证明它们的和与全平坦迹的差异为一个分布，其阿贝尔正则化在每个测地周期处具有三次增长。这些特征值还给出零阶的亏格为零的Fredholm行列式，将完整的大$\ell$展开截断至任意固定阶可得到有限的多重对数表达式。Bargmann-Fock计算确定了重叠的大$\ell$渐近中与$\ell$无关的 prefactor$(2\sinh 2\tau)^{1/4}e^{-\tau/2}$，该渐近对应于比较两个切触平面的 metaplectic 算子的高斯矩阵系数。
英文摘要
We compare two natural Hardy quantizations carried by the unit cosphere bundle of the round two-sphere. Through $S^2\simeq CP^1$, the cosphere bundle is the unit circle bundle of the canonical bundle $O(-2)$, and its Hardy space assembles the section spaces $H^0(CP^1,O(2\ell))$. Through the real-analytic round metric, the imaginary-time exponential map identifies the same cosphere bundle with every Grauert-tube boundary carrying the adapted complex structure, whose Reeb flow is the geodesic flow. Both Hardy spaces are realized in $L^2(SO(3))$, are multiplicity-free under the left action, and select one line in each Peter-Weyl multiplicity space. We compute the normalized overlap of these lines in closed form as $\frac{\sqrt{\binom{2\ell}{\ell}}}{2^\ell}\frac{\sinh^\ellτ}{\sqrt{P_\ell(\cosh 2τ)}}$, with $P_\ell$ the Legendre polynomial. The normalized kernel vectors define an explicit equivariant unitary between the two Hardy spaces, and the squared overlaps are the eigenvalues of the positive trace-class operator $Π_hΠ_τΠ_h$. We derive four exact trace series with the Hardy projectors inserted and show that their sum differs from the full flat trace by a distribution whose Abel regularization has cubic growth at every geodesic period. The eigenvalues also give a genus-zero Fredholm determinant of order zero, and truncating the complete large-$\ell$ expansion at any fixed order gives a finite polylogarithmic expression. A Bargmann-Fock calculation identifies the $\ell$-independent prefactor $(2\sinh 2τ)^{1/4}e^{-τ/2}$ in the large-$\ell$ asymptotic of the overlap with the Gaussian matrix coefficient of a metaplectic operator comparing the two contact planes.
