有托里克簇上从混合极化到实极化的量子化极限
Limits of quantization from mixed to real polarizations on toric varieties
浏览论文内容
中文总结 AI 辅助
该研究针对托里克簇，利用虚时流构造混合极化的单参数族，通过预量子线丛的等变同构，证明量子希尔伯特空间随虚时流收敛到实极化对应的量子希尔伯特空间。
中文摘要 AI 辅助
设$(M, \boldsymbol{\u03c9}, J)$是由Delzant多面体$P$确定的$2n$维托里克簇，其$T^{n}$对称性决定了实极化$\u0394_{\u211d}$。设$K \boldsymbol{\u2282} T^{n}$为子环面，基于Leung与第一作者的构造，$K$作用诱导出混合极化$\u0394_{K}$。本文研究与极化$\u0394_{K}$和$\u0394_{\u211d}$对应的量子希尔伯特空间$\u039d_{K}$与$\u039d_{\u211d}$的关系。从$\u0394_{K}$出发，利用虚时流构造$M$上的单参数混合极化族$\u0394_{K,t}$，在$\u0394_{K}$与$\u0394_{\u211d}$之间插值，满足$\u0394_{K,0}=\u0394_{K}$且$\u0394_{K,t}$当$t \to \u221e$时极限为$\u0394_{\u211d}$。对对应的量子希尔伯特空间$\u039d_{K,t}$，将虚时流提升到预量子线丛，得到$T^{n}$等变同构$\u039d_{K}\u2245\u039d_{K,t}$，最终证明$\u039d_{K,t}$当$t \to \u221e$时收敛到$\u039d_{\u211d}$。
英文摘要
Let $(M, ω, J)$ be a $2n$-dimensional toric variety determined by a Delzant polytope $P$, whose $T^{n}$-symmetry determines a real polarization $\mathcal{P}_{\mathbb{R}}$. Let $K \subset T^{n}$ be a subtorus. By a construction due to Leung and the first author, the $K$-action induces a mixed polarization $\mathcal{P}_{K}$. This paper investigates the relationship between the quantum Hilbert spaces $\mathcal{H}_{K}$ and $\mathcal{H}_{\mathbb{R}}$ associated with the polarizations $\mathcal{P}_{K}$ and $\mathcal{P}_{\mathbb{R}}$. Starting from $\mathcal{P}_{K}$, we use an imaginary-time flow to construct a one-parameter family of mixed polarizations $\mathcal{P}_{K,t}$ on $M$ interpolating between $\mathcal{P}_{K}$ and $\mathcal{P}_{\mathbb{R}}$, with $\mathcal{P}_{K,0}=\mathcal{P}_{K}$ and $\lim_{t\to\infty}\mathcal{P}_{K,t}=\mathcal{P}_{\mathbb{R}}$. For the corresponding quantum Hilbert spaces $\mathcal{H}_{K,t}$, we lift the imaginary-time flow to the prequantum line bundle to obtain a $T^{n}$-equivariant isomorphism $\mathcal{H}_{K}\cong\mathcal{H}_{K,t}$. We finally show that $\mathcal{H}_{K,t}$ converges to $\mathcal{H}_{\mathbb{R}}$ as $t\to\infty$.
