arXiv 2107.094122021-07math.SGmath.DGmath.GT

空间多面体模空间几何量子化中的操作数结构

Operad Structures in Geometric Quantization of the Moduli Space of Spatial Polygons

Yuya Takahashi

备注：: 23 pages, 7 figures. v2: added 7 figures and Remark 4.7; improved expressions in Introduction. To appear in J. Math. Soc. Japan

摘要：

空间多边形的模空间被称为同时具有凯勒和实极化的辛流形。结合Kähler极化和实极化，利用量子希尔伯特空间$\mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}$和$\mathscr{H}_\mathrm{re}$分别构造了算符$\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$和$\mathsf{f}_{\mathsf{re}}$的态射.研究了运算符$\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$和$\mathsf{f}_{\mathsf{re}}$的两个态射之间的关系，证明了在一般情况下，运算符$\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$和$\mathsf{f}_{\mathsf{re}}$的等价性$\dim \mathscr{H} _{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}=\dim \mathscr{H}_\mathrm{re}$.这种运算框架被认为是Kamiyama在特殊情况下证明$\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}=\dim \mathscr{H}_\mathrm{re}$的递归关系方法的发展。

英文摘要：

The moduli space of spatial polygons is known as a symplectic manifold equipped with both Kähler and real polarizations. In this paper, associated to the Kähler and real polarizations, morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ are constructed by using the quantum Hilbert spaces $\mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}$ and $\mathscr{H}_\mathrm{re}$, respectively. Moreover, the relationship between the two morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ is studied and then the equality $\dim \mathscr{H} _{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}=\dim \mathscr{H}_\mathrm{re}$ is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama for proving $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}=\dim \mathscr{H}_\mathrm{re}$ in a special case.

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