标题： 关于紧致曲面上的射影结构的一点注记 显示英文标题

标题： A Note On Projective Structures On Compact Surfaces

摘要： 拓扑曲面上的射影结构支持二维共形场论 (CFT) 的构造，并且在一定程度上简化了技术细节。我们提出一个复解析空间 $\mathcal{P}_g$，它双全纯于 $T^*_{(1,0)} \mathcal{M}_g$，作为亏格为 $g$ 的拓扑曲面射影结构的模空间的候选者。在 $g=1$ 处的显式分析，包括对 $\mathcal{M}_{g=1}$ 中虚构orbifold点纤维的分析以及模群变换下的分析，支持这一提议。 它还表明，$\mathcal{P}_{g=1}$自然解决了仿射结构模空间的 orbifold locus $\mathcal{A}_{g=1}$，该 locus 与$\mathcal{M}_{g=1}$上的 Hodge 椅相关。 对于$g \geq 2$，预计在 orbifold locus $\mathcal{M}_g$的纤维上会有复杂的商运算，我们将其分析留待将来的工作。 物理上，空间$\mathcal{P}_g$表示了空间上局限于紧致亏格为$g$的黎曼曲面的普遍、平稳、手性的流体力学流的束。 显示英文摘要

摘要： Projective structures on topological surfaces support the structure of 2d CFTs with a degree of technical simplification. We propose a complex analytic space $\mathcal{P}_g$ biholomorphic to $T^*_{(1,0)} \mathcal{M}_g$ as a candidate moduli space of the projective structures of the genus $g$ topological surface. Explicit analysis at $g=1$, including of the fibers over the fictitious orbifold loci of $\mathcal{M}_{g=1}$ and of transformations under the modular group, supports this proposal. It also shows that $\mathcal{P}_{g=1}$ naturally resolves the orbifold locus of the affine structure moduli space $\mathcal{A}_{g=1}$ which is related to the Hodge bundle over $\mathcal{M}_{g=1}$. For $g \geq 2$, intricate quotient operations are expected along fibers over the orbifold loci of $\mathcal{M}_g$, whose analysis we leave to future work. Physically, the space $\mathcal{P}_g$ represents the bundle of universal, stationary, chiral hydrodynamic flows spatially confined to compact genus-$g$ Riemann surfaces.