A Note On Projective Structures On Compact Surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929574978256896 |
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| author | Liu, Xiao |
| author_facet | Liu, Xiao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01810 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Note On Projective Structures On Compact Surfaces Liu, Xiao High Energy Physics - Theory Mathematical Physics Complex Variables Differential Geometry 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. |
| title | A Note On Projective Structures On Compact Surfaces |
| topic | High Energy Physics - Theory Mathematical Physics Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2409.01810 |