A Note On Projective Structures On Compact Surfaces

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 $T_{(1,0)}^{\ast }{\mathcal{M}}_g$ 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 $g\ge 2$, 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.