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. Arguments for their existence are reviewed along with their essential properties. We propose a complex analytic manifold $\mathcal{P}_g$ biholomorphic to $T^*_{(1,0)} \mathcal{M}_g$ as a pseudo moduli space of the projective structures of the genus $g$ topological surface. Explicit computations at $g=1$ including the analysis of transformations under the modular group support this proposal, and show that $\mathcal{P}_{g=1}$ naturally resolves the orbifold locus of the affine structure moduli space. For $g \geq 2$, whether $\mathcal{P}_g$ contains redundancy at each value of the complex structure moduli remains open. Physically, the space $\mathcal{P}_g$ represents the bundle of universal, stationary, chiral hydrodynamic flows spatially confined to compact genus-$g$ Riemann surfaces.
Comment: 44 pages, no figure