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 Pg biholomorphic to T^* (₁, ₀) Mg 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 P₆=₁ naturally resolves the orbifold locus of the affine structure moduli space. For g 2, whether Pg contains redundancy at each value of the complex structure moduli remains open. Physically, the space Pg represents the bundle of universal, stationary, chiral hydrodynamic flows spatially confined to compact genus-g Riemann surfaces.