A Note On Projective Structures On Compact Surfaces

TL;DR

This paper proposes a complex analytic space as a moduli space for projective structures on compact surfaces, analyzing the genus 1 case in detail and discussing implications for higher genus surfaces and physical theories.

Contribution

It introduces a candidate moduli space $\

Findings

Supports the proposal with explicit genus 1 analysis.

Shows $\\mathcal{P}_1$ resolves orbifold loci of the affine structure moduli.

Discusses future work for higher genus surfaces.

Abstract

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.