Lie Algebroids and generalized projective structures on Riemann surfaces

TL;DR

This paper explores the structure of generalized projective structures on Riemann surfaces, linking them to Lie algebroids, moduli spaces, and $W_N$-gravity, providing a geometric and algebraic framework for their analysis.

Contribution

It introduces the concept of AGD algebroids as a new class of Lie algebroids related to $W_N$-gravity and describes the moduli space of these structures via cohomology of a BRST complex.

Findings

The space of generalized projective structures forms an affine space over the cotangent bundle of SL(N)-opers.

The moduli space of $W_N$-gravity is realized as a symplectic quotient involving AGD algebroids.

The cohomology of a BRST complex describes the moduli space of these structures.

Abstract

The space of generalized projective structures on a Riemann surface $\Sigma$ of genus g with n marked points is the affine space over the cotangent bundle to the space of SL(N)-opers. It is a phase space of $W_N$-gravity on $\Sigma\times\mathbb{R}$. This space is a generalization of the space of projective structures on the Riemann surface. We define the moduli space of $W_N$-gravity as a symplectic quotient with respect to the canonical action of a special class of Lie algebroids. This moduli space describes in particular the moduli space of deformations of complex structures on the Riemann surface by differential operators of finite order, or equivalently, by a quotient space of Volterra operators. We call these algebroids the Adler-Gelfand-Dikii (AGD) algebroids, because they are constructed by means of AGD bivector on the space of opers restricted on a circle. The AGD-algebroids are particular case of Lie algebroids related to a Poisson sigma-model. The moduli space of the generalized projective structure can be described by cohomology of a BRST-complex.