Abstract. A projective structure on a Riemann surface M is achieved by lifting to a universal covering space of M the ratio of solutions of a linear differential equation defined on the surface. We show that these solutions, and vector fields preserving the projective structure induced on M, define a sheaf of graded Lie algebras. At the same time, we construct a sheaf of graded Lie algebras over the Teichmüller family of Riemann surfaces, which is related to the process of simultaneous uniformization for the family. This structure constitutes a modular supersymmetry, an extended notion of local sl2-symmetry, which is motivated by Grand Unification Theory in theoretical physics.

Introduction

A projective structure on a Riemann surface M is a reduction of its pseudo-group of holomorphic coordinate transformations to the group of fractional linear transformations PL(2,C). On a Riemann surface of genus g > 2, this can be achieved by choosing a projective connection, an element of an affine space over the vector space of globally defined holomorphic quadratic differentials [7]. A new complex atlas on M with projective coordinate transformations is obtained by composing the old coordinates with the ratio of two linearly independent solutions of a differential equation defined on the surface.

A significant fact, proved in [9], is that solutions of such a uniformizing equation must be local spinors, sections of a square root of the tangent bundle of M.