Central extensions of area preserving membrane algebras

Understanding the symmetries of membrane and p-brane theories is important in order to gain some insight into their quantum structure. In the light-cone gauge there is a residual symmetry of the classical theories that corresponds to the subgroup of the diffcomorphisms of the p-brane for which the transformation jacobian is equal to unity. We investigate the possible appearance of anomalies in this algebra, by looking for the most general possible central extension compatible with the Jacobi identity. We show that for membranes the most general central extension is characterized by the space of harmonic one-forms on the membrane, whose dimension is twice the genus of the surface. We generalize this algebra to a hypothetical supersymmetric theory with super-reparametrizations of the membrane surface, and show that it cannot admit any central extension. (orig.)