SO(9, 1) invariant matrix formulation of a supermembrane

Abstract

An $SO(9,1)$ invariant formulation of an 11-dimensional supermembrane is presented by combining an $SO(10,1)$ invariant treatment of reparametrization symmetry with an $SO(9,1)$ invariant $\theta_{R} = 0$ gauge of $\kappa$-symmetry. The Lagrangian thus defined consists of polynomials in dynamical variables (up to quartic terms in $X^{\mu}$ and up to the eighth power in $\theta$), and reparametrization BRST symmetry is manifest. The area preserving diffeomorphism is consistently incorporated and the area preserving gauge symmetry is made explicit. The $SO(9,1)$ invariant theory contains terms which cannot be induced by a naive dimensional reduction of higher dimensional supersymmetric Yang-Mills theory. The $SO(9,1)$ invariant Hamiltonian and the generator of area preserving diffeomorphism together with the supercharge are matrix regularized by applying the standard procedure. As an application of the present formulation, we evaluate the possible central charges in superalgebra both in path integral and in canonical (Dirac) formalism, and we find only the two-form charge $[X^\mu, X^\nu]$.