A perturbative expansion scheme for supermembrane and matrix theory

Abstract

We reconsider the supermembrane in a Minkowski background and in the light-cone gauge as a one-dimensional gauge theory of area preserving diffeomorphisms (APDs). Keeping the membrane tension $T$ as an independent parameter we show that $T$ is proportional to the square of the gauge coupling $g$ of this gauge theory, such that the small (large) tension limit of the supermembrane corresponds to the weak (strong) coupling limit of the APD gauge theory and its SU$(N)$ matrix model approximation. A perturbative linearization of the supersymmetric theory suitable for a quantum mechanical path-integral treatment can be achieved by formulating a Nicolai map for the matrix model, which we work out explicitly to ${\cal O}(g^4)$. The corresponding formulae remain well-defined in the limit $N{\to}\infty$; this result relies on a cancellation of infinities not present for the bosonic membrane, indicating that the $N{\to}\infty$ limit does not exist for the purely bosonic matrix model. Furthermore we show that the map has improved convergence properties in comparison with the usual perturbative expansions because its Jacobian admits an expansion in $g$ with a non-zero radius of convergence. Possible implications for unsolved issues with the matrix model of M theory are also mentioned.