Update on the quantum properties of the supermembrane

Abstract

In this note we summarize some of the quantum properties found since the early 80's until nowdays that characterize at quantum level the spectrum of the supermembrane. In particular we will focus on a topological sector of the 11D supermembrane that, contrary to the general case, has a purely discrete spectrum at supersymmetric level. This construction has been consistently implemented in different types of backgrounds: toroidal and orbifold-type with G2 structure able to lead to a true G2 compactification manifold. This theory has N = 1 supersymmetries in 4D. comment on the relevant points of this construction as well as on its spectral characteristics. We will also make some comments on the quantum properties of some effective formulation of multiple M2's theories recently found. The 11D supermembraneThe 11D supermembrane, also called M2 brane, [1] was discovered in 1987. A year later its hamiltonian formulation in the Light Cone Gauge (L.C.G.) and its matrix regularization by [3] was obtained. The supermembrane is the natural extension of the string in 11D and it was thought to be a fundamental object in 11D. However, the spectral properties of the regularized M2 differed substantially from those of the string as it was shown in a rigorous proof by [2]. Firstly, the theory of the supermembrane is a constrained system highly nonlinear and difficult to solve, in distinction with the harmonic oscillator-type hamiltonian of the string. Secondly, the supermembrane classically contains instabilities that make the scalar potential along directions in the configuration space vanish. They are called 'valleys' and render the classical system unstable. The third and most important difference is that its supersymmetric spectrum is continuous. Let us characterize it in more detail. The hamiltonian of the D = 11 Supermembrane [1] may be defined in terms of maps X μ , μ = 0, . . . , 10, from a base manifold Σ × R onto a target manifold which we will assume to be 11D Minkowski. Σ is a Riemann surface of genus g. σ a , a = 1, 2 are local spatial coordinates over Σ and τ ∈ R represents the worldvolume time. Decomposing X μ and Γ μ accordingly to the standard L.C.G ansatz and solving the constraints, the canonical reduced hamiltonian to the light-cone gauge has the expression