Supermembranes and Supermatrix Models

We briefly review recent developments in the theory of supermembranes and su-permatrix models. In a second part we discuss their interaction with background fields. In particular, we present the full background field coupling for the bosonic case. This is a short summary of the talk at the workshop. A more extended version will appear elsewhere. 1 Supermembranes and matrix models It has been known for some time 1 that certain supersymmetric quantum-mechanical models characterized by the presence of zero-potential valleys and Gauss-type constraints 2 play an important role in the quantization of fundamental supermembranes 3. In the light-cone formulation with a flat target space 4,1 , the supermembrane theory exhibits a residual invariance under area-preserving diffeomorphisms of the membrane surface. This infinite-dimensional group can be truncated in many cases to a finite-dimensional group, e.g. to U(N), such as to lead to a quantum-mechanical model based on a finite number of degrees of freedom. The relevant Hamiltonian equals H = g −1 Tr 1 2 P 2 − 1 4 [X a , X b ] 2 + g θ T γ a [X a , θ]. (1) Here, X, P and θ take values in the Lie algebra of the gauge group and are represented by matrices that span this Lie algebra. Furthermore, these mo-menta and coordinates transform as vectors and spinors under the 'transverse'