Supermatrix Models

Abstract

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 + 1 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' a Talk presented at theVaì encia workshop Beyond the Standard Model; from Theory to Experiment, October 13-17, 1997.

Negative branes, supergroups and the signature of spacetime

“…By a supersymmetric localization argument, the same should be true of supergroup gauge theories, and hence we can learn something about their non-perturbative physics by studying Gaussian supermatrix models. Recently it was observed that, despite earlier claims to the contrary [44], the U(N |M ) supermatrix model is not equivalent to the U(N − M ) matrix model [3], as there are gauge invariant operators whose expectation values are zero in the latter but non-zero in the former. It turns out that even the exact partition functions of these two theories are not the same: the differences appear to arise from subtleties associated to fermionic gauge symmetries [10], which are best resolved by gauge-fixing.…”

“…We note, however, that the second option has an intriguing parallel to section 9.1, in that we remove by hand certain Fourier modes along T 1,1 , in this case those with null momenta. In type IIB language, this corresponds to removing certain tensionless (p, q) string bound states, 44 whereas in M-theory language we remove the massless non-zero modes. This cures the problems with the KK spectrum (9.2), in that the number of KK modes with mass below any fixed threshold is finite.…”

Supermatrix models, loop equations, and duality

“…For instance, Itzykson-Zuber and character formulas for U (n|m) have been obtained in [1]. This had been preceded, in in the beginning of the 1990s, by a few attempts to generalize the well known connection between conventional matrix models and quantum gravity in 2D [2,6,18]. It was soon realized however that supermatrix models could not provide a discrete version of supergravity.…”

“…It was soon realized however that supermatrix models could not provide a discrete version of supergravity. 1 Especially, convincing arguments were given for the equivalence between supermatrix and matrix models when no external fields are involved [2].…”

Supersymmetric matrix models and branched polymers

“…Even in the quantum gravity applications [21,22] as well as in purely combinatorial applications like the Meander problem [23] supersymmetric matrix models have been most useful.…”