Non-perturbative Effective Lagrangians for Super-matrix Models

We discuss d = 1,N = 2 supersymmetric matrix models and exhibit the associated d = 2 collective field theory in the limit of dense eigenvalues. From this field theory we construct, by the addition of several new fields, a d = 2 supersymmetric effective field theory, which reduces to the collective field theory when the new fields are replaced with their vacuum expectation values. This effective theory is Poincare invariant and contains perturbative and non-perturbative information about the associated superstrings. ∗) Work supported in part by DOE under Contract No. DOE-AC02-76-ERO-3071. ∗∗) On sabbatical leave from the Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. ∗∗∗) Invited talk, International Europhysics Conference on High Energy Physics, Marseille, France, July 22-28, 1993. CERN-TH.7013/93 September 1993 The question of how supersymmetry is broken remains one of the most important issues in superstring theory. The advent of d = 1 supersymmetric matrix models[1], which contain nonperturbative information about a class of two-dimensional superstrings, offers the hope that one can identify, and study quantitatively, the exact string mechanism for supersymmetry breaking. To do this, it is essential to construct the complete two-dimensional supersymmetric effective Lagrangian associated with this class of theories. Once that is achieved, one can look for non-perturbative phenomena, such as instantons, that will induce supersymmetry violating operators (spontaneous or explicit) in this effective Lagrangian. This entire program has already been carried out[2], with positive results, in the case of d = 1 bosonic matrix models where the nonperturbative effective Lagrangian was constructed and its fundamental symmetry, a non-compact shift symmetry, ζ → ζ + c in one of its fields ζ , was shown to be broken by instantons in a single eigenvalue of the matrix model. In this talk we will discuss the extension of these results to the supersymmetric case[3]. To be precise, we will construct the two-dimensional supersymmetric effective Lagrangian for the superstrings associated with the d = 1 supersymmetric matrix model. We have also discovered instantons in the eigenvalues of these matrix models, which have all the right characteristics to break supersymmetry. However, the constraints placed on the length of this manuscript mandate that we discuss these instantons and supersymmetry breaking elsewhere. The simplest supersymmetric matrix model which admits non-trivial interactions involves a d = 1,N = 2 supersymmetry. The fundamental variable of the theory is a time-dependent N ×N , Hermitian matrix superfield Φij=Mij(t)+iθ1Ψ1ij(t)+iθ2Ψ2ij + iθ1θ2Fij(t), (1) where θ1 and θ2 are the two anticommuting parameters. The Lagrangian is given by L = ∫