Non critical superstring configuration and Minkowski space time in four dimensions

We extend the non-perturbative time-dependent bosonic string action of [3] to a N = 1 supersymmetric world sheet action with graviton background, and assume a superpotential, function of the time super coordinate. The use of (super)strings for the description of cosmological events suggests that String Theory might become observable, given the huge amount of cosmological data [1]. String Cosmology is generally based on the study of the effective theory in the target space [2], where the effective action is built in such a way that the corresponding equations of motion give the perturbative Weyl invariance conditions on the world sheet of the string. Instead, in [3], a time-dependent bosonic string configuration was found to satisfy Weyl invariance conditions in a non-perturbative way: it was conjectured that this configuration cancels the ressumation of all loops in the beta functions corresponding to Weyl invariance. The corresponding action is given by S = 1 4πα′ ∫ dξ √ γ { γ κ0ημν (X0)2 ∂aX ∂bX ν + α ′ Rφ0 ln(X ) } , (1) where κ0 and φ0 are constants. Weyl invariance is then satisfied for any target space dimension D and any dilaton amplitude φ0. The metric amplitude κ0 is then a function of D and φ0, which is not known precisely, but which has no cosmological relevance. The string configuration (1) describes of a spatially-flat Robertson Walker Universe, with metric ds = −dt + a(t)(d~x), where t is the time in the Einstein frame, and a(t) is the scale factor. For the configuration (1), the scale factor was shown to be a power law a(t) ∝ t D−2 2φ0 , (2) such that, if the following relation holds D = 2− 2φ0, (3) the target space is static and flat (Minkowski Universe). We are interested here in a superstring action with corresponding bosonic part (1), and we consider for this a N = 1 world sheet supersymmetric action, with graviton background, and 1 in which we assume a superpotential Φ, function of X, where X are the supercoordinates on the world sheet. The string supercoordinates of an NSR string are [4] X = X + iθψ + iθψ μ + θθF , (4) where ψ and ψ μ are complex conjugate if the world sheet is Euclidean. The supercovariant derivatives are D = ∂θ + θ∂z and D = ∂θ + θ∂z, where z = (z, θ) denotes the superspace coordinates on the world sheet. The N = 1 supersymmetric world sheet action we consider is S = 1 4πα′ ∫