Conformal invariance beyond the leading order in the supersymmetric nonlinear σ model with dilaton

We calculate the O(α${\mathcal{’}}^3$) contributions to the renormalization-group β functions in the N=1 supersymmetric σ model with a dilaton. At this order both metric and dilaton β functions are found to depend nontrivially on the dilaton field and vanish if the dilaton satisfies ${\mathrm{\nabla }}_{\mathrm{\mu }}$ ${\mathrm{\nabla }}_{\nu }$φ=0. By employing the Curci-Paffuti relation it is shown that such dilaton backgrounds in Ricci-flat spaces ${\mathit{R}}_{\mathrm{\mu }\nu }$=0 satisfy the conformal invariance conditions up to this order. The particular class of Ricci-flat, compact, and orientable manifolds naturally emerge as appropriate internal-space configurations consistent with local scale invariance. We further explore the cosmological consequences of these dilaton configurations. In a Robertson-Walker four-dimensional background we find all dilatons satisfying ${\mathrm{\nabla }}_{\mathrm{\mu }}$ ${\mathrm{\nabla }}_{\nu }$φ=0. Except for the constant and the time-dependent dilaton φ(t)=-2 lnt+λ whose cosmological implications have been already discussed in the literature, additional solutions are found. These may be of relevance beyond leading order and for nonvanishing background values for the antisymmetric tensor ${\mathit{B}}_{\mathrm{\mu }\nu }$. For these solutions, also the cosmic scale factor is at most linear in time therefore giving rise to either a static or a linearly expanding (contracting) universe.