An Alternative to Exact Renormalization and Cosmological Solutions in String Theory

In this work we review the application of a functional method, serving as an alternative to the Wilsonian Exact Renormalization approach, to stringy bosonic $\sigma$-models with metric and dilaton backgrounds on a spherical world sheet [1]. We derive an exact evolution equation for the dilaton with the amplitude of quantum fluctuations, driven by the kinetic term of the two-dimensional world-sheet theory. The linear dilaton conformal field theory, corresponding to a linearly (in cosmic Einstein-frame time) expanding Universe, appears as a trivial fixed point of this equation. With the help of conformal-invariance conditions, we find a logarithmic dilaton as another, exact and non trivial, fixed-point solution. Cosmological implications of our solutions are briefly discussed, in particular the transition (exit) from the expanding Universe of the linear dilaton to the Minkowski vacuum, corresponding to the non-trivial fixed point of our generalised flow. This novel renormalization-group method may therefore offer new insights into exact properties of string theories of physical significance.