Non-linear sigma models and string effective actions

We study the problem of finding the beta functions, and the associated space-time effective actions, for free closed strings propagating in backgrounds. We begin with a general presentation of the classical string theories and briefly discuss their spectra. We then review previous works on the renormalization of the non-linear sigma model paying special attention to the relation between Weyl invariance of two-dimensional field theories and their corresponding beta functions.Next the beta function for the purely bosonic string is calculated explicitly up to three loops, using two different choices for handling the loop integrals. We find that this beta function is derivable from a space-time effective action which also generates the S-matrix scattering amplitudes, at the tree loop level, for the massless modes of the string. Our results are also subjected to a consistency check using renormalization group considerations. We then proceed to show that these bosonic contributions are exactly cancelled by the fermionic contributions which arise when considering the N = 1 supersymmetric sigma model.Finally, the calculation is repeated for the heterotic string and the non-vanishing /^-function is shown to be compatible with an O(a'2) effective action in which there are no terms cubic in the Riemann tensor or the gauge field strength. A renormalization group equation for the gauge field is constructed and its /?-function is shown to vanish at the two loop level. (DX84441)