The leading divergent part of the effective action for the nonlinear sigma model in n dimensions

Abstract

Covariant derivative expansion techniques are generalized to the case of curved space-time. A formal expression for the one-loop effective action for a nonlinear sigma model in n space-time dimensions with a classical background gravitational field is obtained and expanded up to second-to-leading divergent terms. For n = 2 the Weyl invariance conditions of the bosonic string are reproduced. We also give fermion and vector contributions to the one-loop gravitational action for the case of minimal couplings. For n = 4 we determine the scalar loop contribution to the gravity-scalar lagrangian including logarithmically divergent terms for the most general nonlinear model and for no-scale supergravity models, and the leading N (number of matter fields) contributions from fermion and vector loops to the quadratically divergent part of the gravitational action. We also evaluate the leading N chiral-supermultiplet loop contribution to the quadratically divergent part of the gravitino wave-function renormalization. We discuss consistent regularization procedures for divergent integrals and their physical interpretation, as well as the application of our results to superstring-inspired supergravity models.