Renormalization of string functionals

Abstract

We investigate analytic renormalization procedures for functional integrals, corresponding to field theories defined on compact manifolds, which arise, e.g., from string functionals of the Nambu-Schild-Eguchi type. Although these models belong to the nonrenormalizable class of quantum field theories, we prove finiteness for a rectangular string shape up to three-loop level, for circular boundary up to two-loop order, and for a variety of graphs in higher order, thus indicating that the result might hold in general. From the explicit calculation of the two-loop approximation we extract the first model-dependent corrections to the $q\overline{q}$ potential or the Casimir effect. The importance of dilation transformations for the properties of the renormalization procedure are investigated. We prove that under certain conditions, forced by symmetry properties, the association of finite values to divergent series is unique, independent of the regularization procedure.