One-loop regularization of the Polyakov string functional

Abstract

The divergences of the vacuum amplitude for the bosonic Polyakov string are studied at the one-loop level in a modular invariant regularization scheme, characterized by a dimensional cutoff analogous to proper time. As a result, the singular behaviour in the cutoff is not uniform in the range of the modulus variable and this yields a control on the singularities induced by the tachyon and the dilaton. The divergences are those of a sigma model, but the coefficients of the sigma-model counter-terms are different for the sphere and the flat torus.