Supersymmetric regularized path-integral measure in x space.

Abstract

The complete supersymmetric Liouville action for a scalar multiplet coupled to supergravity in two-dimensional x space is obtained by integrating all anomalies, and not only the conformal anomaly. The resulting action is supersymmetric, but supersymmetry is derived, and not merely used as a device to extend the integrated conformal anomaly to the complete supersymmetric Liouville action. The integration variables are obtained from the matter variables not only (as usual) by multiplication by powers of dete(x), but also by addition of terms containing products of matter fields and supergravity fields. The dependence of the action on dete, \ensuremath{\gamma}\ensuremath{\cdot}\ensuremath{\psi}, and S is removed by rescaling and shifting the new matter variables. As a regulator we use a matrix operator which contains the D'Alembertian in each diagonal entry and which is obtained from the matter action by squaring and introducing a ``twist'' operator. Because the regulator contains off-diagonal elements, all three anomalies (Weyl, super-Weyl, and ``auxiliary'') contribute.