Instanton-induced scalar potential for the universal hypermultiplet

We calculate the scalar potential in the gauged N = 2 supergravity with a single hypermultiplet, whose generic quaternionic moduli space metric has an Abelian isometry. This isometry is gauged by the use of a graviphoton gauge field. The hypermultiplet metric and the scalar potential are both governed by the single real potential that is a solution to the 3d (integrable) continuous Toda equation. An explicit solution, controlled by the Eisenstein series E_3/2, is found in the case of the D-instantoncorrected universal hypermultiplet moduli space metric having an U(1) × U(1) isometry, with one of the isometries being gauged.