Summing up D-instantons in N = 2 supergravity

The non-perturbative quantum geometry of the universal hypermultiplet (UH) is investigated in N = 2 supergravity. The UH low-energy effective action is given by the four-dimensional quaternionic non-linear sigma-model having an U(1) × U(1) isometry. The UH metric is governed by the single real pre-potential that is an eigenfunction of the Laplacian in the hyperbolic plane. We calculate the classical pre-potential corresponding to the standard (Ferrara-Sabharwal) metric of the UH arising in the Calabi-Yau compactification of type-II superstrings. The non-perturbative quaternionic metric, describing the D-instanton contributions to the UH geometry, is found by requiring the SL(2,Z) modular invariance of the UH pre-potential. The pre-potential found is unique, while it coincides with the D-instanton function of Green and Gutperle, given by the order-3/2 Eisenstein series. As a by-product, we prove cluster decomposition of D-instantons in curved spacetime. The non-perturbative UH pre-potential interpolates between the perturbative (large CY volume) region and the superconformal (Landau Ginzburg) region in the UH moduli space. We also calculate a non-perturbative scalar potential in the hyper-Kähler limit, when an abelian isometry of the UH metric is gauged in the presence of D-instantons.