N=1 special geometry, mixed Hodge variations and toric geometry

We study the superpotential of a certain class ofN = 1 supersymmetric type II compactications with fluxes andD-branes. We show that it has an important two-dimensional meaning in terms of a chiral ring of the topologically twisted theory on the world-sheet. In the open-closed string B-model, this chiral ring is isomorphic to a certain relative cohomology group V , which is the appropriate mathematical concept to deal with both the open and closed string sectors. The family of mixed Hodge structures on V then implies for the superpotential to have a certain geometric structure. This structure represents a holomorphic, N = 1 supersymmetric generalization of the well-known N = 2 special geometry. It denes an integrable connection on the topological family of open-closed B-models, and a set of special coordinates on the spaceM of vev’s inN = 1 chiral multiplets. We show that it can be given a very concrete and simple realization for linear sigma models, which leads to a powerful and systematic method for computing the exact non-perturbativeN = 1 superpotentials for a broad class of toric D-brane geometries.