N=1 Special Geometry, Mixed Hodge Variations and Toric Geometry

TL;DR

This paper explores the geometric structure of superpotentials in N=1 supersymmetric string compactifications, linking mixed Hodge structures to a generalized special geometry and providing a systematic method for computing superpotentials in toric models.

Contribution

It introduces a geometric framework based on mixed Hodge structures for understanding N=1 superpotentials and offers a practical approach for calculations in toric D-brane geometries.

Findings

Superpotential has a geometric structure related to mixed Hodge structures.

The framework generalizes N=2 special geometry to N=1 cases.

Provides a systematic method for exact superpotential computations in toric models.

Abstract

We study the superpotential of a certain class of N=1 supersymmetric type II compactifications with fluxes and D-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 defines an integrable connection on the topological family of open-closed B-models, and a set of special coordinates on the space \cal M of vev's in N=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-perturbative N=1 superpotentials for a broad class of toric D-brane geometries.