Holomorphic couplings in non‐perturbative string compactifications

Abstract

AbstractIn this review article we present an analysis of several aspects of four‐dimensional, non‐perturbative 𝒩 = 1 compactifications of string theory. Our study focuses on brane dynamics and their effective physics as encoded in the holomorphic couplings of the low‐energy 𝒩 = 1 effective action, most prominently the superpotential W. This article is divided into three parts. In part one we derive the effective action of a spacetime‐filling D5‐brane in generic Type IIB Calabi‐Yau orientifold compactifications. In the second part we invoke tools from string dualities, namely from F‐theory, heterotic/F‐theory duality and mirror symmetry, for a more elaborate study of the dynamics of (p,q) 7‐branes and heterotic five‐branes. In this context we perform exact computations of the complete perturbative effective superpotential, both due to branes and background fluxes. Finally, in the third part we present a novel geometric description of five‐branes in Type IIB and heterotic M‐theory Calabi‐Yau compactifications via a non‐Calabi‐Yau threefold $\hat {Z}_3$, that is canonically constructed from the original five‐brane and the Calabi‐Yau threefold Z3 via a blow‐up. We use the blow‐up threefold $\hat {Z}_3$ to derive open‐closed Picard‐Fuchs differential equations, that govern the complete effective brane and flux superpotential. In addition, we present first evidence to interpret $\hat {Z}_3$ as a flux compactification geometrically dual to the original five‐brane by defining an SU(3)‐structure on $\hat {Z}_3$, that is generated dynamically by the five‐brane backreaction.