Flux compactification on smooth, compact three-dimensional toric varieties

Abstract

Three-dimensional smooth, compact toric varieties (SCTV), when viewed as real six-dimensional manifolds, can admit G-structures rendering them suitable for internal manifolds in supersymmetric flux compactifications. We develop techniques which allow us to systematically construct G-structures on SCTV and read off their torsion classes. We illustrate our methods with explicit examples, one of which consists of an infinite class of toric $$ \mathbb{C}{\mathbb{P}^1} $$ bundles. We give a self-contained review of the relevant concepts from toric geometry, in particular the subject of the classiffication of SCTV in dimensions ≤ 3. Our results open up the possibility for a systematic construction and study of supersymmetric flux vacua based on SCTV.