Flux compactification on smooth, compact three-dimensional toric varieties

TL;DR

This work develops a systematic framework to realize supersymmetric flux vacua by taking the internal six-manifold to be a smooth, compact toric threefold $V_oldsymbol{ Sigma}$ and equipping it with an $SU(3)$ structure. By translating toric data (fans and their fundamental generators) into symplectic-quotient charges $Q^a$ and introducing a globally defined (1,0)-form $K$, the authors construct a family of $J$ and $ ext{Ω}$—hence an $SU(3)$ structure—on $M_6$, and read off the associated torsion classes. They illustrate the method with explicit examples, including $ ext{CP}^3$ and a broad class of toric $ ext{CP}^1$ bundles, deriving their torsion classes and assessing the existence of IIA flux vacua. The approach provides a concrete, algebraic-geometric pathway to classifying and constructing flux vacua on SCTVs, enabling systematic scans and offering potential avenues toward AdS/dS vacua, holography, and extensions to higher dimensions.

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 CP^1 bundles. We give a self-contained review of the relevant concepts from toric geometry, in particular the subject of the classification of SCTV in dimensions less or equal to 3. Our results open up the possibility for a systematic construction and study of supersymmetric flux vacua based on SCTV.

Figures (8)

• Figure 1: A maximal cone in a three-dimensional space $N_{\mathbb{R}}$ with one-dimensional fundamental cone generators $(v_1, v_2, ..., v_n)$.
• Figure 2: A two dimensional fan $\Sigma$ consisting of four maximal cones ($\sigma_i$), eight one-dimensional cones ($\tau_i$) and one zero-dimensional one ($0$).
• Figure 3: A two dimensional toric variety $V_\Sigma$ determined by primitive vectors $v_i\in N$, $i=1,\dots, n$, and its associated weighted circular graph.
• Figure 4: Two-dimensional admissible weighted circular graphs with number of vertices $n\leq6$.
• Figure 5: Two-dimensional fans $\Sigma$ and fundamental generators corresponding to admissible weighted circular graphs with number of vertices $n\leq6$.