Toric Geometry and Enhanced Gauge Symmetry of F-Theory/Heterotic Vacua

TL;DR

This work develops a toric-geometry framework to extract enhanced gauge symmetries in F-theory/heterotic vacua by translating elliptic Calabi–Yau hypersurfaces in toric varieties into reflexive polyhedra data. It presents an explicit algorithm that reads off the gauge algebra $G$ and the number of tensor multiplets from dual polyhedra $\Delta$ and $\nabla$, tying geometry to 6D spectra via $h_{11} = \text{rank}(G) + n_T + 2$ and $n_T = h_{11}(B) - 1$, while accounting for non-toric deformations through a correction term $\delta$. The approach predicts very large non-perturbative gauge groups (up to rank $296$ in 6D and up to $121{,}328$ for elliptic fourfolds) and is applicable to mirrors and to elliptic fourfolds, highlighting both the utility and the subtleties of the toric dictionary, including monodromy and δ-corrections. The results are illustrated with explicit examples, demonstrating how polyhedral data encode gauge content and suggesting directions for extending the method to broader compactifications and dualities. Overall, the paper provides a practical, scalable toolkit for mapping toric polyhedra to gauge content in F-theory vacua.

Abstract

We study F-theory compactified on elliptic Calabi-Yau threefolds that are realised as hypersurfaces in toric varieties. The enhanced gauge group as well as the number of massless tensor multiplets has a very simple description in terms of toric geometry. We find a large number of examples where the gauge group is not a subgroup of E8xE8, but rather, is much bigger (with rank as high as 296). The largest of these groups is the group recently found by Aspinwall and Gross. Our algorithm can also be applied to elliptic fourfolds, for which the groups can become extremely large indeed (with rank as high as 121328). We present the gauge content for two of the fourfolds recently studied by Klemm et al.