Geometric Engineering in Toric F-Theory and GUTs with U(1) Gauge Factors

TL;DR

This paper presents a systematic algorithm for constructing Calabi-Yau elliptic fibrations in toric geometry, specifically applied to SU(5) GUT models with multiple U(1) gauge factors, analyzing their geometric and gauge properties.

Contribution

It introduces a comprehensive method to construct and analyze elliptic fibrations with specified gauge groups and U(1) factors in F-theory using toric geometry, including flatness conditions and matter charge patterns.

Findings

Constructed all SU(5)-tops and their splitting types.

Computed the toric Mordell-Weil group for each top.

Identified conditions for flat elliptic fibrations and U(1) charge patterns.

Abstract

An algorithm to systematically construct all Calabi-Yau elliptic fibrations realized as hypersurfaces in a toric ambient space for a given base and gauge group is described. This general method is applied to the particular question of constructing SU(5) GUTs with multiple U(1) gauge factors. The basic data consists of a top over each toric divisor in the base together with compactification data giving the embedding into a reflexive polytope. The allowed choices of compactification data are integral points in an auxiliary polytope. In order to ensure the existence of a low-energy gauge theory, the elliptic fibration must be flat, which is reformulated into conditions on the top and its embedding. In particular, flatness of SU(5) fourfolds imposes additional linear constraints on the auxiliary polytope of compactifications, and is therefore non-generic. Abelian gauge symmetries arising in toric F-theory compactifications are studied systematically. Associated to each top, the toric Mordell-Weil group determining the minimal number of U(1) factors is computed. Furthermore, all SU(5)-tops and their splitting types are determined and used to infer the pattern of U(1) matter charges.