Geometric Bounds on the 1-Form Gauge Sector

TL;DR

This paper establishes geometric bounds on the discrete 1-form gauge sectors in six-dimensional F-theory compactifications, linking algebraic geometry constraints to physical gauge factors and proposing their role as swampland criteria.

Contribution

It classifies allowed 1-form gauge structures in 6D supergravity from F-theory, deriving bounds on cyclic factors and connecting geometric constraints to physical symmetries.

Findings

Cyclic gauge factors satisfy m ≤ 6.

Bounds derived from elliptic Calabi-Yau geometry.

Physical origin linked to heterotic string worldsheet symmetry.

Abstract

We classify the allowed structures of the discrete 1-form gauge sector in six-dimensional supergravity theories realized as F-theory compactifications. This provides upper bounds on the 1-form gauge factors $\mathbb{Z}_m$ and in particular demands each cyclic factor to obey $m\leq 6$. Our bounds correspond to the universal geometric constraints on the torsion subgroup of the Mordell-Weil group of elliptic Calabi-Yau three-folds. For any F-theory vacua with at least one tensor multiplet, we derive the constraints from the $\mathbb{P}^1$ fibration structure of the base two-fold and identify their physical origin in terms of the worldsheet symmetry of the associated effective heterotic string. The bounds are also extended to the F-theory vacua with no tensor multiplets via a specific deformation of the theory followed by a small instanton transition, along which the 1-form gauge sector is not reduced. We envision that our geometric bounds can be promoted to a swampland constraint on any six-dimensional gravitational theories with minimal supersymmetry and also extend them to four-dimensional F-theory vacua.