The fate of U(1)'s at strong coupling in F-theory

TL;DR

This work provides a geometric criterion to distinguish massless and massive $U(1)$ gauge factors in F-theory by examining codimension-three singularities and their resolutions. The authors leverage a reinterpreted Sen limit to connect weak-coupling IIB data with strong-coupling F-theory geometry, showing that massless $U(1)$s correspond to singularities admitting small, Kahler resolutions, while massive ones persist only with non-Kähler small resolutions. They validate the criterion through both local (branes and images) and global (compact) models, including cases with multiple $U(1)$s and mixed massless/massive sectors. The results offer a concrete geometric framework for identifying $U(1)$ masses and set the stage for future work on fluxes and non-harmonic two-forms associated with massive $U(1)$s.$

Abstract

U(1) gauge symmetries in F-theory are expected to manifest themselves as codimension three singularities of Calabi-Yau fourfolds. However, some of these are known to become massive at strong coupling via the Stückelberg mechanism. In this note, we propose a geometric picture for detecting all U(1)'s, and determining which ones are massive and which ones are massless. We find that massive gauge symmetries show up as codimension three singularities that only admit small, non-Kähler, resolutions. Our proposal passes several highly non-trivial tests, including a case with a non-diagonal mass matrix.