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

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 St\"uckelberg 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-K\"ahler, resolutions. Our proposal passes several highly non-trivial tests, including a case with a non-diagonal mass matrix.