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

Understand

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.