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.
Highlights
This statement begs the following question: Is it possible to detect U(1)’s directly in an F-theory CY fourfold and to discriminate between the massive and massless ones? In this paper, we provide an answer to this question
U(1) gauge symmetries in F-theory are expected to manifest themselves as codimension three singularities of Calabi-Yau fourfolds
Some of these are known to become massive at strong coupling via the Stuckelberg mechanism
Summary
Let us first establish some definitions to set up the notation. We start from a smooth Calabi-Yau fourfold X4 that is an elliptic fibration over the base manifold B3, with a section. This curve carries the bifundamental matter charged under this relative group In this weak coupling limit, we see that this curve worth of conifold singularities admits a small resolution, as it has the form AB = CD. This is consistent with the fact that, in the limit gs → 0, the expected mass goes to zero: at ǫ = 0, there are two independent six-cycles in WE, (y ± s, x1) and (y ± s, x2), that generate two massless U(1)’s by expanding C3. To move away from the weak coupling limit, we add the missing λ s3 term, and export this out of the central fiber This amounts to setting the hypersurface equation to:.
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