Abstract
We study the tuning of U(1) gauge fields in F-theory models on a base of general dimension. We construct a formula that computes the change in Weierstrass moduli when such a U(1) is tuned, based on the Morrison-Park form of a Weierstrass model with an additional rational section. Using this formula, we propose the form of “minimal tuning” on any base, which corresponds to the case where the decrease in the number of Weierstrass moduli is minimal. Applying this result, we discover some universal features of bases with non-Higgsable U(1)s. Mathematically, a generic elliptic fibration over such a base has additional rational sections. Physically, this condition implies the existence of U(1) gauge group in the low-energy supergravity theory after compactification that cannot be Higgsed away. In particular, we show that the elliptic Calabi-Yau manifold over such a base has a small number of complex structure moduli. We also suggest that non-Higgsable U(1)s can never appear on any toric bases. Finally, we construct the first example of a threefold base with non-Higgsable U(1)s.
Highlights
The counting of independent degrees of freedom in the Morrison-Park form is not obvious, since the parameters in the Morrison-Park form may include redundant components
We developed formula (2.27) to count the change in Weierstrass moduli when a U(1) is tuned from the non-Higgsable phase of F-theory on an arbitrary base, if the Weierstrass form can be written in the Morrison-Park form (2.5)
We argued that this formula should be exact for base point free line bundles L parameterizing the MorrisonPark form
Summary
We start from a generic complex d-dimensional base B with anticanonical line bundle (class) −K. For non-Abelian gauge groups, they correspond to local structures on the base B, which are called non-Higgsable clusters (NHC). They appear when f , g and the discriminant ∆ = 4f 3 + 27g2 vanish to certain degrees on some divisors. For some bases with non-Higgsable clusters of high rank gauge groups, taking L = 0 may lead to non-minimal singularities in the total space X that cannot be resolved [5]. We will discuss this issue explicitly for B = F12 in section (2.5).
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