Abstract
IIB compactifications enjoy the possibility to break GUT groups via fluxes without giving mass to the hypercharge gauge field. Although this important advantage has greatly motivated F-theory constructions, no such fluxes have been constructed directly in terms of the M-theory $G_4$-form. In this note, we give a general prescription for constructing hypercharge G-fluxes. By using a stable version of Sen's weak coupling limit, we verify their connection with IIB fluxes. We illustrate the lift of fluxes in a number of examples, including a compact ${\rm SU}(5) \times {\rm U}(1)$ model with explicit realization of doublet-triplet splitting. Finally, we prove an equivalence conjectured in an earlier work as a by-product.
Highlights
The cohomology of a divisor is typically much richer than that of the threefold in which it lives, i.e. ı∗H2(X3, Z) H2(D, Z) with D a generic divisor
Suppose S is a divisor of the form P ≡ A B − C D = 0, a typical glue vector will be a curve
A typical glue vector C ∈ Γ cannot be written as a pulled back two-form by definition, but it will still not be orthogonal to all of the pulled back forms
Summary
Supersymmetric F-theory compactifications to four dimensions require a Calabi-Yau fourfold that is elliptically fibered over a base manifold B3 as part of the defining data. The information on the location of the D7-brane locus, encoded in b4 and b6, is lost completely To properly understand this degeneration it is necessary to consider the whole family of fourfolds described by (2.4). One immediately recognizes this as a Calabi-Yau double cover X3 of B3 This procedure allows us to find the type IIB Calabi-Yau threefold as a submanifold of the (degenerate) F-theory fourfold over = 0. The two rational curves fibered over the subset ∆E ⊂ B3 make up a threefold R3 ≡ WE ∩ {∆E = 0}, that we call the cylinder (even if the proper cylinder is a normalization of R3 [17]) This is the basic object needed to lift fluxes on D7-branes as we will see
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