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Hypercharge flux in F-theory and the stable Sen limit

Andreas P. Braun, Andres Collinucci, Roberto Valandro

TL;DR

This work cungresses a robust framework to realize hypercharge flux in F-theory by lifting Type IIB F_2 fluxes through a stable Sen limit, clarifying the necessity of glue-vector data and non-factorizable four-cycles rather than simple pullbacks. The authors develop explicit prescriptions for constructing G_4 fluxes that keep U(1)_Y massless, apply them to SU(5) GUTs with U(1) restrictions, and demonstrate precise matches between IIB fluxes and F-theory four-form fluxes, including Cartan and hypercharge sectors. They provide concrete realizations in SU(2) and SU(5) setups, derive doublet–triplet splitting via hypercharge flux, and extend the lifting to brane/image-brane and Whitney-brane configurations in smooth Weierstrass models. Overall, the paper offers a concrete, geometrically controlled path to GUT breaking through hypercharge flux in F-theory with explicit flux lifts, matter content, and consistency checks (D3-tadpoles, chiral indices). The results enhance the viability and calculability of F-theory GUTs with massless hypercharge and realistic MSSM-like spectra.

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.

Hypercharge flux in F-theory and the stable Sen limit

TL;DR

This work cungresses a robust framework to realize hypercharge flux in F-theory by lifting Type IIB F_2 fluxes through a stable Sen limit, clarifying the necessity of glue-vector data and non-factorizable four-cycles rather than simple pullbacks. The authors develop explicit prescriptions for constructing G_4 fluxes that keep U(1)_Y massless, apply them to SU(5) GUTs with U(1) restrictions, and demonstrate precise matches between IIB fluxes and F-theory four-form fluxes, including Cartan and hypercharge sectors. They provide concrete realizations in SU(2) and SU(5) setups, derive doublet–triplet splitting via hypercharge flux, and extend the lifting to brane/image-brane and Whitney-brane configurations in smooth Weierstrass models. Overall, the paper offers a concrete, geometrically controlled path to GUT breaking through hypercharge flux in F-theory with explicit flux lifts, matter content, and consistency checks (D3-tadpoles, chiral indices). The results enhance the viability and calculability of F-theory GUTs with massless hypercharge and realistic MSSM-like spectra.

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 -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 model with explicit realization of doublet-triplet splitting. Finally, we prove an equivalence conjectured in an earlier work as a by-product.

Paper Structure

This paper contains 16 sections, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The stable version of Sen's weak coupling limit
  3. Lifting type IIB $\mathcal{F}_2$ fluxes to F-theory $G_4$ fluxes
  4. One brane and its image
  5. Non-abelian gauge group: ${\rm SU}(2)$ stack
  6. Hypercharge flux
  7. General case: $A_1$ singularity
  8. Example: a rigid $\mathbb{P}^1\times \mathbb{P}^1$ in $B_3$
  9. GUT on a $dP_7$ in $B_3$
  10. ${\rm SU}(5)\times {\rm U}(1)$ model
  11. Hypercharge $G_4$-flux
  12. Matter curves and matter surfaces
  13. Massless ${\rm U}(1)$ flux and doublet-triplet splitting
  14. Lifting fluxes for $U(1)$ restriction and smooth Weierstrass model
  15. Brane/Image-brane fluxes
  16. ...and 1 more sections

Figures (1)

  • Figure 1: The fiber components of a fiber of type $I_5$ meet according to the extended Dynkin diagram of ${\rm SU}(5)$. Here we have drawn each $\mathbb{P}^1$ as a line.