G-flux in F-theory and algebraic cycles
Andreas P. Braun, Andres Collinucci, Roberto Valandro
TL;DR
This work presents a global, algebraic construction of $G_4$ fluxes in F-theory by identifying algebraic four-cycles in the Weierstrass model of elliptic Calabi–Yau fourfolds. It shows how to compute the D3-tadpole and induced chirality directly in F-theory, and demonstrates exact agreement with weakly coupled Type IIB results where available. The fluxes are described as $(2,2)$-type with one leg in the fiber and are tied to a coherent-sheaf description, including Pfaffian representations of the Weierstrass equation, thereby unifying flux and geometry. The framework is tested in K3 and fourfold examples with abelian and non-abelian sectors and highlights how fluxes can fix complex-structure moduli while preserving non-abelian gauge symmetry in many cases. Overall, the approach provides a powerful, algebraic toolkit for controlling chirality, tadpoles, and moduli in global F-theory compactifications.
Abstract
We construct explicit G4 fluxes in F-theory compactifications. Our method relies on identifying algebraic cycles in the Weierstrass equation of elliptic Calabi-Yau fourfolds. We show how to compute the D3-brane tadpole and the induced chirality indices directly in F-theory. Whenever a weak coupling limit is available, we compare and successfully match our findings to the corresponding results in type IIB string theory. Finally, we present some generalizations of our results which hint at a unified description of the elliptic Calabi-Yau fourfold together with the four-form flux G4 as a coherent sheaf. In this description the close link between G4 fluxes and algebraic cycles is manifest.