Abstract
We construct G4 fluxes that stabilize all of the 426 complex structure moduli of the sextic Calabi-Yau fourfold at the Fermat point. Studying flux stabilization usually requires solving Picard-Fuchs equations, which becomes unfeasible for models with many moduli. Here, we instead start by considering a specific point in the complex structure moduli space, and look for a flux that fixes us there. We show how to construct such fluxes by using algebraic cycles and analyze flat directions. This is discussed in detail for the sextic Calabi-Yau fourfold at the Fermat point, and we observe that there appears to be tension between M2-tadpole cancellation and the requirement of stabilizing all moduli. Finally, we apply our results to show that even though symmetric fluxes allow to automatically solve several F-term equations, they typically lead to flat directions.
Highlights
The minima of the induced scalar potential are solutions of the F-term equations DI W = 0, I = 1, . . . , h1,3(X)
We show how to construct such fluxes by using algebraic cycles and analyze flat directions
We find that the Fermat sextic fourfold contains the algebraic cycles x2σ(0)
Summary
In this paper we consider M-theory compactified on a CY fourfold X. A non zero flux along internal directions generates a potential for the metric moduli after compactification [17]. This can be understood from the 11d C3 kinetic term G4∧∗G4, which depends on the metric through the Hodge star operator ∗). They are supersymmetric if the vevs of W and Wvanish, i.e. W |min = 0 and W |min = 0 This condition together with (2.4) can be rephrased by saying that the fourform flux must lie in Hp2r,2im(X), i.e. G4 must be a primitive four-form of Hodge type (2, 2). Let us come back to the GVW superpotential that generates the minima condition for the complex structure moduli. The algebraic cycles we consider here are exactly of this type [12]
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