Abstract
D7-brane moduli are stabilized by worldvolume fluxes, which contribute to the D3-brane tadpole. We calculate this contribution in the Type IIB limit of F-theory compactifications on Calabi-Yau four-folds with a weak Fano base, and are able to prove a no-go theorem for vast swathes of the landscape of compactifications. When the genus of the curve dual to the D7 worldvolume fluxes is fixed and the number of moduli grows, we find that the D3 charge sourced by the fluxes grows faster than 7/16 of the number of moduli, which supports the Tadpole Conjecture of ref. [1]. Our lower bound for the induced D3 charge decreases when the genus of the curves dual to the stabilizing fluxes increase, and does not allow to rule out a sliver of flux configurations dual to high-genus high-degree curves. However, we argue that most of these fluxes have very high curvature, which is likely to be above the string scale except on extremely large (and experimentally ruled out) compactification manifolds.
Highlights
Compactification in the language of F-theory, both the complex structure moduli and the D7 moduli appear as complex structure moduli of the F-theory four-fold, despite their seemingly different IIB origin
When the genus of the curve dual to the D7 worldvolume fluxes is fixed and the number of moduli grows, we find that the D3 charge sourced by the fluxes grows faster than 7/16 of the number of moduli, which supports the Tadpole Conjecture of ref
Proving the Tadpole Conjecture would rule out all String Theory compactifications with large numbers of stabilized D7 or complex structure moduli, and rule out de Sitter vacua obtained by uplifting anti de Sitter compactifications using antibranes in warped throats [2], since one needs a large QD3 to avoid instabilities [4]
Summary
We briefly recall the relation between F-theory and Type IIB orientifolds, which will be heavily used . In the simple example of a base B3 ∼= P3[x], the functions η, χ, and ψ are homogeneous polynomials of degree 16, 24, and 8 in the coordinates xi, and the number of D7-brane deformation moduli is given by nD7(P3) =. All of the negative-charge contributions to the tadpole coming from O7-planes and D7-branes are captured in the topology of the four-fold Z4, by its Euler number, χ(Z4), via. This topological quantity gives an upper bound on the positive-charge contributions, which come from 3-form flux, D7-brane worldvolume flux, and mobile D3-branes.
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