Abstract
We examine the mechanism of moduli stabilization by fluxes in the limit of a large number of moduli. We conjecture that one cannot stabilize all complex-structure moduli in F-theory at a generic point in moduli space (away from singularities) by fluxes that satisfy the bound imposed by the tadpole cancellation condition. More precisely, while the tadpole bound in the limit of a large number of complex-structure moduli goes like 1/4 of the number of moduli, we conjecture that the amount of charge induced by fluxes stabilizing all moduli grows faster than this, and is therefore larger than the allowed amount. Our conjecture is supported by two examples: K3 × K3 compactifications, where by using evolutionary algorithms we find that moduli stabilization needs fluxes whose induced charge is 44% of the number of moduli, and Type IIB compactifications on mathbbm{CP} 3, where the induced charge of the fluxes needed to stabilize the D7-brane moduli is also 44% of the number of these moduli. Proving our conjecture would rule out de Sitter vacua obtained via antibrane uplift in long warped throats with a hierarchically small supersymmetry breaking scale, which require a large tadpole.
Highlights
We examine the mechanism of moduli stabilization by fluxes in the limit of a large number of moduli
We conjecture that one cannot stabilize all complex-structure moduli in F-theory at a generic point in moduli space by fluxes that satisfy the bound imposed by the tadpole cancellation condition
Our conjecture is supported by two examples: K3 × K3 compactifications, where by using evolutionary algorithms we find that moduli stabilization needs fluxes whose induced charge is 44% of the number of moduli, and Type IIB compactifications on CP3, where the induced charge of the fluxes needed to stabilize the D7-brane moduli is 44% of the number of these moduli
Summary
We will present analytic arguments as well as evidence based on differential evolution algorithms for this Tadpole Conjecture, and point out some ways in which one could try to prove it Note that this conjecture implies that one cannot stabilize a growing number of moduli with a finite number of fluxes, whose tadpole would stay O(1). The fluxes which stabilize the h2,1 complex-structure moduli of a smooth Calabi-Yau threefold in a Type IIB compactification will give a positive contribution to the D3 brane tadpole that for large h2,1 grows at least linearly with the number of stabilized moduli: Q(D23,1) stabilization > α × h2,1. Conjecture 3a is based on the similarity between the Type IIB and the F-theory equations governing the stabilization of moduli and the contribution of the fluxes to the D3 brane tadpole, and of the similarity between flux compactifications and bubbling solutions that will be described below.
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