The tadpole problem

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 $$ \mathbbm{CP} $$ 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.

How a four-dimensional de Sitter solution remains outside the swampland

We argue that, in the presence of time-dependent fluxes and quantum corrections, four-dimensional de Sitter solutions should appear in the type IIB string landscape and not in the swampland. Our construction considers generic choices of local and non-local quantum terms and satisfies the no-go and the swampland criteria, the latter being recently upgraded using the trans-Planckian cosmic censorship. Interestingly, both time-independent Newton constant and moduli stabilization may be achieved in such backgrounds even in the presence of time-dependent fluxes and internal spaces. However, once the time-dependence is switched off, any four-dimensional solution with de Sitter isometries appears to have no simple effective field theory descriptions and is back in the swampland.

Three-dimensional flux vacua from IIB on co-calibrated G2 orientifolds

We derive the 3D N = 1 superpotential for the closed string sector of type IIB supergravity on toroidal O5 orientifolds with co-calibrated G2 structure and RR background flux. We find that such compactifications can provide full closed string moduli stabilization on supersymmetric $$\hbox {AdS}_3$$ AdS 3 vacua, and once we include brane-supersymmetry-breaking we also find indication for the existence of classical 3D de Sitter solutions. The latter however are rather difficult to reconcile with the “shape” moduli stabilization and flux quantization. We also discuss the possibility of achieving scale separation in $$\hbox {AdS}_3$$ AdS 3 and $$\hbox {dS}_3$$ dS 3 vacua, but such effects seem to be hindered by the geometric flux quantization.

Heterotic complex structure moduli stabilization for elliptically fibered Calabi-Yau manifolds

Holomorphicity of vector bundles can stabilize complex structure moduli of a Calabi-Yau threefold in N = 1 supersymmetric heterotic compactifications. In principle, the Atiyah class determines the stabilized moduli. In this paper, we study how this mechanism works in the context of elliptically fibered Calabi-Yau manifolds where the complex structure moduli space contains two kinds of moduli, those from the base and those from the fibration. Defining the bundle with spectral data, we find three types of situations when bundles’ holomorphicity depends on algebraic cycles exist only for special loci in the complex structure moduli, which allows us to stabilize both of these two moduli. We present concrete examples for each type and develop practical tools to analyze the stabilized moduli. Finally, by checking the holomorphicity of the four-flux and/or local Higgs bundle data in F-theory, we briefly study the dual complex structure moduli stabilization scenarios.

Landscape of modular symmetric flavor models

We study the moduli stabilization from the viewpoint of modular flavor symmetries. We systematically analyze stabilized moduli values in possible configurations of flux compactifications, investigating probabilities of moduli values and showing which moduli values are favorable from our moduli stabilization. Then, we examine their implications on modular symmetric flavor models. It is found that distributions of complex structure modulus τ determining the flavor structure are clustered at a fixed point with the residual ℤ3 symmetry in the SL(2, ℤ) fundamental region. Also, they are clustered at other specific points such as intersecting points between |τ|2 = k/2 and Re τ = 0,±1/4,±1/2, although their probabilities are less than the ℤ3 fixed point. In general, CP-breaking vacua in the complex structure modulus are statistically disfavored in the string landscape. Among CP-breaking vacua, the values Re τ = ±1/4 are most favorable in particular when the axio-dilaton S is stabilized at Re S = ±1/4. That shows a strong correlation between CP phases originated from string moduli.

An update on moduli stabilization with antibrane uplift

It was recently shown that in warped compactifications based on a Klebanov-Strassler throat there is a light complex structure field, governing the size of the throat and the redshift at its tip. We show that after uplift of the cosmological constant by an anti-D3 brane at the tip of the throat, the contribution to supersymmetry breaking coming from the new light field is large. We work out the mass scales, in particular the condition for this field to be heavier than the Kähler modulus. We check that for the range of parameters relevant for the destabilization we find agreement with de Sitter swampland conjecture. Adding matter fields on distant branes, we discuss the effects on supersymmetry breaking in the observable sector. A hierarchically small scale of supersymmetry breaking translates generically into large values of localized D3 charges in the manifold.

G 4 flux, algebraic cycles and complex structure moduli stabilization

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.