We describe a method for finding flux vacua of type IIB string theory in which the Gukov-Vafa-Witten superpotential is exponentially small. We present an example with ${W}_{0}\ensuremath{\approx}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$ on an orientifold of a Calabi-Yau hypersurface with $({h}^{1,1},{h}^{2,1})=(2,272)$, at large complex structure and weak string coupling.

Vacua with Small Flux Superpotential

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 cancelation 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 x K3 compactifications, where by using evolutionary algorithms we find that moduli stabilization needs fluxes whose induced charge is (slightly smaller than) 1/2 of the number of moduli, and Type IIB compactifications on CP^3 where the induced charge is equal to the number of (D7-brane) moduli. Proving our conjecture would rule out in particular de Sitter vacua obtained via antibrane uplift in long warped throats with a hierarchically small supersymmetry breaking scale, which require a large tadpole.

The Tadpole Problem

We compute the flux-induced F-term potential in 4d F-theory compactifications at large complex structure. In this regime, each complex structure field splits as an axionic field plus its saxionic partner, and the classical F-term potential takes the form $V = Z^{AB} \rho_A\rho_B$ up to exponentially-suppressed terms, with $\rho$ depending on the fluxes and axions and $Z$ on the saxions. We provide explicit, general expressions for $Z$ and $\rho$, and from there analyse the set of flux vacua, for an arbitrary number of fields. We identify two families of vacua with all complex structure fields fixed and a flux contribution to the tadpole $N_{\rm flux}$ which is bounded. In the first and most generic one, the saxion vevs are bounded from above by a power of $N_{\rm flux}$. In the second their vevs may be unbounded and $N_{\rm flux}$ is a product of two arbitrary integers, unlike what is claimed by the Tadpole Conjecture. We specialise to type IIB orientifolds, where both families of vacua are present, and link our analysis with several results in the literature. We finally illustrate our findings with several examples.

/pdf/f-theory-flux-vacua-at-large-complex-structure-rtbhkutczb.pdf

F-theory flux vacua at large complex structure

We propose a new mechanism of (geometric) moduli stabilisation in type IIB/F-theory four-dimensional compactifications on Calabi–Yau manifolds, in the presence of 7-branes, that does not rely on non-perturbative effects. Complex structure moduli and the axion-dilaton system are stabilised in the standard way, without breaking supersymmetry, using 3-form internal fluxes. Kahler class moduli stabilisation utilises perturbative string loop corrections, together with internal magnetic fields along the D7-branes world-volume leading to Fayet-Iliopoulos D-terms in the effective supergravity action. The main ingredient that makes the stabilisation possible at a de Sitter vacuum is the logarithmic dependence of the string loop corrections in the large two-dimensional transverse volume limit of the 7-branes.

/pdf/perturbative-moduli-stabilisation-in-type-iib-f-theory-18xlioc4je.pdf

Perturbative moduli stabilisation in type IIB/F-theory framework

We propose a new mechanism of (geometric) moduli stabilisation in type IIB/F-theory four-dimensional compactifications on Calabi-Yau manifolds, in the presence of 7-branes, that does not rely on non-perturbative effects. Complex structure moduli and the axion-dilaton system are stabilised in the standard way, without breaking supersymmetry, using 3-form internal fluxes. Kahler class moduli stabilisation utilises perturbative string loop corrections, together with internal magnetic fields along the D7-branes world-volume leading to Fayet-Iliopoulos D-terms in the effective supergravity action. The main ingredient that makes the stabilisation possible at a de Sitter vacuum is the logarithmic dependence of the string loop corrections in the large two-dimensional transverse volume limit of the 7-branes.

/pdf/perturbative-moduli-stabilisation-in-type-iib-f-theory-4pybpo5aog.pdf

Hajime Otsuka

Papers

On the flux vacua in F-theory compactifications