Smearing and Unsmearing KKLT AdS Vacua

TL;DR

This paper provides a ten-dimensional description of supersymmetric AdS4 vacua with gaugino condensates, confirming effective field theory results and exploring conditions for scale separation and localized solutions.

Contribution

It derives supersymmetry equations for type II AdS4 vacua with gaugino condensates using generalized complex geometry and analyzes their solutions in detail.

Findings

Supersymmetry requires a conformal Calabi-Yau manifold and specific fluxes.

The cosmological constant is linked to gaugino condensate and cycle volume.

Exponential scale separation is possible with suppressed (0,3) flux components.

Abstract

Gaugino condensation on D-branes wrapping internal cycles gives a mechanism to stabilize the associated moduli. According to the effective field theory, this gives rise, when combined with fluxes, to supersymmetric AdS$_4$ solutions. In this paper we provide a ten-dimensional description of these vacua. We first find the supersymmetry equations for type II AdS$_4$ vacua with gaugino condensates on D-branes, in the framework of generalized complex geometry. We then solve them for type IIB compactifications with gaugino condensates on smeared D7-branes. We show that supersymmetry requires a (conformal) Calabi-Yau manifold and imaginary self-dual three-form fluxes with an additional (0,3) component. The latter is proportional to the cosmological constant, whose magnitude is determined by the expectation value of the gaugino condensate and the stabilized volume of the cycle wrapped by the branes. This confirms, qualitatively and quantitatively, the results obtained using effective field theory. We find that exponential separation between the AdS and the KK scales seems possible as long as the three-form fluxes are such that their (0,3) component is exponentially suppressed. As for the localized solution, it requires going beyond SU(3)-structure internal manifolds. Nevertheless, we show that the action can be evaluated on-shell without relying on the details of such complicated configuration. We find that no "perfect square" structure occurs, and the result is divergent. We compute the four-fermion contributions, including a counterterm, needed to cancel these divergencies.