From ten to four and back again: how to generalize the geometry

TL;DR

This paper develops a four-dimensional supergravity framework for warped flux compactifications, incorporating non-perturbative effects like instantons, which deform the internal geometry and influence moduli stabilization.

Contribution

It derives a superpotential and supersymmetry conditions that include non-perturbative corrections, linking ten-dimensional geometry with four-dimensional effective theories.

Findings

Non-perturbative effects induce a quantum deformation of the internal geometry.

Smeared instantons help understand KKLT-like AdS vacua from 10D perspective.

Non-smeared instantons destabilize Calabi-Yau structure into generalized complex geometry.

Abstract

We discuss the four-dimensional N=1 effective approach in the study of warped type II flux compactifications with SU(3)x SU(3)-structure to AdS_4 or flat Minkowski space-time. The non-trivial warping makes it natural to use a supergravity formulation invariant under local complexified Weyl transformations. We obtain the classical superpotential from a standard argument involving domain walls and generalized calibrations and show how the resulting F-flatness and D-flatness equations exactly reproduce the full ten-dimensional supersymmetry equations. Furthermore, we consider the effect of non-perturbative corrections to this superpotential arising from gaugino condensation or Euclidean D-brane instantons. For the latter we derive the supersymmetry conditions in N=1 flux vacua in full generality. We find that the non-perturbative corrections induce a quantum deformation of the internal generalized geometry. Smeared instantons allow to understand KKLT-like AdS vacua from a ten-dimensional point of view. On the other hand, non-smeared instantons in IIB warped Calabi-Yau compactifications 'destabilize' the Calabi-Yau complex structure into a genuine generalized complex one. This deformation gives a geometrical explanation of the non-trivial superpotential for mobile D3-branes induced by the non-perturbative corrections.