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

TL;DR

The paper develops a Weyl-invariant four-dimensional N=1 framework for warped type II flux compactifications with SU(3)×SU(3) structure, showing that a holomorphic off-shell superpotential together with a conformal Kähler potential reproduces all ten-dimensional SUSY equations via D- and F-flatness. It clarifies the role of warp factor in coupling moduli, connects to established GVW results in special truncations, and demonstrates how non-perturbative corrections deform internal generalized geometry, enabling KKLT-like AdS vacua through smeared instantons and lifting of D3-brane moduli via localized instantons. The work provides a unified 4D-10D perspective on flux vacua, generalized geometry, and non-perturbative effects, with concrete implications for moduli stabilization and the geometry of mobile D-branes. Overall, it highlights the central role of generalized geometry in understanding the full implications of warping and non-perturbative physics in string compactifications.

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.