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

Abstract

We discuss the four-dimensional N = 1 effective approach in the study of warped type II flux compactifications with SU (3) × 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.

Smeared versus localised sources in flux compactifications

“…The T-duality chain that relates these BPS solutions to each other is quite straightforward if one T-dualises the smeared GKP solution on a torus 11 . To obtain the solutions with a Ricci-flat internal space one either takes the T-duality circle along the orientifold (going down in dimension) or on the torus along a cycle without H-flux.…”

“…One way to incorporate the changes that arise upon localising a source is through "warped effective field theory" [7][8][9][10][11][12][13][14][15][16][17][18], in which one derives the correction to the fourdimensional effective action. Another way, which we are pursuing in this paper, is to work directly with the ten-dimensional equations of motion.…”

Dimensional reduction for D3-brane moduli

“…It is also worth discussing the relationship of (4.31) to the D3-brane EOM in the electric formalism as given in (2.14). In static gauge, 16 this is 33) where the · · · include higher powers of ∂ µ Y / m , which we did not calculate in the magnetic framework. The first terms, which involve the metric and its derivatives, are manifestly identical in the electric and magnetic formalisms, so we are left to compare the terms involving the potential/flux.…”

“…One challenge for the construction of an effective theory in GKP backgrounds is that the metric becomes a warped product between the internal and external spaces, complicating the identification of the degrees of freedom. The supersymmetry of the background along with the fact that scaling the warp factor can be removed with a 10D diffeomorphism has allowed [16][17][18][19] to derive many aspects of the effective theory without a direct dimensional reduction.…”

DWSB in heterotic flux compactifications

“…18 Here, for simplicity, we do not consider the gauge bundle contribution to the Kähler potential. 19 The overall factor in (6.19) has been fixed by reproducing (6.17) and (6.19) following the approach of [54], which combines the domain-wall arguments, analogous to the ones originally used in [53], and the use of superconformal supergravity in four dimensions. 20 Notice that, in fact, the (3, 0)-form Ω appearing in (6.17) and (6.19) has no fixed normalization and only matches the Ω used in the rest of the paper (normalized as Ω ∧Ω = i8volM ) up to a overall constant.…”