On brane back-reaction and de Sitter solutions in higher-dimensional supergravity

We argue that the problem of finding lower-dimensional de Sitter solutions to the classical field equations of higher-dimensional supergravity necessarily requires understanding the back-reaction of whatever localized objects source the bulk fields. However, we also find that most of the details of the back-reacted solutions are not important for determining the lower-dimensional curvature. We find, in particular, a classically exact expression that, for a broad class of geometries, directly relates the curvature of the lower-dimensional geometry to asymptotic properties of various bulk fields near the sources. Specializing to codimension-two sources, we find that the contribution involving the asymptotic behaviour of the warp factor (which has a definite sign for most supergravities and so is usually used to infer a preference for anti-de Sitter geometries) is precisely canceled by the contribution of the sources themselves (that are left out in earlier treatments). We identify which combination of bulk fields survives this cancelation, and so controls the sign of the lower-dimensional geometry, for several supergravities in 6, 10 and 11 dimensions. Our results show precisely why explicit 4D de Sitter solutions to 6D supergravity evade general no-go theorems. As an application we show that all classical compactifications of Type IIB supergravity (and F-theory) to 8 dimensions are 8D-flat if they involve only the metric and the axio-dilaton sourced by codimension-two sources, extending earlier results to include warped solutions and more general source properties.