On Brane Back-Reaction and de Sitter Solutions in Higher-Dimensional Supergravity

TL;DR

The paper derives a general relation for the lower-dimensional curvature $R$ in warped, higher-dimensional supergravity compactifications, showing that $R$ is given by a sum of four terms $I+II+III+IV$, where $I$ is the bulk on-shell action, $II$ is a total-derivative warp-factor boundary term, $III$ is the localized source action, and $IV$ vanishes without space-filling fluxes.For codimension-two sources, the authors demonstrate that the boundary warp term $II$ cancels with the source contribution $III$ via careful near-source matching, so $R$ is determined solely by the on-shell bulk action $S_{ m on-shell}$, which in scale-invariant theories reduces to a total derivative and thus to near-source boundary data.They provide explicit expressions for the on-shell bulk action in 11D, 10D IIA, and 10D IIB supergravities, illustrating how the surviving combination of bulk fields near sources fixes the sign and magnitude of the lower-dimensional curvature, and they apply the framework to show that all classical Type IIB/F-theory compactifications to 8 dimensions with only metric and axio-dilaton and codimension-two sources are 8D-flat.These results explain why no-go theorems for de Sitter solutions can be evaded in certain higher-dimensional theories and highlight that full back-reaction details are often unnecessary; instead, the near-source asymptotics of a specific bulk-field combination suffice to constrain the lower-dimensional curvature.

Abstract

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.