On the fakeness of fake supergravity

TL;DR

The paper clarifies the relationship between fake supergravity and true 5D, N=2 gauged supergravity by examining curved BPS-domain walls. It shows that curved walls require nontrivial hypermultiplet scalars, while vector-only walls must be flat due to very special geometry, and it demonstrates that fake supergravity can describe true supergravity only locally along a domain-wall flow after two fixes: an adapted scalar-coordinates system and an SU(2) gauge choice. The authors derive and compare the BPS equations and scalar potentials in both formalisms, identifying precise compatibility conditions and illustrating how, in adapted coordinates, the vector and hypermultiplet sectors acquire symmetric forms. The results illuminate the limits of fake supergravity and provide a concrete framework for embedding or mapping fake-wall solutions into true supergravity along a chosen flow, with broader implications for related dimensions and compactifications.

Abstract

We revisit and complete the study of curved BPS-domain walls in matter-coupled 5D, N=2 supergravity and carefully analyse the relation to gravitational theories known as "fake supergravities". We first show that \emph{curved} BPS-domain walls require the presence of non-trivial hypermultiplet scalars, whereas walls that are solely supported by vector multiplet scalars are necessarily \emph{flat}, due to the constraints from very special geometry. We then recover fake supergravity as the effective description of true supergravity where one restricts the attention to the flowing scalar field of a given BPS-domain wall. In general, however, true supergravity can be simulated by fake supergravity at most \emph{locally}, based upon two choices: (i) a suitable adapted coordinate system on the scalar manifold, such that only one scalar field plays a dynamical role, and (ii) a gauge fixing of the SU(2) connection on the quaternionic-Kahler manifold, as this connection does not fit the simple formalism of fake supergravity. Employing these gauge and coordinate choices, the BPS-equations for both vector and hypermultiplet scalars become identical to the fake supergravity equations, once the line of flow is determined by the full supergravity equations.