On the nonlinear KK reductions on spheres of supergravity theories

TL;DR

This paper develops general nonlinear Kaluza-Klein reduction ansätze for supergravity theories compactified on spheres, providing explicit formulas for metrics, scalars, and fermions, and extends previous results to broader classes of compactifications.

Contribution

It introduces a universal metric ansatz for sphere compactifications with scalar deformations, and formulates the fermionic ansatz involving a matrix U, advancing the understanding of supergravity reductions.

Findings

Derived a universal metric ansatz with scalar-induced deformations.

Provided explicit fermionic ansatz involving the matrix U.

Obtained the full metric and vielbein ansatz for 10d supergravity on AdS5×S5.

Abstract

We address some issues related to the construction of general Kaluza-Klein (KK) ans\"atze for the compactification of a supergravity (sugra) theory on a sphere $S_m$. We first reproduce various ans\"atze for compactification to 7d from the ansatz for the full nonlinear KK reduction of 11d sugra on $AdS_7\times S_4$. As a side result, we obtain a lagrangian formulation of 7d ${\cal N}=2$ gauged sugra, which so far had only a on-shell formulation, through field equations and constraints. The $AdS_7\times S_4$ ansatz generalizes therefore all previous sphere compactifications to 7d. Then we consider the case when the scalars in the lower dimensional theory are in a coset $Sl(m+1)/SO(m+1)$, and we keep the maximal gauge group $SO(m+1)$. The 11-dimensional sugra truncated on $S_4$ fits precisely the case under consideration, and serves as a model for our construction. We find that the metric ansatz has a universal expression, with the internal space deformed by the scalar fluctuations to a conformally rescaled ellipsoid. We also find the ansatz for the dependence of the antisymmetric tensor on the scalars. We comment on the fermionic ansatz, which will contain a matrix $U$ interpolating between the spinorial $SO(m+1)$ indices of the spherical harmonics and the $R$-symmetry indices of the fermionic fields in the lower dimensional sugra theory. We derive general conditions which the matrix $U$ has to satisfy and we give a formula for the vielbein in terms of $U$. As an application of our methods we obtain the full ansatz for the metric and vielbein for 10d sugra on $AdS_5\times S_5$ (with no restriction on any fields).