Four-dimensional N=4 SO(4) Gauged Supergravity from D=11

TL;DR

This work provides an explicit non-linear Kaluza-Klein Ansatz embedding the bosonic sector of $D=4$ $N=4$ $SO(4)$ gauged supergravity into $D=11$ supergravity on $S^7$, enabling exact uplift of all 4D bosonic solutions to 11D. The authors express the internal $S^7$ as a foliation by two $S^3$ factors and derive a complete metric and 4-form Ansatz, including the two $SU(2)$ gauge sectors, dilaton and axion, and their couplings. A key result is that the standard $N=4$ $SO(4)$ theory has effectively a single genuine gauge coupling, with an apparent two-parameter freedom removable by field redefinitions, while the Freedman–Schwarz model corresponds to a singular limit of the standard theory, in which the internal geometry degenerates to $ ext{R} imes S^3 imes S^3$ and the two couplings become independent parameters that cannot be rescaled away. The paper also clarifies the relation between these four-dimensional theories and higher-dimensional reductions, showing that the Freedman–Schwarz model can be obtained via a singular $S^7$-to-$ ext{R} imes S^3 imes S^3$ limit, connected to an $S^3 imes S^3$ reduction from $D=10$. Overall, the explicit Ansatz makes practical the uplift of 4D solutions to 11D and sheds light on the structure and limits of related gauged supergravities.

Abstract

We construct the complete and explicit non-linear Kaluza-Klein Ansatz for deriving the bosonic sector of the standard N=4 SO(4) gauged four-dimensional supergravity from the reduction of D=11 supergravity on S^7. This provides a way of interpreting all bosonic solutions of the four-dimensional gauged theory as exact solutions in eleven-dimensional supergravity. We discuss certain limiting forms of the Kaluza-Klein reduction, and compare them with related forms in the Freedman-Schwarz N=4 SU(2)xSU(2) gauged theory. This leads us to the result that the Freedman-Schwarz model is in fact a singular limiting case of the standard SO(4) gauged supergravity. We show that in this limit, our Ansatz for getting the SO(4) gauged theory as an S^7 reduction from D=11 indeed reduces to an S^3 x S^3 reduction from D=10, which makes contact with previous results in the literature. We also show that there is no distinction to be made between having equal or unequal values for the gauge coupling constants $g$ and $\tilde g$ of the two SU(2) gauge-group factors in the standard N=4 SO(4) gauged supergravity, whilst by contrast the ratio of $g$ to $\tilde g$ is a non-trivial parameter of the Freedman-Schwarz model.