Dimensional reduction of eleven dimensional supergravity on SU(2) group manifold and N=4 Gauged supergravity

Abstract

The consistency at the level of equations of motion from the dimensional reduction giving rise to four-dimensional N = 4, half-maximal gauged supergravity from the unique supergravity in eleven dimensions is thoroughly veri ed in this study. Dimensional reductions are said to be consistent at the level of equations of motion if and only if substitutions of reduction ansatze in higher dimensional equations of motion yield all equations of motion for lower dimensional theory. Apart from some special cases, a Kaluza-Klein sphere reduction is generally not consistent. However, there exists an alternative way to obtain a consistent dimensional reduction. The guaranteed-consistent reduction, known as the Scherk- Schwarz reduction, can be found by performing the dimensional reduction on a group manifold of some particular Lie group. From the fact that the SU(2) group manifold is topologically a three-dimensional sphere S³, the Kaluza-Klein reduction involving S³ can be obtained from a group manifold reduction via replacing the S³ in the reduction ansatz by an SU(2) group manifold. N = 4 gauged supergravity in four dimensions is obtained from a seven-dimensional sphere S⁷ reduction ansatze by writing the S7 as S³ S³ S¹ and replacing each S³ by the SU(2) group manifold while the consistency of seven-dimensional sphere S⁷ reduction exists at the level of equations of motion. Apart from the main SO(4) gauged theory, another related N = 4 gauged supergravity in four dimensions, the Freedman-Schwarz model, can be obtained by using a one-way map between the two versions of N = 4 gauged supergravity. Moreover, the consistent reduction ansatze allow solutions in four-dimensional N = 4 SO(4) gauged supergravity theory to be embedded in eleven-dimensional supergravity.