Embedding the SU(3) sector of SO(8) supergravity in D=11

Abstract

The SU(3)-invariant sector of maximal supergravity in four dimensions with an SO(8) gauging is uplifted to D = 11 supergravity. In order to do this, the SU(3)-neutral sector of the tensor and duality hierarchies of the D = 4 N = 8 supergravity is first worked out. The consistent D = 11 embedding of the full, dynamical SU(3) sector is then expressed at the level of the D = 11 metric and three-form gauge field in terms of these D = 4 tensors. The redundancies introduced by this approach are eliminated at the level of the D = 11 four-form field strength by making use of the D = 4 duality hierarchy. Our results encompass previously known truncations of D = 11 supergravity down to sectors of SO(8) supergravity with symmetry larger than SU(3), and include new ones. In particular, we obtain a new consistent truncation of D = 11 supergravity to minimal D = 4 N = 2 gauged supergravity. The SU(3)-invariant sector of SO(8) supergravityLet us start by reviewing some aspects of the SU(3) sector of SO(8)-gauged supergravity. We choose a triangular, or Iwasawa, parametrisation for the (SU(3)-invariant truncation of the) E 7(7) /SU(8) coset representative. Since previous literature often chooses the unitary gauge for the coset, we believe that our presentation has some intrinsic value even if the material that is covered (the Lagrangian in section 2.1, the further subsectors in 2.3, and the vacuum structure in 2.4) is mostly review. The SU(3)-invariant, restricted tensor and duality hierarchies worked out in section 2.2 are new.

Universal properties of Kaluza-Klein gravitons

“…A convenient starting point for our analysis is the local geometries recently presented in [18]. In that reference, the full, dynamical SU(3)-invariant sector of D = 4 N = 8 SO(8)-gauged supergravity [9] was uplifted to D = 11 using the consistent truncation formulae of [34].…”

“…In that reference, the full, dynamical SU(3)-invariant sector of D = 4 N = 8 SO(8)-gauged supergravity [9] was uplifted to D = 11 using the consistent truncation formulae of [34]. In particular, the results of [18] provide a unified treatment for all the D = 11 AdS 4 solutions that uplift from critical points of D = 4 N = 8 SO(8)-gauged supergravity with at least SU(3) symmetry. In order to simplify the calculations, we will focus on two disjoint further subsectors with symmetries G 2 and SU(4) c larger than SU(3).…”

“…The G 2 -invariant sector of the D = 4 SO(8) supergravity contains an SL(2, R)/SO(2) dilaton-axion pair (ϕ, χ), in the notation of appendix A. The D = 11 uplift of this sector was given in section 3.2.3 of [18]. The warp factor and internal d = 7 geometry that feature in (1.1) are given by…”

“…With the understanding that ϕ, φ depend on the independent fields χ, ζ,ζ, the former can be conveniently used to parametrise the SU(4) c -invariant sector, as the resulting expressions are more compact. The embedding of this sector into the D = 11 warp factor and internal metric reads [25], in the notation of [18],…”

“…In order to narrow down and simplify the problem, we will further focus on the SU(3)-invariant sector of these gaugings. This sector has been explicitly uplifted to the respective higher-dimensional supergravities [18,19,20], and the corresponding vacua have been charted [21,10,20]. The list of AdS vacua in this sector across all three gaugings of interest has been summarised for convenience in table 1.…”