7-dimensional \( \mathcal{N} \) = 2 consistent truncations using SL(5) exceptional field theory

Received: February 7-dimensional Ludwig-Maximilians-Universitat Munchen 0 1 2 0 Open Access , c The Authors 1 Theresienstra e 37 , 80333 Munchen , Germany 2 Arnold Sommerfeld Center for Theoretical Physics, Department fur Physik We show how to construct seven-dimensional half-maximally supersymmetric consistent truncations of 11-/10-dimensional SUGRA using SL(5) exceptional eld theory. Such truncations are de ned on generalised SU(2)-structure manifolds and give rise to seven-dimensional half-maximal gauged supergravities coupled to n vector multiplets and thus with scalar coset space R+ ArXiv ePrint: 1612.01692 Flux compacti cations; String Duality; Supergravity Models; Superstring - 2 consistent truncations using SL(5) exceptional eld theory O(n). The consistency conditions for the quadratic constraint when the section condition is satis ed. 1 Introduction Overview of exceptional eld theory Spinor bilinears and SU(2) structure Spinor convention Spinor bilinears Properties of the spinor bilinears Reformulating the SL(5) EFT 4.1 Intrinsic SU(2)-torsion Doublet and closure conditions The half-maximal embedding tensor 5.5 Intrinsic torsion and the T -tensor Reducing the external covariant derivative Reducing the scalar potential Reducing the kinetic terms Scalar kinetic terms Gauge kinetic terms and topological term Reducing the SUSY variations A Scalar potential in terms of spinors A.1 Determined connections B SUSY variations of the gravitino Representations in the intrinsic SU(2)-torsion Explicit expressions for the intrinsic SU(2)-torsion Intrinsic torsion in terms of spinors Supersymmetry variation of the gravitino Scalar potential Kinetic terms Consistent truncations to half-maximal gauged supergravity Decomposition of supergravity elds De ning the truncation Truncation Ansatz Scalar truncation Ansatz Fermion, gauge eld and external metric truncation Ansatz Consistency conditions and the embedding tensor Finding consistent truncations of higher-dimensional supergravity to yield lowerdimensional theories is a notoriously di cult problem. By a consistent truncation we those of the initial higher-dimensional theory. Because of the non-linearity of the equations such consistent truncation Ansatze are generically hard to nd [1], unless the manifolds. As a result the truncation has the same number of supersymmetries as the higher-dimensional theory. Recently, the Scherk-Schwarz set-up has been generalised using double eld theory (DFT) [3{6] and exceptional eld theory (EFT) [7{9], as well as generalised geometry [10{13]. These theories are O(D; D)- and Ed(d)-manifest extensions (or reformulations gauge and gravitational elds on an equal footing, see [14{17] for earlier work in this direction. They thus naturally include uxes in the Scherk-Schwarz set-up. As a result, a an otherwise remarkable set of consistent truncations on spheres, in particular S7 [30] and S4 [31, 32] of 11-dimensional SUGRA and S5 [33] for IIB SUGRA, can be understood as such generalised Scherk-Schwarz Ansatze [23, 28]. With this set-up it has been possible to derive and study a variety of new consistent truncations [38].1 Because such generalised Scherk-Schwarz truncations are de ned on genRAs when used in DFT, or their respective generalised geometry analogues. While it is possible to de ne a further truncation of the maximal gauged SUGRA to half-maximal ones, corresponding to the reduction of EFT to DFT, see e.g. [20, 42], there are of course type II theories but require either the heterotic SUGRA or 11-dimensional SUGRA. The purpose of this paper is to initiate the study of consistent truncations which break coset space Mscalar = 1The closely-related approach of [39{41] has also been fruitful in nding consistent truncations. shown in [43].2 In order to break half of the supersymmetry, the internal manifold must have generIn particular, the embedding tensor is encoded in the generalised Lie derivative acting is su cient for the gaugings to ful l the quadratic constraint. group acting on the sections we keep in the truncation. The linear symmetry group is in the case of exceptional eld theory just Ed(d), or in the case considered here SL(5). This is considered.3 However, when we consider truncations on generalised parallelisable spaces Scherk-Schwarz reductions lead to gauged SUGRAs with global symmetry group Ed(d). On the other hand, when the background is not generalised parallelisable, as we will be considering in this paper, the group acting on the space of sections can be much larger To emphasise this point, let us consider the more familiar example of general relativity in d + 4-dimensions on a product manifold so that its linear symmetry group is GL(4). When performing a truncation on T 4, one obtains d-dimensional gravity O(9). The duality group O(3; 19) acts of course on the space of sections de ning the truncation on K3, i.e. the 22 harmonic forms. The linear symmetry group of the interna (...truncated)