Consistent S3 reductions of six-dimensional supergravity

Abstract

We work out the consistent AdS 3 × S 3 truncations of the bosonic sectors of both the six-dimensional N = (1, 1) and N = (2, 0) supergravity theories. They result in inequivalent three-dimensional half-maximal SO(4) gauged supergravities describing 32 propagating bosonic degrees of freedom apart from the non-propagating supergravity multiplet. We present the full non-linear Kaluza-Klein reduction formulas and illustrate them by explicitly uplifting a number of AdS 3 vacua. Conclusions 24A S 3 harmonics and identities 25 theories of concern and the dualities they enjoy. Notably, these are not truncations in an effective field theory sense, with the massive Kaluza-Klein towers integrated out, yet every solution of the lower-dimensional theory lifts to a solution of the higher-dimensional theory. They are of particular importance in holographic applications, ensuring the validity of lower-dimensional supergravity computations, such as holographic correlators and renormalization group (RG) flows [4]. This work deals with consistent sphere compactifications in the context of AdS 3 × S 3 , one of the central examples in the AdS/CFT correspondence [5] in which supergravity techniques have been successfully employed [6][7][8][9][10][11] in order to unravel the structure of the dual two-dimensional conformal field theories. Generic consistent S 3 truncations in (super)gravity have been discussed in [12][13][14], where the full non-linear Kaluza-Klein Ansätze were constructed for a higherdimensional theory that comprises the field content of the bosonic string. The resulting lowerdimensional theories are SO(4) gauged (super)gravities carrying gauge fields, a 2-form, and scalar fields whose potential do not admit any stationary points. In the particular case of AdS 3 × S 3 , the higher-dimensional theory is D = 6, N = (1, 0) supergravity coupled to a single tensor multiplet that carries an anti-selfdual 2-form. In contrast to the higher-dimensional examples, the 2-forms in the resulting three-dimensional theory are auxiliary and can be integrated out, giving rise to an additional contribution to the scalar potential. This amended potential turns out to support a stable supersymmetric AdS 3 vacuum [15], corresponding to the supersymmetric AdS 3 × S 3 solution of the D = 6 theory. The non-linear Kaluza-Klein Ansätze can be confirmed by direct computation.More recently, new techniques have emerged for a more systematic understanding of consistent truncations within exceptional field theory (ExFT) and generalized geometry [16][17][18][19][20][21], see also [22,23] in the context of double field theory. Using the reformulation of D = 6, N = (1, 0) supergravity as an exceptional field theory based on the group SO(4,4) [24], the non-linear Kaluza-Klein Ansätze from [12,15] can straightforwardly be reproduced from the generalized Scherk-Schwarz twist matrices U in this framework. In this paper, we will extend the consistent S 3 truncations to the full N = (1, 1) and N = (2, 0) supergravities in six dimensions. The relevant framewor...

Microstate geometries from gauged supergravity in three dimensions

“…As will become evident, we will find the results in [1] for N = (2, 0) supergravity immensely useful in extending the results of [31] to obtain the three-dimensional supergravity corresponding to N = (1, 0) supergravity coupled to two tensor multiplets. The relevant supergravity in three dimensions is fully defined by its amount of supersymmetry, the scalar coset, the gauge symmetry and the gauge couplings as defined by an embedding tensor.…”

“…There is now an extensive literature [1,[26][27][28][29][30][31] on how this compactification leads to N = 8 (16 supersymmetries) gauged supergravity in three dimensions. The gauge group is SO(4) ∼ = SO(3) + × SO(3) − and comes from the isometries of S 3 ; the scalar coset becomes:…”

“…In particular, the precise relationship between N = (2, 0) supergravity coupled to one anti-self-dual tensor multiplet in six dimensions and the three-dimensional N = 8, SO (8,4) supergravity was recently laid out in [1]. The construction of superstrata usually takes place in the less supersymmetric, N = (1, 0) theories in six dimensions.…”

“…Note that we are using the anti-diagonal form of η because it is far more convenient in describing the degrees of freedom and in expressing the gauging. For ε = −1 we have G = SO(5, 4) (in the conventions of [1]) and for ε = 1 we have G = SO (4,5). While this might seem a trivial notational distinction, it is related to the self-duality or anti-self-duality of the additional tensor multiplet.…”

“…One of the remarkable features that has become evident in recent constructions of asymptotically-AdS superstrata [5,7,17] is that most of the interesting physics of superstrata is encoded in a three-dimensional space-time, K. Indeed, the six-dimensional space-time of a superstratum naturally decomposes into the S 3 surface around the branes, the radial coordinate, r, and the common directions, (t, y) along the branes. 1 The manifold, K, is the geometry described by the coordinates (t, y, r) that are complementary to the S 3 .…”