119187953
Consistent N=8$$ \mathcal{N}=8 $$ truncation of massive IIA on S6
A bstractMassive type IIA supergravity is shown to admit a consistent truncation on the six-sphere to maximal supergravity in four dimensions with a dyonic ISO(7) gauging. We obtain the complete, non-linear embedding of all the D = 4 fields into the IIA metric and form potentials, and show its consistency. We first rewrite the IIA theory in an SO(1, 3) × SL(7)-covariant way. Then, we employ an N=8$$ \mathcal{N}=8 $$ SL(7)-covariant restriction of the D = 4 tensor hierarchy in order to find the full embedding. The redundant D = 4 degrees of freedom introduced by the tensor hierarchy can be eliminated by writing the embedding in terms of the field strengths and exploiting the restricted duality hierarchy. In particular, closed expressions for the Freund-Rubin term are found using this technique which reveal a pattern valid for other truncations. Finally, we show that the present N=8$$ \mathcal{N}=8 $$ truncation of massive IIA on S6 and the N=2$$ \mathcal{N}=2 $$ truncation obtained when S6 is equipped with its nearly-Kähler structure, overlap in the N=1$$ \mathcal{N}=1 $$, G2-invariant sector of the former.
115687498
Gauged maximally extended su - pergravity in seven dimensions
A rolled metallized film capacitor is encased between a pair of box halves (13 and 13') each having a pair of dovetailed slots (36 and 37) formed in opposed walls so that when the box halves are assembled about a capacitor blank (10) the slots in the respective box halves are aligned with each other. The box halves are mounted in abutting relation about a pair of parallel wires (11 and 12), and then heat fusible metal (46 and 47) is deposited in the aligned slots to lock the box halves together while bonding the wires (11 and 12) to opposite ends of the capacitor blank (10). When assembled, the boxed capacitor with the laterally extending leads may be readily machine inserted or otherwise assembled in a circuit module, such as a printed circuit board.
14415042
Consistent nonlinear KK reduction of 11d supergravity on AdS7×S4 and self-duality in odd dimensions
Abstract We show that there exists a consistent truncation of 11 dimensional supergravity to the `massless' fields of maximal ( N =4) 7 dimensional gauged supergravity. We find the complete expressions for the nonlinear embedding of the 7 dimensional fields into the 11 dimensional fields, and check them by reproducing the d =7 susy transformation laws from the d =11 laws in various sectors. In particular we determine explicitly the matrix U which connects the Killing spinors to the gravitinos in the KK ansatz, and the dependence of the 4-index field strength on the scalars. This is the first time a complete nonlinear KK reduction of the original d =11 supergravity on a nontrivial compact space has been explicitly given. We need a first order formulation for the 3 index tensor field A ΛΠΣ in d =11 to reproduce the 7 dimensional result. The concept of `self-duality in odd dimensions' is thus shown to originate from first order formalism in higher dimensions. For the AdS-CFT correspondence, our results imply that one can use 7d gauged supergravity (without further massive modes) to compute certain correlators in the d =6 (0,2) CFT at leading order in N . This eliminates an ambiguity in the formulation of the correspondence.
123187026
Gauged N = 8 supergravity in five dimensions☆
Abstract We construct gauged N = 8 supergravity theories in five dimensions. Instead of the twenty-seven vector fields of the ungauged theory, the gauged theories contain fifteen vector fields and twelve second-rank antisymmetric tensor fields satisfying self-dual field equations. The fifteen vector fields can be used to gauge any of the fifteen-dimensional semisimple subgroups of SL(6, R ), specially SO( p , 6− p ) for p = 0, 1, 2, 3. The gauged theories also have a physical global SU(1,1) symmetry which survives from the E 6(6) symmetry of the ungauged theory. This SU(1,1) for the SO(6) gauging is presumably related to that of the chiral N = 2 theory in ten dimensions. In our formalism we maintain a composite local USp(8) symmetry analogous to SU(8) in four dimensions.
119201330
Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions
A bstractMotivated by the fact that there exists a continuous one-parameter family of gauged SO(8) supergravities, possible eleven-dimensional origins of this phenomenon are explored. Taking the original proof of the consistency of the truncation of 11D supergravity to SO(8) gauged supergravity as a starting point, a number of critical issues is discussed, such as the preferred electric-magnetic duality frame in four dimensions and the existence of dual magnetic gauge fields and related quantities in eleven dimensions. Some of those issues are resolved but others seem to point to obstructions in embedding the continuous degeneracy in 11D supergravity. While the final outcome of these efforts remains as yet inconclusive, several new results are obtained. Among those is the full non-linear ansatz for the seven-dimensional flux expressed in terms of the scalars and pseudoscalars of 4D supergravity, valid for both the S7 and the T7 truncations without resorting to tensor-scalar duality.
82627589
Spheres, generalised parallelisability and consistent truncations
We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten- and eleven-dimensional supergravity. In all cases the reduction manifold admits a "generalised parallelisation" with a frame algebra with constant coefficients. The consistent truncation then arises as a generalised version of a conventional Scherk-Schwarz reduction with the frame algebra encoding the embedding tensor of the reduced theory. The key new result is that all round-sphere $S^d$ geometries admit such generalised parallelisations with an $SO(d+1)$ frame algebra. Thus we show that the remarkable consistent truncations on $S^3$, $S^4$, $S^5$ and $S^7$ are in fact simply generalised Scherk-Schwarz reductions. This description leads directly to the standard non-linear scalar-field ansatze and as an application we give the full scalar-field ansatz for the type IIB truncation on $S^5$.
53842478
IIB supergravity and the E6(6) covariant vector-tensor hierarchy
A bstractIIB supergravity is reformulated with a manifest local USp(8) invariance that makes the embedding of five-dimensional maximal supergravities transparent. In this formulation the ten-dimensional theory exhibits all the 27 one-form fields and 22 of the 27 two-form fields that are required by the vector-tensor hierarchy of the five-dimensional theory. The missing 5 two-form fields must transform in the same representation as a descendant of the ten-dimensional ‘dual graviton’. The invariant E6(6) symmetric tensor that appears in the vector-tensor hierarchy is reproduced. Generalized vielbeine are derived from the supersymmetry transformations of the vector fields, as well as consistent expressions for the USp(8) covariant fermion fields. Implications are discussed for the consistency of the truncation of IIB supergravity compactified on the five-sphere to maximal gauged supergravity in five space-time dimensions with an SO(6) gauge group.
2064779
Consistent Type IIB Reductions to Maximal 5D Supergravity
We use exceptional field theory as a tool to work out the full non-linear reduction ansaetze for the AdS5 ˆ S 5 compactification of IIB supergravity and its non-compact counterparts in which the sphere S 5 is replaced by the inhomogeneous hyperboloidal space H p,q . The resulting theories are the maximal 5D supergravities with gauge groups SOpp,qq. They are consistent truncations in the sense that every solution of 5D super- gravity lifts to a solution of IIB supergravity. In particular, every stationary point and every holographic RG flow of the scalar potentials for the compact and non-compact 5D gaugings directly lift to solutions of IIB supergravity.
121808806
Dynamics of dimensional reduction
In d-dimensional unified theories that, along with gravity, contain an antisymmetric tensor field of rank s-1, preferential compactification of d-s or of s space-like dimensions is found to occur. This is the case in 11-dimensional supergravity where s = 4.
15524041
AdS4/CFT3 squashed, stretched and warped
We use group theoretic methods to calculate the spectrum of short multiplets around the extremum of = 8 gauged supergravity potential which possesses = 2 supersymmetry and SU(3) global symmetry. Upon uplifting to M-theory, it describes a warped product of AdS4 and a certain squashed and stretched 7-sphere. We find quantum numbers in agreement with those of the gauge invariant operators in the = 2 superconformal Chern-Simons theory recently proposed to be the dual of this M-theory background. This theory is obtained from the U(N) × U(N) theory through deforming the superpotential by a term quadratic in one of the superfields. To construct this model explicitly, one needs to employ monopole operators whose complete understanding is still lacking. However, for the U(2) × U(2) gauge theory we make a proposal for the form of the monopole operators which has a number of desired properties. In particular, this proposal implies enhanced symmetry of the U(2) × U(2) ABJM theory for k = 1,2; it makes its similarity to and subtle difference from the BLG theory quite explicit.
18008297
On the nonlinear KK reductions on spheres of supergravity theories
Abstract We address some issues related to the construction of general Kaluza–Klein (KK) ansatze for the compactification of a supergravity (sugra) theory on a sphere S m . We first reproduce various ansatze for compactification to 7D from the ansatz for the full nonlinear KK reduction of 11D sugra on AdS 7 ×S 4 . As a side result, we obtain a lagrangian formulation of 7D N =2 gauged sugra, which so far had only a on-shell formulation, through field equations and constraints. The AdS 7 ×S 4 ansatz generalizes therefore all previous sphere compactifications to 7D. Then we consider the case when the scalars in the lower dimensional theory are in a coset Sl(m+1)/SO(m+1) , and we keep the maximal gauge group SO(m+1) . The 11-dimensional sugra truncated on S 4 fits precisely the case under consideration, and serves as a model for our construction. We find that the metric ansatz has a universal expression, with the internal space deformed by the scalar fluctuations to a conformally rescaled ellipsoid. We also find the ansatz for the dependence of the antisymmetric tensor on the scalars. We comment on the fermionic ansatz, which will contain a matrix U interpolating between the spinorial SO(m+1) indices of the spherical harmonics and the R -symmetry indices of the fermionic fields in the lower dimensional sugra theory. We derive general conditions which the matrix U has to satisfy and we give a formula for the vielbein in terms of U . As an application of our methods we obtain the full ansatz for the metric and vielbein for 10D sugra on AdS 5 ×S 5 (with no restriction on any fields).
8803261
Consistent Kaluza-Klein truncations via exceptional field theory
A bstractWe present the generalized Scherk-Schwarz reduction ansatz for the full supersymmetric exceptional field theory in terms of group valued twist matrices subject to consistency equations. With this ansatz the field equations precisely reduce to those of lower-dimensional gauged supergravity parametrized by an embedding tensor. We explicitly construct a family of twist matrices as solutions of the consistency equations. They induce gauged supergravities with gauge groups SO(p, q) and CSO(p, q, r). Geometrically, they describe compactifications on internal spaces given by spheres and (warped) hyperboloides Hp,q , thus extending the applicability of generalized Scherk-Schwarz reductions beyond homogeneous spaces. Together with the dictionary that relates exceptional field theory to D = 11 and IIB supergravity, respectively, the construction defines an entire new family of consistent truncations of the original theories. These include not only compactifications on spheres of different dimensions (such as AdS5 × S5), but also various hyperboloid compactifications giving rise to a higher-dimensional embedding of supergravities with non-compact and non-semisimple gauge groups.
13304016
Maximally supersymmetric solutions of ten- and eleven-dimensional supergravities
We classify (up to local isometry) the maximally supersymmetric solutions of the eleven- and ten-dimensional supergravity theories. We find that the AdS solutions, the Hpp-waves and the flat space solutions exhaust them.
15553883
Consistent Kaluza-Klein reductions for general supersymmetric AdS solutions
For the most general supersymmetric solutions of type IIB supergravity consisting of a warped product of $Ad{S}_{5}$ with a five-dimensional manifold ${M}_{5}$, we construct an explicit consistent Kaluza-Klein reduction on ${M}_{5}$ to minimal $D=5$ gauged supergravity. Thus, any solution of the gauged supergravity can be uplifted on ${M}_{5}$ to obtain an exact solution of type IIB supergravity. We also show that for general $Ad{S}_{4}\ifmmode\times\else\texttimes\fi{}S{E}_{7}$ solutions, where $S{E}_{7}$ is a seven-dimensional Sasaki-Einstein manifold, and for a general class of supersymmetric solutions that are a warped product of $Ad{S}_{4}$ with a seven-dimensional manifold ${N}_{7}$, there is an analogous consistent reduction to minimal $D=4$ gauged supergravity.
35297248
String Theory Origin of Dyonic N=8 Supergravity and Its Chern-Simons Duals.
We clarify the higher-dimensional origin of a class of dyonic gaugings of D=4  N=8 supergravity recently discovered, when the gauge group is chosen to be ISO(7). This dyonically gauged maximal supergravity arises from consistent truncation of massive IIA supergravity on S^6, and its magnetic coupling constant descends directly from the Romans mass. The critical points of the supergravity uplift to new four-dimensional anti-de Sitter space (AdS4) massive type IIA vacua. We identify the corresponding three-dimensional conformal field theory (CFT3) duals as super-Chern-Simons-matter theories with simple gauge group SU(N) and level k given by the Romans mass. In particular, we find a critical point that uplifts to the first explicit N=2 AdS4 massive IIA background. We compute its free energy and that of the candidate dual Chern-Simons theory by localization to a solvable matrix model, and find perfect agreement. This provides the first AdS4/CFT3 precision match in massive type IIA string theory.
120581473
Massive N=2a Supergravity in Ten-Dimensions
A new version of the non-chiral N = 2a supergravity in ten dimensions is obtained, in which the two-index tensor field of the theory “eats” the single vector field and acquires a mass in a Higgs-type mechanism. The new theory, although it contains no fundamental vectors, bears many formal resemblances to gauged supergravities (in particular, the recently constructed F(4) theory in six dimensions). The scalar potential has no extrema, but nevertheless the classical equations of motion admit a wide variety of spontaneous compactifications, many to four dimensions.
118558186
Symplectic deformations of gauged maximal supergravity
A bstractWe identify the space of symplectic deformations of maximal gauged supergravity theories. Coordinates of such space parametrize inequivalent supergravity models with the same gauge group. We apply our procedure to the SO(8) gauging, extending recent analyses. We also study other interesting cases, including Cremmer-Scherk-Schwarz models and gaugings of groups contained in SL(8, ℝ$$ \mathbb{R} $$) and in SU*(8).
96426319
Dyonic ISO(7) supergravity and the duality hierarchy
A bstractMotivated by its well defined higher dimensional origin, a detailed study of D = 4 N=8$$ \mathcal{N}=8 $$ supergravity with a dyonically gauged ISO7=SO7⋉ℝ7$$ \mathrm{I}\mathrm{S}\mathrm{O}(7)=\mathrm{SO}(7)\ltimes {\mathbb{R}}^7 $$ gauge group is performed. We write down the Lagrangian and describe the tensor and duality hierarchies, focusing on an interesting subsector with closed field equations and supersymmetry transformations. We then truncate the N=8$$ \mathcal{N}=8 $$ theory to some smaller sectors with N=2$$ \mathcal{N}=2 $$ and N=1$$ \mathcal{N}=1 $$ supersymmetry and SU(3), G2 and SO(4) bosonic symmetry. Canonical and superpotential formulations for these sectors are given, and their vacuum structure and spectra is analysed. Unlike the purely electric ISO(7) gauging, the dyonic gauging displays a rich structure of vacua, all of them AdS. We recover all previously known ones and find a new N=1$$ \mathcal{N}=1 $$ vacuum with SU(3) symmetry and various non-supersymmetric vacua, all of them stable within the full N=8$$ \mathcal{N}=8 $$ theory.
120651374
The Embedding of Gauged $N=8$ Supergravity Into $d=11$ Supergravity
Abstract We investigate several aspects of the embedding of gauged N = 8 supergravity into d = 11 supergravity and give the full nonlinear metric ansatz for this embedding. This allows us to rederive the solutions with SO(7) ± and SU(4) − symmetry directly from the critical points of the d = 4 theory as well as a new solution with G 2 invariance and N = 1 supersymmetry. We discuss the geometrical aspects of our results and their implications for the interpretation of spontaneously broken solutions.
17042692
New Vacua of Gauged N=8 Supergravity
We analyze a particular SU(2) invariant sector of the scalar manifold of gauged N=8 supergravity in five dimensions, and find all the critical points of the potential within this sector. The critical points give rise to Anti-de Sitter vacua, and preserve at least an SU(2) gauge symmetry. Consistent truncation implies that these solutions correspond to Anti-de Sitter compactifications of IIB supergravity, and hence to possible near-horizon geometries of 3-branes. Thus we find new conformal phases of softly broken N=4 Yang--Mills theory. One of the critical points preserves N=2 supersymmetry in the bulk and is therefore completely stable, and corresponds to an N=1 superconformal fixed point of the Yang--Mills theory. The corresponding renormalization group flow from the N=4 point has c_{IR}/c_{UV} = 27/32. We also discuss the ten-dimensional geometries corresponding to these critical points.
14082098
N=2 supersymmetric RG flows and the IIB dilaton
We show that there is a non-trivial relationship between the dilaton of IIB supergravity, and the coset of scalar fields in five-dimensional, gauged N=8 supergravity. This has important consequences for the running of the gauge coupling in massive perturbations of the AdS/CFT correspondence. We conjecture an exact analytic expression for the ten-dimensional dilaton in terms of five-dimensional quantities, and we test this conjecture. Specifically, we construct a family of solutions to IIB supergravity that preserve half of the supersymmetries, and are lifts of supersymmetric flows in five-dimensional, gauged N=8 supergravity. Via the AdS/CFT correspondence these flows correspond to softly broken N=4, large N Yang–Mills theory on part of the Coulomb branch of N=2 supersymmetric Yang–Mills. Our solutions involve non-trivial backgrounds for all the tensor gauge fields as well as for the dilaton and axion.
120400112
d = 11 supergravity with local SU(8) invariance
The transformation rules and field equations of d = 11 supergravity are given in a form which is manifestly covariant under local SU(8) transformations. In this formulation the SO(1, 10) local Lorentz group is replaced by SO(1, 3) × SU(8). The spinless fields are described by covariant quantities which constitute the 56 and 133 representations of E7(+7).
7298296
The maximal D = 4 supergravities
All maximal supergravities in four space-time dimensions are presented. The ungauged Lagrangians can be encoded in an E_7(7)\Sp(56,R)/GL(28) matrix associated with the freedom of performing electric/magnetic duality transformations. The gauging is defined in terms of an embedding tensor which encodes the subgroup of E_7(7) that is realized as a local invariance. This embedding tensor may imply the presence of magnetic charges which require corresponding dual gauge fields. The latter can be incorporated by using a recently proposed formulation that involves tensor gauge fields in the adjoint representation of E_7(7). In this formulation the results take a universal form irrespective of the electric/magnetic duality basis. We present the general class of supersymmetric and gauge invariant Lagrangians and discuss a number of applications.
18603303
Gauged Supergravities, Tensor Hierarchies, and M-Theory
Deformations of maximal supergravity theories induced by gauging non-abelian subgroups of the duality group reveal the presence of charged M-theory degrees of freedom that are not necessarily contained in supergravity. The relation with M-theory degrees of freedom is confirmed by the representation assignments under the duality group of the gauge charges and the ensuing vector and tensor gauge fields. The underlying hierarchy of these gauge fields is required for consistency of general gaugings. As an example gauged maximal supergravity in three space-time dimensions is presented in a version where all possible tensor fields appear.
13880201
Gauge Theories, Duality Relations and the Tensor Hierarchy
We compute the complete 3- and 4-dimensional tensor hierarchies, i.e. sets of p-form fields, with 1 ≤ p ≤ D, which realize an off-shell algebra of bosonic gauge transformations. We show how these tensor hierarchies can be put on-shell by introducing a set of duality relations, thereby introducing additional scalars and a metric tensor. These so-called duality hierarchies encode the equations of motion of the bosonic part of the most general gauged supergravity theories in those dimensions, including the (projected) scalar equations of motion. We construct gauge-invariant actions that include all the fields in the tensor hierarchies. We elucidate the relation between the gauge transformations of the p-form fields in the action and those of the same fields in the tensor hierarchy.
6217736
Consistent supersymmetric Kaluza-Klein truncations with massive modes
We construct consistent Kaluza-Klein reductions of D = 11 supergravity to four dimensions using an arbitrary seven-dimensional Sasaki-Einstein manifold. At the level of bosonic fields, we extend the known reduction, which leads to minimal N = 2 gauged supergravity, to also include a multiplet of massive fields, containing the breathing mode of the Sasaki-Einstein space, and still consistent with N = 2 supersymmetry. In the context of flux compactifications, the Sasaki-Einstein reductions are generalizations of type IIA SU(3)-structure reductions which include both metric and form-field flux and lead to a massive universal tensor multiplet. We carry out a similar analysis for an arbitrary weak G2 manifold leading to an N = 1 supergravity with massive fields. The straightforward extension of our results to the case of the seven-sphere would imply that there is a four-dimensional Lagrangian with N = 8 supersymmetry containing both massless and massive spin two fields. We use our results to construct solutions of M-theory with non-relativistic conformal symmetry.
118476654
Tri-Sasakian consistent reduction
A bstractWe establish a universal consistent Kaluza-Klein truncation of M-theory based on seven-dimensional tri-Sasakian structure. The four-dimensional truncated theory is an $ \mathcal{N} $ = 4 gauged supergravity with three vector multiplets and a non-abelian gauge group, containing the compact factor SO(3). Consistency follows from the fact that our truncation takes exactly the same form as a left-invariant reduction on a specific coset manifold, and we show that the same holds for the various universal consistent truncations recently put forward in the literature. We describe how the global symmetry group SL(2, $ \mathbb{R} $) × SO(6, 3) is embedded in the symmetry group E7(7) of maximally supersymmetric reductions, and make the connection with the approach of Exceptional Generalized Geometry. Vacuum AdS4 solutions spontaneously break the amount of supersymmetry from $ \mathcal{N} $ = 4 to $ \mathcal{N} $ = 3, 1 or 0, and the spectrum contains massive modes. We find a subtruncation to minimal $ \mathcal{N} $ = 3 gauged supergravity as well as an $ \mathcal{N} $ = 1 subtruncation to the SO(3)-invariant sector. We also show that a reduction on the homogeneous space N010 enhances the universal tri-Sasakian truncation with a Betti vector multiplet.
119288709
Type IIB supergravity on squashed Sasaki-Einstein manifolds
We provide a consistent $ \mathcal{N} = 4 $ Kaluza-Klein truncation of type IIB supergravity on general 5-dimensional squashed Sasaki-Einstein manifolds. Our reduction ansatz keeps all and only the supergravity modes dual to the universal gauge sector of the associated conformal theories, via the gauge/gravity correspondence. The reduced 5-dimensional model displays remarkable features: it includes both zero-modes as well as massive iterations of the Kaluza-Klein operators on the internal manifold; it contains tensor fields dual to vectors charged under a non-abelian gauge group; it has a scalar potential with a non-supersymmetric AdS vacuum in addition to the supersymmetric one.
35701935
Universal Kaluza-Klein reductions of type IIB to N = 4 supergravity in five dimensions
We construct explicit consistent Kaluza–Klein reductions of type IIB supergravity on HK4 ×S1, where HK4 is an arbitrary four-dimensional hyper-Kähler manifold, and on SE5, an arbitrary five-dimensional Sasaki–Einstein manifold. In the former case we obtain the bosonic action of D = 5 N = 4 (ungauged) supergravity coupled to two vector multiplets. For the SE5 case we extend a known reduction, which leads to minimal D = 5 N = 2 gauged supergravity, to also include a multiplet of massive fields, containing the breathing mode of the SE5. We show that the resulting D = 5 action is also consistent with N = 4 gauged supergravity coupled to two vector multiplets. This theory has a supersymmetric AdS5 vacuum, which uplifts to the class of supersymmetric AdS5 × SE5 solutions, that spontaneously breaks N = 4 to N = 2, and also a non-supersymmetric AdS5 vacuum which uplifts to a class of solutions first found by Romans.
119207421
Consistent massive truncations of type-IIB supergravity on Sasaki-Einstein manifolds
Recent work on holographic superconductivity and gravitational duals of systems with nonrelativistic conformal symmetry have made use of consistent truncations of D = 10 and D = 11 supergravity retaining some massive modes in the Kaluza-Klein tower. In this paper we focus on reductions of type-IIB supergravity to five dimensions on a Sasaki-Einstein manifold, and extend these previous truncations to encompass the entire bosonic sector of gauged D = 5, N = 2 supergravity coupled to massive multiplets up to the second Kaluza-Klein level. We conjecture that a necessary condition for the consistency of massive truncations is to only retain the lowest modes in the massive trajectories of the Kaluza-Klein mode decomposition of the original fields. This is an extension of the well-known result that consistent truncations may be obtained by restricting to the singlet sector of the internal symmetry group.
6197309
Nearly Kähler reduction
We consider compactification of type IIA supergravity on nearly Kahler manifolds. These represent a simple class of SU(3) structure manifolds which includes S6 and 3. We exhibit for the first time an explicit reduction ansatz in this context, obtaining an = 2 gauged supergravity in 4d with a single vector and hypermultiplet. We verify that supersymmetric solutions of the 4d theory lift to 10d solutions. Along the way, we discuss questions related to encountering both electric and magnetic charges in the 4d theory.
119102457
Fermions and D = 11 supergravity on squashed Sasaki-Einstein manifolds
We discuss the dimensional reduction of fermionic modes in a recently found class of consistent truncations of D = 11 supergravity compactified on squashed seven-dimensional Sasaki-Einstein manifolds. Such reductions are of interest, for example, in that they have (2 + 1)-dimensional holographic duals, and the fermionic content and their interactions with charged scalars are an important aspect of their applications. We derive the lower-dimensional equations of motion for the fermions and exhibit their couplings to the various bosonic modes present in the truncations under consideration, which most notably include charged scalar and form fields. We demonstrate that our results are consistent with the expected supersymmetric structure of the lower dimensional theory, and apply them to a specific example which is relevant to the study of (2 + 1)-dimensional holographic superconductors.
118431220
Fermions and type IIB supergravity on squashed Sasaki-Einstein manifolds
We discuss the dimensional reduction of fermionic modes in a recently found class of consistent truncations of type IIB supergravity compactified on squashed five-dimensional Sasaki-Einstein manifolds. We derive the lower dimensional equations of motion and effective action, and comment on the supersymmetry of the resulting theory, which is consistent with N = 4 gauged supergravity in d = 5, coupled to two vector multiplets. We compute fermion masses by linearizing around two AdS5 vacua of the theory: one that breaks N = 4 down to N = 2 spontaneously, and a second one which preserves no supersymmetries. The truncations under consideration are noteworthy in that they retain massive modes which are charged under the U(1) R-symmetry, a feature that makes them interesting for applications to condensed matter phenomena via gauge/gravity duality. In this light, as an application of our general results we exhibit the coupling of the fermions to the type IIB holographic superconductor, and find a consistent further truncation of the fermion sector that retains a single spin-1/2 mode.
118600049
Supersymmetric massive truncations of type IIB supergravity on Sasaki-Einstein manifolds
Motivated by recent interest in applications of the AdS/CFT correspondence to condensed matter applications involving fermions, we present the supersymmetric completion of the recent massive truncations of IIB supergravity on Sasaki-Einstein manifolds. In particular, we reduce the fermionic sector of IIB supergravity to obtain five-dimensional N = 2 supergravity coupled to one hypermultiplet and one massive vector multiplet. The supersymmetry transformations and equations of motion are presented and analyzed. Finally, a particularly interesting truncation to N = 2 supergravity coupled to a single hypermultiplet is presented which is the supersymmetric completion of the recently constructed bosonic theory dual to a 3 + 1 dimensional system exhibiting a superconducting phase transition.
8513117
Exploiting N=2 in consistent coset reductions of type IIA
Abstract We study compactifications of type IIA supergravity on cosets exhibiting SU ( 3 ) structure. We establish the consistency of the truncation based on left-invariance, providing a justification for the choice of expansion forms which yields gauged N = 2 supergravity in four dimensions. We explore N = 1 solutions of these theories, emphasizing the requirements of flux quantization, as well as their non-supersymmetric companions. In particular, we obtain a no-go result for de Sitter solutions at string tree level, and, exploiting the enhanced leverage of the N = 2 setup, provide a preliminary analysis of the existence of de Sitter vacua at all string loop order.
119280480
All homogeneous N = 2 M-theory truncations with supersymmetric AdS4 vacua
A bstractWe study consistent truncations of M-theory to gauged N = 2 supergravity in four dimensions, based on a large class of SU(3)-structures in seven dimensions. We show that the gauging involves isometries of the vector multiplet scalar manifold as well as the Heisenberg algebra and a special isometry of the hyperscalar manifold. As a result, non-abelian gauge groups and new non-trivial scalar potentials are generated. Then we specialize to all homogeneous SU(3)-structures supporting supersymmetric AdS4 vacua. These are the Stiefel manifold V5,2, the Aloff-Wallach spaces N (k, l), the seven-sphere (seen as SU(4)/SU(3) or Sp(2)/Sp(1)) and the M110 and Q111 coset spaces. For each of these cases, we describe in detail the N = 2 model and discuss its peculiarities.
118594356
Supergravity instabilities of non-supersymmetric quantum critical points
Motivated by the recent use of certain consistent truncations of M-theory to study condensed matter physics using holographic techniques, we study the SU(3)-invariant sector of four-dimensional, gauged supergravity and compute the complete scalar spectrum at each of the five non-trivial critical points. We demonstrate that the smaller SU(4)− sector is equivalent to a consistent truncation studied recently by various authors and find that the critical point in this sector, which has been proposed as the ground state of a holographic superconductor, is unstable due to a family of scalars that violate the Breitenlohner–Freedman bound. We also derive the origin of this instability in 11 dimensions and comment on the generalization to other embeddings of this critical point which involve arbitrary Sasaki–Einstein seven manifolds. In the spirit of a resurging interest in consistent truncations, we present a formal treatment of the SU(3)-invariant sector as a U(1) × U(1) gauged supergravity theory coupled to one hypermultiplet.
119224823
Supersymmetric Janus solutions in five and ten dimensions
We explicitly truncate $ \mathcal{N} = 8 $ gauged supergravity in five dimensions to its SU(3)-invariant sector with dilaton and axion fields. We show that this truncation has a solution which is identical to the super Janus constructed in $ \mathcal{N} = 2 $ gauged supergravity in five dimensions. Then we lift the solution of the SU(3)-invariant truncation to type IIB supergravity by employing the consistent truncation ansatz. We show that the lifted solution falls into a special case of the supersymmetric Janus solutions constructed in type IIB supergravity. Additionally, we also prove that the lifted solution provides a particular example of the consistent truncations of type IIB supergravity on Sasaki-Einstein manifolds.
118492339
Consistent truncation of d = 11 supergravity on AdS4 × S7
A bstractWe study the system of equations derived twenty five years ago by B. de Wit and the first author [Nucl. Phys. B281 (1987) 211] as conditions for the consistent truncation of eleven-dimensional supergravity on AdS4 × S7 to gauged $ \mathcal{N} = 8 $ supergravity in four dimensions. By exploiting the E7(7) symmetry, we determine the most general solution to this system at each point on the coset space E7(7)/SU(8). We show that invariants of the general solution are given by the fluxes in eleven-dimensional supergravity. This allows us to both clarify the explicit non-linear ansätze for the fluxes given previously and to fill a gap in the original proof of the consistent truncation. These results are illustrated with several examples.
119158860
AdS4 solutions of massive IIA from dyonic ISO(7) supergravity
A bstractExplicit formulae are given for the consistent truncation of massive type IIA supergravity on the six-sphere to the SU(3)-invariant sector of D = 4 N=8$$ \mathcal{N}=8 $$ supergravity with dyonic ISO(7) gauging. These formulae are then used to construct AdS4 solutions of massive type IIA via uplift on S6 of the critical points of the D = 4 supergravity with at least SU(3) symmetry. We find a new N=1$$ \mathcal{N}=1 $$ solution with SU(3) symmetry, a new non-supersymmetric solution with SO(6) symmetry, and recover previously known solutions. We quantise the fluxes, calculate the gravitational free energies of the solutions and discuss the stability of the non-supersymmetric ones. Among these, a (previously known) G2-invariant solution is found to be stable.
59335510
Consistent 4-form fluxes for maximal supergravity
A bstractWe derive new ansätze for the 4-form field strength of D = 11 supergravity corresponding to uplifts of four-dimensional maximal gauged supergravity. In particular, the ansätze directly yield the components of the 4-form field strength in terms of the scalars and vectors of the four-dimensional maximal gauged supergravity — in this way they provide an explicit uplift of all four-dimensional consistent truncations of D = 11 supergravity. The new ansätze provide a substantially simpler method for uplifting d = 4 flows compared to the previously available method using the 3-form and 6-form potential ansätze. The ansatz for the Freund-Rubin term allows us to conjecture a ‘master formula’ for the latter in terms of the scalar potential of d = 4 gauged supergravity and its first derivative. We also resolve a long-standing puzzle concerning the antisymmetry of the flux obtained from uplift ansätze.
4682703
New vacua for type II string theory
Abstract Lorentz-invariant expectation values for antisymmetric tensor field strengths in Calabi-Yau compactification of IIA string theory are considered. These are found to impart magnetic and/or electric charges to the dilation hypermultiplet. This results in a potential which can have supersymmetric minima at zero coupling or at conifold points in the moduli space. The latter occurs whenever the dilaton charge is aligned with that of the light black hole at the conifold. It is shown that there is a flat direction extending from the conifold along which there is a black hole condensate whose strength is of order the string coupling g s . It is speculated that these new vacua correspond to string compactification on generalized Calabi-Yau spaces which have c 1 = 0 but are not Kahler.
14411532
Gauged Six-Dimensional Supergravity from Massive Type IIA String Theory
We obtain the complete nonlinear Kaluza-Klein ansatz for the reduction of the bosonic sector of massive type IIA supergravity to the Romans F(4) gauged supergravity in six dimensions. The latter arises as a consistent warped S{sup 4} reduction. (c) 1999 The American Physical Society.
53538737
Generalised geometry from the ground up
A bstractExtending previous work on generalised geometry, we explicitly construct an E7(7)-valued vielbein in eleven dimensions that encompasses the would be scalar bosonic degrees of freedom after reduction to four dimensions of D = 11 supergravity, by identifying new “generalised vielbeine” in eleven dimensions associated with the dual 6-form potential and the dual graviton. By maintaining full on-shell equivalence with the original theory at every step our construction furnishes the non-linear ansatz for the dual (magnetic) 7-form flux for any non-trivial compactification of D = 11 supergravity, complementing the known non-linear ansätze for the metric and the 4-form flux. A preliminary analysis of the generalised vielbein postulate for the new vielbein components reveals tantalising hints of new structures beyond D = 11 supergravity and ordinary space-time covariancxe, and also points to the possible D = 11 origins of the embedding tensor. We discuss the extension of these results to E8(8).
53127406
E d ( d ) × R + generalised geometry , connections and M theory
We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a d-dimensional manifold for all d ≤ 7. The theory is based on an extended tangent space which admits a natural Ed(d) × R+ action. The bosonic degrees of freedom are unified as a “generalised metric”, as are the diffeomorphism and gauge symmetries, while the local O(d) symmetry is promoted to Hd, the maximally compact subgroup of Ed(d). We introduce the analogue of the Levi-Civita connection and the Ricci tensor and show that the bosonic action and equations of motion are simply given by the generalised Ricci scalar and the vanishing of the generalised Ricci tensor respectively. The formalism also gives a unified description of the bosonic NSNS and RR sectors of type II supergravity in d − 1 dimensions. Locally the formulation also describes M-theory variants of double field theory and we derive the corresponding section condition in general dimension. We comment on the relation to other approaches to M theory with Ed(d) symmetry, as well as the connections to flux compactifications and the embedding tensor formalism.
53723555
Supergravity as generalised geometry II: Ed(d) × ℝ+ and M theory
A bstractWe reformulate eleven-dimensional supergravity, including fermions, in terms of generalised geometry, for spacetimes that are warped products of Minkowski space with a d-dimensional manifold M with d ≤ 7. The reformulation has an Ed(d) × ℝ+ structure group and it has a local $ {{\widetilde{H}}_d} $ symmetry, where $ {{\widetilde{H}}_d} $ is the double cover of the maximally compact subgroup of Ed(d). The bosonic degrees for freedom unify into a generalised metric, and, defining the generalisked analogue D of the Levi-Civita connection, one finds that the corresponding equations of motion are the vanishing of the generalised Ricci tensor. To leading order, we show that the fermionic equations of motion, action and supersymmetry variations can all be written in terms of D. Although we will not give the detailed decompositions, this reformulation is equally applicable to type IIA or IIB supergravity restricted to a (d − 1)-dimensional manifold. For completeness we give explicit expressions in terms of $ {{\widetilde{H}}_4} $ = Spin(5) and $ {{\widetilde{H}}_7} $ = SU(8) representations for d = 4 and d = 7.
28121295
Exceptional form of D=11 supergravity.
Eleven-dimensional supergravity reveals large exceptional symmetries upon reduction, in accordance with the U-duality groups of M theory, but their higher-dimensional geometric origin has remained a mystery. In this Letter, we show that D=11 supergravity can be extended to be fully covariant under the exceptional groups E(n(n)), n=6, 7, 8. Motivated by a similar formulation of double field theory we introduce an extended "exceptional spacetime." We illustrate the construction by giving the explicit E(6(6)) covariant form: the full D=11 supergravity, in a 5+6 splitting of coordinates but without truncation, embeds into an E(6(6)) covariant 5+27 dimensional theory. We argue that this covariant form likewise comprises type IIB supergravity.
119001157
Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB
We present the details of the recently constructed $E_{6(6)}$ covariant extension of 11-dimensional supergravity. This theory requires a 5+27 dimensional spacetime in which the `internal' coordinates transform in the $\bar{\bf 27}$ of $E_{6(6)}$. All fields are $E_{6(6)}$ tensors and transform under (gauged) internal generalized diffeomorphisms. The `Kaluza-Klein' vector field acts as a gauge field for the $E_{6(6)}$ covariant `E-bracket' rather than a Lie bracket, requiring the presence of two-forms akin to the tensor hierarchy of gauged supergravity. We construct the complete and unique action that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. The theory is subject to covariant section constraints on the derivatives, implying that only a subset of the extra 27 coordinates is physical. We give two solutions of the section constraints: the first preserves GL(6) and embeds the action of the complete (i.e. untruncated) 11-dimensional supergravity; the second preserves GL(5) x SL(2) and embeds complete type IIB supergravity. As a by-product, we thus obtain an off-shell action for type IIB supergravity.
119266419
Exceptional field theory. II. E$_{7(7)}$
We introduce the exceptional field theory for the group E 7 ( 7 ) , based on a ( 4 + 56 ) -dimensional spacetime subject to a covariant section condition. The “internal” generalized diffeomorphisms of the coordinates in the fundamental representation of E 7 ( 7 ) are governed by a covariant “E-bracket,” which is gauged by 56 vector fields. We construct the complete and unique set of field equations that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. Among them are featured the non-Abelian twisted self-duality equations for the 56 gauge vectors. We discuss the explicit solutions of the section condition describing the embedding of the full, untruncated 11-dimensional and type IIB supergravity, respectively. As a new feature compared to the previously constructed E 6 ( 6 ) formulation, some components among the 56 gauge vectors descend from the 11-dimensional dual graviton but nevertheless allow for a consistent coupling by virtue of a covariantly constrained compensating 2-form gauge field.
6790887
The maximal D = 5 supergravities
Abstract The general Lagrangian for maximal supergravity in five spacetime dimensions is presented with vector potentials in the 27 ¯ and tensor fields in the 27 representation of E 6 ( 6 ) . This novel tensor–vector system is subject to an intricate set of gauge transformations, describing 3 ( 27 − t ) massless helicity degrees of freedom for the vector fields and 3 t massive spin degrees of freedom for the tensor fields, where the (even) value of t depends on the gauging. The kinetic term of the tensor fields is accompanied by a unique Chern–Simons coupling which involves both vector and tensor fields. The Lagrangians are completely encoded in terms of the embedding tensor which defines the E 6 ( 6 ) subgroup that is gauged by the vectors. The embedding tensor is subject to two constraints which ensure the consistency of the combined vector–tensor gauge transformations and the supersymmetry of the full Lagrangian. This new formulation encompasses all possible gaugings.
46984312
Uplifting non-compact gauged supergravities
A bstractWe provide the M-theory uplift of de Sitter vacua of SO(5,3) and SO(4,4) gaugings of maximal supergravity in 4 dimensions. We find new non-compact backgrounds that are squashed hyperboloids with non-trivial flux for the 3-form potential. The uplift requires a new non-linear ansatz for the 11-dimensional metric and for the 3-form potential that reduces to the known one leading to the 7-sphere solution in the case of the SO(8) gauging.
118956521
Superconformal Chern-Simons theories
A Lagrangian description of a maximally supersymmetric conformal field theory in three dimensions was constructed recently by Bagger and Lambert (BL). The BL theory has SO(4) gauge symmetry and contains scalar and spinor fields that transform as 4-vectors. We verify that this theory has OSp(8|4) superconformal symmetry and that it is parity conserving despite the fact that it contains a Chern-Simons term. We describe several unsuccessful attempts to construct theories of this type for other gauge groups and representations. This experience leads us to conjecture the uniqueness of the BL theory. Given its large symmetry, we expect this theory to play a significant role in the future development of string theory and M-theory.
118234305
D2-brane Chern-Simons theories: F -maximization = a-maximization
A bstractWe study a system of N D2-branes probing a generic Calabi-Yau three-fold singularity in the presence of a non-zero quantized Romans mass n. We argue that the low-energy effective cN=2$$ \mathcal{N}=2 $$ Chern-Simons quiver gauge theory flows to a superconformal fixed point in the IR, and construct the dual AdS4 solution in massive IIA supergravity. We compute the free energy F of the gauge theory on S3 using localization. In the large N limit we find F = c (nN )1/3a2/3, where c is a universal constant and a is the a-function of the “parent” four-dimensional N=1$$ \mathcal{N}=1 $$ theory on N D3-branes probing the same Calabi-Yau singularity. It follows that maximizing F over the space of admissible R-symmetries is equivalent to maximizing a for this class of theories. Moreover, we show that the gauge theory result precisely matches the holographic free energy of the supergravity solution, and provide a similar matching of the VEV of a BPS Wilson loop operator.
15926321
Maximal supergravity from IIB flux compactifications
Abstract Using a recently proposed group-theoretical approach, we explore novel gaugings of maximal supergravity in four dimensions with gauge group embeddings that can be generated by fluxes of IIB string theory. The corresponding potentials are positive without stationary points. Some allow domain wall solutions which can be elevated to ten dimensions. Appropriate truncations describe type-IIB flux compactifications on T 6 orientifolds leading to non-maximal, four-dimensional, supergravities.