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.

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abstract: A bstractWe study the system of equations derived twenty five years ago by B. de Wit and the first author [ (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.

title: AdS4 solutions of massive IIA from dyonic ISO(7) supergravity
abstract: 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.

title: Consistent 4-form fluxes for maximal supergravity
abstract: 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.

title: 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.

title: Gauged Six-Dimensional Supergravity from Massive Type IIA String Theory
abstract: 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.

title: Generalised geometry from the ground up
abstract: 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).

title: E d ( d ) × R + generalised geometry , connections and M theory
abstract: 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.

title: Supergravity as generalised geometry II: Ed(d) × ℝ+ and M theory
abstract: 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.

title: Exceptional form of D=11 supergravity.
abstract: 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.

title: Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB
abstract: 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.

title: Exceptional field theory. II. E$_{7(7)}$
abstract: 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.

title: 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.

title: Uplifting non-compact gauged supergravities
abstract: 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.

title: Superconformal Chern-Simons theories
abstract: 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.

title: D2-brane Chern-Simons theories: F -maximization = a-maximization
abstract: 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.

title: 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.