<p>Maximal supergravities in ten and eleven dimensions admit consistent truncations on particular spheres to maximal supergravities in lower dimensions. Concurrently, the truncation to singlets under any subgroup of the sphere isometry group leads to consistent truncations with less or no supersymmetry. We review the relation between these truncations in the framework of exceptional field theory. As an application, we derive three new <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math>-invariant solutions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mo>=</mo><mn>11</mn></math> supergravity. Their geometry is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>×</mo><msub><mi mathvariant="normal">Σ</mi><mn>7</mn></msub></mrow></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Σ</mi><mn>7</mn></msub></math> is a deformed seven-sphere, preserving SO(7) isometries.</p>

France

Bastien Duboeuf

Australia

Michele Galli

United Kingdom

Emanuel Malek

Henning Samtleben

Consistent truncations and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math>-invariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math> solutions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mo>=</mo><mn>11</mn></math> supergravity

We show how to use Exceptional Field Theory to efficiently compute n-point couplings of all Kaluza-Klein modes for vacua that can be uplifted from maximal gauged supergravities to 10/11 dimensions via a consistent truncation. Via the AdS/CFT correspondence, these couplings encode the n-point functions of holographic conformal fields theories. Our methods show that these n-point couplings are controlled by the n-point invariant of scalar harmonics of the maximally symmetric point of the truncation, allowing us to show that infinitely-many n-point couplings vanish for any vacua of the truncation, even though they may be allowed by the remnant symmetry group of the vacua. This gives new results even for the maximally supersymmetric AdS5 × S 5, AdS4 × S 7 and AdS7 × S 4 vacua of string and M-theory, where we prove old conjectures about the vanishing of n-point extremal and near-extremal couplings.

Germany

Cubic and higher-order supergravity couplings for AdS vacua using Exceptional Field Theory

We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to D ≥ 5 dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted Y , that provides exactly the necessary properties to also describe compactifications to D = 4 dimensions. We classify “M-exact” E 7-algebroids and show that this precisely matches the form of the generalised tangent space of E 7(7) × ℝ+-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids.

Belgium

Ondřej Hulík

Fridrich Valach

Daniel Waldram

Y-algebroids and E 7(7) × ℝ+-generalised geometry

We use E 8(8) Exceptional Field Theory to construct the consistent truncation of IIB supergravity on S 3 × S 3 × S 1 to maximal 3-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 16 gauged supergravity containing the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (4, 4) AdS3 vacuum. We explain how to achieve this by adding a 7-form flux to the S 1 reduction of the dyonic E 7(7) truncation on S 3 × S 3 previously constructed in the literature. Our truncation Ansatz includes, in addition to the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (4, 4) vacuum, a host of moduli breaking some or all of the supersymmetries. We explicitly construct the uplift of a subset of these to construct new supersymmetric and non-supersymmetric AdS3 vacua of IIB string theory, which include a range of perturbatively stable non-supersymmetric 10-d vacua. Moreover, we show how the supersymmetric direction of the moduli space of AdS3 vacua of six-dimensional gauged supergravity studied in [1] is compactified upon lifting to 10 dimensions, and find evidence of T-duality playing a role in global aspects of the moduli space. Along the way, we also derive the form of 3-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 16 gauged supergravity in terms of the embedding tensor and rule out a 10-/11-dimensional origin of some 3-dimensional gauged supergravities.

Camille Eloy

Adding fluxes to consistent truncations: IIB supergravity on AdS3 × S 3 × S 3 × S 1

We construct and analyze the domain-wall solution in <math><mi>D</mi><mo>=</mo><mn>11</mn></math> supergravity connecting the <math><mi>N</mi><mo>=</mo><mn>1</mn></math>, <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>×</mo><msubsup><mi>S</mi><mtext>squashed</mtext><mn>7</mn></msubsup></mrow></math> vacuum to the <math><mi>N</mi><mo>=</mo><mn>8</mn></math>, <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>×</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mtext>round</mtext></mrow><mrow><mn>7</mn></mrow></msubsup></mrow></math> vacuum. This domain wall describes the holographic renormalization group flow from an <math><mi>Sp</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>×</mo><mi>Sp</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> symmetric UV fixed point to the SO(8) symmetric IR fixed point. It breaks all supersymmetries which are (partially) restored at its endpoints. We show how recent techniques from exceptional field theory allow us to compute the quadratic couplings of all Kaluza-Klein (KK) fluctuations around the domain wall background, encoding all two-point correlators along the holographic RG flow.

Holographic RG flow from the squashed to the round <math><msup><mi>S</mi><mn>7</mn></msup></math>

We show how to use Exceptional Field Theory to compute the full Kaluza-Klein spectra of 10- and 11-dimensional supergravity around deformations of backgrounds of maximal gauged supergravity by scalar modes that do not form part of the consistent truncation. This includes deformations of AdS4 × S 7 and AdS5 × S 5 by modes that are not part of the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 8 supermultiplet. As an application, we compute the full Kaluza-Klein spectrum of the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 and   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 0 squashed S 7. In this example, all conformal dimensions are captured by a universal formula in terms of the Casimir operators and additional quantum numbers which organise the spectrum.

Kaluza-Klein spectrometry beyond consistent truncations: the squashed S 7

We show how to construct consistent truncations of 10-/11-dimensional supergravity to 3-dimensional gauged supergravity, preserving various amounts of supersymmetry. We show, that as in higher dimensions, consistent truncations can be defined in terms of generalised G-structures in Exceptional Field Theory, with G ⊂ E 8(8) for the 3-dimensional case. Differently from higher dimensions, the generalised Lie derivative of E 8(8) Exceptional Field Theory requires a set of “covariantly constrained” fields to be well-defined, and we show how these can be constructed from the G-structure itself. We prove several general features of consistent truncations, allowing us to rule out a higher-dimensional origin of many 3-dimensional gauged supergravities. In particular, we show that the compact part of the gauge group can be at most SO(9) and that there are no consistent truncations on a 7-or 8-dimensional product of spheres such that the full isometry group of the spheres is gauged. Moreover, we classify which matter-coupled   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  ≥ 4 gauged supergravities can arise from consistent truncations. Finally, we give several examples of consistent truncations to three dimensions. These include the truncations of IIA and IIB supergravity on S 7 leading to two different   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 16 gauged supergravites, as well as more general IIA/IIB truncations on H  p,7−p . We also show how to construct consistent truncations of IIB supergravity on S 5 fibred over a Riemann surface. These result in 3-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 gauged supergravities with scalar manifold   <math> <mi>M</mi> <mo>=</mo> <mfrac> <mrow> <mi>SO</mi> <mfenced> <mn>6</mn> <mn>4</mn> </mfenced> </mrow> <mrow> <mi>SO</mi> <mfenced> <mn>6</mn> </mfenced> <mo>×</mo> <mi>SO</mi> <mfenced> <mn>4</mn> </mfenced> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mi>SU</mi> <mfenced> <mn>2</mn> <mn>1</mn> </mfenced> </mrow> <mrow> <mi>S</mi> <mfenced> <mrow> <mi>U</mi> <mfenced> <mn>1</mn> </mfenced> <mo>×</mo> <mi>U</mi> <mfenced> <mn>2</mn> </mfenced> </mrow> </mfenced> </mrow> </mfrac> </math>  $$ \mathcal{M}=\frac{\mathrm{S}\mathrm{O}\left(6,4\right)}{\mathrm{S}\mathrm{O}(6)\times \mathrm{SO}(4)}\times \frac{\mathrm{S}\mathrm{U}\left(2,1\right)}{\mathrm{S}\left(\mathrm{U}(1)\times \mathrm{U}(2)\right)} $$  with a ISO(3) × U(1)4 gauging and for hyperboloidal Riemann surfaces contain   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (2, 2) AdS3 vacua.

Consistent truncations to 3-dimensional supergravity

Using exceptional generalised geometry, we classify which five-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 gauged supergravities can arise as a consistent truncation of 10-/11-dimensional supergravity. Exceptional generalised geometry turns the classification into an algebraic problem of finding subgroups G  S  ⊂ USp(8) ⊂ E6(6) that preserve exactly two spinors. Moreover, the intrinsic torsion of the G  S  structure must contain only constant singlets under G  S , and these, in turn, determine the gauging of the five-dimensional theory. The resulting five-dimensional theories are strongly constrained: their scalar manifolds are necessarily symmetric spaces and only a small number of matter multiplets can be kept, which we completely enumerate. We also determine the largest reductive and compact gaugings that can arise from consistent truncations.

Grégoire Josse

Michela Petrini

The higher-dimensional origin of five-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 gauged supergravities

We provide the first holographic evidence for the existence of a nonsupersymmetric conformal manifold arising from exactly marginal but supersymmetry-breaking deformations of a superconformal three-dimensional field theory. In particular, we construct a 2-parameter nonsupersymmetric deformation of a supersymmetric AdS nongeometric background in type IIB string theory. We prove that the nonsupersymmetric backgrounds are perturbatively stable and also do not suffer from various nonperturbative instabilities. Finally, we argue that diffeomorphism symmetry protects our solutions against higher-derivative string corrections.

Italy

Alfredo Giambrone

Spain

Adolfo Guarino

Colin Sterckx

Mario Trigiante

Holographic evidence for nonsupersymmetric conformal manifolds

We study a one-parameter family of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 anti-de Sitter vacua with U(1)2 symmetry of gauged four-dimensional maximal supergravity, with dyonic gauge group [SO(6) × SO(1, 1)] ⋉ ℝ12. These backgrounds are known to correspond to Type IIB S-fold solutions with internal manifold of topology S 1 × S 5. The family of AdS4 vacua is parametrized by a modulus χ. Although χ appears non-compact in the four-dimensional supergravity, we show that this is just an artefact of the four-dimensional description. We give the 10-dimensional geometric interpretation of the modulus and show that it actually has periodicity of   <math> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </math>  $$ \frac{2\pi }{T} $$ , which is the inverse radius of S 1. We deduce this by providing the explicit D = 10 uplift of the family of vacua as well as computing the entire modulus-dependent Kaluza-Klein spectrum as a function of χ. At the special values χ = 0 and χ =   <math> <mfrac> <mi>π</mi> <mi>T</mi> </mfrac> </math>  $$ \frac{\pi }{T} $$ , the symmetry enhances according to U(1)2 → U(2), giving rise however to inequivalent Kaluza-Klein spectra. At χ =   <math> <mfrac> <mi>π</mi> <mi>T</mi> </mfrac> </math>  $$ \frac{\pi }{T} $$ , this realizes a bosonic version of the “space invaders” scenario with additional massless vector fields arising from formerly massive fields at higher Kaluza-Klein levels.