Consistent Truncation Research Articles

Physical content of quadratic gravity

We have recently undergone an analysis of gravitational theories as defined in first order formalism, where the metric and the connection are treated as independent fields. The physical meaning of the connection field has historically been somewhat elusive. In this paper, a complete spin analysis of the torsionless connection field is performed, and its consequences are explored. The main properties of a hypothetical consistent truncation of the theory are discussed as well.

Two infinite families of resonant solutions for the Gross-Pitaevskii equation

We consider the two-dimensional Gross-Pitaevskii equation describing a Bose-Einstein condensate in an isotropic harmonic trap. In the small coupling regime, this equation is accurately approximated over long times by the corresponding nonlinear resonant system whose structure is determined by the fully resonant spectrum of the linearized problem. We focus on two types of consistent truncations of this resonant system: first, to sets of modes of fixed angular momentum, and second, to excited Landau levels. Each of these truncations admits a set of explicit analytic solutions with initial conditions parametrized by three complex numbers. Viewed in position space, the fixed angular momentum solutions describe modulated oscillations of dark rings, while the excited Landau level solutions describe modulated precession of small arrays of vortices and antivortices. We place our findings in the context of similar results for other spatially confined nonlinear Hamiltonian systems in recent literature.

Supersymmetric AdS7 and AdS6 vacua and their minimal consistent truncations from exceptional field theory

We show how to construct supersymmetric warped AdS7 vacua of massive IIA and AdS6 vacua of IIB supergravity, using “half-maximal structures” of exceptional field theory. We use this formalism to obtain the minimal consistent truncations around these AdS vacua.

On the de Sitter swampland criterion

A new swampland criterion has recently been proposed. As a consequence, it forbids the existence of de Sitter solutions in a low energy effective theory of a quantum gravity. However, there exist classical de Sitter solutions of ten-dimensional (10d) type II supergravities, even though they are unstable. This appears at first sight in contradiction with the criterion. Beyond possible doubts on the validity of these solutions, we propose two answers to this apparent puzzle. A first possibility is that the known 10d solutions always exhibit an energy scale of order or higher than a Kaluza–Klein scale, as we argue. A corresponding 4d low energy effective theory would then differ from the usual consistent truncations, and as we explain, would not admit a de Sitter solution. This would reconcile the existence of these 10d de Sitter solutions with the 4d criterion. A second, alternative possibility is to have a refined swampland criterion, that we propose. It forbids to have both the existence and the stability of a de Sitter solution, while unstable solutions are still allowed.

Pauli reductions of supergravities in six and five dimensions

The dimensional reduction of a generic theory on a curved internal space such as a sphere does not admit a consistent truncation to a finite set of fields that includes the Yang-Mills gauge bosons of the isometry group. In rare cases, for example the $S^7$ reduction of eleven-dimensional supergravity, such a consistent "Pauli reduction" does exist. In this paper we study this existence question in two examples of $S^2$ reductions of supergravities. We do this by making use of a relation between certain $S^2$ reductions and group manifold $S^3=SU(2)$ reductions of a theory in one dimension higher. By this means we establish the non-existence of a consistent $S^2$ Pauli reduction of five-dimensional minimal supergravity. We also show that a previously-discovered consistent Pauli reduction of six-dimensional Salam-Sezgin supergravity can be elegantly understood via a group-manifold reduction from seven dimensions.

The CST bounce universe model — A parametric study

A bounce universe model with a scale-invariant and stable spectrum of primordial density perturbations was constructed using a consistent truncation of the D-brane dynamics from Type IIB string theory. A coupling was introduced between the tachyon field and the adjoint Higgs field on the D3-branes to lock the tachyon at the top of its potential hill and to model the bounce process, which is known as the Coupled Scalar and Tachyon Bounce (CSTB) Universe. The CSTB model has been shown to be ghost free, and it fulfils the null energy condition; in addition, it can also solve the Big Bang cosmic singularity problem. In this paper we conduct an extensive follow-up study of the parameter space of the CSTB model. In particular we are interested in the parameter values that can produce a single bounce to arrive at a radiation-dominated universe. We further establish that the CSTB universe is a viable alternative to inflation, as it can naturally produce a sufficient number of e-foldings in the locked inflation epoch and in the post-bounce expansion to overcome the four fundamental limitations of the Big Bang cosmology, which are flatness, horizon, homogeneity and singularity, resulting in a universe of the current size.

Magnetically-charged supersymmetric flows of gauged mathcal{N}=8 supergravity in five dimensions

We study magnetically-charged supersymmetric flow equations in a consistent truncation of gauged mathcal{N}=8 supergravity in five dimensions. This truncation gives gauged mathcal{N}=2 supergravity coupled to two vector multiplets and two hypermultiplets. We derive magnetically-charged flow equations of scalar fields from vector and hypermultiplets. It turns out that there could be only up to two nontrivial scalar fields out of eight from the hypermultiplets. Along the way, we recover some known AdS3 solutions of the flow equations.

On the U(1)2-Invariant Sector of Dyonic Maximal Supergravity

Recently a new maximally supersymmetric, dyonically gauged supergravity in four-dimenions has been constructed. This theory admits several supersymmetric AdS solutions, and a Chern- Simons-matter dual theory has been proposed for a solution with an unbroken SU(3) symmetry. The gravity side and the field theory side of the free energy computation show nice agreements at superconformal point. With a view to a study of non-conformal deformations, in this paper we study a consistent truncation of the full theory where SU(3) is broken to U(1)$^2$. We elucidate the geometry of the scalar sector, and also present the superpotentials in both $\mathcal{N} = 2$ and $\mathcal{N} = 1$ supergravity formulation.

Quantum phase space trajectories with application to quantum cosmology

We develop an approach to quantum dynamics based on quantum phase space trajectories. The latter are built from a unitary irreducible representation of the symmetry group of the respective classical phase space. We use a quantum action functional to derive the basic equations. In principle, our formulation is equivalent to the Hilbert space formulation. However, the former allows for consistent truncations to reduced phase spaces in which approximate quantum dynamics can be derived. We believe that our approach can be very useful in the domain of quantum cosmology and therefore, we use the cosmological phase space example to establish the basic equations of this formalism.

Beyond a = c : gravitational couplings to matter and the stress tensor OPE

We derive constraints on the operator product expansion of two stress tensors in conformal field theories (CFTs), both generic and holographic. We point out that in large N CFTs with a large gap to single-trace higher spin operators, the stress tensor sector is not only universal, but isolated: that is, leftlangle TTmathcal{O}rightrangle =0 , where mathcal{O}ne T is a single-trace primary. We show that this follows from a suppression of leftlangle TTmathcal{O}rightrangle by powers of the higher spin gap, Δgap, dual to the bulk mass scale of higher spin particles, and explain why leftlangle TTmathcal{O}rightrangle is a more sensitive probe of Δgap than a − c in 4d CFTs. This result implies that, on the level of cubic couplings, the existence of a consistent truncation to Einstein gravity is a direct consequence of the absence of higher spins. By proving similar behavior for other couplings leftlangle T{mathcal{O}}_1{mathcal{O}}_2rightrangle where {mathcal{O}}_i have spin si ≤ 2, we are led to propose that 1/Δgap is the CFT “dual” of an AdS derivative in a classical action. These results are derived by imposing unitarity on mixed systems of spinning four-point functions in the Regge limit. Using the same method, but without imposing a large gap, we derive new inequalities on these three-point couplings that are valid in any CFT. These are generalizations of the Hofman-Maldacena conformal collider bounds. By combining the collider bound on TT couplings to spin-2 operators with analyticity properties of CFT data, we argue that all three tensor structures of 〈TTT〉 in the free-field basis are nonzero in interacting CFTs.

The 10d uplift of the GPPZ solution

We present the uplift of the GPPZ solution of the five-dimensional maximal supergravity to ten dimensions. The five dimensional solution involves two real scalar fields, with one of them encoding holographically the (norm of the complex) supersymmetri mathcal{N}=1 mass deformation and the other the real part of the gaugino condensate. We embed this solution in a consistent truncation of D = 5 maximal supergravity which involves two complex scalars dual to the complex mass deformations and the complex gaugino condensate, and a U(1) gauge field dual to the U(1)R current, and uplift it to ten dimensions. The ten dimensional solution is completely explicit, with all fields given in terms of elementary functions. The metric and the axion-dilaton agree with those of a partial uplift of the GPPZ flow by Pilch and Warner. We analyze the asymptotics and the singularity structure of the ten dimensional solution. The uplifted solution is singular, but the singularity is milder than that of the five dimensional solution, and there is conformal frame in which the metric is only singular at one point of S5. We compare the asymptotics of the 10d solution with that of the Polchinski-Strassler and Freedman-Minahan solutions, and find agreement with Freedman-Minahan and disagreement with Polchinski-Strassler In particular, we infer that while the Polchinski-Strassler 10d fields satisfy the correct boundary conditions, they do not solve the field equations near the boundary.

Chromo-natural inflation in Supergravity

We present supergravity realizations of chromo-natural inflationary models. We show that by using superpotentials with “imaginary” holomorphic functions of the inflaton one can obtain effective theories of inflation that are also consistent truncations of the original supergravity models.

Supersymmetric solutions of the cosmological, gauged, \u2102 magic model

We construct supersymmetric solutions of theories of gauged mathcal{N} = 1, d = 5 supergravity coupled to vector multiplets with a U(1)R Abelian (Fayet-Iliopoulos) gauging and an independent SU(2) gauging associated to an SU(2) isometry group of the Real Special scalar manifold. These theories provide minimal supersymmetrizations of 5-dimensional SU(2) Einstein-Yang-Mills theories with negative cosmological constant. We consider a minimal model with these gauge groups and the “magic model” based on the Jordan algebra J3ℂ with gauge group SU(3) × U(1)R, which is a consistent truncation of maximal SO(6)-gauged supergravity in d = 5 and whose solutions can be embedded in Type IIB Superstring Theory. We find several solutions containing selfdual SU(2) instantons, some of which asymptote to AdS5 and some of which are very small, supersymmetric, deformations of AdS5. We also show how some of those solutions can be embedded in Romans’ SU(2) × U(1)-gauged half-maximal supergravity, which was obtained by Lu, Pope and Tran by compactification of the Type IIB Superstring effective action. This provides another way of uplifting those solutions to 10 dimensions.

Conformal bootstrap at large charge

We consider unitary CFTs with continuous global symmetries in d > 2. We consider a state created by the lightest operator of large charge Q ≫ 1 and analyze the correlator of two light charged operators in this state. We assume that the correlator admits a well-defined large Q expansion and, relatedly, that the macroscopic (thermodynamic) limit of the correlator exists. We find that the crossing equations admit a consistent truncation, where only a finite number N of Regge trajectories contribute to the correlator at leading nontrivial order. We classify all such truncated solutions to the crossing. For one Regge trajectory N = 1, the solution is unique and given by the effective field theory of a Goldstone mode. For two or more Regge trajectories N ≥ 2, the solutions are encoded in roots of a certain degree N polynomial. Some of the solutions admit a simple weakly coupled EFT description, whereas others do not. In the weakly coupled case, each Regge trajectory corresponds to a field in the effective Lagrangian.

Universal formula for the holographic speed of sound

We consider planar hairy black holes in five dimensions with a real scalar field in the Breitenlohner–Freedman window and derive a universal formula for the holographic speed of sound for any mixed boundary conditions of the scalar field. As an example, we numerically construct the most general class of planar black holes coupled to a single scalar field in the consistent truncation of type IIB supergravity that preserves the SO(3)×SO(3) R-symmetry group of the gauge theory. For this particular family of solutions, we find that the speed of sound exceeds the conformal value. From a phenomenological point of view, the fact that the conformal bound can be violated by choosing the right mixed boundary conditions is relevant for the existence of neutron stars with a certain mass-size relationship for which a large value of the speed of sound codifies a stiff equation of state. In the way, we also shed light on a puzzle regarding the appearance of the scalar charges in the first law. Finally, we generalize the formula of the speed of sound to arbitrary dimensional scalar-metric theories whose parameters lie within the Breitenlohner–Freedman window.

Hairy black holes and duality in an extended supergravity model

We consider a D = 4, mathcal{N}=2 gauged supergravity with an electromagnetic Fayet-Iliopoulos term. We restrict to the uncharged, single dilaton consistent truncation and point out that the bulk Lagrangian is self-dual under electromagnetic duality. Within this truncation, we construct two families of exact hairy black hole solutions, which are asymptotically AdS4. When a duality transformation is applied on these solutions, they are mapped to two other inequivalent families of hairy black hole solutions. The mixed boundary conditions of the scalar field correspond to adding a triple-trace operator to the dual field theory action. We also show that this truncation contains all the consistent single dilaton truncations of gauged mathcal{N}=8 supergravity with a possible ω-deformation.

Complete D=11 embedding of SO(8) supergravity

The truncation formulae of $D=11$ supergravity on $S^7$ to $D=4$ ${\cal N} =8$ SO(8)-gauged supergravity are completed to include the full non-linear dependence of the $D=11$ three-form potential $\hat A_{(3)}$ on the $D=4$ fields, and their consistency is shown. The full embedding into $\hat A_{(3)}$ is naturally expressed in terms of a restricted version, still ${\cal N} =8$ but only SL(8)--covariant, of the $D=4$ tensor hierarchy. The redundancies introduced by this approach are removed at the level of the field strength $\hat F_{(4)}$ by exploiting $D=4$ duality relations. Finally, new expressions for the full consistent truncation formulae are given that are, for the first time, explicit in all $D=4$ fields.

Black holes with halos

We present new AdS4 black hole solutions in gauged supergravity coupled to vector and hypermultiplets. We focus on a particular consistent truncation of M-theory on the homogeneous Sasaki–Einstein seven-manifold M111, characterized by the presence of one Betti vector multiplet. We numerically construct static and spherically symmetric black holes with electric and magnetic charges, corresponding to M2 and M5 branes wrapping non-contractible cycles of the internal manifold. The novel feature characterizing these nonzero temperature configurations is the presence of a massive vector field halo. Moreover, we verify the first law of black hole mechanics and we study the thermodynamics in the canonical ensemble. We analyze the behavior of the massive vector field condensate across the small-large black hole phase transition and we interpret the process in the dual field theory.

Generalised Scherk-Schwarz reductions from gauged supergravity

A procedure is described to construct generalised Scherk-Schwarz uplifts of gauged supergravities. The internal manifold, fluxes, and consistent truncation Ansatz are all derived from the embedding tensor of the lower-dimensional theory. We first describe the procedure to construct generalised Leibniz parallelisable spaces where the vector components of the frame are embedded in the adjoint representation of the gauge group, as specified by the embedding tensor. This allows us to recover the generalised Scherk-Schwarz reductions known in the literature and to prove a no-go result for the uplift of ω-deformed SO(p, q) gauged maximal supergravities. We then extend the construction to arbitrary generalised Leibniz parallelisable spaces, which turn out to be torus fibrations over manifolds in the class above.

Ten-dimensional origin of Minkowski vacua in [formula omitted] supergravity

Maximal supergravity in four dimensions admits two inequivalent dyonic gaugings of the group SO(4)×SO(2,2)⋉T16. Both admit a Minkowski vacuum with residual SO(4)×SO(2)2 symmetry and identical spectrum. We explore these vacua and their deformations. Using exceptional field theory, we show that the four-dimensional theories arise as consistent truncations from IIA and IIB supergravity, respectively, around a Mink×4S3×H3 geometry. The IIA/IIB truncations are efficiently related by an outer automorphism of SL(4) ⊂ E7(7). As an application, we give an explicit uplift of the moduli of the vacua into a 4-parameter family of ten-dimensional solutions.