Consistent Truncation Research Articles

Hidden Kac-Moody structures in the fermionic sector of five-dimensional supergravity

We study the supersymmetric quantum dynamics of the cosmological models obtained by reducing $D=5$ supergravity to one timelike dimension. This consistent truncation has fourteen bosonic degrees of freedom, while the quantization of the homogeneous gravitino field leads to a $2^{16}$--dimensional fermionic Hilbert space. We construct a consistent quantization of the model in which the wave function of the Universe is a $2^{16}$--component spinor %\textcolor{red}{of Spin(24,8)} depending on fourteen continuous coordinates, which satisfies eight Dirac-like wave equations (supersymmetry constraints) and one Klein-Gordon-like equation (Hamiltonian constraint). The fermionic part of the quantum Hamiltonian is built from operators that generate a $2^{16}$-dimensional representation of the (infinite-dimensional) maximally compact sub-algebra $K(G_2^{++})$ of the rank-4 hyperbolic Kac--Moody algebra $G_2^{++}$. The (quartic-in-fermions) squared-mass term $\widehat \mu^2$ entering the Klein-Gordon-like equation has several remarkable properties: (i) it commutes with the generators of $K(G_2^{++})$; and (ii) it is a quadratic polynomial in the fermion number $N_F \sim \overline\Psi \Psi$, and a symplectic fermion bilinear $C_F \sim \Psi C\Psi$. Some aspects of the structure of the solutions of our model are discussed, and notably the Kac-Moody meaning of the operators describing the reflection of the wave function on the fermion-dependent potential walls ("quantum fermionic Kac-Moody billiard").

The higher-dimensional origin of five-dimensional mathcal{N} = 2 gauged supergravities

Using exceptional generalised geometry, we classify which five-dimensional 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 GS ⊂ USp(8) ⊂ E6(6) that preserve exactly two spinors. Moreover, the intrinsic torsion of the GS structure must contain only constant singlets under GS, 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.

A dilaton-axion model for string cosmology

The generic scale-invariant theory of an axion and a dilaton coupled to gravity in d-dimensions is generalized to a ‘universal’ one-axion model with two dilatons that reproduces itself under consistent dimensional-reduction/truncation. Flat FLRW cosmologies are shown to correspond to trajectories of a three-dimensional autonomous dynamical system, which we analyse with a focus on accelerated cosmic expansion, deriving the precise swampland bounds that exclude eternal acceleration. We also show that for two sets of values of its three independent parameters, the model is a consistent truncation of maximal ‘massive’ supergravity theories arising from string/M-theory; for these maximal-supergravity parameter values the FLRW cosmologies include some with a transient de Sitter-like phase, but not the recurring de Sitter-like phase or eternal cosmic acceleration that is possible for other parameter values.

Positive Energy and Non-SUSY Flows in ISO(7) Gauged Supergravity

We consider maximal gauged supergravity in 4D with the ISO(7) gauge group, which arises from a consistent truncation of massive IIA supergravity on a six-sphere. Within its G2-invariant sector, the theory is known to possess a supersymmetric AdS extremum, as well as two non-supersymmetric ones. In this context, we provide a first-order formulation of the theory by making use of the Hamilton–Jacobi (HJ) formalism. This allows us to derive a positive energy theorem for both non-supersymmetric extrema. Subsequently, we also find novel non-supersymmetric domain walls (DWs) interpolating between the supersymmetric extremum and each of the other two. Finally, we discuss a perturbative HJ technique that may be used in order to solve for curved DW geometries.

Exceptional Algebroids and Type IIA Superstrings

AbstractWe study exceptional algebroids in the context of warped compactifications of type IIA string theory down to n dimensions, with . In contrast to the M‐theory and type IIB case, the relevant algebroids are no longer exact, and their locali moduli space is no longer trivial, but has 5 distinct points. This relates to two possible scalar deformations of the IIA theory. The proof of the local classification shows that, in addition to these scalar deformations, one can twist the bracket using a pair of 1‐forms, a 2‐form, a 3‐form, and a 4‐form. Furthermore, we use the analysis to translate the classification of Leibniz parallelisable spaces (corresponding to maximally supersymmetric consistent truncations) into a tractable algebraic problem. We finish with a discussion of the Poisson–Lie U‐duality and examples given by tori and spheres in 2, 3, and 4 dimensions.

Curved-space Dirac description of elastically deformed monolayer graphene is generally incorrect

Undistorted monolayer graphene has energy bands which cross at protected Dirac points. It elastically deforms, and much research has assumed the Dirac description persists, now in a curved space and coupled to a gauge field related to lattice strain. We show this is incorrect by using a real space gradient expansion to study how the Dirac equation derives from the tight-binding model. Generic spatially varying hopping functions give rise to large magnetic fields which spoil the truncation in derivatives. In the perturbative regime, the only consistent truncation to Dirac is one with nontrivial gauge field but in flat space. One can instead fine-tune the magnetic field to be small, and we derive the resulting differential condition that the hopping functions must satisfy to yield a consistent truncation to Dirac in curved space. We consider whether mechanical effects might impose this fine tuning but find this is not the case for a simple elastic membrane model.

N = 3 conformal supergravity in four dimensions

In this paper we derive the action for N = 3 conformal supergravity in four space-time dimensions. We construct a density formula for N = 3 conformal supergravity based on the superform action principle. Finally, we embed the N = 3 Weyl multiplet in the density formula to obtain the invariant action for N = 3 conformal supergravity. There are two inequivalent embeddings by changing a particular coefficient from real to imaginary. They lead to invariant actions, which will either be the supersymmetrization of the Weyl square term or the Pontryagin density in the eventuality of gauge fixing to Poincaré supergravity. As a consistency check of our formalism, we will show that the supersymmetrization of the Pontryagin density is a total derivative. We will demonstrate this for purely bosonic terms. We will also present the complete action for the supersymmetrization of Weyl square term. We also discuss consistent truncation of N = 4 Weyl multiplet to N = 3 Weyl multiplet and use it for a robust check of our results using the earlier known results in N = 4 conformal supergravity.

Consistent truncations around half-maximal AdS5 vacua of 11-dimensional supergravity

We use exceptional field theory to systematically study all possible consistent truncations around any M-theory half-maximal vacua of the warped product form AdS5 × M 3 × S 2 × S 1, with M 3 a three-dimensional manifold. We show that, for any of these vacua, only truncations with at most three vector multiplets are consistent. Furthermore, the possible gaugings are restricted to be either SU(2) × U(1) or ISO(3) × U(1) where, in the first case, the U(1) factor can have different embeddings inside the global symmetry group SO(5, n), where n ⩽ 3 equals the number of vector multiplets. This rules out the possibility of any other gauging arising as a consistent truncation around the aforementioned M-theory vacua. Our analysis shows that of the many flows from half-maximal to quarter-maximal AdS5 vacua constructed in five-dimensional supergravity in Bobev (2018 J. High Energy Phys. 6 86), only those corresponding to an adjoint mass deformation in the dual SCFT can be uplifted to 11-dimensional supergravity. The other flows are five-dimensional artefacts without a higher-dimensional origin. Furthermore, consistent truncations with vector multiplets exist only if the vacuum satisfies certain conditions, which we derive.

Dimensional reduction of higher derivative heterotic supergravity

Higher derivative couplings of hypermultiplets to 6D, N = (1, 0) supergravity are obtained from dimensional reduction of 10D heterotic supergravity that includes order α′ higher derivative corrections. Reduction on T4 is followed by a consistent truncation. In the resulting action the hyperscalar fields parametrize the coset SO(4, 4)/(SO(4) × SO(4)). While the SO(4, 4) symmetry is ensured by Sen’s construction based on string field theory, its emergence at the field theory level is a nontrivial phenomenon. A number of field redefinitions in the hypermultiplet sector are required to remove several terms that break the SO(4) × SO(4) down to its SO(4) diagonal subgroup in the action and the supersymmetry transformation rules. Working with the Lorentz Chern-Simons term modified 3-form field strength, where the spin connection has the 3-form field strength as torsion, is shown to simplify considerably the dimensional reduction.

On the search for multicenter AdS black holes from M-theory

We study the effective potentials for various probe branes surrounding AdS4 black holes with massive halos in consistent truncations of M-theory on the Sasaki-Einstein7 manifolds Q111 and M111. These probes are either M2 branes extended in spacetime or “particle-like” probes such as internally wrapped M2 branes and, upon reduction to type IIA String theory, D6 branes corresponding to baryon operators in the dual Chern-Simons theory. We find both global and local minima of the potential outside the horizon, indicating the existence of stable and metastable multicenter AdS black holes in the extreme mass ratio regime, at fixed temperature and charges. For the planar case, we also find an instability towards nucleation of spacetime-filling M2 branes. With this analysis, we address some open questions on the holographic description of glassy phases of matter.

Homotopy Transfer and Effective Field Theory II: Strings and Double Field Theory

AbstractWe continue our study of effective field theory via homotopy transfer of ‐algebras, and apply it to tree‐level non‐Wilsonian effective actions of the kind discussed by Sen in which the modes integrated out are comparable in mass to the modes that are kept. We focus on the construction of effective actions for string states at fixed levels and in particular on the construction of weakly constrained double field theory. With these examples in mind, we discuss closed string theory on toroidal backgrounds and resolve some subtle issues involving vertex operators, including the proper form of cocycle factors and of the reflector state. This resolves outstanding issues concerning the construction of covariant closed string field theory on toroidal backgrounds. The weakly constrained double field theory is formally obtained from closed string field theory on a toroidal background by integrating out all but the ‘doubly massless’ states and homotopy transfer then gives a prescription for determining the theory's vertices and symmetries. We also discuss consistent truncation in the context of homotopy transfer.

Triality and the consistent reductions on AdS3 \xd7 S3

We study compactifications on AdS3×S3 and deformations thereof. We exploit the triality symmetry of the underlying duality group SO(4,4) of three-dimensional supergravity in order to construct and relate new consistent truncations. For non-chiral D = 6, mathcal{N} 6d = (1, 1) supergravity, we find two different consistent truncations to three-dimensional supergravity, retaining different subsets of Kaluza-Klein modes, thereby offering access to different subsectors of the full nonlinear dynamics. As an application, we construct a two-parameter family of AdS3 × M3 backgrounds on squashed spheres preserving U(1)2 isometries. For generic value of the parameters, these solutions break all supersymmetries, yet they remain perturbatively stable within a non-vanishing region in parameter space. They also contain a one-parameter family of mathcal{N} = (0, 4) supersymmetric AdS3 × M3 backgrounds on squashed spheres with U(2) isometries. Using techniques from exceptional field theory, we determine the full Kaluza-Klein spectrum around these backgrounds.

Taxonomy of brane gravity localisations

Generating an effective theory of lower-dimensional gravity on a submanifold within an original higher-dimensional theory can be achieved even if the reduction space is non-compact. Localisation of gravity on such a lower-dimensional worldvolume can be interpreted in a number of ways. The first scenario, Type I, requires a mathematically consistent Kaluza-Klein style truncation down to a theory in the lower dimension, in which case solutions purely within that reduced theory exist. However, that situation is not a genuine localisation of gravity because all such solutions have higher-dimensional source extensions according to the Kaluza-Klein ansatz. Also, there is no meaningful notion of Newton’s constant for such Type I constructions.Types II and III admit coupling to genuinely localised sources in the higher-dimensional theory, with corresponding solutions involving full sets of higher-dimensional modes. Type II puts no specific boundary conditions near the worldvolume aside from regularity away from sources. In a case where the wave equation separated in the non-compact space transverse to the worldvolume admits a normalisable zero mode, the Type III scenario requires boundary conditions near the worldvolume that permit the inclusion of that zero mode in mode expansions for gravitational wave fluctuations or potentials. In such a case, an effective theory of lower-dimensional gravity can emerge at sufficiently large worldvolume distance scales.This taxonomy of brane gravity localisations is developed in detail for linearised perturbations about a background incorporating the vacuum solution of Salam-Sezgin theory when embedded into ten-dimensional supergravity with a hyperbolic non-compact transverse space. Interpretations of the Newton constant for the corresponding Type III localisation are then analysed.

A Systematic Approach to Consistent Truncations of Supergravity Theories

Exceptional generalised geometry is a reformulation of eleven/ten-dimensional supergravity that unifies ordinary diffeomorphisms and gauge transformations of the higher-rank potentials of the theory in an extended notion of diffeormorphisms. These features make exceptional generalised geometry a very powerful tool to study consistent truncations of eleven/ten-dimensional supergravities. In this article, we review how the notion of generalised G-structure allows us to derive consistent truncations to supergravity theories in various dimensions and with different amounts of supersymmetry. We discuss in detail the truncations of eleven-dimensional supergravity to N=4 and N=2 supergravity in five dimensions.

Exceptional Algebroids and Type IIB Superstrings

AbstractIn this note we study exceptional algebroids, focusing on their relation to type IIB superstring theory. We show that a IIB‐exact exceptional algebroid (corresponding to the group , for ) locally has a standard form given by the exceptional tangent bundle. We derive possible twists, given by a flat ‐connection, a covariantly closed pair of 3‐forms, and a 5‐form, and comment on their physical interpretation. Using this analysis we reduce the search for Leibniz parallelisable spaces, and hence maximally supersymmetric consistent truncations, to a simple algebraic problem. We show that the exceptional algebroid perspective also gives a simple description of Poisson–Lie U‐duality without spectators and hence of generalised Yang–Baxter deformations.

Janus solutions from dyonic ISO(7) maximal gauged supergravity

We give a large class of new supersymmetric Janus solutions in dyonic ISO(7) maximal gauged supergravity encompassing all known solutions appearing recently. We consider SO(3) invariant sector in which the gauged supergravity admits four supersymmetric AdS4 vacua with N = 1, 1, 2, 3 unbroken supersymmetries and G2, SU(3), SU(3) × U(1) and SO(4) symmetries, respectively. We find Janus solutions preserving N = 1, 2, 3 supersymmetries and interpolating between AdS4 vacua and the N = 8 SYM phase. These solutions can be interpreted as conformal interfaces between SYM and various conformal phases of the dual field theory in three dimensions. The solutions can also be embedded in massive type IIA theory by a consistent truncation on S6. We also find a number of singular Janus solutions interpolating between N = 8 SYM or AdS4 vacua and singularities or between singular geometries. It turns out that, apart from the SYM phase on D2-branes, all the singularities arising in these solutions are unphysical in both four- and ten-dimensional frameworks.

Criticality of lower dimensional AdSd black holes

In lower dimensions, charged AdS black holes in an extended phase space, where the cosmological constant is interpreted as the thermodynamic pressure, are typically absent of liquid/gas phase transitions. We investigate the criticality of lower dimensional charged, dilatonic, asymptotically AdS (CDAdS$_d$) black holes generated from consistent truncations of RNAdS$_{d+2}$ black objects. We demonstrate that CDAdS$_{d}$ black holes in $d<4$ can exhibit rich van der Waals behavior and confirm that the associated critical exponents match those expected from mean field theory.

Wrapped NS5-branes, consistent truncations and In\xf6n\xfc-Wigner contractions

We construct consistent Kaluza-Klein truncations of type IIA supergravity on (i) Σ2 × S3 and (ii) Σ3 × S3, where Σ2 = S2/Γ, ℝ2/Γ, or ℍ2/Γ, and Σ3 = S3/Γ, ℝ3/Γ, or ℍ3/Γ, with Γ a discrete group of symmetries, corresponding to NS5-branes wrapped on Σ2 and Σ3. The resulting theories are a D = 5, mathcal{N} = 4 gauged supergravity coupled to three vector multiplets with scalar manifold SO(1, 1) × SO(5, 3)/(SO(5) × SO(3)) and gauge group SO(2) × (SO(2) {ltimes}_{Sigma_2} ℝ4) which depends on the curvature of Σ2, and a D = 4, mathcal{N} = 2 gauged supergravity coupled to one vector multiplet and two hypermultiplets with scalar manifold SU(1, 1)/U(1) × G2(2)/SO(4) and gauge group ℝ+ × ℝ+ for truncations (i) and (ii) respectively. Instead of carrying out the truncations at the 10-dimensional level, we show that they can be obtained directly by performing Inönü-Wigner contractions on the 5 and 4-dimensional gauged supergravity theories that come from consistent truncations of 11-dimensional supergravity associated with M5-branes wrapping Σ2 and Σ3. This suggests the existence of a broader class of lower-dimensional gauged supergravity theories related by group contractions that have a 10 or 11-dimensional origin.

Cubic action in double field theory

We study target space theory on a torus for the states with NL+NR=2 through Double Field Theory. The spin-two Fierz-Pauli fields are not allowed when all spatial dimensions are non-compact. The massive states provide both non-vanishing momentum and winding numbers in the target space theory. To derive the cubic action, we provide the unique constraint for NL≠NR compatible with the integration by part. We first make a correspondence of massive and massless fields. The quadratic action is gauge invariant by introducing the mass term. We then proceed to the cubic order. The cubic action is also gauge invariant by introducing the coupling between the one-form field and other fields. The massive states do not follow the consistent truncation. One should expect the self-consistent theory by summing over infinite modes. Hence the naive expectation is wrong up to the cubic order. In the end, we show that the momentum and winding modes cannot both appear for only one compact doubled space.

Supersymmetry and superstrata in three dimensions

We analyze the supersymmetry transformations of gauged SO(4) supergravity coupled to extra hypermultiplets in three dimensions, and find large families of smooth BPS solutions that preserve four supersymmetries. These BPS solutions are part of the consistent truncation of some families of six-dimensional superstrata. From the three-dimensional perspective, these solutions give rise to “smoothly-capped BTZ” geometries. We show how the twisting of the spin connection, the holomorphy of the fields, and the Chern-Simons connections all play an essential role in the existence of these supersymmetric solutions. This paper also closes the circle on the consistent truncation of superstrata, showing precisely how every feature of the superstratum enters into the three-dimensional BPS structure.