Dual Graviton Research Articles

Magnetic charges for the graviton

Symmetries and conserved charges are investigated for linearised gravity and its dual formulation in terms of the dual graviton field. Conserved charges are constructed for the dual graviton theory that are associated with invariances of the dual graviton theory. These invariances arise for gauge parameters that are certain generalised Killing tensors. These electric-type charges for the dual graviton are then shown to give magnetic-type charges for the graviton. One of the magnetic-type gravitational charges obtained in this way is the linearisation of the gravitational charge in d ≥ 5 dimensions that arises as a central charge in the supersymmetry algebra and is carried by Kaluza-Klein monopoles. Solutions of linearised gravity carrying the magnetic gravitational charges are discussed. The application of the approach used here to other gauge theories is discussed.

Higher dualisations of linearised gravity and the {A}_1^{+++} algebra

The non-linear realisation based on {A}_1^{+++} is known to describe gravity in terms of both the graviton and the dual graviton. We extend this analysis at the linearised level to find the equations of motion for the first higher dual description of gravity that it contains. We also give a systematic method for finding the additional fields beyond those in the non-linear realisation that are required to construct actions for all of the possible dual descriptions of gravity in the non-linear realisation. We show that these additional fields are closely correlated with the second fundamental representation of {A}_1^{+++} .

On-shell electric-magnetic duality and the dual graviton

Using on-shell amplitude methods, we explore four-dimensional electric-magnetic duality and its double copy. We show explicitly that the on-shell scattering amplitudes know about ``dual'' photons (and dual gravitons), that the off-shell photon propagator double-copies to the graviton propagator, and that the magnetic part of the propagator is essential for the double copy to hold. We also show that there is an equivalent gravitomagnetic part of the graviton propagator, which is essential in giving rise to solutions with either angular momentum or NUT charge. Furthermore, we comment on the so-called Weinberg paradox, which states that scattering amplitudes involving the mixing of electric and magnetic monopoles cannot be Lorentz invariant, and would seem to preclude the existence of the 't Hooft--Polyakov (topological) monopole. We trace this paradox to the magnetic part of the propagator, showing that it can be eliminated if one restricts the scenario to proper orthochronous Lorentz transformations. Finally, we compute the fully relativistic cross section for arbitrary spin dyons using the recently formulated on-shell duality transformation, and we show that this is always fully Lorentz invariant.

A master exceptional field theory

We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

Higher spins from exotic dualisations

At the free level, a given massless field can be described by an infinite number of different potentials related to each other by dualities. In terms of Young tableaux, dualities replace any number of columns of height hi by columns of height D − 2 − hi, where D is the spacetime dimension: in particular, applying this operation to empty columns gives rise to potentials containing an arbitrary number of groups of D − 2 extra antisymmetric indices. Using the method of parent actions, action principles including these potentials, but also extra fields, can be derived from the usual ones. In this paper, we revisit this off-shell duality and clarify the counting of degrees of freedom and the role of the extra fields. Among others, we consider the examples of the double dual graviton in D = 5 and two cases, one topological and one dynamical, of exotic dualities leading to spin three fields in D = 3.

The non-linear dual gravity equation of motion in eleven dimensions

We derive the non-linear dual graviton equation of motion in eleven dimensions in the context of E theory.

Gravity, dual gravity and A1+++

We construct the nonlinear realisation of the semidirect product of the very extended algebra [Formula: see text] and its vector representation. This theory has an infinite number of fields that depend on a space–time with an infinite number of coordinates. Discarding all except the lowest level field and coordinates the dynamics is just Einstein’s equation for the graviton field. We show that the gravity field is related to the dual graviton field by a duality relation and we also derive the equation of motion for the dual gravity field.

Exotic branes and mixed-symmetry potentials II: Duality rules and exceptional p-form gauge fields

Abstract In $U$-duality-manifest formulations, supergravity fields are packaged into covariant objects such as the generalized metric and $p$-form fields $\mathcal A_p^{I_p}$. While a parameterization of the generalized metric in terms of supergravity fields is known for $U$-duality groups $E_n$ with $n\leq 8$, a parameterization of $\mathcal A_p^{I_p}$ has not been fully determined. In this paper, we propose a systematic method to determine the parameterization of $\mathcal A_p^{I_p}$, which necessarily involves mixed-symmetry potentials. We also show how to systematically obtain the $T$- and $S$-duality transformation rules of the mixed-symmetry potentials entering the multiplet. As the simplest non-trivial application, we find the parameterization and the duality rules associated with the dual graviton. Additionally, we show that the 1-form field $\mathcal A_1^{I_1}$ can be regarded as the generalized graviphoton in the exceptional spacetime.

A note on the double dual graviton

The (free) graviton admits, in addition to the standard Pauli-Fierz description by means of a rank-two symmetric tensor, a description in which one dualizes the corresponding -curvature tensor on one column to get a -tensor, where D is the spacetime dimension. This tensor derives from a gauge field with mixed Yound symmetry called the ‘dual graviton’ field. The dual graviton field is related non-locally to the Pauli-Fierz field (even on-shell), in much the same way as a p -form potential and its dual -form potential are related in the theory of an abelian p -form. Since the Pauli-Fierz field has a Young tableau with two columns (of one box each), one can contemplate a double dual description in which one dualizes on both columns and not just on one. The double dual curvature is now a -tensor and derives from a gauge field with mixed Young symmetry, the ‘double dual graviton’ field. We show, however, that the double dual graviton field is algebraically and locally related to the original Pauli-Fierz field and, so, does not provide a truly new description of the graviton. From this point of view, it plays a very different role from the dual graviton field obtained through a single dualization. We also show that these equations can be obtained from a variational principle in which the variables to be varied in the action are (all) the components of the double-dual field as well as an auxiliary field with Young symmetry. By gauge fixing the shift symmetries of this action principle, one recovers the Pauli-Fierz action. Our approach differs from the interesting approach based on parent actions and covers only the free, sourceless theory. Similar results are argued to hold for higher spin gauge fields.

The Dual Graviton in Duality Covariant Theories

AbstractWe review (and elaborate on) the ‘dual graviton problem’ in the context of duality covariant formulations of M‐theory (exceptional field theories). These theories require fields that are interpreted as components of the dual graviton in order to build complete multiplets under the exceptional groups E, . Circumventing no‐go arguments, consistent theories for such fields have been constructed by incorporating additional covariant compensator fields. The latter fields are needed for gauge invariance but also for on‐shell equivalence with and type IIB supergravity. Moreover, these fields play a non‐trivial role in generalized Scherk‐Schwarz compactifications, as for instance in consistent Kaluza‐Klein truncation on AdS4 × S7.

Electric-Magnetic Duality in Gravity and Higher-Spin Fields

Over the past two decades, electric-magnetic duality has made significant progress in linearized gravity and higher spin gauge fields in arbitrary dimensions. By analogy with Maxwell theory, the Dirac quantization condition has been generalized to both the conserved electric-type and magnetic-type sources associated with gravitational fields and higher spin fields. The linearized Einstein equations in $D$ dimensions, which are expressed in terms of the Pauli-Fierz field of the \textit{graviton} described by a 2nd-rank symmetric tensor, can be dual to the linearized field equations of the \textit{dual graviton} described by a Young symmetry $(D-3, 1)$ tensor. Hence, the dual formulations of linearized gravity are written by a 2nd-rank symmetric tensor describing the Pauli-Fierz field of the dual graviton in $D=4$, while we have the Curtright field with Young symmetry type $(2,1)$ in $D=5$. The equations of motion of spin-$s$ fields ($s>2$) described by the generalized Fronsdal action can also be dualized to the equations of motion of dual spin-$s$ fields. In this review, we focus on dual formulations of gravity and higher spin fields in the linearized theory, and study their SO(2) electric-magnetic duality invariance, twisted self-duality conditions, harmonic conditions for wave solutions, and their configurations with electric-type and magnetic-type sources. Furthermore, we briefly discuss the latest developments in their interacting theories.

The graded differential geometry of mixed symmetry tensors

We show how the theory of $\mathbb{Z}_2^n$ -manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed.

Dual actions for massless, partially-massless and massive gravitons in (A)dS

We provide a unified treatment of electric–magnetic duality, at the action level and with manifest Lorentz invariance, for massive, massless as well as partially-massless gravitons propagating in maximally symmetric spacetimes of any dimension n>3. For massive and massless fields, we complete previous analyses that use parent-action techniques by giving dual descriptions that enable direct counting of physical degrees of freedom in the flat and massless limit. The same treatment is extended to the partially-massless case, where the duality has been previously discussed in covariant form only at the level of the equations of motion. The nature of the dual graviton is therefore clarified for all values of the mass and of the cosmological constant.

E11 and the non-linear dual graviton

The non-linear dual graviton equation of motion as well as the duality relation between the gravity and dual gravity fields are found in E theory by carrying out E11 variations of previously found equations of motion. As a result the equations of motion in E theory have now been found at the full non-linear level up to, and including, level three, which contains the dual graviton field. When truncated to contain fields at levels three and less, and the spacetime is restricted to be the familiar eleven dimensional space time, the equations are equivalent to those of eleven dimensional supergravity.

On the different formulations of the E11 equations of motion

The nonlinear realization of the semi-direct product of [Formula: see text] with its vector representation leads to equation of motions for the field graviton, three-form, six-form, dual graviton and the level four fields which correctly describe the degree of freedom of 11-dimensional supergravity at the linearized level. The equations with one derivative generically hold as equivalence relations and are often duality relations. From these equations, by taking derivatives, one can derive equations that are equations of motion of the familiar kind. The entire hierarchy of equations is [Formula: see text] invariant and the construction does not require any steps beyond [Formula: see text]. We review these past developments with an emphasis on the features that have been overlooked in hep-th:1703.01305, whose alternative approach we also comment on.

E11, Romans theory and higher level duality relations

From the underlying nonlinear realisation, we compute the complete E[Formula: see text] invariant equations of motion in eleven dimensions, at the linearised level, up to and including level four in the fields. Thus, we include the metric, the three and six forms, the dual graviton and three fields at level four. The fields are linked by a set of duality equations, which are first-order in derivatives and transform into each other under the E[Formula: see text] symmetries. From these duality relations, we deduce second-order equations of motion, including those for the usual supergravity fields. As a result the on-shell degrees of freedom are those of the eleven-dimensional supergravity. We also show that the level four fields provide an eleven-dimensional origin of Romans theory and lead to a novel duality relation.

Electron-Phonon Interactions, Metal-Insulator Transitions, and Holographic Massive Gravity.

Massive gravity is holographically dual to "realistic" materials with momentum relaxation. The dual graviton potential encodes the phonon dynamics, and it allows for a much broader diversity than considered so far. We construct a simple family of isotropic and homogeneous materials that exhibit an interaction-driven metal-insulator transition. The transition relates to the formation of polarons-phonon-electron quasibound states that dominate the conductivities, shifting the spectral weight above a mass gap. We characterize the polaron gap, width, and dispersion.

IIB supergravity and the E6(6) covariant vector-tensor hierarchy

IIB 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.

Kac-Moody and Borcherds symmetries of six-dimensional chiral supergravity

We investigate the conjectured infinite-dimensional hidden symmetries of six-dimensional chiral supergravity coupled to two vector multiplets and two tensor multiplets, which is known to possess the $F_{4,4}$ symmetry upon dimensional reduction to three spacetime dimensions. Two things are done. (i) First, we analyze the geodesic equations on the coset space $F_{4,4}^{++}/K(F_{4,4}^{++})$ using the level decomposition associated with the subalgebra $\mathfrak{gl}(5)\oplus \mathfrak{sl}(2)$ of $F_{4,4}^{++}$ and show their equivalence with the bosonic equations of motion of six-dimensional chiral supergravity up to the level where the dual graviton appears. In particular, the self-duality condition on the chiral $2$-form is automatically implemented in the sense that no dual potential appears for that $2$-form, in contradistinction with what occurs for the non chiral $p$-forms. (ii) Second, we describe the $p$-form hierarchy of the model in terms of its $V$-duality Borcherds superalgebra, of which we compute the Cartan matrix.

Generalised vielbeins and non-linear realisations

We briefly review why the non-linear realisation of the semi-direct product of a group with one of its representations leads to a field theory defined on a generalised space-time equipped with a generalised vielbein. We give formulae, which only involve matrix multiplication, for the generalised vielbein, the Cartan forms and their transformations. We consider the generalised space-time introduced in 2003 in the context of the non-linear realisation of the semi-direct product of E(11) and its first fundamental representation. For this latter theory we give explicit expressions for the generalised vielbein up to and including the levels associated with the dual graviton in four, five and eleven dimensions and for the IIB theory in ten dimensions. We also compute the generalised vielbein, up to the analogous level, for the non-linear realisation of the semi-direct product of very extended SL(2) with its first fundamental representation, which is a theory associated with gravity in four dimensions.