^{1}

^{1}

^{2}

^{3}

^{1}

^{*}

^{1}

^{2}

^{3}

Edited by: Ashkbiz Danehkar, Harvard-Smithsonian Center for Astrophysics, United States

Reviewed by: Yurii M. Zinoviev, Institute for High Energy Physics, Russia; Ioannis Papadimitriou, Korea Institute for Advanced Study, South Korea

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Starting from the dual Lagrangians recently obtained for (partially) massless spin-2 fields in the Stueckelberg formulation, we write the equations of motion for (partially) massless gravitons in (A)dS in the form of twisted-duality relations. In both cases, the latter admit a smooth flat limit. In the massless case, this limit reproduces the gravitational twisted-duality relations previously known for Minkowski spacetime. In the partially-massless case, our twisted-duality relations preserve the number of degrees of freedom in the flat limit, in the sense that they split into a decoupled pair of dualities for spin-1 and spin-2 fields. Our results apply to spacetimes of any dimension greater than three. In four dimensions, the twisted-duality relations for partially massless fields that appeared in the literature are recovered by gauging away the Stueckelberg field.

Electric-magnetic duality, the symmetry of vacuum Maxwell equations under the exchange of electric and magnetic fields that interchanges dynamical equations with Bianchi identities, has counterparts in other physical systems, including supersymmetric field theories, linearised gravity and free higher-spin gauge theories. In supersymmetric Yang-Mills theories, electric-magnetic duality—see Olive and West [

If the dimension of spacetime is bigger than four, electric-magnetic duality actually links different descriptions of the same physical system. For linearised gravity on a flat background in _{ab} with a description in terms of an irreducible mixed-symmetry tensor _{a1 … an−3|b}, completely antisymmetric in its first

Recently Boulanger et al. [

In this Letter we focus on the massless and partially-massless cases and formulate the field equations derived from the actions of Boulanger et al. [

In the case of a partially-massless spin-2 field [

Twisted-duality relations are interesting for many reasons. In particular they relate, for a pair of dual theories, the Bianchi identities of one system to the field equations of the dual one, and vice versa. In the present work, we show that the field equations of two dual theories are formulated as a twisted-duality equation, although we note that the latter is not obtained from a variational principle that is manifestly spacetime covariant. Forgoing the latter requirement, for linearised Einstein theory around flat spacetime Bunster et al. [

As for our conventions, we work on constant-curvature spacetimes with either negative or positive cosmological constant Λ. We denote the number of spacetime dimensions by _{n} one has σ = 1, while σ = −1 for dS_{n}. The commutator of covariant derivatives gives _{ab} is the background (A)dS_{n} metric. The symbols ϵ_{a1 ⋯ an} and

In the Fierz-Pauli formulation for a massless spin-2 field around a maximally-symmetric spacetime of dimension

It is invariant, up to a total derivative, under the gauge transformations^{1}

The primary gauge-invariant quantity for the Fierz-Pauli theory is given by

It possesses the same symmetries as the components of the Riemann tensor,

and obeys the differential Bianchi identity

The field equations derived from the Lagrangian

where weak equalities are used throughout this paper to indicate equalities that hold on the surface of the solutions to the equations of motion. More precisely, defining

By virtue of the differential Bianchi identity for the curvature, one also finds that, on-shell, the curvature has vanishing divergence:

To summarize, the important equations in this section are 2.4, 2.5, and 2.6. The latter relation was derived from the Lagrangian

We start from the dual formulation of the massless spin-2 theory as given by the Lagrangian

This Lagrangian describes the propagation of the same degrees of freedom as the Fierz-Pauli one in Equation 2.1. It has been built in two steps: a Lagrangian depending only on the field Ŷ_{ab|c} (that is a traceless combination of the components of the spin connection) is obtained by eliminating the vielbein from the first-order formulation of linearised gravity in (A)dS. The full Lagrangian 2.9 then results from the Stueckelberg shift

The Lagrangian 2.9 possesses as many differential gauge symmetries as the Lagrangian obtained in Boulanger et al. [

We now define the following quantities

together with their various non-vanishing traces

Further introducing the traceless tensor _{ab}|^{cd} encoding the traceless projection of _{ab}|^{cd},

we find that _{ab}|^{cd} is invariant under the following gauge transformations:

Finally, the traceless tensor

is also found to be gauge invariant.

As in Boulanger et al. [^{2}

The corresponding curvatures are obtained from the previous gauge-invariant tensors

In components, the curvature tensors read

where the ellipses denote terms that are necessary to ensure ^{C}_{a[n−2]|bc} and ^{T}_{a[n−1]|bc} on the two-column Young tableaux of types [

Indeed, tracelessness of _{bc}|^{de} and _{ab}^{c} implies that the Hodge dual tensors

The two curvatures are linked via the following differential Bianchi identities:

These are equivalent to the following two identities:

The equations of motion for the dual gauge fields _{a[n−3]|b} and _{a[n−2]|b} derived from the Lagrangian

The field equations 2.26 and 2.27 can easily be obtained by starting from the field equations of the Lagrangian ^{abc|}_{d}^{ab|}_{c} in terms of their Hodge duals _{a[n−3]|b} and _{a[n−3]|b} in terms of their Hodge duals _{a[n−2]|b} and _{a[n−2]|b}, respectively. More in details, the left-hand sides of the field equations derived from

and the gauge invariant tensors

The field equations 2.26 and 2.27 imply the tracelessness of the curvatures:

In fact, from a result in representation theory of the orthogonal group—see the theorem on p. 394 of Hamermesh [

The curvature for the field

Upon using the first and second differential Bianchi identities 2.23 and 2.24, we also find the following two relations that are true on shell:

These equations, together with 2.30, imply that the divergences of the curvature ^{C} vanish on shell:

To summarise, the important equations of this section are the equations of motion 2.30 and the Bianchi identities 2.22, 2.23 and 2.24. In the following section we will relate them to the field equations and the Bianchi identities of the Fierz-Pauli formulation via a twisted-duality relation.

The twisted-duality relations for the massless spin-2 theory around (A)dS backgrounds are

As usual for twisted-duality relations, the Bianchi identities in a formulation of the theory are mapped to the field equations of the dual formulation, and vice versa, as we now explain in details.

First, the algebraic Bianchi identity 2.22 for the left-hand side of the twisted-duality relation 2.34 implies that the trace of _{ab|cd} vanishes on-shell, which is the field equation 2.6 in the metric formulation. The converse is true: If one takes the trace of the relation 2.34, the right-hand side vanishes by virtue of the algebraic Bianchi identity 2.4. This implies that the trace of the left-hand side of 2.34 vanishes, which enforces the field equation 2.30 in the dual formulation.

Second, starting again from the twisted-duality equation 2.34, the differential Bianchi identity 2.24 on the second column of ^{C} combined with the Bianchi differential identity 2.5 imply the on-shell vanishing of ^{T}, that is, 2.31. Using this result, the differential Bianchi identity 2.23 on the first column of ^{C} gives the first equation of 2.32 that implies in its turn, via 2.34, the field equation 2.8 in the metric formulation of the massless spin-2 theory. The converse is also true: acting on the twisted-duality relation 2.34 with ∇^{a} gives identically zero, from the right-hand side and as a consequence of the differential Bianchi identity 2.5 for the curvature in the metric formulation of linearised gravity around (A)dS. This implies the first field equation 2.33 for the dual graviton. Moreover, acting on 2.34 with ∇_{d} and antisymmetrising over the three indices {

Third, the twisted-duality relation 2.34 exactly reproduces, in the limit where the cosmological constant goes to zero, the twisted-duality relations given by Hull [

We consider the Stueckelberg Lagrangian for a partially-massless, symmetric spin-2 field in which both signatures are allowed (making AdS manifestly non-unitary at the classical level):

where the partially massless theory really appears in the limit

The last two lines in the expression 3.1 are new terms in comparison with the Lagrangian for a strictly massless spin-2 field in (A)dS, see 2.1. In the limit 3.2, the Lagrangian

The quantity

is invariant under the gauge transformations with parameter ξ_{a}, but not under the gauge transformations with parameter ϵ. A fully gauge-invariant quantity is provided by the antisymmetrised curl of _{ab}. Indeed, defining

we have that _{[a}_{b]c}. We further define the derived quantity ^{ab|mn} as follows:

It possesses the symmetries of the components of the Riemann tensor, like _{ab|cd} in the massless case. The second line of the above expression is identically vanishing in the limit 3.2, so that ^{ab|mn} is indeed a composite object purely built out of the gauge-invariant quantity ∇_{[a}_{b]c}. The writing that we adopted in 3.6 facilitates the relation between _{ab|cd} and _{ab|cd}. The interest in defining 3.6 rests in the fact that the field equations for _{ab} read

As a consequence, the field equations for _{ab} imply that the curvature _{ab|cd} is traceless on-shell, as it was for _{ab|cd} in the strictly massless case.

The Noether identities associated with the gauge parameter ξ_{a} give the left-hand side of the field equations for the vector _{a}:

The non-vanishing of the covariant divergence of _{ab} is also related to the Bianchi identity

where the gauge-invariant quantity _{a} reads

so that the field equations for _{a} imply that the curvature

We now consider the dual formulation of the partially-massless spin-2 theory that is described by the Lagrangian

A Lagrangian depending only on the field _{abc|d} has first been obtained by solving the equations of motion given by the variation of the vielbein in a first-order formulation of the partially-massless theory. In analogy with the massless case, the additional field ^{abc} has then been introduced by a Stueckelberg shift.

Starting from 3.11 one can define the following quantities

together with the successive traces

In a similar manner to the massless case, we introduce the traceless tensor _{ab}|^{cd} according to

and we find that the tensors _{ab}|^{cd} and

Also in this case, we then express

The curvature tensor for

We also define the curvature

In order to invert this relation, we first compute

and take the trace of the above relation, which produces

Inserting this relation back in 3.21 gives

Explicitly, we have

which is gauge invariant under [

The curvatures obey the following algebraic Bianchi identities

which means that ^{C}_{a[n−2]|bc} and

The left-hand sides of the equations of motion derived from the Lagrangian 3.11 are given by

Combining with what we obtained above, the field equations therefore imply

The Bianchi identities read

In terms of the curvatures

By taking a trace of the Bianchi identity and using the field equations, one therefore deduces that

The twisted duality that mixes the field equations and Bianchi identities of the two dual theories, the one for

This equation plays the same role as 2.34 in the strictly massless case.

What is new in the partially massless case compared to the massless case is that the flat limit of 3.35 is not enough to describe all degrees of freedom of a partially massless field. In fact, the twisted-duality relation 3.35 also induces a duality relation between the curvatures _{a} and contracting the result with ϵ^{a[n−1]d}. One then uses 3.31 and the trace of 3.9, taking into account that, on shell, the traces of the four curvatures _{ab|cd} and

where we stress that 3.35 and 3.36 are equivalent for

Now, taking the flat limit of _{a[n−3]|b}, _{ab}) and (_{a[n−3]}, _{a}). Both together, they propagate the correct degrees of freedom for a partially massless spin-2 field in the flat limit, as was found and discussed in section 4.3 of Boulanger et al. [

where _{ab} = 2∂_{[a}_{b]} are the field strengths for _{b[n−3]} and _{b}, respectively. In the flat limit, these latter quantities are gauge invariant, therefore the gradient ∂_{b} on both sides of the above relation 3.37 can be stripped off to give, up to an unessential coefficient that can be absorbed into a redefinition of _{a[n−3]}, the usual electric-magnetic duality between a 1-form and its dual (

As a consistency check for the second duality relation 3.36, one can start from the twisted-duality relation 3.35 and this time take the curl of

We then use the Bianchi identities 3.32 and 3.9 and take a trace, taking into account the field equation 3.29, which allows us to obtain the relation

which is fully consistent with 3.36.

Finally, we come back to the twisted-duality relation 3.36 and gauge fix to zero both _{a} and _{a[n−3]} since they are Stueckelberg fields as long as λ is different from zero. In these gauges for the dual formulations, our second twisted-duality relation 3.36 becomes

while the first twisted-duality relation 3.35 is just its curl, as one can readily check. This duality relation makes immediate contact with the one proposed for the specific case _{[a}_{b]|c}, the curl of the dual potential _{b|c} = _{c|b}. Note that, once the Stueckelberg fields _{a} and _{a[n−3]} have been set to zero, one cannot take a smooth flat limit any longer in the sense that physical degrees of freedom are lost in the flat limit.

The advantage of our Stueckelberg formulation for the twisted-duality relation is that the identification of the helicity degrees of freedom is manifest and does not require any specific system of coordinates to be seen. In the original Stueckelberg formulation, _{ab} and _{a} carry the helicity two and one degrees of freedom, and the twisted-duality relations 3.35 and 3.36 identify these degrees of freedom with the dual fields _{a[n−2]|b} and _{a[n−3]}, respectively, in a manifestly covariant way.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We performed or checked several computations with the package xTras [

^{1}Indices enclosed between (square) round brackets are (anti)symmetrised, and dividing by the number of terms involved is understood (strength-one convention). Moreover, we will use a vertical bar to separate groups of antisymmetrised indices, see e.g., Equation 2.3.

^{2}We substitute groups of antisymmetrised indices with a label denoting the total number of indices, e.g., ϵ_{a1⋯an}≡ϵ_{a[n]}. Moreover, repeated indices denote an antisymmetrisation, e.g., A_{a}B_{a}≡A_{[a1}B_{a2]}.