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
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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
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
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
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
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
we find that
Finally, the traceless tensor
is also found to be gauge invariant.
As in Boulanger et al. [
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
Indeed, tracelessness of
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
The field equations 2.26 and 2.27 can easily be obtained by starting from the field equations of the Lagrangian
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
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
Second, starting again from the twisted-duality equation 2.34, the differential Bianchi identity 2.24 on the second column of
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
we have that
It possesses the symmetries of the components of the Riemann tensor, like
As a consequence, the field equations for
The Noether identities associated with the gauge parameter ξ
The non-vanishing of the covariant divergence of
where the gauge-invariant quantity
so that the field equations for
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
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
and we find that the tensors
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
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
where we stress that 3.35 and 3.36 are equivalent for
Now, taking the flat limit of
where
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
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
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,
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 [
1Indices 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.
2We substitute groups of antisymmetrised indices with a label denoting the total number of indices, e.g., ϵa1⋯an≡ϵa[n]. Moreover, repeated indices denote an antisymmetrisation, e.g., AaBa≡A[a1Ba2].