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REVIEW article

Front. Phys., 09 January 2019
Sec. High-Energy and Astroparticle Physics
This article is part of the Research Topic Electric-Magnetic Duality in Gravitational Theories View all 8 articles

Electric-Magnetic Duality in Gravity and Higher-Spin Fields

  • High Energy Astrophysics Division, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, United States

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 graviton described by a 2nd-rank symmetric tensor, can be dual to the linearized field equations of the 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.

1. Introduction

Electric-magnetic duality basically evolved from the invariance of Maxwell's equations [1], which led to the hypothesis about magnetic monopoles in electromagnetism [2], and the introduction of magnetic-type sources to gauge theories [37]. Electric-magnetic duality contributed to the development of Yang-Mills theories [811], p-form gauge fields [1214], supergravity theories [15, 16], and string theories [1719]. Generalizations of this duality emerged as weak-strong (Montonen-Olive) duality in Yang-Mills theories [7]. The duality principle was refined in supersymmetry [20] and was extended to N=4 supersymmetric gauge theories [21]. The N=4 Super Yang-Mills theory has an exact SL(2, ℤ) symmetry, typically referred to as S-duality, that includes a transformation known as weak-strong duality [22], and can be understood as electric-magnetic duality in certain circumstances such as the Coulomb branch. Strong-weak duality exchanges the electrically charged string with the magnetically charged soliton in the heterotic string theory [2325]. Electric-magnetic duality was shown in N=1 supersymmetric gauge theories [26]. Compatifications of type II string theory (or M-theory) on a torus have an E7 symmetry, referred to as U-duality, which includes the T-duality O(6, 6, ℤ) and the S-duality SL(2, ℤ) [27]. S-duality of the N=4 Super Yang-Mills theory and S- and U-dualities of type II string and M-theory correspond to exact symmetries of the full consistent quantum theory [27, 28]. U-duality (see e.g., [2931]) provides nonperturbative insights in M-theory [32], and also in string theories [33, 34]. The electric-magnetic duality principle also permeated into gravitational theories [3543], supergravity [4446], higher-spin field theories [38, 4749], holographic dictionary [5052], hypergravity [53], and partially-massless and massive gravity on (Anti-) de Sitter space [5456].

Gravitational electric-magnetic duality originally appeared in wave solutions of general relativity [5759]. The Bianchi identities for the Weyl curvature provide some dynamical equations for perturbations in cosmological models [6066], which are analogous with Maxwell's equations [67, 68]. In general relativity, the Riemann curvature is decomposed into the Ricci curvature and the Weyl curvature. The Ricci curvature defined by the Einstein equations corresponds to the local dynamics of spacetime, while the nonlocal characteristics of gravitation, such as Newtonian force and gravitational waves, are encoded in the Weyl curvature [6670]. Based on this analogy, the electric and magnetic parts of the Weyl curvature tensor have been called gravitoelectric and gravitomagnetic fields [64, 68, 7174], though their spatial, symmetric 2-rank tensor fields make them dissimilar to the electric and magnetic vector fields of the U(1) gauge theory for spin-1 fields.

Analogies between gravitation and electromagnetism led to propose gravitationally magnetic-type (also called gravitomagnetic) mass almost a half-century ago [7578]. The solutions of the geodesic equations for the Taub–NUT [79, 80] metric also pointed to the magnetic-type mass, whose characteristic could be identical with the angular momentum [81]. Moreover, the gravitational lensing by bodies with gravitomagnetic masses, as well as the predicted spectra of magnetic-type atoms have been explained in Taub–NUT space [82]. The effects of the gravitomagnetic mass on the motion of test particles and the propagation of electromagnetic waves have been studied in Kerr-Newman-Taub-NUT spacetime [83]. Furthermore, the gravitational field equations and dual curvature tensors, which characterize the dynamics of gravitomagnetic matter (dual matter), as well as the relationship between the dynamical theories of gravitomagnetic mass (dual mass) and gravitoelectric mass (ordinary mass) have been elaborated among the NUT solutions, the spin-connection and vierbein [84, 85]. Finally, energy-momentum and angular momentum of both electric- and magnetic-type masses have been formulated for spin-2 fields [86].

The linearized formulations of gravity describe the graviton using a 2-rank symmetric tensor hμν, which can be dualized to a gauge field h~μ1···μD-3ν with mixed symmetry (D − 3, 1), the so-called dual graviton. The dual gravity with mixed symmetry tensors first appeared in 1980 [35], and mixed symmetry (2, 1) tensors were shown to be gauge fields [35]. It was also argued that a free massive mixed symmetry (2, 1) field in D = 4 could be dual to a free massive spin-2 field (massive graviton) and a massive spin-0 field (Higgs boson) that may illuminate “magnetic-like” aspects of gravity [87]. Following these earlier works [35, 87], mixed symmetry (D − 3, 1) was first suggested as a candidate for the dual graviton in D dimensions in 2000 [37]. Using the “parent” first-order action at the linearized level, it was then demonstrated that the formulations of the dual graviton with mixed symmetry (D − 3, 1) and the graviton are equivalent and have the same bosonic degrees of freedom [44]. Considering the Riemann curvature and its dual curvature, the linearized Einstein equations and Bianchi identities were found to hold on dual gravity with mixed symmetry (D − 3, 1) [38]. From the “parent” first-order action introduced in [44], the familiar “standard” second-order formulation of the dual graviton with mixed symmetry (D − 3, 1) in D dimensions was then obtained in [47], which was found to be the Curtright action [35] in D = 5 [through this paper, we consider the “standard” second-order formulations of dual gravity and spin-s fields, while the “parent” first-order actions are briefly presented in Appendix 1 (Supplementary Material)]. The dual formulation of linearized gravity have been further investigated by several authors [40, 41, 43, 46, 8890]. In the linearized theory, the magnetically charged D − 4 branes arise from Kaluza-Klein monopole, which are indeed gravitationally magnetic-type sources (0-brane in D = 4), can naturally be coupled with the gauge field of the dual graviton ([37]; see also [38, 91]). In fact, the dual graviton in D = 4 becomes the Pauli-Fierz field based on a symmetric tensor h~μν, while in D = 5 we get the Curtright [35] field h~μνρ with Young symmetry (2, 1) [40]. Generally, the spin-2 field described by a rank-2 symmetric tensor hμν is dualized to the linearized dual gravitational field h~μ1···μD-3ν described by a (D − 2)-rank tensor with Young symmetry type (D − 3, 1) [37, 38, 43]. It was also demonstrated that spin-2 fields can be dualized to two different theories:1 dual graviton with (D − 3, 1) and double-dual graviton with (D − 3, D − 3) [38, 96]. It was understood that mixed symmetry (D − 3, 1) fields can possess a Lorentz-invariance action principle. Although the equations of motion for mixed symmetry (D − 3, D − 3) fields could maintain the Lorentz group SO(D − 1, 1) [38], some authors argued that they may not have a Lorentz-invariance action in D > 4 [40]. Moreover, the graviton, the dual graviton and the double-dual graviton in 4 dimensions can be interchanged by a kind of the S-duality of D = 4 Maxwell theory [88]. In the context of the Macdowell-Mansouri formalism, the strong-weak duality for linearized gravity also corresponds to small-large duality for the cosmological constant [36] (see also [97] for duality with a cosmological constant). The dual formulations of spin-2 fields in D dimensions have also been generalized to spin-s fields whose dual fields are formulated using Young symmetry type (D-3,1,···,1s-1) [38, 47, 49, 89, 96, 98].

Dirac's relativistic wave solutions [99] for massive particles were the first theory extended to higher spins in 1939 [100], which provided the foundation for the Lagrangian formulations of massive, bosonic fields with arbitrary spin [101]. The free and interacting theories of massless, bosonic gauge fields with spins higher than 2 (the so-called Fronsdal [102] field) have been investigated later [102106]. In particular, dual representations of spin-s fields (s ≥ 2) were formulated using exotic higher-rank gauge fields of the Lorentz group SO(D − 1, 1) in the linearized theory [38, 48]. It was found that dual formations of spin-s fields can be described by a tensor with mixed Young symmetry type with one column with D − 3 boxes and s−1 columns with one box [47], which turns out to be the Pauli-Fierz field (s = 2), the Fronsdal field (s ≥ 3) in D = 4, and the Curtright field (s = 2) in D = 5. Nonlinear theories of massless, totally symmetric spin-s fields were also discussed in the (Anti-) de Sitter space, (A)dS [107]. It was shown that nonlinear representations of dual spin-s gauge theories can partially be implemented with a Killing vector [38]. The conserved electric- and magnetic-type sources associated with spin-s fields and their corresponding dual fields were also derived, in analogy with electromagnetism [91]. Recently, the twisted self-duality conditions, which were previously discussed in dual formulations of linearized gravity [43, 108], were also illustrated in higher spin gauge fields, which maintain an SO(2) electric-magnetic invariance in D = 4 [49].

In what follows we evaluate general aspects of electric-magnetic duality in gravity and higher spin field theories (s ≥ 2) in arbitrary dimensions (D ≥ 4), and advocate their formulations in linearized theories, their couplings with electric- and magnetic-type sources, and their harmonic conditions for wave propagation. In section 2, we study dual formulations of the spin-2 field (dual graviton). We then extend them to the spin-3 field (section 3) and higher spin fields (section 4). Section 5 discusses the interacting theories of the dual graviton and spin-s fields.

1.1. Notations and Conventions

To describe tensors with mixed symmetry types, we employ Young tableaux where a box in the Young diagram is associated with each index of a tensor (see [41, 96, 109, 110] for full details). A vector field (e.g., Aμ) has Young symmetry type (1) described by a single-box tableau yes. A 2nd rank anti-symmetric tensor, e.g., the Maxwell field Fμν, has Young symmetry type (2), which is represented by two boxes arranged in a column, yes. A 2nd rank symmetric tensor, e.g., the graviton field hμν, has Young symmetry type (1, 1), which is represented by two boxes arranged in a row, yes. Accordingly, the Curtright field, which describes the dual gravition h~μνρ in D = 5, is a 3rd rank tensor with Young symmetry type (2, 1), and is represented by the Young diagram yes. The Riemann tensor Rμνρλ and the Weyl tensor Cμνρλ are 4th rank tensors whose algebraic properties correspond to Young symmetry (2, 2), i.e., the diagram yes. We utilize the notations in which the Greek alphabet refers to covariant spacetime indices (e.g., Aμ, hμν), and the Latin alphabet refers to light-cone gauge indices (e.g., Ai, hij), which is also used for the E11s l1 non-linear realization in section 2.3 (e.g., hab, Aa1a2a3).

Throughout this paper, the graviton is denoted by a 2nd rank tensor hμν, whereas the dual graviton is shown by a (D − 2)-rank tensor h~μ1···μD-3ν with mixed symmetry (D − 3, 1) in D dimensions. A 3nd rank symmetric tensor, e.g., the typical spin-3 field hμνρ, has mixed symmetry type (1, 1, 1), which is represented by three boxes arranged in a row, yes. The spin-s fields (s > 2) are denoted by a s-th rank symmetric tensor hμ1···μs with mixed symmetry type (1,···,1s), which is represented by s boxes arranged in a row, while their dual field are represented by a (D − 2 + s)-th rank tensor h~μ1···μD-3ν1···νs-1 with mixed symmetry (D-3,1,···,1s-1) in D dimensions. The electric- and magnetic-type sources for spin-s fields in D = 4 are denoted by s-th rank energy-momentum tensors Tμ1···μs and T~μ1···μs, respectively. The Riemann curvatures for spin-s fields are represented by 2s-rank tensors Rμ1ν1···μsνs. The dual Riemann curvatures for spin-s fields in D = 4 are denoted by R~μ1ν1···μsνs. We write formulations according to the convention 8πGN(D)=1=c, unless the physical constants specified (GN(D) is the gravitational constant in D dimensions and c is the speed of light). We utilize the round brackets enclosing indices for symmetrization, hμν = hνμ = h(μν), and the square brackets for antisymmetrization, Fμν = −Fνμ = F[μν].

2. Dual Graviton

Free gauge theories in D-dimensional flat space are typically described by a particular irreducible tensor representation of the little group SO(D − 2) [37]. In the view point of the little group SO(D − 2), free abelian gauge theories can be dualized to a number of mixed symmetry tensors (see e.g., [38, 96]). The dual formulations for the free theory F present symmetries in the equations of motion, which transform the field F into its Hodge dual fields F (left dual), F (right dual), and F (left-right dual) [96]. For symmetric tensors, we have F = F.

The 1-form potential Aμ of the free Maxwell field strength Fμν ≡ 2∂Aν] in D dimensions possesses physical gauge degrees of freedom Ai in the little group SO(D − 2) representation, which can be dualized to a dual (D − 3)-form potential as follows [38]:

A˜j1···jD3=ϵj1···jD3iAi,    (1)
Ai=1(D3)!ϵij1···jD3A˜j1···jD3.    (2)

While Aμ is a single-box Young tableau, yes, its dual field is a single-box tableau yes in D = 4, a tableau with two boxes in a column yes in D = 5, and a tableau with D − 3 boxes in a column in arbitrary D dimensions.

For spin-2 fields such the graviton hμν, there are two different dual linearized formulations in generic D-dimensional spacetime [38, 40] (and infinitely many off-shell dualities [92, 93]). In the first dual formulation, the physical gauge graviton hij in D dimensions is dualized to a dual field h~i1···iD-3j in the little group SO(D − 2) representation as follows [38]:

h˜i1···iD3j=ϵi1···iD3khjk,    (3)
hjk=1(D3)!ϵi1···iD3jh˜i1···iD3k,    (4)

which has the following properties:

h˜i1···iD3j=h˜[i1···iD3]jh˜i1···iD3|j, h˜[i1···iD3j]=0.    (5)

The first dual graviton h~α1···αD-3μ is described by a tensor with Young symmetry type (D − 3, 1). In D = 4, the dual graviton is a symmetric 2nd rank tensor h~μν with the Young tableau having two boxes in a raw, yes, while it is a mixed symmetry (2, 1) tensor in D = 5 with the Young diagram yes (the so-called Curtright field). The dual graviton h~ possesses the equations of motion, which are equivalent to gravity and describe the same gravitational degrees of freedom of the graviton h [44]. The equivalence between the different formulations can be demonstrated by using their light-cone gauge fields.

In the second dual formulation, the physical light-cone gauge graviton hij in D dimensions is dualized to a dual field ĥi1···iD − 3j1···jD − 3as follows [38]:

h^i1···iD3j1···jD3=ϵi1···iD3mϵj1···jD3nhmn,    (6)
hmn=1(D3)!(D3)!ϵi1···iD3mϵj1···jD3nh^i1···iD3j1···jD3,    (7)

having the following properties:

h^i1···iD3j1···jD3=h^[i1···iD3][j1···jD3],h^[i1···iD3j1]j2···jD3=0.    (8)

The second dual graviton ĥα1···αD − 3β1···βD − 3 (also called double-dual graviton) is a tensor with Young symmetry type (D − 3, D − 3). In D = 4, it yields a symmetric 2nd rank tensor ĥμν with Young symmetry type (1, 1). In D = 5, it becomes a mixed symmetry (2, 2) tensor with the Young diagram yes whose algebraic properties are similar to the typical (spin-2) Riemann tensor Rμνρλ.

It can easily be seen that the physical light-cone dual field on both indices, ĥi1···iD − 3j1···jD − 3, can be written using the physical light-cone dual field on one index h~i1···iD-3j,

h^i1···iD3j1···jD3=12(ϵi1···iD3kh˜j1···jD3k+ϵj1···jD3lh˜i1···iD3l).    (9)

The first physical dual field h~ is recovered from the physical second dual field ĥ, so the equations of motion written for the Hodge dual field ĥ are equivalent to the field h~. However, it was argued that the second dual field ĥ in D > 4 may not have a Lorentz-invariant action [40] (though Lorentz covariant field equations were given in Hull [38]). We also note that for a traceless field (hμμ = 0), there is no second dual formulation, as it cannot be dualized on both indices. Therefore, we only consider the first dual formulation of gravity (also its equivalences for spin-s fields) throughout this paper.

For the typical (spin-2) linearized Riemann curvature Rαβμν, one gets the following dual linearized Riemann curvature in D-dimensional spacetime for the first and second dual formulations, respectively [38]:

R˜α1···αD2μν=12ϵα1···αD2αβRαβμν,    (10)
R^α1···αD2β1···βD2=14ϵα1···αD2μνϵβ1···βD2ρλRμνρλ,    (11)

where the Riemann tensor Rαβμν is of Young symmetry type (2, 2) ≡ yes, the first dual Riemann tensor R~α1···αD-2μν has Young symmetry type (D − 2, 2), and the second dual Riemann tensor R^α1···αD-2β1···βD-2 has mixed symmetry type (D − 2, D − 2).

2.1. D = 4: The Pauli-Fierz Field

The Pauli-Fierz action [100, 111], describing the equations of motion of a free massless spin-2 field hμν in D-dimensional Minkowski spacetime ημν=diag(-,+,···,+D-1) reads explicitly

SPF=c316πGN(D)dDxLPF[hμν],    (12)

where the Lagrangian LPF is quadratic in the first derivatives of the spin-2 field hμν and its covariant trace (hhαα):

LPF[hμν]=12(αhμναhμν2μhμναhαν                       +2hμνhμναhαh),    (13)

and is invariant under the gauge transformation

δξhμν=2(μξν),    (14)

where ξμ is an arbitrary vector field. The above gauge transformation is irreducible.

The action (13) has the following equations of motion:

δLPF[hμν]δhμν=ααhμνμαhαν+μνhημνααh=0.    (15)

The symmetric tensor hμν is associated with a free massless spin-2 particle (graviton), and its linearized Riemann curvature Rμναβ is defined as (e.g., [41, 43, 108]),

Rμναβ=2[α[μhν]β],    (16)

with the following properties

Rμναβ=R[μν]αβ, Rμναβ=Rμν[αβ], R[μνα]β=0,    (17)

and fulfills the Bianchi identity ∂Rμν]αβ = 0 and the linearized Einstein equations Rμν = 0, where RμνRμανβηαβ is the linearized Ricci tensor. We can also define the scalar curvature as RRμμ.

From the definition (16) and the equations of motion (15), it follows the Euler-Lagrange variation:

δLPF[hμν]δhμν=Rμν12ημνRGμν,    (18)

where Gμν is the linearized Einstein tensor, and fulfills the contracted Bianchi identity.

The linearized Riemann curvature tensor Rμναβ of the Pauli-Fierz field hμν in D dimensions can be split into the Weyl curvature tensor and the expression formulated by the linearized Ricci tensor and scalar curvature:

Rμναβ=Cμναβ+4D2R[α[μηβ]ν]2(D1)(D2)Rημ[αηβ]ν,    (19)

where the Weyl tensor Cμναβ is traceless, and has mixed symmetry type (2, 2) ≡ yes, so

Cμναβ=Cνμαβ, Cμναβ=Cμνβα, Cμναβ=Cαβμν,    (20)
Cμ[ναβ]=0, Cμνμρ=0.    (21)

The Weyl tensor has D(D + 1)(D + 2)(D − 3)/12 independent components (0, 10, and 35 components for D = 3, 4, and 5, respectively).

The Weyl curvature tensor contains electric and magnetic components, the so-called gravitoelectric and gravitomagnetic fields [64, 66, 68], which are encoded by covariant and time derivatives of the Pauli-Fierz field hμν (see Equations 27, 28), and defined as follows:

Eμν=CμανβuαuβCμ0ν0, Bμν=C˜μανβuαuβC˜μ0ν0,    (22)

where C˜μναβ=12ϵμνρλCρλαβ is a Hodge dual of the Weyl tensor, so Bμν=12ϵμαρλCρλνβuαuβ=12ϵμρλCρλν0 (the sign convention similar to Maxwell theory 2). Here, ϵμρλ is taken to be the spatial permutation tensor (ϵμνρuρ=0) with ϵμναβ = 2uϵν]αβ − 2ϵμν[αuβ], so ϵμνα=ϵμναβuβ and ϵμνβ=-ϵμναβuα (taking uμuμ=-1 and uμημν=uν in Minkowski spacetime).

The electric and magnetic parts of the Weyl tensor are both symmetric and traceless:

Eμν=E(μν), Bμν=B(μν), Eμμ=0, Bμμ=0.    (23)

In D = 4, the Weyl tensor has ten independent components, and its electric and magnetic parts have five independent components each.

From (16), one obtains

Rμν=12ρρhμν+ρ(μhν)ρ12νμhρρ,    (24)
R=μνhμνμμhνν,    (25)

so the linearized Weyl tensor can be written in terms of the Pauli-Fierz field hμν:

Cμναβ=2[μ[αhβ]ν]+2(D1)(D2)ημ[αηβ]ν                  ×(ρσhρσρρhσσ)                2D2([αρhρ[μ+[μρhρ[αρρh[α[μ                [α[μhρρ)ηβ]ν].    (26)

Accordingly, the electric and magnetic parts of the linearized Weyl tensor can be formulated using twice covariant and/or time derivatives of the Pauli-Fierz field hμν in the instantaneous rest-space of an observer moving with the relativistic velocity uμ in Minkowski spacetime ημν,

Eμν[h]=12[μνh002((μhν)0)+(hμν)]                    1(D1)(D2)(uμuν+ημν)(ρσhρσρρhσσ)                    1(D2)[12ρρhμν(μρhν)ρ+12μνhρρ                   +((μuν)hρρ)(ρhρ(μuν))+ρρh0(μuν)ρhρ0(μuν)                    12ρρh00ημν+(ρhρ0)ημν12(hρρ)ημν],    (27)
Bμν[h]=12ϵρλ(μ(ρhν)λ)+12ϵρλ(μν)ρh0λ                  1(D1)(D2)ϵρλ(μηρν)uλ(γσhγσγγhσσ)                  +12(D2)[ϵρλ(μην)λσσh0ρϵρλ(μhν)ρσσuλ                  +ϵρλ(μσhν)σρuλϵρλ(μρην)λσhσ0                  +ϵρλ(μν)σhσρuλϵρλ(μ(σhσρην)λ)                  ϵρλ(μν)ρuλhσσ+ϵρλ(μ(ρην)λhσσ)],    (28)

where primes denote time derivatives, i.e., (Tμ···)=uααTμ···, and ϵμρλ is the spatial permutation tensor (ϵμνρuρ=0). It is seen that the above equations are symmetric, traceless (Eμμ = 0 = Bμμ) and spatial (Eμνuν=0=Bμνuν). While the gravitoelectric field Eμν is associated with time/covariant derivatives of time/covariant derivatives of the Pauli-Fierz field hμν, the gravitomagnetic field Bμν is introduced by covariant curls and rotations of time/covariant derivatives of the Pauli-Fierz field hμν (the covariant curl is defined as curl(h)μνϵρλ(μρhν)λ=ϵμρλ[ρhνλ]).

The linearized Riemann curvature tensor Rμναβ can also be decomposed as

Rμναβ=Cμναβ+4S[μ[αην]β],    (29)

where Sμν is the linearized Schouten tensor that is defined in D dimensions as

Sμν=1D2(Rμν12(D1)Rημν),    (30)

whose curl is called the Cotton tensor:

Cμαβ=2[αSμβ],    (31)

which satisfies

Cμνρ=Cμ[νρ],C[μνρ]=0.    (32)

Substituting Equation (30) into Equation (29), the Riemann decomposition Equation (19) is easily recovered.

The Pauli-Fierz theory of linearized gravity in D = 4 is dualized to the Pauli-Fierz action S=12d4xLPF[h~μν] being formulated in terms of the symmetric massless 2nd-rank tensor h~μν=h~νμ=h~(μν) (dual graviton) and its covariant trace (h˜h˜αα)[40, 42, 43]. In such a way, the symmetric tensor hμν is replaced with h~μν in the Pauli-Fierz action (13). Similarly, we can define the dual Riemann curvature R~μναβ for the Pauli-Fierz field h~μν as R˜μναβ=2[α[μh˜ν]β], which is decomposed into the dual Ricci tensor R~μνR~μανβηαβ and the dual Weyl tensor C~μναβ.

There is an SO(2) electric-magnetic invariance between the Riemann curvature and its dual curvature, which is called the “twisted self-duality conditions”[38, 41, 43, 49, 88]:

R[h]=R˜[h˜],     R˜[h˜]=R[h].    (33)

The gravitoelectric and gravitomagnetic fields of the Pauli-Fierz field hμν and its dual field h~μν also demonstrate the twisted self-duality conditions[43, 49]:

Eμν[h]=B˜μν[h˜],       Bμν[h]=E˜μν[h˜],    (34)

where μν and B~μν are the electric and magnetic parts of the (dual) Weyl tensor C~μναβ associated with the dual Pauli-Fierz field h~μν (dual graviton), and are both symmetric and traceless.

The twisted self-dual conditions (Equations 33, 34) can be expressed in a duality-symmetric way [49]:

=S,       Eμν=SBμν,       Bμν=SEμν.    (35)

with matrix notations

=(R[h]R˜[h˜]),       =(R[h]R˜[h˜]),    (36)
Eμν=(Eμν[h]E˜μν[h˜]),       Bμν=(Bμν[h]B˜μν[h˜]),    (37)
S=(0110),    (38)

which imply that the equations of motion for the dual graviton h~μν are fully equivalent to those for the graviton hμν under SO(2) electric-magnetic duality rotations.

Using the first dual expression (Equation 10), the dual Riemann curvature R~μναβ in D = 4 is defined by

R˜αβμν=12ϵαβρλRρλμν,    (39)

which enjoys the following properties

R˜[μνα]β=0, [ρR˜μν]αβ=0.    (40)

The equations of motion are invariant under the duality transformations [42]

Rμναβ=cosαRμναβ+sinαR˜μναβ,    (41)
R˜μναβ=sinαRμναβ+cosαR˜μναβ.    (42)

where the Riemann curvature Rμναβ and its dual curvature R~μναβ are transferred into each other under the electric-magnetic duality rotations (Equations 41, 42).

It is useful to see the electric-magnetic duality rotations in terms of the electric and magnetic parts of the linearized Riemann curvature tensor (see e.g., [112]):

Eμν=Rμ0ν0, Bμν=R˜μ0ν0.    (43)

While the equations of motion are satisfied, the rotations in (41) and (42) are equivalent to

Eμν=cosαEμνsinαBμν,    (44)
Bμν=sinαEμν+cosαBμν.    (45)

This SO(2) rotation in gravity is analogous to the duality transformations of Maxwell fields [113, 114] in which the free Maxwell field strength Fμν is transformed into its dual field strength F~μν=12ϵμναβFαβ. Using the definitions (16), it can be shown that the graviton hμν is rotated into the dual graviton h~μν under the transformations (41) and (42), so they are invariant under the electric-magnetic duality.

2.2. D = 5: The Curtright Field

The equations of motion of a free massless spin-2 field h~[μν]ρh~μν|ρ (dual graviton) in 5-dimensional flat spacetime can be described using the following action [35, 41, 47, 105]

L[h˜μν|ρ]=14[γh˜μν|ργh˜μν|ρ4γh˜γμ|λρh˜ρμ|λ                               +4h˜λ μ|λγρh˜γμ|ρ2γh˜ρ μ|ργh˜λμ|λ                               +2σh˜σμ|γρh˜ρμ|γ+2γh˜λγ|λσh˜ρσ|ρ].    (46)

It is also convenient to use a Lagrangian consisting of the Curtright field strength tensor F̄[μνλ]αF̄μνλ|α, the so-called Curtright action [35, 40] 3

L[h˜μνλ]=112(F¯μνρ|λF¯μνρ|λ3F¯μνλ|λF¯μνρ|ρ,)    (47)

where the Curtright field strength tensor F̄μνλα is a tensor with mixed symmetry type (3, 1) ≡ yes defined as

F¯[μνρ]λμh˜νρλ+ρh˜μνλ+νh˜ρμλ=3[μh˜νρ]λ,    (48)

and h~μνλ is the Curtright [35] field with mixed symmetry type (2, 1) ≡ yes,

h˜μνρ=h˜[μν]ρh˜μν|ρ,      h˜[μνρ]=0.    (49)

The Curtright action (47) is invariant under the following gauge transformation [35, 37, 38, 40, 115]

δσ,αh˜μνρ=2([μσν]ρ+[μαν]ρραμν),    (50)

where σμν and αμν are symmetric and antisymmetric arbitrary tensors, respectively:

σμν=σνμ=σ(μν),      αμν=ανμ=α[μν].    (51)

The Curtright field F̄μνλα is transformed as [43]:

δαF¯μνλρ=6ρ[μανλ],    (52)

although it is not gauge invariant. The gauge transformations for σμν and αμν are reducible, so we get δσσμν = 6∂σν) and δσαμν = 2∂σν], where σμ is an arbitrary vector field. The gauge symmetry for σμ is irreducible [40].

The Curtright field h~μνλ is associated with the linearized dual Riemann curvature tensor [37, 40, 43]:

R˜[μνρ]=[αβ]2[αF¯[μνρ]=β]6[α[μh˜νρ],β]    (53)

and the linearized dual Ricci curvature tensor and the dual vector curvature, respectively,

R˜[μν]α=ηβρR˜[μνρ][βα],    (54)
R˜μ=ηναR˜[μν]α.    (55)

The linearized dual Riemann curvature tensor has the Young tableau symmetry (3, 2) ≡ yes, and the linearized dual Ricci curvature tensor has the Young tableau symmetry (2, 1) ≡ yes.

The linearized dual Riemann curvature satisfies the “Bianchi identity”[38, 88]

[λR˜μνρ]αβ=0.    (56)

We can also define the linearized “dual Einstein tensor” for the Curtright field [37, 40, 43, 88] through

G˜[μν]α=R˜[μν]αR˜[μην]α,    (57)

where G~[μν]α is the dual Einstein tensor with Young symmetry type (2, 1), and fulfills the “contracted Bianchi identities”

μG˜[μν]α=0,        αG˜[μν]α=0.    (58)

The doubly contracted Bianchi identity yields μR~μ=0. The linearized dual Einstein equations are G~[μν]α=0, which are equivalent to R~μνα=0.

The action (47) is gauge-invariant, and provides the following equations of motion [40]:

δL[h˜μνρ]δh˜μνρ=3(R˜[μν]ρ+ημ[νR˜ρ])=0,    (59)

which satisfy the contracted Bianchi identities (58).

The linearized dual Riemann curvature of the Curtright field h~μνρ is decomposed into a traceless part (dual 5-D “Weyl” tensor), and terms containing the (dual 5-D) Schouten tensor:

R˜μνρ=αβC˜μνρ+αβ3S˜[μν[αηρ]β],    (60)

where the dual Schouten tensor S~μνα is defined as [43]:

S˜μνα=R˜μνα13R˜[μην]α.    (61)

The dual 5-D Weyl tensor C~μνραβ has 35 independent components.

One can also define the Cotton tensor, the curl of the Schouten tensor, as

C˜μναβ=2[αS˜μνβ],    (62)

which is traceless C~μναβηνβ=0, and contains both mixed symmetry types (2, 2) and (3, 1), and satisfies

C˜[μναβ]=0,      [γC˜μν=αβ]0.    (63)

It was shown in Curtright [35], Aulakh et al. [105], Labastida and Morris [115], Hull [37], and Hull [38] that the physical degrees of freedom of the dual graviton in D = 5 are massless, and its dynamical variables are represented by spatial, transverse and traceless components h~ijk of the Curtright field h~μνρ, which have the Young symmetry similar to h~μνρ, and transform in the little group SO(D − 2). There are the spatial dual Einstein tensor G~ijk and the spatial dual Riemann curvature R~ijrsm which are constructed by the spatial tensor h~ijk. Moreover, the dynamical variables of the standard graviton, which is dual to the Curtright field h~μνρ in D = 5, are given by spatial, transverse and traceless components hij of the Pauli-Fierz field hμν, which is symmetric similar to hμν. Similarly, there are the spatial Einstein tensor Gij and the spatial Riemann curvature Rijrs which are constructed by the spatial tensor hij.

Let us consider the electric and magnetic fields of the standard graviton, which are constructed from the spatial Riemann curvature [43]:

Eijmn[h]=14ϵijrsRrslkϵlkmn,Bmni[h]=12ϵmnlkR0ilk,    (64)

where Eijmn[h] has the Young symmetry yes, Bmni[h] and has the Young symmetry yes.

We also build the electric and magnetic fields of the dual graviton from the spatial dual Einstein tensor and the spatial dual Riemann curvature, respectively [43]:

E˜ijm[h˜]=G˜ijm,      B˜ijmn[h˜]=12ϵmnlkR˜0ijlk,    (65)

where E˜ijm[h˜] has the Young symmetry yes, B˜ijmn[h˜] and has the Young symmetry yes.

The electric fields Eijmn[h] and E˜ijm[h˜] are double-traceless and traceless, respectively, which correspond to G00 = 0 and G~0j0=0. There are similar properties for the magnetic fields Bmni[h] and B˜ijmn[h˜]:

Eijmn[h]ηjnηim=0,       E˜ijm[h˜]ηjm=0,    (66)
Bmni[h]ηni=0,      B˜ijmn[h˜]ηjnηim=0.    (67)

The electric fields, Eijmn[h] and E˜ijm[h˜], and the magnetic fields, Bmni[h] and B˜ijmn[h˜], have the following identities

E[ijm]n[h]=0,     E˜[mni][h˜]=0    (68)
B[mni][h]=0,      B˜[ijm]n[h˜]=0,    (69)

which are equivalent to the Ricci conditions R0j = 0 and R~0jm = 0.

The Bianchi identities imply that the electric fields, Eijmn[h] and E˜ijmn[h˜], and magnetic fields, Bmni[h] and B˜ijmn[h˜], are similarly transverse:

iEijmn[h]=0,      iE˜ijm[h˜]=0,    (70)
mBmni[h]=0,    mB˜ijmn[h˜]=0.    (71)

The electric fields Eijmn[h] and E˜ijm[h˜] transform under the mixed symmetries (2,2) and (2,1), while the magnetic fields Bmni[h] and B˜ijmn[h˜] transform under the mixed symmetries (2,1) and (2,2). From the twisted self-duality conditions [43], it follows that

Eijmn[h]=B˜ijmn[h˜],         Bijm[h]=E˜ijm[h˜],    (72)

or in a duality-symmetric way

E=SB,    B=SE,    (73)

with matrix notations

E=(Eijmn[h]E˜ijm[h˜]),   B=(Bijm[h]B˜ijmn[h˜]),    (74)

where the SO(2) duality matrix S is defined by Equation (38). Therefore, the electric and magnetic fields of the standard graviton are fully equivalent to the magnetic and electric fields of the dual graviton in D = 5 under SO(2) electric-magnetic duality rotations (see [43] for constraint and dynamical formulas).

2.3. D = 11: Mixed Symmetry (8, 1) Field

In the case of D = 11 spacetime, which are relevant to maximal supergravity/M-theory, the dual graviton defined by the first dual representation is associated with a tensor field h~a1···a8|b of Young symmetry (8, 1). Interestingly, the conjectured hidden symmetry E11 of M-theory is decomposed into GL(11) subalgebras: a mixed symmetry (8, 1) tensor (dual graviton), a 3-form, a 6-form, and other mixed symmetry fields [44]. It was shown that the dynamics and hidden symmetries of gravity are strongly linked to Lorentzian Kac–Moody algebras [116]. Moreover, the Weyl groups of infinite-dimensional Kac–Moody algebras (see [117] for review) are compatible with the groups E11(=E8+++) [118120] and E10(=E8++) [121, 122], which contain information about the gravitational electric-magnetic duality, as well as the structures of M-theory (see [123] for review).

The Dynkin diagram of the Lorentzian Kac-Moody algebra E11=E8+++ is depicted below (see e.g., [44, 124126]):

yes

which encodes underlying hidden symmetries of M-theory [44, 124] (the same for E10=E8++ [127]). The various exceptional Lie groups (E6, …, E10) are embedded in the infinite-dimensional Kac–Moody algebra E11, i.e., E6 ⊂ ··· ⊂ E10E11. The infinite-dimensional Kac–Moody algebra E11 (also E10) can be constructed from E8 through extensions of E8 with three extra nodes (two extra nodes in E10), so we derive the Lorentzian Kac–Moody algebra E8+++=E11 (also E8++=E10) [117].

The Kac–Moody algebra E11 can be decomposed into the GL(11) subalgebras by deleting the node 11 in the E11 Dynkin diagram (see the above diagram) [44, 46, 128]. The decomposition subalgebras correspond to the low level E11 tensor fields in the non-linear realization of E11 and its vector, E11s l1,

Level:0123456Field:hab,Aa1a2a3,Aa1···a6,h˜a1···a8|b,Aa1···a9|b1b2b3,Aa1···a10|b1b2,Aa1···a11|b,.    (76)

The zero- and third-level fields, hab and h~a1···a8|b, correspond to the graviton and the dual graviton in D = 11, respectively. The first-level bosonic 3-form field, Aα1α2α3 is related to non-gravitational degrees of freedom of M-theory. The second-level 6-form field, Aa1···a6, has equations of motion, which can be dual to those of the 3-form field Aa1a2a3, and provides another way to describe the degrees of freedom of Aa1a2a3. These fields lead to at least one spacetime coordinate for each in the E11s l1 non-linear realization [125, 129],

Level 0: habxa,Level 1: Aa1a2a3xa1a2,Level 2: Aa1···a6xa1···a5,Level 3: h˜a1···a8|bxa1···a8,xa1···a7|b,    (77)

The spacetime coordinate xa, which corresponds to the graviton, constructs the spacetime curvature for the dynamics of gravity. It is seen that the 3-form field, Aa1a2a3, and 6-form field, Aa1···a6, are associated with the 2-form and 5-form coordinates, respectively. The mixed symmetry (8, 1) tensor field, h~a1···a8|b, which is dual to the graviton in D = 11, is related to the 8-form coordinate and the mixed symmetry (7, 1) coordinate. The dual graviton field and its equation of motion possess the following identities [130]

h˜[a1···a8|b]=0,E[a1···a8|b](2)=0,

where the equations of motion Eα1···α8|μ(2) is of Young symmetry type (8, 1), and the numerical superscript describes the number of spacetime derivatives (see [129, 130] for more details of the E11 equations of motion), i.e., in empty space

Ea1···b8|(2)b14[c[ch˜a1···a8]b]=0,    (78)

while the equations of motion for the graviton are [129]:

Ea(2)b2[aωbc|c]=0,    (79)

where ωαβ|γ is defined as (hab in the E11s l1 non-linear realization) [129]:

ωab|cah(bc)+bh(ac)+ch[ab].    (80)

The equations of motion for the non-gravitational fields Aα1α2α3 and Aα1···α6 are also given by [129, 130]

Ea1a2a3(2)b[bAa1a2a3]=0,     Ea1···a6(2)b[bAa1···a6]=0.    (81)

There is a unique E11 invariant equation with one derivative, which is obtained from the non-linear realization [129], and it contains the graviton and its dual field:

Eab|c(1)ωab|c14ϵabd1···d9[d1h˜d2···d9]c0.    (82)

The non-gravitational fields Aa1a2a3 and Aa1···a6 possess a unique E11 invariant equation with one derivative (see [126, 131] for details):

Ea1···a4(1)[a1Aa2a3a4]12(4!)ϵa1···a4b1···b7b1Ab2···b7=0.    (83)

Variations of the E11 equations of motion (82 ) and (83) result in Eab(2) and Ea1a2a3(2), with the duality relations of Ea1···a8|b(2) and Ea1···a6(2), respectively [129]:

Ea1a4(1)=0Eab(1)|c0           Ea1a2a3(2)=0 Ea(2)b=0  Ea1a6(2)=0 Ea1a8|b(2)=0    (84)

From the unique E11 invariant Equation (82), Eab(1)|c0, we get the linearized Einstein equation, Ea(2)b=0, and the equation of motion for the 11-dimensional dual graviton, Ea1···a8|b(2)=0. While from the unique E11 invariant Equation (83), one deduces the equations of motion Ea1a2a3(2)=0, and its dual equations of motion Ea1···a6(2)=0. Therefore, the algebra E11 yields a network of equations of motion, including the typical equations of motion in general relativity.

2.4. Generalized Dual Spin-2 Fields

As mentioned at the beginning of section 2, the Pauli-Fierz field associated with the spin-2 field hαβ is dual to a field dualized on one index (dual graviton) or on both indices (double-dual graviton) [38, 96]. We consider only its dualization on one index since it presents equations of motion equivalent to the linearized Einstein equations and its action satisfies Lorentz-invariance [40]. Hence, the generalized, free massless, dual spin-2 fields h~[α1···αD-3]βh~α1···αD-3|β in arbitrary dimensions (D ≥ 4) can be given by Young symmetry type (D − 3, 1) having the Young diagram [38, 43]:

yes

which possesses the following properties

h˜α1···αD3β=h˜[α1···αD3]βh˜α1···αD3|β,     h˜[α1···αD3β]=0,    (86)

and has (D − 3)(D + 1)!/3(D − 2)! components in D dimensions [41].

The action associated with the equations of motion of free massless, dual spin-2 fields h~α1···αD-3β in D-dimensional Minkowski spacetime explicitly reads [41, 47, 105]4

L[h˜α1···αD3|β]       =12(D3)![γh˜α1...αD3|βγh˜α1...αD3|β            2(D3)γh˜γα2...αD3|βρh˜ρα2...αD3|β           +2(D3)h˜λ α2...αD3|λγρh˜γα2...αD3|ρ           (D3)γh˜ρ α2...αD3|ργh˜λα2...αD3|λ           +(D3)(D4)σh˜σα2...αD3|γρh˜ρα2...αD2αD3|γ          +(D3)(D4)γh˜λγα3...αD3|λσh˜ρσα3...αD3|ρ].    (87)

It may also be written in terms of a field strength tensor F̄α1αD-2|β, in a similar manner of the Curtright action (47), as follows [132134]5:

L[h˜α1αD3|β]=12(D2)!(F¯α1...αD2|βF¯α1...αD2|β                                                     (D2) F¯α1...αD3λ|λF¯α1...αD3ρ| ρ) ,    (88)

where the “generalized Curtright” field strength tensor F̄α1αD-2|β, a general form of (48), is a tensor with Young symmetry type (D − 2, 1) with the diagram:

yes

defined as

F¯α1...αD2|β(D2)[α1h˜α2···αD2]β.    (90)

For D = 4 and 5, the Lagrangian (87) recovers the Pauli-Fierz (13) formulated by h~μν and the Curtright action (46), respectively. For D = 5, the Lagrangian (88) also becomes the Curtright action (47).

The action (87) is invariant under the following gauge transformation [38, 41, 134]

δχ,αh˜α1···αD3|β=(D3)([α1χα2···αD3]β                               +[α1αα2···αD3]β(1)D3βαα1···αD3),    (91)

where χα1···αD − 4 and αα1···αD − 3 are mixed symmetry type (D − 4, 1) and antisymmetric (D − 3)-rank tensors, respectively:

yes

The action (88) is also invariant under the gauge transformation (91) (see [132, 133]). The field strength tensor F̄α1αD-2|β defined by (90) is not gauge invariant, but it is transformed as

δαF¯α1...αD2|β=(D2)(D3)β[α1αα2···αD2].    (93)

The gauge transformations for χα1···αm−1 = χ[α1···αm−1 and αα1···αm = α[α1···αm] (m = D − 3 at the initial level) are reducible to an arbitrary mixed symmetry type (m − 2, 1) tensor χα1···αm−2 = χ[α1···αm−2 and an arbitrary antisymmetric (m − 1)-rank tensor αα1···αm−1 = α[α1···αm−1] for m > 3 (see [115] for irreducible representations):

δχχα1···αm1|β=(m1)([α1χα2···αm1]β                             +[α1αα2···αm1]ββαα1α2···αm1),    (94)
δχαα1···αm=(m)[α1αα2···αm].    (95)

For m = 3, they are reduced to an arbitrary symmetric tensor σμν = σ(μν) and an arbitrary antisymmetric tensor αμν = α[μν],

δχχα1α2|β=2([α1σα2]β+[α1αα2]ββαα1α2),    (96)
δχαα1α2α2=3[α1αα2α2].    (97)

The gauge transformations for σμν and αμν are also reducible to an arbitrary vector field σμ,

δσσμν=6(μσν),       δσαμν=2[μσν],    (98)

The gauge symmetry for σμ is irreducible.

The linearized dual Riemann curvature tensor in D dimensions is defined as

R˜α1···αD2μ2ν2=12ϵα1···αD2μ1ν1Rμ1ν1μ2ν2,    (99)

which is of Young symmetry (D − 2, 2):

yes

Similarly, the D-dimensional dual Einstein equations are

G˜α1···αD3|μ=0,    (101)

where the D-dimensional dual Einstein tensor G~α1···αD-3|μ is of mixed symmetry type (D − 3, 1) with the Young diagram:

yes

The D-dimensional dual Ricci tensor is defined by

R˜α1···αD3|μ2R˜α1···αD2|μ2ν2ηαD2ν2,    (103)

where the dual Ricci tensor is of Young symmetry (D − 3, 1).

2.5. Gravitationally Magnetic-type Source

The introduction of magnetic monopoles to Maxwell theory requires that the magnetic charge qm is related to the electric charge qe through the so-called Dirac quantization condition [1, 2]

qeqm=n12ħc,     n,    (104)

which is invariant under the SO(2) electric-magnetic duality rotations,

qeqm,     qmqe.    (105)

A singularity of the magnetic field or a Dirac string is unobservable in the spin-1 field if the condition (104) is imposed on the magnetic charge.

Similarly, there is a quantization condition for the spin-2 field such as gravity [91]:

pμp˜μ=nħc34GN(4),     n.    (106)

where the quantities pμ and p~μ are 4-momenta of linearized diffeomorphisms associated with the ordinary (electric-type) and magnetic-type matters, respectively. For a massive particle, we get pμ = muμ and p~μ=m~ũμ, where m is the ordinary (electric-type) mass, uμ is the 4-velocity of the electric-type source, m~ is the magnetic-type mass, and ũμ is the 4-velocity of the magnetic-type source.

It is important to note that the Dirac quantization condition (Equation 104) is applied to the charges in the spin-1 case, whereas the spin-2 quantization condition (Equation 106) is imposed on 4-momenta, not the mass. In the rest frame, we get a quantization condition:

EE˜=nħc54GN(4),       n.    (107)

where E is the ordinary (electric-type) energy, and is the magnetic-type energy. Hence, we get the energy quantization condition [91, 135, 136] (see also [77, 78, 137] for a quantization condition for ordinary mass and gravitomagnetic mass).

The Einstein equations coupled to the gravitating source are

Gμν=8πGN(4)c4Tμν,    (108)

which are equivalent to

Rμν=8πGN(4)c4(Tμν12ημνTρρ),    (109)

where Tμν is the energy-momentum tensor of the ordinary (electric-type) matter.

In section 2.1, it was shown that the Pauli-Fierz field equations are duality invariant in D = 4, so the Einstein tensor G~μν for the dual gravity h~μν can be coupled to the magnetic-type energy-momentum tensor T~μν as follows:

G˜μν=8πGN(4)c4T˜μν,    (110)

or equivalently to the dual Ricci curvature:

R˜μν=8πGN(4)c4(T˜μν12ημνT˜ρρ).    (111)

The electric-magnetic duality rotations (Equations 41, 42) of the curvatures imply that there are the SO(2) rotations of the electric-type and magnetic-type energy-momentum tensors:

Tμν=cosαTμν+sinαT˜μν,    (112)
T˜μν=sinαTμν+cosαT˜μν.    (113)

Hence, the Einstein equations are invariant under the electric-magnetic duality in D = 4.

Considering D-dimensional spacetime, the linearized Einstein equations become

Gμν=8πGN(D)c4Tμν,    (114)

which are equivalent to [38]:

Rμρνληρλ=8πGN(D)c4(Tμν+1D2ημνTρρ),    (115)

where Tμν is the energy-momentum tensor of the electric-type source.

The conservation laws for the electric-type and magnetic-type sources impose that νTμν=0 and νT~μν=0, so the motion of a massive particle should follow straight lines. For a massive particle, one can write

Tμν=muμdτδ(4)[xw(τ)]wντ,    (116)
T˜μν=m˜u˜μdτδ(4)[xw˜(τ)]w˜ντ,    (117)

where wν and w~ν represent the electric- and magnetic-type worldlines, respectively, uμ = dwμ/ds and ũμ=dw~μ/ds are the 4-velocities of the electric- and magnetic-type sources, and m and m~ the electric- and magnetic-type masses, respectively. The quantity s is spacelike, and τ is timelike. From Equations (116, 117), one obtains Tμν=δ(3)(x-w(x0))uμuν/u0 and T~μν=δ(3)(x-w~(x0))ũμũν/ũ0.

Dirac strings in D = 4 can be constructed as follows. Let decompose the linearized Riemann curvature Rμναβ into the massless part and the massive part,

Rμναβ=rμναβ+mμναβ.    (118)

The massless part rμναβ satisfies the cyclic and Bianchi identities (rνα = 0 and ∂rμν]αβ = 0), and the massive part mμναβ is coupled to the magnetic-type source

mμναβ=12ϵμνρσ[α(M˜ρσβ]+δ[ρβ]M˜σ]λλ),    (119)

where M~αμν defines the magnetic-type source as

16πGN(4)c4T˜μν=αM˜αμν,    (120)

and M~αβμ=M~[αβ]μM~αβ|μ is a tensor with the mixed symmetry type (2, 1).

There is a symmetric tensor hμν that satisfies

rμναβ=2[α[μhν]β].    (121)

If we define yαβμ=ϵαβγλγhλμ, where yαβμ=y[αβ]μ is a mixed symmetry type (2, 1) tensor (αyαβμ=0), the linearized Riemann curvature Rμναβ can be rewritten as

Rμναβ=12ϵμνρσ[α(yρσβ]+δ[ρβ]yσ]λλ               +M˜ρσβ]+δ[ρβ]M˜σ]λλ).    (122)

Defining Yμνα=yμνα+M~μνα (where Yμνα = Y[μν]α) yields

Rμναβ=12ϵμνρσ[α(Yρσβ]+δ[ρβ]Yσ]λλ).    (123)

The tensor M~αβ|μ defined by (120) can be written using a Dirac string yμ(τ, θ) attached to the magnetic-type source, yμ(τ,0)=w~μ(τ) in (117). One can take [91]

M˜αβ|μ=16πGN(4)c4m˜u˜μdτdθδ(4)[xy(τ,θ)]                 ×(yαθyβτyατyβθ).    (124)

From Equations (117) and (124), it can be seen that the divergence of M~αβ|μ is identical with T~μν (by a factor of 16πGN(4)/c4). The momentum m~ũμ of the magnetic-type source by its mass m~ and 4-velocity ũμ is therefore conserved.

Similarly, the linearized Einstein equations of the dual graviton h~α1···αD-3|β in D dimensions with the magnetic-type source read

G˜α1···αD3|β=8πGN(D)c4T˜α1···αD3|β,    (125)

where G~α1···αD-3|β is the dual Einstein tensor, T~α1α2···αD-3|β is the energy-momentum tensor of the magnetic-type source in D-dimensional spacetime, both G~α1···αD-3|β and T~α1α2···αD-3|β are tensors with the mixed symmetry type (D − 3, 1).

The linearized dual Einstein Equations (125) are equivalent to

R˜α1ρα2λ···αD3μηρλ=8πGN(D)c4(T˜α1···αD3μ                                      +(D3)2ημ[α1T˜α2···αD3]ρρ),    (126)

where R~α1α2···αD-2μν is the linearized dual Riemann curvature defined by Equation (10).

In the little group SO(D − 2), the Einstein tensor and its dual tensor in D dimensions satisfy (see [38, 138] for further discussions):

Gij=8πGN(D)c41(D3)!ϵk1···kD3iT˜k1k2···kD3|j    (127)
G˜i1···iD3|j=8πGN(D)c4ϵi1···iD3kTjk,    (128)

If the magnetic-type energy-momentum tensor satisfies

T˜i1i2···iD3|j=[i1M2···iD3]j,    (129)

where Mi2···iD − 3j is an arbitrary tensor with the mixed symmetry type (D − 4, 1), and the electric-type energy-momentum tensor can be written as follows:

Tij=1(D3)!ϵk1···kD3i[k1Mk2···kD3]j,    (130)

so the magnetic-type source is correlated to the Hodge dual of the electric-type source:

T˜i1···iD3|j=ϵi1···iD3kTjk,    (131)
Tij=1(D3)!ϵk1···kD3iT˜k1···kD3j.    (132)

The magnetic-type energy-momentum tensor T~i1···iD-3|j can be a source for the gravitational Bianchi identity [38]

R[ijm]n=1(D3)!ϵijmk1k2···kD3T˜k1k2···kD3|n.    (133)

Similarly, the energy-momentum tensor Tij is a source for the Bianchi identity of the dual Riemann curvature [38]

R˜[i1i2···iD1]j=ϵi1···jD1kTjk.    (134)

It can be seen that the Einstein Equations (127, 128), and the Bianchi identities (133) and (134) are invariant under the twisted self-dual conditions discussed in section 2.1. We can express their electric-magnetic duality transformations using matrix notations

=S,     T=ST.    (135)

where the SO(2) electric-magnetic duality matrix S is defined by Equation (38), 𝔗 and ℜ represent the D-dimensional energy-momentum tensor and Ricci curvature associated with both the electric-type and magnetic-type sources, respectively,

T=(TijT˜n1···nD3|m),     =(RijR˜n1···nD3|m),    (136)

and 𝔗 and ℜ represent their corresponding Hodge isomorphisms,

T=(ϵn1···nD3lTml1(D3)!ϵk1···kD3iT˜k1k2···kD3|j),=(ϵn1···nD3lRml1(D3)!ϵk1···kD3iR˜k1k2···kD3|j).    (137)

It means that the equations of motion for the D-dimensional dual graviton h~i1···iD-3|j are equivalent to those for the graviton hij under the SO(2) electric-magnetic invariance.

2.6. Spin-2 Harmonic Condition

In this section, we introduce the harmonic coordinate condition, which makes it possible to propagate gravitational waves (ρρhμν=0) in linearized gravity. We will see that the harmonic condition is equivalent to divergenceless and nonvanishing curl of the electric and magnetic parts of the linearized Weyl tensor encoded by the Pauli-Fierz field hμν in the instantaneous rest-space of an observer moving in flat spacetime (defined by Equations 27, 28).

We consider the vacuum Einstein equations in the linearized flat condition (Rμν = 0 and R = 0), which imply

Rμν=12ρρhμν+ρ(μhν)ρ12νμhρρ=0,    (138)
R=μνhμνμμhνν=0,    (139)
Cμναβ=2[μ[αhβ]ν].    (140)

Equation (138) presents a second-order equation for the Pauli-Fierz field hμν. To have a wave equation for the Pauli-Fierz field, ρρhμν=0, it is required to have a constraint, the so-called De Donder gauge condition [103]:

Dμνhμν12μhρρ=0.    (141)

Substituting Equation (141) into (138) leads to the wave equation ρρhμν=0.

We can have some constraints along a four-velocity vector uμ in the rest-space of a local co-moving observer in Minkowski spacetime:

Rμνuμ=12ρρh0ν+12ρ(hνρ)   +12νρh0ρ12ν(hρρ)=0,    (142)
Rμνuμuν=12ρρh00+(ρh0ρ)12(hρρ)=0,    (143)
uμDμ=ρh0ρ12(hρρ)=0.    (144)

From Equation (140), the electric and magnetic parts of the linearized Weyl tensor in the vacuum Einstein equations become

Eμν[h] =12[μνh002((μhν)0)+(hμν)],    (145)
Bμν[h]=12ϵρλ(μ(ρhν)λ)+12ϵρλ(μν)ρh0λ.    (146)

Taking a covariant divergence from the gravitoelectric field (145) and the gravitomagnetic field (146), and using constraints (Equations 142 –144), one obtains the covariant divergenceless condition for the gravitoelectric and gravitomagnetic fields:

μEμν=0=μBμν.    (147)

In Minkowski spacetime, the covariant derivative of a spatial, traceless symmetric 2nd-rank tensor is decomposed into the time derivative along with a four-velocity vector uμ and the spatial derivative, μEμν=-uμ(Eμν)+DμEμν (see e.g., [66]). As the gravitoelectric and gravitomagnetic tensor are spatial (Eμνuν=0=Bμνuν), one gets uμ(Eμν)=0=uμ(Bμν), so we get the spatial divergenceless condition (DμEμν=0=DμBμν), which were previously proven for gravitational waves in the 1+3 covariant formalism [66, 139, 140].

It can easily be verified that the covariant curl and distortion of the gravitoelectric field (145) and the gravitomagnetic field (146) do not vanish under the De Donder gauge condition and the linearized flat condition:

curl(E)μν0curl(B)μν,      (ρEμν)0(ρBμν),    (148)

where the covariant curl is defined as curl(E)μνϵρλ(μρEν)λ. The nonvanishing covariant curl and distortion conditions are equivalent to the nonzero, spatial curls and distortions of Eμν and Bμν for gravitational waves in the 1 + 3 covariant formalism [66, 67, 141]: curl(E)μν ≠ 0 ≠ curl(B)μν and DρEμν ≠ 0 ≠ DρBμν, where the spatial curl is defined using the spatial derivative Dμ as curl(E)μνϵρλ(μDρEν)λ. From Equations (147, 148), the wave equation of the Pauli-Fierz field hμν in empty space, ρρhμν=0, corresponds to ρρEμν=0=ρρBμν in the linearized theory, which are equivalent to D2Eμν-(Eμν)=0=D2Bμν-(Bμν) in the 1 + 3 covariant formalism [60, 66]. Hence, the De Donder gauge condition in the vacuum Einstein equations allows the propagation of the spin-2 waves, ρρhμν=0, and imposes divergenceless and nonvanishing-curl of the gravitoelectric and gravitomagnetic fields encoded by the Pauli-Fierz field (Equations 27 and 28) of the linearized theory in Minkowski spacetime.

To satisfy the De Donder gauge condition (141), the Pauli-Fierz field could be assumed to be transverse, traceless, and spatial (e.g., [39]):

νhμν=0, hρρ=0, uνhμν=0,    (149)

which is called the transverse–traceless gauge condition, and allows the propagation of a massless, spin-2 field in Minkowski flat spacetime.

In the case of the transverse–traceless gauge condition (149), the harmonic, massless spin-2 field of the wave equations (see e.g., [39]) in the momentum space mμ=(|m|,m) reads

hμν(m)=h+(m)eμν+(m)+h×(m)eμν×(m),    (150)

where h+ and h× are the helicity states associated with the + and × polarization states of the Pauli-Fierz field hμν (graviton), eμν+=xμxν-yμyν and eμν×=xμyν+yμxν are the + and × polarization tensors, xμ=(0,x^) and yμ = (0, ŷ) are spatial 4-vectors satisfying x^·x^=1=ŷ·ŷ and x^·ŷ=0 with x^·m^=0= ŷ·m^ making an orthogonal triad ŷ=x^×m^ (i.e., x^=-ŷ×m^) where m^=m/|m|.

Since the equations of motion for the Pauli-Fierz field hμν are fully equivalent to the dual Pauli-Fierz field h~μν under SO(2) electric-magnetic duality rotations, we have the following expression for the dual Pauli-Fierz field h~μν in D = 4:

h˜μν(z)=h˜+(m)eμν+(m)+h˜×(m)eμν×(m),    (151)

where h~+ and h~× are the helicity states associated with the + and × polarization states of the dual Pauli-Fierz field h~μν(dual graviton).

Considering the first dual formulation, and taking a 2-D permutation ϵμν from the expression (150) yields h˜μν(m)=h+(m)ϵμρeρν+(m)+h×(m)ϵμρeρν×(m). As the spatial vectors x^ and ŷ making an orthogonal triad with m^, one gets ϵμρxρ(m)=yμ(m) and ϵμρyρ(m)=xμ(m) in the momentum space mμ (equivalent to x^ × m^ = ŷ and ŷ×m^=-x^), so we get ϵμρeρν+(m)=eμν×(m) and ϵμρeμν×(m)=eμν+(m):

h˜μν(m)=h+(m)eμν×(m)h×(m)eμν+(m),    (152)

by comparison with Equation (150), the dual helicity states in the momentum space have the following rotations:

h+(m)=h˜×(m),      h×(m)=h˜+(m),    (153)

which is similar to Equation (34). We see that the + and × helicity states of the graviton hμν and the dual graviton h~μν are fully equivalent to each others under the following twisted self-dual condition in the momentum space:

h+(m)=Sh×(m),      h×(m)=Sh+(m).    (154)

with S defined by (38), and matrix notations of the helicity states:

h+(m)=(h+(m)h˜+(m)),        h×(m)=(h×(m)h˜×(m)),    (155)

which is analogous with the twisted self-dual conditions (35). We can also show that the helicity states of the graviton hμν is rotated into those of the dual graviton h~μν under the duality transformations:

h+(m)=cosαh+(m)sinαh×(m),    (156)
h×(m)=sinαh+(m)+cosαh×(m).    (157)

Thus, the + and × helicity states of wave equations for the dual graviton (ρρh~μν=0) are respectively rotated to the × and + helicity states of gravitational waves (ρρhμν=0) under SO(2) electric-magnetic duality rotations in D = 4, and are duality invariant under the twisted self-dual condition.

3. Spin-3 Field and Its Dual Fields

We now introduce the spin-3 field, as a step toward the spin-s generalization. The typical spin-3 field in D-dimensional spacetime is represented by a symmetric 3rd rank tensor hμνρ = hνρ) with Young symmetry (1, 1, 1) and the diagram yes. The equations of motion of hμνρ (the ordinary spin-3 field) in D-dimensional flat spacetime can be described using the Fronsdal action [102] for the spin-3 case

Sspin-3=12dDxL[hμνρ],    (158)

where the Lagrangian L[hμνρ] is as follows [47, 142, 143]

L[hμνρ]=23[λhμνρλhμνρ3μhμνρλhλνρ                                +6hλλμνρhνρμ3λhμμνλhρρν                                 32λhλμμνhνρρ].    (159)

The Fronsdal Lagrangian (159) is a generalization of the Pauli-Fierz theory on the spin-3 field, and is invariant under the gauge transformation

δξhμνρ=3(μξνρ),    (160)

where the gauge parameter ξμν = ξ(μν) is an arbitrary symmetric 2rd rank field.

By analogy with the Einstein equations, one can write the spin-3 Einstein equations in D dimensions (see [91] for the generalized Einstein equations):

Gμ1μ2μ3=0,    (161)

where the spin-3 Einstein tensor Gμ1μ2μ3 = G(μ1μ2μ3) is a traceless (Gμρρ=0), symmetric tensor with the Young diagram yes. The tensor Gμ1μ2μ3 can be written in terms of the so-called Fronsdal [102] tensor Fμ1μ2μ3 for the spin-3 case as follows

Gμ1μ2μ3=Fμ1μ2μ332η(μ1μ2Fμ3)ρρ,    (162)

where the spin-3 Fronsdal tensor Fμ1μ2μ3 = F(μ1μ2μ3) is a symmetric tensor with the diagram yes, defined by

Fμ1μ2μ3=ρρhμ1μ2μ33(μ1ρhμ2μ3)ρ+3(μ1μ2hμ3)ρρ,    (163)

which is related to the spin-3 Ricci tensor

Rμ1ν1|μ2μ3Rμ1ν1μ2ν2μ3ν3ην2ν3=2[ν1Fμ1]μ2μ3,    (164)

where the spin-3 Ricci tensor Rμ1ν12μ3 is a 4th-rank tensor with the mixed symmetry (2, 1, 1) having the Young diagram yes, and the spin-3 Riemann tensor Rμ1ν1μ2ν2μ3ν3 is a 6th-rank tensor with the mixed symmetry (2, 2, 2) having the Young diagram yes.

For spin-3 fields, we can define the first dual representation in a similar manner of the dual definitions for the spin-2 field, i.e., Equations (3, 4) [38, 89, 98]. The first dual field of the spin-3 field hμνρ = hνρ) in D-dimensional spacetime is dualized on one index only. For the physical gauge spin-3 field hijk, we have in the little group SO(D − 2) [38]:

h˜i1···iD3jk=ϵi1···iD3lhljk,    (165)
hjkl=1(D3)!ϵi1···iD3jh˜i1···iD3kl,    (166)

with the following properties [38]:

h˜i1···iD3jk=h˜[i1···iD3]jkh˜i1···iD3|jk,      h˜[i1···iD3j]k=0.    (167)

The first dual spin-3 field h~α1···αD-3μν is then described by a tensor with Young symmetry (D − 3, 1, 1) having the Young diagram:

yes

For spin-3 fields, we may also define the second dual formulation in a similar manner of the spin-2 dual field (6) and (7)6. In the second dual formulation, the physical gauge spin-3 field hijk in D-dimensional spacetime is dualized on two indices:

h^i1···iD3j1···jD3k=ϵi1···iD3mϵj1···jD3nhmnk,    (169)
hmnl=1(D3)!(D3)!ϵmi1···iD3ϵnj1···jD3h^i1···iD3j1···jD3l,    (170)

having the following properties:

h^i1···iD3j1···jD3k=h^[i1···iD3][j1···jD3]k,   h^[i1···iD3|j1]j2···jD3k=0.    (171)

The second dual spin-3 field ĥα1···αD − 3β1···βD − 3ρ is described by a tensor with the mixed symmetry (D − 3, D − 3, 1) having the Young diagram:

yes

It might be possible to define a third dual field of the spin-3 field, which is dualized on all three indices. The physical gauge spin-3 field hijk in D dimensions can be dualized in the little group SO(D − 2) as follows:

hi1···iD3j1···jD3k1···kD3=ϵi1···iD3mϵj1···jD3nϵk1···kD3lhmnl,    (173)
hmnl=1(D3)!(D3)!(D3)!(ϵi1···iD3mϵj1···jD3n                   ×ϵk1···kD3lhi1···iD3j1···jD3k1···kD3),    (174)

having the following properties:

hi1···iD3j1···jD3k1···kD3=h[i1···iD3][j1···jD3][k1···kD3],                  h[i1···iD3|j1]j2···jD3k1···kD3=0.    (175)

The third dual spin-3 field hα1···αD-3β1···βD-3γ1···γD-3 is described by a tensor with the mixed symmetry (D − 3, D − 3, D − 3).

In particular, we notice that there are two different dual formulations for the spin-2 field in section 2, whereas three different dual formulations for the spin-3 field exist. In D = 4, all three dual formulations correspond to a symmetric 3rd rank tensor h~μνρ=h~(μνρ) with the Young diagram yes. Thus, we get only one dual formulation of the spin-3 field in 4-dimensional spacetime.

The second dual physical gauge field ĥ and the third dual physical gauge field h can be written in terms of the first dual physical gauge field h~ and second dual physical gauge field ĥ, respectively, and one can recover the first dual physical gauge field h~ from the second and third dual physical gauge field in the little group SO(D − 2). Nevertheless, the action principle of the second field h~ and third dual field h in D dimensions higher than 4 may not follow from a Lorentz-invariance action (similar to the argument made for the spin-2 field [40]), so the first dual field formulation, which is dualized on one index only, is considered in the following parts of this section.

The Fronsdal theory for the spin-3 case in D = 4 is dualized to a Fronsdal action S=d4xL[h~μνρ] where the Lagrangian L[h~μνρ] is formulated in terms of a symmetric massless 2nd-rank tensor h~μνρ=h~(μνρ) (dual spin-3 field), similar to Equation (159) with the gauge transformation δξh˜μνρ=3(μξ˜νρ) where ξ~μν is a arbitrary, symmetric 2nd rank tensor.

3.1. D = 5: Mixed Symmetry (2, 1, 1) Field

The dual field of the spin-3 field hμνλ in 5-dimensional spacetime is represented by h~αβ|μν, which is a tensor with Young symmetry (2, 1, 1) with the diagram:

yes

and satisfies:

h˜αβμν=h˜[αβ](μν)h˜αβ|μν, h˜[αβμ]ν=0.    (177)

The equations of motion of a free massless spin-3 field h~αβ|μν in 5-dimensional Minkowski spacetime ημν = diag(−, +, +, +, +) is represented by the following action principle [47] 7

L[h˜μν|ρσ]=13[λh˜μν|ρσλh˜μν|ρσ2λh˜λν|ρσμh˜μν|ρσ                               2ρh˜μν|ρσλh˜μν|λσ+8h˜λν|λσμρh˜μν|ρσ                              +2h˜μν|λλρσh˜μν|ρσ4ρh˜λν|λσρh˜μν|μσ                               λh˜μν|ρρλh˜μν|σσ4ρh˜ρν|μμσh˜λν|λσ                              +σh˜σμ|ρργh˜γμ|αα+2σh˜σν|λσρh˜ρν|λσ                              +2γh˜λγ|λσρh˜μρ|μσ],    (178)

which satisfies the gauge symmetry

δχ,φh˜μν|ρσ=2(2[μχν]ρσ                         +[μφν]ρ|σ+[μφν]σ|ρ2(ρφμν|σ)    )    ,    (179)

where χμνρ = χνρ) is an arbitrary symmetric tensor with Young symmetry (1, 1, 1) ≡ yes, and φμν|ρ = φ[μν]ρ is an arbitrary tensor with Young symmetry (2, 1) ≡ yes (see also gauge transformations in [47, 143]). The gauge transformations for χμ|νρ and φμν|ρ are reducible:

δσχμνρ=3(μσνρ),    (180)
δσ,αφμν|ρ=2([μσν]ρ+[μαν]ρραμν).    (181)

The parameters σμν = σ(μν) and αμν = α[μν] are symmetric and antisymmetric arbitrary tensors, respectively, and their gauge transformations are reducible:

δσσμν=6(μσν),       δσαμν=2[μσν].    (182)

The parameter σμ is an arbitrary vector field, and its gauge transformation is irreducible.

3.2. D > 5: Young Symmetry (D − 3, 1, 1) Fields

The D-dimensional spin-3 field hαβρ can be dualized to a free massless spin-2 fields h~[α1···αD-3]μνh~α1···αD-3|μν with Young symmetry type (D − 3, 1, 1) defined by (Equation 165, 166), and satisfying

h˜α1α1D3|μν=h˜[α1···αD3]|μν=h˜α1···αD3|(μν),                        h˜[α1···αD3|μ]ν=0,    (183)
h˜α1···αD3|μνημν=0,         h˜α1···αD3|μνηα1α2ηα3α4=0.    (184)

This is associated with the dualization on one index only, which provides equations of motion with a Lorentz invariant action. The equations of motion of free massless, dual spin-3 fields h~α1···αD-3|βρ in D-dimensional flat spacetime are represented by the action principle [47]:

L[h˜α1...αD3|μν] =23(D3)![ρh˜α1αD3|μνρh˜α1αD3|μν        (D3)ρh˜ρα2αD3|μνσh˜σα2αD3|μν        2ρh˜α1αD3|ρμσh˜α1αD3|σμ        +4(D3)h˜γα2αD3|γμρσh˜ρα2αD3|σμ        +2h˜α1αD3|γγσρh˜α1αD3|σρ        2(D3)ρh˜γα2αD3|γμρh˜σα2αD3|σμ        ρh˜α1αD3|σσρh˜α1αD3|γγ                2(D3)σh˜ρα2αD3|ρσγh˜γα2αD3|ββ        +12(D3)σh˜σα2αD3|ρργh˜γα2αD3|ββ        +(D3)(D4)σh˜σα2αD3|γμρh˜ρα2αD3|γμ        +(D3)(D4)ρh˜γρα3αD3|γμσh˜λσα3αD3|λμ],          (185)

which satisfies the gauge symmetry

δχ,φh˜α1···αD3=|μν2(D3)([α1χα2...αD3]|μν                                  +[α1φα2...αD3](μ|ν)(μφα1...αD3|ν)),    (186)

where χα1…αD − 4|μν = χ[α1…αD − 4]μν = χα1…αD − 4(μν) is an arbitrary tensor with the mixed symmetry (D − 4, 1, 1), and φα1…αD − 3 = φ[α1…αD − 3 is an arbitrary tensor with the mixed symmetry (D − 3, 1). The gauge transformations for χα1…αD − 4|μν and φα1…αD − 3 are reducible.

The action (185) is also invariant under the following gauge symmetry [47]:

δχ,φh˜α1···αD3=|μν(D3)([α1χα2...αD3]|μν                                  +2(μχ[α1...αD4|αD3]ν)),    (187)

where χα1…αD − 4|μν is a tensor with the mixed symmetry (D − 4, 1, 1), and its gauge transformation is reducible.

In a similar way of the Curtright action (47) and the generalized Curtright action (88), one may write the action of the dual spin-3 field h~α1···αD-3|μν using a “spin-3 Curtright” field strength tensor F̄α1αD-2|μν defined as

F¯α1...αD2|μν(D2)[α1h˜α2···αD2]μν.    (188)

is a tensor with Young symmetry type (D − 2, 1, 1):

yes

The spin-3 Curtright action should be invariant under the gauge symmetry (186), however, the field strength tensor F̄α1αD-2|μν is not gauge invariant. The field strength tensor F̄α1αD-2|μν has the following transformation

δαF¯α1...αD2|μν=2(D2)(D3)(μ[α1φα2...αD2|ν).    (190)

Note that the Curtright field strength tensor F̄α1αD-2|μν defined by Equation (188) should be not mistaken with the Fronsdal tensor F~α1···αD-3|μν defined by (198) in this section.

The spin-3 dual Riemann tensor defined as

R˜α1···αD2μ2ν2μ3ν3=12ϵα1···αD2μ1ν1Rμ1ν1μ2ν2μ3ν3,    (191)

is a tensor with Young symmetry (D − 2, 2, 2) and the diagram:

yes

For a free, massless dual spin-3 field h~α1···αD-3|μν, one can write the spin-3 dual Einstein equations similar to the spin-2 case [38]:

G˜α1···αD3|μν=0,    (193)

where the spin-3 dual Einstein tensor G~α1···αD-3|μν is of Young symmetry type (D − 3, 1, 1), and traceless (G~α1···αD-3|μνημν=G~α1···αD-3|μνηαD-3ν=0) with the following definition:

G˜α1···αD3|μν=F˜α1···αD3|μν12[2(D3)η[α1(μF˜α2···αD3]ρ|ν)ρ                             +ημνF˜α1···αD3|ρρ],    (194)

where both the spin-3 dual Fronsdal tensor F~α1···αD-3|μν and the spin-3 dual Einstein tensor G~α1···αD-3|μν are of Young symmetry type (D − 3, 1, 1), and the diagram

yes

The spin-3 dual Ricci tensor can also be defined by

R˜α1···αD2|μ2μ3R˜α1···αD2|μ2ν2μ3ν3ην2ν3,    (196)

where the spin-3 dual Ricci tensor is of Young symmetry (D − 2, 1, 1), and defined in term of the dual Fronsdal tensor F~α1···αD-3|μν as follows:

R˜α1···αD2|μν=(D2)[α1F˜α2···αD2]μν.    (197)

The dual Ricci tensor R~α1···αD-2|μν is antisymmetrized in α1···αD − 2 and symmetrized in μν, i.e., R~α1···αD-2|μν=R~[α1···αD-2]μν=R~α1···αD-2(μν).

The Fronsdal tensor F~α1···αD-3|μ1μ2 can be written in term of the dual spin-3 field h~α1···αD-3|μν as follows:

F˜α1···αD3|μν=ρρh˜α1···αD3|μν(D3)[α1ρh˜α2···αD3]ρ|μν                          2(μρhα1···αD3|ν)ρ                           +2[(D3)[α1(μh˜α2···αD3]ρ|ν)ρ                            +12μνh˜α1···αD3|ρρ].    (198)

The dual Einstein tensor G~α1···αD-3|μν and the dual Fronsdal tensor F~α1···αD-3|μν for the spin-3 field are antisymmetrized in indices α1···αD − 3 and symmetrized in indices μν, i.e., G~α1···αD-3|μν=G~[α1···αD-3]μν=G~α1···αD-3(μν) and F~α1···αD-3|μν=F~[α1···αD-3]μν=F~α1···αD-3(μν).

3.3. Spin-3 Magnetic-Type Source

The spin-3 field (hμνρhνρ)) can be coupled to the spin-3 electric-type source TμνρTνρ) [38, 91]. For the spin-3 Riemann tensor Rμ1ν1μ2ν2μ3ν3, one can write the field equations in D-dimensional spacetime as follows

Rμ1ν1|μ2μ3Rμ1ν1μ2ν2μ3ν3ην2ν3=12[ν1Tμ1]μ2μ3,    (199)

where the spin-3 Ricci tensor Rμ1ν12μ3 = R[μ1ν12μ3 = Rμ1ν12μ3) is a tensor with mixed symmetry (2, 1, 1), defined by (164), and Tμνρ=T(μνρ) is a traceless (Tμρρ=0), symmetric 3rd rank tensor defined by removing the traces Tμρρ from the energy-momentum tensor Tμνρ:

Tμ1μ2μ3Tμ1μ2μ334η(μ1μ2Tμ3)ρρ.    (200)

By analogy with the Einstein equations, one can write an equivalence of Equation (199):

Gμ1μ2μ3=Tμ1μ2μ3,    (201)

which can be considered as the spin-3 Einstein equations with the spin-3 electric-type source (see [91] for spin-s generalization of the Einstein equations).

The dual Einstein tensor (193) for the dual spin-3 field can be coupled to the spin-3 magnetic-type source:

G˜α1···αD3|μν=T˜α1···αD3|μν,    (202)

where the spin-3 dual stress-energy tensor T˜α1···αD3|μν is a traceless (T˜α2···αD3ρ|νρ=0), tensor with Young symmetry type (D − 3, 1, 1) and defined as

T˜α1···αD3|μνT˜α1···αD3|μν(D3)2η[α1(μT˜α2···αD3]ρ|ν)ρ,    (203)

where T˜α1···αD3|μν is the spin-3 dual stress-energy, and has Young symmetry type (D − 3, 1, 1). Both T˜α1···αD3|μν and T˜α1···αD3|μν are antisymmetrized in α1···αD − 3, and symmetrized in μν.

The dual Einstein tensor (202) is equivalent to

R˜α1···αD2|μ2μ3R˜α1···αD2μ2ν2μ3ν3ην2ν3=12[α1T˜α2···αD2]μ1μ2,    (204)

where the spin-3 dual Riemann tensor R~α1···αD-2μ2ν2μ3ν3 is a tensor with Young symmetry type (D − 2, 2, 2), and the spin-3 dual Ricci tensor R~α1···αD-2|μ2μ3R~α1···αD-2μ2ν2μ3ν3 is a tensor with Young symmetry type (D − 2, 1, 1), defined by (197).

In the little group SO(D − 2), we may write the following relations

Gi1i2i3=1(D3)!ϵj1···jD3i1T˜j1···jD3|i2i3,    (205)
G˜j1···jD3|i1i2=ϵj1···jD3i3Ti1i2i3.    (206)

It can easily verify that the spin-3 Einstein equations (201) and (202) are invariant under the twisted self-dual conditions 𝔊=S𝔊 and 𝔗=S𝔗 if we employ the following matrix notations

T=(Ti1i2i3T˜j1···jD3|i1i2),          G=(Gi1i2i3G˜j1···jD3|i1i2),    (207)
T=(ϵj1···jD3i3Ti1i2i31(D3)!ϵj1···jD3i1T˜j1j2···jD3|i2i3),G=(ϵj1···jD3i3Gi1i2i31(D3)!ϵj1···jD3i1G˜j1j2···jD3|i2i3),    (208)

and the matrix S is defined by Equation (38).

Following Equations (116) and (117), for a spin-3 charged particle in D = 4, the spin-3 stress-energy tensors, which are coupled to its field, read

Tμνρ=mUνρdτδ(4)[xw(τ)]wμτ,    (209)
T˜μνρ=m˜U˜νρdτδ(4)[xw˜(τ)]w˜μτ,    (210)

where m is the electric-type spin-3 charge, and m~ is the magnetic-type spin-3 charge, U〈μν〉u〈μuν〉 denotes the traceless part of uμuν, and Ũ〈μν〉 ≡ ũ〈μũν〉 the traceless part of ũμũν (the angle brackets denote the traceless part; see e.g., [66, 144] for the same notation). For example, one defines traceless parts of symmetric tensors Sαβ and Sαβρ in 4-dimensional flat spacetime as follows

Sμν(η(μαην)β14ημνηαβ)Sαβ,    (211)
Sμνρ(η(μαην)β14ημνηαβ)Sαβρ,    (212)

so the traceless part of uμuν is defined by

Uμνuμuν(η(μαην)β14ημνηαβ)uαuβ.    (213)

Equations (209, 210) can be rewritten as

Tμνρ=muμUνρu0δ(3)[xw(x0)],    (214)
T˜μνρ=m˜u˜μU˜νρu0δ(3)[xw˜(x0)].    (215)

It can be easily varified that these stress-energy tensors are conserved.

One can write a quantization condition for the spin-3 field in D = 4 similar to the Dirac quantization condition (104) and the spin-2 quantization condition (106) as follows (see [91] for spin-s fields):

PμνP˜μν=n12ħc,     n,    (216)

where the conserved spin-3 charges are defined by

Pμν=mUμν,    P˜μν=m˜U˜μν.    (217)

We now consider the construction of a Dirac string in the spin-3 theory in D = 4. The spin-3 Riemann curvature Rμ1ν1μ2ν2μ3ν3 can be split into the chargeless and charged terms,

Rμ1ν1μ2ν2μ3ν3=rμ1ν1μ2ν2μ3ν3+mμ1ν1μ2ν2μ3ν3.    (218)

The chargeless term rμ1ν1μ2ν2μ3ν3 is of mixed symmetry type (2, 2, 2), and satisfies the cyclic and Bianchi identities. The charged term mμ1ν1μ2ν2μ3ν3 is of mixed symmetry type (2, 2, 2), and may be coupled to the spin-3 magnetic-type source

mμ1ν1μ2ν2μ3ν3=12ϵμ1ν1ρσ([μ2[μ3M˜ρσν2]ν3]                                                        +[μ2δ[ρν2][μ3|M˜σ]λ|ν3]λ).    (219)

The tensor M~αβ|μν, which represents the spin-3 magnetic-type source, is of Young symmetry type (2, 1, 1) ≡ yes. The notations [ | and | ] separate different antisymmetrized indices from each other, e.g., 4[μ1|[μ2|S|ν1]|ν2]4[μ1[μ2Sν1]ν2].

The chargeless term rμ1ν1μ2ν2μ3ν3 may be rewritten in terms of covariant derivatives of a symmetric tensor hμ1μ2μ3:

rμ1ν1μ2ν2μ3ν3=2[μ3|[μ2[μ1hν1]ν2]|ν3].    (220)

Let us define yαβ|μρ=ϵαβγλγhλμρ, where yαβ|μρ=y[αβ]μρ=yαβ(μρ) is of Young symmetry type (2, 1, 1) tensor (αyαβ|μρ=0), the spin-3 Riemann curvature Rμ1ν1μ2ν2μ3ν3 can be defined by

Rμ1ν1μ2ν2μ3ν3=12ϵμ1ν1ρσ[μ2|[μ3|(yρσ|ν2]|ν3]+δ[ρ|ν2]yσ]λ|λ|ν3]                                     +M˜ρσ+|ν2]|ν3] δ[ρ|ν2]M˜σ]λ|λν3]).    (221)

If we take Yαβ|μν=yαβ|μν+M~αβ|μν (where Yμνα = Y[μν]α), it is obtained

Rμ1ν1μ2ν2μ3ν3=12ϵμ1ν1ρσ[μ2[μ3(Yρσν2]ν3]+δ[ρν2]Yσ]λ|ν3]λ).    (222)

A Dirac string yμ(τ, θ) can be connected to the spin-3 magnetic-type source M~αβ|μν, yμ(τ,0)=w~μ(τ) in (210). By analogy with (124), one can write

M˜αβ|μν=m˜U˜μνdτdθδ(4)[xy(τ,θ)](yαθyβτyατyβθ),    (223)

where the tensor M~αβ|μν defines the spin-3 magnetic-type source as

T˜μνρ=αM˜αμ|νρ,    (224)

so the divergence of M~αμ|νρ is identical with T~μνρ, and the Dirac string of a spin-3 magnetic-type point source is conserved. One can prove the conserved charges associated with a spin-3 electric-type point source (see [91] for complete proof of the spin-s conserved charges with magnetic-type and electric-type point sources).

3.4. Spin-3 Harmonic Condition

The spin-2 harmonic coordinate condition can be generalized to the spin-3 field hμνλ that allows the propagation of a spin-3 particle in empty space. To have a wave equation for the spin-3 fields in empty space, ρρhμνλ=0, it is necessary to eliminate the last two of the spin-3 Fronsdal equation (163):

Dμνρhμνρμhνρρ=0,    (225)

which is invariant under the gauge transformation (160):

δξDμν=3ρρξμν.    (226)

Equation (225) is the De Donder gauge condition for the spin-3 field hμνλ. To have the De Donder gauge condition, one way is to make the spin-3 field hμνλ transverse and traceless:

ρhμνρ=0, hμρρ=0, uμhμνρ=0,    (227)

which imply that the gauge parameter ξμν is a symmetric, traceless 2-th rank tensor.

The transverse–traceless spin-3 field hμνλ, which is projected along a four-velocity vector uλ in the rest-space of a co-moving observer, may be written in terms of its helicity state vectors hλ+ and hλ×, and the polarization tensors eμν+ and eμν× in the momentum space mμ=(|m|,m):

hμνλ(m)=eμν+(m)hλ+(m)+eμν×(m)hλ×(m),    (228)

where the polarization tensors are

eμν+=xμxνyμyν,       eμν×=xμyν+yμxν,    (229)

while xμ=(0,x^) and yμ = (0, ŷ) are spatial 4-vectors with x^·x^=ŷ·ŷ=1 and x^·ŷ=0 with x^·m^= ŷ·m^=0 , and form an orthogonal triad ŷ=x^×m^, where m^=m/|m|.

Similarly, the transverse–traceless dual spin-3 field h~μνλ, which is projected along a four-velocity vector ũλ in the rest-space of a co-moving observer, may be written in terms of its helicity states h~λ+ and h~λ×, and the polarization tensors in D = 4:

h˜μνλ(m)=eμν+(m)h˜λ+(m)+eμν×(m)h˜λ×(m),    (230)

Taking a permutation ϵμα from Equation (228) according to the first dual formulation definition, one can easily find in the momentum space:

hμ+(m)=h˜μ×(m),       hμ×(m)=h˜μ+(m),    (231)

which means that the helicity state vectors of the spin-3 field are respectively rotated to those of its dual field under SO(2) duality rotations in D = 4, and demonstrate the twisted self-dual condition, similar to what were shown the graviton and the dual graviton in section 2.6.

4. Higher-Spin Fields and their Dual Fields

We now consider the general spin-s fields (s ≥ 4) and its dual formulations. The spin-s fields in D-dimensional spacetime can be represented by a symmetric s-th rank tensor hμ1μ2···μs = h(μ1μ2···μs) with the Young diagram:

yes

which is double-traceless for s ≥ 4 [102]:

hμ1μ2μ3μ4···μsημ1μ2ημ3μ4=0.    (233)

The action of the free, massless spin-s fields hμ1μ2···μs in arbitrary dimensions (D ≥ 4) reads:

Sspin-s=12dDxL[hμ1μ2···μs],    (234)

where L[hμ1μ2···μs] is the Fronsdal Lagrangian [102]:

L[hμ1μ2···μs]=(s1)s[ρhμ1···μsρhμ1···μs                                            sρhρμ1···μs1λhλμ1···μs1                                            +s(s1)hλλμ1···μs2ρσhρσμ1···μs2                                            12s(s1)ρhλλμ1···μs2ρhσσμ1···μs2                                            14s(s1)(s2)λhρρλμ1···μs3                                            ×νhσσνμ1···μs3].    (235)

The Fronsdal Lagrangian (235) is invariant under the gauge transformation [102]

δξhμ1μ2···μs=s(μ1ξμ2···μs).    (236)

The gauge parameter ξμ2···μs−1 is a symmetric tensor.

The equations of motion for free, massless spin-s fields satisfy the generalized spin-s Einstein equations [91, 102]

Gμ1μ2···μs=0,    (237)

where the spin-s Einstein tensor Gμ1μ2···μs = G(μ1μ2···μs) is a double-traceless (Gμ1μ2···μsημ1μ2ημ3μ4=0), symmetric tensor.

The spin-s Einstein tensor Gμ1μ3···μs can be written using the Fronsdal tensor Fμ1···μs as

Gμ1···μs=Fμ1···μss(s1)4η(μ1μ2Fμ3···μs)ρρ,    (238)

where the Fronsdal tensor Fμ1···μs = F(μ1···μs) is a symmetric tensor, and defined by [49, 102]

Fμ1···μs=ρρhμ1···μss(μ1ρhμ2···μs)ρ                       +s(s1)2(μ1μ2hμ3···μs)ρρ.    (239)

The linearized spin-s Riemann tensor Rμ1ν1μ2ν2···μsνs is a (2s)-th rank tensor with Young symmetry (2,···,2s) and the diagram:

yes

The linearized spin-s Ricci tensor,

Rμ1ν1μ2ν2···μs2νs2|μs1μsRμ1ν1μ2ν2···μs2νs2μs1νs1μsνsηνs1νs,    (240)

is a (2s−2)-th rank tensor with Young symmetry type

yes

and obtained by taking covariant derivatives from the Fronsdal tensor Fμ1···μs as follows

Rμ1ν1μ2ν2···μs2νs2|μs1μs               =2s2[ν1|[ν2|···[νs2|F|μ1]|μ2]···|μs2]μs1μs.    (242)

Here, the notations [ | and | ] such as 4∂[ν1|[ν2|S|μ1]|μ2] imply that covariant derivatives are antisymmetrized on ν1μ1 and ν2μ2 separately, i.e., yαβ|μρ=y[αβ]μρ=yαβ(μρ). The equations of motion of a free, massless spin-s field hμ1μ2···μs satisfy Rμ1ν1μ2ν2···μs−2νs−2s−1μs = 0, which is equivalent to the spin-s Einstein equations (237).

Substituting the Fronsdal definition (239) into the Ricci definition (242) yields [49]

Rμ1ν1μ2ν2···μs2νs2|μs1μs          =2s2[ρρ[ν1|[ν2|···[νs2|h|μ1]|μ2]···|μs2]μs1μs          μs[ν1|[ν2|···[νs2|ρh|μ1]|μ2]···|μs2]μs1ρ          μs1[ν1|[ν2|···[νs2|ρh|μ1]|μ2]···|μs2]μsρ         +μs1μs[ν1|[ν2|···[νs2|h|μ1]|μ2]···|μs2]ρρ].    (243)

For the spin-s field hμ1μ2···μs, there might be s different dual formulations (also infinite chains of dualities [94, 95]). Considering the fact that spin-s fields are double-traceless (s ≥ 4), it is problematic to dualize on multiple indices. Thus, we consider only the first dual formulation of the spin-s field in D-dimensional spacetime, which is dualized on one index only. For the physical gauge spin-s field, we can write

h˜i1···iD3|j2···js=ϵi1···iD3j1hj1j2···js,    (244)
hj1j2···js=1(D3)!ϵj1i1···iD3h˜i1···iD3|j2···js.    (245)

The dual spin-s field h~α1···αD-3|μ2···μs is of Young symmetry (D-3,1,···,1s-1) having the Young diagram:

yes

and double-traceless for s ≥ 4 [47]:

h˜α1···αD3|μ2μ3μ4μ5···μsημ2μ3ημ4μ5=0,    (247)
h˜α1α2α3α4···αD3|μ2···μsηα1α2ηα3α4=0.    (248)

It has the following properties:

h˜α1···αD3|μ2···μs=h˜[α1···αD3]μ2···μs,           h˜[α1···αD3μ2]μ3···μs=0,    (249)
h˜α1···αD3|μ2···μs=h˜α1···αD3(μ2···μs).    (250)

In D = 4, the dual spin-s field is symmetric, h~μ1μ2···μs=h~(μ1μ2···μs). The Fronsdal action (234) for the spin-s field hμ1μ2···μs in D = 4 is dual to the Fronsdal action S=12d4xL[h~μ1···μs], which is formulated in terms of the dual spin-s field h~μ1μ2···μs, i.e., hμ1···μsh~μ1···μs in the action (234).

4.1. D > 4: Generalized Dual Fronsdal Fields

The Fronsdal field hμ1μ2···μs is dual to a field dualized on one index. The dual Fronsdal field h~α1···αD-3|μ2···μs is a double-traceless, tensor with Young symmetry (D-3,1,···,1s-1). It is a symmetric tensor in D = 4, whose equations of motion can be described by the Fronsdal Lagrangian (235). In D > 4, the equations of motion of free massless, dual spin-s field h~α1···αD-3|μ2···μs explicitly reads [47]

[h˜α1αD3|β2βs]=(s1)s(D3)![ρh˜α1αD3|β2βsρh˜α1αD3|β2βs                                   (D3)ρh˜ρα2αD3|β2βsσh˜σα2αD3|β2βs                                   (s1)ρh˜α1αD3|ρβ3βsσh˜α1αD3|σβ3βs                                   +2(s1)(D3)h˜γα2αD3|γβ3βsρσh˜ρα2αD3|σβ3βs                                   +(s1)(s2)h˜α1αD3|γ γβ4βsσρh˜α1αD3|σρβ4βs                                   (s1)(D3)ρh˜γα2αD3|γβ3βsρh˜σα2αD3|σβ3βs                                   12(s1)(s2)ρh˜α1αD3|σσβ4βsρh˜α1αD3|γ γβ4βs                                   (s1)(s2)(D3)σh˜ρα2αD3|ρσ β4βsγh˜γα2αD3|λλβ4βs                                   +14(s1)(s2)(D3)σh˜σα2αD3|ρ ρβ4βsγh˜γα2αD3|λλβ4βs                                   +12(s1)(D3)(D4)σh˜σα2αD3|λβ3βsρh˜ρα2αD3|λβ3βs                                   +12(s1)(D3)(D4)ρh˜γρα3αD3|γβ3βsσh˜λ    σα3αD3|λβ3βs                                   14(s1)(s2)(s3)γh˜α1...αD3|ρ ργβ5βsσh˜α1...αD3|λλσβ5βs],    (251)

which is invariant under the gauge symmetry [47]:

δχh˜α1...αD3=|β2···βs(D3)[[α1χα2...αD3]|β2···βs                                   +(s1)(β2χ[α1...αD4|αD3]β3···βs)]    (252)

where χα1…αD − 42···βs = χ[α1…αD − 4]β2···βs = χα1…αD − 42···βs) is an arbitrary tensor with the mixed symmetry (D-4,1,···,1s-1),

yes

such

χα1...αD4|β2···βs=χ[α1...αD4]β2···βs=χα1...αD4(β2···βs).    (254)

The gauge transformations for χα1…αD − 42···βs are reducible.

The action (251) also has the following gauge symmetry

δχ,φh˜α1...αD3|β2···βs       =(s1)(D3)[[α1χα2...αD3]|β2···βs           +[α1φα2...αD3](β2|β3···βs)(β2φα1...αD3|β3···βs)],    (255)

where χα1…αD − 42···βs is an arbitrary tensor with the mixed symmetry (D-4,1,···,1s-1), and φα1…αD − 33···βs = φ[α1…αD − 3 is an arbitrary tensor with the mixed symmetry (D-3,1,···,1s-2). The gauge transformations for χα1…αD − 42···βs and φα1…αD − 33···βs are reducible.

One may define the spin-s generalized Curtright action of the dual spin-s field h~α1···αD-3|β2···βs using a field strength tensor F̄α1αD-2|β2···βs as (see Appendix 1 in Supplementary Material):

L[h˜α1···αD3|β2···βs]=(s1)2s(1(D2)!F¯α1···αD2|β1···βs1                                                               ×F¯α1···αD2|β1···βs1                                                               +(s2)(s1)(D3)!Fσγα1···αD4|β1···βs1                                                               ×Fσβ1α1···αD4|γβ2···βs1                                                               +(s3)(D2)(D+s4)(D3)!F¯α1···αD3σ|β1···βs2σ                                                               ×F¯α1···αD3γ|β1···βs2γ                                                               +(s2)(D2)(D+s4)(D3)!Fσγα1···αD4|β1···βs2σ                                                               ×Fρβ1α1···αD4|γβ2···βs2ρ),    (256)

where the “spin-s Curtright” field strength tensor F̄α1αD-2|β2···βs defined as

F¯α1...αD2|β2···βs(D2)[α1h˜α2···αD2]β2···βs,    (257)

is a tensor with Young symmetry type (D-2,1,···,1s-1):

yes

The spin-s Curtright field strength tensor F̄α1αD-2|β2···βs is transferred as follows

δαF¯α1...αD2|β2···βs=2(D2)(D3)(β2[α1φα2...αD2|β3···βs).    (259)

However, it is not gauge invariant under the gauge symmetry (252).

One can define the spin-s dual Riemann tensor by

R˜α1···αD2|μ2ν2···μsνs=12ϵα1···αD2μ1ν1Rμ1ν1μ2ν2···μsνs,    (260)

which has Young symmetry (D-2,2,···,2s-1):

yes

The spin-s dual Ricci tensor is defined by

R˜α1···αD2|μ2ν2···μs2νs2|μs1μs        R˜α1···αD2|μ2ν2···μs2νs2μs1νs1μsνsηνs1νs,    (262)

with Young symmetry (D-2,2,···,2,1,1s-1) and the diagram:

yes

Similarly, there is the spin-s dual Einstein tensor G~α1···αD-3|β2βs with Young symmetry:

yes

which is defined by

G˜α1···αD3|β2...βs=F˜α1···αD3|β2...βs(s1)4[2(D3)η[α1β2                                                       ×F˜α2···αD3]ρ|β3...βsρ                                                       +(s2)ηβ2β3F˜α1···αD3|ρβ4...βsρ],    (265)

where F~α1···αD-3|β2βs is the spin-s dual Fronsdal tensor [47]

F˜α1···αD3|β2...βs = ρρh˜α1···αD3|β2...βs(D3)                           ×[α1ρh˜ρα2···αD3]|β2...βs                         (s1)(β2ρhα1···αD3|β3...βs)ρ                         +(s1)(D3)[α1(β2h˜ρα2···αD3]|β3...βs)ρ                          +(s1)(s2)2(β2β3h˜α1···αD3|ρβ4...βs)ρ.    (266)

The spin-s dual Fronsdal tensor F~α1···αD-3|β2βs is of Young symmetry type (D-3,1,···,1s-1). In D = 4, the definition (266) recovers Equation (239).

The spin-s dual Ricci tensor (262) can be defined in term of the dual spin-s Fronsdal tensor F~α1···αD-3|β2βs:

R˜α1···αD2|μ2ν2···μs2νs2|μs1μs=2s3(D2)[ν2|···[νs2|                                                  ×[α1F˜α2···αD2]|μ2]···|μs2]μs1μs.    (267)

Substituting the dual Fronsdal definition (266) into the dual Ricci definition (267) presents

R˜α1αD2|μ2ν2μs2νs2|μs1μs      =2s3(D2)[ρρ[ν2|[νs2|[α1h˜α2αD2]|μ2]|μs2]μs1μs          (D3)[ν2|[νs2|[α1ρh˜ρα2αD3]|μ2]|μs2]μs1μs          μs[ν2|[νs2|ρ[α1h˜α2αD2]|μ2]|μs2]μs1ρ          μs1[ν2|[νs2|ρ[α1h˜α2αD2]|μ2]|μs2]μsρ          +(D3)μs[ν2|[νs2|ρ[α1h˜ρα2αD2]|μ2]|μs2]μs1ρ          +(D3)μs1[ν2|[νs2|ρ[α1h˜ρα2αD2]|μ2]|μs2]μsρ          +μs1μs[ν2|[νs2|[α1h˜α2αD2]|μ2]|μs2]ρρ].    (268)

4.2. Spin-s Magnetic-Type Sources

We now introduce the coupling between the spin-s fields hμ1μ2···μs = h1μ2···μs) and spin-s electric-type sources, and then make a generalization to the dual spin-s fields and spin-s magnetic-type sources.

The linearized spin-s Ricci tensor Rμ1ν1μ2ν2μ3ν3 can be coupled to a traceless (Tμ1μ2···μsημ1μ2ημ3μ4=0), symmetric tensor in D-dimensional spacetime:

Rμ1ν1μ2ν2···μs2νs2|μs1μs=12[ν1|···[νs2|T|μ2]···|μs2]μs1μs.    (269)

The tensor Tμ1···μs is defined by

Tμ1···μsTμ1···μss4η(μ1μ2Tμ3···μs)ρρ,    (270)

where Tμ1···μs = T(μ1···μs) is the spin-s electric-type stress-energy tensor.

The spin-s Einstein equations with the spin-s electric-type source read [91]:

Gμ1···μs=Tμ1···μs.    (271)

The dual spin-s Einstein tensor defined by (265) can be coupled to the spin-s magnetic-type source:

G˜α1···αD3|β2...βs=T˜α1···αD3|β2...βs,    (272)

equivalently,

R˜α1···αD2|μ2ν2···μs2νs2|μs1μs              =12[ν2|···[νs2|[α1T˜α2···αD2]|μ2]···|μs2]μs1μs,    (273)

where T˜α1···αD3|β2···βs is a double-traceless (T˜α1···αD3|β2···βsηα1α2ηα3α4=0=T˜α1···αD3|β2...βsηβ2β3ηβ4β5), tensor with Young symmetry type (D-3,1,···,1s-1) and defined as

T˜α1···αD3|μ2···μsT˜α1···αD3|μ2···μs                                                           (s1)(D3)4η[α1(μ2T˜α2···αD3]ρ|μ3···μs)ρ,    (274)

and T~α1···αD-3|μ2···μs is the spin-3 magnetic-type stress-energy, and has Young symmetry type (D-3,1,···,1s-1).

By analogy with the Dirac quantization condition (104), the spin-s quantization condition in D = 4 is [91]

Pμ1···μs1P˜μ1···μs1=n12ħc,     n,    (275)

where Pμ1···μs−1 = mUμ1···μs−1 is the conserved spin-s electric-type charge, and P~μ1···μs-1=m~Ũμ1···μs-1 is the conserved spin-s magnetic-type charge, m is the electric-type spin-s charge, and m~ is the magnetic-type spin-s charge, Uμ2···μsuμ2···uμs the traceless part of uμ2···uμs, and Ũμ2···μs ≡ ũμ2···ũμs the traceless part of ũμ2···ũμs.

The stress-energy electric-type and magnetic-type tensors in D = 4 can be written as

Tμ1μ2···μs=mUμ2···μsdτδ(4)[xw(τ)]wμ1τ                   =muμ1Uμ2···μsu0δ(3)[xw(x0)],    (276)
T˜μ1μ2···μs=m˜U˜μ2···μsdτδ(4)[xw(τ)]w˜μ1τ                   =m˜u˜μ1U˜μ2···μsu0δ(3)[xw˜(x0)].    (277)

A Dirac string in the 4-D spin-s theory can be constructed as follow. Let us consider the decomposition of the spin-s Riemann curvature Rμ1ν1···μsνs, it can be split into the chargeless part and the charged part,

Rμ1ν1···μsνs=rμ1ν1···μsνs+mμ1ν1···μsνs.    (278)

Both rμ1ν1···μsνs and mμ1ν1···μsνs are of mixed symmetry type (2,···,2s). The tensor rμ1ν1···μsνs satisfies the cyclic and Bianchi identities, and can be written in terms of covariant derivatives of a symmetric tensor hμ1μ2μ3:

rμ1ν1···μsνs=2[μs|···[μ2|[μ1hν1]|ν2]···|νs],    (279)

The charged part mμ1ν1···μsνs can be defined in terms of the spin-s magnetic-type source:

mμ1ν1μsνs= 12ϵμ1ν1ρσ[[μ2|[μs|M˜ρσ|ν2]|νs]                   +[μ2δ[ρν2][μ3|[μs|M˜σ]λ| |ν3]|νs]λ],    (280)

where the tensor M~αβ|μ2···μs is of Young symmetry type (2,1,···,1s-1):

yes

If we define

yαβ|μ2···μs=ϵαβρμ1ρhμ1···μs,    (282)

where the tensor yαβ|μ2···μs is of Young symmetry type (2,1,···,1s-1) tensor,

yαβ|μ2···μs=y[αβ]μ2···μs=yαβ(μ2···μs),           αyαβ|μ2···μs=0,    (283)

the spin-s Riemann curvature Rμ1ν1···μsνs can be rewritten as

Rμ1ν1μsνs=  12ϵμ1ν1ρσ[μs|[μ2|[yρσ|ν2]|νs]+δ[ρ|ν2]yσ]λ|λ|ν3]|νs]                                  +M˜ρσ|ν2]|ν3]|νs]+δ[ρ|ν2]M˜σ]λ|λ|ν3]|νs]].    (284)

Assuming Yαβ|μ2···μs=yαβ|μ2···μs+M~αβ|μ2···μs, one has

Rμ1ν1μ2ν2μ3ν3=12ϵμ1ν1ρσ[μ2|···[μs|                        ×(Yρσ|ν2]···|νs]+δ[ρν2]Yσ]λ|λν3]···|νs]).    (285)

We can now connect a Dirac string yμ(τ, θ) to the spin-s magnetic-type source M~αβ|μ2···μs, yμ(τ,0)=w~μ(τ) by analogy with (124):

M˜αβ|μ2···μs=m˜U˜μ2···μsdτdθδ(4)[xy(τ,θ)]                       ×(yαθyβτyατyβθ),    (286)

The divergence of M~αβ|μ2···μs is equal to T~μ1···μs:

T˜μ1···μs=αM˜αμ1|μ2···μs.    (287)

Hence, the Dirac string of a spin-s magnetic-type point source is conserved (see also [91] for spin-s electric-type source).

4.3. Higher-Spin Harmonic Condition

We now introduce a harmonic coordinate condition for the generalized Fronsdal equation (239) that makes the linearized wave equation for spin-s fields by eliminating the last two terms of (239).

To have a wave equation for the spin-s fields in empty space, it is required to have the generalized De Donder gauge condition [103]:

Dμ2···μsρhμ2···μsρs12μ2hμ3···μsρρ=0.    (288)

Substituting Equation (288) into (239) leads to the wave equation ρρhμ1···μs=0.

It can be seen that Equation (288) is also invariant under the gauge transformation

δξDμ2···μs=sρρξμ2···μs.    (289)

It was imposed that the spin-s field has the double-traceless condition (233) [102]. To make the De Donder gauge condition (288), one may require that the spin-s field is transverse and traceless (see [145] for proof):

ρhμ2μ2···μsρ=0,          μ2hμ3···μsρρ=0.    (290)

Having the transverse–traceless gauge condition, the gauge parameter ξμ2···μs is a symmetric, traceless (s − 1)-th rank tensor. Similar to massless spin-2 fields described in section 2.6, massless, spin-s fields have only two polarization states with the ±s helicity states [146]. However, massive, spin-s fields have 2s+1 polarization states [147, 148].

5. Discussion: Interacting Theories

Consistency of the interacting theory for a free field can traditionally be determined from coupling deformations of the gauge transformations of the free theory. The BRST formalism [149152] was originally developed to examine whether the deformed gauge transformations can be constructed. Inclusion of auxiliary fields (antifields and ghosts) led to the BRST-antifield formalism [153157] that allowed us to assess consistency of interactions of the free theory with other theories and itself. This involves studying consistent deformations of the master equation [158, 159], which contains gauge transformation structures, corresponding local reducibilities, as well as stationary surfaces (inc. equations of motion) of each field. The master equation of the free theory is associated with the BRST differential. The coupling-order deformations of the master equation of the interacting theory by means of the BRST differential allow us to evaluate all the requirements for consistency of the interacting theory (see reviews [160167] ). The action principle of the free theory is the integral of the Lagrangian over the manifold that corresponds to the propagation of perturbations of the free field in the spacetime. Nevertheless, the action of the interacting theory associated with observables, which are invariant under the gauge transformations, may make consistent interactions with themselves or other fields of other free theories. A detail study of consistency of interactions of the free theory of the fields with itself can deduce whether the interacting theory has consistent deformations.

The action principle of a massless, spin-2 field particle is described by the Pauli-Fierz action [100, 111], which corresponds to the linearized Einstein equations in Minkowskian flat spacetime. The Pauli-Fierz action is free of ghost excitations, and its field equations are algebraically consistent. The interacting theory of gravity has only one single massless spin-2 field [168170], and there are no consistent interactions among multiple interacting massless spin-2 fields [171], i.e., no spin-2 analogy of Yang-Mills theory. However, the action of a single, massless spin-2 field hμν coupled to a positive-definite dynamical metric is gμν is not invariant under the hμν gauge transformation, so it is inconsistent [172]. Similarly, cross-interactions among multiple massless spin-2 fields are inconsistent in a positive-definite metric [171], while consistency is mathematically possible in a negative-definite metric (negative-energy) [171, 173]. In the case of direct coupling to matter, the locality and consistency conditions require that the interacting theory includes all nonlinear terms [168, 169, 174], otherwise the spin-2 field of the linearized theory remains ghost-like in empty flat space [170]. Despite the inconsistency of linear, massless spin-2 interactions, the nonlinear theory of a massive spin-2 field provides ghost-free consistent coupling to matter [175177]. A spin-2 field coupled to matter or nonlinear self-interaction leads to the general covariance of the interacting theory [169]. Constructing the interacting multi-graviton model and coupling a single spin-2 field to a dynamical metric require a fully non-linear representation of the interacting theory for gravity.

It was proven that the dual formulations of linearized gravity in D = 5, the Curtright Field, does not have consistent interactions involving two derivatives of the fields under local, Poincaré invariance [40]. The dual formulations of linearized gravity in D > 5 do not possess consistent, local deformations [178], so the dual graviton of the linearized theory does not have any self-interaction. However, the dual graviton can be coupled to other theories, such as topological BF theories [132, 179]. To overcome the inconsistency of the linearized dual gravity, one may consider a non-linear theory of the dual graviton, which incorporates a topological (Chern-Simons like) term for the original graviton, which leads to consistent nonlinear deformations, and is manifestly invariant due to a dynamical metric for the dual graviton [134]. In particular, the properties of the Kac-Moody algebra E11 (see section 2.3) are associated with spacetime covariance and supergravity covariant spectra. In the case of D = 11 supergravity, the Kac-Moody algebra (AD-3+++ for pure gravity) applies to pure gravity (as well as maximal supergravity), so the dual graviton cannot be coupled to the original Einstein-Hilbert action, and it doubles degrees of freedom. Nevertheless, the dual graviton in the E11 algebra can be coupled with a topological term containing the original metric without any doubling of degrees of freedom [134]. The interacting theory of dual graviton in D ≥ 5 [43] can also be constructed in de Sitter space (dSn) [180], similar to the nonlinear implementation of gravitational duality with a cosmological constant in dS4 [181]. Thus, it is necessary to formulate a nonlinear representation of dual gravity (see [134, 182] for nonlinear dual graviton), which is dual to nonlinear Einstein gravity, containing both the original and dual metrics, and is fully gauge covariant.

It has been shown that a massless spin-3 (also s > 3) particle described by a linear gauge field cannot have a self-interaction in D = 4 [183, 184]. Although first-order deformations of spin-3 fields are possible [183], self-interactions encounter some difficulties: the first-order deformations do not obey the Jacobi identity [185], and the commutator of two gauge transformations cannot be written [184]. Similarly, the commutator between two gauge transformations of a linear spin-s gauge field (>3) cannot be constructed in D = 4, so self-interactions of a spin-s field do not exist in 4-D flat space [184]. Using the BRST-antifield formalism, the first-order deformations of a spin-3 gauge field in D = 4, defined by a non-abelian symmetric rank-3 tensor hμνρah(μνρ)a, were built under the assumptions of parity and Poincaré invariance, local and perturbative deformation of the free theory in Minkowski space, which correspond to the cubic vertex of Berends, Burgers, and van Dam [183] involving three derivatives of the fields, but the theory is again inconsistent at second order [186]. In D = 5 Minkowski spacetime, the second-order deformations of spin-3 fields can be obtained, while the first-order deformations involve the generalized de Wit–Freedman connection [187]. In D > 4 flat spacetime, the structure constants given by a completely antisymmetric tensor can make a cubic vertex at second order involving five derivatives of the fields, but the analysis will be complicated due to tedious five-derivative calculations [186]. Hence, the interacting theory of linearized spin-3 particles is problematic in flat space D ≥ 4, and only feasible under some certain assumptions involving 3 derivatives at first order and 5 derivatives at second order in spacetime dimensions higher than 4.

The generalizations of the Coleman–Mandula “no-go” theorem [188] to supersymmetry [189] and higher spin gauge theories [190192] indicate that the interacting theory of linearized higher spin fields (s > 2) are incompatible in flat spacetime, since their conserved currents are associated with a free theory, which does not permit to have a nontrivial S-matrix [190]. The no-go theorem allows us to have only gauge fields of spin-1 and spin-2 perturbations around a flat background. However, the restrictions imposed by the no-go theorem on higher spin fields (s > 2) can be overcome in the nonlinear theory of spin-s perturbations around the anti de Sitter (AdSd) spacetime of arbitrary d dimensions [193]. It is argued that the interacting theory of nonlinear, massless bosonic fields of arbitrary spins s > 1 can be formulated in d-dimensional AdS space [194196]. Nonlinear spin-s fields with a non-vanishing cosmological constant in D = 4 were shown to have consistent interaction in the cubic-order [197, 198], and in the first-order in the Weyl 0-forms [199], and also in all orders [200203] (the so-called Vasiliev higher-spin theory). Consistent interactions of nonlinear spin-s fields were also formulated in 3-D AdS spacetime [204], in AdS5 spacetime [205, 206], and in AdSd space with arbitrary d dimensions [207]. In particular, it was shown that irreducible massless mixed symmetry fields in AdSd are decomposed into certain reducible massless flat fields in the flat limit [208], which permit to have consistent deformations for generic higher-spin fields in the AdS space. It was demonstrated that all spin-s consistent deformations are explicitly made using Vasiliev's unfolded equations in D = 4, which are expressed in terms of a gauge connection and a (twisted)-adjoint matter field, while the Vasiliev gauge connection is found to be related to the spin-s Fronsdal fields [209]. Therefore, the nonlinear, massless higher spin gauge theories can have consistent interactions around the AdS vacuum solution with the non-zero cosmological constant in the Vasiliev higher-spin theory [210].

It is well known that the compactification of M-theory or supergravity on (d + 1)-dimensional AdS spacetime is dual to conformal field theory (CFT) in d dimensions, which is called the AdSd + 1/CFTd correspondence or Maldacena duality [211]. This correspondence indicates that the chiral operators of N=4 Super Yang–Mills theory (CFT observables) in D = 4 correspond to those of Kaluza–Klein Type IIB supergravity on AdS5×S5 (where S5 is a 5-dimensional compact space) [212, 213]. We know that electric-magnetic duality in the bulk of AdS4 can relate holographically derived deformations of the boundary CFT3 [50]. It was also argued that gravity theories in AdS4 are holographically dual to two CFTs in D = 3 with different parity: the Dirichlet CFT3 with the graviton source and a dual CFT3 with a dual graviton source at the non-linear level [52]. The holographic properties of electric-magnetic duality in gravity also provide some insights into non-linear aspects of M-theory (see [51] for review). The “yes-go” Vasiliev higher-spin theory of spin-s perturbations in the AdS background is naturally conjectured in the higher-spin holography [214216], which gives evidence for duality between Vasiliev's higher-spin fields and the free field theory of N massless scalar fields [217], and holographic dualities between Vasiliev's higher-spin fields and O(N) vector models [215, 216] (the so-called higher spin/vector model duality; see [218, 219] for review). The interacting theory of Vasiliev's higher-spin fields may reveal the hidden origin of AdSd + 1/CFTd correspondence (see [220222] for more discussions). Nevertheless, we do not yet have any full nonlinear representation of the spin-s Fronsdal fields.

Author Contributions

The author confirms being the sole contributor of this work and approved it for publication. The author presented some mathematical notations and expressions slightly different from the original versions, and provided few new results (e.g., Equations 27 and 28 in section 2.1, a part of section 2.6).

Funding

A part of this work was supported by the EU contract MRTN-CT-2004-005104 Constituents, Fundamental Forces and Symmetries of the Universe at University of Craiova in 2008.

Conflict of Interest Statement

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

Acknowledgments

AD acknowledges hospitality from the University of Craiova, where a part of this work was carried out, and thanks C. Hull, P. West, M. Vasiliev, G. Barnich, and M. Henneaux for helpful comments and valuable discussions.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2018.00146/full#supplementary-material

Footnotes

1. ^In particular, it was shown that there are infinitely many off-shell dualities of the graviton [38, 92, 93], and infinite chains of dualities for spin-s fields [94, 95].

2. ^The magnetic part of the Weyl curvature defined here has the opposite sign compared to the definition often found in the literature of the 1+3 covariant formalism, see e.g., [6466, 68].

3. ^The Curtright action could also be expressed by Equations (73, 74) of [89].

4. ^Also, see the action defined by Equations (77) and (78) in [89], and Appendix 1 in Supplementary Material for the parent first-order action.

5. ^Ref. [134] presented a similar generalized Curtright action via the Einstein-Hilbert assumptions.

6. ^For traceless spin-3 fields, it is impossible to dualize on two indices, so there is no second dual formulation.

7. ^The 5-dimensional dual spin-3 action could be expressed by Equations (75) and (76) of [89].

References

1. Dirac PAM. Quantised singularities in the electromagnetic field. Proc Roy Soc Lond A. (1931) 133:60–72.

Google Scholar

2. Dirac PA. The Theory of Magnetic Poles. Phys Rev. (1948) 74:817–30.

Google Scholar

3. 't Hooft G. Magnetic monopoles in unified gauge theories. Nucl Phys B. (1974) 79:276–84.

Google Scholar

4. Polyakov AM. Particle spectrum in quantum field theory. JETP Lett. (1974) 20:194.

Google Scholar

5. Polyakov AM. Particle spectrum in quantum field theory. ZhETF Pis ma Redaktsiiu (1974) 20:430.

6. Monastyrskiǐ MI, Perelomov AM. Concerning the existence of monopoles in gauge field theories. JETP Lett. (1975) 21:43.

Google Scholar

7. Montonen C, Olive D. Magnetic monopoles as gauge particles? Phys Lett B. (1977) 72:117–20.

Google Scholar

8. Julia B, Zee A. Poles with both magnetic and electric charges in non-Abelian gauge theory. Phys Rev D. (1975) 11:2227–32.

Google Scholar

9. Corrigan E, Olive DI, Fairlie DB, Nuyts J. Magnetic monopoles in SU(3) gauge theories. Nucl Phys B. (1976) 106:475–92.

Google Scholar

10. Goddard P, Nuyts J, Olive D. Gauge theories and magnetic charge. Nucl Phys B. (1977) 125:1–28.

Google Scholar

11. Actor A. Classical solutions of SU(2) Yang-Mills theories. Rev Mod Phys. (1979) 51:461–526.

Google Scholar

12. Nepomechie RI. Magnetic monopoles from antisymmetric tensor gauge fields. Phys Rev D. (1985) 31:1921–4.

PubMed Abstract | Google Scholar

13. Teitelboim C. Gauge invariance for extended objects. Phys Lett B. (1986) 167:63–8.

Google Scholar

14. Teitelboim C. Monopoles of higher rank. Phys Lett B. (1986) 167:69–72.

Google Scholar

15. Seiberg N, Witten E. Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory. Nucl Phys B. (1994) 426:19–52.

Google Scholar

16. Intriligator K, Seiberg N. Lectures on supersymmetric gauge theories and electric-magnetic duality. Nucl Phys B Proc Suppl. (1996) 45:1–28.

Google Scholar

17. Banks T, Dine M, Dijkstra H, Fischler W. Magnetic monopole solutions of string theory. Phys Lett B. (1988) 212:45–50.

Google Scholar

18. Harvey JA, Liu J. Magnetic monopoles in N=4 supersymmetric low-energy superstring theory. Phys Lett B. (1991) 268:40–6.

Google Scholar

19. Gauntlett JP, Harvey JA, Liu JT. Magnetic monopoles in string theory. Nucl Phys B. (1993) 409:363–81.

Google Scholar

20. Witten E, Olive D. Supersymmetry algebras that include topological charges. Phys Lett B. (1978) 78:97–101.

Google Scholar

21. Osborn H. Topological charges for N = 4 supersymmetric gauge theories and monopoles of spin 1. Phys Lett B. (1979) 83:321–6.

Google Scholar

22. Vafa C, Witten E. A strong coupling test of S-duality. Nucl Phys B. (1994) 431:3–77.

Google Scholar

23. Sen A. Strong-weak coupling duality in four-dimensional string theory. Int J Mod Phys A. (1994) 9:3707–50.

Google Scholar

24. Sen A. Strong-weak coupling duality in three dimensional string theory. Nucl Phys B. (1995) 434:179–209.

Google Scholar

25. Font A, Ibáñez LE, Lüst D, Quevedo F. Strong-weak coupling duality and non-perturbative effects in string theory. Phys Lett B. (1990) 249:35–43.

Google Scholar

26. Seiberg N. Electric-magnetic duality in supersymmetric non-Abelian gauge theories. Nucl Phys B. (1995) 435:129–46.

Google Scholar

27. Hull CM, Townsend PK. Unity of superstring dualities. Nucl Phys B. (1995) 438:109–37.

Google Scholar

28. Hull CM. Conformal non-geometric gravity in six dimensions and M-theory above the Planck energy. Class Quant Grav . (2001) 18:3233–9. doi: 10.1088/0264-9381/18/16/313

CrossRef Full Text | Google Scholar

29. Schwarz JH. Lectures on superstring and M theory dualities. Nucl Phys B Proc Suppl. (1997) 55:1–32.

Google Scholar

30. Obers NA, Pioline B. U-duality and M-theory. Phys Rep. (1999) 318:113–225.

Google Scholar

31. Mizoguchi S, Schröder G. On discrete U-duality in M-theory. Class Quant Grav . (2000) 17:835–70. doi: 10.1088/0264-9381/17/4/308

CrossRef Full Text | Google Scholar

32. Townsend P. Four lectures on M-theory. In: Gava E, Masiero A, Narain KS, Randjbar-Daemi S, Shafi Q, editors. High Energy Physics and Cosmology. Proceedings, Summer School (ICTP). New York, NY: World Scientific Publishing Co. (1997). p. 385.

33. Sen A. An Introduction to Non-perturbative String Theory. Cambridge: Lectures, Isaac Newton Institute and DAMTP (1998).

Google Scholar

34. Álvarez-Gaumé L, Zamora F. Duality in quantum field theory (and string theory). In: Trends in Theoretical Physics CERN-Santiago de Compostela-La Plata Meeting, AIP Conference Proceeding. Vol. 419, Nashville (1998). p. 1–53.

35. Curtright T. Generalized gauge fields. Phys Lett B. (1985) 165:304–8.

Google Scholar

36. Nieto JA. S-duality for linearized gravity. Phys Lett A. (1999) 262:274–81.

Google Scholar

37. Hull CM. Strongly coupled gravity and duality. Nucl Phys B. (2000) 583:237–59. doi: 10.1016/S0550-3213(00)00323-0

CrossRef Full Text | Google Scholar

38. Hull CM. Duality in gravity and higher spin gauge fields. JHEP (2001) 9:027. doi: 10.1088/1126-6708/2001/09/027

CrossRef Full Text | Google Scholar

39. Casini H, Montemayor R, Urrutia LF. Duality for symmetric second rank tensors. II. The linearized gravitational field. Phys Rev D. (2003) 68:065011. doi: 10.1103/PhysRevD.68.065011

CrossRef Full Text | Google Scholar

40. Bekaert X, Boulanger N, Henneaux M. Consistent deformations of dual formulations of linearized gravity: A no-go result. Phys Rev D. (2003) 67:044010. doi: 10.1103/PhysRevD.67.044010

CrossRef Full Text | Google Scholar

41. Bekaert X, Boulanger N. Tensor gauge fields in arbitrary representations of GL(D, ). Duality and poincaré lemma. Commun Math Phys. (2004) 245:27–67. doi: 10.1007/s00220-003-0995-1

CrossRef Full Text | Google Scholar

42. Henneaux M, Teitelboim C. Duality in linearized gravity. Phys Rev D. (2005) 71:024018. doi: 10.1103/PhysRevD.71.024018

CrossRef Full Text | Google Scholar

43. Bunster C, Henneaux M, Hörtner S. Twisted self-duality for linearized gravity in D dimensions. Phys Rev D. (2013) 88:064032. doi: 10.1103/PhysRevD.88.064032

CrossRef Full Text | Google Scholar

44. West P. E11 and M theory. Class Quant Grav . (2001) 18:4443–60. doi: 10.1088/0264-9381/18/21/305

CrossRef Full Text | Google Scholar

45. de Wit B. Electric-magnetic dualities in supergravity. Nucl Phys B Proc Suppl. (2001) 101:154–71. doi: 10.1016/S0920-5632(01)01502-X

CrossRef Full Text | Google Scholar

46. West P. Very extended E8 and A8 at low levels, gravity and supergravity. Class Quant Grav . (2003) 20:2393–405. doi: 10.1088/0264-9381/20/11/328

CrossRef Full Text | Google Scholar

47. Boulanger N, Cnockaert S, Henneaux M. A note on spin-s duality. JHEP (2003) 6:060. doi: 10.1088/1126-6708/2003/06/060

CrossRef Full Text | Google Scholar

48. Bekaert X, Boulanger N. On geometric equations and duality for free higher spins. Phys Lett B. (2003) 561:183–90. doi: 10.1016/S0370-2693(03)00409-X

CrossRef Full Text | Google Scholar

49. Henneaux M, Hörtner S, Leonard A. Twisted self-duality for higher spin gauge fields and prepotentials. Phys Rev D. (2016) 94:105027. doi: 10.1103/PhysRevD.94.105027

CrossRef Full Text | Google Scholar

50. de Haro S, Gao P. Electric-magnetic duality and deformations of three-dimensional conformal field theories. Phys Rev D. (2007) 76:106008. doi: 10.1103/PhysRevD.76.106008

CrossRef Full Text | Google Scholar

51. de Haro S, Petkou AC. Holographic aspects of electric-magnetic dualities. J. Phys. Conf. Ser. (2008) 110:102003. doi: 10.1088/1742-6596/110/10/102003

CrossRef Full Text | Google Scholar

52. de Haro S. Dual gravitons in AdS4/CFT3 and the holographic Cotton tensor. JHEP (2009) 1:042. doi: 10.1088/1126-6708/2009/01/042

CrossRef Full Text | Google Scholar

53. Bunster C, Henneaux M, Hörtner S, Leonard A. Supersymmetric electric-magnetic duality of hypergravity. Phys Rev D. (2014) 90:045029. doi: 10.1103/PhysRevD.90.045029

CrossRef Full Text | Google Scholar

54. Hinterbichler K. Manifest duality invariance for the partially massless graviton. Phys Rev D. (2015) 91:026008. doi: 10.1103/PhysRevD.91.026008

CrossRef Full Text | Google Scholar

55. Boulanger N, Campoleoni A, Cortese I. Dual actions for massless, partially-massless and massive gravitons in (A)dS. Phys Lett B. (2018) 782:285–90. doi: 10.1016/j.physletb.2018.05.046

CrossRef Full Text | Google Scholar

56. Boulanger N, Campoleoni A, Cortese I, Traina L. Spin-2 twisted duality in (A)dS. Front. Phys. 6:129. doi: 10.3389/fphy.2018.00129

CrossRef Full Text

57. Pirani FA. Invariant formulation of gravitational radiation theory. Phys Rev. (1957) 105:1089–99.

Google Scholar

58. Penrose R. A spinor approach to general relativity. Annals Phys. (1960) 10:171–201.

Google Scholar

59. Bel L. Radiation states and the problem of energy in general relativity. Cahiers Phys. (1962) 16:59–80.

Google Scholar

60. Hawking SW. Perturbations of an expanding universe. Astrophys J. (1966) 145:544.

Google Scholar

61. Campbell WB, Morgan T. Debye potentials for the gravitational field. Physica (1971) 53:264–88.

Google Scholar

62. Stewart JM. Perturbations of friedmann-robertson-walker cosmological models. Class Quant Grav . (1990) 7:1169–80.

Google Scholar

63. Ehlers J. Contributions to the relativistic mechanics of continuous media. Gen Rel Grav. (1993) 25:1225–66.

Google Scholar

64. Ellis GFR, van Elst H. Cosmological models (Cargèse lectures 1998). In: Lachièze-Rey M, editor. NATO Advanced Science Institutes (ASI) Series C. vol. 541 of NATO Advanced Science Institutes (ASI) Series C (1999). p. 1–116.

Google Scholar

65. Tsagas CG, Challinor A, Maartens R. Relativistic cosmology and large-scale structure. Phys Rep. (2008) 465:61–147. doi: 10.1016/j.physrep.2008.03.003

CrossRef Full Text | Google Scholar

66. Danehkar A. On the significance of the weyl curvature in a relativistic cosmological model. Mod Phys Lett A. (2009) 24:3113–127. doi: 10.1142/S0217732309032046

CrossRef Full Text | Google Scholar

67. Ellis GFR, Hogan PA. The electromagnetic analogue of some gravitational perturbations in cosmology. Gen Rel Grav. (1997) 29:235–44.

Google Scholar

68. Maartens R, Bassett BA. Gravito-electromagnetism. Class Quant Grav . (1998) 15:705–17.

Google Scholar

69. Ellis GFR, Dunsby PKS. Newtonian evolution of the weyl tensor. Astrophys J. (1997) 479:97–101.

Google Scholar

70. Hofmann S, Niedermann F, Schneider R. Interpretation of the Weyl tensor. Phys Rev D. (2013) 88:064047. doi: 10.1103/PhysRevD.88.064047

CrossRef Full Text | Google Scholar

71. Jantzen RT, Carini P, Bini D. The many faces of gravitoelectromagnetism. Annals Phys. (1992) 215:1–50.

Google Scholar

72. McIntosh CBG, Arianrhod R, Wade ST, Hoenselaers C. Electric and magnetic Weyl tensors: classification and analysis. Class Quant Grav . (1994) 11:1555–64.

Google Scholar

73. Bonnor WB. The electric and magnetic Weyl tensors. Class Quant Grav . (1995) 12:499–502.

Google Scholar

74. Bonnor WB. The magnetic weyl tensor and the van stockum solution. Class Quant Grav . (1995) 12:1483–9.

Google Scholar

75. Dowker JS, Roche JA. The gravitational analogues of magnetic monopoles. Proc Phys Soc. (1967) 92:1–8.

Google Scholar

76. Schwinger J. Sources and magnetic charge. Phys Rev. (1968) 173:1536–44.

Google Scholar

77. Schwinger J. A magnetic model of matter. Science (1969) 165:757–61.

PubMed Abstract | Google Scholar

78. Zee A. Gravitomagnetic pole and mass quantization. Phys Rev Lett. (1985) 55:2379–81.

PubMed Abstract | Google Scholar

79. Taub AH. Empty space-times admitting a three parameter group of motions. Annals Math. (1951) 53:472–90.

Google Scholar

80. Newman E, Tamburino L, Unti T. Empty-space generalization of the schwarzschild metric. J Math Phys. (1963) 4:915–23.

Google Scholar

81. Zimmerman RL, Shahir BY. Geodesics for the NUT metric and gravitational monopoles. Gen Rel Grav. (1989) 21:821–48.

Google Scholar

82. Lynden-Bell D, Nouri-Zonoz M. Classical monopoles: newton, NUT space, gravomagnetic lensing, and atomic spectra. Rev Mod Phys. (1998) 70:427–45.

Google Scholar

83. Bini D, Cherubini C, Jantzen RT, Mashhoon B. Gravitomagnetism in the Kerr-Newman-Taub-NUT spacetime. Class Quant Grav . (2003) 20:457–68. doi: 10.1088/0264-9381/20/3/305

CrossRef Full Text | Google Scholar

84. Shen JQ. Gravitational analogues, geometric effects and gravitomagnetic charge. Gen Rel Grav. (2002) 34:1423–35. doi: 10.1023/A:1020082903104

CrossRef Full Text | Google Scholar

85. Shen JQ. The dual curvature tensors and dynamics of gravitomagnetic matter. Annalen Phys. (2004) 516:532–53. doi: 10.1002/andp.200410093

CrossRef Full Text | Google Scholar

86. Barnich G, Troessaert C. Manifest spin 2 duality with electric and magnetic sources. JHEP (2009) 1:030. doi: 10.1088/1126-6708/2009/01/030

CrossRef Full Text | Google Scholar

87. Curtright TL, Freund PGO. Massive dual fields. Nucl Phys B. (1980) 172:413–24.

Google Scholar

88. Hull CM. Symmetries and compactifications of (4,0) conformal gravity. JHEP (2000) 12:007. doi: 10.1088/1126-6708/2000/12/007

CrossRef Full Text | Google Scholar

89. de Medeiros PF, Hull CM. Exotic tensor gauge theory and duality. Commun Math Phys. (2003) 235:255–73. doi: 10.1007/s00220-003-0810-z

CrossRef Full Text | Google Scholar

90. Bekaert X, Boulanger N. Massless spin-two field S-duality. Class Quant Grav . (2003) 20:S417–23. doi: 10.1088/0264-9381/20/12/306

CrossRef Full Text | Google Scholar

91. Bunster C, Portugues R, Cnockaert S, Henneaux M. Monopoles for gravitation and for higher spin fields. Phys Rev D. (2006) 73:105014. doi: 10.1103/PhysRevD.73.105014

CrossRef Full Text | Google Scholar

92. Boulanger N, Cook PP, Ponomarev D. Off-shell Hodge dualities in linearised gravity and E 11. JHEP (2012) 9:89. doi: 10.1007/JHEP09(2012)089

CrossRef Full Text | Google Scholar

93. Boulanger N, Ponomarev D. Frame-like off-shell dualisation for mixed-symmetry gauge fields. J Phys A Math Gen. (2013) 46:214014. doi: 10.1088/1751-8113/46/21/214014

CrossRef Full Text | Google Scholar

94. Boulanger N, Sundell P, West P. Gauge fields and infinite chains of dualities. JHEP (2015) 9:192. doi: 10.1007/JHEP09(2015)192

CrossRef Full Text | Google Scholar

95. Bergshoeff EA, Hohm O, Penas VA, Riccioni F. Dual double field theory. JHEP (2016) 6:26. doi: 10.1007/JHEP06(2016)026

CrossRef Full Text | Google Scholar

96. Francia D, Hull CM. Higher-spin gauge fields and duality. In: Higher Spin Gauge Theories: Proceedings, 1st Solvay Workshop (Brussels) (2004). p. 35–48.

Google Scholar

97. de Medeiros P. Massive gauge-invariant field theories on spaces of constant curvature. Class Quant Grav . (2004) 21:2571–93. doi: 10.1088/0264-9381/21/11/004

CrossRef Full Text | Google Scholar

98. de Medeiros P, Hull CM. Geometric second order field equations for general tensor gauge fields. JHEP (2003) 5:019. doi: 10.1088/1126-6708/2003/05/019

CrossRef Full Text | Google Scholar

99. Dirac PAM. Relativistic Wave Equations. Proc Roy Soc Lond A. (1936) 155:447–59.

Google Scholar

100. Fierz M, Pauli W. On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc Roy Soc Lond A. (1939) 173:211–32.

Google Scholar

101. Singh LP, Hagen CR. Lagrangian formulation for arbitrary spin. I. The boson case. Phys Rev D. (1974) 9:898–909.

Google Scholar

102. Fronsdal C. Massless fields with integer spin. Phys Rev D. (1978) 18:3624–9.

Google Scholar

103. de Wit B, Freedman DZ. Systematics of higher-spin gauge fields. Phys Rev D. (1980) 21:358–67.

Google Scholar

104. Vasiliev MA. 'gauge' form of description Of massless fields with arbitrary Spin. Yad Fiz. 1980;32:855–861. [Sov. J. Nucl. Phys. 32:439(1980)].

105. Aulakh CS, Koh IG, Ouvry S. Higher spin fields with mixed symmetry. Phys Lett B. (1986) 173:284–8.

Google Scholar

106. Koh IG, Ouvry S. Interacting gauge fields of any spin and symmetry. Phys Lett B. (1986) 179:115–8.

Google Scholar

107. Vasiliev MA. Higher spin gauge theories in various dimensions. Fortsch Phys. (2004) 52:702–17. doi: 10.1002/prop.200410167

CrossRef Full Text | Google Scholar

108. Bunster C, Henneaux M, Hörtner S. Gravitational electric-magnetic duality, gauge invariance and twisted self-duality. J Phys A Math Gen. (2013) 46:214016. doi: 10.1088/1751-8113/46/21/214016

CrossRef Full Text | Google Scholar

109. Hamermesh M. Group Theory and Its Application to Physical Problems. Dover Publications (1989).

Google Scholar

110. Henneaux M, Lekeu V, Leonard A. Chiral tensors of mixed Young symmetry. Phys Rev D. (2017) 95:084040. doi: 10.1103/PhysRevD.95.084040

CrossRef Full Text | Google Scholar

111. Pauli W, Fierz M. Über relativistische feldgleichungen von teilchen mit beliebigem spin im elektromagnetischen feld. Helv Phys Acta (1939) 12:297.

112. Deser S, Nepomechie RI. Electric-magnetic duality of conformal gravitation. Phys Lett A. (1983) 97:329–32.

Google Scholar

113. Deser S, Teitelboim C. Duality transformations of Abelian and non-Abelian gauge fields. Phys Rev D. (1976) 13:1592–7.

Google Scholar

114. Deser S. COMMENT: Off-shell electromagnetic duality invariance. J Phys A Math Gen. (1982) 15:1053–4.

Google Scholar

115. Labastida JMF, Morris TR. Massless mixed-symmetry bosonic free fields. Phys Lett B. (1986) 180:101–6.

Google Scholar

116. Damour T, Henneaux M. E10, BE10 and arithmetical chaos in superstring cosmology. Phys Rev Lett. (2001) 86:4749–52. doi: 10.1103/PhysRevLett.86.4749

PubMed Abstract | CrossRef Full Text | Google Scholar

117. Henneaux M, Persson D, Spindel P. Spacelike singularities and hidden symmetries of gravity. Living Rev Rel. (2008) 11:1. doi: 10.12942/lrr-2008-1

PubMed Abstract | CrossRef Full Text | Google Scholar

118. Schnakenburg I, West P. Kac-Moody symmetries of IIB supergravity. Phys Lett B. (2001) 517:421–8. doi: 10.1016/S0370-2693(01)01044-9

CrossRef Full Text | Google Scholar

119. Englert F, Houart L. Script G+++ invariant formulation of gravity and M-theories: exact BPS solutions. JHEP (2004) 1:002. doi: 10.1088/1126-6708/2004/01/002

CrossRef Full Text | Google Scholar

120. Englert F, Houart L. G+++ Invariant Formulation of Gravity and M-Theories: Exact Intersecting Brane Solutions. JHEP (2004) 5:059. doi: 10.1088/1126-6708/2004/05/059

CrossRef Full Text | Google Scholar

121. Damour T, Nicolai H. Higher-order M-theory corrections and the Kac moody algebra E10. Class Quant Grav . (2005) 22:2849–79. doi: 10.1088/0264-9381/22/14/003

CrossRef Full Text | Google Scholar

122. Damour T, Hanany A, Henneaux M, Kleinschmidt A, Nicolai H. Curvature corrections and Kac Moody compatibility conditions. Gen Rel Grav. (2006) 38:1507–28. doi: 10.1007/s10714-006-0317-y

CrossRef Full Text | Google Scholar

123. Englert F, Houart L. The emergence of fermions and the E(11) content. In: Quantum Mechanics of Fundamental Systems: The Quest for Beauty and Simplicity: Claudio Bunster Festschrift (Valdivia) (2009). p. 125–46.

Google Scholar

124. Englert F, Houart L, Taormina A, West P. The symmetry of M-theories. JHEP (2003) 9:020. doi: 10.1088/1126-6708/2003/09/020

CrossRef Full Text

125. West P. A brief review of E theory. Int J Mod Phys A. (2016) 31:1630043. doi: 10.1142/S0217751X1630043X

CrossRef Full Text | Google Scholar

126. Tumanov AG, West P. E11 in 11D. Phys Lett B. (2016) 758:278–85. doi: 10.1016/j.physletb.2016.04.058

CrossRef Full Text | Google Scholar

127. Englert F, Henneaux M, Houart L. From very-extended to overextended gravity and M-theories. JHEP (2005) 2:070. doi: 10.1088/1126-6708/2005/02/070

CrossRef Full Text | Google Scholar

128. Lambert ND, West PC. Coset symmetries in dimensionally reduced bosonic string theory. Nucl Phys B. (2001) 615:117–32. doi: 10.1016/S0550-3213(01)00415-1

CrossRef Full Text | Google Scholar

129. West P. On the different formulations of the E11 equations of motion. Mod Phys Lett A. (2017) 32:1750096. doi: 10.1142/S0217732317500961

CrossRef Full Text | Google Scholar

130. Tumanov AG, West P. E11, Romans theory and higher level duality relations. Int J Mod Phys A. (2017) 32:1750023. doi: 10.1142/S0217751X17500233

CrossRef Full Text | Google Scholar

131. West P. Generalised geometry, eleven dimensions and E11. JHEP (2012) 2:18. doi: 10.1007/JHEP02(2012)018

CrossRef Full Text | Google Scholar

132. Bizdadea C, Saliu SO, Stanciu-Oprean L. Dual linearized gravity coupled to Bf-Type topological field theories in D = 7. Mod Phys Lett A. (2012) 27:1250137. doi: 10.1142/S0217732312501374

CrossRef Full Text | Google Scholar

133. Bizdadea C, Miauta MT, Saliu SO, Toma M. Consistent interactions between dual formulations of linearised gravity in terms of massless tensor fields with mixed symmetries (k,1) and (2,2). Rom J Phys. (2013) 58:459–68.

Google Scholar

134. Boulanger N, Hohm O. Nonlinear parent action and dual gravity. Phys Rev D. (2008) 78:064027. doi: 10.1103/PhysRevD.78.064027

CrossRef Full Text | Google Scholar

135. Mazur PO. Spinning cosmic strings and quantization of energy. Phys Rev Lett. (1986) 57:929–32.

PubMed Abstract | Google Scholar

136. Mazur PO. Mazur replies. Phys Rev Lett. (1987) 59:2380.

PubMed Abstract

137. Mueller M, Perry MJ. Constraints on magnetic mass. Class Quant Grav . (1986) 3:65–9.

Google Scholar

138. Bergshoeff EA, de Roo M, Kerstan SF, Kleinschmidt A, Riccioni F. Dual gravity and matter. Gen Rel Grav. (2009) 41:39–48. doi: 10.1007/s10714-008-0650-4

CrossRef Full Text | Google Scholar

139. Dunsby PKS, Bassett BACC, Ellis GFR. Covariant analysis of gravitational waves in a cosmological context. Class Quant Grav . (1997) 14:1215–22.

Google Scholar

140. Hogan PA, Ellis GFR. Propagation of information by electromagnetic and gravitational waves in cosmology. Class Quant Grav . (1997) 14:A171–88.

Google Scholar

141. Maartens R, Ellis GFR, Siklos STC. Local freedom in the gravitational field. Class Quant Grav . (1997) 14:1927–36.

Google Scholar

142. Baekler P, Boulanger N, Hehl FW. Linear connections with a propagating spin-3 field in gravity. Phys Rev D. (2006) 74:125009. doi: 10.1103/PhysRevD.74.125009

CrossRef Full Text | Google Scholar

143. Boulanger N, Kirsch I. Higgs mechanism for gravity. II. Higher spin connections. Phys Rev D. (2006) 73:124023. doi: 10.1103/PhysRevD.73.124023

CrossRef Full Text | Google Scholar

144. Maartens R. Covariant velocity and density perturbations in quasi-Newtonian cosmologies. Phys Rev D. (1998) 58:124006.

Google Scholar

145. Mikhailov A. Notes on Higher Spin Symmetries. Report number: NSF-ITP-01-181, ITEP-TH-66-01 (2002).

146. Weinberg S. Photons and gravitons in perturbation theory: derivation of Maxwell's and Einstein's equations. Phys Rev. (1965) 138:988–1002.

Google Scholar

147. Raynal J. Multipole expansion of a two-body interaction in helicity formalism and its applications to nuclear structure and nuclear reaction calculations. Nucl Phys A. (1967) 97:572–92.

Google Scholar

148. Gastmans R, Troost W, Wu TT. Production of heavy quarkonia from gluons. Nucl Phys B. (1987) 291:731–45.

Google Scholar

149. Becchi C, Rouet A, Stora R. The abelian Higgs Kibble model, unitarity of the S-operator. Phys Lett B. (1974) 52:344–6.

Google Scholar

150. Becchi C, Rouet A, Stora R. Renormalization of the abelian Higgs-Kibble model. Commun Math Phys. (1975) 42:127–62.

Google Scholar

151. Becchi C, Rouet A, Stora R. Renormalization of gauge theories. Annals Phys. (1976) 98:287–321.

Google Scholar

152. Tyutin IV. Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism. PN Lebedev Physical Institute Preprint, LEBEDEV-75-39 (1975).

153. Batalin IA, Vilkovisky GA. Gauge algebra and quantization. Phys Lett B. (1981) 102:27–31.

154. Batalin IA, Vilkovisky GA. Quantization of gauge theories with linearly dependent generators. Phys Rev D. (1983) 28:2567–82.

Google Scholar

155. Batalin IA, Vilkovisky GA. Erratum: quantization of gauge theories with linearly dependent generators. Phys Rev D. (1984) 30:508.

Google Scholar

156. Batalin IA, Vilkovisky GA. Closure of the gauge algebra, generalized lie equations and Feynman rules. Nucl Phys B. (1984) 234:106–24.

Google Scholar

157. Batalin IA, Vilkovisky GA. Existence theorem for gauge algebra. J Math Phys. (1985) 26:172–84.

Google Scholar

158. Barnich G, Henneaux M. Consistent couplings between fields with a gauge freedom and deformations of the master equation. Phys Lett B. (1993) 311:123–9.

Google Scholar

159. Henneaux M. Consistent interactions between gauge fields: the cohomological approach. Contemp Math. (1998) 219:93.

Google Scholar

160. Henneaux M. Lectures on the antifield-BRST formalism for gauge theories. Nucl Phys B Proc Suppl. (1990) 18:47–105.

Google Scholar

161. Fisch JML, Henneaux M. Homological perturbation theory and the algebraic structure of the antifield-antibracket formalism for gauge theories. Commun Math Phys. (1990) 128:627.

Google Scholar

162. Gomis J, París J, Samuel S. Antibracket, antifields and gauge-theory quantization. Phys Rep. (1995) 259:1–145.

Google Scholar

163. Barnich G, Brandt F, Henneaux M. Local BRST cohomology in the antifield formalism: I. General theorems. Commun Math Phys. (1995) 174:57–91.

Google Scholar

164. Barnich G, Brandt F, Henneaux M. Local BRST cohomology in the antifield formalism: II. Application to Yang-Mills theory. Commun Math Phys. (1995) 174:93–116.

165. Fuster A, Henneaux M, Maas A. BRST quantization: a short review. Int J Geom Meth Mod Phys. (2005) 2:939–64. doi: 10.1142/S0219887805000892

CrossRef Full Text

166. Rahman R. Higher Spin Theory - Part I. PoS.ModaveVIII:004 (2012).

167. Danehkar A. On the cohomological derivation of Yang-Mills theory in the antifield formalism. JHEP Grav Cosmol. (2017) 3:368–87. doi: 10.4236/jhepgc.2017.32031

CrossRef Full Text | Google Scholar

168. Ogievetsky VI, Polubarinov IV. Interacting field of spin 2 and the einstein equations. Annals Phys. (1965) 35:167–208.

Google Scholar

169. Wald RM. Spin-two fields and general covariance. Phys Rev D. (1986) 33:3613–25.

PubMed Abstract | Google Scholar

170. Hindawi A, Ovrut BA, Waldram D. Consistent spin-two coupling and quadratic gravitation. Phys Rev D. (1996) 53:5583–96.

PubMed Abstract | Google Scholar

171. Boulanger N, Damour T, Gualtieri L, Henneaux M. Inconsistency of interacting, multi-graviton theories. Nucl Phys B. (2001) 597:127–71. doi: 10.1016/S0550-3213(00)00718-5

CrossRef Full Text | Google Scholar

172. Aragone C, Deser S. Consistency problems of spin-2-gravity coupling. Nuovo Cim B. (1980) 57:33–49.

173. Cutler C, Wald RM. A new type of gauge invariance for a collection of massless spin-2 fields. I. Existence and uniqueness. Class Quant Grav. (1987) 4:1267–78.

Google Scholar

174. Deser S. Self-interaction and gauge invariance. Gen Rel Grav. (1970) 1:9–18.

Google Scholar

175. Hassan SF, Rosen RA. Bimetric gravity from ghost-free massive gravity. JHEP (2012) 2:126. doi: 10.1007/JHEP02(2012)126

CrossRef Full Text

176. Hassan SF, Schmidt-May A, von Strauss M. Proof of consistency of nonlinear massive gravity in the Stückelberg formulation. Phys Lett B. (2012) 715:335–9. doi: 10.1016/j.physletb.2012.07.018

CrossRef Full Text | Google Scholar

177. Hassan SF, Schmidt-May A, von Strauss M. On consistent theories of massive spin-2 fields coupled to gravity. JHEP (2013) 5:86. doi: 10.1007/JHEP05(2013)086

CrossRef Full Text | Google Scholar

178. Bekaert X, Boulanger N, Cnockaert S. No self-interaction for two-column massless fields. J Math Phys. (2005) 46:012303. doi: 10.1063/1.1823032

CrossRef Full Text | Google Scholar

179. Bizdadea C, Cioroianu EM, Danehkar A, Iordache M, Saliu SO, Săraru SC. Consistent interactions of dual linearized gravity in D=5: couplings with a topological BF model. Eur Phys J C. (2009) 63:491–519. doi: 10.1140/epjc/s10052-009-1105-0

CrossRef Full Text | Google Scholar

180. Julia BL. Electric-magnetic duality beyond four dimensions and in general relativity. In: Differential geometry and physics. Proceedings, 23rd International Conference (Tianjin) (2005). p. 266–72.

Google Scholar

181. Julia B, Levie J, Ray S. Gravitational duality near de Sitter space. JHEP (2005) 11:025. doi: 10.1088/1126-6708/2005/11/025

CrossRef Full Text | Google Scholar

182. Tumanov AG, West P. E11 and the non-linear dual graviton. Phys Lett B. (2018) 779:479–84. doi: 10.1016/j.physletb.2018.02.015

CrossRef Full Text | Google Scholar

183. Berends FA, Burgers GJH, van Dam H. On spin three self interactions. Z Phys C. (1984) 24:247–54.

Google Scholar

184. Berends FA, Burgers GJH, Van Dam H. On the theoretical problems in constructing interactions involving higher-spin massless particles. Nucl Phys B. (1985) 260:295–322.

Google Scholar

185. Bengtsson AKH. Gauge invariance for spin-3 fields. Phys Rev D. (1985) 32:2031–6.

PubMed Abstract | Google Scholar

186. Bekaert X, Boulanger N, Cnockaert S. Spin three gauge theory revisited. JHEP (2006) 1:052. doi: 10.1088/1126-6708/2006/01/052

CrossRef Full Text | Google Scholar

187. Boulanger N, Leclercq S, Cnockaert S. Parity-violating vertices for spin-3 gauge fields. Phys Rev D. (2006) 73:065019. doi: 10.1103/PhysRevD.73.065019

CrossRef Full Text | Google Scholar

188. Coleman S, Mandula J. All possible symmetries of the S matrix. Phys Rev. (1967) 159:1251–6.

Google Scholar

189. Haag R, Łopuszański JT, Sohnius M. All possible generators of supersymmetries of the S-matrix. Nucl Phys B. (1975) 88:257–74.

Google Scholar

190. Maldacena J, Zhiboedov A. Constraining conformal field theories with a higher Spin symmetry. J Phys A Math Gen. (2013) 46:214011. doi: 10.1088/1751-8113/46/21/214011

CrossRef Full Text | Google Scholar

191. Alba V, Diab K. Constraining conformal field theories with a higher spin symmetry in d>3 dimensions. JHEP (2016) 3:44. doi: 10.1007/JHEP03(2016)044

CrossRef Full Text | Google Scholar

192. Alba V, Diab K. Constraining conformal field theories with a higher spin symmetry in d=4. arXiv:1307.8092 [hep-th]. (2013). Available online at: https://arxiv.org/abs/1307.8092

Google Scholar

193. Bekaert X, Cnockaert S, Iazeolla C, Vasiliev MA. Nonlinear higher spin theories in various dimensions. In: Higher Spin Gauge Theories: Proceedings, 1st Solvay Workshop (Brussels) (2004). p. 132–97.

Google Scholar

194. Vasiliev MA. Extended higher-spin superalgebras and their realizations in terms of quantum operators. Fortsch Phys. (1988) 36:33–62.

Google Scholar

195. Vasiliev MA. Equations of motion of interacting massless fields of all spins as a free differential algebra. Phys Lett B. (1988) 209:491–7.

Google Scholar

196. Lopatin VE, Vasiliev MA. Free massless bosonic fields of arbitrary spin in D-dimensional de sitter space. Mod Phys Lett A. (1988) 3:257–70.

Google Scholar

197. Fradkin ES, Vasiliev MA. On the gravitational interaction of massless higher-spin fields. Phys Lett B. (1987) 189:89–95.

Google Scholar

198. Fradkin ES, Vasiliev MA. Cubic interaction in extended theories of massless higher-spin fields. Nucl Phys B. (1987) 291:141–71.

Google Scholar

199. Vasiliev MA. Consistent equations for interacting massless fields of all spins in the first order in curvatures. Annals Phys. (1989) 190:59–106.

Google Scholar

200. Vasiliev MA. Consistent equations for interacting gauge fields of all spins in 3+1 dimensions. Phys Lett B. (1990) 243:378–82.

Google Scholar

201. Vasiliev MA. Algebraic aspects of the higher-spin problem. Phys Lett B. (1991) 257:111–8.

Google Scholar

202. Vasiliev MA. Properties of equations of motion of interacting gauge fields of all spins in 3+1 dimensions. Class Quant Grav . (1991) 8:1387–417.

Google Scholar

203. Vasiliev MA. More on equations of motion for interacting massless fields of all spins in 3+1 dimensions. Phys Lett B. (1992) 285:225–34.

Google Scholar

204. Prokushkin SF, Vasiliev MA. Higher-spin gauge interactions for massive matter fields in 3D AdS space-time. Nucl Phys B. (1999) 545:385–433.

Google Scholar

205. Vasiliev MA. Cubic interactions of bosonic higher spin gauge fields in AdS5. Nucl Phys B. (2001) 616:106–62. doi: 10.1016/S0550-3213(01)00433-3

CrossRef Full Text | Google Scholar

206. Alkalaev KB, Vasiliev MA. N=1 supersymmetric theory of higher spin gauge fields in AdS5 at the cubic level. Nucl Phys B. (2003) 655:57–92. doi: 10.1016/S0550-3213(03)00061-0

CrossRef Full Text | Google Scholar

207. Vasiliev MA. Nonlinear equations for symmetric massless higher spin fields in (A)dSd. Phys Lett B. (2003) 567:139–51. doi: 10.1016/S0370-2693(03)00872-4

CrossRef Full Text | Google Scholar

208. Brink L, Metsaev RR, Vasiliev MA. How massless are massless fields in AdS d. Nucl Phys B. (2000) 586:183–205. doi: 10.1016/S0550-3213(00)00402-8

CrossRef Full Text | Google Scholar

209. Boulanger N, Kessel P, Skvortsov E, Taronna M. Higher spin interactions in four-dimensions: vasiliev versus Fronsdal. J Phys A Math Gen. (2016) 49:095402. doi: 10.1088/1751-8113/49/9/095402

CrossRef Full Text | Google Scholar

210. Didenko VE, Skvortsov ED. Elements of Vasiliev theory. arXiv:1401.2975 [hep-th]. (2014). Available online at: https://arxiv.org/abs/1401.2975

Google Scholar

211. Maldacena J. The Large-N limit of superconformal field theories and supergravity. Int J Theor Phys. (1999) 38:1113.

Google Scholar

212. Witten E. Anti-de Sitter space and holography. Adv Theor Math Phys. (1998) 2:253–91.

Google Scholar

213. Gubser SS, Klebanov IR, Polyakov AM. Gauge theory correlators from non-critical string theory. Phys Lett B. (1998) 428:105–14.

Google Scholar

214. Sezgin E, Sundell P. Massless higher spins and holography. Nucl Phys B. (2002) 644:303–70. doi: 10.1016/S0550-3213(02)00739-3

CrossRef Full Text | Google Scholar

215. Klebanov IR, Polyakov AM. AdS dual of the critical O( N) vector model. Phys Lett B. (2002) 550:213–9. doi: 10.1016/S0370-2693(02)02980-5

CrossRef Full Text | Google Scholar

216. Sezgin E, Sundell P. Holography in 4D (super) higher spin theories and a test via cubic scalar couplings. JHEP (2005) 7:044. doi: 10.1088/1126-6708/2005/07/044

CrossRef Full Text | Google Scholar

217. Giombi S, Yin X. Higher spin gauge theory and holography: the three-point functions. JHEP (2010) 9:115. doi: 10.1007/JHEP09(2010)115

CrossRef Full Text | Google Scholar

218. Giombi S, Yin X. The higher spin/vector model duality. J Phys A Math Gen. (2013) 46:214003. doi: 10.1088/1751-8113/46/21/214003

CrossRef Full Text | Google Scholar

219. Jevicki A, Jin K, Ye Q. Perturbative and non-perturbative aspects in vector model/higher spin duality. J Phys A Math Gen. (2013) 46:214005. doi: 10.1088/1751-8113/46/21/214005

CrossRef Full Text | Google Scholar

220. Vasiliev MA. Symmetries and invariants in higher-spin theory. In: Brink L, Henneaux M, Vasiliev MA, editors. Higher Spin Gauge Theories. Proceedings, International Workshop. Singapore; New York, NY: World Scientific Publishing Co. (2017). p. 1–15. doi: 10.1142/10142

CrossRef Full Text | Google Scholar

221. Vasiliev MA. Current interactions and holography from the 0-form sector of nonlinear higher-spin equations. JHEP (2017) 10:111. doi: 10.1007/JHEP10(2017)111

CrossRef Full Text | Google Scholar

222. Vasiliev MA. On the local frame in nonlinear higher-spin equations. JHEP (2018) 1:62. doi: 10.1007/JHEP01(2018)062

CrossRef Full Text | Google Scholar

Keywords: electric-magnetic duality, gravitation, dual graviton, higher-spin fields, gauge fields

Citation: Danehkar A (2019) Electric-Magnetic Duality in Gravity and Higher-Spin Fields. Front. Phys. 6:146. doi: 10.3389/fphy.2018.00146

Received: 15 March 2018; Accepted: 04 December 2018;
Published: 09 January 2019.

Edited by:

Osvaldo Civitarese, National University of La Plata, Argentina

Reviewed by:

Marika Taylor, University of Southampton, United Kingdom
Peter C. West, King's College London, United Kingdom

Copyright © 2019 Danehkar. 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.

*Correspondence: Ashkbiz Danehkar, ashkbiz.danehkar@cfa.harvard.edu

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.