REVIEW published: 09 January 2019 doi: 10.3389/fphy.2018.00146

Electric-Magnetic Duality in Gravity and Higher-Spin Fields

Ashkbiz Danehkar*

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

Edited by: fields. In this review, we focus on dual formulations of gravity and higher spin fields in Osvaldo Civitarese, the linearized theory, and study their SO(2) electric-magnetic duality invariance, twisted National University of La Plata, Argentina self-duality conditions, harmonic conditions for wave solutions, and their configurations Reviewed by: with electric-type and magnetic-type sources. Furthermore, we briefly discuss the latest Marika Taylor, developments in their interacting theories. University of Southampton, United Kingdom Keywords: electric-magnetic duality, gravitation, dual graviton, higher-spin fields, gauge fields Peter C. West, King’s College London, United Kingdom 1. INTRODUCTION *Correspondence: Ashkbiz Danehkar Electric-magnetic duality basically evolved from the invariance of Maxwell’s equations [1], which [email protected] led to the hypothesis about magnetic monopoles in electromagnetism [2], and the introduction of magnetic-type sources to gauge theories [3–7]. Electric-magnetic duality contributed to the Specialty section: development of Yang-Mills theories [8–11], p-form gauge fields [12–14], supergravity theories This article was submitted to [15, 16], and string theories [17–19]. Generalizations of this duality emerged as weak-strong High-Energy and Astroparticle (Montonen-Olive) duality in Yang-Mills theories [7]. The duality principle was refined in Physics, supersymmetry [20] and was extended to N = 4 supersymmetric gauge theories [21]. The N = 4 a section of the journal Z Frontiers in Physics Super Yang-Mills theory has an exact SL(2, ) symmetry, typically referred to as S-duality, that includes a transformation known as weak-strong duality [22], and can be understood as electric- Received: 15 March 2018 magnetic duality in certain circumstances such as the Coulomb branch. Strong-weak duality Accepted: 04 December 2018 Published: 09 January 2019 exchanges the electrically charged string with the magnetically charged soliton in the heterotic string theory [23–25]. Electric-magnetic duality was shown in N = 1 supersymmetric gauge Citation: theories [26]. Compatifications of type II string theory (or M-theory) on a torus have an E Danehkar A (2019) Electric-Magnetic Z 7 Duality in Gravity and Higher-Spin symmetry, referred to as U-duality, which includes the T-duality O(6,6, ) and the S-duality Fields. Front. Phys. 6:146. SL(2, Z)[27]. S-duality of the N = 4 Super Yang-Mills theory and S- and U-dualities of type II doi: 10.3389/fphy.2018.00146 string and M-theory correspond to exact symmetries of the full consistent quantum theory [27, 28].

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U-duality (see e.g., [29–31]) provides nonperturbative insights in demonstrated that the formulations of the dual graviton with M-theory [32], and also in string theories [33, 34]. The electric- mixed symmetry (D − 3, 1) and the graviton are equivalent and magnetic duality principle also permeated into gravitational have the same bosonic degrees of freedom [44]. Considering theories [35–43], supergravity [44–46], higher-spin field theories the Riemann curvature and its dual curvature, the linearized [38, 47–49], holographic dictionary [50–52], hypergravity [53], Einstein equations and Bianchi identities were found to hold and partially-massless and massive gravity on (Anti-) de Sitter on dual gravity with mixed symmetry (D − 3,1) [38]. From space [54–56]. the “parent” first-order action introduced in [44], the familiar Gravitational electric-magnetic duality originally appeared “standard” second-order formulation of the dual graviton with in wave solutions of general relativity [57–59]. The Bianchi mixed symmetry (D − 3,1) in D dimensions was then obtained identities for the Weyl curvature provide some dynamical in [47], which was found to be the Curtright action [35] in equations for perturbations in cosmological models [60–66], D = 5 [through this paper, we consider the “standard” second- which are analogous with Maxwell’s equations [67, 68]. In general order formulations of dual gravity and spin-s fields, while the relativity, the Riemann curvature is decomposed into the Ricci “parent” first-order actions are briefly presented in Appendix 1 curvature and the Weyl curvature. The Ricci curvature defined (Supplementary Material)]. The dual formulation of linearized by the Einstein equations corresponds to the local dynamics of gravity have been further investigated by several authors [40, 41, spacetime, while the nonlocal characteristics of gravitation, such 43, 46, 88–90]. In the linearized theory, the magnetically charged as Newtonian force and gravitational waves, are encoded in the D−4 branes arise from Kaluza-Klein monopole, which are indeed Weyl curvature [66–70]. Based on this analogy, the electric and gravitationally magnetic-type sources (0-brane in D = 4), can magnetic parts of the Weyl curvature tensor have been called naturally be coupled with the gauge field of the dual graviton gravitoelectric and gravitomagnetic fields [64, 68, 71–74], though ([37]; see also [38, 91]). In fact, the dual graviton in D = 4 their spatial, symmetric 2-rank tensor fields make them dissimilar becomes the Pauli-Fierz field based on a symmetric tensor h˜ν , to the electric and magnetic vector fields of the U(1) gauge theory while in D = 5 we get the Curtright [35] field h˜νρ with for spin-1 fields. Young symmetry (2,1) [40]. Generally, the spin-2 field described Analogies between gravitation and electromagnetism by a rank-2 symmetric tensor hν is dualized to the linearized led to propose gravitationally magnetic-type (also called dual gravitational field h˜ ν described by a (D − 2)-rank gravitomagnetic) mass almost a half-century ago [75–78]. 1 D−3 tensor with Young symmetry type (D − 3,1) [37, 38, 43]. It The solutions of the geodesic equations for the Taub–NUT was also demonstrated that spin-2 fields can be dualized to two [79, 80] metric also pointed to the magnetic-type mass, whose different theories:1 dual graviton with (D − 3, 1) and double- characteristic could be identical with the angular momentum dual graviton with (D − 3, D − 3) [38, 96]. It was understood [81]. Moreover, the gravitational lensing by bodies with that mixed symmetry (D − 3, 1) fields can possess a Lorentz- gravitomagnetic masses, as well as the predicted spectra of invariance action principle. Although the equations of motion for magnetic-type atoms have been explained in Taub–NUT space mixed symmetry (D − 3, D − 3) fields could maintain the Lorentz [82]. The effects of the gravitomagnetic mass on the motion group SO(D − 1, 1) [38], some authors argued that they may of test particles and the propagation of electromagnetic waves not have a Lorentz-invariance action in D > 4 [40]. Moreover, have been studied in Kerr-Newman-Taub-NUT spacetime [83]. the graviton, the dual graviton and the double-dual gravitonin4 Furthermore, the gravitational field equations and dual curvature dimensions can be interchanged by a kind of the S-duality of D = tensors, which characterize the dynamics of gravitomagnetic 4 Maxwell theory [88]. In the context of the Macdowell-Mansouri matter (dual matter), as well as the relationship between the formalism, the strong-weak duality for linearized gravity also dynamical theories of gravitomagnetic mass (dual mass) and corresponds to small-large duality for the cosmological constant gravitoelectric mass (ordinary mass) have been elaborated among [36] (see also [97] for duality with a cosmological constant). The the NUT solutions, the spin-connection and vierbein [84, 85]. dual formulations of spin-2 fields in D dimensions have also been Finally, energy-momentum and angular momentum of both generalized to spin-s fields whose dual fields are formulated using electric- and magnetic-type masses have been formulated for s−1 spin-2 fields [86]. Young symmetry type (D − 3, 1, ,1)[38, 47, 49, 89, 96, 98]. The linearized formulations of gravity describe the graviton Dirac’s relativistic wave solutions [99] for massive particles using a 2-rank symmetric tensor h , which can be dualized to ν were the first theory extended to higher spins in 1939 [100], ˜ − a gauge field h1D−3ν with mixed symmetry (D 3,1), the which provided the foundation for the Lagrangian formulations so-called dual graviton. The dual gravity with mixed symmetry of massive, bosonic fields with arbitrary spin [101]. The free tensors first appeared in 1980 [35], and mixed symmetry (2,1) and interacting theories of massless, bosonic gauge fields with tensors were shown to be gauge fields [35]. It was also argued that spins higher than 2 (the so-called Fronsdal [102] field) have been a free massive mixed symmetry (2, 1) field in D = 4 could be dual investigated later [102–106]. In particular, dual representations to a free massive spin-2 field (massive graviton) and a massive of spin-s fields (s ≥ 2) were formulated using exotic higher-rank spin-0 field (Higgs boson) that may illuminate “magnetic-like” gauge fields of the Lorentz group SO(D − 1, 1) in the linearized aspects of gravity [87]. Following these earlier works [35, 87], theory [38, 48]. It was found that dual formations of spin-s fields mixed symmetry (D − 3,1) was first suggested as a candidate for the dual graviton in D dimensions in 2000 [37]. Using the 1In particular, it was shown that there are infinitely many off-shell dualities of the “parent” first-order action at the linearized level, it was then graviton [38, 92, 93], and infinite chains of dualities for spin-s fields [94, 95].

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s can be described by a tensor with mixed Young symmetry type with one column with D − 3 boxes and s − 1 columns with one tensor h1s with mixed symmetry type (1, ,1), which is represented by s boxes arranged in a row, while their dual field 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 are represented by a (D − 2 + s)-th rank tensor h1D−3ν1νs−1 (s = 2) in D = 5. Nonlinear theories of massless, totally s−1 symmetric spin-s fields were also discussed in the (Anti-) de Sitter with mixed symmetry (D − 3, 1, , 1) in D dimensions. The electric- and magnetic-type sources for spin-s fields in D = space, (A)dS [107]. It was shown that nonlinear representations of dual spin-s gauge theories can partially be implemented with 4 are denoted by s-th rank energy-momentum tensors T1s ˜ a Killing vector [38]. The conserved electric- and magnetic-type and T1s , respectively. The Riemann curvatures for spin-s sources associated with spin-s fields and their corresponding fields are represented by 2s-rank tensors R1ν1sνs . The dual dual fields were also derived, in analogy with electromagnetism Riemann curvatures for spin-s fields in D = 4 are denoted by ˜ [91]. Recently, the twisted self-duality conditions, which were R1ν1sνs . We write formulations according to the convention previously discussed in dual formulations of linearized gravity (D) (D) 8πGN = 1 = c, unless the physical constants specified (GN [43, 108], were also illustrated in higher spin gauge fields, which is the gravitational constant in D dimensions and c is the speed maintain an SO(2) electric-magnetic invariance in D = 4[49]. of light). We utilize the round brackets enclosing indices for In what follows we evaluate general aspects of electric- symmetrization, hν = hν = h(ν), and the square brackets magnetic duality in gravity and higher spin field theories (s ≥ for antisymmetrization, Fν = −Fν = F[ν]. 2) in arbitrary dimensions (D ≥ 4), and advocate their formulations in linearized theories, their couplings with electric- 2. DUAL GRAVITON and magnetic-type sources, and their harmonic conditions for wave propagation. In section 2, we study dual formulations of the Free gauge theories in D-dimensional flat space are typically spin-2 field (dual graviton). We then extend them to the spin- described by a particular irreducible tensor representation of 3 field (section 3) and higher spin fields (section 4). Section 5 the little group SO(D − 2) [37]. In the view point of the little discusses the interacting theories of the dual graviton and spin-s group SO(D − 2), free abelian gauge theories can be dualized fields. 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 1.1. Notations and Conventions dual fields ⋆F (left dual), F⋆ (right dual), and ⋆F⋆ (left-right dual) To describe tensors with mixed symmetry types, we employ [96]. For symmetric tensors, we have ⋆F = F⋆. Young tableaux where a box in the Young diagram is associated The 1-form potential A of the free Maxwell field strength with each index of a tensor (see [41, 96, 109, 110] for full details). Fν ≡ 2∂[Aν] in D dimensions possesses physical gauge degrees A vector field (e.g., A) has Young symmetry type (1) described of freedom Ai in the little group SO(D − 2) representation, which by a single-box tableau . A 2nd rank anti-symmetric tensor, can be dualized to a dual (D − 3)-form potential as follows [38]: e.g., the Maxwell field Fν , has Young symmetry type (2), which A˜ = ǫ Ai, (1) is represented by two boxes arranged in a column, . A 2nd j1jD−3 j1jD−3i

i 1 ij1jD−3 rank symmetric tensor, e.g., the graviton field h , has Young A = ǫ A˜ j j . (2) ν (D − 3)! 1 D−3 symmetry type (1, 1), which is represented by two boxes arranged in a row, . Accordingly, the Curtright field, which describes While A is a single-box Young tableau, , its dual field is a the dual gravition h˜νρ in D = 5, is a 3rd rank tensor with Young single-box tableau in D = 4, a tableau with two boxes in a symmetry type (2, 1), and is represented by the Young diagram column in D = 5, and a tableau with D − 3 boxes in a column . The Riemann tensor R and the Weyl tensor C are νρλ νρλ in arbitrary D dimensions. 4th rank tensors whose algebraic properties correspond to Young For spin-2 fields such the graviton hν , there are two different symmetry (2,2), i.e., the diagram . We utilize the notations dual linearized formulations in generic D-dimensional spacetime [38, 40] (and infinitely many off-shell dualities [92, 93]). In in which the Greek alphabet refers to covariant spacetime indices the first dual formulation, the physical gauge graviton hij in D (e.g., A, hν ), and the Latin alphabet refers to light-cone gauge dimensions is dualized to a dual field h˜ in the little group indices (e.g., A , h ),which isalso used for the E ⊗ l non-linear i1 iD−3j i ij 11 s 1 SO(D − 2) representation as follows [38]: b realization in section 2.3 (e.g., ha , Aa1a2a3 ). Throughout this paper, the graviton is denoted by a 2nd ˜ k hi1iD−3j = ǫi1iD−3 hjk, (3) rank tensor hν , whereas the dual graviton is shown by a (D − 1 i1iD−3 ˜ 2)-rank tensor h˜ with mixed symmetry (D − 3,1) in hjk = ǫ jhi1iD−3k, (4) 1D−3ν (D − 3)! D dimensions. A 3nd rank symmetric tensor, e.g., the typical spin-3 field hνρ , has mixed symmetry type (1,1,1), which is which has the following properties: represented by three boxes arranged in a row, . The ˜ ˜ ˜ ˜ spin-s fields (s > 2) are denoted by a s-th rank symmetric hi1iD−3j = h[i1iD−3]j ≡ hi1iD−3|j, h[i1iD−3j] = 0. (5)

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˜ The first dual graviton hα1αD−3 is described by a tensor with where the Riemann tensor Rαβν is of Young symmetry type Young symmetry type (D − 3,1). In D = 4, the dual graviton ˜ (2,2) ≡ , the first dual Riemann tensor Rα1αD−2ν has is a symmetric 2nd rank tensor h˜ν with the Young tableau having two boxes in a raw, , while it is a mixed symmetry Young symmetry type (D − 2,2), and the second dual Riemann ˆ tensor Rα1αD−2β1βD−2 has mixed symmetry type (D − 2, D − 2). (2,1) tensor in D = 5 with the Young diagram (the 2.1. D = 4: The Pauli-Fierz Field so-called Curtright field). The dual graviton h˜ possesses the The Pauli-Fierz action [100, 111], describing the equations of equations of motion, which are equivalent to gravity and describe motion of a free massless spin-2 field hν in D-dimensional the same gravitational degrees of freedom of the graviton h D−1 [44]. The equivalence between the different formulations can be Minkowski spacetime η = diag(−, +, , +) reads explicitly demonstrated by using their light-cone gauge fields. ν In the second dual formulation, the physical light-cone 3 c D gauge graviton hij in D dimensions is dualized to a dual field SPF = d xLPF[hν ], (12) ˆ 16πG(D) hi1iD−3j1jD−3 as follows [38]: N

ˆ m n where the Lagrangian LPF is quadratic in the first derivatives of hi i j j = ǫi i ǫj j hmn, (6) 1 D−3 1 D−3 1 D−3 1 D−3 the spin-2 field h and its covariant trace (h ≡ h α): 1 ν α i1iD−3 j1jD−3 ˆ hmn = ǫ mǫ nhi i j j , (7) (D − 3)!(D − 3)! 1 D−3 1 D−3 1 α ν ν α LPF[hν ] = − ∂αhν ∂ h − 2∂h ∂ hαν having the following properties: 2 ν α +2h∂∂νh − ∂αh∂ h , (13) hˆ = hˆ , hˆ = 0. i1 iD−3j1 jD−3 [i1 iD−3][j1 jD−3] [i1 iD−3j1]j2 jD−3 (8) and is invariant under the gauge transformation ˆ The second dual graviton hα1αD−3β1βD−3 (also called double- dual graviton) is a tensor with Young symmetry type (D − 3, D − δξ hν = 2∂(ξν), (14) 3). In D = 4, it yields a symmetric 2nd rank tensor hˆν with Young symmetry type (1,1). In D = 5, it becomes a mixed where ξ is an arbitrary vector field. The above gauge transformation is irreducible. symmetry (2, 2) tensor with the Young diagram whose The action (13) has the following equations of motion: algebraic properties are similar to the typical (spin-2) Riemann L tensor Rνρλ. δ PF[hν ] α ν α α = ∂α∂ h − ∂∂ hαν + ∂∂νh − ην ∂α∂ h = 0. It can easily be seen that the physical light-cone dual field on δhν ˆ both indices, hi1iD−3j1jD−3 , can be written using the physical (15) ˜ The symmetric tensor hν is associated with a free massless spin- light-cone dual field on one index hi1iD−3j, 2 particle (graviton), and its linearized Riemann curvature Rναβ ˆ 1 k ˜ l ˜ is defined as (e.g., [41, 43, 108]), hi i j j = ǫi i h + ǫj j h . 1 D−3 1 D−3 2 1 D−3 j1 jD−3k 1 D−3 i1 iD−3l αβ [α β] (9) Rν = −2∂ ∂[hν] , (16) The first physical dual field h˜ is recovered from the physical second dual field hˆ, so the equations of motion written for the with the following properties Hodge dual field hˆ are equivalent to the field h˜. However, it was argued that the second dual field hˆ in D > 4 may not Rναβ = R[ν]αβ , Rναβ = Rν[αβ], R[να]β = 0, (17) have a Lorentz-invariant action [40] (though Lorentz covariant field equations were given in Hull [38]). We also note that for a and fulfills the Bianchi identity ∂[ρRν]αβ = 0 and the linearized αβ traceless field (h = 0), there is no second dual formulation, as Einstein equations Rν = 0, where Rν ≡ Rανβ η is the it cannot be dualized on both indices. Therefore, we only consider linearized Ricci tensor. We can also define the scalar curvature the first dual formulation of gravity (also its equivalences for as R ≡ R . spin-s fields) throughout this paper. From the definition (16) and the equations of motion (15), it For the typical (spin-2) linearized Riemann curvature Rαβν , follows the Euler-Lagrange variation: one gets the following dual linearized Riemann curvature in D- δLPF[hν ] 1 dimensional spacetime for the first and second dual formulations, = Rν − ηνR ≡ Gν , (18) respectively [38]: δhν 2

1 αβ where Gν is the linearized Einstein tensor, and fulfills the R˜ α α ν = ǫα α Rαβν , (10) 1 D−2 2 1 D−2 contracted Bianchi identity. ˆ 1 ν ρλ The linearized Riemann curvature tensor Rναβ of the Pauli- Rα1αD−2β1βD−2 = ǫα1αD−2 ǫβ1βD−2 Rνρλ, (11) 4 Fierz field hν in D dimensions can be split into the Weyl

Frontiers in Physics | www.frontiersin.org 4 January 2019 | Volume 6 | Article 146 Danehkar Duality in Gravity and Higher-Spin Fields curvature tensor and the expression formulated by the linearized Accordingly, the electric and magnetic parts of the linearized Ricci tensor and scalar curvature: Weyl tensor can be formulated using twice covariant and/or time derivatives of the Pauli-Fierz field hν in the instantaneous rest- ν ν 4 [ ν] 2 ν R αβ = C αβ + R[α ηβ] − Rη [αηβ] , space of an observer moving with the relativistic velocity u in D − 2 (D − 1) (D − 2) (19) Minkowski spacetime ην , where the Weyl tensor Cναβ is traceless, and has mixed symmetry type (2,2) ≡ , so 1 0 ′ ′′ Eν [h] =− ∂∂ν h0 − 2 ∂ hν + hν 2 ( )0 1 ρσ ρ σ C = −C , C = −C , C = C , (20) − (uuν + ην ) ∂ρ ∂σ h − ∂ ∂ρ hσ ναβ ναβ ναβ νβα ναβ αβν (D − 1) (D − 2) 1 1 ρ ρ 1 ρ − ∂ ∂ρ hν − ∂ ∂ hν ρ + ∂∂ν hρ C[ναβ] = 0, C νρ = 0. (21) (D − 2) 2 ( ) 2 ρ ′ ρ ′ ρ ρ The Weyl tensor has D(D + 1)(D + 2)(D − 3)/12 independent + ∂(uν)hρ − ∂ hρ(uν) + ∂ ∂ρ h0(uν) − ∂ hρ0∂(uν) components (0, 10, and 35 components for D = 3,4, and 5, 1 ρ 0 ρ ′ 1 ρ ′′ − ∂ ∂ρ h0 ην + ∂ hρ0 ην − hρ ην , (27) respectively). 2 2 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 1 ρ λ ′ 1 ρ λ Bν [h] = − ǫρλ( ∂ hν) + ǫρλ(∂ν)∂ h0 derivatives of the Pauli-Fierz field hν (see Equations 27, 28), and 2 2 defined as follows: 1 ρ λ γσ γ σ − ǫρλ η ν u ∂γ ∂σ h − ∂ ∂γ hσ (D − 1) (D − 2) ( ) α β α β Eν = Cανβ u u ≡ C0ν0, Bν = −C˜ ανβ u u ≡ −C˜ 0ν0, 1 λ σ ρ ρ σ λ (22) + ǫρλ(ην) ∂ ∂σ h0 − ǫρλ(hν) ∂ ∂σ u 2(D − 2) ˜ = 1 ǫ ρλ where Cναβ 2 νρλC αβ is a Hodge dual of the Weyl σ ρ λ ρ λ σ 0 1 ρλ α β 1 ρλ + ǫρλ(∂ hν)σ ∂ u − ǫρλ(∂ ην) ∂σ h tensor, so Bν = − 2 ǫαρλC νβ u u = − 2 ǫρλC ν0 (the 2 σ ρ λ σ ρ λ ′ sign convention similar to Maxwell theory ). Here, ǫρλ is + ǫρλ(∂ν)∂ hσ u − ǫρλ( ∂ hσ ην) ρ taken to be the spatial permutation tensor (ǫνρu = 0) with ρ λ σ ρ λ σ ′ β − ǫρλ(∂ν)∂ u hσ + ǫρλ( ∂ ην) hσ , (28) ǫναβ = 2u[ǫν]αβ − 2ǫν[αuβ], so ǫνα = ǫναβ u and ǫνβ = α ν ν −ǫναβ u (taking u u = −1 and u η = u in Minkowski ′ α spacetime). where primes denote time derivatives, i.e., (T) = u ∂αT, ρ The electric and magnetic parts of the Weyl tensor are both and ǫρλ is the spatial permutation tensor (ǫνρu = 0). symmetric and traceless: It is seen that the above equations are symmetric, traceless ν ν (E = 0 = B ) and spatial (Eν u = 0 = Bν u ). While E = E , B = B , E = 0, B = 0. ν (ν) ν (ν) (23) the gravitoelectric field Eν is associated with time/covariant In D = 4, the Weyl tensor has ten independent components, and derivatives of time/covariant derivatives of the Pauli-Fierz field its electric and magnetic parts have five independent components hν , the gravitomagnetic field Bν is introduced by covariant each. curls and rotations of time/covariant derivatives of the Pauli- Fierz field hν (the covariant curl is defined as curl(h)ν ≡ From (16), one obtains ρ λ [ρ λ] ǫρλ(∂ hν) = ǫρλ∂ hν ). 1 ρ ρ 1 ρ The linearized Riemann curvature tensor Rναβ can also be Rν = − ∂ ∂ρhν + ∂ ∂(hν)ρ − ∂ν ∂hρ , (24) 2 2 decomposed as

ν ν ν ν [ ν] R = ∂∂ν h − ∂ ∂hν , (25) R αβ = C αβ + 4S [αη β], (29) so the linearized Weyl tensor can be written in terms of the where Sν is the linearized Schouten tensor that is defined in D Pauli-Fierz field hν : dimensions as 2 Cν = − 2∂[∂ h ν] + η η ν αβ [α β] − ( − ) [α β] 1 1 (D 1) D 2 Sν = Rν − Rην , (30) ρσ ρ σ D − 2 2(D − 1) × ∂ρ∂σ h − ∂ ∂ρhσ 2 ρ [ [ ρ ρ [ − ∂[α∂ hρ + ∂ ∂ hρ[α − ∂ ∂ρh[α whose curl is called the Cotton tensor: D − 2 [ ρ ν] −∂[α∂ hρ ηβ] . (26) C αβ = 2∂[αS β], (31) 2The magnetic part of the Weyl curvature defined here has the opposite sign which satisfies compared to the definition often found in the literature of the 1 + 3 covariant formalism, see e.g., [64–66, 68]. Cνρ = C[νρ], C[νρ] = 0. (32)

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Substituting Equation (30) into Equation (29), the Riemann The equations of motion are invariant under the duality decomposition Equation (19) is easily recovered. transformations [42] The Pauli-Fierz theory of linearized gravity in D = 4 is 1 4 L ˜ R′ = cos αR + sin αR˜ , (41) dualized to the Pauli-Fierz action S = 2 d x PF[hν ] being ναβ ναβ ναβ formulated in terms of the symmetric massless 2nd-rank tensor R˜ ′ = − sin αR + cos αR˜ . (42) ˜ ˜ ˜ ναβ ναβ ναβ hν = hν = h(ν) (dual graviton) and its covariant trace ˜ ˜ α (h ≡ hα )[40, 42, 43]. In such a way, the symmetric tensor hν where the Riemann curvature Rναβ and its dual curvature ˜ is replaced with h˜ν in the Pauli-Fierz action (13). Similarly, we Rναβ are transferred into each other under the electric-magnetic can define the dual Riemann curvature R˜ ναβ for the Pauli-Fierz duality rotations (Equations 41, 42). ˜ αβ [α ˜ β] It is useful to see the electric-magnetic duality rotations in field hν as R˜ ν = −2∂ ∂[hν] , which is decomposed into αβ the dual Ricci tensor R˜ ν ≡ R˜ ανβ η and the dual Weyl tensor terms of the electric and magnetic parts of the linearized Riemann C˜ ναβ . curvature tensor (see e.g., [112]): There is an SO(2) electric-magnetic invariance between the ˜ Riemann curvature and its dual curvature, which is called the Eν = R0ν0, Bν = −R0ν0. (43) “twisted self-duality conditions” [38, 41, 43, 49, 88]: While the equations of motion are satisfied, the rotations in (41) R[h] = −⋆R˜ [h˜], R˜ [h˜] = ⋆R[h]. (33) and (42) are equivalent to E′ E B The gravitoelectric and gravitomagnetic fields of the Pauli-Fierz ν = cos α ν − sin α ν , (44) ˜ ′ field hν and its dual field hν also demonstrate the twisted Bν = sin αEν + cos αBν . (45) self-duality conditions [43, 49]: This SO(2) rotation in gravity is analogous to the duality Eν[h] = B˜ ν[h˜], Bν [h] = −E˜ ν [h˜], (34) transformations of Maxwell fields [113, 114] in which the free Maxwell field strength Fν is transformed into its dual field ˜ ˜ ˜ 1 αβ where Eν and Bν are the electric and magnetic parts of the strength Fν = 2 ǫναβ F . Using the definitions (16), it can be ˜ (dual) Weyl tensor Cναβ associated with the dual Pauli-Fierz shown that the graviton hν is rotated into the dual graviton h˜ν field h˜ν (dual graviton), and are both symmetric and traceless. under the transformations (41) and (42), so they are invariant The twisted self-dual conditions (Equations 33, 34) can be under the electric-magnetic duality. expressed in a duality-symmetric way [49]: 2.2. D = 5: The Curtright Field ⋆ ˜ R = S R, Eν = −SBν , Bν = SEν . (35) The equations of motion of a free massless spin-2 field h[ν]ρ ≡ h˜ν|ρ (dual graviton) in 5-dimensional flat spacetime can be with matrix notations described using the following action [35, 41, 47, 105]

⋆ R[h] ⋆ R[h] 1 R = , R = , (36) L[h˜ ] = − ∂γ h˜ν|ρ∂ h˜ − 4∂ h˜γ |λ∂ρh˜ R˜ [h˜] ⋆R˜ [h˜] ν|ρ γ ν|ρ γ ρ|λ 4 |λ γ ρ γ |ρ λ + 4h˜λ ∂ ∂ h˜γ |ρ − 2∂ h˜ρ ∂γ h˜ |λ σ |γ ρ γ |λ σ ρ Eν[h] Bν [h] +2∂ h˜ ∂ h˜ + 2∂ h˜ ∂ h˜ . E = , B = , (37) σ ρ|γ γ λ σ |ρ ν E˜ [h˜] ν B˜ [h˜] ν ν (46)

0 −1 It is also convenient to use a Lagrangian consisting of the S = , (38) 1 0 Curtright field strength tensor F[νλ]α ≡ Fνλ|α, the so-called Curtright action [35, 40] 3 which imply that the equations of motion for the dual graviton h˜ν are fully equivalent to those for the graviton hν under SO(2) 1 νρ|λ λ νρ| L[h˜νλ] = − Fνρ|λF − 3Fνλ| F ρ , (47) electric-magnetic duality rotations. 12 Using the first dual expression (Equation 10), the dual Riemann curvature R˜ ναβ in D = 4 is defined by where the Curtright field strength tensor Fνλα is a tensor with mixed symmetry type (3,1) ≡ defined as 1 ρλ R˜ αβν = ǫαβ Rρλν, (39) 2 ˜ ˜ ˜ ˜ which enjoys the following properties F[νρ]λ ≡ ∂hνρλ + ∂ρhνλ + ∂ν hρλ = 3∂[hνρ]λ, (48)

3 R˜ [να]β = 0, ∂[ρR˜ ν]αβ = 0. (40) The Curtright action could also be expressed by Equations (73, 74) of [89].

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and h˜νλ is the Curtright [35] field with mixed symmetry type The action (47) is gauge-invariant, and provides the following equations of motion [40]: (2,1) ≡ ,

δL[h˜ ] ˜ ˜ ˜ ˜ νρ = ˜ [ν]ρ + η[ν ˜ ρ] = hνρ = h[ν]ρ ≡ hν|ρ, h[νρ] = 0. (49) 3(R R ) 0, (59) δh˜νρ The Curtright action (47) is invariant under the following gauge transformation [35, 37, 38, 40, 115] which satisfy the contracted Bianchi identities (58). The linearized dual Riemann curvature of the Curtright field ˜ ˜ δσ ,αhνρ = 2(∂[σν]ρ + ∂[αν]ρ − ∂ραν ), (50) hνρ is decomposed into a traceless part (dual 5-D “Weyl” tensor), and terms containing the (dual 5-D) Schouten tensor: where σν and αν are symmetric and antisymmetric arbitrary νρ νρ [ν ρ] tensors, respectively: R˜ αβ = C˜ αβ + 3S˜ [αη β], (60)

σ = σ = σ , α = −α = α . (51) ν ν (ν) ν ν [ν] where the dual Schouten tensor S˜να is defined as [43]: The Curtright field Fνλα is transformed as [43]: 1 S˜να = R˜ να − R˜ [ην]α. (61) 3 δαFνλρ = −6∂ρ∂[ανλ], (52) ˜ although it is not gauge invariant. The gauge transformations The dual 5-D Weyl tensor Cνραβ has 35 independent components. for σν and αν are reducible, so we get δσ σν = 6∂(σν) and One can also define the Cotton tensor, the curl of the Schouten δσ αν = 2∂[σν], where σ is an arbitrary vector field. The gauge tensor, as symmetry for σ is irreducible [40]. The Curtright field h˜νλ is associated with the linearized dual ν ν C˜ αβ = 2∂[αS˜ β], (62) Riemann curvature tensor [37, 40, 43]: νβ which is traceless C˜ ναβ η = 0, and contains both mixed ˜ [νρ] [νρ] [ ˜νρ] R [αβ] = 2∂[αF β] = 6∂[α∂ h β], (53) symmetry types (2,2) and (3,1), and satisfies and the linearized dual Ricci curvature tensor and the dual vector ν C˜ ναβ = 0, ∂ γ C˜ αβ = 0. (63) curvature, respectively, [ ] [ ]

βρ It was shown in Curtright [35], Aulakh et al. [105], Labastida R˜ ν α = η R˜ νρ βα , (54) [ ] [ ][ ] and Morris [115], Hull [37], and Hull [38] that the physical ˜ να ˜ R = η R[ν]α. (55) degrees of freedom of the dual graviton in D = 5 are massless, and its dynamical variables are represented by spatial, transverse The linearized dual Riemann curvature tensor has the Young ˜ ˜ and traceless components hijk of the Curtright field hνρ , which ˜ tableau symmetry (3,2) ≡ , and the linearized dual Ricci have the Young symmetry similar to hνρ , 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 curvature tensor has the Young tableau symmetry (2, 1) ≡ . ˜ constructed by the spatial tensor hijk. Moreover, the dynamical The linearized dual Riemann curvature satisfies the “Bianchi variables of the standard graviton, which is dual to the Curtright identity” [38, 88] field h˜νρ in D = 5, are given by spatial, transverse and traceless components hij of the Pauli-Fierz field hν , which is symmetric ∂[λR˜ νρ]αβ = 0. (56) similar to hν . Similarly, there are the spatial Einstein tensor Gij We can also define the linearized “dual Einstein tensor” for the and the spatial Riemann curvature Rijrs which are constructed by Curtright field [37, 40, 43, 88] through the spatial tensor hij. Let us consider the electric and magnetic fields of the standard G˜ [ν]α = R˜ [ν]α − R˜ [ην]α, (57) graviton, which are constructed from the spatial Riemann curvature [43]: where G˜ [ν]α is the dual Einstein tensor with Young symmetry type (2,1), and fulfills the “contracted Bianchi identities” 1 rslk 1 lk Eijmn[h] = ǫijrsR ǫ , Bmni[h] = ǫ R0i , (64) 4 lkmn 2 mnlk α ∂ G˜ [ν]α = 0, ∂ G˜ [ν]α = 0. (58)

where Eijmn[h] has the Young symmetry , Bmni[h] and has The doubly contracted Bianchi identity yields ∂ R˜ = 0. The linearized dual Einstein equations are G˜ = 0, which are [ν]α the Young symmetry . equivalent to R˜ να = 0.

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We also build the electric and magnetic fields of the dual 2.3. D = 11: Mixed Symmetry (8,1) Field graviton from the spatial dual Einstein tensor and the spatial dual In the case of D = 11 spacetime, which are relevant to maximal Riemann curvature, respectively [43]: supergravity/M-theory, the dual graviton defined by the first dual ˜ representation is associated with a tensor field ha1a8|b of Young ˜ ˜ 1 lk E˜ijm[h] = G˜ ijm, B˜ijmn[h] = ǫ R˜ 0ij , (65) symmetry (8,1). Interestingly, the conjectured hidden symmetry 2 mnlk E11 of M-theory is decomposed into GL(11) subalgebras: a mixed symmetry (8, 1) tensor (dual graviton), a 3-form, a 6-form, where E˜ [h˜] has the Young symmetry , B˜ [h˜] and has ijm ijmn and other mixed symmetry fields [44]. It was shown that the the Young symmetry . dynamics and hidden symmetries of gravity are strongly linked to Lorentzian Kac–Moody algebras [116]. Moreover, the Weyl ˜ The electric fields Eijmn[h] and E˜ijm[h] are double-traceless groups of infinite-dimensional Kac–Moody algebras (see [117] +++ and traceless, respectively, which correspond to G00 = 0 and for review) are compatible with the groups E11(= E8 )[118– ˜ ++ G0j0 = 0. There are similar properties for the magnetic fields 120] and E10(= E8 ) [121, 122], which contain information ˜ Bmni[h] and B˜ijmn[h]: about the gravitational electric-magnetic duality, as well as the structures of M-theory (see [123] for review). jn im ˜ jm Eijmn[h]η η = 0, E˜ijm[h]η = 0, (66) The Dynkin diagram of the Lorentzian Kac-Moody algebra +++ ni ˜ jn im E11 = E8 is depicted below (see e.g., [44, 124–126]): Bmni[h]η = 0, B˜ijmn[h]η η = 0. (67) ˜ 11 The electric fields, Eijmn[h] and E˜ijm[h], and the magnetic fields, • B B˜ ˜ +++ | mni[h] and ijmn[h], have the following identities E11 = E ≡ • − • − • − • − • − • − • − • − • − • (75) 8 10 9 8 7 6 5 4 3 2 1 E [h] = 0, E˜ [h˜] = 0 (68) [ijm]n [mni] which encodes underlying hidden symmetries of M-theory [44, B B˜ ˜ ++ [mni][h] = 0, [ijm]n[h] = 0, (69) 124] (the same for E10 = E8 [127]). The various exceptional Lie groups (E6,..., E10) are embedded in the infinite-dimensional which are equivalent to the Ricci conditions R0j = 0 and Kac–Moody algebra E11, i.e., E6 ⊂ ⊂ E10 ⊂ E11. R˜ = 0. 0jm The infinite-dimensional Kac–Moody algebra E11 (also E10) The Bianchi identities imply that the electric fields, E [h] ijmn can be constructed from E8 through extensions of E8 with E˜ ˜ B B˜ ˜ and ijm[h], and magnetic fields, mni[h] and ijmn[h], are three extra nodes (two extra nodes in E10), so we derive the +++ ++ similarly transverse: Lorentzian Kac–Moody algebra E8 = E11 (also E8 = E10) [117]. i i ˜ ∂ Eijmn[h] = 0, ∂ E˜ijm[h] = 0, (70) The Kac–Moody algebra E11 can be decomposed into m m ˜ ∂ Bmni[h] = 0, ∂ B˜ijmn[h] = 0. (71) the GL(11) subalgebras by deleting the node 11 in the E11 Dynkin diagram (see the above diagram) [44, 46, 128]. The ˜ The electric fields Eijmn[h] and E˜ijm[h] transform under the mixed decomposition subalgebras correspond to the low level E11 tensor symmetries (2,2) and (2,1), while the magnetic fields Bmni[h] and fields in the non-linear realization of E11 and its vector, E11 ⊗s l1, Level:0 1 2 3 4 5 6 b ˜ (76) Field: ha , Aa1a2a3 , Aa1a6 , ha1a8|b, Aa1a9|b1b2b3 , Aa1a10|b1b2 , Aa1a11|b, . . . .

The zero- and third-level fields, h b and h˜ , correspond B˜ [h˜] transform under the mixed symmetries (2,1) and (2,2). a a1a8|b ijmn = From the twisted self-duality conditions [43], it follows that to the graviton and the dual graviton in D 11, respectively. The first-level bosonic 3-form field, Aα1α2α3 is related to non- ˜ ˜ Eijmn[h] = B˜ijmn[h], Bijm[h] = −E˜ijm[h], (72) gravitational degrees of freedom of M-theory. The second-level 6-form field, Aa1a6 , has equations of motion, which can be dual or in a duality-symmetric way to those of the 3-form field Aa1a2a3 , and provides another way to describe the degrees of freedom of A . These fields lead to at E = −SB, B = SE, (73) a1a2a3 least one spacetime coordinate for each in the E11 ⊗s l1 non-linear with matrix notations realization [125, 129],

Eijmn[h] Bijm[h] b a E = , B = , (74) Level 0: ha ↔ x , E˜ [h˜] B˜ [h˜] ijm ijmn Level 1: Aa1a2a3 ↔ xa1a2 , where the SO(2) duality matrix S is defined by Equation (38). Level 2: A ↔ x , Therefore, the electric and magnetic fields of the standard a1a6 a1a5 (77) graviton are fully equivalent to the magnetic and electric fields of ˜ Level 3: ha1a8|b ↔ xa1a8 , xa1a7|b, the dual graviton in D = 5 under SO(2) electric-magnetic duality . rotations (see [43] for constraint and dynamical formulas). .

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a (1)|c . The spacetime coordinate x , which corresponds to the graviton, From the unique E11 invariant Equation (82), Eab = 0, we get constructs the spacetime curvature for the dynamics of gravity. It (2)b the linearized Einstein equation, Ea = 0, and the equation is seen that the 3-form field, A , and 6-form field, A , are (2) a1a2a3 a1a6 of motion for the 11-dimensional dual graviton, E = 0. associated with the 2-form and 5-form coordinates, respectively. a1a8|b While from the unique E11 invariant Equation (83), one deduces The mixed symmetry (8,1) tensor field, h˜ , which is dual to a1a8|b the equations of motion E(2) = 0, and its dual equations of = a1a2a3 the graviton in D 11, is related to the 8-form coordinate and (2) the mixed symmetry (7, 1) coordinate. The dual graviton field and motion Ea1a6 = 0. Therefore, the algebra E11 yields a network its equation of motion possess the following identities [130] of equations of motion, including the typical equations of motion in general relativity. ˜ (2) h[a a |b] = 0, E = 0, 1 8 [a1a8|b] 2.4. Generalized Dual Spin-2 Fields As mentioned at the beginning of section 2, the Pauli-Fierz field where the equations of motion E(2) is of Young symmetry α1α8| associated with the spin-2 field hαβ is dual to a field dualized type (8, 1), and the numerical superscript describes the number on one index (dual graviton) or on both indices (double-dual of spacetime derivatives (see [129, 130] for more details of the graviton) [38, 96]. We consider only its dualization on one index E11 equations of motion), i.e., in empty space since it presents equations of motion equivalent to the linearized Einstein equations and its action satisfies Lorentz-invariance 1 (2) b [c ˜ b] [40]. Hence, the generalized, free massless, dual spin-2 fields E | ≡ − ∂ ∂[cha1a8] = 0, (78) a1 b8 4 ˜ ˜ h[α1 αD−3]β ≡ hα1 αD−3|β in arbitrary dimensions (D ≥ 4) can while the equations of motion for the graviton are [129]: be given by Young symmetry type (D − 3, 1) having the Young diagram [38, 43]: (2)b bc| Ea ≡ −2∂[aω c] = 0, (79) where ω is defined as (h b in the E ⊗ l non-linear αβ|γ a 11 s 1  realization) [129]: ˜ ≡ − hα1 αD−3|β D 3 boxes  (85)   ωab|c ≡ −∂ah(bc) + ∂bh(ac) + ∂ch[ab]. (80)   The equations of motion for the non-gravitational fields Aα α α  1 2 3 which possesses the following properties and Aα1α6 are also given by [129, 130]

(2) (2) ˜ ˜ ˜ ˜ b b hα1 αD−3β = h[α1 αD−3]β ≡ hα1 αD−3|β , h[α1 αD−3β] = 0, Ea1a2a3 ≡ ∂ ∂[bAa1a2a3] = 0, Ea1a6 ≡ ∂ ∂[bAa1a6] = 0. (81) (86) and has (D − 3)(D + 1)!/3(D − 2)! components in D dimensions There is a unique E11 invariant equation with one derivative, which is obtained from the non-linear realization [129], and it [41]. contains the graviton and its dual field: The action associated with the equations of motion of ˜ free massless, dual spin-2 fields hα1 αD−3β in D-dimensional 1 Minkowski spacetime explicitly reads [41, 47, 105]4 (1) d1d9 ˜ . E ≡ ω | − ǫ ∂ h = 0. (82) ab|c ab c 4 ab [d1 d2 d9]c L ˜ [hα1 αD−3|β ] The non-gravitational fields Aa a a and Aa a possess a unique 1 2 3 1 6 1 E11 invariant equation with one derivative (see [126, 131] for γ ˜α1...αD−3|β ˜ = − ∂ h ∂γ hα1...αD−3|β details): 2(D − 3)! ˜γ α2...αD−3|β ρ ˜ − 2(D − 3)∂γ h ∂ hρα2...αD−3|β (1) 1 b b 1 7 α2...αD−3|λ γ ρ Ea a ≡ ∂[a1 Aa2a3a4] − ǫa1a4 ∂b1 Ab2b7 = 0. (83) + 2(D − 3)h˜ ∂ ∂ h˜ 1 4 2(4!) λ γ α2...αD−3|ρ γ ˜ α2...αD−3|ρ ˜λ − (D − 3)∂ hρ ∂γ h α2...αD−3|λ Variations of the E11 equations of motion (82) and (83) result in ˜σ α2...αD−3|γ ρ ˜ (2) (2) (2) (2) + (D − 3)(D − 4)∂σ h ∂ hρα ...α α |γ E and E , with the duality relations of E and E , 2 D−2 D−3 ab a1a2a3 a1a8|b a1 a6 ˜ γ α3...αD−3|λ σ ˜ρ respectively [129]: +(D − 3)(D − 4)∂γ hλ ∂ h σ α3...αD−3|ρ . (87) . (1) = ↔ (1)|c = Ea1a4 0 Eab 0 It may also be written in terms of a field strength tensor ↓ ↓ Fα ...α − |β , in a similar manner of the Curtright action (47), as (2) (2)b 1 D 2 Ea1a2a3 = 0 Ea = 0 (84) ↔ 4Also, see the action defined by Equations (77) and (78) in [89], and Appendix 1 (2) (2) E = 0 E = 0 in Supplementary Material for the parent first-order action. a1 a6 a1a8|b

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5 follows [132–134] : The gauge transformations for χα1αm−1|β = χ[α1αm−1]β and α = α (m = D − 3 at the initial level) are 1 α1αm [α1αm] L ˜ α1...αD−2|β − [hα1 αD−3|β ] = − Fα1...αD−2|β F reducible to an arbitrary mixed symmetry type (m 2, 1) tensor 2(D − 2)! χ = χ and an arbitrary antisymmetric (m − α1αm−2|β [α1αm−2]β λ α1...αD−3ρ| α = α > − (D − 2)Fα1...αD−3λ| F ρ , 1)-rank tensor α1αm−1 [α1αm−1] for m 3 (see [115] for irreducible representations): (88) δ χ = − ∂ χ where the “generalized Curtright” field strength tensor χ α1αm−1|β (m 1) [α1 α2αm−1]β +∂ α − ∂ α , (94) Fα1...αD−2|β , a general form of (48), is a tensor with Young [α1 α2αm−1]β β α1α2αm−1 symmetry type (D − 2,1) with the diagram:

δχ αα1αm = (m)∂[α1 αα2αm]. (95)

 For m = 3, they are reduced to an arbitrary symmetric tensor  σ = σ and an arbitrary antisymmetric tensor α = α , F ≡ D − 2 boxes  (89) ν (ν) ν [ν] α1...αD−2|β    δχ χα1α2|β = 2 ∂[α1 σα2]β + ∂[α1 αα2]β − ∂β αα1α2 , (96)    δχ αα α α = 3∂[α αα α ]. (97) defined as  1 2 2 1 2 2 The gauge transformations for σν and αν are also reducible to ≡ − ∂ ˜ Fα1...αD−2|β (D 2) [α1 hα2αD−2]β . (90) an arbitrary vector field σ, For D = 4 and 5, the Lagrangian (87) recovers the Pauli- δσ σν = 6∂(σν), δσ αν = 2∂[σν], (98) Fierz (13) formulated by h˜ν and the Curtright action (46), respectively. For D = 5, the Lagrangian (88) also becomes the The gauge symmetry for σ is irreducible. Curtright action (47). The linearized dual Riemann curvature tensor in D The action (87) is invariant under the following gauge dimensions is defined as transformation [38, 41, 134] 1 ˜ = ǫ 1ν1 ˜ Rα1αD−22ν2 α1αD−2 R1ν12ν2 , (99) δχ,αhα1αD−3|β = (D − 3)(∂[α1 χα2αD−3]β 2 D−3 +∂[α1 αα2αD−3]β − (−1) ∂β αα1αD−3 ), which is of Young symmetry (D − 2,2): (91) where χ and α are mixed symmetry type (D − α1αD−4|β α1αD−3  4,1) and antisymmetric (D − 3)-rank tensors, respectively:  ˜ ≡ −  Rα1αD−22ν2 D 2 boxes  . (100)     χ ≡ −  α1αD−4|β D 4 boxes  ,     Similarly, the D-dimensional dual Einstein equations are

 G˜ = 0, (101)  α1 αD−3|  (92) ˜ where the D-dimensional dual Einstein tensor Gα1 αD−3| is of  mixed symmetry type (D − 3, 1) with the Young diagram:  α ≡ −  [α1αD−3] D 3 boxes  .     ˜ ≡ −  Gα1 αD−3| D 3 boxes  . (102)   The action (88) is also invariant under the gauge transformation  (91) (see [132, 133]). The field strength tensor Fα ...α |β defined 1 D−2  by (90) is not gauge invariant, but it is transformed as  The D-dimensional dual Ricci tensor is defined by δαFα1...αD−2|β = −(D − 2)(D − 3)∂β ∂[α1 αα2αD−2]. (93) ˜ ˜ αD−2ν2 Rα1αD−3|2 ≡ Rα1αD−2|2ν2 η , (103) 5Ref. [134] presented a similar generalized Curtright action via the Einstein- Hilbert assumptions. where the dual Ricci tensor is of Young symmetry (D − 3,1).

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2.5. Gravitationally Magnetic-type Source for the dual gravity h˜ν can be coupled to the magnetic-type The introduction of magnetic monopoles to Maxwell theory energy-momentum tensor T˜ ν as follows: requires that the magnetic charge qm is related to the electric (4) charge qe through the so-called Dirac quantization condition 8πG G˜ ν = N T˜ ν , (110) [1, 2] c4 1 qeqm = n hc , n ∈ Z , (104) or equivalently to the dual Ricci curvature: 2 (4) which is invariant under the SO(2) electric-magnetic duality ν 8πGN ν 1 ν ρ R˜ = T˜ − η T˜ ρ . (111) rotations, c4 2 qe → qm, qm → −qe. (105) The electric-magnetic duality rotations (Equations 41, 42) of the curvatures imply that there are the SO(2) rotations of the A singularity of the magnetic field or a Dirac string is electric-type and magnetic-type energy-momentum tensors: unobservable in the spin-1 field if the condition (104) is imposed on the magnetic charge. ′ T = cos αTν + sin αT˜ ν , (112) Similarly, there is a quantization condition for the spin-2 field ν ˜ ′ ˜ such as gravity [91]: Tν = − sin αTν + cos αTν . (113)

hc 3 Hence, the Einstein equations are invariant under the electric- p p˜ = n , n ∈ Z. (4) (106) magnetic duality in D = 4. 4GN Considering D-dimensional spacetime, the linearized Einstein equations become where the quantities p and p˜ are 4-momenta of linearized diffeomorphisms associated with the ordinary (electric-type) and (D) 8πGN magnetic-type matters, respectively. For a massive particle, we get Gν = Tν , (114) c4 p = mu and p˜ =m ˜ u˜ , where m is the ordinary (electric- type) mass, u is the 4-velocity of the electric-type source, m˜ is the which are equivalent to [38]: magnetic-type mass, and u˜ is the 4-velocity of the magnetic-type source. (D) ρλ 8πGN 1 ρ It is important to note that the Dirac quantization condition Rρνλη = Tν + ηνT ρ , (115) c4 D − 2 (Equation 104) is applied to the charges in the spin-1 case, whereas the spin-2 quantization condition (Equation 106) is where Tν is the energy-momentum tensor of the electric-type imposed on 4-momenta, not the mass. In the rest frame, we get a source. quantization condition: The conservation laws for the electric-type and magnetic-type ν ν sources impose that ∂ Tν = 0 and ∂ T˜ ν = 0, so the motion hc 5 EE˜ = n , n ∈ Z. of a massive particle should follow straight lines. For a massive (4) (107) 4GN particle, one can write

∂wν where E is the ordinary (electric-type) energy, and E˜ is the T = mu dτδ(4) x − w(τ) , (116) ν ∂τ 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). ∂w˜ ν T˜ =m ˜ u˜ dτδ(4) x −w ˜ (τ) , (117) The Einstein equations coupled to the gravitating source are ν ∂τ (4) where wν and w˜ ν represent the electric- and magnetic-type ν 8πG ν G = N T , (108) worldlines, respectively, u = dw /ds and u˜ = dw˜ /ds are c4 the 4-velocities of the electric- and magnetic-type sources, and which are equivalent to m and m˜ the electric- and magnetic-type masses, respectively. The quantity s is spacelike, and τ is timelike. From Equations (3) 0 (4) (116, 117), one obtains Tν = δ x −w (x ) uuν/u0 and ν 8πGN ν 1 ν ρ R = T − η T ρ , (109) ˜ (3) 0 c4 2 Tν = δ x − w˜ (x ) u˜ u˜ ν/u˜ 0. Dirac strings in D = 4 can be constructed as follows. Let ν where T is the energy-momentum tensor of the ordinary decompose the linearized Riemann curvature Rναβ into the (electric-type) matter. massless part and the massive part, In section 2.1, it was shown that the Pauli-Fierz field equations ν are duality invariant in D = 4, so the Einstein tensor G˜ Rναβ = rναβ + mναβ . (118)

Frontiers in Physics | www.frontiersin.org 11 January 2019 | Volume 6 | Article 146 Danehkar Duality in Gravity and Higher-Spin Fields

The massless part rναβ satisfies the cyclic and Bianchi identities The linearized dual Einstein Equations (125) are equivalent to (r[να]β = 0 and ∂[ρrν]αβ = 0), and the massive part mναβ is coupled to the magnetic-type source (D) ρλ 8πGN R˜ α ρα λα η = T˜ α α 1 2 D−3 c4 1 D−3 1 ˜ ρσ [ρ ˜ σ ]λ mναβ = ǫνρσ ∂[α M β] + δ β]M λ , (119) (D − 3) ρ 2 + η[α T˜ α α ]ρ ,(126) 2 1 2 D−3 where M˜ αν defines the magnetic-type source as ˜ where Rα1α2αD−2ν is the linearized dual Riemann curvature (4) defined by Equation (10). 16πGN α T˜ ν = ∂ M˜ αν , (120) In the little group SO(D − 2), the Einstein tensor and its c4 dual tensor in D dimensions satisfy (see [38, 138] for further discussions): and M˜ αβ = M˜ [αβ] ≡ M˜ αβ| is a tensor with the mixed symmetry type (2,1). 8πG(D) There is a symmetric tensor hν that satisfies N 1 k1kD−3 Gij = ǫ iT˜ | (127) c4 (D − 3)! k1k2 kD−3 j αβ [α β] rν = −2∂ ∂[hν] . (121) 8πG(D) ˜ = N ǫ k Gi1iD−3|j 4 i1iD−3 Tjk , (128) αβ αβγ λ αβ [αβ] c If we define y = ǫ ∂γ hλ, where y = y is a αβ mixed symmetry type (2,1) tensor (∂αy = 0), the linearized If the magnetic-type energy-momentum tensor satisfies Riemann curvature Rναβ can be rewritten as ˜ Ti1i2iD−3|j = −∂[i1 M2iD−3]j, (129) 1 ρσ [ρ σ ]λ Rναβ = ǫνρσ ∂[α y β] + δ β]y λ 2 where Mi i j is an arbitrary tensor with the mixed symmetry ˜ ρσ [ρ ˜ σ ]λ 2 D−3 +M β] + δ β]M λ . (122) type (D − 4, 1), and the electric-type energy-momentum tensor can be written as follows: Defining Yνα = yνα + M˜ να (where Yνα = Y[ν]α) yields 1 k1kD−3 Tij = − ǫ i∂ M , (130) (D − 3)! [k1 k2 kD−3]j 1 ρσ [ρ σ ]λ Rναβ = ǫνρσ ∂[α Y β] + δ β]Y λ . (123) 2 so the magnetic-type source is correlated to the Hodge dual of the The tensor M˜ αβ| defined by (120) can be written using a Dirac electric-type source: string y(τ, θ) attached to the magnetic-type source, y(τ,0) = ˜ k w˜ (τ) in (117). One can take [91] Ti1iD−3|j = ǫi1iD−3 Tjk, (131) 1 (4) k1 kD−3 ˜ Tij = ǫ iTk1kD−3j. (132) 16πGN (4) (D − 3)! M˜ αβ| = m˜ u˜ dτdθδ x − y(τ, θ) c4 ∂yα yβ ∂yα ∂yβ The magnetic-type energy-momentum tensor T˜ can be a × − . (124) i1iD−3|j ∂θ ∂τ ∂τ ∂θ source for the gravitational Bianchi identity [38] From Equations (117) and (124), it can be seen that the 1 k1k2kD−3 ˜ ˜ R[ijm]n = ǫijm Tk1k2kD−3|n. (133) divergence of M˜ αβ| is identical with Tν (by a factor of (D − 3)! (4) 4 16πGN /c ). The momentum m˜ u˜ of the magnetic-type source by its mass m˜ and 4-velocity u˜ is therefore conserved. Similarly, the energy-momentum tensor Tij is a source for the Similarly, the linearized Einstein equations of the dual Bianchi identity of the dual Riemann curvature [38] ˜ graviton hα1 αD−3|β in D dimensions with the magnetic-type ˜ k source read R[i1i2iD−1]j = ǫi1jD−1 Tjk. (134)

(D) 8πGN It can be seen that the Einstein Equations (127, 128), and the G˜ α α |β = T˜ α α |β , (125) 1 D−3 c4 1 D−3 Bianchi identities (133) and (134) are invariant under the twisted self-dual conditions discussed in section 2.1. We can express ˜ ˜ where Gα1αD−3|β is the dual Einstein tensor, Tα1α2αD−3|β is their electric-magnetic duality transformations using matrix the energy-momentum tensor of the magnetic-type source in D- notations ˜ ˜ dimensional spacetime, both Gα1αD−3|β and Tα1α2αD−3|β are tensors with the mixed symmetry type (D − 3,1). R = S⋆R, T = S⋆T. (135)

Frontiers in Physics | www.frontiersin.org 12 January 2019 | Volume 6 | Article 146 Danehkar Duality in Gravity and Higher-Spin Fields where the SO(2) electric-magnetic duality matrix S is defined spacetime: by Equation (38), T and R represent the D-dimensional energy- 1 ρ 1 ρ ′ momentum tensor and Ricci curvature associated with both the Rνu = − ∂ ∂ρh0ν + ∂ hνρ electric-type and magnetic-type sources, respectively, 2 2 1 ρ 1 ρ ′ + ∂ν∂ h0ρ − ∂ν hρ = 0, (142) 2 2 Tij Rij ν 1 ρ ρ ′ 1 ρ ′′ T = , R = , (136) Rνu u = − ∂ ∂ρh00 + ∂ h0ρ − hρ = 0, (143) T˜ | R˜ | 2 2 n1 nD−3 m n1 nD−3 m ρ 1 ρ ′ u D = ∂ h0ρ − hρ = 0. (144) 2 and ⋆T and ⋆R represent their corresponding Hodge isomorphisms, From Equation (140), the electric and magnetic parts of the linearized Weyl tensor in the vacuum Einstein equations become

l 1 ǫ T ′ ′′ n1 nD−3 ml Eν [h] = − ∂∂νh00 − 2 ∂(hν)0 + hν , (145) ⋆T = 1 , 2  ǫk1kD−3 T˜  i k1k2kD−3|j 1 ′ 1 (D − 3)! = − ǫ ∂ρ λ + ǫ ∂ ∂ρ λ l (137) Bν [h] ρλ( hν) ρλ( ν) h0 . (146)  ǫn1nD−3 Rml  2 2 ⋆R = 1 . k1kD−3 ˜  ǫ iRk1k2kD−3|j  Taking a covariant divergence from the gravitoelectric field (D − 3)! (145) and the gravitomagnetic field (146), and using constraints   (Equations 142 –144), one obtains the covariant divergenceless It means that the equations of motion for the D-dimensional condition for the gravitoelectric and gravitomagnetic fields: ˜ dual graviton hi i − |j are equivalent to those for the graviton 1 D 3 ∂ Eν = 0 = ∂ Bν. (147) hij under the SO(2) electric-magnetic invariance. In Minkowski spacetime, the covariant derivative of a spatial, 2.6. Spin-2 Harmonic Condition traceless symmetric 2nd-rank tensor is decomposed into the time derivative along with a four-velocity vector u and the spatial In this section, we introduce the harmonic coordinate condition, ′ derivative, ∂E = −u E + DE (see e.g., [66]). As the which makes it possible to propagate gravitational waves ν ν ν ν = ∂ρ∂ = gravitoelectric and gravitomagnetic tensor are spatial (Eνu ( ρhν 0) in linearized gravity. We will see that ν ′ ′ 0 = Bν u ), one gets u Eν = 0 = u Bν , so we get the harmonic condition is equivalent to divergenceless and the spatial divergenceless condition (D Eν = 0 = D Bν ), nonvanishing curl of the electric and magnetic parts of the which were previously proven for gravitational waves in the 1 + 3 linearized Weyl tensor encoded by the Pauli-Fierz field hν in the instantaneous rest-space of an observer moving in flat spacetime covariant formalism [66, 139, 140]. (defined by Equations 27, 28). It can easily be verified that the covariant curl and distortion of We consider the vacuum Einstein equations in the linearized the gravitoelectric field (145) and the gravitomagnetic field (146) do not vanish under the De Donder gauge condition and the flat condition (Rν = 0 and R = 0), which imply linearized flat condition:

1 ρ ρ 1 ρ curl(E)ν = 0 = curl(B)ν, ∂(ρEν) = 0 = ∂(ρBν), Rν = − ∂ ∂ρhν + ∂ ∂ hν ρ − ∂ν∂hρ = 0, (138) 2 ( ) 2 (148) ν ν where the covariant curl is defined as curl(E) ≡ ǫ ∂ρE λ. R = ∂∂νh − ∂ ∂hν = 0, (139) ν ρλ( ν) The nonvanishing covariant curl and distortion conditions are Cν = −2∂[∂ h ν]. (140) αβ [α β] equivalent to the nonzero, spatial curls and distortions of Eν and Bν for gravitational waves in the 1 + 3 covariant formalism Equation (138) presents a second-order equation for the Pauli- [66, 67, 141]: curl(E)ν = 0 = curl(B)ν and DρEν = 0 = DρBν , where the spatial curl is defined using the spatial Fierz field hν . To have a wave equation for the Pauli-Fierz field, ρ λ ρ derivative D as curl(E)ν ≡ ǫρλ D Eν . From Equations ∂ ∂ρhν = 0, it is required to have a constraint, the so-called De ( ) (147, 148), the wave equation of the Pauli-Fierz field hν in empty Donder gauge condition [103]: ρ ρ ρ space, ∂ ∂ρhν = 0, corresponds to ∂ ∂ρEν = 0 = ∂ ∂ρBν in 2 ′′ the linearized theory, which are equivalent to D Eν − Eν = 1 ′′ D ν ρ = 2 − + ≡ ∂ hν − ∂hρ = 0. (141) 0 D Bν Bν in the 1 3 covariant formalism [60,66]. 2 Hence, the De Donder gauge condition in the vacuum Einstein ρ equations allows the propagation of the spin-2 waves, ∂ ∂ρhν = Substituting Equation (141) into (138) leads to the wave equation 0, and imposes divergenceless and nonvanishing-curl of the ρ ∂ ∂ρhν = 0. gravitoelectric and gravitomagnetic fields encoded by the Pauli- We can have some constraints along a four-velocity vector u Fierz field (Equations 27 and 28) of the linearized theory in in the rest-space of a local co-moving observer in Minkowski Minkowski spacetime.

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To satisfy the De Donder gauge condition (141), the Pauli- which is analogous with the twisted self-dual conditions (35). Fierz field could be assumed to be transverse, traceless, and spatial We can also show that the helicity states of the graviton hν (e.g., [39]): is rotated into those of the dual graviton h˜ν under the duality ν ρ ν transformations: ∂ hν = 0, hρ = 0, u hν = 0, (149) ′ = α − α which is called the transverse–traceless gauge condition, and h+(m) cos h+(m) sin h×(m), (156) ′ allows the propagation of a massless, spin-2 field in Minkowski h×(m) = sin αh+(m) + cos αh×(m). (157) flat spacetime. In the case of the transverse–traceless gauge condition (149), Thus, the + and × helicity states of wave equations for the dual ρ the harmonic, massless spin-2 field of the wave equations (see e.g., graviton (∂ ∂ρh˜ν = 0) are respectively rotated to the × and + ρ [39]) in the momentum space m = (|m |, m ) reads helicity states of gravitational waves (∂ ∂ρhν = 0) under SO(2) = + + × × electric-magnetic duality rotations in D 4, and are duality hν (m) = h (m)eν(m) + h (m)eν(m), (150) invariant under the twisted self-dual condition. where h+ and h× are the helicity states associated with the + and × polarization states of the Pauli-Fierz field hν (graviton), 3. SPIN-3 FIELD AND ITS DUAL FIELDS + × e = xxν − yyν and e = xyν + yxν are the + and × ν ν We now introduce the spin-3 field, as a step toward the spin- polarization tensors, x = (0, xˆ) and y = (0, yˆ) are spatial 4- vectors satisfying xˆ x ˆ = 1 =y ˆ y ˆ and xˆ y ˆ = 0 with xˆ m ˆ = 0 = s generalization. The typical spin-3 field in D-dimensional = yˆ m ˆ making an orthogonal triad yˆ =x ˆ ×m ˆ (i.e., xˆ = −ˆy ×m ˆ ) spacetime is represented by a symmetric 3rd rank tensor hνρ where mˆ =m /|m |. h(νρ) with Young symmetry (1, 1, 1) and the diagram . The equations of motion of hνρ (the ordinary spin-3 field) in D- Since the equations of motion for the Pauli-Fierz field hν dimensional flat spacetime can be described using the Fronsdal are fully equivalent to the dual Pauli-Fierz field h˜ under ν action [102] for the spin-3 case SO(2) electric-magnetic duality rotations, we have the following expression for the dual Pauli-Fierz field h˜ν in D = 4: 1 D Sspin-3 = d xL[hνρ ], (158) ˜ ˜+ + ˜× × 2 hν (z) = h (m)eν(m) + h (m)eν(m), (151)

where the Lagrangian L[hνρ] is as follows [47, 142, 143] where h˜+ and h˜× are the helicity states associated with the + and × polarization states of the dual Pauli-Fierz field h˜ (dual ν 2 λ νρ λνρ L[hνρ ] = − ∂λhνρ∂ h − 3∂ hνρ ∂λh graviton). 3 Considering the first dual formulation, and taking a 2-D λ ν ρ λ ρν + 6hλ ∂ ∂ hνρ − 3∂λh ν∂ hρ permutation ǫν from the expression (150) yields h˜ν (m) = + ρ + × ρ × 3 λ νρ h (m)ǫ eρν(m) + h (m)ǫ eρν(m). As the spatial vectors xˆ − ∂λh ∂νh ρ . (159) and yˆ making an orthogonal triad with mˆ , one gets ǫ ρx (m) = 2 ρ y (m) and ǫ ρy (m) = −x (m) in the momentum space m ρ The Fronsdal Lagrangian (159) is a generalization of the Pauli- (equivalent to xˆ ×m ˆ =y ˆ and yˆ ×m ˆ = −ˆx), so we get Fierz theory on the spin-3 field, and is invariant under the gauge ǫ ρe+ (m) = e× (m) and ǫ ρe× (m) = −e+ (m): ρν ν ν ν transformation + × × + h˜ν (m) = h (m)e (m) − h (m)e (m), (152) ν ν δξ hνρ = 3∂(ξνρ), (160) by comparison with Equation (150), the dual helicity states in the where the gauge parameter ξν = ξ(ν) is an arbitrary symmetric momentum space have the following rotations: 2rd rank field. By analogy with the Einstein equations, one can write the spin- h+(m) = h˜×(m), h×(m) = −h˜+(m), (153) 3 Einstein equations in D dimensions (see [91] for the generalized which is similar to Equation (34). We see that the + and × Einstein equations): helicity states of the graviton hν and the dual graviton h˜ν G = 0, (161) are fully equivalent to each others under the following twisted 1 2 3 self-dual condition in the momentum space: where the spin-3 Einstein tensor G123 = G(123) is a ρ traceless (Gρ = 0), symmetric tensor with the Young diagram h+(m) = −Sh×(m), h×(m) = Sh+(m). (154) . The tensor G123 can be written in terms of the so-called Fronsdal [102] tensor F for the spin-3 case as with S defined by (38), and matrix notations of the helicity states: 123 follows h+(m) h×(m) + × 3 ρ h (m) = + , h (m) = × , (155) = − η h˜ (m) h˜ (m) G123 F123 (12 F3)ρ , (162) 2

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ˆ where the spin-3 Fronsdal tensor F123 = F(123) is a The second dual spin-3 field hα1αD−3β1βD−3ρ is described by a symmetric tensor with the diagram , defined by tensor with the mixed symmetry (D − 3, D − 3, 1) having the Young diagram: ρ ρ ρ F123 = ∂ρ∂ h123 − 3∂(1 ∂ h23)ρ + 3∂(1 ∂2 h3)ρ , (163) which is related to the spin-3 Ricci tensor  hˆ ≡ D − 3 boxes . (172) ≡ ην2ν3 = − ∂ α1αD−3β1βD−3ρ  R1ν1|23 R1ν12ν23ν3 2 [ν1 F1]23 , (164)   where the spin-3 Ricci tensor R ν | is a 4th-rank tensor with 1 1 2 3   the mixed symmetry (2, 1, 1) having the Young diagram ,  It might be possible to define a third dual field of the spin-3 field, and the spin-3 Riemann tensor R1ν12ν23ν3 is a 6th-rank tensor which is dualized on all three indices. The physical gauge spin- with the mixed symmetry (2, 2, 2) having the Young diagram 3 field hijk in D dimensions can be dualized in the little group . SO(D − 2) as follows:

For spin-3 fields, we can define the first dual representation in m n l h˘ = ǫ ǫ ǫ h , a similar manner of the dual definitions for the spin-2 field, i.e., i1iD−3j1jD−3k1kD−3 i1 iD−3 j1 jD−3 k1kD−3 mnl (173) Equations (3, 4) [38, 89, 98]. The first dual field of the spin-3 field hνρ = h(νρ) in D-dimensional spacetime is dualized on one 1 i1iD−3 j1jD−3 index only. For the physical gauge spin-3 field h , we have in the h = ǫ mǫ n ijk mnl (D − 3)!(D − 3)!(D − 3)! little group SO(D − 2) [38]: k1kD−3 ˘ × ǫ lhi1iD−3j1jD−3k1kD−3 , (174) ˜ l hi1iD−3jk = ǫi1iD−3 hljk, (165) 1 i1iD−3 ˜ having the following properties: h = ǫ jh , (166) jkl (D − 3)! i1 iD−3kl ˘ ˘ hi1iD−3j1jD−3k1kD−3 = h[i1iD−3][j1jD−3][k1kD−3], with the following properties [38]: ˘ (175) h[i1iD−3|j1]j2jD−3 k1kD−3 = 0. ˜ ˜ ˜ ˜ hi i jk = h[i i ]jk ≡ hi i |jk, h[i i j]k = 0. 1 D−3 1 D−3 1 D−3 1 D−3 ˘ (167) The third dual spin-3 field hα1αD−3β1βD−3γ1γD−3 is described ˜ The first dual spin-3 field hα1αD−3ν is then described by a tensor by a tensor with the mixed symmetry (D − 3, D − 3, D − 3). with Young symmetry (D − 3, 1, 1) having the Young diagram: 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 hνρ = h(νρ) with the Young diagram . Thus, we hα1 αD−3|ν D 3 boxes  . (168)  get only one dual formulation of the spin-3 field in 4-dimensional  spacetime.  ˆ  The second dual physical gauge field h and the third dual  ˘ For spin-3 fields, we may also define the second dual formulation physical gauge field h can be written in terms of the first dual in a similar manner of the spin-2 dual field (6) and (7)6. In the physical gauge field h˜ and second dual physical gauge field hˆ, second dual formulation, the physical gauge spin-3 field hijk in respectively, and one can recover the first dual physical gauge D-dimensional spacetime is dualized on two indices: field h˜ from the second and third dual physical gauge field in the little group SO(D − 2). Nevertheless, the action principle of the ˆ m n ˜ ˘ hi1iD−3j1jD−3k = ǫi1iD−3 ǫj1jD−3 hmnk, (169) second field h and third dual field h in D dimensions higher than 1 4 may not follow from a Lorentz-invariance action (similar to the i1iD−3 j1jD−3 ˆ h = ǫm ǫn h , mnl (D − 3)!(D − 3)! i1 iD−3j1 jD−3l argument made for the spin-2 field [40]), so the first dual field (170) formulation, which is dualized on one index only, is considered in the following parts of this section. having the following properties: The Fronsdal theory for the spin-3 case in D = 4 is dualized 4 to a Fronsdal action S = d xL[h˜νρ ] where the Lagrangian ˆ ˆ ˆ h = h , h[i i |j ]j j = 0. L ˜ i1 iD−3j1 jD−3k [i1 iD−3][j1 jD−3]k 1 D−3 1 2 D−3 k [hνρ ] is formulated in terms of a symmetric massless 2nd-rank (171) ˜ ˜ tensor hνρ = h(νρ) (dual spin-3 field), similar to Equation (159) ˜ ˜ ˜ 6For traceless spin-3 fields, it is impossible to dualize on two indices, so there is no with the gauge transformation δξ hνρ = 3∂(ξνρ) where ξν is a second dual formulation. arbitrary, symmetric 2nd rank tensor.

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3.1. D = 5: Mixed Symmetry (2,1,1) Field 3.2. D > 5: Young Symmetry (D − 3,1,1) The dual field of the spin-3 field hνλ in 5-dimensional spacetime Fields ˜ is represented by hαβ|ν , which is a tensor with Young symmetry The D-dimensional spin-3 field hαβρ can be dualized to a free (2,1,1) with the diagram: ˜ ˜ massless spin-2 fields h[α1 αD−3]ν ≡ hα1 αD−3|ν with Young symmetry type (D − 3,1,1) defined by (Equation 165, 166), and satisfying h˜αβ|ν ≡ , (176) ˜ ˜ ˜ hα1...α1D−3|ν = h[α1αD−3]|ν = hα1αD−3|(ν), ˜ (183) and satisfies: h[α1αD−3|]ν = 0,

˜ = ˜ ≡ ˜ ˜ = (177) ˜ ν ˜ α1α2 α3α4 hαβν h[αβ](ν) hαβ|ν , h[αβ]ν 0. hα1αD−3|νη = 0, hα1αD−3|νη η = 0. (184) This is associated with the dualization on one index only, which ˜ The equations of motion of a free massless spin-3 field hαβ|ν in provides equations of motion with a Lorentz invariant action. 5-dimensional Minkowski spacetime ην = diag(−, +, +, +, +) The equations of motion of free massless, dual spin-3 fields is represented by the following action principle [47] 7 ˜ hα1 αD−3|βρ in D-dimensional flat spacetime are represented by the action principle [47]: 1 L ˜ λ ˜ν|ρσ ˜ ˜λν|ρσ ˜ [hν|ρσ ] = − ∂ h ∂λhν|ρσ − 2∂λh ∂ hν|ρσ L ˜ 3 [hα1...αD−3|ν] ˜ν|ρσ λ ˜ ˜ ν|λσ ρ ˜ 2 − 2∂ρh ∂ hν|λσ + 8hλ ∂ ∂ hν|ρσ ρ ˜α1αD−3|ν ˜ = − ∂ h ∂ρhα1αD−3|ν ˜ν|λ ρ σ ˜ ρ ˜ ν|λσ ˜ 3(D − 3)! + 2h λ∂ ∂ hν|ρσ − 4∂ hλ ∂ρh ν|σ ˜ρα2αD−3|ν σ ˜ ρ λ ν|σ ρ ν|λσ − (D − 3)∂ρh ∂ hσ α2αD−3|ν − ∂λh˜ν|ρ ∂ h˜ σ − 4∂ h˜ρν| ∂σ h˜λ ˜α1αD−3|ρ σ ˜ σ |ρ γ α σ ν|λσ ρ − 2∂ρh ∂ hα1αD−3|σ + ∂σ h˜ ρ∂ h˜γ |α + 2∂σ h˜ ∂ h˜ρν|λσ + 4(D − 3)h˜ α2αD−3|γ ∂ρ∂σ h˜ γ |λσ ρ γ ρα2αD−3|σ +2∂γ h˜λ ∂ h˜ ρ|σ , (178) α α − |γ σ ρ + 2h˜ 1 D 3 γ ∂ ∂ h˜α α |σρ 1 D−3 − − ∂ρ ˜ α2αD−3|γ ∂ ˜σ which satisfies the gauge symmetry 2(D 3) hγ ρh α2αD−3|σ ˜ σ ρ ˜α1αD−3|γ − ∂ρhα1αD−3|σ ∂ h γ ν ν δ h˜ = 2 2∂[ χ ] ˜ α2αD−3|ρσ γ ˜ β χ,ϕ |ρσ ρσ − 2(D − 3)∂σ hρ ∂ hγ α2αD−3|β 1 [ ν] [ ν] ν| ˜σ α2αD−3|ρ γ ˜ β +∂ ϕ ρ|σ + ∂ ϕ σ |ρ − 2∂(ρϕ σ ) , (179) + (D − 3)∂σ h ρ∂ hγ α α |β 2 2 D−3 ˜σ α2αD−3|γ ρ ˜ + (D − 3)(D − 4)∂σ h ∂ hρα2αD−3|γ where χνρ = χ(νρ) is an arbitrary symmetric tensor with ˜ ρα3αD−3|γ σ ˜λ Young symmetry (1,1,1) ≡ , and ϕν|ρ = ϕ[ν]ρ is an +(D − 3)(D − 4)∂ρhγ ∂ h σ α3αD−3|λ , arbitrary tensor with Young symmetry (2,1) ≡ (see also (185) gauge transformations in [47, 143]). The gauge transformations which satisfies the gauge symmetry for χ|νρ and ϕν|ρ are reducible: ˜α1...αD−3 [α1 α2...αD−3] δχ,ϕh |ν = 2(D − 3) ∂ χ |ν δσ χνρ = 3∂ σ , (180) ( νρ) [α1 α2...αD−3] α1...αD−3| +∂ ϕ (|ν) − ∂(ϕ ν) , δσ ,αϕν|ρ = 2(∂[σν]ρ + ∂[αν]ρ − ∂ραν ). (181) (186)

The parameters σν = σ(ν) and αν = α[ν] are symmetric where χα ...α |ν = χ[α ...α ν = χα ...α (ν) is an arbitrary and antisymmetric arbitrary tensors, respectively, and their gauge 1 D−4 1 D−4] 1 D−4 tensor with the mixed symmetry (D − 4,1,1), and ϕα ...α | = transformations are reducible: 1 D−3 ϕ[α1...αD−3] is an arbitrary tensor with the mixed symmetry (D −

3, 1). The gauge transformations for χα1...αD−4|ν and ϕα1...αD−3| δσ σν = 6∂(σν), δσ αν = 2∂[σν]. (182) are reducible. The action (185) is also invariant under the following gauge The parameter σ is an arbitrary vector field, and its gauge symmetry [47]: transformation is irreducible. α ...α − [α α ...α − ] δχ h˜ 1 D 3 |ν = (D − 3) ∂ 1 χ 2 D 3 |ν 7The 5-dimensional dual spin-3 action could be expressed by Equations (75) and [α1...αD−4|αD−3] (76) of [89]. + 2∂(χ ν) , (187)

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˜ where χα1...αD−4|ν is a tensor with the mixed symmetry (D − where both the spin-3 dual Fronsdal tensor Fα1αD−3|ν and the ˜ 4, 1, 1), and its gauge transformation is reducible. spin-3 dual Einstein tensor Gα1 αD−3|ν are of Young symmetry In a similar way of the Curtright action (47) and the type (D − 3,1,1), and the diagram generalized Curtright action (88), one may write the action of ˜ the dual spin-3 field hα1 αD−3|ν using a “spin-3 Curtright” field strength tensor Fα1...αD−2|ν defined as F˜  α1αD−3|ν ≡ D − 3 boxes  . (195) G˜  ≡ − ∂ ˜ α1 αD−3|ν  Fα1...αD−2|ν (D 2) [α1 hα2αD−2]ν . (188)   is a tensor with Young symmetry type (D − 2,1,1):   The spin-3 dual Ricci tensor can also be defined by

R˜ ≡ R˜ ην2ν3 , (196)  α1αD−2|23 α1αD−2|2ν23ν3   F ≡ D − 2 boxes  . (189) where the spin-3 dual Ricci tensor is of Young symmetry α1...αD−2|ν   (D − 2, 1, 1), and defined in term of the dual Fronsdal tensor  ˜ Fα1αD−3|ν as follows:    ˜ = − − ∂ ˜  Rα1αD−2|ν (D 2) [α1 Fα2αD−2]ν. (197)  The spin-3 Curtright action should be invariant under the gauge ˜ The dual Ricci tensor Rα1αD−2|ν is antisymmetrized in symmetry (186), however, the field strength tensor Fα1...αD−2|ν is α α − and symmetrized in ν, i.e., R˜ α α |ν = not gauge invariant. The field strength tensor Fα1...αD−2|ν has the 1 D 2 1 D−2 ˜ ˜ following transformation R[α1αD−2]ν = Rα1αD−2(ν). ˜ The Fronsdal tensor Fα1αD−3|12 can be written in term of δ ν = − − − ∂(∂ ϕ ν) ˜ αFα1...αD−2| 2(D 2)(D 3) [α1 α2...αD−2| . (190) the dual spin-3 field hα1 αD−3|ν as follows:

Note that the Curtright field strength tensor Fα α |ν defined ˜ ν ρ ˜ ν ρ ˜ ν 1... D−2 Fα1αD−3| =∂ρ∂ hα1αD−3| − (D − 3)∂[α1 ∂ hα2αD−3]ρ| by Equation (188) should be not mistaken with the Fronsdal ( ν)ρ ˜ − 2∂ ∂ρ hα1αD−3| tensor Fα1αD−3|ν defined by (198) in this section. The spin-3 dual Riemann tensor defined as ( ˜ ν)ρ + 2 (D − 3)∂[α1 ∂ hα2αD−3]ρ| 1 1 1ν1 ν ˜ ρ R˜ = ǫ R , (191) + ∂ ∂ hα α − |ρ . (198) α1αD−22ν23ν3 α1αD−2 1ν12ν23ν3 2 1 D 3 2 is a tensor with Young symmetry (D − 2,2,2) and the diagram: ˜ The dual Einstein tensor Gα1 αD−3|ν and the dual Fronsdal ˜ tensor Fα1αD−3|ν for the spin-3 field are antisymmetrized in indices α1 αD−3 and symmetrized in indices ν, i.e., G˜ = G˜ = G˜ and  α1αD−3|ν [α1αD−3]ν α1αD−3(ν)  F˜ = F˜ = F˜ . ˜ ≡ −  α1αD−3|ν [α1αD−3]ν α1αD−3(ν) Rα1αD−22ν23ν3 D 2 boxes  . (192)   3.3. Spin-3 Magnetic-Type Source The spin-3 field (hνρ ≡ h(νρ)) can be coupled to the spin-   3 electric-type source Tνρ ≡ T(νρ) [38, 91]. For the spin-3   Riemann tensor R ν ν ν , one can write the field equations For a free, massless dual spin-3 field h˜ , one can write 1 1 2 2 3 3 α1 αD−3|ν in D-dimensional spacetime as follows the spin-3 dual Einstein equations similar to the spin-2 case [38]: 1 ν2ν3 T ˜ = R1ν1|23 ≡ R1ν12ν23ν3 η = ∂[ν1 1]23 , (199) Gα1 αD−3|ν 0, (193) 2

˜ where the spin-3 Ricci tensor R ν | = R ν = where the spin-3 dual Einstein tensor Gα1 αD−3|ν is of Young 1 1 2 3 [ 1 1] 2 3 ν R is a tensor with mixed symmetry (2,1,1), defined by symmetry type (D − 3,1,1), and traceless (G˜ α α |ν η = 1ν1(23) 1 D−3 T T T ρ ˜ αD−3ν (164), and νρ = (νρ) is a traceless ( ρ = 0), symmetric Gα1 αD−3|ν η = 0) with the following definition: ρ 3rd rank tensor defined by removing the traces Tρ from the energy-momentum tensor Tνρ : ˜ ν ˜ ν 1 ( ˜ ν)ρ Gα1αD−3| = Fα1αD−3| − 2(D − 3)η[α1 Fα2αD−3]ρ| 2 3 ν ρ T ρ + η F˜ , (194) 123 ≡ T123 − η( T )ρ . (200) α1αD−3|ρ 4 1 2 3

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By analogy with the Einstein equations, one can write an Following Equations (116) and (117), for a spin-3 charged equivalence of Equation (199): particle in D = 4, the spin-3 stress-energy tensors, which are coupled to its field, read G = T , (201) 1 2 3 1 2 3 ∂w T = mU dτδ(4) x − w(τ) , (209) νρ νρ ∂τ which can be considered as the spin-3 Einstein equations with ∂w˜ the spin-3 electric-type source (see [91] for spin-s generalization T˜ =m ˜ U˜ dτδ(4) x −w ˜ (τ) , (210) νρ νρ ∂τ of the Einstein equations). The dual Einstein tensor (193) for the dual spin-3 field can be where m is the electric-type spin-3 charge, and m˜ is the magnetic- coupled to the spin-3 magnetic-type source: type spin-3 charge, Uν ≡ uuν denotes the traceless part of uuν, and U˜ ν ≡u ˜ u˜ ν the traceless part of u˜ u˜ ν (the angle ˜ T˜ Gα1 αD−3|ν = α1αD−3|ν, (202) brackets denote the traceless part; see e.g., [66, 144] for the same notation). For example, one defines traceless parts of symmetric T˜ tensors Sαβ and Sαβρ in 4-dimensional flat spacetime as follows where the spin-3 dual stress-energy tensor α1αD−3|ν is a T˜ νρ traceless ( α2αD−3ρ| = 0), tensor with Young symmetry type α β 1 αβ (D − 3,1,1) and defined as Sν ≡ η ην − ην η Sαβ , (211) ( ) 4 (D − 3) α β 1 αβ T˜ ν ≡ ˜ ν − η ( ˜ ν)ρ Sνρ ≡ η( ην) − ην η Sαβρ , (212) α1αD−3| Tα1αD−3| [α1 Tα2αD−3]ρ| , 4 2 (203) ˜ so the traceless part of uuν is defined by where Tα1αD−3|ν is the spin-3 dual stress-energy, and has Young symmetry type (D − 3,1,1). Both T˜ α α |ν 1 D−3 α β 1 αβ T˜ Uν ≡ uuν ≡ η( ην) − ην η uαuβ . (213) and α1αD−3|ν are antisymmetrized in α1 αD−3, and 4 symmetrized in ν. The dual Einstein tensor (202) is equivalent to Equations (209, 210) can be rewritten as u U = νρ δ(3) − ν2ν3 1 Tνρ m x w(x0) , (214) R˜ α α | ≡ R˜ α α ν ν η = ∂[α T˜α α ] , u 1 D−2 2 3 1 D−2 2 2 3 3 2 1 2 D−2 1 2 0 ˜ ˜ (204) ˜ uUνρ (3) ˜ Tνρ =m ˜ δ x − w˜ (x0) . (215) where the spin-3 dual Riemann tensor Rα1αD−22ν23ν3 is a u0 tensor with Young symmetry type (D − 2,2,2), and the spin-3 ˜ ˜ It can be easily varified that these stress-energy tensors are dual Ricci tensor Rα1αD−2|23 ≡ Rα1αD−22ν23ν3 is a tensor with Young symmetry type (D − 2,1,1), defined by (197). conserved. In the little group SO(D − 2), we may write the following One can write a quantization condition for the spin-3 field in relations D = 4 similar to the Dirac quantization condition (104) and the spin-2 quantization condition (106) as follows (see [91] for spin-s 1 fields): G = ǫj1jD−3 T˜ , (205) i1i2i3 − i1 j1jD−3|i2i3 (D 3)! ν 1 PνP˜ = n hc , n ∈ Z, (216) ˜ i3 T Gj1jD−3|i1i2 = ǫj1jD−3 i1i2i3 . (206) 2 where the conserved spin-3 charges are defined by It can easily verify that the spin-3 Einstein equations (201) and ν ν (202) are invariant under the twisted self-dual conditions G = Pν = mUν, P˜ =m ˜ U˜ . (217) S⋆G and T = S⋆T if we employ the following matrix notations We now consider the construction of a Dirac string in the spin- 3 theory in D = 4. The spin-3 Riemann curvature R ν ν ν T G 1 1 2 2 3 3 T = i1i2i3 , G = i1i2i3 , (207) can be split into the chargeless and charged terms, T˜ G˜ j1jD−3|i1i2 j1jD−3|i1i2 R1ν12ν23ν3 = r1ν12ν23ν3 + m1ν12ν23ν3 . (218)

i3 T The chargeless term r1ν12ν23ν3 is of mixed symmetry type ǫj1jD−3 i1i2i3 ⋆T = 1 , (2,2,2), and satisfies the cyclic and Bianchi identities. The j1jD−3 T˜  ǫ i j j j − |i i  (D − 3)! 1 1 2 D 3 2 3 charged term m1ν12ν23ν3 is of mixed symmetry type (2,2,2), (208) i3 and may be coupled to the spin-3 magnetic-type source  ǫj1jD−3 Gi1i2i3  ⋆G = 1 j1jD−3 ˜ , 1  ǫ i1 Gj1j2jD−3|i2i3  3ν3 [3 ˜ ρσ ν3] − m1ν12ν2 = ǫ1ν1ρσ ∂[2 ∂ M ν2] (D 3)! 2   [ρ [3| ˜ σ ]λ|ν3] and the matrix S is defined by Equation (38). +∂[2 δ ν2]∂ M λ . (219)

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The tensor M˜ αβ|ν, which represents the spin-3 magnetic-type Equation (225) is the De Donder gauge condition for the spin-3 field h . To have the De Donder gauge condition, one way is to source, is of Young symmetry type (2,1,1) ≡ . The νλ make the spin-3 field hνλ transverse and traceless: notations [ | and | ] separate different antisymmetrized indices [2 ν2] ρ ρ from each other, e.g., 4∂[1|∂[2|S|ν1]|ν2] ≡ 4∂[1 ∂ Sν1] . ∂ hνρ = 0, hρ = 0, u hνρ = 0, (227) The chargeless term r1ν12ν23ν3 may be rewritten in terms of covariant derivatives of a symmetric tensor h123 : which imply that the gauge parameter ξν is a symmetric, traceless 2-th rank tensor. 2ν2 [2 ν2] r1ν1 3ν3 = −2∂[3|∂ ∂[1 hν1] |ν3]. (220) The transverse–traceless spin-3 field hνλ, which is projected along a four-velocity vector uλ in the rest-space of a co-moving αβ| αβγ λ αβ| [αβ] Let us define y ρ = ǫ ∂γ hλρ, where y ρ = y ρ = observer, may be written in terms of its helicity state vectors αβ αβ| h+ and h×, and the polarization tensors e+ and e× in the y (ρ) isofYoungsymmetrytype(2,1,1)tensor(∂αy ρ = 0), λ λ ν ν momentum space m = (|m |, m ): the spin-3 Riemann curvature R1ν12ν23ν3 can be defined by + + × × hνλ(m) = eν(m)hλ (m) + eν(m)hλ (m), (228) 1 ρσ [ρ σ ]λ| R ν ν ν = ǫ ν ρσ ∂[ |∂[ | y |ν ]|ν ] + δ |ν ]y λ|ν ] 1 1 2 2 3 3 2 1 1 2 3 2 3 2 3 where the polarization tensors are ρσ [ρ σ ]λ| +M˜ |ν ]|ν ] + δ |ν ]M˜ λ|ν ] . (221) 2 3 2 3 + × eν = xxν − yyν, eν = xyν + yxν , (229) If we take Y = y + M˜ (where Y = Y ), it αβ|ν αβ|ν αβ|ν να [ν]α while x = (0, xˆ) and y = (0, yˆ) are spatial 4-vectors with is obtained xˆ x ˆ =y ˆ y ˆ = 1 and xˆ y ˆ = 0 with xˆ m ˆ =y ˆ m ˆ = 0 , and form an orthogonal triad yˆ =x ˆ ×m ˆ , where mˆ =m /|m |. 3ν3 1 [3 ρσ ν3] [ρ σ ]λ|ν3] R ν ν = ǫ ν ρσ ∂[ ∂ Y ν ] + δ ν ]Y λ . ˜ 1 1 2 2 2 1 1 2 2 2 Similarly, the transverse–traceless dual spin-3 field hνλ, (222) which is projected along a four-velocity vector u˜ λ in the rest- A Dirac string y(τ, θ) can be connected to the spin-3 magnetic- space of a co-moving observer, may be written in terms of its ˜ ˜+ ˜× type source Mαβ|ν , y (τ,0) =w ˜ (τ) in (210). By analogy with helicity states hλ and hλ , and the polarization tensors in D = 4: (124), one can write ˜ + ˜+ × ˜× hνλ(m) = eν(m)hλ (m) + eν(m)hλ (m), (230)

(4) ∂yα yβ ∂yα ∂yβ M˜ αβ|ν =m ˜ U˜ ν dτdθδ x − y(τ, θ) − , Taking a permutation ǫ from Equation (228) according to ∂θ ∂τ ∂τ ∂θ α (223) the first dual formulation definition, one can easily find in the momentum space: where the tensor M˜ αβ|ν defines the spin-3 magnetic-type source as + ˜× × ˜+ h (m) = h (m), h (m) = −h (m), (231) α T˜ νρ = ∂ M˜ α|νρ , (224) which means that the helicity state vectors of the spin-3 field so the divergence of M˜ α|νρ is identical with T˜ νρ, and the Dirac are respectively rotated to those of its dual field under SO(2) string of a spin-3 magnetic-type point source is conserved. One duality rotations in D = 4, and demonstrate the twisted self-dual can prove the conserved charges associated with a spin-3 electric- condition, similar to what were shown the graviton and the dual type point source (see [91] for complete proof of the spin-s graviton in section 2.6. conserved charges with magnetic-type and electric-type point sources). 4. HIGHER-SPIN FIELDS AND THEIR DUAL FIELDS 3.4. Spin-3 Harmonic Condition The spin-2 harmonic coordinate condition can be generalized We now consider the general spin-s fields (s ≥ 4) and its dual to the spin-3 field hνλ that allows the propagation of a spin-3 formulations. The spin-s fields in D-dimensional spacetime can particle in empty space. To have a wave equation for the spin-3 be represented by a symmetric s-th rank tensor h12s = fields in empty space, ∂ ∂ρh = 0, it is necessary to eliminate ρ νλ h(12s) with the Young diagram: the last two of the spin-3 Fronsdal equation (163): s boxes ρ ρ Dν ≡ ∂ hνρ − ∂hνρ = 0, (225) h12s ≡ , (232) which is invariant under the gauge transformation (160): which is double-traceless for s ≥ 4[102]:

D ρ 12 34 δξ ν = 3∂ρ∂ ξν. (226) h1234s η η = 0. (233)

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The action of the free, massless spin-s fields h12s in arbitrary isa(2s − 2)-th rank tensor with Young symmetry type dimensions (D ≥ 4) reads: s boxes 1 D L Sspin-s = d x [h12s ], (234) R ≡ , (241) 2 1ν12ν2s−2νs−2|s−1s L where [h12s ] is the Fronsdal Lagrangian [102]: and obtained by taking covariant derivatives from the Fronsdal

tensor F1s as follows (s − 1) L ρ 1s [h12s ] = − ∂ρh1s ∂ h s R1ν12ν2s−2νs−2|s−1s ρ λ1s−1 s−2 − s∂ hρ1s−1 ∂λh = −2 ∂[ν1|∂[ν2| ∂[νs−2|F|1]|2]|s−2]s−1s . (242) λ1s−2 ρ σ + s(s − 1)hλ ∂ ∂ hρσ 1s−2 Here, the notations [ | and | ] such as 4∂[ν1|∂[ν2|S|1]|2] imply 1 λ ρ σ 1s−2 that covariant derivatives are antisymmetrized on ν11 and ν22 − s(s − 1)∂ρh λ ∂ hσ 1 s−2 [ν2 2] ν2 2 2 ν2 2 separately, i.e., 4∂[ν1 ∂ S1] = ∂ν1 ∂ S1 − ∂ν1 ∂ S1 − 1 ν2 2 2 ν2 λ ρ ∂1 ∂ Sν1 +∂1 ∂ Sν1 . The equations of motion of a free, − s(s − 1)(s − 2)∂ h ρλ1s−3 4 massless spin-s field h12s satisfy R1ν12ν2s−2νs−2|s−1s = σ ν1s−3 0, which is equivalent to the spin-s Einstein equations (237). × ∂ν hσ . (235) Substituting the Fronsdal definition (239) into the Ricci definition (242) yields [49] The Fronsdal Lagrangian (235) is invariant under the gauge transformation [102] R1ν12ν2s−2νs−2|s−1s = −2s−2 ∂ ∂ρ∂ ∂ ∂ h δξ h12s = s∂(1 ξ2s). (236) ρ [ν1| [ν2| [νs−2| |1]|2]|s−2]s−1s − ∂ ∂ ∂ ∂ ∂ρh The gauge parameter ξ is a symmetric tensor. s [ν1| [ν2| [νs−2| |1]|2]|s−2]s−1ρ 2s−1 ρ The equations of motion for free, massless spin-s fields satisfy − ∂s−1 ∂[ν1|∂[ν2| ∂[νs−2|∂ h|1]|2]|s−2]sρ the generalized spin-s Einstein equations [91, 102] ρ + ∂s−1 ∂s ∂[ν1|∂[ν2| ∂[νs−2|h|1]|2]|s−2]ρ . (243) G12s = 0, (237) For the spin-s field h12s , there might be s different dual formulations (also infinite chains of dualities [94, 95]). where the spin-s Einstein tensor G = G is a 12s (12s) Considering the fact that spin-s fields are double-traceless (s ≥ double-traceless (G η12 η34 = 0), symmetric tensor. 12s 4), it is problematic to dualize on multiple indices. Thus, we The spin-s Einstein tensor G can be written using the 13s consider only the first dual formulation of the spin-s field in D- Fronsdal tensor F as 1s dimensional spacetime, which is dualized on one index only. For the physical gauge spin-s field, we can write s(s − 1) ρ G1s = F1s − η(12 F3s)ρ , (238) 4 ˜ j1 hi1iD−3|j2js = ǫi1iD−3 hj1j2js , (244) 1 where the Fronsdal tensor F = F is a symmetric i1iD−3 ˜ 1 s ( 1 s) hj j j = ǫj hi i |j j . (245) tensor, and defined by [49, 102] 1 2 s (D − 3)! 1 1 D−3 2 s

ρ ρ ˜ The dual spin-s field hα α − | is of Young symmetry (D − F1s = ∂ρ∂ h1s − s∂(1 ∂ h2s)ρ 1 D 3 2 s s−1 s(s − 1) ρ + ∂ ∂ h ρ . (239) 3, 1, , 1) having the Young diagram: 2 ( 1 2 3 s) s boxes The linearized spin-s Riemann tensor R1ν12ν2sνs is a (2s)-th s rank tensor with Young symmetry (2, , 2) and the diagram:  ˜ ≡ − hα1αD−3|2s D 3 boxes , s boxes   ≡ R1ν12ν2sνs .   (246)  and double-traceless for s ≥ 4[47]: The linearized spin-s Ricci tensor, ˜ 23 45 hα1αD−3|2345s η η = 0, (247) νs−1νs R ν ν − ν − | − ≡ R ν ν − ν − − ν − ν η , 1 1 2 2 s 2 s 2 s 1 s 1 1 2 2 s 2 s 2 s 1 s 1 s s ˜ ηα1α2 ηα3α4 = (240) hα1α2α3α4αD−3|2s 0. (248)

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It has the following properties: where χα1...αD−4|β2βs = χ[α1...αD−4]β2βs = χα1...αD−4(β2βs) is an s−1 − ˜ ˜ ˜ arbitrary tensor with the mixed symmetry (D 4, 1, ,1), hα1αD−3|2s = h[α1αD−3]2s , h[α1αD−32]3s = 0, (249) s boxes

˜ ˜ hα1αD−3|2s = hα1αD−3(2s). (250)  χ ≡ D − 4 boxes , α1...αD−4|β2βs  ˜  In D = 4, the dual spin-s field is symmetric, h12s =  h˜ . The Fronsdal action (234) for the spin-s field (12s)  h in D = 4 is dual to the Fronsdal action S =  (253) 12s  1 4 L ˜ such  2 d x [h1s ], which is formulated in terms of the dual spin-s field h˜ , i.e., h → h˜ in the action (234). 12s 1s 1s χα1...αD−4|β2βs = χ[α1...αD−4]β2βs = χα1...αD−4(β2βs). (254)

4.1. D > 4: Generalized Dual Fronsdal The gauge transformations for χα1...αD−4|β2βs are reducible. Fields The action (251) also has the following gauge symmetry

The Fronsdal field h12s is dual to a field dualized on ˜ δ ˜α1...αD−3 one index. The dual Fronsdal field hα1αD−3|2s is a double- χ,ϕh |β2βs s−1 [α1 α2...αD−3] traceless, tensor with Young symmetry (D − 3, 1, ,1). It is = (s − 1)(D − 3) ∂ χ |β2βs a symmetric tensor in D = 4, whose equations of motion [α1 α2...αD−3] α1...αD−3| can be described by the Fronsdal Lagrangian (235). In D > +∂ ϕ (β2|β3βs) − ∂(β2 ϕ β3βs) , (255) 4, the equations of motion of free massless, dual spin-s field ˜ hα1αD−3|2s explicitly reads [47]

− (s 1) ρ α α − |β β L[h˜α α |β β ] = − ∂ h˜ 1 D 3 2 s ∂ρh˜α α |β β 1 D−3 2 s s(D − 3)! 1 D−3 2 s ˜ρα2αD−3|β2βs σ ˜ − (D − 3)∂ρh ∂ hσ α2αD−3|β2βs ˜α1αD−3|ρβ3βs σ ˜ − (s − 1)∂ρh ∂ hα1αD−3|σβ3βs ˜ α2αD−3|γβ3βs ρ σ ˜ + 2(s − 1)(D − 3)hγ ∂ ∂ hρα2αD−3|σβ3βs ˜α1αD−3|γ σ ρ ˜ β4βs + (s − 1)(s − 2)h γβ4βs ∂ ∂ hα1αD−3|σρ ρ ˜ α2αD−3|γβ3βs ˜σ − (s − 1)(D − 3)∂ hγ ∂ρh α2αD−3|σβ3βs

1 σβ β ρ α α − |γ − (s − 1)(s − 2)∂ρh˜α α |σ 4 s ∂ h˜ 1 D 3 γβ β 2 1 D−3 4 s ˜ α2αD−3|ρσ γ ˜ λβ4βs − (s − 1)(s − 2)(D − 3)∂σ hρ β4βs ∂ hγ α2αD−3|λ

1 σ α α − |ρ γ λβ β + (s − 1)(s − 2)(D − 3)∂σ h˜ 2 D 3 ρβ β ∂ h˜γ α α |λ 4 s 4 4 s 2 D−3

1 σ α α − |λβ β ρ + (s − 1)(D − 3)(D − 4)∂σ h˜ 2 D 3 3 s ∂ h˜ρα α |λβ β 2 2 D−3 3 s

1 ρα α − |γβ β σ λ + (s − 1)(D − 3)(D − 4)∂ρh˜γ 3 D 3 3 s ∂ h˜ σ α α |λβ β 2 3 D−3 3 s 1 γ ˜α1...αD−3|ρ ˜ λσβ5βs − (s − 1)(s − 2)(s − 3)∂ h ργβ β ∂σ hα ...α |λ , (251) 4 5 s 1 D−3

where χα1...αD−4|β2βs is an arbitrary tensor with the mixed which is invariant under the gauge symmetry [47]: s−1

symmetry (D − 4, 1, ,1), and ϕα1...αD−3|β3βs = ϕ[α1...αD−3] s−2 δ h˜α1...αD−3 =(D − 3) ∂[α1 χ α2...αD−3] χ |β2βs |β2βs is an arbitrary tensor with the mixed symmetry (D − 3, 1, ,1). The gauge transformations for χ and ϕ +(s − 1)∂ χ [α1...αD−4|αD−3] , (252) α1...αD−4|β2βs α1...αD−3|β3βs (β2 β3βs) are reducible.

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One may define the spin-s generalized Curtright action of s−1 ˜ which has Young symmetry (D − 2, 2, ,2): the dual spin-s field hα1 αD−3|β2βs using a field strength tensor F as (see Appendix 1 in Supplementary Material): α1...αD−2|β2βs s boxes

(s − 1)2 1  L ˜ [hα1αD−3|β2βs ] = − Fα1αD−2|β1βs−1  s (D − 2)! R˜ ≡ D−2 boxes . α1αD−2|2ν2sνs   × F α1αD−2|β1βs−1  (s − 2)  + F  σ γ α1αD−4|β1βs−1  (261) (s − 1)(D − 3)!  The spin-s dual Ricci tensor is defined by × Fσβ1α1αD−4|γβ2βs−1 (s − 3)(D − 2) ˜ σ Rα1αD−2|2ν2s−2νs−2|s−1s + Fα1αD−3σ |β1βs−2 (D + s − 4)(D − 3)! ˜ νs−1νs ≡ Rα1αD−2|2ν2s−2νs−2s−1νs−1sνs η , (262) α α − γ |β β − × F 1 D 3 1 s 2 γ s−1 (s − 2)(D − 2) σ + Fσ γ α α − |β β − with Young symmetry (D − 2, 2, ,2,1,1) and the diagram: (D + s − 4)(D − 3)! 1 D 4 1 s 2 ˜ ρβ α α − |γβ β − Rα1αD−2|2ν2s−2νs−2|s−1s × F 1 1 D 4 2 s 2 ρ , (256) s boxes

 where the “spin-s Curtright” field strength tensor F α1...αD−2|β2βs  defined as ≡ D − 2 boxes . (263)  

˜  Fα ...α |β β ≡ (D − 2)∂[α hα α ]β β , (257)  1 D−2 2 s 1 2 D−2 2 s   ˜ Similarly, there is the spin-s dual Einstein tensor Gα1 αD−3|β2...βs with Young symmetry: s−1 is a tensor with Young symmetry type (D − 2, 1, ,1): s boxes  s boxes ˜ ≡ − Gα1 αD−3|β2...βs D 3 boxes ,       (264) ≡ −   Fα1...αD−2|β2βs D 2 boxes .   which is defined by  (s − 1)  G˜ β2...βs =F˜ β2...βs − 2(D − 3)η β2  α1αD−3| α1αD−3| [α1  (258) 4  ˜ β3...βsρ  × Fα2αD−3]ρ| The spin-s Curtright field strength tensor Fα1...αD−2|β2βs is transferred as follows β2β3 ˜ β4...βsρ + (s − 2)η Fα1αD−3|ρ , (265) ˜ where Fα1αD−3|β2...βs is the spin-s dual Fronsdal tensor [47] β2βs (β2 β3βs) δαFα1...αD−2| = −2(D − 2)(D − 3)∂ ∂[α1 ϕα2...αD−2| . (259) F˜ β2...βs = ∂ ∂ρ h˜ β2...βs − (D − 3) However, it is not gauge invariant under the gauge symmetry α1αD−3| ρ α1αD−3| ρ ˜ β2...βs (252). × ∂[α1 ∂ hρα2αD−3]| One can define the spin-s dual Riemann tensor by (β2 β3...βs)ρ − (s − 1)∂ ∂ρ hα1αD−3| (β2 ˜ β3...βs)ρ + (s − 1)(D − 3)∂[α1 ∂ hρα2αD−3]| (s − 1)(s − 2) 1 (β2 β3 ˜ β4...βs)ρ 1ν1 + ∂ ∂ hα α |ρ . (266) R˜ α α | ν ν = ǫα α R ν ν ν , (260) 1 D−3 1 D−2 2 2 s s 2 1 D−2 1 1 2 2 s s 2

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˜ T˜ The spin-s dual Fronsdal tensor Fα1αD−3|β2...βs is of Young where α1αD−3|β2βs is a double-traceless s−1 T˜ α1α2 α3α4 T˜ β2β3 β4β5 ( α1αD−3|β2βs η η = 0 = α1αD−3|β2...βs η η ), symmetry type (D − 3, 1, ,1). In D = 4, the definition (266) s−1 recovers Equation (239). tensor with Young symmetry type (D − 3, 1, , 1) and defined The spin-s dual Ricci tensor (262) can be defined in term of as ˜ the dual spin-s Fronsdal tensor Fα1αD−3|β2...βs : T˜ 2s ˜ 2s α1αD−3| ≡ Tα1αD−3| ˜ s−3 Rα1αD−2|2ν2s−2νs−2|s−1s = − 2 (D − 2)∂[ν2| ∂[νs−2| (s − 1)(D − 3) − η (2 T˜ 3s)ρ, ˜ [α1 α2αD−3]ρ| × ∂[α1 Fα2αD−2]|2]|s−2]s−1s . 4 (267) (274) ˜ Substituting the dual Fronsdal definition (266) into the dual Ricci and Tα1αD−3|2s is the spin-3 magnetic-type stress-energy, definition (267) presents s−1 and has Young symmetry type (D − 3, 1, ,1). By analogy with the Dirac quantization condition (104), the R˜ α1αD−2|2ν2s−2νs−2|s−1s spin-s quantization condition in D = 4is[91] = −2s−3(D − 2) ∂ ∂ρ ∂ ∂ ∂ h˜ ρ [ν2| [νs−2| [α1 α2αD−2]|2]|s−2]s−1s 1 ˜ 1s−1 Z ρ ˜ P1s−1 P = n hc , n ∈ , (275) − (D − 3)∂[ν2| ∂[νs−2|∂[α1 ∂ hρα2αD−3]|2]|s−2]s−1s 2 ρ ˜ − ∂ ∂[ν | ∂[ν |∂ ∂[α hα α ]| ]| ] ρ s 2 s−2 1 2 D−2 2 s−2 s−1 where P1s−1 = mU1s−1 is the conserved spin-s electric- ρ ˜ ˜ 1s−1 ˜ 1s−1 − ∂s−1 ∂[ν2| ∂[νs−2|∂ ∂[α1 hα2αD−2]|2]|s−2]sρ type charge, and P =m ˜ U is the conserved spin- ρ ˜ s magnetic-type charge, m is the electric-type spin-s charge, and + (D − 3)∂s ∂[ν2| ∂[νs−2|∂ ∂[α1 hρα2αD−2]|2]|s−2]s−1ρ m˜ is the magnetic-type spin-s charge, U ≡ u u ρ ˜ 2 s 2 s + (D − 3)∂ − ∂[ν | ∂[ν − |∂ ∂[α hρα α − ]| ]| − ] ρ ˜ s 1 2 s 2 1 2 D 2 2 s 2 s the traceless part of u2 us , and U2s ≡u ˜ 2 ˜us the ˜ ρ traceless part of u˜ ˜u . + ∂s−1 ∂s ∂[ν2| ∂[νs−2|∂[α1 hα2αD−2]|2]|s−2]ρ . (268) 2 s The stress-energy electric-type and magnetic-type tensors in 4.2. Spin-s Magnetic-Type Sources D = 4 can be written as

We now introduce the coupling between the spin-s fields (4) ∂w1 T12s = mU2s dτδ x − w(τ) h12s = h(12s) and spin-s electric-type sources, and ∂τ then make a generalization to the dual spin-s fields and spin-s u1 U2s (3) magnetic-type sources. = m δ x −w (x0) , (276) u The linearized spin-s Ricci tensor R can be coupled 0 1ν12ν23ν3 (4) ∂w˜ 1 to a traceless (T η 1 2 η 3 4 = 0), symmetric tensor in T˜ =m ˜ U˜ dτδ x − w(τ) 1 2 s 12s 2s ∂τ D-dimensional spacetime: ˜ ˜ u1 U2s (3) 1 =m ˜ δ x − w˜ (x0) . (277) R ν ν ν | = ∂[ν | ∂[ν |T| ]| ] . u0 1 1 2 2 s−2 s−2 s−1 s 2 1 s−2 2 s−2 s−1 s (269) A Dirac string in the 4-D spin-s theory can be constructed as T The tensor 1s is defined by follow. Let us consider the decomposition of the spin-s Riemann curvature R ν ν , it can be split into the chargeless part and s ρ 1 1 s s T ≡ T − η T ρ , (270) 1 s 1 s 4 ( 1 2 3 s) the charged part, = + where T1s = T(1s) is the spin-s electric-type stress-energy R1ν1sνs r1ν1sνs m1ν1sνs . (278) tensor. Both r1ν1sνs and m1ν1sνs are of mixed symmetry type The spin-s Einstein equations with the spin-s electric-type s source read [91]: (2, ,2). The tensor r1ν1sνs satisfies the cyclic and Bianchi G = T . (271) identities, and can be written in terms of covariant derivatives of 1s 1s a symmetric tensor h : The dual spin-s Einstein tensor defined by (265) can be coupled 123 to the spin-s magnetic-type source: r1ν1sνs = −2∂[s| ∂[2|∂[1 hν1]|ν2]|νs], (279)

G˜ α α |β β = T˜α α |β β , (272) 1 D−3 2... s 1 D−3 2... s The charged part m1ν1sνs can be defined in terms of the spin-s magnetic-type source: equivalently, 1 ˜ ρσ R˜ m1ν1sνs = ǫ1ν1ρσ ∂[2| ∂[s|M |ν2]|νs] α1αD−2|2ν2s−2νs−2|s−1s 2 1 [ρ ˜ σ ]λ| = ∂[ν | ∂[ν |∂[α T˜α α ]| ]| ] , (273) + ∂[2 δ ν2]∂[3| ∂[s|M |ν3]|νs]λ , (280) 2 2 s−2 1 2 D−2 2 s−2 s−1 s

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˜ where the tensor Mαβ|2s is of Young symmetry type Substituting Equation (288) into (239) leads to the wave equation s−1 ρ ∂ρ∂ h1s = 0. (2, 1, ,1): It can be seen that Equation (288) is also invariant under the gauge transformation s boxes δ D = s∂ ∂ρξ . (289) ˜ ξ 2s ρ 2s Mαβ|2s ≡ . (281) It was imposed that the spin-s field has the double-traceless If we define condition (233) [102]. To make the De Donder gauge condition (288), one may require that the spin-s field is transverse and αβ| αβρ1 y 2s = ǫ ∂ρh1s , (282) traceless (see [145] for proof): where the tensor y is of Young symmetry type αβ|2s ∂ρ = ∂ ρ = s−1 h22sρ 0, 2 h3sρ 0. (290) (2, 1, ,1) tensor, Having the transverse–traceless gauge condition, the gauge parameter ξ is a symmetric, traceless (s − 1)-th rank yαβ| = y[αβ] = yαβ , ∂ yαβ| = 0, 2 s 2s 2s (2s) α 2s tensor. Similar to massless spin-2 fields described in section 2.6, (283) massless, spin-s fields have only two polarization states with the the spin-s Riemann curvature R can be rewritten as 1ν1sνs ±s helicity states [146]. However, massive, spin-s fields have 2s+1 polarization states [147, 148]. 1 ρσ [ρ σ ]λ| R ν ν = ǫ ν ρσ ∂ | ∂ | y |ν |ν + δ |ν y λν |ν 1 1 s s 2 1 1 [ s [ 2 2] s] 2] 3] s] 5. DISCUSSION: INTERACTING THEORIES ˜ ρσ [ρ ˜ σ ]λ| + M |ν2]|ν3]|νs] + δ |ν2]M λ|ν3]|νs] . (284) Consistency of the interacting theory for a free field can ˜ Assuming Yαβ|2s = yαβ|2s + Mαβ|2s , one has traditionally be determined from coupling deformations of the gauge transformations of the free theory. The BRST formalism

3ν3 1 [149–152] was originally developed to examine whether the R ν ν = ǫ ν ρσ ∂[ | ∂[ | 1 1 2 2 2 1 1 2 s deformed gauge transformations can be constructed. Inclusion of ρσ [ρ σ ]λ| auxiliary fields (antifields and ghosts) led to the BRST-antifield × Y |ν2]|νs] + δ ν2]Y λν3]|νs] . (285) formalism [153–157] that allowed us to assess consistency of interactions of the free theory with other theories and itself. This We can now connect a Dirac string y(τ, θ) to the spin-s involves studying consistent deformations of the master equation magnetic-type source M˜ , y(τ,0) =w ˜ (τ) by analogy αβ|2s [158, 159], which contains gauge transformation structures, with (124): corresponding local reducibilities, as well as stationary surfaces (inc. equations of motion) of each field. The master equation ˜ ˜ (4) Mαβ|2s =m ˜ U2s dτdθδ x − y(τ, θ) of the free theory is associated with the BRST differential. The coupling-order deformations of the master equation of the ∂yα yβ ∂yα ∂yβ × − , (286) interacting theory by means of the BRST differential allow us to ∂θ ∂τ ∂τ ∂θ evaluate all the requirements for consistency of the interacting theory (see reviews [160–167]). The action principle of the free ˜ ˜ The divergence of Mαβ|2s is equal to T1s : theory is the integral of the Lagrangian over the manifold that corresponds to the propagation of perturbations of the free field ˜ α ˜ T1s = ∂ Mα1|2s . (287) in the spacetime. Nevertheless, the action of the interacting theory associated with observables, which are invariant under the Hence, the Dirac string of a spin-s magnetic-type point source is gauge transformations, may make consistent interactions with conserved (see also [91] for spin-s electric-type source). themselves or other fields of other free theories. A detail study 4.3. Higher-Spin Harmonic Condition of consistency of interactions of the free theory of the fields with itself can deduce whether the interacting theory has consistent We now introduce a harmonic coordinate condition for the deformations. generalized Fronsdal equation (239) that makes the linearized The action principle of a massless, spin-2 field particle is wave equation for spin-s fields by eliminating the last two terms described by the Pauli-Fierz action [100, 111], which corresponds of (239). to the linearized Einstein equations in Minkowskian flat To have a wave equation for the spin-s fields in empty space, spacetime. The Pauli-Fierz action is free of ghost excitations, and it is required to have the generalized De Donder gauge condition its field equations are algebraically consistent. The interacting [103]: theory of gravity has only one single massless spin-2 field [168– s − 1 170], and there are no consistent interactions among multiple D ρ ρ 2s ≡ ∂ h2sρ − ∂2 h3sρ = 0. (288) interacting massless spin-2 fields [171], i.e., no spin-2 analogy 2

Frontiers in Physics | www.frontiersin.org 24 January 2019 | Volume 6 | Article 146 Danehkar Duality in Gravity and Higher-Spin Fields of Yang-Mills theory. However, the action of a single, massless identity [185], and the commutator of two gauge transformations spin-2 field hν coupled to a positive-definite dynamical metric cannot be written [184]. Similarly, the commutator between is gν is not invariant under the hν gauge transformation, two gauge transformations of a linear spin-s gauge field (> so it is inconsistent [172]. Similarly, cross-interactions among 3) cannot be constructed in D = 4, so self-interactions of multiple massless spin-2 fields are inconsistent in a positive- a spin-s field do not exist in 4-D flat space [184]. Using definite metric [171], while consistency is mathematically the BRST-antifield formalism, the first-order deformations of possible in a negative-definite metric (negative-energy) [171, a spin-3 gauge field in D = 4, defined by a non-abelian a a 173]. In the case of direct coupling to matter, the locality symmetric rank-3 tensor hνρ ≡ h(νρ), were built under and consistency conditions require that the interacting theory the assumptions of parity and Poincaré invariance, local and includes all nonlinear terms [168, 169, 174], otherwise the spin- perturbative deformation of the free theory in Minkowski space, 2 field of the linearized theory remains ghost-like in empty which correspond to the cubic vertex of Berends, Burgers, and flat space [170]. Despite the inconsistency of linear, massless van Dam [183] involving three derivatives of the fields, but the spin-2 interactions, the nonlinear theory of a massive spin-2 theory is again inconsistent at second order [186]. In D = 5 field provides ghost-free consistent coupling to matter [175– Minkowski spacetime, the second-order deformations of spin-3 177]. A spin-2 field coupled to matter or nonlinear self- fields can be obtained, while the first-order deformations involve interaction leads to the general covariance of the interacting the generalized de Wit–Freedman connection [187]. In D > 4 theory [169]. Constructing the interacting multi-graviton model flat spacetime, the structure constants given by a completely and coupling a single spin-2 field to a dynamical metric require antisymmetric tensor can make a cubic vertex at second order a fully non-linear representation of the interacting theory for involving five derivatives of the fields, but the analysis will be gravity. complicated due to tedious five-derivative calculations [186]. It was proven that the dual formulations of linearized Hence, the interacting theory of linearized spin-3 particles is gravity in D = 5, the Curtright Field, does not have problematic in flat space D ≥ 4, and only feasible under some consistent interactions involving two derivatives of the fields certain assumptions involving 3 derivatives at first order and under local, Poincaré invariance [40]. The dual formulations 5 derivatives at second order in spacetime dimensions higher of linearized gravity in D > 5 do not possess consistent, than 4. local deformations [178], so the dual graviton of the linearized The generalizations of the Coleman–Mandula “no-go” theory does not have any self-interaction. However, the dual theorem [188] to supersymmetry [189] and higher spin gauge graviton can be coupled to other theories, such as topological theories [190–192] indicate that the interacting theory of BF theories [132, 179]. To overcome the inconsistency of linearized higher spin fields (s > 2) are incompatible in flat the linearized dual gravity, one may consider a non-linear spacetime, since their conserved currents are associated with a theory of the dual graviton, which incorporates a topological free theory, which does not permit to have a nontrivial S-matrix (Chern-Simons like) term for the original graviton, which [190]. The no-go theorem allows us to have only gauge fields leads to consistent nonlinear deformations, and is manifestly of spin-1 and spin-2 perturbations around a flat background. invariant due to a dynamical metric for the dual graviton However, the restrictions imposed by the no-go theorem on [134]. In particular, the properties of the Kac-Moody algebra higher spin fields (s > 2) can be overcome in the nonlinear E11 (see section 2.3) are associated with spacetime covariance theory of spin-s perturbations around the anti de Sitter (AdSd) and supergravity covariant spectra. In the case of D = 11 spacetime of arbitrary d dimensions [193]. It is argued that +++ supergravity, the Kac-Moody algebra (AD−3 for pure gravity) the interacting theory of nonlinear, massless bosonic fields of applies to pure gravity (as well as maximal supergravity), so arbitrary spins s > 1 can be formulated in d-dimensional AdS the dual graviton cannot be coupled to the original Einstein- space [194–196]. Nonlinear spin-s fields with a non-vanishing Hilbert action, and it doubles degrees of freedom. Nevertheless, cosmological constant in D = 4 were shown to have consistent the dual graviton in the E11 algebra can be coupled with a interaction in the cubic-order [197, 198], and in the first-order topological term containing the original metric without any in the Weyl 0-forms [199], and also in all orders [200–203] (the doubling of degrees of freedom [134]. The interacting theory so-called Vasiliev higher-spin theory). Consistent interactions of dual graviton in D ≥ 5 [43] can also be constructed of nonlinear spin-s fields were also formulated in 3-D AdS in de Sitter space (dSn) [180], similar to the nonlinear spacetime [204], in AdS5 spacetime [205, 206], and in AdSd implementation of gravitational duality with a cosmological space with arbitrary d dimensions [207]. In particular, it was constant in dS4 [181]. Thus, it is necessary to formulate a shown that irreducible massless mixed symmetry fields in AdSd nonlinear representation of dual gravity (see [134, 182] for are decomposed into certain reducible massless flat fields in the nonlinear dual graviton), which is dual to nonlinear Einstein flat limit [208], which permit to have consistent deformations for gravity, containing both the original and dual metrics, and is fully generic higher-spin fields in the AdS space. It was demonstrated gauge covariant. that all spin-s consistent deformations are explicitly made using It has been shown that a massless spin-3 (also s > 3) particle Vasiliev’s unfolded equations in D = 4, which are expressed described by a linear gauge field cannot have a self-interaction in terms of a gauge connection and a (twisted)-adjoint matter in D = 4 [183, 184]. Although first-order deformations of field, while the Vasiliev gauge connection is found to be related spin-3 fields are possible [183], self-interactions encounter some to the spin-s Fronsdal fields [209]. Therefore, the nonlinear, difficulties: the first-order deformations do not obey the Jacobi massless higher spin gauge theories can have consistent

Frontiers in Physics | www.frontiersin.org 25 January 2019 | Volume 6 | Article 146 Danehkar Duality in Gravity and Higher-Spin Fields interactions around the AdS vacuum solution with the non-zero for more discussions). Nevertheless, we do not yet have any full cosmological constant in the Vasiliev higher-spin theory nonlinear representation of the spin-s Fronsdal fields. [210]. It is well known that the compactification of M-theory or AUTHOR CONTRIBUTIONS supergravity on (d + 1)-dimensional AdS spacetime is dual to conformal field theory (CFT) in d dimensions, which is The author confirms being the sole contributor of this work called the AdSd+1/CFTd correspondence or Maldacena duality and approved it for publication. The author presented some [211]. This correspondence indicates that the chiral operators mathematical notations and expressions slightly different from of N = 4 Super Yang–Mills theory (CFT observables) in D = the original versions, and provided few new results (e.g., 4 correspond to those of Kaluza–Klein Type IIB supergravity Equations 27 and 28 in section 2.1, a part of section 2.6). 5 5 on AdS5 × S (where S is a 5-dimensional compact space) [212, 213]. We know that electric-magnetic duality in the bulk FUNDING of AdS4 can relate holographically derived deformations of the boundary CFT3 [50]. It was also argued that gravity theories A part of this work was supported by the EU contract MRTN-CT- in AdS4 are holographically dual to two CFTs in D = 3 with 2004-005104 Constituents, Fundamental Forces and Symmetries different parity: the Dirichlet CFT3 with the graviton source and of the Universe at University of Craiova in 2008. a dual CFT3 with a dual graviton source at the non-linear level [52]. The holographic properties of electric-magnetic duality in ACKNOWLEDGMENTS gravity also provide some insights into non-linear aspects of M-theory (see [51] for review). The “yes-go” Vasiliev higher- AD acknowledges hospitality from the University of Craiova, spin theory of spin-s perturbations in the AdS background where a part of this work was carried out, and thanks C. Hull, is naturally conjectured in the higher-spin holography [214– P. West, M. Vasiliev, G. Barnich, and M. Henneaux for helpful 216], which gives evidence for duality between Vasiliev’s higher- comments and valuable discussions. spin fields and the free field theory of N massless scalar fields [217], and holographic dualities between Vasiliev’s higher-spin SUPPLEMENTARY MATERIAL fields and O(N) vector models [215, 216] (the so-called higher spin/vector model duality; see [218, 219] for review). The The Supplementary Material for this article can be found interacting theory of Vasiliev’s higher-spin fields may reveal the online at: https://www.frontiersin.org/articles/10.3389/fphy. hidden origin of AdSd+1/CFTd correspondence (see [220–222] 2018.00146/full#supplementary-material

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Frontiers in Physics | www.frontiersin.org 30 January 2019 | Volume 6 | Article 146