aa r X i v : . [ h e p - t h ] F e b Self-dual models in D = 2 + 1 from dimensionalreduction D. Dalmazi ∗ UNESP - Campus de Guaratinguet´a - Departamento de FisicaCEP 12516-410 - Guaratinguet´a - SP - Brazil.
February 11, 2021
Abstract
Here we perform a Kaluza-Klein dimensional reduction of Vasiliev’s first-order de-scription of massless spin-s particles from D = 3 + 1 to D = 2 + 1 and derive first-orderself-dual models describing particles with helicities ± s for the cases s = 1 , ,
3. In thefirst two cases we recover known (parity singlets) self-dual models. In the spin-3 case wederive a new first order self-dual model with a local Weyl symmetry which lifts the trace-less restriction on the rank-3 tensor. A gauge fixed version of this model corresponds toa known spin-3 self-dual model. We conjecture that our procedure can be generalized toarbitrary integer spins. ∗ [email protected] Introduction
Elementary massless particles of spin-s in D = 3 + 1 have well defined helicities ± s however,it is not possible to write down local Lagrangians for helicity eigenstates, they can not belocally separated. The situation is different in D = 2 + 1 where massive spin-s particles withhelicity + s or − s can be described by local Lagrangians (parity singlets), also called self dualmodels. In the present work the symbol SD ( s ) j stands for a self-dual model of helicity s and of j -th order in derivatives. Those models are irreducible representations of the Poincare groupin D = 2 + 1. The Maxwell-Chern-Simons theory and the linearized topologically massivegravity [1] correspond respectively to SD ( ± and SD ( ± .Remarkably, self dual models of opposite helicities + s and − s can be “soldered” into aconsistent (ghost free) parity doublet as in the spin-1 [2] and spin-2 [3, 4] cases and morerecently for spin-3/2 [5] and spin-3 [6]. In particular, the soldering of second order spin-2self dual models, as defined in [7], gives rise to the well known Fierz-Pauli [8] massive spin-2theory while the soldering of linearized topologically massive gravities leads to the linearizednew massive gravity (NMG) of [9]. The fine tuned coefficients necessary in [8, 9] in order tohave a ghost free theory are automatically produced by the soldering procedure.In the case of higher spin models ( s >
2) the soldering procedure may furnish interestinghints about the much less known higher spin geometry. In the spin-3 case the soldering of SD ( ± (or SD ( ± ) has produced a sixth order parity doublet model which seems to be anatural spin-3 generalization of linearized NMG. It reinforces the naturalness of the restricted(traceless) symmetry δh µνρ = ∂ ( µ ξ νρ ) as opposed to the non restricted one, see [6]. We believethat similar solderings of SD ( ± s )2 s and SD ( ± s )2 s − into parity doublets for s ≥ D = 3 + 1 massive models is not clearsince those higher derivative models are unitary only in D = 2 + 1 just like NMG.On the other hand, if we stick to lower order self dual models other technical challengesshow up. Indeed, it took us some time until we were able to overcome the spin-2 barrier forthe soldering procedure. The main obstacle is the typical presence of auxiliary fields in higherspin theories. It is not yet clear how to include the auxiliary fields in the soldering procedure.Fortunately, thanks to the trading of auxiliary fields into higher derivatives, the higher orderself dual models used in [6] do not have auxiliary fields. We still do not know how to solder SD ( ± j with j = 1 , , , s = 1 , / , , s equivalent self-dual models SD ( ± s ) j with j = 1 , , . . . , s . In thecases s = 1 , / , SD ( s ) j to SD ( s ) j +1 from bottom to top via a Noethergauge embedding procedure (NGE). However, at s = 3 although we can go from SD (3) j to SD (3) j +1 for j = 1 , , j = 5 [11], we can not connect SD (3)4 , which contains a vectorauxiliary field, with SD (3)5 which only depends on a totally symmetric rank-3 tensor. We hope2hat if we start from a more general first order model SD (3)1 we might be able to overcome the4th order barrier and follow the row of models continuously until SD (3)6 .In the present work we have been able to derive a new first order spin-3 self dual model,which generalizes the previous known model in the literature [12], via a Kaluza-Klein (KK)dimensional reduction of the first order Vasiliev [13] description of massless spin-s models in D = 3+1. The dimensional reduction gives rise to a pair of opposite helicities self-dual modelsthat we can decouple via simple linear transformations (section 4), this is how the apparentparadox of deriving helicity eigenstates (parity singlets) from four dimensional models (parityinvariants) is solved.In sections 2 and 3 we introduce our notation and basic ideas in the simpler cases of s = 1and s = 2 where we reproduce known models in the literature. In section 4 we obtain the new s = 3 self dual model and in section 5 we draw our conclusions. The Maxwell theory can be written in a first order version with the help of an antisymmetricfield Y [ MN ] . In D = 3 + 1-dimensions we have S Max = Z d x h Y [ MN ] Y [ MN ] − Y [ MN ] ( ∂ M e N − ∂ N e M ) i . (1)Here we use capital Latin letters to denote the 3+1-dimensional indices, ( M, N, . . . = 0 , , , µ, ν, . . . = 0 , , x D = x ≡ y in a circle of radius R = 1 /m and keep only one massive mode. Thisis a known [14, 15] method of obtaining massive models from massless ones. The fields aredecomposed as Y [ MN ] ( x α , y ) → Y [ µν ] ( x, y ) = p mπ y [ µν ] ( x ) cos myY [ µD ] ( x, y ) = p mπ C µ ( x ) sin mye µ ( x, y ) = p mπ e µ ( x ) cos mye D ( x, y ) = p mπ φ ( x ) sin my (2)Integrating over y in (1) ranging from 0 to 2 π/m we obtain an action in D = 2 + 1 whoseLagrangian is a first order version of the Maxwell-Proca theory, L MP = 14 y [ µν ] y [ µν ] − y [ µν ] ( ∂ µ g ν − ∂ ν g µ ) + 12 C µ C µ + m C µ g µ . (3)where we have introduced the U (1) invariant vector field g µ = e µ + ∂ µ φ/m . The gaugesymmetry ( δe µ , δφ ) = ( ∂ µ Λ , − m Λ) is inherited from δe M = ∂ M Λ in (1). Integrating over C µ and introducing another gauge invariant vector field without loss of generality, Throughout this work we use the metric η µν = ( − , + , + , · · · , +) and the notation: ( αβ ) = ( αβ + βα ) / αβ ] = ( αβ − βα ) / [ µν ] ≡ m ǫ µνρ f ρ , (4)after using ǫ -identities we have, L MP = m f ρ E ρµ g µ − m f ρ − m g ρ . (5)Where E µν = ǫ µνρ ∂ ρ . On one hand, if we integrate over f ρ we obtain the Maxwell-Procamodel in terms of g µ . Notice that we can always fix the unitary gauge φ = 0, even at actionlevel, and treat g µ as an elementary vector field. On the other hand, we can decouple the ± L MP = L (1) SD ( m, e + µ ) + L (1) SD ( − m, e − µ ) . (6)where e ± µ = ( f µ ± g µ ) / √ L (1) SD is the spin-1 self-dual model suggested long ago in [16], L (1) SD ( m, e ) = m e µ E µν e ν − m e µ . (7)which describes massive particles of helicity | m | /m . Massless spin-2 particles are commonly described by the linearized Einstein-Hilbert (EH)theory in terms of a symmetric rank-2 tensor field. This theory can be written in a first orderversion using a non-symmetric rank-2 tensor, e MN , and a mixed symmetry rank-3 tensor Y [ AB ] M as in the spin-2 case of the Vasiliev’s formulation of massless spin-s particles [13],using the notation of [17] we have S s =2 = 12 Z d x h Y [ AB ] M Y [ AM ] B − Y A Y A − Y [ AB ] M ( ∂ A e BM − ∂ B e AM ) i , (8)where Y B = η MA Y [ AB ] M . The action S s =2 is invariant under the following gauge transforma-tions: δe AB = ∂ A ξ B + ω [ AB ] , (9) δY [ BA ] M = ∂ M ω [ AB ] + η AM ∂ C ω [ BC ] − η BM ∂ C ω [ AC ] (10) ǫ µνλ ǫ αβγ = − η µα η νβ η λγ + η µβ η να η λγ − η µβ η νγ η λα + η µγ η νβ η λα − η µγ η να η λβ + η µα η νγ η λβ ǫ µνα ǫ γρα = δ µρ δ νγ − δ µγ δ νρ ; ǫ µνα ǫ γνα = − δ µγ ; ǫ µνα ǫ µνα = − Y [ MA ] B lead to the linearized Einstein-Hilbert theory in terms of e ( MN ) . The KK dimensional reduction of S s =2 has been carried outin [18] but for the sake of comparison with the more involved spin-3 case we reproduce heresome formulae in a convenient notation. Compactfying the spatial dimension y = x D in acircle as in the spin-1 case, the fields and the gauge parameters are redefined according to: Y [ AB ] M ( x, y ) → Y [ αβ ] µ = p mπ y [ αβ ] µ ( x ) cos myY [ αβ ] D = p mπ y [ αβ ] ( x ) sin myY [ αD ] µ = p mπ C µα ( x ) sin myY [ αD ] D = p mπ C α ( x ) cos my , (11) e MN ( x, y ) → e µν ( x, y ) = p mπ e µν ( x ) cos mye µD ( x, y ) = p mπ U µ ( x ) sin mye Dν ( x, y ) = p mπ S ν ( x ) sin mye DD ( x, y ) = p mπ φ ( x ) cos my , (12) ξ M ( x, y ) → ( ξ µ ( x, y ) = p mπ ǫ µ ( x ) cos myξ D ( x, y ) = p mπ ǫ ( x ) sin my , (13) ω [ MN ] ( x, y ) → ( ω [ µν ] ( x, y ) = p mπ ω [ µν ] ( x ) cos myω [ µD ] ( x, y ) = p mπ ω µ ( x ) sin my , (14)where C µν is an arbitrary rank-2 tensor without symmetry. Once again we introduce a dualnon symmetric rank-2 field f ρµ via the Levi-Civita tensor, without loss of generality, y [ αβ ] µ ≡ m ǫ ραβ f ρµ . (15)After integrating over the cyclic coordinate x D = y we obtain a 3D massive action whoseLagrangian is given by L (2) m = − m f ργ E ρβ e γβ + m f αβ f βα − f ) + y [ µν ] ∂ µ U ν − C µν C νµ + C C µν (cid:0) m e νµ + ∂ ν S µ − y [ νµ ] (cid:1) − C µ + m C µ (cid:18) U µ − ǫ µαβ f αβ (cid:19) (16)Where U µ = U µ − ∂ µ φ/m . Integrating over the C-fields we are able to write down theLagrangian in a simpler form: L (2) m = − m f ∗ ργ E βρ e ∗ βγ + m (cid:2) f ∗ αβ f ∗ βα − ( f ∗ ) (cid:3) + m (cid:2) e ∗ αβ e ∗ βα − ( e ∗ ) (cid:3) (17)with 5 ∗ µν = f µν − m ∂ µ K ρ + ǫ µνρ U ρ (18) e ∗ µν = e µν + 1 m ∂ µ S ρ − m ∂ µ U ρ + ǫ µνρ K ρ . (19)where we have introduced the dual K ρ field via the invertible map: y [ µν ] = m ǫ µνρ K ρ . (20)Finally, after a simple rotation we can disentangle the helicity ± L (2) m = L (2) SD ( m, e + ) + L (2) SD ( − m, e − ) , (21)where e ( ± ) µν = ( e ∗ µν ± f ∗ µν ) / √
2. We obtain the spin-2 self-dual model of Aragone and Khoudeir[19] with helicity 2 | m | /m , i.e., L (2) SD ( m, e ) = m e ργ E βρ e βγ + m (cid:2) e αβ e βα − e (cid:3) (22)As a double check of our final Lagrangian, it is easy to show that (17) and consequently (21),is invariant under the gauge symmetries associated with the parameters (13) and (14), i.e., δf αβ = ǫ αβρ ω ρ − ∂ α ω β /m ; δK ρ = − ω ρ (23) δe αβ = ǫ αβρ ω ρ + ∂ α ǫ β ; δS ρ = − m ǫ ρ − ω ρ (24) δU ρ = − ω ρ + ∂ ρ ǫ ; δφ = m ǫ , (25)where ω ρ ≡ − ǫ ρµν ω µν /
2. The symmetries follow from the gauge invariance of the compositefields f ∗ µν and e ∗ µν . This will not be true in the spin-3 case as will see in the next section. Wecan turn the composite fields into elementary ones after fixing the unitary gauge at actionlevel: ( K ρ , U ρ , S ρ , φ ) = (0 , , , . (26)Once f ∗ µν and e ∗ µν are considered elementary fields, no symmetry is left in (17). The fact thatwe are allowed to fix (26) at action level is grounded on the “completeness” criterion, see [20].Namely, the 10 gauge conditions (26) uniquely (completely) determine the same number ofgauge parameters: ( ω ρ , ω ρ , ǫ ρ , ǫ ). The Vasiliev’s model [13] for a massless spin-3 particle, in the notation of [17], is given by6 s =3 = Y [ MN ]( RS ) Y [ MN ]( RS ) − Y [ MN ]( RS ) Y [ RN ]( MS ) Y [ RM ]( NR ) Y S )[ SN ]( M Y [ MN ]( RS ) ∂ M e N ( RS ) (27)The bars remind us of the traceless conditions: η RS Y [ MN ]( RS ) = 0 = η RS e N ( RS ) (28)The 4D action corresponding to (27) is invariant under the gauge symmetries: δe N ( RS ) = ∂ N ξ ( RS ) + ω N ( RS ) , (29)( − δY [ MN ]( RS ) = ∂ R ω [ MN ] S + ∂ S ω [ MN ] R + η MR ∂ A ω [ NA ] S + η MS ∂ A ω [ NA ] R − η NR ∂ A ω [ MA ] S − η NS ∂ A ω [ MA ] R , (30)where ω [ MN ] S = ( ω M ( NS ) − ω N ( MS ) ) / η RS ξ ( RS ) = 0 , (31) η RS ω M ( RS ) = 0 = η MR ω M ( RS ) (32)and also ω M ( RS ) + ω R ( SM ) + ω S ( MR ) = 0 . (33)The dimensional reduction from 3 + 1 to 2 + 1 is performed as before with the notation: Y [ MN ]( AB ) ( x, y ) → Y [ µν ]( αβ ) ( x, y ) = p mπ y [ µν ]( αβ ) ( x ) cos myY [ µν ]( αD ) ( x, y ) = p mπ y α [ µν ] ( x ) sin myY [ µD ]( αβ ) ( x, y ) = p mπ C µ ( αβ ) ( x ) sin myY [ µD ]( αD ) ( x, y ) = p mπ C µα ( x ) cos my , (34) e M ( AB ) ( x, y ) → e µ ( αβ ) ( x, y ) = p mπ e µ ( αβ ) ( x ) cos mye µ ( αD ) ( x, y ) = p mπ U µα ( x ) sin mye D ( αβ ) ( x, y ) = p mπ S ( αβ ) ( x ) sin mye D ( Dα ) ( x, y ) = p mπ φ α ( x ) cos my , (35) ξ ( AB ) ( x, y ) → ( ξ ( αβ ) ( x, y ) = p mπ ǫ ( αβ ) ( x ) cos myξ ( αD ) ( x, y ) = p mπ ǫ α ( x ) sin my , (36) ω M ( AB ) ( x, y ) → ( ω µ ( αβ ) ( x, y ) = p mπ ω µ ( αβ ) ( x ) cos myω α ( Dβ ) ( x, y ) = p mπ ω αβ ( x ) sin my , (37)7here C µν , U µν and ω µν are non symmetric rank-2 tensors, the last one is traceless: η αβ ω αβ = 0 . (38)Now several words are in order before we proceed. The reader may be missing theparameters ω D ( DD ) , ω D ( Dα ) , ω α ( DD ) , ξ ( DD ) . The first one vanishes as one can see by fixing( M, R, S ) = (
D, D, D ) in (33). The other ones are not independent quantities, due to thetraceless conditions and (33) we have ( ω D ( Dα ) , ω α ( DD ) , ξ ( DD ) ) =( − ω α , − ω α , − η µν ǫ ( µν ) ) where ω α ≡ η µν ω µ ( να ) and ω α ≡ η µν ω α ( µν ) . From (33) we also have the traceless condition (38), ω α = − ω α / ω µ ( να ) + ω ν ( αµ ) + ω α ( µν ) = 0 . (39)For the accounting of the number of independent gauge parameters we notice that the 10constraints (39) allow us to write the 18 gauge parameters ω µ ( να ) in terms of a non symmetrictraceless rank-2 tensor Ω µν with 8 independent degrees of freedom. Indeed, using ǫ -identitiesone can can show: ω α ( βγ ) = (cid:2) ǫ ραβ (cid:0) ǫ νλρ ω λ ( νγ ) (cid:1) + ǫ ραγ (cid:0) ǫ νλρ ω λ ( νβ ) (cid:1)(cid:3) /
3. This suggests thatwe can always rewrite ω α ( βγ ) in terms of a traceless rank-2 tensor. Indeed, ω α ( βγ ) = ǫ ραβ Ω ργ + ǫ ραγ Ω ρβ . (40)solves (39) if η ρβ Ω ρβ = 0. In particular, ω µ = − ǫ µαβ Ω αβ only depends on the 3 antisymmetriccomponents Ω [ αβ ] .After integrating over the cyclic coordinate y we have a massive spin-3 theory in D = 2 + 1dimensions whose Lagrangian is given by L (3) m = y [ µν ]( αβ ) ∂ µ ˜ e ν ( αβ ) + 13 (cid:2) y µν ]( αβ ) − y [ µν ]( αβ ) y [ µα ]( νβ ) + y µν ] (cid:3) + 49 y αβ y βα + 23 y µ [ αβ ] − y µ [ αβ ] y α [ µβ ] + 2 y µ [ αβ ] ∂ α ˜ U αµ + L C µν + L C µ ( νρ ) (41)where˜ e µ ( αβ ) = e µ ( αβ ) + ∂ µ S ( αβ ) /m + η αβ ( e µ + ∂ µ S/m ) ; e µ = η αβ e µ ( αβ ) ; S = η αβ S ( αβ ) , (42) y [ µν ] = η αβ y [ µν ]( αβ ) ; y µβ = η να y [ µν ]( αβ ) ; ˜ U µν = U µν − ∂ µ φ ν /m , (43) L C µ ( νρ ) = 23 C µ ( νρ ) − C µ ( νρ ) C ν ( µρ ) − C µ + 89 C µ C µ + m C µ ( νρ ) g µ ( νρ ) , (44) L C µν = − C µν C νµ + 49 C − C µν T µν . (45)with 8 µ ( αβ ) = ˜ e µ ( αβ ) + 49 m (cid:0) η µα y β + η µβ y α − y β [ µα ] − y α [ µβ ] (cid:1) (46) T µν = m ˜ U µν + 49 y νµ + 43 y [ νµ ] ; y µ = η αβ y α [ βµ ] ; C µ = η αβ C µ ( αβ ) ; C β = η µα C µ ( αβ ) , (47)Performing the Gaussian integrals over C µν and C µ ( νρ ) in (41) amounts to the replacement: L C µν + L C µ ( νρ ) → (cid:0) T µν T νµ − T (cid:1) + 3 m (cid:20) g µ ( αν ) g α ( µν ) − g µ g µ + g µ g µ (cid:21) , (48)where g µ = η αβ g µ ( αβ ) and g β = η µα g µ ( αβ ) . The Lagrangian (41), using (48), is invariant underthe gauge transformations:( − δy [ µν ]( αβ ) = m η µα (cid:2) ω [ νβ ] + 3 ω ( νβ ) (cid:3) − m η να (cid:2) ω [ µβ ] + 3 ω ( µβ ) (cid:3) + ( α ↔ β )+ η µα ∂ λ (cid:2) ω ν ( λβ ) − ω λ ( νβ ) (cid:3) − η να ∂ λ (cid:2) ω µ ( λβ ) − ω λ ( µβ ) (cid:3) + ( α ↔ β )+ ∂ α (cid:2) ω µ ( νβ ) − ω ν ( µβ ) (cid:3) + ∂ β (cid:2) ω µ ( να ) − ω ν ( µα ) (cid:3) , (49)( − δy α [ µν ] = m [ ω ν ( µα ) − ω µ ( να ) ] + 3 m η µα ω ν − η να ω µ )+ 2 ∂ α ω [ µν ] + 2 (cid:0) η µα ∂ λ ω [ νλ ] − η να ∂ λ ω [ µλ ] (cid:1) (50) δφ µ = m ǫ µ + ω µ / δU νβ = ω νβ + ∂ ν ǫ β , (51) δS ( αβ ) = − mǫ ( αβ ) − ω ( αβ ) ; δe µ ( αβ ) = ∂ µ ǫ ( αβ ) + ω µ ( αβ ) , (52)where the gauge parameters must satisfy (38) and (39), or (40).Although, g µ ( αβ ) is not fully gauge invariant, the reader can check from (50), (51) and (52)that δg µ ( αβ ) only depends upon derivatives of ω µν , the gauge parameters ǫ ( αβ ) and ω µ ( αβ ) dropout. Analogously, based on (49) and (51) we are led to replace y [ µν ]( αβ ) by a new field whosegauge transformations only depend upon derivatives of ω µ ( αβ ) without any dependence on ǫ µ or ω [ µν ] , namely˜ y [ µν ]( αβ ) = y [ µν ]( αβ ) + m (cid:2) η µα (cid:0) U [ νβ ] + 3 U ( νβ ) (cid:1) − η να (cid:0) U [ µβ ] + 3 U ( µβ ) (cid:1) + ( α ↔ β ) (cid:3) (53) U µν = U µν − ∂ µ φ ν m (54)Similar to the spin-1 and spin-2 cases, see (4) and (15), in order to have a more symmetricaction with respect to the ± y [ µν ]( αβ ) ≡ m ǫ ρµν f ρ ( αβ ) , (55) y α [ µν ] ≡ m ǫ ρµν K ρα , (56)The non symmetric rank-2 tensor K µν plays a similar role as U µν . Now we can write downthe parity doublet Lagrangian (41), using (38), in a quite symmetric form, L (3) m = − m f ρ ( αβ ) E ρµ G µ ( αβ ) − m f ρ E ρµ G µ + 2 m h f µ ( αβ ) f α ( µβ ) − f µ + 4 f µ i + 3 m h G µ ( αβ ) G α ( µβ ) − G µ + 4 G µ i + 8 m (cid:0) K [ νβ ] + 3 K ( νβ ) (cid:1) (cid:16) ∂ ν f β + ∂ ν f β − ∂ β f ν − ∂ α f ν ( αβ ) (cid:17) (57)+ m (cid:0) U [ νβ ] + 3 U ( νβ ) (cid:1) (cid:16) ∂ ν G β + ∂ ν G β − ∂ β G ν − ∂ α G ν ( αβ ) (cid:17) − m U − m K + 2 m U ǫ µνρ ∂ µ K νρ + Kǫ µνρ ∂ µ U νρ ] , where we have made another invertible field redefinition G µ ( αβ ) ≡ g µ ( αβ ) − η αβ g µ / m G and m f terms acquire the same form, see (48). No similar field redefinition wasnecessary in the previous spin-1 and spin-2 cases where the mass square terms have naturallyappeared in a symmetric form.Now we can easily decouple the +3 and − L (3) m = L (3) SD ( m, e + , λ + ) + L (3) SD ( − m, e − , λ − ) , (58)where L (3) SD ( m, e, λ ) = − m e ρ ( αβ ) E ρµ e µ ( αβ ) − m e ρ E ρµ e µ + m (cid:2) e µ ( αβ ) e α ( µβ ) − e µ + 4 e µ (cid:3) + m λ + 2 m λ ǫ µνρ ∂ µ λ νρ + 2 m (cid:0) λ [ νβ ] + 3 λ ( νβ ) (cid:1) (cid:2) ∂ ν e β + ∂ ν e β − ∂ β e ν − ∂ α e ν ( αβ ) (cid:3) , (59)with λ = η µν λ µν and e ± µ ( νρ ) = 4 f µ ( νρ ) ± G µ ( νρ ) √ λ ± µν = 3 U µν ± K µν √ , (60)Inspired by the first order spin-3 self-dual model of [12], we have further simplified (59) via e µ ( νρ ) → e µ ( νρ ) + 2 m ∂ µ λ νρ − (cid:0) ǫ βµν λ βρ + ǫ βµρ λ βν (cid:1) , (61)with λ µν = λ µν − η µν λ/
3. Consequently, the derivative couplings λ∂e are replaced by nonderivative ones leading to our main result, a new spin-3 self dual model:10 (3) SD ( m, e, A, φ ) = − m e ρ ( αβ ) E ρµ e µ ( αβ ) − m e ρ E ρµ e µ + m (cid:2) e µ ( αβ ) e α ( µβ ) − e µ + 4 e µ (cid:3) − m ( e µ + 5 e µ ) A µ + 3 m A µ + m A µ E µν A ν − m φ ∂ µ A µ − m φ , (62)where A ± µ ≡ − ǫ µνρ λ νρ ± ; φ ± ≡ λ ± (63)The doublet Lagrangian can be written as L (3) m = L (3) SD ( m, e + , A + , φ + ) + L (3) SD ( − m, e − , A − , φ − ) . (64)Taking into account the several field redefinitions that we have carried out so far, the compositefields in (64) are given by e ± µ ( αβ ) = 4 f µ ( αβ ) ± G µ ( αβ ) + 2 η αβ A ± µ − η µβ A ± α − η µα A ± β , (65) f µ ( αβ ) = − m ǫ ρνµ y [ µν ]( αβ ) + 2 m ∂ µ K αβ − (cid:2) ǫ νµα ( U [ νβ ] + 3 U ( νβ ) ) + ( α ↔ β ) (cid:3) , (66) G µ ( αβ ) = e (0) µ ( αβ ) + ∂ µ S ( αβ ) m −
43 ( ǫ ρµα K ρβ + ǫ ρµβ K ρα )+ 49 ( η µα ǫ βλρ + η µβ ǫ αλρ − η αβ ǫ µλρ ) K λρ + 2 m ∂ µ U ( αβ ) (67)where U µν = U µν − η µν U / K µν = K µν − η µν K/ A ± µ and φ ± .Remarkably, they only depend on one vector parameter Λ µ . In the +3 helicity case we have: δφ = − m ∂ µ Λ µ ; δA µ = 32 Λ µ + 12 m ǫ µαβ ∂ α Λ β , (68)Λ µ ≡ ω µ − ǫ µαβ ω [ αβ ] . (69)From (65)-(67) and the gauge transformations (50)-(52) we have δe µ ( αβ ) = η αβ Λ µ − ( η αµ Λ β + η βµ Λ α ) / m (cid:2) ǫ νµα ( ∂ ν Λ β + ∂ β Λ ν ) + ( α ↔ β ) (cid:3) + ∂ µ m (cid:20) η αβ ∂ · Λ − ∂ α Λ β − ∂ β Λ α (cid:21) . (70)which imply In formulae (68)-(72) we suppress the upper index +. For the general case ± ± µ = ± ω µ − ǫ µαβ ω [ αβ ] and m → ± m in δA ± µ and δφ ± ( η αβ e µ ( αβ ) ) = δ e µ = 2 Λ µ , (71) δ ( η αβ e α ( βµ ) ) = δ e µ = − Λ µ − m (cid:18) (cid:3) Λ µ + 13 ∂ µ ∂ · Λ (cid:19) (72)After extensive use of ǫ -identities we have checked that (62) is invariant under the Weyltransformations (68) and (70). For the opposite helicity we replace ( e µ ( αβ ) , A µ , φ, Λ µ ) → ( e − µ ( αβ ) , A − µ , φ − , Λ − µ ). This shows that the original doublet theory (58) is invariant under thegauge transformations associated with the 6 gauge parameters ( ω [ µν ] , Ω [ µν ] ) correspondingto linear combinations of (Λ µ , Λ − µ ). The gauge symmetries associated with the remaining 19independent gauge parameters ( ǫ µ , ǫ ( α,β ) , ω ( αβ ) , Ω ( αβ ) ) are automatically implemented throughthe gauge invariant composite fields (65), (66) and (67). As expected, all symmetries (49)-(52)hold true in the massive doublet theory (58).It is remarkable that the symmetries associated with ( ω [ µν ] , Ω [ µν ] ) have become a non trivialdynamical symmetry involving second order time derivatives. Notice, in particular, that (70)depends upon the symmetric combination ∂ ( µ Λ ν ) while in δA µ only ∂ [ µ Λ ν ] appears, so there isno way of combining e µ ( αβ ) with A µ and its derivatives into a Weyl invariant tensor.Regarding the particle content of (62), how can we be sure that it correctly describeshelicity ± e ± µ = 0 , (73)uniquely determine, see (69), (71) and footnote 3, the six parameters ( ω [ µν ] , Ω [ µν ] ) thus, satis-fying the “completeness” criterium of [20]. Therefore (73) can be fixed at action level withoutloosing relevant field equations. After breaking those symmetries, from (51) we see that φ µ = 0uniquely fix ǫ µ . Analogously, S ( αβ ) = 0 = U ( αβ ) = K ( αβ ) uniquely fix ǫ ( αβ ) , ω ( αβ ) and K ( αβ ) .All those gauges amount to replace the composite field e µ ( αβ ) by an elementary traceless field e µ ( αβ ) ( η αβ e µ ( αβ ) = 0) in (62) so reproducing the model [12].Alternatively, instead of (73), we might have fixed a ± µ ≡ A ± µ ∓ ǫ µαβ ∂ α e β ± / (4 m ) = 0 and getrid of the auxiliary fields A µ and φ (which decouples) while keeping the traces e ± µ = 0. Noticethat δa ± µ = 3 Λ ± µ / a ± µ = 0 uniquely fix the three parameters Λ ± µ , soit can be fixed at action level without problems. The price we pay for eliminating A ± µ is thepresence of third order terms in derivatives of e ± µ . Here we have shown how to obtain first order spin-s self dual models (parity singlets) in D = 2 + 1 via Kaluza-Klein dimensional reduction of the Vasiliev’s [13] first order action for Recall that ω µ = − ǫ µαβ Ω αβ . D = 3 + 1. We have explicitly worked out the cases s = 1 , , s = 1 , s = 3 weobtain a new self dual model (62) invariant under the non trivial local Weyl transformations(68) and (70). After fixing the gauge e µ = 0 we recover the model of [12].There are several questions about the Weyl symmetry that we are currently addressing.Namely, its importance for the definition of the self dual model itself and for the introductionof cubic and higher order self interacting vertices for spin-3 particles in D = 2 + 1.Even the origin of the symmetry is not yet clear. Usually, the KK dimensional reduc-tion leads to Stueckelberg fields that can eliminated via an unitary gauge such that no lo-cal symmetry is left over, since the number of independent gauge parameters equals thenumber of new fields. For instance, in the spin-1 case e D is gauged away by the U (1)symmetry while in the spin-2 case the 10 fields ( e µD , e Dµ , e DD , y [ µν ] D ) can be eliminatedby the 10 gauge parameters ( ǫ µ , ω µ , ǫ, ω µ ). However, in the spin-3 case the 25 indepen-dent parameters ( ξ ( µD ) , ξ ( αβ ) , ω µ ( νD ) , ω µ ( αβ ) ) are not enough to eliminate the 33 new fields( e D ( Dµ ) , e D ( αβ ) , e µ ( νD ) , y [ µν ]( α D ) , e µ , ǫ ργµ y [ ργ ]( αα ) ), there are 8 exceeding fields. They correspondto the necessary auxiliary fields ( φ ± , A ± µ ) which are linear combinations of the antisymmetriccomponents of the tensors e µ ( νD ) and y [ µν ]( α D ) . Therefore, all remaining new fields must beeliminated, including the vectors ( e µ , ǫ ργµ y [ ργ ]( αα ) ), or equivalently e ± µ . However, we have notbeen able to foresee that the Weyl symmetry associated with the elimination of e ± µ would bedynamically realized. In the s = 1 and s = 2 cases we are able to define composite fieldsinvariant under all gauge transformations. This is apparently not possible in the spin-3 case.A similar situation occurs in the KK dimensional reduction of massless limite of NMG from D = 2 + 1 to D = 1 + 1, see [21].We stress that differently from the s = 1 and s = 2 cases, for s = 3 it is not clear whichnew fields are pure gauge and could be eliminated right from the start in order to avoid theseveral field redefinitions that we have done. In particular, some of the new fields correspondto ( A ± µ , φ ± ) and must be kept in the off-shell formulation of the theory.Hopefully the analysis of the next case ( s = 4) will clarify the issue of dynamical versustrivial Stueckelberg symmetries. In the general integer spin-s case the action of [13] in D = 3+1depends on two fields S [ y [ MN ]( A ··· A s − ) , e M ( A ··· A s − ) ]. After the dimensional reduction, the fieldredefinitions (4),(15) and (55) will be replaced by y [ µ,ν ]( α ··· α s − ) = m ǫ ρµν f ρ ( α ··· α s − ) . So thehelicity eigenstates will be linear combinations of f ρ ( α ··· α s − ) and e ρ ( α ··· α s − ) . This is now inprogress. We point out that s = 3 is still special. The next case s = 4 is more promisingregarding an arbitrary integer spin generalization.Finally, we hope that the new model (62) will allow us to run the successive Noether gaugeembeddings in order to derive the whole sequence of self dual models SD (3) j ( j = 1 , , · · · , SD (3)4 . The issue of soldering of opposite helicities and the connectionwith D = 3 + 1 massive models as well as the generalization of (62) and the Weyl symmetry Recall that ω µ ( νD ) = ω µν is traceless, see (38), and the 10 constraints (33) reduce the 18 variables ω µ ( αβ ) to 8 independent parameters.
13o curved space backgrounds is also under investigation.
This work is partially supported by CNPq (grant 306380/2017-0).
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