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31
UDC 530.1; 539.1
On cubic AdS interactions of mixed-symmetry higher spins
L. Lopez
Scuola Normale Superiore and INFN Piazza dei Cavalieri 7, 56126 Pisa, Italy
E-mail: luca.lopez@sns.it
The problem of finding consistent cubic AdS interactions of massless mixed-symmetry higher-spin fields is recast into a system of partial differential equations that can be solved for given spins of the particles entering the cubic vertices. For simplicity, we consider fields with two families of indices for which some examples of interactions are explicitly discussed.
Keywords: Gauge Symmetry, Field Theories in Higher Dimensions, Space-Time Symmetries.
1 Introduction
Understanding the relation between String Theory (ST) and higher-spin (HS) field theories is currently regarded as a key step in order to gain new insights into both.1 One of the main obstacles in pursuing this goal is our limited knowledge of the behavior of mixed-symmetry fields2 in AdS background. These types of fields naturally appear in ST and are believed to underlie most of its extraordinary features. Although a consistent theory of interacting massless symmetric HS fields has been known for a long time [4], when it comes to mixed-symmetry fields the situation is completely different. So much so that an understanding of the free theory has been attained only very recently [5-14], while the interactions problem have been little investigated so far [10,15-18].
The key differences between flat and AdS backgrounds show up as soon as massless mixed-symmetry fields are considered. Indeed, while unitary gauge fields in flat space possess one gauge parameter for each rectangular block present in the corresponding Young tableaux (YT), their AdS counterparts only possess the gauge parameter whose diagram lacks one box in the upper rectangular block [19]. As result, in general an AdS massless field propagates more degrees of freedom than the corresponding flat-space one. The exact pattern of flat-space gauge fields associated to a single AdS one was first conjectured by Brink, Metsaev and Vasiliev (BMV) in [20] and was then proved in [12,21]
These notes are aimed at setting the stage for a systematic study of the cubic interactions of massless mixed-symmetry fields in AdS. For simplicity, we restrict our attention to fields with two families of indices which describe the most general representations of the AdS isometry group SO(d — 1, 2) up to d = 6. Moreover, we focus on those portions of the couplings that
do not contain divergences, traces or auxiliary fields (here denoted by TT parts for brevity). The reason is that, besides being simpler to determine, they suffice to encode the on-shell content of the theory. Relying on the Noether procedure in the ambient-space formalism, we show that this problem is equivalent to finding polynomial solutions of a set of linear partial differential equations (PDEs), along the lines of [22-24]. Although rather complicated, the latter can be explicitly solved for given spins of the particles entering the cubic vertices and we describe in some detail the interactions between the simplest hook gauge field and two spin-1 gauge fields as an example. The resulting AdS interactions present a highest-derivative part dressed by a chain of lower-derivative vertices size by corresponding powers of the cosmological constant. This structure is very similar to the one found by Fradkin and Vasiliev (FV) [25] for the gravitational interactions of totally-symmetric HS fields in AdS. In the FV system, the highest-derivative part has exactly the same form as the flat-space vertices which can be singled out considering a proper (non-singular) flat limit [26]. However, as far as mixed-symmetry fields are concerned, the situation is in principle subtler due to the non-conservation of degrees of freedom. A correct analysis requires the introduction of appropriate Stückelberg fields according to the BMV pattern in order to guarantee that the same amount of gauge symmetry be present on both sides. We postpone this issue for a future work.
The organization of this paper is the following: Section 2 is devoted to the formulation of the free theory of massless mixed-symmetry HS fields in the ambientspace formalism. In Section 3 we show how to recast the consistency condition for the TT parts of the interactions into a system of PDEs. The entire discussion focuses on the two-family case. Finally, we solve the system describing the interactions between the simplest
1See [1,2] for recent reviews on HS fields.
2 See [3] and references therein for mixed-symmetry fields in flat space.
hook gauge field and two spin-1 gauge fields.
2 Ambient-space formalism for HS fields
The ambient-space formalism [27, 28] regards the
d
hypersurface X2 = —L2 embedded into a (d + 1)-dimensional flat-space. In our convention the ambient metric is n = ( —, +, ..., +)> so that AdS is actually Euclidean.
Focusing on the region X2 < 0, there exists an isomorphism between multi-symmetric tensor fields in AdS, ;...; Pl---PsK > and corresponding ambient-
space fields, $Ml---Mn;---; P1 — PsK > that satisfy the homogeneity and tangentiality (HT) conditions
Homogeneity : (X • dx — A) $(X, U) = 0 , (1)
Tangentiality : X • dum $(X, U) = 0 . (2)
Here, m = 1,..., k runs over the number of independent families of symmetrized indices, while the deA
quadratic Casimir, i.e. the AdS mass. As it will prove very convenient in describing the interactions, we have introduced the auxiliary-variable notation for the ambient-space fields
$(X U) = —j--------7 $M1-MS1 ;••• ; Pi-Psk (X)
s1 • •••j
x (U 1)Ml ••• (U 1)Msi ••• (UK)Pl ••• (UK)Psk, (3)
where s1 > • • • > sK . Since no definite symmetry properties between different families of indices are enforced, these fields define reducible tensors.
In the following we restrict the attention to mass-less fields, namely to short representations associated with gauge symmetries3 [19]
¿(0) $(X,U) = Um• dx Em(X,U), (4)
X • dum En(X,U) = 0 . (5)
However, as we have anticipated not all these reducible gauge symmetries can be preserved in AdS. Indeed, the compatibility with the HT conditions (1,2) restricts
A
ponents of Em. More precisely, it translates into the following set of equations
(X • dx — A — 1) Em (X, U) = 0, (6)
(A+1) Em(X,U) — U1 • dum E1 (X, U) = 0. (7)
Henceforth we focus on the the two-family case, for which the system (7) becomes
Let us analyze separately two cases:
1) A = s1 — 2. In this case, one can substitute eq. (8) into eq. (9), ending up with
(A + 2 — s1) (A + 2 — s2) E2 = U1 dU2 U2 dU1 E2. (10)
At this point, one can decompose the reducible gauge E2
E2 = E (U2 • du 1 )S2-1-n ^¡s1+,-1-nM , (n)
n=0
U1 • du2 E2^--^} =0 , (12)
where the labels {s1, s2} identify the structure of the corresponding YT
S1
11 I'M <13>
S2
Plugging the expansion (11) into eq. (10), all the irreducible components, except the ones with n satisfying
(A + 2 — s1) (A + 2 — s2) = (s1 — n) (s2 — n), (14)
A
n
quence, no gauge symmetry is allowed and the corresponding fields are massive.4 On the other hand, assuming A € Z, the solutions to eq. (14) together with the corresponding gauge parameters are given by
• n = A + 2
E2 = (U2 • du 1 )S2-~A E2S1+S2_A-3,A+2}, (15)
where —2 < A < s2 — 3.
• n = s1 + s2 — A — 2
E2 = (U2 • du 1 )A+1-S1 JE{2A+1,S1+S2-A-2} , (16)
s1 — 1 < A < s1 + s2 — 2 E1
E2
2) A = s1 — 2. In this case, eqs. (8,9) imply
E1 = JE{1S1_1jS2}, E2 =0. (17)
s1 = s2 = s
E!^ s} is no longer admissible and therefore no gauge symmetry survives.
A
ing for gauge symmetries is 2 s2 + 1, for s1 = s2 and
(A + 2 — s1) E1 = U2 • du 1 E2, (A + 2 — s2) E2 = U1 • du2 E1.
(8)
(9)
3Requiring constraints on the gauge parameter that are less stringent than (5) other symmetries would be allowed [12,24]. These are associated to (non-unitary) partially-massless fields [29-32] that we will not consider in this paper.
4 Here by massive fields we mean the ones that do not possess any gauge symmetry. Partially-massless symmetries that we have mentioned above can only appear for integer values of A.
2 s, for s1 = s2 = s • For each of these values, only one irreducible gauge parameter, associated to one of the irreducible components of $, can be preserved.5 As a consequence, if one fixes the degree of homogeneity A, the corresponding reducible field will contain only one massless (unitary or non-unitary) irreducible
s2
be massive or partially-massless. However, as already mentioned, due to the tangentiality constraint (5) these partially-massless fields do not have the corresponding gauge symmetries and we will not consider them in the following. In order that the massless component be the one described by the YT {s1, s2} (13), one can choose either A = s1 — 2orA = s2 — 3. In the former case, the corresponding gauge parameters are given by (17), while in the latter case eqs. (8,15) lead to
A = s2 — 3, E1 E2
U2 • du 1 E2
{ s 1 ,s 2 1}
p 2
E{ s 1 ,s 2 1} •
¿(0) Ф1(Х,и) = U1 • dx JÊ{1S1-1,S2}(X,U) •
(19)
The corresponding homogeneity conditions (1) and (6), written in operatorial form, read
(X • dx — U1 (X • dx — U1 •
du 1 +2^1(X,U ) =
dU 1 ) E{s1-1,S2}(X,U)
0,
0.
S(3) =y dd+1 Xj( V—X2 — Lj C(Y,Z) x Фl(Xl, U1) Ф2(X2, U2) Фз(Xs, Us)
where C is an arbitrary function of nine variables
JU3 , Z211
Y1 = dU\ • ; Y1 = dU2 • ; Y2 = • дХз
Y3 = dU3 • , Z1 = dU2 • d
Z212 = du2 ; Z3U = ^21
For instance, a vertex of the form
<% du\
Z3 = dU,2 du
$M1M2! N dMl dN dp
corresponds to the choice
c = y1 y2 y2 z11 .
(22)
(23)
(24)
(18)
A = s — 2
tary AdS massless fields [12]. On the other hand, as
s = s2 = s
A
A=s—3
3 Two-familiy HS cubic interactions
In this section we describe how to translate the consistency condition for the TT parts of the interactions into a system of PDEs. For simplicity, we focus on couplings involving one mixed-symmetry gauge field (denoted by $ 1) and two totally-symmetric gauge fields (denoted by $2 and $3). Moreover, we also assume that $ 1 have s 1 > s2 mid A = s 1 — 2, so that its irreducible component {s 1, s2} is a unitary massless field with gauge transformations given by (17)
The insertion of the delta function in (21) is aimed at removing the ambiguities related to the divergent radial integral. As a consequence, total-derivative terms arising in the gauge variation no longer vanish, but give a contribution that can be cast in the form
V—X2 — LjdXM = — V—X2 — L L Xm . (25)
Here, the auxiliary variable S encodes the derivatives of the delta function according to the rule
— l = (R dR)” ¿(R — L
= ¿(Д — L) — L
(26)
In order to compensate these terms, the cubic vertices are to be amended by additional total-derivative contributions. These terms contain a lower number of derivatives compared to the initial vertices and are weighted by proper powers of L-2, or equivalently, of (5/L. This is the ambient-space counterpart of what happens in the intrinsic formulation: the replacement of ordinary derivatives by covariant ones requires the inclusion of additional lower-(covariant)derivative vertices in the Lagrangian.
Whenever a massless field takes part in the interactions, the corresponding vertices must be compatible with its gauge symmetries. Following the Noether procedure, gauge consistency can be studied order by order in the number of fields, and at the cubic level it translates into the condition
¿(0) S(3) « 0,
(27)
(20)
—1,S2 }
Neglecting divergences, traces, on-shell vanishing terms and auxiliary fields, the most general expression for the TT parts of the cubic vertices reads [23,24]
where « means equivalence modulo the free field equations, divergences and traces. In our notation, the requirement (27) is equivalent to imposing
(21)
[ C (Y,Z) , U1 • ÔX1] [ C (Y,Z )
[ C (Y,Z )
U2 • Us •
dX2
dX3 ]
U1 = 0
U2 = 0
U3 = 0
f (Y,Z ) U1 • ÖU
0,
U1 = 0 '
(28)
5 In general, the number of irreducible gauge parameters associated to the S2 + 1 (or s + 1) components of a reducible field with
si > s2 (or si = s2 = s) is 2 s2 + 1 (or 2 s) . The reason is that, except for the totally-symmetric component (and also for the
rectangular one when si = s2) that possesses only one gauge parameter, all the others have two gauge parameters. However, only
one of the two is associated to unitary AdS massless fields [191.
1
0
where the right-hand side of the first equation appears due to the irreducibility of the gauge parameter. Using the Leibniz rule together with the HT conditions (2,5,20) and the identity (25), one can recast eqs. (28) into a system of three PDEs
1) Y3dZ11 - Y2dZ11 + 2T (Yl2dY12 - 2 Y2dY2 +2 Y3dY3 + Z2l2dZ12 — Z3ldZ21) dY11
+ SZ32ldZ11 dY12] C(Y,Z)
= (Y!dY11 + Z22dz12 + Z321 dZ21 ) f (Y Z) :
2) YlldZ11 + Yl2 dZ21 — Y3dZ
+ 2L (2 YlldY11 — 2 Y3dY3 + Yl2 dY12
— Z2l2d
2 dZ1
— Z3ldz21
dY2
C (Y, Z ) = 0 ,
3) Y2dZ — YlldZ11 — Yl2dZ12
+ 2L (2 Y2dY2 — 2 YlldY11 — dY12
+ Z3ldz2
+ Z2l2dZ1
dY3
C (Y, Z ) = 0 .
C
= ai (Yl1 )2 Yl2 Y2 Y3 + «2 Yl1 Yl2 Y3 Z311 + a3 Y1 Y2 Y2 Z211 + a4 (Y/)2 Y3 Z321
+ «5 (Y1 )2 Y2 Z12 + ae (Y1 )2 Y2 Zi
+ a7 Yl2 Zl1 Z3n + a8 Yl Z2n Zf1
+ ag Yi1 Z12 Z11,
(30)
associated to the nine possible Lorentz-invariant TT couplings one can start with. Similarly, one can expand f as
f = bi (Yi)2 Y2 Y3 + 62 Y1 Y2 Zi1 + 63 (Yi)2 Zi + 64 Y11Y3 Z11 + 65 Z11 Z11. (31)
Plugging the ansatze (30,31) into eqs. (29) and solving the resulting linear system, one can express all the unknown coefficients a mid 6j in terms of three independent ones 01, a2 mid a3 . Plugging these solutions back into (30), one ends up with C = 01 C1 + a2 C2 + a3 C3 , where
Ci
C2
(Yl1)2 Yl2 Y2 Y3 + 2 L (Yl1)2 Yl2 Zi
= (Y1 )2 Yi2 Y2 Y3 — 2L (2 (Y1 )2 Y3 Z:
+ 3 Yi1 Y2 Y2 Z11 + Y1 Y2 Y3 Z11)
2
21
3
Y 2 7 il 7 ll + 2 Y l 7 ll Z2l Yl Z2 Z3 +2 Yl Z2 Z3
C3
(29)
= (Yi1)2 Y3 Z321 — (Yi1)2 Y2Z12
+ Yi1 Yi2 (Y, Z11 — Y3 Z11)
Finding the general solutions to such a system is a nontrivial task and we leave this issue for a forthcoming paper [33]. However, for given spins of the fields, all Cf
the variables (22) with arbitrary coefficients. In this way, eqs. (29) reduce to a system of linear equations for the coefficients, which can be easily solved, for instance, with Mathematica. Besides the interactions of the massless irreducible component {s1, s2} , these solutions contain also the vertices associated to the remaining s2 ones {s1 + s2 —n, n}, n = 0,..., s2 — 1. In order to single the former types of interactions, one has to further project the solutions onto the corresponding YT {s1,s2}•
Example: {2,1} — 1 — 1
As an example, let us consider the interactions between the simplest hook gauge field and two spin-1 gauge fields, i.e. the triple {2,1} —1 — 1. In this case,
C
efficients
+
3 ^ y l 2 L Yl
r/12 zll 7ll ry2l Z2 Z3 — Z2 Z3
(32)
Finally, acting on the solutions (32) with the projector onto the hook YT {2,1}
Y{2,i} = 1 — 3 U2 • du 1U1 • du2
(33)
3 {2, 1}—1—1
C1
C2 C3
C{2-1}-1-1 = (Yll)2 (Y2 Z2l2 — Y3 Z32i)
+ YiYi2 (Y3 Z311 — Y2 Z211)
+
3 A
2 L
Ô vW ylW2l ryl2 r7ll\ L Yl (Z2 Z3 — Z2 Z3 ) ■
(34)
Hence in the end one is left with only one consistent coupling involving a three-derivative part which is dressed by a one-derivative AdS tail.
Acknowledgement
I am very grateful to E. Joung and M. Taronna for collaborations on the topics presented in this note. I would like to thank also A. Sagnotti for reading the manuscript and for useful suggestions. Finally, I thank E.D. Skvortsov and Y.M. Zinoviev for discussions. The present research was supported in part by Scuola Normale Superiore, by INFN (I.S. TV12) and by the MIUR-PRIN contract 2009-KHZKRX.
References
[1] Bekaert X. and Boulanger N. and Sundell P. 2010 [arXiv: 1007.0435]
[2] Sagnotti A. 2011 [arXiv:1112.4285]
[3] Campoleoni A. and Francia D. and Mourad J. and Sagnotti A. 2009 Nucl.Phys. B815 289-367; 2010 Nucl.Phys. B828 405-514
[4] Vasiliev M. A. 1989 Annals Phys. 190 59-106; 2003 Phys.Lett. B567 139-151
[5] Zinoviev Y. M. 2002 [hep-th/0211233]
[6] de Medeiros P. 2004 Class.Quant.Grav. 21 2571-2593
[7] Burdik C. and Reshetnyak A. 2012 J.Phys.Conf.Ser. 343 012102
[8] Alkalaev K. B. and Shaynkman O. V. and Vasiliev M. A. 2005 JHEP 0508 069
[9] Zinoviev, Y. M. 2009 Nucl.Phys. B821 21-47; 2010 Nucl.Phys. B826 490-510
[10] Alkalaev K. B. 2011 JHEP 1103 031
[11] Skvortsov E. D. 2009 J.Phys. A42 385401
[12] Alkalaev K. B. and Grigoriev M. 2010 Nucl.Phys. B835 197-220; 2011 Nucl.Phys. B853 663-687
[13] Grigoriev M. 2012 [arXiv:1204.1793]
[14] Campoleoni A. and Francia D. 2012 [arXiv: 1206.5877]
[15] Zinoviev Y. M. 2011 JHEP 1103 082
[16] Boulanger N. and Skvortsov E. D. and Zinoviev Y. M. 2011 J.Phys. A44 415403
[17] Zinoviev Y. M. 2012 Class.Quant.Grav. 29 015013
[18] Boulanger N. and Skvortsov E. D. 2011 JHEP 1109 063
[19] Metsaev R. R. 1995 Phys.Lett. B354 B354; 1997 Lect.Notes Phys. 524 331-340
[20] Brink L. and Metsaev R. R. and Vasiliev M. A. 2000 Nucl.Phys. B586 183-205
[21] Boulanger N. and lazeolla C. and Sundell P. 2009 JHEP 0907 013; 2009 JHEP 0907 014
[22] Taronna M. 2012 JHEP 1204 029
[23] Joung E. and Taronna M. 2011 Nucl.Phys. B861 145-174
[24] Joung E. and Lopez L. and Taronna M. 2012 JHEP 1207 041; 2012 [arXiv;1207.5520]
[25] Fradkin E. S. and Vasiliev M. A. 1987 Nucl.Phys. B291 141; 1987 Phys.Lett. B189 89-95
[26] Boulanger N. and Leclercq S. and Sundell P. 2008 JHEP 0808 056
[27] Fronsdal C. 1979 Phys.Rev. D20 848-856
[28] Biswas T. and Siegel W. 2002 JHEP 0207 005
[29] Deser S. and Nepomechie R. I. 1984 Annals Phys. 154 396; 1983 Phys.Lett. B132 321
[30] Deser S. and Waldron A. 2001 Phys.Rev.Lett. 87 031601; 2001 Phys.Lett. B508 347-353; 2001 Nucl.Phys. B607 577-604:2001 Phys.Lett. B501 134-139; 2001 Phys.Lett. B513 137-141; 2003 Nucl.Phys. B662 379-392
[31] Skvortsov E. D. and Vasiliev M. A. 2006 Nucl.Phys. B756 117-147
[32] Francia D. and Mourad J. and Sagnotti A. 2008 Nucl.Phys. B804 383-420
[33] Joung E. and Lopez L. and Taronna M. to appear
Received 01.10.2012
Л. Jlonec
КУБИЧНЫЕ ВЗАИМОДЕЙТСВИЯ ПОЛЕЙ ВЫСШИХ СПИНОВ СО СМЕШАННЫМ ТИПОМ СИММЕТРИИ В ПРОСТРАНСТВЕ АДС
Проблема нахождения согласованного кубичного взаимодействия безмассовых полей высших спинов со смешанным типом симметрии в пространстве АдС сводится к системе дифференциальных уравнений в частных производных, которые могут быть решены для частиц с заданными значениями спина, входящих в кубичную вершину. Для простоты рассматриваются поля с двумя семействами индексов, для которых обсуждаются в явном виде некоторые примеры взаимодействия.
Ключевые слова: калибровочная симметрия, теория поля в высших измерениях, пространственно-временные симметрии.
Лопес, Л.
Seuola Normale Superiore and INFN.
Piazza dei Cavalieri 7, 56126 Pisa, Italy.
E-mail: luca.lopez@sns.it
