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Physics Letters B 590 (2004) 95-104

www. elsevier. com/locate/physletb

Massive totally symmetric fields in AdSd

R.R. Metsaev

Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky prospect 53, 119991 Moscow, Russia Received 3 March 2004; received in revised form 17 March 2004; accepted 21 March 2004 Available online 28 April 2004 Editor: P.V. Landshoff

Abstract

Free totally symmetric arbitrary spin massive bosonic and fermionic fields propagating in AdSd are investigated. Using the light cone formulation of relativistic dynamics we study bosonic and fermionic fields on an equal footing. Light-cone gauge actions for such fields are constructed. Interrelation between the lowest eigenvalue of the energy operator and standard mass parameter for arbitrary type of symmetry massive field is derived. © 2004 Elsevier B.V. All rights reserved.

1. Introduction

A study of higher spin theories in AdS space-time has two main motivations (see, e.g., [1,2]): firstly to overcome the well-known barrier of N < 8 in d = 4 supergravity models and, secondly, to investigate if there is a most symmetric phase of superstring theory that leads to the usual string theory as a result of a certain spontaneous breakdown of higher spin symmetries. Another motivation came from conjectured duality of a conformal N = 4 SYM theory and a theory of type IIB superstring in AdS5 x S5 background [3]. Recent discussion of this theme in the context of various limits in AdS superstring may be found in [4]. As is well known, quantization of GS superstring propagating in flat space is straightforward only in the light-cone gauge. It is the light-cone gauge that removes unphysical degrees of freedom explicitly and

E-mail address: metsaev@lpi.ru (R.R. Metsaev).

reduces the action to quadratical form in string coordinates. The light-cone gauge in string theory implies the corresponding light-cone formulation for target space fields. In the case of strings in AdS background this suggests that we should study a light-cone form dynamics of target space fields propagating in AdS space-time. It is expected that AdS massive fields form spectrum of states of AdS strings. Therefore understanding a light-cone description of AdS massive target space fields might be helpful in discussion of various aspects of AdS string dynamics. This is what we are doing in this Letter.

Let us first formulate the main problem we solve in this Letter. Fields propagating in AdSd space are associated with positive-energy unitary lowest weight representations of SO(d — 1,2) group. A positive-energy lowest weight irreducible representation of the SO(d — 1, 2) group denoted as D(E0, h), is defined by E0, the lowest eigenvalue of the energy operator, and by h = (h\,..., hv), v = [^y^L which is the highest weight of the unitary representation of the SO(d — 1)

0370-2693/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.03.057

group. The highest weights hi are integers and half-integers for bosonic and fermionic fields, respectively. In this Letter we investigate the fields whose E0 and h are given by

E0 >h1 + d — 3, h = (h1, 0,...,0), Eo > hi + d — 3,

d is even, d is odd.

The fields in (1.1) are massive bosonic fields while the ones in (1.2) are fermionic fields. The massive fields in (1.1), (1.2) with h1 > 1, are referred to as totally symmetric fields.1 In manifestly Lorentz covariant formulation the bosonic (fermionic) totally symmetric massive representation is described by a set of the tensor (tensor-spinor) fields whose SO(d — 1, 1) space-time tensor indices have the structure of the respective Young tableauxes with one row. Covariant actions for the bosonic totally symmetric massive fields in AdSd space were found in [10].2 Fermionic massive totally symmetry AdSd fields with arbitrary E0 and h have not been described at the field theoretical level so far.3 In this Letter we develop a light-cone gauge formulation for such fields at the action level. Using a new version of the old light-cone gauge formalism in AdS space developed in [7], we describe both the bosonic and fermionic fields on an equal footing. Since, by analogy with flat space, we expect that a quantization of the Green-Schwarz AdS superstring with Ramond-Ramond charge will be straightforward only in the light-cone gauge [14] it seems that from the stringy perspective of AdS/CFT

1 We note that the case h = (1,0,...,0) corresponds to spin one massive field, the case h = (2,0,...,0) is the massive spin two field. The labels hi are the standard Gelfand-Zeitlin labels. They are related with Dynkin labels hD by formula: (hD,hD,...,hD—VhD) = (h1 — h2,h2 — h3,...,hv—l —

hv,hv—1 + hv).

2 Massive self-dual spin fields in AdS'3 were investigated in [8]. Spin two massive fields were studied in [17]. Discussion ofmassive totally symmetric fields in (A)dSd, d ^ 4, may be found in [9].

3 Group theoretical description ofvarious massive representation via oscillator method may be found, e.g., in [11]. Lorentz covariant equations of motion for AdS5 self-dual massive fields with special values of E0 were discussed in [12]. Light cone actions for AdS5 self-dual massive fields with arbitrary values of E0 were found in [13].

correspondence the light-cone approach is the fruitful direction to go.

2. Light-cone gauge action and its global symmetries

In this section we present new version of the old light cone formalism developed in [7]. Let $(x) be a bosonic arbitrary spin field propagating in AdSd space. If we collect spin degrees of freedom in a ket-vector |$> then a light-cone gauge action for $ can be cast into the following 'covariant form' [7]4

□ = 29 + d— + 9? + dj.

An operator A does not depend on space-time coordinates and their derivatives. This operator acts only on spin indices collected in ket-vector |$>. We call the operator A the AdS mass operator.

We turn now to discussion of global so(d — 1,2) symmetries of the light-cone gauge action. The choice of the light-cone gauge spoils the manifest global symmetries, and in order to demonstrate that these global invariances are still present one needs to find the Noether charges which generate them.5 Noether charges (or generators) can be split into kinematical and dynamical generators. For x+ = 0 the kinematical generators are quadratic in the physical field |$>, while the dynamical generators receive corrections in interaction theory. In this Letter we deal with free fields. At a quadratic level both kinematical and dynamical generators have the following standard representation in terms of the physical light cone

4 We use parametrization of AdSd space in which ds2 = (—dx? + dx2 + dxJ_j + dz2)/z2. Light-cone coordinates in ± directions are defined as x± = (xd—1 ± x °)/V2 and x+ is taken to be a light-cone time. We adopt the conventions: 3' = dj = d/dx', dz = d/dz, d± = = d/dx^, z = xd—2 and use indices ', j = l,...,d — 3; I,J = l,...,d — 2. Vectors of so(d — 2) algebra are decomposed as XI = (X',Xz).

5 These charges play a crucial role in formulating interaction vertices in field theory. Application of Noether charges in formulating superstring field theories may be found in [15].

field [7]

G= I dx-dd-2x id+$\G\0).

Representation for the kinematical generators in terms of differential operators G acting on the physical field \$) is given by

P' = di, P + = d+,

D = x+ P - + x-d + + x'd1 +

j+- = x+p- -x-d+,

J+i = x+d' - x'd+,

Jij = x'dj - xjdi + Mij,

K+ = -i(2 x+x~ +xJ xJ)d+ +x+D,

Kl = --(2x+x~ +xJxJ)di

+ x'D + M'JxJ + Mi-x+

while a representation for the dynamical generators takes the form

{QAdS) is eigenvalue of the second order Casimir operator of the so(d — 1,2) algebra for the representation labelled by D(E0, h):

— {QAdS) = E0(E0 + 1 — d)

+ J2 ha(ha — 2a + d — 1), (2.17)

while Bz is z-component of so(d — 2) algebra vector B1 which satisfies the defining equation6

[BI,BJ ] + (M 3)[/ |J ]

/ 1 2 N(N — 1) + 2 \ + i (fiAds) " -m2 - 2 -\mij « 0.

(2.18)

Here we use sign & to write instead of equations Xlfi) = 0 simplified equations X & 0. As was noted the operator B1 transforms in vector representation of the so(d — 2) algebra

29+ 2z2d+

(2.10) (2.11)

l'lx+x-+x^D~ 1 1 1

J-i = x-d' - x'P- + M-i,

K~ = --(2 x+x~ +xf)P~ +x~D + —x'dJM

xi 1 2 z9+L J 9+

where M-i = -M'- and

.;rdJ 1

M~' = Mu '

d+ 2zd+

[Mz',A\

(2.12)

(2.13)

Operators A, B, MIJ are acting only on spin degrees of freedom of wave function |0). MIJ = M'j, Mzl are spin operators of the so(d — 2) algebra

[MIJ,MKL] = SJKMIL + 3 terms, (2.14)

while the operators A and B admit the following representation

A = 1MIJMIJ + d[d~ 2) - <eAds>

+ 2Bz + MziMzi B = Bz + MziMzi.

(2.15)

(2.16)

[b',mjk] = SIJBK - SIKBJ.

(2.19)

Making use of the formulas above-given one can check that the light-cone gauge action (2.1) is invariant with respect to the global symmetries generated by so(d — 1, 2) algebra taken to be in the form &g№) = Glfi). To summarize a procedure of finding light cone description consists of the following steps:

(i) choose form of realization of spin degrees of freedom of the field l$);

(ii) fix appropriate representation for the spin operator MIJ;

(iii) using formula (2.17) evaluate an eigenvalue of the Casimir operator;

(iv) find solution to the defining equations for the operator B1 (2.18).

Now following this procedure we discuss bosonic and fermionic fields in turn.

6 We use the notation (M3)^7! = \M1KMklMlj - (I

J), M2 = MIJ MIJ. Throughout this Letter we use the convention N = d - 2.

3. Bosonic fields

where the coefficients as', bs' depend on E0, s and s'

To discuss field theoretical description of massive AdS field we could use so(d — 1) totally symmetric traceless tensor field &Iv'Is, I = 1', 1,...,d — 2. Instead of this we prefer to decompose such field into traceless totally symmetric tensors of so(d — 2) algebra , I = 1,...,d — 2; s ' = 0, 1,...,s:

0h ■■■i, = j2® $Iv"Is'. (3.1)

As usual to avoid cumbersome tensor expressions we introduce creation and annihilation oscillators aI and aI

[aI,aJ] = SIJ, a110>=0, (3.2)

and make use of ket-vectors |$s > defined by l$s'>=aI1 • • • aIs'$I1~Is' (x)|0>. (3.3)

The |$s,> satisfies the following algebraic constraints

(aa— s' )$s'>=0, aa = aI aI, (3.4)

aI aI |$s,>=0, tracelessness. (3.5)

Eq. (3.4) tells us that |$s > is a polynomial of degree s' in oscillator aI. Tracelessness of the tensor fields $IvIsr is reflected in (3.5). The spin operator MIJ of the so(d — 2) algebra for the above defined fields |$s,> takes then the form

MIJ = a1 aJ -aJa1.

We are going to connect our spin s field with the unitary representation labelled by D(E0, h) (1.1) whose h1 is identified with spin value s:7

h1 = s.

Eigenvalue of the second order Casimir operator for totally symmetric representation under consideration is given then by (see (1.1), (2.17), (3.7))

— <ÔAdS> = Eo(Eo + 1 — d) + s(s + d — 3). (3.8)

An action of the operator B1 on the physical fields |^s/> is found to be

B1 |<M = as'AI— 1 |0s'—1> + bs> a1 |0s'+1>, (3.9)

7 This identification can be proved rigorously by exploiting the

procedure of Section 6 in Ref. [13].

as' = b(E0,s,s' — 1), bs' = b(E0,s,s'), (3.10) and a function b(E0,s,s') is defined to be

b(E0,s, s') = ((s — s')(s + s' + N — 1)(E0 — s' — N)

x (E0 + s' — 1)(2s' + N)—1)1/2.

(3.11)

Operator A^, which appears in definition of the operator BI (3.9) is given by

As' = a1 —

2- I a2 a'

2s' + N — 2 '

2 II a2 = aI aI.

(3.12)

Let us outline procedure of derivation above-mentioned results. Most difficult problem is to find solution to the defining equation (2.18). Fortunately, for the case of totally symmetric fields this equation simplifies due to the following relation for MIJ :

(M3)^ = (-V+(iV-2f-3))M"

(3.13)

This formula can be checked directly by using representation for MIJ given in (3.6). Plugging (3.13) in (2.18) we get the following simplified form of defining equation

[BI,BJ] + (<QAdS> — M2 — 2N + 2)MIJ « 0.

(3.14)

Applying this equation to |^s > and using (3.9) we get a relationship for the coefficients as' and bs'

as'+1bs' = (s — s ')(s + s ' + N — 1)(Eo — s' — N)

x (Eo + s' — 1)(2s' + N)—1.

(3.15)

Now we exploit a requirement the operator BI be Hermitian8

<f = [BIf ||0>,

(3.16)

with respect to scalar product defined by

<f m = J2<*s'№s'>. (3.17)

8 We use anti-Hermitian representation for generators of so(d — 1,2) algebra (2.2). This implies that the spin operator MIJ should be anti-Hermitian, while the operators A and BI should be Hermitian.

This requirement gives the relation as' = b*'_ 1. Making use of this relation in (3.15) we arrive at the final solution given in (3.10). Some helpful formulas to evaluate commutator [BI ,BJ ] are given by

a;,J - (I ^ J) = 0 aI,aJ - (I ^ J) = mij,

aiaJ/- (I ^ J) = -

4. Fermionic fields

2s ' + N Ir

2s' + N - 2

(3.18)

(3.19)

(3.20)

Light cone action for fermionic fields takes the following form

oferm _

Ч.О. =

/^(^l^(n-^)l^- (4.1)

This action is invariant with respect to transformations 8 —> = Gferm|—>, where differential operators Gferm are obtainable from the ones for bosonic fields ((2.3)-(2.12)) by making there the following substitution x~ —y x~ -jTj-. In addition to this in expressions for generators in ((2.3)-(2.12)) we should use the spin operator MIJ suitable for fermionic fields. Defining Eq. (2.18) for the operator BI does not change. Before to discuss concrete form of the spin operators MIJ and BI we should fix a field theoretical realization of spin degrees of freedom collected in — >.

To discuss field theoretical description of massive AdS field we could used totally symmetric traceless and y-transversal tensor-spinor field &I1 "'Isa, I = 1', 1,...,d — 2 which corresponds to irreducible spin s + 1/2 representation of so(d — 1) algebra. Instead of this we prefer to decompose such field into traceless totally symmetric tensor-spinor fields of so(d — 2)

algebra ф11 '

■Lia

I = 1,...,d - 2; s' = 0, 1,...,s:9

^h -ha = ф1

9 Details of such a decomposition may be found in Appendix B. The —I1'"Is'a are obtainable from 2[d/2] Dirac tensor-spinor fields of so(d — 1,1) algebra: \jj = jK^K^VODirac- We use 2d/2] x 2d/2]-Dirac y-matrices: {Ya,Yb} = 2nab, nab = (—1, +1,...,+1), a,b = 0,1,...,d — 1. In light cone frame we use a decomposition ya = y + , y ~, y!, where y ^ = ± y°)/V2.

I = 1,...,d — 2.

As before to avoid cumbersome expressions we exploit creation and annihilation oscillators aI and aI (3.2) and make use of ket-vectors s'> defined by

\fs,>=ah ■■■aIs'ф11 -Is'a(x)|0>.

Here and below spinor indices are implicit. The | —s satisfies the following algebraic constraints

(aa- s ' )Ws'>=0,

Y;a1 \fs'> = 0,

a 1 \fs,> = 0,

Y —transversality, tracelessness.

Eq. (4.4) tells us that \fs') is a polynomial of degree s' in oscillator a1. Tracelessness of the tensor-spinor field фIl 'Is'a is reflected in (4.6). Realization of spin operator MIJ on the space of the ket-vectors \fs') is given by

MIJ = M" +

k7/4(KV-KV).

MlJ = a'aJ - aJa',

We are going to connect our massive tensor-spinor field with unitary representation labelled by D(E0, h) (1.1) whose h1 is related with integer s:

h=s + -.

Eigenvalue of the second order Casimir operator for totally symmetric fermionic representation is given then by (see (1.2), (2.17), (4.8))

-<ÔAdS> = E0(E0 - 1 - N) + s(s + N) N(N + 1)

An action of the operator B1 on the physical fermionic fields \ fs<) is found to be

B1 \fs'>=qs' yI'\fs'>+as' Äs'-1\fs'-1)

+ b s' a1 \fs'+1>

(4.10)

As before the coefficients qs', as', bs' turn out to be functions of E0, s, s' and are given by

2 s + N 2(2 У + N)

as' = f(E0,s,s' - 1), bs' = f(E0,s,s' ),

(4.11)

(4.12)

(4.13)

•V a

where we use the notation

f(Eo,s,s' )

= l(s - s')(s + s' + N)(E0 - s' - N -

X^Eo+J'-^CJ' + AO"1^ • (4.14)

Operators A!s, and yj, which enter definition of basic operator B1 (4.10) are given by

Y!- = y! -

2~ ! a2a1

(ya)y!

2s ' + N 2(ya)â ! 2s' + N - 2 '

ya = y!a!.

(4.15)

Now let us outline procedure derivation of these results. We start with general representation for B1 given in (4.10) and the problem is to find coefficients qs', asi, bsr which satisfy the defining Eq. (2.18). To this end we evaluate expression for (M3)[I J]

3\[!\J ]

!J I ' J

s' [s' + N-- +

2N(N - 3) + 5

N(N - 3) + 3

2s'+ N - 1 -j, -3-(ya)SIJ,

S!J = Y!aJ - YJa!,

(4.16)

where a sign & indicates that (4.16) is valid by module of terms which are equal to zero by applying to l^s'), i.e., to derive (4.16) we use constraints (4.4)-(4.6). After this using Eqs. (2.18), (4.9), (4.16) we find solution for qs' and product as'+1bs'. Exploiting then the hermicity condition for the operator B1 (3.16), (3.17) we arrive at the solution for qs', as', bs' given above. Some helpful formulas to evaluate commutator [B1 ,BJ ] are given by

Aj,Aj,-1 - (! ^ J) = 0,

y, yJA = -

MjJ + 2y

2s' + N - 2

2s' + N - 1 -jr

+ 4-::--(ya)S ,

(2s' + N-2)2 7 J

(4.17)

(4.18)

Al,aJ - (! ^ J) = MjJ -

(ya)S' 2s' + N '

a1AJ,- (! ^ J)=-

2s'+ N+ 2 2 s' + N

2s ' + N 2s' + N

(4.19)

(4.20)

5. Interrelation between lowest energy value E0 and mass parameter m

In previous sections we have expressed our results in terms of lowest eigenvalue of energy operator E0. Because sometimes formulation in terms of the standard mass parameter m is preferable we would like to derive interrelation between E0 and m. Before to going into details let of first present our results.

Given massive fields with massive parameter m corresponding to unitary representation labelled by D(E0, h) we find the following relationship between E0 and m (for even AdS space-time dimension d; v = (d — 2)12)

■ + Am2+[hk-k +

for bosonic fields,

E0 = m + hk - k - 2 + d,

for fermionic fields, (5.2)

where a number k is defined from the relation

h1 = ... = hk> hk+1 > hk+2 > hv > 0. (5.3)

We remind that for bosonic fields the labels ha are integers while for fermionic fields the ha are half-integers. We note that relations (5.1), (5.2) are valid also for those massive fields in odd-dimensional AdSd whose h(d—1)/2 = 0.

Now let us outline procedure of derivation these results. Let <^=0 be massless field in AdSd. As was demonstrated in [5,6] the massless fields associated with unitary representation labelled by D(Em=0, h) should satisfy the equation of motion

D2 - Em=0 (Em=0 + 1 - d) + J2 ho

where V2 is a covariant D'Alembertian operator in satisfies an equation

AdSd and Em=0 is given by Em=0 = hk - k - 2 + d.

Eq. (5.4) reflects the well-known fact that equations of motion for AdS massless field involve mass-like term which is expressible in terms of Em=0. As is well known to discuss gauge invariant description of massive fields one introduces the set offields including some fields ^m1" which we shall refer to as leading field plus Goldstone fields (sometimes referred to as Stueckelberg fields). By definition, the structure of Lorentz indices of the leading massive field ^m1" is the same as the one for massless field <m=0. If we impose on the leading field ^m1" and the Goldstone fields an appropriate covariant Lorentz gauge and tracelessnes conditions then for the leading field one gets the equation

(D2 — m2 — E0m=0(E0m=0 + 1 — d) + ¿haUm1.

One other hand an analysis of Refs. [5,6] implies that the leading field satisfies the following equations of motion

(v2 — E0(E0 + 1 — d) + ¿ha^m1... = 0. (5.7)

Comparison of Eqs. (5.6) and (5.7) gives a relationship between E0 and m

m2 = Eq(Eq + 1 — d) — E0m=0 (E0m=0 + 1 — d). (5.8)

Solution to this equation corresponding to positive values of E0 is given in (5.1).

The same arguments can be applied to the fermi-onic fields. In this case equation for massless field in AdSd is given by (see the second Ref. in [5])

Ya^DßL + E™ + = 0, (5.9)

where DfL is a covariant derivative with respect to local Lorentz rotation, e!a is an inverse of the vielbein elt\ and E0m=0 is given in (5.5). Because on the one hand the leading massive tensor-spinor field ^m^1". taken to be in appropriate covariant Lorentz gauge

yae>:DßL +m + E™=° + W1" = 0,

1 — d —)fm

(5.10)

and on the other hand this equation should be repre-sentable as (see [5])

yaeZDßL + E0 + ^-)tZl-=^ (5.11)

we find the relation for m

Z7 77m =0

m = E0 - E0 ,

(5.12)

which together with (5.5) leads to (5.2).

In the AdS/CFT correspondence the E0 is connected with dimension of conformal operator as E0 = A. The A forbosonic massive totally anti-symmetric and bosonic massive symmetric spin two fields were evaluated in Refs. [16,17]. Our results coincide with the ones obtained in these references. For instance, for the case of bosonic massive totally symmetric fields we have h1 = s, k = 1 and this leads to

d - 1 d - 5

E0 = A = —— +Jm2+[s +

(5.13)

For the case of s = 2 this is result of Ref. [17].

For fermionic massive totally symmetric fields we have h\= s + \ and formula (5.2) leads to

E0 = A = m + s + d -

(5.14)

For particular value of s = 1 (Rarita-Schwinger field) appropriate A was evaluated in Refs. [18,19]. Note that our result taken to be for s = 1 differs from the one obtained in these references. The reason for this is that in this Letter we use normalization of mass parameter such that the point m = 0 corresponds to massless fields (see Eqs. (5.9), (5.10)). In Refs. [18,19] another normalization was used.

Because while derivation of our results for E0 we used arguments based on gauge invariant formulation for massive fields our relations (5.1), (5.2) are not applicable to the scalar field and spin one-half field. Conformal dimension for these fields are well known (see [20,21]).

6. Conclusions

The results presented here should have a number of interesting applications and generalizations, some of which are: (i) in this Letter we develop light cone formulation for massive totally symmetric fields. It would be interesting to extend such formulation to the study of massive mixed symmetry fields (see Refs. [22-24]) and then to apply such formulation to the study of AdS/CFT correspondence along the line of Refs. [25,26]. (ii) As is well known massive and mass-less fields can be connected via procedure of dimensional reduction. Procedure of dimensional reduction AdSd ^ AdSd—1 was developed in [8] (for discussion of alternative reductions see [27,28]). It would be interesting to find mass spectra of AdS massive modes upon dimensional reduction of massless AdS fields.

Acknowledgements

This work was supported by the INTAS project 0351-6346, by the RFBR Grant No. 02-02-17067, and RFBR Grant for Leading Scientific Schools, Grant No. 1578-2003-2.

Appendix A. Interrelation between new and old light cone formulations

In this appendix we explain interrelation of basic defining equation for the operator BI and representation for the operators A and B (2.15), (2.16) with the old defining equations given in Ref. [7]. Defining equations for the operators A and B take the form [7]

2{ Mz',A} — [ [Mz',A],A] = 0,

[Mz', [Mzj,A]] + {M'L,MLj} = —2SjjB,

[A,M'j ] = 0.

d(d — 2)

— <QAdS>,

(A.1) (A.2)

We note that Eq. (A.4) tells us that the operators A is invariant with respect to so(d — 3) rotations. Eq. (A.3) is because of the second order Casimir operator of so(d — 1,2) algebra is diagonal in irreps labelled by D(E0, h). Eqs. (A.1), (A.2) are consequences of commutators of the so(d — 1, 2) algebra. Introducing

new operator B by relation

B = B + Mz'Mz',

and plugging the representation for the operator A given in (A.3) in Eq. (A.2) we cast the above given system of equations into the following form

[[Mz',B], B] + (m

d2 — 5d + 8

Mz' = 0,

Mz', [Mzj,B]] + &'jB = 0,

(A.6) (A.7)

1 2 d(d — 2) ~ • • A = -mjj + 4 - <eAds> +2b + mzlmzl,

(A.8) (A.9)

[B ,M'j] = 0.

While deriving of these formulas we use commutation relations of so(d — 2) algebra spin operators MIJ = M'j ,Mz' given in (2.14) and exploit the relation

MIJMIJ = M'j M'j + 2Mz' Mz'.

(A.10)

Our basic observation is that if we introduce the quantities

Bz = B,

b' = [B ,Mz'\

(A.11)

then the operator BI = Bi,Bz transforms as a vector under rotation generated by so(d — 2) algebra. Indeed the second relation in (A.11) and Eqs. (A.7), (A.9) can be rewritten as

[Mz',Bj ] = S'jBz, [Mz',Bz] = —B', [M'j,Bz ] = 0. (A.12)

These commutation relations together with the ones following from so(d — 3) covariance

[M'j,Bk ] = SjkB' — S'kBj, (A.13)

imply that the operator BI is indeed transformed in vector representation of so(d — 2) algebra (cf. (2.19)). All that remains is to analyse Eq. (A.6). Making use of (A.11) we can rewrite (A.6) as

Bz,Bi + M

+ ( <ÔAds>--M^-

1 2 d2 — 5d + 8\ 7, A ^Mzl= 0.

(A.14)

Now taking into account that a state obtainable by acting the spin operator Mzj on wave function l<) should also belong to l<) we conclude that constraint (A.14) should commute with Mzj. This gives a new constraint

[B{,Bj] + (m-

3\[i\j ]

( 1 2 d2 - 5d + 8\

+ (<ÔAdS)--M2--

\Mij = 0.

(A.15)

Constraints (A.14), (A.15) are collected into the ones given in (2.18). Thus we proved that the old light cone formalism of Ref. [7] is equivalent to the one used in this Letter. Remarkable features of new formalism are (i) appearance of so(d — 2) vector B1 which was hidden in old formalism; (ii) manifest so(d — 2) invariance of defining equations for B1.

Appendix B. Decompositions of tensor-spinor fields

Here we wish to describe an decomposition of totally symmetric tensor-spinor field №h-1^ transforming in irreps of the so(d — 1) algebra in terms of tensor-spinor fields lfs') which are irreps of the so(d — 2) algebra. Consider a generating function

) = №I1"-Isaal1 ■■■aIs l0), a1 = (a1 ',aI), (B.1) which satisfies the constraint

a1 aI\& )=s),

r'a' } = 0,

a1 aI\& ) = 0,

where to define r -transversality constraint we use r1 -symbols defined by relations

f1 = yi,

p1 ,= | i(d—2)/2Y1 ■■■Yd —2,

d is even, d is odd.

We start with analysis of the second constraint in (B.2) which can be considered as the second order differential equation with respect to oscillator variable

Obvious solution to this equation is found to be

& (a1 ',a')) = cos(max ') | ^(a1)

sin(«a1 ' ), , , ,, +-(ay)),

where \&s) and \&s—1> are rank s and s — 1 traceful tensor-spinors, i.e., they are reducible representations of the so(d — 2) algebra. The solution (B.5) reflects well know fact that symmetric traceless rank s tensor of so(d — 1) algebra can be decomposed into symmetric rank s and s — 1 traceful tensors of so(d — 2) algebra. The \&s), \&s—1) satisfy the constraints (a1 a1 — s)\&s) = 0, (a1 à1 — s + 1)\&s—1) =0. Now plugging (B.5) into the third constraint in (B.2) we find a relation

\&s—1) = aIYIr1 '\&s ).

Taking into account (B.6) and (B.5) we get a relation )= exp(a1 'aI yI r1' )№), (B.7)

which tells us that traceless and r-transversal tensor-spinor field ) (see constraints (B.2)) is expressible in terms of one tracefull tensor-spinor l№s). This tracefull tensor-spinor field in turn canbe decomposed into traceless tensor-spinors fields of so(d — 2) algebra

lf s')

) = J2®lt s'), s'

s ' = s,s — 2,s — 4,...,s — 2[s/2]. (B.8)

The traceless tensor-spinor fields of so(d — 2) algebra li?s') satisfy by definition the constraints

a Ia1 \f s') = 0,

(aa — s ' M s>) = 0.

Because the tensor-spinor lifrs') does not satisfy Y -transversality constraint this field is a reducible representation of the so(d — 2) algebra. It can be decomposed into two irreducible representations of the so(d — 2) algebra by formulas

. . . M (ya)(yg)

IVv) = ( 1 ~ 2^/ _ 2 ^

\fs'—1) = Ya\fs>).

(B.10)

' + a?) |&(a1 V)) = 0, «2 = aIaI. (B 4) so(d — 1, 2) algebra.

It is the set of traceless and Y -transversal tensor-spinor fields lfs') that we used to formulate light cone action for massive fermionic representations of

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