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ELSEVIER Physics Letters B 643 (2006) 205-212

www.elsevier.com/locate/physletb

Gauge invariant formulation of massive totally symmetric fermionic fields in (A)dS space

R.R. Metsaev

Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky prospect 53, Moscow 119991, Russia Received 6 September 2006; received in revised form 1 November 2006; accepted 2 November 2006 Available online 10 November 2006 Editor: N. Glover

Abstract

Massive arbitrary spin totally symmetric free fermionic fields propagating in d-dimensional (anti-)de Sitter space-time are investigated. Gauge invariant action and the corresponding gauge transformations for such fields are proposed. The results are formulated in terms of various mass parameters used in the literature as well as the lowest eigenvalues of the energy operator. We apply our results to a study of partial masslessness of fermionic fields in (A)dSd, and in the case of d = 4 confirm the conjecture made in the earlier literature. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

Conjectured duality [1] of conformai N = 4 SYM theory and superstring theory in AdS$ x S5 Ramond-Ramond background has led to intensive study of field (string) dynamics in AdS space. By now it is clear that in order to understand the conjectured duality better it is necessary to develop powerful approaches to study of field (string) dynamics in AdS space. Light-cone approach is one of the promising approaches which might be helpful to understand AdS/CFT duality better. As is well known, quantization of Green-Schwarz super-strings propagating in flat space is straightforward only in the light-cone gauge. Since, by analogy with flat space, we expect that quantization of the Green-Schwarz AdS superstring with Ramond-Ramond flux [2] will be straightforward only in a light-cone gauge [3] we believe that from the stringy perspective of AdS/CFT correspondence the light-cone approach to field dynamics in AdS is a fruitful direction to go. Light-cone approach to dynamics of massive fields in AdS space was developed in [4,5] and a complete description of massive arbitrary spin bosonic and fermionic fields in AdS5 was obtained in [6].

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

Unfortunately, this is not enough for a complete study of the AdS/CFT correspondence because in order to apply the light-cone approach to study of superstring in AdS space we need a light-cone formulation of field dynamics in AdS5 x S5 Ramond-Ramond background. Practically useful and self-contained way to give a light-cone gauge description is to start with a Lorentz covariant and gauge invariant description of field dynamics in AdS5 x S5 Ramond-Ramond background and then to impose the light-cone gauge. Our experience led us to conclusion that the most simple way to develop light-cone approach in AdS5 x S5 space is to start with gauge invariant description of fermionic fields. It turns out, however, that gauge invariant description of massive fermionic fields (with fixed but arbitrary spin) even in AdS5 is still not available in the literature. In this Letter we develop Lagrangian Lorentz covariant and gauge invariant formulation1 for massive totally symmetric arbitrary spin fermionic fields in (A)dSd space. We believe that our results will be helpful to find a gauge invariant description of arbitrary spin fields in AdS5 x S5 case. Our approach allows us to study fermionic fields in AdSd space and dSd space on an equal footing. In this Letter we apply our results to study of par-

1 Sometimes, a gauge invariant approach to massive fields is referred to as a

Stueckelberg approach.

0370-2693/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.11.002

tial masslessness of fermionic fields in (A)dSd. For d = 4 our results confirm the conjecture made in Ref. [7].

Before proceeding to the main theme of this Letter let us mention briefly the approaches which could be used to discuss gauge invariant action for fields in (A)dS. Since the works [8-11] devoted to massless fields in AdSd various descriptions of massive and massless arbitrary spin fields in (A)dS have been developed. In particular, an ambient space formulation was discussed in [12,13] and various BRST formulations were studied in [14-17]. The frame-like formulations of free fields which seems to be the most suitable for formulation of the theory of interacting fields in (A)dS was developed in [18,19]. Other interesting formulations of higher spin theories were also discussed recently in [20-23]. In this Letter we adopt the approach of Ref. [24] devoted to the bosonic fields in (A)dS. This approach turns out to be the most useful for our purposes.

2. Gauge invariant action of massive fermionic field in

In d-dimensional (A)dSd space the massive totally symmetric arbitrary spin fermionic field is labelled by one mass parameter and by one half-integer spin label s + 2, where s > 0 is an integer number. To discuss Lorentz covariant and gauge invariant formulation of such field we introduce Dirac complex-valued tensor-spinor spin s' + | fields of the so(d — 1, 1) Lorentz algebra f A1-As'a, s' = 0,1,...,s (where A = 0, 1,...,d — 1 are flat vector indices of the so(d — 1, 1) algebra), i.e. we start with a collection of the tensor-spinor fields

Ai...As; a

In order to obtain the gauge invariant description of a massive field in an easy-to-use form, let us introduce a set of the creation and annihilation operators aA, Z and aA, Z defined by the relations

[a A,aB ] = nAB,

a A|0> = 0,

[Z ,Z] = 1,

Z |0> = 0, (2.2)

where nAB is the mostly positive flat metric tensor. The oscillators aA, aA and Z, Z transform in the respective vector and scalar representations of the so(d — 1,1) Lorentz algebra. The tensor-spinor fields (2.1) can be collected into a ket-vector |f> defined by

if > = J2 Zs—s'ifs

|fs'> = aA1 ■ ■ ■ aAs' f ^A' a(x)|0>.

Here and below spinor indices are implicit. The ket-vector |fy> (2.4) satisfies the constraint

(aAaA — s')|fs'>=0, s ' = 0, 1,...,s, (2.5)

which tells us that |fy> is a degree s' homogeneous polynomial in the oscillator aA. The tensor-spinor field |fy> is subjected the basic algebraic constraint

ya a21 fs'> = 0, s ' = 0, 1,...,s,

ya = yAa A,

a 2 = aAa A,

which tells us that |fy> is a reducible representation of the Lorentz algebra so(d — 1,1).3 Note that for s' = 0,1,2 the constraint (2.6) is satisfied automatically. In terms of the ket-vector |f > (2.3) the algebraic constraints (2.5), (2.6) take the form

(aa + NZ — s)|f > =0, yaa2|f > = 0,

Nz = ZZ,

AA aa = aAaA.

(2.8) (2.9)

Eq. (2.7) tells us that |f > is a degree s homogeneous polynomial in the oscillators aA, Z.

Lagrangian for the massive fermionic field in (A)dSd space we found takes the form

L — Lder + Lm

(2.10)

where Lder stands for a derivative depending part of L, while Lm stands for a mass part of L:4

ie—1Lder =<f |L|f >,

ie—lLm = <f |^|f >.

(2.11) (2.12)

The standard first-derivative differential operator L which enters Lder (2.11) is given by

L = D — aDya — yaaD + yapya + ^ yaaDa2

+1 a2yaaD —1 a2Da2, where we use the notation

A A A A

ya = yAaA, ya = yAaA,

a2 = aAaA, a2 = aAaA,

P = yADA, aD = aADA,

aD = aADA,

Da = e'ad

(2.13)

(2.14)

(2.15)

and e'A stands for inverse vielbein of (A)dSd space, while DM stands for the Lorentz covariant derivative

L>M = d+ - uABM

(2.16)

2 We use oscillator formulation [10,25] to handle the many indices appearing for arbitrary spin fields. It can also be reformulated as an algebra acting on the symmetric-spinor bundle on the manifold M [21]. Note that the scalar oscillators Z, Z arise naturally by a dimensional reduction [15,21] from flat space. Our oscillators aA, aA, Z, Z are respective analogs of dx^, ddu, du of Ref. [21] dealing, among other thing, with massless fermionic fields in (A)dS. We thank A. Waldron for pointing this to us.

3 Important constraint (2.6) was introduced for the first time in [9] while study of massless fermionic fields in AdS4. This constraint implies that the field |fs'> being reducible representation of the Lorentz algebra so(d — 1, 1) is

! 1 ! 1 ! 3 decomposed into spin s + ^, s — j, s — 2 irreps of the Lorentz algebra. Various Lagrangian formulations in terms of unconstrained fields in flat space and (A)dS space may be found e.g. in [26-30].

4 The bra-vector <f | is defined according the rule <f | = (|f >)^y0.

The a>AB is the Lorentz connection of (A)dSd space, while a

spin operator MAB forms a representation of the Lorentz algebra so(d — 1,1):

MAB = MAB + 1 yAB,

MAB —aAaB — aBaA,

— 2YaYb - YBYA).

(2.17)

We note that our derivative depending part of the Lagrangian Lder (2.11) is nothing but a sum of the Lagrangians of Ref. [9] for the tensor-spinor fields (2.1).

We now proceed with discussion of the mass operator M (2.12). The operator M is given by

M = ( 1 — YaYa — 1 a2a2j m i + m YaZ — 1 a2ZYa

— ( ZYa — 2YaÇa2 )m4,

(2.18)

where operators m 1, m4 do not depend on the y -matrices and a-oscillators, and take the form

2s + d — 2 2s + d — 2 — 2NZ K

2s + d — 3 — Nz

F(K, s, Nz)

(2.19)

(2.20)

y2s + d — 4 — 2NZ

Function F(k, s, Nz) depends on a mass parameter k, spin s and operator Nz, and is given by

F(k, s,NZ) = к2 + ^s +

— Nz

(2.21)

F is restricted to be positive and throughout this Letter, unless otherwise specified, we use the convention5:

I— 1 for AdS space,

0 for flat space, (2.22)

+1 for dS space.

The mass parameter k is a freedom of our solution, i.e. gauge invariance allows us to find Lagrangian completely by module of mass parameter as it should be for the case of massive fields. Now we discuss gauge symmetries of the action

S= / ddx£.

(2.23)

To this end we introduce parameters of gauge transformations eA1"As'a, s' = 0,1 ,...,s — 1 which are y-traceless (for s' > 0) Dirac complex-valued tensor-spinor spin s' + 2 fields of the so(d — 1,1) Lorentz algebra, i.e. we start with a collection of the tensor-spinor fields

s —1

^0fA1.J-'a, YAeAA2...As' = 0, for s' > 0. (2.24)

5 Thus our Lagrangian gives description of massive fermionic fields in (A)dS space and flat space on an equal footing. Discussion of massive fermionic fields in flat space in framework of BRST approach may be found in [26,31].

As before to simplify our expressions we use the ket-vector of gauge transformations parameter

= £ Zs—1—s' |

|es/) — aA1 ■■■aAs' fA1..As' a(x)|0).

The ket-vector |e) satisfies the algebraic constraints

(aa + Nz — s + 1)|e) =0, Y«|e) =0.

(2.25) (2.26)

(2.27)

(2.28)

The constraint (2.27) tells us that the ket-vector \e) is a degree s — 1 homogeneous polynomial in the oscillators aA, Z, while the constraint (2.28) respects the y-tracelessness of \e).

Now the gauge transformations under which the action (2.23) is invariant take the form

) = (aD + A)M, A — ÇÂ1 + YaÄ 2 + a2Ä3Z,

(2.29)

(2.30)

where operators Zii, A2, Zi3 do not depend on the y-matrices and a-oscillators, and take the form

, 2s + d — 3 — Nz ^ 1/2

A1 = ( ^^--—F(k, s, Nz)

2s + d — 4 — 2NZ

2s + d — 2

(2s + d — 2 — 2NZ)(2s + d — 4 — 2NZ)

A 3 = —

2s + d — 3 — Nz

:F(K,s,Nz)

(2.31)

(2.32)

(2.33)

v(2s + d — 4 — 2NZ)3'

and F is defined in (2.21). Thus we expressed our results in terms of the mass parameter k . Since there is no commonly accepted definition of mass in (A)dS we relate our mass parameter k with various mass parameters used in the literature. One of the most-used definitions of mass, which we denote by mD, is obtained from the following expansion of mass part of the Lagrangian:

ie lCm = {fs|mD|^s) + ■•

(2.34)

where dots stand for terms involving s' < s, and for contribution which vanishes while imposing the constraint Y&lf}. Comparing (2.34) with (2.18), (2.19) leads then to the identification

к = mD.

(2.35)

Another definition of mass parameter for fermionic fields in AdSd [5], denoted by m, can be obtained by requiring that the value of m = 0 corresponds to the massless fields. For the case of spin s + 2 field in AdSd the mass parameter m is related with mD as

mD = m + s +

for AdSd,

(2.36)

where m > 0 corresponds to massive unitary irreps of the so(d — 1,2) algebra [5,12]. Below we demonstrate that natural generalization of (2.36) which is valid for both AdS and

dS spaces is given by

mo = m + V—^^ + 2

for (A)dSd.

(2.37)

Since sometimes in the case of AdS the formulation in terms of the lowest eigenvalue of energy operator E0 is preferable we now express our results in terms of E0. To this end we use the relation found in [5]:

m = E0 - s — d + - for AdSd.

(2.38)

Making use then (2.35), (2.36) we get for the case of AdSd the desired relations

к = Eo -

F = ( Eo - j - d + 2 + nA i Eo + j - 2 - Nz

3. Limit of massless fields in (A)dSd

(2.39)

(2.40)

while the relations (3.4), (3.5) lead to the gauge transformation S\fm=°) = (aD + |q_i>, (3.7)

which is noting but the standard gauge transformation of mass-less field in (A)dS space. Thus the Lagrangian for massless spin s + 1 fermionic field in (A)dSd space takes the form6

ie-1£ = [fm=0\L + Mm=0\fm=0)

where |^m=0} is given by (3.2), (2.4), while the operators L and Mm=0 are defined by (2.13) and (3.6) respectively. The remaining ket-vectors |^s-1},..., |^0} in (2.3) decouple in the massless limit and they describe spin s - | massive field, i.e. in the massless limit the generic field } is decomposed into two decoupling systems—one massless spin s + 2 field and one massive spin s - 2 field. Adopting (2.34) for spin s - 2 field we find mass of the massive spin s - 2 field: mD = */-0(s + (d - 2)/2).

In previous section we presented the action for the massive field. In limit as the mass parameter m tends to zero our Lagrangian leads to the Lagrangian for massless field in (A)dSd. Let us discuss the massless limit in detail. To realize limit of massless field in (A)dSd we take (see (2.35), (2.37))

mo ^ V-e(s + -y-) ^^ m ^ 0. (3.1)

We now demonstrate that this limit leads to appearance of the invariant subspace in } (2.3) and this invariant subspace, denoted by |^m=°}, is given by the leading (s' = s) term in (2.3):

\fm=0) =\fs

All that is required is to demonstrate that in the limit (3.1), the ket-vector |^m=0} satisfies the following requirements:

(i) |^m=0} is invariant under action of the mass operator M;

(ii) the gauge transformation of the ket-vector |^m=0} becomes the standard gauge transformation of massless field. To this end we note that an action of the mass operator M on |^s} and the gauge transformation of |^s} take the form

M\^s) = ( 1 - yaya - — a2a2 \rni(0)\^J

+ (ya - 1 а2уа^m4(0)\fs-i), S\fs ) = (aD + yaz\2(0))\ej-i) +a2ZÎ3(0)\ej-2),

where m 1,4(0) (3.3) and ¿2,3(0) (3.4) stand for m 1,4 (2.19), (2.20) and ¿2,3 (2.32), (2.33) in which we set Nz = 0. Taking into account

lim m4(0) = 0,

lim Л3(0) = 0,

we see that if m = 0 then the ket-vector |^s} is indeed invariant under action of the mass operator M and a realization of the mass operator on the ket-vector |^s} takes the form

Mm=0 = V-? (s + ^

1 - yaya - 1 a2a2

4. Partial masslessness of fermionic fields in (A)dSd

Here we apply our results to study of partial masslessness7 of fermionic fields in (A)dSd. We confirm conjecture of Ref. [7] for d = 4 and obtain a generalization to the case of arbitrary d > 4. In this section we assume that the в (2.22) takes the values ±1. We start our discussion of partial masslessness of fermionic fields with simplest case of (see also Ref. [34]).

4.1. Massive spin 5/2 field

Such field is described by ket-vectors |ф2), |ф0) (see (2.3)). For spin 5/2 field there is one critical value of mD which leads to appearance of partial massless field. For this critical value of mD the generic field | ф) is decomposed into one partial massless field and one massive spin 2 field. To demonstrate this we consider the gauge transformations (2.29),

5|ф2> = aD|ei) + yaZh^ki) +a2 Лз(О)ко), 5Wi)=aD|eo) + ii(0)|ei) + yai2(1)|eo). 5|фо) = ^1(1)ко),

where ¿\1,2,3(«) are given in (2.31)-(2.33) in which we set s = 2 and argument n stands for an eigenvalue of the operator Nz. The critical value of mD is obtained from the requirement of decoupling of the field |^0}. This requirement amounts to the equation 1(1) = 0 which leads to the critical value 2

mD(0) =

1 + ■

For this value of mD the generic field } is decomposed into two decoupling systems—one partial massless field described

6 Our Lagrangian (3.8) is a generalization to d-dimensions of the Lagrangian of Ref. [9] for massless field in (A)dS4. Alternative Lagrangian descriptions of massless fermionic fields in (A)dSd may be found in [11,21].

7 Partial masslessness was discovered in [32]. Recent discussion of this theme and to some extent complete list of references may be found in [33].

by |f2>, |f 1> and one massive spin 1 field described by |f0>. We proceed with discussion of partial masslessness for

4.2. Massive spin 7/2 field

Spin 7/2 field |f > is described by ket-vectors |f3>, |f2>, |f1>, |f0> (see (2.3)). The gauge transformations (2.29) for these ket-vectors take the form

8^3) = aD|€2> + YaA2me2)+a2A3me1), 8^2) = aD|61) + Â1(0)|e2) + Ya^2(1)|^l)

+ 2a2^3(1)|e0), 8f 1) = aD|f0) + AÏ1(1)|e1) + Ya^2(2)|€o),

8|f0) = A 1(2)|60),

where expressions for A 1,2,3(n) are given in (2.31)-(2.33) in which we set s = 3 and argument n stands for an eigenvalue of the operator Nç. For the spin 7/2 field there are two critical values of mD. For each critical value of mD the generic field } is decomposed into one partial massless field and one massive field. We consider these critical values in turn.

First critical value of mD is obtained from the requirement of decoupling of the field |^0} (see (4.8)). This requirement amounts to the equation AA i(2) = 0 which leads to the critical value

mD(0) = —e

1 + ■

For this value of mD the generic field |f > is decomposed into two decoupling systems—one partial massless field described by |f3>, |f2>, |f 1> and one massive spin 1 field |f0>.

The second critical value of mD is obtained from the requirement of decoupling of the fields |f1>, |f0> (see (4.6), (4.7)). This requirement amounts to the equations A 1(1) = 0, A3 (1) = 0 which lead to the critical value

mD(1) = —e

2 +■

(4.10)

For this mD the generic field |f > is decomposed into one partial massless field described by |f3>, |f2> and one massive spin 2 field described by |f 1>, |f0>. We finish with partial massless-ness for

4.3. Massive arbitrary spin s + 2 field

Such field is described by ket-vectors |fs'>, s' = 0,1,...s. Gauge transformations (2.29) for these ket-vectors take the form

5|fs'>=aDks'—1> + A1(s — s' — 1)ks'>

+ YaA2(s — s')|es'—1) + (s — s '+ 1)a2^3(s — s')|es/—2),

(4.11)

where A1i2,3(n) are given in (2.31)-(2.33) and argument n stands for an eigenvalue of the operator Nz. For values s' = 0, 1,s (4.11) we use the convention |e-2} = |e-1} = |es} = 0.

For the spin s + 2 field there are s — 1 critical values of mD, denoted by mD(n), n = 0, 1,...,s — 2 (the case of n = s — 1 leads to massless field and was considered in Section 3). For each mD(n) the generic field |f > is decomposed into one partial massless field and one spin n + 1 massive field. To find mD(n) we note that the requirement of decoupling of the fields |fn>,..., |f0>, n = 0,...,s — 2 amounts to equations A 1(s — n — 1) = 0, A3 (s — n — 1) = 0. Solution to these equations8

m2D(n) = —0[n + 1 +

d — 4

(4.12)

is in agreement with conjecture made in Ref. [7] for the case of d = 4. Thus we confirmed conjecture of Ref. [7] and obtained mD(n) for d > 4. For each mD(n) the generic field }

is decomposed into two decoupling systems—one partial mass-

less field |^par} described by |^s},..., |^n+1}, and one massive spin n + 2 field ^"1} described by |^n},..., |^0}. This is to say that by decomposing } (2.3) into the respective ket-vectors

№ — E zs—s'

s/=n+1

№v)—E zs—^ f

(4.13)

one can make sure that if mD = mD(n) then the mass part of the Lagrangian (2.12) is factorized

ie—1Cm = f W|M|fW) + f mviMlfW,

(4.14)

For values mD = mD(n) the gauge transformations (2.29) are also factorized, while Lder (2.11) is factorized for arbitrary

5. Uniqueness of Lagrangian for massive fermionic field

We now demonstrate that the Lagrangian and gauge transformations are uniquely determined by requiring that the action be gauge invariant. We formulate our statement. Suppose the derivative depending part of the Lagrangian is given by (2.11), while the derivative depending part of gauge transformations (2.29) is governed by aD-term. Suppose the gauge field \ f) (2.3) and the gauge transformations parameter \e) (2.25) satisfy the respective constraints (2.8), (2.28). Then we state that the mass operator M given in (2.18) and the operator A which enters gauge transformations (2.29) are uniquely determined

8 We note that the key point is not positivity or even reality of the mass part of action (2.12), but rather stability of the energy and unitarity of the underlying physical representations. For bosons a negative mass term is allowed in AdS (the Breitenlohner-Freedman bound, [35]), while partially massless fermions even have an imaginary mass term in their actions but are still stable and unitary in dS. Partially massless fermions are not unitary in AdS (see Refs. [7,34,36]).

9 mD - and m-masses of the field |^par} are given by: mp = -J-0(n + 1 + (d — 4)/2), m = s/-d(n + 1 — s), while for the field |^mv} we get mp = V=ê(s + (d — 2)/2), m = V—ë(s — n + 1).

by the following requirements: (i) the action be gauge invariant; (ii) there are no invariant subspaces in } under action of gauge transformations. Here we outline prove of this statement. We start with general form of the mass operator M and the operator ¿:

M = mi + Yam2Y01 + a2m?,a2 + Yam4 + a2msya

— m^Ya — Yam^ct + a m^ + m6a A = A1 + ycA2 + a2 ¿3,

where m1,...,6 and ¿1,2,3 do depend on y-matrices and a-oscillators, and are given by

mi = m i,

m4 = m 4<t,

ms = m5\, 2

m6 = m6Z , Ai = ZA i,

m2 = m 2,

tt m4 = Zm 4,

tt m5 = Zm 5, t >.2 ~t m6 = Z m6,

A2 = A 2,

m3 = m 3,

A3 = A31.

Operators m 1,...,6 and A1i2,3 depend only on Nz (2.9). m[ 6 stand for hermitian conjugated of m 1,...,6. Since m 1i2,3 are her-mitian, m 1 2 3 = m 1i2,3, and depend only on the hermitian operator Nz the operators m 1i2,3 are real-valued functions of Nz from the very beginning. These properties of m 1,...,6 and A1i2,3 and expressions for M (5.1), A (5.2) are obtained by requiring that:

(i) M and ¿ commute with the spin operator of the Lorentz algebra MAB (2.17) and satisfy the commutators [aa + Nz, M] = 0, [aa + N^¿1 = ¿;

(ii) M does not involve terms like a2yaf1 and f2a2ya, where f1,2 are polynomial in the oscillators (such terms in view of (2.8) do not contribute to Lm);

(iii) ¿ does not involve terms like a2yaf3, f4y a, where f3,4 are polynomial in the oscillators (the f3-terms lead to violation of constraint (2.8) for gauge transformed field (2.29), while the f4-terms in view of (2.28) do not contribute to } (2.29));

(iv) M and hermitian conjugated of M satisfy the relation Mf = -y °My 0.

Thus all that is required is to find dependence of the operators m 1,...,6 and ¿^2,3 on Nz. We now demonstrate that this dependence can be determined by requiring that the action be gauge invariant. We evaluate the variation of the action (2.10), (2.23) under gauge transformations (2.29), (5.2),

SS =-i/ddxe{f |(DX1 + yaa DX2 + aDX3

+ yaDX4 + a2a DX5 + a2^X6 + aDyaX7 + a2aDX8 + aDX9 + Y(0) + ya(Y(1) + Y^) + a2Y(2)) |e}+h.c., (5.6)

where we use the notation

X3 = —(2s + d — 4 — 2Nz)A2 + mi, X4 = (2s + d — 4 — 2Nz)A2 + m2, i

X5 = - (2s + d — 4 — 2Nz)A2 + 2m3, i

X6 = — (2s + d — 4 — 2NZ)A3 + m5,

(5.1) X7 = (2s + d — 4 — 2NZ)A3 + m4,

(5.2) Xs = m6, X9 = 2ml, Y(0) = miAi — (2s + d — 2NZ)mtA2,

Y(i) = miA2 + (2s + d — 2 — 2NZ)m2A2 + m4Ai — 2mlA3 — 2 (2s + d — 2 — 2NZ)mi^A3,

(5.3) Y(2) = miA3 + 2m2A3 + 2(2s + d — 4 — 2NZ)m3A3

+ m4A2 + (2s + d — 4 — 2NZ)m5A2, Y(A)dS =—T(2s + d — 3 — 2Nz)(2s + d — 4 — 2Nz).

Requiring this variation to vanish gives the equations

Xa = 0, a = 1,...,9,

Y(0) = 0,

Y(1) + Y(A)dS = 0,

Y(2) = 0.

Solution to Eq. (5.18) is easily found to be

m1 = (2s + d - 4 - 2Nz)¿2, m4 = -(2s + d - 4 - 2Nz)¿3,

m2 = —mi,

m 4 = Ai,

m3 =--mi,

3 4 i,

ml =--Ai,

m5 = m4,

m6 = mt = 0,

(5.s) (5.9)

(5.10)

(5.11)

(5.12)

(5.13)

(5.14)

(5.15)

(5.16)

(5.17)

(5.18)

(5.19)

(5.20)

(5.21)

(5.22)

(5.23)

(5.24)

i.e. Eqs. (5.18) allow us to express the operators ma, a = 1,..., 5, entirely in terms of the operators ¿2, ¿3 which enter the gauge transformations. Moreover, the expressions for m4 (5.22) and m4 (5.24) imply the relation

At = —(2s + d — 4 — 2NZ)A3.

(5.25)

We now analyze (5.i9). Inserting mi (5.22) and m4 (5.24) in (5.i9) we cast (5.i9) in the form

Ai ((2s + d — 6 — 2NZ)A+ — (2s + d — 2 — 2NZ)A2) = 0,

(5.26)

where A2 = A2(NZ) and A+ = A2(NZ + i). Solution to (5.26) A i = 0 for all Nz (or A i = 0 for some particular eigenvalues of Nz ) leads to massless fields (or partial massless fields), i.e. such solution leads to appearance of invariant subspaces in \ f ). We are not interested in such solution and assume A i = 0 for all Nz . Eq. (5.26) allows us then to find dependence of the operator A2 on Nz :

X, = A, — mi

X2 = —Ai — 2m5,

(2s + d — 2 — 2NZ)(2s + d — 4 — 2NZ)

(5.27)

where A2(0) is dimensionfull parameter not depending on Nz . Inserting A2 (5.27) in (5.21) one can make sure that (5.21) is satisfied automatically. All that remains then is to solve Eq. (5.20). Making use of (5.22)-(5.25) and (5.27) one can make sure that (5.20) amounts to the equation10

Z(Nz) — Z(Nz — 1)

(2s + d — 3 — 2Nz)(A2(0))2 (2s + d — 2 — 2NZ)2(2s + d — 4 — 2NZ)2 0

— - (2s + d — 3 — 2Nz) = 0, (5.28)

where we use the notation

Z(Nz) = (2s + d — 4 — 2Nz)(Nz + 1)(A3)2. (5.29)

The relation (5.29) implies a condition Z(—1) = 0. This condition and (5.28) lead to the initial condition

(2s + d — 3)(A2(0))2 0

Z(0) =-(—--, 2(0))-t + -(2s + d — 3). (5.30)

(2s + d — 2)2(2s + d — 4)2 4 ( )

Eq. (5.28) and the initial condition (5.30) allow us to find Z(Nz) uniquely and taking into account (5.29) we obtain

2 2s + d - 3 - NZ

(A 3 )2 = ■ Z

2s + d — 4 — 2NZ

(A2(0))2

(2s + d — 2)2(2s + d — 4 — 2NZ)2 + 4

(5.31)

Thus we satisfied all equations imposed on M and A by the requirement of gauge invariance of the action and the expressions (5.22)-(5.25), (5.27), (5.31) determine M and A uniquely. In view of the first relation in (5.22) and (5.27) the A2(0) is real-valued and introducing the mass parameter k (which is assumed to be positive) by relation

A2(0) = (2s + d — 2)k,

(5.32)

we arrive at the expressions for M and A given in Section 2.

To summarize, we found the gauge invariant action for the fermionic fields in (A)dSd. All that remains to construct action for fermionic fields in AdS5 x S5 Ramond-Ramond background is to add appropriate dependence of S5-coordinates and take into account contribution of Ramond-Ramond background fields.11 The result will be reported elsewhere.

Acknowledgements

This work was supported by the INTAS project 03-51-6346, by the RFBR Grant No. 05-02-17654, RFBR Grant for Leading

10 It is easy to demonstrate that making use of field redefinitions, the phase factors of A3 can be normalized to be equal to —1. Therefore in (5.28) and below A3 is assumed to be real-valued and negative. Relations (5.5) and Eq. (5.25) imply then that A1 is real-valued and positive.

11 Study of some leading contributions of Ramond-Ramond background fields to mass operator of the bosonic fields in AdS5 x S5 Ramond-Ramond background may be found in [37]. Precise form of mass operator for bosonic fields is still to understood.

Scientific Schools, Grant No. 1578-2003-2 and Russian Science Support Foundation.

Appendix A. Notation and commutators of oscillators and covariant derivative

We use 2[d/2] x 2[d/2] Dirac gamma matrices yA in d-dimensions, {YA,YB} = 2nAB, yA^ = Y0YAY0, where nAB is mostly positive flat metric tensor and flat vectors indices of the so(d — 1, 1) algebra take the values A, B = 0, 1,...,d — 1. To simplify our expressions we drop nAB in scalar products, i.e. we use XaYa = nABXAYB .Indices fi,v = 0, 1,...d — 1 stand for indices of space-time base manifold.

We use the algebra of commutators for operators that can be constructed out the oscillators aA, aA (2.2) and derivative DA (2.15), (2.16) (see also Appendix A in Ref. [4]). Starting with

râ â T D CZ cfABC _ ,ABC , ,BAC

[oa,°bj = ^AB de, m =—m + m ,

BC _ 0V,3C MA = eAMB ,

where dA = e'A, = 8/8x,x, and i2ABC is a contorsion tensor we get the basic commutator

[Da,Db ] = MabcDc + 1RABCDMCD = MabcDc + 0MAB,

and RABCD is a Riemann tensor which for (A)dSd geometry takes the form

rabcd = e {nACnBD — nADnBC)- (A.3)

The spin operator MAB is given in (2.17). For flexibility in (A.2) and below we present our relations for a space of arbitrary geometry and for (A)dSd space. Using (A.2) and the commutators

[DA,aB] = —MABCaC, [Da, aB] = — MABCaC, [Da,Yb ] = —MabcYc, (A.4)

we find straightforwardly

[DA,a2] = 0, [a 2,Da] = 0, [DA,Ya] = 0, (A.5)

[a 2,aD] = 2a D, [ya ,aD] = D,

{D,Ya} = 2aD, (A.6)

D2 = DaDa + MaabDb +1 YabRabcdMcd

= DaDa + MaabDb + °-YABMAB, (A.7)

[aD, aD] = DaDa + maabDb — 1RABCDMABMCD

= DaDa + MaabDb — -MABMAB, (A.8)

1 A„,B t,abcd A/,CD

[D, aD] = -YAaBRABCDM

= ^ Ya[ aa +

— a2Ya I,

[aD, D ] = — KAa bRabcdMc

= 0 aa +

— yaa

(A.10)

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