Mixed-symmetry massless gauge fields in AdS(5)

0.18 MB·English

by  K. B. Alkalaev

FIAN/TD/02/05
Mixed-symmetry massless gauge fields in AdS
5
K.B. Alkalaev
I.E.Tamm Department of Theoretical Physics, P.N.Lebedev Physical Institute,
Leninsky prospect 53, 119991, Moscow, Russia
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n Abstract
a
J
4 Free AdS5 mixed-symmetry massless bosonic and fermionic gauge fields
of arbitrary spins are described by using su(2,2) spinor language. Manifestly
1
covariant action functionals are constructed and field equations are derived.

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v
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1 Introduction
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0
/
h We continue the study of the five-dimensional higher-spin gaugetheory [1, 2,3, 4,5].
t
- Our goal is to formulate manifestly covariant gauge invariant actions describing the
p
e free-field dynamics of massless mixed-symmetry bosonic and fermionic gauge fields
h
of arbitrary spins s =(s ,s ). Our treatment of AdS mixed-symmetry gauge fields
: 1 2 5
v is based on the frame-like approach proposed in [6, 7]. But instead of the (spinor)-
i
X tensor language used in [6, 7], we use a spinorial description of AdS higher-spin
5
r fields based on the well-known fact that the AdS algebra o(4,2) is isomorphic
a 5
to su(2,2). Therefore, o(4,2) (spinor)-tensor fields can be described equivalently
as su(2,2) multispinors. The main advantage of the spinorial description is that
bosonic and fermionic fields of any spin s =(s ,s ) can be considered uniformly.
1 2
Although the d-dimensional analysis in [6, 7] certainly includes the AdS case,
5
reformulating these results in the spinor language may be interesting in several as-
pects. In general, such a reformulation is motivated by the desire to make a step
towards a supersymmetric nonlinear higher-spin gauge theory, which is of interest in
the context of higher-spin version of the AdS /CFT correspondence (see [8, 9] for
5 4
a review). The nonlinear equations of motion for AdS totally symmetric bosonic
d
fields and the underlying higher-spin gauge algebras are now constructed [10, 11],
but extending them to general mixed-symmetry fields remains the open problem.
On the other hand, in the case of AdS higher-spin dynamics, there is a real possi-
5
bility to construct nonlinear theory of higher-spin fields of all symmetry types. In-
deed, in five dimensions, one benefits from using the isomorphism o(4,2) ∼ su(2,2).
1
In particular, in the spinor language, (supersymmetric) higher-spin algebras were
identified as certain star-product algebras with su(2,2) spinor generating elements
[12, 13]. There also exist manifestly covariant formulations of free AdS higher-
5
spin dynamics1 [1, 2, 3, 4] and N = 0,1 (supersymmetric) action functionals that
describe cubic interactions of totally symmetric AdS fields [2, 5].
5
The paper is organized as follows. In Sec. 2, we describe the background AdS
5
geometry in the spinor notations [2]. In Sec. 3, we consider AdS higher-spin
5
gauge fields of mixed-symmetry type and describe their multispinor and (spinor)-
tensor forms. In Sec. 4, we introduce higher-rank tensors as functions of auxiliary
spinor variables [2] and construct gauge transformations and linearized higher-spin
curvatures. In Sec. 5, we construct manifestly gauge invariant higher-spin actions.
In Sec. 6, we derive equations of motions and discuss constraints for extra fields.
We make concluding remarks in Sec. 7.
2 AdS background geometry in the spinor nota-
5
tions
Gravitational fields in AdS are identified with 1-form connection that takes values
5
in the AdS algebra su(2,2)
5
Ω(x) = dxn Ω α t β , (2.1)
n β α
where t β are basis elements of su(2,2) and Ωα = 0 2. The su(2,2) gauge field in
α α
(2.1) decomposes into a frame field and a Lorentz spin connection. This splitting
can be performed in a manifestly su(2,2) - covariant manner by introducing a com-
pensator field [22, 2], an antisymmetric bispinor Vαβ = −Vβα. The compensator is
1
normalized such that V Vβγ = δ β and V = ε Vγρ. The Lorentz subalgebra
αγ α αβ αβγρ
2
is identified with the stability algebra of the compensator. This allows defining the
frame field Eαβ and Lorentz spin connection ωα(2) as [2]
Eαβ = DVαβ ≡ dVαβ +Ωα Vγβ +Ωβ Vαγ , EαβV = 0 ,
γ γ αβ
(2.2)
λ
ωα = Ωα + EαγV ,
β β γβ
2
where λ is a cosmological parameter, λ2 > 0. We note that because the compensator
is Lorentz-invariant, we regard Vαβ as a symplectic metric that allows raising and
lowering spinor indices in a Lorentz-covariant way: Xα = VαβX , Y = YβV .
β α βα
1Thelight-coneequationsofmotionforAdS masslessfieldsofarbitraryspinswereconstructed
d
in[14,15,16]andthelight-coneactionsforgenericAdS5 mixed-symmetrymasslessfieldswerecon-
sideredin[17]. Also,there arevariousmanifestly covariantLagrangianformulationsfor particular
examples of AdS mixed-symmetry gauge fields [18, 19, 20, 21].
d
2 Throughout the paper we work within the mostly minus signature and use the notations
α,β,γ =1÷4 for su(2,2)spinor indices, m,n=0÷4 for worldindices, a,b,c=0÷4 for tangent
Lorentz so(4,1) vectorindices and A,B,C =0÷5 for tangentso(4,2) vector indices. We also use
condensednotations fora setofsymmetric spinorindices α(k)≡α1...αk. Indices denotedby the
same letter are assumed to be symmetrized as XαYα ≡Xα1Yα2 +Xα2Yα1.
2
The AdS field strength corresponding to gauge field (2.1) has the form
5
R β = dΩ β +Ω γ ∧Ω β (2.3)
α α α γ
and the background AdS space is described by 1-form field Ωα = (hαβ,ωα(2)),
5 0β 0
which satisfies the zero-curvature condition [23]
R β(Ω ) = 0 . (2.4)
α 0
3 Higher-spin fields
To describe a spin-(s ,s ) gauge field, we introduce a pair of mutually conjugate
1 2
su(2,2)traceless multispinors symmetric in the lower and upper indices [1, 2, 3, 4, 6]
3,
Ωα(m) (x)⊕Ωβ(n) (x), Ωα(m−1)γ (x)δρ = 0 , (3.1)
β(n) α(m) β(n−1)ρ γ
which are 1-forms
Ωα(m) (x) = dxnΩ α(m) (x) (3.2)
β(n) n β(n)
with
m = s +s −1 , n = s −s −1 . (3.3)
1 2 1 2
To decompose representations (3.1) of the AdS algebra su(2,2)into representations
5
ofitsLorentzsubalgebra,weuseofthecompensatorVαβ. Theresultofthereduction
is given by
s1−s2−1
Ωα(s1+s2−1) (x) = ωα(s1+s2−1)γ(t),γ(s1−s2−t−1)(x)V ,
β(s1−s2−1) β(s1−s2−1),γ(s1−s2−1)
t=0
X
(3.4)
where the condensed notation V ≡ V V ...V is introduced. The
α(k),β(k) α1β1 α2β2 αkβk
Lorentz algebra irreducible components
ωα(s1+s2+t−1),β(s1−s2−t−1)(x) , 0 ≤ t ≤ s −s −1 (3.5)
1 2
satisfy the Young symmetry condition
ωα(s1+s2+t−1),αβ(s1−s2−t−1)(x) = 0 (3.6)
and contractions with V are zero,
αβ
ωα(s1+s2+t−1),β(s1−s2−t−1)(x)V = 0 . (3.7)
αβ
Accordingtotheanalysisof[1,2],multispinorswith|m−n| = 0correspondtototally
symmetric spin-s bosonicfields andareself-conjugate. Otherfieldswith|m−n| ≥ 1
1
are described by a pair of mutually conjugate multispinors and correspond either to
3The complex conjugation operation is defined by the rule, X¯ = XβC , Y¯α = CαβY ,
α βα β
where the bar denotes complex conjugation and Cαβ = −Cβα and C = −C are some real
αβ βα
matrices such that C Cβγ =δ β [2].
αγ α
3
totally symmetric fermionic spin-s fields, |m−n| = 1 [3, 4], or to mixed-symmetry
1
bosonic and fermionic fields, |m−n| ≥ 2 [2,4]. We note that mixed-symmetry gauge
fields necessarily occur in the spectrum of N ≥ 2 extended five-dimensional higher-
spin gauge superalgebras, while N ≤ 1 (super)algebras describe totally symmetric
fields [13].
Torelatethespinorand(spinor)-tensorformsofmixed-symmetry fielddynamics,
we examine the o(4,1) (spinor)-tensor cousins of multispinor fields (3.5)-(3.7) at
s 6= 0. The result is that a collection of o(4,1) gauge fields is represented by
2
complex-valued (spinor)-tensor fields of the form
ωa(s1−1),b(s2+t) = dxnω a(s1−1),b(s2+t) , 0 ≤ t ≤ s −s −1 (3.8)
n 1 2
for bosonic mixed-symmetry fields and
wα|a(s1−1),b(s2+t) = dxnw α|a(s1−1),b(s2+t) , α = 1÷4 , 0 ≤ t ≤ s −s −1 (3.9)
n 1 2
for fermionic mixed-symmetry fields (α denotes a five-dimensional Dirac spinor in-
dex). In both cases, fields (3.8) and (3.9) have the Young symmetry property and
are traceless (bosons) or gamma-transverse (fermions).
In accordance with the nomenclature in [6], fields (3.8) and (3.9) with the pa-
rameter t ≥ 1 are called extra fields. Fermionic field (3.9) at t = 0 is called the
physical field. To classify the bosonic field in (3.8) with t = 0, we decompose it into
real and imaginary parts as
ωa(s1−1),b(s2) = ω a(s1−1),b(s2) +iω a(s1−1),b(s2) . (3.10)
1 2
Using the five-dimensional Levi-Civita symbol, one of the fields, ω or ω , can be
1 2
dualized into a field with one index in the third row, for example,
ωa(s1−1),b(s2) = ω a(s1−1),b(s2) +iǫabcdeω a(s1−2) b(s2−1) , (3.11)
1 2 c, d,e
wherethedualthree-rowfieldωa(s1−1),b(s2),e istracelessandhastheYoungsymmetry
2
property. We call the real part Reωa(s1−1),b(s2) of field (3.11) the physical field. The
imaginary part Imωa(s1−1),b(s2) of this field is called the auxiliary field. In fact, any
bosonic field (3.8) can be represented as a pair of real fields with one of them having
one index in a third row. The resulting collection of real Lorentz-covariant fields
is described by three-row o(4,1) Young tableaux arising as a decomposition of a
certain o(4,2) three-row Young tableau [2, 6].
4 Higher-spin linearized curvatures
The analysis of linearized curvatures in this section is close to the analysis in the
previous papers on totally symmetric fields [2, 3]. It turns out that the general
form of gauge transformations for mixed-symmetry fields remains intact except for
an additional dependence on the spin s and the appearance of a nonzero operator
2
T for bosonic nonsymmetric fields (see (4.18), (4.20)). This last feature is not
0
4
typical for bosonic systems and is a reflection of an implicit presence the Levi-Civita
symbol in the definition of real bosonic components of complex-valued fields (3.8).
The calculation of the bosonic operator T is the main result in this section.
0
We introduce auxiliary commuting variables a and bβ transforming under the
α
fundamental and the conjugate fundamental representations of su(2,2). It is conve-
nient to represent higher-spin fields (3.2) as functions of auxiliary variables
Ω(a,b|x) = Ωα(s1+s2−1) (x)a bβ(s1−s2−1) , (4.1)
β(s1−s2−1) α(s1+s2−1)
where
a = a ···a , bβ(n) = bβ1 ···bβn . (4.2)
α(m) α1 αm
The corresponding five-dimensional linearized higher-spin curvature is given by
∂ ∂
R(a,b|x) = dΩ(a,b|x)+Ω α (bβ −a )∧Ω(a,b|x) , (4.3)
0 β ∂bα α∂a
β
where is the background 1-form connection Ω α satisfies the zero-curvature condi-
0 β
tion (2.4). The linearized (Abelian) higher-spin transformation are
δΩ(a,b|x) = D ξ(a,b|x) , (4.4)
0
where the background covariant derivative is given by
∂ ∂
D = d+Ω α (bβ −a ) . (4.5)
0 0 β ∂bα α∂a
β
Condition (2.4) implies that δR(a,b|x) = 0. The Bianchi identities have the form
D R(a,b|x) = 0 . (4.6)
0
In the subsequent analysis, we use two sets of the differential operators in aux-
iliary variables [2],
∂ ∂
S− = a Vαβ , S+ = bα V , S0 = N −N (4.7)
α∂bβ ∂a αβ b a
β
and
1 ∂2 1
T− = , T+ = a bα , T0 = (N +N +4) , (4.8)
4∂a ∂bα α 4 a b
α
where
∂ ∂
N = a and N = bα . (4.9)
a α∂a b ∂bα
α
With (4.7) and (4.8), the irreducibility conditions for Ω(a,b) are reformulated as
T−Ω(a,b) = 0 , (S0 +2s )Ω(a,b) = 0 . (4.10)
2
5
As demonstrated in Sec. 3, the higher-spin gauge field Ω decomposes into Lorentz
subalgebra representations in accordance with formula (3.4). In terms of operators
(4.7) and (4.8), formula (3.4) is rewritten as
s1−s2−1
Ω(a,b|x) = (S+)t ωt(a,b|x), (4.11)
t=0
X
where
ωt(a,b|x) = ωα(s1+s2+t−1),β(s1−s2−t−1)(x)a b (4.12)
α(s1+s2+t−1) β(s1−s2−t−1)
are Lorentz-covariant gauge fields (3.5). The irreducibility conditions in (3.6) and
(3.7) become
S−ωt(a,b) = 0 , T−ωt(a,b) = 0 . (4.13)
Higher-spin gauge symmetry (4.4) requires the bosonic and fermionic Lorentz-
covariant higher-spin curvatures rt and gauge transformations to be given by
rt = Dωt +T −ωt+1 +λT0ωt +λ2T +ωt−1 , (4.14)
δωt = Dξt +T −ξt+1 +λT0ξt +λ2T+ξt−1 , (4.15)
where 0-forms ξt are Lorentz-covariant gauge parameters and D is the background
Lorentz-covariant derivative
∂ ∂
D = d+wα (a +b ) . (4.16)
0β α∂a α∂b
β β
The operators T−, T + and T 0 have the form
∆2 ∂
T + = (1− ) hα a , (4.17)
β α
(S0)2 ∂b
β
∆ ∂ ∂ 2 ∂ ∂
T 0 = − hα (b −a + (b )a ) , (4.18)
β α α γ α
S0 ∂b ∂a S0 −2 ∂a ∂b
β β γ β
1 ∂ ∂ ∂ ∂
T− = hα ((2−S0)b +b (b −a )
β α γ α α
1−S0 ∂a ∂a ∂b ∂a
β γ β β
(4.19)
1 ∂ ∂
+ (b )2a ) ,
γ α
S0 −3 ∂a ∂b
γ β
where the parameter ∆ takes the values
2s , for bosons,
∆ = 2 (4.20)
2s +1, for fermions,
( 2
6
and satisfy the relations
{T 0,T−} = {T 0,T+} = 0,
(T −)2 = 0 , (T +)2 = 0 , (4.21)
D2 +λ2{T −,T+}+λ2(T 0)2 = 0 .
We note that the coefficients in (4.17)-(4.19) can be changed by field redefinitions
of the form ωt = C(t,s)ωt with C 6= 0.
e
5 Higher-spin action
Before the considering actions for arbitrary mixed-symmetry gauge fields, we exam-
ine the case of the simplest nonsymmetric bosonic field of spin (2,1) described by
1-form Ωα(2)(x). Up to total derivative terms, the action functional has the unique
form
S(2,1) = hα ∧Rβγ ∧R¯ , (5.1)
2 β γα
ZM5
where hα is the background AdS frame field and the curvature is
β 5
Rα(2) = D Ωα(2) ≡ DΩα(2) +λhα ∧Ωγα . (5.2)
0 γ
The equations of motion resulting from the action (5.1) are
H α ∧Rγβ +H β ∧Rγα = 0 (5.3)
2 γ 2 γ
plus the complex-conjugated equations for Ω¯ . We note that these bosonic equa-
αβ
tions are of 1-st order, which makes them similar to fermionic equations. But as
discussed inSec. 3, thereal andimaginaryparts ofthe complex-valued field Ωα(2)(x)
are regarded as physical and auxiliary fields (3.11), with the auxiliary field being
expressed by virtue of its equation of motion in terms of first derivatives of the
physical field. To describe this mechanism in more detail, we consider the tensor
form of action (5.1). According to (3.10), the o(4,1) field isomorphic to Ωα(2)(x) is
ω[ab] = ωab +iωab . (5.4)
1 2
The corresponding linearized curvature and gauge transformations have the forms
iλ iλ
Rab = Dωab − ǫabcdeh ∧ω , δωab = Dξab − ǫabcdeh ξ , (5.5)
c de c de
2 2
where D is the background Lorentz-covariant derivative, ξab is a 0-form complex
gauge parameter, and ha is the background frame field. We note that the terms in
(5.5) involving the Levi-Civita symbol are in fact the operator T 0 expressed in the
spinor notation by formula (4.18). The Bianchi identities are
iλ
DRab − ǫabcdeh ∧R = 0 . (5.6)
c de
2
7
The action has a form analogous to (5.1)
S(2,1) = ǫ he ∧Rab ∧R¯cd , (5.7)
2 abcde
ZM5
where R¯cd is complex-conjugate curvature (5.5). The equations of motion are
H c ∧R −H c ∧R = 0 , H d=ef h ∧h (5.8)
a cb b ca ab a b
plus the complex-conjugate equations.
To clarify the dynamical content of these equations, we regard the real or imag-
inary part of the field ωab given by (5.4) as dualized auxiliary field, for example,
1
ωab = ωab , ωab = ǫabcdeω , (5.9)
1 1 2 λ 2cde
where ωab and ωabc are the physical and auxiliary fields with antisymmetric indices
1 2
and the factor λ−1 is introduced to express the fact that mass dimensions of the
physical and auxiliary fields are different. These fields can be unified into a single
o(4,2) field Ω[ABC] [6]. It can be shown that the action (5.7) can be rewritten as
1
S(2,1) = ǫ hEVF ∧RABM ∧RCDN V V (5.10)
2 λ2 ZMd ABCDEF M N
In this form, the action coincides with the action for the AdS ”hook” field ex-
d
plicitly studied in [6]. We note that the flat limit of action (5.7) (or, equivalently,
(5.10)) yields the dual description of the spin-2 field, which precisely corresponds
to Curtright action [26]. Another comment is that described procedure for unifying
dynamical and auxiliary fields into a single complex-valued field was used to study
the so-called odd-dimensional self-duality for massive antisymmetric tensor fields in
Minkowski space [27].
5.1 Action for nonsymmetric AdS gauge fields
5
In what follows, we construct free actions describing mixed-symmetry bosonic and
fermionic gauge fields in AdS . The case of totally symmetric fields was considered
5
in [2, 3].
As in [2, 3, 6], we seek mixed-symmetry field action functionals in the form
S(s1,s2) = Hˆ ∧Rs1,s2(a ,b )∧R¯s1,s2(a ,b )| , (5.11)
2 1 1 2 2 ai=bi=0
ZM5
where Rs1,s2 is linearized higher-spin curvature (4.3) and Hˆ is the 1-form differential
operator
∂2 ∂2
Hˆ = α(p,q)h ˆb +β(p,q)hαβ aˆ
αβ∂a ∂a 12 ∂bα∂bβ 12
(cid:16) 1α 2β 1 2
(5.12)
∂2 ∂2
+γ(p,q)h β cˆ +ζ(p,q)h β cˆ (cˆ )2s2 .
α β 12 α β 21 12
∂a ∂b ∂a ∂b
2α 1 1α 2 (cid:17)
8
Here h β is the background frame field and the coefficients α, β, γ and ζ are
α
functions of operators
ˆ
p = aˆ b , q = cˆ cˆ , (5.13)
12 12 12 21
where
∂2 ∂2
aˆ = V , ˆb = Vαβ ,
12 αβ∂a ∂a 12 ∂bα∂bβ
1α 2β 1 2
(5.14)
∂2 ∂2
cˆ = , cˆ = .
12 ∂a ∂bα 21 ∂a ∂bα
1α 2 2α 1
These functions are responsible for various types of index contractions between the
frame field and curvatures. The action is invariant under complex conjugation S¯ =
2
S when the coefficients α, β, γ and ζ are real.
2
Because the general variation of the linearized curvatures is δR = D δΩ and
0
because the action is formulated in an AdS covariant way, integrating by parts
5
yields the variation
δS(s1,s2) = D Hˆ ∧δΩ(a ,b )∧R¯(a ,b ))| + c.c. part . (5.15)
2 0 1 1 2 2 ai=bi=0
ZM5
The derivative D produces the frame field each time it hits the compensator
0
D Vαβ = hαβ. Taking D hαβ = 0, h β = h Vβγ into account and using the
0 0 α αγ
notation Hαβ = Hβα = hα ∧hβγ, we find
γ
∂2 ∂2
D Hˆ = ρ H β cˆ +ρ H β cˆ
0 1 α ∂a ∂b β 12 2 α ∂a ∂b β 21
2α 1 1α 2
(cid:16)
(5.16)
∂2 ∂2
+ρ H ˆb +ρ Hαβ aˆ (cˆ )2s2 ,
3 αβ∂a ∂a 12 3 ∂bα∂bβ 12 12
1α 2β 1 2 (cid:17)
where
1 ∂
ρ = 1+p −2γ(p,q)+(α+β)(p,q) ,
1
2 ∂p
(cid:16) (cid:17)(cid:16) (cid:17)
1 ∂
ρ = 1+p −2ζ(p,q)+(α+β)(p,q) , (5.17)
2
2 ∂p
(cid:16) (cid:17)(cid:16) (cid:17)
1 ∂
ρ = q ζ(p,q)−γ(p,q) .
3
2 ∂p
(cid:16) (cid:17)
For the trivial solution ρ = 0, the covariant derivative of Hˆ vanishes, D Hˆ = 0,
i 0
and the corresponding action functional is a total derivative. It follows from (5.17)
that ρ = 0 whenever
i
(α+β)(p,q) = 2γ(p,q) , ζ(p,q) = γ(p,q) . (5.18)
Clearly, by adding total derivatives with the coefficients satisfying (5.18), we can
always set γ = 0 and β = 0 in the action (5.11), (5.12).
9
Generally, action (5.11) does not describe massless higher-spin fields, because
there are too many nonphysical dynamical variables associated with the extra fields.
To eliminate the corresponding degrees of freedom, we must fix the operator Hˆ in
an appropriate form by virtue of the decoupling condition [24, 25, 6]. It requires the
variation of the quadratic action with respect to the extra fields is identically zero,
δS(s1,s2)
2 ≡ 0 . (5.19)
δωt>0
To analyze the extra field decoupling condition, we observe that all gauge fields
of the extra type can be combined into a single irreducible su(2,2) tensor ξ(a,b)
satisfying (N − N − 2s − 2)ξ(a,b) = 0. Then, the variation of the extra fields
a b 2
becomes
δΩextra(a,b) = S+ξ(a,b) (5.20)
and the extra field decoupling condition (5.19) amounts to
∂ ∂
− (qρ )+ρ = 0,
2 3
∂p ∂q
(cid:16) (cid:17)
∂ ∂ (5.21)
− ρ = 0,
3
∂p ∂q
(cid:16) (cid:17)
ρ +ρ = 0.
1 3
Modulo total derivative contributions (5.18), the general solution to the system
(5.21) is
(p+q)s1−s2−1
γ(p,q) = 0, β(p,q) = 0, ζ(p,q) = ζ(0) ,
q
1
α(p,q) = −ζ(0)(s −s −1) dτ(pτ +q)s1−s2−2 = (5.22)
1 2
Z0
s1−s2−2 (s −s −1)!
= −ζ(0) 1 2 pkqs1−s2−k−2 .
(k +1)!(s −s −k −2)!
k=0 1 2
X
The factor q−1 appearing in ζ(p,q) can be removed by redefining ζ(p,q) → qζ(p,q).
This operation does remove the singularity because the last term in the operator
Hˆ in (5.12) contains the combination cˆ (cˆ )2s2, which is always q(cˆ )2s2−1 by the
21 12 12
definition of q in (5.14). An overall factor ζ(0) in front of the action of a given
spin (s ,s ) cannot be fixed from the analysis of the free action and represents the
1 2
residual ambiguity in the coefficients.
10