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arXiv:hep-th/0210278v2 10 Feb 2003

DFPD/02/TH/25

ULB-TH-02/31

Consistent deformations of dual formulations of

linearized gravity: A no-go result

Xavier Bekaerta, Nicolas Boulangerb,1and Marc Henneauxb,c

aDipartimento di Fisica, Universit` a degli Studi di Padova, Via F. Marzolo 8,

I-35131 Padova, Italy

bPhysique Th´ eorique et Math´ ematique, Universit´ e Libre de Bruxelles, C.P. 231,

B-1050, Bruxelles, Belgium

cCentro de Estudios Cient´ ıficos, Casilla 1469, Valdivia, Chile

Abstract

The consistent, local, smooth deformations of the dual formulation

of linearized gravity involving a tensor field in the exotic representa-

tion of the Lorentz group with Young symmetry type (D − 3,1) (one

column of length D − 3 and one column of length 1) are systemati-

cally investigated. The rigidity of the Abelian gauge algebra is first

established. We next prove a no-go theorem for interactions involving

at most two derivatives of the fields.

1“Chercheur F.R.I.A., Belgium”

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1Introduction

The electric-magnetic duality is one of the most fascinating symmetries of

theoretical physics. Recently [1], dual formulations of linearized gravity [2]

have been systematically investigated with M-theory motivations in mind

[3, 4] (see also [5]). These dual formulations involve tensor fields in “ex-

otic” representations of the Lorentz group characterized by a mixed Young

symmetry type. There exist in fact three different dual formulations of lin-

earized gravity in generic spacetime dimension D. The first one is the familiar

Pauli-Fierz description based on a symmetric tensor hµν. The second one is

obtained by dualizing on one index only and involves a tensor Tλ1λ2···λD−3µ

with

Tλ1λ2···λD−3µ= T[λ1λ2···λD−3]µ,

T[λ1λ2···λD−3µ]= 0

(1.1)

(1.2)

where square brackets denote antisymmetrization with strength one. Finally,

the third one is obtained by dualizing on both indices and is described by

a tensor Cλ1···λD−3µ1···µD−3with Young symmetry type (D − 3,D − 3) (two

columns with D −3 boxes). Although one can write equations of motion for

this theory which are equivalent to the linearized Einstein equations, these

do not seem to follow (when D > 4) from a Lorentz-invariant action principle

in which the only varied field is Cλ1···λD−3µ1···µD−3. For this reason, we shall

focus here on the dual theory based on Tλ1λ2···λD−3µ.

The purpose of this paper is to determine all the consistent, local, smooth

interactions that this dual formulation admits. It is well known that the only

consistent (local, smooth) deformation of the Pauli-Fierz theory is - under

quite general and reasonable assumptions - given by the Einstein theory (see

[6] and the more recent works [7, 8] for systematic analyses). Because dual-

ization is a non-local process, one does not expect the Einstein interaction

vertex to have a local counterpart on the dual Tλ1λ2···λD−3µ-side. This does

not a priori preclude the existence of other local interaction vertices, which

would lead to exotic self-interactions of “spin-2” particles. Our main – and

somewhat disappointing – result is, however, that this is not the case.

The first instance for which Tλ1λ2···λD−3µtransforms in a true exotic rep-

resentation of the Lorentz group occurs for D = 5, where one has

T[αβ]γ≃

β γ

α

.

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The action of this dual theory is given in [2](see also [9, 10, 11]). We shall

explicitly investigate the T[αβ]γ-case in this paper and comment on general

gauge fields Tλ1λ2···λD−3µat the end.

Our precise result is that the free field dual theory based on Tλ1λ2µ, ad-

mits no consistent local deformation which (i) is Lorentz-invariant; and (ii)

contains no more than two derivatives of the field [i.e., the allowed interaction

terms under consideration contains at most ∂2T or (∂T)2]. No restriction is

imposed on the polynomial degree of the interaction. Our result confirms

previous unsuccessful attempts [2, 1, 12]. We also demonstrate the rigid-

ity, to first-order in the deformation parameter, of the algebra of the gauge

symmetries without making any assumption on the number of derivatives.

Besides their occurrence in dual formulations of linearized gravity, tensor

fields in exotic representations of the Lorentz group arise in the long-standing

related problem of constructing consistent interactions among particles with

higher spins [13, 14, 15, 16, 17]. A further motivation for the analysis of

exotic higher spin gauge fields come from recent developments in M-theory,

where a matching between the D = 11 supergravity equations [18] and the

E10|+10/K(E10) coset model equations (K(E10) being the maximal compact

subgroup of the split form of E10|+10of E10) was exhibited up to height 30

in the E10roots [19] (the relevance of E10in the supergravity context was

indicated much earlier in [20]). One possibility for going beyond this height

would be to introduce additional higher spin fields, most of which would

be in exotic representations of the Lorentz group. Indeed, a quick argument

shows that such fields might yield the exponentials associated with the higher

height E10-roots – if they can be consistently coupled to gravity, an unsolved

problem so far. The introduction of such additional massless fields would

also be in line with what one expects from string theory (in the high energy

limit where the string tension goes to zero [21]). The same motivations come

from the covariant coset construction of [22] where D = 11 supergravity is

conjectured to provide a non-linear realization of E11. The dual tensor field

Tλ1λ2···λ8µhas actually already been identified in connection with both the

E11[22] and the E10roots [19, 23]. Note that mixed symmetry fields appear

also in the models of [24, 25].

In order to investigate the consistent, local, smooth deformations of the

theory, we shall follow the cohomological approach of [26], based on the anti-

field formalism [27, 28, 29]. An alternative, Hamiltonian based deformation

point of view has been developed in [30]. One advantage of the cohomological

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approach, besides its systematic aspect, is that it minimizes the work that

must be done because most of the necessary computations are either already

in the literature [31] or are direct extensions of existing developments car-

ried out for 1-forms [32, 33], p-forms [34] or gravity [8, 35] (see also [36, 37]

for recent developments on the 1-form-p-form case). To a large extent, our

no-go theorem is obtained by putting together, in a standard fashion, var-

ious cohomological computations which have an interest in their own right

and which have been already published or can be obtained through by-now

routine techniques.

2 The free theory

2.1 Lagrangian, gauge symmetries

As stated above, we first restrict the explicit analysis to the case of a tensor

T with 3 indices, T = Tαβµ, which is dual to linearized gravity in D = 5 (but

we shall carry the analysis without specifying D, taken only to be stricly

greater than 4, D > 4, so that the theory carries local degrees of freedom).

The symmetry properties read

Tαβγ= T[αβ]γ,T[αβ]γ+ T[βγ]α+ T[γα]β= 0.(2.1)

As shown in [38, 39], the appropriate algebro-differential language for dis-

cussing gauge theories involving exotic representations of the Lorentz group

is that of multiforms, or more accurately, that of hyperforms2[41, 38]. Mul-

tiforms were discussed recently in [40] as an auxiliary tool for investigating

questions concerning N-complexes associated with higher spin gauge the-

ories. It turns out that hyperforms have been introduced much earlier in

the mathematical literature by Olver in the analysis of higher order Pfaffian

systems with integrability criteria (Olver, unpublished work [41]). We shall

not use here the language of multiforms or hyperforms, however, because the

relevant tensors involve only a few indices.

The Lagrangian for the gauge tensor field Tλ1λ2µreads

L = −1

12

?

F[αβγ]δF[αβγ]δ− 3F

ξ

[αβξ]F[αβλ]

λ

?

,(2.2)

2Hyperforms are in irreps of the general linear group, while multiforms sit in reducible

ones.

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where F is the tensor

F[αβγ]δ= ∂αT[βγ]δ+ ∂βT[γα]δ+ ∂γT[αβ]δ≡ 3∂[αTβγ]δ. (2.3)

The gauge invariances are

δσ,αT[αβ]γ= 2(∂[ασβ]γ+ ∂[ααβ]γ− ∂γααβ),(2.4)

where σαβand ααβare arbitrary symmetric and antisymmetric tensor fields.

The tensor F is invariant under the σ-gauge symmetries, but not under

the α-ones. To get a completely gauge-invariant object, one must take one

additional derivative. The tensor

E[αβδ][εγ]≡1

2(∂εF[αβδ]γ− ∂γF[αβδ]ε) (2.5)

is easily verified to be gauge invariant. Moreover its vanishing implies that

T[αβ]γis pure-gauge [38]. The most general gauge invariant object depends

on the field Tαβµand its derivatives only through the “curvature” E[αβδ][εγ]

and its derivatives. It is convenient to define the Ricci-like tensor E[αβ]γand

its trace :

E[αβ]γ= ηεδE[αβδ][εγ],Eα= ηβγE[αβ]γ. (2.6)

The equations of motion are then

δL

δT[αβ]γ

= 3[E[αβ]γ+ ηγ[αEβ]] = 0.(2.7)

Because the action is gauge-invariant, the equations of motion fulfill the

“Bianchi identities”

∂α(E[αβ]γ+ ηγ[αEβ]) ≡ 0.(2.8)

One easy way to check these identities is to observe that one has

δL

δT[µν]ρ

≡ ∂λGλµνρ

(2.9)

where the tensor Gλµνρis completely antisymmetric in its first three indices,

Gλµνρ= G[λµν]ρ. Explicitly,

Gλµνρ=3

2

?

∂[λTµν]ρ− ηρλ∂[µTνα]

α− ηρµ∂[νTλα]

α− ηρν∂[λTµα]

α

?

. (2.10)

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The gauge symmetries (2.4) are reducible. Indeed,

δ˜ σ, ˜ αT[αβ]γ≡ 0(2.11)

when

˜ σαβ= 6∂(αγβ),˜ ααβ= 2∂[αγβ]

(2.12)

where γαare arbitrary fields. There is no further local reducibility identity.

The problem of introducing (smooth) consistent interactions is that of

smoothly deforming the Lagrangian (2.2),

L → L + gL1+ g2L2+ ···,(2.13)

the gauge transformations (2.4),

δσ,αT[αβ]γ= (2.4) + g δ(1)

σ,αT[αβ]γ+ g2δ(2)

σ,αT[αβ]γ+ ···(2.14)

and the reducibility relations (2.12) in such a way that (i) the new action

is invariant under the new gauge symmetries; and (ii) the new gauges sym-

metries reduce to zero on-shell when the gauge parameters fulfill the new

reducibility relations. By developing these requirements order by order in

the deformation parameter g, one gets an infinite number of consistency con-

ditions, one at each order.

We shall impose the further requirement that the first order vertex L1

be Lorentz-invariant. Under this sole condition (together with consistency),

we show that one can always redefine the fields and the gauge parameters

in such a way that the gauge structure is unaffected by the deformation (to

first order in g). That is, the gauge transformations remain abelian and the

reducibility relations remain unchanged (“rigidity of the gauge algebra”).

We next restrict the deformations to contain at most two derivatives of the

fields, as the original free Lagrangian. This still leave a priori an infinite

number of possibilities, of the schematic form Tk(∂T)2where k is arbitrary (a

term Tl∂2T is of course equivalent to Tl−1(∂T)2upon integration by parts).

We show, however, that within this infinite class, there is no non-trivial

deformation. Any deformation can be redefined away by a local change of

field variables.

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2.2BRST differential

As shown in [26], the first-order consistent local interactions correspond to

elements of the cohomology HD,0(s|d) of the BRST differential s modulo the

spacetime exterior derivative d, in maximum form degree D and in ghost

number 0. That is, one must compute the general solution of the cocycle

condition

sa + db = 0,(2.15)

where a is a D-form of ghost number zero and b a (D − 1)-form of ghost

number one, with the understanding that two solutions a and a′of (2.15)

that differ by a trivial solution

a′= a + sm + dn (2.16)

should be identified as they define the same interactions up to field redefini-

tions. The cochains a, b, etc that appear depend polynomially on the field

variables (including ghosts and antifields) and their derivatives up to some

finite order (“local polynomials). Given a non trivial cocycle a of HD,0(s|d),

the corresponding first-order interaction vertex L1is obtained by setting the

ghosts equal to zero.

According to the general rules, the spectrum of fields and antifields is

given by

• the fields T[αβ]γ, with ghost number zero and antifield number zero;

• the ghosts S(αβ)and A[αβ]with ghost number one and antifield number

zero;

• the ghosts of ghosts Cαwith ghost number two and antifield number

zero, which appear because of the reducibility relations;

• the antifields T∗[αβ]γ, with ghost number minus one and antifield num-

ber one;

• the antifields S∗(αβ)and A∗[αβ]: ghost number minus two and antifield

number two;

• the antifields C∗αwith ghost number three and antighost number three.

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The antifield number is also called “antighost number”. Since the theory

at hand is a free theory, the BRST differential takes the simple form

s = δ + γ(2.17)

The decomposition of s into δ plus γ is dictated by the antifield number :

δ decreases the antifield number by one unit, while γ leaves it unchanged.

Combining this property with s2= 0, one concludes that

δ2= 0, δγ + γδ = 0, γ2= 0.(2.18)

A grading is associated to each of these differentials : γ increases by one unit

the “pure ghost number” denoted puregh while δ increases the “antighost

number” antigh by one unit. The ghost number gh is defined by

gh = puregh − antigh.(2.19)

The action of the differentials γ and δ on all the fields of the formalism is

displayed in the following array which indicates also the pureghost number,

antighost number, ghost number and grassmannian parity of the various

fields :

Zγ(Z)

γT[αβ]γ

6∂(αCβ)

2∂[αCβ]

0

0

0

0

0

δ(Z)

0

0

0

0

puregh(Z)

0

1

1

2

0

0

0

0

antigh(Z)

0

0

0

0

1

2

2

3

gh(Z)

0

1

1

2

−1

−2

−2

−3

parity

0

1

1

0

1

0

0

1

T[αβ]γ

S(αβ)

A[αβ]

Cα

T∗[αβ]γ

S∗αβ

A∗αβ

C∗α

3[E[αβ]γ+ ηγ[αEβ]]

−∂γ(T∗[γα]β+ T∗[γβ]α)

−3∂γ(T∗[γα]β− T∗[γβ]α)

6∂µS∗µα+ 2∂µA∗µα

where γT[αβ]γ= 2(∂[αSβ]γ+ ∂[αAβ]γ− ∂γAαβ).

It is convenient to perform a change of variables in the antigh = 2 sector

in order for the Koszul-Tate differential to take a simpler expression when

applied on all the antifields of antigh ≥ 2. We define

C∗αβ= 3S∗αβ+ A∗αβ. (2.20)

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It leads to the following simple expressions

δC∗αβ

δC∗µ

= −6∂γT∗[γα]β,

= 2∂νC∗νµ.

(2.21)

(2.22)

2.3Strategy

To compute HD,0(s|d), one proceeds as in [32, 33] : one expands the cocycle

condition sa + db = 0 according to the antifield number. To analyse this re-

sulting equations, one needs to know the cohomological groups H(γ), H(γ|d)

in strictly positive antighost number, H(δ|d) and Hinv(δ|d).

3 Standard results

Of the cohomologies just listed, some are already known while some can be

computed straightforwardly.

3.1Cohomology of γ

The cohomology of γ (space of solutions of γa = 0 modulo trivial cobound-

aries of the form γb) has been explicitly worked out in [31] and turns out to

be generated by the following variables,

• the antifields and all their derivatives, denoted by [Φ∗],

• the undifferentiated ghosts of ghosts Cµ,

• the following “field strength” components of the ghosts A[αβ]: HA

∂[αAβγ](but not their derivatives, which are exact),

[αβγ]≡

• the T-field strength components defined in (2.5) and all their derivatives

denoted by [E[αβγ][δε]].

Therefore, the cohomology of γ is isomorphic to the algebra

?

f

?

[E[αβγ][δε]],[Φ∗],Cµ,HA

[αβγ]

??

(3.1)

of functions of the generators. The ghost-independent polynomials α([E[αβγ][δε]],[Φ∗])

are called “invariant polynomials”.

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Comments

Let

variables Cµand HA

basis, hence for any γ-cocycle α

?

ωI?

Cµ,HA

[αβγ]

[αβγ]. Any element of H(γ) can be decomposed in this

??

be a basis of the algebra of polynomials in the

γα = 0⇔α = αI([E[αβγ][δε]],[Φ∗]) ωI?

Cµ,HA

[αβγ]

?

+ γβ(3.2)

where the αIare invariant polynomials. Furthermore, αIωIis γ-exact if and

only if all the coefficients αIare zero

αIωI= γβ,⇔αI= 0,for all I.(3.3)

Another useful property of the ωIis that their derivatives are γ-exact and

thus, in particular,

dωI= γˆ ωI

(3.4)

for some ˆ ωI.

3.2General properties of H(γ|d)

The cohomological space H(γ|d) is the space of equivalence classes of forms

a such that γa+db = 0, identified by the relation a ∼ a′⇔ a′= a+γc+df.

We shall need properties of H(γ|d) in strictly positive antighost (= antifield)

number.To that end, we first recall the following theorem on invariant

polynomials (pure ghost number = 0) :

Theorem 3.1 In form degree less than n and in antifield number strictly

greater than 0, the cohomology of d is trivial in the space of invariant poly-

nomials.

The argument runs as in [32, 33], to which we refer for the details.

Theorem 3.1, which deals with d-closed invariant polynomials that involve

no ghosts (one considers only invariant polynomials), has the following useful

consequence on general γ-mod-d-cocycles with antigh > 0.

Consequence of Theorem 3.1

If a has strictly positive antifield number (and involves possibly the ghosts),

the equation γa + db = 0 is equivalent, up to trivial redefinitions, to γa = 0.

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That is,

γa + db = 0,

antigh(a) > 0

?

⇔

?

γa′= 0,

a′= a + dc

.(3.5)

Thus, in antighost number > 0, one can always choose representatives of

H(γ|d) that are strictly annihilated by γ. Again, see [32, 33].

3.3Characteristic cohomology H(δ|d)

We now turn to the groups H(δ|d), i.e., to the solutions of the condition

δa + db = 0 modulo trivial solutions of the form δm + dn. As shown in

[32], these groups are isomorphic to the groups H(d|δ) of the characteristic

cohomology, describing ordinary and higher order conservation laws (i.e., n-

forms built out of the fields and their derivatives that are closed on-shell).

Without loss of generality, one can assume that the solution a of δa+db = 0

does not involve the ghosts, since any solution that vanishes when the ghosts

are set equal to zero is trivial [42]. By applications of the results and methods

of [32], one can establish the following theorems (in HD

degree and q the antighost (= antifield) number) :

q(δ|d), D is the form

Theorem 3.1 The cohomology groups HD

strictly greater than 3,

q(δ|d) vanish in antifield number q

HD

q(δ|d) = 0 for q > 3. (3.6)

Theorem 3.2 A complete set of representatives of HD

antifields C∗µconjugate to the ghost of ghosts, i.e.,

3(δ|d) is given by the

δaD

3+daD−1

2

= 0 ⇒ aD

3= λµC∗µdx0∧dx1∧...∧dxD−1+δbD

4+dbD−1

3

(3.7)

where the λµare constants.

Theorem 3.3 In antighost number 2, the general solution of

δaD

2+ daD−1

1

= 0(3.8)

reads, modulo trivial terms,

aD

2= C∗µνtµνρxρdx0∧ dx1∧ ... ∧ dxD−1

(3.9)

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where tµνρis an arbitrary, completely antisymmetric, constant tensor, tµνρ=

t[µνρ]. If one considers cochains a that have no explicit x-dependence (as it

is necessary for constructing Poincar´ e-invariant Lagrangians), one thus find

that the cohomological group HD

2(δ|d) vanishes.

Comment

The cycle C∗µis associated to the conservation law d∗G ≈ 0 for the

(D−3)-form∗G dual to G[λµν]ρ(the equations of motion read ∂λGλµνρ≈ 0 ).

The cycle C∗µνtµνρxρis associated to the conservation law ∂λIλσµνρ≈ 0 where

Iλσµνρis equal to the tensor Gλσµνxρ+ 3ηλµTνρσ− 3ηλµησνTαρ

antisymmetrized in the three indices µ, ν, ρ and in the pair λ, σ. The above

theorems provide a complete description of HD

these groups are finite-dimensional. In contrast, the group HD

is related to ordinary conserved currents, is infinite-dimensional since the

theory is free. It is not computed here.

αcompletely

k(δ|n) for k > 1 and show that

1(δ|d), which

3.4Invariant characteristic cohomology : Hinv(δ|d)

The crucial result that underlies all consistent interactions deals not with the

general cohomology of δ modulo d but rather with the invariant cohomology

of δ modulo d. The group Hinv(δ|d) is important because it controls the

obstructions to removing the antifields from a s-cocycle modulo d, as we

shall see explicitly below.

The central theorem that gives Hinv(δ|d) in antighost number ≥ 2 is

Theorem 3.1 Assume that the invariant polynomial ap

k = antifield number) is δ-trivial modulo d,

k(p = form-degree,

ap

k= δµp

k+1+ dµp−1

k

(k ≥ 2). (3.10)

Then, one can always choose µp

k+1and µp−1

k

to be invariant.

Hence, we have Hn,inv

Theorem 3.2 and Hn,inv

cochains), by Theorem 3.3.

The proof of this theorem proceeds exactly as the proofs of similar the-

orems established for vector fields [33], p-forms [34] or gravity [8]. We shall

therefore skip it.

k

(δ|d) = 0 for k > 3 while Hn,inv

(δ|d) vanishes (in the space of translation-invariant

3

(δ|d) is given by

2

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4Rigidity of the gauge algebra

We can now proceed with the derivation of the cohomology of s modulo d in

form degree D and in ghost number zero. A cocycle of H0,D(s|d) must obey

sa + db = 0(4.1)

(besides being of form degree D and of ghost number 0). To analyse (4.1), we

expand a and b according to the antifield number, a = a0+a1+...+ak, b =

b0+b1+...+bk, where, the expansion stops at some finite antifield number [33].

We recall [26] (i) that the antifield-independent piece a0is the deformation of

the Lagrangian; (ii) that a1, which is linear in the antifields T∗[αβ]γcontains

the information about the deformation of the gauge transformations of the

fields, given by the coefficients of T∗[αβ]γ; (iii) that a2contains the information

about the deformation of the gauge algebra (the term C∗

C∗

functions appearing in the commutator of two gauge transformations, while

the term T∗T∗CC gives the on-shell terms) and about the deformation of the

reducibility functions (terms containing the ghosts of ghosts and the antifields

conjugate to the ghosts); and (iv) that the ak(k > 3) give the information

about the deformation of the higher order structure functions, which appear

only when the algebra does not close off-shell. Thus, if one can show that

the most general solution a of (4.1) stops at a1, the gauge algebra is rigid :

it does not get deformed to first order.

Writing s as the sum of γ and δ, the equation sa + db = 0 is equivalent

to the system of equations δai+ γai−1+ dbi−1 = 0 for i = 1,···,k, and

γak+ dbk= 0.

AfA

BCCBCCwith

A≡ S∗αβ,A∗αβand CA≡ Sαβ,Aαβgives the deformation of the structure

4.1Terms ak, k > 3

To begin with, let us assume k > 3. Then, using the consequence of theorem

3.1, one may redefine akand bkso that bk= 0, i.e., γak= 0. Then, ak= αJωJ

(up to trivial terms), where the αJare invariant polynomials and where the

{ωJ} form a basis of the algebra of polynomials in the variables Cµ and

HA

γak= 0 , we get dγbk−1= 0 i.e. γbk−1+ dmk−1= 0; and then, thanks again

to the consequence of theorem 3.1, bk−1can also be assumed to be invariant,

[αβγ]. Acting with γ on the second to last equation and using γ2= 0 ,

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bk−1= βJωJ. Substituting these expressions for akand bk−1in the second to

last equation, we get :

δ[αJωJ] + d[βJωJ] = γ(...).(4.2)

This equation implies

[δαJ+ dβJ]ωJ= γ(...) (4.3)

because the exterior derivative of a ωJis equivalent to zero in H(γ). Then,

as dicussed at the end of subsection 3.1, this leads to :

δαJ+ dβJ= 0 ,∀ J. (4.4)

If the antifield number of αJis strictly greater than 3, the solution is trivial,

thanks to our results on the cohomology of δ modulo d : αJ = δµJ+ dνJ.

Furthermore, theorem 3.1 tells us that µJ and νJ can be chosen invariants.

This is the crucial place where we need theorem 3.1. Thus ak = (δµJ+

dνJ)ωJ= s(µJωJ) + d(νJωJ) ± νJdωJ. The last term νJdωJis equal to

νJsˆ ωJand differs from the s-exact term s(±νJˆ ωJ) by the term ±δνJˆ ωJ,

which is of lowest antifield number. Trivial redefinitions enable one to set

ak to zero. Once this is done, βJ must satisfy dβJ = 0 and is then exact

in the space of invariant polynomials, βJ = dρI, and bk−1can be removed

by appropriate trivial redefinitions. One can next repeat the argument for

antifield number k − 1, etc, until one reaches antifield number 3. This case

deserves more attention, but what we can stress already now is that we can

assume that the expansion of a in sa + db = 0 stops at antifield number 3

and takes the form a = a0+ a1+ a2+ a3with b = b0+ b1+ b2. Note that

this result is independent of any condition on the number of derivatives or

of Lorentz invariance. These requirements have not been used so far. The

crucial ingredient of the proof is that the cohomological groups Hinv

which control the obstructions to remove akfrom a, vanish for k > 3.

k (δ|d),

4.2Computation of a3

We have now the following descent:

δa1+ γa0+ db0 = 0 ,

δa2+ γa1+ db1 = 0 ,

δa3+ γa2+ db2 = 0 ,

(4.5)

(4.6)

(4.7)

(4.8) γa3 = 0 .

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We write a3 = αIωIand b2 = βIωI. Proceeding as before we find that a

necessary (but not sufficient) condition for a3to be a non-trivial solution of

(4.7), so that a2exists, is that αIbe a non-trivial element of Hn

Theorem 3.2 imposes then αI∼ C∗µ. We then have to complete this αIwith

an ωIof ghost number 3 in order to build a candidate αIωIfor a3. There

are a priori a lot of possibilities to achieve this, but if one demands Lorentz

invariance, only two possibilities emerge:

3(δ|d). The

a3 = C∗µHµαβHανρHβ

a3 = C∗µεµνρλσHνρλCσ,

νρ,(4.9)

(4.10)

where we recall that Hµαβ≡ ∂[µAαβ]∈ H(γ) at ghost number one, and Cσ

∈ H(γ) at ghost number two. Since C∗µhas antighost number three [i.e.

ghost number (−)3], we indeed have two ghost-number-zero a3candidates.

The first is quartic in the fields and, if consistent, would lead to a quartic

interaction vertex. The second is cubic and, if consistent, would lift to an a0

which breaks PT invariance.

However, neither of these candidates can be lifted all the way to a0. Both

get obstructed at antighost number one: a2exists, but there is no a1that

solves (4.6) (given a3and the corresponding a2). The computation is direct

but not very illuminating and so will not be reproduced here.

4.3 Computation of a2

Continuing with our analysis, we set a3= 0 and get the system

δa1+ γa0+ db0 = 0 ,

δa2+ γa1+ db1 = 0 ,

(4.11)

(4.12)

(4.13) γa2 = 0 .

Now a2has to be found in Hn

of translation-invariant deformations), as shown in theorem 3.3. We can thus

conclude that there is no possibility of deforming the free theory to obtain an

interacting theory whose gauge algebra is non-abelian. To obtain this no-go

result we just asked for locality, Lorentz invariance and the assumption that

the deformed theory reduces smoothly to the free one as the deformation

parameter goes to zero.

2(δ|d), but this latter group vanishes (in the space

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