Consistent deformations of dual formulations of linearized gravity: A no-go result

Abstract

The consistent, local, smooth deformations of the dual formulation of linearized gravity involving a tensor field in the exotic representation of the Lorentz group with Young symmetry type (D-3,1) (one column of length D-3 and one column of length 1) are systematically 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. Comment: Reference added. Version to appear in Phys. Rev. D

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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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