On the uniqueness of D=11 interactions among a graviton, a massless gravitino and a three-form. I: Pauli-Fierz and three-form

0.35 MB·English

by  E. M. Cioroianu

D
On the uniqueness of = 11 interactions among
a graviton, a massless gravitino and a three-form.
9
I: Pauli-Fierz and three-form
0
0
2
E. M. Cioroianu∗, E. Diaconu†, S. C. Sararu‡
n
Faculty of Physics, University of Craiova,
a
J 13 Al. I. Cuza Street Craiova, 200585, Romania

3
2

]
h Abstract
t
- Cross-couplings between a massless spin-two field (described in the
p
free limit by the Pauli-Fierz action) and an Abelian three-form gauge
e
h field inD=11 areinvestigated intheframework of thedeformation the-
[ ory based on local BRST cohomology. These consistent interactions are
obtained on the grounds of smoothness in the coupling constant, local-
1
ity, Lorentz covariance, Poincar´e invariance, and the presence of at most
v
two derivatives in the interacting Lagrangian. Our results confirm the
1
uniquenessoftheeleven-dimensionalinteractionsbetweenagraviton and
5
a three-form prescribed by General Relativity.
6
3 PACS number: 11.10.Ef
.
1
0 1 Introduction
9
0
A key point in the development of the BRST formalism was its cohomological
:
v understanding, which allowed,among others,a useful investigationof many in-
i
X terestingaspectsrelatedtotheperturbativerenormalizationproblem[1]–[5],the
anomaly-trackingmechanism [5]–[10], the simultaneous study of localand rigid
r
a invariancesofagiventheory[11]aswellasthereformulationoftheconstruction
ofconsistentinteractionsingaugetheories[12]–[16]intermsofthe deformation
theory [17]–[21] or, actually, in terms of the deformation of the solution to the
masterequation. The impossibilityofcross-interactionsamongseveralEinstein
(Weyl)gravitons,seeRef. [22](orrespectivelyRef.[23]), andofcross-couplings
among different Einstein gravitons in the presence of matter fields [22, 24]–[27]
has recently been shown by means of cohomological arguments. In the same
context the uniqueness of D =4, N =1 supergravity was proved in Ref. [28].
∗e-mailaddress: [email protected]
†e-mailaddress: [email protected]
‡e-mailaddress: [email protected]
1
On the other hand, D = 11, N = 1 supergravity [29, 30] has regained a
central role with the advent of M-theory, whose QFT (local) limit it is. Of
the many special properties of D = 11, N = 1 supergravity, one of the most
strikingis that itforbids a cosmologicalterm. The proofof this resulthas been
done in Ref. [31] using a combined technique — the standard Noether current
method and a cohomological approach. It is known that the field content of
D =11, N =1 supergravityis quite simple; it comprises a graviton,a massless
Majoranaspin-3/2field,andathree-formgaugefield. The analysisofallpossi-
ble interactions in D =11 related to this field content necessitates the study of
cross-couplingsinvolvingeachpairofthesesortsoffieldsandthentheconstruc-
tionofsimultaneousinteractionsamongallthe three types offields. One ofthe
mostefficientandmeanwhileelegantapproachestothe problemofconstructing
consistentinteractions in gaugefield theories1 is that basedon the deformation
technique [17] combined with local BRST cohomology [32, 33]. This approach
relies oncomputing the deformations of the solutionto the master equation for
the interacting theory with the help of the ‘free’ BRST cohomology. Our main
aim is to constructall consistent interactions in D =11 that can be addedto a
freetheorydescribingaPauli-Fierzgraviton,amasslessRarita-Schwingergrav-
itino, and an Abelian three-form gauge field from the deformation of the ‘free’
solution to the master equation such that the interactions satisfy some general
and quite natural assumptions (smoothness in the coupling constant, locality,
Lorentz covariance,Poincar´einvariance, and preservation of the differential or-
der of the free field equations at the level of the coupled theory). One of the
finaloutcomesof this procedurewill be the quest forthe uniqueness ofD =11,
N = 1 SUGRA. In order to organize the results as logical as possible, to ex-
poseindetailthe cohomologicalaspectsinvolved,and(lastbutnotleast)make
variouscommentsonandcomparisonswith otherresults fromthe literaturewe
chosetosplitourworkintofourmainparts. Thefirstthreearededicatedtothe
construction of consistent interactions that involve only two of the three types
of fields under considerations: i) a graviton and a three-form (present paper);
ii) a three-form and massless gravitini [34]; iii) massless gravitini and a gravi-
ton [35]. The fourth and last part [36] will put the things together and present
what happens when all these fields are present: what new vertices appear, how
consistentare those obtainedfrom the previoussteps, and how does the overall
coupled theory looks like.
In this workweimplement the firstof the four steps explainedinthe above,
namely we analyze the cross-couplings between a massless spin-two field (de-
scribed in the free limit by the Pauli-Fierz action [37, 38]) and an Abelian
three-form gauge field in eleven spacetime dimensions. The cross-interactions
are obtained under the hypotheses of smoothness of the interactions in the
coupling constant, locality, Poincar´e invariance, Lorentz covariance, and the
presence of at most two derivatives in the Lagrangian of the interacting the-
ory (the same number of derivatives like in the free Lagrangian). Our results
1By‘consistent’ wemeanthat theinteractingtheory preservesboththefieldcontent and
thenumberofindependent gaugesymmetriesofthefreeone.
2
are obtained in the context of the deformation of the solution to the master
equation.
We compute the interaction terms to order two in the coupling constant.
In this way we obtain that the first two orders of the interacting Lagrangian
resulting from our setting originate in the development of the full interacting
Lagrangian(in eleven spacetime dimensions)
2
˜= √g R 2λ2Λ + h−A,
L λ2 − L
(cid:0) (cid:1)
where the cross-coupling part reads as
1
Lh−A =−2 4!√gF¯µνρλF¯µνρλ+λqǫµ1...µ11A¯µ1µ2µ3F¯µ4...µ7F¯µ8...µ11,
·
withg =detg ,Λthecosmologicalconstant,λthecouplingconstant,andqan
µν
arbitrary, real constant. Consequently, we show the uniqueness of interactions
describedby ˜. TheaboveinteractingLagrangianforΛ=0isapartofD =11,
L
N =1 SUGRA Lagrangian. We note that the gravitonsector is allowedat this
stagetoincludeacosmologicalterm,unlikeD =11,N =1SUGRA.Thisisnot
a surprise since it is the simultaneous presence of all fields (supplemented with
massless gravitini) that ensures the annihilation of the cosmological constant,
as it will be made clear in Ref. [36].
This paper is organizedin six sections. In section 2 we construct the BRST
symmetry of the free model, consisting in a Pauli-Fierz and an Abelian three-
form gauge field. Section 3 briefly addresses the deformation procedure based
onBRSTsymmetry. Insection4wecomputethefirsttwoordersoftheinterac-
tionsbetweenthemasslessspin-twofieldandanAbelianthree-formgaugefield.
Section 5 is devoted to analyzing the deformed theory obtained in the previous
section. In this contextwe obtaina possible candidate that describes the inter-
actingtheorytoallordersinthecouplingconstant. Section6isdedicatedtothe
investigation of the uniqueness of interactions described by the candidate em-
phasizedin the previous section. The last section exposes the main conclusions
on this paper.
2 Free model: Lagrangian formulation and BRST
symmetry
OurstartingpointisrepresentedbyafreeLagrangianaction,writtenasthesum
between the linearized Hilbert-Einstein action (also known as the Pauli-Fierz
action)andtheactionforanAbelianthree-formgaugefieldinelevenspacetime
dimensions
1
SL[h ,A ] = d11x (∂ h )(∂µhνρ)+(∂ hµρ)(∂νh )
0 µν µνρ −2 µ νρ µ νρ
Z (cid:18)
1 1
(∂ h)(∂ hνµ)+ (∂ h)(∂µh) F Fµνρλ
µ ν µ µνρλ
− 2 − 2 4!
· (cid:19)
3
d11x h+ A . (1)
≡ L L0
Z
(cid:0) (cid:1)
Throughoutthepaperweworkwiththeflatmetricof‘mostlyminus’signature,
σ = (+ ). In the above h denotes the trace of the Pauli-Fierz field,
µν
−···−
h= σ hµν, and F denotes the field-strength of the three-form gauge field
µν µνρλ
(F ∂ A ). The notation [µ...ν] (respectively (µ...ν)) signifies anti-
µνρλ [µ νρλ]
≡
symmetry (respectively symmetry) with respect to all indices between brackets
withoutnormalizationfactors(i.e.,theindependenttermsappearonlyonceand
arenotmultipliedbyoverallnumericalfactors). Thetheorydescribedbyaction
(1) possesses an Abelian generating set of gauge transformations
δ h =∂ ǫ , δ A =∂ ε , (2)
ǫ,ε µν (µ ν) ǫ,ε µνρ [µ νρ]
wherethegaugeparametersǫΓ1 ǫµ,εµν arebosonicfunctions, withthelast
≡{ }
set completely antisymmetric. We observe that if in (2) we make the transfor-
mations
ε ε(θ) =∂ θ , (3)
µν → µν [µ ν]
then the gauge variation of the three-form identically vanishes
δ A 0. (4)
ε(θ) µνρ ≡
Moreover,if in (3) we perform the changes
θ θ(φ) =∂ φ, (5)
µ → µ µ
with φ an arbitrary scalar field, then the transformed gauge parameters from
(3) identically vanish
(θ(φ))
ε 0. (6)
µν
≡
Meanwhile, there is no nonvanishing local transformation of φ that annihilates
θ(φ) of the form (5), and hence no further local reducibility identity. All these
µ
allow us to conclude that the generating set of gauge transformations given
in (2) is off-shell, second-stage reducible. It is obvious that the accompanying
gauge algebra is Abelian.
In order to construct the BRST symmetry for (1) we introduce the field,
ghost, and antifield spectra
ΦΓ0 =(hµν,Aµνρ), Φ∗Γ0 =(h∗µν,A∗µνρ) (7)
ηΓ1 =(ηµ,Cµν), ηΓ∗1 =(η∗µ,C∗µν), (8)
ηΓ2 =(Cµ), ηΓ∗2 =(C∗µ), (9)
ηΓ3 =(C), η∗ =(C∗). (10)
Γ3
The fermionic ghosts ηΓ1 respectively correspondto the bosonicgauge parame-
tersǫΓ1 from(2),thebosonicghostsforghostsηΓ2 areassociatedwiththefirst-
stage reducibility parameters θ in (3), while the fermionic ghost for ghost for
µ
ghost ηΓ3 is present due to the second-stage reducibility parameter φ from (5).
4
The star variables represent the antifields of the corresponding fields/ghosts.
Their Grassmann parities are obtained via the standard rule of the BRST
method ε(χ∗)= ε χΓ +1 mod2, where we employed the notations
Γ
χΓ = (cid:0)Φ(cid:0)Γ0,η(cid:1)Γ1,η(cid:1)Γ2,ηΓ3 , χ∗ = Φ∗ ,η∗ ,η∗ ,η∗ . (11)
Γ Γ0 Γ1 Γ2 Γ3
Sinceboththe(cid:0)gaugegenerators(cid:1)andtheredu(cid:0)cibilityfunctionsf(cid:1)orthismodel
are field-independent, it follows that the BRST differential s reduces to
s=δ+γ, (12)
where δ is the Koszul-Tate differential and γ denotes the exterior longitudinal
derivative. The Koszul-Tate differential is graded in terms of the antighost
number(agh,agh(δ)= 1,agh(γ)=0)andenforcesaresolutionofthealgebra
−
ofsmoothfunctionsdefinedonthestationarysurfaceoffieldequationsforaction
(1), C∞(Σ), Σ : δSL/δΦα0 = 0. The exterior longitudinal derivative is graded
0
in terms of the pure ghost number (pgh, pgh(γ) = 1, pgh(δ) = 0) and is
correlated with the original gauge symmetry via its cohomology in pure ghost
numberzerocomputedinC∞(Σ),whichisisomorphictothealgebraofphysical
observables for this free theory. These two degrees of the generators (7)–(10)
from the BRST complex are valued as
pgh ΦΓ0 =0, pgh ηΓk =k, (13)
pgh(cid:0)Φ∗Γ0(cid:1)=0, pgh(cid:0)ηΓ∗k(cid:1)=0, (14)
agh(cid:0)ΦΓ0(cid:1)=0, agh(cid:0)ηΓk(cid:1)=0, (15)
agh(cid:0)Φ∗Γ0(cid:1)=1, agh(cid:0)ηΓ∗k(cid:1)=k+1, (16)
(cid:0) (cid:1) (cid:0) (cid:1)
for k =1,3. The actions of the differentials δ and γ on the generatorsfrom the
BRST complex are given by
1
δh∗µν =2Hµν, δA∗µνρ = ∂ Fµνρλ, (17)
λ
3!
δη∗µ = 2∂ h∗µν, δC∗µν = 3∂ A∗µνρ, (18)
ν ρ
− −
δC∗µ = 2∂ C∗µν, δC∗ = ∂ C∗µ, δχΓ =0, (19)
ν µ
− −
γχ∗ =0, γh =∂ η , γA =∂ C , (20)
Γ µν (µ ν) µνρ [µ νρ]
γη =0, γC =∂ C , γC =∂ C, γC =0. (21)
µ µν [µ ν] µ µ
In the above Hµν = Kµν 1σµνK is the linearized Einstein tensor, with Kµν
− 2
and K the linearized Ricci tensor and respectively the linearized scalar curva-
ture, both obtained from the linearized Riemann tensor K = 1∂ h
µναβ 2 [µ ν][α,β]
via its trace and respectively double trace: K = σνβK and respectively
µα µναβ
K =σµασνβK .
µναβ
The BRST differential is known to have a canonical action in a structure
namedantibracketanddenotedbythesymbol(,)(s =(,S)),whichisobtained
· ·
by considering the fields/ghosts respectively conjugated to the corresponding
antifields. ThegeneratoroftheBRSTsymmetryisabosonicfunctionalofghost
5
number zero, which is solution to the classicalmaster equation (S,S)=0. The
full solution to the master equation for the free model under study reads as
Sh,A =SL+ d11x h∗µν∂ η +A∗µνρ∂ C +C∗µν∂ C +C∗µ∂ C .
0 (µ ν) [µ νρ] [µ ν] µ
Z
(cid:0) ((cid:1)22)
The solution to the master equation encodes all the information on the gauge
structure of a given theory.
3 Deformation of the solution to the master equa-
tion: a brief review
Webeginwitha“free”gaugetheory,describedbyaLagrangianactionSL ΦΓ0 ,
0
invariant under some gauge transformations δǫΦΓ0 = ZΓΓ01ǫΓ1, i.e. δδΦSΓ0L0Z(cid:2)ΓΓ01 =(cid:3)
0, and consider the problem of constructing consistent interactions among the
fields ΦΓ0 such that the couplings preserve the field spectrum and the original
numberofgaugesymmetries. Thismatteris addressedbymeansofreformulat-
ingtheproblemofconstructingconsistentinteractionsasadeformationproblem
of the solution to the master equation corresponding to the “free” theory [17].
Such a reformulationis possible due to the fact that the solution to the master
equation contains all the information on the gauge structure of the theory. If
an interacting gauge theory can be consistently constructed, then the solution
S to the master equation associated with the “free” theory, (S,S) = 0, can be
deformed into a solution S¯
S S¯=S+λS +λ2S + =S+λ dDxa+λ2 dDxb+ (23)
1 2
→ ··· ···
Z Z
of the master equation for the deformed theory
S¯,S¯ =0, (24)
suchthatboththeghostandantifi(cid:0)eldsp(cid:1)ectraoftheinitialtheoryarepreserved.
Equation (24) splits, according to the various orders in the coupling constant
(deformation parameter) λ, into a tower of equations:
(S,S) = 0 (25)
2(S ,S) = 0 (26)
1
2(S ,S)+(S ,S ) = 0 (27)
2 1 1
(S ,S)+(S ,S ) = 0 (28)
3 1 2
.
.
.
Equation (25) is fulfilled by hypothesis. The next equation requires that
the first-order deformation of the solution to the master equation, S , is a co-
1
cycleofthe“free”BRSTdifferentials,sS =0. However,onlycohomologically
1
6
nontrivialsolutions to(26)shouldbe takenintoaccount,since the BRST-exact
ones can be eliminated by some (in general nonlinear) field redefinitions. This
means that S pertains to the ghost number zero cohomological space of s,
1
H0(s), which is generically nonempty because it is isomorphic to the space of
physicalobservablesofthe“free”theory. Ithasbeenshown(byofthe triviality
of the antibracket map in the cohomology of the BRST differential) that there
are no obstructions in finding solutions to the remaining equations, namely
(27)–(28), etc. However, the resulting interactions may be nonlocal and there
might even appear obstructions if one insists on their locality. The analysis of
these obstructions can be done with the help of cohomologicaltechniques.
4 Consistent interactions between the Pauli-Fierz
field and an Abelian three-form gauge field
4.1 Standard material: basic cohomologies
The aim of this section is to investigate the cross-couplings that can be intro-
duced between a Pauli-Fierz field and an Abelian three-form gauge field. This
matter is addressed in the context of the antifield-BRST deformation proce-
dure described in the above and relies on computing the solutions to equations
(26)–(28), etc., with the help of the BRST cohomology of the free theory. The
interactionsareobtainedunderthefollowing(reasonable)assumptions: smooth-
nessinthedeformationparameter,locality,Lorentzcovariance,Poincar´einvari-
ance,andthepresenceofatmosttwoderivativesinthe interactingLagrangian.
‘Smoothnessin the deformationparameter’refersto the fact thatthe deformed
solutionto the master equation, (23), is smooth in the coupling constantλ and
reduces to the original solution, (22), in the free limit λ = 0. The requirement
ontheinteractingtheorytobePoincar´einvariantmeansthatonedoesnotallow
anexplicitdependenceonthespacetimecoordinatesinto thedeformedsolution
to the master equation. The requirement concerning the maximum number of
derivativesallowedto enter the interactingLagrangianisfrequently imposed in
the literature at the level of interacting theories; for instance, see the case of
cross-interactions for a collection of Pauli-Fierz fields, Ref. [22], the couplings
between the Pauli-Fierz and the massless Rarita-Schwinger fields, Ref. [28], or
the direct cross-interactionsfor a collection of Weyl gravitons,Ref. [23]. Equa-
tion (26), which we have seen that controls the first-order deformation, takes
the local form
sa=∂ mµ, gh(a)=0, ε(a)=0, (29)
µ
for some local mµ, and it shows that the nonintegrated density of the first-
orderdeformationpertainstothelocalcohomologyofthefreeBRSTdifferential
in ghost number zero, a H0(sd), where d denotes the exterior spacetime
∈ |
differential. The solutionto (29)is unique upto s-exactpiecesplus divergences
a a+sb+∂ nµ, (30)
µ
→
7
with gh(b) = 1, ε(b) = 1, gh(nµ) = 0, and ε(nµ) = 0. At the same time,
−
if the general solution of (29) is found to be completely trivial, a =sb+∂ nµ,
µ
then it can be made to vanish a=0.
In order to analyze equation (29), we develop a according to the antighost
number
I
a= a , agh(a )=i, gh(a )=0, ε(a )=0, (31)
i i i i
i=0
X
and assume, without loss of generality, that decomposition (31) stops at some
finitevalueofI. ThiscanbeshownforinstancelikeinAppendixAofRef.[22].
Replacing decomposition (31) into (29) and projecting it on the various values
of the antighost number by means of (12), we obtain the tower of equations
µ
(I)
γa = ∂ m , (32)
I µ
µ
(I−1)
δa +γa = ∂ m , (33)
I I−1 µ
µ
(i−1)
δa +γa = ∂ m , 1 i I 1, (34)
i i−1 µ
≤ ≤ −
µ µ
(i) (i)
where m are some local currents, with agh m = i. Moreover,
(cid:18) (cid:19)i=0,I (cid:18) (cid:19)
accordingtothegeneralresultfromRef.[22]intheabsenceofcollectionindices,
equation (32) can be replaced in strictly positive antighost numbers by
γa =0, I >0. (35)
I
Due to the second-ordernilpotency ofγ (γ2 =0), the solutionto (35)is unique
up to γ-exact contributions
a a +γb , agh(b )=I, pgh(b )=I 1, ε(b )=1. (36)
I I I I I I
→ −
Meanwhile,ifitturns outthat a reducesto γ-exacttermsonly,a =γb , then
I I I
it can be made to vanish, a =0. In other words, the nontriviality of the first-
I
order deformation a is translated at its highest antighost number component
into the requirement that a HI(γ), where HI(γ) denotes the cohomology
I
∈
of the exterior longitudinal derivative γ in pure ghost number equal to I. So,
inorderto solveequation(29) (equivalentwith(35)and(33)–(34)), we needto
compute the cohomology of γ, H(γ), and, as it will be made clear below, also
the local cohomology of δ, H(δ d).
|
Using the results on the cohomology of γ in the Pauli-Fierz sector [22] as
wellasdefinitions(20)and(21),wecanstatethatH(γ)isgeneratedontheone
hand by χ∗, F , and K , together with their spacetime derivatives and,
Γ µνρλ µναβ
on the other hand, by the undifferentiated ghost for ghost for ghost C as well
by the ghosts η and their first-order derivatives ∂ η . So, the most general
µ [µ ν]
(andnontrivial)solutionto (35) canbe written, uptoγ-exactcontributions,as
ah,A =α ([F ],[K ],[χ∗ ])ωI C,η ,∂ η , (37)
I I µνρλ µναβ ∆ µ [µ ν]
(cid:0) (cid:1)
8
wherethenotationf([q])meansthatf dependsonq anditsderivativesuptoa
finiteorder,whileωI denotestheelementsofabasisinthespaceofpolynomials
withpureghostnumber I inthe correspondingghostforghostforghost,Pauli-
Fierz ghosts and their antisymmetrized first-order derivatives. The objects α
I
(obviously nontrivial in H0(γ)) were taken to have a finite antighost number
and a bounded number of derivatives, and therefore they are polynomials in
the antifields χ∗, in the linearized Riemann tensor K and in the field-
Γ µναβ
strength of the three-form F as well as in their subsequent derivatives.
µνρλ
They are required to fulfill the property agh(α ) = I in order to ensure that
I
the ghost number of a is equal to zero. Due to their γ-closeness, γα = 0,
I I
and to their polynomial character, α will be called invariant polynomials. In
I
antighost number equal to zero the invariant polynomials are polynomials in
the linearized Riemann tensor, in the field-strength of the Abelian three-form,
and in their derivatives.
Inserting (37) in (33), we obtain that a necessary(but not sufficient) condi-
tionfor the existenceof(nontrivial)solutionsa isthatthe invariantpolyno-
I−1
mials α are (nontrivial) objects from the local cohomology of the Koszul-Tate
I
differential H(δ d) in antighost number I >0 and in pure ghost number zero,
|
µ µ µ
(I−1) (I−1) (I−1)
δα =∂ j , agh j =I 1, pgh j =0. (38)
I µ
! − !
WerecallthatthelocalcohomologyH(δ d)iscompletelytrivialinbothstrictly
|
positive antighostand pure ghost numbers (for instance, see Theorem 5.4 from
Ref. [32] and also Ref. [33]). Using the fact that the Cauchy order of the free
theory under study is equal to four, the general results from Refs. [32, 33],
accordingto which the local cohomologyof the Koszul-Tatedifferential in pure
ghostnumberzeroistrivialinantighostnumbersstrictlygreaterthanitsCauchy
order, ensure that
H (δ d)=0, J >4, (39)
J
|
where H (δ d) denotes the local cohomology of the Koszul-Tate differential in
J
|
antighost number J and in pure ghost number zero. It can be shown that
any invariant polynomial that is trivial in H (δ d) with J 4 can be taken
J
| ≥
to be trivial also in Hinv(δ d). (Hinv(δ d) denotes the invariant characteristic
J | J |
cohomologyin antighostnumber J — the local cohomologyof the Koszul-Tate
differential in the space of invariant polynomials.) Thus:
µ µ
(J) (J)
α =δb +∂ c , agh(α )=J 4 α =δβ +∂ γ , (40)
J J+1 µ J J J+1 µ
≥ ⇒
(cid:18) (cid:19)
µ
(J)
with both β and γ invariantpolynomials. Results (39) and (40) yield the
J+1
conclusion that
Hinv(δ d)=0, J >4. (41)
J |
By proceeding in the same manner like in Refs. [22] and [39], it can be proved
that the spaces (H (δ d)) and Hinv(δ d) are spanned by
J | J≥2 J | J≥2
H (δ d),H(cid:0)inv(δ d): (cid:1) (C∗), (42)
4 | 4 |
9
H (δ d),Hinv(δ d): (C∗µ), (43)
3 | 3 |
H (δ d),Hinv(δ d): (C∗µν,η∗µ). (44)
2 | 2 |
In contrast to the groups (H (δ d)) and Hinv(δ d) , which are finite-
J | J≥2 J | J≥2
dimensional, the cohomology H (δ d) in pure ghost number zero, known to
1 | (cid:0) (cid:1)
be related to global symmetries and ordinary conservation laws, is infinite-
dimensional since the theory is free. Fortunately, it will not be needed in the
sequel.
The previous results on H(δ d) and Hinv(δ d) in strictly positive antighost
| |
numbers are important because they control the obstructions to removing the
antifieldsfromthefirst-orderdeformation. Basedonformulas(39)–(41),onecan
successively eliminate all the pieces of antighost number strictly greater than
fourfromthenonintegrateddensityofthefirst-orderdeformationbyaddingonly
trivial terms. Consequently, one can take (without loss of nontrivial objects)
I 4 into the decomposition(31). (The proofofthis statementcanbe realized
≤
like in Appendix C fromRef. [40].) In addition,the last representativereads as
in (37), where the invariant polynomial is necessarily a nontrivial object from
Hinv(δ d) or from H (δ d) for J =1.
J | 2≤J≤4 1 |
(cid:0) (cid:1)
4.2 First-order deformation
Assuming I =4, the nonintegrateddensity of the first-order deformation, (31),
becomes
ah,A =ah,A+ah,A+ah,A+ah,A+ah,A. (45)
0 1 2 3 4
We can further decompose a in a natural manner as
ah,A =ah+ah−A+aA, (46)
whereah containsonlyfields/ghosts/antifieldsfromthePauli-Fierzsector,ah−A
describesthe cross-interactionsbetweenthe twotheories(soit effectivelymixes
both sectors), and aA involves only the three-form gauge field sector. The
componentah iscompletelyknown[22]andindividuallysatisfiesanequationof
the type (29). It admits a decomposition similar to (45)
ah =ah+ah+ah, (47)
0 1 2
where
1
ah = η∗µην∂ η , (48)
2 2 [µ ν]
ah = h∗µρ (∂ ην)h ην∂ h , (49)
1 ρ µν − [µ ν]ρ
(cid:0) (cid:1)
10