Conjecture on Hidden Superconformal Symmetry of N=4 Supergravity

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by  Sergio Ferrara

Conjecture on Hidden Superconformal Symmetry of N=4 Supergravity
Sergio Ferrara1,2, Renata Kallosh3, and Antoine Van Proeyen4
1Physics Department, Theory Unit,
CERN CH 1211, Geneva 23, Switzerland
2INFN - Laboratori Nazionali di Frascati,
Via Enrico Fermi 40, 00044 Frascati, Italy
3Stanford Institute for Theoretical Physics and Department of Physics,
Stanford University, Stanford, CA 94305
4Instituut voor Theoretische Fysica, KU Leuven,
Celestijnenlaan 200D, B-3001 Leuven, Belgium
2 (Dated: September4, 2012)
1
WearguethattheobservedUVfinitenessofthe3-loopextendedsupergravitiesmaybeamanifes-
0
2 tationofahiddenlocalsuperconformalsymmetryofsupergravity. WefocusontheSU(2,2|4)dimen-
sionless superconformal model. In Poincaré gauge where the compensators are fixed to φ2 =6MP2
p
this model becomes a pure classical N=4 Einstein supergravity. We argue that in N=4 the higher-
e derivative superconformal invariants like φ−4W2W¯2 and the consistent local anomaly δ(lnφW2)
S
are not available. This conjecture on hidden local N=4 superconformal symmetry of Poincaré su-

3 pergravity may be supported by subsequentloop computations.

]
h
I. INTRODUCTION gravity. Namely, we will show that the 3-loop finiteness
-t of pure1 N=4 supergravity [8], taken together with the
p
absence of a candidate for a consistent local N=4 su-
e
The purpose of this note is to address the following perconformalanomaliessuggeststhatthe principle oflo-
h
[ issue: what if extended supergravity is perturbatively cal superconformal symmetry may control the quantum
finite? Even if it is true (which of course we do not properties of the gravitational theory, in the same way
1
know at present) why could it be important? Is it possi- as the principle of non-abelian gauge symmetry controls
v
ble that the conjectured perturbative UV finiteness may the quantum properties of the standard model.
8
reveal some hidden symmetry of gravity? Here we pro-
1
4 poseaconjecturethatsuchahidden symmetrymaybean We are not usedto think of a local 4-dimensionalcon-
0 N=4 local superconformal symmetry. If the 4-loop N=4 formalsymmetryasareliablegaugesymmetry,wherethe
. supergravity is UV divergent, this conjecture will be in- gauge-fixingand the ghosts structure support the BRST
9
0 validated and, if it is UV finite, the conjecture will be symmetryandthecomputationsconfirmtheformalprop-
2 supported. erties of the path integral. The common expectation is
1 thatthissymmetrymaybeunreliablebecauseofanoma-
: Starting from the early days of supergravity, the su- lies. Therefore we cannot use it for investigation of di-
v
perconformal calculus was a major tool for constructing vergences in the usual Einstein gravity.
i
X new Poincaré supergravity models, see for example [1]
and the recent book [2], which describes in detail the The N=4 local superconformal symmetry may be an
r
a superconformal origin of N=1, 2 supergravities, includ- example of an anomaly-free theory, and therefore it is
ing the role of the compensators and the gauge-fixing tempting to study possible implications of the local su-
of the superconformal models down to super Poincaré. perconformal symmetry, starting with this case. In par-
Extended N 4 supergravity models were developed in ticular,the absenceofthe 3-loopUVdivergencesinpure
[3–6], startin≤g with superconformal symmetry. N=4 supergravity [8] may be interpreted as a manifes-
tation of the superconformalsymmetry of the un-gauge-
N> 4 supergravity models do not have an underlying fixed version of this theory.
superconformalsymmetry. Thisisrelatedtothefactthat
there are no matter multiplets, only pure supergravity We will discuss possible implications of the conjecture
multiplets are available. In particular, there is no super- ofhiddenN=4superconformalsymmetryfortheall-loop
symmetric extension of the square of the Weyl tensor in UV properties of N 4 supergravities 2. In N>4, in ab-
≥
N>4, as shown in [7].
Here we would like to suggest a possibility that the
superconformally symmetric model underlying N=4 su- 1 OuranalysisisnotvalidforthecaseofN=4supergravityinter-
pergravity is not just a tool, but a major feature of a acting with matter, studied in [9]. These models have a 1-loop
consistentperturbative supersymmetrictheoryinvolving UVdivergence [10].
2 Some early hints about the possibilityof UV finiteness of N≥5
2
sence of duality anomalies [11], the duality current con- Now letus look for the consequencesofour conjecture
servation argument can be used towards the UV finite- that the local (super)conformal symmetry is fundamen-
ness of the perturbative supergravity [13]. In N=4 case, tal, instead of Poincaré (super)gravity.
wherethere isa 1-loopglobalU(1)duality anomaly[11],
1. We know that the first UV divergence that was
one mighthavesome concernsregardingthe explanation
predicted in pure gravity (not taking into account the
ofthe3-loopUVfinitenessinpureN=4supergravity[12].
conformal predictions, but only general covariance) at
However, we will argue below that in the underlying su-
the 2-loop level [16] is given by the cube of the Weyl
perconformal N=4 model the local superconformal sym-
tensor
metry is anomaly free.
1
ΓN=0 κ2 d4x√ gC λδC αβC µν . (5)
2 ∼ ǫ − µν λδ αβ
Z
II. CONFORMAL COMPENSATOR IN N=0
Here the on-shellcondition is R=R =0 and C =
SUPERGRAVITY µν µνλδ
R .
µνλδ
2. The actual computation was performed in [17],
Considera modelof puregravity,N=0,promotedto a whichdemonstratedthat2-loopgravityisindeedUVdi-
local Weyl conformal symmetry: vergent:
Sconf = 12 d4x√−g ∂µφ∂νφgµν + 16φ2R . (1) ΓN2 =0 = (4κπ2)42280890 1ǫ d4x√−gCµνλδCλδαβCαβµν .
Z (cid:18) (cid:19) Z
(6)
The field φ is referred to as a conformal compensator.
Various aspects of this toy model of gravity with a Weyl ThisfinalizedaconvincingstoryoftheUVinfinitiesin
compensator field (1) were studied over the years [15]. pureN=0quantumgravity. Thereisnoreasontoexpect
The actionis conformalinvariantunder the followinglo- that the 3-loop counterterm as well as all higher loop
cal Weyl transformations: order 1 UV divergences, will not show up.
ǫ
g′ =e−2σ(x)g , φ′ =eσ(x)φ. (2) Now assume that we use the underlying conformal
µν µν
model with local conformal symmetry. It is easy to pro-
Thegaugesymmetry(2)withonelocalgaugeparameter motethe2-loopUVdivergencetotheformofaconformal
can be gauge fixed. We may choose the unitary gauge invariant:
φ2 = 6 (3) d4x√ gφ−2CµνλδCλδαβCαβµν . (7)
κ2 −
Z
Note that one has to take a scalar field with ghost-like Upon gauge-fixing it will produce the candidate for the
sign for the kinetic term to obtain the right kinetic term 2-loop divergence. Thus, even if we would use the em-
for the graviton. This does not lead to any problems beddingofgravityintoamodelwithconformalsymmetry
since this field disappears after the gauge fixing and the byintroducinganextrascalar-compensator,itwouldnot
action (1) reduces to the Einstein action, which is not help us to forbid the 2-loop UV divergence in the N=0
conformally invariant anymore: supergravity.
1
Sconf = d4x√ gR. (4)
gauge−fixed 2κ2 −
Z A. N=0 supergravity with matter
In this action, the transformation (2) does not leave the
Einstein action invariant any more. The R-term trans-
forms with derivatives of σ(x), which in the action (1) If we would add some additional matter to our super-
were compensated by the kinetic term of the compen- conformalN=0toymodel,wewouldhavetoconsiderthe
sator field and the weight was compensated by φ2 term 1-loop conformal counterterm independent on the com-
which is not present in the gauge-fixed action anymore. pensator field, proportional to the square of the Weyl
But the general covariance is still the remaining local tensor
symmetry of the action.
1
ΓN=0 d4x√ gC Cλδµν . (8)
1 ∼ ǫ − µνλδ
Z
andinthetopologicallytrivialbackgroundthiscountert-
and not only N=8 supergravity were given in[14] based on the erm is
observationthatgenerictheoriesofquantumgravitybasedonthe
Einstein-Hilbertaction may be better behaved in UV at higher 2 1
ΓN=0 d4x√ g R Rµν R2 . (9)
loopsthansuggestedbynaivepowercounting. 1 ∼ ǫ − µν − 3
Z (cid:16) (cid:17)
3
The coefficient in front depends on the matter content. and is based on the fact that in N=2 there is a local
The reasonfor its absence in pure gravity was explained superconformal calculus and there are chiral multiplets
using the background field method in [18] by the fact with arbitrary Weyl weight [21], in particular the nega-
that it is proportional to classical equations of motion tive powers of the compensator multiplet, that can be
when the rhs of Einstein equation, Tmat is vanishing. used for building higher derivative superconformal in-
µν
This means that the relevantdivergence can be removed variants. Moreover, various examples of superconformal
by change of variables. In [19] it was shown explicitly higher derivative invariants in the N=2 model are pre-
that in pure gravity there is a choice of the parameters sented in [22] and recently used for comparison with on
a,b of the general covariance gauge-fixing condition, of shellsuperspacecountertermsin[23]. The simplestN=2
the type aD hµν bD h=0 which makes both the UV superconformal version of R4 corresponding to minimal
µ µ
divergences R R−µν and R2 a,b-gauge-dependent, and pure N=2 supergravity is given by the following chiral
µν
vanishing at certain values of a,b. superspace integral [23]
Thus, in the exceptional case of pure gravity without
matter there is a 1-loop UV finiteness, the same is valid λ d4θ W2T W2 . (13)
for all pure supergravities without matter. However, in S2 S2 !!
Z
presenceofmatteringravityaswellasinsupergravities,
the 1-loop UV divergences that are present are defined
The N=2 superconformal calculus allows to use the chi-
by the matter part of the energy momentum tensor. ral multiplets S−2 as well as any higher negative power
1 S−2n forbuilding higherandhigherderivativeinvariants
ΓN=0 = d4x√ g α(Tmat)2+β(Tmat)2 . (10)
1 ǫ − µν in the N=2 supergravity [22]. Thus the hidden local su-
Z (cid:16) (cid:17) perconformalN=2 symmetry does not lead to a particu-
whereαandβ dependonthemattercontentofthegiven
lar restriction on N=2 supergravity counterterms.
model.
III. N=1, N=2, N=4 SUPERGRAVITY B. N=4 supergravity
A. N=1,2
1. The prediction was made in [24] that the 3-loop
divergence of the form
The generic N=1,2 supergravity models were derived
1
bygauge-fixingtheN=1,2superconformalalgebra,start- ΓN=4 κ4 d4x√ gC C CαβγδCα˙β˙γ˙δ˙ (14)
3 ∼ ǫ − αβγδ α˙β˙γ˙δ˙
ing with SU(2,21),SU(2,22), respectively, see [2] and
Z
| |
references therein. The known facts are:
is expected since the relevantcandidate countertermhas
1. InN=1,2supergravitiesthe predictionwasmadein all required non-linear symmetries of N=4 supergravity,
[20] that the 3-loop divergence of the form including the SU(1,1) SO(6) duality.
×
1
ΓN=1,2 κ4 d4x√ gC C CαβγδCα˙β˙γ˙δ˙ 2. The recent computation in [8] revealed that
3 ∼ ǫ − αβγδ α˙β˙γ˙δ˙
Z
(11)
ΓN=4 =0 . (15)
is possible. 3
2. There were no computations of the 3-loop UV di-
vergence in N=1, N=2 supergravity so far. The computations of UV loop divergences in [25] and
in [8] are based on the information about the tree am-
The prediction in (11) was based on local supersym- plitudes and on the unitarity method. Therefore these
metry, associated with Poincaré N=1,2 supergravity. computations seem to shed some light on all version of
extendedsupergravity,whichatthetreelevelareequiva-
The superconformal embedding prediction would re-
lent. Suchversionsarerelatedbyvariousclassicalduality
quireusto providethesuperconformalembedding ofthe
transformations. We proceed from here by suggesting a
term in (11). The question is: is there a N=1,2 super-
conjecture of a hidden superconformal symmetry, which
conformal generalization of the expression
these computations may have revealed.
d4x√−gφ−4CαβγδCα˙β˙γ˙δ˙CαβγδCα˙β˙γ˙δ˙ , (12) We will now proceed with the analysis based on our
Z conjecturethatthe localsuperconformalsupersymmetry
which is the gravity part of the full N=1,2 superconfor- may controlthe UV divergences of N=4 Poincaré super-
mal higher derivative invariant. The answer is positive gravity.
4
IV. N=4 SUPERCONFORMAL SYMMETRY symmetry, special supersymmetry, SU(4), U(1), confor-
AND SUPERGRAVITY mal boosts, Abelian gauge transformation on the vector
fieldsandspinorequationofmotiononψi,whichweshow
in the last term in X . The explicit expressions are
EOM
Here we follow [3,4] andspecifically [5], where the de- derived in [5]. The parameters of all these transforma-
tails on N=4 case have been workedout. To derive N=4 tions,whichforman‘openalgebra’,arebilinearinǫ (x),
1
supergravity from the superconformal model based on ǫ (x). The constraints on the curvatures lead to certain
2
theSU(2,24)gradedalgebrarequiresanumberofrather relations between the gauge fields so that some of them
|
complicated steps. We will only describe here the ones are not independent anymore.
that are relevant for our purpose, referring the reader to
the original papers [3–5].
To derive the action of a pure N=4 Poincaré super- A. Superconformal coupling of vector multiplets to
gravity one has to start with 6 (wrong sign) metric N=4 the Weyl multiplet
vector multiplets interacting with the N=4 Weyl gravi-
tational multiplet. The Abelian vector multiplet action
with the correct sign of the metric invariant under rigid The N=4 superconformal Lagrangian of the vector
N=4 supersymmetry is multiplets interacting with the Weyl multiplet is given
in eq. (3.16)in [5] andtakes a full page. We will present
1 1
F Fµν ψ¯iγ ∂ψ ∂ φ ∂µφij , (16) herethebosonicpartoftheactionforthe6compensating
µν i µ ij
− 4 − · − 2
vector multiplets, with the wrong sign of kinetic terms,
with i,j = 1,...,4. For the 6 compensator vector multi- which is relatively simple.
plets (I,J =1,...,6) we take
−14FµIνηIJFµJν−ψ¯iIηIJγ∂ψiJ−21∂µφIijηIJ∂µφijJ , (17) e−1Lbso.cs. =−41Fµ+νIηIJFµ+νJφ1−Φφ2 − 14DµφIijηIJDµφijJ
where φij =(φij)∗ =εijkℓφkℓ andthe constantrealmet- 1 1 Φ∗
ric η is diagonal and has 6 negative eigenvalues, 1. F+Iη TµνφijJ T φijJη TµνφklJ
IJ − − µν IJ ij Φ − 2 µνij IJ kl Φ
ThisactionisinvariantunderglobalSU(4). Thesixneg-
asatitvoerseiogfenavcaolunefosrpmoainltstyomwmaredtsryt,haesroelxepolafiφnIiejdasincothmepteony- − 418φIijηIJφijJ EklEkl+4DaφαDaφα−12fµµ
model above. In case of pure N=4 supergravity without (cid:16) (cid:17)
matter multiplets all scalars from the 6 N=4 supercon- 1
+ φI η φklJDij +h.c. (19)
formal vector multiplets are the compensator scalars, as 8 ij IJ kl
we will see below.
Here
To derivethe N=4pure supergravityactiononestarts
with 6 such vector multiplets and couple them to the Φ=φ1+φ2 , Φ∗ =φ φ , φαφ =1 . (20)
fields of conformal N=4 supergravity. There is a deriva- 1− 2 α
tive Da = eµaDµ, which is covariant with respect to all The conformal boost gauge field fµa is a function of a
superconformalsymmetriesofSU(2,24). Meanwhile
µ curvature
| D
is covariant under Lorentz, Weyl, SU(4) and U(1) sym-
metries. The S- and K-covariantization is performed in 1
f µ = R(ω) . (21)
[5] explicitly. µ −6
The rigid supersymmetry algebra Q,Q leads to The scalars φ with α = 1,2 transform as a doublet
α
{ }
translation P, so it is necessary to convert it into gen- under SU(1,1). The constraint(20) and the U(1) gauge
eral coordinate transformations to describe the coupling invariance reduce the 2 complex variables φ to 2 real
α
withgravity. Thisandanalogousstepsrequiresomecon- fields, so that they are in SU(1,1) coset space.
U(1)
straintsonthecurvaturesaswellasintroductionoffields,
in additionto gaugefields above,to closethe algebra,so The fields T ,Ekl,Dij belong to the gravita-
µνij kl
that, after all tional Weyl multiplet. This action, supplemented by all
fermionic terms, leads to equations of motion of fermion
[δ (ǫ ),δ (ǫ )]=δcov(ξµ)+δ (ǫab)+δ (ǫi)+δ (ηi)+ partner of the compensator scalars ψiI in the rhs of the
Q 1 Q 2 G M Q 3 S
commutator of two supersymmetries (18), which we de-
noted by X .
δ (λi )+δ (λ )+δ (λa )+δ (λ)+X (18) EOM
SU(4) j U(1) T K K A EOM
TheactionislinearinDij ,soitisconvenientforthe
kl
Therhsofthiscommutatordependsonacombinationof purpose of future gauge-fixing to use the reparametriza-
all local symmetries: general covariant, Lorentz, super- tionofthe 36variablesφ I intermsofthe 36ϕ I(x)
ij M
≡
5
ϕI (x),ϕI (x) , so that M =1,...,6: of motion leads to a sign conversion of the kinetic term
{ m m+3 }
forthevectorsfromthe6vectormultiplets,theybecome
φI (x)=ϕI (x)βm +iϕI (x)αm , (22) physical vectors with the correct sign kinetic term.
ij m ij m+3 ij
where αm and βm with m = 1,2,3 are SU(2) SU(2) To summarize, the 6 vector multiplets at the super-
numerical matrices introduced in [26]. × conformal stage all have wrong kinetic terms since the
N=4 scalar partners play the role of conformal compen-
sators. When scalars are gauge-fixed to eliminate the
local Weyl D-symmetry (dilatation), the Einstein grav-
V. POINCARÉ GAUGE ity arises. The 6 quartets of spinors (6 4 4 = 96
× ×
components) from the vector gauge multiplets are elimi-
natedby the combinationof16 gaugeconditionsof local
The superconformal action has unbroken local K-, D-
S-supersymmetry (special supersymmetry) and the field
and S-symmetries which are not present in supergravity
equationsofthe auxiliaryfermionsinthe Weylmultiplet
and must be gauge-fixed to convert the superconformal
(80 components).
action into a supergravity one.3 This is done the same
wayasinthetoyexampleabove,namely,thescalarcom- Thevectorsfromthe 6vectormultiplets areconverted
pensator dependent term in front of R is designed to in- into physical vectors of supergravity, when the auxiliary
troduceaPlanck massinto conformaltheorywhichorig- field T of the Weyl multiplet is excluded on its equa-
µνij
inally, before gauge-fixing, has no dimensionful parame- tions of motion. The action becomes that of pure N=4
ters. To fix the local dilatation D one can take supergravity in κ2 = 1 units where the local U(1) sym-
metry is still present. The bosonic part is
6
φI η φijJ = . (23)
ij IJ −κ2
1 1
e−1Lbos = R(ω)+ D φαDaφ +
This provides the Einstein curvature term in the action sg 4 2 a α
1 φ +φ
− 214φIijηIJφijJR ⇒ 4κ12R (24) +4Fµ+νIηIJF+Jµν φ11−φ22 +h.c. (27)
andexplainswhythediagonalmetricη hassixnegative Note that the actions are supersymmetric after adding
IJ
values. The S- and K- local symmetries are fixed by the fermionic part. This means that the variation of
taking the actionvanishesforarbitraryfieldconfigurations: the
fields do not satisfy any equations. The statement that
b =0 , ψ J =0 . (25) themultipletsare’onshell’isastatementonthealgebra
µ i
oftransformations,andthatonedependsonspecificfield
The fact that our 6 vector multiplets have a wrong sign equations. Thus no other invariant can be constructed
kinetic terms is in agreement with the fact that the with these on shell multiplets, since this would change
scalars are conformal compensators. As long as ϕI (x) the field equations. This is why the inverse powers of
m
and ϕI (x) with m = 1,2,3 are present, there is also the vector multiplet (or a logarithmic function) can’t be
m+3
a local SU(4) symmetry. So, we can use the 15 param- used to construct other invariant actions, see a further
eters from SU(4) together with 20+1 conditions: field discussion of this in Sec. VI.
equationsofDij andthedilatationgaugementionedin
kℓ
(27), to take the scalars in eq. (22) constant,
1 Triangular U(1) gauge-fixing
ϕ I(x)= δ I , (26)
M M
2κ
to remove these 36 variables. The local U(1) gauge we take4 is
The remaining important steps include the elimina-
Im(φ φ )=0 (28)
tionoftheauxiliaryfieldsoftheWeylmultiplet,Ekl and 1− 2
T . The field Ekl turned out to be proportional to Our choiceis motivated by the triangulardecomposition
µνij
fermion bilinears, which changes the fermionic part of of the SL(2,R) matrix of our model. We start with the
the action. However, the role of T is extremely im- SU(1,1) matrix defined in [5]
µνij
portant: the procedure of its exclusion on its equations ∗
φ φ
U = 1 2∗ (29)
φ φ
(cid:18) 2 1 (cid:19)
3 This is a precise analog of 3 gauge symmetries in the SU(3)×
SU(2)×U(1)modelwhichhavebeengauge-fixedintheunitary 4 Therelated construction isdiscussedin[6],withoutmakingex-
gaugewhereW± andZ aremassivevectormesons. plicitchoiceoftheU(1)gauge.
6
We switch to the SL(2,R) basis and get InthislocalU(1)gaugetheN=4supergravityisaninter-
mediateversionbetweentheCSFmodel[26]andtheone
S = U −1 = Re(φ1+φ2) −Im(φ1+φ2) (30) given in [27] and in full details in [28]. Specifically, the
A A Im(φ1 φ2) Re(φ1 φ2) scalarsZ arethesameasin[28],however,thevectorsare
(cid:18) − − (cid:19)
related to the ones in [28] by a duality transformation.
where
1 1 1 ThuswefindthatournewgaugewhichprovidesaCSF
= (31) N=4supergravitymodel[26]directlyfromthesupercon-
A √2 i i
(cid:18)− (cid:19) formal model is nice and simple, comparative to other
We define an independent variable τ as versions of N=4 supergravity.
φ +φ
1 2
τ =τ +iτ i (32)
1 2
≡ φ φ
1 2
−
VI. HIGHER DERIVATIVE
which parametrizes the coset space SL(2,R). We take SUPERCONFORMAL ACTIONS IN N=4 MODEL
U(1)
1 1
φ = (1 iτ) φ = (1+iτ) (33)
1 2
2√τ2 − −2√τ2 Thereis only onetype ofpossible mattermultiplets in
N=4 supersymmetry, i.e. N=4 Maxwell multiplets (and
Inthisnotationwithτ =e−2ϕandτ =χthetriangular
2 1 thenon-abelianversion). Thescalarφ hasWeylweight
decomposition of the SL(2,R) matrix is clear ij
w = 1 and therefore it is used to gauge-fix local dilata-
e−ϕ χeϕ tion by the φIijηIJφijJ = −κ62 condition. The fact that
S = (34) the algebra does not close on these N=4 Maxwell multi-
0 e+ϕ
(cid:18) (cid:19) pletsimpliesthatwecannotusethemanymoreinfurther
When these values of φ are inserted into the supercon- tensor calculus for N=4. Moreover, the local supercon-
α
formal action (3.16) or the partially gauge-fixed (4.18) formal symmetry algebra is closed on the Weyl multi-
of [5], we get the CSF N=4 supergravity model [26] the plet. Since there are no other multiplets in that case, it
bosonic part of which is isnotpossibletoconstructtheN=4superconformalver-
sion of C4 shown in eq. (12), which requires a superfield
1 1 ∂τ∂τ¯ withtheconformalweightw = 4whichinthePoincaré
e−1LCSF = 2R− 4(Imτ)2 gaugebecomesκ4. Itwouldrequ−ireanN=4superconfor-
mal version of the bosonic expression (φI η φijJ)−2C4,
1 ij IJ
+ δ iτF+IF+Jµν +h.c. (35) which does not exist. Including supercovariant deriva-
4 IJ µν tives D = eµD with w(eµ) = +1 can only increase
a a µ a
(cid:2) (cid:3) the positiveconformalweightw ofthe correspondingsu-
perconformal invariant, which requires higher negative
powers of a compensator.
Bergshoeff, de Roo, de Wit U(1) gauge
ThesituationintheN=4caseisinsharpcontrastwith
N=1,2 cases where there are chiral superfields of arbi-
traryconformalweightw[21],whichcanbeusedtobuild
The choice in [4, 5] is
the superconformal invariants. Moreover, according to
Im φ =0 (36) eq. (C.2)in[22]onecantakeanarbitraryfunctionofthe
1
chiral compensator superfield (φ) and construct such
and the independent variable is defined as a negative conformal weight suGperfield out of the com-
pensators in the N=2 superconformal case. In Poincaré
φ
Z 2 (37) gauge such a superfield (φ) = φ−2n will provide the
≡ φ1 increasing powers of gravGitationalcoupling φ−2n κ2n.
⇒
Here the scalars
ToexplainwhyinN=4superconformaltheoryitisnot
1 Z possibletoproducesuperinvariantactionswitharbitrary
φ = , φ = , (38)
1 2
1 Z 2 1 Z 2 functionofsuperfieldsconsideranexample: theoff-shell
−| | −| |
chiral multiplet
parametrizepthe cosetspace SU(1,1)p. The 6 vector multi-
U(1)
plets havea correctsignkinetic terms. The theory has a (z,χL,F) (40)
globaldualitysymmetrySU(1,1) SO(6)inheritedfrom
× We want to construct an action S = d2θ (z). To find
thesuperconformalN=4model. Thescalarcouplingsare
G
the components, we obtain the fermion component of
∂Z∂Z¯ by calculating one susy transformaRtion on the lowest
(39) G ′
− (1 Z 2)2 component (z). Thisgives (z)χL. Afurthertransfor-
−| | G G
7
mationgives(for the componentthat willbe integrated) change of variables in the path integral of the kind per-
formedin[33], which allowsto proveanequivalence the-
′ ′′
(z)F (1/2) (z)χ¯LχL (41) oremfor the S-matrix in arbitrarygauges,may be inval-
G − G
idated in presence of anomalies.
This transforms to γµ∂ [ ′(z)χ ] and thus gives a good
µ L
invariant action. G For example, in the simple case ofSU(2) gaugemodel
we may be interestedin transverserenormalizablegauge
However, consider now that we would have only the ∂µA m = 0 or in the unitary gauge Bm = 0. To find
µ
on-shell multiplet, e.g. for a massless multiplet. Then the relation between these two gauges one may look at
F =0. ThealgebraonχLleadsthentothefieldequation a more general class of gauges like a∂µAµm+bBm =0.
γµ∂µχL = 0. The second susy transformation as above In the unitary gauge the theory is not renormalizableoff
′′
leads to (1/2) (z)χ¯LχL Thus the superfield wouldbe shell, however, if the equivalence theorem
− G
G(z)+θ¯LG′(z)χL−(1/2)G′′(z)θ¯LθLχ¯LχL (42) h|S|i|a,b =h|S|i|a+δa,b+δb (44)
This is a superfield for any (z). However, the integral is valid, the physical observable are the same as the
G
d2θ gives the last component, which transforms under ones in renormalizable gauge (with account of some de-
susy to pendence on gauge-fixing of renormalizationprocedure).
R
Alsotheproofofunitarityintherenormalizablegaugeis
γµχL∂µ ′(z) (43) based on validity of (44).
G
This is not a total derivative (missing a term propor- Localsymmetry anomaly may invalidate the a,b inde-
tional to the field equation, but that we cannot use to pendence of physical observables. Instead of equivalence
have an invariant action). This illustrates that the mul- we have a relation
tiplet calculus can only be used for off-shell multiplets.
We provide a more detailed discussion and relation to S = S +X Λα(x,φi,δa,δb) (φi)
a,b a+δa,b+δb α
N=2 deformation in models with higher derivatives in h| |i| h| |i| h A i
Z
(45)
the Appendix.
Here (φi)istheconsistentanomalydependingonvar-
α
Beforewe takeseriouslya predictiononhigherderiva- ious fiAelds φi of the model, and Λα(x,φi,δa,δb) is a spe-
tive superinvariants following from the local N=4 super- cific change of variables, leaving the classical action in-
conformal theory, we have to study the situation with variant, but effectively changing the gauge-fixing condi-
anomalies. The local anomalies for N=1 superconformal tion,with examplesgivenin[33]. X is anumericalvalue
theories were studied in [29] and in [30, 31]. Our N=4 in front of a candidate anomaly, which may vanish, in
superconformal model of 6 (wrong sign) compensators case of cancellation, or not, depending on the model.
interacting with the Weyl multiplet, upon gauge-fixing,
Thus,inthecontextoflocalanomalieswhichmayexist
leads to pure N=4 supergravitywith Einstein curvature,
and destroy the quantum consistency of the model, we
without the square of the curvature in the action. The
will look at possible candidates for anomalies given by
local anomalies of this model will be discussed below.
expressions like
δ Γ(φi)= d4xΛα(x) (φi) (46)
Λ α
A. Superconformal anomalies A
Z
where Λα(x) corresponds to all gauge symmetries of a
Localsuperconformalanomalieswere studied indetail given model.
in N=1 case in [29]. It was explained there that the
Thereare2conditionsforananomalytobe fatalfora
consistent anomalies can be constructed using the Wess-
gaugetheory,i. e. tomakequantumtheoryinconsistent.
Zumino method [32]. In gauge theories the method al-
lows to construct terms Γ(Φ,Aµ), whose variation takes I. The candidate consistent anomaly (46) should be
a form of a consistent anomaly δΛΓ(Φ,Aµ), which does available according to local symmetries of the model
notdependonthecompensatorfieldΦ. Latertherelated
work was performed in a somewhat different context in II. The numerical coefficient in front of a candidate
[30] based on [31]. We are interested in local symmetry anomaly,whichisdue tocontributionfromvariousfields
anomalies,which in gauge theory examples may be fatal of the model, should not cancel, X =0 in (45).
6
andleadtoaquantuminconsistenttheory. Forexample,
the triangle local chiral symmetry anomaly in standard
model, if not compensated, means that the physical ob-
N=1 case
servables in the unitary gauge do not coincide with the
physical observables in the renormalizable gauge. The The symmetries of N=1 superconformal models in-
8
clude and
ǫ(x), η(x), λD(x), λT(x) (47) δΓST(φ,W2)=2(c a) d8zE−1 δΣW Wαβγ+c.c.,
αβγ
− R
i.e. local Q-supersymmetry, local S-supersymmetry, Z (52)
Weyl local conformal symmetry, local chiral U(1) sym- where under the superconformal transformations the
metry,respectively,andofcourse,generalcovarianceand compensator superfield transforms as
Lorentz symmetry.
φ(x,θ) eΣ(x,θ)φ(x,θ) (53)
IncaseofN=1superconformalmodelsthecorrespond- →
ing δΛΓ(φ,W2) was given in [29] in eq. (5.7). The inte- Therefore,whenits vev is non-vanishing,one may try to
grated form of the anomaly is given by a local action in define the Goldstone superfield, which according to [30]
eq. (5.6) in [29] is ‘dimensionless’ and transforms by a superfield shift
ΓdWG(φ,W2) (48) δlnφ(x,θ) δΣ(x,θ) (54)
→
Here φ is the compensator superfield of a Weyl weight and therefore
w =1,andW isaWeylsuperfieldofconformalweight
αβγ
w = 3/2. The variation of (48) produces a consistent δlnφ(x,θ)W2(x,θ)=δΣ(x,θ)W2(x,θ) (55)
anomaly. Atthelinearlevelthisactionisassociatedwith
the F-component of the chiral superfield Therefore,thelocaldilatationsofthesupermultipletlnφ
are different from those of a multiplet with a particu-
ΓdWG(φ,W2)=(lnφWαβγWαβγ)F +... (49) lar Weyl weight. For example, for the chiral multiplet
(φ = Z,χ,F ) of the Weyl weight w the superconfor-
Terms with ... involve important corrections, re- { }
mal transformations, given for example in eq. (16.33) in
quired for locally superconformal action. The superfield
[2]) have some w-dependent terms like
lnφW Wαβγ seems to have a Weyl weight w =3, ex-
αβγ
ceptthatthelnφdoesnothaveauniformscalingweight δZ =w(λ +iλ )Z+... (56)
D T
w = 0, which leads to complication and modification of
the scale, chiral and S-supersymmetry transformations. whereλ (x)isalocaldilatationandλ (x)isalocalchi-
D T
Nevertheless, the consistent exact non-linear expression ral transformation. The same for χ,F, there are terms
for N=1 superconformal anomaly in the form (46) depending on w. These w-dependent terms are replaced
by different transformations when the fields do not scale
δΓdWG(φ,W2) (50) homogeneously under local dilatations. These transfor-
mations for lnφ can be inferred from (54) where
was established in [29] and given in eq. (5.7) there.
It has terms with all local parameters in (47), i. e.
Σ= λ +iλ ,√2η,0 (57)
there is aWeyllocalconformalsymmetry anomaly,local D T
chiral U(1) symmetry anomaly, local S-supersymmetry n o
andcorrespondingchangesinthesuperconformalderiva-
anomaly and local Q-supersymmetry anomaly, all pro-
tives. The standard superconformal action for the mul-
portionalto eachother: either allof them or none. Note
tiplet, given for example in eq. (16.33) in [2]) is not
that the analysis in [29] was not based on specific com-
superconformal invariant anymore due to these correc-
putations of anomalies, it was an analysis based on con-
tions to the superconformal transformations. However,
sistencyoftheanomaliesinN=1superconformalmodels.
itssuperconformalvariationdoesnotdependonthecom-
The candidate consistent anomaly (46) is available, the
pensator. As a result, the complete non-linear expres-
coefficient X in (45) is model-dependent.
sion for anomaly in eq. (5.7) in [29] is different from
It may be useful also to bring up here the relevant δΓST(φ,W2) in (52). The expression for the anomaly
discussion of the N=1 superconformal anomaly in [30, in (52) depends on manifestly Q-supersymmetric super-
31]. The corresponding gauge-independent5 part of the fields and gives the impression that only Weyl, chiral
anomaly is given by and S-supersymmetry anomalies are consistent. Mean-
while, the extra terms in δΓdWG(φ,W2) involve also the
ΓST(φ,W2)=2(c a) d8zE−1 lnφW Wαβγ +c.c. Q-supersymmetry anomaly, and therefore the complete
αβγ
− R non-linear expression for N=1 superconformal anomaly
Z
(51) is not given in terms of superfields with manifest Q-
supersymmetry, but in eq. (5.7) in [29].
Thus, a complete expression in eq. (5.7) in [29]
5 Weomittermsin[30]proportional toRandGαβ˙ superfieldsas for the superconformal anomaly δΓdWG(φ,W2) of N=1
theydependonthegauge-fixingconditionandmayberemoved, superconformal models contains local scale, chiral, S-
asshownforexamplein[19]. supersymmetry and Q-supersymmetry anomaly. It is
9
generatedbythe superconformalvariationofthe expres- tensor. Thelinearizedversionofit6isgivenineq. (3.17)
sionΓdWG(φ,W2). This is aconstructionofaconsistent of [4]. The complete non-linear action for the N=2 su-
anomaly which we intend to generalize to the N=4 case. perconformal case is given in eq. (5.18) of [4].
N=4 case The existence of this 1-loop counterterm is in agree-
ment with N=4 supergravity analysis, as well as ac-
For the N=4 superconformal anomaly the actions of
tual computations in [10]. In components is starts with
the type (49), (51) are not available. The reason is the C2 +... which corresponds to R2 and R2 terms, as
µνλδ µν
same as we have already explained with regard to can-
explainedineqs. (8),(9). ThesevanishforpureN=4su-
didate counterterms. In N=4 case the generalization of
pergravity,correspondingtothe modelwith6N=4com-
the N=1 case of arbitrary functions of a chiral compen-
pensators, since only in pure supergravity R=R =0.
satorlike (φ)=φ−2n arenotavailable,suchsuperfields µν
G In presence of matter multiplets, the counterterm has
can’t be used to provide invariant actions. Indeed, the
terms which do not vanish on shell, like T2 and T2.
compensatingmultipletsareinthiscasethevectormulti- µν
pletswhosetransformationscloseonlyusingspecificfield The one-loopN=4 square of the Weyl multiplet coun-
equations. Therefore, one cannot manipulate with these tertermissuperconformalbyitself,itdoesnotneedN=4
multiplets, as we explained in the beginning of this sec. compensators since it has a correctWeyl weight. This is
VI. This excludes also a possibility to use (φ) = lnφ why it escapes the problem with negative power of com-
G
forbuildingsuperinvariants. Thereforethereisnosuper- pensators, which is present for all N=4 superconformal
symmetric versionof lnφ(R R∗)2 (for chiralanomaly). invariants, starting with 3 loops. They need φ−2(L−1)
It is available in N=1 and N−=2 superconformal theories corresponding to κ2(L−1). Clearly, for L=1 there is no
but not available in N=4. The N=4 superconformally such dependence on a compensator.
invariantversionof ln(φI η φijJ)(R R∗)2 is not avail-
able. ij IJ − 2. Now we apply our method to N=4 conformal su-
pergravity interacting with some N=4 vector multiplets
The restrictions ofN=4 superconformalsymmetryare [35]. This model is believed to be renormalizable but
significantly stronger than the ones for N=1,2. Q- hasghosts. Thesuperconformalcountertermcorrespond-
supersymmetry has a limited restriction on N-extended ing to the square of the Weyl tensor is not excluded
supergravity counterterms, and suggests that for N- and it is not vanishing. It is not proportional to the
extended supergravity the L = N geometric on shell equations of motion of conformal supergravity interact-
counterterms are available, the same prediction follows ing with any number of vector multiplets. Thus the
from N=1,2 superconformal models. However, for N=4, renormalizableUV divergences,proportionalto the part
the symmetries allow only the local classical action and of conformal supergravity classical action are expected.
protect the model from anomalies and counterterms. And since again, this particular unique superinvariant
has the proper Weyl weight, the action does not depend
ThissupportsourconjecturethatN=4superconformal
oncompensators. Thereforethiscountertermescapesthe
models are quantum mechanically consistent and there-
problem with negative power of compensators.
forewemaytrustthe analysisofcandidatecounterterms
based on N=4 superconformal symmetry, which predict Itisratherinterestingtoseehowallfactsknownabout
the UV finiteness of perturbative theory. thesevariousN=4modelsfallintoplace. Ourconjecture
therefore,is thatnew computationswillcontinuetosup-
port the N=4 superconformal symmetry of the model
underlying pure N=4 supergravity.
B. Can we falsify our arguments using more
VII. DISCUSSION
general N=4 models?
We have discussed here the pure N=4 Poincaré super-
1. Consider N=4 supergravity interacting with some
gravity,whichis agauge-fixedversionofthe correspond-
numbernofN=4vectormultiplets. Thesuperconformal
ing N=4 superconformaltheory, the details of which, in-
un-gauge-fixed version of this model is described in [5,
cluding the action in eq. (3.16), are given by de Roo in
6]. It correspond to the model which we present in eq.
[5]. Herewehaveexplainedshortlytheimportantdetails
(19) where η has six negative eigenvalues as well as n
IJ
of the gauge-fixing to N=4 Poincaré supergravity at the
positive eigenvalues.
There is a 1-loop UV divergence in case of N=4 SG
+ N=4 vector multiplets, see for example [10]. There is
also a corresponding counterterm in the underlying su- 6 The complete non-linear bosonic action was recently derived in
perconformal theory, it contains the square of the Weyl [34]byintegratingovertheN=4Yang-Millsfields.
10
simple level of the bosonic part of the theory, as well as Theanomalycandidaterequirestousethesuperfieldlnφ,
the role of the conformal compensators, six vector mul- thelogarithmofthecompensatorfield,forconstructinga
tiplets with the wrong sign kinetic terms. In particular, consistentanomaly. ButitisnotpossibleinN=4,forthe
we have explicitly presented a triangular gauge for the samereasonasthenegativepowersofφarenotavailable
localU(1)symmetryinwhichthe superconformalmodel as building blocks for superinvariants.
[5] becomes a pure N=4 supergravity model [26].
This observation provides the simplest possible expla-
We argued that the N=4 superconformal action in [5] nationofthecomputationin[8]whereR4 UVdivergence
is unique and that the symmetry does not admit higher inN=4L=3supergravitywasfoundtocancel. Notethat
derivative actions. The argument about the uniqueness if this is the true explanation, it would mean also that
of the N=4 superconformal model is based on the open no other higher loop UV divergences are predicted by
gauge algebra of the SU(2,24) superconformal symme- the N=4 superconformal theory. Therefore our conjec-
try7, which requires the eq|uations of motion for the ture is falsifiable, as soon as the UV properties of the
fermionpartnerofthecompensator. ThisalloweddeRoo 4-loop N=4 supergravity will be known, they will either
in [5] to reconstruct the action consistent with the open confirm or invalidate our conjecture.
algebra. Ourargumentaboutthe uniquenessoftheN=4
TheconjecturedsuperconformalsymmetryofN=4su-
superconformal theory is related to the absence of the
pergravity supports UV finiteness arguments for N 4
higherderivativesuperconformalinvariants. Suchinvari-
≥
supergravities. For these models the UV finiteness ar-
ants require the presence of negative conformal weight
gument is associated with the Noether-Gaillard-Zumino
superfields, constructed from conformal compensators,
deformeddualitycurrentconservation[13]andwithlocal
whichcanbeusedinbuildingnewsuperconformalinvari-
supersymmetry deformed by the presence of the higher
ants. Since in N=4 the only matter multiplets that are
derivative superinvariant [23]. Both arguments require
available to serve as conformal compensators are vector
the existence of the Born-Infeld type deformation of ex-
multipletswiththeopenalgebra,theydonotprovidethe
tended supergravities [23, 38]. In the particular case of
negative weight superfields which will allow to make the
N=4 superconformalgeneralizationof d4x√ gφ−4C4 N=4 suchaBorn-Infeldtype deformationis not possible
− according to our current best understanding of super-
counterterms,whereC istheWeyltensor. Thereforethe
R4 UV divergences are forbidden by thRe N=4 supercon- conformalsymmetry,whichisasupportingargumentfor
theUVfinitenessofthe N>4models. IftheN=8Born-
formalsymmetry ofthe un-gauge-fixedtheory,assuming
Infeldsupergravitywouldbeavailable,onewouldbeable
that we have all tools available for such a construction.
to derive the N=4 one by supersymmetry truncation, in
WehavepresentedintheAppendixthedetaileddiscus- conflict with superconformal symmetry.
sionofthedifficultieswiththe“bottomuporderbyorder
Ifourconjecturethatthelocalsuperconformalsymme-
attempts” to construct the corresponding higher deriva-
tryexplainsthe3-loopUVfinitenessinN=4isconfirmed
tive N=4 supergravity invariants. One may try to start
by the 4-loop case, it will give us a hint that the models
from the known on shell N=4 superspace [40] candidate
withsuperconformalsymmetrywithoutanydimensionful
counterterms[24,39]anddeformtheclassicalsupersym-
parametersmayserveasabasisforconstructingaconsis-
metry toreachthe agreementwith anexactdeformation
tent quantum theory where M appears in the process
of classical theory studied in the N=2 theory [23]. The Pl
of gauge-fixing spontaneously broken Weyl symmetry.
problemis the absenceofa clearguiding principle inthe
N=4case. Thereforeonecanviewthecomputationin[8]
as an indication that such a deformation may be indeed
impossible since we already have all tools available and
they do not produce higher order genuine supersymmet- ACKNOWLEDGMENTS
ric invariants.
We have analyzedthe situation with N=4 localsuper-
We are grateful to Z. Bern, R. Roiban, M. Roček, I.
conformal anomalies based on earlier detailed studies of
Tyutin for useful discussions and to A. Tseyltin for a
consistent anomalies in N=1 superconformal theories in
significant impact on the development of this investiga-
[29] and in [30, 31]. We argued that there is no gener-
tion and to A. Linde and G. ’t Hooft for the stimulating
alization of local superconformal N=1 anomalies to the
discussions of the potential role of the superconformal
N=4case,thereasonbeingthesameasforcounterterms.
symmetry in physics. The work of S.F. is supported by
the ERC Advanced Grant no. 226455, Supersymmetry,
Quantum Gravity and Gauge Fields (SUPERFIELDS).
The work of R.K. is supported by SITP and NSF grant
7 Notethatthealgebra isclosedonthe Weyl multiplet,therefore
PHY-0756174. The work of A.V.P. is supported in part
alllocalsymmetrytransformationsarefixed. Meanwhilethefact
that thealgebra isopen onthe N=4Maxwell multipletmaybe by the FWO - Vlaanderen, Project No. G.0651.11, and
also related to the need to use an infinite number of auxiliary in part by the European Science Foundation Holograv
fieldstoclosethealgebra[36]. Network.