Moduli anomalies and local terms in the operator product expansion
Received: May
Moduli anomalies and local terms in the operator product expansion
Adam Schwimmer 0 1 2
Stefan Theisen 0 1
0 14467 Golm , Germany
1 Rehovot 76100 , Israel
2 Weizmann Institute of Science
Local terms in the Operator Product Expansion in Superconformal Theories with extended supersymmetry are identi ed. Assuming a factorized structure for these terms their contributions are discussed.
Anomalies in Field and String Theories; Conformal Field Theory; Extended
Supersymmetry
1 Introduction 2 3 4
N = 2 in d = 4
N = (2; 2) in d = 2
Conclusions
A Nonzero structure constant implies nonzero charge
B The Maxwell case
C Free theory with moduli in d = 2
view [1{5]. We will call \semilocal" in a generic sense those terms which do not have the
full analytic structure in momentum space expected in a correlator with singularities in all
kinematical invariants. The semilocal terms have a singularity (logarithmic or powerlike)
in one kinematical variable while the dependence on the other invariants is polynomial.
We will distinguish these terms from the ordinary local terms which are polynomial in all
invariants and which appear in speci c correlators. In position space locality means that
all operator insertion points coincide while in semilocal correlators only some of them do.
The generic situation which shows the full analytic structure in momentum space is when
they are all di erent.
We will use alternatively the operatorial or the covariant (i.e. based on the analytic
structure of the correlators) language and will verify their compatibility in each case. We
start by de ning the local terms in the Operator Product Expansion (OPE) and necessary
conditions for their appearance.
{ 1 {
In a CFT the information about the structure constants can be reconstructed from the
operator product expansion which we will use in the following: given two primaries O1; O2
with scaling dimensions d1 and d2, their OPE is
0. In momentum space the singularity produces
an analytic function of the kinematical invariant p2 with a branch point at p2 = 0, which
in general has both an imaginary and a real part. A special situation arises when the
y). Then, in momentum space, we have a purely
HJEP07(218)
real (polynomial) dependence on p2, and in con guration space the singular function is
\local". For such a situation to occur, the following conditions have to be satis ed:
a) the dimensions should full l d1 + d2
d3 = integer
d (the spacetime dimension),
following from the above form of the local singular function;
b) the ordinary structure constant/OPE coe cient should vanish. In position space
for noncoincident points the form of the correlator is
xed. In momentum space
a regularization is needed. In particular powers of momenta should be extracted
leaving logarithmic dependence. Generically there are two logarithms left and the
ambiguity in their arguments would completely \mask" the nonleading, semilocal
structure which is a polynomial multiplying one logarithm. Therefore the semilocal
terms have a well de ned meaning only if the dependence implied by a nonvanishing
structure constant is not there.
While the momentum dependence involving O1; O2 will be polynomial, the singular
behaviour in the correlators is obtained by singularities of O3 with other operators.
We remark a basic di erence between the ordinary terms in the OPE and the local
ones. For the ordinary OPE terms coming from a unique three point function there is a
relation between the three possible orderings. In contrast to this, the local terms in the
OPE for the three possible orderings are independent: they produce the three operator
correlators through convolutions with di erent two point functions.
A basic issue which will be important in our discussion is whether the local term in
the OPE can be removed and its e ect in the various correlators absorbed in a rede nition
in the ordinary framework of the CFT. Alternatively this reduces to the question whether
the local term can be formulated in a universal regularization independent way.
To illustrate this setup and the issue of universality we review the well known case of
the Zamolodchikov metric on the conformal manifold in d = 2. This was studied a long
time ago in [6]. We will denote moduli, which are exactly marginal operators, by Mi. They
have dimension d, which we assume to be even, and their structure constants vanish by
their requirement of being moduli.
In the present paper global properties of the moduli space do not play an essential
role and we will not discuss them. We will limit the range of moduli couplings to the
vicinity of a
ducial CFT where the moduli space does not have singularities. In this
range, where the conformal deformation of the CFT by adding the moduli multiplied by
{ 2 {
couplings is well de ned, we will treat the couplings as particular x independent values
of the sources. They are local coordinates on the moduli space which is also referred to
as the conformal manifold. If the spacetime dependent sources are denoted by J i(x), the
conformal deformation is obtained by adding
S =
sources of the moduli are denoted by J i, the anomaly is
HJEP07(218)
Gij (J ) is the Zamolodchikov metric. The universal information contained in (1.3), like in
every type B anomaly [8, 9], is the logarithmically divergent counterterm
Mi(p1)Mj (p2) =
4 ikj Mk(p1 + p2)
{ 3 {
(1.2)
(1.3)
(1.4)
(1.6)
(1.7)
which contains semilocal correlators of the moduli, the singularity in momentum space
being inherited from the two point function.
Taking three functional derivatives with respect to J and Fourier transforming,
as proposed in [6]. In con guration space this corresponds to
4
Mi(x)Mj (y) =
2(x
y) ikj Mk(x)
(1.8)
In the previous argument we used universal features of the logarithmically divergent
counterterm. This will be part of our approach, i.e. we will always start with the semilocal
term in the correlator which has all the symmetries and analyticity properties of the theory
and derive from it the local contribution to the OPE needed to reproduce it. In particular
the transformation properties under source reparametrizations re ect the covariance of the
Zamolodchikov metric de ned by the counterterm.
We now discuss the issue of the universality of the local term in the OPE above. The
key is the behaviour of this term under source reparametrization, which is a symmetry of the
theory. The term found is normalized by the Christo el connection of the Zamolodchikov
metric. It transforms inhomogenously under reparametrizations of the sources which
suggests that it is not universal. Indeed, using Gaussian normal coordinates at a given point in
moduli space, the connection can be put to zero. This shows that the semilocal term which
it represents can be obtained from the ordinary set up of the theory without the need of
local terms in the OPE. Moreover in [10] an explicit procedure in a special regularization is
given which shows how one could recover the reparametrization invariant information
contained in the Zamolodchikov metric from correlators calculated without using additional
local terms in the OPE. An example where some local contributions to the OPE can be
removed while others not was discussed recently in [11].
The situation in d = 4 is similar: we start from the anomaly [7]
B =
1
(1.9)
J i +
ijkr J j r J j . This re ects the counterterm (in
at spacetime)
1
3
g
factor) that p2 is replaced by (p2)2 and one has a fourdimensional function in (1.8). The
generalization of these equations to arbitrary even dimensions is obvious.1 Again, the local
term in the OPE of two moduli is proportional to the Christo el connection and removable
through a source reparametrization. It is therefore not universal.
In this paper we will apply the same logic to identify new local contributions to the
OPE involving currents and moduli. However, the terms which we single out cannot be
removed and are therefore universal. This is a consequence of supersymmetry2 i.e. we will
study N = (2; 2) superconformal theories in d = 2 and N = 2 SCFTs in d = 4. The
common feature we will nd is that the local terms in the OPE will be normalized by the
Zamolodchikov metric itself and therefore cannot be removed by source reparametrization.
1but not so the generalization of (1.9) which can, however, be worked out case by case.
2In the present paper we use extended supersymmetry, but some results may also follow from e.g. N = 1
superconformal symmetry in d = 4. We thank Z. Komargodski for pointing out this possibility.
{ 4 {
HJEP07(218)
Once the additional terms in the OPE are identi ed we study whether factorization
can be used for the local terms in a manner analogous to the decomposition in terms of the
ordinary structure constants. The local terms generate classes of contributions to certain
correlators of the theory in addition to the usual one. We check various requirements for
these contributions, in particular their consistency with supersymmetry. While for (2; 2)
theories in d = 2 we have a complete and consistent construction, in d = 4 we have to face
an ambiguity in the separation of certain correlators into ordinary contributions (the result
of combining the usual three point structures) and the ones produced through factorization
from the local terms.
The paper is organized as follows: in section 2 we will review the structure of moduli
anomalies for N = 2 superconformal theories in four dimensions and identify the
logarithmically divergent counterterm involving currents. We identify a contribution to the trace
anomaly originating in the correlator of a U(
1
)R current and moduli and applying the logic
outlined above we identify the local term in the OPE of two moduli giving the current.
In section 3 we discuss N = (2; 2) superconformal theories in d = 2. In the general
expression for the superconformal anomalies we identify a contribution to the U(
1
) anomaly
and we determine the local contribution in the OPE needed to reproduce it. Using
special features of d = 2 we construct the full anomalous part of the e ective action which
incorporates terms obtained through factorization from the local additions to the OPE.
In section 4, where we present our conclusions, we discuss the possibility of using the
local terms in the OPE in a factorized manner, the consequences of such an assumption
and the consistency checks needed.
In appendix A we give a proof for the absence of ordinary structure constants of moduli
and conserved currents. In appendix B we work out explicitly the local contributions for the
simplest example of an N = 2 theory in d = 4: the free Maxwell gauge supermultiplet. We
identify in a Feynman diagram calculation the contribution of redundant operators which
leads to the semilocal structures we nd. This suggests that in a diagrammatic calculation
the local terms in the OPE can be replaced by redundant (i.e. vanishing on shell) operators.
We check the consistency of these factorized contributions by calculating the anomalous
four moduli correlator consistent with supersymmetry in this model as a sum of factorized
and ordinary contributions. In appendix C we work out a similar eld theoretic model
realizing the structures we found for the N = (2; 2) theories in two dimensions.
2
N
= 2 in d = 4
In this section we discuss fourdimensional N = 2 superconformal theories with moduli.
Superconformal theories in d = 2 with (2; 2) supersymmetry, which have several special,
simplifying features, will be discussed in the next section.
Conformal eld theories with N = 2 supersymmetry have an SU(2) U(
1
) Rsymmetry,
of which the U(
1
) factor is anomalous. The basic result of this section is that this anomalous
U(
1
) Rcurrent appears as a local term in the M M operator product. This follows from
the structure of a counterterm related to a type B Weylanomaly. This counterterm is
required by supersymmetry. In appendix B we will verify some of our general results and
{ 5 {
a
+ (c
a)W
W
claims by looking at pure N = 2 U(
1
) gauge theory where all features appear at one loop
order and can be explicitly computed.
We gauge the global symmetries and couple the CFT to an external metric and gauge
elds, the sources for the energy momentum tensor and for the Rsymmetry currents,
respectively. And, of course, we also have the sources for the moduli. The anomalous
Ward identities are then most succinctly incorporated in the e ective action, which is the
nonlocal functional of the sources obtained by integrating out the CFT. It necessarily
violates some of the symmetries and the anomaly A is the variation of the generating
functional under these transformations. Here the anomalous symmetries are superWeyl
transformations, which are parametrized by a chiral super eld , whose lowest component
HJEP07(218)
j =
+ i
parametrizes Weyl ( ) and U(
1
)R transformations ( ).
The superWeyl anomaly is therefore the variation of the e ective action with the chiral
super eld parameter
expansion of the above expression is [7, 13]3
B =
1
+ K
3In [7] the U(
1
)R and SU(2)R gauge elds were set to zero.
~
H
R
R
1
3
1
3
R g
R g
{ 6 {
+ i Gi r^ r^ J i r J
j
r^ r^ J j r J
+ 2A
F
F
+2 F
+F
r K r
(2.2)
r A
A r
Here A is the Kahler connection, de ned as
A
=
i
2
j
and F
its eld strength which depends only on the Kahler metric and is therefore invariant
under Kahler shifts and covariant under holomorphic coordinate changes on the conformal
manifold. F is the
eld strength of the U(
1
) gauge
eld A and H that of the SU(2)
gauge
eld.
Were it not for supersymmetry, many terms in the component expression
would be cohomologically trivial and could be dropped, but as it is obvious from the (three
irreducible) superspace expressions, supersymmetry demands that they accompany the
cohomologically nontrivial terms. Supersymmetry is also responsible for the appearance
of the target space Riemann tensor in the fourth line. In a bosonic theory it could be
replaced by an arbitrary tensor with the correct symmetries and would still be a
nontrivial solution to the WessZumino consistency condition. But N = 2 supersymmetry
requires that this tensor is the Riemann tensor. In the general form this anomaly rst
appeared in [14] and we therefore refer to it as the Osborn anomaly.
One can write down a nonlocal action, both in superspace and in components, whose
superWeyl variation reproduces (2.1), but one is faced with the same problem as for the
ordinary Weyl anomaly in four dimensions that this \Riegert" action does not have the
correct analyticity properties [15]. It therefore di ers from the unknown `true' e ective
action by unknown nonlocal Weyl invariant terms.
The anomaly polynomial is invariant under a combined ( eld dependent) superWeyl
transformation and a Kahler transformation if their parameters are related as [7]
~ =
1
24a
F
invariance of A
1
24 a A
variation of the e ective action.
This is easy to verify for the superspace action (2.1) and can also be veri ed for the
component expression (2.2). It is readily observed that every term in the rst three lines
with a bare gauge eld A , i.e. not appearing in the gauge invariant combination F ,
has a counterpart in the last two lines if we replace A
!
1
24a A . This re ects the
under a joint gauge and Kahler transformation with (2.4). In
the \Conclusions" section we will reformulate this symmetry in terms of the Kahler shift
Let us now analyse the anomaly polynomial (2.2) further. Consider the rst two terms
in the last line. The second one vanishes for
= const: and therefore will not contribute
to the following argument. The rst term is a type B Weylanomaly and corresponds to
a counterterm, in the same way as was described in the Introduction. As such it contains
information about nonlocal terms in correlation functions. Taking functional derivatives
with respect to J i, J  and A , one nds the correlator
hMi(k1) M (k2) j ( k1
where Gi is evaluated for constant sources. We have de ned q = k1 + k2 and r = k1
Of course the same counterterm also generates correlation functions of one current and an
{ 7 {
(2.3)
(2.4)
arbitrary number of moduli, always via the currentcurrent twopoint function. The term
cannot come from an ordinary three point function since, as we show in appendix A, the
moduli being neutral under U(
1
)R the structure constant vanishes. This indicates that the
U(
1
) Rcurrent j appears in a contact term in the M M operator product. Since it is
proportional to the Zamolodchikov metric it cannot be removed by a reparametrization of
the conformal manifold. The fact that the Zamolodchikov metric appears is a consequence
of supersymmetry. If it were not for supersymmetry, the counterterm which is responsible
for this correlator could be omitted and one could adopt a scheme where there are no local
terms in the operator product of two moduli.
normalized to R A j , we nd the local terms in the M M OPE
If we normalize the U(
1
)R current such that the coupling in the microscopic theory is
y) j (y)
y) j (x) + : : :
(2.6)
There could be other local terms in this operator product, but they do not contribute to
the three point function with the Rsymmetry current. We will give their speci c form for
the particular case of the free Maxwell theory in appendix B.
Once we formulated the local term in operatorial language we can translate it into
a covariant one: the local term in the OPE will give a contribution to any correlator
involving moduli by coupling the moduli to the U(
1
)R current. The correlators of
Rcurrents are represented by terms in the e ective action containing its source A . Therefore
the contribution of the local term in the OPE to correlators with moduli is obtained by
replacing A
1
in any term in the generating functional by 24c A . This is the general
formulation of factorization we are using. The normalization follows from comparing the
last term in the last line with the third term in the rst line of (2.2).
One might wonder to what extend factorization determines the form of the anomaly
polynomial. An explicit calculation in N = 2 superMaxwell theory shows that the
counterterm in (2.2) which involves the Riemann tensor on the conformal manifold (the \Osborn
anomaly"), is not completely accounted for by factorization. The same calculation however
shows that without the local term in the above operator product the Riemann tensor would
not appear, but it is required by N = 2 supersymmetry.
More generally if the anomaly polynomial were given completely by factorization all
the terms would contain the combination A + 214c A . This is clearly not the case for (2.2)
which contains invariant eld strengths of A
without the corresponding terms constructed
from A . This seems to be dictated by supersymmetry, because there is no way to
super4 We will elaborate on this point in the concluding section, but
symmetrize e.g.
F
F
~ .
already draw the partial conclusion that while the factorized contributions of the moduli
are needed, the typical situation is that they come together with ordinary contributions
obtained ignoring the local terms. An interesting connection appears in the explicit
example discussed in appendix B: the local terms in the OPE can be replaced by including
4At least as long as we use only chiral multiplets to represent the sources Ji. This might be reminiscent of
the situation in d = 2 where the coupling of chiral multiplets to a target space B eld cannot be accomplished
o shell. To do this one has to use semichiral multiplets. We did not pursue the generalization of this
possibility to d = 4.
{ 8 {
in the moduli \redundant operators" in the covariant calculation of the correlators. Here
\redundant" means operators in a lagrangian CFT which vanish if one uses the equations
of motion.
3
N
symmetry. In this case the anomaly is5
We begin with a review of the basic features of moduli anomalies in N = (2; 2)
superconformal theories [7]. Extended supersymmetry implies additional global symmetries which
in this case are the U(
1
)A
U(
1
)V Rsymmetries. We can choose to preserve either one
of the two U(
1
) factors. The second factor then belongs to the multiplet of anomalous
currents. Due to this choice we have two possible types of theories.
For concreteness, we will only discuss the theory which preserves the U(
1
)A
RB =
F
(3.1)
In this expression, whose superspace version will be given later, K is the Kahler potential
on the conformal manifold, a real function of the sources J and J and
A
=
i
2
is again the Kahler connection. F
is the eld strength of the U(
1
)A gauge eld A . Under
local U(
1
)V transformations it transforms as
A
=
=
A ,
V
and
F
= 2 r V .
parametrizes local Weyl transformations and c is the
Virasoro central charge.6 The relative coe cients in (3.1) are dictated by supersymmetry.
The invariance of the e ective action (nonlocal and local terms) under the axial (U(
1
)A)
gauge transformation A
is part of the de nition of the theory. As a consequence
the terms
and
F , which can be obtained by the gauge variation of A A
and A A respectively, remain cohomologically nontrivial since the addition of these local
terms to the e ective action would violate the U(
1
)A symmetry, i.e. the de nition of the
theory. Note also a very special feature of d = 2: the chiral anomaly can be seen not
only in odd parity correlators like in all even dimensions but also as a \clash" between
conservation in the even parity vectorvector and axialaxial correlators when the vector
and axial currents are related by a duality transformation.
We now analyse the
term of the anomaly polynomial (3.1). It represents
an anomaly in the correlator of the U(
1
)V current and at least one modulus and one
antimodulus or, equivalently, a noninvariance of the corresponding terms in the e ective
action under a vector gauge transformation of the gauge eld A . The momentum space
structure of the term which leads to this anomaly is qq2q , where q is the momentum
carried by the axial current to which A is coupled. By an argument similar to the one in
5For simplicity we only consider chiral primary moduli. For the general case, which includes also twisted
chiral primary moduli, we refer to [7], where further details of the notation can also be found.
6Di eomorphism invariance requires c = cL = cR, i.e. absence of a gravitational anomaly.
{ 9 {
d = 4 which is discussed in appendix A, such a contribution cannot come from a
modulusantimoduluscurrent coupling since the moduli are neutral under the Rsymmetries. In this
case the semilocal structure involves not a logarithm like in d = 4 but the characteristic
q12 pole multiplied by a polynomial in the momenta of the moduli. In order to reproduce
it we need to assume the existence of a local term in the OPE
y) ~ (y)
y) ~ (x)
where ~ is the anomalous vector current. We could use instead equivalently the
nonanomalous axial current j related to ~ by a duality transformation. Combining then the
local term in the OPE with the correlator of two vector currents:
sources and therefore it is universal. It leads through factorization to classes of calculable
contributions to the e ective action.
These factorized contributions can be calculated
following the rule analogous to the one used in d = 4 i.e. wherever the gauge eld A
is coupled to the axial current we should replace it by the combination A
+ 6c
A .
This combination is manifest in the anomaly polynomial (3.1) and comparing the terms
it is clear that the anomaly involving the moduli is reproduced.
The combination which appears is invariant under a joint transformation of the Kahler
by f (J ) + f (J ) and a vector gauge transformation of A
f ). The consistency between the combinations selected by
potential which generates A
with parameter
=
3ci (f
dimensional theory.
factorization and the invariance of the anomaly polynomial is a special feature of the two
Since the local terms in the OPE are factorized we treat this e ective coupling on
equal terms with A , i.e. for every term in the e ective action involving A
we can get a
term involving correlators of moduli by the above replacement. As discussed in section 2
in d = 4 the typical situation is that both factorized local OPE contributions and ordinary
ones are needed to reproduce the total supersymmetric expressions.
In d = 2 due the special kinematical features we are able to give a complete description
of the anomalous part of the e ective action and to check including only the factorized
contributions we get an answer consistent with supersymmetry and the Kahler structure.
We proceed now to construct the anomalous part of the e ective action. We start with
the rst building block involving the Zamolodchikov anomaly, i.e. eq.(1.3). Generically
a typeB trace anomaly is induced by a logarithmically divergent term, but the anomaly
itself appears in the Weyl variation of a
nite correlator which involves the sources in the
divergent term and the metric which is coupled to the energy momentum tensor [8, 9].
For a general type B anomaly there are no closed expressions for the
nite part to all
orders in the external sources. Even for the standard ctrace anomaly in d = 4 the nite,
non local correlator, whose Weyl variation is the anomaly, is known only in the leading
order i.e. a correlator of three energy momentum tensors [15]. For the case considered
here, i.e. the nite part reproducing the Zamolodchikov anomaly in d = 2 containing any
number of moduli and external metric perturbations, the problem can be exactly solved.
Let us start with the rst correlator which contributes to the nite part: a correlator of two
moduli and one energy momentum. In a convenient basis the correlator has the kinematical
decomposition
hMi(k1)Mj (k2)T ( q)i = A(q2; k12; k22)(
q
2
energymomentum tensor) determines the B and C amplitudes in terms of the twopoint
function of the moduli
Q(k2)
hMi(k)Mj ( k)i
evaluated at k12 and k22 respectively, while the Weyl transformation (trace of energy
momentum) Ward identity determines the A amplitude in terms of the trace of the energy
momentum tensor, i.e. the anomaly which we denote by B:
it is traceless as seen from the explicit decomposition. That part contains through the two
point correlator of moduli the logarithmic divergence. It follows that due to the very special
kinematical features of d = 2, the three point function of two moduli and one energy
momentum tensor splits into a nonanomalous part and a completely explicit anomalous part
represented by the A amplitude. We remark the q12 structure in the anomalous part which
is surprising, since a priori one would expect singularities combining the three kinematical
invariants. Once this lowest correlator is understood it is easy to write the result for any
number of energy momentum tensors and moduli by simply making the result covariant
in spacetime and using covariance under source reparametrizations for the moduli. The
result for the anomalous part of the e ective action which has the correct Zamolodchikov
trace anomaly is:
where we have de ned
Wa =

(3.7)
(3.8)
(3.9)
(3.10)
This can be combined in a single expression with the Polyakov trace anomaly since they
have the same structure, i.e. 1= .
The supersymmetrization is now straightforward by replacing the scalar curvature in
the Polyakov anomaly and the Zamolodchikov anomaly with their superspace
generalizations in a single linear combination. The relative normalization is
xed by the linear
combination selected through factorization for the gauge components since the superspace
curvature contains the gauge eld A while the superspace Zamolodchikov anomaly contains
the Kahler U(
1
) eld A. Then the anomalous part of the e ective action in superspace is
W =
c Z
48
Its component expansion is (3.2). That (3.12) follows from (3.11) can be checked using
R =
! K + f + f and superWeyl transformations
is a chiral super eld which parametrizes
superWeyl transformations; its lowest component is
+ i
and R is the curvature chiral
super eld, whose top component contains the Ricci scalar (also denoted by R) and the
j =
eld strength F . The sources J i are chiral super elds and K is a real function of
the sources, the Kahler potential on the conformal manifold. For further details on the
geometry of N = (2; 2) supergravity we refer to [16]. The anomalous e ective action in
superconformal gauge was given in [7]. The symmetry under a joint Kahler shift and
a correlated Weyl transformation, which acts on the anomaly polynomial, is promoted
here to a symmetry of the anomalous part of the e ective action. We remark that the
anomalous part as we de ned it through factorization, contains a local Weyl invariant piece
R d2x R d
4 EK2. To this we should add the fully Weyl invariant nonlocal contribution.
There is an additional freedom we have since the Weyl invariance is anomalous: one is
allowed to add local Weyl nonivariant functionals of the external sources respecting all the
other symmetries:
where we have de ned F =
F . Note that it contains the gauge eld and the Kahler
connection only in the combination A
= A
+ 6c
The last term in (3.15) represents correlators of moduli which are only induced through
the factorization assumption. The local term in the OPE which through factorization
It is instructive to have the anomalous e ective action also in components:
W =
Z
d2xpg
G
1
R
F
1
r A
1
4
3
+
2 c G K
G
1
c
96
G
1
R + F
1
F
1
8
K R
K
1
r A
r A
Z
1
4
Z
R
K
1
4
produced the A dependent terms above was de ned in terms of Gi, but after translating
it to the covariant formalism we ended up with an explicit dependence on A . Since
the
eld strength F corresponding to A , which is the pullback of the Kahler form,
contains only Gi, it is clear that in order to recover the original information we should
impose a gauge invariance of A. Such a gauge invariance is induced by a Kahler shift
K ! K + f (J ) + f (J ) i.e.
i
2
A
=
f )
We are therefore led to study the behaviour of the anomalous part of the e ective
action under a Kahler shift.
Since the e ective action is by construction invariant
under a joint transformation by a Kahler shift and a Weyl transformation with
the transformation under a Kahler shift can easily be calculated, simply replacing in the
anomaly calculation
by f . The result of the Kahler shift can be absorbed by the
folI(J; J ) ! I(J; J ) + 23 c K(J; J )(f (J ) + f (J )).
lowing changes in the local Weyl noninvariant term (3.14): H(J ) ! H(J )
In summary in this class of theories through factorization the local terms in the OPE
produce contributions to the e ective action consistent with (2; 2) supersymmetry, but
the Weyl anomalous part of the e ective action is not invariant under a Kahler shift, its
variation being local.
4
Conclusions
In [7] the behaviour of the anomaly polynomial under a Kahler shift was studied. In this
section, for the discussion of the implications of factorization, we
nd it convenient to
discuss the behaviour under a Kahler shift of the e ective action itself. In this way we
are able to isolate universal features of the terms generated by factorization which are not
invariant under Kahler shift.
We will consider terms in the e ective action which depend on the moduli through a
Kahler potential K. When the N = 2 theory is the result of a compacti cation from a six
dimensional theory on a Riemann surface and K has an ab initio geometric meaning [17, 18],
this is the case for the full e ective action. For a generic N = 2 theory K is de ned by
the moduli trace anomalies and therefore we are really discussing only the Weyl anomalous
part of the e ective action. One expects a \Kahler shift invariance" for the transformation
K(J; J ) ! K(J; J ) + f (J ) + f (J )
This transformation induces on the pulled back universal U(
1
) Kahler form a gauge
transformation
A
! A
+
f )
i
2
This Kahler shift can give a nonvanishing result. We will treat it as an anomaly with the
understanding that it originates just in those terms in the e ective action which depend
on K. Then like for any other anomaly one should look for nontrivial solutions of the
(3.16)
= 6c f ,
14 f (J ) and
(4.1)
(4.2)
is omitted, being cohomologically trivial in superspace. It is an open question if there are
possible additional terms in the anomaly equation in which some of the dependence on the
N = 2 supergravity multiplet elds is replaced by a dependence on some elds derived
from K itself. We will discuss this aspect in more detail below.
The Kahler shift anomaly has the special feature that there are counterterms present
with the same structure as (4.3) with arbitrary chiral coe cient functions and the anomaly
shifts these coe cients. These terms, though local, are chiral and therefore cannot cancel
the Kahler shift produced starting from K and therefore this feature does not change the
way we treat the anomaly.
The Kahler shift anomaly and Weyl anomalies have WessZumino type consistency
conditions which follow from the commutativity of the two transformations:
This equation xes a0 in terms of the moduli contribution to the Weyl anomaly,
This condition is equivalent to the invariance of the anomaly polynomial under a joint
transformation with correlated f and
[7].
On the other hand b0 is left un xed since the expression it multiplies is Weyl invariant.
It follows immediately from the above discussion that if there are additional terms in the
Kahler shift anomaly polynomial they should be Weyl invariant since there is no term
which could match its Weyl variation.
We now come to the role of the local terms in the OPE. Through factorization for every
term in the e ective action involving A , we should get a corresponding term with 214c A
replacing A . We will discuss the implications for just the Kahler shift anomaly polynomial:
2
3
~
H
R
1
2
1
Kahler shifts terms with the same structure as those in (4.4) which contain f
f . For
a general N = 2 theory in d = 4 their normalization is however incompatible with
the relative normalization obtained by the WessZumino condition. This implies that
the local terms in the OPE, while contributing to the anomalous correlators, do not
account for the complete answer. For this we need to add the ordinary contributions.
In appendix B we describe the explicit check of a similar situation for the Maxwell
supermultiplet: the Osborn anomaly is completely
xed by N = 2 supersymmetry
in terms of the Riemann tensor computed from the Zamolodchikov metric. This was
obtained as the sum of two terms, one representing the factorized contributions of the
local terms in the OPE and the other one the ordinary contribution. Interestingly the
two terms had even di erent index structures and only their sum gave the Riemann
tensor of the Zamolodchikov metric.
2) If we want to replace more than one A , we should limit ourself to the anomalous term
involving three A
which generate the U(
1
)R chiral anomaly. Using factorization,
terms depending on A in the anomaly polynomial could generate terms in the Kahler
shift anomaly polynomial. We will assume in the following discussion that the Weyl
anomaly polynomial is \complete" i.e. the new anomalies suggested by factorization
will appear only in the Kahler shift. This can always be achieved by adding variations
of local counterterms. Then replacing two A
we get a new term in the Kahler
f )F
~
F
anomaly polynomial (f
N
= 2 supersymmetry, i.e. obtained from an appropriate superspace expansion.
If the supersymmetrization turns out to be impossible, the factorized contribution
. This term should be made compatible with
should be cancelled by an ordinary term.
Finally, by replacing all three A
we have the new term (f
should be supersymmetrized and the previous discussion applies. We remark that in order
that the structures discussed under this point would appear, one needs at least four (real)
moduli: otherwise the contributions vanish or are cohomologically trivial. In summary
in d = 4, while the factorized local OPE contributions are needed, they seem to act
always together with the ordinary terms and their normalization therefore does not have
unambiguous predictive power.
In d = 2 the situation is di erent. Due to the speci c two dimensional kinematical
simpli cations and (2; 2) supersymmetry, the Weyl anomalous part determines completely
the Kahler shift anomalies in this component of the e ective action. The Weyl
anomalous part of the e ective action can be separated unambiguously from the Weyl invariant
part and it depends on an explicit combination of the curvature super eld and the Kahler
potential. The normalization of this combination is determined by factorization and
therefore the Kahler shift anomalies can be understood to follow entirely from Weyl anomalies
combined with factorization. One cannot exclude of course that the Weyl invariant part
of the e ective action produces under the Kahler shift an additional contribution with the
f )F
F
~ . This again
same anomaly structure but it is a consistent assumption that the Weyl invariant part is
also Kahler shift invariant.
Finally we would like to comment on the possible role of local terms in the OPE in the
conformal bootstrap. For theories with extended supersymmetries the local terms should
be included as additional couplings to the usual conformal blocks. The constraints following
from crossing symmetry should give interesting relations between the contributions of local
terms and the ordinary ones.
Acknowledgments
HJEP07(218)
This work is supported in part by ICORE program of the Planning and Budgeting
Committee and the Israel Science Foundation (grant number 1937/12) and by GIF  the
GermanIsraeli Foundation for Scienti c Research and Development (grant number 1265).
We thank the Galileo Galilei Institute for Theoretical Physics for hospitality; S.T. also
acknowledges the hospitality extended to him at the Instituto de F sica Teorica (IFT) in
Madrid. We have bene tted from discussions with D. Butter, Z. Komargodski, S. Kuzenko
and M. Rocek.
A
Nonzero structure constant implies nonzero charge
Consider in d = 4 the correlator of a conserved current J (z) with two dimension four
operators M1(x) and M2(y). We assume that M1; M2 are not orthogonal to each other but
we do not assume anything about their charge under J . From conformal invariance the
coordinate dependence is completely xed [19] for noncoinciding coordinates
hM1(x)M2(y)J (z)i = c~
1
1
(z
(x
y)6 (z
x)2 (z
x)
x)4
x $ y
except for the structure constant c~. The OPE between J (z) and M1(x) can be extracted
from the above correlator assuming that the representation holds also when one coordinate
approaches another. We put x = 0 and z in nitesimally close to 0 while y is kept xed
with the component in the direction
chosen to be 0. Then the OPE has the form
(A.1)
(A.2)
(A.3)
(A.4)
Continuing to Minkowski space we obtain
J (z)M1(0)
z
c~ z4 M1(0)
T Q(t)M1(0)
c~sign(t) M1(0)
[Q; M1(0)]
c~M1(0)
where Q is the charge operator R d3zJ0(t; ~z) and T is time ordering. Considering the
relation above for t =
we nd
i.e. M is necessarily charged if the structure constant c~ is not zero. The manipulations above
are completely equivalent to the standard use of conformal and gauge Ward identities.
The Maxwell case
A simple toy model is fourdimensional N = 2 supersymmetric U(
1
) gauge theory. A useful
reference is appendix B of [20] whose notation we follow in this appendix. The eld content
are the gauge eld, a SU(2)R doublet of Weyl spinors and a complex scalar. There is also
a SU(2)R triplet of auxiliary elds. They play no role in our analysis. The action is
S =
d4x
F
F
+
F
~
F
+ i i
The fermions carry U(
1
)R charge +1 while the scalar has charge +2. All other elds are
HJEP07(218)
neutral. The U(
1
)R current is therefore
j =
i
)
This theory has a complex modulus, i.e. an exactly marginal operator,
M =
i
2
1
8
F + F +
+ i i
where F
= F
i F~. This operator is neutral under U(
1
)R and one might expect that
the hM M j i correlation function vanishes. But this is not quite true and it has, in
fact, an imaginary part.
Note that the last two terms in (B.3) vanish onshell. The
reason why we are not allowed to set these redundant operators to zero is supersymmetry.
As we will show they contribute in an essential way to the threepoint function. When
inserted into a Feynman diagram they cancel a propagator, but the diagram still retains a
nontrivial analytic structure. In the fourmoduli correlator which we will compute below,
the redundant operators contribute in a similar way and their contribution is required in
order to get the result which is consistent with supersymmetry. On the other hand it is
also clear that their contribution to the twopoint function is completely real and therefore
for the Zamolodchikov metric only the gauge eld part of the moduli is relevant.
In this free eld theory the hM M j i correlation function is given by triangle diagrams.
Only the fermions and the scalar contribute and among the di erent possible contractions
those where the propagator between the M and M insertions is cancelled, have an
imaginary part. This implies a local term in the M (x) M (y) operator product expansion which
is proportional to the current and the hj j i twopoint function is responsible for the
logarithm.
(B.1)
(B.2)
(B.3)
Explicit calculation of the oneloop triangle diagram gives
where q = k1 + k2 and r = k1
k2.
hM (k1) M (k2) j ( k1
Some comments/observations are in order: (i) The correlation function is not gauge
anomalous. This is consistent with (2.2) where all moduli dependent terms with
are
cohomologically trivial. (ii) It follows from the calculation that the logarithmic divergence
is due to the cancelled propagators. Those contractions where this does not happen, do not
contribute. (iii) The kinematical structure of the diagrams corresponds to R d4x F
This re ects the general structure of contact terms in this simplest example of a free theory.
F
.
We can also compute the logarithmically divergent part of the fourpoint function.
For the nonsupersymmetric theory, where the modulus consists only of the spin one part
in (B.3), this was done by Osborn [21]. His result cannot be cast into the form dictated
by N = 2 supersymmetry, which contains the target space Riemann tensor (cf. [7], or the
fourth line of eq. (2.2)), which for the Zamolodchikov metric g
is the single source in this case and 2 its imaginary part. The di erence between
these two expressions is proportional to
2 (r
r
)
2
5 jr
r
j
2
(B.5)
where now
is the uctuation around a constant value of the source and we have only
kept terms up to O( 4). This di erence must be accounted for by spin 0 and spin 1/2
contributions of M and M via the cancelled propagator argument. In each case, there are
two Feynman diagrams which contribute7
(B.6)
(B.7)
2
384
2
384
For the spin zero part these two diagrams evaluate to
m12 m22 + m12 m23 + m12 m24 + m22 m23 + m22 m24 + m32 m24
while for the fermions one computes
2 m12 m22 + 2 m32 m24
m12 m23
m12 m24
m22 m23
m22 m24
(s + u)(m12 + m22 + m32 + m42) + s2 + s u + u2 log 2 + nite
+ (s + u)(m12 + m22 + m32 + m42)
s
2
4su
u
2 log 2 + nite
7The spin 0 and 1/2 parts of M do not contribute nonlocal parts to M 3 M or M 4 correlators. As for the
other orderings around the box digram, there are always at least three cancelled propagators and therefore
the Cutkosky rules give zero imaginary part and therefore no logarithm.
We have expressed the amplitudes in terms of an independent set of kinematical invariants
mi2 = ki2 ;
s = (k1 + k3)2 ;
u = (k1 + k4)2
Their sum is proportional to
m12 m22 + m32 m24
2(m12 m23 + m12 m24 + m22 m23 + m22 m24)
+ 2(s + u)(m12 + m22 + m32 + m42)
2(s2 + u2)
5 s u
which is precisely the kinematical structure derived from (B.5). The overall normalization
can be xed by an appropriate rescaling of the source .
We remark that analysing the above diagrams in terms of the OPE we identify two
additional local terms speci c to this model which contribute: denoting by the S(x)
(x) (x)
the dimension two scalar operator and by
the (conserved) vector operator which di ers
from j only by the relative sign between the bosonic and fermionic contributions to (B.2)
and which can be shown to have vanishing twopoint function with j , we nd
M (x)M (y)
y) (3j (y)
(y))
y) (3j (x)
(x))
Z
d2x d4 E
+
E J M + c.c.
=
Z
d2x d4 E (1 + J + J )
A useful reference for at (2; 2) superspace is chapter 12 of [22]. For curved superspace
we follow [16]. With the help of results obtained there, we nd the following component
turbed superspace action R
satis es D+
D
and D
primary operator
eld, the deformed action is
(x) 4(x
y)S(y)
gauge
to SU(4).
The coupling of the Maxwell N = 4 supermultiplet to the N = 4 conformal supergravity
background was studied in [23{25]. In this case the complex modulus belongs to the
current multiplet and therefore the AdS2 Zamolodchikov metric is a consequence of N = 4
supersymmetry. The term which interests us, i.e. the mixing of the Kahler form with the
eld of the Rsymmetry current, cannot appear due to the gauge eld belonging
C
Free theory with moduli in d = 2
We want to study the free (2,2) SCFT of a single twisted chiral super eld
with
unper.
= ( ; +;
; F ) being twisted chiral means that it
= D
= 0 and the complex conjugate relations D+
= D
= 0. Here
are at superspace covariant derivatives. We deform the theory by a chiral
M = D+
D
We then couple the deformed CFT to U(
1
)A supergravity. If J is the chiral source
super(B.8)
(B.9)
(B.10)
(C.1)
(C.2)
action
Z
d2xpg
i
2
$
r
+ (1 + J + J )
+ (1 + J + J )
A
+ 12
A
Here we have set the gravitini to zero. The auxiliary scalar in the gravity multiplet drops
out. We have de ned the Dirac spinor
= (
; +)T and J is now the lowest component
of the source super eld. The other components are set to zero as is the auxiliary
eld F
contained in
; it vanishes onshell. A is the U(
1
)A gauge eld of the SUGRA multiplet
and A is the Kahler connection computed from the potential8
K =
ln(1 + J + J )
Note in (C.3) the relative factor 12 in the coupling to the U(
1
)current j
. It is 6c for c = 3, the central charge of the twisted chiral multiplet and we see
that besides the coupling to gravity, the fermions couple precisely to the combination of the
U(
1
)A and the Kahler connection which was discussed in section 3. From the action we can
also read o the moduli operators as the coe cients of the sources. In a at background
= 0) they are
(C.3)
(C.4)
=
(C.5)
+ i
The fermionic contribution vanishes onshell i.e. it is redundant and will only contribute
via the cancelled propagator argument, already familiar from the discussion of the free
Maxwell theory. The bosonic part accounts for the ordinary contributions to correlators.
As in the free Maxwell theory, only this nonredundant part contributes to the logarithmic
divergence of the hM M i twopoint function and therefore to the Zamolodchikov metric. If
we expand the action around constant moduli, J =
+ J and compute hM M i, we nd
that it is proportional to (1 +
+ ) 2, from the normalization of the kinetic term of
and the fact that the oneloop diagram which computes it has two propagators. This is
the Kahler metric derived from (C.4). Again as in the N = 2 Maxwell theory in d = 4, the
redundant piece of M is responsible for the nonvanishing of hM M j i and the cancelled
propagator localizes the M (x)M (y) operator product on the U(
1
) current.
If we integrate out
and
, we recover the nonlocal e ective action. It is easy to
integrate out the fermions. They can be rescaled to eliminate the (1 + J + J ) factor. What
is left are free fermions coupled to an external gauge eld A
A
+ 6c
A . This leads to a term
Z
1
(C.6)
in the e ective action, in agreement with (3.15).
8The relation with the more familiar Kahler potential for the metric on the upper halfplane, K =
ln(
) is established with the coordinate transformation J = i
1
2
.
It is special to this simple model that the microscopic action R d4 e K(J;J)
formally
depends on the sources through the Kahler potential on the moduli space. However the
explicit expansion in components shows that due to the fact that the scalar elds without
derivatives acting on them are not legal operators, the actual dependence on the sources
does not necessarily re ect the coupling of the potential. For this model therefore one can
see explicitly that while the Weyl anomalous part of the action is de ned by the Kahler
potential with its potentially anomalous shift invariance, the couplings of the Weyl invariant
part e ectively do not contain anymore the Kahler potential.
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momentum space, JHEP 03 (2014) 111 [arXiv:1304.7760] [INSPIRE].
renormalisation, functions and anomalies, JHEP 03 (2016) 066 [arXiv:1510.08442]
[3] A. Bzowski, P. McFadden and K. Skenderis, Renormalised 3point functions of stress tensors
and conserved currents in CFT, arXiv:1711.09105 [INSPIRE].
[4] A. Dymarsky et al., Scale invariance, conformality and generalized free elds, JHEP 02
(2016) 099 [arXiv:1402.6322] [INSPIRE].
[5] Y. Nakayama, On the realization of impossible anomalies, arXiv:1804.02940 [INSPIRE].
[6] D. Kutasov, Geometry on the space of conformal eld theories and contact terms, Phys. Lett.
B 220 (1989) 153 [INSPIRE].
(1976) 45 [INSPIRE].
[7] J. Gomis et al., Anomalies, conformal manifolds and spheres, JHEP 03 (2016) 022
[8] S. Deser, M.J. Du and C.J. Isham, Nonlocal conformal anomalies, Nucl. Phys. B 111
[arXiv:1611.03101] [INSPIRE].
JHEP 10 (2013) 151 [arXiv:1307.7586] [INSPIRE].
[9] S. Deser and A. Schwimmer, Geometric classi cation of conformal anomalies in arbitrary
dimensions, Phys. Lett. B 309 (1993) 279 [hepth/9302047] [INSPIRE].
[10] D. Friedan and A. Konechny, Curvature formula for the space of 2D conformal eld theories,
JHEP 09 (2012) 113 [arXiv:1206.1749] [INSPIRE].
[11] J. Gomis et al., Shortening anomalies in supersymmetric theories, JHEP 01 (2017) 067
[12] S.M. Kuzenko, SuperWeyl anomalies in N = 2 supergravity and (non)local e ective actions,
[13] D. Butter, B. de Wit, S.M. Kuzenko and I. Lodato, New higherderivative invariants in
N = 2 supergravity and the GaussBonnet term, JHEP 12 (2013) 062 [arXiv:1307.6546]
HJEP07(218)
Zamolodchikov metric, JHEP 12 (2017) 140 [arXiv:1710.03934] [INSPIRE].
general dimensions, Annals Phys. 231 (1994) 311 [hepth/9307010] [INSPIRE].
rings in 4d N = 2 SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
Lett. B 561 (2003) 174 [hepth/0302119] [INSPIRE].
Mathematical Society, U.S.A. (2003). [23] I.L. Buchbinder, N.G. Pletnev and A.A. Tseytlin, \Induced" N = 4 conformal supergravity,
[1] A. Bzowski , P. McFadden and K. Skenderis , Implications of conformal invariance in [2] A. Bzowski , P. McFadden and K. Skenderis , Scalar 3point functions in CFT: [hepth/9605009] [INSPIRE].
[14] H. Osborn , Weyl consistency conditions and a local renormalization group equation for general renormalizable eld theories , Nucl. Phys. B 363 ( 1991 ) 486 [INSPIRE].
[15] J. Erdmenger and H. Osborn , Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions , Nucl. Phys. B 483 ( 1997 ) 431 [16] S.J. Gates , Jr., M.T. Grisaru and M.E. Wehlau , A Study of general 2D, N = 2 matter coupled to supergravity in superspace , Nucl. Phys. B 460 ( 1996 ) 579 [ hep th/9509021] [17] N. Seiberg , Y. Tachikawa and K. Yonekura , Anomalies of duality groups and extended [23] I.L. Buchbinder , N.G. Pletnev and A.A. Tseytlin , \ Induced" N = 4 conformal supergravity , Phys. Lett. B 717 ( 2012 ) 274 [arXiv: 1209 .0416] [INSPIRE].
[24] F. Ciceri and B. Sahoo , Towards the full N = 4 conformal supergravity action , JHEP 01