Shortening anomalies in supersymmetric theories
Received: December
Shortening anomalies in supersymmetric theories
Jaume Gomis 1 2 4 8 9 10
Zohar Komargodski 1 2 4 6 9 10
Hirosi Ooguri 1 2 3 4 7 9 10
Nathan Seiberg 1 2 4 5 9 10
Yifan Wang 0 1 2 4 9 10
Princeton 1 2 4 9 10
NJ 1 2 4 9 10
U.S.A. 1 2 4 9 10
Open Access 1 2 4 9 10
c The Authors. 1 2 4 9 10
0 Joseph Henry Laboratories, Princeton University
1 Caltech , Pasadena, CA 91125 , U.S.A
2 Rehovot 76100 , Israel
3 Kavli IPMU, WPI, University of Tokyo
4 Waterloo , Ontario, N2L 2Y5 , Canada
5 School of Natural Sciences, Institute for Advanced Study
6 Weizmann Institute of Science
7 Walter Burke Institute for Theoretical Physics
8 Perimeter Institute for Theoretical Physics
9 Princeton , NJ 08540 , U.S.A
10 Kashiwa , 778583 , Japan
We present new anomalies in twodimensional N theories. They obstruct the shortening conditions of chiral and twisted chiral multiplets at coincident points. This implies that marginal couplings cannot be promoted to background super elds in short representations. Therefore, standard results that follow from N = (2; 2) spurion analysis are invalidated. These anomalies appear only if supersymmetry is enhanced beyond N = (2; 2). These anomalies explain why the conformal manifolds of the K3 and T 4 sigma models are not Kahler and do not factorize into chiral and twisted chiral moduli spaces and why there are no N = (2; 2) gauged linear sigma models that cover these conformal manifolds. We also present these results from the point of view of the Riemann curvature of conformal manifolds.
Anomalies in Field and String Theories; Conformal Field Theory; Supersym

Dedicated to John Schwarz on his 75th birthday
1 Introduction 2 3 4
N = (4; 4) conformal manifolds
Riemannian curvature
Supercurrent bundle
A The WessZumino perspective
B Special geometry relation
C Fourpoint function involving supercurrents
D SU(2)out selection rules
Supersymmetric contact terms and the shortening anomaly
Curvature of conformal manifold and factorization
Conformal Field Theories (CFT's) often come in a continuous family labeled by their
exactly marginal couplings. This family, known as the conformal manifold M, is endowed
with a canonical metric, the Zamolodchikov metric [1].
The Zamolodchikov metric is
determined by the twopoint functions of the exactly marginal operators. The general
Riemannian structure of conformal manifolds was rst discussed in [2, 3].
Such conformal manifolds appear in theories with extended symmetries, such as
suThese conformal manifolds also play an important role in the AdS/CFT correspondence
and on string worldsheets. In the former case, the conformal manifold of the boundary
CFT maps to the space of vacua in the bulk Antide Sitter space (AdS). In the latter
case, the conformal manifold of the worldsheet theory maps to the space of solutions of the
equations of motion in spacetime.
One of the fundamental properties of the conformal manifold of twodimensional
(2; 2) superconformal eld theories (SCFT's) is that it factorizes locally2 into the product
N =
1Some general arguments about when such families may exist can be found in [4].
2There are examples where the conformal manifold is modded out by a discrete symmetry, which prevents
it from being a product globally [5, 6]. We thank D. Morrison for a useful discussion about this point.
of two Kahler manifolds,
Coordinates of Mc and Mtc are the coupling constants of the exactly marginal operators
constructed from the dimension ( 1 ; 12 ) operators in the chiral and twisted chiral rings [7] of
2
linear sigma model with a CalabiYau threefold as its target space, the factorization (1.1)
was proven in [8] by combining string theory worldsheet SCFT and target space arguments.3
worldsheet view points, see [9, 10] and [8, 11{13], respectively.
More recently, in [14] the Weyl anomaly on the conformal manifold of N
= (2; 2)
theories was used to rederive the factorization (1.1). The argument in [14] applies to
tion made in [14] was that the coupling constants parameterizing Mc and Mtc could be
promoted to supersymmetric dimension (0; 0) background chiral and twisted chiral
multiplets, in the spirit of [15]. Factorization of the conformal manifold (1.1) then followed
from the classi cation of anomalies of the partition function under superWeyl
transformations. Alternatively, one can also easily provide an argument for factorization (1.1) in the
spirit of [16].
torus T n SCFT is locally [17],
On the other hand, it is wellknown that the conformal manifold of the ndimensional
while the conformal manifold of the K3 SCFT is locally [2] ,
(see also [19]),
for some n.4 These examples appear to be at odds with the factorization (1.1) of the
conforcase in (1.2), these conformal manifolds do not factorize locally into a product of Kahler
manifolds. In fact, they are not even Kahler manifolds. This is in spite of the fact that the
Another way of presenting this puzzle is the following. Normally, when a particular
global symmetry of a theory implies some special properties, extending that symmetry
does not ruin those properties. Here, we see a counterexample to that. Speci cally, if
3For SCFT's realizing CalabiYau compacti cations, Mc and Mtc parameterize the moduli space of
complex structure and complexi ed Kahler class.
that it should factorize as in (1.1) . This conclusion turns out to be wrong. Later we will
extended supersymmetry.
factorization mentioned above, the OPE has a pole5
expansion (OPE) between a chiral multiplet O and a twisted chiral multiplet O~ of scaling
dimensions ( 12 ; 12 ) does not have poles. On the other hand, in all the counterexamples to
J++(x2) +
where the symbol j on the lefthand side picks up the bottom component of each multiplet.
The operator J++ appearing on the righthand side must be both chiral and twisted chiral,
and with scaling dimensions (1; 0) and Rcharges (2; 0). J++ is the bottom component of
a multiplet implies that the Rsymmetry is enhanced and that the SCFT enjoys enlarged
SCFT, the current J++ enlarges the Rsymmetry from U(1) to SU(2).6
Let us imagine that we deform the SCFT with exactly marginal operators
A powerful idea [15] is to promote
and ~ to background chiral and twisted chiral
superelds. We will see that whenever the OPE between O and O~ is singular as in (1.5), then
the couplings cannot be promoted to such background super elds due to an anomaly! We
can either promote the s to background chiral super elds or the ~s to background twisted
chiral super elds, but we cannot do both simultaneously. To our knowledge, this type of
obstruction has not been discussed before.
Before we explain how this comes about let us explain the physics of promoting
couplings to background
elds. The e ective action as a function of the background
satis es the required symmetries when the operators do not have contact terms that spoil
those symmetries. For example, if a conserved current is not conserved at coincident points
then the background e ective action will not be gauge invariant. Another example is if in a
CFT the operator equation T
e ective action fails to depend just on the conformal class of the background metric.
Similarly, in supersymmetric theories, for the e ective action to depend on
and twisted chiral super elds, the operators O and O~ have to obey their de ning equations
and ~ as chiral
O = 0 ;
D+O~ = D
O~ = 0
, as if we are in Lorentzian signature.
state in the SCFT.
5Throughout this note we study the theories in Euclidean space. However, we use Lorentzian signature
notation with coordinates x
etc., which are complex conjugates of each other. The reason for using this
notation is that we use complex conjugation notation on the charged chiral objects like the odd coordinates
6It is important that the current J++ which enlarges the Rsymmetry corresponds to a normalizable
not only at separated points (which holds by de nition) but also at coincident points.
Loosely speaking, one could say that the equations (1.7) have to be obeyed o shell. As
with all anomalies, our shortening anomaly can be understood as the failure of the
partition function to be invariant under certain background eld transformations. The operator
equations (1.7) at coincident points would be a consequence of the partition function being
invariant under certain background super eld transformations, akin to the standard way
conservation laws follow from the invariance of the partition function under background
eld transformations.
When the partition function is not invariant under these
back
eld transformations, we encounter an anomaly. This point of view is elaborated
in appendix A.
We will show that the OPE (1.5) induces contact terms of the form
D+(1)O(Z1) O~(Z2)
D+(2)O(Z1) O~(Z2)
and that it is impossible to tune both of these contact terms away (the notation will be
explained in section 3.) Therefore, the e ective action does not depend on the background
coupling constants as if they were chiral and twisted chiral super elds. Some of the
background couplings have to sit in long multiplets. We call this phenomenon a shortening
anomaly.7 The discussion above is reminiscent of the clash between conservation of the
vector and axial current in theories with fermions, where by adjusting counterterms either
symmetry can be preserved, but not both simultaneously.
As is standard in such situations, where some operator equations are violated at
coincident points, when we turn on nontrivial backgrounds for
and ~ then the contact
terms (1.8) lead to nontrivial operator equations. Depending on which counterterms we
choose, one of the equations below has to be true:
For constant couplings, where only the bottom components of
operators O and O~ remain short, as they should.
The obstruction to promoting both chiral and twisted chiral couplings to short
multiplets (which exists only if supersymmetry is enhanced) invalidates the conclusions found
in [14] for such theories.8 We therefore conclude that factorization (1.1) breaks down only
7We would like to emphasize that the operators O, O~ remain short in the standard situation where
the couplings are constant. Indeed, ; ~ have vanishing beta functions, since the operators O; O~ have no
operators to combine with (see [20] and also earlier literature, e.g. [11]). In particular, (1.5) does not induce
a beta function. The shortening anomaly is in the background
elds, not in the operators. The operator
equations are modi ed only in nontrivial con gurations for the background super elds (see equation (1.9)).
8The analysis in the spirit of [16] is also invalidated by this anomaly since an implicit assumption in
such an analysis is that it is possible to preserve the shortening conditions of all the dimension ( 12 ; 12 ) chiral
and twisted chiral operators both at separate and coincident points.
and ~ are turned on, the
does not factorize.10
ifolds (1.2) (with n > 2 even), (1.3) and (1.4) are indeed endowed with such an operator
of Rcharge (2; 0) and have enhanced supersymmetries, thus resolving the paradox.9
This phenomenon is similar to a familiar situation in supergravity. The target space
= 2
supergravity with hypermultiplets has a quaternionic target space, which is not Kahler
(see [22] for a review of the scalar manifolds of supergravities in various dimensions).
supergravity theory because it has a multiplet including a graviphoton and a gravitino,
that resolves our puzzle, which includes a conserved spin 32 current, plays a similar role to
the graviphoton multiplet in this supergravity analog. For a related discussion see [23].
We can relate our discussion to supergravity more directly if we view our 2d models as
worldsheet theories for strings and we study their spacetime description. The low energy
scalar manifold indeed takes the factorized form (1.1). On the other hand, compacti cations
on manifolds leading to enhanced spacetime supersymmetry, such as T 6 or K3
As with all 't Hooftlike anomalies, our analysis leads to theorems about the
nonsupersymmetric RG
ow in which all the infrared marginal couplings are realized as chiral
or twisted chiral couplings along the ow, then it is guaranteed that the corresponding
coupling constants in the infrared are in short representations and hence there is no shortening
anomaly. Therefore, the conformal manifold would have to factorize into a chiral Kahler
manifold and a twisted chiral Kahler manifold. Therefore, one can immediately conclude
that there cannot exist an N = (2; 2) RG
ow that realizes the full conformal manifold of
the K3 SCFT. Indeed, constructions of gauged linear sigma models (GLSM's) [24] which
lead to subspaces of the K3 conformal manifold are known (see for example [18, 25]), but
it has never been possible to embed the K3 SCFT in a UV completion that covers the
full conformal manifold. The same holds for T 4 but not for T 2, which does not have a
the complete T 2 SCFT conformal manifold.11 We now see that the obstruction for K3 and
T 4 is due to an anomaly in the infrared that must be matched in the ultraviolet.12
9Such an operator does not exist for the T 2 SCFT. And indeed, the T 2 conformal manifold (1.2)
10The conformal manifold of the worldsheet SCFT is a subspace of the supergravity scalar manifold.
(1; 1; 1; 3) with a superpotential W
3. The twisted chiral coupling is realized by the complexi ed FI parameter and the chiral coupling by the
single complex parameter in G3(X).
conformal manifold of sigma models on K3 or T 4.
It is interesting to relate this discussion to recent developments concerning the S2
compactify such theories on S2 and they compute, respectively, the Kahler potentials of
the chiral and twisted chiral deformations [28, 30] (see also [14, 31]). However, this
statement is not meaningful, if the total space does not factorize as in (1.1)! Therefore, when
(One can understand it again as being due to the failure of spurion analysis.) Because of
manifold, therefore, what one can extract from the sphere partition function is at best the
Kahler potential on some Kahler submanifolds of the conformal manifold. An alternative
sphere compacti cation that utilizes the extended supersymmetry may exist and it may
probe the full conformal manifold.
shortening anomaly.
We also provide a complementary perspective on the anomalies by studying the
Riework [8, 19, 32]. The study of the mixed chiral and twisted chiral exactly marginal
curvature components leads us to establish a factorization theorem: the conformal manifold
conserved current of Rcharges (2; 0), precisely the same operator J++ responsible for the
The outline of the paper is as follows. In section 2 we set the stage by discussing the
OPE of chiral and twisted chiral super elds. In section 3 we show that if a particular
short representation J++ appears in the OPE, then one inevitably
nds an obstruction to
imposing the shortening conditions simultaneously on both the chiral and twisted chiral
super elds. In section 4 the same result is established by analyzing the Riemann
curvaperconformal algebra and prove that the conformal manifold of such theories indeed takes
the form (1.4). In this section we further show that the extended supercharges acquire
nontrivial holonomies when transported around the conformal manifold.
In appendix A, we discuss the shortening anomaly from the WessZumino point of
view. Some of the calculations of the various curvatures are presented in appendices B, C,
OPE of chiral and twisted chiral operators in N
= (2; 2) SCFT's
operator can be realized as the top component of a chiral multiplet O with U(1)+
Rcharges (1; 1) or as the top component of a twisted chiral multiplet O~ with Rcharges
(1; 1). These multiplets obey the shortening conditions
D+O = 0
D+O~ = 0
O = 0 ;
O~ = 0 :
Our aim is to determine to what extent these shortening conditions can be maintained as
we explore the conformal manifold of an N = (2; 2) SCFT.
Monitoring the shortening conditions (2.1) under an exactly marginal perturbation
leads us to analyze the contact terms in
; I ) are points in superspace. In this section we determine the
operator product expansion (OPE) of a chiral and a twisted chiral multiplets, whose top
components yield marginal operators, leaving the analysis of contact terms to the
follo
In supersymmetry, it is often useful to employ spurion analysis. E.g., the coupling
constants of chiral operators O are promoted to background chiral multiplets [15] . This
procedure makes sense only if the operator equation D
O = 0 is respected also at
coincident points (loosely speaking, we can say that O is chiral o shell). This is because when
we write the partition function depending on some background
elds, by taking
derivatives with respect to the background
elds, we can probe the correlation functions of the
corresponding operators both at separated and at coincident points. By de nition,
operators equations are always obeyed at separated points, but they may fail at coincident
points. The famous (continuous) 't Hooft anomalies arise when a conservation equation
is not obeyed at coincident points. As a consequence, the partition function depending
on the associated background elds does not obey the naively expected equations (in the
famous case of the chiral anomaly, the partition function is not gauge invariant). Similarly,
we can couple twisted chiral background
elds to twisted chiral operators as long as the
corresponding shortening conditions are valid o shell.
A general method that guarantees that some conservation equation is obeyed also at
coincident points is to construct a regularization obeying the conservation equation. This
automatically tunes the contact terms in the infrared to zero. Such a regularization may not
exist, if there is a genuine anomaly in the conservation equation. In the next section we will
establish the existence of some contact terms that violate the shortening conditions (2.1)
at coincident points. Therefore, the standard arguments relying on the selection rules of
ows that contain all the infrared chiral and twisted chiral couplings cannot exist.
SCFT in two dimensions when it has an operator of dimensions (1; 0) and Rcharges (2; 0).
Unitarity implies that this operator, which we denote by J++, is a conserved current,
that the supersymmetry is enhanced.
A typical (but not the only) example of supersymmetry enhanced by such an extra
encode the contributions of nonchiral operators and descendants. For simplicity,
we are suppressing some coe cients that we will make more explicit later. In particular,
a hyperKahler manifold M , there is a unique holomorphic 2form
2 H2;0(M ), and the
extra Rcurrent is expressed as J++ =
ab +a +b. Chiral and twisted chiral operators are
Oj = kab +
O~j = kabgbc cd
are fermions. The OPE (2.5)
where k 2 H1;1(M ), gab is the Kahler metric on M , and
then follows from
More generally, one can show that the existence of J++ of dimensions (1; 0) and
Rcharges (2; 0) alone is su cient for the pairing (2.4) and therefore the OPE (2.5) follows
commute with the U(1)+
group must be larger than just U(1)+
Rsymmetry in an N = (2; 2) SCFT, the Rsymmetry
U(1) and the dimension ( 1 ; 1 ) chiral and twisted
chiral operators must furnish a representation of it. Therefore, J++ cannot act trivially
and hence some pairing as in (2.4) and (2.5) must be present.
Unitarity and supersymmetry further imply that the extra Rcurrent J++ is the bottom
component of a very short multiplet, which we denote by J++. It obeys
extra Rcurrent as
where Oj denotes the bottom component of the operator O. By taking OPE's of both sides
of the equation with another twisted chiral operator O~(x3) (or, in other words, using the
symmetry of the OPE coe cients), we nd
J++(x3) +
U(1)+ U(1)
D+ J++ = D
J++ = D
J++ = 0 :
For convenience, we summarize the Rcharges of various objects used in the following:
Our nal goal is to monitor the shortening conditions (2.1) as we explore the
conforsupersymmetry manifest. Introducing coordinates (x; ; ) (suppressing the +;
The chiral coordinates y
D =
D =
fD; Dg = 2i@ :
D y = 0
D y = 0
(c; c), O~ is (c; c) and J++ is simultaneously (c; c) and (c; c).
We want to determine the dependence of threepoint correlators on the superspace
position of the operators.
We start with the supertranslation Ward identities.
two points in superspace (x1; 1; 1) and (x2; 2; 2), we can de ne two independent even
linear invariants:13
i 12 12 = y1
2i 2 1 = y1
Supertranslation invariance implies that correlators depend on the position of operators
through zi j and z i j . The coordinates we have de ned are rather convenient. Indeed, if
the ith operator is chiral, the correlators depend on zi j only, while if it is antichiral they
depend on z i j .
Our correlator of interest is:
; I ). The shortening conditions,
hO(Z1) O~(Z2) J ++(Z3)i ;
D+O = D
D+O~ = D
O = 0 ;
O~ = 0 ;
D+ J ++ = D
J ++ = D
J ++ = 0 ;
together with supertranslational and rotational invariance imply that the correlator (2.13)
depends on z1+3+, z2+3+ and z1 2 . A subtlety in SCFT's that must be taken into account is
the existence of a superconformal invariant X(Z1; Z2; Z3) constructed out of three points
in superspace [33]. Superconformal invariance and nilpotency of X(Z1; Z2; Z3) imply that
the most general correlator consistent with superconformal invariance is given by
hO(Z1) O~(Z2) J ++(Z3)i =
[1 + a X(Z1; Z2; Z3)] :
id ; d ; d ), where g = exp
i(xP + Q +
Q) . This yields the supertranslation invariants ( 12
2). Note that z1 2 = 12 i 12 12 and z1 2 =
hO(Z1) O~(Z2) J ++(Z3)i =
Moreover, using that
we obtain our desired OPE:
hJ++(Z2) J ++(Z3)i =
The superspace correlators we have constructed obey, by construction, the shortening
conditions (2.14) at separated points. Our next task is to study the shortening
conditions at coincident points (2.3) and determine whether counterterms can be adjusted so
that the shortening conditions for chirals and twisted chirals (2.1) can be both
simultatherefore conclude that the correlator at separated points is
Supersymmetric contact terms and the shortening anomaly
The OPE (2.18) may lead to some contact terms (2.3).14 To understand these we need
to study the superspace derivatives (the derivatives with respect to the second argument
follow from these)
= @++@
log(jx12j2) =
log(jx12j2) =
Our strategy to compute (3.1) is to extend the wellknown formula, @++
to Green's function in superspace. This line of inquiry makes manifest an inherent
ambiguity in de ning the derivatives (3.1) in superspace. In order to de ne these derivatives
we need to specify the behavior of Green's function in superspace for the leftmovers, and
di erent choices yield di erent answers. This ambiguity is a manifestation of the fact that
bution.15 In fact, we will nd an ambiguity of 1+2 12
+ (2)(z12) in de ning 1=z1 2 . This is
14A related discussion appeared in [34], where it clari ed the need for a contact term, which had been
found earlier in [35, 36]. A more modern discussion of that problem appeared in [37].
15On the other hand, the pole 1=x
on the 2plane of (x++; x
) can be extended to an unambiguous
distribution because R d2xf (x++; x
superspace, however, is akin to 1=(x
distribution de ned via 1=(x
)2 and cannot be extended to an unambiguous distribution. The
is wellde ned for any smooth function f . The pole 1=z1 2 in
@ (1=x
) corresponds to a particular choice of regularization.
the addition of (x).16 As a result, the derivatives (3.1) su er from certain ambiguities.
However, we will nd that, whichever choice we make for these ambiguities, we cannot
remove the contact terms for both chiral and twisted chiral operators simultaneously.
We can compute (3.1) starting from the Green's functions in superspace
D+(1)D(1) log(z1+2+z1 2 ) =
D+(1)D(1) log(z1+2+z1 2 ) = 4
Physically, (3.3) and (3.4) can be interpreted as the SchwingerDyson equations or the
Ward identities for the shift symmetry of a chiral multiplet and a twisted chiral multiplet
respectively. But these are not the only choices. Consider the following identity
D(1)z1 2 = D(1) a log(z1+2+z1 2 ) + b log(z1+2+z1 2 ) ;
= 2 ib (2)(z12) 1+2 :
fD+; D+g = 2i@++ :
Amongst these, there are distinguished canonical choices
Preserves chiral Ward identity
Preserves twisted chiral Ward identity
Symmetric violation of Ward identities
We note that there is no choice of a that preserves simultaneously the chiral and twisted
chiral Ward identity. Even though the derivatives are subject to some ambiguities, we now
proceed to unambiguously establish our shortening anomaly.
and D+(1) 1=z1 2 .
is proportional to
i ) di er by distributions proportional to (x).
17Note that equations (3.3) and (3.4) only x D+(1) 1=z1 2
and D+(1) 1=z1 2
up to terms
propor
Anomalies arise when we have an operator that should satisfy an operator equation,
such as (2.1), and one nds that such an equation is only correct at separated points while
at coincident points there are various contact terms.18 Establishing the anomaly amounts
to showing that these contact terms cannot be removed by rede ning the scheme since
scheme rede nitions change the theory by contact terms.
the most general OPE between a chiral and a twisted chiral super eld in a theory with an
enhanced Rsymmetry, now allowing for supersymmetric contact terms, is given by
J++(Z2) + r (2)(z12) 1+2 1+2 J++(Z2) ;
where r is a scheme dependent constant, which can be shifted by changing the scheme.
In order to establish our shortening anomaly we must show that it is not possible to
tune the contact term r such that the shortening conditions for a chiral and a twisted chiral
multiplet can be maintained simultaneously. Since the contact term in (3.9) is annihilated
by D(1) and D(2), these shortening equations are automatically preserved. Acting with
D+(1) and D+(2) we get using (3.6)19
D+(1)O(Z1) O~(Z2)
D+(2)O(Z1) O~(Z2)
2 iC a (2)(z12) 1+2 J++(Z1) + r (2)(z12) 1+2 J++(Z1) :
Preserving the chiral and twisted chiral shortening conditions along the conformal manifold
requires tuning the coe cient of the supersymmetric counterterm to obey
twisted chiral :
r = 2 iCb ;
r =
cannot simultaneously preserve the chiral and twisted chiral shortening conditions along
the conformal manifold. By tuning the contact term, we can either preserve the chiral or
the twisted chiral constraint, but not both. This is our shortening anomaly.
The fact that we cannot preserve both constraints simultaneously is analogous to the
situation in two dimensions with vector and axial anomalies [39], where contact terms
cannot remove both anomalies simultaneously. For a complementary derivation of our
shortening anomaly in the cohomological approach based on the WessZumino consistency
conditions, see appendix A.
The contact terms we encountered lead to an operatorial violation of the shortening
equations upon deforming the theory by background super eld sources for the exactly
18By turning on suitable background elds the ambiguities in the contact terms can be described as an
ambiguity in adding local counterterms constructed out of the operators in the theory and the background
elds. When such background elds are present, the problem with contact terms can be uplifted to problems
at separated points.
marginal operators (as in (2.2)). The shortening conditions become
ators to full edged background chiral and twisted chiral super elds is impossible whenever
the theory includes the operator J++. We emphasize that the violations in (3.12) depend
on the fermionic components of the multiplet to which the couplings have been promoted.
Curvature of conformal manifold and factorization
A CFT with exactly marginal operators comes equipped with additional structure beyond
the metric on the conformal manifold M. Operators in the CFT are sections of vector
bundles over M. A canonical example of this is the set of exactly marginal operators,
which are sections of the tangent bundle T M. Transporting operators in the vector bundle
along M leads to operator mixing, which is governed by a connection on the vector
bundle [3, 40, 41]. The curvature of these connections captures geometrical and topological
data about the vector bundle of operators in the CFT.
The curvature of the tangent bundle T M can be used to prove theorems about the
conformal manifold M. The computation in [8] of the curvature of T M in twodimensional
guments, was used to prove that the conformal manifold of such SCFT's factorizes (1.1).
The result in [8] follows from the vanishing of the mixed chiral and twisted chiral curvature
components, which implies that the holonomy group is the direct product of two
commuting subgroups, which in turn implies factorization (1.1). In this section we establish the
and only if the SCFT has a conserved current with Rcharges (2; 0).
We now proceed to compute the mixed components of the Riemann tensor of M using
CFT techniques, in the spirit of [3, 19, 41, 42]. The tangent bundle T M comes equipped
with a metric compatible connection, whose curvature we would like to nd. The curvature
of this connection is determined by a certain fourpoint function of the exactly marginal
operators, which we denote by Ui. The formula for the curvature can be written as [42]20
Rijkl =
) hUi(0) Uk(y) Ul(1) Uj (1)ic :
The symbol `RV' stands for the prescription where we cut out small discs around the xed
operators and remove the powerlaw divergent terms.21 The subscript c stands for the
20Note that our de nition of the curvature includes an overall normalization of 4 compared to that of [42].
This is the convention in which the special geometry relation takes the standard form. In addition, we de ne
as usual O(1) = limx!1 x
2 O O(x) while keeping all the other insertions
angular dependence and thus this singularity vanishes upon integrating over the angles. We recall that
dimension (1; 1) operators do not appear in the OPE since a nonzero OPE coe cient would lead to a
connected correlator, which is de ned by subtracting the three possible disconnected
contributions to the fourpoint function. As shown in [41], an explicit formula for the curvature
can be given in terms of the CFT data: spectrum of operators and OPE coe cients.
further exploited in [19] we can show that the only nontrivial mixed chiral and twisted
chiral moduli fourpoint function that needs to be studied is
= @y++ @y
= @y++ @y
where we have denoted the exactly marginal operators constructed from operators in the
chiral and twisted chiral ring by
Fi =
1 Z
F~~i =
Using the superconformal Ward identities introduced in [8], the fourpoint function of
interest (4.2) can be expressed in terms of the fourpoint function of chiral and twisted
chiral operators of dimension ( 1 ; 1 )
We are now ready to compute the curvature using (4.1). It follows from (4.4) that
the independent components of the Riemann curvature tensor are Rik~`~j and Ri`~k~j . The
connected component prescription in (4.1) can be extended to both sides of (4.4), and
therefore, by pulling the operator located at x to in nity we arrive at
Integrating by parts in (4.1) and remembering that we are integrating over the complex
plane with disks around the punctures removed, the answer reduces to contour integrals
around the punctures. In order to get a nonzero contribution to the curvature, the function
c z=0;w=1
22More singular terms are removed by the prescription in (4.1). One can verify that there is no
contribution from in nity.
where J++ has Rcharges ( 2; 0). Using
J++(w) O~k~(z) Oi(1) =
around y = 0.
demonstrates that there is no contribution to the curvature from the contour integral
In conclusion, we have shown that there is a nontrivial component of the mixed
only if the SCFT has a current with Rcharges (2; 0). In such a SCFT, the Riemann
curvature is given by
and the conformal manifold M no longer factorizes.
= (4; 4) conformal manifolds
by S+A++, and the SU(2)R currents by J++ ), where
(
is an important example of extended supersymmetry. In this section, we use the
forgive a purely
eldtheoretic derivation that the local geometry of the coset is (1.4). We
This gives a geometric perspective on our shortening anomaly.
and A are doublet indices for the
algebra, respectively.
We use the convention 12 =
21 = 1 for the invariant tensors
; AB and their
inverses. For the right moving sector, we use dotted indices. We denote the weight 12 ; 12
are exactly marginal operators which preserve the N
= (4; 4) superconformal
symmeTheir twopoint functions are
F AA_ =
hOi _ (x)Oj _ (y)i =
hFiAA_ (x)FjBB_ (y)i =
where ij is the Zamolodchikov metric.
Riemannian curvature
RiAA_;jBB_ ;kCC_ ;`DD_ =
log jy j FiAA_ (0) FjBB_ (y) FkCC_ (1) F`DD_ (1)E : (5.4)
2 D
(anti)chiral multiples are Fi11_ (Fi22_ ), while those for (anti)twisted chiral multiples are
As shown in the previous section, there are nonzero curvature components in mixed
Ri11_;j12_;k21_;`22_ = Ri11_;j21_;k12_;`22_ =
using the Bianchi identity, we can also determine23
Ri21_;j12_;k11_;`22_ =
On the other hand, the curvatures in the purely chiral directions are controlled by the
Ri11_;j22_;k11_;`22_ = ij k` + i` jk
where CiJk are the chiral ring coe cients.
between the curvature components (5.5), (5.6), and (5.7)
Ri11_;j22_;k11_;`22_ + Ri21_;j12;k11_;`22_ + Ri21_;j22_;k11_;`12_ = 0 :
This allows us to determine
Ri11_;j22_;k11_;`22_ =
Comparing this with (5.7), we obtain as a byproduct the following constraint on the chiral
ring of any N = (4; 4) SCFT
CiIkCjJ`gIJ =
would be nice to verify that the cohomology ring of M satis es the constraint (5.10).24
Combining these results, the full Riemannian curvature is
RiAA_;jBB_ ;kCC_ ;`DD_ =
AB CD + ( AC BD + AD BC ) A_B_ C_ D_ :
This implies that the conformal manifold of an N
= (4; 4) SCFT is locally the coset
M =
Supercurrent bundle
RiCC_ jDD_ A B =
identities to be
RiCC_ ;jDD_ ; _ A_; _B_ =
The nontrivial SU(2)out
SU(2)out holonomies shown in these curvatures means that it is
Since the tangent bundle T M is a tensor product of the left and right supercurrent
bundles and the bundle of weight
chiral primaries, the curvature tensor for the
supercurrents computed here can be combined with the curvatures for the chiral primaries
computed in [19] to reproduce the Riemann curvature (5.11) on M.
Although M is not Kahler, it has Kahler submanifolds. In fact, the maximal Kahler
submanifold of a quaternionicKahler manifold is middledimensional [44]. In our case,
the maximal Kahler submanifold of M is locally
S =
24A curious observation is that (5.10) implies that there is a uniform bound on chiral ring coe cients
(squared) associated with the Rcharge (1; 1) chiral primaries. The bound takes the schematic form C2 <
up, say, as at the conifold point.
) DS A+++(0) FiCC_ (y) FjDD_ (1)S B+++(1)E :
hS+A++(0) FiCC_ (y) FjDD_ (1)S+B++(1)ic =
The yintegral in (5.13) can then be performed
AC BD
RiCC_ ;jDD_ ; A; B =
If we only turn on the marginal couplings associated to chiral multiplets and explore that
submanifold of M, then there is no shortening anomaly, and the argument of [14] leading
associated to chiral or twisted chiral operators corresponds to the submanifold (5.17) .
We thank Kevin Costello, David Morrison, Kyriakos Papadodimas, Ronen Plesser, Adam
Schwimmer, Stefan Theisen, and Edward Witten for useful discussions. J.G.'s research
was supported in part by the Perimeter Institute for Theoretical Physics. Research at
Perimeter Institute is supported by the Government of Canada through Industry Canada
and by the Province of Ontario through the Ministry of Research and Innovation. Z.K.
is supported in part by an Israel Science Foundation center for excellence grant and by
the ICORE program of the Planning and Budgeting Committee and the Israel Science
Foundation (grant number 1937/12). Z.K. is also supported by the ERC STG grant 335182
and by the United StatesIsrael BSF grant 2010/629. H.O. is supported in part by U.S.
Department of Energy grant DESC0011632, by the Simons Investigator Award, by the
World Premier International Research Center Initiative, MEXT, Japan, by JSPS
GrantinAid for Scienti c Research C26400240, and by JSPS GrantinAid for Scienti c Research
on Innovative Areas 15H05895. N.S. was supported in part by DOE grant DESC0009988.
Y.W. was supported by the NSF grant PHY1620059 and by the Simons Foundation Grant
#488653. H.O. thanks the hospitality of the Institute for Advanced Study and Harvard
University, where he spent his sabbatical in 2015  2016, and of the Aspen Center for
Physics, which is supported by the National Science Foundation grant PHY1066293. N.S.
thanks the hospitality of the Weizmann Institute of Science during the completion of this
The WessZumino perspective
Anomalies arise when we have some operator that should satisfy an operator equation, e.g.
@ j = 0 or T
points while at coincident points there are various contact terms. The essence is to show
that these contact terms cannot be removed by rede ning the scheme. Indeed, scheme
rede nitions change the theory by various contact terms and so we need to demonstrate
that the anomaly is invariant under scheme rede nitions. A convenient way to establish
it is to introduce background
elds for the various operators. Then scheme rede nitions
correspond to adding new local terms to the action, which depend on these background
elds and also, possibly, on the operators in the theory.
We would like to examine the operator [Q+; Oj], which is normally zero if O is chiral.
To this end we couple a background
eld to this operator. A standard procedure is to
couple the super eld O to a background
eld in the superpotential but then we do not
have a source for the redundant operator. Therefore we will couple O to a source in the
Kahler potential. We add a corresponding term for a twisted chiral super eld O:
Now we imagine computing the partition function Z[A; B] (with A; B super elds). It
is useful to tabulate their charges
0.26 So the standard expectation is that
The standard expectation is that the partition function would not actually depend on
Z[A; B] = Z[A + D+
Z[A; B] = Z[A; B + D+
. This should be viewed, for example, in analogy with
Z[g ] = 0 for
the conformal anomaly case. What we would like to test is whether we can respect (A.3)
whether the shortening of the background multiplets is consistent with supersymmetry.
The general principles that we reviewed above tell us that for in nitesimal
Z[A; B] should be a local functional of the sources and operators in the theory i.e.
log Z[A; B] =
1 ( 1=2; 1=2)
( 1=2; 1=2)
with Llocal some local function of the couplings and operators. The right hand side in (A.5)
is restricted by demanding that it is supersymmetric and also by demanding that it obeys
the WessZumino consistency conditions [45].
Let us assume that the partition function is invariant under the
namely, O~ obeys the twisted chiral shortening conditions at both separated and coincident
points. We can then write the variation under
as follows (the formula for the variation
under + is analogous)
log Z[A; B] =
some constant. Equation (A.6) respects supersymmetry (because it is a R d4
integral), and it is consistent with the Rsymmetry (D+D B carries Rcharges (0; 0) and
carries Rcharges ( 2; 1) and therefore D
carries ( 2; 0) and thus it exactly
cancels the Rcharge of J++). Furthermore, (A.6) obeys the WessZumino consistency
26This standard expectation follows from the fact that these conditions hold \o shell," namely there is a
regularization where this is true. Technically, it means that there are no cohomologically nontrivial contact
terms in correlation functions of these redundant operators.
condition since D+D B is invariant under B ! B + D+
+. Therefore, (A.6)
does not violate the fact that the partition function is postulated to be invariant under
of the local term
We now proceed to prove that (A.6) is cohomologically nontrivial. If (A.6) were
cohomologically trivial then one could add a local term to log Z[A; B] such that the right
hand side of (A.6) would vanish while retaining supersymmetry and invariance under
transformations. It is clear (by integration by parts and using (2.7)) that the
could cancel the right hand side of (A.6). However, (A.7) spoils the invariance of the
partition function under
transformations. One can easily verify that indeed the right
hand side of (A.6) is physical as long as we insist on supersymmetry and invariance under
To summarize let us make some comments
in (A.6) is nonzero, then it turns out that we may not be able to respect both (A.3)
and (A.4). In other words, we cannot embed the coupling constants of chiral and
twisted chiral operators into short multiplets. At least some of the couplings have to
be in longer multiplets.
2. We can view the and
transformations as analogous to U(1)A and U(1)V
transformations in 2d electrodynamics. If we preserve one we must give up on the
other, but choosing which one to preserve is at our discretion. Therefore, the situation
is very similar to the way the usual chiral 't Hooft anomalies arise [39].
Note that from equation (A.6) we can immediately write the anomaly in operatorial
formalism. This is because
couples to D+O and so we nd
But since our partition function is invariant under
transformations and hence depends
Hence in a \fermionic background" for the twisted chiral coupling, the operator O ceases
to be chiral.
It is now straightforward to make contact with the analysis in the bulk of the paper.
Our discussion in this appendix has shown that there may be an anomaly with coe cient
and that it would manifest itself as (A.9). Comparing with (3.12) we thus see that this
coe cient is nonzero whenever the OPE coe cient in (3.9) is nonzero. Furthermore, the
analysis in this appendix sheds light on the choices we could make in (3.11). Indeed, we
could have chosen whether to postulate that the partition function preserves (A.3), (A.4),
or none of the two. As in all cases with anomalies, these various choices are related to
each other by adding counterterms to the action, e.g. the one we discussed in (A.7). By
choosing the coe cient of this counterterm carefully, we can change the scheme from the
transformations are obeyed to the one where
transformations are obeyed.
Special geometry relation
formations generated by chiral primary elds is determined in terms of the Zamolodchikov
geometry relation. Below we will give a simple derivation of this relation using (4.1) (our
derivation is valid for any central charge c).
Fi(1)Fj (y)Fk(1)F`(0)
= @y++ @y
)hOi(1)Oj (y)Ok(1)O`(w)ic
The curvature is then computed by integration by parts, with nonvanishing
contribuin the limit limy!0 @w++ @w
)hOi(1)Oj (y)Ok(1)O`(w)ic
We can act with the derivatives on the prefactor jy
wj2 and thus reduce the problem to
studying the y ! 0 limit of hOi(1)Oj (y)Ok(1)O`(0)ic. There is a contribution from the
unit operator in the t and u channel as well as a contribution from the (2; 2) chiral primaries,
Rk`ij =
yli!m0hOi(1)Oj (y)Ok(1)O`(0)ic = gij gk` + gi`gkj
where gIJ is the metric associated with the Rcharge (2; 2) chiral primaries.
Fourpoint function involving supercurrents
Let us focus on the connected fourpoint function
hS+A++(x) FiCC_ (y) FjDD_ (z)S+B++(w)ic :
Since the fourpoint function is holomorphic in z and w, it is determined completely by
the poles in the OPE between the supercurrents and other insertions. We rst consider
the singularities in x, and denote the polar terms in (x
z) by I1; I2; I3
From the OPE of the N = 4 supercurrents,
S+A++(x)S+B++(w) = AB
ful analysis of the integrand shows that such contributions are absent for this particular fourpoint function.
Using again (C.5) and also
where we have dropped the disconnected pieces and also the terms involving the SU(2)R
currents since Fi are SU(2)R singlets. It is then easy to obtain using the OPE between T
I1 =
I1 =
Finally from the OPE between S+A++ and FjDD_ we get
and following the same steps as above we obtain
I3 =
Putting together (C.4), (C.8) and (C.10) while taking the limit w ! 1, we arrive at
S+A++(z)F BB_ (w) = 2 @w++ @
I2 = 2 h@y++
S+A++(z)O _ (w) =
Similarly looking at the OPE between S+A++ and FiCC_
I2 =
I3 =
= 2
hS+A++(0) FiCC_ (y) FjDD_ (1)S+B++(1)ic
AC BD
Fi11_ (0) Fk11_ (y) F`22_ (1) Fj22_ (1) +
Fi21_ (0) Fk11_ (y) F`12_ (1) Fj22_ (1)
Fi21_ (0) Fk11_ (y) F`22_(1) Fj12_ (1) = 0;
function of the exactly marginal operators FiAA_ FjBB_ FkCC_ F`DD_ respects SU(2)out Ward
2@x++ Oi1(x)Q2+2Ok1(y)O`2(z)Q1+1Oj2(1) ;
Q2+1Oi1(x)Q2+2Ok1(y)O`2(z)Oj1(1) ;
SU(2)out selection rules
In this appendix we prove that
identities.28
Q2+2Oi1(x)Q2+2Ok1(y)Q1+1O`2(z)Q1+1Oj2(1)
Q2+1Oi1(x)Q2+2Ok1(y)Q1+2O`2(z)Q1+1Oj2(1)
Q2+1Oi1(x)Q2+2Ok1(y)Q1+1O`2(z)Q1+2Oj2(1)
Moreover, from (C.5) we can derive
Q2+1Oi1(x)Ok1(y)O`2(z)Q1+2Oj2(1) =
Q2+2Oi1(x)Ok1(y)O`2(z)Q1+1Oj2(1) :
Putting together (D.2), (D.3), (D.4) and (D.5), we obtain
Q2+2Oi1(x)Q2+2Ok1(y)Q1+1O`2(z)Q1+1Oj2(1) + Q2+1Oi1(x)Q2+2Ok1(y)Q1+2O`2(z)Q1+1Oj2(1)
+ Q2+1Oi1(x)Q2+2Ok1(y)Q1+1O`2(z)Q1+2Oj2(1) = 0;
which leads to (D.1).
Open Access.
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Here we see that certain correlation functions respect the SU(2)out invariance. This is analogous to the bonus
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