Sphere partition functions and the Zamolodchikov metric
Efrat Gerchkovitz
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1
3
Jaume Gomis
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1
2
Zohar Komargodski
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1
3
Open Access
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1
c The Authors.
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1
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Waterloo
,
Ontario, N2L 2Y5
,
Canada
1
Rehovot 76100
,
Israel
2
Perimeter Institute for Theoretical Physics
3
Weizmann Institute of Science
We study the finite part of the sphere partition function of ddimensional Conformal Field Theories (CFTs) as a function of exactly marginal couplings. In odd dimensions, this quantity is physical and independent of the exactly marginal couplings. In even dimensions, this object is generally regularization scheme dependent and thus unphysical. However, in the presence of additional symmetries, the partition function of evendimensional CFTs can become physical. For twodimensional N = (2, 2) supersymmetric CFTs, the continuum partition function exists and computes the Kahler potential on the chiral and twisted chiral superconformal manifolds. We provide a new elementary proof of this result using Ward identities on the sphere. The Kahler transformation ambiguity is identified with a local term in the corresponding N = (2, 2) supergravity theory. We derive an analogous, new, result in the case of fourdimensional N = 2 supersymmetric CFTs: the S4 partition function computes the Kahler potential on the superconformal manifold. Finally, we show that N = 1 supersymmetry in four dimensions and N = (1, 1) supersymmetry in two dimensions are not sufficient to make the corresponding sphere partition functions welldefined functions of the exactly marginal parameters.

Conformal manifolds and sphere partition functions
Sphere partition functions of superconformal field theories
N = (2, 2) theories and supersymmetric twospheres
The S2 partition function
An explicit regularization
N = 1 field theories on S4
N = 2 field theories on S4
The Kahler potential from the foursphere
neighborhood of the reference conformal field theory p, there is a family of CFTs. We will
refer to this family of CFTs as the conformal manifold, and denote it by S
admits a natural Riemannian metric, the Zamolodchikov metric gij (p), given by [1]
Conformal manifolds and sphere partition functions
Suppose that in a ddimensional CFT, denoted by p, there are exactly marginal operators
1 Z
hOi(x)Oj (0)ipS =
1We choose a normalization that will make later formulae simpler.
S and the operators Oi represent vector fields in S
. Thus, under a change of variables
Consider an observable O. Suppose we are interested in computing this observable as
1 Z
It is very important that (1.4) is ambiguous. Indeed, even if we solve the reference
we still need the integrated correlation functions in order to compute (1.4). Integrated
correlation functions generically have ultraviolet divergences. They need to be regularized
by introducing a cutoff and appropriate counterterms need to be added to the effective
action so that the correlation functions are finite in the continuum limit. Therefore,
computations of various observables via (1.4) may depend on the regularization scheme, and
consequently be subject to ambiguities.
Let us now analyze these ambiguities in some detail. The formalism where the
couallows to easily classify all such ambiguities. Different regularization schemes simply differ
by local terms of dimension equal or smaller than d in the effective action. In order to
preserve Ward identities, one may need to finetune some of the counterterms.
Let us briefly describe a few general pedagogical examples. The finite counterterm
with Cij any symmetric tensor. This counterterm accounts for a power divergent piece
Our primary interest in this paper lies in CFTs on curved spaces. A conformal field
theory can be placed canonically on any conformallyflat manifold, in particular, on Sd (in
on the conformal manifold S [2]. In even dimensions, the local counterterm
follows from conformal Ward identities. Indeed, to leading order in the expansion (1.4)
we get R ddxhL(0)Oi(y)Oj (x)i. This correlator is infrared finite but has an ultraviolet
tuning the local term (1.6). Finally, another example of a counterterm is
this case we can use the stereographic map). Since there are no infrared divergences, we
can study the partition function ZSd . We can repeat this procedure at any point on the
conformal manifold. This provides an interesting probe of the conformal manifold, ZSd (p).
While infrared divergences are absent, ultraviolet divergences remain. As in our
disd = 2n :
log ZS2n = A1(i)(rUV )2n + A2(i)(rUV )2n2 + . . . + An(i)(rUV )2
d = 2n + 1 :
log ZS2n+1 = B1(i)(rUV )2n+1 + B2(i)(rUV )2n1
where r is the radius of Sd. The powerlaw divergent terms correspond to counterterms of
in even dimensions and
in odd dimensions. Therefore, all the powerlaw divergent terms in (1.8) and (1.9) can be
tuned to zero in the continuum limit.
In even dimensions, the sphere partition function has a logarithmic dependence on the
radius (see (1.8)), which cannot be canceled by a local counterterm. It is associated to the
Weyl anomaly. The variation of the partition function under a Weyl transformation with
in the Weyl anomaly vanish on the sphere, see [3]). However, this violates the WessZumino
with the usual Atype anomaly [3], which is therefore independent of exactly marginal
deformations.2
This is necessary for its interpretation as a monotonic function under
as it can be removed by the local counterterm3
In summary, we have shown that the only physical data in the continuum limit of ZS2n
is the Aanomaly, which is independent of the exactly marginal parameters.
2We thank L. Di Pietro and A. Schwimmer for discussions.
has a natural normalization via radial quantization.
For odd dimensions, absent additional restrictions on the counterterms, we have seen
above that all the Bi are ambiguous and can be tuned to zero (a logarithmic term is absent
because one cannot write an appropriate local anomaly polynomial in odd dimensions.)
only conceivable dimensionless counterterm would be a gravitational ChernSimons term
Moreover, it has to have an imaginary coefficient due to CPT symmetry. Hence, the
choice of regularization scheme that preserves coordinate invariance.4 It measures the
finite entanglement entropy across a S2n1 in IR2n,1 [11].5
. Start at an
arbitrary point on the conformal manifold and expand to second order
ddx ghOii +
ddx g
ddy ghOi(x)Oj (y)i . (1.14)
Conformal Ward identities on the sphere imply that
hOii = 0 .
Since this is true at every point on the conformal manifold, we conclude that the continuum
conformal field theory sphere partition function is independent of exactly marginal
parameters. As we will see, this argument cannot be repeated in even dimensions because there
are finite counterterms. Indeed, we will see examples where the sphere partition function
does depend on exactly marginal parameters.
We can understand this simplicity of the sphere partition function in odd dimensions
from another point of view. The integrated twopoint function of exactly marginal
operators in the last term of (1.14) is ultraviolet divergent. The singularity arises from the
domain where x y. The twopoint function on the sphere can be obtained from the
corresponding twopoint function in flat space by a stereographic map. Hence, the integrated
twopoint function in (1.14) is proportional to
ddyg
4The imaginary part is more subtle. Only its fractional part is well defined. See, for example, the
discussions in [810]. We will not comment any further on the imaginary part of F2n+1.
5The entanglement entropy provides another way to see that the finite part in even dimensions is
ambiguous while in odd dimensions it is physical. Indeed, it is straightforward to write finite local counterterms on
the entangling surface of evendimensional spheres, while in odd dimensions this is impossible. For example,
absolute value renders it nonlocal, while without the absolute value symbol it is not consistent with the
vacuum being a pure state. We thank S. Pufu for discussing this with us.
where d(x, y) is the SO(d + 1) invariant distance on the sphere Sd. This can be regulated
in a region where it converges, analytically continuing to the region of interest at the end,
0
for d = 2n + 1 .
Therefore the integral vanishes in odd dimensions, and since we have shown that there
are no finite counterterms in odd dimensions, the answer computed in any other choice of
regularization scheme would yield the same result. The fact that the second term in the
expansion (1.14) vanishes holds true around any point on the conformal manifold. This is
consistent with the independence of the partition function of exactly marginal couplings.
Note that the same regularization scheme yields a nonzero answer in even dimensions. This
will be important below.
In summary, we have shown that the (real part of the) finite part of ZS2n+1 in the
continuum limit is physical and unambiguous, and is independent of the exactly marginal
parameters of the conformal field theory. This is a necessary requirement for the finite part
of ZS2n+1 to serve as a candidate monotonic function under renormalization group flows
Sphere partition functions of superconformal field theories
Our analysis of the ambiguities of the sphere partition function of a CFT followed from
assuming that the partition function can be regulated in a diffeomorphism invariant way. We
then classified all the diffeomorphism invariant counterterms and determined their influence
on the sphere partition function. Starting in this section, we pursue an analogous analysis
for superconformal field theories (SCFTs). Imposing that the sphere partition function of
SCFTs can be regulated in a supersymmetric way, we show that the partition function of
SCFTs with various amounts of supersymmetry and in various dimensions have (not
unexpectedly) a more restricted set of ambiguities. This makes the sphere partition function of
such SCFTs interesting, rich observables: the sphere partition functions of some of these
theories are known to compute the Kahler potential on the conformal manifold [14, 15].
We now outline the logic of our arguments. Consider a ddimensional SCFT and place
it on the sphere by stereographic projection. Such a SCFT is by definition invariant
under the corresponding superconformal algebra. As we explained in section 1, however, the
partition function of a CFT suffers from ultraviolet divergences that arise when exploring
the dependence of the partition function on the conformal manifold S
. We now assume
that we can regulate the SCFT partition function while preserving the subalgebra of the
superconformal algebra that closes into the superisometries of the ddimensional
supersymmetric sphere, which projects out all conformal generators of Sd. This is the general
supersymmetry algebra of a massive supersymmetric theory on the ddimensional sphere.
In the supersymmetric context, the local counterterms that parametrize the
ambiguities of the partition function are now supergravity counterterms. In supersymmetric
theories, the coordinates that parametrize the conformal manifold are the bottom
component of an apropriate supersymmetry multiplet. Therefore we must consider the most
sponding supergravity theory. The supergravity theory that must be considered is found
by embedding the rigid supersymmetric ddimensional sphere as a supersymmetric
background of ddimensional offshell supergravity, in the framework put forward by [16]. By
constructing marginal supergravity counterterms in a given supergravity theory, we can
determine the ambiguities and the leftover physical content of the partition function of
SCFTs on Sd.
Let us briefly summarize our main results explained in the rest of the paper. In
manifold of such theories is a Kahler manifold, and locally takes a direct product form
Sc Stc, where Sc, and Stc are the chiral and twisted chiral manifolds, respectively.
These theories can be placed on supersymmetric S2 in two different ways while preserving
four supercharges [15, 1719]. For each choice there is a corresponding twodimensional
of [17, 19], where the corresponding supersymmetry algebras were labeled by SU(21)A
and SU(21)B. Imposing that the SCFT partition function preserves SU(21)B, we find
that there is a finite Btype supergravity counterterm [20]
sure. A similar Atype supergravity counterterm can be constructed for a SCFT preserving
S2 background they give rise to the following Kahler ambiguities in the partition function
of counterterms, then the partition functions may differ by (2.2). This means that the
twosphere partition functions ZA and ZB are generally not functions, but rather sections:
the transitions between different patches may involve holomorphic functions. Our explicit
construction of these counterterms gives a microscopic realization of the Kahler ambiguities
implied by the proof [15] that the exact twosphere partition function [1719] computes
the Kahler potential K on the conformal manifold Sc Stc [14, 15]
ZSA2 = eKtc
ZSB2 = eKc .
For some followup work see e.g. [21, 22].
One wellknown way in which the partition function can turn into a section is if there
are t Hooft anomalies. Then, the partition function transforms nontrivially under gauge
transformations. Ordinary derivatives of the partition function then yield gauge
noninvariant quantities (which can be fixed by the BardeenZumino procedure [23], or
equivalently, anomaly inflow [24]). Here we encounter a reminiscent situation. The partition
function becomes a nontrivial section. As a result, various derivatives of the partition
function need to be supplemented with an appropriate connection that for holomorphic
nice to understand if the analogy with ordinary anomalies is deeper than these superficial
In section 3 we also give a new elementary proof of (2.3) that follows from
supersymmetry Ward identities and therefore does not require the existence of a Lagrangian
description or localization of the partition function. We also show that the twosphere
parthat changes the finite part of the partition function arbitrarily, similar to what we found
for evendimensional nonsupersymmetric CFTs.
give rise to a Kahler manifold [25]. We argue that in this case the partition function on the
foursphere does not have a preferred, unambiguous nontrivial continuum limit. We show
Evaluated on S4, this shifts the finite part of the partition function by an arbitrary function
ways (one of them being supersymmetric localization and the other by using an explicit
supersymmetric regularization of the second term in (1.14)) that
of the moduli.
ZS4 = eK/12 ,
where K is the Kahler potential on the space of exactly marginal deformations S. That
is, the foursphere partition function becomes physical if the counterterms are restricted
partition function is again a section rather than a function, where the transition functions
are holomorphic plus antiholomorphic in the moduli. Note the sign difference of the
exponent (2.5) with respect to (2.3).
One can therefore, in principle, use the localization computation on S4 [26] to
comcan also derive (2.5) by generalizing the argument of [15] (i.e. geometrically deforming the
equations should be along the lines of [28]). One may also try to extend to four
dimensions our supersymmetry Ward identity proof in section 3. It would also be very nice to
counterterm (2.1), perhaps using the tools of [29, 30]. Finally, it would be interesting to
geometric structure beyond Kahler (additional geometric structure is known to exist in
In this paper we have confined ourselves to discussing conformal field theories and
their sphere partition functions. However, some of our results can be relevant for the
interpretation of partition functions of gapped theories as well. For example, if we study
a massive theory with a gap at the scale M on a manifold with typical scale R much
larger than the the scale associated to the gap R
1/M , then the partition function
corresponding to local terms in the action for the background fields. Therefore, supergravity
counterterms of the type discussed in this paper can be useful to understand the leading
terms in the expansion of partition functions of massive supersymmetric theories.
Rsymmetry group. Exactly marginal operators are superconformal descendants of
operators in the chiral and twisted chiral rings which carry charge (2, 0) and (0, 2) under
U(1)V U(1)A. We can view the coupling constants for these operators as charge (0, 0)
background chiral superfields and charge (0, 0) background twisted chiral superfields,
rethe metric in the chiral and twisted chiral directions.6
and the Zamolodchikov metric takes a factorized form Sc Stc: locally it is the sum of
Such conformal field theories can be placed on S2 by a stereographic transformation.
Killing spinors and , which satisfy
6One can derive this as follows. Regardless of supersymmetry, in any twodimensional CFT, promoting
where we omitted the usual Atype anomaly, as we have already discussed it in the introduction. Ckl
is some symmetric tensor that depends on the exactly marginal parameters. Ckl is proportional to the
Zamolodchikov metric on the space of CFTs. This is because the momentum space twopoint function
of exactly marginal operators looks like p2 log(p2), which clearly contains a rescaling anomaly. Such a
nonlinear sigma model (3.1) is supersymmetrized as usual, by replacing the twoderivative term with
a Kahler function for the background superfields. Hence, the total space of exactly marginal couplings
is Kahler. Since the top component of the product of a chiral field with twisted chiral fields is a total
derivative, this leads to a factorized Zamolodchikov metric.
the Rsymmetry group are as follows
U(1)V U(1)A
1
1
1
1
The equations (3.2) have an eightcomplexdimensional space of solutions on S2. These
furnish four doublets of the SU(2) isometry group of the twosphere.
The superconformal algebra on S2 can be linearly realized on supermultiplets. For our
A superconformal invariant can be constructed from the top component of a chiral multiplet
of Weyl weight 1 (rV = 2)
I =
d2xg F ,
since in this case F = im (m), and the variation integrates to zero.
The twisted chiral multiplet transformations are given by [15]
I =
d2xg G
is a superconformal invariant, since G = im ( Pm) im (P+m).
The superconformal invariants (3.5)(3.7) represent the exactly marginal operators that
are descendants of operators in the chiral and twisted chiral rings. The antichiral and
antitwistedchiral transformations and invariants are constructed similarly.
The superconformal superalgebra just described admits interesting massive
subalgetheories can be placed on S2 while respecting these subalgebras [15, 1719].
A regularization scheme can be thought of as a deformation of the theory by some
massive sector. Hence, in order to understand the possible ways of regularizing the partition
function ZS2 while preserving some supersymmetry, we need to study these massive
subalgebras in detail. We will mainly focus on the subalgebras that preserve four supercharges,
but will also pay some attention to one subalgebra that preserves two supercharges.
sphere. They are generated by supercharges in the superconformal algebra that close
into the SU(2) isometry of the twosphere and a U(1) subgroup of the U(1)V U(1)A
superconformal Rsymmetry. The algebras that preserve U(1)V and U(1)A have been
denoted by SU(21)A and SU(21)B in [17, 19]. They are mapped to each other by the
One can also study massive subalgebras preserving only two supercharges. We do not
carry out an exhaustive analysis of this case, but only discuss one example: the massive
into the SU(2) isometries of S2, thus breaking the Rsymmetry group completely.
This supersymmetry algebra is obtained by restricting the conformal
Killing spinors on the twosphere (3.2) to obey
so that = + + and = + + satisfy
In stereographic coordinates, where ds2 =
dimensional space of solutions is given by
tion of (3.5) and (3.7) to these conformal Killing spinors.
This supersymmetry algebra is obtained by restricting the conformal
Killing spinors on the twosphere (3.2) to obey
m =
m =
=
m =
m =
The SU(21)B supersymmetry transformations are the restriction of (3.5) and (3.7)
to these conformal Killing spinors.
OSp(12): massive N
= (1, 1). The superconformal N
= (1, 1) superalgebra is
=
two supercharges. The chiral and twisted chiral multiplets become reducible, each
breaking up into two N = (1, 1) real scalar multiplets.
The S2 partition function
the formal expansion (1.4) contains divergences, we need to decide which symmetries our
regulator preserves. Below we will analyze the consequences of the assumption that the
superconformal algebra discussed above.
Let us begin by assuming that the physics at coincident points is SU(21)A
invariant. That means that we are allowed to use SU(21)A Ward identities, even at coincident
points. With this assumption, the first observation we make is that the chiral multiplet
invariant (3.5) is SU(21)Aexact. By taking
= 0 in (3.5) we have that
F =
fore, the S2 partition function is independent of the chiral couplings constants.
already shows that the twosphere partition function is not completely ambiguous. The
independence of the chiral couplings implies that there is no local SU(21)A supergravity
counterterm that could reduce on S2 to a function of the bottom components of the chiral
The twisted chiral multiplet invariant (3.7) is more interesting. It is not exact with
respect to the SU(21)A subalgebra of the superconformal algebra. Indeed, in SU(21)A
the SU(21)A Killing spinor equation (3.8). The SU(21)Ainvariant partition function may
thus depend on the twisted chiral moduli.
Even though the top component of the twisted chiral multiplet G is not exact, its
integral is almost so. From (3.7) we see that
If + had been nowhere vanishing, then the integrated top component would have been
+ =
=
d2xg G = 4rY () = 4rY (S) .
We note that the insertion of the bottom component of a twisted chiral multiplet Y is
invariant under this supersymmetry transformation if the operator is inserted at the North
Pole of the twosphere (and if also vanishes at the North Pole).
We can focus on the physics near the point where the spinor vanishes as follows.
and z
= 0), after some simple algebra we arrive at
This expression is only valid away from the North Pole.
We rewrite the supersymmetric invariant as
+ 2iz
d2xg G =
d2xg G +
d2xg G ,
where DN = {xx2
R2} and DS = {xx2
R2}. Using the expression (3.17), which is
regular in DS, we get
d2xg G =
d2xg z
ishes and the entire contribution from DS comes from the second, total derivative term.
Ultimately we will take the limit that R 0, so that the contribution from DN is
vanishingly small. We thus need to evaluate the total derivative term in (3.19). Using polar
coordinates, Stokes theorem, and the identities
+2 =
4r2 1 + 4zrz2 0
d2xg G = 2r limR0
1 + 4Rr22 = 4rY (0) = 4rY (N ) .
An analogous analysis for the top component of the antitwisted chiral multiplet yields
Therefore, the integrated twisted chiral (antitwisted chiral) multiplet top component inside
a correlation function inserts the bottom component of the twisted chiral (antitwisted
chiral) multiplet at the North (South) Pole of the twosphere. We will now use these
formulae to prove that ZSA2 = exp(Ktc).
Differentiating the partition function twice with respect to twisted chiral moduli we get
ij log ZSA2 =
d2xg Gi(x)
d2yg Gj (y)i .
Using (3.21), (3.22) we arrive at
In completing the proof we have used a supersymmetry Ward identity relating the twopoint
ambiguity which will be discussed in detail in the next subsection,
ZSA2 = eKtc .
ZSB2 = eKc .
If we assume that the physics at coincident points is SU(21)B invariant, the analysis
is similar to what we have done above, essentially only exchanging the role of chiral and
twisted chiral multiplets. The twisted chiral invariant (3.7) is SU(21)Bexact. The chiral
invariant (3.5) is not SU(21)Bexact, again, because one cannot construct it as a variation
of a fermion without encountering some zero of a Killing spinor. Repeating the steps above
ij ln ZSB2 = (4r)2hi(N )j (S)i = (2r)4hFi(N )F j (S)i = ij Kc .
Thus, up to a holomorphic ambiguity which we will discuss below,
OSp(12) described in the previous subsection. The exactly marginal operators are given by
superconformal invariant is (3.5), but with F now being real. Let us now consider the sphere
partition function regulated in an OSp(12) invariant way. One can attempt to repeat the
analysis performed for SU(21)A and SU(21)B. However, we find that the integrated
twopoint function of the top component of a real scalar multiplet does not reduce to the
analysis. While it is still true that the top component is not supersymmetry exact (it is of
derivation of (3.24) we used in a crucial way that the insertion of the bottom component of
involving an arbitrary function of the moduli, which also sit in bottom components of
SCFTs on S2 is scheme dependent and thus ambiguous.
We will reach a very similar
Before we turn to the holomorphic ambiguity afflicting the partition functions ZSA2,B, let
us make some comments. We discussed what happens if the theory is regulated in a manner
that preserves SU(21)A,B. But what if we could find a regulator that preserves a bigger
symmetry group, i.e. the full superconformal group which has eight supercharges on S2?
Clearly, such a regulator cannot exist. The reason is simply that if it had existed, using
two different subgroups of the superconformal symmetry group, we would have arrived
at two different results (3.25), (3.27). In fact, if all the eight supercharges are available
to us, then one can formally prove that the chiral invariant (3.5) and the twisted chiral
invariant (3.7) are both superconformal exact. Hence, one would be led to the conclusion
that the partition function is independent of the exactly marginal parameters. The fact that
group cannot be preserved. This is reminiscent of an anomaly.
The Kahler potential admits the following wellknown ambiguity
supergravity counterterm that when evaluated on the twosphere captures the Kahler
ambiguity (3.28). For example, if we describe the same conformal field theory in two
different duality frames, the partition functions may differ by some holomorphic (plus
antiholomorphic) function of the moduli. This means that the partition function is a section
on the conformal manifold.
We proceed to construct this local supergravity counterterm explicitly.
of the U(1) Rsymmetry that is gauged (vector or axial), and are mapped into each other
under the Z2 mirror automorphism. The rigid limits of these two supergravities correspond
to preserving SU(21)A or SU(21)B on the twosphere.
We begin with the SU(21)B theory. The ambiguity (3.28) depends on the couplings
to the exactly marginal operators from the chiral ring. They sit in background chiral
multiplets of vanishing U(1)A Rcharge. The supergravity counterterm supersymmetrizes
the product of the Ricci scalar curvature R with the sum of an arbitrary holomorphic
function and an antiholomorphic function of chiral multiplets
d2xg R
The relevant local counter term is obtained from the SU(21)B supergravity action7
density superspace measure and R is a chiral superfield whose bottom component is a
complex scalar auxiliary field that belongs to the SU(21)B supergravity multiplet. The
spacetime scalar curvature R sits in the top component of the superfield R. When (3.30)
is evaluated in the SU(21)B supersymmetric twosphere background and the superfields
establishes the Kahler ambiguity for the SU(21)B theory.
Similarly, for the SU(21)A theory one has to consider
twisted chiral density superspace measure. F is a twisted chiral superfield that contains
as its lowest component a complex scalar auxiliary field that belongs to the
SU(21)Asupergravity multiplet and the scalar curvature R in its top component. The rigid limit
of this coupling evaluated on supersymmetric backgrounds was recently considered in [37].
This supergravity counterterm evaluated in the SU(21)A supersymmetric twosphere
back
Hence, the finite piece of the twosphere partition function has a holomorphic plus
an antiholomorphic ambiguity. The pieces that cannot be shifted away by a holomorphic
and an antiholomorphic function of the exactly marginal parameters are physical and
calculable, as long as our regularization scheme respects SU(21)A or SU(21)B, as explained
in the previous subsection.
contains the graviton, the gravitino and a real auxiliary field B. The superspace description
of this N = (1, 1) supergravity counterterm is
7We thank N. Seiberg for a discussion.,KetovES
component [38]. This supergravity counterterm evaluated in the OSp(12)
supersymmetric twosphere background yields the marginal counterterm (3.32). Therefore, the sphere
function of nonsupersymmetric CFTs, for which the S2 partition function does not have a
preferred nontrivial continuum limit. As we shall see in section 4, we will reach the same
An explicit regularization
Let us for a moment forget about supersymmetry and examine a little more closely the
proportional to the double integral
ddxg Oi(x)
ddyg Oj (y)i .
hOi(x)Oj (y)iSd =
d(x, y) =
x y
The twopoint function on Sd can be obtained from the one in flat space by a Weyl
transsufficiently large and positive the integral converges and evaluates to
hRSd ddxg Oi(x) RSd ddyg Oj (y)i = gij vol(Sd)vol(Sd1) (d/2)(d/2+ ) .
2d+1 ( )
Setting d = 2, in the
0 limit we get that
d2xg Oi(x)
In the case without any supersymmetry, this procedure of regularizing (3.34) by continuing
the dimension of the operator is just one of many possible regularization schemes and since
the answer is not renormalization scheme invariant, there is no particularly special meaning
see that (3.38) gives an incorrect answer in some situations! Indeed, we have shown above
that, if we preserve SU(21)A, the partition function is independent of the chiral couplings.
However (3.38) does not distinguish the chiral from the twisted chiral couplings.
The resolution is, of course, that the procedure of analytically continuing the dimension
of chiral operators is inconsistent with SU(21)A. This is because if one changes the
dimension of a chiral operator in the superpotential, one necessarily breaks U(1)V and therefore
SU(21)A. However, this procedure is perfectly SU(21)A invariant for the twisted chiral
operators, and since we have already explained that the answer is unique, (3.38) must give
the right result in this case. Indeed, (3.38) is completely consistent with the formula derived
in (3.24), which implies (3.25). The mirror symmetric statement is that this regularization
scheme preserves SU(21)B for chiral operators while it breaks it for twisted chiral
operators, and therefore leads to (3.27). We will use this derivation via analytic continuation
in the conformal dimension of the exactly marginal operators again in section 5, when we
discuss fourdimensional N = 2 SCFTs.
= 1 field theories on S4
superconformal field theories on their exactlymarginal couplings. The exactlymarginal
conformal theories [39]. These theories have a U(1)R Rsymmetry. The exactly marginal
ral ring; they are the top component of a chiral multiplet with Rcharge 2 (and Weyl
weight 3). We note that not all such operators are exactly marginal in fourdimensional
superconformal field theories with a normalizable vacuum state, where superconformal
descendant operators of the type described in the previous section are necessarily exactly
sphere. The superconformal transformations are generated by a Dirac conformal Killing
spinor on S4, which obeys
Under superconformal transformations the components of a chiral multiplet of Weyl weight
w (and Rcharge 23 w) transform as [41]
with the projectors PL = 12 (1 + ) and PR = 12 (1 ), so that PL,R = L,R.
T C, and C is the charge conjugation matrix. We have also defined chiral spinors
of Weyl weight w = 3
I =
d4xgF .
8The origin of this difference is as follows: as explained in [40], a marginal operator can cease to be
to eat our marginal operator. Therefore, it has to be exactly marginal.
the following conformal Killing spinors on S4
This represents a general marginal operator in the superconformal field theory. From now
on, we will only discuss the exactly marginal ones. The parameters multiplying these
operators, the coordinates in the conformal manifold, are realized as the lowest component
of background chiral multiplets with vanishing Rcharge.
which corresponds to choosing
m =
The supersymmetry transformation generated by these spinors anticommute to give the
SO(5) isometry of the foursphere, hence, projecting out the conformal and Rsymmetry
theories on S4 preserving (4.4). The action of this subalgebra on the chiral multiplet (4.3)
is of course induced from the action of the full superconformal group, restricted to the
One can attempt to repeat our analysis in two dimensions and aim to write the
invariant (4.3) as an OSp(14) supersymmetry variation of a fermion modulo a total derivative
and a zero. However, one quickly encounters a geometrical obstruction. One finds an
addicommutator cannot be put to zero on S4 (loosely speaking, the OSp(14) algebra contains
only complexified Killing vectors not commuting with their complex conjugate).
Therefore, the invariant (4.3) is not OSp(14)exact (even if one ignores zeroes of spinors
and total derivatives). In fact, we now prove that the dependence of the sphere partition
therefore analogous to the situation in nonsupersymmetric theories.
real function of the chiral and antichiral multiplets. This counterterm once evaluated on
the OSp(14) invariant foursphere is nonvanishing and explicitly demonstrates that the
dependence on the moduli of the partition function is ambiguous. Denoting chiral fields
2 Z
3
The supergravity multiplet contains, in addition to the graviton and the gravitino, two
measure and is the superfield that contains the square root of the determinant of the
metric in its bottom component. The chiral superfield R has the auxiliary field M as its
lowest component. After setting the gravitino to zero, we get:
R = 6
M 6
2 R +
Manifolds that preserve four supercharges obey the integrability conditions [16]:
2 R bb 2M M = 0 ,
metric sphere (in addition, we have set the gravitino to zero).
a little algebra, we then obtain that (4.6) evaluates to
1 Z
on the supersymmetric sphere. Thus, (4.6) supersymmetrizes the finite counterterm (4.10)
for an arbitrary function of the exactly marginal parameters. This renders the sphere
partition function of fourdimensional N = 1 SCFTs ambiguous.
= 2 field theories on S4
superconformal field theories. These theories have an SU(2)R U(1)R Rsymmetry. An
exactly marginal operator in such a theory is realized as a superconformal descendant of
the bottom component of a chiral multiplet of U(1)R Rcharge 2 (and Weyl weight 2).
superconformal transformations are parametrized by an SU(2)R Rsymmetry doublet of
left chiral conformal Killing spinors i and right chiral conformal Killing spinors i
m
weight w are given by [43]
= /(A i) +
2 Bij
2 Bij
i = 41 ab/ Fab i 2 / Bij jk
1
is invariant, where C is the top component of the multiplet (5.2).
of Weyl weight 0. Note that Weyl weight 0 chiral superfields are irreducible [44], thus, in
We are interested in unraveling the physical content of the sphere partition function of
sphere S4. We describe it below in detail. We will prove that the S4 partition function of
the Kahler potential on the conformal manifold
The OSp(24) supersymmetry transformations are generated by conformal Killing
spinors j and j obeying
where 1jk = i3jk = (1jk), and 3 =
spinors (5.1) with
0 1
ZS4 = eK/12 .
m
j =
m j =
I =
d4xg C
C = ij i/ j + 2wij ij = m(ij imj ) + (2w 4)ij ij ,
such a multiplet of Weyl weight 2. Indeed, it follows from (5.2) that
(1 + w) Bij jk k + (1 w) ab Fabi
(1 + w) Bij jk k + (3 w) ab Fabi ,
9 These correspond to conformal Killing
9We introduce this notation in order to follow the notations of [41].
We can diagonalize equations (5.5) by defining
and bj = j , where PL/R = 12 (1 ).
on the fields of a chiral multiplet gives a representation of the N
= 2 superconformal
algebra on the fields
With the choice of spinors (5.5), it is easy to prove that the vector field produced by two
superconformal transformations
i
2 2im 1
is a Killing vector on S4. Moreover, with the choice of spinors (5.5) the parameters
associated to local dilatations and U(1)R Rsymmetry vanish, while the SU(2)R Rsymmetry is
broken down to SO(2)R
i12i + 1i2i i21i 2i1i = 0 ,
i12i 1i2i i21i + 2i1i = 0 ,
a = i( i12j + 1j 2i + i21j 2j 1i)aij = 1 =
symmetry algebra, whose supercharges close into an SO(5) isometry and an SO(2)R
R
Using the OSp(24) supersymmetry transformations of a chiral multiplet (5.2) we
expect that the top component C of the multiplet with Weyl weight 2 can be written as
three consecutive supersymmetry transformations of a linear combination of fermions in
the multiplet modulo a zero and a total derivative. One could then repeat the argument of
section 3 and arrive at (5.4). Instead, here we follow a closely related strategy, extending to
four dimensions the localization proof of (5.4) presented in [15]. In addition, we derive (5.4)
by an explicit supersymmetric regularization. The two derivations agree.
The Kahler potential from the foursphere
First, we employ the localization computation [26] of the S4 partition function of
Laon vector multiplets with gauge group G and hypermultiplets transforming in a
representation R of G. The partition function can be computed by localizing the functional integral
with respect to a supercharge in OSp(24). For our analysis, the details of the
hypermultiplets, which vanish on the localization saddle points [26], are not important. Therefore,
we can focus on the N = 2 vector multiplets.
ponent of an N = 2 chiral and antichiral multiplet of Weyl weight 2
Ai =
S =
d4xg X
where Ci and Ci are the top components of the composite chiral and antichiral multiplets.
The exactly marginal parameters are the complexified gauge couplings
2 .
1 Z
Calculating the second derivative with respect to the marginal couplings we get
ij log ZS4 =
d4xg Ci(x)
d4yg Cj (y)i = (32r2)2hAi(N )Aj (S)i.
In the final step, we have used supersymmetric localization.
To relate this to the Zamolodchikov metric we can use the supersymmetry
transformations (5.2) to relate the twopoint function of the bottom components (5.15) to the
1
ric. The result is hCi(N )Cj (S)i = (2r)8 gij . Combining all the factors we find
Finally, we need to relate the twopoint function hCi(N )Cj (S)i to the Zamolodchikov
metThis shows (2.5)
ij log ZS4 =
12 gij =
12 ij K .
ZS4 = eK/12 .
We can now compare this result to another derivation. In section 3, equation (3.37), we
have evaluated the integrated twopoint function using regularization by analytic
continuation in the scaling dimension. As we have explained in section 3, this regularization does
not always work. One needs to make sure that it preserves the massive supersymmetry
11The argument below can be carried without referring to a specific microscopic realization. This requires
Plugging d = 4 in (3.37) and taking the limit
0, we find precisely (5.16) (after correctly
normalizing the operators, as in (1.1)). This therefore provides a derivation of (5.16) that
does not depend on localization.
i.e. the gauge coupling. The S4 partition function depends on the masses of the fields in
the adjoint hypermultiplet, and those need to be tuned such that the foursphere partition
function is that of the conformally coupled theory. For the correct, conformal choice of
the mass parameter [45], the instanton contributions to the foursphere partition function
manifestation of a fact that we have explained in detail in section 3; the partition function
is a section rather than a function. This Kahler transformation should be understood as a
local supergravity counterterm, similar to the ones we have found in section 3. We leave
We are very grateful to C. Closset, L. Di Pietro, N. Doroud, T. Dumitrescu, B. Le Floch,
D. Gaiotto, S. Lee, J. Maldacena, V. Narovlansky, J. Polchinski, A. Schwimmer, A. Van
Proeyen for useful discussions. We especially thank N. Seiberg for helpful discussions at
various stages of the project. JG and ZK are grateful to the KITP for its warm hospitality
during the initial stages of this project, which was supported in part by the National
Science Foundation under Grant No. NSF PHY1125915. EG and ZK thank the Perimeter
Institute for its very kind hospitality during the course of this project. Research at the
Perimeter Institute is supported in part by the Government of Canada through NSERC
and by the Province of Ontario through MRI. J.G. also acknowledges further support from
an NSERC Discovery Grant and from an ERA grant by the Province of Ontario. ZK
is supported by the ERC STG grant number 335182, by the Israel Science Foundation
under grant number 884/11, by the United StatesIsrael Binational Science Foundation
(BSF) under grant number 2010/629, and by the ICORE Program of the Planning and
Budgeting Committee and by the Israel Science Foundation under grant number 1937/12.
Any opinions, findings, and conclusions or recommendations expressed in this material are
those of the authors and do not necessarily reflect the views of the funding agencies.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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