Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories

We present some new exact results for general four-dimensional superconformal field theories. We derive differential equations governing the coupling constant dependence of chiral primary correlators. For N=2 theories we show that the Zamolodchikov metric on the moduli space and the operator mixing of chiral primaries are quasi-topological quantities and constrained by holomorphy. The equations that we find are the four-dimensional analogue of the tt* equations in two-dimensions, discovered by the method of"topological anti-topological fusion"by Cecotti and Vafa. Our analysis relies on conformal perturbation theory and the superconformal Ward identities and does not use a topological twist.


Introduction
Supersymmetric field theories in four dimensions are interesting for their possible phenomenological applications and as toy models for the analysis of non-perturbative phenomena in quantum field theory. Supersymmetry allows us to go beyond perturbation theory and to derive exact results, which proved important for the analysis of strongly coupled gauge theories and the discovery of dualities.
In this paper we will derive some new exact results for general four-dimensional superconformal field theories. We will mainly focus on theories with N = 2 supersymmetry, but some of our results are also true for N = 1 theories. A basic example of N = 2 SCFTs in four dimensions is the N = 2 SU (N ) gauge theory with N f = 2N massless quarks in the fundamental representation [1], [2]. Other N = 2 theories in four dimensions include [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. More recently a larger class of related theories has been studied by Gaiotto [13]. They arise by compactifying the six-dimensional superconformal field theory living on N M5 branes on a punctured Riemann surface. Our discussion will be general and not based on any specific theory.
Many superconformal theories in four dimensions are not isolated, but come in continuous families parametrized by coupling constants, which can be freely adjusted without breaking conformal invariance. The set of possible values for these coupling constants often has the structure of a smooth manifold that we call the moduli space M of the conformal field theory. This space plays the role of a "parameter space" and has to be distinguished from the moduli space of vacua, like the Coulomb or Higgs branch, of any specific conformal field theory 1 . Motion along M is generated by perturbing the conformal field theory by marginal operators. In general there may be special points on the moduli space where the conformal field theory admits a weakly coupled Lagrangian description, but at a generic point of M the theory is strongly coupled.
Our main goal is to develop a method that will allow us to probe the interior of the moduli space non-perturbatively. For this we will derive a set of classical differential equations for correlators of BPS operators, when considered as functions of the coupling constants. The value of such correlators in the weak-coupling regions of the moduli space provide "boundary conditions" for these differential equations, whose solutions can then give us the correlators at all values of the coupling. This is potentially useful because it 1 For example in the case of the N = 4 super Yang-Mills the moduli space is the upper half plane parametrized by τ = θ 2π + i 4π , modded out by the action of the SL(2, Z) duality group. may allow us to relate weak and strong coupling results and to probe the interior of the moduli space, where no weakly coupled description is available. The BPS operators that we will study are the chiral primaries with their "chiral ring" multiplication. Additionally we want to understand how superconformal invariance constrains the Zamolodchikov metric on the moduli space.
In two-dimensional N = (2, 2) theories many exact results are known about the correlation functions of BPS operators. By considering the topologically twisted theory it was discovered that such correlators obey differential equations with respect to the coupling constants, called the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations [14], [15], [16].
These equations govern the dependence of correlators with respect to holomorphic deformations of the couplings. While useful in topological field theories, it is not easy to directly apply these equations to the physical quantum field theory since for a deformation to be consistent with unitarity it must be a real linear combination of holomorphic and antiholomorphic deformations.
In [17] a beautiful method, called "topological anti-topological fusion", was developed by Cecotti and Vafa, which allows one to consider simultaneous holomorphic and antiholomorphic deformations of correlators in two-dimensional N = (2, 2) theories. Such correlators can be computed by gluing together a hemisphere on which the the theory is topologically twisted and one where the conjugate twisting is performed. The resulting amplitude turns out to be quasi-topological and constrained to satisfy exact differential equations called the tt * equations. We would like to emphasize that while the derivation was based on the topologically twisted theory, the final equations are valid even in the physical, untwisted quantum field theory.
How much of this structure survives in higher dimensional supersymmetric field theories? At first thought it appears to be difficult to generalize the arguments of [17] to four-dimensional superconformal field theories. Some important ingredients of the twodimensional story, such as the correspondence between Ramond ground states and chiral primaries via spectral flow are missing in four-dimensions. However in [18] it was shown that in the case of two-dimensional N = (2, 2) superconformal field theories, the tt * equations can also be derived from the point of view of standard conformal perturbation theory in the NS sector, without relying on the topological twisting. We will show that with small modifications the same arguments can be applied to four-dimensional N = 2 superconformal field theories.
To summarize our results we find that four-dimensional N = 2 superconformal field theories have rich mathematical structure, comparable to that of N = (2, 2) theories in two dimensions, as far their chiral ring sector is concerned. We find that the Zamolodchikov metric on the moduli space M of a four-dimensional N = 2 theory satisfies constraints similar to those of "special geometry".
For N = 1 and N = 2 theories we show that the 3-point functions C k ij of chiral primary operators vary holomorphically on M and satisfy the WDVV integrability equations. Similarly more general "extremal correlators" of chiral primaries vary holomorphically. Finally for N = 2 theories we find that the operator mixing for chiral primaries is characterized by the fact that such operators are sections of holomorphic vector bundles over M whose curvature is computed by the tt * equations. As we will explain in the main text, these can be considered as a partial nonlinear differential equation relating the 2-and 3-point functions of chiral primaries over the moduli space. While the 2-point functions g kl are not holomorphic, they are computed as solutions of these differential equations, whose coefficients are the holomorphic functions C k ij . In this paper we will focus on the basic derivation of the four-dimensional tt * equations from conformal perturbation theory. We hope to report on possible applications and on the relation to the topologically twisted theory in the future [19].
The plan of the paper is as follows: in section 2 we briefly summarize some basic properties of N = 2 superconformal field theories in four dimensions. In section 3 we discuss aspects of conformal perturbation theory and operator mixing. In section 4 we present our main results for N = 2 theories. In section 5 we describe the proof of the tt * equations. In section 6 we analyze the constraints for the Zamolodchikov metric on the moduli space. In section 7 we consider our results for N = 1 superconformal field theories.
In section 8 we make some observations about N = 4 theories and in section 9 we make some remarks about the space of vacua of such theories. In in section 10 we have some general comments and discuss possible applications and extensions.

N = 2 superconformal field theories in four dimensions
In this section we review some basic facts about four-dimensional N = 2 superconformal field theories. Some useful references for the algebra and its representations are [20], [21].

The algebra and the chiral ring
The N = 2 superconformal algebra consists of the standard conformal generators P µ , K µ , M µν , D, the 8 supercharges Q i a , Q i,ȧ and their superconformal partners S a i , S i,ȧ .
Additionally it contains the generators of the SU (2) R × U (1) R R-symmetry algebra. Here a,ȧ are Lorentz spinor indices and i is an SU (2) R index in the I = 1 2 representation. We work in conventions where the left chiral supercharges Q i a have U (1) R charge equal to −1. Superconformal primary operators 2 are labeled by their conformal dimension ∆, the Lorentz spin (j, j), the SU (2) R "isospin" I and the U (1) R charge R.
We will be interested in chiral primary operators φ which satisfy the following condi- The antichiral operators φ have j = 0, I = 0, ∆ = − R 2 and are annihilated by the supercharges of left chirality Q i a . Notice that we defined the "chiral primaries" as operators which are annihilated by all supercharges of one chirality, or in other words, which are chiral with respect to both N = 1 sub-algebras of the N = 2 theory. This definition implies I = 0.
The 2-point function of chiral primaries defines the Zamolodchikov metric for these Using the U (1) R conservation it is easy to show that the operator product expansion of chiral primaries is non-singular φ i (x)φ j (0) = C i jk φ k (0) + ... and that the operator φ k (0) is also chiral primary of charge R k = R i + R j . In this sense the chiral primary operators form a ring under OPE multiplication, called the "chiral ring" [22]. The constants C k ij are the structure constants of the ring. The chiral ring coefficients are related to the 2-and 3-point functions of chiral primaries by 4 We call conformal primary operators those which are annihilated by the K µ 's. Superconformal primary operators are those which annihilated by all S and S's, which implies that they are also annihilated by the K µ 's. 3 When we write operators at infinity, such as φ j (∞), what we really mean is explained in equation (2.2). 4 Any three points can be mapped to {0, 1, ∞} by a conformal transformation, so the 3-point function is fixed by conformal invariance up to an overall constant.

Superconformal Ward identities
In the rest of the paper we will often use superconformal Ward identities so we briefly review them here. A basic Ward identity is that for a set of local bosonic operators ϕ i and a supercharge Q we have which expresses the "supercharge conservation". For simplicity in this section we will assume that all local operators ϕ i are bosonic 5 .
We will also need a somewhat less familiar identity. First we remind that if ϕ is a scalar 6 conformal primary operator of dimension ∆ then we define Now let us consider the correlator of a superconformal primary operator ϕ with a number of operators ϕ i (not necessarily primary). According to (2.1) we have We multiply this relation by |x| 2∆ where ∆ is the dimension of ϕ and take the limit x → ∞.
The term |x| 2∆ [Q, ϕ](x)ϕ 1 (x 1 )...ϕ n (x n ) goes to zero in that limit, because [Q, ϕ](x) is a conformal primary 7 of dimension ∆ + 1 2 so this term falls off at least as fast as 1 |x| while the other terms have a finite limit according to (2.2) and we have where in this sum there is no term where the supercharge Q acts on φ(∞). So when applying the superconformal Ward identity (2.1) if a superconformal primary is "hidden" at infinity then Q does not act on it. 5 For fermionic operators there may be additional minus signs in the Ward identities. 6 For operators with spin we have to multiply with the inverse 2-point function. The general case is explained in appendix A.

Marginal Deformations and Operator Mixing
In N = 2 superconformal field theories exactly marginal operators preserving supersymmetry must be descendants of scalar chiral primaries of dimension ∆ = 2, U (1) R charge R = 4 and their conjugates. In the rest of the paper we denote chiral primaries of this R-charge by capital letters Φ i , while for chiral primaries of general charge we use the The moduli space has the structure of a complex Kähler manifold. The marginal operators are divided into holomorphic and antiholomorphic ones as 8 In superspace language the two deformations can be written as N = 2 F-terms of the form The 2-point functions of chiral primaries and those of marginal operators can be related by using (2.1) to move the Q's from O i onto O j and then using the supersymmetry algebra {Q i a , Q j,ḃ } = 2δ i j P aḃ . With appropriate normalization of (3.1) we have where the constant of proportionality is a matter of conventions and can be adjusted to simplify certain equations.
where ∂ 2 y is the Laplacian operator with respect to y. This expression implies G ij = 192·g ij . The quantity G ij is the Zamolodchikov metric on the moduli space M. Now we want to analyze how the chiral ring varies as a function of the coupling constants {λ µ } parametrizing M. At first one may think that the coupling constant dependence is simply captured by the fact that the 3-point functions of chiral primaries become moduli-dependent functions C k ij (λ). However the situation is somewhat more complicated due to the operator mixing of chiral primaries with the same quantum numbers under marginal deformations.
To understand this better, let us first notice that at each point on the moduli space we can choose the basis of chiral primaries in an arbitrary way. Under a coupling con- . From this ambiguity it is clear that comparing the chiral ring i.e. the structure constants C k ij or the 3-point functions C ijk at different points on M is not completely straightforward.
Is there a canonical way to choose a basis of operators over the moduli space and to eliminate this ambiguity? As we now explain the answer is negative. Operator mixing is an intrinsic property of the conformal field theory, completely determined by its dynamics, and it is impossible to gauge it away by an appropriate choice of basis.
One way to understand why the operator mixing is unavoidable is the following: under an infinitesimal deformation by a marginal operator O the correlators of the conformal field theory change by 9 1 In general these integrated correlators are divergent due to short distance singularities in the OPE between O and the other insertions. This means that to define the deformed correlators we have to find a way to renormalize the divergent integral (3.4).
The renormalization method that we use is quite natural [23]: to first order in the perturbation we cut out small balls of radius ε around the insertions ϕ i (x i ), and we compute the integrated correlator as a function of the cutoff ε. Then we expand the integrated correlator in powers of 1 ε , throw away all the divergent pieces and keep the finite result as ε → 0. This defines "renormalized" deformed correlators to first order in the deformation.
Going to second order we basically do the same, but we have to be careful about the pieces 9 We included the factor of 1 (2π) 2 in the definition of the marginal deformations in order to simplify certain formulas. that we already subtracted at first order. This approach was discussed in [24], [25], [26] and nicely developed for deformations of two-dimensional conformal field theories in [23], to which we refer the reader for more details.
A careful analysis shows that if we apply this procedure twice to compute the second order perturbation we find that in general the following expression does not vanish where the brackets denote antisymmetrization. The non-vanishing of this expression indicates that the two deformations do not commute, i.e. there is curvature on the space of operators. In other words, the operator mixing is inevitable because there is no way one can consistently renormalize the deformed correlators and at the same time avoid (3.5) 10 .
Because of the operator mixing we should be thinking of chiral primaries as operators taking values in vector bundles over the moduli space of the theory. These bundles have nontrivial curvature. Let us call V R the bundle of the chiral primaries of U (1) R charge R.
Then the chiral ring coefficients describe the OPE multiplication between different bundles Sometimes this discussion is presented in terms of "contact terms" in the OPE between O and the other operators. The contact terms are tuned in such a way as to cancel the divergent parts of the integrated correlator, but may also give finite contributions which are interpreted as the connection on the space of operators [27], [28].
In a sense the contact terms are not an intrinsic notion in a fixed CFT, before we consider perturbing it. In a given CFT only correlators of operators at distinct points are meaningful. The contact terms have to be introduced when we want to consider integrated correlators which, as we discussed, require the use of some renormalization method. The contact terms are then chosen in such a way that they impose our preferred renormalization procedure in all correlators in a consistent way, and allow us to do the integrals such as (3.4) including the coincident points.
On the other hand if we decide to explicitly use our renormalization prescription [26] whenever we deal with deformed correlators, then we will never have to bring two operators at exactly the same point and we do not need to worry about contact terms. Said differently, the contact terms do not have to be included in the renormalization method, but rather they are determined by it. In the rest of the paper we will follow the prescription of [23] which was described above and will work in terms of regularized and renormalized integrated correlators, so we will not have to talk about contact terms anymore. We would like to thank K. Skenderis for discussions about the contact terms.
Each of the V R bundles has nontrivial connection which encodes the operator mixing. For a set of chiral primaries of the same charge φ i which can mix under motion on M we describe the mixing by the connection A µ and we have the covariant derivative and its curvature The connection is determined by the dynamics of the conformal field theory by the following condition where ϕ i are general local operators (not necessarily chiral primary) and the renormalization prescription is the one we mentioned earlier. From now on, whenever we write integrated correlators we will always refer to their "renormalized" values, so we will drop the "renormalized" subscript from the integrals. Mapping the S 4 onto R 4 we find that the curvature can also be written as where again the double integral is supposed to be renormalized by the prescription that we mentioned earlier, see also [23], [18] for more details. From the curvature it is possible to reconstruct the connection A j µi , at least locally. Once this connection has been computed, then it is meaningful to ask what is the dependence of the chiral ring coefficients on the coupling constants.
To give a simple example why all of the previous analysis was necessary, let us consider a theory where we believe that the 3-point functions of chiral primaries are "independent of the coupling constant". According to what we discussed this statement should be interpreted as ∇ µ C k ij = 0 and not ∂ µ C k ij = 0 which is a non-covariant, basis-dependent equation. What this means in practice is that in a theory where F µν = 0, it is impossible to choose a basis such that ∂ µ C k ij = 0 everywhere on the moduli space, while it is still possible to have ∇ µ C k ij = 0. This is the right way to express the notion that the 3-point functions are "independent of the coupling constant".
Let us make a few more comments. In all of the above we have assumed that the conformal dimensions of the operators involved in the correlators do not change as we vary the coupling constants. This assumption was motivated because we are interested in correlation functions of chiral primary operators, whose conformal dimension cannot change as long as they remain "chiral primaries" 11 . If we consider more general operators whose conformal dimensions vary continuously, then the deformed correlators can have logarithmic terms and we have to be more careful in defining the connection. 11 Consider an operator φ which is chiral primary everywhere on the moduli space, so it satisfies ∆ = R 2 everywhere. Can its R-charge be a function of the coupling constants? Since it is an abelian charge one may wonder whether it can change continuously and simultaneously with ∆, preserving the condition ∆ = R 2 . However it is easy to use a superconformal Ward identity to show that the 3-point function Oφφ vanishes, which means that the conformal dimension (and hence R-charge) of a chiral primary cannot change continuously on the moduli space. Notice that the discussion in this paragraph is not about whether chiral primaries can combine into long multiplets and lift from the unitarity bound ∆ = R 2 , which is not excluded by this argument. We have ignored this effect because it can only happen at special points of the moduli space. We would like to thank J. de Boer for discussions on these issues. Finally, the connection defined above is compatible with the Zamolodchikov metric defined by the 2-point functions of chiral primaries since the 3-point function in the integrand vanishes 12 .

Marginal deformations in N = 2 superconformal field theories
We now focus on the main goal of this paper, the analysis of marginal deformations of N = 2 superconformal field theories in four dimensions. The important element is that in these theories the marginal operators are descendants of chiral primaries (3.1). As a result, the variation of the chiral ring under marginal deformations is tightly constrained.
In this section we present the main results of our paper and leave some of the derivations for the next section and the appendices.

Holomorphy of the chiral ring and WDVV equations
We start by analyzing the moduli dependence of 3-point functions of chiral primaries.
We consider a deformation of the theory 13 We denote by ∇ m , ∇ m the derivatives with respect to holomorphic and antiholomorphic marginal operators where A l mk is the connection. First we show that the chiral ring coefficients are (covariantly) holomorphic which is a consequence 14 of the vanishing of the 4-point function This a direct result of the superconformal Ward identity ( Notice that this argument did not depend on the number of chiral primaries inserted in the correlator, since we only needed to "hide" the antichiral operator φ l at infinity.
So it can be used to show that more general "extremal correlators" are also covariantly holomorphic where the U (1) R charges satisfy R n+1 = n i=1 R i . If we have two or more antichiral insertions then we cannot apply a similar argument. Now let us consider the holomorphic dependence of 3-point functions of chiral primaries and show how to derive the four-dimensional analogue of the WDVV equation [14], [15], [16].
14 From the vanishing of the 4-point function follows that ∇ m C ijl = 0. Then using that ∇ m g lk = 0 from (3.8) and C k ij = C ijl g lk we get (4.1).
We can however prove the integrability condition where we assumed that the indices i, j correspond to chiral primaries of ∆ = 2. This is a result of the following relation of 4-point functions This can be derived in similar manner using the superconformal Ward identities by moving the supercharges from O i onto φ j , which they transform into O j (the contribution from φ k can be killed according to the Ward identity in appendix A). Then by acting with a conformal transformation we can switch the positions of the operators φ i and O j leaving the other two unchanged.
Next we consider the associativity of the chiral ring. We start with the following 4-point function of chiral primaries where we take φ i , φ j to be Lorentz scalars. Since the OPE of chiral primaries is nonsingular Associativity of the OPE (or crossing symmetry) implies that the two limits should be the same. Multiplying with the inverse metric we get the desired expression Before we proceed, let us summarize the results we have found so far where the notation C i means that we think of it as a matrix (C i ) l k . These results are identical to those found in two-dimensional superconformal field theories The WDVV equations in N = 2 theories in four dimensions have been discussed (in a different context) in [29], [30], [31], [32]. These papers focused on the prepotential of the low energy effective action on the Coulomb branch of Seiberg-Witten theories, while we study the dependence of correlators on the position on the moduli space (i.e. parameter space) of the CFT. As we show in the next section, in the case of N = 2 superconformal field theories, we can actually perform the double integral of (3.7) and we find the following simple expressions for the curvature 15 where in these equations the curvature operators are acting on the subspace of chiral is proportional to the metric on the moduli space G ij = 192g ij (see (3.3)) and we use a in . These equations, whose proof is presented in section 5, are the main results of our analysis. They were first discovered in the context of two-dimensional N = (2, 2) supersymmetric field theories and called "tt * equations" [17]. The first two equations show that the bundles of chiral primaries V p are holomorphic vector bundles, while the last one allows us to compute the curvature in terms of the chiral ring coefficients C k ij .

Curvature of the supercurrents
Before we continue we have to explain a subtlety. The N = 2 superconformal algebra is invariant under a U (1) automorphism which rotates the left chiral supercharges as Q → e iθ Q and the right chiral ones as Q → e −iθ Q leaving all bosonic generators unchanged 16 .
This means that the phase of the supercharges is ambiguous and as we will see there is nontrivial holonomy for it, under motion on the moduli space. In other words the left chiral supercharges are sections of a line bundle L over the moduli space and the right chiral supercharges Q of its conjugate L.
The curvature of L is the same as that of the left chiral supercurrents G i aµ , since Q i a = d 3 xG i a0 . So we can use the general formula (3.7) for the curvature of the supercurrents to compute the curvature of L where for simplicity we did not write the spinor and flavor indices. The curvature can be computed by similar methods as those used in section 5 and after some work described in appendix D we find the following 17 So we learn that L is a holomorphic vector bundle whose curvature is proportional to the Kähler form of the moduli space. We remind that we are working in conventions where the 2-point function g ij of chiral primaries of ∆ = 2 is related to the 2-point function of marginal operators G ij by G ij = 192 g ij . 16 Notice that under this automorphism the superconformal partners rotate as S → e −iθ S , S → e iθ S. 17 In the first version of this paper the curvature of the supercurrent was off by an overall numerical factor.

tt * equations as differential equations
The set of equations (4.6) and (4.7) can be made less abstract by working in a "holomorphic gauge": from the second equation in (4.7) we see that it is possible to choose the basis of chiral primaries to depend on the coupling constants in such a way that A l jk = 0. In that basis the covariant and ordinary antiholomorphic derivatives coincide ∇ j = ∂ j and equation (4.1) becomes So in this basis the chiral ring coefficients become ordinary holomorphic functions of the moduli (and not just "covariantly holomorphic"). If we know the global structure of the moduli space M and the behavior of the chiral ring coefficients in the weak-coupling limits it may be possible to completely determine them for all values of the coupling.
To proceed we need to notice the following 18  For the special case of chiral primaries of R = 4 one can see that while the φ ′ i 's are a holomorphic section, the φ i 's are those which give holomorphic sections after Q 4 acts on them, which ensures that the marginal operators as defined by (4.10) are indeed holomorphic sections. We have for the 2-point functions In the holomorphic gauge, i.e. in the primed basis, the connection and curvature can be directly written in terms of the metric and hence where we used the fact that the bundles are holomorphic and that the metric and connection are hermitian and compatible 19 . Using (4.11) this implies Putting everything together we find that the integrability equations (4.6) become (4.13) and the last of the tt * equations (4.7) can be written as where it is important to emphasize that these equations hold in the choice of gauge (normalization) described above.
When written in the form (4.9), (4.13) and (4.14), the tt * equations can be considered as partial differential equations relating the 2-and 3-point functions g kl (λ) and C k ij (λ) of chiral primaries over the moduli space. One of their applications is that they may allow us to determine the Zamolodchikov metric g kl (λ) (for the entire tower of chiral primaries) for all values of the coupling. The important point is that while the 2-point functions g kl are not holomorphic, they can be computed as solutions of these classical differential equations whose coefficients are the holomorphic functions C k ij (λ), which are in general easier to determine. 19 From the compatibility of the metric with the connection we have ∇ i g km = 0. Using that in the holomorphic gauge we have A l jk = A l jk = 0 we can also write the compatibility condition as ∂ i g km − A l ik g lm = 0 and then (4.12) follows.

A flat connection
Let us briefly comment on the term g ij g kl 1 + R 4c in the third equation of (4.7), which was absent in the original derivation [17] of the 2-dimensional version of the tt * equations.
There the connection for the Ramond ground states of the topologically twisted theory was computed. In our case we are computing the connection characterizing operator mixing in a (four dimensional) conformal field theory. To check that in our case we do need this extra term we notice that without it we would have nontrivial holonomy (phase) for the identity operator in the CFT (see also [18]), which seems unnatural.
In any case it is not difficult to relate the two results. Let us call W the holomorphic line bundle whose curvature is proportional to the Kähler form on the moduli space And let us redefine each of the bundles of chiral primaries V in the following way V ′ = Then the curvature for the bundles V ′ will be which is the same as the one discussed in [17].
Finally let us mention that the tt * equations, together with (4.6), are equivalent to the statement that one can define an improved connection The improved connection is flat However this connection is not compatible with the Zamolodchikov metric.

Derivation of the four-dimensional tt * equations
Now we show how the four-dimensional tt * equations can be derived from the point of view of conformal perturbation theory. Let us start with the first equation [∇ i , ∇ j ] l k = 0. For this it is sufficient to show that We first act with a conformal transformation to interchange the points at ∞ and 0, so We will now show how to derive the third of the tt * equations (4.7). For that we need to compute the following integrated 4-point function The superconformal Ward identities of the N = 2 CFT allow us to write where ∂ 2 x is the four-dimensional Laplacian with respect to x. A quick explanation of this identity is as follows: we have O j = Q 4 · Φ j . We want to move the Q's away from the operator Φ j (y) using (2.3). The operator φ k is annihilated by the Q's being an antichiral primary and φ l does not contribute according to (2.3), so the only contribution will be when the Q's hit the O i . Then we end up with something of the form From the supersymmetry algebra {Q i a , Q j,ḃ } = 2δ i j P aḃ we see that we get derivatives ∂ x when we anticommute the supercharges. The only Lorentz invariant combination we can Now we use the following conformal Ward identity 20 Using this result and the OPEs between chiral and antichiral primaries (appendix E) we can see that the integrand in (5.2) is smooth as y → 0, so doing the y-integration Now we use two more conformal Ward identities 21

Substituting (5.4) and (5.5) into (5.3) and doing the x integral we find
which is schematically depicted in figure 2. The contribution from the limit r → 0 can be easily evaluated by considering the OPE between the operator at x and the one at 0. We where we did not include any higher order terms since they do not contribute to the integral in the limit r → 0. Notice that the first expression is the chiral ring OPE while the second is explained in appendix E. So the contribution from this limit is Then for the contribution to (5.6) from the r → 1 limit, we notice that there are many cancellations between the integrands at points related by x ↔ y as r → 1, and the only possible nonzero contribution is from the vicinity of the region x = y. There we are allowed to use the OPE between Φ i and Φ j . As we explain in appendix C, the only terms which do contribute are the identity operator and operators in the supermultiplet containing the stress-energy tensor. The contribution to the OPE of these operators is fixed by Ward identities so finally we find the following result Putting everything together we have

The Zamolodchikov metric on the moduli space
The previous results allow us to find constraints for the geometry of the moduli space of general N = 2 superconformal field theories. In this section we show that the Zamolodchikov metric satisfies constraints similar to those of "special geometry". The marginal operators are of the form where C M ij are the chiral ring coefficients between chiral primaries Φ i , Φ j of ∆ = 2 and φ M of ∆ = 4 and g ij is proportional to the metric on the moduli space G ij as G ij = 192g ij (see (3.3)). These are partial differential equations for the Zamolodchikov metric G ij on M, with coefficients given by the holomorphic correlators C M ij and their complex conjugates. These equations are generalizations of special geometry and similar to those describing the Weil-Petersson metric on the moduli space of Calabi-Yau n-folds or those of N = (2, 2) SCFTs in two dimensions with arbitrary central charge [33], [34], [35], [36], [37]. 22 In the first version of this paper this formula contained an incorrect factor which has been fixed now.

N = 1 theories
Let us now explain which of our previous results are true for theories with N = 1 superconformal invariance in four dimensions. The N = 1 SCFT has a U (1) R R-symmetry.
The standard normalization of the U (1) R in N = 1 theories is such that the unitarity bound is ∆ ≥ 3 2 R. Superconformal primary operators saturating this bound are chiral primaries and are annihilated by the supercharges of right chirality Qȧ. We have the structure of a chiral ring as in N = 2 theories and we use the same notation for the 2-and 3-point functions g ij and C ijk and the chiral ring coefficients C k ij . In N = 1 SCFTs marginal deformations preserving supersymmetry are descendants of chiral primaries Φ i of ∆ = 3 and again they split into holomorphic and antiholomorphic All of the results of section 4.1 are also true for N = 1 theories. In particular the 3-point functions of chiral primaries satisfy the following set of equations where the notation C i means that we think of it as a matrix (C i ) l k . We can also show that the following relations are true The last equation implies that we can pick a holomorphic gauge as before and then we have the following equations We can also show that "extremal" correlators have to be holomorphic It would be very interesting to see whether the third of the tt * equations (4.7) can also be derived for N = 1 SCFTs, possibly with some modifications. We hope to report on this in the future [19].

N = 4 theories
Now we discuss how the previous analysis applies to theories with N = 4 superconformal invariance. Certain aspects of the connection on the space of operators for N = 4 theories have been discussed in [38], [39]. An N = 4 theory can also be viewed as an N = 2 theory and has several kinds of short supermultiplets [20], [21]. Superconformal multiplets of the N = 4 algebra are labeled by the conformal dimension ∆, the spin (j, j) and by their SO(6) R R-symmetry representation. It is standard to parametrize it in terms of the complexified gauge coupling with the metric The global structure depends on the action of the S-duality group and cannot be determined by our local analysis. We see from If we have n gauge group factors we will have n corresponding short multiplets Φ i which give marginal operators and then the moduli space is (locally) the product of spaces of the form (8.1). The statement that the gauge group is a direct product is related to the fact that the Φ i 's do not mix among themselves under motion on the moduli space.

Exotic N = 4 theories?
In this speculative paragraph let us mention that it is a logical possibility that exotic In the previous section we found that the moduli space of N = 4 theories is a homogeneous space. Bundles of covariantly constant curvature over homogeneous spaces are called homogeneous bundles and their structure is completely determined by group theory [41], in analogy with the two-dimensional story in [18]. For Lagrangian N = 4 theories this implies that the bundles of chiral primaries are (locally) flat. For "exotic" N = 4 theories, if they exist, we would have nontrivial homogeneous bundles labeled by representations of SO(n). These issues will be discussed in more detail in [19].

A comment on the moduli space of vacua
In this paper the term "moduli space" has been used to refer to the parameter space of the conformal field theory. This has to be distinguished from the moduli space of vacua. The first parametrizes a family of conformal field theories which are continuously connected, while the second is a family of vacua of a given and fixed conformal field theory.
The first is a characteristic of the quantum field theory in the UV and refers to a tuning of the parameters in its Lagrangian, while the second is an IR concept, which is characterized by vacuum expectation values of certain operators.
Notice that in two-dimensional conformal field theory this distinction is not usually made because in two dimensions there cannot be moduli spaces of vacua due to infrared 26 The argument we followed is analogous to the one in [40] generalizing the results of [27] for the moduli space of two-dimensional N = (4, 4) theories.
divergences. The situation is different in higher dimensional conformal field theories where it is possible for operators to get nonzero vevs.
It is also natural to consider the total space, the space of all possible vacua of all continuously connected conformal field theories. This has the structure of a bundle where the base is the moduli space of conformal field theories and the fiber is the space of vacua for the given theory [42]. Perhaps it would be interesting to explore the topology and geometry of this total space in more detail.
In many Lagrangian N = 2 gauge theories this total space is nicely described by the Seiberg-Witten curves [43], [1], which in the case of finite theories depend on the marginal

Discussions
We derived some new exact results for the chiral ring of general N = 2 superconformal field theories and found structures similar to those encountered in two-dimensional N = (2, 2) theories. Our analysis was based on conformal perturbation theory. It would be interesting to study whether our equations can be derived in an alternative way by considering the topologically twisted N = 2 theory [44] and following similar arguments as in [17]. Such an approach might also give results about massive deformations of N = 2 superconformal field theories, which could be useful for applications to theories with less supersymmetry. It might also be interesting to explore the topologically twisted N = 2 superconformal field theories on more general four-manifolds. For such an approach the analysis of [45] might be useful and also [46] for recent discussions.
It would be instructive to apply our somewhat abstract analysis to specific examples, such as the finite N = 2 SU (N ) gauge theories with N f = 2N flavors in the fundamental.
In these theories the chiral ring is generated by chiral primaries of the form Tr(Φ k ) where Φ is the adjoint scalar field and the moduli space is parametrized by the gauge coupling . It is possible to perform perturbative computations in the τ → i∞ limit and then we can use the holomorphy of the chiral ring coefficients and the tt * equations to extend the computation in the interior of the moduli space. In particular it would be interesting to try to connect the τ → i∞ point with the S-dual "infinitely-strongly coupled" limit τ → 1 discussed in [47].
More generally it would be interesting to analyze the rich class of N = 2 superconformal field theories studied by Gaiotto in [13]. These theories have more complicated moduli spaces related to those of punctured Riemann surfaces. Based on this, it was pointed out in [48] that the Zamolodchikov metric on the moduli spaces of such conformal field theories is the Weil-Petersson metric on the Teichmuller space, which is consistent with the equations that we found in section 6. The chiral primary operators of the four-dimensional theory are related to holomorphic differentials on the Riemann surface on which the M5 branes are wrapped. Thus the connection for the chiral primaries that we computed is related to the Gauss-Manin connection for the cohomology elements of the Riemann surface under variation of its complex structure moduli. It would be nice to make these statements more precise.
It might also be useful to explore our results from the six-dimensional point of view. A six-dimensional superconformal field theory can be either wrapped on a Riemann surface giving an N = 2 theory in four dimensions, or on a 4-manifold resulting into a SCFT in 2 dimensions. Both theories obey the tt * equations and it would be nice to understand relations between the two and possible connections with [49], [50].
Another interesting direction is clarifying the bulk interpretation of our results in cases of N = 2 theories with AdS duals. The moduli space of the conformal field theory is mapped to the moduli space of the AdS compactification of supergravity/string theory.
The chiral primaries are related to certain Kaluza-Klein modes in the internal manifold, or possibly other supersymmetric probes such as giant gravitons.
It is worth trying to extend our results in N = 1 superconformal field theories in four dimensions. Upon dimensional reduction on a two-torus they lead to two-dimensional N = (2, 2) supersymmetric theories (not necessarily conformal). Hence one might hope that some degree of control exists even in the parent four-dimensional theory, see also [51]. From the AdS/CFT point of view the structure of the moduli space of N = 1 superconformal field theories was briefly discussed in [52] in relation to a-maximization [53]. It might be possible to explore these questions using methods similar to the ones developed in this paper.
Our analysis has been local on the moduli space. It would be interesting to analyze certain global issues such as the action of the S-duality group on correlators and possible monodromies for the chiral primaries around non-contractible cycles. Of interest would also be to analyze what happens at points where chiral primaries combine into long multiplets and lift [54]. In a sense these seem like the SCFT analogue 27 of "walls of marginal stability".
It would be nice to check that the renormalization method that we adopted to implement conformal perturbation theory is actually consistent with all axioms of conformal field theory and with supersymmetry, especially at higher orders in the perturbation.
Finally let us mention that the connection of operators that we have computed can be interpreted as Berry's phase [58] (and its non-abelian generalization [59]) for a class of states as we now explain. If we define the conformal field theory on S 3 × time then it can be thought of as a quantum system whose Hilbert space is isomorphic to the set of local operators on the plane, via the state-operator map. These states fall into unitary representations of the superconformal algebra. A special class of these states, the ones that are dual to the chiral primary operators, belong to shorter representations. If we adiabatically vary the coupling constants of the theory, then states are subject to the phenomenon known as geometric or Berry's phase. The tt * equations compute the Berry phase for the chiral states. In the AdS bulk this would be the Berry phase of certain quantum states under adiabatic variation of the supergravity moduli. See also [60] and [61], [62], [63], [64]. happens along surfaces of codimension larger than one, so the term "wall" in this case is not so accurate. It would be useful to understand the rules of joining and splitting of such BPS states and possible relations with [55], [56], [57].

Acknowledgments
We would like to thank L. Our starting point is that the CFT has a spin (1/2, 1/2) operator J µ of ∆ = 3. Using the conformal algebra we can compute the norm of the first descendant ∂ µ J µ and we find that it vanishes, which implies the operator equation ∂ µ J µ = 0, so J µ is a conserved current.
We define the corresponding conserved charge by R = d 3 x J 0 . For any local operator ϕ we have the following OPE with no other terms of the form x µ |x| 4 . This can be written as where we compute the integral over a small sphere 28 surrounding the point x. Now consider the general correlator where we assume that all points are distinct. If we keep x k fixed and think of V as a function of z then it is vector field defined on D = R 4 −{x 1 , ..., x n }, which is divergenceless 28 We compute the integral on a sphere of radius ǫ and then take the limit ǫ → 0.
Since J µ has conformal dimension ∆ = 3 we know that V µ (z) falls off at least as 1 |z| 6 at large z, so there is no contribution from the sphere at infinity in (A.2) and we only get contributions from the punctures {x 1 , ..., x n }. Using (A.1) we find which is the desired Ward identity expressing charge conservation. Now we will do the same thing for the supercurrent. The left chiral supercurrent G iµ a is a conformal primary operator of dimension ∆ = 7/2 transforming in the (1, 1/2) representation of the Lorentz group (indices a, µ) and in the I = 1/2 of SU (2) R (index i).

It satisfies the following equations
We can construct conserved fermionic currents by multiplying the supercurrent with a conformal killing spinor ψ a (x) as It is easy to show that ∂ µ j iµ = 0 and the corresponding conserved supercharge is d 3 xj i0 . In R 4 the most general conformal killing spinor is where λ a is an arbitrary constant (1/2, 0) spinor, µȧ an arbitrary constant (0, 1/2) spinor and we use the notation xȧ a = x µ σȧ a µ . If we take λ a = 0 and µȧ = 0 we generate the left chiral supercharges Q i a , while for λ a = 0 and µȧ = 0 we get the right chiral superconformal In a similar way the right chiral supercurrent G µ iȧ generates the right chiral supercharges Q iȧ and the left chiral superconformal charges S a i . The OPE of the supercurrent with a scalar primary operator ϕ has the following form and follow the same steps as before 29 . Being careful about the expansion of j iµ around the punctures and using (A.6) we find the following superconformal Ward identity If we take ψ a (x) to be constant then we find that the usual supercharges can be moved around without any factors n k=1 and if we take ψ a (x) to be proportional to x we find that for the superconformal partners we have n k=1 for any x 0 .
From (A.7) we see that if a field ϕ is a superconformal primary (hence annihilated by the S's) then we can choose the conformal Killing spinor ψ a (x) to vanish at the point of insertion of ϕ and that when applying the superconformal Ward identity (for this special ψ a (x)) the field ϕ does not contribute at all, whether it is annihilated by the Q's or not.
This is the essence of identity (2.3). In two dimensions it was discussed in [65], [14].
For completeness we present the Ward identities for the right chiral supercurrent G ..ϕ n (x n ) = 0 (A.10) 29 We also have to use that G iµ a has ∆ = 7/2, so if it is taken to infinity inside any correlator, it falls off at least as 1 |z| 7 .
for all x 0 .
Appendix B. Some basic properties of N = 2 superconformal theories The supermultiplet containing the stress-energy tensor and the other conserved currents begins with a scalar superconformal primary operator A with ∆ = 2, I = 0, R = 0.
It is a short multiplet of the superconformal algebra [66]. The other conformal primaries in the multiplet are (diagram is from Dolan and Osborn [67]) where the ւ arrows denote the action of the left-chiral supercharges Q i a and ց of Q i,ȧ .
The conformal dimension ∆ and U (1) R charge R of the operators can be seen in the diagram. The fermions ψ i a have spin (1/2, 0) and transform in the I = 1/2 representation of SU (2) R . The U (1) R current is J µ and the SU (2) R currents are I i µ , i = 1, 2, 3. The field H ab is a spin (1, 0) operator singlet under the SU (2) R . The fields G i,µ a (and their conjugates) are the supercurrents and transform in the I = 1/2 of SU (2) R . The stressenergy tensor T µν has spin (1, 1), I = 0 and R = 0. N = 2 SCFTs have two "central charges" c, a defined as coefficients of the trace anomaly on a curved manifold The central charge c is related to the 2-point function of the stress-energy tensor as

Appendix C. Contours
Now let us consider the following quantity In this double integral the variable y lies on a 3-sphere of unit radius while x on a 3-sphere of radius r and we are taking the limit r → 1 − . So let us parametrize IfΩ y 3 =Ω x 3 then the contributions between the two points (Ω y 3 ,Ω x 3 ) and (Ω x 3 ,Ω y 3 ) cancel in the limit r → 1. So one might conclude that the integral vanishes, however the previous argument cannot be applied for the pointΩ y 3 =Ω x 3 where we may have a δ-function-like contribution. To check the contribution to E from the region where x → y we can use the OPE between Φ i and Φ j .
In order to include at once the contribution of all descendants for each primary, it is more convenient to perform the expansion in conformal partial waves in the (ij) → (kl) channel. This means that we can write the 4-point function as where the sum is over all conformal primary (though not necessarily superconformal primary) operators O of dimension ∆ and spin 30 l. The quantities u, v are the conformal cross-ratios which for our configuration of 4-points x, y, 0, ∞ are The function g ∆,l (u, v) is the conformal partial wave. Explicit expressions for these functions can be found in [68].
The other 4-point function φ l (∞)Φ i (x)Φ j (y)φ k (0) can be expanded similarly with the roles of x, y interchanged which is equivalent to the substitution u → u/v, v → 1/v.
Now we use a basic property of the conformal partial waves [68] g ∆,l (u/v, 1/v) = (−1) l g ∆,l (u, v) We can thus finally write the integral that we want to compute as and in this integral the parameters u, v have to be computed from (C.2). Notice that the quantities X ∆,l are independent of dynamical information of the CFT and only depend on kinematics of the conformal group.
After some work one can verify that the only nonzero terms are the following 30 In the OPE of two scalars we do not have any primary operators of spin (j, j) with j = j.
Here we take l = j + j.
The first term corresponds to the identity operator. The second from the conformal partial wave of the special superconformal primary A of ∆ = 2, l = 0 whose supermultiplet contains the stress energy tensor, as explained in the previous appendix 31 . The third term corresponds to the U (1) R current 32 . So all in all we find What is missing now is to calculate the quantities C A ij C Akl and C J ij C Jkl . To proceed we will use the following logic. The Ward identities fix that the conformal partial wave of the stress tensor is weighed by the overall coefficient C T ij C T kl = ∆ i ∆ k 90c g ij g kl (see [68]). Using ∆ i = 2 and ∆ k = R/2 we find In theories with N = 1 and N = 2 superconformal invariance, supersymmetry relates the weight of the conformal partial wave of the stress tensor to the weights of the partial waves of the other members of the superconformal multiplet containing the stress tensor. This can be found in [67] and more recently in [69]. Using their results we find for the and for the operator A These relations are explained in some more detail in the next appendix. Putting everything together we find E = 1 + R 4c g ij g kl 31 We assume that this operator is unique, otherwise there would be two conserved spin-2 currents. This assumption is not correct in the free limit of N = 2 superconformal gauge theories where we also have the Konishi scalar with ∆ = 2, in addition to A. However, when we turn on the coupling the Konishi multiplet gets anomalous dimensions. So we will assume that a generic point on the moduli space of the CFT there is no other scalar with the same quantum numbers as A. It would be interesting to explore this assumption in more detail. 32 If we want the marginal operator to be neutral under the current (otherwise the symmetry would not be preserved after the marginal deformation and it would be "accidental" and not present at a generic point on the moduli space), then only the U (1) R current can appear, since the supercharges carry R-charge and can make the marginal operator neutral even if the chiral primary itself is charged ( supercharges are uncharged under the non R-symmetries).

Appendix D. Conformal block decomposition and supersymmetry
Let us explain how we fix the relative normalization of the conformal partial waves in N = 2 superconformal field theories. We first consider the N = 2 theory as an N = 1 superconformal field theory and then we use the results of [69]. An N = 1 SCFT has a U (1) R R-symmetry. The corresponding current r µ is related to the N = 2 R-currents by where J µ is the N = 2 U (1) R current and I µ 3 one of the SU (2) R currents of the N = 2 theory. The orthogonal linear combination of these two currents plays the role of a flavor current from the N = 1 point of view. The overall normalization of f is not important for us since it will drop out, while the relative coefficients were fixed by requiring the orthogonality 33 of the currents r µ , f µ . Since the chiral primaries are only charged under the current J µ (and not I µ 3 ) we have On the other hand from (D.1), (D.2) we have where by g JJ , g II we denote the 2-point function of the corresponding current. The N = 2 algebra fixes g JJ = 8 g II (see for example [47]) and we find Combining (D.3) with (D.4) we find that C f ij C f kl (D.5) 33 i.e. that the 2-point function r µ f ν = 0. Notice that the currents J µ , I µ 3 are orthogonal.
In [69] the relative weight of conformal partial waves of operators in the same superconformal multiplet in N = 1 SCFTs were presented. The N = 1 R-current r µ is in the same N = 1 superconformal multiplet as the stress tensor. From their results we find that C r ij C rkl = − 5 2 C T ij C T kl and using (D.5) we have Similarly, the flavor current f µ is in the same multiplet as the scalar A of ∆ = 2,R = 0.
From the results of [69] we have C A ij C Akl = −3C f ij C f kl and using (D.5), (D.6) we find The relations (D.6) and (D.7) are the ones that we need.

Appendix E. Curvature of supercurrents
Here we show that the curvature of supercurrents is given by expressions (4.8). The curvature of the supercurrents F L µν can be computed as the relative curvature between the chiral primaries φ k and their first descendants Q I a φ k , since we have 34 ( F µν ) l k = (F µν ) l k + F L µν δ l k where (F µν ) l k is the curvature of the chiral primaries computed in (4.7) and ( F µν ) l k is the curvature of the descendants Q I a that we will try to compute now. To do it we consider the formula (3.7) for operators of the form Q I a φ k . First let us show that the line bundle L of the supercurrents is holomorphic, i.e. that F L ij = F L i j = 0. In (4.7) we showed that (F ij ) l k = 0 so we simply have to show that ( F ij ) l k =0. For this it is sufficient to show the vanishing of the 4-point function Notice that there is no summation over I implied. In the rest of this appendix we simply choose I to be either 1 or 2, the answer does not depend on this choice. Using the Ward 34 We supress the indices I, a in this formula since the curvature is diagonal in I, a.
identity (A.8) we can move the supercharge Q I a away from the point 0. The supercharge Q I a annihilates the operators O i , O j , φ l , so using the algebra we find A = 2∂ z aḃ φ l (z)O i (x)O j (y)φ k (0) = 0 because, as we explained in equation (5.1), the 4-point function appearing above vanishes.
From (3.7) it follows that ( F ij ) l k = 0 and we find Similarly we show that F L i j = 0. Thus we have shown that L is a holomorphic line bundle over the moduli space.  35 Here we are taking advantage of the fact that the curvature of operators Q I a φ k is the same for all a.
For the first term we used that for antichiral primaries {S˙b ,I , [Q I,ḃ , φ m ]} = −8∆φ m (as we explained above, here we are summing overḃ but not I). After plugging the expression for B (and the corresponding one for x ↔ y) into (3.7), the first term in (E.1) gives the curvature (F ij ) l k of the superconformal primaries φ k . The third term in (E.1) is actually zero in the limit z → ∞. Only the second term in (E.1) is relevant that we define as where we dropped a factor of y since it does not contribute in the limit z → ∞. Similarly we define C with x ↔ y.
So the curvature of the supercurrents is given by where the integrals have to be regularized in the way described in the main text. Using the superconformal Ward identity (A.12) to move Q I,ḃ away from z we find that Massaging this expression a little more we find and a similar answer for C with the substitution x ↔ y.
One can check that in the integration over x, y in (E.2) then the expression (E.4) (and the one for C) gives a finite contribution from the region y → 0 and x → 0. Hence to evaluate (E.2) one can integrate by parts and simply pick up the terms from the hemispheres y → 1 and x → 1 − as we did in section 5 and appendix C.
The partial integration leads to the following expression dΩ y 3 |x| 2 |y| 2 (y · ∂ y )(x · ∂ x ) (4(x − y) · ∂ y − 8) |y| 2 |x| 2 φ m (∞)φ i (x)φ j (y)φ k (0) − (x ↔ y) + 8 (y · ∂ y )(x · ∂ y ) |y| 2 |x| 2 φ m (∞)φ i (x)φ j (y)φ k (0) − (x ↔ y) (E.5) As we explained in appendix C, we can evaluate these integrals by considering the conformal partial wave expansion of the 4-point function in the channel (ij) → (kl). Following the analogous steps as in appendix C we find that in this expansion only the conformal partial wave of the operator A and of the U (1) R current J contribute to these integrals and that using ∆ = R/2 and the results from the previous appendices we finally find From U (1) R charge conservation we have R ′ = 2∆ − 4 and unitarity implies ∆ ′ ≥ R ′ 2 or ∆ ′ ≥ ∆ − 2. We find that the term on the RHS of (F.1) goes like 1 |x| 4 when this inequality is saturated ∆ ′ = ∆ − 2 which implies that O is one of the chiral primaries, say φ n . Then it is not difficult to show that D O jk = g kr C * r jp g pn