On exact correlation functions in SU(N) ${\cal N} = 2$ superconformal QCD

We consider the exact coupling constant dependence of extremal correlation functions of ${\cal N} = 2$ chiral primary operators in 4d ${\cal N} = 2$ superconformal gauge theories with gauge group SU(N) and N_f=2N massless fundamental hypermultiplets. The 2- and 3-point functions, viewed as functions of the exactly marginal coupling constant and theta angle, obey the tt* equations. In the case at hand, the tt* equations form a set of complicated non-linear coupled matrix equations. We point out that there is an ad hoc self-consistent ansatz that reduces this set of partial differential equations to a sequence of decoupled semi-infinite Toda chains, similar to the one encountered previously in the special case of SU(2) gauge group. This ansatz requires a surprising new non-renormalization theorem in ${\cal N} = 2$ superconformal field theories. We derive a general 3-loop perturbative formula for 2- and 3-point functions in the ${\cal N} = 2$ chiral ring of the SU(N) theory, and in all explicitly computed examples we find agreement with the tt* equations, as well as the above-mentioned ansatz. This is suggestive evidence for an interesting non-perturbative conjecture about the structure of the ${\cal N} = 2$ chiral ring in this class of theories. We discuss several implications of this conjecture. For example, it implies that the holonomy of the vector bundles of chiral primaries over the superconformal manifold is reducible. It also implies that a specific subset of extremal correlation functions can be computed in the SU(N) theory using information solely from the S^4 partition function of the theory obtained by supersymmetric localization.


Introduction
The tt * equations provide a powerful set of relations between 2-and 3-point functions in the chiral ring of N = 2 theories.They were originally derived with the method of the topological-antitopological fusion in 2d N = (2, 2) theories in [1].In 4d N = 2 superconformal field theories (SCFTs) they were derived using superconformal Ward identities in conformal perturbation theory in [2].
There are important differences between N = 2 chiral rings in two and four dimensions that are reflected in the geometry of the superconformal manifold, as well as the structure and solutions of the tt * equations.For example, 2d N = (2, 2) chiral rings have a spectrum with an upper bound on the scaling dimension [3].On the other hand, it is believed that the N = 2 chiral ring of a generic 4d N = 2 SCFT is freely generated without any upper bound on scaling dimensions.As we pointed out in [4] this feature has important implications for the structure of the tt * equations.
As a more explicit illustration of this fact, in specific 2d theories [1,5] the tt * equations take the form of a periodic Toda chain.In four-dimensional examples, e.g. the SU (2) N = 2 super-Yang-Mills (SYM) theory coupled to 4 hypermultiplets, also known as SU (2) N = 2 superconformal QCD (SCQCD), which was analyzed in [4,6], the tt * equations also reduce to a Toda chain, but in this case the chain is semi-infinite.Periodic and semi-infinite Toda chains have qualitatively different features.The solution of the periodic case is uniquely fixed by unitarity and a small set of perturbative data, but the solution of the semi-infinite case requires complete knowledge of a single member of the chain.The latter does not appear to be uniquely determined by consistency and a few perturbative data.In the context of the SU (2) N = 2 SCQCD theory we proposed [4,6] that the solution can be determined from the exact form of the Zamolodchikov metric on the superconformal manifold, which is the lowest non-trivial member of the Toda chain.In turn, the Zamolodchikov metric, and the exact quantum Kähler potential, are directly related to the S 4 partition function of the theory [7,8], which can be computed efficiently using localization techniques [9].
The clean example of the SU (2) N = 2 SCQCD theory invites us to think more generally about the structure of the tt * equations in four-dimensional theories, the constraints that they impose on the correlation functions of the N = 2 chiral ring, and the independent data needed to determine a physically consistent solution.The precise answer to many of these questions is far from obvious.For instance, already in the general SU (N ) SCQCD theory the tt * equations (in an appropriate gauge) take the form of an infinite set of coupled, nonlinear differential equations for matrix-valued quantities whose size grows indefinitely with the scaling dimension (see equation (1.3) below).In more general N = 2 SCFTs, which possess higher dimensional superconformal manifolds, the structure of the tt * equations is an even more complicated set of partial differential equations.
As a step towards a better understanding of this structure, in this paper we initiate a more detailed study of the tt * equations of the general SU (N ) N = 2 SCQCD theory.First, by an explicit computation of chiral primary 2-and 3-point functions up to 3-loops in perturbation theory, we verify that the matrix-valued tt * equations (1.3) are satisfied up to that order.Second, we investigate a specific non-perturbative ansatz for the complete solution of these equations, which is consistent with the perturbative computations.The precise form of this ansatz will be explained in the next subsection.One of its characteristic properties is that it leads to a drastic reduction of the complicated set of matrix-valued equations (1.3) to a decoupled set of semi-infinite Toda chains (similar to the chains encountered in the SU (2) case).These can be solved recursively from a single member in each chain.
Besides this drastic reduction of the tt * equations the proposed solution has other surprising properties.One of them is the requirement of a novel non-renormalization theorem in this class of N = 2 theories, where orthogonal chiral primary operators in a specific basis do not mix by quantum finite coupling effects.Relatedly, the holonomy of the vector bundles of chiral primaries over the superconformal manifold is required to be reducible.
At the moment, we do not have a proof of the above-mentioned ansatz in gauge theory.Besides the favorable evidence provided by explicit 3-loop computations in perturbation theory, it is encouraging that this ansatz is mathematically a self-consistent way to solve the exact non-perturbative tt * equations.Nevertheless, we do not have an argument that this is the only way to solve the tt * equations and the logical possibility of more complicated alternatives (that we have not yet discovered) remains.We point out some alternatives in the main text.A complete non-perturbative proof of the no-mixing conjecture, the explanation of its physical origin, and its relevance in more general N = 2 theories are some of the interesting open questions that this work is opening up.

Summary of main results
The N = 2 chiral primary fields of the SU (N ) N = 2 SYM theory coupled to 2N hypermultiplets are believed to be freely generated from the product of a finite set of N − 1 generators.In the standard Lagrangian description of the theory these generators are represented as single-trace operators Tr[ϕ +1 ], = 1, . . ., N − 1, where ϕ is the adjoint complex scalar field in the N = 2 vector multiplet.From now on, we will denote the generic chiral primary in this representation as φ K with a multi-index ( The anti-chiral primaries are multi-trace operators of the complex-conjugate field ϕ and will be denoted as φ K .We single out the special chiral primary φ 2 ∝ Tr[ϕ 2 ], which is the single operator with scaling dimension 2 in this family.The supersymmetric descendant O τ = Q 4 • φ 2 of this operator gives the exactly marginal interaction of the theory associated to the complexified coupling constant . As usual, θ is the theta-angle of the theory and g Y M the gauge coupling.
In a specific set of normalization conventions, the so-called holomorphic gauge [4], the OPE coefficients C M KL are 0 or 1, so that Here φ K+L denotes the multi-trace operator : φ K φ L : and the dots represent descendants with higher scaling dimension.In these normalization conventions the tt * equations become a coupled set of matrix partial differential equations for the 2-point function coefficients Here ∆ is the common scaling dimension of the insertions.The index 2 in the notation employed in (1.3) refers to the chiral primary φ 2 .(Further explanations of (1.3) are provided in section 2.) In this paper we evaluate the relevant Feynman diagrams and derive a (3-loop) formula that computes O(g 4 Y M ) corrections to the general 2-point function g KL in the SU (N ) N = 2 chiral ring.We apply this formula in several explicit SU (3) and SU (4) examples and verify that the equations (1.3) are indeed obeyed up to that order.
Moreover, the perturbative results provide highly suggestive evidence that the chiral primary correlators in this class of theories may correspond to a solution of (1.3) of rather special form.The main aim of this paper is to describe the ansatz for this special solution and explore its implications for chiral primary correlators.
Our discussion begins with the special role played by the chiral primary φ 2 ∝ Tr[ϕ 2 ].The tt * equations (1.3) relate the 2-point functions of chiral primaries at a given level of R-charge with those chiral primaries that can be reached by the action of the special chiral primary φ 2 or its conjugate φ 2 .Hence it is natural to consider the operator C 2 (and its conjugate C † 2 ), corresponding to chiral ring OPE multiplication by φ 2 , acting on the vector space of chiral primaries.We first show at tree level that it is possible to construct a basis of chiral primaries that diagonalizes simultaneously the 2-point functions g KL and the action of C 2 . 1ather surprisingly, we find that even after including 3-loop corrections to these correlators, there still exists a basis where the simultaneous diagonalization of 2-point functions and of C 2 is possible.
This encourages us to investigate the possibility that there is a basis of chiral primaries in which the full non-perturbative matrix of 2-point functions remains diagonal simultaneously with the matrix C 2 for all values of the coupling.This possibility lies at the core of the ansatz that we explore in this paper.The assumption that there is a basis in the chiral ring in which different degenerate operators do not mix under conformal perturbation theory has a simple geometric meaning.It is the geometric statement that the gauge connection on the chiral primary vector bundles over the superconformal manifold is reducible.More specifically, at a generic scaling dimension ∆ with degeneracy D the holonomy of the chiral primary vector bundle is, according to this ansatz, not U (D) (as one might have a priori expected), but much smaller, U (1) D .
As we explain in the main text, the next-to-leading order perturbative results in this paper do not allow us to check conclusively the no-mixing properties for all possible 2-point functions.They only allow us to find direct non-trivial evidence of the absence of mixing for degenerate operators that contain 'a different number of Tr ϕ 2 factors'.This leaves open the possibility of a partial mixing in gauge theory, where at generic scaling dimensions the 2point functions are non-perturbatively block-diagonal instead of completely diagonal.In that case, the chiral primary vector bundles over the superconformal manifold would be partially reducible to a product of U (1) line bundles times bundles with a non-abelian connection.We explain when non-abelian factors in the holonomy could in principle appear.
Under the postulate of full reducibility the proposed solution implies that the tt * equations reduce to a decoupled set of semi-infinite Toda chains.Each of these Toda chains, whose explicit form can be found in equation (3.20), can be solved in terms of a single external datum.A notable class of data that we can compute in this way are the 2-point functions of the form (Tr[ϕ 2 ]) n (x)(Tr[ϕ 2 ]) n (0) .Similar to the SU (2) results in [4,6], we find specific predictions for these data in terms of the Zamolodchikov metric of the SU (N ) theory, which is known exactly from supersymmetric localization [8,9].We further show that the no-mixing conjecture allows to extract more information from the S 4 partition function for additional extremal correlation functions.

Outline of the paper
In section 2 we review the basic features of the theory of interest and set up our notation.The precise form of the tt * equations that we analyze is also reviewed here.Section 3 explains the main proposal and how it leads to a recursive solution of the tt * equations in the SU (N ) theory.Starting at tree level we present a linear transformation on the vector space of chiral primaries that diagonalizes simultaneously the 2-point functions and the components of the OPE coefficients C L 2K , and reorganizes the tt * equations into a set of decoupled semiinfinite Toda chains.We propose that the nice properties of this basis continue to hold non-perturbatively at finite coupling.The implications of this proposal are discussed further in section 4, which contains a list of specific predictions for exact correlation functions in chiral ring of the SU (N ) N = 2 SCQCD theory.These predictions are tested non-trivially in perturbation theory in section 5 using a general O(g 4 Y M ) perturbative formula for 2-point functions in the chiral ring derived in appendix A. We conclude in section 6 with a summary of interesting open issues.For the benefit of the reader appendix B contains a supplementary description of the diagonalization of 2-point functions discussed in section 3.
2 Review of the tt * equations in SU (N ) N = 2 SCQCD theory The general properties of the tt * equations in four-dimensional N = 2 SCFTs are reviewed in Ref. [4], whose notation we will mostly follow.In the rest of the paper we will omit many of the technical details, which can be found in [4], and will focus directly on the case of interest: the N = 2 superconformal QCD (SCQCD) theory defined as N = 2 super-Yang-Mills theory with gauge group SU (N ) coupled to 2N massless hypermultiplets in the fundamental representation.The global symmetry group of this theory for generic N is U (2N ) × SU (2) R × U (1) R , where U (2N ) is the flavor symmetry group and SU (2) R × U (1) R is the R-symmetry group.
The N = 2 chiral ring of the SU (N ) N = 2 SCQCD theory is freely generated by the N − 1 single-trace operators with scaling dimension ∆ = + 1.Here, ϕ is the adjoint complex scalar field in the N = 2 vector multiplet.Hence, the generic chiral primary field is a multi-trace product of arbitrary powers of these generators.These fields are neutral under the flavor U (2N ) and the R-symmetry group SU (2) R .They are charged under the U (1) R with R-charge R and scaling dimension 3) The N = 2 SCQCD theory has a single (complex) exactly marginal deformation where O τ is the supersymmetry descendant of the chiral primary φ 2 ∝ Tr[ϕ 2 ].The notation Q 4 • φ 2 denotes the nested (anti)-commutator of four supercharges of left chirality on the field φ 2 .The corresponding exactly marginal coupling that parametrizes the (complex) 1-dimensional superconformal manifold is the complexified gauge coupling constant . In what follows we will employ a specific set of normalization conventions for the chiral primaries φ {n } , where (i) φ 2 adheres to the conventions (2.4)-(2.5),(ii) we require (iii) the remaining generators in (2.1) are chosen with an arbitrary holomorphic normalization factor, and (iv) we require that the non-vanishing OPE coefficients are For general indices K, L, . . . of the form {k }, {l }, . . . the notation K + L in (2.7) denotes the index {k + l } and the OPE coefficient (2.7) implies the operator product expansion This OPE is enough to fix the normalization of all multi-trace chiral primaries in terms of the normalization of the generators (2.1).
As explained in [2,4] it is most appropriate to think of the chiral primary fields φ L as sections in a holomorphic vector bundle V whose base space is the superconformal manifold of the theory.The above set of conventions is a choice that makes the rescaled chiral primary fields e − R L c K φ L (2.9) holomorphic sections of the bundle V. Here, R L is the U (1) R charge of the fields φ L , c = 8 × 192 × c (where c is the central charge of the CFT), and K is the exact Kähler potential of the superconformal manifold.φ L are the chiral primaries in the conventions (2.6)-(2.7).
The reason for the appearance of the factor e − R L c K can be traced back to the choice of normalization conventions for the supercharges, or equivalently to the choice of a section in the holomorphic line bundle associated to the supercurrents.
These choices constitute the so-called holomorphic gauge where a connection A on V compatible with the 2-point function coefficients g KL has components [2] We remind the reader that g KL is defined in (1.4).The notation g KL refers to the components of the inverse matrix of 2-point function coefficients: L vanish by definition in the holomorphic gauge.
The tt * equations express the curvature of this connection.In holomorphic gauge they lead to a set of partial differential equations for the 2-and 3-point function coefficients g KL , C M KL , which have the form where C L 2K denotes the coefficient in the OPE of the chiral primaries φ 2 and φ K .In the conventions (2.7) we set C L 2K = δ L K+2 and (2.12) simplifies to (1.3).Note that in (2.12) we are using the more explicit index notation K ∆ , . . . to keep track of the scaling dimension ∆ of the corresponding chiral primary fields.In this way it is apparent that equation (2.12) is an equation that relates 2-point functions at three different scaling dimensions: ∆ − 2, ∆, and ∆ + 2. Finally, ) to the Zamolodchikov metric of the theory up to an overall constant factor.Hence g 2 is conveniently related [7] to the S 4 partition function of the theory, Z S 4 , which is exactly computable as an (N − 1)-dimensional ordinary integral with the use of supersymmetric localization methods [9].The precise relation between g 2 and Z S 4 in our conventions is (2.13) This equation was first proven in [7].
In the special case of the SU (2) theory, which was the focus of Ref. [6], the chiral ring is freely generated by Tr[ϕ 2 ] only, and then (2.7), (2.12) reduce to the simple recursive set of differential equations where by definition g 0 = 1.These equations can be recast as a semi-infinite Toda chain by setting In the general SU (N ) case, due to the presence of additional chiral ring generators (2.1), the equations (2.12) are instead a complicated set of coupled differential equations for matrixvalued quantities.The appearance of inverse matrices introduces a high level of non-linearity.
In [6] we provided an explicit independent check that the matrix-valued equations (2.12) are satisfied at tree level for any SU (N ) N = 2 SCQCD theory.In this work we provide additional non-trivial independent evidence for the validity of the tt * equations, by computing the first quantum corrections which arise at 3-loops in perturbation theory.
It is clear that a single datum, like the S 4 partition function, is not enough to obtain a full solution of the tt * equations in the SU (N ) theory with N > 2. For example, since the scaling dimensions of the fields appearing in (2.12) are related by an increment of 2, the equations for chiral primaries of even scaling dimension are decoupled from the equations of the chiral primaries of odd scaling dimension.Hence, separate data are needed to solve the tt * equations for the chiral primaries of odd scaling dimension.Moreover, even within the sector of even or odd scaling dimensions, the pattern of increasing degeneracies does not admit an obvious recursive solution as in the simple SU (2) case.In what follows, we propose a surprising reduction of this problem.
At a second stage, we examine how this structure is modified at finite coupling.We argue that the tt * equations (2.12) -seen abstractly as a set of coupled differential equations -admit a class of solutions, in which different chiral primaries in the above basis remain orthogonal for all values of the coupling.This class does not necessarily include the most general solution of the tt * equations and a priori it is not clear whether it includes the physically relevant solution that we seek in the context of the gauge theory.Perturbative evidence in favor of the physical relevance of this restricted class is provided in section 5.

The chiral ring at tree level
The basis of chiral primaries generated by the single-trace operators (2.1) is particularly convenient because the structure constants are very simple -see equation (2.7).Nevertheless, it is straightforward to check that the 2-point functions in this basis are not diagonal.In turn, this means that the tt * equations in (2.12) do not reduce to simple recursive onedimensional chains of equations, but rather they constitute a set of coupled non-linear partial differential equations where various components of the 2-point functions mix nontrivially among themselves.As a result, it makes sense to look for a new basis that diagonalizes the 2-point functions while preserving some of the simplicity of (2.7).Since the only structure constants that appear in the tt * equations are the ones that involve φ 2 , namely C L 2K , it will suffice to look for a basis where these structure constants remain diagonal, i.e. as matrices they have a single non-vanishing element at each row.
Along these lines let us consider first chiral primaries φ K with the defining OPE property These are chiral primaries where the most singular term in the OPE with φ 2 vanishes. 2enceforth, we will refer to these distinguished chiral primaries as "C 2 -primaries".We can construct generic chiral primaries by acting repeatedly on the C 2 -primaries with As an obvious benefit, the structure constants C L 2K are manifestly diagonal in this basis.Hence, for our purposes we would only need to show that the 2-point functions at tree level are diagonal as well.For starters, let us show that In any basis, N = 2 chiral primaries exhibit the OPEs Hence, as we implied already, there are two natural operators acting on the space of chiral primaries Put differently, if we have a chiral primary φ = v K φ K , where v K is an arbitrary vector, then C 2 • φ is the chiral primary that appears in the OPE of φ with φ 2 , and C † 2 • φ is the chiral primary that appears in the OPE of φ with φ 2 , that is In particular, the operator C 2 raises the R-charge of an operator by 4 and the conformal dimension by 2, while C † 2 lowers them by the same amount.We will now show that C † 2 is the adjoint of C 2 with respect to the metric defined by the 2-point functions.Consider the 3-point function where α is a constant and we used ∆ = ∆ + 2 due to R-charge conservation.In the limit while in the limit The combination of equations (3.11), (3.12) verifies the advertised statement: C † 2 is indeed the adjoint of C 2 with respect to the metric induced by the 2-point functions.
There is a second crucial property of the operators C 2 , C † 2 .Their commutator acts as follows The combination [C 2 , C 2 ] Q K satisfies at tree level a nice combinatorial identity that was proven in appendix C of [4] [ Hence, when we plug this identity into (3.13),we find This means that we can regard C 2 and C † 2 as creation and annihilation operators respectively.In other words, we can decompose the space of chiral primaries in terms of representations of this algebra.We start from 'highest weight chiral primaries' annihilated by C † 2 (the C 2primaries), and we build the space of states by acting with C 2 (multiplying by φ 2 , as in (3.2)).Then, the resulting representations must necessarily be orthogonal, i.e. they diagonalize the 2-point functions.At this stage this is easy to verify directly if, say, m < n and where in the last step we used repeatedly the commutator [C 2 , C † 2 ].In the special case m = n, i.e. when two chiral primaries φ, φ are degenerate, we obtain similarly the identity Consequently, if there is a degeneracy in the spectrum of C 2 -primaries, we can choose any orthogonal combination and the orthogonality will be preserved by the C 2 -descendants.As a result, the basis built on C 2 -primaries that is singled out in this section is not unique.At tree level any orthogonal combination of degenerate C 2 -primaries is equally acceptable for our purposes.It is nevertheless useful to keep this freedom in mind in the context of finite coupling effects, which will be discussed shortly.

tt * equations at tree level
It is rather straightforward to show that, in the basis defined in the previous section, the tree level tt * equations decouple into a collection of one-dimensional semi-infinite Toda chains.Indeed, starting with a given arbitrary C 2 -primary φ (0) , let us consider the subsequence of chiral primaries φ (n) = φ n 2 φ (0) .Since the 2-point functions are diagonal at tree level, we can focus on the components Inserting the results of the previous subsection into the tree level version of (2.12), it is easy to see that the G 2n satisfy which is very similar to the one-dimensional chain (2.14) of the SU (2) case (and can be recast as the semi-infinite Toda chain (2.15)).
In the special case, where φ (0) = 1 (the identity operator), the sequence φ (n) = φ n 2 is comprised of the same type of operators that constitute the N = 2 chiral ring in the SU (2) case.For those operators we have G 2n = g 2n , and equation (3.20) is exactly the same semiinfinite Toda chain that was encountered in the SU (2) case (2.14).In this situation the solution (3.24) simplifies to where (x) n is the Pochhammer symbol This relation was noticed empirically and conjectured to hold for the general SU (N ) theory in [4].Amusingly, a similar relation has been proven with direct methods some time ago in appendix A.4 of Ref. [10] for the U (N ) N = 4 SYM theory.The SU (N ) and U (N ) formulae are identical with the suggestive substitution of N 2 − 1 with N 2 (the dimension of the gauge group) inside the Pochhammer symbol (3.25).The formula (3.24) is an interesting generalization to arbitrary N = 2 chiral primary operators.It is equally applicable to chiral primary operators in SU (N ) N = 4 SYM theory, where the tree level 2-and 3-point functions do not receive quantum corrections.

tt * equations at finite coupling: a no-mixing ansatz
Having shown at tree level that the tt * equations decouple into a sequence of independent Toda chains, it is natural to ask if a similar decoupling continues to hold when quantum corrections are taken into account.A sufficient condition for this effect is the requirement that the full non-perturbative 2-point functions remain diagonal in at least one of the bases constructed in the previous subsections, call it φK (recall that the previous tree level construction of the bases based on C 2 -primaries was not unique).This requirement is mathematically consistent from the point of view of the tt * equations.Notice that what we are postulating here is essentially the ability to diagonalise the exact 2-point functions within the holomorphic gauge.
From the point of view of the gauge theory the no-mixing condition postulated by this ansatz appears to be a new non-renormalization theorem in a four-dimensional N = 2 theory.Notice that unlike the non-renormalization theorem in N = 4 SYM [11][12][13][14][15][16][17][18][19][20], this theorem would not fix completely the moduli-dependence of correlation functions in the chiral ring.So far we have not been able to prove it using superconformal Ward identities.If true, this theorem would lead to several non-trivial consequences, which are discussed in detail in the next section.For instance, it would imply geometrically that the gauge connection of the holomorphic chiral primary vector bundles on the N = 2 superconformal manifold are reducible.
In appendix B we also formulate the no-mixing condition in terms of the original multitrace basis φ K of equation (2.2).In that basis, the non-renormalization condition translates into a statement about the coupling constant independence of appropriate ratios of 2-point functions.
In section 5 we put the above ansatz to the test in perturbation theory by computing the first non-trivial quantum corrections to several 2-point functions of chiral primary operators in SU (3) and SU (4) SCQCD.We proceed as high in scaling dimension and gauge group rank as possible, given our current computational limitations with the complicated combinatoric structures at 3-loops.In all cases, we verify the no-mixing ansatz: the 2-point functions remain diagonal, and the decoupled Toda equations (3.20) are explicitly verified.This evidence seems to be highly suggestive.

Implications of the decoupling
In this section, we explore some of the implications of the conjectured no-mixing condition and the related decoupling of the tt * equations.First, we show that, similar to the SU (2) case [4], we can use the decoupled Toda chains to determine exact correlation functions in the N = 2 chiral ring of the SU (N ) theory from a single datum in each decoupled subsector.In particular, we show that the subsequence based on the identity operator is exactly solvable using current knowledge from supersymmetric localization.With the same data we also obtain predictions for the exact form of certain extremal correlators that involve only single-trace operators, e.g.certain single-trace 3-point functions.
Second, we examine the geometric interpretation of this decoupling and show that it implies that the holonomy group on the space of chiral primaries is restricted -assuming full decoupling the holonomy group is a product of abelian factors.
Finally, we present other implications of the no-mixing ansatz on general (not necessarily extremal) integrated correlation functions.
Assuming the no-mixing ansatz of subsection 3.3 this is now an equation that holds nonperturbatively in the SU (N ) theory at finite coupling.The solution is determined recursively from the 2-point function of the C 2 -primary operator φ (0) under consideration and the 2-point function of the unique ∆ = 2 chiral primary φ 2 which is, up to a convention-dependent numerical coefficient, the Zamolodchikov metric.Different choices of the operator φ (0) sample different subsectors of the N = 2 chiral ring and correspond to different solutions of the recursive equations (4.1).
As we noted already in section 2, the Zamolodchikov metric can be obtained from the S 4 partition function using supersymmetric localization [7,9].In the SU (N ) N = 2 SCQCD theory the S 4 partition function can be written as an (N − 1)-dimensional ordinary integral.We refer the reader to the original references for explicit formulae.
To the best of our knowledge, it is not currently known how to compute the general G 0 exactly as a function of the moduli for arbitrary C 2 -primaries.A notable exception is the main subsequence defined by the identity operator, φ (0) = 1.As we pointed out already in subsection 3.2, in this case the 2-point functions G 2n ≡ g 2n satisfy equation (2.14), so the analysis of [4,6] can be repeated almost without changes.The only difference is that the starting point g 2 must be computed from the S 4 partition function for the gauge group SU (N ) instead of SU (2).

Single-trace extremal correlation functions and large-N limits
According to our conjecture, current knowledge of the Zamolodchikov metric also gives exact predictions for certain extremal correlation functions that involve only single-trace chiral primaries.Correlation functions of single-trace operators have an obvious interest in large-N limits.
As an illustrating example, let us consider first such a 3-point function of single-trace operators in the SU (4) theory.It will be shown in the next section that the only C 2 -primary at scaling dimension ∆ = 4 is Our conjecture implies in particular that Moreover, we know that with g 4 being determined from g 2 and the Toda equation as The result (4.8) has the following straightforward generalization.From the analysis of Ref. [4] we can deduce, using superconformal Ward identities, the identity (4.9) The extremal (k+1)-point function in question is independent of the insertions of the operators and equal to a 2-point function for two operators at scaling dimension 2k.Following an argument similar to the one of the previous paragraphs, or equivalently the non-renormalization identities of appendix B, we obtain where the prefactor is evaluated at tree level, and g 2n is determined from the S 4 partition function and the chain (2.14).
It would be interesting to study the behavior of such single-trace correlators further in the large-N limit and explore possible implications in related applications of the AdS/CFT correspondence.We note in passing that, as a simple check of our formalism and the tt * equations (2.12) in the large-N limit, one can easily verify that large-N factorization is an automatic solution of the tt * equations.
More along the lines of the large-N limit, the recent work [21] studied in supergravity the structure of the moduli space of certain supersymmetric AdS 5 vacua, which have the right amount of supersymmetry to be the holographic duals of 4d N = 2 SCFTs.If these theories have a holographic dual, then the moduli space of vacua in supergravity correspond to the conformal manifold of the dual SCFT.It would be interesting to investigate the form of the Zamolodchikov metric in the large-N limit directly from the gauge theory and to compare it with the supergravity results of [21].

Reducible chiral primary bundles
Another consequence of the existence of a holomorphic basis φK that diagonalizes the 2-point functions non-perturbatively is that the vector bundles of chiral primaries are reducible.It is readily seen from equations (2.10)- (2.11), that the connection Â is diagonal in the basis of the operators φK .Consequently, if there are D chiral primaries of scaling dimension ∆ (at arbitrary ∆), the holonomy will be restricted to the subgroup U (1) D ⊂ U (D).A reducible holonomy is a non-trivial condition for the geometry of the chiral primary vector bundles over the superconformal manifold.
The strong version of the no-mixing conjecture proposed in subsection 3.3 states that the 2-point functions are fully diagonalizable non-perturbatively in a holomorphic basis, hence the connection and the associated holonomy are fully reducible.Notice that full reducibility is consistent with the operator product structure that allows us to multiply sections from two chiral primary vector bundles to obtain a section on a third chiral primary vector bundle at the sum of scaling dimensions.Currently we have not excluded the possibility of a consistent weaker version of the nomixing conjecture, where the holonomy is partially reducible to a subgroup that is a product of abelian and non-abelian factors.In the next section, where we provide direct evidence for decoupling in perturbation theory, we verify that 2-point functions do not mix at the quantum level.In all the cases that we have analyzed so far, the degenerate operators are C 2 -descendants of primaries at different scaling dimensions.Interesting subtleties, with potential non-abelian holonomies, could seemingly appear in situations with more than one degenerate C 2 -primary operators.Recall that this was precisely the origin of the non-uniqueness of the basis constructed from the C 2 -algebra in section 3. It is clear that we can build two independent C 2 -primary combinations out of the operators in this list.At the moment, we cannot exclude the possibility that there is no constant linear combination of these two C 2 -primaries that keeps them orthogonal at finite coupling.
Verifying what actually happens would require a perturbative computation at more than 3 loops, which lies beyond our current computational power.Therefore, we cannot currently provide decisive evidence that favors a U (1) 4 holonomy compared to a U (1) 2 ×U (2) holonomy in this sector.

Other implications
The reducibility of the connection has further implications, even for non-extremal correlation functions in the N = 2 chiral ring.Consider the general (n + n)-point function in the N = 2 chiral ring in the diagonal hatted basis φK where the total R-charge of the insertions vanishes.The covariant derivative of this correlation function with respect to the complexified gauge coupling τ expresses by definition [2,4] the renormalized integrated (n + n + 1)-point function where in the last step we assumed the full reducibility of the connection.
A characteristic example of the general relation (4.15) is the case of the covariant derivative of the 3-point function Direct computation of the RHS of equation (4.15) in this case implies the vanishing of the integrated 4-point function Since there is no obvious symmetry reason for this identity, it would be interesting to obtain it with an independent derivation.We suspect that such a derivation might be a useful step towards the ultimate proof of the no-mixing conjecture.

Checks in perturbation theory
In this section we compute the first non-trivial quantum corrections to the 2-point functions of chiral primaries in certain examples in SU (N ) SCQCD.The first non-trivial correction appears diagrammatically at 3-loops.In all examples we find evidence that the connection on the space of chiral primaries is indeed reducible in accordance with the no-mixing proposal of section 3.3.More specifically, using the general 3-loop perturbative formula of appendix A, (A.32), we compute the perturbative matrix of 2-point functions up to conformal dimension ∆ = 8 for SU (3) and ∆ = 6 for SU (4).The explicit computation was performed with Mathematica.We report only these cases at this stage, because as we increase the rank N and the scaling dimension ∆ of the operators, the combinatorics of the general formula (A.32) quickly render the computation slow and impractical.( The first scaling dimension with non-trivial degeneracy is ∆ = 6, where we have the operators Notice that in order to determine whether or not the tt * equations decouple, we need to study 2-point functions up to level 8. Applying the formulae and normalization conventions of appendix A, we find the following results (5.9) The 2 × 2 matrix G 6 is written in the basis Tr[ϕ 2 ] 3 , Tr[ϕ 3 ] 2 , while G 8 is written in the basis Tr[ϕ 2 ] 4 , Tr[ϕ 2 ]Tr[ϕ 3 ] 2 .It is manifest that this basis does not diagonalize the 2-point functions, not even at tree-level.As explained in previous sections, we can diagonalize the 2point functions by constructing the C 2 -primaries.Tr[ϕ 3 ] 2 , in particular, is not a C 2 -primary, as can be easily seen from the tree-level OPEs ) It is then easy to take an appropriate linear combination of the two chiral primaries of dimension 6 that is annihilated by C † 2 .The appropriate bases at scaling dimensions 6 and 8 are then given by the operators where it is easily checked that C † 2 • φ 6 = 0.In the new basis, the 2-point functions become diagonal even when we include the first non-trivial quantum corrections verifying at this order the no-mixing conjecture.
It is also easy to check that the correlators satisfy the appropriate Toda chains (3.20), as explained in the previous sections.Applying the formulae of appendix A, we find the 2-point functions As before, we can find a constant linear rotation that diagonalizes the 2-point functions at tree level by finding the appropriate C 2 -primary combinations.The new basis is given by the operators Once again the no-mixing conjecture is verified.It is also easy to verify that the diagonal components of the above 2-point functions obey (3.20).

Outlook
The observations in this paper suggest the existence of a new interesting class of nonrenormalization theorems in four-dimensional N = 2 superconformal field theories.It would be important to prove these theorems in the SU (N ) N = 2 SCQCD theories, and to clarify whether the holonomy of the chiral primary vector bundles is fully or partially reducible.
We emphasized that full reducibility is a consistent ansatz from the point of view of the tt * equations, which reduces them to an independent set of semi-infinite Toda chains.The nonperturbative solution of the 2-point functions in each of these chains requires a single external datum.It would be interesting to explore techniques that will allow the exact computation of these data generalizing the success of supersymmetric localization on the four-sphere for the Zamolodchikov metric.
We would also like to highlight the efficiency of our results already at tree level.The tree level formulae derived in this paper are also applicable in the same form in the context of chiral primaries in N = 4 SYM theory.
In conclusion, in this paper we have seen that the study of the tt * equations is a powerful guide towards new exact results in four-dimensional quantum field theories.It would be extremely interesting to study the solution of the tt * equations in more general classes of N = 2 superconformal field theories, and to examine the possibility of more general nonrenormalization theorems in N = 2 theories.At face value, the appearance of such theorems in N = 2 theories is rather unexpected.Perhaps there are similar surprises in N = 1 theories as well.It would be interesting to explore this possibility.vector φ K 2 to find a linear combination orthogonal to φ K 1 , then combines φ K 1 and φ K 2 with a third vector φ K 3 to find a linear combination orthogonal to the previous two orthogonal vectors and so on and so forth.The choice of the order of the vectors φ K 1 , . . . in this prescription translates to different linear transformations between the original and the orthogonal bases.
In what follows, we adopt a slight variant of the Gram-Schmidt diagonalization procedure that reproduces the results of section 3.1 based on the C 2 -algebra (3.16).We single out the first vector φ K 1 in the multi-trace basis (2.2) as a chiral primary operator with the maximum number of φ 2 factors.Then, we perform a first linear transformation (φ (1) that leaves φ K 1 unchanged and transforms all the remaining vectors to set φ L φ K 1 = 0 for L = K 1 .A general matrix (M 1 ) L K with these properties takes the form where the indices L i refer to chiral primaries other than φ K 1 and the matrix elements µ L i L j are free for the moment.We will generate non-trivial entries µ L i L j sequentially, hence at this stage we adopt a scheme where µ L i L j = δ L i L j .Then, At the second step we single out a vector φ K 2 (other than φ K 1 ), with the next largest number of φ 2 factors, and repeat the same transformation in the subspace that excludes φ K 1 .Accordingly, we perform a second linear transformation (φ (2) KL we have denoted the 2-point function coefficients in the transformed basis φ K , g for K, L = K 1 .We continue in this fashion until the full diagonalization of the matrix g KL .The complete transformation matrix is M = M D R −1 . . .We encounter the same freedom in this process that we encountered also in section 3.1.When two operators have the same number of φ 2 factors it is unclear which order we should proceed in.
The no-mixing conjecture of section 3.3 postulates that the new section remains holomorphic, i.e. the linear transformation matrix M is a holomorphic function of the moduli KL = g KL .As expected by consistency, these relations are invariant under a holomorphic rescaling of the chiral primary fields.Moreover, they imply that by suitable holomorphic rescalings it is possible to adopt a more specific set of normalization conventions where all the 2-point function coefficients g KL are real.This is the real φ K basis that was aluded to in the main text and was explicit in the perturbative computations.In this basis the complex conjugate of the relations (B.12) implies that the ratio of 2-point functions is also τ -independent.As a result, in the real basis the ratios g (i−1) LK i g (i−1) K i K i are coupling constant independent and their value is fixed at tree level, namely g where we used the fact that the chiral primary φ 2 ∝ Tr[φ 2 ] is the single scaling dimension 2 operator and does not transform.Also, we used the notation M (∆) to denote the tranformation matrix at scaling dimension ∆.Notice that transformation matrices at two different scaling dimensions appear on the RHS of equation (B.14).It is obvious that the dimensionality of the transformation matrices remains the same or increases as the scaling dimension increases, i.e.D 2∆ ≤ D 2(∆+2) .
Let us phrase the precise conditions that guarantee that the transformed OPE coefficients ĈL 2K remain proportional to δ L K+2 .Part of our prescription above is to organize the rows and columns of the transformation matrix M (∆+2) so that its i-th row and column (for i ≤ D 2∆ ) refers to the chiral primary of the i-th row and column of M ∆ after the OPE with φ 2 .It is then straightforward to verify that , i = 1, 2, . . ., D R . (B.15) The above relations from the viewpoint of the tt * equations before the tranformation.Before we end this appendix, we would like to present a slighlty different description of the above relations from the point of the view of the tt * equations in the original basis (2.1).Returning to the tt * equations (1.3) in the multi-trace basis (2.2) we single out the first chiral primary φ K 1 (that takes part in the above diagonalization procedure, see eq. (B.1)), and consider the component of the equations with K = K 1 and L = K 1 It follows easily from the previous discussion that the non-renormalization equations (B.13) set the LHS (connection part) of this equation to zero.The RHS vanishes as a consequence of equations (B.15).Indeed, using these equations  and we can recast the RHS in the form

4. 1
Proposed recursive solution of the tt * equations at finite coupling We begin by rewriting (3.20) in the recursive form

M 2 M 1 (B. 7 )
where D R is the degeneracy of the chiral primary fields with U (1) R charge R. The chiral primaries in the new basis are φK = M L K φ L (B.8) and the matrix of 2-point function coefficients is diagonal φK φL = ĝKL = ĝKK δ KL .(B.9)

13 )
This equation is a statement of non-renormalization formulated in a local patch (based on the holomophic gauge) on the superconformal manifold of the N = 2 SCQCD theory.Diagonalization of C 2 .In the construction of section 3.1 based on the C 2 -algebra (3.16) the simultaneous diagonalization of the OPE coefficient C 2 was automatic.In the above language this property can be formulated as follows.In the original basis (2.2) the normalization conventions guarantee C L 2K = δ L K+2 .After the linear transformation (B.7) we obtain the new OPE coefficients ĈL 2K = (M (∆) ) S K C P 2S (M −1 (∆+2) ) L P = (M (∆)