Constraints on Chiral Operators in N=2 SCFTs

We study certain higher-spin chiral operators in N=2 superconformal field theories (SCFTs). In Lagrangian theories, or in theories related to Lagrangian theories by generalized Argyres-Seiberg-Gaiotto duality ("type A"theories in our classification), we give a simple superconformal representation theory proof that such operators do not exist. This argument is independent of the details of the superconformal index. We then use the index to show that if a theory is not of type A but has an N=2-preserving deformation by a relevant operator that takes it to a theory of this type in the infrared, the ultraviolet theory cannot have these higher-spin operators either. As an application of this discussion, we give a simple prescription to extract the 2a-c conformal anomaly directly from the superconformal index. We also comment on how this procedure works in the holographic limit.


Introduction
In the space of quantum field theories (QFTs), conformal field theories (CFTs) form a special subspace of enhanced symmetry. The resulting conformal symmetry gives rise to important simplifications. For instance, conformal invariance allows us to describe a CFT by a tightly constrained set of data: its spectrum and operator product expansion (OPE) coefficients.
While it is usually difficult to solve a given CFT (by which we mean to derive its spectrum and OPE coefficients), one can use general principles to restrict the space of allowed CFTs. For example, one can study the constraints imposed by associativity of the OPE and find bounds on operator dimensions and OPE coefficients; see, e.g., [1,2].
The restrictions on the space of superconformal field theories (SCFTs) are potentially even more powerful; see, e.g., [3,4] and references therein.
In this note, we describe new constraints (not derived from associativity of the OPE) on the operator spectra of three very broad (and overlapping) classes of four-dimensional N = 2 SCFTs: (A) Theories, T , that have Lagrangian descriptions or theories related to Lagrangian SCFTs by generalized Argyres-Seiberg-Gaiotto duality [5,6]: in the latter case, we gauge a global symmetry G of T in an exactly marginal fashion (adding additional matter or additional interacting SCFTs charged under the gauge group as necessary) and find a dual description in terms of a Lagrangian.
(C) N = 2 SCFTs admitting deformations by relevant couplings that trigger N = 2preserving renormalization group (RG) flows to Lagrangian theories (or, more generally, theories of type A) in the infrared (IR).
Although it is clear that many theories of type A are also theories of class S (the T N theories are a prototypical set of examples), we will not assume that this is the case in general.
Furthermore, as far as we are aware, all known N = 2 SCFTs (whether they are in class S or not) are of type B, with the exception of theories consisting of free hypermultiplets.
Note that one class of type B theories not of type A is the set of (G, G ′ ) (generalized) Argyres-Douglas (AD) SCFTs [7][8][9][10]. 1 Finally, the class of theories of type C is also very broad. For example, it includes all the (G, G ′ ) AD SCFTs. In fact, it is not immediately clear to us if there are any theories that are of type B but not of type A or of type C.
The main claims of our paper are: 1. Theories of type A and C do not have a certain class of higher-spin chiral (and antichiral) primaries. 2 2. If these operators are present in theories of type B, then they necessarily satisfy a non-trivial set of operator relations we will describe in detail. where we have a commutator or anti-commutator depending on whether the SU (2) 1 Lorentz spin, j 1 , is even or odd, and I is an SU (2) R index labeling the two sets of Poincaré supercharges. An interesting subset of the operators we study satisfy an additional con-  When j 1 = 0, the constraint in (1.2) implies that O is a free scalar satisfying the Klein-Gordon equation (it is the N = 2 primary for a multiplet that contains a free U (1) field strength). In non-abelian Lagrangian theories, with vector multiplets involving scalars Φ in the adjoint of the gauge group, the j 1 = 0 primaries that satisfy (1.1) but not (1.2) have the form O = Tr Φ k (with k ≥ 2). Hereafter we will refer to N = 2 primaries with j 1 > 0 satisfying (1.1) as "exotic chiral primaries." 3 In some sense it is surprising that exotic chiral primaries do not exist in such broad classes of theories. 4 Operators satisfying (1.1) and (1.2) with j 1 = 0 are ubiquitous in 2 An argument against the existence of such operators in certain theories of class S was given in [11]. 3 In the classification of [12], operators satisfying (1.1) and not (1.2) are denoted as E r(j 1 ,0) .
Operators satisfying both (1.1) and (1.2) are denoted by D 0(j 1 ,0) . 4 For example, in N = 1 theories it is common to find chiral primaries that have non-zero spin [13].
all known N = 2 SCFTs (with the exception of free hypermultiplet theories): their vevs parameterize the Coulomb branch, and so it might have been natural to imagine that exotic chiral operators describe some other geometrical aspects of theories of type A and C. In order to determine if such operators exist at all, it would be interesting to understand whether there are theories of type B that are not of type A or type C (or, if there are any theories outside of our classification), and, if so, whether one can use the operator relations we derive to prove that these operators do not exist in such theories. One might also attempt to study such operators using the conformal bootstrap.
Let us briefly describe the plan of the rest of the paper. In the next section, we discuss certain short multiplets of the four-dimensional N = 2 superconformal algebra, highlighting our multiplets of interest. We then give a general argument that theories of type A do not have exotic chiral operators. Our proof relies only on superconformal representation theory (and the dynamical assumption of a duality with a Lagrangian theory). In order to set the stage for our discussion of theories of type B and C, we then introduce the superconformal index and a particularly useful limit of the index-the "Coulomb branch" or "chiral U (1) R " index-that captures only contributions from operators of the general type defined in (1.1) and (1.2). We then study a special limit of the chiral U (1) R index and use the structure of the resulting divergences to argue that if theories of type B have exotic chiral operators, then they necessarily satisfy certain operator relations. In the next section we move on to theories of type C and show that exotic chiral operators do not exist in such theories. Finally, we conclude with an application of our results to an index-based computation of the 2a − c conformal anomaly.
2. N = 2 SCFT Generalities and the Chiral U (1) R Sector Any four-dimensional N = 2 SCFT, T , has a conserved and traceless stress tensor along with two conserved supercurrents and conserved currents for the SU (2) R × U (1) R superconformal R symmetry. Such theories typically admit a zoo of short representations (note that we will refer to "semi-short" representations as short representations) with N = 2 primaries that are annihilated by certain combinations of the Poincaré supercharges [12,[14][15][16]. One universal short representation that is present in any T is the multiplet J that contains the aforementioned stress tensor, supercurrents, and U (1) R × SU (2) R currents. In the notation of [12,11,3], 5 the J multiplet is of type C 0(0,0) (it has zero U (1) R × 5 We provide a short summary of these conventions in Appendix A.
SU (2) R charge, is a Lorentz scalar, and has scaling dimension two). Superconformal representation theory allows various short higher spin cousins of this multiplet, C 0(j 1 ,j 2 ) , that contain conserved higher spin currents and are therefore not present unless T has a sector of free fields. 6 If T has a continuous flavor symmetry, G, (i.e., a continuous symmetry that commutes with the N = 2 superconformal algebra), then the corresponding spin one symmetry current is a descendant in a short multiplet of type B 1 . 7 The primary, L IJ G , is a dimension two scalar of SU (2) R spin one and U (1) R charge zero satisfying where the σ A are the SU (2) R Pauli matrices, we see from (2.1) that L 3 G is the real moment map corresponding to G. This fact will be useful for us in our discussion below.
The main focus of our paper is on the set of short multiplets of the N = 2 superconformal algebra whose primaries are annihilated by both sets of anti-chiral Poincaré supercharges. Indeed, such operators are necessarily of the type we discussed in (1.1) where R is the SU (2) R spin, E is the scaling dimension, r is the U (1) R charge, and the implication in (2.2) follows from unitarity. If the operator does not satisfy any additional shortening conditions, it is referred to as being of type E r(j 1 ,0) . Note that the inequality in (2.2) is saturated if and only if the operator also satisfies (1.2)-such operators are referred to as being of type D 0(j 1 ,0) . In a slight abuse of terminology, we will refer to the operators in (2.2) as constituting the "chiral U (1) R " sector of the theory. 8 We will be particularly 6 All Lagrangian N = 2 SCFTs have a special limit of measure zero on their conformal manifolds where free fields emerge, but away from this limit the various C 0(j 1 ,j 2 ) multiplets combine according to the rules given in [12] to become long multiplets with non-zero anomalous dimension. 7 Note that while Lagrangian N = 2 SCFTs necessarily have flavor symmetries, the same is not true in general for interacting theories. For example, the original AD theory in [8] does not have any flavor symmetry. 8 One sometimes refers to primaries that have j 2 = 0 as chiral primaries. Here we only refer to operators annihilated by the full set of anti-chiral supercharges as being in the chiral U (1) R sector of the theory. Note that the superconformal algebra allows other short multiplets charged under U (1) R but not under SU (2) R that have j 2 = 0 but are not annihilated by the anti-chiral supercharges (e.g., the C 0,r(j 1 ,0) multiplets).
interested in understanding the "exotic" chiral U (1) R sector of the theory-namely, those operators satisfying (2.2) with j 1 > 0.
The U (1) R chiral sector operators have two special properties that will be important in what follows: (i) The E r(j 1 ,0) and D 0(j 1 ,0) multiplets cannot combine with other short representations of the N = 2 superconformal algebra to form long representations.
(ii) The E r(j 1 ,0) and D 0(j 1 ,0) operators are singlets under any flavor symmetries of the theory.
The first property follows from the analysis in [12]. We will prove the second property by contradiction. To that end, let us assume that the U (1) R chiral sector operators are charged under some flavor symmetry. Then, since the E r(j 1 ,0) and D 0(j 1 ,0) operators are chiral with respect to an N = 1 ⊂ N = 2 sub-algebra, it must be the case that the chiral-anti-chiral OPE includes terms of the type where a runs over the full set of O with the same N = 2 superconformal quantum numbers, A is the real moment map corresponding to a flavor symmetry generator labeled by A (see the discussion below (2.1)), and The t B matrix in (2.4) is the representation matrix that O (a)α 1 ···α 2j transforms under, when acted on by the symmetry generator labeled by B. It appears in the OPE where j µ, B is the symmetry current corresponding to the symmetry generator B. Now, since L 3 A is part of an SU (2) R triplet and O (a)α 1 ···α 2j is an SU (2) R singlet, we see that T A ab = 0. As a result, from (2.4) and unitarity, (t B ) b a = 0, and so the chiral U (1) R sector operators cannot be charged under any flavor symmetries.

Constraints on Exotic Chiral Operators
In this section we study constraints on the exotic chiral operators we introduced above.
In the first subsection, we apply our previous results to conclude that theories of type A do not have such operators. In the next subsection we introduce the superconformal index and a particularly useful limit of it in order to set the stage for our discussion of theories of type B and C.

Theories of Type A
For Lagrangian SCFTs it is straightforward to prove that there are no exotic chiral operators since such theories have exactly marginal gauge couplings that can be tuned to zero. By property (i) of the previous section, we see that superconformal representation theory prevents the E r(j 1 ,0) and D 0(j 1 ,0) operators from pairing up to form long representations, and so it suffices to study the free theory. At zero coupling, we immediately see that the only possible chiral operators have the form O = Tr Φ k , where Φ is a chiral adjoint, and so j 1 = 0.
Let us now consider a non-Lagrangian theory, T , that we can deform to a Lagrangian one. We assume that such a deformation does not induce an RG flow, and so it must proceed by an exactly marginal gauging of some global symmetry group of T , G 0 ⊂ G T (see, for example, the appendix of [17]). In order to ensure that the gauging is exactly marginal, we may have to add another SCFT, T ′ , with G 0 ≃ G ′ 0 ⊂ G T ′ (T ′ may be a collection of decoupled SCFTs; note that T ′ may just be a set of free hypermultiplets). In the spirit of [5,6] we may then be able to find a duality where L is a Lagrangian theory, and we have identified the gauge group G with the diagonal subgroup of G 0 × G ′ 0 . In general, the remaining flavor symmetry, G L , is just the maximal In writing (3.2), we have used the fact that flavor symmetries are non-anomalous in N = 2 theories since the moment maps satisfy (see, e.g. [18,17]) Let us now study the chiral U (1) R sector of the theories appearing in (3.1). In particular, let us take the limit of zero gauge coupling for the G gauge group, τ G → i∞. In this limit, it is clear that the only effect of the gauging on T and T ′ is to eliminate non gauge-invariant operators. However, by property (ii) in the previous section, we see that the E r(j 1 ,0) and D 0(j 1 ,0) operators are not charged under G. Therefore, such operators persist on the right hand side of (3.1), and we must have where the implication that there are no exotic chiral primaries in T or T ′ follows from the fact that there are no exotic chiral primaries in the Lagrangian sectors L and G.
As a simple example, we can consider the original duality of [5]. In that case, L is the SU (3) theory with six flavors, T is the E 6 theory of [19], G = SU (2), and T ′ is a collection of two hypermultiplets (so G T = E 6 , G T ′ = Sp (2), and G L = U (1) × SU (6)).
By our above reasoning, we see that the chiral U (1) R sector of the pure E 6 theory (without gauging an SU (2) ⊂ E 6 ) is fully described by a non-exotic dimension three operator, O, dual to the Tr Φ 3 operator in L.
Let us again emphasize that while many constructions of the type (3.1) are in class S, we do not need to assume that this is the case more generally for our above logic to hold.
Finally, let us mention that there are large classes of theories in which (3.1) does not hold because (among other possibilities) there are no appropriate symmetries to gauge (such theories are of type B or C but not A). One example is the original AD theory considered in [8] (and many of the generalizations in [10]). To constrain the existence of chiral exotics in such theories, we will need some more powerful tools which we introduce in the next subsection.

The N = 2 Superconformal Index
The superconformal index [20,21]  (other choices lead to equivalent constructions of the index) and define the index to be 9 where p, q, t are complex fugacities, j 1,2 are Cartans for the SU (2) 1,2 isometry, r is the superconformal U (1) R charge, and R is the Cartan of SU (2) R . In (3.5), the trace is understood as being over the full Hilbert space of the theory. However, the only contributions come from operators in R 4 (or, equivalently, the corresponding states on S 3 ) that are annihilated by Q 2− and S 2− ; such operators (or states) necessarily have E = 2j 2 + 2R − r. 10 In a Lagrangian theory, the gauge and matter sectors contribute to the index as follows where the subscripts "V " and " H 2 " refer to contributions from an N = 2 vector multiplet and half hypermultiplet respectively; the superscript "s.l." stands for "single letter" and indicates that we should appropriately exponentiate the corresponding contribution to the index in order to compute the contributions from all short multiplets built out of products of V , H, and their derivatives. For simplicity, we will specialize our Lagrangian discussion to a U (1) N gauge theory with free (uncharged) hypermultiplets (the non-abelian and charged-matter generalizations are straightforward) and we will drop all flavor fugacities.
In this case, we find (3.7) 9 We will use the same letter for both the Cartan generators and the associated charges, hoping that this will not cause confusion. We review our conventions for the superconformal algebra in Appendix A. 10 Note that we can often add various flavor fugacities to (3.5), but we will not do so in what follows since our primary interest is in operators that we have seen are flavor neutral.
where the product is over all the abelian U (1)'s and matter fields, and "P.E." stands for the "plethystic exponential." Via the state-operator map, one can also understand the index as being equivalentmodulo regularization scheme-dependent pre-factors-to a partition function for the theory on S 1 ×S 3 with twisted boundary conditions for the various fields according to their charges [20][21][22][23]. 11 By assigning a chemical potential to each fugacity, x = e −βv x , we can write (3.5) as The corresponding Euclidean path integral, schematically [DΦ]e −S E [Φ,D 4 Φ] , then counts the same 1 8 -BPS states as the index. This will provide us with an interesting alternative perspective below. Note that the index is by construction independent of continuous couplings. As a result, the evaluation of the twisted partition function for the N = 1 chiral multiplets inside the N = 2 vector or hypermultiplets can be carried out in the free limit, where it reduces to one-loop determinant factors for the scalars and fermions [21,23,25]. 12 It is interesting to point out that there is yet another, more geometric, interpretation of the index as corresponding (modulo pre-factors) to the partition function of the theory placed on a Hopf surface, M 4 . The complex structure and holomorphic bundle moduli of M 4 are directly related to the fugacities of the index [26] (see also [27]). 11 In line with [22], the version of the theory on S 1 × S 3 has additional mass terms, coupling the scalars to the background curvature. 12 For the N = 1 vector multiplet contributions one needs to take into account gauge field zero-modes that have to be dealt with exactly.

The Chiral U (1) R Limit
Since we will be interested in analyzing theories of type B-i.e., theories with a Coulomb branch, M C , in R 4 -it is useful for us to ask how we can see such a moduli space from the index (3.7) in the simple case above (see also the general discussion of moduli spaces and the index in [28]). In this example, M C is parameterized by the U (1) Rcharged vector multiplet scalars, Φ a . Therefore, away from the origin of M C , the U (1) R symmetry is broken. As a result, if we wish to study the effects of such vacua, we should take a limit of fugacities pq = t in (3.5) so that the U (1) R charge does not enter the index.
We define Using this index to evaluate the single letter contributions for vector multiplets and halfhypermultiplets, we find that (3.6) becomes The first contribution can be understood as coming from a Φ a scalar with no derivatives acting on it. Note that the hypermultiplet contributions vanish, and our result depends only on operators in the chiral U (1) R sector; hence the subscript " U (1) R " in (3.11). Plugging these results into (3.7), we find unsupressed contributions from all powers of Φ a This divergence describes the opening up of a Coulomb branch.
In fact, this discussion is more general. Indeed, we find the following single letter contributions to the index for any N = 2 SCFT in the above limit: is a character for the spin j 1 representation of SU (2) 1 .
The subscripts denote contributions from the various short representations of the N = 2 superconformal algebra. We review the various properties of these different multiplets in Appendix A. However, suffice it to say that the only contributions to the index in (3.13) come from the chiral U (1) R -sector operators E r(j 1 ,0) and D 0(j 1 ,0) we introduced above.
The divergence of the index in the above limit, as well as the absence of hypermultiplet contributions, have simple physical interpretations in terms of the path integral description.
Note that this construction is useful for theories with Lagrangian limits. In order to proceed, let us first remind the reader that in the for generic values of chemical potentials, the scalars Φ a typically have a mass (since they couple to curvature). Due to this mass term, the theory on curved space has a unique vacuum, as opposed to the moduli space of vacua enjoyed by the theory in flat space.
However, for the limit described in (3.10), we will find bosonic zero modes in the Φ a .
The Lagrangian at quadratic order in the fields includes the following terms for the vector multiplet scalars where ∆ S 3 is the Laplacian on the three-sphere and the factor of 1 comes from the conformal coupling to the curvature of the S 3 , in units of its radius [22]. One can read off the scalar mass from (3.14): Further focusing on the constant mode (s-wave) for the vector multiplet scalars on S 1 ×S 3 , such that j 1 = j 2 = 0, r = −1 and R = 0, produces Clearly, in the limit where pq = t, or equivalently v p + v q − v t = 0 with β fixed, this mass goes to zero and the Coulomb branch opens up. Hence, the divergence in the I U (1) R index for pq = t, (3.12), can be understood as the result of integrating over this zero mode in field space. This insight can be applied more generally. Appropriately tuning the various fugacities can lead to opening up different flat directions; see [28] for a baryonic Higgs-branch example.
We can similarly interpret the absence of hypermultiplet contributions. In the pq = t limit, the bosonic and fermionic contributions to the single-letter hypermultiplet index (3.11) canceled out. In the path integral description, this is suggestive of an additional supersymmetric cancelation between chiral scalars and fermions. It can indeed be seen that is, in the limit of interest the partition function counts 1 4 -BPS states annihilated by both Q 2− and Q 1+ , thus excluding the hypermultiplets.

The Chiral U (1) R Limit
We have seen that the limit of the index (3.10) captures contributions only from Coulomb-branch operators. However, before proceeding to analyze the chiral U (1) R sector operators of interest, we would like to find a better-behaved (finite) limit of the index where most of the theory decouples and only chiral U (1) R sector operators contribute. It is crucial that the exotic chiral operators contribute as well, since we will use their putative contributions to argue that they must be absent in theories of type C and constrained in theories of type B. This limit will also immediately shed light on the order of the divergence in (3.7) and the nature of the divergent contribution in (3.11).
To that end, we consider the limit described in [11] where we neglect corrections of order O(|p|, |q|, |t|) to the various quantities we study. Here, z and w correspond to U (1) R and j 1 fugacities respectively. In this limit, we have with contributions only from states satisfying E = −r and j 2 = −R. As we will see below, there are no SU (2) R -charged states contributing and so we can take z to be a U (1) R fugacity. 13 There this limit was written as We will refer to this limit as the "chiral U (1) R " limit of the index. For a general SCFT we have I s.l.
As a result, the divergence in (3.7) is regulated and (3.23) In particular, taking the additional limit z → 1, once again opens up the Coulomb branch.
We see that the divergence in this limit corresponds to a pole of order N (the complex dimension of the Coulomb branch, dimM C ) due to contributions from the N different Φ a scalars.
For a general Lagrangian SCFT, we have When we write the right hand side of the first equation in (3.25), we work modulo descendants. We can include such constraints via a meromorphic factor, I(z) If there are no additional relations among the f i (O a ), then one can capture the full effect of the constraints via a "wrong-sign" contribution of the form In this case, each constraint contributes a zero to the index in the limit z → 1, and we interpret it as "lifting" a direction of the moduli space. If the number of O a , N a , is larger than the number of f i , N i , then we find a pole of order N a − N i as z → 1, and we interpret the order of this pole as the complex dimension of the U (1) R moduli space. 14 If our theory has exotic chiral operators, we get contributions to the index of the form where the first line in (3.28) comes from N j 1 spin j 1 chiral bosons, and the second line in (3.28) comes from N j ′ 1 spin j ′ 1 exotic chiral fermions. We can implement higher-spin constraints via a meromorphic function, I(z, w), and find the general result (3.29) 14 As an aside, we observe that for type A theories, the limit of the index we are studying is equivalent to the Hilbert series of the Coulomb branch. Various examples of Coulomb branch Hilbert series have been studied in [29]. Note that this relation between the index and the Hilbert series seems to be more robust than the relation between the Hall-Littlewood index and the Hilbert series of the Higgs branch (see e.g., [30]), which only holds for special genus zero classes of type A theories [11].
Before proceeding, let us note that I(z, w) must be such that we have where B and F run over the full set of bosonic and fermionic chiral operators that contribute to the index taking into account all chiral operator relations. Crucially for us below, it follows that in any N = 2 SCFT In other words, the z → 1 limit of the chiral U (1) R index is a non-trivial function of w if and only if we have exotic chiral operators.
Let us next switch to the path integral interpretation. The limit (3.19) involved taking: We found that only vector multiplets contribute (finitely), while the hypermultiplets gave zero for all fields without the need for mutual cancelations. In terms of chemical potentials, and specializing to real and positive p, q, t, (3.32) can be recast into with v p ∼ O(1) and β ≫ 1 .

(3.33)
In order to proceed, consider and focus once again on the masses for the constant modes. Using (3.34) one has Then for the vector multiplet and hypermultiplet scalars, (Φ, Q), for which r = −1, R = 0 and r = 0, R = 1 2 respectively, we obtain Note that the vector multiplet scalars are parametrically lighter than the hypermultiplets.
When we set v z = 0 to get z = 1, we see that M 2 Φ,s = 0, and the Coulomb branch opens up.
For a theory in flat space, one can integrate down to an arbitrary momentum shell and obtain a two-derivative effective Lagrangian. However, for a theory on curved space one can never integrate out states at scales lower than the inverse radius of curvature and remain with an effective two-derivative theory. The reason is that there is no way to separate higher derivative terms in the effective Lagrangian from terms suppressed by the inverse radius of curvature [22]. This picture self-consistently reverts to the flat space intuition when the volume-and hence the radius of curvature-goes to infinity.
We will hence compare the masses (3.36) against the curvature, which for S 1 × S 3and in units of the S 3 radius-simply reads R ∝ 1. For some fixed 0 < v t < 4 this leads to a hierarchy of scales: 15 As a result, in this second limit of chemical potentials, the hypermultiplets can be integrated out and the index will once again only receive contributions from the vector multiplets. Since the index only depends on the parameter β (in units of the S 3 radius) and chemical potentials, and in particular is independent of all other continuous parameters including the RG scale [22], we can conclude that there will be no contributions from the hypermultiplets at any energy.
It may be useful to point out that a rescaling of both the S 1 and S 3 radii by κ, which does not affect the index since their ratio is kept fixed, will not disrupt the scale hierarchy.
Although the curvature changes as R → κ −2 R, the masses (3.36) are defined in units of the S 3 radius and will also scale by M 2 → κ −2 M 2 leaving (3.37) unchanged.

Theories of Type B
We now have the necessary tools to analyze theories of type B. Let us begin by discussing SU (N ) N = 2 superconformal quantum chromodynamics (SCQCD) in more detail. Although we can immediately conclude from the preceding discussion that this theory does not have exotic chiral operators, a careful analysis of SU (N ) SCQCD and its Coulomb branch furnishes some important intuition about the behavior of the index in more general theories. From (3.24) we have (3.38) In the limit z → 1, we find bosonic zero modes corresponding to the opening up of a Coulomb branch, and , are 16 Similar reasoning was used in [28] to study the effect of hypermultiplet zero modes on the superconformal index. respectively (they are part of a classification due to Kodaira [32]).

N = 2 Theory
The values of |W | −1 required to appropriately restrict the partition function are given in Table 1. These are precisely the inverse dimensions of the Coulomb branch operators in these theories, E −1 , and so they can also be understood as arising from the z → 1 17 One interesting check of this story is to note that the Coulomb branch of SU (2) SCQCD is given by the cone C/W SU(2) ≃ C/Z 2 and that therefore we should have |W SU(2) | −1 = 1/2 by our reasoning. Indeed, this result agrees with (3.38) and (3.39).
We see that this argument is quite general and so we expect that for any theory, T , of type B there is a pole as z → 1 of order the complex dimension of the Coulomb branch multiplying a term describing the geometry on M C . 18 Since the theories away from the origin of the Coulomb branch are Lagrangian (with the possible exception of theories living on certain complex co-dimension one or higher sub-manifolds of M C ), this term should be independent of w. Furthermore, we do not expect divergent contributions to the index coming purely from exotic operators since there is no supersymmetric moduli space associated with them. 19 17 This discussion also applies to the new rank one theories in [33], since these latter theories have the same Coulomb branch spectrum as the corresponding MN theories. 18 If the Coulomb branch consists of multiple sub-branches-potentially of different dimensionsthen we expect one such term for each sub-branch. 19 As we will see below, it is possible to relax this assumption at the cost of making some of our arguments more complicated.
In other words, we should have that where d C = dimM C (T ), κ is a constant, and the leading term on the right hand side describes the Lagrangian bulk of the Coulomb branch (if the theory has more than one Coulomb branch, then we have a similar term for each branch). The ellipsis in (3.40) contains various lower-order terms (including any contributions in w).
One immediate consequence of (3.40) is that if a theory of type B has exotic chiral operators, then for each exotic chiral operator, O Aα 1 ···α 2j A , there exist non-negative integers, N A,a,b , and dimensionless constants c b such that Let us now briefly discuss the lower-order divergences in the index as z → 1. A priori, it is possible that sub-leading divergences in 1 − z are w-dependent since there can be co-dimension one or higher sub-manifolds of M C that correspond to non-Lagrangian (or, more generally, to non-type A) theories. In the next section, however, we will rule out any w-dependence in the superconformal indices of theories that can flow to type A theories by turning on a relevant N = 2-preserving coupling.
From the above discussion, it is not too difficult to see that the superconformal index of a type B theory can be written as follows where a i ∈ {1, · · ·, d C , d C + 1, · · ·, d C + n} runs over the Coulomb branch generators, O a i , of dimension E a i (let us note again that we mean the non-nilpotent scalar generators of the U (1) R chiral ring-more precisely, a minimal subset of these generators that give all the infinite-order contributions to the U (1) R chiral ring). If there are Coulomb branches of various dimensions, then d C refers to the maximal dimensional Coulomb branch(es). Note that in type A theories, n = 0 and a i ∈ {1, · · ·, d C }.
In writing (3.42), we have factored out the infinite-order contributions from the Coulomb branch operators, and so the I are non-singular. Furthermore, we take the I to be symmetric in their indices, and if two indices take on the same value, then the corresponding I vanishes (so as not to over-count the contributions from Coulomb branch operators).
For completeness, let us examine the I in more detail. We define the I so that If we include the contribution of O in I a 1 ···a d C (z), then we find a corresponding term of the form where (a i 1 , · · · , a i k ) takes values in the d C ! (d C −k)!k! unordered k-tuples built out of {a 1 , · · · , a d C }. It follows from our discussion that I a 1 ···a d C (z) is independent of the superconformal U (1) R charge when we set z = 1 (the only dependence of (3.43) and (3.44) on the superconformal U (1) R is through the scaling dimension of the chiral operators). Furthermore, I a 1 ···a d C (1) is independent of whether we choose to include the contribution of O in I a 1 ···a d C (1) or not-i.e., the left hand sides of (3.43) and (3.44) are both equal to +1 when evaluated at z = 1 (this matching is guaranteed by the fact that removing a finite number of operator contributions cannot change the leading divergence of the index 20 ). In other words, we see that the various I functions that multiply the leading divergence in (3.42) are well-defined (finite and independent of the ambiguities discussed above) and universal (independent of the superconformal U (1) R ) at z = 1, with Note that the general form of I a 1 ···a d C (1), as a sum over positive contributions from contributing operators, is independent of which minimal subset of generators we choose to use in writing the infinite order contributions in (3.42).
Next, let us proceed to the order d C − 1 terms (i.e., the I in the second sum in (3.47) Clearly, (3.46) and (3.47) are both equal to χ j (w) at z = 1. Therefore, we find that the exotic operator contributions to I a 1 ···a d C −1 (1, w) are unambiguously given by where the n (j) are non-negative integers.
In particular, it is clear that the order d C − 1 contribution to the index depends on w if and only if this is true in the limit z → 1. Furthermore, if the order d C −1 contribution is independent of w, then the order d C − 2 contribution is independent of w if and only if this is true in the limit z → 1 (and so on sequentially down to the finite contributions). Finally, let us note that the divergences multiplying the various I depend on the superconformal U (1) R only through a positive overall factor (times (1−z) −n I) given by the inverse product of (minus) the U (1) R charges of the corresponding Coulomb branch operators.

Theories of Type C
In this subsection, we wish to discuss theories that flow to type A theories when we turn on a deformation by a relevant coupling. Since we want to preserve N = 2 invariance, we can turn on relevant deformations of the prepotential Note that we will not consider deformations by FI terms (in any case, the corresponding U (1) gauge factors are not UV-complete).
Both types of relevant deformations in (3.49) and (3.50) break the U (1) R symmetry of the UV SCFT (but preserve the SU (2) R symmetry). Therefore, if we wish to compute the supersymmetric index of the interpolating theory, we should turn off the fugacity corresponding to U (1) R , i.e., we should take z → 1. It is again useful to study the U (1) R limit of the index. In the UV we find that the index is I a 1 (z, w) + I(z, w) .

(3.51)
This formula for the index is very similar to the one in (3.42), but we allow for the leading term to be w-dependent since we do not assume that the U (1) R moduli space, M U(1) R , parameterized by the vevs of the (non-nilpotent) chiral generators, O a i , is a Coulomb branch (i.e., we allow for a fully interacting theory along the whole moduli space, although we do not know of any such examples). In this case, by reasoning similar to that used in the previous subsection, the leading I functions have the following general form at z = 1 where the n (j) are non-negative integers.
As we discussed in the previous subsection, the index in (3.51) will generally diverge as we take z → 1. However, as we mentioned above, the I functions are finite and do not depend on the (broken) superconformal U (1) R symmetry when we set z = 1. We can therefore associate a universal meaning to (3.52) at all length scales, including in the deep IR. A useful way to extract the I is to consider the following contour integrals around z = 1: We can then pick out w-dependent quantities in the A n+1 by defining Let us now compute the various A n+1 in the UV and compare them to the corresponding quantities in the IR. We begin with A d C (for n + 1 > d C , the A n+1 vanish) The superconformal U (1) R dependence of the above expression is contained in the inverse UV scaling dimensions of the O a i . In the IR we can then compute the same quantity 21 and find where E a i are the (generally different) scaling dimensions of the IR O a i operators. While it is usually true that A UV d C (w) = A IR d C (w), it is also clear that since the E a i , E a i > 0 and the I a 1 ···a d C have the form given in (3.52 . Suppose now that the IR theory is type A. In this case, we know that B IR j i ,d C = 0 for all j i = 0. Therefore, the same must hold in the UV. In other words, (3.52) is independent of w. Proceeding sequentially down the list of B UV,IR j i ,n+1 , we find that if the IR theory is of type A, then the UV theory has B UV j i ,n+1 = 0 for all n. In other words, the UV theory cannot have exotic operators. 22

Examples of type C theories
Let us now consider some particular cases of non-Lagrangian type C theories. For example, we can consider the N f = 2 AD theory discussed in [8]. This theory has a single relevant prepotential coupling. It corresponds to an operator, O, of dimension 4/3. At the superconformal point, the Seiberg-Witten curve is (3.57) Deforming the theory by δL ∼ d 4 θ u 2 O + h.c. is equivalent to deforming (3.57) as follows In the deep IR we recover (in addition to a decoupled U (1)) which is the curve of a free Lagrangian theory and is therefore of type A.
As a result, we conclude that the N f = 2 AD theory does not contain exotic chiral operators. More generally, we can take any of the infinitely many (G, G ′ ) AD SCFTs for G, G ′ = A n , D n , E n [34,9] and sequentially deform them to Lagrangian theories via relevant deformations (the theory in (3.57) is the (A 1 , A 3 ) theory). We explain this in some detail in Appendix B. From our above reasoning, we conclude that these theories cannot have exotic chiral operators. It would be interesting to understand the full set of theories that can be deformed to type A ones-we suspect that this is a very large class. 22 In the discussion so far, we have assumed that the exotic chiral operators give finite contributions to the superconformal index (since they are not associated with a moduli space). If we relax this assumption, we can repeat the discussion around (3.42) and (3.51) with the corresponding additional poles factored out.

Conformal Anomalies and the Index
As we discussed above, the superconformal index is a robust observable in any SCFT, global information about the theory (since they measure T 's response to a background metric). Therefore, it is natural to imagine that there is some relation between the index and the conformal anomalies (see also the recent holographic discussion in [35,36] and the more general analysis in [37]). 23 This idea can be made more concrete in an N = 2 SCFT, T , with a freely generated Coulomb branch (i.e., a Coulomb branch with no relations between the generators; as far as we are aware, all known N = 2 SCFTs are of this type) and a Seiberg-Witten description by recalling that [33,39] 2a where a and c are the anomaly coefficients of T , and the E a are the scaling dimensions of the E r(0,0) operators. We have seen that the E r(0,0) operators contribute to the chiral U (1) R index with a weight determined by their conformal dimension. Thus, it is clear that we should be able to extract the right hand side of (4.1), and therefore 2a − c, from the index.
In order to understand how to make contact with (4.1) using the index, it turns out to be useful to recall how the authors of [39] derived (4.1). The main strategy of [39] was to topologically twist T and couple it to a background metric. Under these conditions, the anomalous shift in the U (1) R -charge of the vacuum becomes 24

2)
23 Note that unlike in the interesting recent work [27], here we have in mind a direct relation between the index-not the M 4 ≃ S 3 × S 1 partition function-and the conformal anomalies. Note also that we are extracting information from the full index, as opposed to the gauge integrand considered e.g. in [38], as this approach is useful even in the context of non-Lagrangian theories. 24 Our conventions for the N = 2 U (1) R charge differ from those in [39]. In particular, we have r here = − 1 2 r there while keeping our conventions for a and c as well as the SU (2) R and N = 1 U (1) R symmetries the same as in [39].
where χ, σ, n, and k G correspond to the Euler character, the signature, the instanton number of the background, and the flavor anomaly respectively. Since we are interested in 2a − c, we will focus on the term in (4.2) proportional to χ.
In an N = 2 SCFT with a Coulomb branch, M C (T ), we can easily compute (4.2) at a generic point on M C (T ). Indeed, since the theory at such a point consists of d C = dimM C (T ) weakly coupled abelian vector multiplets, we simply have that On the other hand, when we approach the (strongly coupled) N = 2 SCFT point it is more difficult to compute (4.2). However, it turns out that the correction to (4.3) from the additional massless matter of the SCFT is encoded in the topologically twisted partition function, Z.
To see this, note that Z reduces to an integral over the (generically abelian) low energy effective theory where [du] and [dq] correspond to the vector multiplets and hypermultiplets that parameterize the abelian moduli space (i.e., these fields are massless everywhere), and the holomorphic measure factors A, B, and C give corrections to the low energy theory arising from matter that becomes massless only at special points on the moduli space. In particular, we have that 2a − c at the superconformal point is just where r is the U (1) R charge of A and E is its scaling dimension. The beautiful observation of the authors in [39,40] is that A can be determined in terms of the data of the low energy theory. In particular, A is just the Jacobian where the u a are the gauge and monodromy-invariant coordinates and the a I are the special electric coordinates. The correction to 2a − c can then be computed since the dimensions of the u a are just the dimensions, E a , of the E r(0,0) multiplets of the SCFT (here we assume that the Coulomb branch is freely generated), while the a I have dimension one. In particular, we have We then see that (4.7) together with (4.5) imply (4.1).
Let us now see how to make contact with (4.1) using the index. Surprisingly, even though our theory is not topologically twisted, there is, as in (4.5), a natural index decomposition of the contributions to 2a − c from weakly coupled vector multiplets that are massless in the bulk of M C and corrections from the SCFT. 25 Just as in [39], we will suppose that the Coulomb branch is freely generated. For simplicity, let us also suppose that the theory is of type A or type C so that we can drop any potential exotic contributions to the index (this result then applies to a broad class of theories including the T N theories and the AD theories with a Seiberg-Witten description; however, the expressions we will derive make sense for any type A or C N = 2 theory). Under these assumptions, the general U (1) R index in (3.42) becomes (4.8) Since the derivation of [39] used properties of the theory on the Coulomb branch, it is natural to take the limit z → 1 in (4.8). In this limit, we find that where the number of degrees of freedom in the bulk of the Coulomb branch, d C , can be read off from the leading divergence of the index-in particular, we see that we should include d C vector multiplets in computing the weakly coupled contribution to 2a − c. This result is as we expect since the theory should behave to first approximation as the abelian theory on M C .
Just as in [39], we would like to see if it is possible to extract the SCFT corrections to 2a − c. Therefore, it is natural to consider the leading corrections to (4.9) 25 Perhaps this is due to some relation between topological twisting and the holomorphic twisting we employ in constructing the index.
In particular, we see that the leading correction precisely captures the scaling dimension of the Jacobian (4.6) that describes the change in coordinates between the free U (1) variables and vevs of SCFT operators.
We can then give a compact formula for 2a − c in terms of contour integrals of the index around the point z = 1. Recalling the definitions in (3.54) 11) it follows that d C = n max = max {n|A n = 0} , Therefore, we arrive at the result

Summary and Conclusions
In this paper we have derived constraints on higher-spin chiral operators in large classes of N = 2 SCFTs. The first family, which we denoted "type A", consisted of theories related to Lagrangian SCFTs by generalized Argyres-Gaiotto-Seiberg duality.
We used this duality in combination with general symmetry arguments to argue that type A theories do not contain the above "exotic" chiral operators in their spectrum.
We then considered a "chiral U (1) R " limit of the N = 2 superconformal index, which only receives contributions from Coulomb branch operators and their cousins, as a tool towards investigating theories of "type B"-generic theories with a Coulomb branch-and theories of "type C", which flow to type A via a deformation by a relevant coupling. In doing so, we elaborated on properties of the chiral U (1) R index from the perspective of its S 1 × S 3 path-integral interpretation. We subsequently used this index to argue that theories of type C do not possess exotic operators.
In the final section, we used the formula 2a − c = 1 2 a (E a − 1 2 ) of [39], for theories of type A or type C with a freely generated Coulomb branch and a Seiberg-Witten description, 26 to give a prescription for reading off 2a − c directly from the chiral U (1) R index.
Our expressions were also valid for examples of such theories with good holographic duals.
Note that our formula in (4.13) is actually well-defined for any type A or C N = 2 SCFT (more generally, we are not aware of any concrete theory for which it is ill-defined).
It would therefore be interesting to study whether our prescription for extracting 2a − c applies directly to theories with a non-freely generated Coulomb branch (assuming such theories exist). If our prescription applies to this class of theories, then we can immediately Finally, it is natural to ask whether, in direct analogy to the above case of 2a − c, one could use the index to also determine the a − c linear combination of conformal anomaly coefficients (see also the discussion in [37]). The latter can often be related to the number of effective vector and hypermultiplets in the theory, which in turn is proportional to the dimension of the Higgs branch (when it exists), a − c = 1 24 (n V − n H ) = − 1 24 dimH. It would then be reasonable to look for a limit of the superconformal index that also counts Higgs-branch operators and attempt to extract a − c along the lines that we have discussed (although we should be careful to note that one must deal with theories that don't have genuine Higgs branches). This is another question that we will leave open as an attractive direction for future research.

Appendix A. N = 2 superconformal multiplet conventions
In this appendix we provide a brief summary of our conventions for quick reference.
We follow Dolan and Osborn [12]. A review of their results 27 can be found in the appendices of [3], to which we point the interested reader for a concise account.
We work with the four-dimensional N = 2 superconformal algebra su(2, 2|2) in terms of a spinorial notation with α = +, − andα =+,−. The purely bosonic subalgebra can be captured by the following expressions: and The Poincaré and superconformal supercharges obey the relations (A.5) These in turn satisfy the hermiticity conditions 27 These are based on earlier work of [14][15][16], which established the classification of unitary representations of su(2, 2|N ) for all N , and studied their unitarity bounds and shortening conditions.
where the operators O above are understood to be in the highest weight representation of A full classification of all such possibilities and the associated short multiplet structure is listed in Table 3. Table 3: The N = 2 superconformal multiplet structure.

Multiplet Unitarity Bounds Shortening
One conventionally incorporates most of the above into multiplets of type C, C, C by allowing for "spin − 1 2 " representations as follows [12,3]: We make use of this convention in the main part of this paper.
Appendix B. RG flow from (G, G ′ ) theories Here we will explain that all the (G, G ′ ) theories defined in [9] are of type C in the sense of subsection 3.6. Let us first review the definition of (G, G ′ ) theories. We consider type IIB string theory on a local Calabi-Yau threefold given by the hypersurface singularity Here W (x 1 , · · · , x 4 ) is a quasi-homogeneous polynomial such that for all ζ ∈ C * . We also assume that dW = 0 if and only if x i = 0. The holomorphic 3-form on the threefold is given by Ω = dx 1 dx 2 dx 3 dx 4 dW up to rescaling. Since the Calabi-Yau threefold is non-compact, it engineers an N = 2 gauge theory in the transverse four dimensions.
If the singularity (B.1) arises at finite distance in the moduli space of some compact Calabi-Yau threefold, we can think of the 4d gauge theory as a rigid supersymmetry limit of some 4d supergravity. In terms of c ≡ 4 i=1 (1 − 2q i ), the necessary and sufficient condition for this is written as 2 − c > 0 [42]. A class of W (x i ) satisfying this condition is given by [9] W (x 1 , · · · , x 4 ) = W G (x 1 , where G, G ′ = A n , D n , E n and The 4d theory engineered by the singularity (B.3) is called the (G, G ′ ) theory [9].
The BPS states of the 4d theory correspond to D3-branes wrapping on supersymmetric compact 3-cycles of the Calabi-Yau threefold. Since all the compact 3-cycles are now shrinking at x i = 0, the 4d BPS states are all massless. This suggests that the (G, G ′ ) theory is a 4d superconformal filed theory of Argyres-Douglas type [7,8]. The transfor- where I W is an ideal of C[x 1 , · · · , x 4 ] generated by all ∂W/∂x i . It is known [43,34] that the number of independent vanishing 3-cycles is equal to dim C R (as a vector space over C). Let us take a set of polynomials {W k } which represent a basis of R. Then the most general deformation is written as for c k ∈ C. We can think of {c k } as a local coordinate of the complex structure moduli near the singular point. In terms of 4d physics, they are interpreted as masses and Coulomb branch parameters of the engineered gauge theory. In particular, c k = 0 corresponds to the superconformal point.
Let us take W k (x i ) to be monomials in x i . There exists Q k ∈ Q + such that for all ζ ∈ C * . The scaling dimension of c k in 4d physics is then evaluated to be [34] [ For any c k there exists c ℓ such that [c k ] + [c ℓ ] = 2, which is consistent with the arguments of [7]. If [c k ] < 1, the parameter c k is interpreted as a coupling constant of the theory.
The corresponding deformation of the theory is given by the Coulomb branch of the 4d theory has a Seiberg-Witten description with the curve W (s, t) = 0 and the 1-form λ = sdt on it [9]. The reason for this is that any monomial containing x 1 or x 2 becomes trivial in R; there is no non-trivial complex structure deformation in the x 1,2 directions. For example, any (A, A) theory has a Seiberg-Witten description. These types of theories were studied in [10] from the viewpoint of M5-branes on a Riemann surface, while their RG-flows investigated in [44]. On the other hand, if W (x i ) cannot be expressed in the form (B.13) (even after changing variables), the corresponding 4d theory does not seem to have a Seiberg-Witten description [9]. For instance, the (D n , D m ) theories for n, m ≥ 4 and (E, E) theories provide such examples.

B.3. RG flow by relevant deformations
We now study RG flows from (G, G ′ ) theories to show that they are of type C in the sense of subsection 3.6. Let us start with the (A n , A m ) theory. The corresponding Calabi-Yau singularity is described by W (x, y, s, t) = x n+1 + y 2 + s m+1 + t 2 = 0 . (B.14) We have q x = 1 n+1 , q s = 1 m+1 , q y = q t = 1 2 , and therefore 2 − c = 2( 1 n+1 + 1 m+1 ). The dimensions of the variables are We now have several massive BPS states whose masses are proportional to (c n−1,0 ) 1/[c n−1,0 ] .
In the deep IR, all the massive degrees of freedom are integrated out. This corresponds to focusing on the region |x| ≪ |c n−1,0 |. Then the IR curve is written as x n−1 + s m+1 = 0 , (B.21) after rescaling x and s. 28 Since this is the curve of the (A n−2 , A m ) theory, we identify that the IR CFT is the (A n−2 , A m ) theory (there is also a decoupled U (1)). Thus, in the case theory is also of type C. Exactly the same argument shows that all the (G, G ′ ) theories are of type C. Our discussion in subsection 3.6 then implies that (G, G ′ ) theories have no exotic chiral operators. 28 To be more precise, we rescale x and s so that c n−1,0 x n−1 +s m+1 = 0 becomes x n−1 +s m+1 = 0 while keeping λ = sdx fixed. This changes the scaling dimensions of x and s.