OPE Selection Rules for Schur Multiplets in 4D $\mathcal{N}=2$ Superconformal Field Theories

We compute general expressions for two types of three-point functions of (semi-)short multiplets in four-dimensional $\mathcal{N}=2$ superconformal field theories. These (semi-)short multiplets are called"Schur multiplets"and play an important role in the study of associated chiral algebras. The first type of the three-point functions we compute involves two half-BPS Schur multiplets and an arbitrary Schur multiplet, while the second type involves one stress tensor multiplet and two arbitrary Schur multiplets. From these three-point functions, we read off the corresponding OPE selection rules for the Schur multiplets. Our results particularly imply that there are non-trivial selection rules on the quantum numbers of Schur operators in these multiplets. We also give a conjecture on the selection rules for general Schur multiplets.


Introduction
The space of four-dimensional N = 2 superconformal field theories (SCFTs) has a rich structure. The best known N = 2 SCFTs are the SU (N c ) gauge theories with N f fundamental matter hypermultiplets where N f = 2N c is satisfied so that the beta function vanishes. While these theories are well-described by Lagrangian, there are many N = 2 SCFTs whose Lagrangian description is not known, such as Argyres-Douglas SCFTs [1][2][3], 1 Minahan-Nemeschanskey theories [14,15], and an infinite series of non-Lagrangian SCFTs of class S [16]. To study general N = 2 SCFTs including these non-Lagrangian theories, we need a technique that relies only on the symmetry and unitarity of SCFTs.
Recently there was important progress in this direction. The authors of [17] showed that the operator product expansions (OPEs) of a special class of BPS local operators are naturally encoded in a two-dimensional chiral algebra. These BPS operators are called "Schur operators" since they contribute to the Schur limit of the superconformal index [17][18][19]. We call the superconformal multiplets including a Schur operator "Schur multiplets." The existence of the associated chiral algebra implies, along with the four-dimensional unitarity and superconformal symmetry, that the "c central charge" of any interacting fourdimensional N = 2 SCFTs is constrained by c ≥ 11 30 [20], which is saturated by the minimal Argyres-Douglas SCFT. 2 Moreover, a similar analysis for N = 2 SCFTs with a flavor symmetry leads to a universal bound involving c and the flavor central charge [17,21]. For more recent works on the associated chiral algebras, see .
Let us briefly sketch how these bounds on the central charges were derived from the chiral algebra analysis. In four-dimensional N = 2 SCFTs, the superconformal symmetry and the unitarity impose strong constraints on superconformal multiplets appearing in OPEs, which we call "selection rules" in the following. In particular, the selection rules 2 ) + · · · , (1.1) were used to derive the central charge bounds mentioned above. Here, B R and C R(j,) are two types of Schur multiplets labeled by the SU (2) R charge R and the spin (j,) of the superconformal primary field, and the ellipses stand for non-Schur multiplets. 3 In particular, the B 1 multiplet is a Schur multiplet including a flavor current, and the C 0(0,0) multiplet is the stress tensor multiplet. The above selection rules (and unitarity) are crucial in deriving the central charge bounds. For example, the bound c ≥ 11 30 was derived by interpreting the reality of the OPE coefficients for the second selection rule in (1.1) in terms of the two-dimensional chiral algebra. 4 This implies that identifying the selection rules for Schur multiplets provides a powerful tool to reveal universal constraints on general N = 2 SCFTs. 2 Our normalization of the four-dimensional central charge is such that a free hypermultiplet has c = 1 12 and a = 1 24 . 3 For the precise definitions ofBR and C R(j,) , see section 2. 4 In this interpretation, it was assumed that C 0( ℓ 2 , ℓ 2 ) for ℓ > 0 are absent in interacting N = 2 SCFTs since they involve higher spin currents.
Moreover, the selection rules are also important in recovering four-dimensional OPEs from the two-dimensional chiral algebra. Indeed, the 4d/2d correspondence of [17] implies that Schur operators with different quantum numbers could correspond to two-dimensional operators with the same quantum numbers. Therefore, it is generically non-trivial to recover four-dimensional OPEs from two-dimensional OPEs. The selection rules, however, strongly constrain Schur multiplets appearing in the four-dimensional OPEs and therefore will be useful for reconstructing the four-dimensional OPEs from the associated chiral algebra.
In this paper, we study the selection rules for C 0(0,0) × O Schur , (1.2) up to non-Schur multiplets, where O Schur is an arbitrary Schur multiplet. Since the Schur operator in the stress tensor multiplet C 0(0,0) maps to the Virasoro stress tensor in the associated chiral algebra, the selection rules for (1.2) are particularly important in the study of the 4d/2d correspondence. In particular, they reveal how the four-dimensional operator associated with a Virasoro primary is related to those of the Virasoro descendants. Indeed we find that, when four-dimensional Schur operators O and O ′ respectively correspond to a Virasoro primary and its descendant in the associated chiral algebra, the SU (2) R charge of O is always smaller than or equal to that of O ′ (see for example (4.31)). Note that the selection rules for C 0(0,0) × C 0(0,0) and C 0(0,0) × B 1 were already identified respectively in [20] and [51], which we generalize to (1.2) for all Schur multiplets O Schur in this paper.
To derive the above selection rules, we study three-point functions of the form where O 1 and O 2 are arbitrary Schur multiplets. Our strategy is to write down the most general ansatz for the three-point functions and then impose the (semi-)shortening conditions corresponding to the Schur multiplets. The same strategy was employed in [20,51,52] to compute several three-point functions. We stress that, since our analysis relies only on the (semi-)shortening conditions which purely follow from the superconformal algebra, our results are applicable to any four-dimensional N = 2 SCFT.
Before studying the selection rules for C 0(0,0) × O Schur , we first apply our strategy to the selection rules for not. This reflects the fact that the sum of the U (1) r charges of the superconformal primary fields in three-point functions is non-vanishing. On the other hand, we find that the sum of the U (1) r charges of Schur operators in these multiplets always vanishes, which suggests that the Schur operators play a central role in Schur multiplets. With this observation, we give a conjecture on the OPE selection rules for general Schur multiplets in section 5.
The outline of this paper is as follow. In section 2, we review the four-dimensional N = 2 superconformal algebra and the (semi-)shortening conditions for the Schur multiplets, and also introduce a useful formalism [52,54,55] to analyze the superconformal three-point functions. In sections 3 and 4, we derive the two types of three-point functions respectively. From these correlation functions, we present the B R 1 × B R 2 selection rule and the C 0(0,0) × O Schur selection rules. section 5 is devoted to conclusions and discussions, where we conjecture more general selection rules between Schur multiplets as a natural generalization of our results. In appendix A, we summarize the nilpotent structure of the Grassmann variables what we call Fierz identities, and appendices B,C, and D are the details of our calculations.

(Semi-)shortening conditions and three-point functions
In this section, we review the four-dimensional N = 2 superconformal algebra and the short multiplets following [56] and introduce a useful formalism constructed in [52,54,55] for the computations of correlation functions of SCFTs. We follow the convention of [17] unless otherwise stated.
A general long multiplet of the four-dimensional N = 2 superconformal algebra is labeled by five eigenvalues of the Cartan subalgebra for the primary state, namely, the conformal dimension ∆, the Lorentz spin (j,), the irreducible representation R of SU (2) R , and the U (1) r charge r. 6 Here, the superconformal primary field is defined as a state annihilated by all conformal supercharges, S α i and Sα i . We denote the superconformal primary field by |∆, r where the parentheses in the scripts such as (i 1 · · · i 2R 1 ) denote the total symmetrization of the indices. Acting Q i α and Qα i on the primary, we can generate 256(2R + 1)(2j + 1)(2 + 1) components of the long multiplet. These long multiplets satisfy 6 We take R so that the Dynkin label for the irreducible SU (2)R representation is 2R. unitarity bounds, where E i and j i are defined by An N = 2 Poincaré supersymmetric field theory has a unitarity bound called the BPS bound, and if a long multiplet saturates the bound it becomes a short multiplet whose number of components is half the original one. The above unitarity bounds for N = 2 SCFTs play a similar role; when the eigenvalues (∆, R, r, j,) saturate the unitarity bounds (2.1) or (2.2), a long multiplet is shortened. If we consider the case of j = 0, ∆ = E 1 − 2, the superconformal primary field |∆ Similarly, for = 0, ∆ = E 2 − 2, the superconformal primary field satisfies the condition These two conditions B 1 andB 2 are called shortening conditions. On the other hand, for ∆ = E 1 , the following condition is possible: Similarly for ∆ = E 2 , we can imposē (2.7) These two conditions C 1 andC 2 are called semi-shortening conditions.
The Schur multiplets are defined as multiplets satisfying a shortening condition or a semishortening condition for each of the chiralities. There are four types of Schur multiplet, which are denoted as B R , D R(0,) ,D R(j,0) , and C R(j,) in the notation of [56]. See table 1 for the definition of the four types and the relations among the quantum numbers of their superconformal primary states.
Some of these Schur multiplets contain important operators. For example, the C 0(0,0) multiplet contains the stress-tensor operator and the SU (2) R × U (1) r conserved current operator.
We can regard the semi-shortening conditions as the conservation equations of these operators. In this paper, we assume that the theory we are considering has a unique stress-tensor multiplet C 0(0,0) . This assumption leads to a constraint on the three-point functions involving

Condition
Conformal dimension and U (1) r charge of the primary two stress tensor multiplets, as discussed later. The C 0(j,) multiplet is a higher-spin generalization of the stress-tensor multiplet C 0(0,0) and contains a higher spin current operator. It is expected that interacting (S)CFTs containing such a higher spin current have a decoupled free sector [57][58][59]. On the other hand, the B R multiplets are half-BPS multiplets whose superconformal primary field is annihilated by both Q α 1 and Qα 2 . As mentioned in section 1, the B 1 multiplet particularly contains a conserved flavor current. Finally, the D 0(0,0) ⊕D 0(0,0) multiplet is an N = 2 free vector multiplet, whose superconformal primary field has the conformal dimension 1, and its (semi-)shortening conditions imply the massless equation of motion.
where O 1 (z 1 ), O 2 (z 2 ), · · · are superfields for general superconformal multiplets. In the rest of this section, we review basic techniques to solve differential equations of this form. We use them in sections 3 and 4 to derive selection rules for the Schur multiplets.
To that end, we first introduce the following chiral and anti-chiral variables for given two points z 1 , z 2 in the superspace: where x μ 12 is anti-chiral for z 1 and chiral for z 2 . Next, we introduce superconformal covariant U (2) R and SU (2) R matrices respectively as where z 12 := (x μ 12 , θ α 12 i ,θα i 12 ). These matrices satisfy the relations whereû ij (z 12 ) := u k i (z 12 )ǫ kj 8 with ǫ kj being the SU (2) R invariant tensor. We also define a spin SL(2, C) covariant matrix (2.12) These matrices (2.9) and (2.12) are building blocks of superconformal two-point functions.
Next, for given different three points z 1 , z 2 , and z 3 in superspace, we introduce the superconformally covariant variable Z 1 = (X 1αα , Θ iα 1 ,Θα 1i ) by 14) The variable Z 1 transforms similarly to z 1 . In particular, the following identity will be important in our calculations below: Similar variables Z 2 and Z 3 are defined as cyclically permuting z 1 , z 2 and z 3 in the above definition. 9 These Z variables play central role in expressing three-point functions. We also 8 Except theseûij (z12), all other SU (2)R indices are raised and lowered as C i = ǫ ij Cj , Ci = ǫij C j . 9 The conformal dimension of (Xαα, Θ iα ,Θα i ) is (1, 1 2 , 1 2 ), and the U (1)r charge is (0, 1 2 , − 1 2 ), respectively.
define SU (2) R matrices using Z 1 as

(Semi-)shortening conditions for three-point functions
Let us now consider a three-point function of three quasi superfields Φ I i (z i ) for i = 1, 2, 3.
We denote their conformal dimension and U (1) r charge by (q i +q i ) and (q i − q i ) respectively.
The subscript I i expresses SU (2) R and SL(2, C) indices collectively. A general three-point where T J 1 I 1 and T J 2 I 2 are some functions composed of (2.9) and (2.12) in the representation of SU (2) R × SL(2, C) specified by I 1 and I 2 . On the other hand, the function H(Z 3 ) satisfies following homogeneity property where a andā are fixed by satisfies the (semi-)shortening conditions. This implies that the (semi-)shortening conditions only constrain the function H(Z 3 ). Therefore, hereafter, we focus on H(Z 3 ). It is easy to where derivativesD jβ , D β j , Q α i andQα i are defined respectively as , Moreover, quadratic derivatives such as D Therefore, the (semi-)shortening conditions are now translated into partial differential equa- While the (semi-)shortening conditions of the first and second superfields, Φ I 1 (z 1 ) and Φ I 2 (z 2 ), are easily expressed as partial differential equations for H(Z 3 ), it is not straightforward to translate the conditions for the third superfield Φ I 3 (z 3 ) into a similar equation for To consider the (semi-)shortening conditions of the third superfield, we change the variable from Z 3 to Z 2 . 10 Indeed, using the cyclicity of z 1 , z 2 and z 3 , the correlation function (2.20) is also expressed as for some function G(Z 2 ). The action of D i 3α andD 3αi on the G(Z 2 ) are given by where the derivatives are now defined by As shown in [52,55], Z 3 and Z 2 are related as (2.31) 10 Here, we can also use Z1 instead of Z2.
Using these relations, we see that the function It is important to consider the third superfield conditions since it is insufficient to fix the function H(Z 3 ) only considering the first and second superfields of the (semi-)shortening conditions in section 4.2.
In the following sections, we will use the above formalism and techniques to study the three-point functions of Schur multiplets.
In this section, we study the most general expressions for three-point functions of two half-BPS Schur multiplets B R and an arbitrary Schur multiplet O I . 11 Our result is particularly consistent with the fusion rules for B R 1 × B R 2 which were first obtained in [53].
The general expression for the three-point function B R 1 B R 2 O I is given by where L (i 1 ···i 2R ) (z) is the superfield of B R multiplet, and the parentheses denote the total symmetrization of the indices. Hereafter, we will often omit the parentheses with the understanding that the indices associated with the same Latin and Greek alphabet letters are always totally symmetrized.
Each of the B R 1 and B R 2 multiplets satisfies two shortening conditions as shown in table 1. As we have mentioned in the previous section, the shortening conditions are translated into differential equations for H(Z 3 ). For the two B R multiplets, the differential equations are written as It is easy to solve (3.3) and (3.4), since these are merely first-order linear equations for Θ orΘ. In contrast, (3.2) and (3.5) contain both X and Θ (orΘ) derivations and therefore are more complicated. However, if we use theZ 3 := (X 3 , Θ 3 ,Θ 3 ) coordinate instead of 11 By definition, SU (2)R irreducible representation R of BR must be R ≥ 1 2 .
, the two equations (3.2) and (3.5) become simpler, because D α i andQα i are expressed in terms ofZ 3 as Indeed, the most general solution to (3.3) and (3.4) is simply expressed in terms of Z 3 as while that of (3.2) and (3.5) is written in terms ofZ 3 as Here, we use the short-hand notation The above two expressions, (3.8) and (3.7), must be equal under the relation (2.16). Therefore, our strategy is to rewrite (3.8) in terms of Z 3 by using (2.16) and restrict the parameters in the expression to be consistent with (3.7). This gives us the most general solution to the equations (3.2)-(3.5).
After solving (3.2)-(3.5), we have to check it also satisfies the (semi-)shortening conditions of the third superfield O I (z 3 ). In this process, we relate H(Z 3 ) to G(Z 2 ) using (2.32) and see if the G(Z 2 ) satisfies the differential equations corresponding to the third set of (semi-) shortening conditions.
Below, we apply this strategy to evaluate the most general expression for the three-point Let us first consider the case of O I in the B R 3 multiplet. In this case, the function H(Z 3 ) and R 3 are constrained by the inequalities 0 In other words, R 3 must be such that Moreover, since the primary of B R is a scalar, so is the H(Z 3 ). Therefore, the function H(Z 3 ) only carries SU (2) R indices. The most general ansatz for H(Z 3 ) is then written as 12 where R ∈ Z ≥0 , and k 1 · · · k 2R 3 are the SU (2) R indices associated with B R 3 . As mentioned above, indices associated with the same Latin and Greek alphabet letters, such as l 1 · · · l 2R 1 , are totally symmetrized. On the other hand, the same function should also be written in terms ofZ 3 . The most general expression in terms ofZ 3 is given by For the above two expressions to be consistent, the coefficients A, B and C have to satisfy some conditions. To identify the conditions, we change the variables fromZ 3 to Z 3 in (3.12) by using (2.16). Using the Fierz identities summarized in appendix A, we see that the conditions for an arbitrary constant A. Up to an overall constant, the function H(Z 3 ) is written as This is the most general expression for H(Z 3 ) satisfying (3.2)-(3.5).
Although it satisfies the shortening conditions of the B R 1 and B R 2 multiplets, it is nontrivial whether the expression (3.14) satisfies the shortening conditions of the third multiplet To check the shortening conditions for B R 3 , let us relate the H(Z 3 ) to G(Z 2 ) using (2.32). Indeed, as reviewed above, the correlation function (3.1) is also written as The function G(Z 2 ) is uniquely fixed by H(Z 3 ) via (2.32), i.e., where with S α n andSα n defined by (2.30). Since (3.14) trivially satisfies the above two equations, the expression (3.16) also satisfies the shortening conditions for the third Schur multiplet B R 3 .
In the rest of this paper, we omit the subscript 3 of X 3 , Θ 3 andΘ 3 in the expression for H(Z 3 ). Similarly, we omit the subscript 2 of X 2 , Θ 2 andΘ 2 in the expression for G(Z 2 ).

3.2
Let us turn to the case of O I in theD R(j,0) multiplet. 13 In this case, the function H(Z 3 ) has dimension −2R + j + 1 and U (1) r charge −j − 1. Since the highest possible degree ofΘ in .7)), the only possible value of j is j = 0, and therefore the U (1) r charge , we see that the most general expression for such H(Z 3 ) is given by where k 1 · · · k 2R 3 are the SU (2) R indices forD R 3 (0,0) . As mentioned at the end of the previous subsection, (X, Θ,Θ) stands for (X 3 , Θ 3 ,Θ 3 ) here. However, when we change the variables from Z 3 toZ 3 , this expression cannot be written in the form of (3.8). 14 This means that there are no solutions to (3.2)-(3.5). Therefore, theD R(j,0) multiplet does not appear in the Note that its conjugate implies that the D R(0,) multiplet also does not appear in the B R 1 × B R 2 fusion. By using the same argument, we can extend our results Let us finally consider the case of O I in the C R 3 (j,) . By using the similar argument in previous section 3.2, = j is necessary for the three-point function to be non-vanishing.
Therefore the function H(Z 3 ) has dimension 2 − 2R + 2j and U (1) r neutral, where we recall

The most general solution to (3.3) and (3.4) is written as
where k i and (β i ,β i ) are the SU (2) R and SL(2, C) indices associated with C R 3 (j,j) . On the other hand, the most general solution to (3.2) and (3.5) is written as 13 Note here that the result for O I in the D R(0,j) multiplet follows from this case by CPT.
For the above two expressions to be consistent, the coefficients have to satisfy some conditions.
Unless R = 0, the conditions are On the other hand, for R = 0 or equivalently R 3 = R 1 + R 2 , all the coefficients have to vanish.
Therefore, the three-point function Next, we consider the semi-shortening conditions for C R 3 (j,) . For that purpose, we relate (2.32). Indeed, the three-point function (3.1) can be rewritten as and the explicit form of G(Z 2 ) becomes In terms of G(Z 2 ), the semi-shortening conditions for C R 3 (j,j) are written as for j > 0 and for j = 0. It is straightforward to check (3.23) satisfies these conditions.
In summary, the function H( is given by, up to an overall constant for any integer or half-integer j ≥ 0 and R 3 < R 1 + R 2 . Therefore, the C R 3 (j,j) multiplet

Selection rule
In the above subsections, we have computed the most general expression for non-vanishing three-point functions of the form B R 1 B R 2 O . From these results, we see that the selection rules for two B R multiplets are written as up to non-Schur multiplets. This is particularly consistent with Eq. (3.44) of [53]. Especially, for R 1 = R 2 = 1, the selection rule is written as Recall here that the stress-tensor multiplet C 0(0,0) has two semi-shortening conditions The most general solution to these two equations are respectively written as where where k i and (β i ,β i ) are respectively the SU (2) R and SL(2, C) indices associated with the third multiplet. The derivations of these functions are given in appendix B.
We see that these are also consistent with the (semi-)shortening conditions for the third Schur multiplet. Indeed, using (2.32), we see that the function G(Z 2 ) corresponding to the above H(Z 3 ) is written as Let us briefly comment on the case of O I in C 0(0,0) . When we assume that there is only one stress tensor in the theory, the corresponding three-point function C 0(0,0) B 1 C 0(0,0) has to be symmetric under the action of Z 2 exchanging the first and the third multiplets. This Z 2 symmetry implies that the function G(Z 2 ) is invariant under (X 2 , Θ 2 ,Θ 2 ) ↔ (−X 2 , −Θ 2 , −Θ 2 ).
Before closing this subsection, let us also make a quick comment on the correlation function C 0(0,0) B R C R−1(j,j) . In CFTs, any correlation function of conformal descendant fields is obtained by differentiating the correlation function of the conformal primary fields. This particularly implies that, when a correlation function of conformal primary fields vanishes, the corresponding descendant correlators also vanish. This, however, is not the case for superconformal descendants in SCFTs. Indeed, when we set all Grassmann variables, θ 1,2,3 ,θ 1,2,3 , to zero in (4.9), the correlation function vanishes. This shows that the correlator of the three superconformal primary fields in C 0(0,0) , B R , and C R−1(j,j) vanishes, while there are non-vanishing correlators involving superconformal descendants. This is a common feature of SCFTs [62].

4.2
C 0(0,0)DR(j,0) O I Let us next consider C 0(0,0)DR(j,0) O I . We denote byN (j 1 ···j 2R )(α 1 ···α 2j ) (z) the superfield of ā D R(j,0) multiplet. The three-point function is written as where j 1 ≥ 0 is an integer or a half-integer, and k i and (β i ,β i ) are respectively the SU (2) R and SL(2, C) indices for the third multiplet. Note that the first expression for the case of O I in the D R(0,j) multiplet is only for j = 1/2. In the case j = 1/2, the function H(Z 3 ) has two independent terms as , in which case the three-point function is written as where J (α 1 ···α 2j ),(α 1 ···α 2 ) (j 1 ···j 2R ) (z) is the superfield in the C R(j,) multiplet. In this case, the H(Z 3 ) has to satisfy the semi-shortening conditions (4.1) and (4.2) associated with C 0(0,0) and the semi-shortening conditions for the other two C R(j,) multiplets. The semi-shortening conditions for the second multiplet, C R(j 1 , 1 ) , are written as Qα (mQα m ′ H I m 1 ···m 2R ) = 0 for j 1 = 0 , (4.20b) while the semi-shortening conditions for the third multiplet, C R ′ (j 2 , 2 ) , are similarly expressed in terms of G(Z 2 ).
We have found that a non-vanishing H(Z 3 ) in (4.18) satisfying all these semi-shortening conditions is possible only for the following two types of correlator 15 : C 0(0,0) C R(j+l 1 ,j) C R(j+l 2 ,j+l 1 +l 2 ) , (4.21) up to charge conjugation, where l 1 and l 2 are non-negative (half-)integers. Note that the function H(Z 3 ) is U (1) r neutral for all these cases. Since general solutions for H(Z 3 ) in these two cases are highly involved, it is beyond the scope of this paper to identify the most general expression for the allowed H(Z 3 ). However, we find a special solution for each of the above two types of correlators, 16 which is sufficient to identify the selection rule.
Let us first focus on (4.21). Unless j = l 1 = 0, our special solution is written as Here (2l + 2) k := (2l+2+k−1)! (2l+2−1)! is the Pochhammer symbol, and γ (and γ ′ when present) are contracted after totally symmetrized as (γγ ′ γ 1 · · · ) . Similarly, m is also contracted after totally symmetrized as (mm 1 m 2 · · · ). On the other hand, in the case j = l 1 = 0, we find a solution with several free parameters: . 15 We will show other type correlation functions does not satisfy semi-shortening conditions in appendix D. 16 Our method is as follows. We first solve all the semi-shortening conditions for smaller spins j = 1/2, 1 and so on to find the explicit expression for H(Z3). We then guess an ansatz (4.23) and (4.29) for general j, and check that the ansatz satisfies the (semi-)shortening conditions for arbitrary j.
Note that, for j = ℓ 1 = R = 0, the three-point function C 0(0,0) C 0(0,0) C 0(l 2 ,l 2 ) has a Z 2symmetry, under the assumption of uniqueness of the stress tensor multiplet. The Z 2 symmetry implies that the function H(Z 3 ) has to be invariant under ( This condition constrains (4.24) as follows.
It must be equal to H(X 3 , Θ 3 ,Θ 3 ), and therefore, l 2 must be a half-integer, otherwise correlation function should vanish.

Selection rules
We here write down the selection rules for C 0(0,0) × O Schur read off from the three-point functions we computed above. Note that all the following rules are only up to non-Schur multiplets.
The C 0(0,0) × B R selection rules are as follows.
• For R > 1, the selection rule is (4.31) • For R = 1, because of the Z 2 -symmetry, the selection rule is which is consistent with [51].
• For R = 1 2 , the rule is The selection rules for C 0(0,0) ×D R(j,0) are written as follows.
• For j − = l 1 ≥ 1 2 , R > 1 2 , (4.36) (4.39) • For = j > 0, R = 1, (4.40) When j is an integer, the stress-tensor multiplet C 0(0,0) in the last term on the righthand side must be excluded by the Z 2 -symmetry, under the assumption of uniqueness of the stress tensor. The cases of j − < 0 are obtained by the charge conjugates of the above ones.

Conclusions and discussions
In this paper, we have computed the three-point functions of the form B R 1 B R 2 O and  [53]. We emphasize that our analysis relies only on the shortening con- In the 4d/2d correspondence of [17], this means that the SU (2) R charge of the four-dimensional ancestor of a two-dimensional operator is always smaller than or equal to those of its Virasoro descendants. Since the SU (2) R symmetry is broken in the associated chiral algebra [17], this relation between the SU (2) R charge and the holomorphic dimension is surprising. 18 See also [64] for a remarkable discussion on reconstructing the SU (2) R -filtration of the chiral algebra.
Another interesting observation is that, in some of the OPE channels allowed by our selection rules, the three-point function of the superconformal primaries vanishes even though those of their descendants do not. As mentioned already, this is a common feature of SCFTs [62]. Indeed, the vanishing of the three-point function of the primaries reflects the fact that the sum of their U (1) r charges is non-vanishing. Therefore, our selection rules for Schur multiplets do not imply the non-vanishing of the three-point function of the corresponding superconformal primaries.
On the other hand, when we focus on the Schur operator in each Schur multiplet, we see that the sum of their U (1) r charges vanishes whenever the corresponding three Schur multi-

Multiplet
Schur operator  for i = 1, 2, and 3. Then we conjecture that the following two conditions are satisfied: Note that these conditions are necessary for the three-point functions of the Schur operators to be non-vanishing. Recognizing (5.1) and (5.2) as principle of selection rule related to fourdimensional N = 2 SCFT whose stress tensor is unique, we recover our all selection rules in section 3.4 and 4.4. We leave the detailed study of this conjecture to future work.

A Fierz identities
In this appendix, we summarize useful identities for Grassmann variables Θ iα andΘα i , which we call Fierz identities. We first introduce the following variables: From the nilpotent structure of Θ andΘ, we see that the following Fierz identities hold 20 Moreover, by using the variables m l , we can expand powers ofX 2 as [20] We can also derive 21 The following Fierz identities are derived by using the Mathematica package grassmann.m [65]: However, this is not consistent with (4.3) unless A = 0, and therefore the correlation function C 0(0,0) B RDR ′ (j,0) has to vanish. From charge conjugation, we see that C 0(0,0) B R D R ′ (0,j) also vanishes. B.2 Let us next turn to the case of O I = C R ′ (j,) . In this case, the function H(Z 3 ) has dimension 2(R ′ − R) + j + and U (1) r charge − j. Recall that | − j| is at most one. Moreover, since charge conjugation exchanges j and, we only need to study the cases of j ≥. Therefore, we assume −1 ≤−j ≤ 0. Then the possible combinations of R ′ and−j are (R ′ ,−j) = (R, −1), , (R+1, 0), (R, 0), and (R−1, 0). We study the most general expression for H(Z 3 ) in each of these cases below.
In this case, the only possible candidate for H(Z 3 ) is This is not consistent with (4.4) unless A = 0.
In this case, the only possible candidate is which is not consistent with (3.5) unless A = 0.
In this case, there are three possible terms in H(Z 3 ); However, this is not consistent with (4.4) unless A = B = C = 0.
The constraint (3.4) is then expressed as and therefore satisfies (4.2). We also see that this expression satisfies (3.5).
In this case, the possible H(Z 3 ) for j > 0 is of the form For this to be consistent with (3.5) and (4.2), we must impose B = −2jA and C = −4ijA.
On the other hand, for j = 0, H(Z 3 ) is given by Using the Fierz identity (A.6), we see that this satisfies all the (semi-)shortening conditions.
Here f i are functions of X and Θ. Note in particular that the superscriptsγ i in f = 1 2 (f γ 1 γ 2 + f γ 2 γ 1 ) while in (C.1) we further symmetrize as in ǫ β(γ 1 f using the function f7 we have just defined, and SU (2)R indices are symmetrized similarly.
However, we see that this does not satisfy the shortening condition (C.2) unless A = 0.
In this case, the possible solution is given by which is not consistent with (4.14) unless A = 0.
In this case, H(Z 3 ) is given by It is straightforward to show that this satisfies (4.2). Let us next consider the semi-shortening condition for the second multiplet. For j > 0, the condition (4.14) reads On the other hand, for j = 0, the condition (4.15) implies C = −4iA. Note that the term proportional to B in (C.10) does not exist for j = 0.
To solve the conditions associated with the third multiplet, let us relate the above H(Z 3 ) to G(Z 2 ) via (2.32). The result is generally written as where C is a free parameter that is present only in the case of j > 0. We see that this expression satisfies the shortening condition (C.2) for arbitrary C. On the other hand, the semi-shortening condition implies C = 0 unless j = 1 2 . For j = 1 2 , the semi-shortening condition does not restrict the value of C.
Note that this expression vanishes if j = 0. Therefore there is no non-trivial solution for For j > 0, the above expression is consistent with (4.2) and (4.14) if and only if C = A−B 2 and D = i(A + B). With these conditions imposed, the function G(Z 2 ) given by (2.32) is written as It is then straightforward to see that A = B = 0 is necessary for this to be consistent with (C.2) and (C.3). Therefore, no non-trivial solution exists in this case.
(C. 19) We see that this expression satisfies the semi-shortening conditions associated with the third multiplet for any j ≥ 0, j 1 ≥ 0 and R ≥ 0.
Since charge conjugation flips the sign of the U (1) r charge, we focus on the case of U (1) r charge −1. Then we see from (4.3) that R ′ = R must hold for a non-trivial solution to exist.