2d small N=4 Long-multiplet superconformal block

We study 2d N=4 superconformal field theories, focusing on its application on numerical bootstrap study. We derive the superconformal block by utilizing the global part of the super Virasoro algebra and set up the crossing equations for the non-BPS long-multiplet 4-point function. Along the way, we build global N=4 superconformal short and long multiplets, and compute all possible 2,3-point functions of long-multiplets that are needed to construct the superconformal blocks and the crossing equations. Since we consider a long-multiplet 4-point function, the number of crossing equations is huge and we expect it to give a strong constraint than the usual superconformal bootstrap analysis, which relies on BPS 4-point functions. In addition, we present an alternative way to derive crossing equations using N=4 superspace and comment on a puzzle.

With this paper, we wish to fill a gap in the literature-CFT in two dimensions with (0, 4) or (4,4) supersymmetry [17], which seems to be the last remaining family of supersymmetric CFTs that has not been explored extensively. 1 At first glance, the infinite dimensional super-Virasoro symmetry [18] could be very constraining and provide a lot of information by itself. However, we are not aware of any literature that worked out the super-Virasoro conformal blocks for N = 2 or higher [19][20][21], and this makes it difficult to use full power of the N = 4 super-Virasoro algebra in the bootstrap analysis.
Still, one can try to use a global part of the superconformal algebra to construct 'smaller' superconformal blocks. More precisely, we will use the fact that the 4-point correlation function of conformal primaries in the long-multiplet is decomposed into bosonic Virasoro conformal blocks, not the super-Virasoro conformal blocks. Since the coefficients in front of each decomposed Virasoro blocks are independent, the set of crossing equations is distinguished from non-supersymmetric 2d CFT 4-point functions and at the same time captures structure of N = 4. This fact was used in [8] to do N = 2 long-multiplet bootstrap analysis. Our goal is to generalize this result to N = 4. As we will find in this paper, the number of crossing equations is larger than that of any numerical bootstrap literature that we are aware of, which makes us confident about the level of precision in using only these small superconformal blocks. Moreover, different from the previous approaches that only analyzed particular BPS sectors of the theory, we set up the crossing equations using generic long-multiplets. Hence, we expect the resulting set of crossing equations to be more comprehensive and constrain the spectrum of the theory.
2d (0, 4) or (4,4) superconformal field theories are interesting in their own right. There are many interesting examples that have N = 4 superconformal symmetry in 2-dimensions. Some of these include K3 (4, 4) theory [16], holographic dual of AdS 3 × S 3 × S 3 × S 1 [22], IR limits of (0, 4) E-string worldsheet theories [23][24][25], a family of (0, 4) theories [26] that originate from class-S theory [27,28], and lastly a huge class of (0, 4) theories from branebox model [29]. A last point that is worth to mention is that the 2d small N = 4 chiral algebra appears in the subsector of 4d N = 4 SYM [30,31], which is at the same time superconformal field theory with algebra psu(2, 2|4). Although we have not attempted to study the implication of our analysis on 4d N = 4 superconformal field theory, it would be very interesting to pursue this direction.
Our paper is organized as follows. In §2, we review the 2d small N = 4 superconformal algebra and construct the supermultiplet using the global part of the superVirasoro alge-bra. In addition, we analyze short-multiplets and decompositions of long-multiplets into short-multiplets. In §3, we compute superconformal blocks, starting from basic building blocks such as 2-point functions, and 3-point functions. A heavy amount of computation is significantly simplified using R-symmetry and Fermion number selection rules. The solution of the system of linear equations for 3-point functions is unique and is expressed in terms of 10 independent constants that match with the counting using the superspace. This provides a strong consistency check of our calculation. With the superconformal blocks, we obtain crossing equations that can be used in the numerical analysis. In §4, an alternative approach to compute superconformal blocks, using N = 4 superspace [33,34,39], was presented. We compute 3-point, and 4-point invariants and using them, construct Nilpotent invariants for superconformal block expansion. Our goal was to use Casimir differential equation to solve the superconformal block, but N = 4 superspace does not seem to fully represent small N = 4 superconformal algebra. As it is not a complete treatment, we pointed out some limitations that we encountered. We conclude the paper with future directions §5. Since 2-point and 3-point function data is huge, we included a part of them in Appendices §B, §C, §D and this submission is also accompanied by a separate Mathematica file that contains all the data.

2d small N = 4 superconformal algebra
In this section, we will provide basic elements that will be used to calculate long-multiplet n-point functions of 2d small N = 4 global long-multiplets. In §2.1, we review 2d small N = 4 superconformal algebra. We will focus on the global part of the super Virasoro algebra. Following the general analysis that was done for d ≥ 3 in [40,41], we build a longmultiplet and short-multiplets in §2.2, along with the decomposition of the long-multiplet into various short-multiplets. It is essential to do short-multiplet analysis even though we are computing long-multiplet 4-point function, since the stress energy tensor lies in one of the short-multiplets. The identification of the multiplet that contains the stress energy tensor is crucial in the bootstrap analysis of central charges, as one needs to compute the 4-point function with stress energy tensor exchanged. Furthermore, we have identified the short-multiplets that contain flavor current operator, which can be used in the bootstrap analysis for 2d CFT with global symmetry.

N = 4 superconformal algebra
Let us review small N = 4 superconformal algebra following [35][36][37]. Other than usual Virasoro algebra generators, due to enhanced supersymmetry, the superconformal algebra contains supersymmetry generators G a r , superconformal symmetry generatorsḠ a r , and SU (2) R R-symmetry current algebra generators T i m , where a, b are a SU (2) R spinor indices, i is SO(3) R vector index, m ∈ Z, and r ∈ Z/2, as we restrict ourselves in the NS sector.
The super-Virasoro algebra generators satisfy following (anti-)commutation relations.
In the following discussion, we will only use the global part of the superconformal algebra to compute 2-point, 3-point, and 4-point functions. It would be far more constraining to use infinite dimensional super-Virasoro algebra when one try to bootstrap two dimensional conformal field theories, but unfortunately, the full recursion relation that leads to the approximate expression for conformal block for extended supersymmetry has not been worked out in the literature. For now, after Zamoldchikov derived recursion relation for Virasoro conformal block [19], only N = 1 super-Virasoro conformal block recursion relation was obtained [20]. In spite of this limitation to use full super-Virasoro symmetry, we expect using only global part of super-Virasoro symmetry and Zamolodchikov recursion relation on conformal blocks should be enough to constrain the system and study spectrum.
Non-trivial (anti-)commutation relations for global part of small N = 4 algebra are

Long-multiplets
Given the algebra, we want to construct N = 4 long-multiplet that are labeled by superconformal primary at the bottom of the multiplet. First of all, define superconformal primary operator O h,r or corresponding state |O h,r to be those annihilated by all positive Fourier modes of super-Virasoro algebra and eigenstate of zero modes of the algebra: where h is conformal weight and r indicates spin r/2 representation of SU (2) R . We will use operator O h,r and state |O h,r interchangeably.
repeatedly on superconformal primary O h,r until it annihilates, one can obtain global long-multiplet L r . Note that by definition of long-multiplet, there is no null-state in the multiplet. Hence, the length of a long-multiplet is purely determined by Fermi-statistics of raising operators. The general structure of the long-multiplet is following: Here the superscripts (n) on each component indicates the number that G orḠ act on O h,r ; we will call half of this number as level k = n/2. As G α −1/2 andḠ α −1/2 are fermionic generators, they annihilate any states after acting twice, hence the level of highest component in the long-multiplet is 4/2 = 2.
To make sure that all the components O (n) h,r of the long-multiplet to be (quasi)conformal primaries, one should modify them properly, checking the (quasi)conformal primary condition: L +1 |O (n) h,r = 0. To illustrate this point clearly, let us explicitly work out the long-multiplet built on φ h,0 , calling it L 0 .
Following diagram shows how to act G −1/2 ,Ḡ −1/2 until they annihilate superconformal primary and complete the multiplet.
Here, G α In other words, each of component operator can be expressed as as ordered action on the state |φ in radial quantization using state/operator correspondence. We will stick to this convention throughout this paper. Some of the operators in the diagram do not satisfy the (quasi)conformal primary condition: L +1 |O (n) h,r = 0, hence one needs to correct the definition of those operators to become a conformal primary for later use. Below, we only present the operators that are modified. Other operators remain same. (2.5) Here Similarly, we write down L r for higher r. For r = 1, we have In other words, each of component operator can be expressed as Next, (quasi)conformal primary condition should be imposed. For simplicity, we will drop [0] assuming the fields shown below are all bottom component of SU (2) R multiplet.
Other components of SU (2) R multiplet can be completed by successively acting SU (2) R raising operator as before. Now, let us write down the most general L r ; here we include all the corrections so the operators below are all (quasi)conformal primaries. To clearly illustrate SU (2) R tensor selection rules, we adopt a new convention for each operator F[r][n], where F is name of an operator(e.g. φ, ψ, . . .), r represents the rank of SU (2) R representation, and n indicates the component of SU (2) R multiplet. We denote n = 0 as its bottom component, as before.
Rather than writing down whole SU (2) R multiplet, we will only present the SU (2) R bottom component of each operator for simplicity. Finally, note that G α andḠ α are both r = 1; we omitted subscript −1/2 on these generators. (2.9)

Short-multiplets
Superconformal algebra determines shortening condition for the long-multiplet. General analysis was done in [40,41] for higher dimension 3 ≤ d ≤ 6. We will use their insights to analyze our case and sometimes adopt their conventions.
By sandwiching between two superconformal primary states |φ h,r and imposing unitarity, one gets This implies the multiplet is shortened when superconformal primary satisfies h = r 2 condition. By looking at the algebra, one can easily see that only this specific type of anti-commutator gives the non-trivial shortening condition that gives zero in the norm of descendants, as it is clear in the explicit calculation given in Appendix §B.
Let us apply this to L 0 , L 1 , L r≥2 , separately. For L 0 , as h[φ] → 0, only the superconformal primary survives φ = 1 (2.12) This is the unit operator of CFT. Let us denote it as A 0 . For L 1 , as h[φ α ] → 1 2 , there is one short-multiplet, as shown below. φ α is twocomponent fermion and ψ, χ are boson. We denote it as A 1 .
This multiplet should be the one that contains flavor current operator. The reason is following. As flavor symmetry commutes with superconformal symmetry, the superconformal multiplet that contains the flavor current operator should place it at the top of the multiplet. One can see the top component of this short-multiplet does not carry SU (2) R index, consistent with the flavor symmetry current operator being R-symmetry neutral. Furthermore, we know that the conformal weight of {ψ, χ} is 1, which is the right dimension of flavor current operator. Also, flavor symmetry current operator can not be in the long-multiplet, as the top-component of any long-multiplet should have conformal weight 2, at least.
For L r≥2 , as h[φ[r ≥ 2]] → r 2 , there is one short-multiplet that appears at the bottom corner. We denote it as A r .
For r = 2, it is natural to think holomorphic stress-energy tensor lives in this short-multiplet as a top component. First, the top component has desired quantum number: (h, r) = (2, 0). Second, as stress energy tensor should commute with global super(conformal)symmetry , it should be on top of multiplet. Furthermore, other components of the multiplet reproduce the desired content of stress-energy multiplet: SU (2) R R-symmetry current operator with SU (2) R rank-2 at the bottom and global super(conformal) currents with SU (2) R rank-1 in the middle. Of course, each operator in the multiplet has expected conformal dimension: 1, 3 2 , 2. In N = 2 superconformal field theory, stress energy tensor lives in N = 2 longmultiplet [8]. The N = 2 long-multiplet is short-multiplet in the point of view of N = 4 theory. Above analysis shows that in N = 4 theory, stress energy tensor should live in the short-multiplet, different from N = 2 case.
There are yet another types of short-multiplets for each L r that come from Fermistatistics of G α Note that different from higher dimension, there is no actual chirality, so it indicates that the multiplet is built by acting only with G α . We denote them as B r andB r .

Decomposition of the Long-multiplets into the Short-multiplets
Similar to higher dimensional superconformal field theory, long-multiplet has a decomposition into short-multiplets. We could see all 2d N = 4 short-multiplets that appear in the decomposition of long-multiplet are 'Short-multiplet at Threshold', in the terminology of [41].
Let us illustrate this point with L 0 , L 1 long-mutiplet.
The green threshold short-multiplet of Moreover, red and yellow operators form L 1 , L 2 short-multiplet that were called as A 1 , A 2 . Finally, it can be checked that the top blue short-multiplet is the short-multiplet of L 2 , L 3 at threshold. Hence, the long-multiplet L 0 , L 1 decomposes when superconformal primary saturates unitarity bound as where we used convention F r [h] for long or short multiplet with rank-r su(2) R representation and conformal weight h. More generally, From this, we can see in 2d N = 4 superconformal algebra, the shortening condition and the kind of short-multiplet that could appear is simpler compared to higher dimension analogue [41]. This is not surprising as there is no non-trivial Lorentz symmetry index, unless combined with Left-moving non-SUSY side, and the R-symmetry algebra is simple in 2d superconformal field theory.
The short-multiplet structures can also be read off from the direct calculation of twopoint function that we will perform in the next section and present in the appendix §B. In short, two-point functions constructed from L 0 ,L 1 ,L 2 have zero at h = 0, h = 1 2 , h = 1, respectively. They are unique zeros for each multiplet and the highest degree is 2, as the G,Ḡ anti-commutator can at most appear twice when we build long-multiplet, due to grassmann nature of the supersymmetry generators.

Superconformal block computation
The main object to study is 4-point function of identical rank-0 long-multiplet L 0 . From now, we will interchangeably use L 0 and Φ i (Z i ) to denote rank-0 long-multiplet. In superspace, Φ i (Z i ) has following expansion with proper SU (2) R index contraction assumed: One way to study 4-point function is to work in superspace, as it provides a natural framework to use the superconformal algebra to fix overall structure of 4-point function and selection rules to classify non-trivial component 4-point functions, such as φ 1 φ 2 φ 3 φ 4 , ψ 1 χ 2 φ 3 φ 4 , τ 1 φ 2 φ 3 φ 4 . However, we found N = 4 superspace has a subtlety that prevented us to use it to compute superconformal blocks. Still, we could proceed to compute component 4-point functions by classical method in computing n-point function and superconformal algebra that we will describe in this section. We will separately discuss N = 4 superspace in the next section, up to the point that we could reach and comment on the subtle point.
We will compute all possible 4-point functions of component operators in L 0 : If we treat different SU (2) R index α = 1, 2 separately, in principle there are 16 4 possible 4-point functions to compute. The number grows tremendously if we include L 0 L 0 L r three point functions. Of course, Fermion number and SU (2) R symmetry selection rules help to restrict the set to a reasonably small subset.
Let us start with the simplest one φ 1 φ 2 φ 3 φ 4 to illustrate the strategy to get conformal block decomposition of general 4-point functions. Here, φ i are identical superconformal primaries of long-multiplet L 0 . Note that although we used different indices to distinguish their positions in superspace for the operators φ i , they are essentially identical supercon-formal primaries with same h: Here I ab (x) is a Lorentz tensor structure. We can decompose each of 4-point function into bosonic blocks: This decomposition is the essential property for long-multiplet 4-point function analysis, since it provides a detour, not to use the unknown N = 4 super-Virasoro conformal blocks. One might think that the coefficients in front of g h O i may be dependent, but this is not the case as can be seen in the explicit computation of 3-point functions shown in the subsequent sub-sections. The independence of the coefficients in the 4-point function decomposition indicates the novelty of our N = 4 study, distinguished from the bootstrap of non-supersymmetric 2d CFT.
Due to Zamolodchikov [19], approximate expression for g h (z) is known and it can be recursively deduced from sl(2) bosonic conformal block Hence, what remains to compute is 3-point function coefficients f φφOn and 2-point function normalization f OnO n . Similarly, it is easy to generalize to any component 4-point functions, p 1 p 2 p 3 p 4 . In general, (3.7) Note that the exchange operators O i , O i can belong to any rank-r supermultiplet L r , not just L 0 where all 4 external operators belong to. We can classify blocks shown in (3.7) in terms of what super-multiplet {O i } belongs to. There are three possible supermultiplets that participate in (3.7); they are L 0 , L 1 , L 2 . As before, the necessary computation reduces down to figure out non-trivial f p 1 p 2 O , f O p 3 p 4 and f OO .

Selection rules
There are two selection rules that we will use frequently in the subsequent sections. 1. Fermion number selection rule, 2. R-symmetry selection rule. For the first selection rule, we assign Fermion number to each operator in L 0 , L 1 , L 2 : For n-point function F 1 . . . F n not to vanish, the sum of fermion number should be even.
Next, in describing R-symmetry selection rules, we will take the general notations that were used in illustrating the primary operators in the general L r multiplet. There are two rules: , the first rule is and the second rule is , the first rule is and the second rule is

2-point functions
Let us start with the simplest case: 2-point function normalization f OO . A simple fact that super(conformal) symmetry generator annihilates vacuum leads to following equation Commuting G α ,Ḡ α to the left generates a set of linear equations for each pair {F 1 , F 2 }.
where factors of (−1) is due to the commutation of F i and fermionic operator G α . f i is 0 for F i boson, and 1 for F i fermion. Here, G α F i andḠ α F i can be computed by utilizing superconformal algebra and is equal to linear combination of F j with known proportionality constants c α j,k ,c α j,k with possible corrections L −1 F i , L −1 L −1 F i as we have seen in the definitions of conformal primaries before.
As 2-point functions are fixed up to normalization constant , the above equation (3.14) becomes By factoring out common denominators, (3.17) becomes from which one can read off the coefficients of We can easily solve the linear system to fix all f F i F j up to three independent constant each of which coming from L 0 , L 1 , L 2 , separately. The three constants can be fixed to 1 in the later computation.
We should obtain all non-trivial 2-point functions of L 0 , L 1 , L 2 . The reason that we do not consider higher rank L r with r > 2 will become clear in the next subsection §3.3 where we discuss 3-point function. In practical computation, because of large number of operators in L 0 , L 1 , L 2 , it would be better to first restrict the set of non-trivial two-point functions by using 2-point function definition, and SU (2) R symmetry selection rules. For instance, 1. φφG α = φψ α + ψ α φ will not give any non-trivial condition as both φψ α and ψ α φ vanish since φ and ψ α have different conformal weight.
2. Equations from φψ 1 G 1 are trivial as it is equal to φ0 + ψ 1 ψ 1 , where the second 2-point function vanishes due to SU (2) R selection rules.
It would be instructive to explicitly work out one non-trivial example that passed the two simple tests above, as we will use this procedure to construct higher n-point functions either. Start from χ 2τ G 1 , where both χ andτ are in L 0 built from conformal weight h superconformal primary φ.
From superconformal algebra, we know χ 2C2 vanishes due to R-symmetry selection rule. For next two terms, we substitute explicit 2-point function formula (3.16) and get So, we obtained one linear equation that relates two 2-point function normalizations. Similarly, we can do the same thing for χ 2τ G 2 , χ 2τḠ1 , χ 2τḠ2 . We automated this procedure in Mathematica to compute all non-trivial 2-point function normalization f F i F j . For simplicity, let us only present those of L 0 . They are fixed up to one constant that we denoted as f φφ . Of course, most of them vanish by definition of 2-point function.

3-point functions
We want to compute 3-point function OPE coefficients f F 1 F 2 F 3 , where F 1 , F 2 ∈ L 0 , F 3 ∈ L 0 , L 1 , L 2 , L 3 , L 4 , L 5 , as F 1 , F 2 are two of external primary operators in the 4-point function and F 3 is an exchanged primary operator that can in principle be in any of F i , although we will find out 3-point function with F 3 ∈ L 3 , L 4 , L 5 vanishes.
n−point correlator calculation becomes extremely complicated from n ≥ 3, as the total number of possible 3 point combination increases tremendously compared to that of 2-point function case.
Hence, we need to introduce more systematic way of selecting non-trivial equations by refining SU (2) R selection rules. We used the rules to construct a linear system for 2-point functions, but as it starts to impose non-trivial constraints starting from 3-point function, we describe the additional procedure here. Before deriving a system of equations from G α ,Ḡ α commutation, it may be more efficient to use SU (2) R generators T + 0 to obtain extra set of equations that prepares a smaller subset of entire set on which we apply G α ,Ḡ α commutation procedure. The method is essentially the same as before with T + 0 replacing G α ,Ḡ α that leads where F i ∈ {φ, ψ, τ, τ,τ , t 0 , t 1 , t 2 , t t , C,C, d}, component primary operators in supermultiplets L 0 , L 1 , L 2 . By commuting T + 0 to the left, we get a set of linear equations As T + 0 is a raising operator of SU (2) R R-symmetry algebra, it is a map from one operator to the same operator with different SU (2) R index. For instance, T + 0 : ψ 1 → ψ 2 . As a result, it will not be as powerful as constraints from equations of G α ,Ḡ α commutation; however, it completely reduces SU (2) R degeneracy and enables us to only consider one component of each SU (2) R multiplet. Especially, this procedure helps us to reduce a significant number of degree of freedom in higher rank representation in L 1 and L 2 . Following table shows how much the number of operators in each L r is reduced after using T + 0 . Let us work out explicitly, for example,

Rank
As three terms in the last line of (3.26) share the same set of conformal weights, (h + 1 2 , h + In this way, we can reduce the number of 3-point functions that we need to treat in G, G commutation equations. Following table shows the reduction of number of independent 3-point functions L 0 L 0 L i after using Fermion number, R-symmetry selection rules and T + 0 commutation. Next, let us work out one simple case from G,Ḡ commutation. Here, f φφt 2 t vanishes due to U (1) R ⊂ SU (2) R selection rule. By change of variables t = z 13 /z 12 , (3.28) reduces to As this should be satisfied for all t > 0, (3.29) is equivalent to (3.30) In this way, by constructing a linear system using all possible non-trivial 3-point function equations, we can solve all f F 1 F 2 F 3 in terms of 10 independent constants. 5 come from F 3 ∈ L 0 : {f φφφ , f φφτ , f φφτ , f φφt 0 , f φφd }, 4 from F 3 ∈ L 1 : {f φφψ , f φφχ , f φφC , f φφC } and 1 from F 3 ∈ L 2 : {f φφt }. This means that all 3-point OPE coefficients can be expressed in terms of f φφf , where φ is a superconformal primary, and f ∈ L r . The counting matches with superspace computation in §4.2. Moreover, the solution set is unique. This provides a strong consistency check of this rather tedious computation. We could check explicitly that all 3-point functions with F 3 ∈ L r with r > 2 vanish, which is not surprising due to the R-symmetry selection rule. We have seen this pattern in the short-multiplet analysis (2.14) too. Hence, we can focus on F 3 ∈ {L 0 , L 1 , L 2 } from now on.

4-point functions
From the above computation, we have gathered all information to construct 4-point function defined in (3.7). It remains then to find the independent set of external 4-points. The strategy is the same as that of lower correlators. Instead, we stop after solving T + 0 equations, which give a set of independent 4-point functions. We could proceed to solve G,Ḡ equations to produce crossing equations, but we can equivalently construct all the 4-point functions as described at the beginning of this section §3 only using 2-point and 3-point functions and also crossing equations that we will describe in the next subsection §3.5.
We construct a linear system of equation commuting T + 0 inside 4-point functions.
Note that different from before, 4-point function coefficient f 1234 [z] is not a constant, but a function of cross-ratio z = z 13 z 24 z 12 z 34 . However, it will not make things complicated, as T + 0 action does not generate any z i dependence C(∂ z i , z i ). After solving all T + 0 equations, we get 4826 4-point functions that will give non-trivial equations from G orḠ commutations.
By using superconformal invariance, we can fix 4-point function of long-multiplet L 0 L 0 L 0 L 0 with first two to be L 0 but last two operators to be superconformal primary φ ∈ L 0 . In other words, in a particular frame:  Consider for example, τ φφφ . By (3.3), it is (3.36)

Crossing equations
By exchanging F 1 and F 3 in F 1 F 2 F 3 F 4 , we get crossing channel F 3 F 2 F 1 F 4 . As we know all possible F 3 F 2 O , O F 1 F 4 , OO , we can compute all 42 crossing channel superconformal blocks that correspond to (3.33).
(3.37) For instance, 1 ↔ 3 crossing channel of (3.36) is (3.38) We have dropped anti-holomorphic part of equations until now, and now we want to restore it. Since we assume that only right moving part is N = 4 supersymmetric, we can simply replace z dependent factors or functions with following rules: where ∆ = h+h 2 . By equating F 1 F 2 F 3 F 4 n and F 3 F 2 F 1 F 4 n for each n = 1, . . . , 42, we arrive at a system of 42 linear equations that can be represented by fourty two 10 × 10 block diagonal F n ij matrices. Most of the matrix component of F n ij is zero, as one can see in (3.36), (3.38). There are 195 independent crossing equations that we need to solve using SDPB. We provide selected few in the Appendix and the complete set of crossing equations is available in the separate Mathematica file.

N = superspace approach
In this section, we explain a separate approach to analyze N = 4 long-multiplet 4-point functions using superspace and Casimir differential equations, generalizing N = 2 superspace approach that was introduced in [8]. We have obtained 3-point, 4-point superconformal invariants, and Nilpotent invariants that are used in the long-multiplet 4-point function expansion and Casimir differential operator that can be used to get the conformal block expansion. Due to a subtle problem in N = 4 superspace, we could not get the final expression for superconformal blocks, but we proceeded as much as possible and pointed out the problem.
In this section, we heavily used Mathematica package 'grassmann.m' developed by Matthew Headrick [38]. For concise presentation, we will drop left-moving non-supersymmetric part of 4-point functions consistently throughout the section and re-introduce in the appropriate place.
We want to study long-multiplet L 0 4-point function In N = 4 superspace, a generic long multiplet is represented as With the explicit superspace expansion (4.1), one can expand the 4-point function in terms of nilpotent superconformal invariants {I i , J j , K k } that we will derive in this section, as where g n is a monomial of {I i , J j , K k } and F n (z) is component 4-point function such as ψ 1 χ 2 φφ . By studying N = 4 superspace 3-point invariants U 123 and 4-point invariants {I i , J j , K k }, one can systematically deduce the expansion.
Each of 4-point function F n (z) can be decomposed into Virasoro conformal blocks labeled by the exchanged conformal primary in one of three long-multiplets: L 0 , L 1 , L 2 . Here, We derived 3-point superspace invariants that are function of three superspace coordinates, invariant under all superconformal transformations. If one start from ansatz that only depends on (super)translation invariants, the main task is to impose inversion invariance that guarantees conformal invariance. We present the detail of the derivation in Appendix §A. The 3-point invariants are   Hence, there are 10 four-point invariants constructed from fermionic bilinears and 1 fourpoint invariant from usual bosonic coordinates in N = 4 superspace. The number 10 matches with the number of independent 3-point OPE coefficients obtained in the previous section §3. 3. Due to grassmann nature, {U i , V j , W k } are nilpotent. This is the reason that one can use those invariants when expanding long-multiplet 4-point functions as it guarantees finite truncation in the superspace expansion. To obtain clear nilpotency relations, we want to convert the basis into special form. To guess the form of nilpotent invariants, first let us take following limits of 4-point invariants:

Nilpotent invariants and their independent combinations
From (4.7), we get some hint to construct a good basis of nilpotent invariants Invariants defined in (4.7) should combine to produce simple limits. Hence, we can guess following combinations and compute their limit in the convenient frame.
As nilpotency condition preserves under conformal transformations, we can use {I i , J i , K i } to figure out the whole expansion of long-multiplet 4-point function. In this frame, (4.2) reduces to Now, we need to get all independent g n (I i , J j , K k ). We start by writing down all possible letters and words and reduce the set by using algebraic relations between them. Some obvious nilpotent relations are following. From now, we will redefine I 1 = (I 1 +I 2 )/2 and I 2 = (I 1 − I 2 )/2.
It is also possible to deduce all non-vanishing g(I i , J j , K k ). We will classify them by the number of letters.
Single Letter : Two Letters : First of all, single-letters are all independent; we can not express any of those in terms of a linear combination of others. There are 10 of them.
There are many two-letters relations between invariants, part of which we wrote down below: These relations reduce the number of two letters from 39 to 20.

Crossing Equations
To write down the crossing equations, we first need to derive the crossing transformed invariants. Crossing acts on {I i , J j , K j } by exchanging (z 1 ,z 1 , θ 1 ,θ 1 ) and (z 3 ,z 3 , θ 3 ,θ 3 ). The crossing symmetry imposes following constraint: The RHS of (4.17) can be rearranged into expansion with the same set of parameters of LHS, since we have seen 41 combinations of nilpotent invariants are linearly independent and span the set of possible 4-point invariants. We could find I t I , J t j , K t k .
) From this, one can deduce the crossing transformed set of nilpotent invariants {S t i }. Given the above information, we are ready to write down crossing equations, starting from 1 − 2, 3 − 4 channel 4-point function: where S i ∈ S. Here we coupled with left-moving non-supersymmetric conformal block that addsz dependence. The crossing channel is The crossing equation is then

Casimir equation
Now, it remains to solve g n (z,z) that take following form. The reason for this particular decomposition is explained around (4.3). By solving g n (z,z), we mean that we solve for c n i with n = 1, . . . , 42, i = 1, . . . , 5 using following set of coupled differential equations [42], Casimir differential equations: where D[I 0 ] is a matrix of differential operators with respect to I 0 and c 2 is a 42 × 42 matrix with constant that depends on h.
We derived quadratic Casimir for N = 4.
(4.24) The way to derive it is to start from the most general ansatz C 2 = i∈b∪f c i G i that is a linear combination of all possible quadratic global generators that is invariant under global N = 4 superconformal algebra and fix coefficients using the algebra, where After moving to the convenient frame x 3 → 0, x 4 → ∞, θ 3 , θ 4 ,θ 3 ,θ 4 → 0, the Casimir operators only act on first two operators of 4-point function Φ 1 Φ 2 φ 3 φ 4 . Hence, we need to get the two particle Casimir operator, similar to [43].

The puzzle
To solve the Casimir equation, we need to know the superspace representation of N = 4 superconformal algebra generators that consist of the quadratic Casimir operator (4.24). For simple notation, let us re-introduce small N = 4 superconformal algebra with outerautomorphism manifest. The global N = 4 superconformal algebra is for i = 1, 2, 3, m, n = 0, ±1 and r = ± 1 2 . Here, α, β indices are that of SU (2) F outerautomorphism of small N = 4 superconformal algebra.
To find the superspace representation of each generator, we start with the most general ansatz and fix the coefficients {p, q, r, s, t, u, v, w, y} in front of each term. (4.28) By using (4.27), we can try to fix the coefficients. However, there is no non-trivial set of solution for the coefficients. As we did not have a superspace representation of each generator, we could not set up the Casimir differential equation that would solve to coefficients in the conformal block expansions.

Discussion
In this paper, we initiated general 2d N = 4 superconformal bootstrap study, using longmultiplet. As we have not specified any other properties of theory, other than N = 4 superconformal symmetry, our analysis is general, but at the same time lack of decorations that could arise from global symmetries and analysis of BPS 4-point functions. This study provides the starting point for numerical bootstrap analysis using the standard methods [44,45]. Also, since our superspace analysis is incomplete, it would be interesting to resolve the problem that we pointed out. Other than these obvious directions, there are several ways to use this set-up by imposing more input depending on the specific theories that preserve N = 4 superconformal symmetry. Different from N = 2 theories, N = 4 theory has stress energy tensor in shortmultiplet. Rather than considering long-multiplet 4-point function L 0 L 0 L 0 L 0 , we can consider short-multiplet 4-point function of L 2 that contains stress energy tensor at the top. Because stress energy tensor is universal ingredient of any CFT [46], this will also provide a general information on N = 4 CFTs. Moreover, we expect L 2 4-point function, though it is BPS, will give different restriction that L 0 4-point function could not impose. Since the length of the multiplet and the number of components are reduced significantly in the BPS multiplet, we expect efficient numerical analysis here.
CFTs with global symmetry will give more stringent bounds, since there is a nontrivial relation between level of Kac-Moody algebra and total central charge. Especially, there is a series of interesting (0, 4) theories with E 8 global symmetry that arise from IR limit of E-string worldsheet gauge theories [23,24]. The gauge theory lives on N D2 brane worldvolume(012 direction); it has finite length(L) in direction 2 and extends between NS5 brane and D8/O8 complex. By taking the L small, there appears 2d O(N ) supersymmetric gauge theory with SO(16) global symmetry. Flowing into deep IR(semi-classical limit or Higgs branch [25]), one expects to get 2d (0, 4) superconformal theory with central charge (c L , c R ) = (6N, 12N ) and global symmetry E 8 . It would be interesting to study this series of CFT labeled by the number of E-strings and it would be also very interesting to see if there is another IR limit that comes from different choice of IR R-symmetry, which was once suggested in [24]. Other big family of (0, 4) theories [26] comes from twisted compactification of class-S theory, and [29] from brane box model, which are another interesting models to study using bootstrap technique.
Lastly, our analysis can be used to study 4d N = 4 SYM or SCFT, as 2d small N = 4 chiral algebra appears in a particular twisted Q−cohomology of 4d N = 4 SCFT [30]. [31] mentioned this fact in their 4d N = 4 numerical bootstrap analysis, but did honest 4d superconformal block computation to construct 4-point function and crossing equations. It would be interesting to use our result to study the 4d N = 4 SCFT as we have much more crossing equations that can give more stringent bounds.
We use following conventions (for α, β = 1, 2 and a = 1, 2, 3): For two doublets ψ α and χ α , Inversion acts on the superspace coordinates by Also, as usual, rigid SUSY with parameters ε α acts as Then θ ij ,θ ij , and A noninfinitesimal SUSY transformation with parameters η and η acts as Then we construct a large superconformal transformation from a translation by (ζ 1 , η 1 , η 1 ) followed by inversion, followed by translation by (ζ 2 , η 2 , η 2 ), followed by dilatation by λ (the dilatation will be implicit). 2 We are grateful to Ori Ganor for sharing his unpublished notes that show preliminary result for the 3-point invariant of N = 4 superspace [47] z → z−η 1 θ+η 1θ +ζ 1 → − After this transformation, we are left with with η 2 and η 2 as above, and We still have dilatation freedom and SU (2) R freedom, and if we also require U (1) invariance, we are left with one overall invariant

B 2-point function normalization
Here, we collected all relevant 2-point function normalization. We also submitted Mathematica files that have the same information.
B.1 L 0 We order and number each component fields of L 0 from bottom component to top component.

C 3-point function normalizations
Since there are too many 3-point functions, here we only present those of φL 0 L 0 , φL 0 L 1 , φL 0 L 2 with φ ∈ L 0 . L 0 , L 1 , L 2 are built from superconformal primary φ, φ α , φ αβ with weight h. The most general results with different weights and L 0 L 0 L 0 , L 0 L 0 L 1 , L 0 L 0 L 2 can be found in the separate Mathematica file that we submitted. Let us describe how to read off the result from our mathematica file. The script consists of the substitution rules for all 3-point coefficients. As we described, they are expressed fully in terms of 10 independent constants. f [i, j, k] indicates 3-point function OPE coefficients of three fields F i , F j , F k , where the i, j indices run in (B.1), and k index runs in (B.1),(B.3),(B.5) in each file. h [1], h [2], h [3] is the conformal weight of superconformal primary of F i , F j , F k , respectively.
where h is the conformal weight for superconformal primary of L 0 , L 1 , L 2 , and h ex is the conformal weight for exchanged operator.
In the mathematica file, we showed all the crossing equations that were obtained by studying long-multiplet 4-point function of L 0 . H is the conformal weight of the superconformal primary of L 0 .