Aspects of Superconformal Multiplets in D>4

We explicitly construct and list all unitary superconformal multiplets, along with their index contributions, in five and six dimensions. From this data, we uncover various unifying themes in the representation theory of five- and six-dimensional superconformal field theories. At the same time, we provide a detailed argument for the complete classification of unitary irreducible representations in five dimensions using a combination of physical and mathematical techniques.


Introduction and Summary
By virtue of their high degree of symmetry, superconformal field theories (SCFTs) are somewhat simpler arenas in which to test and understand general ideas in quantum field theory (QFT) like duality [1,2] and emergent symmetry [3]. Moreover, since the endpoints of supersymmetric (SUSY) renormalisation group (RG) flows are often SCFTs, they constrain the asymptotics of QFT and give rise to striking manifestations of the idea of universality [4].
To construct an SCFT, we start with a superconformal algebra (SCA). In his pioneering work, Nahm showed that these algebras admit a simple classification [5]. This list of allowed SCAs gives rise to important constraints even away from criticality. 1 At a superconformal point, we find the basic building blocks of the theory-the multiplets of local operators 2 -by studying the unitary irreducible representations (UIRs) of the SCA.
These UIRs are of two general types: short representations and long representations.
Short UIRs have primaries that are annihilated by certain non-trivial combinations of the Poincaré supercharges while long representations do not. Moreover, short representations can contribute to the superconformal index [6,7], can realise non-trivial structures like chiral algebras [8,9] and chiral rings that enjoy various non-renormalisation properties, can be used to study the structure of anomalies [10], and can describe the SUSY-preserving relevant and marginal deformations of SCFTs [11,12]. Furthermore, by understanding how short representations recombine to form long representations one can hope, when sufficient symmetry is present, to bootstrap non-trivial correlation functions of local operators and perhaps even whole theories (see [8,13] for important recent progress on this front).
In this paper, we perform the conceptually straightforward, but calculationally nontrivial, task of giving the level-by-level construction of all UIRs for the five-dimensional N = 1 and six-dimensional (1,0) and (2,0) SCAs (these are the only allowed SCAs in five and six dimensions [5]). We also calculate the most general superconformal index associated with these multiplets. Our approach throughout is based on the presentation and conventions of [7,14]. 3 1 For example, Nahm's classification shows that six-dimensional (1, 1) QFTs do not flow to SCFTs at short distances. 2 We should also supplement these degrees of freedom with the non-local operators of the theory. 3 Note that a comprehensive classification of unitary irreducible representations (UIRs) for all SCAs was carried out in [7,[14][15][16][17][18] and further discussed in [12]. However, in this paper we add to these works by giving the level-by-level construction of the corresponding multiplets as well as the resulting superconformal index contributions. Part of this work was already done in [8] for the 6D (2, 0) SCA (but we will provide the full set of multiplets and index contributions for this algebra).
We expect the results assembled here to be useful for more detailed studies of the many still-mysterious SCFTs in five and six dimensions (see, e.g., the theories described in the classic works [19][20][21] and the more recent literature [22]) as well as for more general explorations of the space of SCFTs in these dimensions.
Before delving into technical details, we should note that although the precise construction of the various multiplets depends on the spacetime dimension and amount of supersymmetry, we find various unifying themes in five and six dimensions. For example, we will see that multiplets containing conserved currents or obeying equations of motion cannot take part in recombination rules. 4 In particular, we will show that: (i) Multiplets containing higher-spin currents can never recombine into long multiplets.
This statement is compatible with the fact that theories in 5D and 6D are isolated as SUSY theories [11,12], because it implies that there are no exactly marginal SUSY deformations of (almost) free theories. 5 This situation is unlike the one in four dimensions, where such pairing up is required in the decoupling limit of N = 2 and N = 4 superconformal gauge theories.
(ii) Multiplets containing conserved spin-two currents can never recombine into long multiplets. This statement is also compatible with the isolated nature of 5D and 6D theories [11,12], because it implies that there are no marginal SUSY couplings between general isolated SCFTs. In four dimensions, such recombination is required in coupling isolated interacting N = 2 SCFTs (see, e.g., [2,24] for examples of such couplings between theories and [25] for a more general discussion).
(iii) Flavour-symmetry currents 6 are present in 5D and 6D (1,0) linear multiplets, which cannot recombine into long multiplets (for a discussion in the six-dimensional context, see [26]). This situation is analogous to the one in four-dimensional N = 2 theories. 4 Unlike in 4D, where one can sometimes tune an exactly marginal parameter and short multiplets may recombine into long ones, 5D and 6D superconformal theories are necessarily isolated [11,12]. Therefore, recombination should be understood purely at the level of superconformal representation theory, i.e. how one can write a long multiplet in terms of short multiplets. 5 Here we use [23]. 6 In this paper, we define these to be currents for symmetries that commute with the full SCA. Therefore, these currents do not sit in multiplets with higher-spin symmetries. Note that, as in four dimensions, there are spin-one currents that give rise to charges that commute with the R symmetry and also sit in higherspin multiplets, but the corresponding charges are necessarily part of a larger algebraic structure including supercharges and higher-spin charges. For example, we will see such a current in (2.30). This operator gives rise to a charge that acts on bosons but not on fermions.
(iv) Certain classes of multiplets cannot appear in free SCFTs. 7 These include certain 6D (1,0) B-type and C-type multiplets in Sec. 4 as well as some 6D (2,0) B-type, C-type and D-type multiplets in Sec. 5.
The methodology we use to extract our results is rather general and well established [7,8,12,14,27,28]. Indeed, we use a simple Verma-module construction to obtain all irreducible representations of the full SCA from irreducible representations of its maximal compact subalgebra. The UIRs are labelled by highest weights corresponding to superconformal primaries, from which all descendants are recovered by the action of momentum operators and supercharges. Hence, each UIR is uniquely identified by a string of quantum numbers, which characterises the superconformal primary state. As we described above, there are both long and short multiplets. The short multiplets have null states, which can be consistently deleted (hence the moniker, "short"). A complete classification of short UIRs can be obtained by imposing the condition of unitarity. For special values of the quantum numbers characterising short UIRs, additional null states can occur. The precise enumeration and analysis of all such possibilities using unitarity is an intricate task.
Once all null states have been identified, the Racah-Speiser (RS) algorithm simplifies the multiplet construction and clarifies the origin of equations of motion and conservation equations, whenever these are present. 8 The RS algorithm provides a prescription for the in terms of constraints imposed on operators inside the multiplet [27].
Since we study the 5D N = 1, 6D (1, 0), and 6D (2, 0) SCAs, our presentation is split 7 Throughout this paper we only consider unitary theories. 8 A concise summary of the Racah-Speiser algorithm can be found in App. B of [27].
into three corresponding sections, all of which are largely self-contained. The reader who is familiar with the classification of UIRs and only interested in looking up the results can proceed directly to the relevant tables. Each multiplet is labelled by the quantum numbers designating its superconformal primary and the shortening conditions the latter obeys.
Some multiplets with special values for their quantum numbers admit a distinct physical interpretation; these are dealt with separately. For those interested in the approach employed to obtain our diagrams, we provide a detailed discussion for the case of the 5D N = 1 SCA in Sec. 2, which extends naturally to 6D in Sec. 4 and Sec. 5. Throughout this analysis, special emphasis is put on identifying operator constraints, whenever present.
Each section also contains expressions for recombination rules and indices for the superconformal multiplets under study. A short collection of simple applications arising from the results of our analysis is presented in Sec. 6. Finally, Sec. 3 contains an argument for the complete classification of 5D UIRs, which has been missing from the literature (paying attention to some recent observations made in [29][30][31]).
We also include various appendices. App. A, B and C contain conventions and results which are necessary for our multiplet construction but would shift the focus away from our aim in the main part of the text; SCA conventions for five and six dimensions, the construction of supercharacters and the superconformal index, as well as the relationship between the RS algorithm and the identification of operator constraints. App. D collects the superconformal indices for all 5D and 6D multiplets for quick reference. As the 6D (2,0) refined indices are cumbersome, we only ever write down their Schur limit. However, we also provide a complementary Mathematica file with all the refined superconformal indices in five and six dimensions. Finally, App. E contains the explicit 6D (2,0) spectra, which are too unwieldy to present in Sec. 5.
Note added: While finalising our construction of multiplets in D > 4, we became aware of an upcoming publication ( [32] cited in [12]). This upcoming work promises to be broader in scope than our own and have overlap with some of our constructions. Knowledge of the multiplet structure in five and six dimensional SCFTs, in [32], is essential background material for the results in [10,12]; see also [26,33]. We would like to thank C. Córdova, T. Dumitrescu, and K. Intriligator for relevant correspondence, as well as for pointing out an error in the classification of 5D multiplets in a previous version of this paper.

Multiplets and Superconformal Indices for 5D N = 1
We begin by providing a systematic analysis of all short multiplets admitted by the 5D N = 1 SCA, F (4). This involves a derivation of the superconformal unitarity bounds. By doing so we reproduce the results of [7]. We then proceed to write the complete multiplet spectra and compute their indices. Our notation and conventions for the 5D SCA are provided in App. A.

UIR Building with Auxiliary Verma Modules
The superconformal primaries of the 5D SCA F (4) are designated |∆; l 1 , l 2 ; k , where ∆ is the conformal dimension, l 1 ≥ l 2 > 0 are Lorentz symmetry quantum numbers in the orthogonal basis and k is an R-symmetry label. Each primary is in one-to-one correspondence with a highest weight state of the maximal compact subalgebra so(5) ⊕ so(2) ⊕ su(2) R ⊂ F (4). 9 There are eight Poincaré and eight superconformal supercharges, denoted by Q Aa and S Aa respectively-where a = 1, · · · , 4 is an so(5) Lorentz spinor index and A = 1, 2 an index of su(2) R . One also has five momenta P µ and special conformal generators K µ , where µ = 1, · · · , 5 is a Lorentz vector index. The superconformal primary is annihilated by all S Aa and K µ . A basis for the representation space of F (4) can be constructed by considering the following Verma module A,a (Q Aa ) n A,a µ P nµ µ |∆; l 1 , l 2 ; k hw (2.1) for some ordering of operators, 10 where n = Aa n A,a andn = µ n µ denote the "level" of a superconformal or conformal descendant respectively. In order to obtain UIRs, the requirement of unitarity needs to be imposed level-by-level on the Verma module. This leads to bounds on the conformal dimension ∆.
The highest su(2) R -weight level-one superconformal-descendant states can be expressed in a particularly suitable alternative basis as Λ a 1 |∆; l 1 , l 2 ; k hw , where we define The λ a b are functions of the so(5) quantum numbers and Lorentz lowering operators. The combinations (2.2) have the property that, when acting upon a conformal-primary highest- 9 The quantum numbers labelling the primary are eigenvalues for the Cartans of the maximal compact subalgebra in a particular basis. 10 Any other ordering can be obtained using the superconformal algebra.
They can be uniquely determined by imposing the requirement that all Lorentz raising operators and R-symmetry raising operators annihilate Λ a A |∆; l 1 , l 2 ; k hw and are given in App. A. 4. It turns out that the most stringent unitarity bounds emerge by studying the norms of states constructed by acting with the Λ a 1 s on the superconformal primary. We provide a detailed argument in favour of this fact in Sec. 3.

Ill-defined States
The definition of the Λ generators-as explicitly given in App. A.4-is such that for certain values of the quantum numbers the resulting state is not well defined. Consider e.g. the where H 2 |∆; l 1 , l 2 ; k = l 2 |∆; l 1 , l 2 ; k . The above is clearly ill-defined for l 2 = 0: Although M − 2 |∆; l 1 , 0; k = 0 the factor of 1/l 2 diverges and the norm of (2.3) is indeterminate. However, there exist cases where products of ill-defined Λs can lead to well-defined states, through various cancellations. Hence, one has to perform a delicate analysis of such possibilities through explicit calculation. This phenomenon will be very important in the classification of unitarity bounds below, where one needs to evaluate the norms of all well-defined, distinct (i.e. not related through commutation relations) products of Λs. 11 Unitarity Bounds for l 1 ≥ l 2 > 0 We can calculate the norms of the superconformal descendant states at level one to be For example, in our upcoming discussion of unitarity bounds for l 1 = l 2 = 0, Λ 3 1 |∆; 0, 0; k hw and Λ 2 1 |∆; 0, 0; k hw are individually ill defined, while Λ 2 1 Λ 3 1 |∆; 0, 0; k hw is not. This can in turn lead to the wrong identification of shortening conditions, since ||Λ 2 1 Λ 3 1 |∆; 0, 0; k hw || 2 = B 3 (0, 0, k)B 1 (0, 0, k), whereas ||Λ 1 1 Λ 2 1 |∆; 0, 0; k hw || 2 = B 2 (0, 0, k)B 1 (0, 0, k), with the first one leading to more stringent unitarity bounds.
where we have normalised |∆; l 1 , l 2 ; k hw 2 = 1. Observe that these norms are all of the form Λ a 1 |∆; l 1 , l 2 ; k hw 2 = ∆ − f a (l 1 , l 2 , k) g a (l 1 , l 2 ) =: B a (l 1 , l 2 , k)g a (l 1 , l 2 ) , where g a (l 1 , l 2 ) is a positive-definite rational function in the fundamental Weyl chamber, l 1 ≥ l 2 > 0. Unitarity demands that the norms are positive semidefinite and this imposes a bound on the conformal dimension via the functions B a (l 1 , l 2 , k). The strongest bound on the conformal dimension is provided by B 4 (l 1 , l 2 , k) ≥ 0. When B 4 (l 1 , l 2 , k) > 0 the UIR can be obtained using (2.1). The resulting multiplet is called "long" and labelled L.
When B 4 (l 1 , l 2 , k) = 0 the state is null. This means that the primary obeys the "shortening condition" Λ 4 1 |∆; l 1 , l 2 ; k hw = 0. All such states can be consistently removed from the superconformal representation. The resulting multiplet is "short" and labelled as type A. Since it can be reached from a long multiplet by continuously dialling ∆ it is called a "regular" short multiplet. At higher levels, n a=1 Λ a 1 |∆; l 1 , l 2 ; k hw with n > 1, the norms involve products of B a (l 1 , l 2 , k)s and the strongest bound still comes from B 4 (l 1 , l 2 , k) ≥ 0.
Therefore, there will be no change to the bounds obtained at level one.
Unitarity Bounds for l 1 > l 2 = 0 We now turn to the special case with l 1 > 0, l 2 = 0, where the concept of ill-defined states becomes important. When l 2 = 0 the operator Λ 4 1 is not well defined and we have to omit the level-one state Λ 4 1 |∆; l 1 , 0; k hw from our spectrum. Naively, the strongest bound then arises from the norm of the state Λ 3 1 |∆; l 1 , 0; k hw . However, the level-two state Λ 3 1 Λ 4 1 |∆; l 1 , 0; k hw is actually well defined, as can be explicitly checked. Its norm is proportional to Unitarity Bounds for l 1 = l 2 = 0 The same logic extends to l 1 = l 2 = 0: At level one the only well-defined state is Λ 1 1 |∆; 0, 0; k hw . However, there exist well-defined states at levels two and four, obtained by These give rise to the conditions B 1 (0, 0, k) = 0, B 3 (0, 0, k) = 0 or B 4 (0, 0, k) ≥ 0 and lead to the new set of isolated short multiplets D.
We summarise their properties and list all short multiplets for the 5D SCA in Table 1.

Additional Unitarity Bounds
Finally, there are supplementary unitarity restrictions and associated null states originating from conformal descendants. These have been analysed in detail in [14,28], the results of which we use. Saturating a conformal bound results in a "momentum-null" state, where the corresponding shortening condition is an operator constraint involving momentum analogues of the superconformal Λs [28]. In that reference, a prescription is given for removing the associated states, P µ |∆; l 1 , l 2 ; k hw , from the auxiliary Verma-module construction, again in analogy with the superconformal procedure. 13 However, we will choose not to exclude any momenta from the basis of Verma-module generators (2.1). After using the RS algorithm this choice will allow us to explicitly recover highest weight states corresponding to the operator constraints from the general multiplet structure.
One can combine the conformal and superconformal bounds to predict that operator constraints will appear in the following short multiplets: (2.7) The multiplet D[0, 0; 0] does not belong to this list as it is the vacuum.

Highest Weight Construction through the Auxiliary Verma Module
The Λ basis through which we obtained the unitarity bounds could in principle be used to construct the full multiplet. However, executing this procedure would require knowledge of the full Clebsh-Gordan decomposition for the resulting states. This is a very difficult task to carry out in practice. For that reason we will resort to constructing the highest weights of the superconformal representation using the auxiliary Verma module via the Racah-Speiser algorithm. This greatly simplifies the Glebsch-Gordan decomposition by implementing it at the level of highest weights. 13 Note that this does not mean that all conformal descendants of a particular type should be removed from the set of local operators. Table 1: A list of all short multiplets for the 5D N = 1 SCA, along with the conformal dimension of the superconformal primary and the corresponding shortening condition.

Multiplet Shortening Condition Conformal Dimension
The Λ a 1 in the shortening conditions are defined in (2.2) and (A.17). The first of these multiplets (A) is a regular short representation, whereas the rest (B, D) are isolated short representations. Here Ψ denotes the superconformal primary state for each multiplet.
Having motivated the use of the auxiliary Verma module basis, we will implement the conjectural recipe of [7,27,34] to generate the spectrum; see also App. C of [8]. According to these references, in addition to removing the supercharge associated with the shortening condition, one is instructed to also remove any other supercharge combination that annihilates the auxiliary primary.
To make this point more transparent, let us consider the example of the B[l 1 , 0; k] multiplet. The shortening condition dictates that we remove Λ 1 3 Ψ ⇒ Q 13 Ψ aux , where note that Ψ and Ψ aux have the same quantum numbers. Since l 2 = 0 we have that M − 2 Ψ aux = 0 and therefore Hence we are required to also remove Q 14 from the auxiliary Verma-module basis. Note that (2.8) does not imply Λ 1 4 Ψ = 0 but is merely a prescription for obtaining the correct set of highest weights. One could similarly use any lowering operator of the maximal compact subalgebra. E.g. for k = 0 additional conditions can be generated by acting on the existing ones with R-symmetry lowering operators, 14 resulting in the removal of more combinations of supercharges from the set of auxiliary Verma-module generators. We will mention explicitly the full set of such "absent supercharges" at the beginning of each case in our upcoming analysis.
We emphasise that this is an auxiliary Verma-module construction which leads to the same spectrum in terms of highest weights. If one is interested in the precise form of the operators, the much more involved Λ-basis should be used.

The Procedure
Based on the above ingredients, let us summarise our strategy for constructing the superconformal UIRs: 1. For a given multiplet type, begin with a superconformal primary and consider the highest-weight component of the corresponding irreducible representation of the Lorentz and R-symmetry algebras.
2. Implement the conjecture of [7,8,27,34] to determine all combinations of supercharges which need to be removed from the auxiliary Verma-module basis (2.1).
3. Use the remaining auxiliary Verma-module generators to determine the highest weights for all descendant states. This may result in some of the quantum numbers labelling the highest weight state becoming negative. 4. Apply the Racah-Speiser algorithm to recover a spectrum with only positive quantum numbers. 15 This could result in some states being projected out, while others acquiring a "negative multiplicity". The latter can cancel out against other states with the same quantum numbers but positive multiplicity. 16 Any remaining states with negative multiplicity can be interpreted as operator constraints. This conjectural identification follows [27] and is based on a large number of examples, but can be additionally supported using supercharacters; c.f. App. C.2.
5. In some special cases, the supercharges that have been removed from the auxiliary Verma-module basis anticommute into momentum generators, which should also be removed. This has the effect of projecting out states corresponding to operator constraints. The operator constraints can be restored using the discussion in App. C. 15 The details of the Racah-Speiser algorithm needed for this step can be found in App. C. 16 There are instances when this general procedure leads to ambiguities, i.e. there is more than one choice for performing the cancellations; see also [12]. However, these can be resolved uniquely by the requirement that all highest weight states should be reached by the successive action of allowed supercharges on the superconformal primary. For the examples that we investigate in this article, this phenomenon only appears in the (2,0) SCA (B[c 1 , c 2 , 0; 0, 0], C[c 1 , 0, 0; 0, 0]), in which case the multipet spectra have also been compared to the construction by successive Q-actions.
The spectrum of a given superconformal multiplet can always be obtained following these steps and we believe the results (in all D > 4 SCAs that we have considered) to be correct: They satisfy the expected recombination rules, which have been checked using supercharacters. Of the multiplets that do not appear in recombination rules, for 5D N = 1 and 6D (1, 0) SCAs, we have also explicitly constructed the states using free fields. For 6D (2, 0) the multiplets that do not appear have had the Schur limit of their superconformal indices matched with the results of [8]. Finally, an additional check on the computational implementation of this procedure is via supercharacters, which can be calculated in two ways: Either by evaluating them over the states of a given short multiplet obtained using the RS algorithm or directly using the Weyl character formula; c.f. App. B. Both of these agree for all multiplets listed in our work.

5D N = 1 Multiplet Recombination Rules
For the purposes of listing the recombination rules as well as for explicitly constructing the multiplets, we will find it more convenient to switch to the Dynkin basis for the various quantum numbers. That is, we will use Short multiplets can recombine to form long multiplets when the conformal dimension for the latter approaches the unitarity bound, that is when ∆ + ǫ → 3 2 K + d 1 + d 2 + 4. It can then be checked, using the results that we will present in the following sections, that (2.10) The following multiplets do not appear in a recombination rule: We define the superconformal index with respect to the supercharge Q 14 , in accordance with [7,35]. This is given by where making use of the spin-statistics theorem the fermion number is F = d 2 and the trace is over the Hilbert space of operators of the theory. The states that are counted by this index satisfy δ = 0, where It is easy to see that as a result long multiplets can never contribute to the index. The charges d 1 and 2 3 ∆ + 1 3 (d 1 + d 2 ) appearing in the exponents of (2.12) are eigenvalues for the generators commuting with Q 14 , S 21 and consequently with δ. In practice, this index can be explicitly evaluated as a supercharacter for each of the multiplets constructed below. A detailed construction of characters for superconformal representations is reviewed in App. B.

Long Multiplets
We can now go ahead with the explicit construction of multiplets. Since long multiplets are not associated with any shortening conditions, we can proceed as per (2.1) acting with all supercharges and momenta on the superconformal primary to obtain the unitary superconformal representation.
We will choose to group the supercharges together as Q = (Q A1 , Q A2 ) andQ = (Q A3 , Q A4 ), purely for book-keeping purposes. The explicit quantum numbers of these supercharges are given by (2.14) With this information in hand, it is straightforward to map out their action starting from a superconformal primary, labelled by (K) (d 1 ,d 2 ) : where we have split the actions of Q andQ into two "chains". Since the Dynkin labels are generic, there is no need to implement the RS algorithm. By definiton, these multiplets do not contribute to the superconformal index.

A-type Multiplets
Recall from Table 1 The two remaining cases with d 2 = 0 and d 1 = d 2 = 0 can be obtained by implementing the recipe at the end of Sec. For K = 0 and d 1 , d 2 generic the primary still lies above the unitarity bound for all conformal descendants and there are no momentum-null states. Moreover, one also needs to 17 Once again, we are using a Dynkin basis for the quantum numbers.
remove Q 24 , Q 23 Q 24 and Q 21 Q 22 Q 23 Q 24 from the construction of the respective multiplets for d 1 , d 2 > 0, d 2 = 0 and d 1 = d 2 = 0 using the auxiliary Verma module. The resulting spectrum is no different from setting K = 0 and running the RS algorithm for su(2) R .
The index over the spectrum of all A-type multiplets is given by We may then combine the action of these supercharges to construct the following grid; acting with Q is captured by southwest motion on the diagram, whileQ by southeast motion. This module is well defined for all d 1 ≥ 1 values: This UIR can also be obtained from B[d 1 , 0; K] by setting d 1 = 0. The same arguments as in the B[d 1 , 0; K] case can be applied to study the behaviour of the multiplet for concrete K values. Again, we find that non-cancelling negative-multiplicity states only appear at The superconformal index for all values of d 1 and K > 0 is calculated to be Let us now address the special case with K = 0. For d 1 = 0 this family of B-type multiplets contains higher-spin currents and has a primary corresponding to the symmetric traceless representation of so (5). One finds that Q A3 and Q A4 need to be removed from the auxiliary Verma-module basis (2.1). Therefore the entire representation is built by just acting on the superconformal primary with the set of Qs from (2.14) and is This result seems to contradict (2.7), which predicted the presence of operator constraints. However, note that the absent Qs anticommute into P 5 , which has therefore been implicitly removed from the auxiliary Verma-module generators. This has the effect of projecting out states corresponding to operator constraints; we will henceforth refer to the remaining states as "reduced states".
The operator constraints can actually be restored-c.f. App. C-by utilising the following relationship between characterŝ (2.28) Here theχ and χ correspond to taking a character of the superconformal representation without/with the P 5 respectively. We can appropriately account for the negative contributions to the RHS via negative-multiplicity states. The full multiplet can then be expressed as: An example of the above [3 + d 1 ; d 1 , 0; 0] primary is the operator where φ A is a free hypermultiplet scalar. This object satisfies the generalised conservation equation It is interesting to point out that for d 1 = 1 the superconformal primary is an R-neutral ∆ = 4 conserved current, which is not itself a higher-spin current. The higher-spin currents are instead found as its descendants. This logic also shows that the commutator of the conserved charge, T , associated with the primary has a non-vanishing commutator with the supercharges where T ′ 11 is the charge associated with the level-one descendant current. This commutator explains why we do not refer to T as a flavour symmetry.
The index is of course insensitive to the above discussion; it gives the same answer when evaluated either on (2.27) or (2.29) and that is We recognise that this multiplet contains the R-symmetry current, the supersymmetry current and the stress tensor, as well as their corresponding equations of motion. In particular, we identify: This multiplet also contains three states that do not obey conservation equations.
The index over this multiplet counts just two components of the R-symmetry current, J 11 1 and J 11 4 . It is given by (2.36)

D-type Multiplets
For the D[0, 0; K] multiplet unitarity requires that the conformal dimension of the primary is ∆ = 3K 2 . The null state associated with this condition instructs us to remove Q 11 from the basis of auxiliary Verma-module generators, but due to having d 1 = d 2 = 0 one actually needs to remove the larger set of supercharges Q 1a , a = 1, · · · , 4. In fact, in this case Λ 1 1 Ψ = Q 11 Ψ = 0 and one has the shortening condition which renders the multiplet 1 2 BPS. The action of the remaining supercharges on a primary state where the quantum numbers are kept generic is then (2.38) After setting d 1 = d 2 = 0 and employing RS the full multiplet is For low enough values of K there can be additional reflections. If K = 1 or K = 2 these will correspond to operator constraints, to be discussed below. If K = 3 then the only problematic state will be the level four [0, 0; K − 4], which becomes [0, 0; −1]; this is on the boundary of the Weyl chamber, and hence will also be deleted.
The index for this multiplet is . When K = 1 no additional shortening conditions arise, therefore we may proceed directly using (2.39). One recovers: We can recognise these highest weight states as  The index for this multiplet counts the first operator in the primary, φ 1 along with the P 1 and P 2 conformal descendants. Therefore we have .
This is the single-letter index obtained in [35,36].

5
(2) (0,0) We recognise these fields to be a scalar µ (AB) in the 3 of su(2) R , a symplectic Majorana fermion ψ A a in the 2 of su(2) R , a vector J µ and an R-neutral scalar M. Furthermore the negative-multiplicity state is the equation of motion for the vector current ∂ µ J µ = 0.
This multiplet is also known as the linear multiplet and appeared in the UV symmetryenhancement discussion of [37].
The corresponding index is: .
This concludes our listing of superconformal multiplets for the 5D SCA.

On the Complete Classification of UIRs for the 5D SCA
We will next give an argument that the conditions imposed in Sec. there is no such argument in the literature for 5D; for proofs in 4D and 6D see [15][16][17][18].
Our argument will proceed in three steps. We first use the results of [31] to establish when the Verma module can admit null states (i.e., when the Verma module is reducible).
We then show that only the 5D multiplet types A, B, D, L can be unitary (determining necessity). Finally, we argue that all multiplets of the above type are indeed unitary (determining sufficiency).

Reducibility Conditions for F (4)
We begin by determining the necessary and sufficient conditions for when a representation of the 5D SCA can contain null states (i.e., when the F (4) parabolic Verma module is reducible). This question has recently been revisited in [30,31], the results of which we use.
Towards that end, let us first establish some notation following, e.g., [38].
The 5D SCA, F (4), is a Lie superalgebra for which we choose the following simple-root with corresponding Dynkin diagram and Cartan matrix: The α 1 , α 3 , α 4 ∈ Π 0 are even simple roots, while α 2 ∈ Π 1 is an odd simple root. These have been expressed in an orthogonal basis in terms of ǫ i , for i = 1, 2, 3, and δ such that The simple roots can be used to obtain the even and odd positive roots of F (4) where the signs are not correlated. One also typically defines the set Φ 0 and the set of although it is easy to see that in our case Φ 0 = Φ 0 and Φ 1 = Φ 1 .
The difference between the half-sum of the positive even roots ρ 0 and half-sum of the positive odd roots ρ 1 denotes the Weyl vector, which can be evaluated to be A highest weight representation of F (4), λ, may be expanded in terms of the fundamental weights of the bosonic subalgebra g 0 = su(2) R ⊕ so(5, 2) as where the ω a for a = 1 and a = 2, 3, 4 are the fundamental weights associated with the simple roots of su(2) R and so(5, 2) respectively and the H a are the Cartans. For each of these bosonic subalgebras, the fundamental weights are related to the simple roots via the inverse of the Cartan matrix. For su(2) R and so(5, 2) this allows us to express Since in our discussion thus far we have been labelling representations in terms of their l = su(2) R ⊕ so(5) ⊕ so(2) ⊂ g Dynkin labels, we will also need the so(5) fundamental weightsω which we use to write Now, by definition we have that since H 1,3,4 multiply the fundamental weights of su(2) R and so(5) respectively. We can naturally assign the remaining label to the conformal dimension H 2 + H 3 + 1 2 H 4 = −∆, where the minus sign is there to account for the signature of so(5, 2). This means that Expressing the highest weight using the above in the orthogonal basis Finally, the following two sets have to be defined before proceeding, where we have adapted the definitions of [31] to the case of F (4): and Ψ λ,non−iso := α ∈ Φ + n,0 where Φ n,0 := Φ n ∩ Φ0, with Φ n = Φ + \Φ l . Hence we have where the signs are not correlated.
With this information at hand, we can now apply the criteria of [31] regarding the necessary and sufficient conditions for irreducibility: From the first bullet point above we immediately determine that the module is reducible when Ψ λ,iso = ∅. This phenomenon occurs when eight conditions are satisfied, one for each of the positive odd roots, Note that, according to the second bullet point above, there can exist additional reducible representations only if Ψ λ,non−iso = ∅. This latter condition is equivalent to the following equations with n α ∈ Z >0 : However, the conformal dimensions of these modules are below the unitarity bounds associated with conformal descendants: Operator with spin Hence, these additional modules are not unitary and will not be useful for the discussion of UIRs.
We conclude that F (4) admits reducible modules precisely when the eight level-one norms become null, as predicted by [14].

An Argument for Necessity
For the necessity part of the argument we will begin from the various unitarity bounds for the A, B and D-type multiplets. As shown in Sec. 2.1, these can be expressed as where the f a (d 1 , d 2 , K) are given in (3.15) and we observe that f 4 > f 3 > f 2 > f 1 . We will next show that there can be no UIRs of the 5D SCA except for the ones satisfying (3.18) and ∆ > f 4 , i.e. that there is at least one negative-norm state in the intervals (f 3 , f 4 ), Towards that end, consider the norms of the following well-defined states: When ∆ = f 4 this describes an result in the positivity of F 4 and F 3 , but leads to F 1 being negative. ∆ < 0 is forbidden by unitarity since it would result in a negative conformal dimension for the superconformal primary. If ∆ = f a for a = 1, 2, 3 then, despite the fact that F 4 = 0, the level-one norm of the state Λ 4 1 Ψ will be negative, and is therefore not allowed.
The same logic can be applied to this case when taking d 2 = 0.
This is because the level-one state Λ 4 1 |∆; d 1 , 0; K hw does not exist as it is ill-defined for d 2 = 0, hence there would be no negative-norm state associated with it.
3. d 1 = d 2 = 0 : There is now one additional way to saturate a unitarity bound. The states Λ a=2,3,4 1 |∆; 0, 0; K hw are all ill-defined for these d i values. Therefore one can We conclude that the necessary conditions for unitarity are ∆ ≥ f 4 for d 1 , d 2 = 0; Similarly to (2.30), for a free theory we may write a superconformal primary with

An Argument for Sufficiency
(3.20) 18 In a previous version of this paper, an additional necessary condition for unitarity had been incorrectly We thank the authors of [32] for bringing this to our attention. 19 The relevant Mathematica notebooks can be obtained from the authors upon request.
Therefore, this multiplet must necessarily be unitary.
However, we can argue for the unitarity of other choices of quantum numbers using the recombination rules can be completed into a long multiplet by including the states Ψ L A 2 . We may therefore write for all states in Ψ L A 2 . Their norms can be explicitly calculated and are related by where note that the coefficient on the RHS is always positive.
Since the multiplet A[d 1 , 1; K] can appear in the recombination rule (3.22) as we can use the above argument to conclude that in the ǫ → 0 limit A[d 1 , 0; K + 1] is also unitary. Moreover, one can also write To that end, recall the index of a unitary SCFT is a sum over contributions from individual local operators of the theory transforming in UIRs. Now, the index for a 5D SCFT with gauge group SU(2) and global symmetry group E 2 = SU(2)×U(1) was evaluated in [35]; see also [39]. Moreover, the arguments in [35] suggest that this result can be understood as an index coming from one of Seiberg's unitary 5D SCFTs [20] that appear in the low-energy limit of string theory. Assuming this identification is correct (or, least restrictively, that the index computed in [35] corresponds to an index of some unitary theory with E 2 global symmetry), then it is straightforward to show that A[0, 0; 0] is also a unitary representation.
To understand this last statement, note that the authors of [35] found the index in question admits the following expansion where q is a fugacity associated with the instanton contributions and can be written as a direct sum of two unitary multiplets as in (3.22), it too must be unitary. The remaining two recombination rules of (2.10) 2 + ǫ = f 4 + ǫ, every state has positive norm. Since the norm of a state is a smooth function of ∆, in order for it to become negative as we move arbitrarily far from f 4 , it must first become null.
By definition this scenario would lead to a reducible Verma module. However, in Sec. 3.1, we determined all the values for which such a situation can arise. It is easy to see that f 4 is the largest of these and therefore the norm can never be negative.
We therefore conclude that the multiplets L[∆; and D[0, 0; K] are unitary hence concluding our argument.

Multiplets and Superconformal Indices for 6D (1,0)
We now switch to six dimensions. In this section we provide a systematic analysis of all superconformal multiplets admitted by the 6D N = (1, 0) algebra, osp(8 * |2). 20 Since the method for building UIRs is completely analogous to Sec. 2.1 we will be brief and merely sketch the derivation of the unitarity bounds obtained in detail by [7]. We then proceed to write out the complete set of spectra for superconformal multiplets along with their corresponding superconformal indices.

UIR Building with Auxiliary Verma Modules
The superconformal primaries of the algebra osp(8 * |2) are designated |∆; c 1 , c 2 , c 3 ; K and labelled by the conformal dimension ∆, the Lorentz quantum numbers for su(4) in the Dynkin basis c i and an R-symmetry label K. Each primary is in one-to-one correspondence with a highest weight labeling irreducible representations of the maximal compact subalgebra so(6) ⊕so(2) ⊕su(2) R ⊂ osp(8 * |2). There are eight Poincaré and superconformal supercharges, denoted by Q Aa and S Aȧ -where a,ȧ = 1, · · · , 4 are su(4) (anti)fundamental indices and A = 1, 2 an index of su(2) R . One also has six momenta P µ and special conformal generators K µ , where µ = 1, · · · , 6 is a Lorentz vector index. The superconformal primary is annihilated by all S Aȧ and K µ . A basis for the representation space of osp(8 * |2) can be constructed by considering the following Verma module 20 Some of the multiplets up to spin 2 discussed in this section have been previously constructed using the dual gravity description in [40].
where n = Aa n A,a andn = µ n µ denote the level of a superconformal or conformal descendant respectively. UIRs can be obtained from the above after also imposing the requirement of unitarity level-by-level.
We will now review the conditions imposed by unitarity, starting with the superconformal descendants [7]. For descendants of level n > 0 it suffices to calculate the norms of states in the highest weight of su(2) R , since these provide the most stringent set of unitarity bounds [7,14,18]. 21 Hence the unitarity bounds stem from the study of superconformal descendants arising from the action of supercharges of the form Q 1a . Before proceeding, it is convenient to define the basis A a |∆; The Υ a b are functions of u(4) Cartans and Lorentz lowering operators, the explicit form of which can be found in [7]. They map highest weights to highest weights.
One then conducts a level-by-level analysis of the norms for the superconformal descendants, while also keeping track of whether the states A a |∆; c 1 , c 2 , c 3 ; K hw are well defined when setting c i = 0. This gives rise to a set of regular and isolated short multiplets, obeying certain shortening conditions [7]. The precise form of the A a is unimportant; they have the same quantum numbers as the supercharge Q 1a . The shortening conditions can then be translated into absent generators in the auxiliary Verma module (4.1) and we are instructed to remove the corresponding combinations of supercharges in a straightforward way. These results are summarised in Table 2.
Additional absent supercharges can be obtained when K = 0 by acting on the existing null states with Lorentz and R-symmetry lowering operators. As a result additional supercharges need to be removed from (4.1). These occurrences will be dealt with on a case-by-case basis.
Finally, there can be supplementary unitarity restrictions and associated null states arising from considering conformal descendants. These have been studied in detail in [14,28].
In analogy with the 5D approach, we will choose not to exclude any momenta from the basis of auxiliary Verma-module generators (4.1); with the help of the RS algorithm this will account for states corresponding to equations of motion and conservation equations. 21 In the language of [31], the proof for the 6D SCAs in [18] only deals with the conditions associated with Ψ λ,iso = ∅. We have investigated the possible reducible modules coming from Ψ λ,non−iso = ∅ and, similar to our findings in Sec. 3, all predicted cases are below the unitarity bounds associated with conformal descendants. The A a in the shortening conditions are defined in (4.2) and [7]. The first of these multiplets (A) is a regular short representation, whereas the rest (B, C, D) are isolated short representations. Here Ψ denotes the superconformal primary state for each multiplet.

Multiplet Shortening Condition Conformal Dimension
We can combine the information from the conformal and superconformal unitarity bounds to predict when a multiplet will contain operator constraints. These are found to be and will be treated separately in the following sections. The multiplet D[0, 0, 0; 0] does not belong to this list as it is the vacuum, which is annihilated by all supercharges and momenta.

6D (1,0) Recombination Rules
Short multiplets can recombine into a long multiplet L. This occurs when the conformal dimension of L approaches the unitarity bound, that is when ∆ + ǫ → 2K + 1 2 (c 1 + 2c 2 + 3c 3 ) + 6. We find that (4.4) These identities can be explicitly checked, e.g. by using supercharacters for the multiplets that we discuss below.
A small number of short multiplets do not appear in any recombination rule. These are We define the 6D (1,0) superconformal index with respect to the supercharge Q 14 , in accordance with [7]. This is given by where the fermion number is F = c 1 + c 3 . The states that are counted satisfy δ = 0, where The Cartan combinations ∆− 1 2 K, c 2 and c 1 are generators of the subalgebra that commutes with Q 14 , S 24 and δ. This index can be evaluated as a 6D supercharacter, as discussed in App. B.

Long Multiplets
Long multiplets are generated by the action of all supercharges. We will choose to group them into Q = (Q A1 , Q A2 ) andQ = (Q A3 , Q A4 ), with their individual quantum numbers given by The action of these supercharges on a superconformal primary (K) (c 1 ,c 2 ,c 3 ) is given by

A-type Multiplets
Recall from Table 2 that the A-multiplets obey four kinds of shortening conditions. These result in the removal of the following combinations of supercharges from the basis of Verma module generators (4.1): (4.10) With regards to the spectrum, let us first consider the A[c 1 , c 2 , c 3 ; K] multiplet. The action of the Q-set of supercharges is the same as for the long multiplet (4.9), however since we are also instructed to remove Q 14 theQ-chain becomes The remaining four cases can be obtained straightforwardly in a similar way. The resulting multiplet spectra turn out to be the same as starting with (4.11), substituting for specific c i values and running the RS algorithm.
For K = 0 one should also remove the additional supercharge combinations obtained by acting with the R-symmetry lowering operators on the supercharges mentioned in (4.10) from the basis of auxiliary Verma-module generators. Once again, the resulting spectra are equivalent to starting with the K = 0 multiplets, explicitly setting K = 0 and running the RS algorithm for su(2) R .
The superconformal index for all values of c i and K is given by This index satisfies the following recombination rules

B-type Multiplets
For the B-type multiplets, the supercharges that need to be removed from the basis (4.1) are (4.14) For the first type of multiplet, B[c 1 , c 2 , 0; K], one should also remove Q 14 . Acting on a primary with generic quantum numbers, (K) (c 1 ,c 2 ,c 3 ) , one has the same action as (4.9) for Q. TheQ-chain is modified to We can represent this multiplet on a grid, where acting with Qs corresponds to southwest motion on the diagram, while acting withQ to southeast motion The next multiplet type is B[c 1 , 0, 0; K], where one is instructed to remove Q 12 Q 13 due to the shortening condition. Since c 2 = c 3 = 0, one can deduce that Q 12 Q 14 and Q 13 Q 14 should also be removed. The multiplet spectrum is given by The last B-type multiplet to consider is B[0, 0, 0; K]. Its shortening condition is A 1 A 2 A 3 Ψ = 0 implying that Q 11 Q 12 Q 13 shoud be removed from the auxiliary Verma-module basis. One further deduces that Q 1a Q 1b Q 1c should also be removed from the set of generators since The index over B-type multiplets for all values of c i and K = 0 is given by . That is, we should completely remove the set ofQ supercharges of (4.8) and the spectrum is generated solely by acting with the set of all Qs: This result seems to contradict (4.3), which predicted the presence of operator constraints. However, note that the absent Qs anticommute into P 6 , which has therefore been implicitly removed from the auxiliary Verma-module generators. This has the effect of projecting out states corresponding to operator constraints and hence the spectrum only contains reduced states. The operator constraints can be restored using the dictionary developed in App. C. This can be done using the character expression where the hat indicates a character of the reduced (i.e. P 6 -removed) Verma module. The result is An example of a superconformal primary for a B[0, c 2 , 0; 0] multiplet with c 2 > 0 is where φ A is a free hypermultiplet scalar. It is interesting to point out that for c 1 = 1, c 2 = 0 or c 1 = 0, c 2 = 1 the superconformal primary is not higher spin. The higher-spin currents are instead found as their descendants. Moreover, the superconformal primary in the multiplet of B[0, 1, 0; 0] is a ∆ = 5, R-neutral conserved current. However, as explained around (2.32) in the case of 5D, this is not a flavour current.
The corresponding superconformal index for any c 1 , c 2 ≥ 0 reads . The last B-type multiplet of note is the stress tensor. Ordinarily, the shortening conditions require that we remove Q 1a Q 1b Q 1c from the basis of generators, but since K = 0 we also need to remove contributions obtained by acting repeatedly with R-symmetry lowering operators. The resulting multiplet is therefore We recognise these states as being associated with the fields where + denotes the selfdual part of the operator.
The index over this stress-tensor multiplet is counting three components of the R-symmetry current plus conformal descendants.

C-type Multiplets
The The corresponding superconformal index is Therefore the spectrum for C[0, 0, 0; K] is given by The associated index for K > 1 evaluates to . (4.34) The cases with K = 0, 1 are predicted to contain operator constraints from (4.3) and will be dealt with separately below.
For this class of multiplets, K = 0 and we obtain additional shortening conditions. These can be translated into the requirement that Q Aa for a = 1 be removed from the basis of auxiliary Verma-module generators. The latter then in turn imply that P 23 ∼ {Q 12 , Q 23 }, P 24 ∼ {Q 12 , Q 24 } and P 34 ∼ {Q 13 , Q 24 } have also been removed from the auxiliary Vermamodule basis. In vector notation, these momentum operators correspond respectively to P 3 , P 5 and P 6 . The implication of this fact is that the multiplet construction will only reproduce the reduced states, with the operator constraints having been projected out.
Thus the C[c 1 , 0, 0; 0] multiplet is very simply: The operator constraints can be restored by means of App. C. Implementing this would result in introducing-for each of the three states in (4.35)-the following combinations for a total of nine additional states corresponding to equations of motion. As a side comment, note that this spectrum is not the one we would have obtained had we just set K = 0 in (4.30) and implemented the RS algorithm. As such, the index that one obtains is different to just setting K = 0 in (4.31), and is given by . When c 1 = 0 we recover the free-tensor multiplet. In this case the lowering operator of su(2) R acts on the null-state condition Q 11 Q 12 Ψ aux = 0 to create additional shortening conditions. This translates into the requirement that should be removed from the basis of auxiliary Verma-module generators. The multiplet can be constructed using the remaining supercharges and is given by We identify the states with the following fields  The index for this configuration evaluates to These multiplets are simple to construct since, contrary to the above case, we need not act with the R-symmetry lowering operator. As such, we can take the spectrum of (4. (1) (c 1 ,0,0) which includes conservation equations. It is worth pointing out that the multiplet with c 1 = 1 contains a ∆ = 5, R-neutral conserved current as a level one descendant. However, we see that there is also a spin-2 current with ∆ = 6 at level three.
The corresponding index evaluates to Note that this index is valid for c 1 = 0. This is because setting K = 1 does not introduce any additional shortening conditions. However, when c 1 = 0 this multiplet does not contain currents with spin j ≥ 2, and will be discussed below. (1) (0,0,0) This multiplet contains a conserved K = 1 spin-1 current and a conserved R-neutral spin-3 2 current. These generate additional global and supersymmetry transformations, whichas we will discuss in Sec. 6.2-are necessary for the description of the (2,0) stress-tensor multiplet in the language of the (1,0) SCA.

Its index is then given by
. and hence the multiplet is 1 2 -BPS. Starting with a generic superconformal primary state (K) (c 1 ,c 2 ,c 3 ) , the two chains ob-tained from the action of Qs andQs are (4.47) One then needs to set c i = 0 and run the RS algorithm. Upon doing so, the result is We notice that for K ≤ 2 the above will lead to negative-multiplicity representations via the RS algorithm and hence operator constraints, to be discussed in the next section.
The superconformal index for K > 2 is given by The corresponding index is given by These states can be identified with the operators µ (AB) , ψ A a and the su(2) R -singlet conserved current J µ , with ∂ µ J µ = 0. This multiplet is also known as a linear multiplet and appears in [37,41].
The index for this multiplet is given by This concludes our discussion of the superconformal multiplets for the 6D (1,0) SCA.

Multiplets and Superconformal Indices for 6D (2,0)
We lastly turn to the construction of superconformal multiplets for the 6D (2,0) SCA, osp(8 * |4). Since we now have sixteen Poincaré supercharges, the UIRs will be much larger compared to the ones obtained in Sec. 2 and Sec. 4 and representing them diagrammatically would not be particularly instructive. Similarly, the full expressions for the most general ("refined") superconformal indices are unwieldy. Instead we will choose to detail the multiplet types, their shortening conditions and the "Schur" limit of their indices, although we do include the refined versions of the index in the accompanying Mathematica notebook. 22

UIR Building with Auxiliary Verma Modules
The superconformal primaries of the algebra osp(8 * |4) are designated |∆; c 1 , c 2 , c 3 ; d 1 , d 2 and labelled by the conformal dimension ∆, the Lorentz quantum numbers for su(4) in the Dynkin basis c i and the R-symmetry quantum numbers in the Dynkin basis d i . Each primary is in one-to-one correspondence with a highest weight labeling irreducible representations of the maximal compact subalgebra so(6) ⊕ so(2) ⊕ so(5) R ⊂ osp(8 * |4). There are sixteen Poincaré and superconformal supercharges, denoted by Q Aa and S Aȧ , wherė a, a = 1, · · · , 4 are (anti)fundamental indices of su(4) and A = 1, · · · , 4 a spinor index of so(5) R . One also has six momenta P µ and special conformal generators K µ , where µ is a vector index of the Lorentz group, µ = 1, · · · , 6. The superconformal primary is annihilated by all S Aȧ and K µ . A basis for the representation space of osp(8 * |4) can be constructed 22 A discussion of the null states for the 6D (2,0) SCA, along with the calculation of Schur indices for the various short multiplets, can be found in App. C of [8]. Here we additionally construct the full multiplets with an emphasis on the equations of motion and conservation equations. We also provide the refined indices for all multiplets. where n = Aa n A,a andn = µ n µ denote the level of a superconformal or conformal descendant respectively. In order to obtain UIRs, the requirement of unitarity needs to be imposed level-by-level on the Verma module. This leads to bounds on the conformal dimension ∆.
Starting with the superconformal descendants, the conditions imposed by unitarity can be deduced as follows. In principle, one needs to calculate the norms of superconformal descendants for n > 0. However, since it is sufficient to perform this analysis in the highest weight of the R-symmetry group [7,14,18], the results of Sec. 4.1 can be easily imported to the (2,0) case and we may still use the basis A a i · · · A a j |∆; c 1 , c 2 , c 3 ; d 1 , d 2 hw . 23 We need only convert the Cartan for the highest weight of su(2) R to so(5) R , which is done by simply This gives rise to a group of similar short multiplets, which we collect in Table 3. We will provide more details regarding the generators that are absent from the auxiliary Verma module when discussing individual multiplets. Furthermore, it will be important to clarify which additional absent generators can occur from tuning the R-symmetry quantum numbers, d 1 and d 2 . These turn out to be far more intricate than for (1,0).
The null states arising from conformal descendants are identical to Sec. 4.1. One can combine the information from the conformal and superconformal unitarity bounds to predict when a multiplet will contain operator constraints [14,28]. These special cases are The multiplet D[0, 0, 0; 0, 0] does not belong to this list as it is the vacuum, which is annihilated by all supercharges and momenta.

The 6D (2,0) Superconformal Index
We define the 6D (2,0) superconformal index with respect to the supercharge Q 24 . This is given by where the fermion number is F = c 1 + c 3 . The states that are counted satisfy δ = 0, with The exponents appearing in (5.5) are the eigenvalues for the generators commuting with Q 24 , S 34 and δ. This index can be evaluated as a 6D supercharacter; this is discussed in our App. B and App. C of [8].

The 6D Schur limit
The authors of [8] define the 6D Schur limit by taking t → 1 in (5.5). Under the trace the resulting index can be rewritten as where It can be seen that all exponents in (5.7) commute with the supercharge Q 12 and the 6D Schur index consequently counts operators annihilated by two supercharges. As a result the above index is independent of both e −β and p and the trace can be taken over operators satisfying δ = 0 = δ ′ : The operators contributing to the index in this limit satisfy (5.10)

Long Multiplets
Long multiplets are constructed by acting with all supercharges on a superconformal pri- . This leads to lengthy expressions which, although not presented here, are available from the authors upon request. For book-keeping purposes we will group the supercharges into Q = (Q 2a , Q 3a ) andQ = (Q 1a , Q 4a ) for the remaining of the 6D (2,0) discussion.

A-type Multiplets
Recall from Table 3 that the A-type multiplets obey four kinds of shortening conditions.
These result in the removal of the following supercharges from the basis of Verma module generators in (5.1) The A-type multiplets cannot contain operator constraints for any d 1 or d 2 .
One can arrive at new shortening conditions-and as a result a reduced number of Q-generators-for d 1 = 0, d 2 = 0 and d 1 = d 2 = 0. Consider for instance the case Table 3 the null state reads A 4 Ψ = 0. In the auxiliary Vermamodule basis, this corresponds to Q 14 Ψ aux = 0. However, since d 2 = 0 one also finds that R − 2 Q 14 Ψ aux = 0. This corresponds to having to additionally remove Q 24 from our auxiliary Verma-module basis. Following on from this, when d 1 = 0 the operator R − 1 also annihilates the superconformal primary and two new conditions are obtained: These correspond to also removing Q 34 and Q 44 respectively from the basis of Verma-module generators.
All A-type multiplets have zero contribution to the Schur limit of the 6D (2,0) index.
We provide their spectra in App. E.1.

B-type Multiplets
For the B-type multiplets, the supercharges that need to be removed from the auxiliary Verma-module basis due to null states (  There are three distinct sub-cases that need to be considered when dialling d 1 , d 2 . These are: When d 2 = 1 we find that (R − 2 ) 2 Ψ aux = 0. This leads to the combination Q 23 Q 24 being additionally removed from the basis of auxiliary Verma-module generators [8]. Recall that we are only constructing the auxiliary Verma module with the supercharges Q A1 and Q A2 . Their action on a superconformal primary with generic Dynkin labels The full representation is then built from these chains of supercharges. Clearly since d 1 = d 2 = 0 we will only be using the so (5)  This is the reduced spectrum because, as predicted from the discussion around (5.13), there are no negative-multiplicity states. In order to restore them we use the character relation where the hat denotes a character over the reduced (i.e. P 6 -removed) Verma module.
Since there are many states in the reduced spectrum, reconstructing the full multiplet with the operator constraints included would be rather unwieldy. We will however note that states with c 2 = 0 will pair up with their conservation equation according to This leads to the observation that, for arbitrary values of c 1 and c 2 , we have an infinite family of conserved currents which have higher spin.
A subset of these conserved higher-spin currents are the ones belonging to the multiplet B[0, c 2 , 0; 0, 0]. This multiplet has a superconformal primary in the rank-c 2 symmetric traceless representation of su (4), which corresponds to the higher-spin currents that one expects to find in the free 6D (2, 0) theory.
For example, we may take where I = 1, · · · , 5 is an so(5) R vector index, and Φ I is a free-tensor primary. Therefore, this object satisfies the conservation equation For generic c 1 , c 2 , the Schur index for this type of multiplet is given by

C-type Multiplets
From Table 3 the two distinct C-type multiplets are C[c 1 , 0, 0; d 1 , Upon repeating the null-state analysis, one finds that for generic values of d 1 , d 2 one is required to remove Q 1a for a = 1 and Q 1a Q 1b respectively from the basis of auxiliary Verma-module generators.
One also obtains additional absent auxiliary Verma-module generators for certain values of d 1 and d 2 . The procedure for identifying these is the same as the one presented in Sec. 5.6, so we simply summarise the additional set of Qs that are to be removed: The only non-vanishing Schur indices for the C-multiplets are where in the last column we have converted to Lorentz vector indices. Thus for generic Dynkin labels the actions of the supercharges lead to Upon replacing the actual values of the Dynkin labels and running the RS algorithm, the physical spectrum is given by As in the B[c 1 , c 2 , 0; 0, 0] case, we can restore the negative-multiplicity states by making use of the character relation where a hat denotes the P-reduced character. This will lead to a set of equations of motion.
The index over this multiplet in the Schur limit is (5.28)

D-type multiplets
These multiplets, summarised in Table 3, are the smallest of the osp(8 * |4) algebra. The associated null state is A 1 Ψ = 0, which implies that Q 1a Ψ aux = 0 and the multiplet is thus For generic d 1 , d 2 we have the Q-chain The action of theQ supercharges will be dealt with on a case-by-case basis as we dial d 2 , and provided in App. E.4. The only exception is the case d 2 = 0, which will be detailed below.
The non-zero contributions to the Schur limit of the index are given by The D-type multiplets contain negative-multiplicity states for d 1 +d 2 ≤ 2. These include the free-tensor and stress-tensor multiplet. Recall that in this case we are prescribed to remove Q 2a from the basis of auxiliary Vermamodule generators. This is because A 1 Ψ = Q 11 Ψ = 0 and using R-symmetry lowering operators we find that all Q 1a , Q 2a annihilate the primary. As a result, the multiplet is 1 2 -BPS. The set ofQs consist entirely of Q 3a supercharges. Acting on a generic superconformal The Verma module is then built out of Q 3a , Q 4a and we obtain ∆ 2d 1 The refined index over this multiplet is compact enough to be presented in full  The refined index for this multiplet is calculated to be For this multiplet we are prescribed to remove the combinations Q 2a Q 2b from the basis of Q-generators of the Verma module. Acting on the superconformal primary with the remaining set ofQ supercharges gives rise to The action of the Q supercharges is still the one presented in (5.30). Combining the two, the full module is given by The Verma module is obtained through the set of Q-actions of (5.30) along with the set ofQ-actions of Table 16; the latter can be found in App. E.4. Using these one has This concludes our discussion of 6D (2,0) superconformal multiplets.

Some Initial Applications
We finally mention some brief applications of the results that we have presented thus far, while leaving a more in-depth exploration for future work. We study aspects of flavour symmetries and flavour anomalies. We also rule out the presence of certain multiplets in free theories and provide evidence that certain 6D (1,0) multiplets can consistently combine to form 6D (2,0) multiplets.

Flavour symmetries
Both the 5D N = 1 and 6D (1, 0) algebras admit flavour symmetries. 24 Therefore it is interesting to ask which representations are allowed to transform nontrivially under these symmetries.
For example, in 4D N = 2 theories, it is known that N = 2 chiral operators do not transform under flavour symmetries [42]. Since scalar N = 2 chiral operators parametrise the Coulomb branch (when it exists), this statement is consistent with the fact that these theories do not have any massless matter (besides free vector multiplets) at generic points on the Coulomb branch. Below, we will explore similar physical constraints in 5D and 6D.
Let us begin by considering 6D (1, 0) theories. Many of these theories have tensor branches. Along these moduli spaces, the sp(1) R and flavour symmetries are unbroken, and free-tensor multiplets play an important role. It is therefore reasonable to assume that there are flavour and sp(1) R -neutral superconformal primaries in short multiplets that (partially) describe the physics on this branch of the moduli space.
To understand the last point in more detail, let us study short multiplets with an sp(1) R -neutral primary, O I . Let us further suppose that O I transforms linearly under a representation, R O , of the theory's flavour symmetry group, G. We will study the conditions under which superconformal representation theory forces this representation to be trivial, i.e. R O = 1. One crucial observation for us is that flavour Ward identities require the following leading OPE between the flavour symmetry current, j α ab , associated with G and O I : In this equation, the ellipsis contains less singular terms, and t α O is the matrix for the representation, R O , that O I transforms under. For now we will assume O I is a Lorentz scalar, but we will relax this condition later. One important point that will come into play below is that j α ab in (6.1) is a level-two superconformal descendant of the Sp(1) R spin-1 moment map primary To understand why certain operators are forbidden by the superconformal algebra from transforming linearly under flavour symmetry, it will be important to relate (6.1) to an 24 Here we define such symmetries to be those that commute with the superconformal algebra. See the discussion around (2.32) for an explanation of why flavour symmetries do not sit in multiplets with higher-spin symmetries.
OPE of superconformal primaries. In particular, we will need the fact that where we have used the superconformal algebra and the form of the translated operator J αAB (x) = e −iP ·x J αAB (0)e iP ·x .
(6.4) Using (6.3), we can also deduce that From this discussion, it follows that we can extract the OPE in (6.1) by acting with the special supercharges on the superconformal primary OPE In order to produce the correct leading singularity in (6.6), the superconformal primary OPE must then contain terms of the form This discussion implies the following results: • It is straightforward to check that in the C[0, 0, 0; 0] multiplet there are no operators of the type written on the RHS of (6.7) since there are no sp(1) R spin-1 descendants at level two. Therefore, these multiplets cannot transform linearly under any flavour symmetries in an SCFT. Since adding spin does not allow for such a descendant, we conclude that all the C[c 1 , 0, 0; 0] multiplets cannot transform under nontrivial linear representations of the flavour symmetry.
• Note that the absence of a conformal primary with the quantum numbers of the operator on the RHS of (6.7) is a sufficient but not necessary condition for the multiplet to be flavour neutral. For example, the stress-tensor multiplet contains an sp(1) R current which has the correct quantum numbers. However, in this case, • We can also play the same game in 5D, since the R symmetry is the same, and the flavour moment maps are again superconformal primaries of su(2) R spin-1 multiplets containing the flavour currents. It is straightforward to check that the above argument does not rule out flavour transformations for any unitary representations.
Note that the above discussion on the triviality of certain linear representations does not preclude the possibility that certain multiplets transform non-linearly when we turn on background gauge fields for global symmetries. Indeed, the free-tensor fields, ϕ I ∈ C[0, 0, 0; 0], often must transform non-linearly in order to match nontrivial 't Hooft anomalies for RG flows onto the tensor branch of certain interacting (1, 0) theories [10,43,44] (our discussion below mostly follows [43,44]). For example, the ϕ I multiplets can be used to match a discrepancy in the UV and IR anomalies of the form In this equation, ∆I 8 is the naive change in the anomaly polynomial eight-form between the UV and the IR, Ω IJ a positive-definite symmetric matrix, and the four-form, X I 4 , has a contribution from the flavour symmetry background fields, F i , where c 2 is a Chern class for the background flavour symmetry. In order to make up for the discrepancy in (6.9) there must be a coupling of the tensor field two-form, B I 2 , to X J 4 in the IR effective action where the necessary anomalous variation is generated by requiring that δB I 2 = −πX I 2 , where In particular, we see that the C[0, 0, 0; 0] multiplet transforms since

Supersymmetry Enhancement of 6D (1, 0) Multiplets
An important consistency condition for our 6D N = 1 multiplets is that they can combine to form N = 2 multiplets. For instance, one should be able to construct the additional supersymmetry and R-symmetry currents of the N = 2 SCA from the multiplets of Sec. 4.
It will be instructive to first show that two hypermultiplets and a tensor multiplet of the 6D N = 1 SCA combine to form a single tensor multiplet in N = 2. We need only consider the R symmetry, since the Lorentz quantum numbers are the same for both cases.
We identify the su(2) R ⊂ so ( where the equivalence is again up to the identification d 1 = K, confirming the original expectation. 25 25 For example, note that we find precisely the expected decomposition of the so(5) R currents (recall that these are level two superconformal descendants of the D[0, 0, 0; 2, 0] multiplet) in terms of representations of su(2) R : 10 = 3 × 1 + 2 × 2 + 1 × 3.

Acknowledgements
We Appendix A. The Superconformal Algebra in 5D and 6D In this appendix we collect our notation and conventions for the 5D and 6D SCAs. For a fully detailed account of conformal and superconformal algebras in various dimensions we refer the interested reader to [14,28].

A.1. The Conformal Algebra in D Dimensions
The conformal group in D dimensions is locally isomorphic to SO(D, 2) and generated by 1 2 D(D − 1) Lorentz generators M µν , D momenta P µ and special conformal generators K µ , and the conformal Hamiltonian H. The associated Lie algebra is defined by the commutation relations with all other commutators vanishing.
We note here that the subsequent analysis is performed in the orthogonal basis of quantum numbers, since the Lorentz raising/lowering operators, as well as P µ and K µ have very natural representations in orthogonal root space.

A.2. The 5D Superconformal Algebra
Extending the 5D conformal algebra to include supersymmetry is achieved by adding to our existing set (now with D = 5) the generators of supertranslations Q Aa and superconformal translations S Aa . These are equipped with a Lorentz spinor index a = 1, · · · , 4 and an su(2) R index A = 1, 2. Their associated Clifford algebras are generated by Γ µ andΓ µ , respectively. 26 The collection of bosonic and fermionic generators build the F (4) superconformal algebra, the bosonic part of which is so(5, 2) ⊕ su(2) R .
First, it will be useful to relate M µν to the Lorentz raising (lowering) operators M ± i , associated with the positive (negative) simple roots, and the Cartans H i (the eigenvalues of which are l i ), where i = 1, 2. We do so using the relations The algebra is now extended to include the following commutation relations-see e.g. [14]: The generators of the su(2) R algebra are denoted T m for m = 1, 2, 3. They act on the supercharges according to where σ m are the Pauli matrices. In fact we may compactly write these R-symmetry with the algebra From the above, we may infer that We denote the eigenvalue ofR in the orthogonal basis to be k. The odd elements of the 5D superconformal algebra satisfy where K is the charge-conjugation matrix and ǫ AB is the antisymmetric 2 × 2 matrix such that ǫ 12 = 1. It is straightforward to check that these matrices are given by We have also used the matrix M b a , which is defined by (A.10) The supercharges have the following conjugation relations [14] Q which allows us to rewrite the {Q, S} anticommutator as

A.3. Gamma-Matrix Conventions in 5D
Here we collect our gamma-matrix conventions. For Euclidean so(5) spinors we have These are supplemented by the charge conjugation matrix Note thatΓ 1,2,3,4 = Γ 1,2,3,4 , whileΓ 5 = −Γ 5 . Each set generates the Euclidean Clifford algebra in five dimensions, that is they satisfy the relations and similarly for theΓ matrices.

A.5. The 6D Superconformal Algebras
The superconformal algebra in 6D is osp(8 * |2N ), the bosonic part of which is so(6, 2) ⊕ sp(N ) R . 27 The set of conformal generators is extended to include the generators of supersymmetry Q Aa and superconformal translations S Aȧ . The Lorentz spinor index ranges from a,ȧ = 1, · · · , 4 (the dotted spinor index refers to the fact that it is an antifundamental index), while the sp(N ) R index ranges from A = 1, · · · , 2N . Their associated Clifford algebras are generated by Γ µ andΓ µ , respectively, the conventions of which are detailed in App. A.6.
Since their algebras are very similar, we first list the features that are shared by both N = 1 and N = 2 cases. Due to the fact that spinors in 6D are pseudo-real, they satisfy the reality conditions where Ω AB is the appropriate antisymmetric matrix in 2N dimensions. 28 The commutation relations of the Q and S with the 6D conformal algebra are given by It is helpful to define the Lorentz raising/lowering operators M ± i and Cartans H i -the eigenvalues of which are h i in the orthogonal basis of so(6)-in terms of M µν . The relations are provided below: (A.21) 27 Note that we will carry out this discussion using the so(6) algebra in the orthogonal basis, instead of su(4) algebra in the Dynkin basis. The dictionary between the two will be provided at the end. 28 For N = 1 one has Ω AB = ǫ AB , the 2D antisymmetric matrix. For N = 2, Ω AB is the 4D symplectic matrix with Ω 14 = −Ω 41 = Ω 23 = −Ω 32 = 1 and all other components vanishing.
For a superconformal algebra, all supercharges must have the same chirality [14], which we choose to be positive. Therefore we define the projector P + = 1 2 (1 + Γ 7 ) and have that We may also use the projector P + to define M b a as For this case, the R-symmetry algebra is sp(1) R ≃ su(2) R . Conveniently, all the information we require about su(2) R has already been provided in the 5D N = 1 discussion, specifically (A.4) onwards. We may therefore use the previously-defined R AB and write the {Q, Q † } anticommutator as The expression (4.6) we used for the 6D (1, 0) index is then, in the orthogonal basis, For this case, the R-symmetry algebra is sp where the J i are the Cartans of so(5) R -the eigenvalues of which are j i -and R ± i are the raising/lowering operators. This allows the {Q, Q † } anticommutator to be written as [14] {Q Aa , Hence the δ we used for the 6D (2,0) index is in the orthogonal basis To make contact with the main part of this paper, we convert from the orthogonal bases to the Dynkin bases using the following expressions for the so(6)-to su(4) Dynkin-, so (5) and su(2) orthogonal Cartans respectively: (A.29)

Appendix B. Multiplet Supercharacters
Consider a representation of a Lie algebra g with highest weight λ. The Weyl character formula is given by [45] where W is the Weyl group of the Lie algebra root system and ρ is the half sum of the positive roots Φ + . Note that sgn(w) = (−1) l(w) where l(w) is the length of the Weyl group element, i.e. how many simple reflections it is comprised of.
One can alternatively obtain this formula using a Verma-module construction: Decompose the algebra g as where h corresponds to the Cartan subalgebra and Φ − (Φ + ) are the negative (positive) roots. We construct the Verma module V corresponding to some highest (lowest) weight |λ by considering the space comprised of the states .

(B.3)
The character of the representation labelled by Λ is recovered by summing over the Weyl group action on the roots .

(B.5)
The formulation of the Weyl character formula (B.4) is particularly useful in the context of UIRs of the SCA [8].

B.1. Characters of 5D N = 1 Multiplets
We are now in a position to compute the characters of F (4). 29 Let us consider a representation the highest weight of which has conformal dimension ∆ with so(5) Lorentz quantum numbers (d 1 , d 2 ) and su(2) R quantum numbers K. Note that these are expressed in the Dynkin basis and hence are integer. The highest weight can be decomposed as where ω β i (i = 1, 2) are the fundamental weights associated with the so(5) simple roots β i , while ω α is the fundamental weight associated with the su(2) R simple root α. We may in turn express the fundamental weights in terms of the simple roots (for reasons which will become apparent) by using the Cartan matrix A ij Let us next consider e λ by defining the fugacities The character of a particular representation R is then defined to be where we have included the so(2) Cartan, ∆. For example we can read off the character for the supercharges Q Aa as (recall that we are in the Dynkin basis) We can then apply this to specific representations of the SCA. 29 A summary of this discussion for the case of the (2,0) SCA can be found in App. C of [8].
The polynomial appearing above can be decomposed as f (a, b, q) = Q(a, b, q)P (b, q) since the momentum operators and supercharges commute. Explicitly these functions are The characters χ [d 1 ,d 2 ] (b) and χ [K] (a) can be obtained through their Weyl orbits. As a result one has w(a) K R(w(a)) , (B.14) where the M(b) and R(a) are the products of the characters of negative roots, explicitly As an aside it will be worthwhile to explicitly demonstrate the Weyl group actions appearing in (B.14). In the orthogonal basis the generators of W SO(5) = S 2 ⋉ (Z 2 ) 2 and W SU(2) = S 2 have the form There are eight elements in S 2 ⋉ (Z 2 ) 2 and two in S 2 . These act on the simple roots as where the RHS is a combination of simple roots depending on the particular example. In fact, the simple reflections on the simple roots will generate every other root minus the Cartans of the algebra. For example, the simple roots of so (5) in the orthogonal basis are β 1 = (1, −1) and β 2 = (0, 1). Acting on them with w B 2 produces Similarly, acting with w B 1 will produce −β 1 and β 1 + β 2 respectively. Furthermore, since the Weyl group has a natural action on e λ (i.e. w(e λ ) = e w(λ) ), this action can be directly translated to the fugacities. Following the same example Combining the Weyl groups leads to W SO(5)×SU (2) , which has sixteen elements acting on a and b i . The Q(a, b, q) and P (b, q) are both invariant under the action of any element of this combined Weyl group. Using this fact, the character for long representations can be rewritten in terms of where [[· · · ]] W is shorthand for the Weyl symmetriser.

B.1.2. Short Representations
Consider now the short multiplets of Table 1. In order to calculate their characters, one is instructed [7,28,34] to remove certain combinations of Qs and Ps from the expressions Q(a, b, q) and P (b, q) given in (B.13). 30 Notice that this includes the character for the product over Q Aa but now with the Q 14 contribution removed; hence the Weyl symmetrisation removes descendant states associated with the action of Q 14 . 30 There is a subtlety with removing momentum operators, which will be addressed in the following section.
b. Take the B[d 1 , 0; K] multiplet with K = 0. One is instructed to remove Q 13 and Q 14 and the supercharacter is c. Suppose now we consider the multiplet B[d 1 , 0; 0]. In this case the R-symmetry lowering operator in the Dynkin basis R − also annihilates the superconformal primary and two additional shortening conditions are generated where we remind the reader that M − 2 is a Lorentz lowering operator in the Dynkin basis. Therefore, for the purposes of building the Verma module we can remove both of these from the basis of Verma-module generators. As a result, the modified product over supercharges, now indicated byQ(a, b, q), iŝ Q(a, b, q) = Q(a, b, q) 1 + ab 2 q The last thing to take into account is the possible removal of Ps from P (b, q) when some components of the multiplet correspond to operator constraints. This will be discussed at length in App. C.

B.1.3. The 5D Superconformal Index
The supercharacter for a given multiplet can be readily converted into the superconformal index. The five-dimensional superconformal index, as we have previously defined it in the Dynkin basis in Sec. 2.4, is given by 31 The fermion number in this case is F = 2d 1 + d 2 ≃ d 2 , since d 1 is always integer.
The states that are counted satisfy δ = 0, where In order to make contact between the character of a 5D superconformal representation and this index, one can simply make the following fugacity reparametrisations and introduce a factor of (−1) F . The resulting object is precisely the index since every state without δ = 0 pairwise cancels.

B.2. Characters of 6D (N , 0) Multiplets
Consider a representation, the highest weight of which has conformal dimension ∆ with su(4) quantum numbers (c 1 , c 2 , c 3 ) and R-symmetry quantum numbers R. For N = 1 we have that R = K, the Dynkin label of su(2) R , while for N = 2 the Dynkin labels of so(5) R are R = (d 1 , d 2 ). The highest weight can be decomposed as where the index in ω R i R i is summed over. The ω α i are the fundamental weights associated with the su(4) simple roots α i and ω R are the fundamental weights associated with the simple roots of the R-symmetry algebra, β. 32 For su(2) R β has only one component, while for so(5) R β has two components, (β 1 , β 2 ). Again, we can express the fundamental weights in terms of the simple roots by using the Cartan matrix A ij Similarly, we define the R-symmetry fugacities b using (B.8). For N = 1 we have that The character of a particular representation R is then given by 34) and for N = 1, 2 respectively we have: We can then apply this to specific irreducible representations of the 6D (N , 0) SCA. w(a 1 ) c 1 w(a 2 ) c 2 w(a 3 ) c 3 M(w(a)), (B.38)

B.2.1. Long Representations
We use W R to indicate the Weyl group appropriate for the R symmetry of the (N , 0) SCA.
The M(a) and R (N ,0) (b) are the products of characters of negative roots as defined in Again, we note that both Q(a, b, q) and P (a, q) are invariant under the appropriate Weyl symmetrisations and one can write for N = 2 matching [8], while for N = 1 where [[· · · ]] W denotes the Weyl symmetriser.

B.2.2. Short Representations
Consider now the short multiplets of Table 2 for N = 1 or Table 3 The states that are counted satisfy δ = 0, where We can therefore write the character of a representation as an index via the following fugacity reparametrisations and inserting (−1) F . The resulting object is precisely the index since every state without δ = 0 pairwise cancels.
The 6D (2, 0) superconformal index, as previously defined in (5.3), is given by The states that are counted satisfy δ = 0, where In order to make contact between the character and this index, we make the following fugacity reparametrisations and insert (−1) F . The resulting object is precisely the index since every state previously counted by the character without δ = 0 pairwise cancels.

Appendix C. The Racah-Speiser Algorithm and Operator Constraints
In this appendix we provide the details needed to carry out the RS algorithm for the 5D N = 1, 6D (1, 0) and 6D (2, 0) SCAs. Fortunately, we need only discuss three different algebras, so (5), su(4) and su(2). We will assume some familiarity with the description of the RS algorithm from App. B of [27], the notation of which we use.
First, let us consider so (5). This is the Lorentz Lie algebra for 5D N = 1 and the R-symmetry Lie algebra for 6D (2, 0). The highest weight (λ) identifications resulting from Weyl reflections (σ) are given by We see that λ σ = λ with sign(σ) = − under the following conditions: Therefore representations which satisfy any one of these conditions are labelled by a highest weight on the boundary of the Weyl chamber, in which case they have zero multiplicity and need to be removed. Notice that the conditions (C.2) correspond to the zeros of the Weyl dimension formula for irreducible representations of so (5): Next we consider su(4), the Lorentz Lie algebra of 6D theories. The highest weight identifications resulting from Weyl reflections are The representations we delete are the ones where the following conditions are met: Again, these correspond to the zeros of the Weyl dimension formula for su (4): Lastly, the R-symmetry Lie algebra for the minimally supersymmetric theories is su (2), which involves identifying highest weights via the reflection: We therefore see that the representations we delete are the ones where the following condition is met: Clearly this corresponds to the zero for the Weyl dimension formula of su(2) We may now simply combine the set of Weyl reflections appropriate for our SCA in order to dictate which states survive when building representations. Our method will involve generating all possible highest weight states, even in cases where the Dynkin labels become negative, and then performing the RS algorithm.
It is interesting to note that after implementing the RS algorithm one often encounters pairs of superconformal descendants with exactly the same Dynkin labels but opposite multiplicities. These cancel out to leave behind a much simpler set of superconformal representations with only positive multiplicities. If after performing this step there are also negative representations that have not been cancelled, they are interpreted as constraints for operators in the multiplet [27].
There are instances when this general procedure leads to ambiguities, i.e. there is more than one choice for performing the cancellations; see also [12]. However, these can

C.1. Operator Constraints Through Racah-Speiser
We now further explore this concept. Since R-symmetry quantum numbers will not play a role in this analysis we will simplify our discussion by denoting all of their quantum numbers by R. The only distinction we need to make is between the so(5) and su (4) Lorentz Lie algebras. It will be instructive to proceed by first providing an example and then the result in full generality.
Consider the Lorentz vector representation in 6D, [∆; 0, 1, 0; R], corresponding to an operator O µ . In terms of quantum numbers, its components are given by  34 One has that (P µ ) † = P µ . Then it can be straightforwardly checked that P µ = P 7−µ .
surviving RS being a Dirac equation −[7/2; 0, 0, 1; R]. For c 1 ≥ 2, all states survive RS, leaving a Bianchi identity for a higher p-form field; this is usually endowed with additional constraints, i.e. self-duality in the free-tensor case, c 1 = 2.
The analogous states in 5D are given by

C.2. The Dictionary Between Racah-Speiser and Momentum-Null States
The expressions (C.13) and (C.14) can also be understood from a different perspective.
When the superconformal primary saturates both a conformal and a superconformal bound, certain components in the multiplet satisfy operator constraints. One is then instructed to remove appropriate P µ generators from the auxiliary Verma-module basis [28]. We call the set contained in the resulting module "reduced states".
Following [28], in 5D N = 1 we are instructed to remove P 5 for the multiplets B (C. 16) The operator constraints recovered in this case are equations of motion.
In 5D one can make the analogous identifications: where we have removed P 5 fromχ. Likewise, when we remove P 3 , P 4 and P 5 we obtain: The method for evaluating characters of conformal UIRs presented in [28] makes no distinction between states that do/do not obey constraints due to the above invariance.
This knowledge is useful when the absent generators of the Verma module are such that the removed Qs anticommute into Ps; c.f. Sec. 2.7.1, Sec. 4.6.1 and Sec. 5.6.1. In that case, one is effectively projecting out the states associated with operator constraints from the very beginning. Therefore what is generated under these circumstances is the spectrum of reduced states. It is still possible to reconstruct the full multiplet spectrum using character relations, as e.g. in (C. 15). This lets us recover all the negative-multiplicity states.

E.1. A-type Multiplets
The superconformal primary null-state condition for a generic A-type multiplet is A 4 Ψ = 0.
This corresponds to removing Q 14 from the basis of auxiliary Verma-module generators (5.1). Starting from a superconformal primary given by [c 1 , c 2 , c 3 ; d 1 , d 2 ] with conformal dimension ∆ = 6 + 2(d 1 + d 2 ) + 1 2 (c 1 + 2c 2 + 3c 3 ), we will provide the states at each level l; the conformal dimension of the superconformal descendants will be equal to ∆+ l 2 . Thus the spectrum for A[c 1 , c 2 , c 3 ; d 1 , d 2 ] is given by acting with all Qs andQs, resulting in the two chains, Table 4 and Table 5 respectively. Note that this is a complete description for generic d 1 and d 2 . The reason being that, as described in Sec. 5 It turns out that the same also holds for dialling d 1 and d 2 to zero. Thus the complete spectra for all A-type multiplets can be obtained exclusively by considering the set of Q andQ actions (Table 4 and Table 5) and then substituting in the desired quantum numbers followed by implementing the RS algorithm.

E.2. B-type Multiplets
For generic quantum numbers, the superconformal primary for this multiplet type obeys the null-state condition A 3 Ψ = 0. This corresponds to removing Q 13 and Q 14 from the auxiliary Verma-module basis (5.1). Starting from a generic primary [c 1 , c 2 , c 3 ; d 1 , d 2 ] with conformal dimension ∆ = 4 + 2(d 1 + d 2 ) + 1 2 (c 1 + 2c 2 ), the action of theQ supercharges remains the same as in Table 5, with the difference that one needs to apply RS after setting c 3 = 0. The Q-chain, however, is significantly shorter and given in Table 6. In this case the action of the Q supercharges is the same as in Table 6. However, recall from Sec. 5.6 that we also have the null-state condition (R − 2 ) 2 A 3 A 4 Ψ = 0. Thus we are also prescribed to remove Q 23 Q 24 from theQ-spectrum. The action of theQ supercharges is therefore adjusted and described in Table 7. For this case the action of the Q supercharges is the same as in Table 6, after replacing d 2 = 0 and running the RS algorithm. We are also prescribed to remove Q 23 and Q 24 from theQ-spectrum. The action of theQ supercharges is therefore adjusted and described by Table 8.

E.3. C-type Multiplets
For generic quantum numbers the superconformal primary for this multiplet obeys the nullstate condition A 2 Ψ = 0. This corresponds to removing Q 12 , Q 13 and Q 14 from the basis of auxiliary Verma-module generators. Starting from a generic primary [c 1 , c 2 , c 3 ; d 1 , d 2 ] with conformal dimension ∆ = 2 + 2(d 1 + d 2 ) + c 1 2 , the action of theQ supercharges remains the same as in Table 5, with the difference that one needs to apply RS after setting c 2 = c 3 = 0.
The action of the Q supercharges is given in Table 9. Recall that in this case we should also remove Q 21 Q 22 Q 23 from the basis of auxiliary Verma-module generators. The resulting set ofQ actions are given in Table 10. For this multiplet we also need to remove Q 2a Q 2b for a = b = 1. The resulting set ofQ actions are given in Table 11. For this multiplet we also need to remove Q 2a for a = 1. The resulting set ofQ actions are given in Table 12. This multiplet has the same Q andQ actions as C[c 1 , 0, 0; d 1 , 0] with the difference that one needs to apply RS after setting d 1 = 1. This multiplet contains generalised conservation equations and is small enough for us to detail its entire spectrum in Table 13.  Table 14. This multiplet also contains generalised conservation equations.

E.4. D-type Multiplets
Since the action of the Q supercharges have been given in Sec. 5 For this multiplet we remove Q 2a Q 2b Q 2c from the basis of auxiliary Verma-module generators. We summarise the actions of theQ supercharges in Table 16.