Notes on Superconformal Representations in Two Dimensions

We study global subalgebras of superconformal algebras in two dimensions and their unitary representations. Global superconformal multiplets are decomposed into conformal multiplets using Racah-Speiser algorithm, revealing many essential aspects of superconformal theories such as stress-energy tensor, conserved current, supersymmetric deformation and supersymmetry enhancement. Character formulae for the representations are presented. We further find a collection of conserved charges that are $k$-forms under the R-symmetry, which must be part of the super Virasoro algebra with $\mathcal{N} \geq 3$ supersymmetries.


Introduction and Conclusion
Conformal field theory(CFT) is one of the key ingredients of theoretical physics, with its far-reaching applications from string theory to condensed matter theory. Power of the conformal field theories lies on the abundance of symmetries, which constrain the structure and contents of the theory to a large degree.
The power of symmetries becomes particularly pronounced with introduction of supersymmetries. The superconformal symmetry, considered to be the most general symmetry group in four spacetime dimensions [1], enables a plethora of progress based on 'kinematics' of the theory alone, let aside the 'dynamics'.
The conformal symmetry becomes significantly extended to have an infinite number of generators in two space-time dimensions, known as the Virasoro algebra [2]. In addition to the Virasoro algebra, many extensions of two-dimensional conformal algebra such as Kač-Moody algebra [3,4] or W-algebra [5] have been studied, see also [6,7].
However, the supersymmetric extension of the Virasoro algebra, namely super Virasoro algebra, is increasingly complicated. It has been fully studied only for a relatively fewer number of supersymmetries N ≤ 4 [8][9][10][11][12][13][14][15][16]. For N ≥ 5, the full super Virasoro algebra has not been clearly constructed to the best of our knowledge.
Numerous authors have attempted to construct super Virasoro algebra with a large number N of supercurrents. The results are yet incomplete and do have certain unconventional features. For instances, see [17][18][19][20][21][22][23][24][25]. It is the existence of certain generators, other than what are expected by a straightforward generalization of cases with fewer N , that makes the construction nontrivial.
Under the circumstances, we first study in the present work the structure of global subalgebra of the super Virasoro algebra in two dimensions and its unitary representations. We find that the study of the global subalgebra provides certain concrete implications on the poorly understood super Virasoro algebra.
Recently, the unitary representations of superconformal algebras with any number of supersymmetries N in dimensions 3 ≤ d ≤ 6 were systematically organized in [26]. The key was to decompose each representation of a superconformal algebra into those of a conformal algebra, utilizing the Racah-Speiser algorithm to organize the conformal primaries into representations of Lorentz and R-symmetry groups.
Since the global conformal algebra so (2,2) in two dimensions is highly analogous to its higher-dimensional counterpart so(d, 2), we can maximize utilization of the methodology of [26].
The problem further simplifies in two dimensions because the two-dimensional conformal algebra splits into two copies of Virasoro algebra. The two copies are often referred to as left-moving and right-moving sectors, or also as holomorphic and antiholomorphic sectors. Each copy of Virasoro algebra can be extended to accommodate any number N of supersymmetries that results in super Virasoro algebra. Therefore, one can study multiplets of individual sectors, then simply take direct product of multiplets in each sector to form a full conformal multiplet.
Despite our limitation to the global subalgebra, we find that it contains many essential features of the larger super Virasoro algebra, particularly involving the shortening, or unitary, conditions. Thus, much can be inferred about the full superconformal theory in two dimensions by studying the global subalgebra.
We list some of the most notable results below.
• In two-dimensional global subalgebra of super Virasoro algebra with any number N of supersymmetries, a multiplet that is constant on the conformal manifold, a.k.a. absolutely protected multiplet does not exist. In other words, every short multiplet that saturates the unitarity bound may recombine with another short multiplet to form a long multiplet in the limit of its saturation of unitarity bound. See section 4.1.
• In all global superconformal theory with any numbers (N ,N ) of supersymmetries, holomorphic and anti-holomorphic conserved currents that are supersymmetric, Rneutral, and have spins s = 1, 3 2 , 2, · · · are allowed. In particular, a conserved current with s = 2 identified as the stress-energy tensor and a supersymmetric higher-spin current s = N 2 − 1 for N ≥ 7 are universal in all theories. See section 5.1.
• In all global superconformal theory with (N = 4,N = 4), or with large N = 4 or N = 4 with equal Kač level for two copies of su(2) ⊂ so(4), relevant supersymmetric deformations with conformal dimension ∆ = 3 2 that are Lorentz scalars and R-spinors are allowed. However, their existence is not guaranteed, and in two dimensions the universal mass as defined in [27] does not exist in any superconformal theory. See section 5.2.
• In all global superconformal theory with any numbers (N ,N ) of supersymmetries, a marginal supersymmetric deformation that is a Lorentz scalar and an R-singlet is allowed. In particular, we rediscover that a universal marginal deformation is guaranteed to exist in the large (4,4) superconformal theory. See [28] and section 5.2.
• A super Virasoro algebra with number of supersymmetries N ≥ 5 must contain conserved current operators that are k-forms under the R-symmetry group SO(N ) and have scaling dimensions (L 0 -eigenvalue) k 2 − 1, where k = 3, 4, · · · , N . The current is bosonic when k is even and fermionic when k is odd. (Anti-)commutation relation between the supercurrent and the k-form current must yield, among others, the (k + 1)-form current and the (k − 1)-form current. See section 6.1.
This article is organized as follows. We begin with preliminaries in section 2, where the global subalgebra, and decomposition principle and unitarity condition for its multiplets are explained. Presented in section 3 are lists of unitary multiplets, for global subalgebras with every number of supersymmetries N . Then, we discuss various implications of the results in order. First, we discuss more or less straightforward results in section 4, namely the recombination rules and character formulae for the multiplets. Then, section 5 contains more physically significant applications such as conserved currents, deformations, and supersymmetry enhancements, highlighted by the stress-energy tensor. Finally in section 6, we take a step further to discuss what we can infer about the super Virasoro algebra from our results on the global subalgebra.

Global Superconformal Algebras
We start with a brief review on the global superconformal algebra with an arbitrary number of supercharges in two dimensions. Generic cases are first discussed, followed by a special case of the N = 4 superconformal algebra. We limit ourselves to the left-moving sector, as the algebra of the right-moving sector is identical.

Generic Global Subalgebras
A superconformal algebra with N supercharges is an so(N ) Kač-Moody algebra [29,30]. Its global subalgebra is generated by three Laurent modes L −1 , L 0 , L 1 of the Virasoro operator, two modes G a are so(N ) generators T cd 0 in the vector representation. We work in Neveu-Schwarz sector, so r, s are half-integers while m, n are integers.

N = 4 Global Subalgebras
The case N = 4 where so (4) su(2) × su(2) calls for a special treatment. As found in [12], there exists a one-parameter family of N = 4 superconformal algebra, where we use α to parametrize relative levels of the two su(2)'s. Six so(4) generators are arranged into two mutually commuting sets of su(2) generators T ±i 0 . For the global subalgebra of the superconformal algebra, last three subequations of (2.1) are modified as follows: 4) and the parameter α is related to the su(2) levels by This global subalgebra is named D(2,1;α). When α = 1, D(2,1;α) is osp(4|2), which is precisely what one obtains by putting in N = 4 in the generic algebra (2.1). In doing so, one must be careful with a numerical factor in the relation between generators of so(4) and su(2) × su (2). That is, , which accounts for the extra factor of 2 in the second term of (2.3c).
In subsequent sections, we will usually leave α as a free parameter and refer to D(2,1;α) as large N = 4 global subalgebra, although we will frequently give it the value 1.
Meanwhile, another subalgebra can be obtained from the above by, e.g., taking a limit α → ∞. This leaves us with only one su(2) generated by T i 0 , under which four supercharges transform as two independent sets of spinors G ± r ,Ḡ ± s where r, s = ± 1 2 as before. The algebra is summarized by, in addition to the first three subequations of (2.1), (see [13,14]) where σ i ab are the usual Pauli matrices. This global subalgebra is su(2|1, 1) (see [12]), to which we refer as small N = 4 global subalgebra.

Comment on Super Virasoro Algebras
In two dimensions, super Virasoro algebras with N supercharges are constructed by central charge extension from the corresponding global subalgebras [2]. As a result, the full algebra contains an infinite number of Laurent modes for each of the operators L, G a , T ab , with non-trivial commutation relations such as However, the super Virasoro algebra in general is not completed by the infinite modes of L, G a , and T ab . For example, the super Virasoro algebra for the large N = 4 contains an additional u(1) generator among others [12], and higher-N algebras are expected to contain more extra generators. See [21] for an example.
While the super Virasoro algebras for 2 ≤ N ≤ 4 have been thoroughly studied [8][9][10][11][12][13][14][15][16], those for generic N ≥ 5 are not fully understood to the best of our knowledge. It is one of the main goals of this article to find extra generators that must enter the algebra by studying their global subalgebras, in particular their stress-energy tensor multiplets. See section 6.1 for this account.

Superconformal Multiplets
We now turn to global superconformal multiplets allowed by each global subalgebra with an arbitrary number N of supercharges. Let us first discuss the unitary multiplets of the left-movers G a 1/2 below. Same argument applies to the right-movers as well.
An irreducible superconformal multiplet is fully determined by its superconformal primary V that is annihilated by {G a 1/2 , L 1 }. A superconformal primary furnishes an irreducible representation under the maximally compact bosonic subalgebra sl(2) × so(N ). #1 The left-moving multiplet then consists of the primary V and its superconformal descendants, obtained by consecutive actions of L −1 and G a −1/2 on the primary.
Given any superconformal primary, it is straightforward to build a multiplet with its descendants. However, unitarity conditions impose a bound on the allowed L 0eigenvalue h 0 of the superconformal primary. Let us examine this bound in any N in two dimensions. This argument closely follows that of [31], and we shall thus be brief.
Let us denote the state corresponding to the superconformal primary r ] collectively denotes the highest weight so(N ) Dynkin labels with r = N /2 and α is an index for the representation [R]. The unitarity bound can be obtained by enforcing all first-level components in the multiplet to have non-negative norms. To be more explicit, let us consider a matrix element #2 Since (G a 1/2 ) † = G a −1/2 , all its eigenvalues are required to be non-negative. #1 There is an exception of small N = 4 superconformal algebra where so(4) reduces to su (2). #2 We actually need conjugate indices for the bra, but it is irrelevant as we are only interested in the eigenvalues.
We can proceed with the (anti-)commutation relation (2.1) since the primary ket is annihilated by G 1/2 : (2.2) and (ρ(T 0 ) cd ) βα represents matrix elements of so(N ) generators in the vector and [R] representations. Eigenvalues of the matrix (2.9) can be obtained as in a well known quantum mechanics problem. The result is: where h 1 is the first orthogonal weight. The orthogonal basis will be used exclusively in section 4.2. For later convenience, we present the relation between the Dynkin labels R i and the orthogonal weights h i below: When N is even, When N is odd, When the BPS condition (2.11) is saturated, A bβ;aα acquires a zero eigenvalue. Since (2.11) corresponds to (2.10) with [R ] that yields the strongest bound, this indicates that among many states G a representation of the R-symmetry group, those belonging to the irreducible representation [R 1 +1 R 2 · · · R r ] are null. Then, not only the component [R 1 +1 R 2 · · · R r ] h 0 + 1 2 but also its conformal descendants must be removed from the superconformal multiplet. A systematic procedure of such a removal is discussed in [26]. Such superconformal multiplets with null states are referred to as short multiplets, as opposed to long multiplets. Following the convention of [26], a long multiplet will be denoted as where [R] and h are quantum numbers of its superconformal primary. Similarly a short multiplet will be denoted as In particular, a vacuum multiplet will be denoted as although it is also a short multiplet. The vacuum multiplet is defined as a multiplet whose primary is annihilated by all operators so there are no descendants. All of its quantum numbers vanish.
So far we have discussed the unitarity condition of first-level states only. We conjecture that it alone suffices, because the first-level states impose the strongest bound.
A complication arises in higher dimensions where the Lorentz group is also nonabelian. The strongest unitarity bound for a given primary arises from the descendant whose Dynkin labels for the Lorentz group are small and that for the R-symmetry are large. As a result, when the primary is a Lorentz singlet, second-level unitarity bound can be stronger than that of the first-level, where the states are necessarily Lorentz spinors rather than of smaller Dynkin labels that would have been present (and given stronger bound) for generic primaries. This non-generic phenomenon leads to a diversity of short multiplets, including those with higher-level null states and more interestingly, isolated short multiplets. See [26] for more details. In contrast, all short multiplets in two dimensions are limiting cases of long multiplets, which results in the absence of absolutely protected multiplets. We will discuss these features further in section 4.1.
A superconformal multiplet is decomposed into conformal multiplets [26] that consist of conformal primaries (annihilated by L 1 ) and their descendants (obtained by consecutive actions of L −1 on the primaries). Thus it is useful to express the superconformal multiplet as a collection of conformal primaries whose conformal multiplets make up the superconformal multiplet. We refer to these conformal multiplets as components of the superconformal multiplet. The collection consists of the superconformal primary V and the operators that are obtained by (repeatedly) acting only G a −1/2 's on V. Note that we can effectively set because the L −1 action does not generate a new conformal primary but generates a descendant. Due to this Fermi-Dirac statistic, decomposition into conformal multiplets is finite.
Since conformal primaries are also in the representations of sl(2) × so(N ) [26], it proves convenient to specify each conformal multiplet by the L 0 eigenvalue h and the highest weight so(N ) Dynkin label [R 1 · · · R r ] of the corresponding conformal primary. Thus, [R] h : a conformal multiplet with corresponding primary.
Note that we are using the same notation to refer to superconformal multiplets as to conformal multiplets, except that a letter L, A, or V to indicate the presence of null states is omitted for the latter.
Conformal multiplet decomposition of a superconformal multiplet can be performed by consecutive actions of G a −1/2 on the superconformal primary, and organizing into irreducible representations of the bosonic subalgebra sl(2) × so(N ). This process is best done via Racah-Speiser algorithm, inspired by [32] and thoroughly explained in [26]. In our case of two dimensions this is particularly simple, because G a −1/2 simply act as raising operators for the sl(2) and the only non-trivial part is the R-symmetry so(N ).
Number of operations of G a −1/2 required on the superconformal primary to obtain a particular conformal primary is referred to as level of the component. Thanks to the Fermi-Dirac statistics, the level is bounded from above by N in any superconformal multiplet. In fact, long multiplets are always terminated at the level N while short multiplets are terminated earlier. In two dimensions where G a −1/2 simply raises the L 0 -eigenvalue by 1 2 , any component at level l has an L 0 -eigenvalue h 0 + l 2 where h 0 is that of the superconformal primary.
Note that in all of the discussions so far, and in most of the discussions that will follow, conformal and superconformal multiplets refer to representations of the global subalgebras. In general, multiplets of the full super Virasoro algebras include many global multiplets, as they include the action of all negative modes such as L n<−1 , n<0 on the primary. For example, the vacuum multiplet and the stressenergy tensor multiplet for the global subalgebra belong to the same super Virasoro multiplet.

Supersymmetric Deformations and Conserved Currents
One of the applications of decomposition of superconformal multiplets is to look for possible deformations of CFTs. Following [27], we seek possible deformations of SCFTs in two dimensions in vicinity of RG fixed points by relevant or marginal local operators O. That is, given a superconformal theory, we aim to find a local operator that i) is a Lorentz singlet, ii) has scaling dimension less than or equal to the dimension, which is 2 throughout this paper, iii) is not a total derivative, and iv) is supersymmetric. Note that the operator must reside in an allowed superconformal multiplet of the theory.
In two dimensions, superconformal algebras separate themselves into left-and right-moving sectors. The L 0 -(in the left) andL 0 -(in the right) eigenvalues h 0 and h 0 of an operator sum up to the scaling dimension, and their difference represents the spin of the operator. Thus, to satisfy the condition i) we require h 0 =h 0 , and further for ii), it suffices to look for operators with h 0 =h 0 ≤ 1. The condition iii) suggests to consider only the conformal primaries, as conformal descendants are obtained by applying L −1 ∼ ∂ z to another local operator.
The condition iv) is a little trickier to satisfy. One obvious way for a conformal primary belonging to a superconformal multiplet to be supersymmetric (i.e. to be annihilated by all supersymmetries G −1/2 ) is to be a generic top component of the superconformal multiplet: to reside at the highest level of the multiplet since an application of G −1/2 raises the level by unity. As discussed in the last subsection, a long multiplet always has its generic top component at the level N while short multiplets have them at lower levels, and possibly more than one of them. Every superconformal multiplet possesses at least one generic top component.
Nevertheless, there are components of short superconformal multiplets at the level lower than the generic top component, that however are annihilated by all supersymmetries. Following [27], we refer to them as sporadic top components, and they prove to be very fruitful in discussion of conserved currents in two dimensions.
A sporadic top component is easily identified when, as in [27], there exists a conformal primary whose Dynkin labels match none of those at the next level when added by any of supercharges. In such case, we infer that all conformal primaries at the next level must be produced by acting on other components at the previous level by supercharges. However, note that this is sufficient but not necessary a condition to be a top component. We shall see counterexamples in sections 3.5, 3.7, and 3.9.
Also of our interest are conserved currents. When an operator satisfies the conditions iii) and iv) above but has either h 0 = 0 orh 0 = 0, the supersymmetric operator is annihilated by L 0 ∼ ∂ z orL 0 ∼ ∂z, respectively. In other words, it is conserved. In particular, if such an operator is an R-singlet and has (h 0 ,h 0 ) = (2, 0) it could be Primary Unitarity bound Null component Sporadic top Generic top the holomorphic part of the stress-energy tensor and if it has (h 0 ,h 0 ) = (0, 2) it could be the anti-holomorphic part, expected to exist in all physical theories. Further, to the same multiplet as the stress-energy tensor but at the previous level must belong the supercurrents (in the vector representation of the R-symmetry) and yet at the level below the R-symmetry currents (in the adjoint representation). Meanwhile, a supersymmetric and R-singlet operator with (h 0 ,h 0 ) = (0, 1) or (1, 0) would similarly indicate a flavor current, and those with (h 0 ,h 0 ) = (0, s > 1) or (s > 1, 0) the higher-spin currents.
Supersymmetric deformations and conserved currents will be discussed in sections 5.1 and 5.2, respectively.

List of Multiplets
We tabulate in this section all superconformal multiplets, long or short as explained in section 2.2, for each number of supersymmetries. In this section, we restrict only to the left-moving sector, as the right-moving sector may have a same list of multiplets with its ownN . In doing so, we will explicitly list top components, generic or sporadic as explained in section 2.3, to be discussed in detail in the following sections.

N = 1
Let us begin with the simplest case, N = 1 that contains no R-symmetry. Representation with respect to the R-symmetry group is always trivial: [0]. For any superconformal primary [0] h 0 , the unitarity bound simply becomes When the bound is saturated, the only superconformal descendant is null and the superconformal multiplet consists of only one conformal multiplet. Then a complete list of superconformal multiplets and their top components is as simple as Table 1.

N = 2
The N = 2 supersymmetry has an abelian R-symmetry SO(2) ∼ U (1) under which the two supercharges G ± −1/2 are charged by ±1. A superconformal primary [j 0 ] h 0 has to satisfy the unitarity bound below: where j 0 denotes a U (1) R-charge rather than a Dynkin label.
Note that this bound is weaker than the unitarity bounds put forward by the super Virasoro case [33]. A list of superconformal multiplets, along with their top components, is summarized in Table 2.

Small N = 4
Let us examine the N = 4 superconformal algebra before N = 3 for the reason that will soon be clear. The small N = 4 superconformal algebra has two independent sets of supercharges G ± −1/2 andḠ ± −1/2 in the fundamental representation of SU (2) R-symmetry. #3 The superconformal primary is labelled by the SU (2) Dynkin label as [R 1 ] h 0 , where R 1 is a non-negative integer. For instance, the fundamental representation is labelled by [1]. The small N = 4 superconformal unitarity bound is and as the bound is saturated, two copies of raising operators G + −1/2 andḠ + −1/2 simultaneously annihilate the primary. We summarize the list of small N = 4 multiplets in Table 3. Here, A[1]1 2 is an example where there exist two degenerate top components [0] 1 .

#3
One can think of an extra U (1) that distinguishes the two, which we choose not to because it plays no role other than labelling the two and justifying some of (2.6).

Unitarity
Null

Large N = 4
The large global N = 4 superconformal algebra D(2, 1; α) contains two copies of su(2) algebras. When the free parameter α is set to unity, the R-symmetry group becomes SO(4). #4 The Dynkin labels R 1 and R 2 now correspond to those of two su(2)'s such that the four supercharges are in the representation [1; 1]. Note that [1; 1] is the highest weight of a vector representation when the R-symmetry group becomes SO(4).
The unitarity bound on the conformal weight h of the large N = 4 superconformal primary is given by One can find the list of superconformal multiplets in Table 4. Since the multiplet A[1, 1]1 2 is the first example where a sporadic top component appears, let us pause to examine this. The superconformal multiplet A[1; 1]1 2 can be decomposed into various conformal multiplets as in Figure 1. We see that the null state [2; 2] 1 and the Racah-Speiser trial state constructed by acting the supercharge of weight [1; 1] on the primary have the same quantum numbers. As explained in [26], any RS states involving the supercharge of weight #4 The free parameter α is related to the levels k + = c(1 + α)/(6α) and k − = c(1 + α)/6 of two su(2) current algebras when the global superconformal algebra is promoted to the large N = 4 super Virasoro algebra. See section 6.1 for more details.

Unitarity
Null

N = 3
The N = 3 superconformal algebra is OSp(3|2) with R-symmetry SO(3). Three supercharges transform as [2], namely the vector representation under SO(3). One can label a superconformal primary as [R 1 ] h 0 where the Dynkin label R 1 is a nonnegative integer. Note that we use the SO(3) Dynkin label rather than SU(2), which differ by a factor of 2. The unitarity condition (2.11) is then becomes null. However, since top components differ when R 1 is small, we list them separately in Table 5. In particular, for the primary [0] >0 , [2] 1 2 + consists the first level alone, and as → 0 it becomes null and the multiplet terminates already at the level zero. This phenomenon is universal: for all N the superconformal multiplet V [0] 0 consists of a superconformal primary only, which corresponds to the identity Primary Unitarity bound Null component Sporadic top Generic top  acted by an appropriate supercharge, and thus the null condition is highly non-trivial. For details, see [26]. Since three supercharges of N = 3 superconformal algebra are a subset of the four supercharges of N = 4 one, this is sufficient to argue that [0] 1 in Figure 2 is indeed a sporadic top component as well.

Primary
Unitarity bound Null component Sporadic top

N = 6
Let us examine the N = 6 superconformal algebra before the N = 5 superconformal algebra for the same reason as for the N = 4 and N = 3 algebras. The R-symmetry is now SO(6) SU (4) #5 , under which supercharges transform as a vector [1 0 0]. We label a superconformal primary as [R 1 R 2 R 3 ] h 0 . The unitarity bound for N ≥ 5 shall always be given by (2.11). In this case, it is We tabulate the list in Table 6.
From this subsection, we do not attempt to give a complete list of short multiplets, but skip many of those that are irrelevant to the subsequent sections. Those short multiplets however can be easily reproduced from long multiplets via the procedure described in [26].
It proves useful to examine the short multiplet A[0 1 1] 1 2 in detail. Its decomposition into conformal multiplets is given by  a sporadic top component because none of the supercharges, except the one of the highest weight [1 0 0] that annihilates the primary in short multiplets, can act on the R-singlet to produce the R-vector component at the next level.

Primary Unitarity bound Null component Sporadic top Generic top
[ Figure 4: An N = 6 multiplet with a sporadic top component.

N = 5
The N = 5 superconformal algebra has the SO(5) Sp(4) R-symmetry #6 of which the supercharges are in the vector representation [1 0]. The unitarity provides a bound on the conformal weight h 0 of a superconformal primary [ We tabulate a partial list of N = 5 superconformal multiplets in Table 7. acted by an appropriate combination of two supercharges, and thus the null condition is highly non-trivial. For details, see [26].
#6 SO(5) and Sp(4) Dynkin labels are related by exchange of the two labels. We choose to use SO(5) labels to be coherent with different values of N .
We tabulate a partial list of N = 8 superconformal multiplets in Table 8.
It again proves useful to examine the stress-energy tensor multiplet A[0 0 1 1]1 2 in detail. Its decomposition into conformal multiplets is given in Figure 7.     Figure 1 and Figure 4. We will discuss this universal feature in section 3.10.

N = 7
For the N = 7 superconformal algebra, the R-symmetry is SO (7), under which supercharges transform as a vector [1 0 0]. Labelling the superconformal primary by [ We tabulate a partial list of N = 7 superconformal multiplets in Table 9.
is a sporadic top component. To see this, we use an argument analogous to those in sections 3.5 and 3.7. Comparing Figure 8 to This observation is crucial because this is the only supersymmetric [0 0 0] 2 component one can find in the N = 7 superconformal algebra, which however is required for the existence of stress-energy tensor. Figure 8: An N = 7 multiplet with a sporadic top component.

N ≥ 9
Having worked out up to N = 8, which was necessary to manifest existence of stress-energy tensors in all N , we are ready to generalize the patterns into generic values of N . Under the R-symmetry group SO(N ), N being even or odd, N supercharges transform as the vector [1 0 · · · 0]. The superconformal primary is labelled as ] is the rank of the R-symmetry group. Let us repeat the unitarity condition (2.11) for completeness: We tabulate partial lists of superconformal multiplets in Tables 10 and 11, for even and odd N . A generic pattern is apparent. Figure 10 shows a generic short superconformal multiplet A[0 · · · 0 1 0 · · · 0] with 1 being the k th Dynkin label, decomposed into conformal primaries. This multiplet has two top components, one of which is placed at the level k and the other at the level (N − k). Both top components are R-singlets. Figure 10: A generic short multiplet with primary [∧ k V ] 1 2 for generic N , written in both Dynkin labels and anti-symmetric tensor product notation.
Structure of this decomposition is physically clearer if we interpret the R-representation [0 · · · 0 1 0 · · · 0] as the k th anti-symmetric product of vector representations, denoted

Primary
Unitarity bound Null component Sporadic top  as ∧ k V . This structure is further justified by the fact that the tensor product where the other parts vanish in short multiplets. Note that ∧ 1 V here represents the supersymmetry G a −1/2 that takes a component to the next level. This not only simplifies the notation, but also manifests the fact that midway in the decomposition for SO(2N + 1) (3.12) appear, and also explains clearly why there are two 'towers' of components that both terminate with an R-singlet: a scalar (∧ 0 V ) or a pseudoscalar (∧ N V ). However, the most important role it plays will become clear in section 5.1.2.

Properties of the Multiplets
Given the lists of multiplets, many implications and applications are in order. In this section, we discuss the recombination phenomenon that happens when the conformal weight h of a long multiplet L[R] h hits the unitarity bound. We also present character formulae for both long and short global superconformal multiplets.

Recombination Rules
Decomposition of superconformal multiplets into conformal multiplets makes the recombination rules extremely apparent. For instance, let us consider an example of a long multiplet  (4.1) The above recombination rule is apparently demonstrated in Figure 11.
In fact, this recombination rule generalizes to any long multiplets with generic N . #7 We present a rather wordy proof for this statement.
Consider an arbitrary long multiplet L[R 1 · · · R r ] h of a global subalgebra with any value of N . As h approaches the unitarity bound h 1 2 in accordance with (2.11), its components are classified into two: they either belong to the short multiplet A[R 1 · · · R r ] h 1 /2 , or are null components of the short multiplet. According to Racah-Speiser algorithm, any components at level l have R-symmetry Dynkin labels that can be obtained by adding l different weights of the vector representation to that of the primary. We refer to these l weights as a path from the primary. Then, the null components can again be classified by whether the path includes the highest weight [1 0 · · · 0], or [1; 1] for the large N = 4, or not: 1. Null components whose path from the primary does include the highest weight [1 0 · · · 0], form a short multiplet A[R 1 + 1 · · · R r ]h 2. Null components whose path from the primary does not include the highest weight [1 0 · · · 0] arise only when first k ≥ 1 Dynkin labels R 1 , · · · , R k are zero. In such a case, some of the R-symmetry group's lowering operators annihilate the primary, thus not only the highest weight supercharge but also some lowered supercharges annihilate the primary. See section 3.3.3. of [26].
Such components complicate the argument because they do not seem to be included in the short multiplet A[R 1 + 1 · · · R r ]h 1 2 + 1 2 . However, these components are eliminated among themselves in the Racah-Speiser algorithm, and thus do not appear in the long multiplet. In other words, all components of the long multiplet that becomes null as it hits the unitarity bound fall into enumeration 1 above.
To see how, consider one of such components whose path from the primary includes [0 · · · 0 − 1 1 0 · · · 0] but not [1 0 · · · 0], · · · , [0 · · · − 1 1 0 0 · · · 0]. R 1 = · · · = R k = 0 with sufficiently large k is required for this to be a null component of For this component to appear in the decomposition of long multiplet, that is, to avoid a Dynkin label equal to −1, the path from the primary must also include either [0 · · · 0 1 − 1 0 · · · 0] or [0 · · · 1 − 1 0 0 · · · 0] but not both. Therefore, there is a one-to-one correspondence between components that belong to the enumeration 2: one whose path from the primary includes the former but not the latter and vice versa. However, treatment of Dynkin label equal to −2 by the RS algorithm precisely cancels the two.
Therefore, all components that appear in the long multiplet L[R 1 · · · R r ] h fall into either the short A[R 1 · · · R r ]h 1 2 or another short A[R 1 + 1 · · · R r ]h 1 2 + 1 2 , and we can write the generic recombination rule as follows: (4.2) Non-generic cases are easy to examine because the decompositions only contain few components. We simply state the result. To count the number of physical states, we first combine a left-moving and a rightmoving multiplet into a two-sided superconformal multiplet. A two-sided multiplet can be labeled by two letters with one unbarred and one barred to indicate the leftmoving and the right-moving null states. For instance, a multiplet can be understood as a tensor product of the left-moving multiplet L[R l ] h and the right-moving multiplet A[R r ]h. Components of the two-sided multiplets are also given by tensor products of components in respective sectors, which we denote as with a bar over the right-moving component.
When the recombination phenomenon happens for the left-moving sector, the number of states of each physical two-sided multiplet can be obtained by dimension of the left-moving sector multiplet multiplied by a common factor, which is dimension of the common right-moving sector multiplet. One exception is that the component [0] 0 in the left-moving sector has to be counted as zero. This is because it combines with the right-mover to lead to a conserved current. Conserved current will be discussed in section 5.1, and comparing dimensions in recombination rules has been discussed in detail in [26].
We can see that the number of physical states on both sides of the recombination rules indeed agree level-by-level, with one exception: let us consider a recombination phenomenon where the long multiplet has an R-singlet primary. This long multiplet L[0] h splits into the vacuum multiplet V [0] 0 and a short multiplet A[1 0 · · · 0]1 2 as h → 0. As discussed in the last paragraph, the vacuum multiplet has to be counted as zero. On the other hand, the short multiplet contains a component [0] 1 at the level one that does not appear as a conformal primary in the long multiplet L[0] h→0 , but appears as a conformal descendant of the superconformal primary [0] h→0 . This will be further discussed in the next subsection. To summarize, the primary [0] h→0 and its descendants of the long multiplet are split into the conserved current [0] 0 which is the vacuum multiplet, and [0] 1 and its descendants which appear in the short multiplet.
We find that unlike in higher dimensions, absolutely protected multiplets do not exist. A multiplet is absolutely protected when it does not appear in any of the recombination rules, so its spectrum is constant on the supersymmetric conformal manifold. Deformations of CFT are discussed in section 5.2, see also [27]. This is a consequence of the absence of isolated short multiplets: every short multiplet in two dimensions appears in the unitarity limit of the long multiplet with same quantum numbers.

Character Formulae for the Global Multiplets
Let us present character formulae for unitary representations, both long and short, of a superconformal algebra. A character of a superconformal representation can be defined as where V (h, R) denotes a representation built on a primary of conformal weight h and R-charge Dynkin label R. Here T i 0 are the Cartan generators of the R-symmetry group, and q = e 2πiτ , y = e 2πiz . Although the central charge c is irrelevant to the global subalgebra, we include its contribution to make connection with super Virasoro characters. See section 6.2.
To express superconformal characters, it is rather convenient to use orthogonal basis for the special orthogonal Lie group SO(N ) than the fundamental basis chosen in the previous section. We can find the linear relation between fundamental weights [R 1 · · · R r ] and orthogonal weights [h 1 · · · h r ] of the SO(N ) in (2.12) and (2.13).
Character formulae for long multiplets follow directly from the structure of the multiplets. Given a superconformal primary, which is an irreducible representation of the R-symmetry group SO(N ), superconformal descendants are obtained by applying successively the supercharges G −1/2 and the Virasoro generator L −1 on the primary. The supercharges are in the vector representation of the SO(N ) and their orthogonal weights are The character formula for a long multiplet L[{h i }] h then becomes for odd N with rank r, and for even N with rank r. Here χ o ({h i }) and χ e ({h i }) denote the Weyl character formula for so(N ) with odd and even N respectively [34], where |a ij | denotes determinant of the matrix with indices i, j = 1, 2, · · · , r. The structure of these characters is straightforward. The primary contributes q h− c 24 multiplied by Weyl character formula corresponding to its R weights. The other factors in (4.7) and (4.8) account for the contribution from the superconformal descendants.
We need more elaborations to obtain the superconformal characters for short multiplets A[{h i }] h due to the presence of null states. One might naively remove the factor (1 + y 1 q 1/2 ) that accounts for the highest weight supercharge that produces the null states. However, although it is highly obscured in the Racah-Speiser algorithm, it is not actually the highest weight supercharge that produces the null states, but it is the highest weight representation obtained from the highest weight supercharge acting on the highest weight of the primary.
The short multiplet character can be derived using the recombination rule from section 4.1. We can rewrite the recombination rule (4.2) using the characters of global long multiplets Ch (4.10) Note that the lowest exponent of q in the second short character on the RHS is larger by 1 2 than that in the first. This allows us to write the character of a short multiplet in terms of that of long multiplets perturbatively and to all orders. For simplicity, we omit the arguments (τ, {z j }) of each character.
We can easily incorporate the series of long multiplet characters into the Weyl determinant, using the fact that determinant of a matrix is linear in one particular row. Different long multiplet characters that appear in the series all depend on determinants of the same matrix except for the first row. Thus, we present the character formulae for global short multiplets as follows, for odd N with rank r, and for even N with rank r. Note that due to the Kronecker delta δ i,1 , the only modification from (4.7) and (4.8) is the first row (i = 1) of the matrix in the determinant. This form of short multiplet character resembles the known formula for short super Virasoro multiplet of the small N = 4 algebra [14], see also (6.20) in particular.
Let us work out a simple example of the character formula. Consider the short multiplet A[1; 1]1 2 of the large N = 4 algebra, which has been depicted in Fig. 1. Note that the primary [1; 1] is equivalent to [1 0] in the orthogonal basis. We first focus on the determinant part of (4.13), for which we expand the factor that includes Kronecker delta: (4.14) Identifying each term to Weyl character formula (4.9) and inserting back into (4.13), we have Let us examine the RHS order-by-order. In the order q 0 we have the primary representation, namely χ e ({1, 0}), or [1; 1] 1 2 in terms of fundamental weights. In the next order q 1 2 are all states that can be obtained by operating one of four supercharges on the primary, as would appear at the first level of the long multiplet. However, the LHS is a short multiplet which has the null states corresponding to χ e ({2, 0}), or [2; 2] 1 in terms of fundamental weights. Thus it is subtracted. Then in the next order q 1 , from all states that can be obtained by operating two of four supercharges on the primary, as would appear at the second level of the long multiplet, those that can be obtained by operating one supercharge on the first-level null states χ e ({2, 0}) are subtracted because they are the null states. However, those corresponding to χ e ({3, 0}) do not exist in the long multiplet due to Fermi-Dirac statistics, yet have been subtracted. Therefore, they are added back. Proceeding similarly, and recalling the effect of Virasoro operator 1/(1 − q) that produces conformal descendants, one can confirm that the character formula (4.15) is compatible with Figure 1.
Recall from the end of section 4.1 that there is an exceptional case of recombination rule related to conserved currents, (4.16) When a long multiplet whose primary is an R-singlet approaches the unitarity bound, the corresponding short multiplet V Consider the vacuum multiplet V [0; 0] 0 of the large N = 4 algebra. Its character, by (4.13), turns out to be The (1 − q) factor in the numerator cancels the same factor in the denominator, nullifying the effect of Virasoro operator L −1 . In other words, in this vacuum multiplet, not only are there no conformal primaries other than the superconformal primary [0; 0] 0 , but also there are no conformal descendants that are derived from the conformal primary by L −1 ∼ ∂. This is the manifestation of the conservation law ∂J z = 0 or∂Jz = 0, (4.18) where the holomorphic conservation law holds when the vacuum multiplet belongs to the holomorphic sector of full superconformal multiplet, and vice versa. A superconformal multiplet with the vacuum multiplet V [0] 0 in its (anti-)holomorphic sector is identified as the (anti-)holomorphic conserved current, as discussed in section 2.3 and will be discussed further in section 5.1.
On the LHS of (4.17), however, the primary component [0] h is not a conserved current since its conformal weight is non-zero, and therefore the conformal descendants (L −1 ) n |[0] h exist. These descendants are precisely the extra [0] 1 component, now treated as a conformal primary |[0] 1 and its descendants, that appears in the second short multiplet A[1 · · · 0]1 2 but not in the long multiplet L[0 · · · 0] h as a conformal primary.

Applications
Continuing based on the results of section 3, we discuss various aspects of twodimensional superconformal field theories, including stress-energy tensor, conserved currents, supersymmetric deformations, and supersymmetry enhancement.

Stress-Energy Tensor
Any two-dimensional conformal field theories contain an identity operator and a stress-energy tensor. We present in this subsection that there always exist superconformal multiplets that contain the corresponding states for all N including both small and large N = 4 with any value of the parameter α.
The holomorphic stress-energy tensor T in two dimensions is a global conformal primary of scaling dimension two and spin two. (2.1) also implies that T must be neutral under the R-symmetry, and is a top component in its superconformal multiplet. We can show that there is a unique multiplet that has the holomorphic (anti-holomorphic) stress-energy tensor [0 · · · 0] 2 ⊗ [0 · · · 0] 0 ([0 · · · 0] 0 ⊗ [0 · · · 0] 2 ) as a top component for each N except N = 6 (eachN exceptN = 6). As will be discussed further in section 5.1.2, the N = 6 superconformal algebra has two candidate multiplets that have the stress-energy tensor.
For all N ≥ 3 except for the small N = 4, the holomorphic stress-energy tensor resides in a short multiplet T ≡ AV [0 0 1 0 · · · 0|0 · · · 0] 1 2 ,0 , as depicted in Figure  12. Note that the primary of T transforms in the 3 rd anti-symmetric representation under the R-symmetry group SO(N ). In the present work, such a short multiplet T is referred to as a stress-energy tensor multiplet.
We also observe from Figure 12 that the stress-energy tensor multiplet T has other [0 0 0] 2 Figure 13: holomorphic N = 6 stress tensor multiplet with left-moving sector only. It is reducible into two parts conjugate to each other. We use both the Dynkin label notation and the anti-symmetric tensor product notation.

Remark on the N = 6 Stress-Energy Tensor and Anti-Symmetric Vector Products
The N = 6 superconformal algebra has SO(6) R-symmetry where the third antisymmetric tensor ∧ 3 V is no longer irreducible but decomposes into two irreducible representations should be interpreted as the stress-energy tensor, or even whether both could be the stress-energy tensor or not. We can argue that only one of them should be identified as a true stressenergy tensor. Otherwise any N ≥ 7 superconformal theories would have two stressenergy tensors and thus become invalidated. This is because the N = 7 stress-energy tensor multiplet AV [0 0 2|0 0 0] 1 2 ,0 can reduce to such two N = 6 multiplets T andT simultaneously. See section 5.3 for the detail.
The anti-symmetric tensor product makes this point clearer. In this language, it is apparent that one of the two [0 0 0] 2 in Figure 13 is a pseudoscalar with respect to the R-symmetry group while the other is a genuine scalar. The former is produced from an axial vector at the previous level, which in turn is produced from a 4-form at its previous level. However, for a genuine stress-energy tensor, the algebra requires that R-vector and R-adjoint components corresponding to the supercurrent and the R-current exist at the previous two levels. This implies that, although the multiplet with an anti-self-dual 3-form primary appears to contain a top component [0 0 0] 2 , it cannot be identified as a stress-energy tensor, and the self-dual counterpart can.
Note that this argument applies to every short multiplet A[∧ k V ]1 2 for every N : one of the top components is a scalar, and the other is a pseudoscalar of the Rsymmtry group. Therefore, it matters to distinguish two short multiplets , although they appear to be identical in terms of Dynkin labels. The case N = 6 was special only because the stress-energy tensor was involved.
An additional argument regarding supersymmetry enhancement and decomposition that supports the result here will be presented in section 5.3.

Conserved currents
From Table 1 through Table 11, one can see the presence of top component [0 · · · 0] 1 for all N . It allows for a conserved current [0 · · · 0] 1 ⊗ [0 · · · 0] 0 with spin 1 in any (N ,N ) theory. It is a Lorentz vector, has scaling dimension 1, and commutes with Rsymmetry and supersymmetry generators up to a total derivative. Thus, it qualifies as a flavor current.
Furthermore, a top component [0 · · · 0] s , thus a spin-s conserved current [0 · · · 0] s ⊗ [0 · · · 0] 0 is allowed for every half value of s starting from s = 1. #8 When s = 3 2 , [0 · · · 0]3 2 ⊗ [0 · · · 0] 0 corresponds to an extra supercurrent, which will be discussed in section 5.3. The case s = 2 corresponds to the stress-energy tensor which has been just discussed, and s ≥ 5 2 corresponds to the higher-spin conserved currents. In particular for each N ≥ 7, a supersymmetric higher-spin conserved current of #8 From s = 1 to s = N 2 they appear as top components of some short multiplets. For s > N 2 they appear as generic top components of long multiplets L[0 · · · 0] s− N 2 . Note that in the latter case s can be any real number. See Tables 10 and 11. spin s = N 2 − 1 appears in the stress tensor multiplet, and thus is universal in all theories. In higher dimensions d ≥ 3, presence of the higher-spin currents indicates a locally free theory [35], thus imposing an upper bound on the number of supersymmetries N for interacting theories [26]. In two dimensions this is not necessarily true. One simple example is the three-state Pott's model which is an interacting CFT with W 3 -algebra. We close this subsection with a remark that the above higher-spin current in the stress-energy tensor multiplet may extend the superconformal algebra to a non-linear (super W) algebra [5].

Relevant and Marginal Deformations
We turn into operators leading to relevant or marginal deformations of a given superconformal theory that preserve supersymmetry, as discussed briefly in section 2.3. These operators have to be Lorentz scalars with scaling dimensions ∆ ≤ 2 and supersymmetric. We thus look for top components from both left-moving and right-moving sectors with h 0 =h 0 ≤ 1. The deformation is marginal when the equality holds, and relevant when the inequality is strict.
From Table 1 through Table 11, we can conclude the followings. Therefore, a supersymmetric relevant deformation with ∆ = 3 2 is allowed for all global superconformal theories with any (N ,N ), except that if any of these is 4 it has to be the large N = 4 with α = 1. This relevant deformation however breaks the R-symmetry because it transforms as spinors under both SO(N ) and SO(N ).
In case of the large N = 4 with α = 1, relevant deformations with scaling dimensions ∆ = 1 + α 1+α and ∆ = 1 + 1 1+α are similarly allowed, provided that both sectors have large N = 4 symmetry and share the common value of α. In particular, note that the small N = 4 superconformal algebra (, i.e., α → ∞) does not admit a relevant deformation.
In particular, the marginal deformation [0; 0] 1 ⊗ [0; 0] 1 is guaranteed to exist in a large (4, 4) superconformal theory. It is known that the stress-energy tensor is actually a quasi-primary with respect to the Virasoro symmetry: it is a super Virasoro descendant of the vacuum. Therefore, whenever the stress tensor multiplet AV , where the marginal deformation resides. Therefore, a large (4, 4) superconformal field theory always contains a marginal deformation, and thus exhibits a non-trivial moduli space. This marginal deformation is in fact exactly marginal, and corresponds to moduli of the type IIB string theory on AdS 3 × S 3 × S 3 × S 1 , see [28,36].
This argument can be applied to higher N , but it results in the existence of an irrelevant deformation, which is subject to less interest.
A relevant deformation that resides in the stress tensor multiplet is referred to as a universal mass [27]. It is a deformation that is guaranteed to exist because it appears in the stress tensor multiplet, which breaks the conformal symmetry and often the R-symmetry as well. It results in a deformed super-Poincaré algebra with central or non-central charge extension. Study of the universal mass in higher dimensions have led to many interesting results, see [27] and references therein.
However, the universal mass does not exist in two dimensions. Every relevant deformation in two dimensions resides in a global superconformal multiplet whose primary has conformal weights h =h = 1 4 . It is obvious that this global multiplet itself is not a stress tensor multiplet nor a flavor current multiplet. One can further argue that this multiplet cannot belong to the same super Virasoro multiplet as the global stress tensor multiplet. This is because the stress tensor multiplet is the lowest super Virasoro descendant of the vacuum, and the primary of h =h = 1 4 cannot become another descendant of the vacuum. This implies that a non-central charge extension of Poincaré supersymmetry cannot be smoothly connected to a superconformal symmetry via relevant deformations.
Lists of allowed top components, conserved currents and deformations discussed Case Allowed top component Primary of the multiplet Table 12: List of allowed top components and multiplets in which they reside.

Supersymmetric operators Top component Multiplet in which it resides
Marginal deformation in sections 5.1 and 5.2 can be found in Tables 12 and 13.

Recombination Rules Revisited
In section 4.1 we have discussed recombination rules: how long multiplets decompose into short multiplets as they hit the unitarity bound. There, we considered only one sector. That is, a long multiplet in the left-moving sector was decomposed into short multiplets while the one in the right-moving sector remained unchanged.
Having discussed conserved currents and deformations of CFT, it is fruitful to consider the case where multiplets in the left-moving and the right-moving sectors approach the unitarity bound simultaneously. In particular, we are interested in recombination rules where marginal deformations or flavor currents appear. We write some of the recombination rules for multiplets that have non-generic R-symmetry Dynkin labels but for any value of (N ,N ) below, When a marginal deformation cannot remain marginal beyond the leading order, its scaling dimension should receive quantum corrections, and the corresponding short multiplet, combined with other short multiplets, should be up-lifted to a long multiplet. In other words, a short multiplet that contains an exactly marginal operator must not participate in any of the recombinations (5.2)-(5.4). It leads to a constraint that all marginal deformations become exactly marginal only if they do not break any flavor symmetry. Otherwise, one of (5.2) and (5.3) has to happen.

Supersymmetry Enhancement
It is useful to understand how an N -superconformal multiplet can decompose into various multiplets of fewer superconformal symmetries N < N . Note that, in the language of N superconformal algebra, N supercharges decompose into N supercharges and extra (N − N ) fermionic conserved charges. The R-symmetry algebra decomposes into the R -symmetry algebra corresponding to the N -supersymmetry, flavor symmetry algebra commuting with the R -symmetry algebra, and the remaining off-diagonal generators charged under both R -symmetry and flavor symmetry.
Let us in particular consider the (holomorphic) stress-energy tensor multiplet T (N +1) of the N + 1 superconformal algebra. T (N +1) can split into various Nmultiplets that must include the following multiplets of N superconformal algebra: 1. a stress-energy tensor multiplet T (N ) that has the holomorphic stress-energy tensor, N supercurrents and R-currents.

a short multiplet that has an extra R-neutral supercurrent as a top component.
In order that it be part of the enhanced N + 1 supercurrents, the N off-diagonal R-currents [1 · · · 0] 1 ⊗ [0 · · · 0] 0 should be contained in the same short multiplet.
Conversely, if a theory with N -supersymmetry contains all the multiplets enumerated above with mentioned properties, the theory is enhanced to N + 1.
As an illustration, let us consider the stress tensor multiplet of large N = 4 algebra with an arbitrary value of α, AV [1; 1|0; 0]1 2 ,0 . The multiplet and its decomposition to the N = 3 algebra are described in Figure 1 For a generic N , the stress tensor multiplet of N + 1 superconformal algebra is decomposed into the stress tensor multiplet of N superconformal algebra and an extra supercurrent multiplet thereof: where the number of supersymmetriesN in the right-moving sector is arbitrary and irrelevant.
Conversely, a global superconformal theory with a generic number N of supersymmetries in two dimensions is enhanced to an N +1 theory if and only if it contains the extra supercurrent multiplet AV [∧ 2 V |0]1 2 ,0 . Note that for large N = 4 or N = 6, AV [∧ 2 V |0]1 2 ,0 or AV [∧ 3 V |0]1 2 ,0 on the RHS of (5.6) is reducible to two irreducible parts. Although only one irreducible part corresponds to a genuine extra supercurrent multiplet or stress tensor multiplet (see section 5.1.2), both parts are required in order to enhance the theory into N + 1.
There are several types of non-generic cases of supersymmetry enhancement. Although what happens in each case is highly analogous to the generic case, we make remarks on the differences.

N ≤ 2 to N + 1
For N ≤ 2 where the smaller R-symmetry is abelian, the representation ∧ 3 V does not have a sensible interpretation. However, similar relations to (5.6) hold: is the large one with SO(4) R-symmetry group. Meanwhile, the small N = 4 with SU (2) R-symmetry group can be decomposed into or enhanced from the N = 2 theory, considering that SO(2) ⊂ SU (2). That is, As an application of (5.9), let us discuss new anomalies in (N ,N ) = (2, 2) superconformal theories resolving a long-standing puzzle that, unlike generic (2, 2) superconformal theories, the conformal manifold of a (4, 4) superconformal theory does not factorize into a product of Kähler manifolds [37]. The authors of [37] have shown that whenever the operator product expansion (OPE) between a chiral multiplet AA[1|1]1 the aforementioned anomaly occurs and the factorization fails. Here O andÕ are the primaries of chiral and twisted chiral multiplets while J ++ is a conserved current of spin one and U (1) R-charge two.
One can argue that any (2, 2) superconformal theories have enhanced (4,4) superconformal symmetry if and only if the OPE between O andÕ has a pole. We present a sketch of the proof below.
Consider the genuine irreducible N = 6 stress tensor multiplet denoted by T (6) . Its decomposition into conformal multiplets is depicted in the upper row of Figure 14. Note that the primary is self-dual part of the representation Decomposition of this multiplet into SO(5) R representations according to the general rule (5.5) is depicted in the lower row of Figure 14. This is precisely the stress tensor multiplet T (5)  Therefore, the irreducible N = 6 stress tensor multiplet is identical to the N = 5 stress tensor multiplet. In other words, an N = 5 global superconformal theory is automatically enhanced into an N = 6 theory. 6 Relation to Super Virasoro Algebra

Implications to the Super Virasoro Algebra
Among deformations and conserved currents discussed in the last section, some are guaranteed to exist because they belong to the same multiplet as the stress tensor, which must exist in any superconformal field theories. In this subsection, we will systematically investigate such components and their implications to the super Virasoro algebra, which is not fully understood for generic values of N .
Let us start with the case N = 3, which is the simplest among interesting cases. For concreteness, consider an (N = 3,N ) theory with anyN for the right-moving sector. See the uppermost part of Figure 15, where structure of the stress-tensor multiplet T (3) is depicted. Note that the product with V [0] 0 in the right-moving sector is implied for all multiplets. Furthermore, the multiplet structure contains additional information about the current. Not only does the current possess conformal weight h = 1 2 and transform as an R-singlet, but its anti-commutation relation with the supercurrents G a r must also yield the R-currents T [ab] m . Denoting the current as Γ 0 and matching dimensions and R-indices, one can predict that  must be part of the N = 3 super Virasoro algebra. This is precisely what was found in [8].
Let us proceed to large N = 4. Small N = 4 is not interesting in this respect, as the stress tensor multiplet therein does not contain any extra component other than {L, G, T }, see the second part of Figure 15. For the rest of this subsection, we no longer mention explicitly the right-moving sector, but assume that tensor product with the vacuum multiplet V [0] 0 in the right-moving sector with any number of superchargesN is always implied.
with dimension 1 2 are guaranteed to exist. The first two are identified as an extra supercurrent and extra R-currents that enhance the supersymmetry into N = 6. This has been the topic of section 5.3.3.
Due to peculiarity discussed in section 5.1.2 and 5.3.3, the N = 6 stress tensor multiplet contains no non-trivial components except for the primary, which is a selfdual 3-form under the R-symmetry with dimension 1 2 . Thus, implications to N = 6 super Virasoro algebra are more or less contained in the last paragraph.
Proceeding further, we can conclude that for all N ≥ 7, conserved currents U [a 1 a 2 ···a k ] m(r) (k = 3, 4, · · · , N ) that are k-forms under the R-symmetry with dimensions k 2 − 1 are guaranteed to exist. The mode number is m ∈ Z for k even, thus bosonic currents, and r ∈ Z + 1 2 for k odd, thus fermionic currents. Extending the mode numbers, commutation relations with the supercurrent is schematically written as Note that the conserved currents discussed in this subsection are to be distinguished from those discussed in section 5.1.3, which were supersymmetric and allowed for every half-integer value of spin, but not guaranteed to exist. Only the N -form coincides.

Relation to Super Virasoro Characters in N = 2 and
Small N = 4 Recall that the superconformal multiplets analyzed in section 3 are representations of global subalgebras of the larger super Virasoro algebras. In this sense, super Virasoro multiplets can be decomposed into the global superconformal multiplets. In this subsection, we discuss how the super Virasoro multiplets are decomposed, and compare recombination rules of super Virasoro multiplets and global superconformal multiplets for N = 2 and small N = 4, for which the super Virasoro multiplets have been thoroughly studied.

N = 2
Super Virasoro multiplets in N = 2 have been classified in [33]. A multiplet is long, or massive, if its primary [j] h lies above the unitarity bound, and it is short, or massless, if the primary lies on the unitarity bound. The unitarity bound is described by a set of line segments (see [38]) The line segment labelled by r describes a set of primaries [j] h that would be annihilated by the supercharge G sign(r) −|r| . Note that the global unitarity bound (2.10) corresponds to the line segments r = ± 1 2 extended to infinity. It is known that the N = 2 super Virasoro algebra has an outer automorphism referred to as spectral flow [39]. Although the extended N = 2 algebra that incorporates the spectral flow by one unit has many applications (see e.g. [40]), we do not impose such an invariance here.
Character of an N = 2 super Viasoro multiplet with primary [j] h , massive or massless, is written as (see [41]) Massless: where q = e 2πiτ and y = e 2πiz as in section 4.2. Note that h(j, r) in the massless character ensures that h be on the unitarity bound, where r labels which line segment (h, j) locates on. These multiplets satisfy a recombination rule as h in the massive multiplet approaches the unitarity bound given j: ,j (τ, z) + χ h(j,r)+|r|,j+sign(r) (τ, z) (6.9) Structure of the character can be analyzed as follows. Starting from the primary which contributes q h− c 24 y j , two factors in the numerator take negative-mode fermionic supercharges G ± −n+ 1 2 into effect while those in the denominator take negative-mode bosonic operators L −n , T −n into effect. The extra factor for the massless case simply cancels the operation by G sign(r) − 1 2 , which produces the null states.
Meanwhile, we can write characters of global N = 2 superconformal multiplets enlisted in Table 2 using more general formula (4.8) and (4.13), . (6.11) . (6.16) Every monomial q a y b in (#) satisfies a > |b| 2 , and appears with a positive coefficient. Therefore, a massless super Virasoro multiplet on the segment r = ± 1 2 is decomposed into exactly one short global multiplet with the same primary, and infinitely many long global multiplets. #10 The decomposition of super Virasoro multiplets can be applied to their recombination rule (6.9). Again, cases r ≥ 3 2 and r = 1 2 need to be treated separately. First, for r ≥ 3 2 , super Virasoro multiplet recombination, or branching if we are looking at it in the opposite direction, occurs at h higher than the global unitarity bound, so the same constituent long global multiplets are simply regrouped to form different sets of super Virasoro multiplets: On the other hand, for r = 1 2 where super Virasoro and global unitarity bounds coincide, the super Virasoro multiplet recombination occurs simultaneously as one of its constituent global multiplet also experiences the recombination.
Note that the global long multiplet Ch (g) j 2 ,j (τ, z) only works as a proxy for producing other long multiplets by multiplication with q a y b 's. It itself does not make sense because the unitarity bound has been reached.

Small N = 4
The small N = 4 super Virasoro multiplets are classified similarly to their global counterparts: a multiplet with primary [R] h is bounded by h ≥ R 2 , and is massless, or short, if the bound is saturated and massive, or long, otherwise. Their characters are similar to the case N = 2 except that now each R-representation may contain many states, graded by z. The central charge needs to be discretized by c = 6k (k = 1, 2, · · · ).
Using the orthogonal basis, we define l = R 2 so that the orthogonal weight l can take half-integer values and represents conventional SU (2) spin T 3 0 . The unitarity bound becomes h ≥ l in the NS sector we are working on. Then, the characters are given by #11 (see [13,14]) where 2l + 1 = 1, 2, · · · , k, and (6.20) where 2l + 1 = 1, 2, · · · , k + 1. The factor 1 1−y −1 is often absorbed into the product in the literature, but we choose to write it separately to manifest the appearance of Weyl character formula.
Meanwhile, characters of the global multiplets read #11 Another fugacity for the U (1) charge that distinguishes two copies of SU (2) ⊂ SO(4) may be introduced, as has been in [14]. However, we simply set the fugacity to unity because it plays no role in our argument.
From (6.19)-(6.22) we can infer the recombination rule of (4.5), common to the super Virasoro and global multiplets. Note that this formulae are not directly derived from (4.8) and (4.13), because in the small N = 4 algebra the R-symmetry is SU (2) rather than SO(4), and the supercharges do not form a vector representation thereof. However, essential properties appear in common. In particular, note how the (1 + y ± 1 2 q 1 2 ) 2 factors nullify the action of two supercharges simultaneously.
Decomposition of super Virasoro multiplets into global multiplets is done in a similar manner to N = 2. Massive super Virasoro multiplets are decomposed into long global multiplets only, while massless super Virasoro multiplets are decomposed into short and long global multiplets.
One thing we would like to check is when a massive super Virasoro multiplet hits the unitarity bound to branch into massless multiplets (6.23), if its global constituents also branch into the global constituents of the massless multiplets via (6.24). For this purpose, we only need to consider global constituents of massive multiplets that also saturate the global unitarity bound as the massive multiplet saturates its super Virasoro unitarity bound, and also short global multiplets in the decomposition of massless super Virasoro multiplets. It seems reasonable to believe that the other parts, which remain manifestly long throughout the recombination/branching, would simply regroup among themselves to belong to appropriate super Virasoro multiplets.
Consider the infinite sum in (6.19) and (6.20). As we are only interested in global constituents that hit the unitarity bound as h → l (h = l is automatic for (6.20)), we look for monomials q a− k 4 y b such that a ≤ b. Apart from the product term that only contains monomials q a y b with a ≥ b, we require 0 ≥ (k + 1)m 2 + (2l + 1)m + l − (k + 1)m − l = (k + 1)m m − (1 − 2l + 1 k + 1 ) . (6.25) However, since 0 < 1 k+1 ≤ 2l+1 k+1 ≤ 1, the only way to satisfy (6.25) is by m = 0, which saturates the inequality. Therefore, m = 0 in the infinite sum always leads to global multiplets that are manifestly long even when the super Virasoro multiplet hits the unitarity bound, and m = 0 may contribute to global multiplets that hit their unitarity bound simultaneously with the super Virasoro multiplet.

A Racah-Speiser Algorithm
In this appendix, we review the Racah-Speiser Algorithm that has been used throughout this article. Similarly to [26], our focus will primarily be on how to use the algorithm rather than its construction, and we refer the readers to [42] for the latter. Moreover, we are particularly interested in its application to special orthogonal groups SO(N ).
Racah-Speiser Algorithm, in short, is an efficient algorithm that yields a tensor product of two irreducible representations of a Lie group as a direct sum of irreducible representations, while avoiding overly detailed relations between the weights best described by Clebsch-Gordan coefficients.
Consider two irreducible representations of a Lie group G r with rank r, denoted by their highest weights λ 1 and λ 2 . Our objective is to find irreducible representations Λ a with corresponding multiplicities m i , such that where m a Λ a indicates that Λ a appears m a times in the sum.
Let us denote by {µ a 2 } (a = 1, · · · , dimλ 2 ) the complete set of weights of the irreducible representation λ 2 . Then, consider the set of weights {λ 1 + µ a 2 } (a = 1, · · · , dimλ 2 ). Each weight belongs to one of three categories, and contributes to the RHS of (A.1), after appropriate treatments: • Some of the weights live in the Weyl chamber, that is, all of their Dynkin labels are non-negative. These weights do not require any special treatment, and each of them contributes to the RHS of (A.1) as a highest weight of an irreducible representation Λ a with multiplicity 1.
• Some of the weights do not live in the Weyl chamber, however, can be brought into one by a series of reflections. Here, the reflection is defined as a shift by the Weyl vector ρ, followed by a Weyl reflection and then a negative shift by the same Weyl vector ρ. Using fundamental Dynkin labels, the i th Weyl reflection σ i of a weight [R 1 · · · R r ] can be written as where A ij denotes the Cartan matrix of g r . Incorporating the shifts by the Weyl vector ρ = [1 · · · 1], the i th reflection is summarized as Note that there are r different reflections, where r is the rank of the Lie group. Number of reflections required to bring a weight into the Weyl chamber is defined modulo 2. Then, each of the weights that belongs to this category contributes to the RHS of (A.1) after being brought into the Weyl chamber via W reflections, with multiplicity (−1) W .
• Finally, some multiplets are reflected into itself. By inspection of (A.3), one can see that a weight belongs to this category if and only if at least one of its fundamental Dynkin labels equals −1. Considering the −1 factor that entails each reflection, these weights naturally have vanishing contribution to the RHS of (A.1).
To summarize, each weight in the set {λ 1 + µ a 2 } whose Dynkin labels do not include −1, contributes to the RHS of (A.1) by an irreducible representation whose highest weight equals itself, or a reflection of itself. If an odd number of reflection is required in the process, it contributes negatively, cancelling a positive but identical contribution from another weight. At the end, the cancellation is always complete and there is no remaining contribution with a negative multiplicity. Note that exchanging the role of two representations l 1 and l 2 gives the same result.
Throughout this article, the Racah-Speiser Algorithm is always performed in a special orthogonal group SO(N ), with one of the two multiplicands being the vector representation that represents the supercharges. In such cases, the minimum Dynkin label that can appear in any of the weights in the set {λ 1 + µ a 2 } is −2, and the classification and treatment of the weights become extremely simple. That is, whenever there is a Dynkin label equal to −1 we dispose the weight, and else if the k th label is −2, we add the k th row of the Cartan matrix to the weight in accordance with (A.3) to bring the k th label to 0 then to cancel with an identical weight. Of course, if the reflection produces a −1 for another label, we dispose the weight.
Let us take the product [0 1 0]⊗[1 0 0] in SO(7) as an example. The weight system of the highest weight representation [1 0 0] is given by When [0 1 0] is the primary of a multiplet, the product with vector [1 0 0] represents states at the first level of the multiplet. The first piece [1 1 0] represents null states when the unitarity bound is saturated, then we can identify the remaining parts as in agreement with (3.11) and the discussion followed.