Studying superconformal symmetry enhancement through indices

In this note we classify the necessary and the sufficient conditions that an index of a superconformal theory in $3\leq d \leq 6$ must obey for the theory to have enhanced supersymmetry. We do that by noting that the index distinguishes a superconformal multiplet contribution to the index only up to a certain equivalence class it lies in. We classify the equivalence classes in $d=4$ and build a correspondence between ${\cal N} = 1$ and ${\cal N}>1$ equivalence classes. Using this correspondence, we find a set of necessary conditions and a sufficient condition on the $d=4$ ${\cal N} = 1$ index for the theory to have ${\cal N}>1$ SUSY. We also find a necessary and sufficient condition on a $d=4$ ${\cal N}>1$ index to correspond to a theory with ${\cal N}>2$. We then use our results to study some of the $d=4$ theories described by Agarwal, Maruyoshi and Song, and find that the theories in question have only ${\cal N} = 1$ SUSY despite having rational central charges. In $d=3$ we classify the equivalence classes, and build a correspondence between ${\cal N} = 2$ and ${\cal N}>2$ equivalence classes. Using this correspondence, we classify all necessary or sufficient conditions on an ${\cal N}=1-3$ superconformal index in $d=3$ to correspond to a theory with higher SUSY, and find a necessary and sufficient condition on an ${\cal N} = 4$ index to correspond to an ${\cal N}>4$ theory. Finally, in $d=6$ we find a necessary and sufficient condition for an ${\cal N} = 1$ index to correspond to an ${\cal N}=2$ theory.


Introduction and summary of results
Superconformal theories have been extensively studied in the past 40 years. This has partly been because their enhanced symmetries allow for various exact computations to be performed in these theories, even at strong coupling, so that they provide useful windows into strong coupling physics. Superconformal field theories exist in 2 ≤ d ≤ 6 dimensions, and in 3 ≤ d ≤ 6 dimensions a superconformal algebra is essentially defined by the number of spacetime dimensions d and the amount of supersymmetry N . Superconformal theories exist for N = 1 − 6, 8 in 3 dimensions, N = 1 − 4 in 4 dimensions, N = 1 in 5 dimensions and N = (1, 0), (2, 0) in 6 dimensions. In the last 25 years superconformal theories without a known Lagrangian (non-Lagrangian theories) became an important research subject. Interest in the non-Lagrangian superconformal theories was first kindled by the seminal paper [1] and since then this topic was vastly developed. In the absence of a Lagrangian one needs some quantity to describe the theory, and one of the most convenient options is the superconformal index. The superconformal index was first defined by Römelsberger [2] for N = 1 d = 4 theories and since then became a popular tool to study the content of superconformal theories in various spacetime dimensions [3]. Unfortunately, the index captures the short multiplets only up to recombination, so it is impossible to restore the full short multiplet content from the index; one can determine only the net difference between the number of multiplets that recombine together into a long multiplet. Therefore, one can put all the multiplets into certain equivalence classes according to their contribution to the index; this idea was first suggested in [4] for the d = 4 N = 2 theories and then developed for the N = 1 case in [5].
The superconformal index might not fully reflect properties of the theory; for example in [6] many N = 2 Argyres-Douglas theories were found at the end of an RG flow, triggered by a d = 4 N = 1 massive deformation. If one would compute the index in this case, one would get an N = 1 index for a theory that in fact has N = 2 SUSY. In this note we study how one can find SUSY enhancement through the index by studying properties of the superconformal index in 3 to 6 dimensions. In other words, we want to find necessary and/or sufficient conditions on a low-SUSY index to correspond to a theory with a higher amount of SUSY. It is reasonable to expect that certain restrictions on the index exist, because if SUSY enhancement happens, the original multiplets should somehow combine to the new, enhanced ones.
There are various ways in which one could enhance supersymmetry, and we focus on two of them: coupling a theory to an exactly marginal deformation (EMD), and triggering an RG flow by a massive deformation. In 3 dimensions only N = 1, 2 admit EMDs, and in N = 2 theories a deformation can be exactly marginal if and only if it doesn't break any flavor symmetry [7]. However, all N > 2 theories have an N = 2-preserving massive deformation; for N = 3, 4 theories it lies in the flavor current multiplet, while for N > 4 theories it lies in the stress-tensor multiplet. Thus, N = 2 theories can be enhanced to N > 2 ones by an EMD; one can also possibly obtain an enhanced SUSY by triggering an N = 2-preserving RG flow.
In N = 1, 2 four-dimensional superconformal theories the number of EMDs is not fixed, pure N = 3 theories do not admit SUSY-preserving EMD and in N = 4 there is always one and only one SUSY-preserving EMD. Thus, some theories may have their SUSY enhanced by EMD; in particular, N = 1 can be enhanced to N = 2 − 4, a famous example involves the EMDs that break N = 4 to N = 1 [8]. Index does not change under an EMD, so if some theory can be deformed to a theory with higher amount of SUSY by an EMD, the index can't obey any sufficient conditions for enhancement. This also means there cannot exist a necessary and sufficient condition on an N = 1 index that would show us N = 1 → N = 2, 3, 4 SUSY enhancement. Extra constraints on index, when part of the enhanced SUSY is visible in the UV, are discussed in appendix A. 4. In 5 dimensions there is only one superconformal algebra and SUSY enhancement is impossible. In 6 dimensions one cannot have relevant or marginal deformation that would preserve superconformal theory; however we still list a necessary and sufficient condition on SUSY enhancement through index, as one can obtain N = (1, 0) index for N = (2, 0) theory from e.g. stringy construction.
The index as a function does not depend on the RG flow. This means that one can compute it in the UV, then trigger an RG flow and track it to the IR where the superconformal symmetry is hopefully restored. However, the R-charge is not guaranteed to remain the same along the flow, so the function one gets in the IR is not necessarily the IR index and one must do a-maximization to find the real index in the IR 1 . On the other hand, unlike in the EMD case the index in the IR may obey sufficient conditions, and this was the case e.g. for all the theories studied by Maruyoshi and Song in [6] when they were able to compute the index (we should notice that the SUSY enhancement conditions suggested in [6] are not precise, as we will discuss below).
One particularly interesting method to obtain new SCFTs after applying massive deformations was described by Maruyoshi and Song in [6] (the first use of this method, to the best of our knowledge, was done by Gadde et al. in [9]). The method is implemented as follows: let us suppose we have a d = 4 N = 2 SCFT T with a non-Abelian flavor symmetry F . First we shall add a superpotential W = Tr M µ, where M is an N = 1 chiral multiplet transforming in the adjoint of F , and µ is the moment map operator of T . Then we give a nilpotent vacuum expectation value (vev) to the chiral multiplet M as M = ρ(σ + ). The nilpotent vev is specified by the embedding ρ : su(2) → f, where f is the Lie algebra of F . For F = SU (N ), SO(N ) the embedding is classified by the partition of N (and partition of 2N for Sp(N )). This procedure defines the theory T IR [T , ρ] in the infrared fixed point, that is generically N = 1, but may have higher SUSY.
This note is composed as follows. In section 2 we list the SUSY enhancement conditions on the d = 4 index. We show that there is a necessary and sufficient condition for SUSY enhancement on N > 1 indices. We also show there is one sufficient and a set of necessary conditions on the N = 1 index for SUSY enhancement, and we list them for N = 1 → N = 2, 3, 4. Section 3 is dedicated to using some of the results we obtained in the previous section. We first describe the massive deformation method of [6] and then study several of the theories obtained in [10] using the methods we developed for d = 4 indices, to show how different necessary and sufficient conditions on N = 1 → N = 2 enhancement work out in different cases; we prove that several theories found by [10] have only N = 1 SUSY despite having rational central charges.
In section 4 we list the SUSY enhancement conditions on the d = 3 index. We show there is a necessary and sufficient condition for SUSY enhancement on N > 4 indices. We demonstrate there is a sufficient and a set of necessary conditions for SUSY enhancement on N = 3 index, we list them for N = 3 → N = 4 − 8. We show there is a set of necessary conditions for SUSY enhancement on N = 2 index and list them for N = 2 → N = 3 − 8. Finally, for N = 1 index we show there is one necessary condition on SUSY enhancement and describe it.
In section 5 we formulate a necessary and sufficient condition on N = 1 → N = 2 SUSY enhancement in d = 6.
Appendices contain derivations of the enhancement conditions and related computations. In appendix A we present various derivations for the d = 4 case. In subsection A.1 we set the notation and then list how N = 2 multiplets decompose into N = 1 multiplets under SUSY reduction, in subsection A.2 we describe how the relation between equivalence classes in N = 1 and N > 1 may be built, and in subsection A.3 we work out this relation between N = 1 and N = 2 equivalence classes in details. In subsection A.4 we work out the relation between N = 1 and N = 2 equivalence classes with flavor taken into account.
In appendix B derivations for the d = 3 case are presented. In subsection B.1 we set the notation, in subsection B.2 we describe multiplets content in N = 2, define N = 2 index and shortly describe multiplet content in N > 2 algebras. In subsection B.4 we derive conditions on N = 2 index for SUSY enhancement and in subsection B.5 we do the same for N = 3 index.

Index and SUSY enhancement conditions in 4 dimensions
All conditions on SUSY enhancement from the index can be divided into sufficient, necessary and sanity checks. A sufficient condition shows that the index at hand in fact must correspond to a theory with a higher amount of SUSY (or a free theory). Such a condition can only be related to the presence of an extra supercurrent multiplet. Necessary conditions stem from the requirement on lower-SUSY multiplets to be able to recombine into higher-SUSY multiplets in the case of SUSY enhancement. Finally, sanity checks are employed to verify that the function we study is in fact a superconformal index; they check that index contributions do not violate unitarity bounds.
Obviously, the most desired SUSY enhancement condition would be both necessary and sufficient. For such a condition to exist, we need the extra supercurrent multiplet to be protected. It turns out, that this is the case for the N = 2 superconformal algebra; a spin-( 1 2 , 1) supercurrent multiplet D 1 2 ,(0,0) is protected [11] and therefore one can write a necessary and sufficient condition on an N = 2, 3 index to correspond to a higher-SUSY theory. If one defines the N = 2 index as where r, R 2 are Cartans of SU (2) r × U (1) R 2 R-symmetry, j,j are the charges under SU (2) L × SU (2) R and the trace is taken over all operators, then the coefficient α in front of the term t 5/3 v −1/3 (x + 1 x ) (that is related to D 1 2 ,(0,0) ) must be nonnegative. If α > 0, then there is N = 2 → N = 2 + α SUSY enhancement or the theory is free; the reverse is also true.
The N = 1 superconformal algebra has very few protected multiplets (and it doesn't protect conserved current or conserved supercurrent multiplets), so we cannot have a necessary and sufficient condition, and must carry out a more thorough analysis, to which we dedicate the rest of the section. The N = 1 superconformal index in 4 dimensions was first defined by Römelsberger; we will use the definition similar to [12]: where only the operators in the kernel of H contribute, H = 1 2 {Q † , Q}, Q is a specific supercharge with U (1) R charge 1 and SU (2) L charge − 1 2 , J 3 is the spin under SU (2) L ,J 3 is the spin under SU (2) R part of SO(4), and R = R + 2J 3 with R being the generator for U (1) R R-symmetry. For example, according to our definition the contribution of the N = 1 stress-tensor multiplet is given by Let us notice that all N = 1 single-multiplet contributions look like where χ j is an SU (2) spin-j character; for almost all multiplets the sum contains only one term (the exception being free-field multiplets). One can notice that if one multiplies the index by (1 − t/x)(1 − tx), one gets an infinite series in t, that loosely speaking is an infinite series in the scaling dimensions of the N = 1 multiplets' bottom components. Therefore it is easier to deal with the corrected index I corr = (1 − t/x)(1 − tx)(I − 1) and search for certain powers of t, x (we subtract 1 because there is no Verma module related to the trivial representation of the superconformal symmetry). We will consider the corrected index from now on, and we will also drop the corr subscript and omit the word "corrected" for brevity.
In the subsections below we list the results without derivations, some details can be found in appendices A.2, A.3; our notation for N = 1 multiplets is defined in appendix A.1 and for N = 2, 4 it is consistent with [11].

SUSY enhancement and restrictions on N=1 indices for enhancement to N=2
Sufficient condition: If the coefficient in front of the t 7 3 (x+ 1 x ) term is positive and equals k, we have either SUSY enhanced to at least N = k + 1 or a free theory; the reverse is not necessarily true. This happens, because the [ 1. If we know that the N = 1 theory in question possesses a global symmetry F with dimension dim(F ) and kS 0 matter multiplets, then either the coefficient in front of t 4 3 is at least dim(F ) − 3 − k or there is no N = 1 → N = 2 SUSY enhancement (there might be N = 1 → N = 3, 4 enhancement). This is due to the fact that in an N = 2 theory the flavor current lies in the multiplet with the moment map operator that contributes to the N = 1 index as t 4 3 , and the only entity that contributes to the index as −t 4 3 is the left free chiral multiplet. As at most 3 N = 1 flavor currents can become N = 2 R-currents and lie in the N = 2 stress-tensor multiplet (in addition to the N = 1 R-current), if the t 4 3 coefficient is less than dim(F ) − 3 − k, we have not enough moment map operators and can't have SUSY enhancement. As contributions to t 4 3 term also come from N = 2 EMDs and N = 2 conserved supercurrent, it is not impossible to have the t 4 3 coefficient be bigger than dim(F ).

If there is a term in the index
3 , then each of the coefficients in front of (−1) 2j+1 t 2 ) should be greater than α, or N = 1 → N = 2 SUSY enhancement cannot happen. This is because of all N = 2 multiplets, only E can contribute to N = 1 index these terms, and the expression we described above counts the number of E 3R+6 2 ,(0,j) . This condition is used most often forj = 0 case, for which it can be simply reformulated as "If there is a term nt R in the index with 2 3 < R < 4 3 , then the coefficients in front of the −t R+1/3 (x + 1 x ), t R+2/3 terms should not be smaller than n".
As there is a relation between N = 1 and N = 2 equivalence classes (see appendix A.3), one cannot have any other necessary conditions on SUSY enhancement through the index without making some assumptions about the theory.
Free theory and sanity checks: 3 and a negative sign, the index does not correspond to a unitary N = 1 theory.
It is worth noticing, that in the scenario of N = 1 SUSY getting enhanced to N = 2 after an exactly marginal deformation the sufficient condition cannot be satisfied. This is because the EMD does not change the index and if the index manifestly shows the presence of a spin-3 2 conserved current for N = 2 theory, then it should also be present for the original N = 1 theory, which signals a contradiction. However, we can use the sufficient condition to check the SUSY enhancement for the theories that are the endpoints of an RG flow, as during the RG flow the R-charge can change. Similar reasoning also precludes us from having any kind of sufficient conditions for N = 1 → N = 3, 4 SUSY enhancement after EMD.
Conditions with additional assumptions If one makes the additional assumption (consistent with all known examples) that N = 2 theories do not have "spinning Coulomb branch operators" (namely E r,(0,j) multiplets with j > 0), then one can obtain an additional necessary condition for SUSY enhancement. It turns out, that if there is a term in the index t 2+R χ j (x) with 2j−2 3 >R > 2j−4 3 and j ≥ 1, then there is no SUSY enhancement. Moreover, coefficients in front of the terms t R , −t R+1/2 (x + 1/x) for 4 3 > R > 2 3 should be equal. Using this fact, one can also simplify necessary condition 3.

SUSY enhancement and restrictions on N=1 indices for enhancement to N=3
We list restrictions on the corrected index that can be useful for studying N = 1 → N = 3 SUSY enhancement. The obtained results are largely similar to the previous case, so we list only those that are new for N = 3. One can also take in consideration that for a theory to have N = 3, 4 SUSY the central charges a and c must match [13]. Sufficient condition is same as above.

Necessary conditions
1. If the coefficient in front of the t 2 term is negative and less than −8, we cannot have SUSY enhanced to N = 3. This is because this term gets contributions from S 2 multiplets (with a positive sign) andĤ (0,0) (with a negative sign). The latter hosts N = 1 conserved flavor currents; as N = 3 theories do not admit conserved flavor currents, the N = 1 index of an N = 3 theory can have at most 9 − 1 = 8 conserved flavor currents, that are in fact N = 3 R-currents.
2. If the coefficient in front of the t 4/3 term is less than 2, we cannot have a pure N = 3 theory. This is because this term gets contributions from S 4/3 multiplets (and only from them) with a positive sign, and the N = 3 stress-tensor multiplet has 2 such multiplets in its N = 1 decomposition. It can be greater than 2 due to contributions of B [0;0],R 3 =12,(0,0) multiplets (an N = 3 multiplet that is canceled under the action of all Q ; in terms of N = 2 multiplets B [0;0],R 3 ,(0,j) → E R 3 6 ,(0,j) ⊕E R 3 6 + 1 2 ,(0,j± 1 2 ) ⊕E R 3 6 +1,(0,j) ). It could be theoretically less than 2 due to contributions of free field multiplets, but a pure N = 3 theory cannot have free fields.
3. If the coefficient in front of t 4/3 is α > 2, then the coefficient in front of [−t 5/3 (x+1/x)] should be at least 2α − 2, or we cannot have an N = 3 theory. This is because −t 5/3 (x + 1/x) gets contributions from S 5/3(0, 1 2 ) multiplets (and only from them) with a negative sign, and the only multiplets allowed in an N = 3 theory that have S 5/3(0, 1 2 ) in their decomposition are the N = 3 stress-tensor multiplet, B [0;0],12,(0,0) and B [0;0],18,(0, 1 2 ) . The stress-tensor has 2 S 5/3(0, 1 2 ) and 2 S 4/3 in its N = 1 decomposition, B [0;0],12,(0,0) has 2 S 5/3(0, 1 2 ) and 1 S 4/3 , while B [0;0],18,(0, 1 2 ) has only 1 S 5/3(0, 1 2 ) . 4. If there is a term αt γ in the index with 2 3 < γ < 4 3 , then there also should be terms corresponds to an S γ multiplet, that only can lie in an . Similarly to section 2.1, in the scenario of N = 1 SUSY getting enhanced to N = 3 after an exactly marginal deformation we cannot have the sufficient condition satisfied. There also is a generalization of condition 4 that puts a restriction on N = 1 index terms of for general j; it comes from the fact that only S multiplets in N = 1 can contribute to these terms, and they pop up only in some special N = 3 multiplets. However, this rule is very cumbersome to formulate, so we do not write it out explicitly. There is a general equivalence relation between N = 3 and N = 1 equivalence classes built as in appendix A.3, but it is even more cumbersome.
Conditions with additional assumptions Similarly to the previous subsection, if one makes the additional assumption (consistent with all known examples) that N = 2 theories do not have "spinning Coulomb branch operators" (namely E r,(0,j) multiplets with j > 0), then one can rule out existence of B [0;0],R 3 ,(0,j) multiplets for all R 3 , j. Therefore, under this assumption for index to correspond to the pure N = 3 theory, its corrected expansion should be 2t 4/3 −2t 5/3 (x+ 1 x )+O(t 2 ) and shall not contain any other terms of the t 2+R χ j (x) kind for 2j−2 3 >R ≥ 2j−4 3 .

SUSY enhancement and restrictions on N=1 indices for enhancement to N=4
We list restrictions on the corrected index that can be useful for studying N = 1 → N = 4 SUSY enhancement. Unlike for the N = 2, 3 cases, the R-symmetry group does not have a U (1), so the multiplet content of the theory is greatly restricted. For example, the only multiplet with bottom component scaling dimension less than 3 that can appear in an interacting N = 4 theory is the stress-tensor multiplet. Using this knowledge, one can find the following restrictions: Sufficient condition is same as above.

Necessary conditions
1. The corrected N = 1 index expansion for an interacting N = 4 theory should look like Here n is the number of N = 4 free field multiplets, α corresponds to the number of multiplets. We also used the fact that the N = 4 stress-tensor multiplet decomposes into N = 1 multiplets as 2. Terms in the index must feature t in powers of n 3 with n integer. This is because the N = 4 R-symmetry group does not have a U (1) subgroup, so all operators in short representations have integer or half-integer scaling dimension.
3. If there is a term of the form t 2+R χ j (x) withR = 2j−3 3 , then the theory can't have N = 4 SUSY. This is because such a term can only lie in an N = 2 E 2R/3,(0,j) multiplet, which cannot appear in a decomposition of an N = 4 multiplet. Terms withR = 2j−4 3 can appear in a free theory only.
One also can write an analogue of condition 3 for terms with 2j 3 >R ≥ 2j−2 3 , but it turns out to be extremely cumbersome and not very useful. It is possible that there are more necessary conditions on N = 1 index for enhancement to N = 4, since the relation between N = 2 and N = 1 equivalence classes does not generalize easily for N = 4; it would be interesting to work out the details of such a relation in detail.

AMS and SUSY enhancement
In this section we will study theories that appear at the end of an RG flow, triggered by giving a nilpotent vev to a gauge singlet M coupled to the moment map operator µ via W = TrµM ; this idea was introduced in [9] and developed in [6,10,[14][15][16][17]. A proper introduction into the subject can be found in [6], and one particular example is studied in detail in [14], we will provide a short recap of their ideas in the subsection below. In subsection 3.2 we will use the SUSY-enhancement checks we found above for several theories studied in [10].

Description of the method
Let us assume we have a d = 4 N = 2 SCFT T with non-Abelian flavor symmetry F ′ , and R-symmetry SU (2) r × U (1) R . Due to the presence of flavor symmetry the theory has a certain number ofB 1 multiplets whose lowest component µ is a scalar that has charge 1 under the SU (2) r Cartan I 3 and is neutral under the U (1) R charge R 2 . In agreement with the notation of [14], we introduce J + = 2I 3 , J − = R 2 ; µ has charges (2, 0) under (J + , J − ), and M has charges (0, 2) (we will use this notation from now on). The whole procedure now goes on as follows: 1. We deform T by adding an N = 1 chiral multiplet M transforming in the adjoint representation of F , and the superpotential coupling W = TrµM . This superpotential breaks the supersymmetry to N = 1; the new R-symmetry is R = 2 3 J + + 1 3 J − and there also is residual symmetry F 0 = 1 2 (J + − J − ) that becomes a global symmetry of the N = 1 theory. This superpotential is by itself a marginally irrelevant term and in the IR theory simply decouples into the original one and the free chiral multiplets we added in the beginning.
2. Now we give a nilpotent vev to M . According to the Jacobson-Morozov theorem, any nilpotent element of a semi-simple Lie algebra f is given via an embedding ρ : su(2) → f as ρ(σ + ) (here and below σ + , σ 3 correspond to the related Cartan subalgebra). Under such an embedding the adjoint representation of f decomposes into adj → (2) and R j is a representation of the residual flavor algebra h under the embedding ρ. The vev breaks J − , but preserves J − − 2ρ(σ 3 ). We denote the SU (2) used in the embedding as SU (2) ρ .
For classical Lie algebras every embedding is specified by a certain partition of N [n 1 , . . . n k ] : n i ≥ n i+1 , n i = N , where N is defined by the flavor algebra F ′ . In a generic case not all partitions correspond to an embedding, we cover that in more details below; the systematic treatment of the problem can be found in [18]. As a certain number n k can appear in the partition more than once, we introduce the shorthand notation for the partition [n m 1 1 , . . . n m k k ], where m i correspond to the number of times n i appears in the partition. Note that this is different from the notation in [10], our n k corresponds to their k and our m k corresponds to their n k . For example, one of the possible partitions of 13 is [3, 3, 2, 2, 2, 1] that can be rewritten in our notation as [3 2 , 2 3 , 1]. For exceptional Lie algebras there is a classification of nilpotent orbits in [18] and the adjoint decomposition for all E 6 and E 7 nilpotent orbits has been worked out in [19] and [20], respectively.
3. After Higgsing, the superpotential becomes where j is the spin from the adjoint decomposition, j 3 is the σ 3 eigenvalue and f labels the representation under the new flavor symmetry group F . However, due to flavor current non-conservation the components of the superpotential with j 3 = j combine with the current and become non-BPS; therefore the corresponding M multiplets decouple. Thus we get that only M multiplets of the M j,−j,f kind will stay, and they are coupled to µ j,j,f ; the related superpotential is 4. If we start from a Lagrangian theory with matter fields in the fundamental rep of F ′ , the quark content of the theory can be read off from the partition. If the partition is [n m 1 1 , . . . n m k k ], then there are m 1 quarks that lie in the dimension-n 1 representation of SU (2) ρ , . . . , and m k quarks that lie in the dimension-n k representation of SU (2) ρ . At the end of the RG flow they will have (J + , J − ) charges (1, 1 − n 1 ), (1, 1 − n 2 ), . . . , (1, 1 − n k ).

5.
From the quark content one can simply calculate the meson content of the theory.
Mesons lie in the adjoint of F ′ , and we can find the adjoint decomposition from the tensor product of 2 fundamental reps (or fundamental and antifundamental for SU (N )). For SO (Sp) flavor group the tensor product will contain symmetric and antisymmetric part, and the adjoint will be antisymmetric (symmetric) respectively, so we will need to keep track of that.
6. Breaking J − and adding the superpotential also affects the anomaly coefficients for J + , J − , so we need to recalculate them accordingly. Anomalies can be expressed in terms of meson operators according to [10] and [14] as Here N M is the total number of operators that got a vev and d(M −j,j ) = 2j + 1, a U V , c U V correspond to the central charges in the UV, k f denotes the flavor central charge, sums are taken over M j,−j , and I y is the embedding index.
7. The R-symmetry in the IR will be given by some mixture of J + , J − : R IR (ǫ) = 1+ǫ 2 J + + 1−ǫ 2 J − (we assume there is no global accidental symmetry in the IR). The exact value of epsilon can be determined by a-maximization: we need to maximize Tr R IR (ǫ) (3.9) 8. After doing a-maximization we need to check whether all scalar chiral operators satisfy unitarity bounds (whether they have R IR ≥ 2 3 ). If this is not the case for some of the operators, this means that they become free and decouple. Therefore, we need to subtract them according to [21] (note that in [21]ã is defined as 3 32 a) and maximize the new a: a n (ǫ) = a n−1 (ǫ) 10) a n−1 is the current trial a, M runs over operators with different R-charges such that R(M ) ≤ 2 3 , #(M ) is the number of operators with given R-charge and R(M ) is the R-charge under the current trial R-symmetry. One should notice that our expression disagrees with [21] by a factor of 1 48 that corresponds to subtraction of an N = 1 multiplet (that is coupled to M ) from the central charge. After doing the subtraction we should recalculate ǫ and check whether any new operators hit the unitarity bound; this should be repeated until all operators in the theory have R(M ) ≥ 2 3 . Apart from the usual suspects of M j,f we introduced in 3., operators of the kind Tr φ k (where φ is a chiral superfield in the adjoint of the gauge group), that are related to the Coulomb branch can also decouple (see [14]). It is also possible that the quark operators q n , q n φ k will decouple; however, gauge-invariant quark operators should be subtracted only if they can't be expressed as a derivative of the superpotential. In fact, in all cases we considered there was no need to subtract quark operators; we are not sure whether this is always the case. 9. c can be computed in a similar fashion; c is given by Tr R IR (ǫ) (3.11) and subtracting decoupled operators goes on as After all these manipulations we obtain the correct a, c and R-symmetry. 10. We can now compute the N = 1 index of the resulting theory. For example, the index of a theory that lies at the end of the RG flow of SQCD with SU (N ) gauge and SU (2N ) flavor group for a general partition and no quark operator decoupling is given by Here t and x are the usual N = 1 index fugacities, ξ is a fugacity for the residual ) is a product that is taken over all M that were added in the superpotential and did not decouple.
is taken over non-zero roots of SU (N ) and corresponds to the vector multiplets contribution; z α should be read as z α i i . The notation Γ(z ±a x) is a shorthand for Γ(z a x)Γ(z −a x), w∈R is taken over the weights of the fundamental representation of SU (N ) and J q + , J q − are quark charges under J + , J − . The expression is altered slightly for different gauge groups.

Unfortunately, straightforward index calculation is very computationally involved
with complexity growing for the larger groups (so Mathematica is unable to compute relevant index terms for SO (20) and larger groups). An alternative is to recall the definition of the index through the plethystic exponential. Using this definition, we can rewrite the index as where the prefactors are as in (3.13), and the integral is taken over the gauge group.
In the exponent I q corresponds to the quark contribution, and I φ , I g are related to the gluon contribution; they are given by where χ f (g) is the character for the fundamental representation of the gauge group, χ f (f ) is the character for the fundamental representation of the quark flavor group (and f as an overall prefactor corresponds to the fundamental rep of some group), χ a (g) is the character for the adjoint representation of the gauge group (and a as an overall prefactor corresponds to the adjoint rep), and r corresponds to the quark representation under SU (2) ρ . One can use the formulas from [12] to see that the integration gives same result as (3.13).
To proceed with the computation, we can use: where S n α (g) is n-th symmetric power of the representation α in the group g. This strongly reminds us of the expressions for the index and allows us to express the contributions of I φ , I g in a simpler form. The quark contributions are, however, harder to account for; e.g. the contribution at order f 2 , g 2 looks like 1 2 . Such terms can be expressed as a sum of plethysms, but in cases we encountered it was often easier to handle these cases manually. For brevity we will denote them as S n f (f ) ⊗ S n f (g).
As an illustration of the method we described above, let us analyze a theory that was studied in detail in [14] and discuss it thoroughly. Let us start with an N = 2 supersymmetric SU (2) gauge theory with N f = 4 fundamental hypermultiplets and SO (8) global symmetry, and add to it a superpotential W = Tr M µ. Next we give a nilpotent vev to M by ρ(σ + ); ρ : su(2) → so(8) that is described by a partition [7; 1]. This means that we fully break the flavor symmetry and the adjoint of SO(8) decomposes as 28

SUSY enhancement checking
Using the index computation procedure we have derived in subsection 3.1, and the SUSY enhancement conditions from section 2.1, we were able to compute the following index series expansions for different theories studied by [10]. The theories in question lie at the end of an RG flow from N = 2 SQCDs with a flavor group F : One can notice the absence of a t 4 3 term, even though the corresponding coefficient according to section 2.1 should be at least (3 + 1) − 3 = 1 for the SUSY enhancement to happen. Therefore, there is no SUSY enhancement in this case.
2. F = SO(8) with [5,3] partition. The theory has U (1) symmetry in the IR, and the corrected index expansion is given by should be also present in the theory and contribute to the index as t 78/51 (x 2 + 1 + 1 x 2 ). This contribution cannot be canceled by contributions of another multiplets and is not present. Therefore, this theory does not exhibit N = 1 → N = 2 SUSY enhancement.
3. F = SO(8) with [7,1] partition (the case discussed in section 3.1). The theory has U (1) symmetry in the IR, and the corrected index expansion is given by One can notice that unlike in the previous case, the low-t terms combine into the contribution of N = 2 E multiplet (t 4/5 −t 17/15 (x+ 1 x )+t 22/15 correspond to E 6/5,(0,0) ), so the necessary conditions are satisfied, and we got the Coulomb branch operator E 6/5,(0,0) . Moreover, there is a term t 7/3 (x+ 1 x ) that signals the presence of a conserved supercurrent and SUSY is enhanced to N = 2. Indeed, the obtained index correspond to the (A 1 , A 2 ) Argyres-Douglas theory, which was first shown by Maruyoshi and Song in [14]. The U (1) conserved current becomes part of the R-symmetry, so there are no conserved current multiplets in the theory.

Index and SUSY enhancement conditions in 3d
Similarly to the d = 4 case, all conditions can be divided into necessary, sufficient and sanity checks. A protected supercurrent multiplet (and, therefore, a necessary and sufficient condition) appears only in N > 3 multiplets and in these cases it is very easy to check whether there is SUSY enhancement. For example, let us define N = 4 index as where the trace is taken over all Verma module states (but only those that lie in the kernel of δ contribute); δ = 1 2 {Q † , Q}, Q is a specific supercharge, that has SO(4) R charge (1; 1) and spin − 1 2 , and ∆ is the scaling dimension. Under such a definition for SUSY to be enhanced from N = 4 to N = 4 + α the coefficient in front of xy 0 term (that is related to supercurrent multiplet B (1;1) ) in the index series expansion should be α and vice versa.

N=1 index
3d N = 1 superconformal theories are very special, because N = 1 superconformal algebra does not contain any R-symmetry. The only short multiplets are the free field multiplets and conserved current multiplets; superconformal index looks more like a Witten index and is a number I that, after taking free fields into account, tracks the difference between the number of conserved fermionic and conserved bosonic currents. If one assumes there is no free sector and only one stress-tensor in the theory, then using the fact that the only conserved fermionic current allowed in non-free theory is the supercurrent, one can derive necessary condition on I to possibly correspond to N > 1 SUSY; this condition is very weak and is of very doubtful usefulness, but we list it for the sake of completeness. It turns out that I ≥ 1 for theory to be able to have N = 2 (with its value depending on the dimension of the flavor symmetry group), I ≥ 2 for possibility of enhancement to N = 3, I ≥ 4 for possible enhancement to N = 4. For N > 4 we can use the fact, that N > 4 algebras cannot have conserved flavor currents and fix I precisely: for possible enhancement to N = 5 I should be equal to 6, for N = 6 I = 10 and for N = 8 I = 21.

N=2
Proper discussion and derivation can be found in the appendix B.4, here we will simply define basic concepts and list the results. For N = 2 theories one can define index as I 0 (x) = Tr (−1) F x ∆+j , where j is the SO(3) spin; all N = 2 multiplets contribute to the index as ±x α /(1 − x 2 ). Therefore, one can consider corrected index I c = (1 − x 2 )(I 0 − 1) = k a k x k (a's are the coefficients). The corrected index will be, loosely speaking, a power series in the scaling dimensions of superconformal multiplets bottom components. If we disregard free fields contribution, lowest in x term should have x in power greater than 1 2 ; for any SUSY enhancement to happen we need index contain only terms with k integer or half-integer. There are no sufficient conditions on SUSY enhancement, as the supercurrent multiplet contribution to the index is same to the contribution of another multiplet. The necessary conditions on the coefficients a k of the corrected index for different cases are as follows: N = 3. In this case a 1 + a 2 + 2 ≥ 0 and a 1 should be equal to the dimension of the flavor group in the enhanced-SUSY theory. N = 4. In this case a 1 + a 2 + 5 ≥ 0 and a 1 should be equal to the dimension of the flavor group in the enhanced-SUSY theory. N = 5. In this case a 1 = 1, a 2 ≥ −3 and a k should be even for non-integer k. N = 6. In this case a 1 = 4, a 2 ≥ −9 and a k should be even for non-integer k. N = 8. 3 In this case a 1 = 10, a 2 ≥ −15 and a k should be divisible by 4 for non-integer k.

N=3
N = 3 index uses the same fugacities as N = 2 does, so one can simply consider contributions of N = 3 multiplets to the N = 2 index and derive from them the conditions on SUSY enhancement B.5. Most of the conditions are similar to N = 2 case, but there is a sufficient enhancement condition: If in the corrected index I c = k a k x k the coefficient −a 2 is greater than a 1 , then there is N = 3 → N = 3 − a 2 − a 1 SUSY enhancement.
The necessary conditions on the coefficients a k of the corrected index for different cases are as follows (derivation can be found in the appendix B.5): N = 4. In this case the dimension of the flavor group in the enhanced-SUSY theory should be either a 1 − 1 or a 1 . N = 5. In this case a 1 = 1, a 2 ≥ −3 and a k should be even for non-integer k. N = 6. In this case a 1 = 4, a 2 ≥ −9 and a k should be even for non-integer k. N = 8. In this case a 1 = 10, a 2 ≥ −15 and a k should be divisible by 4 for non-integer k.

A.1 N = 2 to N = 1 multiplet decomposition and index
In this section we will list how all N = 2 superconformal multiplets decompose into N = 1 multiplets. Our notation for N = 2 multiplets is as in [11], and we denote N = 1 long multiplet as L, multiplets with semi-BPS shortening under Q (Q) as H R 1 ,(j,j) (H), and multiplets with symmetric semi-BPS shortening asĤ (j,j) . Multiplets with BPS shortening are denoted by S R 1 ,(0,j) and free field multiplets are denoted asŜ,S; we drop spin charges for scalars for brevity. As in the general case there will be many long multiplets in the decomposition, we will not bother to write out the R-charge and spins for them precisely. We start with the general case and then discuss special cases where the general formulas can't be formally applied. In the last subsection we list the N = 1 multiplets contributions to the index for reference.

A.1.1 General case
Moving from smaller to bigger, the multiplets decompose as follows: Finally let us move to the semi-BPS shortened multiplets. Here forĈ the first line corresponds to the multiplets spawning from |BC , the second -to the ones spawning from Q|BC , the third -fromQ|BC , and the fourth -from QQ|BC : For C the first line corresponds to |BC and Q|BC , the second -toQ|BC , the thirdto QQ|BC , and the fourth -toQ 2 |BC and QQ 2 |BC : (A. 16) As C and B lie in the same equivalence classes, the decomposition rules in the equivalence class language are the same.

A.1.2 Special cases
Now let us discuss the special cases when either r is small or we have some conservation at hand; we will simply list all interesting results.
1.B r : for r ≤ 1 the decomposition formula (A.1) doesn't work (it gives a negative number of long multiplets). In these cases the decomposition goes as follows: 2. D r,(0,j) . The decomposition formula (A.3) doesn't work for r ≤ 1 2 . D 1 2 ,(0,j) contains conserved higher-spin currents, forj = 0 it is the multiplet that contains the conserved supercharges, and forj > 0 it can only appear in a free theory. The decomposition happens as follows: 3. D 0,(0,j) : this is a generalized vector multiplet that decomposes as D 0,(0,j) →Ŝ (0,j) ⊕Ŝ (0,j+ 1 2 ) ; cases withj > 0 are possible only in a free theory. 4. B r, R 2 2 ,(0,j) . The equation (A.5) technically stops working when r = 0, however, in this case the restrictions are enhanced and one gets E instead. 5.Ĉ 0,(j,j) . Here the usual formula (A.9) doesn't work and we get instead For j =j = 0 we get the stress-tensor multiplet, otherwise it is a free theory multiplet.
Looking at these results, one can notice that for an N = 1 theory to be enhanced to N = 2, the N = 1 BPS multiplets should come in some sets. Therefore, there should be some restrictions on the N = 1 index that would determine whether a theory possesses higher supersymmetry.

A.1.3 N = 1 index
For reference, we list here the corrected index contributions for different N = 1 multiplets; they differ from the usual ones by a factor of (1 − t/x)(1 − tx). Our normalization is in agreement with the one in [12], with the exception of considering the left index instead of right and a different definition of SU (2) character Conserved current multipletŝ Other multiplets H R,(j,j) → 0 (A.33)

A.2 Relation between N = 1 and N > 1 equivalence classes
Let us describe the general algorithm that establishes the relation between the equivalence classes. First of all, let us consider a general N = 1 equivalence class (a detailed discussion can be found in appendix A.3). It corresponds to a set of N = 1 multiplets and is described byR 1 ,j and a plus or minus sign. This means one can classify the terms in the index according to the value ofR 1 . For an N = 2 multiplet entryR 1 can be determined as R 1 = 4r+R 2 3 + 2j, for N = 3R 1 = 8a+4b+R 3 9 + 2j, and for N = 4R 1 = 3k+2p+q 3 + 2j. Here r, a, b, k, p, q are weights in some representation (so they might be negative), and the formulas are obtained from the multiplets decomposition rules.
A general N = k equivalence class, on the other side, corresponds to a set of N = k multiplets and is described by p 1 . . . p k−1 that are related to the SU (k) weights of the bottom component of the multiplet,j, a plus or minus sign, and (not always) the charge under the U (1) R R k . Under SUSY reduction an N = k multiplet M breaks into many N = 1 multiplets (and each M entry breaks into many N = 1 multiplet entries) that can spawn either from the bottom component of M or from the states that are created by the action of Q 2 , . . . , Q k ,Q 2 , . . . ,Q k on the bottom component. As the N = 1 index counts only the short multiplets that are canceled under the action of Q 1 , only terms with maximal (or maximal and next to maximal, if we consider a multiplet with BPS shortening)R 1 in the given N = k multiplet entry may contribute. Now let us notice, that after N = k → N = 1 SUSY reduction the non-residual supercharges Q 2 , . . . , Q k will have charge − 1 3 under R 1 andQ 2 , . . . ,Q k will have charge 1 3 . As Q i have spin j = 1 2 ,j = 0 andQ i have j = 0,j = 1 2 , underR 1 = R 1 + 2j Q 2 , . . . , Q k will have charge 2 3 andQ 2 , . . . ,Q k will have charge 1 3 . One can also notice that N = 1 multiplets that are spawned not from the bottom component of M can be obtained by the action of non-residual supercharges on the bottom component. This means that the N = 1 multiplets that spawn not from the bottom component contribute to the index terms with biggerR 1 . Therefore one can relate the N = 1 equivalence class with certainR 1 ,j to a set of N = k multiplets, whose bottom components have the sameR 1 ,j.
Finally, let us discuss the content of SU (k) representations. A representation is described by the highest weight, the other weights can be obtained by subtracting the simple roots from it according to the well-known rules. One can notice that according to our definition ofR 1 , if we subtract the root that corresponds to the first line of the Cartan matrix (2 for N = 2, (2, −1) for N = 3, (2, −1, 0) for N = 4),R 1 will decrease; subtracting other roots won't changeR 1 . This means that for each SU (2) representation (in an N = 2 multiplet) there always will be one term with maximalR 1 under N = 2 → N = 1 reduction. For SU (3) [a; b] representation there will be (b+ 1) terms, and for SU (4) [k; p; q] there will be (p + 1)(q + 1)(1 + p+q 2 ) terms. An important implication for us is that an N = 2 multiplet with bottom component r R 2 ,(j,j) will contribute 1 term to the N = 1 index with R 1 = 4r+R 2 3 +2j (and some more with largerR 1 ). An N = 3 multiplet with bottom component [a; b] R 3 ,(j,j) will contribute b+1 terms to the N = 1 index withR 1 = 8a+4b+R 3 9 +2j. An N = 4 multiplet with bottom component [k; p; q] (j,j) will contribute (p + 1)(q + 1)(1 + p+q 2 ) terms to the N = 1 index withR 1 = 3k+2p+q 3 + 2j. Using this knowledge, one can prove that with the exception of N = 1 protected multiplets, it is always possible to express the contribution of a certain N = 1 equivalence class to the N = 1 index as a sum of contributions of an (infinite) set of N = 2, N = 3 multiplets and the only conditions that can be imposed on the N = 1 index can be related to the N = 1 protected multiplets.
The statement above contains an important loophole. In the infinite set of higher-SUSY multiplets that are used to express the contribution of an N = 1 multiplet to the index some multiplets may not appear due to prohibitions on the theory content. These prohibitions are related to the conserved currents: we cannot have any spin > 2 conserved current in a non-free theory; we can have only one stress-tensor, and the number of conserved spin-3 2 currents is also fixed. If we somehow know the flavor symmetry, we also can constrain the relevant term in the index. This idea is used to obtain a sufficient condition on SUSY enhancement from the N = 1 index, as well as a necessary and sufficient condition on SUSY enhancement from the N = 2, 3 index. N = 4 theories do not have a U (1) R symmetry (which immediately requires the N = 1 index of an N = 4 theory to have only terms with t k/3 , where k is an integer). Moreover, the lowest in t contribution of a multiplet with a bottom component SU (4) representation [k; 0; 0] to the N = 1 index will always have integerR 1 , and the lowest in t contributions of a multiplet with a bottom component SU (4) representation [k; p; q] for p + q > 0 have the absolute value of the coefficient at least 3. Therefore, one can expect many more necessary conditions on an N = 1 index for it to correspond to an N = 4 theory.

A.3 N = 2 to N = 1 equivalence classes reduction relation
There are six kinds of N = 2 multiplets that are canceled by the Q 1 supercharge (not counting the special case of free field multiplets); in Dolan and Osborn notation [11] they can be denoted as E, B,B, D,Ĉ and C. Index equivalence classes were first noticed by Gadde et al. [4] for N = 2 case, and for N = 1 equivalence classes were suggested in [5]. We will adopt their notation in general.
According to Gadde et al., a general N = 2 left index equivalence class, when there are no shortening conditions onQ, can be described by three parameters (related to fugacities) r = r+j,R 2 = R 2 +2j,j, and a ± sign (corresponding to a positive or negative contribution to the index): whereĈ r,(− 1 2 ,j) denotes D r+ 1 2 ,(0,j) Finally, for E multiplets that weren't discussed in [4], we can simply define the equivalence class (every E lies in a class of its own) as [e, R 2 2 ,j] ± . For N = 1 the situation is simpler: in the long multiplet decomposition into shorts, exactly two multiplets will contribute to the left index. The equivalence classes are described by two parametersR = R + 2j,j, and a sign ±: Here H R,(− 1 2 ,j) denotes S R+1,(0,j) . This description leaves outŜ, but as each of these multiplets lies in its own equivalence class and is a free field multiplet, we can disregard them for the time being. The equivalence class contribution to the index is given by To do the decomposition, one should use our results from appendix A.1 and the multiplets tables from [11] for N = 2; one should also remember that the N = 1 R-charge is expressed through the N = 2 R-charge as The scaling dimension restriction on the index can be reformulated asR 2 ≥ 2j or R 2 = −2 (this case corresponds to BPS shortening onQ, it can also be written asj = − 1 2 and is considered independently in [4]); equality corresponds to the cancellation of the multiplet's bottom component by one of theQ. For N = 1 scaling dimensions restrictions requireR ≥ 2(j+2j) 3 orR = 4j−2 3 ; the latter scenario corresponds to the free field multiplet. From now on we disregard the contribution of free field multiplets to the index, as they require special attention and footnotes in the analysis, while it is easy to subtract them from the index manually.
The reduction rules are slightly different for five different cases and can be formulated as 1.R 2 > 2j. In this case the shortening conditions restrict only Q and notQ, the reduction goes on as Whenj = 0, the equivalence classes withj = − 1 2 that appear in the reduction should be disregarded.

A.4 N = 2 to N = 1 equivalence classes reduction with flavor
When an N = 2 theory is written as an N = 1 theory, its R-symmetry contains an N = 1 flavor symmetry F . This symmetry may or may not be present in the N = 1 theory we start from, but when it is, it gives us additional constraints on SUSY enhancement. We will write a general index term as t 2+R 1 y F χj(x), where F is the fugacity corresponding to the flavor.
Sufficient conditions: 1. If the index contains a term αt 5 3 y − 1 3 χ 1 2 (x) (α being the overall coefficient), then there either is N = 1 → N = 2 + α enhancement, or flavor is not lifted to the R-symmetry. This is because of all N = 2 multiplets, only D 1 2 ,(0,0) can contribute to this term.
2. If the coefficient in front of t 7 3 (x + 1 x )y − 2 3 is positive, we have N = 1 → N = 2 enhancement; this is because this term with a positive sign corresponds to a conserved spin- 3 2 current (that comes from the N = 2 stress-tensor multiplet). If we have the coefficient less than −1, this means that the theory has more than one stress-tensor (it has several sectors that do not talk to each other).
Necessary conditions: 1. If the index contains terms with F >R 1 + 2 or F ≤ −2 −R 1 2 , SUSY enhancement cannot happen. This is because in this scenario there are no N = 2 multiplets that could give such a contribution to the index; their existence is prohibited by the restrictions the N = 2 superconformal invariance places on the scaling dimension.
2. If the index contains terms that have non-integerR 1 − F , the SUSY enhancement cannot happen. This is because in this scenario the N = 2 multiplet should have an entry with 2 · I 3 non-integer, which does not make sense.
3. If the coefficient in front of a term (−1) 2j+1 t 2+ 2j 3 y 1+ 2j 3 χ j (x) for j > 1 2 is positive, we have a free theory; the reverse is not necessary true. This is because the only N = 1 multiplet that can contribute to this term with such a sign isĤ (0,j) that can only appear in a free theory.

B.1 Superconformal symmetry in 3 dimensions
First of all, let us briefly set the notation. 3-dimensional superconformal algebra with 2N supercharges has R-symmetry SO(N ), and a conformal group SO (3,2). Commutation relations for the supercharges are given by Here we define M β α as Superconformal multiplet can be defined by its bottom component and the set of shortening conditions it obeys; bottom component is described by the charges under R-symmetry [r i ], its spin j, and its scaling dimension ∆. Shortening conditions can be divided into BPS (which can be written as Q|BC = 0 and require j = 0) and semi-BPS (which prohibit one state with spin j − 1 2 at the second level of the multiplet). For N > 8 corresponding interacting superconformal field theories cannot exist; for N = 1 R-symmetry group is trivial and so is the analysis (we discuss this case in section 4.1); therefore we are restricted to 2 ≤ N ≤ 8 cases. N = 2 case is very different from the rest, because in this case SO(2) R = U (1) R and we have independent Q,Q, so we will use different notation for it.

B.2 N=2 multiplets
In this case we can have shortening conditions on Q,Q independently and R-charge R 2 can be negative. The multiplet content is quite similar to d = 4, N = 1 superconformal theories and is given by: 1. BPS-semiBPS shortening. There are only two such multipletsŜ,S with j = 0, R 2 = ± 1 2 , ∆ = 1 2 that are the free chiral multiplets; they are absolutely protected.
3. BPS shortening. The bottom component of these multiplets (S R 2 ,S R 2 ) has ∆ = ±R 2 , j = 0, |R 2 | > 1 2 . These multiplets host deformations and are protected for 1 2 < |R 2 | < 2. S 1 will be important in our studies because this multiplet appears in the decomposition of many higher-SUSY stress-tensor multiplets; we will call these multiplets BPS multiplets for short.
When the long multiplet hits the unitarity bound, there are several possibilities for its decomposition: 1. R 2 = 0 (we consider R 2 > 0 for brevity, R 2 < 0 case is similar). In this case long multiplet decomposes as: 2. R 2 = 0. In this case long multiplet decomposes as

B.2.1 Index and equivalence classes
One can define N = 2 index as where trace is taken over all Verma module states that lie in the kernel of δ; δ = 1 2 {Q † , Q}, Q is a specific supercharge, that has U (1) R charge 1 and spin − 1 2 ; ∆ is the scaling dimension of the state. In practice that means that various multiplets will contribute as follows: Using these results and the decomposition rules, one can define N = 2 equivalence classes according to their contributions to the index: 1−x 2 , so one can introduce the notion of a corrected index I corr = (1 − x 2 )(I − 1) and make one-to-one correspondence between equivalence classes contents and index terms. One can also notice that [R 2 ] − equivalence classes are empty for R 2 < 0 (after taking into account free chirals), so index terms with x α for α < 2 give full number of S α multiplets.

B.3 N>2
In this case supercharges lie in the N representation of SO(N ) and shortening conditions are one-sided. The multiplet content is similar for different N , we will describe it for N = 3. For N = 3 R-symmetry is SO(3) and unitarity bound for long multiplet is ∆ = j + R 2 + 1 (R-charges of supercharges are 2,0,−2): 1. BPS shortening. These multiplets have bottom component scaling dimension ∆ = 2. semi-BPS shortening. These multiplets have bottom component scaling dimension ∆ = R 2 + j + 1, we denote them as C R j .
3. Long multiplets. We denote these multiplets as L R,∆ j The decomposition rules are as follows:

B.4 N=2 index and SUSY enhancement
The only way to be certain that we have SUSY enhancement is to see the presence of conserved spin-3/2 current that lies inĤ 1 2 multiplet 4 . This multiplet can recombine with ; this means thatĤ ( 1 2 ) and H −1(0) contribute to the index with a different sign. However, it is not the only possibility for H −1(0) to recombine into a long multiplet: it also can recombine with S −3(0) into L 2 −1,(0) . This means that S −3(0) andĤ ( 1 2 ) contribute to the index in a very same way and we cannot extract sufficient SUSY enhancement condition from the unflavored d = 3 N = 2 index.
However, there still are necessary conditions on SUSY enhancement from index. The most obvious one is that for the (corrected) index to correspond to N > 2 theory it must have all terms in it to be powers of t 1/2 , as N > 2 R-symmetry groups do not have U (1) R part and cannot contribute to the index other terms. Apart from that, the necessary conditions appear due to the following reasons: 1. Protected multiplets and related equivalence classes. There are only three N = 2 protected multiplets that can contribute to the N > 2 index, one of which is a free field multiplet. The two remaining multiplets are S 1 (contributes as t) and S 3/2 (contributes as t 3/2 ). When the SUSY enhancement happens, S 1 can combine with conserved current and conserved supercurrent multiplets, and some of the flavor currents may become R-symmetry currents. From that we can deduce the following conditions on the coefficients a 1 , a 2 in front of t k ; Here we list conditions on exact enhancement (e.g. N = 3 should be read as N = 2 enhanced to N = 3, but not N = 4) and also make use of the fact that there is a sufficient condition on the number of N = 2 conserved current multiplets (the index term t 2 tracks number of exactly marginal deformations minus number of conserved current multiplets) and that N = 5 − 8 theories cannot have flavor symmetry. We denote flavor symmetry group in the enhanced-SUSY theory with F .
As t 3/2 contribution cannot be related to any conserved current multiplet, the SUSY enhancement conditions we can derive from it are much weaker. In fact, for enhancement to N = 3, 4 one cannot derive any SUSY enhancement condition for this term, while for larger SUSY they are as follows: After working out the decomposition rules one can notice that N = 5 [R 1 , R 2 ] multiplet will have R 2 + 1 N = 2 S ∆ multiplets in its decomposition (and no other S, ∆ is defined according to the (B.25) for each of the N ), N = 6 [R 1 , R 2 , R 3 ] will contain (R 2 + 1)(R 3 + 1) N = 2 S ∆ (and no other S). Finally, N = 8 [R 1 , R 2 , R 3 , R 4 ] will contain d(R 2 , R 3 , R 4 ) N = 2 S ∆ and no other S; d(R 2 , R 3 , R 4 ) is the dimension of the SO(6) rep with highest weight [R 2 , R 3 , R 4 ]. This means that for the t n , n ∈ N contributions we can find a direct one-to-one correspondence between N = 2 S n and higher-SUSY B [n,0,...,0] multiplets.
As B [n,0,0,1] multiplet has 4 N = 2 S n+1/2 multiplets in its N = 2 decomposition, we deduce that for enhancement to N = 8 to be possible, N = 2 index coefficients a k should be divisible by 4 for non-integer k.
B.5 N=3 index d = 3 N = 3 SUSY algebra has R-symmetry group SO(3) R , the charges Q,Q have scaling dimension 1 2 , spin j = 1 2 and charges 2, 0, −2 under the SO(3) R . The multiplet content of the theory was described above in B.3, below we will list the N = 3 → N = 2 decomposition rules for various multiplets: is the stress-tensor multiplet. N = 3 index is written in the terms of the very same fugacities as N = 2, so one can simply consider contributions of N = 3 multiplets to the N = 2 index. The only way to be certain that we have SUSY enhancement is to see the presence of conserved spin-3/2 current that lies in C 0 0 multiplet. This multiplet can recombine with B 4 into a long multiplet, so there is a sufficient condition on SUSY enhancement that stems from the limitation on the number of such multiplets. From (B.32), (B.34) and results in B.4 one can see that if −a 2 > a 1 , then there is at least N = 3 → N = 3 − a 2 − a 1 SUSY enhancement. The necessary conditions that stem from the absence of one-to-one correspondence between the N = 3 and N > 3 equivalence classes are similar to what we had in N = 2 case due to the fact that all N = 3 multiplets contribute to the index as ±t R + O(t R+1/2 ) and there is a one-to-one correspondence between N = 2 S multiplets and N = 3 B multiplets as well as between N = 2 H and N = 3 C. The necessary condition on t 3/2 coefficient also stays the same.