On the Protected Spectrum of the Minimal Argyres-Douglas Theory

Despite the power of supersymmetry, finding exact closed-form expressions for the protected operator spectra of interacting superconformal field theories (SCFTs) is difficult. In this paper, we take a step towards a solution for the"simplest"interacting 4D $\mathcal{N}=2$ SCFT: the minimal Argyres-Douglas (MAD) theory. We present two results that go beyond the well-understood Coulomb branch and Schur sectors. First, we find the exact closed-form spectrum of multiplets containing operators that are chiral with respect to any $\mathcal{N}=1\subset\mathcal{N}=2$ superconformal subalgebra. We argue that this"full"chiral sector (FCS) is as simple as allowed by unitarity for a theory with a Coulomb branch and that, up to a rescaling of $U(1)_r$ quantum numbers and the vanishing of a finite number of states, the MAD FCS is isospectral to the FCS of the free $\mathcal{N}=2$ Abelian gauge theory. In the language of superconformal representation theory, this leaves only the spectrum of the poorly understood $\bar{\mathcal{C}}_{R,r(j,\bar j)}$ multiplets to be determined. Our second result sheds light on these observables: we find an exact closed-form answer for the number of $\bar{\mathcal{C}}_{0,r(j,0)}$ multiplets, for any $r$ and $j$, in the MAD theory. We argue that this sub-sector is also as simple as allowed by unitarity for a theory with a Coulomb branch and that there is a natural map to the corresponding sector of the free $\mathcal{N}=2$ Abelian gauge theory. These results motivate a conjecture on the full local operator algebra of the MAD theory.


Introduction
Inspired by the success of the 2D CFT program initiated by Belavin, Polyakov, and Zamolodchikov (BPZ) almost forty years ago [1], many proposals have been put forward to solve CFTs in D > 2 (e.g., with various insights building on ideas in [2] and related works). A slightly more modest, but nonetheless daunting, objective is to find the exact spectrum of operators in such a higher-dimensional CFT. Even more modestly still, finding exact closed-form spectra of "protected" operators of interacting superconformal field theories (SCFTs) in D > 2 is also an important open problem.
In this paper, we make progress toward this last goal in the case of the "simplest" interacting 4D N = 2 SCFT: the minimal Argyres-Douglas (MAD) theory discovered a decade after BPZ's groundbreaking work [3,4]. 1 In many ways, the MAD theory is the 1 Although we will review in what sense this theory is "simplest," we note that it does not have an N = 2 Lagrangian. See [5] for algorithms that in principle allow one to obtain short multiplet spectra in the large-N limit of certain N = 2 Lagrangian theories that are closely related to N = 4 super Yang-Mills. We thank Let us briefly introduce some of our claims in more detail. In order to first set the stage, let us recall the quintessential short multiplet of the 4D N = 2 superconformal algebra (SCA): the N = 2 chiral multiplet. Such a representation has an N = 2 chiral primary, O, satisfying where i is an SU(2) R fundamental index,α is a right Lorentz index, and theQ iα are the four anti-chiral supercharges of the N = 2 SCA. Here O is neutral under SU(2) R but transforms under the U(1) r ⊂ U(1) r × SU(2) R superconformal R symmetry (we will denote this latter charge as r(O) = 0). In the language of [12], O is a primary of anĒ r(0,0) multiplet 5 (see also the equivalent taxonomies in [13]).
In the case of the MAD theory, the spectrum of these multiplets is well understood: the various N = 2 chiral operators form a ring that is freely generated with generator O having r(O) = 6/5. In particular, we have all O n (with r = 6n/5) and so we have onē E 6n/5(0,0) multiplet for every n ≥ 1. We say that an N = 2 chiral operator is non-trivial in the N = 2 chiral ring if, for any i (or combination thereof) where O ′ is any well-defined local operator. Otherwise, a chiral operator is said to be trivial in the N = 2 chiral ring. Hoping no confusion arises, we will write this condition as O = 0 (even though O may be non-vanishing in the local operator Hilbert space). Note that trivial N = 2 chiral ring operators cannot be primaries ofĒ r multiplets.
More generally, it is interesting to study less protected operators than those in (1.1).
One natural class of operators to study are those that are annihilated by half the anti-chiral supercharges. Without loss of generality, we can consider O annihilated byQ 1  5 We mostly follow the notation of [12] for short multiplets; however, the sign of the r charge is flipped: On the other hand, as in [12], R is the SU (2) R weight (and can therefore be negative) except when R ≥ 0 labels the multiplet. In this case, R is the SU (2) R spin of the superconformal primary (or, equivalently, the R weight for the SU (2) R highest-weight component of the superconformal primary).
for any well-defined local operator O ′ . When (1.4) is violated, we will write O = 0 to indicate triviality in the FCS (even though O can be non-vanishing in the local operator Hilbert space).
Which multiplets can operators such as O of (1. 3

) and (1.4) sit in? A quick look at
representations of the SCA in [12] reveals the following: We define the FCS to be the set of the above multiplets (along with the corresponding chiral OPEs). See Appendix A for some properties of the FCS in general 4D N = 2 SCFTs.
Note that in the MAD theory it is already known that there are neitherD norB multiplets (e.g., see [9]).
The list of multiplets in (1.5) can be understood heuristically as follows. These are the only representations that obey shortening conditions that are linear in the various supercharges and involve no contractions of supercharge spin indices. All other short representations (i.e., theĈ andC ⊕ C multiplets) obey shortening conditions that are quadratic in the supercharges or conditions involving contractions of supercharge spins (so-called "semishortening" conditions in the nomenclature of [12]). As a result, such multiplets cannot house a non-trivial element of the FCS because then they would necessarily obey additional shortening conditions.
Given this groundwork, we can state our first main result: in the MAD theory there are noB multiplets (and, by CPT, there are no B representations either). As a result, the MAD FCS is as simple as allowed by unitarity and the existence of a Coulomb branch Our argument in favor of (1.6) revolves around the RG flow to the MAD theory studied in [14] involving N = 1 in the UV accidentally enhanced to N = 2 in the IR. We start by constructing the naive N = 1 chiral ring generators in the UV. Our key idea is to then demand that these operators form a part of an N = 2 FCS in the IR. Checking this fact amounts to a representation theoretical version of the more analytic arguments on chiral rings in [15]. 7 6 Antichiral operators satisfying the conjugate conditions sit in FCS := E ⊕ D ⊕B ⊕ B. 7 We should also note that we find the same FCS generators as the authors in [16], who used an approach The vanishing of theB multiplets then leads to our second result. Indeed, we argue that it is then possible to compute the exact spectrum ofC 0,r(j,0) multiplets for any r and j from the superconformal index (since the index can, a priori, involve highly non-trivial cancellations, this statement is not obvious).
These latter multiplets are of interest for two main reasons. First, theC R,r(j,j) multiplets are the last unknown pieces of the MAD short multiplet spectrum. While we restrict ourselves to the subset of multiplets with R =j = 0, our results shed the first nonperturbative (in index fugacities and quantum numbers) light on this sector. The second reason for being interesting is that they arise in the OPEs of FCS and Schur operators.
These latter operators sit in multiplets closely related to those in (1.5) The Schur multiplets are well understood since they turn out to be isomorphic (in a sense defined in [17]) to operators in a 2D vertex operator algebra (VOA).
The main point is that our result on theC 0,r(j,0) spectrum shows that these multiplets act as an OPE-induced bridge between the Schur sector and the FCS. This realization motivates us to conjecture general constraints on the MAD local operator algebra (including for long multiplets).
In the next section, we build the FCS of the MAD theory. We conclude with a proof of our result on chiral operators. We then proceed to considerC operators. Afterwards, we turn our attention to the free Abelian gauge theory and show that both its FCS and C 0,r(j,0) sectors are closely related to those of the MAD theory. Using this fact, along with our two main results as inspiration, we then proceed to our conjecture.

The full chiral ring: beyond the Coulomb branch operators
All interacting 4D N = 2 SCFTs are believed to possess an N = 2-preserving Coulomb branch. 8 As discussed in the introduction, this moduli space is parameterized by (SU(2) Rinspired by [15] and studied an N = 2 RG flow with broken U (1) r based on one-flavor N = 2 SU (2) SQCD.
We extend this result by proving that there are noB multiplets. In addition, our proof involves an RG flow with manifest U (1) r × SU (2) R Cartans. As a result, operator mixing is fully under control in our flow and provides a proof of the claim in [16]. 8 We define the Coulomb branch to be the set of vacua in which the superconformal U (1) r is spontaneously broken, and the SU (2) R is unbroken. Vacua on the Coulomb branch necessarily contain at least one N = 2 free vector multiplet, but they can also contain additional degrees of freedom at generic points (e.g., as in the case of N = 4 SYM).
neutral) N = 2 chiral operators (i.e., operators annihilated by all four anti-chiral supercharges) that are, in the nomenclature of [12], N = 2 superconformal primaries ofĒ r multiplets. 9 The corresponding N = 2 (Coulomb branch) chiral ring is infinite and, in the simplest theories, has a single generator.
In the case of the MAD theory, the single generator of the chiral ring has ∆ = r = 6/5 (here ∆ is the scaling dimension) and will be denoted as O 6/5 ∈Ē 6/5 . The MAD Coulomb branch chiral ring, R C , is then described as where we take arbitrary linear combinations of the O 6n/5 .
In what follows, we will be interested in studying more general operators that are chiral with respect to an N = 1 ⊂ N = 2 superconformal subalgebra. In fact, we will see that the MAD theory has, in a sense we will make precise, the simplest set of such operators allowed by unitarity and the existence of a Coulomb branch.
To understand this statement, we first fix an N = 1 subalgebra. Without loss of generality, we consider the subalgebra generated by where we write the quantum numbers as (j,j) R,r , with j the left spin,j the right spin, R the SU(2) R weight, 10 and r the U(1) r charge (note that R − r is a flavor symmetry of this subalgebra). 11 Then, using the orthogonal algebra generated by we see that anĒ r multiplet contains three chiral primaries with respect to the N = 1 subalgebra in (2.2) (note that F ⊥ := R + r is a flavor symmetry of the orthogonal subalgebra).
In particular, these primaries embed as follows In principle, the superconformal algebra allowsĒ r(j,0) multiplets to have j = 0, but a careful analysis of locality shows that j = 0 in 4D N = 2 SCFTs [18] (as conjectured and verified in various classes of theories in [19]). Therefore, we drop the spin from the label of these multiplets. 10 Note that SU (2) Lorentz spins j,j ∈ 1 2 N are non-negative, while the SU (2) R weight, R ∈ 1 2 Z, can be either positive or negative. Therefore (j,j) R,r actually represents a Lorentz multiplet with specific R, r charge. 11 To get a superconformal algebra, we should also include S 2α ∼ S 1 α andS 1α ∼S 2 α .
where the leftmost operator is the N = 2 chiral primary, and the remaining operators are level-one and level-two superconformal descendants. Note that O r,α and O ′ r are not in the N = 2 chiral ring (they are not annihilated byQ 2 α ), but they are in the N = 1 chiral ring defined by (1.3). In fact, they are non-trivial elements of the N = 1 chiral ring since the structure of theĒ r multiplet does not allow them to beQ 1α -descendants of another operator in that multiplet. For the MAD theory, we have r = 6 5 n with n ∈ Z >0 . We now arrive at the main claim of this section: Claim 1: The only operators in the MAD theory that are non-trivial chiral ring elements with respect to an N = 1 ⊂ N = 2 sub-algebra are the three N = 1 chiral operators in eachĒ 6n/5 multiplet (n ≥ 1).
A quick scan of the allowed unitary superconformal representations in [12] shows that, besides theĒ r multiplets, the only other multiplets that can host N = 1 chiral operators are of typeB R,r(j,0) ,D R(j,0) , andB R ; however, recall from [9] that the MAD theory has neitherB R norD R(j,0) multiplets.
The putative N = 1 chiral operators inB R,r(j,0) multiplets can be described as SU(2) R highest-weight primaries and descendants at levels one and two where superscripts denote SU(2) R fundamental weights, and the α i are fundamental left spin indices. 12 As a result, we have the following corollary: Corollary 1: The MAD theory has noB R,r(j,0) multiplets (by CPT the same is true for B R,−r(0,j) ).
Defining, as in (1.5) of the introduction, the set ofĒ ⊕B ⊕B ⊕D multiplets to be the "full chiral sector" of a 4D N = 2 SCFT, we see that 12 Note that acting withQ 2 α gives additional N = 1 chiral operators. However, these are trivial in the N = 1 chiral ring. Indeed, consider Q 2 α , O 11···1 α1···α2j . Now recall that the shortening condition of the primary is Acting with an SU (2) R -lowering operator, we see that is trivial in the N = 1 chiral ring.

Corollary 2:
The full chiral sector (FCS) of the MAD theory, written in (1.6), is as simple as possible for a unitary theory with a Coulomb branch. 13 Moreover, if all unitary interacting theories have a Coulomb branch, then the FCS of the MAD theory is as simple as possible for a unitary interacting theory. 14

A proof of Claim 1
Let us now give a proof of Claim 1 and therefore also of Corollaries 1 and 2. To that end, our strategy will be to use the N = 1 Lagrangian of [14] (as modified in [21]; see also [22] for related discussions). In particular, we consider the N = 1 SU(2) gauge theory with matter content in table 1 and superpotential where we have suppressed couplings in front of the terms in W . We have also defined φ 2 := φ a φ a , φq ′ q ′ := φ a (q ′ q ′ ) a , and φqq := φ a (qq) a with the SU(2) adjoint index, a = 1, 2, 3, summed over. Note that the UV theory is written in terms of the N = 1 supercharges in (2.2) (the remaining N = 2 supercharges are emergent in the IR).
Our algorithm for proving Claim 1 can be summarized as follows: 1. Write down all possible naive N = 1 chiral ring generators in the UV theory.
3. Argue that any N = 1 chiral ring operators (including composites built from generators) cannot sit inB R,r(j,0) multiplets.
As is clear from the above list, the existence of a unitary N = 2 superconformal algebra in the IR and the resulting imposition of step 2 is the crucial part of our algorithm. Note that the role of the superpotential is indirect in step 2: it simply controls the conserved R-symmetry quantum numbers. In step 3, the rough form of the superpotential plays a brief but important role we will describe below. 13 To understand the importance of unitarity, consider the free N = 2 Abelian gauge theory with fields of wrong statistics (e.g., see the discussions in [20] for more general theories along these lines). In this case, the chiral ring generator, ϕ, is nilpotent: ϕ 2 = 0. Therefore, the full chiral sector of this non-unitary theory consists of operators in a singleD 0(0,0) multiplet. Of course, one may object to this theory having a Coulomb branch since the moduli space is just a point. On the other hand, its dynamics is that of an abelian gauge theory (albeit with wrong statistics). 14 This is the chiral ring equivalent of the minimality of the MAD Schur sector proven in [9]. Table 1: The charges of primaries of all N = 1 (chiral) UV fields. The first column of charges give the representation under the SU(2) gauge group. Here I IR 3 refers to the N = 2 superconformal SU(2) R weight in the IR (i.e., I IR 3 → R), and r IR refers to the N = 2 superconformal U(1) r charge in the IR (i.e., r IR → r), which is determined through amaximization. These quantum numbers are visible in the UV description (hence the power of this RG flow), and the superpotential has quantum numbers r IR (W ) = I IR 3 (W ) = 1.
To that end, let us first implement step 1. Since any trace over SU(2) generators can be written in terms of δ ab ∼ Tr(T a T b ) and ǫ abc ∼ Tr([T a , T b ]T c ), chiral ring generators will contain at most three adjoints. 15 Proceeding in this way results in the twenty generators 15 Note that we do not demand that the operators obey classical relations. To understand this statement, recall that in the N = 1 construction above, we have a conserved U (1) r symmetry, r, and a conserved SU (2) R Cartan, R. Consider a particular chiral operator, O 1 with r(O 1 ) = r 1 and R(O 1 ) = R 1 . Since all chiral gauge-invariant operators have R ≥ 0 and R > 0 if r ≤ 0, there are a finite number of chiral operators with the same r and R quantum numbers. SupposeÔ A has r(Ô A ) = r 1 and R(Ô A ) = R 1 , and supposeÔ A is a product of chiral gauge invariant operators with each factor involving at most three adjoints. Let us denote S 1 as the set of all suchÔ A . Then, we have the quantum operator relation where "| · · · |" is the number of elements in the enclosed set, h i are the implicit couplings in the superpotential (2.7), τ is the holomorphic gauge coupling, and Λ is the dynamical scale. Here the n 1A are constants, while the a 1A are functions of the couplings that vanish in the UV limit, i.e. a 1A (0, 0, 0, 0, 0) = 0 (the classical chiral ring relation is of the form O 1 = A n iAÔA ). Starting with O 1 involving four or fewer fields and working iteratively in the number of fields, we see that the chiral ring generators in the quantum theory must involve traces over either two or three fields (even though the classical chiral ring relations are modified). 2 3 Table 2: The result of step 1 of our algorithm: the naive UV chiral ring generators. Note that qq ′ := ǫ ij q i q ′ j . We use δ ab to write the remaining generators involving gauge nonsinglets in the first line (e.g., qqλ := δ ab (qq) a λ αb ) and ǫ abc to write the generators in the Next, we implement step 2 of our algorithm. Clearly, any of the generators must either sit in anĒ multiplet as in (2.4) or aB multiplet as in (2.5).
Let us study the N = 1 chiral ring generators that can sit inĒ first. Any such operator has R ≤ 1 and j ≤ 1/2, as one can see from (2.4). Of the operators in table 2, only X, M, qq ′ , φλ α , λ 2 , φ 2 , φqq, φqq ′ , and φq ′ q ′ satisfy these constraints. Note that φ 2 violates a unitarity bound and decouples. Moreover, to be in anĒ multiplet, X, λ 2 , φqq, φqq ′ , and φq ′ q ′ would need to be level-two descendants of primaries with r = 8/5, r = 2, r = 2, r = 7/5, and r = 4/5 respectively. Since there are no such N = 2 chiral operators in the MAD theory with these values of r (see (2.1)), we conclude that the only generators sitting 16 At tree level, a subset of generators are trivial in the chiral ring. These are φ 2 , φq ′ q ′ , φqq, φqq ′ , qqφλ α , and qq ′ φλ α , as one can see from contracting various fields with ∂ X W = ∂ M W = ∂ q W = 0. Furthermore, where M is the primary, φλ α is the level-one descendant, and qq ′ is the level-two descendant described in (2.4) with r = 6/5.
Next, let us discuss generators that can potentially sit in theB multiplets. The primary in (2.5) satisfies r > 1 + j (if r = 1 + j, we have aD multiplet, which we know is absent [9]), while the descendants satisfy r > j. We see that φq ′ q ′ do not satisfy this condition. As a result, we conclude these operators are trivial in the IR FCS. 17 The remaining operators are Since r ≤ 1 + j, we see that these operators cannot be primaries of aB multiplet. We can then immediately see that qq ′ φλ α is not a generator. Indeed, if it is a level-one descendant, then the primary has r = 7/5, R = 1, and j = 0 (j = 1 is ruled out because r < 2). By SU(2) R and Lorentz spin, it must be of the form of a linear combination of Mqq ′ and then the primary has R = j = 1/2 and must be of the form M n φλ α . However, this operator is not a primary: it is a descendant of M n+1 ∈Ē 6(n+1)/5 .
Let us now study the remaining generators in (2.11). To that end, the primary of aB multiplet containing these operators must take the form are (potentially) composite operators built from (2.9) and (2.11), respectively (we do not include factors of qq ′ φλ α in O 2 , since it is, at best, built from operators contributing to O 1 ).
A crucial observation is that the charge F ⊥ := R + r commutes with the supercharges used to relate theB members of the FCS in (2.5) (i.e., it is a flavor symmetry of the algebra in (2.3)). Furthermore, for operators in (2.9), we have F ⊥ = 6 5 , while, for all the operators in (2.11) except for qq ′ φλ α , one finds F ⊥ ≥ 7/5 and F ⊥ ∈ 6 5 Z. Therefore, if the primary is purely made out of operators from (2.9)-namely O 2 is trivial-then all operators in the multiplet must satisfy F ⊥ ∈ 6 5 Z. Since this is in contradiction with the fact that F ⊥ ∈ 6 5 Z in (2.11), O 2 must be non-trivial. Now, the multiplet generated by a primary of the form O 1 O 2 (with O 1,2 = 1) has minimal charge F ⊥ ≥ 6/5 + 7/5 = 13/5, 18 which rules out most operators in (2.11), except for We can repeat the same procedure to rule them out. In particular all operators in (2.12) satisfy 3 ≥ F ⊥ ≥ 14/5. Therefore, the multiplet generated by the primary O 1 O 2 has F ⊥ ≥ 14/5 + 6/5 = 4. This logic then rules out all operators in (2.12).
To conclude, all operators in (2.11) are trivial in the chiral ring (with the possible exception of qq ′ φλ α which is either trivial or else is not a generator of the FCS and satisfies qq ′ φλ α ∼ (qq ′ )(φλ α ); we will rule out this latter possibility shortly). We present an alternate proof based on a case-by-case analysis in Appendix B.
As an additional consistency check, note that M cannot sit in aB multiplet because it has R = 0. 19 Since φλ α has R = 1/2, it can only sit as a primary ofB, but r = 7/10 < 1 + 1/2 precludes this. Finally, qq ′ has R = 1, so it can only sit as a primary or a level-one descendant. However, r = 1/5 < 1 precludes its being a primary. If it were a level-one descendant, the primary would have to be φλ α on U(1) r grounds. Fortunately, this scenario is already ruled out.
Therefore, to summarize, we learn that the IR FCS is generated by the three operators of (2.9) living in theĒ 6 More generally, it is useful to find constraints among the operators in (2.9). For example, from the discussion in [23], we expect both φλ α and qq ′ to be nilpotent. Of course, φλ α is nilpotent since it is a fermion. Using (2.7), it is easy to see that qq ′ is as well. Indeed, classically in the N = 1 chiral ring, we have where we have used our above analysis to show that X vanishes in the IR FCS.
We can ask if (2.13) can be modified in the quantum theory. By U(1) r conservation, any putative quantum corrections would have r = 2/5. Our analysis of N = 2 superconformal representation theory and its consequences are fully non-perturbative in nature and so the operators we ruled out cannot appear. Moreover, any correction cannot involve M or φλ α either since there is no negative U(1) r charged chiral operator to compensate for their larger U(1) r charge. As a result, we see that (2.13) is exact. In particular, we see that qq ′ is minimally nilpotent in the quantum theory (it vanishes at quadratic order).
Another useful constraint is that in the IR FCS. To understand this statement, note that this operator has R = 3/2 and r = 9/10. Since R > 1, it can only sit in aB multiplet. Clearly, it cannot be a primary or a level-one descendant since [Q 1α , ((φλ α )(qq ′ ))] = (qq ′ ) 2 = 0 in the chiral ring. If it is a level-two descendant, then the primary has r = 19/10 and R = 1/2, which means it is of the form Mφλ α (but this is a level-one descendant ofĒ 12/5 ).
Now we can ask if it is at all possible to construct a primary of aB multiplet. Note that our reasoning above implies in the FCS. As a result, M n (φλ α ) 2 and M n qq ′ cannot be primaries of aB multiplet. To construct such a primary, we must therefore use exactly one φλ α . However, by (2.14) such a primary must not involve qq ′ . The only option remaining is M n φλ α , but this is a level-one descendant of anĒ multiplet. 21 We have therefore established Claim 1 and the attendant Corollaries 1 and 2. In particular, Corollary 2 implies that, as promised, there are noB or B multiplets in the MAD theory.
identicalĒ r multiplets. Note, however, that if we have multipleĒ r multiplets, such operators can appear in the OPE. 21 In fact, one can easily see that (φλ α ) 2 and M qq ′ have the same quantum numbers R = 1 and r = 7/5.
In the next section we will use the absence of these multiplets, combined with the superconformal index, to make exact statements about the multiplicities ofC 0,r(j,0) (and C 0,−r(0,j) ) multiplets for all possible r and j.
3. Exact results on the semi-short spectrum: beyond the Schur sector Combining the results of the last section with those of [9], let us summarize the known part of the MAD short multiplet spectrum. First, we list the number of multiplets containing FCS operators but no Schur operators 22 NĒ r = δ r,6n/5 , (n ∈ Z + ), NB R,r(j,0) = 0 .
Finally, let us list the number of multiplets containing operators only in the Schur sector.
The number ofĈ multiplets is given by the following generating function [9] x R(R+2) In order to fully determine the short multiplet spectrum of the MAD theory, all that remains is to fix the multiplicity ofC R,r(j,j) multiplets. Finding the full spectrum of these multiplets is an involved problem that we will return to [24]. Interestingly, we will see in later sections that these multiplets act as a kind of "glue" that link the Schur and chiral sectors (we will make this notion more precise via a conjecture).
In this section, we will argue that it is possible to determine the exact spectrum of C 0,r(j,0) multiplets (and we will interpret this fact physically in section 5). In particular we Therefore, we may expect a relation between them where a, b ∈ C are constants. Such a relation must exist because they can only appear in the level-two Q 1 α descendant of M 2 . This logic implies (φλ α ) 2 and M qq ′ cannot be two independent operators in the chiral ring. It would be interesting to determine the coefficients a and b. This computation is beyond the scope of this paper. 22 Multiplicities of conjugate multiplets are fixed by CPT to be equal to those we list.

(3.4)
Our main tool for deriving (3.4) is the superconformal index of the MAD theory [14].
However, we must first overcome certain ambiguities in the index in order to find the precise spectrum of our multiplets of interest. To understand this point, let us first recall the definition of the superconformal index where the trace is over the Hilbert space of local operators, (−1) F is the fermion number, j,j are the left and right Lorentz spins, r is the U(1) r charge, and R is the SU(2) R weight introduced before. The superconformal index counts the contributions of short multiplets up to recombination (and long multiplets do not contribute to the index). Below, it turns out to be more convenient to use another set of fugacities u, y, τ which are related to p, q, t through p = τ 3 y , q = τ 3 /y , u = pq/t . (3.6) A big advantage of working with u, y, τ is that we can study the index perturbatively in τ and exactly in u and y.
The index of the SCFT is given by the sum of contributions from various short multiplets. For the MAD theory, the only short multiplets are of typeĈ,Ē, andC (the conjugate E and C multiplets do not contribute to the index). Therefore, we can write the index as where the leading contribution is from the vacuum. In order to furnish the decomposition, let us first recall the index contributions from the individualC,Ĉ, andĒ multiplets of interest IĈ R(j,j) x − x −1 is the character of the spin-j representation of

SU(2).
Our goal here is to decompose the superconformal index into the contributions from various multiplets, Ξ, and thus compute the multiplicities, N Ξ . ForĒ andĈ multiplets, the multiplicities can be read off unambiguously from the Coulomb branch and Macdonald indices respectively 23 (the latter statement follows non-trivially from the considerations in [9]). The results are given in (3.1) and (3.3). The next step is to find the multiplicities of theC multiplets. Before setting up the computation, let us first remark on a few subtleties.
In general, given (3.5) and some multiplet of type Ξ, there are several potential obstacles to finding the precise number of such multiplets, N Ξ : 1. Index cancellations due to the fact that there are multiplets, Ξ ′ i , that can combine with Ξ to form a long multiplet. In this case, even if the short multiplets have not recombined to a long multiplet (i.e., the long multiplet is at its unitarity bound), the index contribution of Ξ is canceled.
2. Cancelling index contributions can also be generated by multiplets that mimic the index contributions of the Ξ ′ i (even if they cannot recombine with Ξ to form a long multiplet).
3. The contribution of Ξ may potentially be the same as the contribution of a multiplet with different quantum numbers. 4. In practice, we need to expand the index in powers of fugacities, but it may be difficult to disentangle the leading contribution from Ξ and subleading contributions from other multiplets.
Let us analyse the case of interest, Ξ =C 0,r(j,0) . First, note that our results in the previous section rule out obstacle 1. Indeed, such multiplets can only recombine as follows (e.g., see [25]) where A is a long multiplet. However, we have shown there are noB multiplets. 24 Let us now consider the remaining obstacles 2-4 simultaneously. As discussed above, since we know the multiplicity of allĒ andĈ multiplets, we can subtract their contributions 23 These are special limits of the superconformal index [8]. 24 Note that theB R,r(j,0) multiplet contributes to the index in the same way as a would-bē C R−1/2,r−1/2(j,−1/2) multiplet. This fact explains the index cancelation that results from the recombination in (3.9).
to the index. Therefore, we need only consider to what extent otherC multiplets can lead to obstacles 2-4. To see these observables do not pose a problem, consider the leading-order in τ index contribution from aC R,r(j,j) multiplet (3.8) (3.10) Sincej, R ≥ 0, we see that anyC R,r(j,j) contributions with R = 0 or j = 0 will be subleading in τ compared to theC 0,r(j,0) index contributions IC 0,r(j,0) = (−1) 2j+1 τ 6 (1 − u)u r χ j (y) + · · · . (3.11) Moreover, these contributions clearly distinguish different r and j. As a result, we see that we can use the index to unambiguously extract the multiplicities of theC 0,r(j,0) multiplets.
Then the index can be written as . (3.17) Note that A 1 and A 2 can also be obtained explicitly, but we do not write down the corresponding complicated expressions here. Using the residue theorem at the z = 0, ±u 1 10 poles, we can evaluate the integral and find As discussed above, in order to solve for N C 0,r(j,0) , we must subtract the contributions fromĈ andĒ multiplets in (3.3) and (3.1) respectively. To that end, theĈ index contributions take the form while theĒ contributions take the form Subtracting these contributions (along with the contribution of the identity operator, as summarized in (3.7)) from the index (3.18) yields theC contribution .  Therefore, we have arrived at our main claim! As we will discuss later, these multiplets appear in OPEs involving onlyĒ 6 5 n andĈ 0(0,0) multiplets. 26 Before concluding this section, let us note that (3.21) also provides a check of our results in the previous section since we can immediately conclude that there are noB 1/2,r(j,0) multiplets. Indeed, generalB multiplets contribute as (3.25) Therefore, we see that multiplets of the formB 1/2,r(j,0) are unambiguously captured by the index at O(τ 3 ) once theĒ multiplets have been subtracted (this statement is related to the fact that these multiplets do not recombine). Indeed, since there is no O(τ 3 ) term in (3.21), these multiplets cannot be present.
In the next section, we will compare (3.22) with the theory at generic points on the MAD Coulomb branch (i.e., the free N = 2 Abelian gauge theory). This comparison will build intuition that we will use in the subsequent section to understand the universality of the spectrum of operators we are discussing.

Comparison with the free vector
In [9], we saw that the MAD theory shares an infinite amount of Schur sector OPE data with the free N = 2 vector multiplet. In this section, we will extend these observations to the full chiral sector (see section 2) and the part of the semi-short spectrum described in section 3. 26 One can proceed further and find that, at the next order It is easy to check that the first line is the contribution from the multiplets in (3.22), while the second line comes fromC 0,r(j, 1 2 ) orC 1/2,r+1/2(j,0) multiplets whose multiplicities are subject to the condition Purely at the level of the index, we are unable to determine the multiplicities unambiguously due to the relation IC R,r(j,j) + IC = 0. It would be interesting to resolve these ambiguities using further physical input. However, even at the above level of precision, we will soon see that (3.24) is consistent with the quantum numbers of operators appearing in OPEs involving onlyĒ 6n 5 andĈ 0(0,0) multiplets.

The full chiral sector
Let us first consider the FCS. Any such operator takes the form φ n , φ n λ 1 α , or φ n λ 1 λ 1 for any n ≥ 1 and α = 1, 2. 27 Clearly, the first type of operator is a primary ofĒ n (if n > 1; otherwise, φ is the primary of aD 0(0,0) multiplet). The second and third types of operators are level one and two descendants ofĒ n (if n > 1; otherwise, they are level one and two descendants of aD 0(0,0) multiplet). Therefore, just as in the MAD case, there are nô B ⊕B ⊕ B multiplets: It is then easy to see that, up to a U(1) r rescaling and the additional equations of motion in theD 0(0,0) multiplet, the FCS of the MAD theory and the FCS of the free vector are isospectral. In particular, equations of motion project out the R = 1 states inD 0(0,0) (i.e., we set the D and F auxiliary fields to zero) and constrain other components of the multiplet.
To begin, let us write the free vector index in terms of the u, τ, y variables of (3.6) As in our analysis of the MAD theory, we should focus on the term at order τ 6 . Recall that the leading-order contribution in τ from aĈ R(j,j) multiplet is τ 6(1+j+R) . As a result, to capture the contributions from the semi-short multiplets in question, we should subtract the contribution from the stress tensor multiplet and theĈ 0(1,0) multiplet (with Schur operator We should also subtract the contribution of theD 0(0,0) ⊕ D 0(0,0) multiplet (all other Schur multiplets are of typeĈ R(j,j) [9]) Finally, we should subtract the contributions from the Coulomb branch operators,Ē n (with n ≥ 2) Therefore, the contributions involvingC are IC = −u(1 + uχ 1 (y))τ 6 + · · · . (4.6) Note that the y-independent term corresponds to while the y-dependent term corresponds to In particular, we see that theC 0,r(j,0) ⊕ C 0,−r(0,j) spectrum of the MAD theory and the free vector are in one-to-one correspondence for j = 0. Note that for j = 1,C 0,1,(1,0) hits a unitarity bound and is actually aĈ 0(1,0) multiplet (we therefore included it in (4.3)).
However, it is natural to include this multiplet in the map betweenC ⊕ C sectors of the MAD theory and the free vector. Indeed, we then get (as in the case of the FCS) a simple one-to-one map, up to the vanishing of a finite number of states (the additional null states inĈ 0(1,0) ). Moreover, we expect MADC ⊕ C multiplets to be sources ofĈ multiplets in the IR (since U(1) r is broken).
Let us examine these operators more carefully. To that end, theC 0,n(0,0) multiplet has a primary arising from the normal-ordered product of a chiral operator with the stress tensor multiplet primaryC where φ n is theĒ n primary (D 0(0,0) if n = 1), and φ † φ is the dimension two primary of C 0(0,0) . On the other hand, theC 0,n(1,0) multiplets take the form where the coefficient γ can be fixed by demanding that the above operator is annihilated by the S ℓ δ supercharges. Up to a shift by a descendant, (4.10) appears in the normal-ordered product of φ m and the level-twoĒ n−m+1 descendant, ǫ ij Q i α , Q j β , φ n−m+1 .
As a result, we see that there is a simple map between the set of MADC 0,r(j,0) multiplets and those of the free U(1) theory. 28 Indeed, consider the j = 0 multiplets first (they cannot mix with the j = 1/2 multiplets under RG flow). In the case of the MAD theory, the minimal U(1) r charge of aC 0,r(0,0) primary is r min = 6/5, while it is r min = 1 in the case of the free vector. The U(1) r charge quantization (the difference in r between successiveC 0,r(0,0) multiplets) is δr = 6/5 for the MAD theory and δr = 1 for the free vector. Therefore, we need only apply a r → 5 6 r rescaling in order to find a map between the j = 0 sectors. Consider now the j = 1 sector. Here, r min = 2 × 6/5 − 1 = 7/5 in the MAD theory and r min = 2 × 1 − 1 = 1 in the free vector theory (where we includeĈ 0 (1,0) in this discussion; otherwise, r min = 2). The charge quantization is as before (δr = 6/5 for MAD and δr = 1 for the free vector). Therefore, to find a map between sectors, we need to rescale r min → 5 7 r min and δr → 5 6 δr. Since U(1) r is broken in any flow to the free vector multiplet, we expect these mappings to be mappings of sectors rather than of individual operators (i.e., there will be mixing). We comment on these ambiguities in the next section.
More generally, we expect that in a unitary N = 2 SCFT with a Coulomb branch, the 28 At the special points on the MAD moduli space where free hypers appear, the map is still simple: the role of the stress tensor is played by the sum of the free vector and free hyper stress tensors.
following normal-ordered products are non-zero where O and O ′ are non-trivialĒ N = 2 chiral primaries that can acquire non-vanishing vevs, O , O ′ = 0. The main reason we expect the above normal-ordered products to not vanish is that the entireĈ 0(0,0) multiplet can be unambiguously tracked along any N = 2preserving RG flow since it contains the stress tensor [26,27] (see also [28] and [29] for 4D N = 1 and 5D N = 1 discussions respectively). Since O and O ′ can also be tracked along an RG flow to the Coulomb branch, 29 we can track the above OPEs to the Coulomb branch where they are guaranteed to be non-vanishing as in the single vector multiplet case discussed in the previous section. In particular, the normal-ordered product does not vanish in the IR. Since the non-vanishing normal-ordered product in the IR must come from a non-trivial operator in the UV, we arrive at the following claim 3031 : Claim 2: In a unitary 4D N = 2 SCFT with a Coulomb branch, we will have at least onē C 0,r(0,0) multiplet for all r corresponding toĒ r multiplets with primaries that can take a vev on the Coulomb branch. Moreover, given any two Coulomb branch multiplets,Ē r and E r ′ , we will have at least oneC 0,r 1 +r 2 −1(1,0) multiplet.
As a result, we see that the MAD theory has the simplestC 0,r(j,0) spectrum allowed by unitarity and the existence of a Coulomb branch. If all interacting 4D N = 2 SCFTs have a Coulomb branch, claim 2 can be upgraded to a claim on all interacting 4D N = 2 SCFTs, and the MAD theory would have the simplestC 0,r(j,0) spectrum of any interacting 4D N = 2 SCFT. 29 This statement holds when there is a Seiberg-Witten description or some generalization thereof that allows one to compute the mixing; see [30] for some examples. 30 Another argument follows from the fact that J = 0 on the Coulomb branch (and, of course, similarly on any Higgs or mixed branch). Therefore, we expect OJ ∼ O J = 0 in the deep IR. For more details on how to track J to the IR, see [27]. 31 In the above argument, unitarity or the existence of a Coulomb branch is crucial. Indeed, consider the free N = 2 Abelian gauge theory with wrong statistics. In this case, O = ϕ (since ϕ n with n > 1 vanish by Fermi statistics), and J = ϕ † ϕ. Therefore, the normal-ordered product clearly vanishes, OJ = 0. Next, note that where we have used the fact that the Lorentz triplet combination of ǫ ij λ i α λ j β vanishes by Bose statistics (recall that the non-unitary gauginos transform as bosons). Therefore, our above argument does not apply to non-unitary theories (including, presumably, the more general ones in [20]).

A conjecture on the structure of the MAD local operator algebra
A defining property of the free N = 2 vector multiplet SCFT is that any operator in the theory can be built from products of the various component fields-φ, λ i α , F αβ , and conjugates-along with derivatives. Said slightly differently, any local operator in this theory, O, is in the (n, m)-fold OPE of theD 0(0,0) ⊕ D 0(0,0) multiplets O ∈D 0(0,0) × · · · ×D 0(0,0) × D 0(0,0) × D 0(0,0) :=D ×n 0(0,0) × D ×m 0(0,0) , ∀O ∈ H FreeVect. , (6.1) where H FreeVect. is the Hilbert space of local operators. Note that, to produce the full set of local operators in the theory, we must consider arbitrarily large n and m (this statement follows from SU(2) R and U(1) R covariance of the OPE). Again, for the same reasons as in the free vector, we will have to consider n an m to be arbitrarily large in order to obtain all local operators.
Therefore, we see that all local operators at any point on the Coulomb branch of the MAD theory (including points where a single massless hypermultiplet appears) are tightly constrained. It is then natural to ask what happens to the local operator algebra at the origin of the moduli space (i.e., in the MAD theory itself).
Since there are no other allowedC 0,r(j,0) multiplets (when, a priori, there could have been infinitely many), we find an infinite amount of evidence for the following conjecture 33 : 32 Recall that the hermitian conjugate of aB multiplet is anotherB multiplet. This latter multiplet may or may not be the same as the original multiplet (in the case of the free hyper, it is not). 33 In fact, our result in (3.24) is also compatible with the conjecture below, sinceC 0,r(j,1/2) appears in OPEs involving justĒ andĈ 0(0,0) (see [31] for a discussion of the relevant selection rules).
As in the case of the theories discussed above, we should consider arbitrarily many products if we wish to generate any local operator of the theory. This conjecture has the added benefit of directly generalizing the situations described around (6.1) and (6.2) on the Coulomb branch of the MAD theory. It would be interesting to understand if we can generalize the statement of this conjecture to apply to other 4D N = 2 SCFTs (e.g., to other AD theories that heuristically look like abelian gauge theories with mutually non-local massless matter).

Conclusions
In this paper, we have taken two steps toward finding the exact spectrum of short multiplets in the MAD theory: we completed the description of the spectrum of chiral operators by showing the MAD theory has noB multiplets. We then used this fact to find the exact spectrum ofC 0,r(j,0) multiplets. We also showed that these results have precise counterparts on the MAD Coulomb branch, and we showed that the MAD theory is indeed maximally simple from these perspectives.
The methods employed in this paper can be straightforwardly generalized to large classes of AD theories that admit N = 1 Lagrangian descriptions in the UV. It would be interesting to do this and to understand how our results interact with RG flows between these theories.
As a bonus, these results combine to suggest a conjecture on the global structure of the MAD operator algebra. Clearly, it would be interesting to prove this conjecture (or else to find operators constituting a counterexample), generalize it to other classes of 4D N = 2 SCFTs, and to understand its implications for the conformal bootstrap 35 and QFT more generally. We hope to return to these questions soon.
Claim: A non-trivial N = 1 chiral operator, O, should sit in an N = 2 multiplet whose SU(2) R highest-weight superconformal primary, O SCP , is also a non-trivial N = 1 chiral operator; in such a multiplet, O can either be the superconformal primary, O SCP , or its SU(2) R highest-weight (Q 1 α ) n descendant (where n = 1, 2). α . Anti-commutingQ 1 α past Q 1α can be done at the cost of introducing another term involving ∂ αα (this follows from the N = 1 SUSY algebra). Note that in order to have ∆ = 2R + r, we need an additionalQ 1 α present (since the net contribution of Q 1α , the firstQ 1 α , and any other supercharges does not decrease ∆ − 2R − r relative to the primary; note that this logic excludes the possible appearance of a second Q 1α ). SinceQ 1α commutes with ∂ αα , we have aQ 1 α -exact term. This is a contradiction. As a result, the superconformal primary should have a state, O SCP , satisfying (A.4)

Proof
It is easy to see this state is highest SU(2) R weight. Indeed, suppose it is not. Then, applying an SU(2) R raising operator, R + , leads to a state with ∆ < 2R + r, which is a violation of unitarity.
Let us now examine which descendants O can correspond to. First, note that we need only consider states obtained by an action of supercharges on the SU(2) R highest-weight state, O SCP . Otherwise, the primary state has 2R + r − ∆ < 0, and we are back to the situation described in the first paragraph of this proof.
Next, let us consider N = 1 chiral descendants obtained by acting with Poincaré supercharges on O SCP . Clearly, 2R + r = ∆ = 1 2 is only satisfied by Q 1 α . Therefore, we can get candidate descendant states O ∼ (Q 1 α ) n O SCP , where n = 1, 2 (since these states satisfy (A.4)). These states are, by construction, highest SU(2) R weight.
In fact, there are no other candidate states: if we act with aQ 1 α supercharge we are back in the situation described in the first paragraph. On the other hand, acting with Q 1α orQ 2 α (which all have ∆ > 2R + r) requires acting withQ 1 α in order to have any hope of obtaining an operator satisfying (A.4). Therefore, we are again back to the situation in the first paragraph.
We therefore conclude that a non-trivial N = 1 chiral operator can appear at most in the following positions where O 11···1 α 1 ···α 2j is the highest SU(2) R weight superconformal primary operator in a multiplet of the FCS:Ē ⊕D ⊕B ⊕B (this list can easily be checked via (A.5)). Note that not all of the states in (A.6) are realized in particular multiplets, because some of the descendants in (A.6) may be affected by the shortening conditions.
• X: It cannot be a level-two descendant because then it would be in anĒ 13/5 multiplet. Suppose it is a level-one descendant in aB multiplet. Then, in this case, the primary would have j = 1/2 and r = 11/10 < 1 + 1/2, which would be inconsistent.
• φqq: If it is a primary, r = 1 + j = 1 and we are in aD multiplet (but no such multiplet exists). If it is a level-two descendant, we are in anĒ 2 multiplet. If it is a level-one descendant, then the primary has R = 1/2, r = 3/2, and j = 1/2. This is again aD multiplet.
• qqφλ α : If it is a level-one descendant, then the primary has R = 1, r = 2, and j = 1 and would be of typeD (since r = 1 + j); similarly, j = 0 is disallowed. If it is a level-two descendant, then the primary has R = 1/2, r = 5/2, j = 1/2. On SU(2) R grounds, this must take the form M n φλ α , but this has r = 5/2.
• φλ α λ β : If it is a level-one descendant, then the primary has R = 1/2, r = 17/10, and j = 1/2 (it cannot have j = 3/2 because r < 5/2). By SU(2) R and spin considerations, this can only correspond to M n φλ α . However, this has r = 17/10. This cannot be a level-two descendant since the primary has R = 0 and would be of typeĒ (which we have already ruled out).
• qqλ α : If it is a level-one descendant, then the primary has R = 1, r = 9/5, and j = 0 (j = 1 is ruled out since 9/5 < 1+1). Based on the R weight and spin, such a primary can only be of the form M n qq ′ or M n (φλ α ) 2 . However, these operators have r = 9/5. If qqλ α is a level-two descendant, then the primary has R = 1/2, r = 23/10, and j = 1/2. On SU(2) R grounds, the only possibility is M n φλ α , but this has r = 23/10.
• qq ′ λ α : If it is a level-one descendant, then the primary has R = 1, r = 6/5, and j = 0 (j = 1 is ruled out since 6/5 < 1 + 1). Based on the R weight and spin, such a primary can only be of the form M n qq ′ or M n (φλ α ) 2 . However, these operators have r = 6/5. If it is a level-two descendant, the primary has R = 1/2, r = 17/10 and j = 1/2. The only possibility is M n φλ α , but this has r = 17/10.