Conformal manifolds in four dimensions and chiral algebras

In this section we suppose that we start with a collection of N isolated SCFTs and gauge a diagonal H symmetry with coupling g

$$\begin{eqnarray}&&\delta W=g\cdot {\rm{\Phi }}\displaystyle \sum _{i=1}^{N}{\mu }_{i},\end{eqnarray} \tag{ 2.7 }$$

where Φ is the adjoint field, and the ${\mu }_{i}$ are the H moment maps of the different isolated sectors, ${{ \mathcal T }}_{i}$. We then want to know what happens to the chiral algebra.

In very general terms, when we turn on a non-zero gauge coupling, many of the multiplets that were short at g = 0 pair up and become long multiplets (according to the rules of [18]). As a result, the chiral algebra that appears at infinite Zamolodchikov distance when g = 0 typically becomes much smaller when we are at finite Zamolodchikov distance and non-zero gauge coupling. For operators built out of the Schur gauginos and moment maps, this pairing up can be explicitly worked out at small gauge coupling⁷ via the following (anti)commutators

$$\begin{eqnarray}\{{\tilde{Q}}_{2\dot{-}},{\tilde{\lambda }}_{2\dot{+}}\} & = & \{{Q}_{-}^{1},{\lambda }_{+}^{1}\}=F=g\displaystyle \sum _{i=1}^{N}{\mu }_{i},\ \ \{{\tilde{Q}}_{2\dot{-}},{\lambda }_{+}^{1}\}=\{{Q}_{-}^{1},{\tilde{\lambda }}_{2\dot{+}}\}=0,\\ [{Q}_{-}^{1},{D}_{+\dot{+}}] & = & g{\tilde{\lambda }}_{\dot{+}}^{1}=g{\tilde{\lambda }}_{2\dot{+}},\ \ [{\tilde{Q}}_{2\dot{-}},{D}_{+\dot{+}}]=g{\lambda }_{+}^{1},\ \ [{Q}_{-}^{1},{D}_{+\dot{+}}]=g{\tilde{\lambda }}_{\dot{+}}^{1}=g{\tilde{\lambda }}_{2\dot{+}},\\ [{\tilde{Q}}_{2\dot{-}},{D}_{+\dot{+}}] & = & g{\lambda }_{+}^{1},\ \ [{Q}_{-}^{1},{\mu }_{I}]=[{\tilde{Q}}_{2\dot{-}},{\mu }_{I}]=0.\end{eqnarray} \tag{ 2.8 }$$

The last two commutators amount to imposing current conservation. Most of the operators built out of these degrees of freedom pair up at non-zero gauge coupling.

The important points for us below are that

• Interacting theories with fermionic chiral algebra generators must have $| \chi | \geqslant 3$ since these operators come in pairs and cannot form a stress tensor composite.
• We do not require that the chiral algebra is invariant as we explore finite diameter subregions of the conformal manifold (although there is considerable evidence that in many theories this invariance holds).

When the conformal manifold has regions in which $N\geqslant 4$ isolated SCFTs, ${{ \mathcal T }}_{i}$ ($i=1,\cdots ,N$), weakly coupled to a gauge field appear (we denote g = 0 as a strict decoupling point), it is straightforward to see that $| \chi | \geqslant 3$ by a direct argument in superconformal representation theory¹¹ . Indeed, consider the Schur conformal primary operators at ${ \mathcal O }({q}^{2})$. On general grounds, these operators must be in multiplets of type ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$, ${\hat{{ \mathcal B }}}_{2}$, ${{ \mathcal D }}_{0(\mathrm{0,1})}\oplus {\bar{{ \mathcal D }}}_{0(\mathrm{1,0})}$, ${{ \mathcal D }}_{1(\mathrm{0,0})}\oplus {\bar{{ \mathcal D }}}_{1(\mathrm{0,0})}$, or ${{ \mathcal D }}_{\tfrac{1}{2}\left(0,\tfrac{1}{2}\right)}\oplus {\bar{{ \mathcal D }}}_{\tfrac{1}{2}\left(\tfrac{1}{2},0\right)}$ [17].

For the purposes of our argument, it suffices to focus on the ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$, ${\hat{{ \mathcal B }}}_{2}$, and ${{ \mathcal D }}_{1(\mathrm{0,0})}\oplus {\bar{{ \mathcal D }}}_{1(\mathrm{0,0})}$ multiplets. These degrees of freedom can only recombine into long multiplets via [18]

$$\begin{eqnarray}&&{\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}\oplus {{ \mathcal D }}_{1(\mathrm{0,0})}\oplus {\bar{{ \mathcal D }}}_{1(\mathrm{0,0})}\oplus {\hat{{ \mathcal B }}}_{2}.\end{eqnarray} \tag{ 4.1 }$$

Clearly, the $N(N-1)/2$ flavor-singlet operators ${{ \mathcal O }}_{{ab}}={\mu }_{a}^{I}{\mu }_{{bI}}\in {\hat{{ \mathcal B }}}_{2}$ with $a\ne b$ are in short multiplets when g = 0 (here $I=1,\cdots ,| H|$ is an adjoint index). By (4.1), these multiplets can only recombine if there are an equal number of (flavor-singlet) ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$ multiplets. These latter multiplets contain a stress tensor, and there are N such multiplets available to pair up with the ${{ \mathcal O }}_{{ab}}$ multiplets when we turn the gauge coupling on. As a result, there are $N(N-3)/2\geqslant 2$ unpaired ${{ \mathcal O }}_{{ab}}$ multiplets. In particular, it follows that $| \chi | \geqslant 3$ as promised. Note that this is a strict lower bound at any point on the conformal manifold since these multiplets can disappear from the short sector only by pairing up with new conserved stress tensor multiplets (which come with an equal number of short ${\hat{{ \mathcal B }}}_{2}$ and ${{ \mathcal D }}_{1(\mathrm{0,0})}\oplus {\bar{{ \mathcal D }}}_{1(\mathrm{0,0})}$ multiplets).

Before we conclude this sub-section, let us discuss some subtleties related to free fields. To that end, we suppose, as part of our collection of isolated theories at g = 0, we have free hypermultiplets transforming in ${n}_{1}\geqslant 1$ copies of a representation, R₁, of H. In this case, we define the collection of n₁ copies of R₁ to form a single isolated sector, ${{ \mathcal T }}_{1}$. The reason we do not count each of the n₁ copies of R₁ as forming different sectors is that the corresponding flavor singlet ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$ multiplet used in our argument above involves multiplets in all n₁ copies of R₁.

To understand this statement, consider $H={SU}(2)$ SCQD with N_f = 4. We have eight fundamental SU(2) doublets, Q_a^Ai, and an SO(8) flavor symmetry (here A = 1, 2 is an ${SU}{(2)}_{R}$ index, a = 1, 2 is a fundamental H gauge index, and $i=1,\cdots ,8$ is a fundamental SO(8) index). This theory has thirty-seven short ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$ multiplets at g = 0

$$\begin{eqnarray}&&{J}_{\lambda }\sim {{\rm{\Phi }}}^{I}{{\rm{\Phi }}}_{I}^{\dagger },\ \ {J}^{{ij}}\sim {\epsilon }_{{AB}}{\epsilon }^{{ab}}{Q}_{a}^{{Ai}}{Q}_{b}^{{Bj}},\end{eqnarray} \tag{ 4.2 }$$

where ${{\rm{\Phi }}}^{I}$ is the chiral primary of the SU(2) vector multiplet. The only flavor singlet ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$ operator arising from the hypermultiplet sector is ${J}_{m}={\sum }_{i=1}^{8}{J}^{{ii}}$, and so it is sensible to define the collection of n₁ copies of R₁ as a single sector.

Note that when we turn on $g\ne 0$, the only remaining short operator in this collection is

$$\begin{eqnarray}&&J={J}_{\lambda }-{J}_{m}.\end{eqnarray} \tag{ 4.3 }$$

The other thirty-six short ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$ multiplets pair up with an equal number of ${\hat{{ \mathcal B }}}_{2}$ multiplets. In particular, consider the flavor singlet operator

$$\begin{eqnarray}&&{J}_{\perp }={\kappa }_{\lambda }{J}_{\lambda }+{J}_{m},\end{eqnarray} \tag{ 4.4 }$$

where ${\kappa }_{\lambda }\ne 0$ is a constant chosen so that $\langle {J}_{\perp }(x)J(0)\rangle =0$ at g = 0. This operator pairs up with the flavor singlet ${\mu }^{{ij}}{\mu }_{{ij}}$ operator to form a long multiplet, where

$$\begin{eqnarray}&&{\mu }^{{ij}}={\epsilon }^{{ab}}{Q}_{a}^{1i}{Q}_{b}^{1j}.\end{eqnarray} \tag{ 4.5 }$$

When we turn on small $g\ne 0$, this statement implies

$$\begin{eqnarray}&&{\mu }^{{ij}}{\mu }_{{ij}}=0,\end{eqnarray} \tag{ 4.6 }$$

at the level of the chiral ring, while, at the level of the ${\mu }^{{ij}}(x){\mu }_{{ij}}(0)$ OPE, we have

$$\begin{eqnarray}&&{\mu }^{{ij}}(x){\mu }_{{ij}}(0)=C\cdot {x}^{{\rm{\Delta }}-4}\cdot {({Q}^{1})}^{2}{({\tilde{Q}}_{2})}^{2}{J}_{\perp }+\cdots ,\end{eqnarray} \tag{ 4.7 }$$

where ${\rm{\Delta }}\gt 4$. In particular, the term appearing on the rhs of (4.7) arises because ${\mu }^{{ij}}{\mu }_{{ij}}$ pairs up with J_⊥.

The above argument does not work when the g = 0 point consists of a decoupled gauge multiplet and a single isolated SCFT sector, ${ \mathcal T }$. Instead, we will take a more indirect route in this case (note that, as we will see in the next sub-section, the argument in this sub-section generalizes to arbitrary numbers of sectors; moreover, we do not assume that each isolated SCFT has a single flavor-singlet ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$ multiplet; instead we allow for one ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$ multiplet that contains the genuine stress tensor of the sector—i.e., the operator that gives rise to the generator of translations in the sector—and an arbitrary number of other ${\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}$ multiplets). To that end, we first show that ${ \mathcal T }$ has a moduli (sub)space parameterized by vevs of the holomorphic moment maps. From this fact, we will be able to show that if $| \chi | \lt 3$, then ${a}_{4d}\geqslant {c}_{4d}$ (note that we add the subscript '4d' to a solely for unity of notation) along the conformal manifold of the gauged theory. Next, assuming the behavior described in [14] and (1.2) (see the introduction for a discussion of this point), we see that the partition function of the resulting chiral algebra lacks an essential singularity in the $q\to 1$ limit. Finally, we argue that the absence of this singularity implies the existence of fermionic chiral algebra generators and hence, by one of the bullet points in section 2.1, $| \chi | \geqslant 3$.

We begin by proving that in ${ \mathcal T }$, the ${{ \mathcal O }}^{n}={({\mu }^{I}{\mu }_{I})}^{n}\in {\hat{{ \mathcal B }}}_{2n}$ operators transform in short multiplets for all $n\gt 0$. In other words, we prove that ${{ \mathcal O }}^{n}\ne 0$ in the chiral ring of ${ \mathcal T }$. For n = 1, this follows from the unitarity bound of [2]. Indeed, if it were not the case, then (4.16) of [2] implies

$$\begin{eqnarray}&&\displaystyle \frac{\dim H}{{c}_{4d}}=\displaystyle \frac{24{h}^{\vee }}{{k}_{4d}}-12=-6,\end{eqnarray} \tag{ 4.8 }$$

and so ${c}_{4d}\lt 0$, which contradicts four-dimensional unitarity. More generally, suppose that there is some maximal $\hat{n}\geqslant 1$ such that ${{ \mathcal O }}^{\hat{n}-1}$ is in a short multiplet of ${ \mathcal T }$. As a result, ${{ \mathcal O }}^{\hat{n}}$ is in a long multiplet of type¹²

$$\begin{eqnarray}&&{\hat{{ \mathcal C }}}_{2\hat{n}-2(\mathrm{0,0})}\oplus {{ \mathcal D }}_{2\hat{n}-1(\mathrm{0,0})}\oplus {\bar{{ \mathcal D }}}_{2\hat{n}-1(\mathrm{0,0})}\oplus {\hat{{ \mathcal B }}}_{2\hat{n}}.\end{eqnarray} \tag{ 4.9 }$$

Let us denote the (would be) Schur operators in the first three sub-multiplets of (4.9) as ${{ \mathcal O }}_{+\dot{+}}^{\prime }$, ${{ \mathcal O }}_{\dot{+}}^{\prime\prime }$, and ${{ \mathcal O }}_{+}^{\prime\prime }$ respectively.

Now, consider working at arbitrarily small $g\ne 0$. The theory still possesses a short stress tensor multiplet, and this multiplet has an associated Schur operator of the form

$$\begin{eqnarray}&&{J}_{+\dot{+}}^{11}={({\lambda }_{+}^{1})}^{I}{({\tilde{\lambda }}_{2\dot{+}})}_{I}-{J}_{1+\dot{+}}^{11},\end{eqnarray} \tag{ 4.10 }$$

where, at g = 0, ${J}_{1+\dot{+}}^{11}$ is the Schur operator in the stress tensor multiplet of ${ \mathcal T }$. Given this operator, let us define

$$\begin{eqnarray}&&{\tilde{{ \mathcal O }}}_{+\dot{+}}={\tilde{\kappa }}_{\lambda }{({\lambda }_{+}^{1})}^{I}{({\tilde{\lambda }}_{2\dot{+}})}_{I}+{J}_{1+\dot{+}}^{11},\end{eqnarray} \tag{ 4.11 }$$

where ${\tilde{\kappa }}_{\lambda }\ne 0,-1$ is a constant chosen so that $\langle {J}_{+\dot{+}}^{11}(x){\tilde{{ \mathcal O }}}_{+\dot{+}}(0)\rangle =0$ (at g = 0). Clearly, ${\tilde{{ \mathcal O }}}_{+\dot{+}}$ is in a long multiplet for small $g\ne 0$ since it pairs up as follows

$$\begin{eqnarray}&&{\tilde{{ \mathcal O }}}_{+\dot{+}}\oplus {\tilde{{ \mathcal O }}}_{+}\oplus {\tilde{{ \mathcal O }}}_{\dot{+}}\oplus (1+{\kappa }_{\lambda }){ \mathcal O },\end{eqnarray} \tag{ 4.12 }$$

where ${{ \mathcal O }}_{+}=(1+{\kappa }_{\lambda }){\mu }^{I}{({\lambda }_{+}^{1})}_{I}$, ${{ \mathcal O }}_{\dot{+}}=(1+{\kappa }_{\lambda }){\mu }^{I}{({\tilde{\lambda }}_{2\dot{+}})}_{I}$, and ${ \mathcal O }={\mu }^{I}{\mu }_{I}$. Moreover, we have that

$$\begin{eqnarray}&&{{ \mathcal O }}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{+\dot{+}}\oplus {{ \mathcal O }}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{+}\oplus {{ \mathcal O }}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{\dot{+}}\oplus (1+{\kappa }_{\lambda }){{ \mathcal O }}^{\hat{n}}.\end{eqnarray} \tag{ 4.13 }$$

It then follows that ${{ \mathcal O }}_{+\dot{+}}^{\prime }-{(1+{\kappa }_{\lambda })}^{-1}{{ \mathcal O }}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{+\dot{+}}$ is in a short multiplet for small $g\ne 0$. However, this statement leads to a contradiction, because, by dialing g arbitrarily small but non-zero, we find that the anomalous dimension of ${{ \mathcal O }}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{+\dot{+}}$ is parametrically smaller than the anomalous dimension of¹³   ${{ \mathcal O }}_{+\dot{+}}^{\prime }$. As a result, we see that, as claimed,

$$\begin{eqnarray}&&{{ \mathcal O }}^{n}\ne 0\ \ \ \forall n\gt 0,\end{eqnarray} \tag{ 4.14 }$$

in the chiral ring of the isolated theory¹⁴ , ${ \mathcal T }$.

As a final aside, note that, a priori, we might have imagined that ${{ \mathcal O }}^{\hat{n}}$ could be identically zero, i.e., it might not even be possible to define this operator to be in a long multiplet as in (4.9). However, such a statement would be in contradiction with (4.13), since ${\tilde{{ \mathcal O }}}_{+\dot{+}}$ must recombine to become part of a long multiplet for $g\ne 0$, and therefore ${{ \mathcal O }}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{+\dot{+}}$ should also be part of a long multiplet at small (but non-zero) coupling (in particular, this statement must hold when we take the leading corrections in g into account).

Now we must use some physical intuition. Indeed, since ${ \mathcal O }$ is an ${SU}{(2)}_{R}$-charged but $U{(1)}_{R}$-neutral chiral operator that is not nilpotent, it is reasonable to associate a SUSY moduli space, ${\hat{{ \mathcal M }}}_{{ \mathcal O }}\subset {{ \mathcal M }}_{{SU}{(2)}_{R}}$, with its vevs (here ${{ \mathcal M }}_{{SU}{(2)}_{R}}$ contains any normal directions to generic points on ${\hat{{ \mathcal M }}}_{{ \mathcal O }}$ that also break ${SU}{(2)}_{R}$ but preserve $U{(1)}_{R}$). Given this picture, we expect that for generic $\langle { \mathcal O }\rangle \ne 0$, the IR theory is just a collection of free hypermultiplets. In some theories, these free hypermultiplets might necessarily be accompanied by free vector multiplets as well (see, e.g., [23]). This insight allows us to compute a bound on the value of ${a}_{4d}-{c}_{4d}$ for the isolated theory

$$\begin{eqnarray}&&{a}_{4d,{ \mathcal T }}-{c}_{4d,{ \mathcal T }}\geqslant -\displaystyle \frac{1}{24}\dim {{ \mathcal M }}_{{SU}{(2)}_{R}}.\end{eqnarray} \tag{ 4.16 }$$

Note that (4.16) follows from the definition of our moduli space: it is a space of vacua in which (i) ${SU}{(2)}_{R}$ is spontaneously broken, but $U{(1)}_{R}$ is not, and (ii) at generic points, the theory consists of a set of free hypermultiplets possibly with additional vector multiplets. As a result, (4.16) follows from linear $U{(1)}_{R}$ 't Hooft anomaly matching. In particular, the above inequality is saturated if we have a genuine Higgs branch (i.e., no vector multiplets at generic points).

We can now show that, in the gauged theory, either ${a}_{4d}\geqslant {c}_{4d}$ along the conformal manifold, or $| \chi | \geqslant 3$. Indeed, suppose ${a}_{4d}\lt {c}_{4d}$. In this case, the gauged theory would still possess a non-trivial moduli space of the type described above with ${\hat{{ \mathcal B }}}_{R}$ generators parameterizing it. As discussed in the bullet points of the first part of section 2, this conclusion means we would have $| \chi | \geqslant 3$. In particular, if we would like a theory with $| \chi | \lt 3$, we must have that

$$\begin{eqnarray}&&{a}_{4d}\geqslant {c}_{4d},\end{eqnarray} \tag{ 4.17 }$$

along the conformal manifold.

In the rest of this sub-section, we show that $| \chi | \geqslant 3$ even in the case of ${a}_{4d}\geqslant {c}_{4d}$. Note first that, assuming the partition function has the behavior described in [14] and (1.2), it is straightforward to show that if ${a}_{4d}\geqslant {c}_{4d}$ then the Schur index lacks an essential singularity as $q\to 1$. Let us see if we can find a chiral algebra with $| \chi | =1$ that is compatible with this asymptotic behavior. Since the stress tensor always exists, such an algebra must be a Virasoro algebra. If there are no null vectors (besides the trivial one involving the derivative of the identity) in the vacuum Virasoro module, then the partition function is

$$\begin{eqnarray}&&{ \mathcal I }={\chi }_{(\mathrm{1,1})}={\rm{P}}.{\rm{E}}.\,\left(\displaystyle \frac{{q}^{2}}{1-q}\right),\end{eqnarray} \tag{ 4.18 }$$

where the plethystic exponential of a function of variables ${x}_{1},\cdots ,{x}_{i}$ is defined as

$$\begin{eqnarray}&&{\rm{P}}.{\rm{E}}.(f({x}_{1},\cdots ,{x}_{i}))\equiv \exp \left(\displaystyle \sum _{n=1}^{\infty }{n}^{-1}f({x}_{1}^{n},\cdots ,{x}_{i}^{n})\right).\end{eqnarray} \tag{ 4.19 }$$

Therefore, (4.18) clearly has an essential singularity as $q\to 1$. This result implies that the corresponding Schur index also has an essentail singularity and therefore ${a}_{4d}\lt {c}_{4d}$.

Let us then consider cases with non-trivial null vectors. From the general formula for the Kac determinant in (7.28) of [24] with h = 0 (since we are considering the Virasoro vacuum representation) as well as (7.31) of that same reference, we find that a non-trivial null vector exists only if

$$\begin{eqnarray}&&{c}_{2d}=13-6(t+{t}^{-1}),\ \ t=\displaystyle \frac{1\pm s}{1\pm r}=\displaystyle \frac{p}{p^{\prime} }\gt \displaystyle \frac{3}{2}\ \mathrm{or}\ t\lt \displaystyle \frac{2}{3},\end{eqnarray} \tag{ 4.20 }$$

for integers r, s, p, and $p^{\prime}$. Without loss of generality, we can take $p,p^{\prime} \gt 1$ relatively prime. Note that if p and $p^{\prime}$ have different sign, then ${c}_{2d}\gt 0$ and ${c}_{4d}\lt 0$, which violates four-dimensional unitarity. Also, if p = 1 or $p^{\prime} =1$, then we have a partition function as in (4.18). Finally, for $t=\tfrac{3}{2}$ or $t=\tfrac{2}{3}$, we have that ${c}_{2d}=0$ and so by our discussion below (2.3), ${c}_{4d}=0$ as well. However, this statement violates four-dimensional unitarity, so we need not consider this case.

Now, according to (8.17) of [24] (we set the overall normalization of the character so that the vacuum contributes unity) we have

$$\begin{eqnarray}&&{ \mathcal I }={\chi }_{(\mathrm{1,1})}=\displaystyle \frac{{q}^{-{(p-p^{\prime} )}^{2}/4{pp}^{\prime} }}{(q;q)}\displaystyle \sum _{n\in {\bf{Z}}}({q}^{{(2{pp}^{\prime} n+p-p^{\prime} )}^{2}/4{pp}^{\prime} }-{q}^{{(2{pp}^{\prime} n+p+p^{\prime} )}^{2}/4{pp}^{\prime} }),\end{eqnarray} \tag{ 4.21 }$$

From the modular transformation properties of the vacuum character, we see that the Schur index (4.21) behaves in the limit $q\equiv {{\rm{e}}}^{-\beta }\to 1$ as

$$\begin{eqnarray}&&{ \mathcal I }\sim {{\rm{e}}}^{-\tfrac{4{\pi }^{2}}{\beta }\left({h}_{\min }-\displaystyle \frac{{c}_{2d}}{24}\right)+{ \mathcal O }(1)}={{\rm{e}}}^{\displaystyle \frac{{\pi }^{2}}{6\beta }\left(1-\displaystyle \frac{6}{{pp}^{\prime} }\right)+{ \mathcal O }(1)},\end{eqnarray} \tag{ 4.22 }$$

where ${h}_{\min }=\tfrac{1-{(p-p^{\prime} )}^{2}}{4{pp}^{\prime} }$ is the smallest holomorphic dimension of a primary state in the $(p,p^{\prime} )$ minimal model (see [8] for a recent discussion of consequences of this fact in the context of 4D/2D relations). Since p and $p^{\prime}$ are relatively prime and t is in the range described in (4.20), we see that ${pp}^{\prime} \geqslant 10$ (${pp}^{\prime} =10$ in the case of the Yang-Lee model that is related to the $({A}_{1},{A}_{2})$ theory). Therefore, we see that there is always an essential singularity in (4.22) and so ${a}_{4d}\lt {c}_{4d}$. This inequality means that the four-dimensional theory cannot have ${a}_{4d}\geqslant {c}_{4d}$ if the corresponding chiral algebra has only one generator¹⁵ .

We therefore require at least one more generator to cancel the essential singularity in the Schur index and realize ${a}_{4d}\geqslant {c}_{4d}$. However, if there are only two generators, one of them is the stress tensor and the other must be a bosonic operator. Since adding a bosonic generator does not cancel the above essential singularity in the vacuum Virasoro character, we conclude that ${a}_{4d}\lt {c}_{4d}$ even in the case of $| \chi | =2$. Therefore, we need three or more generators of the chiral algebra (including fermions) for the four-dimensional theory to have ${a}_{4d}\geqslant {c}_{4d}$.

Let us now study the case of $N\geqslant 2$ isolated SCFT sectors, ${{ \mathcal T }}_{i}$ ($i=1,\cdots ,N$), at g = 0 with decoupled gauge fields for a diagonal H global symmetry of the ${{ \mathcal T }}_{i}$. We will generalize the discussion of the previous subsection for N = 1. To that end, consider the N operators ${{ \mathcal O }}_{i}={\mu }_{i}^{I}\left({\sum }_{a=1}^{N}{\mu }_{{aI}}\right)$. We claim that, in the chiral ring of the collection of the isolated ${{ \mathcal T }}_{i}$

$$\begin{eqnarray}&&{{ \mathcal O }}_{i}^{n}\ne 0,\ \ \forall n\gt 0,i=1,\cdots ,N.\end{eqnarray} \tag{ 4.23 }$$

From this result, it follows (as in the N = 1 case discussed previously) that each of the isolated ${{ \mathcal T }}_{i}$ SCFTs have moduli spaces, ${{ \mathcal M }}_{{SU}{(2)}_{R}}$, where ${SU}{(2)}_{R}$ is spontaneously broken but $U{(1)}_{R}$ is not¹⁶ . The argument we used in the single sector case starting with (4.16) (but now for all N isolated sectors) applies directly to the case of N sectors since we have a non-trivial ${{ \mathcal M }}_{{SU}{(2)}_{R}}$. We again conclude that the chiral algebra must have at least three generators.

To justify (4.23), we first note that when we turn on small $g\ne 0$, only one stress tensor multiplet remains short. This multiplet has the following Schur operator

$$\begin{eqnarray}&&{J}_{+\dot{+}}^{11}={({\lambda }_{+}^{1})}^{I}{({\tilde{\lambda }}_{2\dot{+}})}_{I}-\displaystyle \sum _{i=1}^{N}{J}_{i+\dot{+}}^{11},\end{eqnarray} \tag{ 4.25 }$$

where (at g = 0) the ${J}_{i+\dot{+}}^{11}$ are the Schur operators in the stress tensor multiplets of the isolated ${{ \mathcal T }}_{i}$ theories. We can form N additional linearly independent combinations of the operators appearing on the rhs of (4.25)

$$\begin{eqnarray}&&{\tilde{{ \mathcal O }}}_{a+\dot{+}}={\kappa }_{a,\lambda }{({\lambda }_{+}^{1})}^{I}{({\tilde{\lambda }}_{2\dot{+}})}_{I}+\displaystyle \sum _{i=1}^{N}{\kappa }_{a,i}{J}_{i+\dot{+}}^{11},\end{eqnarray} \tag{ 4.26 }$$

where $a=1,\cdots ,N$ runs over the linear combinations. The constants ${\kappa }_{a,\lambda }$ and ${\kappa }_{a,i}$ in (4.26) are chosen so that $\langle {\tilde{{ \mathcal O }}}_{a+\dot{+}}(x){J}_{+\dot{+}}^{11}(0)\rangle =\langle {\tilde{{ \mathcal O }}}_{a+\dot{+}}(x){\tilde{{ \mathcal O }}}_{b+\dot{+}}(0)\rangle =0$ at g = 0. As in the N = 1 case, we can again check that all N of the operators in (4.26) must pair up with linearly independent combinations of the N operators ${{ \mathcal O }}_{i}={\mu }_{i}^{I}\left({\sum }_{a=1}^{N}{\mu }_{{aI}}\right)$.

Therefore, to justify (4.23), we can, in analogy with the N = 1 discussion in the previous subsection, suppose that for some particular ${{ \mathcal O }}_{i}$ there is a minimal $\hat{n}\gt 0$ such that ${{ \mathcal O }}_{i}^{\hat{n}}=0$ in the chiral ring of the collection of the isolated ${{ \mathcal T }}_{i}$. Clearly, this operator is in a long multiplet of the following type (the case in which this operator identically vanishes can be ruled out as in the case of a single isolated SCFT described in the previous section)

$$\begin{eqnarray}&&{\hat{{ \mathcal C }}}_{2\hat{n}-2(\mathrm{0,0})}\oplus {{ \mathcal D }}_{2\hat{n}-1(\mathrm{0,0})}\oplus {\bar{{ \mathcal D }}}_{2\hat{n}-1(\mathrm{0,0})}\oplus {\hat{{ \mathcal B }}}_{2\hat{n}}.\end{eqnarray} \tag{ 4.27 }$$

Now, we study the operator ${{ \mathcal O }}_{i}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{+\dot{+}}^{\prime }$ in the theory with $g\ne 0$, where ${\tilde{{ \mathcal O }}}_{+\dot{+}}^{\prime }$ is the appropriate linear combination of the ${\tilde{{ \mathcal O }}}_{a+\dot{+}}$ in (4.26) that pairs up with ${{ \mathcal O }}_{i}$. From this discussion, we have

$$\begin{eqnarray}&&{{ \mathcal O }}_{i}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{+\dot{+}}^{\prime }\oplus {{ \mathcal O }}_{i}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{+}^{\prime }\oplus {{ \mathcal O }}_{i}^{\hat{n}-1}{\tilde{{ \mathcal O }}}_{\dot{+}}^{\prime }\oplus {{ \mathcal O }}_{i}^{\hat{n}}.\end{eqnarray} \tag{ 4.28 }$$

As in the N = 1 case, we again find a contradiction since ${{ \mathcal O }}_{i}^{\hat{n}}$ is already in a long multiplet at g = 0. In particular, we conclude that

$$\begin{eqnarray}&&{{ \mathcal O }}_{i}^{n}\ne 0,\ \ \forall n\gt 0,i=1,\cdots ,N,\end{eqnarray} \tag{ 4.29 }$$

in the chiral ring of the collection of the ${{ \mathcal T }}_{i}$. Therefore, as discussed below (4.23), we see that $| \chi | \geqslant 3$.

In this sub-section we further check the identification in (5.11) by explicitly constructing the Schur operators, up to ${ \mathcal O }({q}^{4})$, that survive for small but non-zero gauge coupling²² . As we will see, all chiral algebra generators of ${ \mathcal O }({q}^{4})$ and lower pair up to form long multiplets, with the exception of the the ${SU}{(2)}_{R}$ current (at ${ \mathcal O }({q}^{2})$) and the two fermionic generators (at ${ \mathcal O }({q}^{4})$). Moreover, it is straightforward to check, using the Macdonald index of the $({A}_{1},{A}_{3})$ theory [5] and the Macdonald analog of (5.2), that the four-dimensional fermionic primaries corresponding to ${{\rm{\Phi }}}^{\pm }$ should be of type ${\hat{{ \mathcal C }}}_{\tfrac{3}{2}\left(\tfrac{1}{2},0\right)}\oplus {\hat{{ \mathcal C }}}_{\tfrac{3}{2}\left(0,\tfrac{1}{2}\right)}$ (the corresponding Schur operators have charge $\pm \tfrac{1}{2}$ under ${M}^{\perp }={j}_{1}-{j}_{2}$, which is in turn mapped to charge $\pm \tfrac{1}{2}$ under A in the chiral algebra).

To check the above statements, we first consider the ${ \mathcal O }({q}^{2})$ Schur operators at g = 0

$$\begin{eqnarray} & & {{ \mathcal O }}_{{ab}}={\mu }_{a}^{I}{\mu }_{{bI}}\in {\hat{{ \mathcal B }}}_{2},\ \ {{ \mathcal O }}_{+a}={\lambda }_{+}^{1I}{\mu }_{{aI}}\in {\bar{{ \mathcal D }}}_{1(\mathrm{0,0})},\ \ {{ \mathcal O }}_{\dot{+}a}=\mathrm{Tr}{\tilde{\lambda }}_{2\dot{+}}^{I}{\mu }_{{aI}}\in {{ \mathcal D }}_{1(\mathrm{0,0})},\\ & & {{ \mathcal O }}_{+\dot{+}}={\lambda }_{+}^{1I}{\tilde{\lambda }}_{2\dot{+}I}\in {\hat{{ \mathcal C }}}_{0(\mathrm{0,0})},\ \ {J}_{a+\dot{+}}^{11}\in {\hat{{ \mathcal C }}}_{0(\mathrm{0,0})},\end{eqnarray} \tag{ 5.13 }$$

where $a,b=1,2,3$ denote the particular $({A}_{1},{A}_{3})$ factor. Note that the Joseph ideal constraint in each $({A}_{1},{A}_{3})$ theory implies that ${{ \mathcal O }}_{{aa}}=0$. As a result, this constraint leaves three ${{ \mathcal O }}_{{ab}}$ operators with $a\ne b$. Using the variations in (2.8), it straightforward to check that when we turn on a small gauge coupling, all operators in (5.13) pair up via

$$\begin{eqnarray}&&{\hat{{ \mathcal C }}}_{0(\mathrm{0,0})}\oplus {{ \mathcal D }}_{1(\mathrm{0,0})}\oplus {\bar{{ \mathcal D }}}_{1(\mathrm{0,0})}\oplus {\hat{{ \mathcal B }}}_{2},\end{eqnarray} \tag{ 5.14 }$$

with the exception of the overall ${SU}{(2)}_{R}$ Schur operator that is preserved along the conformal manifold

$$\begin{eqnarray*}&&{J}_{+\dot{+}}^{11}={{ \mathcal O }}_{+\dot{+}}-\displaystyle \sum _{a=1}^{3}{J}_{a+\dot{+}}^{11}.\end{eqnarray*}$$

Next, at ${ \mathcal O }({q}^{3})$, we have the following Schur generators for g = 0

$$\begin{eqnarray} & & {{ \mathcal O }}_{+++}={f}_{{IJK}}{\lambda }_{+}^{1I}{\lambda }_{+}^{1J}{\lambda }_{+}^{1K}\in {\bar{{ \mathcal D }}}_{1(\mathrm{1,0})},\ \ {{ \mathcal O }}_{\dot{+}++}={f}_{{IJK}}{\lambda }_{+}^{1I}{\lambda }_{+}^{1J}{\tilde{\lambda }}_{2\dot{+}}^{K}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{2},0\right)},\\ & & {{ \mathcal O }}_{\dot{+}\dot{+}+}={f}_{{IJK}}{\tilde{\lambda }}_{2\dot{+}}^{I}{\tilde{\lambda }}_{2\dot{+}}^{J}{\lambda }_{+}^{1K}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(0,\displaystyle \frac{1}{2}\right)},\ \ {{ \mathcal O }}_{\dot{+}\dot{+}\dot{+}}={f}_{{IJK}}{\tilde{\lambda }}_{2\dot{+}}^{I}{\tilde{\lambda }}_{2\dot{+}}^{J}{\tilde{\lambda }}_{2\dot{+}}^{K}\in {{ \mathcal D }}_{1(\mathrm{0,1})},\\ & & {{ \mathcal O }}_{++a}={f}_{{IJK}}{\lambda }_{+}^{1I}{\lambda }_{+}^{1J}{\mu }_{a}^{K}\in {\bar{{ \mathcal D }}}_{\displaystyle \frac{3}{2}\left(\displaystyle \frac{1}{2},0\right)},\ \ {{ \mathcal O }}_{\dot{+}+a}={f}_{{IJK}}{\lambda }_{+}^{1I}{\tilde{\lambda }}_{2\dot{+}}^{J}{\mu }_{a}^{K}\in {\hat{{ \mathcal C }}}_{1(\mathrm{0,0})},\\ & & {{ \mathcal O }}_{\dot{+}\dot{+}a}={f}_{{IJK}}{\tilde{\lambda }}_{2\dot{+}}^{I}{\tilde{\lambda }}_{2\dot{+}}^{J}{\mu }_{a}^{K}\in {{ \mathcal D }}_{\displaystyle \frac{3}{2}\left(0,\displaystyle \frac{1}{2}\right)},\ \ {{ \mathcal O }}_{+{ab}}={f}_{{IJK}}{\lambda }_{+}^{1I}{\mu }_{a}^{J}{\mu }_{b}^{K}\in {{ \mathcal D }}_{2(\mathrm{0,0})},\\ & & {{ \mathcal O }}_{\dot{+}{ab}}={f}_{{IJK}}{\tilde{\lambda }}_{2\dot{+}}^{I}{\mu }_{a}^{J}{\mu }_{b}^{K}\in {\bar{{ \mathcal D }}}_{2(\mathrm{0,0})},\ \ {{ \mathcal O }}_{{abc}}={f}_{{IJK}}{\mu }_{a}^{I}{\mu }_{b}^{J}{\mu }_{c}^{K}\in {\hat{{ \mathcal B }}}_{3},\\ & & {{ \mathcal O }}_{\dot{+}+++}={\lambda }_{+}^{1I}{D}_{+\dot{+}}{\lambda }_{+}^{1I}\in {\hat{{ \mathcal C }}}_{0(\mathrm{1,0})},\ \ {{ \mathcal O }}_{\dot{+}\dot{+}++}={\lambda }_{+}^{1I}{D}_{+\dot{+}}{\tilde{\lambda }}_{2\dot{+}I}\in {\hat{{ \mathcal C }}}_{0\left(\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right)},\\ & & {{ \mathcal O }}_{\dot{+}\dot{+}\dot{+}+}={\tilde{\lambda }}_{2\dot{+}}^{I}{D}_{+\dot{+}}{\tilde{\lambda }}_{2\dot{+}I}\in {\hat{{ \mathcal C }}}_{0(\mathrm{0,1})},\ \ {{ \mathcal O }}_{\dot{+}+[{ab}]}={\mu }_{[a}^{I}{D}_{+\dot{+}}{\mu }_{b]I}\in {\hat{{ \mathcal C }}}_{1(\mathrm{0,0})},\\ & & {{ \mathcal O }}_{\dot{+}++a}={\lambda }_{+}^{1I}{D}_{+\dot{+}}{\mu }_{{aI}}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{2},0\right)},\ \ {{ \mathcal O }}_{\dot{+}\dot{+}+a}={\tilde{\lambda }}_{2\dot{+}}^{I}{D}_{+\dot{+}}{\mu }_{{aI}}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(0,\displaystyle \frac{1}{2}\right)},\end{eqnarray} \tag{ 5.15 }$$

where f_IJK are the SU(2) structure constants. We see that anti-symmetry implies that ${{ \mathcal O }}_{{abc}}\sim {{ \mathcal O }}_{123}$. Similarly, the ${{ \mathcal O }}_{+{ab}}$ and ${{ \mathcal O }}_{\dot{+}{ab}}$ operators are anti-symmetric in a and b. As a result, there are sixteen fermionic and sixteen bosonic Schur generators at g=0. It is straightforward to check that when we turn on a small but non-zero gauge coupling, all of these operators pair up via

$$\begin{eqnarray} & & {\hat{{ \mathcal C }}}_{1(\mathrm{0,0})}\oplus {{ \mathcal D }}_{2(\mathrm{0,0})}\oplus {\bar{{ \mathcal D }}}_{2(\mathrm{0,0})}\oplus {\hat{{ \mathcal B }}}_{3},\\ & & {\hat{{ \mathcal C }}}_{0(\mathrm{1,0})}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{2},0\right)}\oplus {\bar{{ \mathcal D }}}_{1(\mathrm{1,0})}\oplus {\bar{{ \mathcal D }}}_{\displaystyle \frac{3}{2}\left(\displaystyle \frac{1}{2},0\right)},\\ & & {\hat{{ \mathcal C }}}_{0(\mathrm{0,1})}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(0,\displaystyle \frac{1}{2}\right)}\oplus {{ \mathcal D }}_{1(\mathrm{0,1})}\oplus {{ \mathcal D }}_{\displaystyle \frac{3}{2}\left(0,\displaystyle \frac{1}{2}\right)},\\ & & {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{2},0\right)}\oplus {\hat{{ \mathcal C }}}_{1(\mathrm{0,0})}\oplus {\bar{{ \mathcal D }}}_{\displaystyle \frac{3}{2}\left(\displaystyle \frac{1}{2},0\right)}\oplus {\bar{{ \mathcal D }}}_{2(\mathrm{0,0})},\\ & & {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(0,\displaystyle \frac{1}{2}\right)}\oplus {\hat{{ \mathcal C }}}_{1(\mathrm{0,0})}\oplus {{ \mathcal D }}_{\displaystyle \frac{3}{2}\left(0,\displaystyle \frac{1}{2}\right)}\oplus {{ \mathcal D }}_{2(\mathrm{0,0})},\\ & & {\hat{{ \mathcal C }}}_{0\left(\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right)}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(0,\displaystyle \frac{1}{2}\right)}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{2},0\right)}\oplus {\hat{{ \mathcal C }}}_{1(\mathrm{0,0})}.\end{eqnarray} \tag{ 5.16 }$$

In particular, there are no new chiral algebra generators at ${ \mathcal O }({q}^{3})$ for small but non-zero gauge coupling, and the only contribution to the Schur index at this order comes from the descendant, ${\partial }_{+\dot{+}}{J}_{+\dot{+}}^{11}$.

Finally, let us discuss the ${ \mathcal O }({q}^{4})$ Schur generators at g = 0

$$\begin{eqnarray} & & {{ \mathcal O }}_{\dot{+}\dot{+}+++}={f}_{{IJK}}{\tilde{\lambda }}_{2\dot{+}}^{I}{D}_{+\dot{+}}{\lambda }_{+}^{1J}{\lambda }_{+}^{1K}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(1,\displaystyle \frac{1}{2}\right)},{{ \mathcal O }}_{++\dot{+}\dot{+}\dot{+}}={f}_{{IJK}}{\lambda }_{+}^{1I}{D}_{+\dot{+}}{\tilde{\lambda }}_{2\dot{+}}^{J}{\tilde{\lambda }}_{2\dot{+}}^{K}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{2},1\right)},\\ & & {{ \mathcal O }}_{\dot{+}\dot{+}\dot{+}+++}={\lambda }_{+}^{1I}{D}_{+\dot{+}}^{2}{\tilde{\lambda }}_{2\dot{+}I}\in \ {\hat{{ \mathcal C }}}_{0(\mathrm{1,1})},\ \ \ {{ \mathcal O }}_{+++\dot{+}\dot{+}a}={D}_{+\dot{+}}^{2}{\mu }_{a}^{I}{\lambda }_{+I}^{1}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(1,\displaystyle \frac{1}{2}\right)},\\ & & {{ \mathcal O }}_{\dot{+}\dot{+}\dot{+}++a}={D}_{+\dot{+}}^{2}{\tilde{\lambda }}_{2\dot{+}}^{I}{\mu }_{{aI}}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{2},1\right)},\ \ {{ \mathcal O }}_{\dot{+}+++a}={f}_{{IJK}}{D}_{+\dot{+}}{\lambda }_{+}^{1I}{\lambda }_{+}^{1J}{\mu }_{a}^{K}\in {\hat{{ \mathcal C }}}_{1(\mathrm{1,0})},\\ & & {{ \mathcal O }}_{+\dot{+}\dot{+}\dot{+}a}={f}_{{IJK}}{D}_{+\dot{+}}{\tilde{\lambda }}_{2\dot{+}}^{I}{\tilde{\lambda }}_{2\dot{+}}^{J}{\mu }_{a}^{K}\in {\hat{{ \mathcal C }}}_{1(\mathrm{0,1})},\ \ {{ \mathcal O }}_{\dot{+}\dot{+}++a}={f}_{{IJK}}{D}_{+\dot{+}}{\lambda }_{+}^{1I}{\tilde{\lambda }}_{2\dot{+}}^{J}{\mu }_{a}^{K},\\ & & {{ \mathcal O }}_{\dot{+}\dot{+}++a}^{\prime }={f}_{{IJK}}{\lambda }_{+}^{1I}{D}_{+\dot{+}}{\tilde{\lambda }}_{2\dot{+}}^{J}{\mu }_{a}^{K}\in {\hat{{ \mathcal C }}}_{1\left(\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right)},\ \ {{ \mathcal O }}_{++\dot{+}\dot{+}{ab}}={D}_{+\dot{+}}^{2}{\mu }_{a}^{I}{\mu }_{{bI}}\in {\hat{{ \mathcal C }}}_{1\left(\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right)},\\ & & {{ \mathcal O }}_{++\dot{+}{ab}}={f}_{{IJK}}{\lambda }_{+}^{1I}{\mu }_{a}^{J}{D}_{+\dot{+}}{\mu }_{b}^{K}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{3}{2}\left(\displaystyle \frac{1}{2},0\right)},\ \ {{ \mathcal O }}_{\dot{+}\dot{+}+{ab}}={f}_{{IJK}}{\tilde{\lambda }}_{2\dot{+}}^{I}{\mu }_{a}^{J}{D}_{+\dot{+}}{\mu }_{b}^{K}\in {\hat{{ \mathcal C }}}_{\displaystyle \frac{3}{2}\left(0,\displaystyle \frac{1}{2}\right)},\\ & & {{ \mathcal O }}_{+\dot{+}{abc}}={f}_{{IJK}}{\mu }_{a}^{I}{\mu }_{b}^{J}{D}_{+\dot{+}}{\mu }_{c}^{K}\in {\hat{{ \mathcal C }}}_{2(\mathrm{0,0})}.\end{eqnarray} \tag{ 5.17 }$$

Naively, we have twenty-six fermionic generators and twenty-seven bosonic generators. However, six bosonic and six fermionic constraints come from applying (4.15) of [5] in each $({A}_{1},{A}_{3})$ sector. In particular, we have

$$\begin{eqnarray} & & {J}_{a+\dot{+}}^{11}{{ \mathcal O }}_{{ab}}={J}_{a+\dot{+}}^{11}\mathrm{Tr}[{\mu }_{a}{\mu }_{b}]=\displaystyle \frac{1}{\sqrt{2}}{f}_{{ABC}}{\mu }_{b}^{A}{\mu }_{a}^{B}{D}_{+\dot{+}}{\mu }_{a}^{C}\sim {{ \mathcal O }}_{+\dot{+}{baa}},\\ & & {J}_{a+\dot{+}}^{11}{{ \mathcal O }}_{+a}={J}_{a+\dot{+}}^{11}\mathrm{Tr}[{\mu }_{a}{\lambda }_{+}^{1}]=\displaystyle \frac{1}{\sqrt{2}}{f}_{{ABC}}{({\lambda }_{+}^{1})}^{A}{\mu }_{a}^{B}{D}_{+\dot{+}}{\mu }_{a}^{C}\sim {{ \mathcal O }}_{++\dot{+}{aa}},\\ & & {J}_{a+\dot{+}}^{11}{{ \mathcal O }}_{\dot{+}a}={J}_{a+\dot{+}}^{11}\mathrm{Tr}[{\mu }_{a}{\tilde{\lambda }}_{2\dot{+}}]=\displaystyle \frac{1}{\sqrt{2}}{f}_{{ABC}}{({\tilde{\lambda }}_{2\dot{+}})}^{A}{\mu }_{a}^{B}{D}_{+\dot{+}}{\mu }_{a}^{C}\sim {{ \mathcal O }}_{\dot{+}\dot{+}+{aa}}.\end{eqnarray} \tag{ 5.18 }$$

Note that the case a = b is not a new constraint (indeed this follows from the Joseph ideal constraints, ${{ \mathcal O }}_{{aa}}=0$). The final three bosonic relations can be seen in the form of the Macdonald index in [5]. In particular, according to table 1 of [5], there is only one singlet of SU(2) at $O({q}^{2}{t}^{2})$. The Joseph ideal constraint eliminates ${\partial }_{+\dot{+}}^{2}({\mu }_{a}^{I}{\mu }_{{aI}})$, which leaves ${D}_{+\dot{+}}^{2}{\mu }_{a}^{I}{\mu }_{{aI}}={{ \mathcal O }}_{\dot{+}\dot{+}++{aa}}$, and ${({J}_{a+\dot{+}}^{11})}^{2}$. Therefore, we must have three relations (one in each $({A}_{1},{A}_{3})$ sector) of the form

$$\begin{eqnarray}&&\kappa {({J}_{a+\dot{+}}^{11})}^{2}={{ \mathcal O }}_{\dot{+}\dot{+}++{aa}},\end{eqnarray} \tag{ 5.19 }$$

where κ is a constant. As a result, we are left with twenty fermionic generators and eighteen bosonic generators. When we turn on the gauge coupling, all the bosonic generators pair up with eighteen fermions via

$$\begin{eqnarray} & & {\hat{{ \mathcal C }}}_{0(\mathrm{1,1})}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(1,\displaystyle \frac{1}{2}\right)}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{2},1\right)}\oplus {\hat{{ \mathcal C }}}_{1\left(\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right)},\\ & & {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(1,\displaystyle \frac{1}{2}\right)}\oplus {\hat{{ \mathcal C }}}_{1(\mathrm{1,0})}\oplus {\hat{{ \mathcal C }}}_{1\left(\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right)}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{3}{2}\left(\displaystyle \frac{1}{2},0\right)},\\ & & {\hat{{ \mathcal C }}}_{\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{2},1\right)}\oplus {\hat{{ \mathcal C }}}_{1(\mathrm{0,1})}\oplus {\hat{{ \mathcal C }}}_{1\left(\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right)}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{3}{2}\left(0,\displaystyle \frac{1}{2}\right)},\\ & & {\hat{{ \mathcal C }}}_{1\left(\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right)}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{3}{2}\left(\displaystyle \frac{1}{2},0\right)}\oplus {\hat{{ \mathcal C }}}_{\displaystyle \frac{3}{2}\left(0,\displaystyle \frac{1}{2}\right)}\oplus {\hat{{ \mathcal C }}}_{2(\mathrm{0,0})}.\end{eqnarray} \tag{ 5.20 }$$

It is easy to check that only one linear combination of the ${{ \mathcal O }}_{++\dot{+}{ab}}$ and one linear combination of the ${{ \mathcal O }}_{\dot{+}\dot{+}+{ab}}$ remain un-lifted. These operators are precisely the generators of type ${\hat{{ \mathcal C }}}_{\tfrac{3}{2}\left(\tfrac{1}{2},0\right)}\oplus {\hat{{ \mathcal C }}}_{\tfrac{3}{2}\left(0,\tfrac{1}{2}\right)}$ we are looking for (they cancel the ${ \mathcal O }({q}^{4})$ contributions from the stress tensor multiplet: ${\partial }_{+\dot{+}}^{2}{J}_{+\dot{+}}^{11}$ and ${({J}_{+\dot{+}}^{11})}^{2}$).