aainstitutetext: Yau Mathematics Science center, Tsinghua University, Beijing,
100084, Chinabbinstitutetext: Department of Mathematics, Tsinghua University,
Beijing, 100084, China

# A study of $\mathcal{N}=1$ SCFT derived from $\mathcal{N}=2$ SCFT: index and
chiral ring

Dan Xie a Wenbin Yan

###### Abstract

One can derive a large class of new $\mathcal{N}=1$ SCFTs by turning on
$\mathcal{N}=1$ preserving deformations for $\mathcal{N}=2$ Argyres-Dougals
theories. In this work, we use $\mathcal{N}=2$ superconformal indices to get
indices of $\mathcal{N}=1$ SCFTs, then use these indices to derive chiral
rings of $\mathcal{N}=1$ SCFTs. For a large class of $\mathcal{N}=2$ theories,
we find that the IR theory contains only free chirals if we deform the parent
${\cal N}=2$ theory using the Coulomb branch operator with smallest scaling
dimension. Our results provide interesting lessons on studies of
$\mathcal{N}=1$ theories, such as $a$-maximization, accidental symmetries,
chiral ring, etc.

##  1 Introduction

The study of 4d $\mathcal{N}=1$ supersymmetric field theory is quite important
for understanding general quantum field theory. On one hand these theories
have richer dynamics than theories with more supersymmetries
Intriligator:1995au , yet on the other hand these theories have minimal
supersymmetry and so one can still get interesting exact results. We are
particularly interested in 4d $\mathcal{N}=1$ superconformal field theory
(SCFT). These theories can be realized as the IR fixed points of
asymptotically free gauge theories with $\mathcal{N}=1$ supersymmetry
Seiberg:1994pq , or obtained by using geometric method such as brane
construction Giveon:1998sr ; Witten:1997ep , 6d $(2,0)$ construction
Xie:2013gma , holography Xie:2019qmw , and etc. These constructions provide us
with lots of interesting $\mathcal{N}=1$ SCFTs, and some properties of them
can be found using the powerful string theoretical methods Witten:1997ep ;
Xie:2013rsa .

It is important to further enlarge the space of $\mathcal{N}=1$ SCFTs. It was
shown in Xie:2019aft  that one can derive a large class of new $\mathcal{N}=1$
SCFTs by turning on $\mathcal{N}=1$ preserving relevant deformations of
$\mathcal{N}=2$ SCFT. The $\mathcal{N}=2$ SCFTs used in Xie:2019aft  are the
so-called strongly coupled Argyres-Douglas theories which consist of a variety
of available deformations, i.e. there are a large number of operators with
scaling dimension $\Delta({\cal O})<3$, which we can use to turn on
$\mathcal{N}=1$ preserving deformation 111See Benini:2009mz  for the
construction of $\mathcal{N}=1$ SCFT by using marginal operators of
$\mathcal{N}=2$ SCFT. . This is in contrast with $\mathcal{N}=2$ Lagrangian
theories or the theories considered in Benini:2009mz , where the only Coulomb
branch operators with dimension less than three is the marginal operator. The
power of the construction studied in Xie:2019aft  is that one can learn
interesting properties about the resulting $\mathcal{N}=1$ SCFTs using results
of $\mathcal{N}=2$ SCFTs, such as the central charges $a,c$, the scaling
dimension of the chiral operators, etc. These properties are often not easy to
compute for $\mathcal{N}=1$ SCFTs.

The purpose of this note is to further study $\mathcal{N}=1$ SCFTs constructed
in Xie:2019aft  by using the $\mathcal{N}=1$ superconformal index (index for
short), which can be derived using known indices of parent $\mathcal{N}=2$
theories. The $\mathcal{N}=1$ index is quite useful to learn the phase
structure of IR $\mathcal{N}=1$ SCFT. The Schur index of a large class of
$\mathcal{N}=2$ AD theories was found for Cordova:2015nma ; Song:2017oew ;
Xie:2019zlb , and recently even the full index has been found for some AD
theories Maruyoshi:2016tqk ; Maruyoshi:2016aim ; Agarwal:2016pjo ;
Agarwal:2018ejn . We will show that it is straightforward to get IR
$\mathcal{N}=1$ index once the UV $\mathcal{N}=2$ index is known.

Once we determine the IR $\mathcal{N}=1$ index, we can use it to probe the
phase structure of IR theory, and we find the following results:

  1. 1.

If the IR theory is an interacting SCFT, one can get the information on the
chiral ring from the single letter index. This confirms the conjecture
proposed in Buican:2016hnq ; Xie:2016hny  for the simplest case.

  2. 2.

In many cases, the IR theory contains just free chirals. For example, if we
turn on the deformation using the $\mathcal{N}=2$ Coulomb branch operators
with the smallest scaling dimension of the $(A_{1},A_{2N})$ theory, the
resulting IR theory is just a free chiral! We found a large class of RG flows
such that the IR $\mathcal{N}=1$ theories consist of just free chirals.

  3. 3.

We also have the intermediate cases where the IR theory consists of free
chirals plus an interacting theory. We find that often the IR theory may have
accidental symmetry.

This paper is organized as follows: in section 2 we review the $\mathcal{N}=1$
preserving deformation of $\mathcal{N}=2$ SCFT, and the way to derive
$\mathcal{N}=1$ index from the index of the parent $\mathcal{N}=2$ theory.
Section 3 discusses information on $\mathcal{N}=1$ chiral ring extracted from
the index; Section 4 discusses the case where IR theory is just free chirals;
Section 5 discusses other interesting phenomenon such as the decoupled sector
and accidental symmetry; finally a conclusion is given in section 6.

##  2  ${\cal N}=1$ index from ${\cal N}=2$ index

###  2.1 $\mathcal{N}=1$ SCFT from $\mathcal{N}=2$ SCFT

We first review some representation theory results of 4d $\mathcal{N}=2$
superconformal algebra. The bosonic symmetry group of a general
$\mathcal{N}=2$ SCFT is $SO(2,4)\times SU(2)_{R}\times U(1)_{r}\times G_{F}$,
where $SO(2,4)$ is the conformal group, $SU(2)_{R}\times U(1)_{r}$ is the $R$
symmetry group which exists for every $\mathcal{N}=2$ SCFT, and $G_{F}$ are
other global symmetry groups commuting with the superconformal group which
could be absent for some theories. A highest weight representation of the
${\cal N}=2$ superconformal group is labelled by
$|\Delta,R,r,j_{1},j_{2}\rangle$, where $\Delta$ is the scaling (conformal)
dimension, $r$ is $U(1)_{r}$ charge, $R$ is $SU(2)_{R}$ spin, $j_{1}$ and
$j_{2}$ are left and right spin. These states could also carry quantum numbers
of flavor symmetry group $G_{F}$. Representation theory of $\mathcal{N}=2$
superconformal group in $3+1$d has been studied in Dolan:2002zh . and short
representations are completely classified in Dolan:2002zh . Three short
representations that we are interested in are ${\cal E}_{r,(0,0)}$ multiplets
containing Coulomb branch operators, $\hat{\cal B}_{R}$ multiplets containing
Higgs branch operators, and supercurrent multiplet $\hat{\cal C}_{0,(0,0)}$.
Their respective BPS conditions are listed below.

$\begin{split}&\text{Coulomb~{}branch operators}:~{}~{}~{}~{}{\bar{\cal
E}}_{r,(0,0)},~{}~{}R=0,~{}~{}\Delta=r,\\\ &\text{Higgs~{}branch
operators}:~{}~{}~{}~{}~{}~{}~{}~{}\hat{{\cal
B}}_{R},~{}~{}~{}~{}~{}~{}~{}r=j_{1}=j_{2}=0,~{}~{}\Delta=2R,\\\
&\text{Supercurrent}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{{\cal
C}}_{0,(0,0)},~{}~{}r=R=0,~{}~{}\Delta=2.\\\ \end{split}$

For example, if the theory has nontrivial flavor symmetry, $\hat{{\cal
B}}_{1}$ multiplets contains conserved currents for the flavor symmetry group
$G_{F}$, and transforms in the adjoint representation of $G_{F}$. The
$\mathcal{N}=1$ subalgebra is generated by the supercharge $Q_{1\alpha}$, and
the corresponding generator of the ${\cal N}=1$ $R$ symmetry is
$\hat{R}_{\mathcal{N}=1}={2\over 3}\hat{r}+{4\over 3}\hat{I}_{3}$ 333Here
$\hat{r}$ is the generator for $\mathcal{N}=2$ $U(1)_{r}$ symmetry, and
$\hat{I}_{3}$ is the generator of the Cartan subalgebra of Lie algebra
associated with $SU(2)_{R}$ symmetry. In our convention a letter with hat
means the generator and the same letter without hat means the eigenvalue of
the generator.. The other global symmetry group in $\mathcal{N}=1$ description
is generated by $\hat{J}=2\hat{r}-2\hat{I}_{3}$ which commutes with the
supercharge $Q_{1\alpha}$ 444In our notation $(Q_{1\alpha},Q_{2\alpha})$ are
spin $SU(2)$ doublet with $I_{3}(Q_{1})=-{1\over 2},I_{3}(Q_{2})={1\over 2}$,
and $U(1)_{r}$ charges are $R(Q_{1})=R(Q_{2})=-{1\over 2}$..

One can engineer a large class of 4d $\mathcal{N}=2$ SCFTs by using either 6d
$(2,0)$ construction Gaiotto:2009we ; Xie:2012hs  or Type IIB string theory on
three-fold singularity Xie:2015rpa . The great advantage of these
constructions are that one can naturally determine the Coulomb branch spectrum
and the Higgs branch structure of these theories.

The soft breaking terms of 4d $\mathcal{N}=2$ SCFT are classified in
Xie:2019aft . In this paper, we are interested in $\mathcal{N}=1$ preserving
deformation by using Coulomb branch operators which form a freely generated
chiral ring with finitely many generators obtained from the Seiberg-Witten
(SW) geometry in ${\cal N}=2$ SCFTs. The $\mathcal{N}=2$ $\bar{\cal
E}_{r(0,0)}$ multiplet containing the Coulomb branch operator consists of
three $\mathcal{N}=1$ chiral multiplets $({\cal O},\lambda_{\alpha},S)$, and
only multiplet ${\cal O}$ 555When there is no ambiguity, we will use the same
symbol to denote the highest weight primary operator, its corresponding
multiplet and superfield to avoid clutter. will give new $\mathcal{N}=1$
preserving deformations. If we start with a $\bar{\cal E}_{r_{0},(0,0)}$
multiplet with $U(1)_{r}$ charge $r_{0}$ of ${\cal N}=2$ theory ${\cal
T}^{{\cal N}=2}$, we can use its buttom $\mathcal{N}=1$ chiral multiplet
${\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}$ to deform our theory,

$\delta S=\lambda\int d^{2}\theta{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}+c.c$
(1)

The resulting IR $\mathcal{N}=1$ SCFT will be denoted as ${\cal T}^{{\cal
T}^{{\cal N}=2}}[{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}]$ to indicate its
${\cal N}=2$ parent theory and the deformation or as ${\cal T}[{\cal
O}_{\bar{\cal E}_{r_{0},(0,0)}}]$ when the parent theory is apparent. The
candidate generator of the $U(1)_{R}$ symmetry for the IR theory ${\cal
T}[{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}]$ is

$\hat{R}_{IR}={2\over r_{0}}\hat{r}+({2-{2\over r_{0}}})\hat{I}_{3}.$ (2)

We have to emphasize that the IR theory ${\cal T}[{\cal O}_{\bar{\cal
E}_{r_{0},(0,0)}}]$ might have accidental symmetry so that the correct
generator of the $U(1)_{R}$ symmetry would be different from the above one,
and one can find the true generator of the IR $U(1)_{R}$ symmetry using the
$a$-maximization procedure Intriligator:2003jj .

Some key properties of ${\cal T}[{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}]$ can
be found as follows (consider only Coulomb type deformations 1 and assume that
the candidate generator 2 is the true generator of the IR $U(1)_{R}$ symmetry
of ${\cal T}[{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}]$):

  1. 1.

Central charges: One can compute the IR central charge as follows:

$\begin{split}&a_{N=1}=(a_{N=2}-c_{N=2})[{27\over 16}x^{3}-{9\over
4}x]+(2a_{N=2}-c_{N=2})[{27\over 32}xy^{2}],\\\
&c_{N=1}=(a_{N=2}-c_{N=2})[{27\over 16}x^{3}-{15\over
4}x]+(2a_{N=2}-c_{N=2})[{27\over 32}xy^{2}].\end{split}$

Here $x={2\over r_{0}},y=2-{2\over r_{0}}$.

  2. 2.

Index: The explicit form of $\mathcal{N}=2$ Schur index of many interesting
theories is known Cordova:2015nma ; Song:2017oew ; Xie:2019zlb , and this
index is actually invariant under RG flow, and can be used to get some useful
information of IR theory Buican:2016hnq . Details are discussed in the next
subsection.

  3. 3.

Chiral ring: The $\mathcal{N}=1$ chiral operator ${\cal O}_{\bar{\cal
E}_{r_{0},(0,0)}}$ which is used to deform our theory satisfies a chiral ring
relation ${\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}=0$ in the IR theory
Xie:2016hny ; Buican:2016hnq .

  4. 4.

Chiral spectrum: One can get some information of chiral operators of ${\cal
T}[{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}]$ from parent $\mathcal{N}=2$ theory.
For an $\mathcal{N}=2$ chiral operator $\bar{\cal E}_{r,(0,0)}$ with
$U(1)_{r}$ charge $r$, the scaling dimension of its three $\mathcal{N}=1$
chiral multiplets after deformation using ${\cal O}_{\bar{\cal
E}_{r_{0},(0,0)}}$ are

$[{\cal O}_{\bar{\cal E}_{r,(0,0)}}]={3r\over
r_{0}},~{}~{}[\lambda_{\alpha}^{\bar{\cal E}_{r,(0,0)}}]={{3\over
2}r_{0}-3+3r\over r_{0}},~{}~{}~{}[S_{\bar{\cal
E}_{r,(0,0)}}]={3(r_{0}+r-2)\over r_{0}}.$ (3)

If $2<r_{0}<3$, the minimal scaling dimension of a chiral scalar operator is
$\Delta_{min}={3r_{min}\over r_{0}}$; if $1<r_{0}\leq 2$, we have
$\Delta_{min}={3(r_{0}+r_{min}-2)\over r_{0}}$. In particular, some of these
chiral operators are relevant, and one can use them to further deform ${\cal
T}[{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}]$ and flow to possibly new
$\mathcal{N}=1$ SCFTs, although these deformations would break the ${\cal
N}=1$ $U(1)_{R}$ symmetry, and we have little to say about IR theory. The
flavor symmetry of UV $\mathcal{N}=2$ theory is not broken by the Coulomb
branch type $\mathcal{N}=1$ preserving deformations 1, so we do know the
existence of $\hat{\cal C}_{(0,0)}$ type multiplets in ${\cal T}[{\cal
O}_{\bar{\cal E}_{r_{0},(0,0)}}]$ from the ${\cal N}=2$ multiplets containing
the flavor symmetry current of the UV theory. If $\mathcal{N}=2$ theory also
have $\hat{B}_{R}$ type operators, one also get chiral operators $X_{R}$ in
${\cal T}[{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}]$ with scaling dimension
${3\over 2}(2-{2\over r_{0}})R$. If there is operator with $R=1$, then the
operator with smallest scaling dimension in the IR theory ${\cal T}[{\cal
O}_{\bar{\cal E}_{r_{0},(0,0)}}]$ would be $X_{1}$ with scaling dimension
$3-{3\over r_{0}}$ for $1<r_{0}\leq 2$.

  5. 5.

Exact marginal deformations: Firstly If the UV $\mathcal{N}=2$ theory has a
multiple number of Coulomb branch operators with the same $U(1)_{r}$ charge
$r_{0}$, the IR theory ${\cal T}[{\cal O}_{{\cal E}_{r_{0},(0,0)}}]$ might
have exact marginal deformations. Secondly if the UV $\mathcal{N}=2$ theory
has a $\mathcal{N}=2$ exact marginal operator, the $S$ component of it would
be exact marginal in the IR $\mathcal{N}=1$ SCFT. Finally if $r_{0}=2$, then
the $\hat{{\cal B}}_{2}$ type operator might also give exact marginal
deformations of ${\cal T}[{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}]$.

  6. 6.

Inherited $\mathcal{N}=1$ duality: If the 4d $\mathcal{N}=2$ theory has an
exact marginal deformation and has different duality frames, the IR theory
would also have different duality frames. There are many $\mathcal{N}=2$
theories whose duality frames are known Xie:2016uqq ; Xie:2017vaf ;
Xie:2017aqx , and using the $\mathcal{N}=1$ preserving deformations, we get a
large class of new Seiberg-like duality for $\mathcal{N}=1$ SCFTs.

###  2.2 The ${\cal N}=2$ superconformal index and its Schur limit

As we discussed in last subsection, one can get the index of IR
$\mathcal{N}=1$ SCFT by using the known result of $\mathcal{N}=2$ index. Here
we review the relation between the indexes of UV $\mathcal{N}=2$ and IR
$\mathcal{N}=1$ theory. The ${\cal N}=2$ superconformal index Kinney:2005ej ;
Romelsberger:2005eg  with respect to the $SU(2,2|2)$ supercharge
$Q_{1\dot{-}}$ is defined as

${\cal I}^{{\cal N}=2}={\rm
Tr}(-1)^{F}p^{j_{1}+j_{2}+r}q^{-j_{1}+j_{2}+r}t^{R-r}e^{-\beta\delta_{1\dot{-}}},$
(4)

where
$\delta_{1\dot{-}}=2\\{\tilde{Q}_{1\dot{-}},\tilde{Q}_{1\dot{-}}^{\dagger}\\}=\Delta-2j_{2}-2R-r$.
$\Delta$, $j_{1}$, $j_{2}$, $R$, $r$ are conformal dimension, two Lorentz
spins, $SU(2)_{R}$ and $U(1)_{r}$ respectively which are Cartans of the
$SU(2,2|2)$. Notice that our $r$ is $-r$ in Gadde:2011uv  so that the BPS
condition of ${\cal N}=2$ chiral multiplet $\bar{\cal E}_{r(j_{1},0)}$ is
$\Delta=r$666We call $\bar{\cal E}_{r(j_{1},0)}$ chiral multiplet because its
highest weight state is annihilated by $\tilde{Q}_{i\dot{\alpha}}$..
Supersymmetry ensures that the trace is only over BPS states satisfying
$\delta_{1\dot{-}}=0$, i.e. state satisfying $\Delta-2j_{2}-2R-r=0$. Note that
the highest weight primary operators of $\hat{{\cal B}}_{R}$ and $\bar{\cal
E}_{r(j_{1},0)}$ multiplets and other deformations used in this paper preserve
$\tilde{Q}_{1\dot{-}}$ supercharge, hence they all contribute to the index.
Information on ${\cal N}=2$ BPS multiplets and their superconformal indices
are summarized in appendix A.

Unfortunately the full superconformal indices of a large portion of ${\cal
N}=2$ theories are unknown yet, however, in many such theories certain limits
of the full indices can still be computed exactlyCordova:2015nma ;
Song:2015wta ; Song:2016yfd ; Buican:2017uka ; Song:2017oew ; Xie:2019zlb .
One of such example is the Schur index which is the full index at $t=q$ limit.
In this limit, only BPS operators in Schur sector which are annihilated by
both $Q_{1+}$ and $\tilde{Q}_{1\dot{-}}$ (i.e. operators satisfy
$\delta_{1+}=\Delta-2j_{1}-2R+r=0$ and
$\delta_{1\dot{-}}=\Delta-2j_{2}-2R-r=0$) contribute to the index, and the
dependence on $p$ is automatically dropped out. The $\bar{\cal
E}_{r}=\bar{\cal E}_{r(0,0)}$ multiplets containing Coulomb branch operators
do not contribute in the Schur limit, while the $\hat{{\cal B}}_{R}$
multiplets still contribute. Multiplets which contribute to the Schur index
are also summarized in appendix A.

####  2.2.1 The ${\cal N}=1$ index from ${\cal N}=2$ index

Once we deform the theory with $\lambda\int d^{2}\theta{\cal O}_{\bar{\cal
E}_{r_{0},(0,0)}}+c.c$ which preserves the $\tilde{Q}_{1\dot{-}}$ supercharge,
the infrared theory has only ${\cal N}=1$ superconformal symmetry. One can
still define the index with respect to $\tilde{Q}_{1\dot{-}}$, but only the IR
$U(1)_{R}$ symmetry generated by
$\hat{R}_{IR}=\frac{2}{r_{0}}\hat{r}+\left(2-\frac{2}{r_{0}}\right)\hat{I}_{3}$
is preserved in the IR fixed point. If we substitute
$t=(pq)^{1-\frac{1}{r_{0}}}$ in the definition ${\cal N}=2$ index 4, we get

${\rm
Tr}(-1)^{F}p^{j_{1}+j_{2}+\frac{R_{IR}}{2}}q^{-j_{1}+j_{2}+\frac{R_{IR}}{2}}e^{-\beta\delta_{1\dot{-}}},$
(5)

which is exactly the definition of ${\cal N}=1$ superconformal index
Kinney:2005ej ; Romelsberger:2005eg  with respect to $Q_{1\dot{-}}$. This
substitution is a generalization of the procedure in Gadde:2010en .

Our strategy to get the ${\cal N}=1$ index of the IR theory is therefore
working out the ${\cal N}=2$ index for the theory we would like to deform
first, then pick the ${\cal N}=1$ preserving deformation and make the
corresponding substitution in $t$ fugacity and obtain the ${\cal N}=1$ index
of the IR theory. Once we get the ${\cal N}=1$ index, we can obtain the
information on the IR theory.

In Buican:2016hnq  authors proposed another limit to deal with theories of
which only the Schur index is known. They took the limit $t=q$ and
$p=q^{\frac{1}{r_{0}-1}}$ in the full index 4,

${\cal I}_{S}^{{\cal N}=1}={\cal I}^{{\cal
N}=2}\left(p=q^{\frac{1}{r_{0}-1}},q,t=q\right)={\rm
Tr}(-1)^{F}q^{-\frac{r_{0}-2}{r_{0}-1}j_{1}+\frac{r_{0}}{r_{0}-1}j_{2}+\frac{r_{0}}{2(r_{0}-1)}R_{IR}}e^{-\beta\delta_{1\dot{-}}}.$
(6)

Since the index under $t=q$ limit (which is the Schur limit) is independent of
$p$, ${\cal I}^{{\cal N}=1}_{S}$ should be the same as the Schur index of the
UV ${\cal N}=2$ theory. On the other hand, since the deformation preserve the
supercharge $Q_{1-}$, ${\cal I}^{{\cal N}=1}_{S}$ is a good index for the IR
${\cal N}=1$ theory albeit the fugacity $q$ labels an unconventional
combination of the Cartans of the ${\cal N}=1$ superconformal algebra.
Although the $\bar{\cal E}_{r,(0,0)}$ type multiplets which represent Coulomb
branch operators do not contribute to the Schur index, some valuable
information on their relations in the deformed theory can still be extracted
from ${\cal I}^{{\cal N}=1}_{S}(q)={\cal I}^{{\cal N}=2}_{Schur}(q)$.

##  3 Interacting $\mathcal{N}=1$ SCFTs and their chiral ring

In this section, we will use the known result of full $\mathcal{N}=2$ index to
get the full index of the IR $\mathcal{N}=1$ theory after deformation. The
full index for $\mathcal{N}=2$ $(A_{1},A_{k})$ and $(A_{1},D_{k})$ theories
have been worked in Maruyoshi:2016tqk ; Maruyoshi:2016aim ; Agarwal:2016pjo .
Using the proposal in last section, we can find the full index for the
$\mathcal{N}=1$ SCFT derived from turning on the $\mathcal{N}=1$ preserving
relevant deformation, then one can further derive some part of $\mathcal{N}=1$
chiral ring.

###  3.1 $(A_{1},A_{2N})$ results

####  3.1.1 ${\cal N}=2$ index

The full index of $(A_{1},A_{2N})$ AD theories are given in Maruyoshi:2016tqk
; Maruyoshi:2016aim ; Agarwal:2016pjo  as an integration over elliptic Gamma
functions

$\begin{split}{\cal I}^{(A_{1},A_{2N})}_{{\cal
N}=2}=&\left[\prod_{i=1}^{N}\frac{\Gamma\left(\left(\frac{pq}{t}\right)^{\frac{2(N+i+1)}{2N+3}}\right)}{\Gamma\left(\left(\frac{pq}{t}\right)^{\frac{2i}{2N+3}}\right)}\right]\Gamma\left(\left(\frac{pq}{t}\right)^{\frac{1}{2N+3}}\right)^{N}\\\
&\times\frac{\kappa^{N}}{2^{N}N!}\oint[d\mathbf{z}]\prod_{\alpha\in\Delta}\frac{\Gamma\left(\mathbf{z}^{\alpha}\left(\frac{pq}{t}\right)^{\frac{1}{2N+3}}\right)}{\Gamma(\mathbf{z}^{\alpha})}\prod_{w\in
R}\Gamma\left(\mathbf{z}^{w}\left(\frac{pq}{t}\right)^{\frac{N+1}{2N+3}}t^{\frac{1}{2}}\right)\Gamma\left(\mathbf{z}^{w}\left(\frac{pq}{t}\right)^{\frac{-N}{2N+3}}t^{\frac{1}{2}}\right),\end{split}$
(7)

where $[d\mathbf{z}]=\prod_{i=1}^{N}\frac{dz_{i}}{2\pi iz_{i}}$, $\Delta$ is
the set of roots of $Sp(N)$ and $R$ is the set of weights of the fundamental
representation of $Sp(N)$. The properties of elliptic Gamma function
$\Gamma(z;p,q)$ are summarized in appendix C.

To better read off the generators and relations, we write the indices in the
form of the Plethystic exponential of single letter indices

${\cal I}=\mathrm{PE}\left[{\cal I}^{s.l.}\right],$ (8)

where $\mathrm{PE}\left[f(a,b,c,\cdots)\right]$ is the Plethystic exponential

$\mathrm{PE}\left[f(a,b,c,\cdots)\right]=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}f(a^{n},b^{n},c^{n},\cdots)\right).$
(9)

${\cal I}^{s.l.}$ contains all basic building blocks and their relations in
the spectrum, then the Plethystic exponential automatically take care of the
products of these operators according to correct statistics.

With the integral formula, the expansions of single letter indices up to the
first few orders when $N=1,2,3$ are

${\cal I}^{s.l.}_{(A_{1},A_{2})}={\cal I}_{\bar{{\cal E}}_{\frac{6}{5}}}+{\cal
I}_{\hat{{\cal C}}_{0(0,0)}}-{\cal I}_{\bar{{\cal
B}}_{1,\frac{7}{5}(0,0)}}+\cdots,$ (10) ${\cal I}^{s.l.}_{(A_{1},A_{4})}={\cal
I}_{\bar{{\cal E}}_{\frac{8}{7}}}+{\cal I}_{\bar{{\cal
E}}_{\frac{10}{7}}}+{\cal I}_{\hat{{\cal C}}_{0(0,0)}}-{\cal I}_{\bar{{\cal
B}}_{1,\frac{9}{7}(0,0)}}-{\cal I}_{\bar{{\cal
B}}_{1,\frac{11}{7}(0,0)}}-{\cal I}_{\bar{{\cal
B}}_{1,\frac{13}{7}(0,0)}}+\cdots,$ (11)

and

$\begin{split}{\cal I}^{s.l.}_{(A_{1},A_{6})}=&{\cal I}_{\bar{{\cal
E}}_{\frac{10}{9}}}+{\cal I}_{\bar{{\cal E}}_{\frac{4}{3}}}+{\cal
I}_{\bar{{\cal E}}_{\frac{14}{9}}}+{\cal I}_{\hat{{\cal C}}_{0,(0,0)}}\\\
&-{\cal I}_{\bar{{\cal B}}_{1,\frac{11}{9}(0,0)}}-{\cal I}_{\bar{{\cal
B}}_{1,\frac{13}{9}(0,0)}}-{\cal I}_{\bar{{\cal
B}}_{1,\frac{5}{3}(0,0)}}-{\cal I}_{\bar{{\cal
B}}_{1,\frac{17}{9}(0,0)}}-{\cal I}_{\bar{{\cal
B}}_{1,\frac{19}{9}(0,0)}}+\cdots.\end{split}$ (12)

Here ${\cal I}_{{\cal M}}$ is the index of the ${\cal N}=2$ multiplet ${\cal
M}$ with ${\cal M}$ being ${\cal I}_{\hat{{\cal C}}_{0(0,0)}}$, $\bar{{\cal
E}}_{r}$777We denote $\bar{\cal E}_{r(0,0)}$ as $\bar{\cal E}_{r}$ for short.,
or $\bar{{\cal B}}_{1,r(0,0)}$,

$\begin{split}{\cal I}_{\hat{{\cal
C}}_{0(0,0)}}&=\frac{(t-pq)(p+q-pq/t)}{(1-p)(1-q)},\\\ {\cal I}_{\bar{{\cal
E}}_{r}}&=\left(\frac{pq}{t}\right)^{r}\frac{\left(1-\frac{t}{p}\right)\left(1-\frac{t}{q}\right)}{(1-p)(1-q)},\\\
{\cal I}_{\bar{{\cal
B}}_{1,r(0,0)}}&=\left(\frac{pq}{t}\right)^{r}\frac{(t-pq)\left(1-\frac{t}{p}\right)\left(1-\frac{t}{q}\right)}{(1-p)(1-q)}.\end{split}$
(13)

And in general we conjecture the single letter index of $(A_{1},A_{2N})$
theory always starts with

${\cal I}^{s.l.}_{(A_{1},A_{2N})}=\sum_{i=1}^{N}{\cal I}_{\bar{{\cal
E}}_{\frac{2N+2i+2}{2N+3}}}+{\cal I}_{\hat{{\cal
C}}_{0(0,0)}}-\sum_{i=1}^{2N-1}{\cal I}_{\bar{{\cal
B}}_{1,\frac{2N+2i+3}{2N+3}(0,0)}}+\cdots.$ (14)

From the single letter index one immediately sees that at lower conformal
dimension, the spectrum of $(A_{1},A_{2N})$ AD theories contains the stress
tensor multiplet $\hat{{\cal C}}_{0(0,0)}$ and $N$ $\bar{{\cal E}}_{r_{i}}$
type multiplets with $r_{i}=\frac{2N+2i+2}{2N+3}$ and $1\leq i\leq N$, which
are $N$ generators of the Coulomb branch chiral ring of the theory. One can
also read off relations of these operators as well. Superconformal
representation theory tells us that the tensor product of two $\bar{{\cal
E}}$-type multiplets $\bar{{\cal E}}_{r_{1}}$ and $\bar{{\cal E}}_{r_{2}}$
gives

$\bar{{\cal E}}_{r_{1}}\otimes\bar{{\cal E}}_{r_{2}}\simeq\bar{{\cal
E}}_{r_{1}+r_{2}}\oplus\bar{{\cal B}}_{1,r_{1}+r_{2}-1(0,0)}+\cdots.$ (15)

In terms of index, this means

${\cal I}_{\bar{{\cal E}}_{r_{1}}}\times{\cal I}_{\bar{{\cal
E}}_{r_{2}}}={\cal I}_{\bar{{\cal E}}_{r_{1}+r_{2}}}+{\cal I}_{\bar{{\cal
B}}_{1,r_{1}+r_{2}-1(0,0)}}+\cdots,$ (16)

In the full index, if one expand in terms of conformal dimension, one has

$\begin{split}\mathrm{PE}\left[{\cal I}_{\bar{\cal E}_{r_{1}}}+{\cal
I}_{\bar{\cal E}_{r_{2}}}\right]&=1+{\cal I}_{\bar{\cal E}_{r_{1}}}+{\cal
I}_{\bar{\cal E}_{r_{2}}}+{\cal I}_{\bar{\cal E}_{r_{1}}\otimes\bar{\cal
E}_{r_{2}}}+\cdots,\\\ &=1+{\cal I}_{\bar{\cal E}_{r_{1}}}+{\cal I}_{\bar{\cal
E}_{r_{2}}}+{\cal I}_{\bar{\cal E}_{r_{1}+r_{2}}}+{\cal I}_{\bar{\cal
B}_{1,r_{1}+r_{2}-1(0,0)}}+\cdots,\end{split}$ (17)

then a $-{\cal I}_{\bar{\cal B}_{1,r_{1}+r_{2}-1(0,0)}}$ term in the single
letter index will cancel the ${\cal I}_{\bar{\cal B}_{1,r_{1}+r_{2}-1(0,0)}}$
contribution from tensor product $\bar{\cal E}_{r_{1}}\otimes\bar{\cal
E}_{r_{2}}$ in the full index, leaving only

$\mathrm{PE}\left[{\cal I}_{\bar{\cal E}_{r_{1}}}+{\cal I}_{\bar{\cal
E}_{r_{2}}}-{\cal I}_{\bar{\cal B}_{1,r_{1}+r_{2}-1(0,0)}}\right]=1+{\cal
I}_{\bar{\cal E}_{r_{1}}}+{\cal I}_{\bar{\cal E}_{r_{2}}}+{\cal I}_{\bar{\cal
E}_{r_{1}+r_{2}}}+\cdots,$ (18)

hence the $-{\cal I}_{\bar{\cal B}_{1,r_{1}+r_{2}-1(0,0)}}$ term in the single
letter index implies the relation of operators

$\left(\bar{\cal E}_{r_{1}}\otimes\bar{\cal E}_{r_{2}}\right)_{\bar{\cal
B}_{1,r_{1}+r_{2}-1(0,0)}}=0,$ (19)

where $(\cdots)_{{\cal M}}$ means picking only the ${\cal M}$ multiplet
appeared in the $\cdots$. It is important to notice that each distinct value
of $r_{i}+r_{j}-1$ appears only once in the $-\sum_{i=1}^{2N-1}{\cal
I}_{\bar{{\cal B}}_{1,\frac{2N+2i+3}{2N+3}(0,0)}}$ term of the single letter
index 14 when $r_{i}$ and $r_{j}$ run over all possible $U(1)_{r}$ charges of
the Coulomb branch operators. Therefore if there are two pairs of chiral
multiplets $(\bar{\cal E}_{r_{i}},\bar{\cal E}_{r_{j}})$ and $(\bar{\cal
E}_{r_{k}},\bar{\cal E}_{r_{l}})$ satisfying $r_{i}+r_{j}=r_{k}+r_{l}$, there
is only one relation

$\left(\bar{\cal E}_{r_{i}}\otimes\bar{\cal E}_{r_{j}}\oplus\bar{\cal
E}_{r_{k}}\otimes\bar{\cal E}_{r_{l}}\right)_{\bar{\cal
B}_{1,r_{i}+r_{j}-1(0,0)}}=0.$ (20)

The same is true for multiple pairs of chiral multiplets with the same total
$U(1)_{r}$ charge, in the end there are $2N-1$ relations for $\bar{\cal
E}_{r}$ type multiplets of $(A_{1},A_{2N})$ for $2N-1$ $-{\cal I}_{\bar{\cal
B}_{1,r_{1}+r_{2}-1(0,0)}}$ terms in the single letter index,

$\sum_{r^{\prime}\in{\cal A},r+1-r^{\prime}\in{\cal A}}\left(\bar{\cal
E}_{r^{\prime}}\otimes\bar{\cal E}_{r+1-r^{\prime}}\right)_{\bar{\cal
B}_{1,r(0,0)}}=0,\,\,\,\,\mathrm{for}\,\
r=\frac{2N+5}{2N+3},\,\frac{2N+7}{2N+3},\cdots,\frac{4N+1}{2N+3},$ (21)

where ${\cal
A}=\\{\frac{2N+4}{2N+3},\frac{2N+6}{2N+3},\cdots,\frac{4N+2}{2N+3}\\}$ is the
set of $U(1)_{r}$ charges of $(A_{1},A_{2N})$ theories.

####  3.1.2 ${\cal N}=1$ index and chiral ring

One $\bar{\cal E}_{r}$ multiplet splits into three ${\cal N}=1$ chiral
multiplets whose primary operators are ${\cal O}_{\bar{\cal E}_{r}}$,
$\lambda_{\bar{\cal E}_{r},\alpha}$ and $S_{\bar{\cal E}_{r}}$
respectively888Again we use the same symbol to denote both the multiplet and
its primary operator to avoid complications in notations. Let $\\{\bar{\cal
E}_{r_{1}},\cdots,\bar{\cal E}_{r_{l}}\\}$ be generators of the Coulomb branch
chiral ring of an AD theory, then the number of generators $l$ is also called
the rank of this AD theory. In the following, we will further denote the
primary operators of these multiplets $\\{u_{i}|1\leq i\leq l\\}$ to separate
them from ${\cal O}_{r}$ of a generic $\bar{\cal E}_{r(0,0)}$, and order them
from smallest to largest conformal dimension (i.e. $u_{1}$ is the operator
with the smallest conformal dimension). According to the previous section,
this theory admits relevant ${\cal N}=1$ preserving deformation $\lambda\int
d^{2}\theta{\cal O}_{\bar{\cal E}_{r_{0},(0,0)}}+c.c$ where ${\cal O}$ can be
chosen from $\\{u_{i},u_{i}u_{j}|1\leq i\leq j\leq l\\}$ or any linear
combinations of operators in this set with the same quantum number because all
generators of the Coulomb branch chiral ring has scaling dimension greater
than $1$ and relevant deformation requires the dimension of ${\cal
O}_{\bar{\cal E}_{r_{0}(0,0)}}$ smaller than $3$.

To get the index of the IR theory, we first substitute
$t=(pq)^{1-\frac{1}{r_{0}}}$ in ${\cal N}=2$ single letter indices, then one
can read off the spectrum and constraints at lower dimension, together with
other interesting informations on the IR theory.

${\cal N}=2$ multiplets as ${\cal N}=1$ multiplets: We fist study how various
${\cal N}=1$ multiplets arises from ${\cal N}=2$ multiplets under the
deformation $\lambda\int d^{2}\theta{\cal O}_{\bar{\cal
E}_{r_{0},(0,0)}}+c.c$. A summary of ${\cal N}=1$ BPS multiplets and their
indices are given in appendix B.

The ${\cal N}=2$ $\bar{\cal E}_{r}$ type multiplets decompose into three
${\cal N}=1$ chiral multiplets, and its index of ${\cal I}_{\bar{\cal E}_{r}}$
under the deformation is

$\begin{split}{\cal I}_{\bar{\cal
E}_{r}}\left(p,q,t=(pq)^{1-\frac{1}{r_{0}}}\right)=&\frac{(pq)^{\frac{r}{r_{0}}}}{(1-p)(1-q)}-\frac{(pq)^{\frac{r-1}{r_{0}}+\frac{1}{2}}\left(\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}\right)}{(1-p)(1-q)}+\frac{(pq)^{\frac{r+r_{0}-2}{r_{0}}}}{(1-p)(1-q)}\\\
=&{\cal I}_{\bar{\cal B}_{\frac{2r}{r_{0}}(0,0)}}+{\cal I}_{\bar{\cal
B}_{\frac{2r+r_{0}-2}{r_{0}}({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal
B}_{\frac{2r+2r_{0}-4}{r_{0}}(0,0)}},\end{split}$ (22)

where ${\cal I}_{\bar{\cal
B}_{r(j_{1},0)}}=(-1)^{2j_{1}}\frac{(pq)^{\frac{r}{2}}\chi_{j_{1}}\left(\sqrt{\frac{p}{q}}\right)}{(1-p)(1-q)}$
is the index of chiral multiplet $\bar{\cal B}_{r(j_{1},0)}$999We call
$\bar{\cal B}_{r(j_{1},0)}$ an ${\cal N}=1$ chiral multiplet because its
highest weight state is annihilated by $\tilde{Q}_{\dot{\alpha}}$. and
$\chi_{j}$ is the character of the spin-$j$ representation of $SU(2)$.
Therefore an $\bar{\cal E}_{r_{i}}$ type multiplet after deformation by ${\cal
O}$ decomposes into chiral multiplets ${\cal O}_{i}$, $\lambda_{i,\alpha}$,
$S_{i}$ with ${\cal N}=1$ $R_{IR}$-charge $\frac{2r}{r_{0}}$,
$1+\frac{2(r-1)}{r_{0}}$ and $2+\frac{2(r-2)}{r_{0}}$ respectively.

The index of $\hat{{\cal C}}_{0(0,0)}$ multiplet under the deformation is

$\begin{split}{\cal I}_{\hat{{\cal
C}}_{0(0,0)}}\left(p,q,(pq)^{1-\frac{1}{r_{0}}}\right)=&\frac{-pq-
pq(p+q)+(pq)^{\frac{3}{2}-\frac{1}{r_{0}}}\left(\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}\right)+(pq)^{1+\frac{1}{r_{0}}}}{(1-p)(1-q)}\\\
=&{\cal I}_{\hat{{\cal C}}_{(0,0)}}+{\cal I}_{\hat{{\cal
C}}_{({\frac{1}{2}},{\frac{1}{2}})}}+{\cal I}_{\bar{\cal
C}_{1-\frac{2}{r_{0}}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{3-\frac{2}{r_{0}}({\frac{1}{2}},0)}}.\end{split}$ (23)

Here $\hat{{\cal C}}_{({\frac{1}{2}},{\frac{1}{2}})}$ is the ${\cal N}=1$
stress tensor multiplet, $\bar{\cal C}_{1-\frac{2}{r_{0}}({\frac{1}{2}},0)}$
is a $1/4$-BPS multiplet. $\bar{\cal B}_{3-\frac{2}{r_{0}}({\frac{1}{2}},0)}$
comes from the equations of motion in $\hat{{\cal C}}_{0(0,0)}$ multiplet.
Actually the full $\hat{C}_{0(0,0)}$ multiplet decomposes as

$\hat{C}_{0(0,0)}\simeq\hat{{\cal C}}_{(0,0)}\oplus\hat{{\cal
C}}_{({\frac{1}{2}},{\frac{1}{2}})}\oplus\bar{\cal
C}_{1-\frac{2}{r_{0}}({\frac{1}{2}},0)}\oplus{\cal
C}_{-1+\frac{2}{r_{0}}(0,{\frac{1}{2}})}$ (24)

in terms of ${\cal N}=1$ multiplets.

The index of $\bar{\cal B}_{1,r(0,0)}$ multiplet under the deformation is

$\begin{split}&{\cal I}_{\bar{\cal
B}_{1,r(0,0)}}\left(p,q,(pq)^{1-\frac{1}{r_{0}}}\right)\\\
=&\frac{(pq)^{1+\frac{r-1}{r_{0}}}}{(1-p)(1-q)}-\frac{(pq)^{\frac{3}{2}+\frac{r-2}{r_{0}}}\left(\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}\right)}{(1-p)(1-q)}+\frac{(pq)^{2+\frac{r-3}{r_{0}}}}{(1-p)(1-q)}\\\
&-\frac{(pq)^{1+\frac{r}{r_{0}}}}{(1-p)(1-q)}+\frac{(pq)^{\frac{3}{2}+\frac{r-1}{r_{0}}}\left(\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}\right)}{(1-p)(1-q)}-\frac{(pq)^{2+\frac{r-2}{r_{0}}}}{(1-p)(1-q)}\\\
=&{\cal I}_{\bar{\cal B}_{2+\frac{2r-2}{r_{0}}(0,0)}}+{\cal I}_{\bar{\cal
B}_{3+\frac{2r-4}{r_{0}}({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal
B}_{4+\frac{2r-6}{r_{0}}(0,0)}}+{\cal I}_{\bar{\cal
C}_{\frac{2r}{r_{0}}(0,0)}}+{\cal I}_{\bar{\cal
C}_{1+\frac{2r-2}{r_{0}}({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal
C}_{2+\frac{2r-4}{r_{0}}(0,0)}}.\end{split}$ (25)

$\bar{\cal B}_{1,r(0,0)}$ contains also ${\cal N}=1$ multiplets which do not
contribute to the index, and we will not write the full decomposition.

Chiral ring relations for general deformation: Now we deform $(A_{1},A_{2N})$
theories with ${\cal N}=1$ deformation discussed above, then the ${\cal N}=1$
single letter index of ${\cal T}^{(A_{1},A_{2N})}[{\cal O}_{\bar{\cal
E}_{r_{0}(0,0)}}]$ theory is

$\begin{split}&{\cal I}^{{\cal N}=1,s.l.}_{(A_{1},A_{2N})}={\cal
I}^{s.l.}_{(A_{1},A_{2N})}\left(p,q,(pq)^{1-\frac{1}{r_{0}}}\right)\\\
=&\sum_{i=1}^{N}\left({\cal I}_{\bar{\cal
B}_{\frac{4(N+i+1)}{(2N+3)r_{0}}(0,0)}}+{\cal I}_{\bar{\cal
B}_{\frac{2(2i-1)}{(2N+3)r_{0}}+1({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal
B}_{\frac{4(-N+i-2)}{(2N+3)r_{0}}+2(0,0)}}\right)\\\ &+\left({\cal
I}_{\hat{{\cal C}}_{(0,0)}}+{\cal I}_{\hat{{\cal
C}}_{({\frac{1}{2}},{\frac{1}{2}})}}+{\cal I}_{\bar{\cal
C}_{1-\frac{2}{r_{0}}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{3-\frac{2}{r_{0}}({\frac{1}{2}},0)}}\right)\\\
&-\sum_{i=1}^{2N-1}\left({\cal I}_{\bar{\cal
B}_{\frac{4i}{(2N+3)r_{0}}+2(0,0)}}+{\cal I}_{\bar{\cal
B}_{\frac{2(-2N+2i-3)}{(2N+3)r_{0}}+3({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal
B}_{\frac{4(-2N+i-3)}{(2N+3)r_{0}}+4(0,0)}}\right)\\\
&-\sum_{i=1}^{2N-1}\left({\cal I}_{\bar{\cal
C}_{\frac{2(2N+2i+3)}{(2N+3)r_{0}}(0,0)}}+{\cal I}_{\bar{\cal
C}_{\frac{4i}{(2N+3)r_{0}}+1({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal
C}_{\frac{2(-2N+2i-3)}{(2N+3)r_{0}}+2(0,0)}}\right)+\cdots.\end{split}$ (26)

The first line on the RHS of the equality comes from ${\cal I}_{\bar{\cal
E}_{r}}$’s , the second line comes from ${\cal I}_{\hat{\cal C}_{0(0,0)}}$,
and the third and forth lines come from $-{\cal I}_{\bar{\cal
B}_{1,r(0,0)}}$’s.

From the ${\cal N}=1$ single letter index 26 one can immediately read off the
generators of the chiral ring, which are

• Generators: $u_{i}$, $\lambda_{i,\alpha}$ and $S_{i}$ for $1\leq i\leq N$,

and their $R_{IR}$-charges are $\frac{4(N+i+1)}{(2N+3)r_{0}}$,
$\frac{2(2i-1)}{(2N+3)r_{0}}+1$, and $\frac{4(-N+i-2)}{(2N+3)r_{0}}+2$
respectively. $\hat{\cal C}_{({\frac{1}{2}},{\frac{1}{2}})}$ is the ${\cal
N}=1$ stress tensor multiplet. And the theory has an extra $\bar{\cal
C}_{1-\frac{2}{r_{0}}({\frac{1}{2}},0)}$ from the ${\cal N}=2$ stress tensor
multiplet.

Now we analyze the chiral ring relations from the index 26. The operator
${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}$ we used to deform the theory is the
bottom component of a $\bar{\cal B}_{2(0,0)}$ multiplet in the IR. If it is a
single trace operator ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}=u_{i}$, its index
${\cal I}_{\bar{\cal B}_{2(0,0)}}$ appears as part of the single letter index
26. If it is a composite operator like ${\cal O}_{\bar{\cal
E}_{r_{0}(0,0)}}=u_{i}u_{j}+\cdots$, its index ${\cal I}_{\bar{\cal
B}_{2(0,0)}}$ appears in the full index from terms like ${\cal I}_{\bar{\cal
B}_{r_{i}}}{\cal I}_{\bar{\cal B}_{r_{j}}}$ with $r_{i}$ and $r_{j}$ are
$R_{IR}$ of $u_{i}$ and $u_{j}$ respectively. Notice there is also the
identity

${\cal I}_{\bar{\cal B}_{2(0,0)}}+{\cal I}_{\hat{\cal C}_{(0,0)}}=0,$ (27)

so in the full index the contribution from ${\cal O}_{\bar{\cal
E}_{r_{0}(0,0)}}$ cancels with the contribution from the $U(1)$ current
multiplet $\hat{\cal C}_{(0,0)}$ from the ${\cal N}=2$ stress tensor
multiplet, leading to the relation ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}=0$.
To understand this relation, recall the recombination rule B of the ${\cal
N}=1$ multiplets

${\cal A}^{2}_{0(0,0)}\simeq\hat{\cal C}_{(0,0)}\oplus{\cal
B}_{-2(0,0)}\oplus\bar{\cal B}_{2(0,0)}.$ (28)

Hence the marginal operator ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}$ recombines
with the conserved current multiplet $\hat{\cal C}_{(0,0)}$, and fails to be
exact marginal. Acting $Q_{1\alpha}$ on ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}$
leads to another ${\cal N}=1$ chiral multiplet $\bar{\cal
B}_{3-\frac{2}{r_{0}}({\frac{1}{2}},0)}$101010We pick ${\cal N}=1$
superconformal subalgebra of the ${\cal N}=2$ SCA containing $Q_{2\alpha}$ and
$\tilde{Q}_{1\dot{\alpha}}$.. However, the $-{\cal I}_{\bar{\cal
B}_{3-\frac{2}{r_{0}}({\frac{1}{2}},0)}}$ term in the single letter index
gives 26 a relation $Q_{1\alpha}{\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}=0$. All
in all, the chiral ring relation coming from ${\cal N}=2$ stress tensor
multiplet ${\cal I}_{\hat{{\cal C}}_{0(0,0)}}$ are

• Relation 1: ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}=0$, • Relation 2:
$Q_{1\alpha}{\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}=0$.

These two relations depend on the explicit form of ${\cal O}_{\bar{\cal
E}_{r_{0}(0,0)}}$.

There are also chiral ring relations from the ${\cal N}=2$ relations 21. These
relations are represented by $-{\cal I}_{\bar{\cal B}_{1,r(0,0)}}$ terms in
the ${\cal N}=2$ single letter index, and becomes constraints on $\bar{\cal
B}$ and $\bar{\cal C}$ type multiplets as show in 25. By matching the quantum
number we can see the third line of 26 gives the chiral ring relations

• Relation 3:
$\lambda_{i,\alpha}\lambda^{\alpha}_{j}+u_{i}S_{j}+u_{j}S_{i}+\cdots=0$, •
Relation 4: $S_{i}\lambda_{j,\alpha}+S_{j}\lambda_{i,\alpha}+\cdots=0$, •
Relation 5: $S_{i}S_{j}+\cdots=0$,

where $\cdots$ means any other possible chiral operators with the same
$R_{IR}$ charge. In certain cases, terms in $\cdots$ are necessary because of
constraints from bootstrap methodPoland:2015mta . Each relation sums over all
pairs of $(i,j)$ with $i+j$ remains the same, and the sum of $i+j$ takes all
integers between $2$ and $2N$. The form of relation 3, 4 and 5 is universal
and does not depend on the explicit form of ${\cal O}_{\bar{\cal
E}_{r_{0}(0,0)}}$.

The last line of 26 also gives relations

$\begin{split}&Q_{2}^{\alpha}\left(u_{i}\lambda_{j\alpha}+u_{j}\lambda_{i,\alpha}+\cdots\right),\\\
&Q_{2\alpha}\left(\lambda_{i,\beta}\lambda^{\beta}+u_{i}S_{j}+u_{j}S_{i}+\cdots\right)=0,\\\
&Q_{2}^{\alpha}\left(S_{i}\lambda_{j,\alpha}+S_{j}\lambda_{i,\alpha}+\cdots\right)=0,\\\
\end{split}$ (29)

with $1\leq i\leq j\leq N$. These $\bar{\cal C}$ type multiplets are originate
from tensor product of two $\bar{\cal B}$ type multiplets

$\bar{\cal B}_{r_{1}(0,0)}\otimes\bar{\cal B}_{r_{2}(0,0)}\simeq\bar{\cal
B}_{r_{1}+r_{2}(0,0)}\oplus\bar{\cal
C}_{r_{1}+r_{2}-1({\frac{1}{2}},0)}\oplus\cdots,$ (30)

and

$\bar{\cal B}_{r_{1}(0,0)}\otimes\bar{\cal
B}_{r_{2}({\frac{1}{2}},0)}\simeq\bar{\cal
B}_{r_{1}+r_{2}({\frac{1}{2}},0)}\oplus\bar{\cal
C}_{r_{1}+r_{2}-1(0,0)}\oplus\cdots.$ (31)

Since these are not chiral ring relations we will not discuss them any
further.

Example 1: Consider ${\cal T}^{(A_{1},A_{2})}[u_{1}^{2}]$ which is the
$(A_{1},A_{2})$ theory deformed by ${\cal O}_{\bar{\cal
E}_{r_{0}(0,0)}}=u_{1}^{2}$ with $r_{0}=r(u_{1})=\frac{6}{5}$ Xie:2016hny ;
Buican:2016hnq . Some of the protected operator spectrum are

  * •

Chiral multiplets $u_{1}$, $\lambda_{1,\alpha}$ and $S_{1}$,

  * •

The ${\cal N}=1$ stress tensor multiplet $\hat{{\cal
C}}_{({\frac{1}{2}},{\frac{1}{2}})}$,

  * •

A multiplet of type $\bar{\cal C}_{\frac{1}{6}({\frac{1}{2}},0)}$,

with the chiral ring relation

  * •

$u_{1}^{2}=0$, $u_{1}\lambda_{1,\alpha}=0$,

  * •

$\lambda_{1,\alpha}\lambda_{1}^{\alpha}+u_{1}S_{1}=0$,
$\lambda_{1,\alpha}S_{1}=0$, and $S_{1}^{2}=0$,

The existence of $\bar{\cal C}_{\frac{1}{6}({\frac{1}{2}},0)}$ multiplet and
the first pair of relations was obtained in Buican:2016hnq  by using the Schur
index, while the existence of the second pair of relations was argued in
Xie:2016hny . Our method reproduce the previous result on $(A_{1},A_{2})$ and
can be easily generalized to any AD theories whose full superconformal indices
are known.

Example 2: Now consider ${\cal T}^{(A_{1},A_{4})}[u_{2}]$ with
$r_{0}=r(u_{2})=\frac{10}{7}$. The chiral ring of this theory is generated by

  * •

Chiral multiplets $u_{1}(\frac{12}{5})$, $\lambda_{1\alpha}(\frac{9}{5})$,
$S_{1}(\frac{6}{5})$, $S_{2}(\frac{9}{5})$,

where the number in the bracket is the IR conformal dimension of the
corresponding chiral operator. The index tells us that there is a relation for
the operator with smallest dimension $S_{1}$,

$S_{1}^{2}+u_{1}=0.$ (32)

The $u_{1}$ term in the relation is necessary because
$\Delta(S_{1})=\frac{6}{5}$ is smaller than $1.407$ and constraints from
bootstrapPoland:2015mta  tells us that $S_{1}^{2}$ can never be $0$. Actually
one can see that for any of the IR ${\cal N}=1$ theories obtained by our
method, if the conformal dimension of the lowest chiral operator is smaller
than $1.407$, there is always other operators in the spectrum with the same
conformal dimension twice as the smallest chiral operator, hence our result is
consistent with bootstrap bounds.

####  3.1.3 Exact marginal deformations

If the UV ${\cal N}=2$ theory has multiple deformations with the same quantum
number, the IR theory could have exact marginal deformations. The marginal
operator is in the $\bar{\cal B}_{2(0,0)}$ multiplet and its single letter
index has a leading contribution $pq$ to the full index. However, if the IR
theory has global symmetries, their conserved currents are in $\hat{{\cal
C}}_{(0,0)}$ multiplets which have opposite single letter index comparing to
$\bar{\cal B}_{2(0,0)}$ multiplets. Therefore the coefficient of the $pq$ term
in the full index is the number of marginal operators minus the number of
conserved currents

$\\#~{}pq-\mathrm{term}=\\#~{}\mathrm{marginal~{}operators}-\\#~{}\mathrm{conserved~{}currents}.$
(33)

At a generic point on the conformal manifold when all flavor symmetries are
broken, the coefficient of the $pq$ term is the actual dimension of the
conformal manifold.

For example, in $(A_{1},A_{6})$ theory, $u_{2}^{2}$ and $u_{1}u_{3}$ have the
${\cal N}=2$ $U(1)_{r}$ charge $\frac{22}{9}$, and in general we can deform
the theory with ${\cal O}=au_{1}u_{3}+bu^{2}_{2}$. After imposing the relation
${\cal O}=0$ we still expect one marginal operator in the IR. Since this
theory has no flavor symmetry, this marginal operator can not recombine with
any conserved current, hence remains exact marginal. Computation of the full
${\cal N}=1$ index shows that the coefficient of the $pq$ term is exact $1$,
hence confirm the fact that the theory has one exact marginal operator in the
IR.

In general, $u^{2}_{i}$ for $1<i<N$ has the same $U(1)_{r}$ charge and
conformal dimension with $u_{i-1}u_{i+1}$, $u_{i-2}u_{i+2}$, and etc. If one
deforms the theory with ${\cal
O}_{i}=u^{2}_{i}+a_{1}u_{i-1}u_{i+1}+a_{2}u_{i-2}u_{i+2}+\cdots$ with $1<i<N$,
the IR theory will have $\mathrm{min}(i-1,N-i)$ exact marginal deformations,
where $\mathrm{min}(a,b)$ is the minimum between $a$ and $b$. This leaves a
series of ${\cal N}=1$ theories with conformal manifolds.

One application of such theories is to notice that the IR conformal dimension
of $u_{j}$ with $1\leq j\leq N$ under the deformation ${\cal
O}_{i}=u^{2}_{i}+u_{i-1}u_{i+1}+u_{i-2}u_{i+2}+\cdots$ is
$\frac{3(N+j+1)}{2(N+i+1)}$ therefore the gap in the spectrum is

$\Delta_{j+1}-\Delta_{j}=\frac{3}{2(N+i+1)}\sim\frac{3}{2N},$ (34)

when $N\gg 1$, hence the theory has a dense spectrum in the sense of
Agarwal:2020pol ; Agarwal:2020polWIP  at large $N$. Our construction provides
first examples of ${\cal N}=1$ theory with both a dense spectrum and a
conformal manifold111111We thank Jaewon Song for the discussion..

###  3.2 $(A_{1},A_{2N+1})$ results

####  3.2.1 ${\cal N}=2$ index

The full index of $(A_{1},A_{2N+1})$ AD theories can be computed again using
the method in Maruyoshi:2016tqk ; Maruyoshi:2016aim ; Agarwal:2016pjo . First
few orders of the single letter indices for $N=1,2,3$ are

${\cal I}^{s.l.}_{(A_{1},A_{3})}={\cal I}_{\bar{\cal E}_{\frac{4}{3}}}+{\cal
I}_{\hat{{\cal B}}_{1}}\chi_{1}(a)+{\cal I}_{\hat{{\cal C}}_{0(0,0)}}-{\cal
I}_{\bar{\cal B}_{1,\frac{4}{3}(0,0)}}\chi_{1}(a)-{\cal I}_{\bar{\cal
B}_{1,\frac{5}{3}(0,0)}}-{\cal I}_{\hat{{\cal B}}_{2}}-{\cal I}_{\hat{{\cal
B}}_{3}}\chi_{1}(a)+\cdots,$ (35) $\begin{split}{\cal
I}^{s.l.}_{(A_{1},A_{5})}=&{\cal I}_{\bar{\cal E}_{\frac{5}{4}}}+{\cal
I}_{\bar{\cal E}_{\frac{3}{2}}}+{\cal I}_{\hat{{\cal B}}_{1}}+{\cal
I}_{\hat{{\cal C}}_{0(0,0)}}+{\cal I}_{\hat{{\cal
B}}_{\frac{3}{2}}}(a+\frac{1}{a})-{\cal I}_{\hat{{\cal B}}_{3}}\\\ &+{\cal
I}_{\bar{\cal B}_{1,\frac{5}{4}(0,0)}}-{\cal I}_{\bar{\cal
B}_{1,\frac{3}{2}(0,0)}}-{\cal I}_{\bar{\cal B}_{1,\frac{7}{4}(0,0)}}-{\cal
I}_{\bar{\cal B}_{1,2(0,0)}}-({\cal I}_{\bar{\cal
B}{\frac{3}{2},\frac{3}{2}(0,0)}}+{\cal I}_{\bar{\cal
B}{\frac{3}{2},\frac{5}{4}(0,0)}})(a+\frac{1}{a})+\cdots,\end{split}$ (36)

and

$\begin{split}{\cal I}^{s.l.}_{(A_{1},A_{7})}=&{\cal I}_{\bar{\cal
E}_{\frac{6}{5}}}+{\cal I}_{\bar{\cal E}_{\frac{7}{5}}}+{\cal I}_{\bar{\cal
E}_{\frac{8}{5}}}+{\cal I}_{\hat{{\cal B}}_{1}}+{\cal I}_{\hat{{\cal
C}}_{0(0,0)}}+{\cal I}_{\hat{{\cal B}}_{2}}(a+\frac{1}{a})-{\cal I}_{\hat{\cal
B}_{4}}\\\ &-{\cal I}_{\bar{\cal B}_{1,\frac{6}{5}(0,0)}}-{\cal I}_{\bar{\cal
B}_{1,\frac{7}{5}(0,0)}}-{\cal I}_{\bar{\cal B}_{1,\frac{8}{5}(0,0)}}-{\cal
I}_{\bar{\cal B}_{1,\frac{9}{5}(0,0)}}-{\cal I}_{\bar{\cal
B}_{1,\frac{10}{5}(0,0)}}-{\cal I}_{\bar{\cal B}_{1,\frac{11}{5}(0,0)}}\\\
&-({\cal I}_{\bar{\cal B}_{2,\frac{6}{5}(0,0)}}+{\cal I}_{\bar{\cal
B}_{2,\frac{7}{5}(0,0)}}+{\cal I}_{\bar{\cal
B}_{2,\frac{8}{5}(0,0)}})(a+\frac{1}{a})+\cdots.\end{split}$ (37)

And we conjecture the single letter index of $(A_{1},A_{2N+1})$ theory always
starts with

$\begin{split}{\cal I}^{s.l.}_{(A_{1},A_{2N+1})}=&\sum_{i=1}^{N}{\cal
I}_{\bar{\cal E}_{\frac{N+i+2}{N+2}}}+{\cal I}_{\hat{{\cal B}}_{1}}+{\cal
I}_{\hat{{\cal C}}_{0(0,0)}}+{\cal I}_{\hat{{\cal
B}}_{\frac{N+1}{2}}}(a+\frac{1}{a})\\\ &-{\cal I}_{\hat{\cal
B}_{N+1}}-\sum_{i=0}^{2N-1}{\cal I}_{\bar{\cal
B}_{1,\frac{N+i+3}{N+2}(0,0)}}-\left(\sum_{i=1}^{N}{\cal I}_{\bar{\cal
B}_{\frac{N+1}{2},\frac{N+i+2}{N+2}}}\right)(a+\frac{1}{a})+\cdots.\end{split}$
(38)

In the single letter index, the $\hat{{\cal B}}_{1}$ multiplets contains the
Higgs branch operator (moment map of the flavor symmetry). In $(A_{1},A_{3})$
case the flavor symmetry enhances to $SU(2)$, otherwise the flavor symmetry is
$U(1)$.

From the index one immediately sees that at smaller conformal dimension, the
spectrum of AD theories consists of $\bar{\cal E}_{r(0,0)}$ type multiplets,
the stress tensor multiplet $\hat{{\cal C}}_{0(0,0)}$, $\hat{{\cal B}}_{1}$
multiplets containing the flavor symmetry current and two more $\hat{{\cal
B}}_{\frac{N+1}{2}}$ multiplets charged $\pm 1$ under the $U(1)$ flavor
symmetry. Primary operators of $\hat{{\cal B}}_{1}$ and $\hat{{\cal
B}}_{\frac{N+1}{2}}$ multiplets are generators of the Higgs branch chiral
ring. On the other hand, there are interesting relations as well. From the
$-{\cal I}_{\hat{\cal B}_{N+1}}$ term in the single letter index, there is a
relation on Higgs branch

$\left(\hat{\cal B}^{+}_{\frac{N+1}{2}}\otimes\hat{\cal
B}^{+}_{\frac{N+1}{2}}\oplus\hat{{\cal B}}_{1}^{\otimes N+1}\right)_{\hat{\cal
B}_{N+1}}=0,$ (39)

In terms of generators of the Higgs branch chiral, the relation has the form

$X^{+}_{\frac{N+1}{2}}X^{-}_{\frac{N+1}{2}}+X^{N+1}_{1}=0.$ (40)

Here $X_{1}$ is the bottom component (primary operator) of the $\hat{\cal
B}_{1}$ multiplet, and $X^{\pm}_{\frac{N+1}{2}}$ are the bottom components of
$\hat{\cal B}_{\frac{N+1}{2}}$ with $U(1)$ flavor charges $\pm 1$. Similar to
Coulomb branch case, we use $X$’s to denote primary operators which are
generators of the Higgs branch chiral ring to separate them from the primary
operator of a generic $\hat{\cal B}_{R}$ multiplet.

On the other hand, recall the tensor product decomposition

$\hat{\cal B}_{R}\otimes\bar{\cal E}_{r}\simeq\bar{\cal B}_{R,r}+\cdots,$ (41)

and the bottom component of $\bar{\cal B}_{R,r}$ is simply the product of
bottom components of $\hat{\cal B}_{R}$ and $\bar{\cal E}_{r}$. Therefore
$-{\cal I}_{{\cal B}_{1,\frac{N+3}{N+2}(0,0)}}$ leads to the relation
$\left(\hat{{\cal B}}_{1}\otimes\bar{\cal E}_{\frac{N+3}{N+2}}\right)_{{\cal
B}_{1,\frac{N+3}{N+2}(0,0)}}=0$, i.e. $X_{1}u_{1}=0$. This relations means
that the moment map of the $U(1)$ flavor symmetry and the Coulomb branch
operator with the lowest dimension can not obtain non-zero vev at the same
time. In general, $-{\cal I}_{\bar{\cal B}_{1,r(0,0)}}$ terms in the single
letter index implies the relation

$\hat{{\cal B}}_{1}\otimes\bar{\cal E}_{\frac{N+3}{N+2}}=0,$ (42)

$\hat{{\cal B}}_{1}\otimes\bar{\cal E}_{r}+\sum_{r^{\prime}\in{\cal
A},r+1-r^{\prime}\in{\cal A}}\left(\bar{\cal E}_{r^{\prime}}\otimes\bar{\cal
E}_{r+1-r^{\prime}}\right)_{\bar{\cal B}_{1,r(0,0)}}=0,\
\mathrm{for}\,\,r=\frac{N+4}{N+2},\frac{N+5}{N+2},\cdots,\frac{2N+2}{N+2},$
(43)

$\sum_{r^{\prime}\in{\cal A},r+1-r^{\prime}\in{\cal A}}\left(\bar{\cal
E}_{r^{\prime}}\otimes\bar{\cal E}_{r+1-r^{\prime}}\right)_{\bar{\cal
B}_{1,r(0,0)}}=0,\
\mathrm{for}\,\,r=\frac{2N+3}{N+2},\frac{2N+4}{N+2},\cdots,\frac{4N+2}{N+2},$
(44)

where $(\cdots)_{\bar{\cal B}_{1,r(0,0)}}$ means picking up the ${\cal
B}_{1,r(0,0)}$ in the tensor product decomposition, and ${\cal
A}=\\{\frac{N+3}{N+2},\frac{N+4}{N+2},\cdots,\frac{2N+2}{N+2}\\}$ is the set
of all $U(1)_{r}$-charges of Coulomb branch operators. Last but not least
$-\left(\sum_{i=1}^{N}{\cal I}_{\bar{\cal
B}_{\frac{N+1}{2},\frac{N+i+2}{N+2}}}\right)(a+\frac{1}{a})$ in the single
letter index indicates relations between $\hat{\cal B}_{\frac{N+1}{2}}^{\pm}$
and $\bar{\cal E}_{r}$

$\left(\hat{\cal B}_{\frac{N+1}{2}}^{\pm}\otimes\bar{\cal E}_{r}\right)_{{\cal
B}_{\frac{N+1}{2},r}}=0,\
\mathrm{for}\,\,r=\frac{N+3}{N+2},\frac{N+4}{N+2},\cdots,\frac{2N+2}{N+2},$
(45)

or in terms of generators of the Higgs and Coulomb branch chiral ring

$X^{\pm}_{\frac{N+1}{2}}u_{i}=0,\,\mathrm{for}\,\ 1\leq i\leq N,$ (46)

which state that operators $X^{\pm}_{\frac{N+1}{2}}$ in the Higgs branch and
Coulomb branch operators $u_{i}$ can not have non zero vev at the same time.

When $N=1$, the relation simplifies to

$\begin{split}&\hat{\cal B}_{1}^{\pm,0}\otimes\bar{\cal E}_{\frac{4}{3}}=0,\\\
&\left(\bar{\cal E}_{\frac{4}{3}}\otimes\bar{{\cal
E}}_{\frac{4}{3}}\right)_{\bar{\cal B}_{1,\frac{5}{3}(0,0)}}=0.\end{split}$
(47)

The first relation tells us that there is no mixed branch in $(A_{1},A_{3})$
theory. However, This is not the case when $N>1$ and mixed branch is allowed
in such cases.

####  3.2.2 ${\cal N}=1$ index and chiral ring

For $(A_{1},A_{2N+1})$ theories, again the relevant ${\cal N}=1$ deformation
${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}$ is chosen from
$\\{u_{i},u_{i}u_{j}\\},~{}i,j=1,\ldots,N$ or any linear combination of such
operators with the same quantum number.

${\cal N}=2$ multiplets as ${\cal N}=1$ multiplets: Beside $\bar{\cal E}_{r}$,
$\hat{\cal C}_{0(0,0)}$ and $\bar{\cal B}_{1,r(0,0)}$ multiplets, there is
also $\hat{\cal B}_{R}$ multiplet in $(A_{1},A_{2N+1})$ theories. After the
${\cal N}=1$ deformation, the index of $\hat{\cal B}_{R}$ multiplet becomes

$\begin{split}{\cal I}_{\hat{\cal
B}_{R}}\left((p,q,(pq)^{1-\frac{1}{r_{0}}}\right)&=\frac{(pq)^{R\left(1-\frac{1}{r_{0}}\right)}}{(1-p)(1-q)}-\frac{(pq)^{R\left(1-\frac{1}{r_{0}}\right)+\frac{1}{r_{0}}}}{(1-p)(1-q)}\\\
&={\cal I}_{\bar{\cal B}_{2R\left(1-\frac{1}{r_{0}}\right)(0,0)}}+{\cal
I}_{\bar{\cal C}_{2(R-1)\left(1-\frac{1}{r_{0}}\right)(0,0)}}.\end{split}$
(48)

Hence a $\hat{\cal B}_{R}$ multiplet decomposes to an ${\cal N}=1$ chiral
multiplet $X$ and an ${\cal N}=1$ $\bar{\cal C}$ type multiplet and other
multiplets which do not contribute to our index. In particular, when $R=1$, we
have

${\cal I}_{\hat{\cal B}_{1}}\left((p,q,(pq)^{1-\frac{1}{r_{0}}}\right)={\cal
I}_{\bar{\cal B}_{2\left(1-\frac{1}{r_{0}}\right)(0,0)}}+{\cal I}_{\hat{\cal
C}_{(0,0)}},$ (49)

which means that the $\hat{\cal B}_{1}$ mutliplet has the following
decomposition in terms of ${\cal N}=1$ multiplets,

$\hat{\cal B}_{1}\simeq{\cal I}_{\hat{\cal C}_{(0,0)}}\oplus\bar{\cal
B}_{2\left(1-\frac{1}{r_{0}}\right)(0,0)}\oplus{\cal
B}_{-2\left(1-\frac{1}{r_{0}}\right)(0,0)},$ (50)

which contains an ${\cal N}=1$ chiral, an ${\cal N}=1$ anti-chiral and an
${\cal N}=1$ conserved current multiplet.

Chiral ring relations:  We first get the ${\cal N}=1$ single letter index of
${\cal T}^{(A_{1},A_{2N+1})}[{\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}]$ theory by
plugging in $t=(pq)^{1-\frac{1}{r_{0}}}$ in equation 38:

$\begin{split}&{\cal
I}^{s.l.}_{(A_{1},A_{2N+1})}\left(p,q,(pq)^{1-\frac{1}{r_{0}}}\right)\\\
=&\sum_{i=1}^{N}{\cal I}_{\bar{\cal
B}_{\frac{2}{r_{0}}\frac{N+i+2}{N+2}(0,0)}}+{\cal I}_{\bar{\cal
B}_{\frac{2}{r_{0}}\frac{N+i+2}{N+2}+1-\frac{2}{r_{0}}({\frac{1}{2}},0)}}+{\cal
I}_{\bar{\cal B}_{\frac{2}{r_{0}}\frac{N+i+2}{N+2}+2-\frac{4}{r_{0}}(0,0)}}\\\
&+{\cal I}_{\hat{{\cal C}}_{(0,0)}}+{\cal I}_{\hat{{\cal
C}}_{({\frac{1}{2}},{\frac{1}{2}})}}+{\cal I}_{\bar{\cal
C}_{1-\frac{2}{r_{0}}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{3-\frac{2}{r_{0}}({\frac{1}{2}},0)}}\\\ &+\left({\cal I}_{\bar{\cal
B}_{2\left(1-\frac{1}{r_{0}}\right)(0,0)}}+{\cal I}_{\hat{\cal
C}_{(0,0)}}\right)+\left({\cal I}_{\bar{\cal
B}_{(N+1)\left(1-\frac{1}{r_{0}}\right)(0,0)}}+{\cal I}_{\bar{\cal
C}_{(N-1)\left(1-\frac{1}{r_{0}}\right)(0,0)}}\right)(a+\frac{1}{a})\\\
&-\left({\cal I}_{\bar{\cal
B}_{2(N+1)\left(1-\frac{1}{r_{0}}\right)(0,0)}}-{\cal I}_{\bar{\cal
C}_{2N\left(1-\frac{1}{r_{0}}\right)(0,0)}}\right)\\\
&-\sum_{i=0}^{2N-1}\left({\cal I}_{\bar{\cal
B}_{\frac{2}{r_{0}}\frac{N+i+3}{N+2}+2-\frac{2}{r_{0}}(0,0)}}+{\cal
I}_{\bar{\cal
B}_{\frac{2}{r_{0}}\frac{N+i+3}{N+2}+3-\frac{4}{r_{0}}({\frac{1}{2}},0)}}+{\cal
I}_{\bar{\cal
B}_{\frac{2}{r_{0}}\frac{N+i+3}{N+2}+4-\frac{6}{r_{0}}(0,0)}}\right)\\\
&-\sum_{i=0}^{2N-1}\left({\cal I}_{\bar{\cal
C}_{\frac{2}{r_{0}}\frac{N+i+3}{N+2}(0,0)}}+{\cal I}_{\bar{\cal
C}_{\frac{2}{r_{0}}\frac{N+i+3}{N+2}+1-\frac{2}{r_{0}}({\frac{1}{2}},0)}}+{\cal
I}_{\bar{\cal
C}_{\frac{2}{r_{0}}\frac{N+i+3}{N+2}+2-\frac{4}{r_{0}}(0,0)}}\right)+\cdots.\end{split}$
(51)

Similarly to previous case, we can extract the generators of the chiral ring

• Generator set 1: $u_{i}$, $\lambda_{i,\alpha}$ and $S_{i}$ for $1\leq i\leq
N$, with $R_{IR}$ charges $\frac{2}{r_{0}}\frac{N+i+2}{N+2}$,
$\frac{2}{r_{0}}\frac{N+i+2}{N+2}+1-\frac{2}{r_{0}}$, and
$\frac{2}{r_{0}}\frac{N+i+2}{N+2}+2-\frac{4}{r_{0}}$, • Generator set 2: $X$
with $R_{IR}$ charge $2-\frac{2}{r_{0}}$, and $X^{\pm}$ with $R_{IR}$ charges
$(N+1)(2-\frac{2}{r_{0}})$, $U(1)$ flavor charge $\pm 1$.

The first set of generators coming from ${\cal N}=1$ decomposition of
$\bar{\cal E}_{r}$ multiplets containing generators of the Coulomb branch
chiral ring, while the second set of generators from the ${\cal N}=1$ chiral
multiplets in the $\hat{\cal B}$ multiplets containing generators of the Higgs
branch chiral ring.

One also gets the following chiral ring relations from different ${\cal N}=2$
multiplets and relations: from ${\cal N}=2$ stress tensor multiplet

• Relation 1: ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}=0$, • Relation 2:
$Q_{1\alpha}{\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}=0$.

From ${\cal N}=2$ Higgs branch relations,

• Relation 3: $X^{+}X^{-}+X^{N+1}+\cdots=0$

where $+\cdots$ means any other operators with the same quantum number. From
${\cal N}=2$ relations 42 and 45

• Relation 4: $Xu_{1}=0$, $X\lambda_{1\alpha}=0$, and $XS_{1}=0$. • Relation
5: $X^{\pm}u_{i}=0$, $X^{\pm}\lambda_{i\alpha}=0$, $X^{\pm}S_{i}=0$, for
$1\leq i\leq N$,

New relation $3$, $4$ and $5$ appear because of the Higgs branch of
$(A_{1},A_{2N+1})$ theory. And last but not least from other ${\cal N}=2$
relations

• Relation 6:
$Xu_{i}+\lambda_{j\alpha}\lambda_{k}^{\alpha}+u_{j}S_{k}+u_{k}S_{j}+\cdots=0$,
• Relation 7:
$X\lambda_{i\alpha}+\lambda_{j\alpha}S_{k}+\lambda_{k\alpha}S_{j}+\cdots=0$, •
Relation 8: $XS_{i}+S_{j}S_{k}=0$,

for $2\leq i\leq N$, and any pairs of $(j,k)$ with $j+k=i$,

• Relation 9:
$\lambda_{j\alpha}\lambda_{k}^{\alpha}+u_{j}S_{k}+u_{k}S_{j}+\cdots=0$, •
Relation 10: $\lambda_{j\alpha}S_{k}+\lambda_{k\alpha}S_{j}+\cdots=0$, •
Relation 11: $XS_{i}+S_{j}S_{k}=0$,

for any pairs of $(j,k)$ with $j+k=N+1,N+2,\cdots,2N$. Relations 6 - 11 which
include both chiral multiplets $X$ and $u$, $\lambda_{\alpha}$ and $S$ are
generalization of chiral ring relation 3, 4 and 5 in $(A_{1},A_{2N})$
theories. Similar to $(A_{1},A_{2N})$ cases, each distinct value of $j+k$
leads to exact one relation and $\cdots$ means one should include all possible
operators with the same quantum number.

###  3.3 $(A_{1},D_{2N+1})$ and $(A_{1},D_{2N})$ theories

We briefly sketch results for $(A_{1},D_{2N+1})$ and $(A_{1},D_{2N})$ theories
as they are similar to $(A_{1},A_{2N+1})$ theories. Using the index formula in
Agarwal:2016pjo  one can read off the ${\cal N}=2$ chiral ring generators and
relations.

For $(A_{1},D_{2N+1})$ theories, generators of chiral rings are

  * •

Three $\hat{\cal B}_{1}$ multiplets: $\hat{\cal B}_{1}^{0}$, $\hat{\cal
B}_{1}^{+}$, and $\hat{\cal B}_{1}^{-}$ which contain moment map $X^{0}$,
$X^{\pm}$ of the flavor $SU(2)$,

  * •

The stress tensor multiplet $\hat{\cal C}_{0(0,0)}$,

  * •

$N$ $\bar{\cal E}_{r}$ multiplets: $\bar{\cal E}_{\frac{2N+2i}{2N+1}}$, for
$1\leq i\leq N$ which contains $N$ Coulomb branch operators $u_{i}$.

${\cal N}=2$ relations are

• Higgs branch chiral ring relation: $\left(\hat{\cal
B}_{1}^{+}\otimes\hat{\cal B}_{1}^{-}\oplus\hat{\cal
B}_{1}^{0}\right)_{\hat{\cal B}_{2}}=0$, or in terms of moment maps
$X^{+}X^{-}+(X^{0})^{2}=0$, • Mixed branch relation 1: $\left(\hat{\cal
B}_{1}^{\pm}\otimes\bar{\cal E}_{\frac{2N+2i}{2N+1}}\right)_{\bar{\cal
B}_{1,\frac{2N+2i}{2N+1}(0,0)}}=0$, or in terms of operators $X^{\pm}u_{i}=0$,
for $1\leq i\leq N$, • Mixed branch relation 2: $\left(\hat{\cal
B}_{1}^{0}\otimes\bar{\cal E}_{\frac{2N+2}{2N+1}}\right)_{\bar{\cal
B}_{1,\frac{2N+2}{2N+1}(0,0)}}=0$, or in terms of operators $X^{0}u_{1}=0$, •
Other relations: $\left(\hat{\cal B}_{1}^{0}\otimes\bar{\cal
E}_{r_{k}}\oplus\sum_{r_{i},r_{j}}\bar{\cal E}_{r_{i}}\otimes\bar{\cal
E}_{r_{j}}\right)_{\bar{\cal B}_{1,r(0,0)}}=0$, for
$r=\frac{2N+4}{2N+1},\cdots,\frac{6N-1}{2N+1}$.

One can then read off the corresponding ${\cal N}=1$ chiral ring generators
and relations after deforming the theory with ${\cal O}_{\bar{\cal
E}_{r_{0}(0,0)}}$.

For $(A_{1},D_{2N})$ theory, the story is similar except that the generators
of the Higgs branch chiral ring are enlarged and have four $\hat{\cal B}_{1}$
multiplets which contain moment maps of $SU(2)\times U(1)$ flavor symmetry,
and four $\hat{\cal B}_{\frac{N}{2}}$ multiplets whose bottom components
transform as $\mathbf{2}_{1}\oplus\mathbf{2}_{-1}$ under the flavor
$SU(2)\times U(1)$.

###  3.4 Examples

We summarize properties of some examples of ${\cal N}=1$ theories deformed
from AD theories. The $a$ and $c$ anomalies are shown in table 1. The IR
theory${\cal T}^{(A_{1},A_{4})}[u_{2}]$ has the smallest $a$ and $c$ anomalies
in all cases.

${\cal N}=2$ theory | deformation ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}$ | $r_{0}$ | $a({\cal T},{\cal O})$ | $c({\cal T},{\cal O})$
---|---|---|---|---
$(A_{1},A_{2})$ | $u_{1}^{2}$ | $\frac{12}{5}$ | $\frac{263}{768}\approx 0.342$ | $\frac{271}{768}\approx 0.353$
$(A_{1},A_{4})$ | $u_{2}$ | $\frac{10}{7}$ | $\frac{644}{2000}\approx 0.316$ | $\frac{683}{2000}\approx 0.342$
$u^{2}_{1}$ | $\frac{16}{7}$ | $\frac{281}{512}\approx 0.744$ | $\frac{389}{512}\approx 0.760$
$u_{1}u_{2}$ | $\frac{18}{7}$ | $\frac{1013}{1296}\approx 0.782$ | $\frac{1031}{1296}\approx 0.796$
$u_{2}^{2}$ | $\frac{20}{7}$ | $\frac{6369}{8000}\approx 0.796$ | $\frac{6469}{8000}\approx 0.809$
$(A_{1},A_{6})$ | $u_{3}$ | $\frac{14}{9}$ | $\frac{234}{343}\approx 0.682$ | $\frac{3891}{5488}\approx 0.709$
$u^{2}_{1}$ | $\frac{20}{9}$ | $\frac{37107}{32000}\approx 1.160$ | $\frac{37707}{32000}\approx 1.178$
$u_{1}u_{2}$ | $\frac{22}{9}$ | $\frac{1179}{968}\approx 1.218$ | $\frac{2391}{1936}\approx 1.235$
$u_{1}u_{3}+u_{2}^{2}$ | $\frac{8}{3}$ | $\frac{2559}{2048}\approx 1.250$ | $\frac{2591}{2048}\approx 1.265$
$u_{2}u_{3}$ | $\frac{26}{9}$ | $\frac{44379}{35152}\approx 1.262$ | $\frac{22443}{17576}\approx 1.277$
$u_{3}^{2}$ | $\frac{28}{9}$ | $\frac{110871}{87808}\approx 1.263$ | $\frac{112047}{87808}\approx 1.276$
$(A_{1},A_{8})$ | $u_{3}$ | $\frac{16}{11}$ | $\frac{1563}{2048}\approx 0.763$ | $\frac{1627}{2048}\approx 0.794$
$u_{4}$ | $\frac{18}{11}$ | $\frac{695}{648}\approx 1.072$ | $\frac{713}{648}\approx 1.100$
$u_{1}^{2}$ | $\frac{24}{11}$ | $\frac{1213}{768}\approx 1.579$ | $\frac{1229}{768}\approx 1.600$
$u_{1}u_{2}$ | $\frac{26}{11}$ | $\frac{29091}{17576}\approx 1.655$ | $\frac{29429}{17576}\approx 1.674$
$u_{1}u_{3}+u_{2}^{2}$ | $\frac{28}{11}$ | $\frac{1335}{784}\approx 1.703$ | $\frac{1349}{784}\approx 1.720$
$u_{1}u_{4}+u_{2}u_{3}$ | $\frac{30}{11}$ | $\frac{5189}{3000}\approx 1.730$ | $\frac{5239}{3000}\approx 1.746$
$u_{2}u_{4}+u^{2}_{3}$ | $\frac{32}{11}$ | $\frac{28527}{16384}\approx 1.741$ | $\frac{28783}{16384}\approx 1.757$
Table 1: $a$ and $c$ anomalies of AD theory ${\cal T}$ with ${\cal N}=1$ deformation ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}$. $r_{0}$ is the $U(1)_{r}$ charge of ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}$. We include only deformations which do not make any chiral multiplets break unitarity bound. ${\cal N}=2$ theory |  | deformation
---
${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}$
|  $\\#$ of
---
independent
chirals
independent chirals
$(A_{1},A_{2})$ | $u_{1}^{2}$ | 4 |  $u_{1}(1)$, $\lambda_{1\alpha}(\frac{7}{6})$, $S_{1}(\frac{4}{3})$, $u_{1}S_{1}(\frac{7}{3})$
$(A_{1},A_{4})$ | $u_{2}$ | 5 |  $u_{1}(\frac{8}{5})$, $\lambda_{1\alpha}(\frac{6}{5})$, $S_{1}(\frac{4}{5})$, $u_{1}S_{1}(\frac{12}{5})$, $S_{2}(\frac{6}{5})$
$u^{2}_{1}$ | $\infty$ |  |  $u_{1}$, $\lambda_{1\alpha}$, $S_{1}$, $u_{1}S_{1}$
---
$u^{n}_{2}$, $u^{n}_{2}\lambda_{2\alpha}$, $u^{n}_{2}S_{2}$,
$u_{1}u^{n}_{2}$, $u^{n}_{2}S_{1}$, $u_{1}u^{n}_{2}S_{1}$,
$u_{1}u^{n}_{2}S_{2}$,
$u^{n}_{2}\lambda_{1\alpha}$,
$u_{2}^{n}(\lambda_{1\alpha}\lambda_{2\beta}-\frac{1}{2}\epsilon_{\alpha\beta}\lambda_{1\gamma}\lambda_{2}^{\gamma})$,
$S_{1}\lambda_{2\alpha}$
$u_{1}u_{2}$ | $\infty$ |  |  $u_{1}^{m}$, $u_{1}^{m}\lambda_{1\alpha}$, $u_{1}^{m}S_{1}$,
---
$u_{2}^{n}$, $u_{2}^{n}\lambda_{2\alpha}$, $u^{n}_{2}S_{2}$,
$u_{1}^{m}S_{2}$, $u^{n}_{2}S_{1}$,
$u_{1}\lambda_{2\alpha}$,
$\lambda_{1\alpha}\lambda_{2\beta}-\frac{1}{2}\epsilon_{\alpha\beta}\lambda_{1\gamma}\lambda_{2}^{\gamma}$,
$S_{1}\lambda_{2\alpha}$
$u_{2}^{2}$ | $\infty$ |  |  $u_{1}^{m}$, $u_{1}^{m}\lambda_{1\alpha}$, $u_{1}^{m}S_{1}$,
---
$u_{2}$, $\lambda_{2\alpha}$, $S_{2}$, $u_{2}S_{2}$,
$u_{1}^{m}u_{2}$, $u_{1}^{m}S_{2}$, $u_{1}^{m}u_{2}S_{1}$,
$u^{m}_{1}u_{2}S_{2}$,
$u^{m}_{1}\lambda_{2\alpha}$,
$u_{1}^{m}(\lambda_{1\alpha}\lambda_{2\beta}-\frac{1}{2}\epsilon_{\alpha\beta}\lambda_{1\gamma}\lambda_{2}^{\gamma})$,
$S_{1}\lambda_{2\alpha}$
Table 2: List of independent chirals of AD theory ${\cal T}$ with ${\cal N}=1$
deformation ${\cal O}_{\bar{\cal E}_{r_{0}(0,0)}}$. The number inside bracket
after each operator is its $R_{IR}$ charge.

Another interesting example is ${\cal T}^{(A_{1},A_{6})}[u_{2}]$ with
$r_{0}=\frac{4}{3}$. Under this deformation, the naive $R_{IR}$ charge of
$u_{1}$ is $\frac{2}{3}$ which is the $R_{IR}$ charge of a free chiral. All
the other chiral multiplet has $R_{IR}$ charge above the unitarity bound
$\frac{2}{3}$. The $a$ and $c$ anomalies are

$\begin{split}a({\cal T}^{(A_{1},A_{6})}[u_{2}])&=\frac{93}{256}=a({\cal
T}^{(A_{1},A_{2})}[u^{2}_{1}])+\frac{1}{48},\\\ c({\cal
T}^{(A_{1},A_{6})}[u_{2}])&=\frac{101}{256}=c({\cal
T}^{(A_{1},A_{2})}[u^{2}_{1}])+\frac{1}{24}.\end{split}$ (52)

This indicates that the IR theory ${\cal T}^{(A_{1},A_{6})}[u_{2}]$ is ${\cal
T}^{(A_{1},A_{2})}[u^{2}_{1}]$ plus a free chiral multiplet. This statement
can also be checked by matching the ${\cal N}=1$ index. Similarly, the IR
theory ${\cal T}^{(A_{1},A_{10})}[u_{3}]$ is ${\cal
T}^{(A_{1},A_{4})}[u_{1}u_{2}]$ plus a free chiral.

In some cases, only finitely many chiral multiplets remain in the IR. We
summarize some results on independent chiral multiplets in table 2.

##  4 Free chirals in the IR

In this section we study a set of particular deformation which leads to free
chiral multiplets in the IR. In many cases the Schur index of the ${\cal N}=2$
theory is known and we can obtain the ${\cal N}=1$ index of the deformed
theory using the method described in section 2.2.1. We see that the IR Schur
indices always look like products of ${\cal N}=1$ chiral multiplets.

###  4.1 $(A_{1},G)$ theories

####  4.1.1 $(A_{1},A_{2N})$: one free chiral

In $(A_{1},A_{2N})$ theories, the Coulomb branch operator $u_{1}$ has
dimension $\frac{2N+4}{2N+3}$. After deforming with ${\cal O}_{\bar{\cal
E}_{r_{0}(0,0)}}=u_{1}$, the ${\cal N}=1$ full index is computed by
substituting $t=(pq)^{1-\frac{1}{r_{0}}}=(pq)^{\frac{1}{2N+4}}$ into formula 7
and the result is 121212If one set $t=(pq)^{\frac{1}{2(N+2)}}$ naively in 7,
the result will be zero because of $\Gamma(pq)$ term in the index. To get the
correct result, we first set $t=(pq)^{\frac{1}{2(N+2)}+\epsilon}$ in 7,
compute the integral using the residue formula and then take the
$\epsilon\rightarrow 0^{+}$ limit.

$\Gamma((pq)^{\frac{1}{N+1}}),$ (53)

This is the same index of a free chiral multiplet with $U(1)_{R}$ charge
${2\over N+1}\leq\frac{2}{3}$. Since the naive $R_{IR}$ charge of the chiral
multiplet $S_{1}$ is ${2\over N+1}$, and we suspect that the IR theory of this
particular deformation is just this single free chiral multiplet.

Let us now look at the Shur index of the $(A_{1},A_{2N})$ theory Song:2017oew
; Xie:2019zlb , and they take the following simple form:

$\mathrm{PE}\left[\frac{q^{2}-q^{2N+2}}{(1-q)(1-q^{2N+3})}\right],$ (54)

which equals to the ${\cal N}=1$ index with the substitution $p\rightarrow
q^{\frac{1}{r_{0}-1}}=q^{2N+3}$, which we might call it a reduced
$\mathcal{N}=1$ index with just one fugacity $q$. Let’s now check that the
above index is equal to that of a free chiral with $U(1)_{R}$ charge ${2\over
N+1}$. Recall the ${\cal N}=1$ index of a chiral multiplet with $U(1)_{R}$
charge $r$ is

${\cal I}^{{\cal
N}=1}_{chiral}(p,q)=\Gamma((pq)^{\frac{r}{2}})=\mathrm{PE}\left[\frac{(pq)^{\frac{r}{2}}-(pq)^{1-\frac{r}{2}}}{(1-p)(1-q)}\right].$
(55)

After replacing $p\rightarrow q^{\frac{1}{r_{0}-1}}$, we have

${\cal I}^{{\cal
N}=1}_{chiral}(q^{\frac{1}{r_{0}-1}},q)=\mathrm{PE}\left[\frac{(q^{\frac{r_{0}}{r_{0}-1}})^{\frac{r}{2}}-(q^{\frac{r_{0}}{r_{0}-1}})^{1-\frac{r}{2}}}{(1-q^{\frac{1}{r_{0}-1}})(1-q)}\right]=\mathrm{PE}\left[\frac{q^{2}-q^{2N+2}}{(1-q)(1-q^{2N+3})}\right]$
(56)

Here we use the fact that $r_{0}={2N+4\over 2N+3},~{}r={2\over N+2}$. The use
of Shur index also indicates that the IR theory is just a free chiral
131313See Bolognesi:2015wta  for the discussion for small $N$..

${\cal N}=2$ $u_{i}$ | $u_{i}$ | $\lambda_{i,\alpha}$ | $S_{i}$
---|---|---|---
$\frac{2N+2+2i}{2N+3}$ | $\frac{2(N+1+i)}{N+2}$ | $\frac{N+2i+1}{N+2}$ | $\frac{2i}{N+2}$
Table 3: Conformal dimensions of ${\cal N}=1$ chiral multiplets from the
${\cal N}=2$ ${\cal E}_{r}$ multiplets. The first column is the dimension of
the $\mathcal{N}=2$ Coulomb branch operator, while subsequent columns are
dimensions of each ${\cal N}=1$ chirals in the IR limit. In the table, $i$
runs from $1$ to $N$.

The naive $R_{IR}$-charge of three ${\cal N}=1$ chiral multiplets of ${\cal
T}^{(A_{1},A_{2N})}[u_{1}]$ from ${\cal N}=2$ $\bar{\cal E}_{r(0,0)}$
multiplets are listed in table 3. From the table one sees immediately that
chirals $S_{i}$ with $1\leq i\leq\lfloor\frac{N+2}{3}\rfloor$141414$\lfloor
x\rfloor$ means the largest integer which is no greater than $x$. violate the
unitarity bound. One common procedure is to assume that operators which
violate the unitarity bound become free, and there is an extra $U(1)$ symmetry
which couples only with these operators and the correct IR $U(1)_{R}$ charge
should include this $U(1)$ so these operators have the correct R-charge for a
free chiral Kutasov:2003iy . The true $a$ and $c$ anomalies should be
replacing the naive contributions of chirals violating the unitarity bound
with contributions of free chirals,

$\begin{split}a^{true}&=a^{trial}-a(M_{R})+a^{free}\\\
c^{true}&=c^{trial}-c(M_{R})+c^{free}\end{split}$ (57)

where $M_{R}$ is the multiplet violating the unitarity bound, $a(M_{R})$ and
$c(M_{R})$ are the anomaly contribution from $M_{R}$ with naive R-charge $R$,
and $a^{free}=\frac{1}{48}$ and $c^{free}=\frac{1}{24}$ are anomalies of a
free chiral.

Direct computation shows that the naive $a$ and $c$ anomalies of ${\cal
T}^{(A_{1},A_{2N})}[u_{1}]$ is

$\begin{split}a({\cal
T}^{(A_{1},A_{2N})}[u_{1}])&=-\frac{3}{16}\frac{N(N^{2}-2N-2)}{(N+2)^{3}},\\\
c({\cal
T}^{(A_{1},A_{2N})}[u_{1}])&=-\frac{1}{8}\frac{N(N^{2}-5N-5)}{(N+2)^{3}},\end{split}$
(58)

which are the same as the $a$ and $c$ anomalies of the $S_{1}$ chiral with the
naive $R_{IR}$ listed in table 3. This implies that in the IR, $S_{1}$ becomes
free, and the correct $a$ and $c$ anomalies should be,

$\begin{split}a^{true}({\cal
T}^{(A_{1},A_{2N})}[u_{1}])&=-\frac{3}{16}\frac{N(N^{2}-2N-2)}{(N+2)^{3}}-a(S_{1})+\frac{1}{48}=\frac{1}{48},\\\
c^{true}({\cal
T}^{(A_{1},A_{2N})}[u_{1}])&=-\frac{1}{8}\frac{N(N^{2}-5N-5)}{(N+2)^{3}}-c(S_{1})+\frac{1}{24}=\frac{1}{24},\end{split}$
(59)

which is the anomalies for a theory of one free chiral multiplet. In the IR,
we only see a single free chiral multiplet, and one might be curious about the
fate of other chiral multiples which simply disappear in the index and $a,c$
anomaly. We will use the chiral ring relation to explain why these extra
chirals vanish.

For example, if we deform $(A_{1},A_{4})$ theory with ${\cal O}_{\bar{\cal
E}_{r_{0}(0,0)}}=u_{1}$ and ignore the unitarity bound temporarily, the IR
theory has the chiral ring relations

$\begin{split}&u_{1}=0,\,\lambda^{a}_{1}=0,\\\
&u_{1}S_{1}+\lambda_{1\alpha}\lambda^{\alpha}_{1}+u_{2}=0,\\\
&\lambda_{1\alpha}S_{1}+\lambda^{a}_{2}=0,\\\ &S^{2}_{1}+S_{2}=0.\end{split}$
(60)

From the chiral ring relation, we see that $u_{2},S_{2},\lambda_{2}$ can be
expressed in terms of the polynomial of $S_{1}$ and so is no longer the
generators of the chiral ring. One can similarly show that only $S_{1}$
survives in the chiral ring.

####  4.1.2 $(A_{1},D_{2N+1})$: three free chirals

The smallest Coulomb branch operator $u_{1}$ of $(A_{1},D_{2N+1})$ theory has
dimension $\frac{2N+2}{2N+1}$, while it Schur index is

$\begin{split}&\mathrm{PE}\left[\frac{(q-q^{2N+1})\chi_{1}(z)}{(1-q)(1-q^{2N+1})}\right]\\\
=&\mathrm{PE}\left[\frac{z^{2}q-z^{-2}q^{2N+1}}{(1-q)(1-q^{2N+1})}\right]\mathrm{PE}\left[\frac{q-q^{2N+1}}{(1-q)(1-q^{2N+1})}\right]\mathrm{PE}\left[\frac{z^{-2}q-z^{2}q^{2N+1}}{(1-q)(1-q^{2N+1})}\right]\end{split}$
(61)

where $\chi_{1}(z)$ is the character of the adjoint representation of
$SU(2)$Song:2017oew ; Xie:2019zlb . After deforming the theory with $u_{1}$,
the UV Schur index 61 is the same as the ${\cal N}=1$ index of the IR theory
${\cal T}^{(A_{1},D_{2N+1})}[u_{1}]$ with $p\rightarrow
q^{\frac{1}{r_{0}-1}}=q^{2N+1}$, and it is also the same as three chiral
multiplets with $U(1)_{R}$-charge $\frac{1}{N}$ which have an flavor $SU(2)$
symmetry. Therefore it is quite possible that the IR theory is just a theory
of three free chirals.

${\cal N}=2$ ${\cal E}_{i}$ | $u_{i}$ | $\lambda_{i,\alpha}$ | $S_{i}$ | $X_{a}$
---|---|---|---|---
$\frac{2N+2i}{2N+1}$ | $\frac{2(N+i)}{N+1}$ | $\frac{N+2i}{N+1}$ | $\frac{2i}{N+1}$ | $\frac{1}{N+1}$
Table 4: Conformal dimensions of ${\cal N}=1$ chiral multiplets of ${\cal
T}^{(A_{1},D_{2N+1})}[u_{1}]$. The first column is the dimension of the
$\mathcal{N}=2$ multiplets corresponding to generators of Coulomb branch
chiral ring of $(A_{1},D_{2n+1}$ theory, while subsequent columns are
dimensions of each ${\cal N}=1$ chirals in the IR limit, and the last coulomb
is the R-charge of the chiral from $\hat{{\cal B}}_{1}$ multiplet in the IR
limit. In the table, $i$ runs from $1$ to $N$.

The conformal dimensions from naive $R_{IR}$-charge of chiral multiplets of
${\cal T}^{(A_{1},D_{2N+1})}[u_{1}]$ is listed in table 4. One sees that all
$X_{a}$ from ${\cal N}=2$ moment maps and $S_{i}$’s with $1\leq
i\leq\lfloor\frac{N+1}{3}\rfloor$ violate the unitarity bound. The IR $a$ and
$c$ anomalies are

$\begin{split}a_{{\cal N}=1}&=\frac{9}{32}\frac{N(2N^{2}-2N-1)}{(N+1)^{3}},\\\
c_{{\cal N}=1}&=\frac{3}{32}\frac{N(4N^{2}-10N-5)}{(N+1)^{3}},\end{split}$
(62)

which are exactly three times the $a$ and $c$ anomalies of the $X_{a}$
multiplet with naive R-charge $\frac{1}{N+1}$. This implies that the three
$X_{a}$’s from moment maps become free chirals in the IR, and the correct $a$
and $c$ anomalies of the IR theory should be

$\begin{split}a^{true}_{{\cal
N}=1}&=\frac{9}{32}\frac{N(2N^{2}-2N-1)}{(N+1)^{3}}-\sum_{a}a(X_{a})+\frac{1}{16}=\frac{1}{16},\\\
c^{true}_{{\cal
N}=1}&=\frac{3}{32}\frac{N(4N^{2}-10N-5)}{(N+1)^{3}}-\sum_{a}c(X_{a})+\frac{1}{8}=\frac{1}{8},\end{split}$
(63)

therefore the IR theory is a free theory with three chirals. Note that some
$S_{i}$’s also violate unitarity bound when $N$ large enough, however, chiral
ring relations tell us $S_{i}=0$ for any $i$ therefore none of them appear in
the IR. For example, in the IR theory ${\cal T}^{(A_{1},D_{3})}[u_{1}]$, there
is a chiral ring relation

$X^{+}X^{-}+X^{2}+S_{1}=0$ (64)

among other relations. All $X$’s have $R_{IR}$ charge ${\frac{1}{2}}$ hence
violating unitarity bound, and should be removed from the relation and yields
$S_{1}=0$. Deforming the theory with $u_{1}$ also leads to relation
$u_{1}=\lambda_{1\alpha}=0$, so all the $\mathcal{N}=1$ chiral multiplets from
$\mathcal{N}=2$ multiplet containing $u_{1}$ disappear. One might explain the
disappearance of other $\mathcal{N}=1$ chiral multiplets from ${\cal E}$-type
multiplets in ${\cal T}^{(A_{1},D_{2N+1})}[u_{1}]$ theories using the same
reasoning.

####  4.1.3 $(A_{1},A_{2N+1})$: four free chirals

In $(A_{1},A_{2N+1})$ theory, the conformal dimension of $u_{1}$ is
$\frac{N+3}{N+2}$, and the Schur index can be recast into the product of four
terms

$\begin{split}&\mathrm{PE}\left[\frac{q+q^{2}+(a+a^{-1})q^{\frac{N+1}{2}}-(a+a^{-1})q^{\frac{N+1}{2}+2}-q^{N+1}-q^{N+2}}{(1-q)(1-q^{N+2})}\right]\\\
=&\mathrm{PE}\left[\frac{q-q^{N+2}}{(1-q)(1-q^{N+2})}\right]\mathrm{PE}\left[\frac{q^{2}-q^{N+1}}{(1-q)(1-q^{N+2})}\right]\mathrm{PE}\left[\frac{aq^{\frac{N+1}{2}}-a^{-1}q^{\frac{N+1}{2}+2}}{(1-q)(1-q^{N+2})}\right]\mathrm{PE}\left[\frac{a^{-1}q^{\frac{N+1}{2}}-aq^{\frac{N+1}{2}+2}}{(1-q)(1-q^{N+2})}\right].\end{split}$
(65)

When $N=1$ the theory is the same as $(A_{1},D_{3})$ and the IR theory after
deforming with $u_{1}$ is just the theory three free chiral multiplets because
the second term in the product becomes $1$ in this case.

When $N>1$, the Schur index of ${\cal T}^{(A_{1},A_{2N+1})}[u_{1}]$ (which is
the same as the UV Schur index) looks like the index of four chirals with
$U(1)_{R}$-charge $\frac{2}{N+3}$, $\frac{4}{N+3}$, $\frac{N+1}{N+3}$, and
$\frac{N+1}{N+3}$. However, the IR theory is more subtle because
$\frac{4}{N+3}=\frac{4}{5}>\frac{2}{3}$ for $N=2$, while
$\frac{N+1}{N+3}>\frac{2}{3}$ for $N>3$ which means that naive R-charges of
these chirals do not violate the unitarity bound so one may question that if
these chirals decouple in the IR. Hence we need carefully analyze the $a$ and
$c$ anomalies to gain more information on IR.

The naive IR $a$ and $c$ anomalies of ${\cal T}^{(A_{1},A_{2N+1})}[u_{1}]$ are

$\begin{split}a_{{\cal
N}=1}&=-\frac{3}{8}\frac{N^{3}-4N^{2}-6N+3}{(N+3)^{3}},\\\ c_{{\cal
N}=1}&=-\frac{1}{8}\frac{2N^{3}-20N^{2}-39N-9}{(N+3)^{3}},\end{split}$ (66)

which are the same as the sum of $a$ and $c$ anomalies of four chirals with
naive R-charge, that is

$\begin{split}a_{{\cal
N}=1}&=a(X(\frac{2}{N+3}))+a(S_{1}(\frac{4}{N+3}))+2a(X^{+}(\frac{N+1}{N+3})),\\\
c_{{\cal
N}=1}&=c(X(\frac{2}{N+3}))+c(S_{1}(\frac{4}{N+3}))+2c(X^{+}(\frac{N+1}{N+3})).\end{split}$
(67)

This is strong evidence that all four chirals become free in the IR even
though some of them have naive R-charge which is above the unitarity bound, so
the actual $a$ and $c$ anomalies of the IR theory should be

$\begin{split}a^{true}_{{\cal N}=1}&=a_{{\cal
N}=1}-a(X(\frac{2}{N+3}))-a(S_{1}(\frac{4}{N+3}))-2a(X^{+}(\frac{N+1}{N+3}))+4\times\frac{1}{48}=\frac{1}{12},\\\
c^{true}_{{\cal N}=1}&=c_{{\cal
N}=1}-c(X(\frac{2}{N+3}))-c(S_{1}(\frac{4}{N+3}))-2c(X^{+}(\frac{N+1}{N+3}))+4\times\frac{1}{24}=\frac{1}{6}.\end{split}$
(68)

Notice that when $N=1$, $S_{1}(\frac{4}{N+3})$ has IR R-charge $1$, which
allows a mass deformation and becomes massive, leaving three free chirals in
this case. Although when $N>3$ the $R_{IR}$ charge of $X^{\pm}$ from
$\hat{\cal B}^{\pm}_{\frac{N+1}{2}}$ is $\frac{N+1}{N+3}$ which do not violate
unitarity bound, these operators still become free in the IR.

####  4.1.4 $(A_{1},D_{2N})$: eight free chirals

For $(A_{1},D_{2N})$ theory, the conformal dimension of $u_{1}$ is
$\frac{N+1}{N}$, and the Schur index is

$\mathrm{PE}\left[\frac{(1+\chi_{1}(z))q+(a+a^{-1})\chi_{{\frac{1}{2}}}(z)q^{\frac{N}{2}}-(a+a^{-1})\chi_{{\frac{1}{2}}}(z)q^{\frac{N+2}{2}}-(1+\chi_{1}(z))q^{N}}{(1-q)(1-q^{N})}\right].$
(69)

When $N=2$, the flavor symmetry of the $(A_{1},D_{4})$ theory enhances to
$SU(3)$ and by the same argument as in $(A_{1},D_{2N+1})$ case the IR theory
is a free theory with eight chiral multiplets.

When $N>2$, the Schur index suggests that the IR theory ${\cal
T}^{(A_{1},D_{2N})}[u_{1}]$ has four chirals with naive R-charge
$\frac{2}{N+1}$ and four chirals with naive $U(1)_{R}$-charge $\frac{N}{N+1}$.
Computation shows that the naive ${\cal N}=1$ $a$ and $c$ anomalies are the
same as $a$ and $c$ anomalies from these chirals

$\begin{split}a_{{\cal N}=1}&=4a(X(\frac{2}{N+1}))+4a(X(\frac{N}{N+1})),\\\
c_{{\cal N}=1}&=4c(X(\frac{2}{N+1}))+4c(X(\frac{N}{N+1})).\end{split}$ (70)

By the same argument as the previous section, one concludes that the IR theory
is a theory with eight free chiral multiplets for $N>2$ as well, and the
actual central charges are $a=\frac{1}{6}$ and $c=\frac{1}{3}$. These chirals
come from ${\cal N}=2$ $\hat{B}$ mutliplets which contain generators of the
Higgs branch operators of parent $\mathcal{N}=2$ theory, and the disappearance
of $S_{1}$ operator can be explained using the chiral ring relation
$S_{1}+X^{2}=0$, here $X$ has IR $U(1)_{R}$ charge ${2\over N+1}$.

###  4.2 $(A_{k-1},G)$ theories

We briefly summarize results for $(A_{k-1},G)$ theories, with
$\mathrm{gcd}(k,h^{\vee})=1$ where $h^{\vee}$ is the dual Coxeter number of
$G$. The Schur index of $(A_{k-1},G)$ theories with $G=ADE$ deformed by the
smallest Coulomb branch operator is

$\mathrm{PE}\left[\frac{\sum_{i}q^{m_{i}+1}-\sum_{i}q^{k+m_{i}}}{(1-q)(1-q^{k+h^{\vee}})}\right]=\mathrm{PE}\left[\frac{\sum_{i}q^{m_{i}+1}-\sum_{i}q^{k+h-m_{i}}}{(1-q)(1-q^{k+h^{\vee}})}\right],$
(71)

where $\\{m_{i}\\}$’s are exponents of $G$. The equality uses the fact that
the set $\\{m_{i}\\}$ is the same as $\\{h^{\vee}-m_{i}\\}$. Here we take
$k>h^{\vee}$.

The smallest Coulomb branch operator of $(A_{k-1},G)$ theory has the conformal
dimension $\frac{k+h^{\vee}+1}{k+h^{\vee}}$, therefore the ${\cal N}=1$ index
is the same as the Schur index 71 which also looks like the index
$\mathrm{rank}\,G$ chiral multiplets with the substitution $p=q^{k+h^{\vee}}$.
By the same logic we conjecture that ${\cal T}^{(A_{k-1},G)}[u_{1}]$ is a
theory with $\mathrm{rank}\,G$ free chiral multiplets. This fact is also
confirmed by the central charge computation.

###  4.3 $W^{k^{\prime}}(\mathfrak{g},f)$ theories

Now we consider ${\cal N}=2$ SCFTs corresponds to the vertex operator algebra
$W^{k^{\prime}}(\mathfrak{g},f)$ with
$k^{\prime}=-h^{\vee}+\frac{h^{\vee}}{h^{\vee}+k}$, $h^{\vee}$ dual Coxeter
number of $\mathfrak{g}$ and $f$ labels a nilpotent orbit of $\mathfrak{g}$.
For generic $f$, the Lie group $G$ of $\mathfrak{g}$ has a subgroup
$SU(2)\times G_{F}$ with $G_{F}$ being the flavor group dtermined by $f$.
Under this subgroup the adjoint represetation of $G$ decomposes as

$adj_{G}=\sum_{j}V_{j}\otimes R_{j},$ (72)

where $V_{j}$ is the spin $j$ representation of $SU(2)$ and $R_{j}$ is the
corresponding representation of $G_{F}$. The Schur index is

$\mathrm{PE}\left[\frac{\sum_{j}q^{1+j}\chi_{R_{j}}(z)-q^{k+h^{\vee}}\sum_{j}q^{-j}\chi_{R_{j}}(z)}{(1-q)(1-q^{k+h^{\vee}})}\right].$
(73)

Again the smallest conformal dimension of Coulomb branch operators is
$\frac{k+h^{\vee}+1}{k+h^{\vee}}$. Under this deformation the ${\cal N}=1$
index at $p=q^{k+h^{\vee}}$ limit looks like an index of $\sum_{j}\dim R_{j}$
chirals. Hence we conjecture that the IR theory ${\cal
T}^{W^{k^{\prime}}(\mathfrak{g},f)}[u_{1}]$ is a theory with $\sum_{j}\dim
R_{j}$ free chirals. Again, central charge computation is consistent with the
claim.

##  5 Other phenomenon

###  5.1 Interacting theory plus free chirals

We have seen that the IR theory of deformed $\mathcal{N}=2$ theory could be
interacting ${\cal N}=1$ SCFT, or free theory. Here we study the intermediate
case where the IR theory consists of interacting theory plus free chirals.

Let us first look at the IR theory ${\cal T}^{(A_{1},A_{6})}[u_{2}]$ of
$(A_{1},A_{6})$. According to equation 7, the ${\cal N}=2$ index of
$(A_{1},A_{6})$ theory is

$\begin{split}{\cal I}^{(A_{1},A_{6})}_{{\cal
N}=2}=&\left[\prod_{i=1}^{3}\frac{\Gamma\left(\left(\frac{pq}{t}\right)^{\frac{2(4+i)}{9}}\right)}{\Gamma\left(\left(\frac{pq}{t}\right)^{\frac{2i}{9}}\right)}\right]\Gamma\left(\left(\frac{pq}{t}\right)^{\frac{1}{9}}\right)^{3}\\\
&\times\frac{\kappa^{3}}{2^{3}3!}\oint[d\mathbf{z}]\prod_{\alpha\in\Delta}\frac{\Gamma\left(\mathbf{z}^{\alpha}\left(\frac{pq}{t}\right)^{\frac{1}{9}}\right)}{\Gamma(\mathbf{z}^{\alpha})}\prod_{w\in
R}\Gamma\left(\mathbf{z}^{w}\left(\frac{pq}{t}\right)^{\frac{4}{9}}t^{\frac{1}{2}}\right)\Gamma\left(\mathbf{z}^{w}\left(\frac{pq}{t}\right)^{-\frac{1}{3}}t^{\frac{1}{2}}\right),\end{split}$
(74)

where $\mathbf{z}=\\{z_{1},z_{2},z_{3}\\}$ are fugacities of $Sp(6)$.
Deforming the theory with $u_{2}$ means that we take $t=(pq)^{1/4}$ in the
${\cal N}=2$ index. However, taking this limit naively leads to a
$\Gamma\left(pq\right)$ factor in the index and by definition
$\Gamma\left(pq\right)=0$.

To take the correct ${\cal N}=1$ limit, we use the similar trick as the
previous section. First setting $t=(pq)^{\frac{1}{4}+\epsilon}$, then
performing the integral and taking $\epsilon\rightarrow 0$ limit in the end.
After integrating over $z_{1}$ and $z_{2}$ fugacities and rescaling $z_{3}$,
we find that

$\begin{split}{\cal I}^{{\cal T}^{(A_{1},A_{6})}[u_{2}]}_{{\cal
N}=1}&=\frac{\kappa}{2}\Gamma\left((pq)^{\frac{1}{3}}\right)\frac{\Gamma\left((pq)^{\frac{1}{12}}\right)}{\Gamma\left((pq)^{\frac{1}{6}}\right)}\oint
dz\frac{\Gamma\left((pq)^{\frac{1}{12}}z^{\pm 2}\right)}{\Gamma\left(z^{\pm
2}\right)}\Gamma\left((pq)^{\frac{5}{24}}z^{\pm}\right)\Gamma\left((pq)^{\frac{11}{24}}z^{\pm}\right)\\\
&=\Gamma\left((pq)^{\frac{1}{3}}\right){\cal
I}^{(A_{1},A_{2}),u^{2}_{1}}_{{\cal N}=1},\end{split}$ (75)

which is the ${\cal N}=1$ index of the IR theory ${\cal
T}^{(A_{1},A_{2})}[u_{1}^{2}]$ times the index of the theory of a free chiral
multiplet. This result confirms the previous observation that the IR theory
${\cal T}^{(A_{1},A_{6})}[u_{2}]$ is the IR theory ${\cal
T}^{(A_{1},A_{2})}[u^{2}_{1}]$ theory plus a free chiral multiplet using $a$
and $c$ anomalies. The decoupled free chiral comes from $S_{1}$ whose naive
$R_{IR}$ charge is ${2\over 3}$, which is the same as a free chiral.

It is now easy to look for generalizations: we look for a $\mathcal{N}=1$
preserving deformation using the operator ${\cal O}_{{\cal E}_{r_{0}(0,0)}}$,
such that the $S_{1}$ chiral multiplet becomes free, i.e. we need to solve the
equation

${3(r_{0}+r_{min}-2)\over r_{0}}=1$ (76)

Here $r_{min}$ is the smallest Coulomb branch operator charge. For example, if
we look at $(A_{1},A_{2N})$ theory, and we have $r_{min}={2N+4\over 2N+3}$,
and we have

$r_{0}={3N+3\over 2N+3}$ (77)

There are operators with this $U(1)_{r}$ charge in the Coulomb branch if $N$
is odd.

###  5.2 Accidental symmetry

Interesting things happen when we consider ${\cal T}^{(A_{1},A_{2N})}[u_{2}]$
theory with $N\geq 4$. The $U(1)_{r}$ charge $r_{0}$ of $u_{2}$ is
$\frac{2N+6}{2N+3}$, so we should take $t=(pq)^{\frac{3}{2N+6}}$ in the ${\cal
N}=2$ index to get the index of ${\cal T}^{(A_{1},A_{2N})}[u_{2}]$. We found
that after integrating out $z_{1},z_{2},\cdots,z_{N-1}$ and rescaling $z_{N}$,
the ${\cal N}=1$ index is

$\begin{split}{\cal I}^{{\cal T}^{(A_{1},A_{2N})}[u_{2}]}_{{\cal
N}=1}=&\frac{\kappa}{2}\Gamma\left((pq)^{\frac{2}{N+3}}\right)\Gamma\left((pq)^{\frac{3}{N+3}}\right)\\\
&\times\frac{\Gamma\left((pq)^{\frac{1}{2N+6}}\right)}{\Gamma\left((pq)^{\frac{1}{N+3}}\right)}\oint
dz\frac{\Gamma\left((pq)^{\frac{1}{2N+6}}z^{\pm 2}\right)}{\Gamma\left(z^{\pm
2}\right)}\Gamma\left((pq)^{\frac{2N-1}{4N+12}}z^{\pm}\right)\Gamma\left((pq)^{\frac{2N+5}{4N+12}}z^{\pm}\right)\\\
=&\Gamma\left((pq)^{\frac{2}{N+3}}\right)\Gamma\left((pq)^{\frac{3}{N+3}}\right){\cal
I}^{N}_{{\cal N}=1},\end{split}$ (78)

where we define ${\cal I}^{N}_{{\cal N}=1}$ to be the integral

${\cal I}^{N}_{{\cal
N}=1}=\frac{\kappa}{2}\frac{\Gamma\left((pq)^{\frac{1}{2N+6}}\right)}{\Gamma\left((pq)^{\frac{1}{N+3}}\right)}\oint
dz\frac{\Gamma\left((pq)^{\frac{1}{2N+6}}z^{\pm 2}\right)}{\Gamma\left(z^{\pm
2}\right)}\Gamma\left((pq)^{\frac{2N-1}{4N+12}}z^{\pm}\right)\Gamma\left((pq)^{\frac{2N+5}{4N+12}}z^{\pm}\right).$
(79)

The above index formula suggests the the IR theory contains two free chiral
multiplets and an interacting theory. We try to understand above results using
the index. Assuming that the IR theory ${\cal T}^{(A_{1},A_{2N})}[u_{2}]$
contains two free chiral multiplets (which comes from $S_{1}$ and $S_{2}$),
and the interacting theory whose index is ${\cal I}^{N}_{{\cal N}=1}$, one can
study the generators and relations of this interacting piece by computing the
single letter index from ${\cal I}^{N}_{{\cal N}=1}$. For example, when $N=4$,
the index takes the form

$\displaystyle{\cal I}^{{\cal T}^{(A_{1},A_{8})}[u_{2}]}_{{\cal N}=1}=PE[{\cal
I}_{\bar{\cal B}_{\frac{4}{7}(0,0)}}+{\cal I}_{\bar{\cal
B}_{\frac{8}{7}({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal
B}_{\frac{6}{7}(0,0)}}+{\cal I}_{\hat{\cal C}_{(0,0)}}-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\cal
I}_{\bar{\cal B}_{\frac{16}{7}(0,0)}}-{\cal I}_{\bar{\cal
B}_{\frac{16}{7}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{\frac{18}{7}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{\frac{20}{7}(0,0)}}+{\cal I}_{\hat{\cal
C}_{({\frac{1}{2}},{\frac{1}{2}})}}+\cdots].$ (80)

The naive $R_{IR}$ charge for chiral operators are listed in table 5. From the
index, we see that generators are $S_{1},\lambda_{1}^{\alpha},S_{2}$. Other
chiral operators listed in table 5 are no longer generators of the chiral ring
as we have following chiral ring relation ${\cal
B}=f(\lambda_{1}^{\alpha},S_{1},S_{2})$. There is a contribution from the
conserved current multiplet $\hat{{\cal C}}_{(0,0)}$ in the single letter
index which indicates that there is an accidental $U(1)$ symmetry in the IR.
From the index, we also see that the chiral ring relation for
$S_{1},\lambda_{1}^{\alpha},S_{2}$ takes the following form

$\displaystyle\lambda_{1}^{\alpha}\lambda_{1\alpha}+S_{1}S_{2}^{2}=0$
$\displaystyle\lambda_{1}^{\alpha}S_{1}^{2}=0$
$\displaystyle\lambda_{1}^{\alpha}S_{1}S_{2}=0$
$\displaystyle\lambda_{1}^{\alpha}\lambda_{1\alpha}S_{1}+S_{1}^{2}S_{2}^{2}+S_{1}^{5}=0$
(81)

Next, the index for the interacting part takes the following form

$\displaystyle{\cal I}^{4}_{{\cal N}=1}=PE[{\cal I}_{\bar{\cal
B}_{\frac{8}{7}(0,0)}}+{\cal I}_{\bar{\cal
B}_{\frac{8}{7}({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal
B}_{\frac{10}{7}(0,0)}}+{\cal I}_{\hat{\cal C}_{(0,0)}}-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\cal
I}_{\bar{\cal B}_{\frac{16}{7}(0,0)}}-{\cal I}_{\bar{\cal
B}_{\frac{16}{7}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{\frac{18}{7}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{\frac{20}{7}(0,0)}}+{\cal I}_{\hat{\cal
C}_{({\frac{1}{2}},{\frac{1}{2}})}}+\cdots].$ (82)

Here generators and their $R_{IR}$ charges are $({\cal
O},\lambda_{\alpha},S)=({8\over 7},{8\over 7},{10\over 7})$. They might be
identified with chiral operators of the original theory as $({\cal
O},\lambda_{\alpha},S)=(S_{1}^{2},\lambda_{1}^{\alpha},S_{1}S_{2})$, and the
chiral ring relation takes the following form

$\displaystyle\lambda_{1}^{\alpha}\lambda_{1\alpha}=0,$
$\displaystyle\lambda_{1}^{\alpha}{\cal O}=0,$
$\displaystyle\lambda_{1}^{\alpha}S=0,$ $\displaystyle S^{2}=0.$ (83)

$\mathcal{N}=2$ chiral $u_{i}$ |  $\mathcal{N}=1$ chiral $({\cal O}_{i},\lambda_{i}^{\alpha},S_{i})$)
---|---
$u_{1}$ | $(\left.\frac{12}{7},\frac{8}{7},\frac{4}{7}\right)$
$u_{2}$ | $(2,\frac{10}{7},\frac{6}{7})$
$u_{3}$ | $(\frac{16}{7},\frac{12}{7},\frac{8}{7})$
$u_{4}$ | $(\frac{18}{7},2,\frac{10}{7})$
Table 5: $U(1)_{R}$ charge of various chiral operators of theory ${\cal
T}^{(A_{1},A_{2N})}[u_{2}]$.

We would like to make several remarks for the above series of theories:

  1. 1.

The computation of chiral ring can be generalized to arbitrary $N\geq 4$, and
we get the same chiral ring generators and relations as 83. This suggests the
interacting theory should be the same one for all $N$ although generators for
the chiral ring have different naive $R_{IR}$ charge. Fortunately, there is an
accidental $U(1)$ symmetry and so the true $U(1)_{R}$ symmetry would be the
mixing of the naive $R_{IR}$ symmetry and this accidental $U(1)$ symmetry.

  2. 2.

When $N=4$ or $5$, $S_{2}$ does not violate the unitarity bound. One might
propose that $S_{2}$ does not become free, but we find an inconsistency as
there would be no combination of chiral operators give the third chiral ring
relation of 81 if we just take $S_{1}$ to be free.

  3. 3.

When $N=3$, there are two differences. Before taking the unitarity bound into
consideration, generators for the chiral ring is
$(S_{1},\lambda_{1}^{\alpha},S_{2})=({2\over 3},{7\over 6},1)$, as the index
takes the form

$\displaystyle{\cal I}^{{\cal T}^{(A_{1},A_{6})}[u_{2}]}_{{\cal N}=1}=PE[{\cal
I}_{\bar{\cal B}_{\frac{2}{3}(0,0)}}+{\cal I}_{\bar{\cal
B}_{\frac{7}{6}({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal B}_{1(0,0)}}+{\cal
I}_{\hat{\cal C}_{(0,0)}}-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\cal
I}_{\bar{\cal B}_{\frac{7}{3}(0,0)}}-{\cal I}_{\bar{\cal
B}_{\frac{5}{2}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{\frac{13}{6}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{\frac{8}{3}(0,0)}}+{\cal I}_{\hat{\cal
C}_{({\frac{1}{2}},{\frac{1}{2}})}}+\cdots].$ (84)

Chiral ring relations take the following form

$\displaystyle\lambda_{1}^{\alpha}\lambda_{1\alpha}+S_{1}^{2}S_{2}=0,$
$\displaystyle\lambda_{1}^{\alpha}S_{1}^{2}=0,$
$\displaystyle\lambda_{1}^{\alpha}S_{2}=0,$ $\displaystyle
S_{1}S_{2}^{2}+S_{1}^{4}=0.$ (85)

These relations are different from those listed in 81, and especially the
third relation has been changed, as properties of the IR theory. First there
is no accidental $U(1)$ symmetry because the chiral operator $S_{2}^{2}$ is
marginal (has $R_{IR}$ charge $2$), which can be combined with the conserved
current of $U(1)$ symmetry (one can also see the fact by noticing that there
is no $t^{6}$ term in the full index). Secondly only $S_{1}$ becomes free, and
the index for the interacting theory takes the form

$\displaystyle{\cal I}^{{\cal T}^{(A_{1},A_{2N})}[u_{2}]}_{{\cal
N}=1}=PE[{\cal I}_{\bar{\cal B}_{1(0,0)}}+{\cal I}_{\bar{\cal
B}_{\frac{7}{6}({\frac{1}{2}},0)}}+{\cal I}_{\bar{\cal
B}_{\frac{4}{3}(0,0)}}+{\cal I}_{\hat{\cal C}_{(0,0)}}-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\cal
I}_{\bar{\cal B}_{\frac{7}{3}(0,0)}}-{\cal I}_{\bar{\cal
B}_{\frac{5}{2}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{\frac{13}{6}({\frac{1}{2}},0)}}-{\cal I}_{\bar{\cal
B}_{\frac{8}{3}(0,0)}}+{\cal I}_{\hat{\cal
C}_{({\frac{1}{2}},{\frac{1}{2}})}}+\cdots],$ (86)

and now chiral ring generators might be identified as $({\cal
O},\lambda_{\alpha},S)=(S_{1}^{2},\lambda_{1}^{\alpha},S_{2})$, and chiral
ring relations take the form

$\displaystyle\lambda_{1}^{\alpha}\lambda_{1\alpha}+{\cal O}S=0,$
$\displaystyle\lambda_{1}^{\alpha}{\cal O}=0,$
$\displaystyle\lambda_{1}^{\alpha}S=0,$ $\displaystyle S^{2}=0.$ (87)

##  6 Conclusions

It was shown in Xie:2019aft  that one can get 4d $\mathcal{N}=1$ SCFTs by
turning on $\mathcal{N}=1$ preserving deformations of 4d $\mathcal{N}=2$
Argyres-Douglas theories. AD theories are special in that they contain many
interesting $\mathcal{N}=1$ chiral operators ${\cal O}_{{\cal E}_{r(0,0)}}$
with scaling dimension less than three, and so one can get interesting IR
$\mathcal{N}=1$ SCFTs by turning on tese relevant dimensions. This is in
strong contrast with Lagrangian theory or $T_{N}$ theory Benini:2009mz .

Since the space of 4d AD theories is now amazingly large Xie:2012hs ;
Xie:2015rpa  and one can get a large class of $\mathcal{N}=1$ SCFTs with these
deformations. The space of these SCFTs seem very useful in learning exact
properties of $\mathcal{N}=1$ SCFTs, as many properties of AD theories such as
their Coulomb branch spectrum, Higgs branch spectrum, superconformal index,
and etc have been worked out. Results of parent $\mathcal{N}=2$ SCFTs can thus
be used to learn properties of IR SCFTs. The IR properties of $\mathcal{N}=1$
preserving RG flow is a lot more subtle though, and so it is important to get
some exact results about them. The index of a 4d $\mathcal{N}=1$ SCFT is a
very useful tool. In this paper, we use the known result of $\mathcal{N}=2$
index to get the $\mathcal{N}=1$ index, which is then used to learn properties
of IR theory, such as the phase structure, the chiral ring, the accidental
symmetry, and etc.

We would like to make some further comments on lessons learned about
$\mathcal{N}=1$ dynamics. Here we start with a $\mathcal{N}=2$ SCFT and have a
$\mathcal{N}=1$ preserving RG flow with a candidate IR $U(1)_{R}$ symmetry:

  1. 1.

For some RG flows, many operators violate unitarity bound under the naive
$R_{IR}$ symmetry. It was pointed out that in these cases Kutasov:2003iy ,
such operators become free. However, we find that this is actually quite
subtle. It seems that if the operator with worst violation becomes free, many
other chiral operators simply drop out from the chiral ring. This happens
because if we take a chiral operator free, the chiral ring relation also
changes, and under the modified chiral ring relation, some of the operators
become trivial in the chiral ring. In this paper, we found many models where
the above phenomenon indeed happens. This suggests that the key object seems
to be the chiral ring, as suggested in Collins:2016icw .

  2. 2.

Another subtle issue is the accidental symmetry, as we have shown in some
models where there is no apparent appearance of extra symmetry. The models
studied in this paper shows that one must be very careful about the accidental
symmetry of any studies of $\mathcal{N}=1$ theories using the RG flow. This
issue is often overlooked in the literature and might lead to serious
mistakes.

  3. 3.

The IR phase structure of our RG flow is quite rich: it could be interacting
SCFTs, free theories, interacting theories plus free chirals, and etc. The IR
SCFTs could be isolated or have conformal manifold. We have not found the
appearance of any gapped phase though.

The space of $\mathcal{N}=1$ SCFTs found in Xie:2019aft  is very large and
quite interesting, and these new $\mathcal{N}=1$ theories definitely deserve
further studies.

## Acknowledgements

DX and Wenbin Yan are supported by Yau Mathematical Science Center at Tsinghua
University, the Young overseas high-level talents introduction plan, national
key research and development program of China (NO. 2020YFA0713000), and NNSF
of China with Grant NO: 11847301 and 12047502.

##  Appendix A ${\cal N}=2$ BPS multiplets and their indices

A generic long multiplet ${\cal A}_{R,r(j_{1},j_{2})}^{\Delta}$ of the
$\mathcal{N}=2$ superconformal algebra is generated by the action of the eight
Poincaré supercharges ${\cal Q}$ and $\tilde{{\cal Q}}$ on a superconformal
primary, which by definition is annihilated by all conformal supercharges
${\cal S}$. If some combination of the ${\cal Q}$s also annihilates the
primary, the corresponding multiplet is shorter and the conformal dimensions
of all its members are protected against quantum corrections. The shortening
conditions for the $\mathcal{N}=2$ superconformal algebra were studied in
Dobrev:1985qv ; Dobrev:1985qz ; Dolan:2002zh . We follow the nomenclature of
Dolan:2002zh , except our $U(1)_{r}$ charge $r$ is $-r$ of Dolan:2002zh . The
classification scheme under our notation is summarized in table 6. Let us take
a moment to explain the notation. The state
$|R,r\rangle^{h.w.}_{(j_{1},j_{2})}$ is the highest weight state with
$SU(2)_{R}$ spin $R>0$, $U(1)_{r}$ charge $r$, which can have either sign, and
Lorentz quantum numbers $(j_{1},j_{2})$. The multiplet built on this state is
denoted as $\mathcal{X}_{R,r(j_{1},j_{2})}$, where the letter $\mathcal{X}$
characterizes the shortening condition. The left column of table 6 labels the
condition. A superscript on the label corresponds to the index
$\mathcal{I}=1,2$ of the supercharge that kills the primary: for example
${\cal B}_{1}$ refers to ${\cal Q}_{1\alpha}$. Similarly a “bar” on the label
refers to the conjugate condition: for example $\bar{{\cal B}}_{2}$
corresponds to $\tilde{Q}_{2\,\dot{\alpha}}$ annihilating the state; this
would result in the short anti-chiral multiplet $\bar{{\cal
B}}_{R,r(j_{1},0)}$, obeying $\Delta=2R+r$. Note that conjugation reverses the
signs of $r$, $j_{1}$ and $j_{2}$ in the expression of the conformal
dimension. The highest weight state of $\bar{\cal E}_{r(j_{1},0)}$ multiplet
is annihilated by both $\tilde{Q}_{1\dot{\alpha}}$ and
$\tilde{Q}_{2\dot{\alpha}}$, and we call it the ${\cal N}=2$ chiral multiplet
with BPS condition $\Delta=r$ in our convention of $U(1)_{r}$151515In
Dolan:2002zh , the authors call ${\cal E}_{r(0,j_{1})}$ chiral multiplet. Here
we call $\bar{\cal E}_{r(j_{1},0)}$ chiral to be consistent with the
definition of chiral multiplet in the usual ${\cal N}=1$ convention. .

Shortening Conditions | Multiplet
---|---
${\cal B}_{1}$ | ${\cal Q}_{1\alpha}|R,r\rangle^{h.w.}=0$ | $j_{1}=0$ | $\Delta=2R-r$ | ${\cal B}_{R,r(0,j_{2})}$
$\bar{{\cal B}}_{2}$ | $\tilde{{\cal Q}}_{2\dot{\alpha}}|R,r\rangle^{h.w.}=0$ | $j_{2}=0$ | $\Delta=2R+r$ | $\bar{{\cal B}}_{R,r(j_{1},0)}$
${\cal E}$ | ${\cal B}_{1}\cap{\cal B}_{2}$ | $R=0$ | $\Delta=-r$ | ${\cal E}_{r(0,j_{2})}$
$\bar{\cal E}$ | $\bar{\cal B}_{1}\cap\bar{\cal B}_{2}$ | $R=0$ | $\Delta=r$ | $\bar{\cal E}_{r(j_{1},0)}$
$\hat{{\cal B}}$ | ${\cal B}_{1}\cap\bar{B}_{2}$ |  $r=0$, $j_{1},j_{2}=0$ | $\Delta=2R$ | $\hat{{\cal B}}_{R}$
${\cal C}_{1}$ | $\epsilon^{\alpha\beta}{\cal Q}_{1\beta}|R,r\rangle_{\alpha}^{h.w.}=0$ |  | $\Delta=2+2j_{1}+2R-r$ | ${\cal C}_{R,r(j_{1},j_{2})}$
|  $({\cal Q}_{1})^{2}|R,r\rangle^{h.w.}=0$ for $j_{1}=0$ |  | $\Delta=2+2R-r$ | ${\cal C}_{R,r(0,j_{2})}$
$\bar{\cal C}_{2}$ | $\epsilon^{\dot{\alpha}\dot{\beta}}\tilde{\cal Q}_{2\dot{\beta}}|R,r\rangle_{\dot{\alpha}}^{h.w.}=0$ |  | $\Delta=2+2j_{2}+2R+r$ | $\bar{\cal C}_{R,r(j_{1},j_{2})}$
|  $(\tilde{\cal Q}_{2})^{2}|R,r\rangle^{h.w.}=0$ for $j_{2}=0$ |  | $\Delta=2+2R+r$ | $\bar{\cal C}_{R,r(j_{1},0)}$
| ${\cal C}_{1}\cap{\cal C}_{2}$ | $R=0$ | $\Delta=2+2j_{1}-r$ | ${\cal C}_{0,r(j_{1},j_{2})}$
| $\bar{\cal C}_{1}\cap\bar{\cal C}_{2}$ | $R=0$ | $\Delta=2+2j_{2}+r$ | $\bar{\cal C}_{0,r(j_{1},j_{2})}$
$\hat{{\cal C}}$ | ${\cal C}_{1}\cap\bar{{\cal C}}_{2}$ | $r=j_{1}-j_{2}$ | $\Delta=2+2R+j_{1}+j_{2}$ | $\hat{{\cal C}}_{R(j_{1},j_{2})}$
| ${\cal C}_{1}\cap{\cal C}_{2}\cap\bar{{\cal C}}_{1}\cap\bar{{\cal C}}_{2}$ | $R=0,r=j_{1}-j_{2}$ | $\Delta=2+j_{1}+j_{2}$ | $\hat{{\cal C}}_{0(j_{1},j_{2})}$
${\cal D}$ | ${\cal B}_{1}\cap\bar{{\cal C}_{2}}$ | $-r=j_{2}+1$ | $\Delta=1+2R+j_{2}$ | ${\cal D}_{R(0,j_{2})}$
$\bar{\cal D}$ | $\bar{\cal B}_{2}\cap{{\cal C}_{1}}$ | $r=j_{1}+1$ | $\Delta=1+2R+j_{1}$ | $\bar{\cal D}_{R(j_{1},0)}$
| ${\cal E}\cap\bar{{\cal C}_{2}}$ | $-r=j_{2}+1,R=0$ | $\Delta=-r=1+j_{2}$ | ${\cal D}_{0(0,j_{2})}$
| $\bar{\cal E}\cap{{\cal C}_{1}}$ | $r=j_{1}+1,R=0$ | $\Delta=r=1+j_{1}$ | $\bar{\cal D}_{0(j_{1},0)}$
Table 6: Shortening conditions and short multiplets for the $\mathcal{N}=2$
superconformal algebra.

The superconformal index counts with signs the protected states of the theory,
up to equivalence relations that set to zero all sequences of short multiplets
that may in principle recombine into long multiplets. The recombination rules
for ${\cal N}=2$ superconformal algebra are Dolan:2002zh

$\displaystyle{\cal A}_{R,r(j_{1},j_{2})}^{2R+r+2j_{1}+2}$
$\displaystyle\simeq$ $\displaystyle{\cal C}_{R,r(j_{1},j_{2})}\oplus{\cal
C}_{R+\frac{1}{2},r-\frac{1}{2}(j_{1}-\frac{1}{2},j_{2})}\,,$ (88)
$\displaystyle{\cal A}_{R,r(j_{1},j_{2})}^{2R-r+2j_{2}+2}$
$\displaystyle\simeq$ $\displaystyle\bar{\cal
C}_{R,r(j_{1},j_{2})}\oplus\bar{\cal
C}_{R+\frac{1}{2},r+\frac{1}{2}(j_{1},j_{2}-\frac{1}{2})}\,,$ (89)
$\displaystyle{\cal A}_{R,j_{2}-j_{1}(j_{1},j_{2})}^{2R+j_{1}+j_{2}+2}$
$\displaystyle\simeq$ $\displaystyle\hat{{\cal
C}}_{R(j_{1},j_{2})}\oplus\hat{{\cal
C}}_{R+\frac{1}{2}(j_{1}-\frac{1}{2},j_{2})}\oplus\hat{{\cal
C}}_{R+\frac{1}{2}(j_{1},j_{2}-\frac{1}{2})}\oplus\hat{{\cal
C}}_{R+1(j_{1}-\frac{1}{2},j_{2}-\frac{1}{2})}\,.$ (90)

The ${\cal C}$, $\bar{\cal C}$ and $\hat{\cal C}$ multiplets obey certain
“semi-shortening” conditions, while ${\cal A}$ multiplets are generic long
multiplets. A long multiplet whose conformal dimension is exactly at the
unitarity threshold can be decomposed into shorter multiplets according to
(88,89,90). We can formally regard any multiplet obeying some shortening
condition (with the exception of the ${\cal E}$ ($\bar{\cal E}$) types, and
$\bar{{\cal D}}_{0(j_{1},0)}$ (${\cal D}_{0(0,j_{2})}$) types) as a multiplet
of type ${\cal C}$, $\bar{\cal C}$ or $\hat{\cal C}$ by allowing the spins
$j_{1}$ and $j_{2}$, whose natural range is over the non-negative half-
integers, to take the value $-1/2$ as well. The translation is as follows:

$\displaystyle{\cal C}_{R,r(-\frac{1}{2},j_{2})}\simeq{\cal
B}_{R+\frac{1}{2},r-\frac{1}{2}(0,j_{2})},\quad\bar{\cal
C}_{R,r(j_{1},-\frac{1}{2})}\simeq\bar{\cal
B}_{R+\frac{1}{2},r+\frac{1}{2}(j_{1},0)}\,,$ (91) $\displaystyle\hat{{\cal
C}}_{R(-\frac{1}{2},j_{2})}\simeq{\cal
D}_{R+\frac{1}{2}(0,j_{2})},\qquad\quad\qquad\hat{{\cal
C}}_{R(j_{1},-\frac{1}{2})}\simeq\bar{{\cal D}}_{R+\frac{1}{2}(j_{1},0)}\,,$
(92) $\displaystyle\hat{{\cal C}}_{R(-\frac{1}{2},-\frac{1}{2})}\simeq{\cal
D}_{R+\frac{1}{2}(0,-\frac{1}{2})}\simeq\bar{{\cal
D}}_{R+\frac{1}{2}(-\frac{1}{2},0)}\simeq\hat{{\cal B}}_{R+1}\,.$ (93)

Note how these rules flip statistics: a multiplet with bosonic primary
($j_{1}+j_{2}$ integer) is turned into a multiplet with fermionic primary
($j_{1}+j_{2}$ half-odd), and vice versa. With these conventions, the rules
(88, 89, 90) are the most general recombination rules. The ${\cal E}$ and
$\bar{\cal E}$ multiplets never recombine.

The index of the ${\cal C}$ and ${\cal E}$ type multiplets vanishes
identically (the choice of supercharge with respect to which the index is
computed, ${\cal Q}=\widetilde{\cal Q}_{1\dot{-}}$, breaks the symmetry
between ${\cal C}$ (${\cal E}$) and $\bar{\cal C}$ ($\bar{\cal E}$)
multiplets). The index of all remaining short multiplets can be specified by
listing the index of $\bar{\cal C},\hat{\cal C}$ , $\bar{\cal E}$, ${\cal
D}_{0(0,j_{2})}$, and $\bar{\cal D}_{0(j_{1},0)}$ multiplets,

$\displaystyle{\cal I}_{\bar{\cal C}_{R,r(j_{1},j_{2})}}$ $\displaystyle=$
$\displaystyle-(-1)^{2(j_{1}+j_{2})}p^{j_{2}+r}q^{j_{2}+r}t^{R-r-1}\frac{(t-pq)(t-p)(t-q)}{(1-p)(1-q)}\chi_{j_{1}}\left(\sqrt{\frac{p}{q}}\right)\,,$
$\displaystyle{\cal I}_{{\hat{\cal C}}_{R(j_{1},j_{2})}}$ $\displaystyle=$
$\displaystyle(-1)^{2(j_{1}+j_{2})}p^{j_{1}}q^{j_{1}}t^{R-j_{1}+j_{2}-1}\frac{t-pq}{(1-p)(1-q)}\left(p^{\frac{1}{2}}q^{\frac{1}{2}}t\chi_{j_{1}+{\frac{1}{2}}}\left(\sqrt{\frac{p}{q}}\right)-pq\chi_{j_{1}}\left(\sqrt{\frac{p}{q}}\right)\right)\,,$
$\displaystyle{\cal I}_{\bar{\cal E}_{r(j_{1},0)}}$ $\displaystyle=$
$\displaystyle(-1)^{2j_{1}}p^{r-1}q^{r-1}t^{-r}\frac{(t-p)(t-q)}{(1-p)(1-q)}\chi_{j_{1}}\left(\sqrt{\frac{p}{q}}\right)\,,\,$
$\displaystyle{\cal I}_{\bar{{\cal D}}_{0(j_{1},0)}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{2j_{1}}(pq/t)^{j_{1}+1}}{(1-p)(1-q)}\times$
$\displaystyle\qquad\qquad\left((1+t)\chi_{j_{1}}\left(\sqrt{\frac{p}{q}}\right)-\frac{t}{\sqrt{pq}}\chi_{j_{1}+{\frac{1}{2}}}\left(\sqrt{\frac{p}{q}}\right)-\sqrt{pq}\chi_{j_{1}-{\frac{1}{2}}}\left(\sqrt{\frac{p}{q}}\right)\right)\,,$
$\displaystyle{\cal I}_{{\cal D}_{0(0,j_{2})}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{2j_{2}+1}t^{j_{2}}(t-pq)}{(1-p)(1-q)}\,.$ (94)

where the Schur polynomial $\chi_{j}\left(\sqrt{\frac{p}{q}}\right)$ gives the
character of the spin $j$ representation of $SU(2)$.

The Schur limit is simply $t=q$, and the Schur index of the above short
multiplets are

$\displaystyle{\cal I}_{\bar{\cal C}_{R,r(j_{1},j_{2})}}$ $\displaystyle=$
$\displaystyle 0\,,$ $\displaystyle{\cal I}_{{\hat{\cal C}}_{R(j_{1},j_{2})}}$
$\displaystyle=$
$\displaystyle(-1)^{2(j_{1}+j_{2})}\frac{q^{2+R+j_{1}+j_{2}}}{1-q}\,,$
$\displaystyle{\cal I}_{\bar{\cal E}_{r(j_{1},0)}}$ $\displaystyle=$
$\displaystyle 0\,,\,$ $\displaystyle{\cal I}_{\bar{{\cal D}}_{0(j_{1},0)}}$
$\displaystyle=$
$\displaystyle(-1)^{2j_{1}+1}\frac{q^{j_{1}+1}}{1-q}\,,\quad{\cal I}_{{\cal
D}_{0(0,j_{2})}}=(-1)^{2j_{2}+1}\frac{q^{j_{2}+1}}{1-q}\,.$ (95)

##  Appendix B ${\cal N}=1$ BPS multiplets and their indices

In this appendix we summarize some basic facts about $\mathcal{N}=1$
superconformal representation theory. A generic long multiplet ${\cal
A}^{\Delta}_{r(j_{1},j_{2})}$ is generated by the action of $4$ Poincaré
supercharges ${\cal Q}_{\alpha}$ and ${\widetilde{\cal Q}}_{\dot{\alpha}}$ on
superconformal primary which is by definition is annihilated by all conformal
supercharges $\cal S$. In table 7 we have summarized possible shortening and
semishortening conditions. Following the usual convention, we call $\bar{\cal
B}_{r(j_{1},0)}$ chiral multiplets as their highest weight states are
annihilated by $\tilde{Q}_{\dot{\alpha}}$’s.

Shortening Conditions | Multiplet
---|---
${\cal B}$ | ${\cal Q}_{\alpha}|r\rangle^{h.w.}=0$ | $j_{1}=0$ | $\Delta=-\frac{3}{2}r$ | ${\cal B}_{r(0,j_{2})}$
$\bar{{\cal B}}$ | $\bar{{\cal Q}}_{\dot{\alpha}}|r\rangle^{h.w.}=0$ | $j_{2}=0$ | $\Delta=\frac{3}{2}r$ | $\bar{{\cal B}}_{r(j_{1},0)}$
$\hat{{\cal B}}$ | ${\cal B}\cap\bar{{\cal B}}$ | $j_{1},j_{2},r=0$ | $\Delta=0$ | $\hat{{\cal B}}$
${\cal C}$ | $\epsilon^{\alpha\beta}{\cal Q}_{\beta}|r\rangle_{\alpha}^{h.w.}=0$ |  | $\Delta=2+2j_{1}-\frac{3}{2}r$ | ${\cal C}_{r(j_{1},j_{2})}$
|  $({\cal Q})^{2}|r\rangle^{h.w.}=0$ for $j_{1}=0$ |  | $\Delta=2-\frac{3}{2}r$ | ${\cal C}_{r(0,j_{2})}$
$\bar{{\cal C}}$ | $\epsilon^{\dot{\alpha}\dot{\beta}}\bar{{\cal Q}}_{\dot{\beta}}|r\rangle_{\dot{\alpha}}^{h.w.}=0$ |  | $\Delta=2+2j_{2}+\frac{3}{2}r$ | $\bar{{\cal C}}_{r(j_{1},j_{2})}$
|  $(\bar{{\cal Q}})^{2}|r\rangle^{h.w.}=0$ for $j_{2}=0$ |  | $\Delta=2+\frac{3}{2}r$ | $\bar{{\cal C}}_{r(j_{1},0)}$
$\hat{{\cal C}}$ | ${\cal C}\cap\bar{{\cal C}}$ | $\frac{3}{2}r=(j_{1}-j_{2})$ | $\Delta=2+j_{1}+j_{2}$ | $\hat{{\cal C}}_{(j_{1},j_{2})}$
${\cal D}$ | ${\cal B}\cap\bar{{\cal C}}$ | $j_{1}=0,-\frac{3}{2}r=j_{2}+1$ | $\Delta=-\frac{3}{2}r=1+j_{2}$ | ${\cal D}_{(0,j_{2})}$
$\bar{{\cal D}}$ | $\bar{{\cal B}}\cap{\cal C}$ | $j_{2}=0,\frac{3}{2}r=j_{1}+1$ | $\Delta=\frac{3}{2}r=1+j_{1}$ | $\bar{{\cal D}}_{(j_{1},0)}$
Table 7: Possible shortening conditions for the $\mathcal{N}=1$ superconformal
algebra.

At the unitarity threshold, a long multiplet can decompose into (semi)short
multiplets. The splitting rules are:

$\displaystyle{\cal A}_{r(j_{1},j_{2})}^{2+2j_{1}-\frac{3}{2}r}$
$\displaystyle\simeq$ $\displaystyle{\cal C}_{r(j_{1},j_{2})}\oplus{\cal
C}_{r-1(j_{1}-\frac{1}{2},j_{2})}$ $\displaystyle{\cal
A}_{r(j_{1},j_{2})}^{2+2j_{2}+\frac{3}{2}r}$ $\displaystyle\simeq$
$\displaystyle\bar{{\cal C}}_{r(j_{1},j_{2})}\oplus\bar{{\cal
C}}_{r+1(j_{1},j_{2}-\frac{1}{2})}$ $\displaystyle{\cal
A}_{\frac{2}{3}(j_{1}-j_{2})(j_{1},j_{2})}^{2+j_{1}+j_{2}}$
$\displaystyle\simeq$ $\displaystyle\hat{{\cal C}}_{(j_{1},j_{2})}\oplus{\cal
C}_{\frac{2}{3}(j_{1}-j_{2})-1,(j_{1}-\frac{1}{2},j_{2})}\oplus\bar{{\cal
C}}_{\frac{2}{3}(j_{1}-j_{2})+1,(j_{1},j_{2}-\frac{1}{2})}$

We are using a notation where the ${\cal B}$ and $\bar{{\cal B}}$ type
multiplets are formally identified with special cases of ${\cal C}$ and
$\bar{{\cal C}}$ multiplets, as follows

${\cal C}_{r(-\frac{1}{2},j_{2})}\simeq{\cal B}_{r-1(0,j_{2})}\qquad\bar{{\cal
C}}_{r(j_{1},-\frac{1}{2})}\simeq\bar{{\cal B}}_{r+1(j_{1},0)}\,.$ (96)

We define the ${\tt L}$eft (${\tt R}$ight) equivalence class of the multiplet
${\cal C}_{r(j_{1},j_{2})}(\bar{{\cal C}}_{r(j_{1},j_{2})})$ as the class of
multiplets with the same ${\tt L}$eft (${\tt R}$ight) index. From the
splitting rules, we see that the classes can be labeled as
$[-r+2j_{1},j_{2}]_{(-)^{2j_{1}}}^{{\tt L}}$
$([r+2j_{2},j_{1}]_{(-)^{2j_{2}}}^{{\tt R}})$. Moreover, ${\cal
I}_{[-r+2j_{1},j_{2}]_{-}^{{\tt L}}}^{{\tt L}}=-{\cal
I}_{[-r+2j_{1},j_{2}]_{+}^{{\tt L}}}^{{\tt L}}$ and ${\cal
I}_{[r+2j_{2},j_{1}]_{-}^{{\tt R}}}^{{\tt R}}=-{\cal
I}_{[r+2j_{2},j_{1}]_{+}^{{\tt R}}}^{{\tt R}}$. The expressions for the
indices of the equivalent classes are

$\displaystyle{\cal I}_{[\tilde{r},j_{2}]_{\pm}^{{\tt L}}}^{{\tt L}}$
$\displaystyle=$
$\displaystyle\pm(-)^{2j_{2}+1}\frac{(pq)^{{\frac{1}{2}}(\tilde{r}+2)}\chi_{j_{2}}\left(\sqrt{\frac{p}{q}}\right)}{(1-p)(1-q)}$
$\displaystyle{\cal I}_{[\bar{\tilde{r}},j_{1}]_{\pm}^{{\tt R}}}^{{\tt R}}$
$\displaystyle=$
$\displaystyle\pm(-)^{2j_{1}+1}\frac{(pq)^{{\frac{1}{2}}(\bar{\tilde{r}}+2)}\chi_{j_{1}}\left(\sqrt{\frac{p}{q}}\right)}{(1-p)(1-q)}\,$
$\displaystyle\mathcal{I}^{R}[\tilde{r},j_{2}]_{\pm}^{{\tt L}}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle\mathcal{I}^{{\tt
L}}[\bar{\tilde{r}},j_{1}]_{\pm}^{{\tt R}}$ $\displaystyle=$ $\displaystyle
0\,.$

The situation is slightly more involved for the $\hat{{\cal C}}$ and ${\cal
D}$ type multiplets. Unlike the ${\cal B},{\cal C}$ type multiplets, they
contribute both to $\mathcal{I}^{{\tt L}}$ as well as $\mathcal{I}^{{\tt R}}$.
The indices Dolan:2008qi  for the different types of multiplets are collected
in table 8.

Multiplet | $\mathcal{I}^{{\tt L}}$ | $\mathcal{I}^{{\tt R}}$
---|---|---
$\AA_{r(j_{1},j_{2})}^{\Delta}$ | $0$ | $0$
${\cal C}_{r(j_{1},j_{2})}$ | $\mathcal{I}_{[-r+2j_{1},j_{2}]_{(-)^{2j_{1}}}^{{\tt L}}}^{{\tt L}}$ | $0$
$\bar{{\cal C}}_{r(j_{1},j_{2})}$ | $0$ | $\mathcal{I}_{[r+2j_{2},j_{1}]_{(-)^{2j_{2}}}^{{\tt R}}}^{{\tt R}}$
$\hat{{\cal C}}_{(j_{1},j_{2})}$ | $\mathcal{I}_{[\frac{2}{3}j_{2}+\frac{4}{3}j_{1},j_{2}]_{(-)^{2j_{1}}}^{{\tt L}}}^{{\tt L}}$ | $\mathcal{I}_{[\frac{2}{3}j_{1}+\frac{4}{3}j_{2},j_{1}]_{(-)^{2j_{2}}}^{{\tt R}}}^{{\tt R}}$
${\cal D}_{(0,j_{2})}$ | $\mathcal{I}_{[\frac{2}{3}j_{2}-\frac{4}{3},j_{2}]_{-}^{{\tt L}}}^{{\tt L}}+\mathcal{I}_{[\frac{2}{3}j_{2}-\frac{1}{3},j_{2}-\frac{1}{2}]_{-}^{{\tt L}}}^{{\tt L}}$ | $\mathcal{I}_{[\frac{4}{3}j_{2}-\frac{2}{3},0]_{+}^{{\tt R}}}^{{\tt R}}$
$\bar{{\cal D}}_{(j_{1},0)}$ | $\mathcal{I}_{[\frac{4}{3}j_{1}-\frac{2}{3},0]_{+}^{{\tt L}}}^{{\tt L}}$ | $\mathcal{I}_{[\frac{2}{3}j_{1}-\frac{4}{3},j_{1}]_{-}^{{\tt R}}}^{{\tt R}}+\mathcal{I}_{[\frac{2}{3}j_{1}-\frac{1}{3},j_{1}-\frac{1}{2}]_{-}^{{\tt R}}}^{{\tt R}}$
Table 8: Indices $\mathcal{I}^{{\tt L}}$ and $\mathcal{I}^{{\tt R}}$ of the
various short and semi-short multiplets.

##  Appendix C Elliptic Gamma function

The elliptic Gamma function is a two parameter generalization of the Gamma
function,

$\Gamma(z;p,q)\equiv\prod_{j,k\geq
0}\frac{1-z^{-1}p^{j+1}q^{k+1}}{1-zp^{j}q^{k}}.$ (97)

The reader can consult Diejen:2013 ; Spiridonov:2008RuMaS ;
Spiridonov:2005math  for reviews of the elliptic Gamma function and of
elliptic hypergeometric mathematics. We will also use the standard condensed
notations

$\begin{split}&\Gamma(z_{1},\cdots,z_{k};p,q)\equiv\prod_{j=1}^{k}\Gamma(z_{j};p,q),\\\
&\Gamma(z^{\pm};p,q)\equiv\Gamma(z;p,q)\Gamma(1/z;p,q),\end{split}$ (98)

and sometimes just omit parameters $p$ and $q$ so
$\Gamma(z)\equiv\Gamma(z;p,q)$ without other specifications. There are two
useful identities of elliptic Gamma functions

$\begin{split}&\Gamma(z^{2};p,q)=\Gamma(\pm
z,\pm\sqrt{q}z,\pm\sqrt{p}z,\pm\sqrt{pq}z;p,q),\\\
&\Gamma(z;p,q)\Gamma(pq/z;p,q)=1.\end{split}$ (99)

Moreover, the residue of $\Gamma(z/a;p,q)$ at $z=a$ is

$\lim_{a\rightarrow a}(1-z/a)\Gamma(z/a;p,q)=\frac{1}{(p;p)(q;q)},$ (100)

with the q-Pochhammer symbol $(z;q)$ defined as

$(z;q)\equiv\prod_{k=0}^{\infty}\left(1-zq^{k}\right).$ (101)

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