# Superconformal indices for non-Lagrangian theories in five dimensions

Hee-Cheol Kima,b, Minsung Kima, Sung-Soo Kimc and Gabi Zafrird a Department of
Physics, POSTECH, Pohang 790-784, Korea  b Asia Pacific Center for Theoretical
Physics, Postech, Pohang 37673, Korea  c School of Physics, University of
Electronic Science and Technology of China, No. 2006 Xiyuan Ave, West Hi-Tech
Zone, Chengdu, Sichuan 611731, China  d Haifa Research Center for Theoretical
Physics and Astrophysics, University of Haifa, Haifa 3498838, Israel

###### Abstract

We propose two novel methods for computing the superconformal index of 5d
superconformal field theories that cannot be described by conventional
Lagrangian descriptions under mass deformations. The first approach involves
the use of Higgs branch flows from UV Lagrangian theories, guided by
transitions in 5-brane webs in Type IIB string theory. The second method
employs the relationship between O$7^{+}$-plane and O$7^{-}$-plane with eight
D7-branes, which applies to particular non-Lagrangian theories realized by
brane configurations involving an O$7^{+}$-plane. As a concrete application of
our method, we compute the superconformal indices for all known rank-1 non-
Lagrangian theories, which we also use to identify flavor symmetries and their
global forms at the conformal field theory (CFT) fixed points.

## Introduction

The superconformal index counts local BPS operators in a superconformal field
theory (SCFT) Kinney  _et al._ (2007); Bhattacharya  _et al._ (2008). As the
index is a robust observable protected from any continuous deformations, it
has been utilized to investigate non-perturbative phenomena, including
dualities and symmetry enhancements, across various dimensions. In the case
where the theory admits a weakly-coupled limit, the index can be computed from
the partition function at the limit in radial quantization using
supersymmetric localization techniques. There exist, however, numerous
strongly-coupled isolated theories, such as the 5d rank-1 SCFT derived from
M-theory compactified on a local $\mathbb{P}^{2}$ embedded in a Calabi-Yau
(CY) threefold, that lack such weak coupling limits. For these theories, which
we refer to as non-Lagrangian theories, the usual localization technique
cannot be applied. In particular, at present there is no known way to compute
superconformal indices for these non-Lagrangian theories in five
dimensions.111In lower dimensions, various techniques have been developed to
compute superconformal indices for non-Lagrangian theories. For instance, in
the context of 4d theories, some notable examples can be found in Gadde  _et
al._ (2015); Maruyoshi and Song (2017).

Non-Lagrangian theories have other BPS observables, for example, the partition
function on $S^{1}$ times $\Omega$-deformed $\mathbb{R}^{4}$ Nekrasov (2003);
Nekrasov and Okounkov (2006), which can be computed on the Coulomb branch
through expansions in terms of Coulomb branch moduli parameters. However, we
cannot use these observables or the techniques used for them to calculate the
superconformal index because the index requires integration over the Coulomb
branch parameters. Also, while non-Lagrangian theories can be obtained by
renormalization group (RG) flows from SCFTs that UV complete certain 5d gauge
theories via large mass deformations, we cannot employ such RG flows to
compute the IR index since large mass deformations are not feasible in the
context of the superconformal index.

In this work, we present two approaches for evaluating the superconformal
index of 5d non-Lagrangian theories. The first approach involves analyzing the
RG flows originating from specific gauge theories on the Higgs branch and
examining the corresponding limits of their superconformal indices. In
contrast to mass deformations, Higgs branch RG flows can be implemented at the
level of superconformal index. At the end of Higgs branch RG flow from a UV
theory, the index of the IR SCFT can be extracted from the residue of the UV
index at a pole in the flavor fugacity associated to the Higgs branch
deformation Gaiotto  _et al._ (2013).

Our first strategy to compute the superconformal index of a non-Lagrangian
SCFT $\mathcal{T}_{IR}$ is to use the Higgs branch RG flows as follows. First,
we will identify a gauge theory for a UV SCFT $\mathcal{T}_{UV}$ that can flow
to $\mathcal{T}_{IR}$ via a Higgs branch RG flow, through a careful analysis
of Type IIB $(p,q)$ 5-brane webs for the non-Lagrangian theory. Next, we will
compute the superconformal index of $\mathcal{T}_{UV}$ using localization, and
extract the residue at a specific pole in the index of $\mathcal{T}_{UV}$,
which realizes the RG flow. Using this method, we will compute the
superconformal indices of all rank-1 non-Lagrangian SCFTs, such as the local
$\mathbb{P}^{2}$ theory, often referred to as the $E_{0}$ theory, and the
local $\mathbb{P}^{2}+1{\bf Adj}$ theory Bhardwaj (2020) which we denote as
the $\widehat{E}_{1}$ theory.

The second approach is to use the idea of “freezing” an O7--plane and eight
D7-branes ($\text{O7}^{-}+8\text{D7's}$) to an O7+-plane in Type IIB 5-brane
configurations, which was recently investigated in Hayashi  _et al._ (2023).
This idea can be implemented in the context of partition functions by
carefully adjusting mass parameters or chemical potentials for the global
symmetries associated with the D7-branes. This method offers a complementary
method for computing the partition functions for the theories realized by
brane configurations with an O7+-plane. We extend this technique to calculate
the superconformal indices of non-Lagrangian theories, including the
$\widehat{E}_{1}$ theory.

Our results will be verified by comparing the indices obtained from two
separate techniques or distinct UV theories that yield equivalent non-
Lagrangian SCFTs in the Higgs branch limits, and demonstrating that they
agree. Additionally, we will use our findings to identify the correct flavor
symmetries and their global forms at the UV fixed points.

## 5-brane webs for non-Lagrangian theories

To begin, we consider 5-brane webs for 5d SCFTs on the Coulomb branch and
explore how Higgsing transitions can lead to non-Lagrangian theories. At
rank-1, which means one-dimensional Coulomb branch, there are two non-
Lagrangian SCFTs whose 5-brane webs are depicted in FIG. 1.

Figure 1: Rank-1 non-Lagrangian SCFTs. An empty dot denotes the position of an
O7+ plane and the dotted line denotes its monodromy cut.

We first discuss the $E_{0}$ theory, which is the most basic rank-1 theory
without flavor symmetry, corresponding to M-theory compactified on a local CY
3-fold containing a $\mathbb{P}^{2}$ surface, as depicted in FIG. 1(a). This
theory is related to the SCFT that UV completes an $SU(2)$ gauge theory at the
discrete theta $\theta=\pi$, through an RG-flow triggered by a mass
deformation associated with its $U(1)$ symmetry. This deformation corresponds
to the blow-down transition from a del Pezzo surface dP1 to $\mathbb{P}^{2}$
in geometry Morrison and Seiberg (1997); Douglas  _et al._ (1997).

We now illustrate how Higgs branch RG-flows from two distinct gauge theories
lead to either the $E_{0}$ theory or two copies of it, which will later be
utilized to calculate the superconformal index of the $E_{0}$ theory. Firstly,
we can consider a Higgs branch limit of the $SU(3)_{\kappa}$ gauge theory at a
Chern-Simons (CS) level $\kappa=3$. This theory has a $U(1)$ symmetry that is
enhanced to an $SU(2)$ symmetry at the UV fixed point. By giving a vacuum
expectation value (VEV) to a component of the moment map operator for the
$SU(2)$ flavor symmetry, we can Higgs this theory to the $E_{0}$ theory. In
the brane web, this Higgsing process can be achieved by anchoring two parallel
NS5-branes on a single external 7-brane, as depicted in FIG. 2. By moving the
external 7-brane downward and performing a Hanany-Witten transition Hanany and
Witten (1997), we can observe that the resulting brane web is identical to the
brane web for the $E_{0}$ theory.

Figure 2: Higgs branch transition from the $SU(3)_{3}$ gauge theory (LEFT) to
the $E_{0}$ theory (RIGHT) by attaching two parallel NS5-branes to a single
7-brane denoted by a solid dot.

The second UV theory that we will use for the index computation of $E_{0}$ is
the $SU(3)_{6}$ gauge theory. The 5-brane web for this theory is illustrated
on the left side of FIG. 3. This theory also has a $U(1)$ symmetry which is
enhanced to an $SU(2)$ flavor symmetry at the CFT fixed point. Again, we can
Higgs this theory by giving a VEV to the moment map operator for the $SU(2)$
symmetry. In the IR limit, one obtains two copies of the $E_{0}$ theory which
are decoupled from each other. This Higgs branch RG flow in the brane web is
demonstrated in FIG. 3.

Figure 3: Higgs branch transition from the $SU(3)_{6}$ gauge theory (LEFT) to
a pair of the $E_{0}$ theory (RIGHT) that are separated along the 7-branes.

We now discuss the other rank-1 non-Lagrangian SCFT, $\widehat{E}_{1}$ theory
or the local $\mathbb{P}^{2}+1{\bf Adj}$ theory, which is derived from the 5d
$\mathcal{N}=2$ $SU(2)$ gauge theory with $\theta=\pi$ by integrating out an
instantonic hypermultiplet Bhardwaj (2020) (see Appendix A for the precise
relation). The corresponding 5-brane web Kim  _et al._ (2021) is given in FIG.
1(b).

A UV theory that has a Higgs branch limit to the $\widehat{E}_{1}$ theory is
the SCFT UV completion of the $SU(3)_{\frac{1}{2}}$ gauge theory with a
symmetric hypermultiplet. The 5-brane web for this theory and its Higgsing
process to the brane web for the $\widehat{E}_{1}$ theory are depicted in FIG.
4. As illustrated here, turning on a VEV for the moment map operator of the
$SU(2)$ flavor symmetry and taking the low energy limit lead to the diagram on
the right-hand side which is equivalent to the brane web for the
$\widehat{E}_{1}$ theory through a Hanany-Witten transition Kim  _et al._
(2021).

Figure 4: Transition from the $SU(3)_{\frac{1}{2}}+1{\bf Sym}$ to the
$\widehat{E}_{1}$ theory.

In the following analysis, we will compute the superconformal indices for the
$E_{0}$ theory and the $\widehat{E}_{1}$ theory using these UV gauge theories
and examining their Higgs branch limits.

## Superconformal index and Higgsing

The 5d superconformal index is the Witten index of a 5d SCFT quantized on
$S^{4}$. It is defined as Bhattacharya  _et al._ (2008); Kim  _et al._ (2012)

$\mathcal{I}={\rm
Tr}(-1)^{F}x^{2(j_{r}+R)}y^{2j_{l}}\textstyle\prod_{i}\mu_{i}^{F_{i}}\ ,$ (1)

where $x,y$ are fugacities for the subgroup $SO(5)\times SU(2)_{R}$ of the
superconformal group whose Cartan generators are $j_{r},j_{l},$ and $R$.
$F_{i}$ denotes the flavor charge and $\mu_{i}$ is its fugacity.

The superconformal indices for 5d gauge theories with classical gauge groups
can be computed by employing supersymmetric localization developed in Kim  _et
al._ (2012). This method, however, requires knowledge of the complete
expression for the instanton partition function at every instanton order,
which is essential for integrating over the gauge holonomies. This can be
achieved through an ADHM construction for the instanton moduli space or by
solving the blowup equations, as discussed in Nekrasov (2003); Nekrasov and
Okounkov (2006); Nakajima and Yoshioka (2005a, b). Unfortunately, this
localization approach is not applicable to calculating superconformal indices
for non-Lagrangian theories. The primary reason for this is that the existing
partition functions are expanded in terms of Coulomb branch parameters, which
eventually need to be integrated to yield the superconformal index.
Consequently, at present, there is no known method to calculate the
superconformal index for any non-Lagrangian SCFT.

Now, we introduce a simple method for calculating the superconformal index for
non-Lagrangian SCFTs using the Higgs branch limits of certain UV gauge
theories which we discussed in the previous section. Specifically, we can take
advantage of the fact that the instanton partition functions for the UV gauge
theories we are studying can be explicitly computed using either the ADHM
construction or the blowup approach. This allows us to compute their
superconformal indices using localization. We then take the Higgs branch limit
to the non-Lagrangian theory, which can be realized at the index level by
taking the residue of the UV index at a specific pole, say $\xi=1$, associated
to the moment map operator with a non-zero VEV Gaiotto  _et al._ (2013). This
procedure yields the superconformal index for the IR theory at the end of the
RG flow, which can be written schematically as

$\displaystyle\mathcal{I}[\mathcal{T}_{\rm IR}]=(\mathcal{I}_{\rm
extra})^{-1}\times\operatorname{Res}_{\xi=1}\mathcal{I}[\mathcal{T}_{\rm UV}]\
.$ (2)

Here, $\mathcal{I}_{\rm extra}$ represents the contribution from the Goldstone
modes of the broken flavor symmetry by the VEV. We multiplied the inverse of
$\mathcal{I}_{\rm extra}$ to subtract zero mode contributions under the
Higgsing procedure.

As our first example, let us compute the superconformal index of the $E_{0}$
theory. To this end, we first calculate the index for the UV SCFT which leads
to the $SU(3)_{3}$ gauge theory at low energy. Based on the localization
argument, we can express the superconformal index for the $SU(3)_{3}$ gauge
theory as an integral expression over the gauge holonomies $\alpha_{i}$ as

$\displaystyle\mathcal{I}[SU(3)_{3}]=\frac{1}{3!}\\!\oint\\!\prod_{i=1}^{2}\frac{d\alpha_{i}}{2\pi}\,\mathcal{I}_{\rm
pert}(\alpha_{i};x,y)\\!\cdot\\!\mathcal{I}_{\rm inst}(\alpha_{i},q;x,y)\ ,$
$\displaystyle\mathcal{I}_{\rm
pert}\\!=\\!\prod_{e\in\Delta}\\!\Big{(}2\sin\frac{e(\alpha)}{2}\Big{)}^{2}\\!\cdot\\!\operatorname{PE}\\!\left[\frac{-x(y\\!+\\!1/y)}{(1\\!-\\!xy)(1\\!-\\!x/y)}\\!\sum_{e\in\Delta}\\!e^{ie(\alpha)}\\!\right]\\!,$
$\displaystyle\mathcal{I}_{\rm inst}=Z_{{\rm inst}}(\alpha_{i},q;x,y)\cdot
Z_{{\rm inst}}(-\alpha_{i},q^{-1};x,y)\ ,$ (3)

where PE[$f$] stands for the plethystic exponential of a single letter index
$f$, $\Delta$ denotes the roots of the gauge group, and the contour encloses
the unit circles $|e^{i\alpha_{i}}|=1$. $\mathcal{I}_{\rm pert}$ is the
perturbative contribution and the two $Z_{\rm inst}$ factors in
$\mathcal{I}_{\rm inst}$ are the instanton partition functions capturing BPS
spectrum of the $SU(3)_{3}$ theory on $\Omega$-deformed
$S^{1}\times\mathbb{R}^{4}$ expanded by the instanton number fugacity $q$ and
$q^{-1}$, respectively, which can be calculated using the standard ADHM
construction for $SU(3)$ instantons.

By collecting all the ingredients, we find the superconformal index for the 5d
SCFT UV completion of the $SU(3)_{3}$ theory as

$\displaystyle\mathcal{I}[SU(3)_{3}]=1+\chi_{\bf 3}(q^{1/2})x^{2}+\chi_{\bf
2}(y)\big{[}1+\chi_{\bf 3}(q^{1/2})\big{]}x^{3}$
$\displaystyle\\!+\big{[}\chi_{\bf 3}(y)\big{(}1+\chi_{\bf
3}(q^{1/2}))+1+\chi_{\bf 5}(q^{1/2})\big{]}x^{4}$
$\displaystyle\\!+\big{[}\chi_{\bf 2}(y)\big{(}1\\!+\\!2\chi_{\bf
3}(q^{1/2})+\chi_{\bf 5}(q^{1/2})\big{)}$ $\displaystyle\quad+\chi_{\bf
4}(y)\big{(}1+\chi_{\bf 3}(q^{1/2})\big{)}\big{]}x^{5}+\mathcal{O}(x^{6})\,,$
(4)

where $\chi_{\bf r}(a)$ stands for the ${\bf r}$-dimensional representation of
the $SU(2)$ symmetry, e.g., $\chi_{\bf 2}(a)\\!=a+\\!a^{-1}$. The term at
$x^{2}$-order corresponds to the contribution from the moment map operators,
which in turn reflects the enhancement of the flavor algebra to
$\mathfrak{su}(2)$ at the CFT fixed point Bashkirov (2012); Bergman  _et al._
(2014). Also, we note that the index is expressed in the odd-dimensional
characters for the flavor algebra. This is because the instanton partition
function is expanded by $q$ and thus the index includes only terms with
integral power of $q$. This implies that the flavor symmetry group is, in
fact, $SU(2)/\mathbb{Z}_{2}=\\!SO(3)$ rather than $SU(2)$ Benetti Genolini and
Tizzano (2021); Apruzzi  _et al._ (2022).

The IR index for the $E_{0}$ theory can be obtained by tracing the Higgs
branch limit of the index of the $SU(3)_{3}$ gauge theory. This limit
corresponds to the moment map operator which develops a pole at $qx^{2}=1$.
Therefore, to take the Higgs branch limit, we evaluate the residue of the UV
index at $qx^{2}=1$. By doing so, we obtain the superconformal index for the
$E_{0}$ theory,

$\displaystyle\mathcal{I}[E_{0}]=(\mathcal{I}_{\rm
extra})^{-1}\times\operatorname{Res}_{qx^{2}=1}\mathcal{I}[SU(3)_{3}]$
$\displaystyle\quad=1+\chi_{\bf 2}(y)x^{3}+\big{[}1+\chi_{\bf
3}(y)\big{]}x^{4}+\big{[}\chi_{\bf 2}(y)+\chi_{\bf 4}(y)\big{]}x^{5}$
$\displaystyle\quad+\big{[}-1+\chi_{\bf 3}(y)+\chi_{\bf
5}(y)\big{]}x^{6}+\big{[}\chi_{\bf 4}(y)+\chi_{\bf 6}(y)\big{]}x^{7}$
$\displaystyle\quad+\big{[}1+\chi_{\bf 3}(y)+2\chi_{\bf 5}(y)+\chi_{\bf
7}(y)\big{]}x^{8}+\mathcal{O}(x^{9})\ ,$ (5)

where

$\displaystyle\mathcal{I}_{\rm
extra}=\operatorname{PE}\left[\frac{x^{2}}{(1-xy)(1-x/y)}\right]\ .$ (6)

Note that there is no term at $x^{2}$-order, which suggests that the IR theory
lacks any conserved current. This result is indeed consistent with the absence
of flavor symmetry in the $E_{0}$ theory.

The same result can be obtained by Higgsing from another UV theory, namely,
the $SU(3)_{6}$ gauge theory. The superconformal index of the UV SCFT leading
to the $SU(3)_{6}$ gauge theory under a mass deformation can also be expressed
as the same form in (Superconformal index and Higgsing), but the instanton
contributions are different. In this case, the instanton moduli space has no
known ADHM construction222For the $SU(3)$ gauge theories with Chern-Simons
level $\kappa>3$, the $U(k)$ holonomy integral in the usual ADHM quantum
mechanics of $k$ instantons exhibits higher order poles at asymptotic
infinity. This indicates that the usual ADHM construction cannot provide
consistent UV completions for the instantons in these theories., but we can
instead compute the instanton partition function using the blowup equations as
presented in Kim  _et al._ (2021). We compute the index of the 5d SCFT UV
completion of the $SU(3)_{6}$ gauge theory as

$\displaystyle\mathcal{I}[SU(3)_{6}]=1+\chi_{\bf 3}(q)x^{2}+\chi_{\bf
2}(y)\big{(}1+\chi_{\bf 3}(q)\big{)}x^{3}$
$\displaystyle\quad+\big{[}1+\chi_{\bf 2}(y)\chi_{\bf 2}(q)+\chi_{\bf
3}(y)(1+\chi_{\bf 3}(q))+\chi_{\bf 5}(q)\big{]}x^{4}$
$\displaystyle\quad+\big{[}\chi_{\bf 2}(y)(1+2\chi_{\bf 3}(q)+\chi_{\bf
5}(q))+\chi_{\bf 2}(q)(1+\chi_{\bf 3}(y))\big{]}x^{5}$
$\displaystyle\quad+\chi_{\bf 4}(y)\big{(}1+\chi_{\bf
3}(q)\big{)}x^{5}\\!+\\!\mathcal{O}(x^{6})\ .$ (7)

The result shows that the flavor symmetry of the $SU(3)_{6}$ gauge theory at
the CFT fixed point is $SU(2)$.

By evaluating the residue of this index at $q^{2}x^{2}=1$, one can compute the
superconformal index of the IR theory, which consists of two decoupled $E_{0}$
theories, as we previously illustrated in FIG. 3 using 5-brane webs. Thus, we
arrive at the following relation:

$\displaystyle\left(\mathcal{I}[E_{0}]\right)^{2}$
$\displaystyle=(\mathcal{I}_{\rm
extra})^{-1}\times\operatorname{Res}_{q^{2}x^{2}=1}\mathcal{I}[SU(3)_{6}]\ .$
(8)

We verified that this relation agrees well with the index of the $E_{0}$
theory given in (Superconformal index and Higgsing) up to the $x^{9}$-order,
which provides a strong evidence for our result.

Next, we shall compute the superconformal index of the $\widehat{E}_{1}$
theory using the Higgsing procedure described in FIG. 4. To begin, we first
compute the index of the $SU(3)_{\frac{1}{2}}+1{\bf Sym}$ theory at the UV
fixed point. The superconformal index of this UV theory can be written as

$\displaystyle\mathcal{I}\big{[}SU(3)_{\frac{1}{2}}\\!+\\!1{\bf
Sym}\big{]}\\!=\\!\frac{1}{3!}\\!\oint\\!\prod_{i=1}^{2}\frac{d\alpha_{i}}{2\pi}\,\mathcal{I}_{\rm
pert}\cdot\mathcal{I}_{\rm inst}\ ,$ (9) $\displaystyle\mathcal{I}_{\rm
pert}\\!=\\!\prod_{e\in\Delta}\\!\Big{(}2\sin\frac{e(\alpha)}{2}\Big{)}^{2}\cdot\operatorname{PE}\bigg{[}\frac{-x(y+1/y)}{(1\\!-\\!xy)(1\\!-\\!x/y)}\sum_{e\in\Delta}\\!e^{ie(\alpha)}$
$\displaystyle\quad\quad+\\!\frac{x}{(1\\!-\\!xy)(1\\!-\\!x/y)}\\!\sum_{w\in{\bf
Sym}}\\!(e^{iw(\alpha)+i\mu}+e^{-iw(\alpha)-i\mu})\bigg{]}\\!,$
$\displaystyle\mathcal{I}_{\rm inst}=Z_{{\rm inst}}(\alpha_{i},\mu,q;x,y)\cdot
Z_{{\rm inst}}(-\alpha_{i},-\mu,q^{-1};x,y)\ ,$

where $\mu$ denotes the chemical potential for the $U(1)$ flavor symmetry of
the symmetric hypermultiplet. Here, the instanton partition function can be
calculated using the ADHM construction for $SU(3)$ instantons coupled to
additional matter in the form of a symmetric hypermultiplet, which we
summarize in Appendix B. By evaluating the integral, we find

$\displaystyle\mathcal{I}\\!\big{[}SU(3)_{\frac{1}{2}}\\!+\\!1{\bf
Sym}\big{]}\\!=\\!1+\big{[}\chi_{\bf 3}(a)+\chi_{\bf 3}(b)\big{]}x^{2}$ (10)
$\displaystyle\ \ +\big{[}\chi_{\bf 2}(y)(1+\chi_{\bf 3}(a)+\chi_{\bf
3}(b))+\chi_{\bf 4}(a)\chi_{\bf 3}(b)\big{]}x^{3}$ $\displaystyle\ \
+\big{[}\chi_{\bf 3}(y)(1+\chi_{\bf 3}(a)+\chi_{\bf 3}(b))+\chi_{\bf
2}(y)\chi_{\bf 4}(a)\chi_{\bf 3}(b)\big{]}x^{4}$ $\displaystyle\ \
+\big{[}2\chi_{\mathbf{5}}(a)+\chi_{\mathbf{5}}(b)+\chi_{\mathbf{3}}(a)\chi_{\mathbf{3}}(b)+3\big{]}x^{4}+\mathcal{O}(x^{5}),$

where $a\equiv qe^{i\mu/2}$ and $b\equiv\sqrt{q}e^{-5i\mu/4}$. This result
enables us to identify the precise global flavor symmetry group of the UV CFT
as $SU(2)_{a}\times SO(3)_{b}$, which also confirms the $\mathfrak{so}(4)$
flavor algebra in this theory predicted in Bhardwaj (2021).

We now take the Higgs branch limit by giving a VEV to the moment map operator
for the $SO(3)_{b}$ flavor symmetry. To achieve this at the level of the
index, we compute the residue of the UV index at the pole $b^{2}x^{2}=1$. The
result then leads to the superconformal index for the $\widehat{E}_{1}$ theory
as follows:

$\displaystyle\mathcal{I}[\widehat{E}_{1}]$
$\displaystyle\\!=\\!1\\!+\\!\widehat{\chi}_{\bf
3}\,x^{2}\\!+\\!\big{[}\chi_{\bf 2}(y)(1\\!+\\!\widehat{\chi}_{\bf
3})\\!+\\!\widehat{\chi}_{\bf 4}\big{]}x^{3}\\!+\\!\mathcal{O}(x^{4}),$ (11)

where $\widehat{\chi}_{\bf r}\\!\equiv\\!{\chi}_{\bf r}(a)$ with $a$ being the
fugacity for the flavor symmetry. The coefficient $\widehat{\chi}_{\bf
3}=a^{2}\\!+\\!1\\!+\\!a^{-2}$ at $x^{2}$-order captures the contribution of
the conserved current multiplet. From this, we can conclude that the naive
$U(1)_{a}$ flavor symmetry group is enhanced to $SU(2)_{a}$ at the CFT fixed
point.

We can try to test (11) by comparing the result against a different
construction of the 5d SCFT. One such realization is through the dimensional
reduction of the 6d $A_{2}$ $(2,0)$ theory with an outer automorphism twist
Bhardwaj (2020). While we cannot compute the index using this realization, we
can understand some general features of its BPS spectrum. In particular, we
can argue that the basic BPS multiplets in this theory should be the $SU(2)$
conserved current multiplets and a Higgs branch operator in the ${\bf 4}$ of
the $SU(2)$ flavor symmetry and whose ground state is in the ${\bf 4}$ of
$SU(2)_{R}$. This indeed matches what we observe in (11). See Appendix A for a
more detailed analysis.

We have computed the above index for $\widehat{E}_{1}$ up to $x^{3}$-order
using the instanton partition function of the UV $SU(3)_{\frac{1}{2}}+1{\bf
Sym}$ theory up to 3-instantons. However, to precisely determine higher order
terms of the $x$-expansion, we need higher instanton results, which our
computational resources unfortunately cannot currently accommodate. In what
follows, as an alternative approach, we will introduce another technique to
compute the superconformal index of specific theories engineered by brane webs
with an O7+-plane. By employing this method, we will improve the order of the
index computation for the $\widehat{E}_{1}$ theory.

## The freezing branes and $\widehat{E}_{1}$ theory

For theories involving an O7+-plane in their brane configuration, such as
$SU(N)$ theory with symmetric matter or $SO(N)$ theory, an alternative yet
novel approach for computing physical observables was proposed in Hayashi  _et
al._ (2023). This method suggests that the O7+-plane contribution can be
retrieved from “freezing” an O7--plane and eight D7-branes. It is a particular
tuning of the hypermultiplet masses with some special values such that most of
the contributions from the eight fundamental hypermultiplets (D7-branes)
cancel each other and what remains, combined with the contribution of an O7--
plane, yields the contribution of an O7+-plane. For instance, the
$SU(N)+1\mathbf{Sym}$ theory can be effectively treated as the
$SU(N)+1\mathbf{AS}+8\mathbf{F}$ theory by such freezing procedure. This was
explicitly checked at the level of the Seiberg-Witten curves Hayashi  _et al._
(2023) and can be extended to the instanton partition functions Kim  _et al._
. It should be noted that not all physical observables associated with an
O7+-plane can be obtained from freezing O7${}^{-}+8$D7s. Only for some
observables like the prepotential or index function, it would provides an
effective way of computing them. See Kim  _et al._ for more details on its
limitation.

It is also worth noting that the freezing is applicable to non-Lagrangian
theories. An illustrative example discussed in Hayashi  _et al._ (2023) is the
$\widehat{E}_{1}$ theory, which can be seen as a local
$\mathbb{P}^{2}+1\mathbf{AS}+8\mathbf{F}$ theory with appropriately adjusted
mass parameters. Since the antisymmetric hypermultiplet decouples here, the
$\widehat{E}_{1}$ theory can be effectively obtained by freezing the
$Sp(1)+7\mathbf{F}$ theory.

We accordingly implement this freezing to compute the index for the
$\widehat{E}_{1}$ theory. Recall that the superconformal index of the UV SCFT
of $Sp(1)+7\mathbf{F}$ is given by a gauge holonomy integral expression as Kim
_et al._ (2012); Hwang  _et al._ (2015),

$\displaystyle\mathcal{I}[Sp(1)\\!+\\!7\mathbf{F}]=\frac{1}{2!}\oint\frac{d\alpha}{2\pi}\mathcal{I}_{\mathrm{pert}}\cdot\mathcal{I}_{\mathrm{inst}}\,,$
(12)
$\displaystyle\mathcal{I}_{\mathrm{pert}}=\Big{(}2\sin\frac{\alpha}{2}\Big{)}^{2}\cdot\operatorname{PE}\bigg{[}\frac{-x(y\\!+\\!1/y)}{(1\\!-\\!xy)(1\\!-\\!x/y)}\sum_{e\in\Delta}e^{ie(\alpha)}$
$\displaystyle\
+\frac{x}{(1\\!-\\!xy)(1\\!-\\!x/y)}\sum_{l=1}^{7}\sum_{w\in\mathbf{F}}\big{(}e^{iw(\alpha)+i\mu_{l}}\\!+\\!e^{-iw(\alpha)-i\mu_{l}}\big{)}\bigg{]},$
$\displaystyle\mathcal{I}_{\mathrm{inst}}=Z_{\mathrm{inst}}(\alpha,\mu_{l},q;x,y)\cdot
Z_{\mathrm{inst}}(-\alpha,-\mu_{l},q^{-1},x,y)\,,$

where $\mu_{l}$ are the chemical potentials for fundamental hypers, and
$Z_{\mathrm{inst}}$ is the instanton partition function which can be computed
using the ADHM construction Kim  _et al._ (2012); Hwang  _et al._ (2015). We
have carried out the explicit integration and computed this index up to
$x^{10}$-order. The detailed result can be found in Appendix C.

Next, we proceed with the freezing procedure acting on this index, which
eventually gives rise to the index of the $\widehat{E}_{1}$ theory. To
accomplish this, we adjust the flavor fugacities, following the process
discussed in Hayashi  _et al._ (2023); Kim  _et al._ , as

$\displaystyle\begin{aligned} &e^{i\mu_{1,2}^{*}}\mapsto ax^{\pm
1/2}\,,\quad&&e^{i\mu_{3,4}^{*}}\mapsto ay^{\pm 1/2}\,,\\\
&e^{i\mu_{5,6}^{*}}\mapsto-ax^{\pm
1/2}\,,\quad&&e^{i\mu_{7,8}^{*}}\mapsto-a^{\pm 1}y^{1/2}\,,\end{aligned}$ (13)

where $a$ is the fugacity for the flavor symmetry of the $\widehat{E}_{1}$
theory. We then obtain with this tuning the superconformal index of the
$\widehat{E}_{1}$ theory,

$\displaystyle\mathcal{I}[Sp(1)\\!+\\!7\mathbf{F}]\overset{\mu_{l}\rightarrow\mu_{l}^{*}}{\mapsto}\operatorname{PE}\bigg{[}\frac{-x(a^{2}+a^{-2})}{(1-xy)(1-x/y)}\bigg{]}\cdot\mathcal{I}[\widehat{E}_{1}]\,,$
$\displaystyle\mathcal{I}[\widehat{E}_{1}]\\!=\\!1\\!+\\!\widehat{\chi}_{\bf
3}x^{2}\\!+\\!\Big{[}\chi_{\mathbf{2}}(y)(1+\widehat{\chi}_{\bf
3})\\!+\\!\widehat{\chi}_{\bf 4}\Big{]}x^{3}$ $\displaystyle\
+\\!\Big{[}\chi_{\mathbf{3}}(y)(1+\widehat{\chi}_{\bf
3})\\!+\\!\chi_{\mathbf{2}}(y)\big{(}\widehat{\chi}_{\bf
2}\\!+\\!\widehat{\chi}_{\bf 4}\big{)}+\\!\widehat{\chi}_{\bf
5}\\!+\\!2\Big{]}x^{4}$ $\displaystyle\
+\\!\Big{[}\chi_{\mathbf{4}}(y)(1\\!+\\!\widehat{\chi}_{\bf
3})\\!+\\!\chi_{\mathbf{3}}(y)(\widehat{\chi}_{\bf
2}\\!+\\!\widehat{\chi}_{\bf
4})\\!+\\!\chi_{\mathbf{2}}(y)(2\\!+\\!2\widehat{\chi}_{\bf
3}\\!+\\!\widehat{\chi}_{\bf 5})$ $\displaystyle\quad+\\!\widehat{\chi}_{\bf
2}+\widehat{\chi}_{\bf 4}+\widehat{\chi}_{\bf
6}\Big{]}x^{5}+\mathcal{O}(x^{6})\,.$ (14)

This perfectly agrees with the result in (11) and further improves it up to
$x^{5}$-order.

## Summary

We have introduced novel method to compute the superconformal indices of 5d
non-Lagrangian SCFTs. Our methods involve using Higgs branch RG flows from a
UV Lagrangian theory or employing a freezing procedure on orientifold 7-brane
in brane constructions. By applying these techniques, we are able to calculate
the superconformal indices for rank-1 non-Lagrangian SCFTs, namely the $E_{0}$
theory and the $\widehat{E}_{1}$ theory, given in (Superconformal index and
Higgsing) and (The freezing branes and $\widehat{E}_{1}$ theory),
respectively.

## Acknowledgments

We are grateful to Hirotaka Hayashi, Kimyeong Lee, Xiaobin Li, Satoshi Nawata,
Jaewon Song, and Futoshi Yagi for their insightful discussions and comments.
SK thanks the hospitality of POSTECH where part of this work was done. HK and
MK are supported by Samsung Science and Technology Foundation under Project
Number SSTF-BA2002-05 and by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (2023R1A2C1006542). SK is
supported by the NSFC grant No. 12250610188.

##  Appendix A Comparison of the $\widehat{E}_{1}$ index with a 6d realization

We can try to test our result for the index of the $\widehat{E}_{1}$ theory by
comparing against expectations from other constructions. One such
construction, is the realization of the $\widehat{E}_{1}$ theory from the
twisted compactification of the $A_{2}$ $6d$ $(2,0)$ theory on a circle.
Indeed, this was how the $\widehat{E}_{1}$ theory was originally found in
Bhardwaj (2020).

To illustrate how the $\widehat{E}_{1}$ theory arises in such a construction,
it is convenient to start first with a simpler example of the reduction of the
$A_{1}$ 6d $(2,0)$ theory on a circle. In this case, we expect to get, at low-
energies, the 5d maximally supersymmetric $SU(2)$ gauge theory with discrete
theta angle $\theta=0$ Douglas (2011); Lambert  _et al._ (2011); Tachikawa
(2011) (we shall henceforth use the shorthand notation $SU(2)_{\theta}$ for
the $SU(2)$ gauge theory with theta angle $\theta$). We can think of this
theory also as a minimally supersymmetric 5d $SU(2)$ gauge theory with an
adjoint hypermultiplet. In this viewpoint, we have an $SU(2)$ flavor symmetry,
rotating the hyper, which is the commutant of the $SU(2)$ R-symmetry of 5d
minimal SUSY in the $USp(4)$ R-symmetry of 5d maximal SUSY. When performing
the circle reduction, we have the freedom of incorporating holonomies in
flavor symmetries on the circle. Of particular interest is the holonomy inside
this $SU(2)$ subgroup of the $USp(4)$ R-symmetry group, which is viewed as a
flavor symmetry in minimal SUSY. This will lead to a 5d theory with minimal
SUSY.

When the holonomy is small, it simply leads to the adjoint hyper acquiring a
mass. This leads to the 5d version of the $\mathcal{N}=2^{*}$ theory, which in
the deep IR reduce to a pure minimally supersymmetric 5d $SU(2)_{0}$ gauge
theory. Of course, this holds when the holonomy is small, but what happens
when the holonomy becomes large? To better understand this, it is convenient
to take a closer look at the BPS operators in the theory. Our starting point
is the 6d $A_{1}$ $(2,0)$ theory. The main BPS multiplet in this theory is the
$D_{1}[0,0,0]^{(0,2)}_{4}$, where we use the notation of Cordova  _et al._
(2019), which is the $(2,0)$ energy-momentum tensor multiplet. Its ground
state is a scalar in the ${\bf 14}$ of $USp(4)$. We would be interested in
what happens to this state when we dimensionally reduce. For this, it is
convenient to first decompose $USp(4)\rightarrow SU(2)_{R}\times SU(2)_{F}$,
with $SU(2)_{R}$ the R-symmetry of the minimally supersymmetric case. Under
this decomposition we have that: ${\bf 14}_{USp(4)}\rightarrow 1+{\bf
2}_{SU(2)_{R}}{\bf 2}_{SU(2)_{F}}+{\bf 3}_{SU(2)_{R}}{\bf 3}_{SU(2)_{F}}$.
Here we shall only consider the last term, which corresponds to the shortest
BPS multiplet we get in 5d from this 6d multiplet333This follows from the
shortening conditions, where acting on the ground state with the supercharge
annihilates the state where the Dynkin indices of the $USp(4)$ representations
of the supercharge and the ground state are summed Cordova  _et al._ (2019).
In other words, trying to raise the weight of the $USp(4)$ representation of
the ground state by acting with the supercharge annihilates the state. If we
now look at the highest weight representation under $SU(2)_{R}$ in the
decomposition, then it will obey a similar shortening condition as trying to
raise its $SU(2)_{R}$ state is tantamount to raising its $USp(4)$ highest
weight.. This operator is simply the conserved current multiplet of the 5d
$SU(2)_{F}$ symmetry.

When we reduce on a circle, we get the 5d operator, as well as all its Kaluza-
Klein excitations. An interesting aspect in 5d gauge theories resulting from
the compactification of 6d SCFTs is that they generally retain some knowledge
on the KK tower through instanton particles, see for instance Douglas (2011);
Lambert  _et al._ (2011); Tachikawa (2015). As such we would want to keep
track on the entire KK tower. For this purpose, we introduce two fugacities,
$f$ and $q$, with the first one associated with the holonomy for $SU(2)_{F}$
and the second to the radius of the circle. Note then, that the first keep
track of the mass to the adjoint hyper, while the second keeps track of the
mass of the KK modes. In the 5d gauge theory, these are associated with
background central charges to the $SU(2)_{F}$ symmetry and the $U(1)$
instanton symmetry of the gauge theory, respectively, justifying their
designation as fugacities. All in all, we can write the charges of the
expected states in 5d as:

$\displaystyle\Big{(}1\\!+\\!f\\!+\\!\frac{1}{f}\Big{)}\\!\\!\\!\sum^{\infty}_{n=-\infty}\\!\\!q^{n}\\!\rightarrow\\!\Big{(}1\\!+\\!f\\!+\\!\frac{1}{f}\Big{)}\\!\Big{(}1\\!+\\!q\\!+\\!\frac{1}{q}\\!+\\!q^{2}\\!+\\!\frac{1}{q^{2}}\\!+\\!\cdots\\!\Big{)}$
(15)

Note that all these states carry the charges of conserved currents in 5d. Next
consider the case where we reduce on a circle of finite radius but without a
holonomy. In that case, all states charged under $q$ acquire a mass, and the
currents associated with them are no longer conserved. We then end up with the
$SU(2)_{F}$ conserved current of the 5d maximally supersymmetric gauge theory.

If we next introduce a small holonomy, then we break $SU(2)_{F}$ to $U(1)$,
and correspondingly the currents charged under $f$ will no longer be
conserved. This describes the regime where we get in 5d the minimally
supersymmetric $SU(2)_{0}$ gauge theory. However, if we continue to increase
the holonomy, we will eventually reach a state where $m_{f}=m_{q}$ and the
states with charges $\frac{f}{q}$ become massless. At this point we expect to
get a new 5d theory due to the presence of additional massless degrees of
freedom. Said new theory should have an $SU(2)$ global symmetry, due to the
extra currents, be minimally supersymmetric and support a mass deformation
leading to the 5d gauge theory. Indeed there is a theory that ticks all these
boxes: the $E_{1}$ SCFT. Therefore, it is natural to expect that at this
regime of the compactification data we get the $E_{1}$ 5d SCFT.

What happens if we increase $m_{f}$ further? The $E_{1}$ SCFT is known to
support only a single mass deformation sending it to the 5d $SU(2)_{0}$ gauge
theory for both positive and negative mass. As such we should again get the
gauge theory, at least as long as $m_{f}$ does not increase too much. The
expected picture of the resulting low-energy theory in these cases is shown in
figure 5. We could continue exploring other possible phase transition though
that would not be of interest to us here. We also note that there could be in
principle phase transitions due to massless matter coming from longer
multiplets, though again we shall not pursue this here.

Figure 5: A schematic summery of the expected low-energy theory for the
compactification of the 6d $(2,0)$ $A_{1}$ theory. The vertical axis stands
for the inverse of the compactification radius, which sets the scale for the
KK masses. The horizontal axis stands for the mass deformation associated with
the holonomy in the $USp(4)$ R-symmetry. We use $\mathcal{N}=2$
$SU(2)_{\theta}$ for the maximally 5d supersymmetric $SU(2)$ gauge theory with
the listed $\theta$ angle, and $SU(2)_{\theta}$ for the minimally
supersymmetric one. Here, we do not explore the full parameter space, rather
contenting ourselves with some region around the maximal SUSY case. Domains
lying outside of said region, where the behavior remains unexplored, are
symbolized with a ”?”.

Now we are going to turn to the case of actual interest to us, the circle
reduction of the $A_{2}$ $(2,0)$ theory. This theory is known to have a
$\mathbb{Z}_{2}$ discrete symmetry associated with the outer automorphism of
$A_{2}$, and we shall be mainly interested in the case where the reduction is
done with a twist in said symmetry. In this case we get, at low-energies, the
5d maximally supersymmetric $SU(2)_{\pi}$ gauge theory Tachikawa (2011). This
is again if we compactify without an $SU(2)_{F}$ holonomy, so maximal SUSY is
preserved. Next we would be interested in the result when we also include this
holonomy. Again when the holonomy is small, it will simply imply a mass term
for the adjoint hyper and we expect to get the 5d minimally supersymmetric
$SU(2)_{\pi}$ gauge theory. We would want to know what happens when the
holonomy becomes large.

To understand this, we again turn to look at the BPS multiplets in the 6d
SCFT. Now we have two of them. One is still the energy-momentum tensor
multiplet, in the $D_{1}[0,0,0]^{(0,2)}_{4}$ of the $(2,0)$ superconformal
group, while the other is in the $D_{1}[0,0,0]^{(0,3)}_{6}$ 444For a generic
$(2,0)$ theory of type $G$, there would be $rank(G)$ such operators in the
$D_{1}[0,0,0]^{(0,d_{i})}_{2d_{i}}$, for $d_{i}$ the dimensions of the
invariant polynomials of $G$.. We shall refer to them as $V_{2}$ and $V_{3}$,
respectively. Their ground state are again scalars, with the $V_{2}$ one being
in the ${\bf 14}$ of $USp(4)$ as before, while the $V_{3}$ one being in the
${\bf 30}$ of $USp(4)$. When decomposed into the $SU(2)_{R}\times SU(2)_{F}$
subgroup, we have that: ${\bf 30}_{USp(4)}\rightarrow 1+{\bf
2}_{SU(2)_{R}}{\bf 2}_{SU(2)_{F}}+{\bf 3}_{SU(2)_{R}}{\bf 3}_{SU(2)_{F}}+{\bf
4}_{SU(2)_{R}}{\bf 4}_{SU(2)_{F}}$, where again we would only be interested in
the last state, which carries the highest weight under $SU(2)_{R}$.

When reduced, $V_{2}$ would give us the same states as before, including the
KK tower, so we turn now to $V_{3}$. One interesting property of this
multiplet is that it is odd under the $\mathbb{Z}_{2}$ symmetry we twist by.
As such it essentially obeys anti-periodic boundary conditions on the circle,
and its KK masses are fractional compared with those of the periodic fields.
If we write in terms of fugacities we have:

$\displaystyle\Big{(}1\\!+\\!f\\!+\\!\frac{1}{f}\Big{)}\\!\\!\\!\sum^{\infty}_{n=-\infty}\\!\\!q^{n}\\!\rightarrow\\!\Big{(}1\\!+\\!f\\!+\\!\frac{1}{f}\Big{)}\\!\Big{(}1\\!+\\!q\\!+\\!\frac{1}{q}\\!+\\!q^{2}\\!+\\!\frac{1}{q^{2}}\\!+\\!\cdots\\!\Big{)}$
$\displaystyle\Big{(}f^{\frac{1}{2}}\\!+\\!\frac{1}{f^{\frac{1}{2}}}\\!+\\!f^{\frac{3}{2}}\\!+\\!\frac{1}{f^{\frac{3}{2}}}\Big{)}\\!\\!\sum^{\infty}_{n=-\infty}\\!\\!q^{n+\frac{1}{2}}$
(16)
$\displaystyle\quad\rightarrow\\!\Big{(}f^{\frac{1}{2}}\\!+\\!\frac{1}{f^{\frac{1}{2}}}\\!+\\!f^{\frac{3}{2}}\\!+\\!\frac{1}{f^{\frac{3}{2}}}\Big{)}\Big{(}q^{\frac{1}{2}}\\!+\\!\frac{1}{q^{\frac{1}{2}}}\\!+\\!q^{\frac{3}{2}}\\!+\\!\frac{1}{q^{\frac{3}{2}}}\\!+\\!\cdots\\!\Big{)}\,,$

where the first term are the states due to $V_{2}$, while the second are from
$V_{3}$. In the latter, we have the charges in the ${\bf 4}$ of $SU(2)_{F}$
which multiplies the KK tower, which now carries fractional powers in $q$ due
to the twist. We also note that the first term has 5d superconformal charges
of a conserved current, while the other has those of a Higgs branch operator
whose lowest component is a scalar in the ${\bf 4}$ of $SU(2)_{R}$.

Next we inquire what happens as we increase the value of the holonomy. As
before, we expect to get the 5d $SU(2)_{\pi}$ gauge theory as long as the
holonomy isn’t big enough that additional states in the KK tower become
massless. This occurs first when $3m_{f}=m_{q}$, in which case the states with
the charges $\frac{q^{\frac{1}{2}}}{f^{\frac{3}{2}}}$ becomes massless. We
note, though, that no additional state in the conserved current multiplet
becomes massless. As such, the global symmetry should remain $U(1)$. We again
expect that in this regime we get a new minimal SUSY theory with a $U(1)$
global symmetry and that possess a mass deformation leading to the 5d gauge
theory. Like in the previous case, such a theory indeed exists, which is the
so-called the $\tilde{E}_{1}$ SCFT. This theory appears as the UV completion
of the 5d $SU(2)_{\pi}$ gauge theory.

What happens if we continue to increase the holonomy? As in the previous case,
we now trigger the same mass deformation but in the opposite direction.
However, unlike the previous case, the low-energy theory one gets for the
$\tilde{E}_{1}$ SCFT depends on the sign of the mass deformation, with one
sign giving the 5d gauge theory while the other giving the $E_{0}$ SCFT. Since
we got to this theory from the gauge theory side, the other direction must
lead to the $E_{0}$ theory. Therefore, we conclude that once the holonomy
increases we should get the $E_{0}$ SCFT at low-energies.

What happens if we continue to increase the holonomy? The next point where we
get massless matter is when $m_{f}=m_{q}$. However, now the massless matter we
get are the $\frac{q}{f}$ ones from the current multiplet, but also
$\frac{q^{\frac{1}{2}}}{f^{\frac{1}{2}}}$ and
$\frac{q^{\frac{3}{2}}}{f^{\frac{3}{2}}}$ from $V_{3}$. As we now have extra
conserved currents, we should get an $SU(2)$ global symmetry, and so expect to
get a new 5d theory with $SU(2)$ global symmetry and a deformation leading to
the $E_{0}$ 5d SCFT. This is the $\widehat{E}_{1}$ 5d SCFT. We also notice
that we get additional massless states from $V_{3}$. These should provide a
Higgs branch operator in the ${\bf 4}$ of the $SU(2)$, and whose ground state
is a scalar in the ${\bf 4}$ of $SU(2)_{R}$. Such a state should contribute to
the index as $\chi_{\bf 4}(c)x^{3}$. This is precisely what we observe in
(11), with this state and the $SU(2)$ conserved currents being the first low-
lying states we observe. The fact that we can identify the origin of these
operators, using a different realization of the $\widehat{E}_{1}$ theory, is
an indication in favor of the correctness of (11).

Figure 6: A schematic summery of the expected low-energy theory for the
compactification of the 6d $(2,0)$ $A_{2}$ theory with a twist in its outer
automorphism symmetry. Here we use the same notations as in Figure 5.

To complete the discussion, we can again ask what happens if we further
increase the holonomy? Again, this will trigger the mass deformation of the
$\widehat{E}_{1}$ SCFT. Like in the $E_{1}$ case, this mass deformation is in
an $SU(2)$ global symmetry and so the Weyl group of $SU(2)$ relates the
positive and negative mass deformations. As such increasing the holonomy
should again yield the $E_{0}$ SCFT as the low-energy theory. Figure 6
provides a summary of the different theories we get in this case. We can in
principle consider increasing the mass deformation further, though we shall
not pursue this here.

##  Appendix B Partition functions of $SU(3)_{\frac{1}{2}}+1{\bf Sym}$

We summarize the detailed computations for the partition functions of
$SU(3)_{\frac{1}{2}}+1\mathbf{Sym}$ theory. We first compute the instanton
partition function on $\Omega$-deformed $S^{1}\times\mathbb{C}^{2}$ using the
ADHM construction for the instanton moduli space which generalizes the
discussions in Nekrasov (2003); Nekrasov and Okounkov (2006); Hwang  _et al._
(2015). The instanton partition function is expressed as a series
$Z_{\mathrm{inst}}=\sum_{k}q^{k}Z_{k}$ expanded by the instanton number
fugacity $q$, and $Z_{k}$ is the $k$-instanton partition function. In the case
of $SU(N)_{\kappa}$ gauge theories, $Z_{k}$ takes a contour integral
expression as

$\displaystyle Z_{k}=\frac{1}{k!}\oint\left[\prod_{I=1}^{k}\frac{du_{I}}{2\pi
i}\right]e^{-\kappa\sum_{I=1}^{k}u_{I}}Z_{\mathrm{vec}}\prod_{\mathbf{R}}Z_{\mathbf{R}}\,,$
(17)

where $\kappa$ is the Chern-Simons level, $Z_{\mathrm{vec}}$ and
$Z_{\mathbf{R}}$ are the contributions from the vector multiplet and the
hypermultiplets in representation $\mathbf{R}$, respectively.
$Z_{\mathrm{vec}}$ is given by

$\displaystyle Z_{\mathrm{vec}}$
$\displaystyle=\frac{\prod_{I,J}2\sinh(\frac{u_{IJ}^{-}+2\epsilon_{+}}{2})\cdot\prod_{I\neq
J}2\sinh(\frac{u_{IJ}^{-}}{2})}{\prod_{I,J}2\sinh(\frac{u_{IJ}^{-}+\epsilon_{1,2}}{2})\cdot\prod_{I,j}2\sinh(\frac{\pm(u_{I}-a_{j})+\epsilon_{+}}{2})}\,,$
(18)

and the contribution from a symmetric hypermultiplet is given by

$\displaystyle
Z_{\mathbf{Sym}}\\!\\!=\\!\\!\frac{\prod_{I,j}\\!2\sinh(\frac{u_{I}+a_{j}+m_{1}}{2})\\!\prod_{I\leq
J}\\!2\sinh(\frac{\pm(u_{IJ}^{+}+m_{1})-\epsilon_{\\!-\\!}}{2})\\!}{\prod_{I<J}\\!2\sinh(\frac{\pm(u_{IJ}^{+}+m_{1})-\epsilon_{+}}{2})}.$
(19)

Here, $u_{IJ}^{\pm}=u_{I}\pm u_{J}$, $\epsilon_{1,2}$ are $\Omega$-deformation
parameters with $\epsilon_{\pm}=\frac{\epsilon_{1}\pm\epsilon_{2}}{2}$,
$a_{j}$ are $SU(N)$ gauge holonomies satisfying $\sum_{j=1}^{N}a_{j}=0$, and
$m_{1}$ is the mass parameter of the symmetric hyper. The contour integral in
(17) is evaluated through the Jeffrey-Kirwan (JK) residue prescription Hwang
_et al._ (2015).

However, the integral in (17) for $SU(3)_{\frac{1}{2}}\\!+\\!1\mathbf{Sym}$
exhibits poles at the infinity $u_{I}=\pm\infty$ with degree higher than 1.
Treating these higher degree poles poses a subtle issue, and currently, we
lack a suitable method to handle such poles in contour integrals. To avoid
this subtlety, we introduce an additional antisymmetric hypermultiplet ($\bf
AS$) and first compute the instanton partition function of the
$SU(3)_{0}+1\mathbf{Sym}+1\mathbf{AS}$ theory. The contour integral receives
an additional contribution from the antisymmetric hyper with mass $m_{2}$ as

$Z_{\mathbf{AS}}\\!\\!=\\!\\!\frac{\prod_{\begin{subarray}{c}I,j\end{subarray}}2\sinh(\frac{u_{I}+a_{j}+m_{2}}{2})\\!\prod_{I<J}\\!2\sinh(\frac{\pm(u_{IJ}^{+}+m_{2})-\epsilon_{-}}{2})\\!}{\prod_{I\leq
J}2\sinh(\frac{\pm(u_{IJ}^{+}+m_{2})-\epsilon_{+}}{2})}\ .$ (20)

This theory with $1{\bf AS}$ is a 5d KK theory and the theory
$SU(3)_{\frac{1}{2}}+1\mathbf{Sym}$ which we are interested in can be obtained
by integrating out the antisymmetric matter.

One can evaluate the contour integral of the
$SU(3)_{0}+1\mathbf{Sym}+1\mathbf{AS}$ theory using the JK-residue
prescription, and we checked that the resulting instanton partition function
becomes a solution to the blowup equation introduced in Kim  _et al._ (2021)
up to 3-instanton order. To compute the instanton partition function of the
$SU(3)_{\frac{1}{2}}+1\mathbf{Sym}$ theory, we take the limit
$m_{2}\to\infty$. Then we find

$\displaystyle\lim_{m_{2}\rightarrow\infty}Z_{\rm
inst}[SU(3)_{0}+1\mathbf{Sym}+1\mathbf{AS}]$ (21)
$\displaystyle\to\operatorname{PE}\left[\frac{-x^{2}e^{-5i\mu/2}}{(1-xy)(1-x/y)}\right]\cdot
Z_{{\rm inst}}[SU(3)_{\frac{1}{2}}+1\mathbf{Sym}]$

and

$\displaystyle Z_{{\rm
inst}}[SU(3)_{\frac{1}{2}}+1\mathbf{Sym}]=\operatorname{PE}\left[\sum_{k=1}^{\infty}q^{k}\mathcal{Z}_{k}\right]\,,$
(22)

with the first few $\mathcal{Z}_{k}$ being

$\displaystyle\mathcal{Z}_{1}\\!=\\!\big{(}e^{-5i\mu/2}\\!+\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{3i\mu/2}\big{)}x^{2}\\!+\\!\Big{[}\chi_{\mathbf{2}}(y)\big{(}e^{-5i\mu/2}\\!-\\!\chi_{\mathbf{3}}^{A_{2}}e^{-i\mu/2}\\!+\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{3i\mu/2}\big{)}\\!+\\!e^{i\mu/2}\\!-\\!\chi_{\mathbf{3}}^{A_{2}}e^{5i\mu/2}\Big{]}x^{3}$
(23) $\displaystyle\
+\\!\Big{[}\chi_{\mathbf{3}}(y)\big{(}e^{-5i\mu/2}\\!-\\!\chi_{\mathbf{\mathbf{3}}}^{A_{2}}e^{-i\mu/2}\\!+\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{3i\mu/2}\big{)}\\!+\\!\chi_{\mathbf{2}}(y)\big{(}(\chi_{\mathbf{8}}^{A_{2}}\\!+\\!2)e^{i\mu/2}\\!-\\!\chi_{\mathbf{3}}^{A_{2}}e^{5i\mu/2}\\!+\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{-3i\mu/2}\big{)}\\!-\\!\big{(}\chi_{\overline{\mathbf{6}}}^{A_{2}}\\!+\\!3\chi_{\mathbf{3}}^{A_{2}}\big{)}e^{-i\mu/2}\Big{]}x^{4}$
$\displaystyle\
+\\!\Big{[}\chi_{\mathbf{4}}(y)\big{(}e^{-5i\mu/2}\\!-\\!\chi_{\mathbf{3}}^{A_{2}}e^{-i\mu/2}\\!+\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{3i\mu/2}\big{)}\\!+\\!\chi_{\mathbf{3}}(y)\big{(}(\chi_{\mathbf{8}}^{A_{2}}\\!+\\!2)e^{i\mu/2}\\!-\\!\chi_{\mathbf{3}}^{A_{2}}e^{5i\mu/2}\\!+\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{-3i\mu/2}\big{)}$
$\displaystyle\quad~{}\\!-\\!\chi_{\mathbf{2}}(y)\big{(}(\chi_{\mathbf{15}}^{A_{2}}\\!+\\!2\chi_{\overline{\mathbf{6}}}^{A_{2}}\\!+\\!4\chi_{\mathbf{3}}^{A_{2}})e^{-i\mu/2}\\!+\\!\chi_{\mathbf{6}}^{A_{2}}e^{3i\mu/2}\big{)}\\!+\\!\big{(}\chi_{\mathbf{10}}^{A_{2}}\\!+\\!3\chi_{\mathbf{8}}^{A_{2}}\\!+\\!1\big{)}e^{i\mu/2}\\!+\\!\big{(}\chi_{\overline{\mathbf{15}}}^{A_{2}}\\!+\\!2\chi_{\overline{\mathbf{3}}}^{A_{2}}\big{)}e^{-3i\mu/2}\Big{]}x^{5}+\mathcal{O}(x^{6})\,,$
$\displaystyle\mathcal{Z}_{2}\\!=\\!e^{i\mu}x^{2}\\!+\\!\left(\chi_{\mathbf{2}}(y)e^{i\mu}\\!+\\!e^{-2i\mu}\\!+\\!e^{4i\mu}\right)x^{3}\\!+\\!\Big{[}\chi_{\mathbf{3}}(y)e^{i\mu}\\!+\\!\big{(}\chi_{\mathbf{2}}(y)\big{(}e^{4i\mu}\\!+\\!e^{-2i\mu}\\!-\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{2i\mu}\big{)}\\!+\\!e^{i\mu}\\!-\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{-i\mu}\big{)}\Big{]}x^{4}$
$\displaystyle\
+\\!\Big{[}\chi_{\mathbf{4}}(y)e^{i\mu}\\!+\\!\chi_{\mathbf{3}}(y)\big{(}e^{4i\mu}\\!+\\!e^{-2i\mu}\\!-\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{2i\mu}\big{)}\\!+\\!\chi_{\mathbf{2}}(y)\big{(}(\chi_{\mathbf{8}}^{A_{2}}\\!+\\!3)e^{i\mu}\\!+\\!\chi_{\mathbf{3}}^{A_{2}}e^{3i\mu}\\!-\\!\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{-i\mu}\big{)}\\!-\\!\big{(}\chi_{\mathbf{6}}^{A_{2}}\\!+\\!3\chi_{\overline{\mathbf{3}}}^{A_{2}}\big{)}e^{2i\mu}\Big{]}x^{5}\\!+\\!\mathcal{O}(x^{6})\,,$
$\displaystyle\mathcal{Z}_{3}\\!=\\!e^{3i\mu/2}x^{3}\\!+\\!\chi_{\mathbf{2}}(y)e^{3i\mu/2}x^{4}\\!+\\!\Big{[}\chi_{\mathbf{3}}(y)e^{3i\mu/2}\\!-\\!\big{(}\chi_{\overline{\mathbf{3}}}^{A_{2}}e^{-i\mu/2}\\!+\\!e^{3i\mu/2}\big{)}\Big{]}x^{5}\\!+\\!\mathcal{O}(x^{6})\,,$

where $x=e^{-\epsilon_{+}}$, $y=e^{-\epsilon_{-}}$, $\mu=im_{1}$, and
$\chi_{\mathbf{R}}^{A_{2}}$ is the character of representation $\mathbf{R}$ in
$\mathfrak{su}(3)$ gauge algebra.

Next, we compute the superconformal index using the expression in (9).
However, our current computational resources impose limitations on our ability
to compute the instanton partition function only up to 3-instantons. This
result may not be sufficient for us to obtain the superconformal index beyond
the third order in $x$-expansion. To improve our computational capacity, we
first identify the flavor symmetry of the SCFT, which turns out to be
$SU(2)_{a}\times SO(3)_{b}$, using the result up to $x^{2}$-order. Then we
exploit the fact that the index must form representations of the flavor
symmetry. This approach enables us to compute the higher-order terms up to
$x^{6}$-order as follows:

$\displaystyle\mathcal{I}[SU(3)_{\frac{1}{2}}\\!+\\!1\mathbf{Sym}]\\!=\\!1\\!+\\!(\chi_{\mathbf{3}}(a)\\!+\\!\chi_{\mathbf{3}}(b))x^{2}$
(24)
$\displaystyle\\!+\\!(1\\!\\!+\\!\\!\chi_{\mathbf{3}}(a)\\!\\!+\\!\\!\chi_{\mathbf{3}}(b))(\chi_{\mathbf{2}}(y)x^{3}\\!\\!+\\!\\!\chi_{\mathbf{3}}(y)x^{4}\\!\\!+\\!\\!\chi_{\mathbf{4}}(y)x^{5}\\!\\!+\\!\\!\chi_{\mathbf{5}}(y)x^{6})$
$\displaystyle\\!+\\!\chi_{\mathbf{4}}(a)\chi_{\mathbf{3}}(b)(x^{3}\\!+\\!\chi_{\mathbf{2}}(y)x^{4}\\!+\\!\chi_{\mathbf{3}}(y)x^{5}\\!+\\!\chi_{\mathbf{4}}(y)x^{6})$
$\displaystyle\\!+\\!(2\chi_{\mathbf{5}}(a)\\!+\\!\chi_{\mathbf{5}}(b)\\!+\\!\chi_{\mathbf{3}}(a)\chi_{\mathbf{3}}(b)\\!+\\!3)x^{4}\\!+\\!3\chi_{\mathbf{2}}(y)x^{5}$
$\displaystyle\\!+\\!\chi_{\mathbf{2}}(y)[2\chi_{\mathbf{5}}(a)\\!+\\!\chi_{\mathbf{5}}(b)\\!+\\!(2\chi_{\mathbf{3}}(a)\\!+\\!1)\chi_{\mathbf{3}}(b)\\!+\\!3\chi_{\mathbf{3}}(a)]x^{5}$
$\displaystyle\\!+\\![(\chi_{\mathbf{6}}(a)\\!+\\!2\chi_{\mathbf{4}}(a)\\!+\\!\chi_{\mathbf{2}}(a))\chi_{\mathbf{3}}(b)\\!+\\!\chi_{\mathbf{4}}(a)\chi_{\mathbf{5}}(b)\\!-\\!\chi_{\mathbf{2}}(a)]x^{5}$
$\displaystyle\\!+\\!\chi_{\mathbf{3}}(y)[3\chi_{\mathbf{5}}(a)\\!+\\!2\chi_{\mathbf{5}}(b)\\!+\\!(\chi_{\mathbf{3}}(a)\\!+\\!1)(3\chi_{\mathbf{3}}(b)\\!+\\!4)\\!+\\!2]x^{6}$
$\displaystyle\\!+\\!\chi_{\mathbf{2}}(y)[(2\chi_{\mathbf{6}}(a)\\!+\\!5\chi_{\mathbf{4}}(a)\\!+\\!3\chi_{\mathbf{2}}(a))\chi_{\mathbf{3}}(b)\\!-\\!\chi_{\mathbf{2}}(a)]x^{6}$
$\displaystyle\\!+\\![\chi_{\mathbf{2}}(y)\chi_{\mathbf{4}}(a)(2\chi_{\mathbf{5}}(b)\\!+\\!1)\\!+\\!(2\chi_{\mathbf{5}}(a)\\!+\\!\chi_{\mathbf{3}}(a)\\!+\\!5)\chi_{\mathbf{3}}(b)]x^{6}$
$\displaystyle\\!+\\![(\chi_{\mathbf{7}}(a)\\!+\\!2\chi_{\mathbf{3}}(a))(\chi_{\mathbf{5}}(b)\\!+\\!3)\\!+\\!\chi_{\mathbf{7}}(b)\\!+\\!2\chi_{\mathbf{5}}(a)]x^{6}\\!\\!+\\!\\!\mathcal{O}(x^{7}).$

Obviously, the higher instanton contributions to this result involves some
guess work relying on representations of the $SU(2)_{a}\times SO(3)_{b}$
flavor symmetry. Nevertheless, we will confirm this result by Higgsing to the
index of the $\widehat{E}_{1}$ theory, and comparing it with another result
obtained by a freezing procedure, which we will carry out below.

We compute the superconformal index of the $\widehat{E}_{1}$ theory by taking
Higgs branch limit corresponding to extracting residue at the pole
$bx^{2}\\!=\\!1$ of the above index. The result is

$\displaystyle\mathcal{I}[\widehat{E}_{1}]=\frac{\operatorname{Res}_{bx^{2}=1}\mathcal{I}[SU(3)_{\frac{1}{2}}+1\mathbf{Sym}]}{\mathcal{I}_{\mathrm{extra}}\cdot\mathcal{I}_{\mathrm{extra}}^{\prime}},$
(25)

where

$\displaystyle\mathcal{I}_{\mathrm{extra}}^{\prime}$
$\displaystyle\\!=\\!\operatorname{PE}\bigg{[}\frac{x(a^{3}\\!+\\!a\\!+\\!a^{-1}\\!+\\!a^{-3})}{(1\\!-\\!xy)(1\\!-\\!x/y)}\bigg{]}$
(26)

corresponds to contributions from free hypermultiplets that decouple from the
CFT.

##  Appendix C $\widehat{E}_{1}$ index from freezing

We now discuss the freezing procedure to calculate the superconformal index of
the $\widehat{E}_{1}$ theory from the index of the $Sp(1)+7\mathbf{F}$ theory.
The superconformal index of $Sp(1)+7\mathbf{F}$ was previously computed in Kim
_et al._ (2012); Hwang  _et al._ (2015). We further improve those results and
compute higher order terms as follows:

$\displaystyle\mathcal{I}[Sp(1)\\!+\\!7\mathbf{F}]=1\\!+\\!\chi_{\mathbf{248}}^{E_{8}}x^{2}\\!+\\!(\chi_{\mathbf{248}}^{E_{8}}\\!+\\!1)(\chi_{\mathbf{2}}(y)x^{3}\\!+\\!\chi_{\mathbf{3}}(y)x^{4}\\!+\\!\chi_{\mathbf{4}}(y)x^{5}\\!+\\!\cdots\\!+\\!\chi_{\mathbf{9}}(y)x^{10})\\!+\\!(\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!1)x^{4}$
(27)
$\displaystyle\\!+\\!\chi_{\mathbf{2}}(y)(\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!\chi_{\mathbf{248}}^{E_{8}}\\!+\\!1)x^{5}\\!+\\!(\chi_{\mathbf{3}}(y)(\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!2\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!2\chi_{\mathbf{248}}^{E_{8}}\\!+\\!2)\\!+\\!\chi_{\mathbf{1763125}}^{E_{8}}\\!+\\!\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!2\chi_{\mathbf{248}})x^{6}$
$\displaystyle\\!+\\!(\chi_{\mathbf{4}}(y)(2\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!2\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!4\chi_{\mathbf{248}}^{E_{8}}\\!+\\!2)\\!+\\!\chi_{\mathbf{2}}(y)(\chi_{\mathbf{4096000}}^{E_{8}}\\!+\\!\chi_{\mathbf{1763125}}^{E_{8}}\\!+\\!\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!2\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!4\chi_{\mathbf{248}}\\!+\\!2))x^{7}$
$\displaystyle\\!+\\!(\chi_{\mathbf{5}}(y)(2\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!3\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!2\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!5\chi_{\mathbf{248}}^{E_{8}}\\!+\\!4)\\!+\\!\chi_{\mathbf{3}}(y)(2\chi_{\mathbf{4096000}}^{E_{8}}\\!+\\!2\chi_{\mathbf{1763125}}^{E_{8}}\\!+\\!\chi_{\mathbf{779247}}^{E_{8}}\\!+\\!3\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!3\chi_{\mathbf{27000}}^{E_{8}}))x^{8}$
$\displaystyle\\!+\\!(\chi_{\mathbf{3}}(y)(\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!8\chi_{\mathbf{248}}^{E_{8}}+3)\\!+\\!\chi_{\mathbf{79143000}}^{E_{8}}\\!+\\!\chi_{\mathbf{4096000}}^{E_{8}}\\!+\\!\chi_{\mathbf{2450240}}^{E_{8}}\\!+\\!\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!3\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!2\chi_{\mathbf{248}}^{E_{8}}\\!+\\!3)x^{8}$
$\displaystyle\\!+\\!(\chi_{\mathbf{6}}(y)(3\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!3\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!2\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!7\chi_{\mathbf{248}}^{E_{8}}\\!+\\!4)\\!+\\!\chi_{\mathbf{4}}(y)(3\chi_{\mathbf{4096000}}^{E_{8}}\\!+\\!\chi_{\mathbf{2450240}}^{E_{8}}\\!+\\!3\chi_{\mathbf{1763125}}^{E_{8}}\\!+\\!2\chi_{\mathbf{779247}}^{E_{8}}\\!+\\!5\chi_{\mathbf{30380}}^{E_{8}}))x^{9}$
$\displaystyle\\!+\\!(\chi_{\mathbf{4}}(y)(7\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!3\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!11\chi_{\mathbf{248}}^{E_{8}}\\!+\\!7)\\!+\\!\chi_{\mathbf{2}}(y)(\chi_{\mathbf{281545875}}^{E_{8}}\\!+\\!\chi_{\mathbf{79143000}}^{E_{8}}\\!+\\!3\chi_{\mathbf{4096000}}^{E_{8}}\\!+\\!\chi_{\mathbf{2450240}}^{E_{8}}\\!+\\!2\chi_{\mathbf{1763125}}^{E_{8}}))x^{9}$
$\displaystyle\\!+\\!\chi_{\mathbf{2}}(y)(2\chi_{\mathbf{779247}}^{E_{8}}\\!+\\!\chi_{\mathbf{147250}}^{E_{8}}\\!+\\!5\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!6\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!2\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!8\chi_{\mathbf{248}}^{E_{8}}\\!+\\!4)x^{9}\\!+\\!\chi_{\mathbf{7}}(y)(3\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!4\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!3\chi_{\mathbf{3875}}^{E_{8}})x^{10}$
$\displaystyle\\!+\\!(\chi_{\mathbf{7}}(y)(8\chi_{\mathbf{248}}^{E_{8}}\\!+\\!6)\\!+\\!\chi_{\mathbf{5}}(y)(5\chi_{\mathbf{4096000}}^{E_{8}}\\!+\\!\chi_{\mathbf{2450240}}^{E_{8}}\\!+\\!4\chi_{\mathbf{1763125}}^{E_{8}}\\!+\\!4\chi_{\mathbf{779247}}^{E_{8}}\\!+\\!\chi_{\mathbf{147250}}^{E_{8}}\\!+\\!9\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!10\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!4\chi_{\mathbf{3875}}^{E_{8}}))x^{10}$
$\displaystyle\\!+\\!(\chi_{\mathbf{5}}(y)(18\chi_{\mathbf{248}}^{E_{8}}\\!+\\!8)\\!+\\!\chi_{\mathbf{3}}(y)(2\chi_{\mathbf{281545875}}^{E_{8}}\\!+\\!\chi_{\mathbf{203205000}}^{E_{8}}\\!+\\!2\chi_{\mathbf{79143000}}^{E_{8}}\\!+\\!\chi_{\mathbf{70680000}}^{E_{8}}\\!+\\!6\chi_{\mathbf{4096000}}^{E_{8}}\\!+\\!3\chi_{\mathbf{2450240}}^{E_{8}}))x^{10}$
$\displaystyle\\!+\\!(\chi_{\mathbf{3}}(y)(4\chi_{\mathbf{1763125}}^{E_{8}}\\!+\\!4\chi_{\mathbf{779247}}^{E_{8}}\\!+\\!2\chi_{\mathbf{147250}}^{E_{8}}\\!+\\!11\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!15\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!7\chi_{\mathbf{3875}}^{E_{8}}\\!+\\!15\chi_{\mathbf{248}}^{E_{8}}\\!+\\!10)\\!+\\!\chi_{\mathbf{2642777280}}^{E_{8}}\\!+\\!9\chi_{\mathbf{248}}^{E_{8}}\\!+\\!2)x^{10}$
$\displaystyle\\!+\\!(\chi_{\mathbf{344452500}}^{E_{8}}\\!+\\!\chi_{\mathbf{281545875}}^{E_{8}}\\!+\\!3\chi_{\mathbf{4096000}}^{E_{8}}\\!+\\!4\chi_{\mathbf{1763125}}^{E_{8}}\\!+\\!3\chi_{\mathbf{779247}}^{E_{8}}\\!+\\!\chi_{\mathbf{147250}}^{E_{8}}\\!+\\!6\chi_{\mathbf{30380}}^{E_{8}}\\!+\\!4\chi_{\mathbf{27000}}^{E_{8}}\\!+\\!2\chi_{\mathbf{2875}}^{E_{8}})x^{10}\\!+\\!\mathcal{O}(x^{11})\,,$

where
$\chi_{\mathbf{r}}^{E_{8}}=\chi_{\mathbf{r}}^{E_{8}}(e^{i\mu_{1}},\cdots,e^{i\mu_{8}})$
are the characters of representation $\mathbf{r}$ of the $E_{8}$ flavor
symmetry at the CFT fixed point, as predicted in Seiberg (1996), and
$e^{i\mu_{8}}\equiv q^{-2}$ is the instanton number fugacity. To arrive at
this result, we used the instanton partition function $Z_{\mathrm{inst}}$ up
to 5-instantons. With this, we can determine the index only up to
$x^{7}$-order. To compute the terms beyond $x^{7}$-order, even though it
requires higher instanton contributions, we use the fact that the terms in the
index must form $E_{8}$ representations. This allows us to uniquely fix all
higher-order terms up to $x^{10}$-order.

We will now proceed with the freezing procedure, which effectively converts
the brane configuration with an O7--plane with 8 D7-branes for the
$Sp(1)+7\mathbf{F}$ theory into that with an O7+-plane and eventually leads to
the brane web for $\widehat{E}_{1}$ as investigated in Hayashi  _et al._
(2023); Kim  _et al._ . In the context of the index, this corresponds to the
specialization of $E_{8}$ fugacities in the index of $Sp(1)+7\mathbf{F}$, as
given by (13). Consequently, we can compute the index for the
$\widehat{E}_{1}$ theory from that for the $Sp(1)+7\mathbf{F}$ by applying the
specialization of $E_{8}$ fugacities. The index of $Sp(1)+7\mathbf{F}$ up to
$x^{10}$-order in (27) enables us to calculate the superconformal index of the
$\widehat{E}_{1}$ theory up to $x^{5}$-order which is summarized in (11). To
obtain the result beyond this order, we need higher instanton computations at
$k>5$ of the $Sp(1)+7\mathbf{F}$ theory, which unfortunately are not currently
available.

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