aainstitutetext: Department of Physics, Technion, Haifa, 32000,
Israelbbinstitutetext: Section de Mathématiques, Université de Genève, 1211
Genève 4, Switzerland

#  $C_{2}$ generalization of the van Diejen model from the minimal
$(D_{5},D_{5})$ conformal matter.

Belal Nazzal b Anton Nedelin<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study superconformal indices of $4d$ compactifications of the $6d$ minimal
$(D_{N+3},D_{N+3})$ conformal matter theories on a punctured Riemann surface.
Introduction of supersymmetric surface defect in these theories is done at the
level of the index by the action of the finite difference operators on the
corresponding indices. There exist at least three different types of such
operators according to three types of punctures with $A_{N},C_{N}$ and
$\left(A_{1}\right)^{N}$ global symmetries. We mainly concentrate on $C_{2}$
case and derive explicit expression for an infinite tower of difference
operators generalizing the van Diejen model. We check various properties of
these operators originating from the geometry of compactifications. We also
provide an expression for the kernel function of both our $C_{2}$ operator and
previously derived $A_{2}$ generalization of van Diejen model. Finally we also
consider compactifications with $A_{N}$-type punctures and derive the full
tower of commuting difference operators corresponding to this root system
generalizing the result of our previous paper.

##  1 Introduction

Intricate connection between supersymmetric gauge theories and integrable
systems plays an important role in modern theoretical physics and mathematics.
From physics point of view whenever some sector of the gauge theory is related
to an integrable system, observables in this sector can be computed using rich
variety of integrability techniques. Canonical example of this situation is
the integrability of $\mathcal{N}=4$ Super Yang-Mills (SYM) theory
Minahan:2002ve ; Beisert:2010jr . In particular it connects calculation of the
planar scaling dimensions of $\mathcal{N}=4$ SYM to the integrable system of
spin chains and allows to compute these scaling dimensions at arbitrary
coupling. On the other hand exploration of such connections can also shed
light on some questions about integrable systems and even lead to construction
of new classes of such systems making this kind of studies interesting from
mathematics point of view.

In our work we are exploring a particular class of connections between six
dimensional $(1,0)$ superconformal field theories and elliptic quantum
mechanics Hamiltonians in the spirit of Bethe/gauge correspondence of Nekrasov
and Shatashvili Nekrasov:2009ui ; Nekrasov:2009rc ; Bullimore:2014awa ;
Hatsuda:2018lnv . In particular we consider four-dimensional theories with
four supercharges obtained by compactification of a $6d$ SCFT on a punctured
Riemann surface. In order for an integrable model to emerge in this setting we
have to introduce surface defects into our $4d$ theory and study its
superconformal index.

This construction was first established in the context of compactifications of
$6d$ $(2,0)$ of ADE type Gaiotto:2012xa ; Lemos:2012ph . In particular the
corresponding $4d$ superconformal indices with the defect were found to be
closely related to the Ruijsenaars-Schneider (RS) elliptic analytic finite
difference operators (A$\Delta$Os). Later these results were extended to many
other cases: class $S_{k}$ $4d$ theories Gaiotto:2015usa , compactifications
of $A_{2}$ and $D_{4}$ minimal $6d$ SCFTs Razamat:2018zel  and
compactifications of rank one E-string theories Nazzal:2018brc . In some of
these cases the obtained A$\Delta$Os were already known in the literature. For
example in E-string compactifications van Diejen (vD) model Diejen ; MR1275485
was observed. But in some cases operators were previously unknown as in the
case of the minimal $6d$ SCFTs compactifications. Study of such novel
operators constitutes an interesting field of research with some initial steps
already taken in this direction Ruijsenaars:2020shk ; minimal_operators .

From the point of view of physics the connection of superconformal indices in
the presence of defects with the integrable quantum mechanics Hamiltonians
allows one to bootstrap index of an arbitrary theory obtained by compactifying
corresponding $6d$ theory on a punctured Riemann surface. In particular if we
know the eigenfunctions of the corresponding A$\Delta$O we can compute the
index of any theory obtained in such compactifications including non-
Lagrnagian theories for which there are no other methods of computing the
index. As mentioned previously, the first setting where these ideas were
tested are $4d$ class-S theories obtained in the compactifications of $6d$
$(2,0)$ theory Gaiotto:2012xa . In this case the corresponding integrable
system was given by elliptic RS model. Eigenfunctions of the elliptic RS
Hamiltonians are not known in general but some of their limits are well
studied in the mathematical literature. These limits were used in order to
prove previously established relations Gadde:2011uv ; Gadde:2009kb ;
Gadde:2011ik  of superconformal indices with the Macdonald and Schur
polynomials allowing one to compute indices of class S non-Lagrangian
theories.

The construction outlined above relies on the intermediate $5d$ layer. For
things to work there should exist an effective $5d$ gauge theory obtained by
compactifying original $6d$ SCFT on a circle with a choice of holonomies for
its global symmetries. In particular different $5d$ compactifications lead to
different types of punctures on the Riemann surfaces used to obtain $4d$
theories. One of the interesting problems related to this fact is the
compilation of the dictionary between known compactifications of various $6d$
SCFTs and elliptic integrable systems. In our previous paper Nazzal:2021tiu
we have considered compactifactions of the $6d$ minimal $(D_{N+3},D_{N+3})$
conformal matter theories Razamat:2019ukg ; Razamat:2020bix . In particular we
have derived A$\Delta$Os corresponding to the intermediate $5d$
$\mathrm{SU}(N+1)$ gauge theory or equivalently the $A_{N}$ type puncture on
the compactification surface. These elliptic A$\Delta$Os appeared to be
previously unknown $A_{N}$ generalizations of vD model. On the other hand
there are at least two more $5d$ effective descriptions of $6d$ SCFTs
corresponding to $\mathrm{USp}(2N)$ and $\mathrm{SU}(2)^{N}$ gauge theories
giving rise to the punctures with the same $C_{N}$ and $(A_{1})^{\otimes N}$
global symmetries. These two descriptions should lead to additional higher-
rank generalizations of the vD model.

In our present paper we follow this line of research and closely study
compactifications of the minimal $(D_{N+3},D_{N+3})$ conformal matter theory
on a Riemann surface with the $C_{N}$-type punctures. In particular we
concentrate on the next to simplest case of $N=2$111The simplest case of
$N=1$, as well as the case of $(A_{1})^{1}$, were studied in our previous
paper Nazzal:2021tiu . As a result we observed two alternative
parametrizations of the standard $BC_{1}$ vD model. and derive corresponding
infinite tower of A$\Delta$Os. We also devote part of the paper to the study
and proof of some remarkable properties of these operators that follow from
the geometry of corresponding compactifications. In particular we prove the
novel kernel property for two operators of different type but same rank. We
also discuss commutation property of obtained operators.

The paper is organised as follows. In section 2 we review our previous results
for $A_{N}$ type operators. In addition to the previous results we also derive
full tower of such A$\Delta$Os which was not obtained previously. In Section 3
we derive in details novel generalization of vD operators corresponding to
$C_{2}$ root system. In section 4 we discuss properties of derived $C_{2}$
A$\Delta$Os. In particular we pay special attention to a new kernel function
for simultaneously $A_{2}$ and $C_{2}$ operators and give fully analytic proof
of the corresponding kernel equation. We also briefly discuss commutation
relations and check them perturbatively in expansion. In section 5 we briefly
summarize our results and discuss plans for the future research further
developing these results. Finally the paper has a number of Appendices
collecting useful formulas as well as technical details of various
calculations in our paper.

##  2 $A_{N}$ operators.

In this section we will briefly review derivation of the $A_{N}$
generalization of the van Diejen operator. So called basic version222See the
definition of the basic operators and their towers below. of this operator has
been derived in our previous paper Nazzal:2021tiu . Here we will extend this
result to the full tower of operators.

Just as in Nazzal:2021tiu  in order to derive A$\Delta$O we start with the
$4d$ three-punctured sphere theory which was first introduced in
Razamat:2020bix . This theory is obtained in the compactification of the $6d$
minimal $(D_{N+3},D_{N+3})$ conformal matter on a sphere with two maximal
$\mathrm{SU}(N+1)$ punctures and one minimal $\mathrm{SU}(2)$ puncture. The
quiver of this theory is shown in Figure 1. In addition to the global
symmetry, the punctures are characterized by the moment map operators. In
particular both $\mathrm{SU}(N+1)$ maximal and $\mathrm{SU}(2)$ minimal
punctures are characterized by $(2N+4)$ mesonic and $2$ baryonic moment maps:

$\displaystyle M_{u}={\bf N+1}^{x}\otimes\left({\bf
2N+4}_{u^{N+3}v^{-N-1}w^{-2}}\oplus{\bf
1}_{\left(uv^{N+1}\right)^{2N+4}}\right)\oplus\overline{{\bf
N+1}}^{x}\otimes{\bf 1}_{\left(u^{N}w^{2}\right)^{2N+4}}\,,$ $\displaystyle
M_{v}={\bf N+1}^{y}\otimes\left({\bf 2N+4}_{v^{N+3}u^{-N-1}w^{-2}}\oplus{\bf
1}_{\left(vu^{N+1}\right)^{2N+4}}\right)\oplus\overline{{\bf
N+1}}^{y}\otimes{\bf 1}_{\left(v^{N}w^{2}\right)^{2N+4}}\,,$ $\displaystyle
M_{w}={\bf 2}^{z}\otimes\left({\bf
2N+4}_{\left(uvw^{-2}\right)^{-N-1}}\oplus{\bf
1}_{\left(wv^{N+1}\right)^{2N+4}}\oplus{\bf
1}_{\left(wu^{N+1}\right)^{2N+4}}\right)\,,$ (2.1)

where $a_{i},\,u,\,v,\,w$ are fugacities of the Cartans of the $6d$ global
$\mathrm{SO}(4N+8)$ symmetry. Subscripts of the moment maps written above
denote their charges w.r.t. to these symmetries.

Further $S$-gluing two such trinions along the maximal punctures we can obtain
four-punctured sphere with zero flux two maximal and two minimal punctures. To
obtain the corresponding $4d$ gauge theory we should just take two copies of
trinion theories shown in Figure 1 and then identify and gauge corresponding
global symmetries of the maximal punctures. Here and everywhere else in the
paper all operations with gauge theories are expressed in terms of the
superconformal indices. In this langauge, the gluing procedure takes the
following form:

$\displaystyle
K_{4}^{A}(x,\tilde{x},z,\tilde{z})=\kappa_{N}\oint\prod\limits_{i=1}^{N}\frac{dy_{i}}{2\pi
iy_{i}}\prod\limits_{i\neq
j}^{N+1}\frac{1}{\Gamma_{e}\left(\frac{y_{i}}{y_{j}}\right)}\bar{K}_{3}^{A}(\tilde{x},y,\tilde{z})K_{3}^{A}(x,y,z)\,.$
(2.2)

where $K^{A}_{3}(x,y,z)$ is the index of the trinion with $y$ being fugacity
of the global $\mathrm{SU}(N+1)$ symmetry of the puncture we glue along and
$\bar{K}_{3}^{A}$ is the index of the conjugated trinion. Finally $\kappa_{N}$
is the usual constant given by:

$\displaystyle\kappa_{N}\equiv\frac{\left(q;q\right)_{\infty}^{N}\left(p;p\right)_{\infty}^{N}}{\left(N+1\right)!}\,.$
(2.3)

Figure 1: (a) $A_{N}$ three-punctured sphere with two maximal and one minimal
puncture. (b) $A_{N}$ four-punctured sphere obtained by S-gluing two three-
punctured spheres

Performing this $S$-gluing operation we obtain the four-punctured sphere
theory shown in Figure 1 with the corresponding superconformal index specified
in (B.1). This theory was previously obtained by us in Nazzal:2021tiu  to
derive basic $A_{N}$ A$\Delta$Os. Now in order to derive the operator we
should close two minimal punctures of this four-punctured sphere. To do it we
should break the global symmetry of the puncture. This can be achieved by
giving a non-trivial vev
$\langle\partial_{12}^{L}\partial_{34}^{K}M\rangle\neq 0$ to the derivatives
of one of the moment map operators. When we close punctures with at least one
of $K$ or $L$ not equal to zero, i.e. vev is space-time dependent, we
effectively insert defect into the theory Gadde:2013dda ; Gaiotto:2012xa . At
the level of the superconformal index, closing the puncture amounts to giving
a corresponding weight to the fugacity of the puncture’s global symmetry. Once
we do it we hit a pole of the index. Then computation of the residue of this
pole results in the superconformal index of the IR theory that the UV theory
flowed to due to the introduction of the vev. In our case we choose to close
$\mathrm{SU}(2)_{z}$ minimal puncture with the defect and
$\mathrm{SU}(2)_{\tilde{z}}$ puncture without, i.e. choosing $L=K=0$ in the
vev. In particular let’s say that we are going to compute A$\Delta$O acting on
the puncture with the moment maps of charges $\tilde{h}_{i}$ and an overall
$\mathrm{U}(1)$ charge

$\displaystyle\tilde{h}\equiv\prod\limits_{i=1}^{2N+6}\tilde{h}_{i}\,.$ (2.4)

For example the moment maps of the $\mathrm{SU}(N+1)_{x}$ puncture of the
trinion shown in Figure 1 correspond to $M_{u}$ operators specified in (2.1)
and have the following charges:

$\displaystyle\tilde{h}_{i}=u^{N+3}v^{-N-1}w^{-2}a_{i},\,\quad
i=1,\dots,2N+4\,,\quad\prod_{i=1}^{2N+4}a_{i}=1\,,$
$\displaystyle\tilde{h}_{2N+5}=\left(uv^{N+1}\right)^{2N+4},\,\quad\tilde{h}_{2N+6}=\left(u^{N}w^{2}\right)^{-2N-4}\,,\quad\tilde{h}=\left(uw^{-1}\right)^{8(N+2)}\,,$
(2.5)

where we have flipped the charge $\tilde{h}_{2N+6}$ of the last moment map
since this parametrization will be more natural for our operators and in this
case all of the moment maps of the maximal puncture transform in the
fundamental representation of $\mathrm{SU}(N+1)_{x}$. We can now notice from
(2.1) that charges of the minimal puncture moment maps are related to the
charges of the moment maps of the maximal punctures by simple relation

$\displaystyle\tilde{h}^{\mathrm{SU}(2)}_{i}=\left(\tilde{h}^{\mathrm{SU}(N+1)}_{i}\right)^{-1}\left(\tilde{h}^{\mathrm{SU}(N+1)}\right)^{\frac{1}{4}}\,,$
(2.6)

So we can express everything in terms of only the moment maps of the maximal
puncture we act on. Now assume we give vevs to the moment maps (mesonic or
baryonic) with charges $\tilde{h}_{i}^{-1}\tilde{h}^{1/4}$ of both
$\mathrm{SU}(2)_{z}$ and $\mathrm{SU}(2)_{\tilde{z}}$. Moment maps we use
should be the same in order to keep total flux of the $6d$ global symmetries
zero. At the level of the index calculations it corresponds to computing the
residue of the index of the four-punctured sphere theory at the pole

$\displaystyle
z=Z_{i;L,M}^{*}=(pq)^{-\frac{1}{2}}\tilde{h}_{i}\tilde{h}^{-\frac{1}{4}}q^{-M}p^{-L},\;\;\;\;\;\;\;\tilde{z}=\tilde{Z}^{*}_{i;0,0}=(pq)^{-\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{\frac{1}{4}}q^{-\tilde{M}}p^{-\tilde{L}}\,,$
(2.7)

where $L,M,\tilde{L},\tilde{M}$ are positive integers corresponding to the
powers of derivatives inside the vev. As we mentioned previously it is enough
to introduce defect only for one of the two punctures. Hence we choose
$\tilde{M}=\tilde{L}=0$ and keep $L\,,M$ general. Then we compute
corresponding residues of the index of the four-punctured sphere theory and
obtain theory for the tube with two maximal $\mathrm{SU}(N+1)$ punctures and a
codimension-two defect. Its superconformal index is given by333Here and
further we often omit some overall factors which are irrelevant for the
derivations of A$\Delta$Os. Because of this we use $\sim$ instead of strict
equality here.:

$\displaystyle K_{(2;i;L,M)}^{A}(x,\tilde{x})\sim\mathrm{Res}_{z\to
Z^{*}_{i;L,M}\,,\tilde{z}\to\tilde{Z}^{*}_{i;0,0}}K_{4}^{A}(x,\tilde{x},z,\tilde{z})$
(2.8)

Finally in order to obtain desired A$\Delta$O we glue our tube with the defect
to an arbitrary Riemmann surface with maximal $\mathrm{SU}(N+1)_{\tilde{x}}$
puncture. As the result of this gluing we expect to obtain action of a finite-
difference operator on the index ${\mathcal{I}}(\tilde{x})$ of this $4d$
$\mathcal{N}=1$ theory:

$\displaystyle{\mathcal{O}}^{(A_{N};h_{k};L,M)}_{x}\cdot{\mathcal{I}}(x)=\kappa_{N}\oint\prod\limits_{j=1}^{N}\frac{d\tilde{x}_{j}}{2\pi
i\tilde{x}_{j}}\prod\limits_{i\neq
j}^{N+1}\frac{1}{\Gamma_{e}\left(\frac{\tilde{x}_{i}}{\tilde{x}_{j}}\right)}K_{(2;i;L,M)}^{A}(x,\,\tilde{x}){\mathcal{I}}(\tilde{x})$
(2.9)

Details of all the calculations summarized above are given in Appendix B. They
result in the following operator:

$\displaystyle{\mathcal{O}}_{x}^{(A_{N};\tilde{h}_{k};M,L)}=\sum\limits_{(\sum_{i=1}^{N+2}{m_{i}}=M)}\sum\limits_{(\sum_{i=1}^{N+2}{l_{i}}=L)}\sum\limits_{(\sum_{i=1}^{N+1}{s_{i}}=M-m_{N+2})}\sum\limits_{(\sum_{i=1}^{N+1}{r_{i}}=L-l_{N+2})}\,C_{L,M}^{\vec{l},\vec{m},\vec{r},\vec{s}}\times$
$\displaystyle\prod\limits_{i=1}^{N+1}\frac{\prod_{b\neq
k}^{2N+6}\prod_{n=0}^{s_{i}-1}\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{b}^{-1}x_{i}^{-1}q^{n-m_{i}}p^{-l_{i}}\right)\prod_{n=0}^{r_{i}-1}\theta_{q}\left((pq)^{\frac{1}{2}}\tilde{h}_{b}^{-1}x_{i}^{-1}q^{s_{i}-m_{i}}p^{n-l_{i}}\right)}{\prod_{j\neq
i}^{N+1}\prod_{n=0}^{m_{j}-1}\theta_{p}\left(q^{n-m_{i}}p^{-l_{i}}x_{j}/x_{i}\right)\prod_{n=0}^{l_{j}-1}\theta_{q}\left(q^{m_{j}-m_{i}}p^{n-l_{i}}x_{j}/x_{i}\right)}$
$\displaystyle\frac{\prod_{n=0}^{M-s_{i}-1}\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{k}^{-1}x_{i}^{-1}p^{r_{i}-l_{i}}q^{n+s_{i}-m_{i}}\right)\prod_{n=0}^{L-r_{i}-1}\theta_{q}\left((pq)^{\frac{1}{2}}\tilde{h}_{k}^{-1}x_{i}^{-1}p^{n+r_{i}-l_{i}}q^{M-m_{i}}\right)}{\prod_{n=0}^{m_{N+2}-1}\theta_{p}\left((pq)^{-\frac{1}{2}}\tilde{h}_{k}\tilde{h}^{-\frac{1}{2}}x_{i}^{-1}p^{-L-l_{i}}q^{n-M-
m_{i}}\right)\prod_{n=0}^{l_{N+2}-1}\theta_{q}\left((pq)^{-\frac{1}{2}}\tilde{h}_{k}\tilde{h}^{-\frac{1}{2}}x_{i}^{-1}p^{n-L-
l_{i}}q^{m_{N+2}-M-m_{i}}\right)}$
$\displaystyle\frac{\prod_{n=0}^{M-m_{N+2}-s_{i}-1}\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{k}^{-1}\tilde{h}^{\frac{1}{2}}x_{i}p^{l_{i}}q^{n+m_{i}}\right)\prod_{n=0}^{L-l_{N+2}-r_{i}-1}\theta_{q}\left((pq)^{\frac{1}{2}}\tilde{h}_{k}^{-1}\tilde{h}^{\frac{1}{2}}x_{i}p^{n+l_{i}}q^{M-m_{N+2}-s_{i}}\right)}{\prod_{n=0}^{m_{i}-1}\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{k}^{-1}\tilde{h}^{\frac{1}{2}}x_{i}p^{L-l_{N+2}+l_{i}}q^{n+M-m_{N+2}}\right)\prod_{n=0}^{l_{i}-1}\theta_{q}\left((pq)^{\frac{1}{2}}\tilde{h}_{k}^{-1}\tilde{h}^{\frac{1}{2}}x_{i}p^{n+L-l_{N+2}}q^{M-m_{N+2}+m_{i}}\right)}$
$\displaystyle\frac{1}{\prod_{j\neq
i}^{N+1}\prod_{n=0}^{r_{i}-1}\theta_{q}\left(q^{m_{j}-m_{i}+s_{i}-s_{j}}p^{n+l_{j}-l_{i}-r_{j}}x_{j}/x_{i}\right)\prod_{n=0}^{s_{i}-1}\theta_{p}\left(q^{n+m_{j}-m_{i}-s_{j}}p^{l_{j}-l_{i}-r_{j}}x_{j}/x_{i}\right)}\times$
$\displaystyle\hskip
284.52756pt\Delta_{q}^{m_{i}-s_{i}}(x_{i})\Delta_{p}^{l_{i}-r_{i}}(x_{i})\,,$
(2.10)

where $C_{L,M}^{\vec{l},\vec{m},\vec{r},\vec{s}}$ are $x$-independent constant
factors given by,

$\displaystyle
C_{L,M}^{\vec{l},\vec{m},\vec{r},\vec{s}}=\frac{\prod_{n=1}^{M-m_{N+2}}\theta_{p}\left(q^{-n}\right)\prod_{n=1}^{L-l_{N+2}}\theta_{q}\left(q^{m_{N+2}-M}p^{-n}\right)}{\prod_{n=1}^{s_{i}}\theta_{p}\left(q^{-n}p^{-r_{i}}\right)\prod_{n=1}^{r_{i}}\theta_{q}\left(p^{-n}\right)\prod_{n=1}^{m_{i}}\theta_{p}\left(q^{-n}p^{-l_{i}}\right)\prod_{n=1}^{l_{i}}\theta_{q}\left(p^{-n}\right)}$
$\displaystyle\frac{\prod_{b\neq
k}^{2N+6}\prod_{n=0}^{m_{N+2}-1}\theta_{p}\left(\tilde{h}_{k}\tilde{h}^{-\frac{1}{2}}\tilde{h}_{b}q^{n-M}p^{-L}\right)\prod_{n=0}^{l_{N+2}-1}\theta_{q}\left(\tilde{h}_{k}\tilde{h}^{-\frac{1}{2}}\tilde{h}_{b}q^{m_{N+2}-M}p^{n-L}\right)}{\prod_{n=1}^{m_{N+2}}\theta_{p}\left(pq\,\tilde{h}_{k}^{-2}\tilde{h}^{\frac{1}{2}}q^{2M-n}p^{2L-l_{N+2}}\right)\prod_{n=1}^{l_{N+2}}\theta_{q}\left(pq\,\tilde{h}_{k}^{-2}\tilde{h}^{\frac{1}{2}}q^{2M}p^{2L-n}\right)}$

Also $\Delta$s are shift operators defined as follows:

$\displaystyle\Delta^{m}_{a}(x_{i})f(x)\equiv f\left(x_{i}\to
a^{m}x_{i}\right)\,,\quad a=q,p\,.$ (2.12)

The operator contains all shifts of the form $q^{m_{i}}p^{l_{i}}x_{i}$ where
$\vec{m}$ and $\vec{l}$ are all possible partitions of length $N+1$ of $M$ and
$L$ correspondingly. At each level, i.e. fixed $M$ and $L$ there are $2N+6$
operators due to $2N+6$ moment maps with the charges
$\tilde{h}_{k}\tilde{h}^{-\frac{1}{4}}$. There are also $2N+6$ other operators
obtained by giving vevs to the flipped moment maps of the charge
$\tilde{h}_{k}^{-1}\tilde{h}^{\frac{1}{4}}$. They have similar form and
properties so we do not present them here. Now we will refer to the case $M=1$
and $L=0$, or vice versa, as basic operators. These basic operators for
$A_{N}$ generalizations of vD model were derived by us previously in
Nazzal:2021tiu . The operators above also reproduce our previous results when
we fix $M=1,\,L=0$ or $M=0,\,L=1$. Further in our paper we will also need the
basic operator obtained by closing flipped moment maps. This operator can be
found in Nazzal:2021tiu  and has the following form:

$\displaystyle{\mathcal{O}}_{x}^{(A_{N};\tilde{h}_{i}^{-1};1,0)}\cdot{\cal
I}(x)\equiv$ $\displaystyle\hskip 42.67912pt\left(\sum\limits_{l\neq
m}^{N+1}A^{(A_{N};\tilde{h}_{i}^{-1};1,0)}_{lm}(x)\Delta_{q}^{-1}(x_{l})\Delta_{q}(x_{m})+W^{(A_{N};\tilde{h}_{i}^{-1};1,0)}\left(x,\tilde{h}\right)\right){\mathcal{I}}(x)\,,$
(2.13)

where the shift part of this operator is given by

$\displaystyle
A^{(A_{N};\tilde{h}_{i}^{-1};1,0)}_{lm}(x)=\frac{\prod\limits_{j=1}^{2N+6}\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{j}^{-1}x_{l}^{-1}\right)}{\theta_{p}\left(\frac{x_{m}}{x_{l}}\right)\theta_{p}\left(q\frac{x_{m}}{x_{l}}\right)}\prod\limits_{k\neq
m\neq
l}^{N+1}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}x_{k}^{-1}\right)\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{1/2}x_{k}\right)}{\theta_{p}\left(\frac{x_{k}}{x_{l}}\right)\theta_{p}\left(\frac{x_{m}}{x_{k}}\right)}\,,$
(2.14)

and the constant part is given by:

$\displaystyle
W^{(A_{N};\tilde{h}_{i}^{-1};1,0)}(x,\tilde{h})=\frac{\prod\limits_{j\neq
i}^{2N+6}\theta_{p}\left(q^{-1}\tilde{h}_{i}\tilde{h}_{j}\tilde{h}^{-1/2}\right)}{\theta_{p}\left(q^{-2}\tilde{h}_{i}^{2}\tilde{h}^{-1/2}\right)}\prod\limits_{k=1}^{N+1}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}x_{k}^{-1}\right)}{\theta_{p}\left((pq)^{-\frac{1}{2}}\tilde{h}_{i}\tilde{h}^{-1/2}q^{-1}x_{k}^{-1}\right)}+$
$\displaystyle\sum\limits_{m=1}^{N+1}\frac{\prod\limits_{j\neq
i}^{2N+6}\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{j}x_{m}\right)}{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{1/2}qx_{m}\right)}\prod\limits_{k\neq
m}^{N+1}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{1/2}x_{k}\right)\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}x_{k}^{-1}\right)}{\theta_{p}\left(q^{-1}\frac{x_{k}}{x_{m}}\right)\theta_{p}\left(\frac{x_{m}}{x_{k}}\right)}\,.$
(2.15)

This constant part is elliptic function in each $x_{i}$ variable with periods
$1$ and $p$. It has poles in the fundamental domain at the following
positions:

$\displaystyle x_{i}=q^{\pm 1}x_{r}\,,\quad
x_{i}=sq^{\pm\frac{1}{2}}P_{i}^{-\frac{1}{2}}\,,\quad
x_{i}=sq^{\pm\frac{1}{2}}p^{\frac{1}{2}}P_{i}^{-\frac{1}{2}}\,,\quad s=\pm
1\,,$ (2.16)

where we defined

$\displaystyle P_{i}\equiv\prod\limits_{j\neq i}^{N}x_{j}\,.$ (2.17)

From expression (2.15) it looks like there are extra poles in the contsnt part
but carefull examination shows that residues at these values of $x$’s are zero
so there are no real poles there. At the poles (2.16) we have the following
residues:

$\displaystyle\mathrm{Res}_{x_{l}=qx_{r}}W^{(A_{N};\tilde{h}_{i}^{-1};1,0)}(x,\tilde{h})$
$\displaystyle=-\frac{qx_{r}}{\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j=1}^{2N+6}\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{j}x_{r}\right)\times$
$\displaystyle\prod\limits_{k\neq l\neq
r}^{N+1}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{1/2}x_{k}\right)\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}x_{k}^{-1}\right)}{\theta_{p}\left(q^{-1}\frac{x_{k}}{x_{r}}\right)\theta_{p}\left(\frac{x_{r}}{x_{k}}\right)}$
$\displaystyle\mathrm{Res}_{x_{l}=q^{-1}x_{r}}W^{(A_{N};\tilde{h}_{i}^{-1};1,0)}(x,h)$
$\displaystyle=\frac{q^{-1}x_{r}}{\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j=1}^{2N+6}\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{j}^{-1}x_{r}^{-1}\right)\times$
$\displaystyle\prod\limits_{k\neq l\neq
r}^{N+1}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{1/2}x_{k}\right)\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}x_{k}^{-1}\right)}{\theta_{p}\left(q^{-1}\frac{x_{r}}{x_{k}}\right)\theta_{p}\left(\frac{x_{k}}{x_{r}}\right)}$
$\displaystyle\mathrm{Res}_{x_{l}=sq^{-\frac{1}{2}}P_{l}^{-\frac{1}{2}}}W^{(A_{N};\tilde{h}_{i}^{-1};1,0)}(x,h)$
$\displaystyle=s\frac{q^{-\frac{1}{2}}P_{l}^{-\frac{1}{2}}}{2\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j=1}^{2N+6}\theta_{p}\left(sp^{1/2}\tilde{h}_{j}P_{l}^{-1/2}\right)\times$
$\displaystyle\prod\limits_{k\neq
l}^{N}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{1/2}x_{k}\right)\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}x_{k}^{-1}\right)}{\theta_{p}\left(sq^{-1/2}P_{l}^{-1/2}x_{k}^{-1}\right)\theta_{p}\left(sq^{-1/2}P_{l}^{1/2}x_{k}\right)}\,,$
$\displaystyle\mathrm{Res}_{x_{l}=sq^{\frac{1}{2}}P_{l}^{-\frac{1}{2}}}W^{(A_{N};\tilde{h}_{i}^{-1};1,0)}(x,\tilde{h})$
$\displaystyle=-s\frac{q^{\frac{1}{2}}P_{l}^{-\frac{1}{2}}}{2\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j=1}^{2N+6}\theta_{p}\left(sp^{1/2}\tilde{h}_{j}P_{l}^{-1/2}\right)\times$
$\displaystyle\prod\limits_{k\neq
l}^{N}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{1/2}x_{k}\right)\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}x_{k}^{-1}\right)}{\theta_{p}\left(sq^{-1/2}P_{l}^{-1/2}x_{k}^{-1}\right)\theta_{p}\left(sq^{-1/2}P_{l}^{1/2}x_{k}\right)}\,,$
$\displaystyle\mathrm{Res}_{x_{l}=sp^{\frac{1}{2}}q^{-\frac{1}{2}}P_{l}^{-\frac{1}{2}}}W^{(A_{N};\tilde{h}_{i}^{-1};1,0)}(x,\tilde{h})$
$\displaystyle=s\frac{q^{-\frac{1}{2}}p^{\frac{3}{2}}\tilde{h}^{-\frac{1}{2}}P_{l}^{\frac{1}{2}}}{2\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j=1}^{2N+6}\theta_{p}\left(s\tilde{h}_{j}P_{l}^{-1/2}\right)\times$
$\displaystyle\prod\limits_{k\neq
l}^{N}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{1/2}x_{k}\right)\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}x_{k}^{-1}\right)}{\theta_{p}\left(sp^{-1/2}q^{-1/2}P_{l}^{-1/2}x_{k}^{-1}\right)\theta_{p}\left(sp^{1/2}q^{-1/2}P_{l}^{1/2}x_{k}\right)}\,,$
$\displaystyle\mathrm{Res}_{x_{l}=sp^{\frac{1}{2}}q^{\frac{1}{2}}P_{l}^{-\frac{1}{2}}}W^{(A_{N};\tilde{h}_{i}^{-1};1,0)}(x,\tilde{h})$
$\displaystyle=-s\frac{q^{\frac{1}{2}}p^{\frac{3}{2}}\tilde{h}^{-\frac{1}{2}}P_{l}^{\frac{1}{2}}}{2\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j=1}^{2N+6}\theta_{p}\left(s\tilde{h}_{j}P_{l}^{-1/2}\right)\times$
$\displaystyle\prod\limits_{k\neq
l}^{N}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}\tilde{h}^{1/2}x_{k}\right)\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{h}_{i}^{-1}x_{k}^{-1}\right)}{\theta_{p}\left(sp^{-1/2}q^{-1/2}P_{l}^{-1/2}x_{k}^{-1}\right)\theta_{p}\left(sp^{1/2}q^{-1/2}P_{l}^{1/2}x_{k}\right)}\,,$
(2.18)

These are completely general expressions for the whole family of $A_{N}$
operators.

##  3 Derivation of $C_{2}$ operators.

In this section we will discuss derivation of the $C$-type rank-$2$ analytic
finite difference operators A$\Delta$O. In our previous paper Nazzal:2021tiu
we have already derived the basic operator for $A_{N}$ generalization of van
Diejen model and the calculation of the full tower of these operators is
summarized in Section 2. Unfortunately full $C_{N}$ generalization is still
out of reach for us due to technical complications but we can concentrate on
the study of rank-$2$ generalizations with $C_{2}$ root system.

We will derive $C_{2}$ operator corresponding to the insertion of the
codimension-two defect due to the nontrivial vev of the holomorphic
derivatives of the moment maps $\langle\partial_{\pm}^{K}M\rangle\neq 0$. To
simplify our calculations of A$\Delta$O we will start with the derivation of
four- and three-punctured spheres with $C_{2}$ type punctures.

The basic $C_{N}$ trinion was already derived by us in Appendix B.5 of
Nazzal:2021tiu  and is shown in Figure 2(b). Generalizations of this trinion
to the cases of three maximal punctures and higher numbers of minimal
punctures can also be found in Kim:2023qbx . In order to derive this trinion
we used a tube theory with one $\mathrm{USp}(2N)$ and one $\mathrm{SU}(N+1)$
maximal punctures shown in Figure 2(a). This theory was first discussed in
Kim:2018bpg . In order to obtain trinion theory shown in Figure 2(b) we start
with the three-punctured sphere with two maximal $\mathrm{SU}(N+1)$ and one
minimal $\mathrm{SU}(2)$ punctures and glue two $A_{N}C_{N}$ tubes to the
maximal punctures. The moment maps of the punctures of the resulting trinion
are given by $(2N+6)$ mesons both for maximal and minimal punctures:

$\displaystyle M_{x}={\bf 2N}_{x}\otimes\left({\bf
2N+6}\right)_{w^{2\frac{(N+2)^{2}}{N+3}}}\,,\qquad M_{y}={\bf
2N}_{y}\otimes\left({\bf 2N+6}\right)_{w^{2\frac{(N+2)^{2}}{N+3}}}\,,$
$\displaystyle M_{z}={\bf 2}_{z}\otimes\left({\bf
2N+6}\right)_{w^{2\frac{(N+2)^{2}}{N+3}}}\,,$ (3.1)

where $w$ is one of the parameters of Cartans of the global
$\mathrm{SO}(4N+8)$ symmetry of the $6d$ minimal conformal matter theory. In
general we will use two types of parametrization of $6d$ global symmetry in
this paper. First one is natural to use in compactifications with $C_{N}$ type
punctures and it is given by

$\displaystyle\tilde{a}_{i}\,,\quad
i=1,\dots,2N+6\,,\quad\prod\limits_{i=1}^{2N+6}\tilde{a}_{i}=1\,,\quad w\,.$
(3.2)

Another parametrization is useful when we work with $A_{N}$ expressions and is
given by:

$\displaystyle a_{i}\,,\quad
i=1,\dots,2N+4\,,\quad\prod\limits_{i=1}^{2N+4}a_{i}=1\,,\quad u\,,v\,,w\,.$
(3.3)

These two parametrizations are related by the following map that we will often
need in our calculations:

$\displaystyle\tilde{a}_{l}=\left(uv\right)^{-N-1}w^{-\frac{2}{N+3}}a_{l}\,,~{}l=1,\dots,2N+4\,,$
$\displaystyle\tilde{a}_{2N+6}=u^{2(N+1)(N+2)}w^{2\frac{N+2}{N+3}}\,,\quad\tilde{a}_{2N+5}=v^{2(N+1)(N+2)}w^{2\frac{N+2}{N+3}}\,.$
(3.4)

Now in order to derive finite difference operators we should proceed in the
same way as in the case of $A_{N}$ operators summarized in Section 2. We start
by taking two $C_{N}$ trinion theories ${\mathcal{T}}^{\mathcal{A}}_{x,y,z}$
where $\mathcal{A}$ is the non-zero flux of the trinion and $x,y,z$ in the
subscript are fugacities of the global symmetries of two maximal and one
minimal punctures correspondingly. Then we take an arbitrary ${\mathcal{N}}=1$
theory obtained in the compactification of the minimal $(D_{N+3},D_{N+3})$
conformal matter theory on the Riemmann surface of genus $g$ with $s$
punctures denoted as ${\mathcal{C}}_{g,s}$ with at least one
$\mathrm{USp}(2N)_{\tilde{x}}$ maximal puncture. Gluing all three surfaces
together along the maximal punctures results in a Riemmann surface of the same
genus and global symmetry fluxes but two extra minimal punctures. We will be
performing all these operations at the level of superconformal indices. There
gluing amounts to identifying global symmetries of the punctures we glue and
gauging it. So for the indices we can write down the following identity after
gluing:

$\displaystyle{\mathcal{I}}\left[{\mathcal{C}}_{g,s}(\tilde{x})\oplus\bar{{\mathcal{T}}}^{\mathcal{A}}_{\tilde{x},y,z_{1}}\oplus{\mathcal{T}}^{\mathcal{A}}_{y,x,z_{2}}\right]=\frac{\left(q;q\right)_{\infty}^{4}\left(p;p\right)_{\infty}^{4}}{2^{4}\cdot
2!\cdot 2!}\oint\prod\limits_{i=1}^{2}\frac{d\tilde{x}_{i}}{2\pi
i\tilde{x}_{i}}\frac{dy_{i}}{2\pi
iy_{i}}\prod\limits_{i=1}^{N}\frac{1}{\Gamma_{e}\left(y_{i}^{\pm
2}\right)}\times$
$\displaystyle\prod\limits_{i<j}^{N}\frac{1}{\Gamma_{e}\left(y_{i}^{\pm
1}y_{j}^{\pm
1}\right)}\prod\limits_{i=1}^{N}\frac{1}{\Gamma_{e}\left(\tilde{x}_{i}^{\pm
2}\right)}\prod\limits_{i<j}^{N}\frac{1}{\Gamma_{e}\left(\tilde{x}_{i}^{\pm
1}\tilde{x}_{j}^{\pm
1}\right)}K_{3}^{C}(x,y,z_{2})\bar{K}_{3}^{C}(\tilde{x},y,z_{1}){\mathcal{I}}\left[{\mathcal{C}}_{g,s}(\tilde{x})\right]\,,$
(3.5)

(a) A tube with one $\mathrm{SU}(N+1)$ and one $\mathrm{USp}(2N)$ puncture.

(b) Trinion with two maximal $\mathrm{USp}(2N)$ and one minimal
$\mathrm{SU}(2)$ puncture.

Figure 2: Tube and trinion with maximal $C_{N}$ punctures.

where ${\mathcal{I}}\left[{\mathcal{C}}_{g,s}\right]$ is the index of a theory
obtained by compactifications of $6d$ theory on the Riemmann surface
${\mathcal{C}}_{g,s}$ and $K_{3}^{C}(x,y,z)$ is the index of the trinion
theory shown in Figure 2(b) while $\bar{K}_{3}^{C}(x,y,z)$ is its conjugate.
Definition and properties of the elliptic $\Gamma$-function
$\Gamma_{e}\left(x\right)$ are given in Appendix A. Then, closing
$\mathrm{SU}(2)_{z}$ and $\mathrm{SU}(2)_{\tilde{z}}$ minimal punctures with
or without introduction of defects we can obtain finite difference operators.
As discussed previously in order to close minimal punctures we have to give
certain weights to corresponding global symmetry fuagcities $z_{1}$ and
$z_{2}$. As a result we get the follwoing identity

$\displaystyle\lim_{z_{1}\to Z_{1}^{*},z_{2}\to
Z_{2}^{*}}{\mathcal{I}}\left[{\mathcal{C}}_{g,s}(x)\oplus\bar{{\mathcal{T}}}^{\mathcal{A}}_{x,y,z_{1}}\oplus{\mathcal{T}}^{\mathcal{A}}_{y,\tilde{x},z_{2}}\right]\sim{\mathcal{O}}\cdot{\mathcal{I}}\left[{\mathcal{C}}_{g,s}(x)\right]\,,$
(3.6)

where ${\mathcal{O}}$ is some finite difference operator, and in case we close
punctures without introducing defects we just obtain the identity operator.

Here for convenience we will not compute the full expression (3.5) directly
right away. Just as for $A_{N}$ operators derived in Section 2 it appears to
be much easier to perform calculation in a slightly different way. First we
derive the index of the four-punctured sphere with zero flux, two maximal
$\mathrm{USp}(2N)$ and two minimal $\mathrm{SU}(2)$ punctures. For this
purpose we perform $S$-gluing of two trinions ${\mathcal{T}}^{AC}_{x,y,z}$
with $\mathrm{SU}(N+1)$ and $\mathrm{USp}(2N)$ maximal punctures and
$\mathrm{SU}(2)$ minimal punctures each. This kind of trinions can be derived
by appropriate gluing of $A_{N}C_{N}$ tube theory shown in Figure 2(a), to one
of the maximal punctures of $A_{N}$ trinion introduced in Razamat:2020bix .
Superconformal index of the resulting four-punctured sphere theory is given
by:

$\displaystyle
K_{4}^{C}(x,\tilde{x},z,\tilde{z})=\kappa_{N}\oint\prod\limits_{i=1}^{N}\frac{dy_{i}}{2\pi
iy_{i}}\prod\limits_{i\neq
j}^{N+1}\frac{1}{\Gamma_{e}\left(\frac{y_{i}}{y_{j}}\right)}\bar{K}_{3}^{AC}(\tilde{x},y,\tilde{z})K_{3}^{AC}(x,y,z)\,.$
(3.7)

where $K^{AC}_{3}(x,y,z)$ is the index of the trinion with $y$ being
$\mathrm{SU}(N+1)$ fugacity and $\kappa_{N}$ is the usual constant given in
(2.3). Geometrically this operation is shown in Figure 5(a).

Next we close on of the two minimal punctures by giving nonzero vev to one of
the $M_{z}$ moment maps given in (3.1). At the level of the index computations
this means that the corresponding global symmetry fugacity should be
consistent with the vev, i.e. in our case we should fix $\tilde{z}$ fugacity
to

$\displaystyle\tilde{z}=\tilde{Z}_{i;K,M}^{*}\equiv(pq)^{-\frac{1}{2}}w^{2\frac{(N+2)^{2}}{N+3}}\tilde{a}_{i}q^{-K}p^{-M}\,,$
(3.8)

where $\tilde{a}_{i}$ can be chosen arbitrary since all the expressions are
symmetric w.r.t. permutations of $\tilde{a}_{i}$. Integers $K$ and $M$
correspond to an order of the derivative of the moment map in $34$ and $12$
planes correspondingly, i.e. we give vev to
$\langle\partial_{12}^{M}\partial_{34}^{K}\widehat{M}_{i}\rangle\neq 0$.
Physically this corresponds to introducing various codimension two defects
into $4d$ theory. For the first $\mathrm{SU}(2)_{\tilde{z}}$ minimal puncture
we choose to close it without introducing any defect, which in turn
corresponds to $K=M=0$ choice in (3.8).

At this value of $\tilde{z}$, the superconformal index (3.7) of the four-
punctured sphere has a pole. Computing the residue at this pole we obtain the
index of the three-punctured sphere:

$\displaystyle
K_{(3;i,0)}^{C}(x\,,y\,,z)\sim\mathrm{Res}_{\tilde{z}\to\tilde{Z}_{i;0,0}^{*}}K_{4}^{C}(x\,,y\,,z\,,\tilde{z})$
(3.9)

Here subscript $(3;i,0)$ refers to the fact that we obtain three-punctured
sphere by closing minimal puncture of the four-punctured sphere choosing
$\tilde{a}_{i}$ for the vev in (3.8) and not introducing a defect, which
corresponds to the choice $K=M=0$ in the same equation.

Calculation of (3.7) and (3.9) results in the theory shown in Figure 3.
Detailed derivations of this section can be found in Appendix C.

Figure 3: Three-punctured sphere theory with two maximal $\mathrm{USp}(2N)$
punctures and one minimal $\mathrm{SU}(2)$ puncture obtained after gluing
(3.7) and closing one minimal $\mathrm{SU}(2)_{\tilde{z}}$ puncture without
defect. Solid and dashed line attached to the gauge node denotes multiplet in
$AS$ and $\overline{AS}$ representation correspondingly.

Now that we have obtained the desired three-punctured sphere theory we are
ready to derive A$\Delta$O. For this purpose we should close the remaining
minimal $\mathrm{SU}(2)_{z}$ puncture by giving nontrivial vev
$\langle\partial_{12}^{M}\partial_{34}^{K}M\rangle\neq 0$ to a derivative of
one of its moment maps. In order to get zero total flux through the resulting
two-punctured sphere we should choose the same moment map as we did closing
$\mathrm{SU}(2)_{\tilde{z}}$ minimal puncture previously. The difference is
that now we have to choose non-trivial derivative that is at least one of $K$
and $M$ numbers is not zero. Physically this corresponds to introducing
codimension two defect into the tube theory with two maximal punctures as
shown in Figure 5(b). At the level of the index we should give the following
value to the $z$-fugacity:

$\displaystyle
z=Z_{i;K,M}^{*}\equiv(pq)^{-\frac{1}{2}}w^{-2\frac{(N+2)^{2}}{N+3}}\tilde{a}_{i}^{-1}q^{-K}p^{-M}\,,$
(3.10)

Then, according to (3.6), capturing the corresponding pole and gluing the
resulting tube with two maximal $\mathrm{USp}(2N)$ punctures to an arbitrary
theory with at least one puncture of this type as shown in Figure 5(c), we
obtain an A$\Delta$O. However, technically it sometimes appears not to be as
straightforward. In particular if we perform these operations with the
expression (C.12) instead of A$\Delta$O we obtain some integral-finite
difference operator. It is highly possible that in fact this operator can be
written in the form of A$\Delta$O. However technical issues make it too
difficult task and we leave it for future investigation.

Here we will instead concentrate on the derivation of next to the lowest rank
$C_{2}$ operator. This case is simpler to analyze. For example, in our trinion
theory shown in Figure 3, when $N=2$ the $\overline{AS}$ multiplet of the
right $\mathrm{SU}(3)$ gauge node becomes just antifundamental, and a chain of
duality transformations can be used to simplify the theory. These calculations
are summarized in Appendix C. As a result we obtain single-node
$\mathrm{SU}(6)$ gauge theory shown in Figure 4 with the corresponding index
specified in (C.13).

Figure 4: Three-punctured sphere theory with two maximal $\mathrm{USp}(4)$ and
one minimal $\mathrm{SU}(2)$ punctures. Dashed line starting and ending on the
gauge $\mathrm{SU}(6)$ node as previously corresponds to the matter in the
$\overline{AS}$ representation.

Finally, we can close $\mathrm{SU}(2)_{z}$ puncture. In particular we give the
following weight consistent with (3.8) to the fugacity $z$:

$\displaystyle
z=Z^{*}_{i;K,0}=(pq)^{-\frac{1}{2}}w^{-\frac{32}{5}}a_{i}^{-1}q^{-K}\,.$
(3.11)

(a) Gluing two trinions.

(b) Closing minimal punctures of the four-punctured sphere.

(c) Gluing tube with the defect to an arbitrary Riemann surface.

Figure 5: On the figures above we summarize all the steps we go through in
order to derive $C_{2}$ A$\Delta$O given in (3.14). First as shown on the
Figure (a) we glue two trinions with two maximal $\mathrm{USp}(4)$ (shown with
green) and one minimal $\mathrm{SU}(2)$ (shown with orange) punctures each.
For one of the trinions we conjugate all the charges in order to perform S
gluing. At the level of the superconformal index this operation is expressed
in (3.7). Next, as shown on the Figure (b), we close two minimal punctures of
the four-punctured sphere. This operation is performed by giving vev
$\langle\partial_{+}^{K}\widehat{M}_{i}\rangle\neq 0$ to holomorphic
derivative of one of the moment map operators with the $U(1)$ charge $h_{i}$.
As the result we obtain tube theory with two maximal $\mathrm{USp}(4)$
punctures and codimension two defect introduced. On the Figure we denote this
defect with the red ring. Finally, as the last step of our algorithm shown on
the Figure (c), we glue this tube with the defect to an arbitrary surface with
at least one maximal $\mathrm{USp}(4)$ puncture. This results in the action of
the set of certain A$\Delta$Os on the index of the original theory.

Performing this closure we obtain the index of the theory for two punctured
sphere with two $\mathrm{USp}(4)$ maximal punctures

$\displaystyle K_{(2;i;K,0)}^{C}(x\,,\tilde{x})\sim\mathrm{Res}_{z\to
Z_{i;K,0}^{*}}K_{(3;i,0)}^{C}(x\,,\tilde{x}\,,z)$ (3.12)

The index itself is specified in (C.18). This time it does not have natural
gauge theory interpretation in case of general $K$ due to the presence of the
codimension two defect.

As a final step of our derivation, we glue the obtained tube with the defect
to an arbitrary $\mathcal{N}=1$ theory with at least one maximal
$\mathrm{USp}(4)$ puncture and obtain A$\Delta$O as follows:

$\displaystyle{\mathcal{O}}^{(C_{2};h_{k};K,0)}_{x}\cdot{\mathcal{I}}(x)\sim\frac{\left(q;q\right)_{\infty}^{2}\left(p;p\right)_{\infty}^{2}}{2^{2}\cdot
2!}\oint\frac{d\tilde{x}_{1,2}}{2\pi
i\tilde{x}_{1,2}}\frac{1}{\Gamma_{e}\left(\tilde{x}_{1,2}^{\pm
2}\right)\Gamma_{e}\left(\tilde{x}_{1}^{\pm 1}\tilde{x}_{2}^{\pm
1}\right)}\times$ $\displaystyle
K_{(2;i;K,0)}^{C}(x,\,\tilde{x}){\mathcal{I}}(\tilde{x})$ (3.13)

Details of this calculation can be found in Appendix C. It leads to the
following expression for A$\Delta$O:

$\displaystyle{\mathcal{O}}^{(C_{2};h_{k};K,0)}_{x}$
$\displaystyle=\sum\limits_{\vec{K}}\sum\limits_{m_{1}=-k_{1,+}}^{k_{1,-}}\sum\limits_{m_{2}=-k_{2,+}}^{k_{2,-}}\tilde{C}^{k}_{\vec{K}}\prod\limits_{l\neq
k}^{10}\prod\limits_{i=1}^{2}\prod\limits_{s_{i}=\pm
1}\prod\limits_{l_{1}=-k_{i,s_{i}}+s_{i}m_{i}}^{2s_{i}m_{i}-1}\theta_{p}\left(q^{l_{1}}x_{i}^{-2s_{i}}\right)^{-1}\times$
(3.14)
$\displaystyle\prod\limits_{l_{2}=-k_{i,s_{i}}}^{s_{i}(k_{i,-}-k_{i,+})-1}\theta_{p}\left(q^{l_{2}}x_{i}^{-2s_{i}}\right)^{-1}\prod\limits_{l_{3}=-k_{2,s_{2}}-s_{1}m_{1}}^{s_{2}m_{2}-s_{1}m_{1}-1}\theta_{p}\left(q^{l_{3}}x_{1}^{s_{1}}x_{2}^{-s_{2}}\right)^{-1}\times$
$\displaystyle\prod\limits_{j\neq
i}^{2}\prod\limits_{l_{4}=-k_{j,s_{j}}}^{k_{i,s_{i}}-k_{j,s_{j}}-1}\theta_{p}\left(q^{l_{4}}x_{i}^{s_{i}}x_{j}^{-s_{j}}\right)^{-1}\prod\limits_{l_{5}=-k_{1,s_{1}}-k_{2,s_{2}}}^{-k_{1,s_{1}}+s_{2}m_{2}-1}\theta_{p}\left(q^{l_{5}}x_{1}^{-s_{1}}x_{2}^{-s_{2}}\right)\times$
$\displaystyle\prod\limits_{l_{6}=K-k_{i,s_{i}}-k_{5}}^{K-k_{i,s_{i}}-1}\theta_{p}\left((pq)^{\frac{1}{2}}h_{k}x_{i}^{-s_{i}}q^{l_{6}}\right)\prod\limits_{l_{7}=K-k_{5}}^{K-k_{5}+k_{i,s_{i}}-1}\theta_{p}\left((pq)^{\frac{1}{2}}h_{k}x_{i}^{s_{i}}q^{l_{7}}\right)^{-1}\times$
$\displaystyle\prod\limits_{l_{8}=-K-k_{i,s_{i}}}^{-K-k_{i,s_{i}}+k_{5}-1}\theta_{p}\left((pq)^{-\frac{1}{2}}h_{k}^{-1}x_{i}^{-s_{i}}q^{l_{8}}\right)\prod\limits_{l_{9}=-k_{i,s_{i}}}^{-k_{i,s_{i}}+k_{6}-1}\theta_{p}\left(p^{-1}q^{l_{9}-1}h^{-\frac{1}{2}}h_{k}x_{i}^{-s_{i}}\right)\times$
$\displaystyle\prod\limits_{l_{10}=-k_{6}}^{k_{i,s_{i}}-k_{6}-1}\theta_{p}\left(q^{-1-l_{10}}h^{-\frac{1}{2}}h_{k}x_{i}^{-s_{i}}\right)^{-1}\prod\limits_{l_{11}=-k_{1,s_{1}}-k_{2,s_{2}}}^{-k_{1,s_{1}}+s_{2}m_{2}-1}\theta_{p}\left(q^{l_{11}}x_{1}^{-s_{1}}x_{2}^{-s_{2}}\right)\times$
$\displaystyle\prod\limits_{l_{12}=-s_{i}m_{i}}^{k_{i,s_{i}}-1}\theta_{p}\left((pq)^{\frac{1}{2}}h_{l}x_{i}^{s_{i}}q^{l_{12}}\right)\prod\limits_{l_{13}=k_{6}-s_{i}m_{i}}^{k_{i,s_{i}}+k_{6}-1}\theta_{p}\left(h^{-\frac{1}{2}}h_{k}x_{i}^{s_{i}}q^{l_{13}}\right)\times$
$\displaystyle\hskip
56.9055pt\prod\limits_{l_{14}=-s_{i}m_{i}}^{K-k_{5}-s_{i}m_{i}-1}\theta_{p}\left((pq)^{\frac{1}{2}}h_{k}x_{i}^{s_{i}}q^{l_{14}}\right)\Delta_{q}^{-m_{1}}(x_{1})\Delta_{q}^{-m_{2}}(x_{2})\,,$

where the constant $\tilde{C}^{k}_{\vec{K}}$ is given by

$\displaystyle\tilde{C}^{k}_{\vec{K}}$ $\displaystyle=\prod\limits_{l\neq
k}\prod\limits_{l_{1}=-K}^{k_{5}-K-1}\theta_{p}\left(h_{l}h_{k}^{-1}q^{l_{1}}\right)\prod\limits_{l_{2}=0}^{k_{6}-1}\theta_{p}\left((pq)^{-\frac{1}{2}}h^{-\frac{1}{2}}h_{l}h_{k}q^{l_{2}}\right)\prod\limits_{l_{3}=-2K}^{-2K+k_{5}-1}\theta_{p}\left(h_{k}^{-2}q^{l_{7}}\right)\times$
(3.15)
$\displaystyle\prod\limits_{l_{4}=K-k_{5}}^{K-k_{5}+k_{6}-1}\theta_{p}\left((pq)^{-\frac{1}{2}}h^{-\frac{1}{2}}h_{k}^{2}q^{l_{5}}\right)^{-1}\prod\limits_{l_{4}=K-k_{6}}^{K-1}\theta_{p}\left((pq)^{\frac{3}{2}}h^{\frac{1}{2}}q^{l_{4}}\right)\times$
$\displaystyle\prod\limits_{l_{6}=-K-k_{6}}^{-K+k_{5}-k_{6}-1}\theta_{p}\left((pq)^{\frac{1}{2}}h^{\frac{1}{2}}h_{k}^{-2}q^{l_{5}}\right)\prod\limits_{l_{7}=-K}^{-K+k_{5}+k_{6}-1}\theta_{p}\left((pq)^{-\frac{1}{2}}h^{-\frac{1}{2}}q^{l_{6}}\right)\,.$

We have also introduced notation $h_{i}$, $i=1,\dots,10$ for $U(1)$ charges of
the moment maps we act on. Notice that according to (3.1) moment maps of both
minimal and maximal punctures have the very same charges, so that in fact

$\displaystyle h_{i}=w^{\frac{32}{5}}\tilde{a}_{i}\,,\qquad
h\equiv\prod\limits_{j=1}^{10}h_{j}=w^{64}\,,$ (3.16)

for both charges of the maximal punctures we act on and minimal punctures we
close. We have also introduced here an overall $U(1)$ charge $h$. The operator
itself is labelled by an index $k$ according to the choice of $U(1)$ charge
$h_{k}$ of the moment map we give vev to in order to close the minimal
puncture and to introduce defects into the theory.

The first sum in the expression is performed over all possible partitions
$\vec{K}=(k_{1},\cdots,k_{6})$ of the integer $K$. Finally the products in
(3.14) should be understood as ordered ones, i.e., one should assume
$\prod_{i=n_{1}}^{n_{2}}$ is just $1$ if $n_{2}<n_{1}$ since there are no
terms in the product. The coefficient $\tilde{C}^{k}_{\vec{K}}$ is chosen so
that the $x$-independent factors of the highest order shift terms
$\Delta_{q}^{\pm K}(y_{1,2})$ are just one.

Notice that the expression above is not the most general since we could have
closed $\mathrm{SU}(2)_{z}$ minimal puncture using general $K$ and $M$
integers in (3.10), but we will concentrate here only on the case of non-
trivial $K$ while putting $M=0$. As we see the operator appears to be quite
complicated and it is hard to analyze it. Instead it is useful to write down
basic operator corresponding to the choice of $K=1$, i.e. the simplest non-
trivial case. Taking into account six possible partitions $\vec{K}$ we can
write down explicit form of A$\Delta$O:

$\displaystyle{\mathcal{O}}^{(C_{2};h_{k};1,0)}_{x}=\sum\limits_{i=1}^{2}\left(A^{(C_{2};h_{k};1,0)}_{i}(x,h)\Delta_{q}(x_{i})+A^{(C_{2};h_{k};1,0)}_{i}(x_{i}\to
x_{i}^{-1},h)\Delta_{q}^{-1}(x_{i})\right)$ $\displaystyle\hskip
85.35826pt+W^{(C_{2};h_{k};1,0)}(x,h)\,,$ $\displaystyle
A^{(C_{2};h_{k};1,0)}_{i}(x,h)=\prod\limits_{j\neq
i}^{2}\frac{1}{\theta_{p}\left(x_{i}^{2}\right)\theta_{p}\left(qx_{i}^{2}\right)\theta_{p}\left(x_{i}x_{j}^{\pm
1}\right)}\theta_{p}\left((pq)^{\frac{1}{2}}h_{k}x_{j}^{\pm
1}\right)\prod\limits_{l=1}^{10}\theta_{p}\left((pq)^{\frac{1}{2}}h_{l}^{-1}x_{i}\right)\,,$
$\displaystyle
W^{(C_{2};h_{k};1,0)}(x,h)=\left[\sum\limits_{i=1}^{2}\prod\limits_{j\neq
i}^{2}\frac{1}{\theta_{p}\left(x_{i}^{2}\right)\theta_{p}\left(q^{-1}x_{i}^{-2}\right)\theta_{p}\left(x_{i}x_{j}^{\pm
1}\right)}\frac{\theta_{p}\left(h^{-\frac{1}{2}}h_{k}x_{i}\right)}{\theta_{p}\left(q^{-1}h^{-\frac{1}{2}}h_{k}x_{i}^{-1}\right)}\times\right.$
$\displaystyle\left.\frac{\prod_{l\neq
k}^{10}\theta_{p}\left((pq)^{\frac{1}{2}}h_{l}x_{i}\right)}{\theta_{p}\left((pq)^{\frac{1}{2}}h_{k}qx_{i}\right)}\prod\limits_{m=1}^{2}\theta_{p}\left((pq)^{\frac{1}{2}}h_{k}x_{m}^{\pm
1}\right)+\left(x_{i}\to x_{i}^{-1}\right)\right]+$
$\displaystyle\prod\limits_{i=1}^{2}\prod\limits_{l\neq
k}^{10}\frac{\theta_{p}\left(h_{l}h_{k}^{-1}q^{-1}\right)\theta_{p}\left((pq)^{\frac{3}{2}}h^{\frac{1}{2}}\right)}{\theta_{p}\left(q^{-2}h_{k}^{-2}\right)\theta_{p}\left((pq)^{\frac{1}{2}}q^{-1}h^{\frac{1}{2}}h_{k}^{-2}\right)}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}h_{k}x_{i}^{\pm
1}\right)}{\theta_{p}\left((pq)^{-\frac{1}{2}}h_{k}^{-1}q^{-1}x_{i}^{\pm
1}\right)}+$ $\displaystyle\prod\limits_{i=1}^{2}\prod\limits_{l\neq
k}^{10}\frac{\theta_{p}\left((pq)^{-\frac{1}{2}}h^{-\frac{1}{2}}h_{k}h_{l}\right)}{\theta_{p}\left((pq)^{-\frac{1}{2}}h^{-\frac{1}{2}}h_{k}^{2}q\right)}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}h_{k}x_{i}^{\pm
1}\right)}{\theta_{p}\left(p^{-1}q^{-1}h^{-\frac{1}{2}}h_{k}x_{i}^{\pm
1}\right)}\,,$ (3.17)

One thing to be noticed is that just as in the case of vD model, the constant
term choice is not unique. What really defines this constant term are the
following properties. First of all it can be noticed that
$W^{(h_{k};1,0)}(x,h)$ is an elliptic function in both $x_{1}$ and $x_{2}$
with periods $1$ and $p$. In the fundamental domain this elliptic function has
poles located at:

$\displaystyle x_{i}=sq^{\pm\frac{1}{2}}\,,\qquad
x_{i}=sq^{\pm\frac{1}{2}}p^{\frac{1}{2}}\,,\quad s=\pm 1\,,$ (3.18)

which are in fact exactly the same as the poles of vD model. Corresponding
residues are given by:

$\displaystyle\mathrm{Res}_{x_{i}=sq^{\frac{1}{2}}}W^{(C_{2};h_{l};1,0)}(x,h)=-s\frac{q^{\frac{1}{2}}\prod\limits_{k=1}^{10}\theta_{p}\left((pq)^{\frac{1}{2}}sq^{-\frac{1}{2}}h_{k}\right)}{2\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j\neq
i}^{2}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}h_{l}x_{j}^{\pm
1}\right)}{\theta_{p}\left(sq^{-\frac{1}{2}}x_{j}^{\pm 1}\right)}\,,$
$\displaystyle\mathrm{Res}_{x_{i}=sq^{-\frac{1}{2}}}W^{(C_{2};h_{l};1,0)}(x,h)=s\frac{q^{-\frac{1}{2}}\prod\limits_{k=1}^{10}\theta_{p}\left((pq)^{\frac{1}{2}}sq^{-\frac{1}{2}}h_{k}\right)}{2\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j\neq
i}^{2}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}h_{l}x_{j}^{\pm
1}\right)}{\theta_{p}\left(sq^{-\frac{1}{2}}x_{j}^{\pm 1}\right)}\,,$
$\displaystyle\mathrm{Res}_{x_{i}=sp^{\frac{1}{2}}q^{\frac{1}{2}}}W^{(C_{2};h_{l};1,0)}(x,h)=-s\frac{ph^{-\frac{1}{2}}\prod\limits_{k=1}^{10}\theta_{p}\left(sh_{k}\right)}{2\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j\neq
i}^{2}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}h_{l}x_{j}^{\pm
1}\right)}{\theta_{p}\left((pq)^{-\frac{1}{2}}sx_{j}^{\pm 1}\right)}\,,$
$\displaystyle\mathrm{Res}_{x_{i}=sp^{\frac{1}{2}}q^{-\frac{1}{2}}}W^{(C_{2};h_{l};1,0)}(x,h)=s\frac{pq^{-1}h^{-\frac{1}{2}}\prod\limits_{k=1}^{10}\theta_{p}\left(sh_{k}\right)}{2\left(p;p\right)_{\infty}^{2}\theta_{p}\left(q^{-1}\right)}\prod\limits_{j\neq
i}^{2}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}h_{l}x_{j}^{\pm
1}\right)}{\theta_{p}\left((pq)^{-\frac{1}{2}}sx_{j}^{\pm 1}\right)}\,.$
(3.19)

This concludes our derivation of the $C_{2}$ A$\Delta$O. All possible details
of these derivations can be found in the Appendix C.

##  4 Properties of the $C_{2}$ operators.

There is a number of interesting properties that the operators we derived
should posses by construction. In this section we discuss checks and, where it
is possible, give proofs of these properties using explicit expressions we
have derived.

The most important but also most complicated property to discuss is the so-
called kernel property of our operators. The main idea behind it is that the
superconformal index of any $\mathcal{N}=1$ theory obtained in the
compactifications of $6d$ minimal $(D_{5},D_{5})$ conformal matter theory is a
kernel function of our operators. The kernel function is defined by the
following mathematical identity:

$\displaystyle{\mathcal{O}}_{z}^{(G_{1};h_{i};r,m)}\cdot{\mathcal{I}}\left[{\mathcal{C}}_{g,s}[z,u]\right]={\mathcal{O}}_{u}^{(G_{2};h_{i};r,m)}\cdot{\mathcal{I}}\left[{\mathcal{C}}_{g,s}[z,u]\right]\,.$
(4.1)

Physically we consider the superconformal index of the theory obtained in the
$6d$ compactification on the Riemmann surface ${\mathcal{C}}_{g,s}[z,u]$ with
at least two maximal punctures parametrized by the fugacities $z$ and $u$
correspondingly. In general there can be as many punctures as we want. The
claim here is that any such superconformal index play the role of the kernel
function for the derived A$\Delta$Os according to (4.1). Namely we can act
with our operators on different punctures and we should always obtain the same
result. This property is expected to hold due to an argument coming from the
geometry of compactification shown in Figure 6. Equivalently it can be
understood from the invariance of the superconformal indices under $S$ duality
transformations.

In principle we can choose any sphere with multiple punctures to check the
kernel property. Natural simplest candidate would be WZW model for the two-
punctured sphere with two maximal $\mathrm{USp}(2N)$ punctures shown in Figure
LABEL:pic:c2c2tube. However, we will check a more interesting and trickier
case where two punctures are of different types. Namely we take the tube
theory shown in Figure 2(a) which has one $\mathrm{SU}(N+1)$ and one
$\mathrm{USp}(2N)$ maximal puncture. Then for $N=2$ the kernel property (4.1)
in this case reads:

$\displaystyle{\mathcal{O}}^{(A_{2};\tilde{h}_{i}^{-1};r,m)}_{y}K_{2}^{AC}(x,y)={\mathcal{O}}^{(C_{2};h_{i};r,m)}_{x}K_{2}^{AC}(x,y)\,,$
(4.2)

where $K_{2}^{AC}(x,y)$ is the index of the tube specified in (C.5) with $x$
and $y$ being fugacities of $\mathrm{USp}(4)$ and $\mathrm{SU}(3)$ punctures
correspondingly. Operators ${\mathcal{O}}^{(A_{2};\tilde{h}_{i};1,0)}_{y}$ and
${\mathcal{O}}^{(C_{2};h_{i};1,0)}_{x}$ are given in (2.13) and (3.17)
correspondingly. Notice that in order for the kernel property to work we need
to close minimal punctures in the same way on two sides of the equation, i.e.
we have to give vev to the same moment map of the minimal puncture. In
particular in $A_{2}$ case we should give vevs to the minimal puncture moment
map of the charge $\tilde{h}_{i}^{-1}\tilde{h}^{\frac{1}{4}}$ where
$\tilde{h}_{i}$ are given in (2.5). At the same time, in $C_{2}$ case minimal
puncture is closed by giving vev to the moment map of charge $h_{i}$ specified
in (3.16). Using the map (3.4) we can see that the only map that matches
between the two is

$\displaystyle\tilde{h}_{10}^{-1}\tilde{h}^{\frac{1}{4}}=v^{-24}w^{-8}=w^{-\frac{32}{5}}\tilde{a}_{10}^{-1}=h_{10}\,.$
(4.3)

If we would like to check the kernel property for other ways of closing
minimal punctures we should consider $A_{2}$ operators (2.10) with the non-
flipped moment maps so that in this case we close the minimal puncture with
the vevs of the moment map with charge
$\tilde{h}_{i}\tilde{h}^{-\frac{1}{4}}$. Because of this complication here we
consider only closing with the vev given to the moment map of charge (4.3).
The proofs for other operators should work identically. Also in our proof we
restrict ourselves to only basic operators (3.17) and (2.13), i.e. we fix
$r=1\,,m=0$ (or equivalently $m=1\,,r=0$ on both sides of the kernel equation
(4.2). We have to do it since higher operators specified in (3.14) and (2.10)
are too complicated for the analysis. However the kernel property (4.2) should
also work for these higher operators as well.

Figure 6: On this Figure we represent the argument in favor of the kernel
property of $\mathcal{N}=1$ indices. For this we consider an index of a theory
obtained in the compactification of the minimal $(D_{5},D_{5})$ conformal
matter theory on a generic Riemann surface with at least two maximal punctures
(shown as colored discs on the Figure) of any types with fugacities $x$ and
$y$ as well as one minimal $\mathrm{SU}(2)$ puncture (shown as red crosses)
with the fugacity $z$. For example in case of the $A_{2}C_{2}$ tube and kernel
property (4.2) one maximal puncture has $\mathrm{USp}(4)$ global symmetry
while the second one has $\mathrm{SU}(3)$ symmetry. Then we close the minimal
puncture by giving space dependent vev
$\langle\partial_{12}^{m}\partial_{34}^{k}M\rangle\neq 0$ to one of the moment
maps of this puncture. As discussed in the paper this corresponds to the
introduction of the codimension-two defect into $4d$ effective description and
results in the action of the A$\Delta$O on one of the maximal punctures of the
index of the theory corresponding to the Riemann surface with only two maximal
punctures. This operation can be performed in different duality frames. In
each frame we obtain operators acting on one of the two punctures. Since the
result of the calculation should not depend on the duality frame we choose we
conclude that the action of two, possibly different operators, on two
different punctures leads to two expressions equal to each other. Hence we
arrive to the kernel property equation (4.1).

Since we precisely know explicit expressions for both operators (2.13) and
(3.17) as well as supposed kernel function (C.5) it is straightforward to
check this kernel identity. Acting with finite difference operators of two
kinds on the kernel we get two algebraic functions. Equality of these
functions implies validity of the kernel property (4.2). In Appendix D we give
details of the analytic proof of this identity.

Another important property of the derived A$\Delta$Os is their commutation
with each other which also directly follows from the S duality of our
compactification construction as shown in Figure 7. Since it does not depend
which duality frame we close the minimal punctures in, it also does not matter
in which order two different A$\Delta$Os act on the index. Hence we conclude
that all of operators we derived should commute with each other:

$\displaystyle\left[{\mathcal{O}}_{x}^{(C_{2};h_{a};K_{1},M_{1})},\,{\mathcal{O}}_{x}^{(C_{2};h_{b};K_{2},M_{2})}\right]=0\,,\quad\forall~{}~{}a,b=1,\ldots,10;~{}K_{1,2}\geq
0;~{}M_{1,2}\geq 0;$ (4.4)

Figure 7: On this Figure we represent the argument in favor of commutation of
A$\Delta$Os derived in the present paper. For this we start with a theory
obtained in compactification of $6d$ minimal $\left(D_{5},D_{5}\right)$
conformal matter on an arbitrary Riemann surface. This surface has one maximal
$\mathrm{USp}(4)$ puncture (orange disc) with fugacity $x$ and two minimal
$\mathrm{SU}(2)$ punctures (red crosses) with fugacities $z_{1}$ and $z_{2}$
correspondingly. Then we introduce codimension two defects into this theory
giving space-dependent vevs to some of the moment maps $M_{h_{1}}$ and
$M_{h_{2}}$ correspondingly. Giving this vev is equivalent to closing puncture
and at the level of the index computations leads to an action of two different
A$\Delta$Os. We can perform this operation in any duality frame and the result
should not depend on the choice of a particular frame. Hence the action of
operators on the index does not depend on their order. Since the
compactification surface, and hence the effective $4d$ descriptions were
chosen arbitrarily we conclude that the operators themselves also commute.

Here $h_{a}$ and $h_{b}$ are moment maps we give vev to in order to close two
minimal punctures and $K_{1,2},M_{1,2}$ are numbers of
holomorphic/antiholomorphic derivatives. Unfortunately it is very hard to
prove or even to check these relations for general operators (3.14) of the
full tower. However we can perform checks of the commutation relations (4.4)
for the basic operators, i.e. for the case when $K_{1,2}$ and $M_{1,2}$ are
taking values $0$ and $1$. In particular we show that:

$\displaystyle\left[{\mathcal{O}}_{x}^{(C_{2};h_{a};1,0)},\,{\mathcal{O}}_{x}^{(C_{2};h_{b};1,0)}\right]=\left[{\mathcal{O}}_{x}^{(C_{2};h_{a};0,1)},\,{\mathcal{O}}_{x}^{(C_{2};h_{b};0,1)}\right]=$
$\displaystyle\hskip
56.9055pt\left[{\mathcal{O}}_{x}^{(C_{2};h_{a};1,0)},\,{\mathcal{O}}_{x}^{(C_{2};h_{b};0,1)}\right]=0\,,\qquad\forall~{}a,b=1,\dots,10\,.$
(4.5)

In Appendix E we check these relations. In particular, using ellipticity
properties of $\theta_{p}\left(x\right)$ function we prove that the action of
the third type of the commutators on an arbitrary trial function is zero.
Hence these commutators are zero themselves. For the first and second type of
commutators it is hard to prove analytically that their action on a trial
function is zero. Instead we check these identities perturbatively in $p$ and
$q$ expansion. These checks performed up to a sufficiently high order suggest
that both of the commutators are indeed zero.

##  5 Discussion and outlook.

In this paper we have considered $4d$ description of the compactification of
the minimal $(D_{5},D_{5})$ conformal matter theory on a Riemann surface with
$\mathrm{USp}(4)$ maximal punctures. Using this $4d$ description and
introducing codimension-two defect in corresponding theory we derive $C_{2}$
generalization of the van Diejen model. In particular we obtain an infinite
set of analytic difference operators acting on the maximal puncture of $C_{2}$
type. Different operators of our set correspond to different ways of closing
minimal punctures on the compactification surfaces. They are organized in a
decuplet of basic operators and an infinite tower of operators on top of each
of the basic ones.

The operators we obtain are supposed to satisfy interesting and important
properties following directly form the geometry of compactifications. First
such property is the commutation of all operators. In our paper, using
combination of residue computations and perturbative expansions we show that
at least all basic operators indeed commute with each other. Second important
property is that the superconformal index of any compactification of the
minimal $(D_{5},D_{5})$ conformal matter theory on a surface with several
$\mathrm{USp}(4)$ punctures is the kernel function of our difference
operators. As an example of such kernel function we considered a tube theory
corresponding to the compactification on a sphere with one $\mathrm{USp}(4)$
and one $\mathrm{SU}(3)$ puncture. In order to prove the kernel property we
should act on two punctures with two different operators: $A_{2}$ operator
derived in Nazzal:2021tiu  from one side and $C_{2}$ operator derived in the
present paper from the other. This fact makes corresponding kernel function
very interesting for the study. In the paper we managed to prove this kernel
property fully analytically for the basic $A_{2}$ and $C_{2}$ operators.

Another result of our paper is derivation of the full tower of $A_{N}$
generalization of van Diejen operator orginating from the compactification of
the minimal $(D_{N+3},D_{N+3})$ conformal matter theory on a Riemann surface
with $\mathrm{SU}(N+1)$ maximal punctures. We have already derived basic
operators of this kind in our previous paper Nazzal:2021tiu . However the
tower of operators was missing and we filled this gap in the present paper.

The present paper is another brick in our program of establishing dictionary
between compactifications of $6d$ SCFTs and integrable analytic difference
operators and studying this dictionary in details. There are plenty of ways
this research can be continued in. In particular there should be separate
difference operator for each possible $5d$ compactification of $6d$ theory, or
equivalently for each possible puncture type from $4d$ prospective. So far
several results have been observed in this frame. First of all $BC_{1}$ van
Diejen model was observed in E-string compactifications in several ways
Nazzal:2018brc ; Nazzal:2021tiu . Also previously unknown $A_{2}$ and $A_{3}$
operators were derived using compactifications of the minimal conformal matter
theories of types $\mathrm{SU}(3)$ and $\mathrm{SO}(8)$ Razamat:2018zel .
Finally we have considered compactifications of the minimal
$(D_{N+3},D_{N+3})$ conformal matter theories on a Riemann surfaces with
$A_{N}$ and $C_{2}$ (in case of $(D_{5},D_{5})$ $6d$ theory) type punctures.
However there are plenty of examples of $6d$ SCFT compactificatoins that are
known up to date Razamat:2022gpm ; Heckman:2015bfa ; DelZotto:2014hpa ;
Bhardwaj:2015xxa ; Hayashi:2015vhy ; Hayashi:2015zka ; Kim:2017toz ;
Razamat:2016dpl ; Kim:2018lfo ; Apruzzi:2019vpe . Our main goal is to extend
our results to more examples of compactifications and ideally establish a
dictionary between them and integrable models. Such dictionary can from one
point of view shed light on physics of $6d$, $5d$ and $4d$ supersymmetric
gauge theories. From the other point of view this research program can lead to
important results in the field of integrable systems.

First of all as a continuation of the present paper it is natural to consider
compactifications of the minimal $(D_{N+3},D_{N+3})$ conformal matter on
Riemann surfaces with other types of punctures. In first place these are of
course punctures of $C_{N}$ type for general $N$. In the present work we
failed to derive explicit form of the corresponding A$\Delta$O because of the
technical difficulties but it is worth trying to overcome them. Second
candidate for the study is $\left(A_{1}\right)^{N}$ puncture. Corresponding
$4d$ description was derived in Razamat:2019ukg . It would be interesting to
derive explicit form of the corresponding difference operators and study their
properties. For example we immediately know examples of the kernel functions
of these operators even before deriving them explicitly.

Second candidate for the studies are non-minimal $(D_{N+3},D_{N+3})$ conformal
matter theories, which are obtained as worldvolume theories of the stack of
$k$ M5 branes probing $D_{N+3}$ singularity Heckman:2015bfa . In this case the
known $5d$ gauge theory, and hence the puncture type on Riemann surface, is a
direct generalization of $\left(A_{1}\right)^{N}$ case of minimal conformal
matter compactification discussed above. It corresponds to the
$\left(A_{k-1}\right)^{2}\times\left(A_{2k-1}\right)^{N}\times\left(A_{k-1}\right)^{2}$
gauge theory in $5d$ and hence the puncture type with the same global
symmetry. Trinion theories with two maximal and one minimal punctures, which
are building blocks in the construction of A$\Delta$Os, are also known
Sabag:2020elc . So construction of the correspodning difference operators
should be straightforward though can happen to be technically complicated.

Finally one more thing to be made in this direction is the study of the rank Q
E-string theory compactifictaions along the same lines of research.
Unfortunately so far corresponding trinion theories were not obtained but the
tube theories were already derived in Hwang:2021xyw ; Pasquetti:2019hxf .
Despite we can not derive corresponding A$\Delta$O using only tube theory in
this case we can naturally conjecture it to be $BC_{n}$ van Diejen model. We
can at least test this conjecture by checking corresponding kernel property.

Another possible direction of the research is related to the study of the
properties of the operators we derived. Most interesting questions here are
related to the eigenfunctions of these operators. Deriving full eigenfunctions
is of course difficult and most probably impossible. Realistically we can try
to derive certain limits of these eigenfunction. In case we succeed these
eigenfunctions can play the same role for indices of $\mathcal{N}=1$ theories
as Macdonald and Schur polynomials played for the indices of $\mathcal{N}=2$
theories Gadde:2011uv ; Gadde:2009kb ; Gadde:2011ik .

It would also be interesting to establish connections of our research program
with other integrable models emerging in supersymmetric gauge theories. For
example integrable systems derived using compactifications of the worldvolume
theory of $k$ M5 branes probing $A_{N-1}$ singularity were shown to be related
to the set of transfer matrices Gaiotto:2015usa ; Maruyoshi:2016caf ;
Ito:2016fpl ; Yagi:2017hmj . It would be interesting to understand if similar
connection emerges in case of $D_{N+3}$ singularity.

Also, a relation of E-string theory to the van Diejen model was observed in
Chen:2021ivd  using quantization of the corresponding Seiberg-Witten curve.
Later, authors of this paper generalized their result to $6d$ $(1,0)$ SCFTs
with $\mathrm{SO}(N)$ gauge group and $(N-8)$ fundamental flavors obtaining
yet another set of difference operators Chen:2021rek . It would be interesting
to clarify precise relation of our construction with the methods and results
of these papers.

## Acknowledgments

We would like to especially thank Shlomo Razamat for numerous discussion,
collaboration on different stages of this projects and for reading the draft
of the paper. AN is also thankful for the hospitality of the Theoretical
Physics Division of Uppsala University where parts of this work were done. The
work of BN is supported in part by Israel Science Foundation under grant no.
2289/18, by a Grant No. I-1515-303./2019 from the GIF, the German-Israeli
Foundation for Scientific Research and Development, by BSF grant no. 2018204.
Research of AN is supported by the Swiss National Science Foundation, Grant
No. 185723.

##  Appendix A Special functions

Here we summarize some definitions and properties of special functions used in
the paper.

Elliptic Gamma function is defined through the following infinite product:

$\displaystyle\Gamma_{e}\left(z\right)\equiv\prod\limits_{k,m=0}^{\infty}\frac{1-p^{k+1}q^{m+1}/z}{1-p^{k}q^{m}z}\,.$
(A.1)

It can be easily seen that the poles of this function are located at the
following values of the argument:

$\displaystyle z=p^{-k}q^{-m}\,,\qquad k,m\in\mathbb{Z}_{\geq 0}\,.$ (A.2)

The following relation will be useful in our calculations:

$\displaystyle\Gamma_{e}\left(\frac{pq}{z}\right)\Gamma_{e}\left(z\right)=1\,.$
(A.3)

Also we will often deal with the elliptic beta integral formula

$\displaystyle\kappa\oint\frac{dz}{4\pi iz}\frac{1}{\Gamma_{e}\left(z^{\pm
2}\right)}\prod\limits_{j=1}^{6}\Gamma_{e}\left(t_{i}z^{\pm
1}\right)=\prod\limits_{i<j}\Gamma_{e}\left(t_{i}t_{j}\right)\,.$ (A.4)

Here $\kappa$ is defined to be

$\displaystyle\kappa=(q;q)_{\infty}(p;p)_{\infty}=\prod_{\ell=0}^{\infty}(1-q^{1+\ell})(1-p^{1+\ell}).$
(A.5)

$A_{N}$ generalization of this formula is

$\displaystyle\frac{\kappa^{N}}{N!}\oint\prod\limits_{i=1}^{N}\frac{dz_{i}}{2\pi
iz_{i}}\prod\limits_{i\neq
j}^{N+1}\Gamma_{e}\left(\frac{z_{i}}{z_{j}}\right)^{-1}\prod\limits_{i=1}^{N+2}\prod\limits_{j=1}^{N+1}\Gamma_{e}\left(s_{i}z_{j}\right)\Gamma_{e}\left(t_{i}z_{j}^{-1}\right)=$
$\displaystyle\prod\limits_{i=1}^{N+2}\Gamma_{e}\left(Ss_{i}^{-1}\right)\Gamma_{e}\left(Tt_{i}^{-1}\right)\prod\limits_{i,j=1}^{N+2}\Gamma_{e}\left(s_{i}t_{j}\right)\,,\qquad\quad\left(T=\prod\limits_{i=1}^{N+2}t_{i}\,,~{}S=\prod\limits_{i=1}^{N+2}s_{i}\right)\,.$
(A.6)

The Theta function is defined as follows:

$\displaystyle\theta_{p}\left(x\right)\equiv\left(x;p\right)_{\infty}\left(x^{-1}p;p\right)_{\infty}\,,$
(A.7)

where $\left(z;p\right)_{\infty}$ is the usual q-Pochhammer symbol defined as
follows:

$\displaystyle\left(x;p\right)_{\infty}=\prod\limits_{k=0}^{\infty}\left(1-xp^{k}\right)\,.$
(A.8)

Following properties of theta function will be useful to us

$\displaystyle\theta_{p}\left(x\right)=\frac{\Gamma_{e}\left(qx\right)}{\Gamma_{e}\left(x\right)}\,,\quad\theta_{p}\left(x^{-1}\right)=-x^{-1}\theta_{p}\left(x\right)\,,\quad\theta_{p}\left(xp^{m}\right)=(-1)^{m}x^{-m}p^{-\frac{1}{2}m(m-1)}\theta_{p}\left(x\right)\,,$
$\displaystyle\hskip
113.81102pt\frac{\Gamma_{e}\left(p^{L}q^{K}x\right)}{\Gamma_{e}\left(x\right)}=\prod_{j=0}^{K-1}\theta_{p}\left(q^{j}x\right)\prod_{j=0}^{L-1}\theta_{q}\left(q^{K}p^{j}x\right)$
(A.9)

We will also use the following duality identity from Spiridonov:2008 :

$\displaystyle V(\underline{t})=\prod\limits_{1\leq j<k\leq
4}^{8}\Gamma_{e}\left(t_{j}t_{k}\right)\Gamma_{e}\left(t_{j+4}t_{k+4}\right)V(\underline{s})\,,$
(A.10)

where

$\displaystyle V(\underline{t})\equiv\kappa\oint\frac{dz}{2\pi
iz}\frac{\prod\limits_{j=1}^{8}\Gamma_{e}\left(t_{j}z^{\pm
1}\right)}{\Gamma_{e}\left(z^{\pm
2}\right)}\,,\qquad\prod\limits_{j=1}^{8}t_{i}=pq\,,\quad|t_{j}|,|s_{j}|<1\,.$
$\displaystyle s_{j}=\rho^{-1}t_{j}\,,\quad j=1,2,3,4;\quad s_{j}=\rho
t_{j}\,,\quad j=5,6,7,8;\quad\rho\equiv\sqrt{\frac{t_{1}t_{2}t_{3}t_{4}}{pq}}$
(A.11)

##  Appendix B Derivation of higher tower $A_{N}$ operators

In this appendix we will give details of the derivation of $A_{N}$ A$\Delta$Os
that we summarize in Section 2. We start with the four-punctured sphere theory
whose quiver is shown in Figure 1. This theory and its superconformal index
were derived in Nazzal:2021tiu . The index is given by the following
expression:

$\displaystyle
K_{4}(x,\tilde{x},z,\tilde{z})=\kappa_{N+2}\oint\prod\limits_{i=1}^{N+2}\frac{dt_{i}}{2\pi
it_{i}}\prod\limits_{i\neq
j}^{N+3}\frac{1}{\Gamma_{e}\left(\frac{t_{i}}{t_{j}}\right)}\times$
$\displaystyle\prod\limits_{i=1}^{N+3}\prod\limits_{j=1}^{N+1}\prod\limits_{k=1}^{2N+4}\Gamma_{e}\left((pq)^{\frac{1}{N+3}}\left(u^{-1}w\right)^{4\frac{N+2}{N+3}}x_{j}^{-1}t_{i}\right)\Gamma_{e}\left((pq)^{\frac{1}{N+3}}\left(uw^{-1}\right)^{2\frac{(N+2)(N+1)}{N+3}}z^{\pm
1}t_{i}\right)$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{N+1}{2(N+3)}}u^{-\frac{(N+1)^{2}}{N+3}}v^{N+1}w^{-2\frac{N+1}{N+3}}a_{k}^{-1}t_{i}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{N+3}v^{-N-1}w^{-2}a_{k}x_{j}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{N+1}{2(N+3)}}u^{-2\frac{(N+1)(N+2)}{N+3}}v^{-2(N+1)(N+2)}w^{-4\frac{N+2}{N+3}}t_{i}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{2N+4}v^{2(N+1)(N+2)}x_{j}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(uvw^{-2}\right)^{-N-1}a_{l}z^{\pm
1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}v^{2(N+1)(N+2)}w^{2N+4}z^{\pm
1}\right)\Gamma_{e}\left((pq)^{\frac{N+1}{2(N+3)}}u^{2\frac{(N+1)(N+2)^{2}}{N+3}}w^{4\frac{(N+2)^{2}}{N+3}}t_{i}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{-2N(N+2)}w^{-4N-8}\tilde{x}_{j}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{-2(N+1)(N+2)}w^{-2N-4}\tilde{z}^{\pm
1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{N+3}}\left(u^{-1}w\right)^{4\frac{N+2}{N+3}}t_{i}\tilde{x}_{j}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{N+3}}\left(uw^{-1}\right)^{2\frac{(N+1)(N+2)}{N+3}}t_{i}\tilde{z}^{\pm
1}\right)\,.$ (B.1)

This is an index of an $\mathrm{SU}(N+3)$ SQCD with $2N+6$ flavors and a
superpotential. Variables $x,\tilde{x},z,\tilde{z}$ are fugacities of the
global symmetry of two maximal $\mathrm{SU}(N+1)_{x,\tilde{x}}$ and two
minimal $\mathrm{SU}(2)_{z,\tilde{z}}$ punctures correspondingly.
We close the two $\mathrm{SU}(2)$ punctures by setting $z$ and $\tilde{z}$ to
the values in (2.7). Performing calculations for general values of the charges
is complicated. So we will perform here the calculation only for one of the
charges fixing
$\tilde{h}_{i}=\tilde{h}_{2N+6}=\left(u^{N}w^{2}\right)^{-2N-4}$.
Corresponding positions of the poles are then given by

$\displaystyle
z=(pq)^{-\frac{1}{2}}\left(wu^{N+1}\right)^{-2N-4}q^{-M}p^{-L},\;\;\;\;\;\;\;\tilde{z}=(pq)^{-\frac{1}{2}}\left(wu^{N+1}\right)^{2N+4}q^{-\tilde{M}}p^{-\tilde{L}}\,.$
(B.2)

Also as discussed in Section 2 we will further consider
$\tilde{L}=\tilde{M}=0$.

The pole in $\tilde{z}$ is an explicit simple pole so we can just take the
residue. The pole in $z$ however, is due to contour pinching. To see this
clearly, it is useful to first perform a Seiberg duality leading to an index
of $SU(N+2)$ gauge thoery with $2N+5$ flavors, with the followng index,

$\displaystyle
K_{(2;2N+6;L,M)}^{A_{N}}\left(x,\tilde{x}\right)=\kappa_{N+1}\oint\prod\limits_{i=1}^{N+1}\frac{dt_{i}}{2\pi
it_{i}}\prod\limits_{i\neq
j}^{N+2}\frac{1}{\Gamma_{e}\left(\frac{t_{i}}{t_{j}}\right)}\times$
$\displaystyle\prod\limits_{i=1}^{N+2}\prod\limits_{j=1}^{N+1}\prod\limits_{k=1}^{2N+4}\Gamma_{e}\left((pq)^{\frac{1}{2(N+2)}}u^{2N+4}x_{j}t_{i}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{-N-3}v^{N+1}w^{2}a_{k}^{-1}\tilde{x}_{j}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(uv^{N+1}\right)^{-2N-4}\tilde{x}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2(N+2)}}u^{2N+4}\tilde{x}_{j}t_{i}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{N+3}{2N+4}}u^{(2N+4)(N+1)}w^{4N+8}t_{i}q^{M}p^{L}\right)\Gamma_{e}\left((pq)^{-\frac{N+1}{2N+4}}u^{-(2N+4)(N+1)}t_{i}q^{-M}p^{-L}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{N+3}{2(N+2)}}u^{-2(N+1)(N+2)}t_{i}\right)\Gamma_{e}\left((pq)^{\frac{N+1}{2(N+2)}}\left(uv\right)^{-N-1}w^{-2}a_{k}t_{i}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{N+1}{2(N+2)}}v^{2(N+1)(N+2)}t_{i}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(u^{N}w^{2}\right)^{-2N-4}\tilde{x}_{j}\right)$

where we have already dropped irrelevant overall factors and substituted
values of $z$ from (B.2). The subscript of the index as usually contains all
the information of the underlying theory. In particular $2$ stands for the
number of punctures. Index $2N+6$ stands for the index of the moment map we
give vev to in order to close the puncture (moment map of charge
$\tilde{h}_{2N+6}\tilde{h}^{-\frac{1}{4}}$ in this case). Finally $(L,M)$
denotes derivative powers in the vev and are the same as $L$ and $M$ integers
in (B.2). As can be seen from the expression above the contour pinching is due
to collision of poles at the following values:

$\displaystyle
t_{i}=(pq)^{-\frac{1}{2(N+2)}}u^{-2(N+2)}x_{i}^{-1}q^{-m_{i}}p^{-l_{i}}\,,\quad
t_{N+2}=(pq)^{\frac{N+1}{2(N+2)}}u^{2(N+1)(N+2)}q^{M-m_{N+2}}p^{L-l_{N+2}}\,,$
$\displaystyle\mathrm{or}$ $\displaystyle
t_{i}=(pq)^{-\frac{1}{2(N+2)}}u^{-2(N+2)}\tilde{x}_{i}^{-1}q^{-m_{i}}p^{-l_{i}}\,,\quad
t_{N+2}=(pq)^{\frac{N+1}{2(N+2)}}u^{2(N+1)(N+2)}q^{M-m_{N+2}}p^{L-l_{N+2}}\,,$
(B.4)

where $m_{i}$ and $l_{i}$ are partitions of $M$ and $L$ respectively. There
are $(N+2)!$ such poles coming from permutations of $t_{i}$ but they all give
the same result and we are ignoring overall factors. It can be checked that
the two lines in (B.4) give the same result so we will consider only the first
set of poles. After computing the residue we get the following expression for
particular partitions $\vec{l},\vec{m}$,

$\displaystyle
K_{\left(2;2N+6;\vec{m},\vec{l}\right)}^{A_{N}}(x,\tilde{x})=\Gamma_{e}\left(pq\right)\prod_{j=1}^{N+2}\frac{\prod_{n=1}^{M-m_{N+2}}\theta_{p}\left(q^{-n}\right)\prod_{n=1}^{L-l_{N+2}}\theta_{q}\left(q^{m_{N+2}-M}p^{-n}\right)}{\prod_{n=1}^{m_{j}}\theta_{p}\left(q^{-n}p^{-l_{j}}\right)\prod_{n=1}^{l_{j}}\theta_{q}\left(p^{-n}\right)}$
$\displaystyle\prod\limits_{i=1}^{N+1}\prod\limits_{k=1}^{2N+4}\prod\limits_{j\neq
i}^{N+1}\Gamma_{e}\left(\frac{\tilde{x}_{i}}{x_{i}}q^{-m_{i}}p^{-l_{i}}\right)\frac{\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{2(N+2)^{2}}q^{M-m_{N+2}}p^{L-l_{N+2}}x_{i}\right)}{\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{2(N+2)^{2}}q^{M+m_{i}-m_{N+2}}p^{L+l_{i}-l_{N+2}}x_{i}\right)}\times$
$\displaystyle\frac{\Gamma_{e}\left((pq)^{-\frac{1}{2}}u^{-2(N+2)^{2}}x_{i}^{-1}q^{-M-m_{i}}p^{-L-l_{i}}\right)}{\Gamma_{e}\left((pq)^{-\frac{1}{2}}u^{-2(N+2)^{2}}x_{i}^{-1}q^{m_{N+2}-M-m_{i}}p^{l_{N+2}-L-l_{i}}\right)}\Gamma_{e}\left(\frac{\tilde{x}_{j}}{x_{i}}q^{-m_{i}}p^{-l_{i}}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{N+3}v^{-N-1}w^{-2}a_{k}x_{i}q^{m_{i}}p^{l_{i}}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(u^{N}w^{2}\right)^{2N+4}x_{i}^{-1}q^{M-m_{i}}p^{L-l_{i}}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{-2(N+2)^{2}}x_{i}^{-1}q^{-m_{i}}p^{-l_{i}}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(uv^{N+1}\right)^{2N+4}x_{i}q^{m_{i}}p^{l_{i}}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{-N-3}v^{N+1}w^{2}a_{k}^{-1}\tilde{x}_{i}^{-1}\right)\Gamma_{e}\left(q^{m_{N+2}-M}p^{l_{N+2}-L}\left(vu^{-1}\right)^{2(N+1)(N+2)}\right)\times$
$\displaystyle\frac{\Gamma_{e}\left(\frac{x_{j}}{x_{i}}q^{-m_{i}}p^{-l_{i}}\right)}{\Gamma_{e}\left(\frac{x_{j}}{x_{i}}q^{m_{j}-m_{i}}p^{l_{j}-l_{i}}\right)}\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{2(N+2)^{2}}q^{M-m_{N+2}}p^{L-l_{N+2}}\tilde{x}_{i}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(u^{N}w^{2}\right)^{-2N-4}\tilde{x}_{i}\right)\Gamma_{e}\left(pq\left(u^{N+1}w\right)^{4N+8}q^{2M-m_{N+2}}p^{2L-l_{N+2}}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(uv^{N+1}\right)^{-2N-4}\tilde{x}_{i}^{-1}\right)\Gamma_{e}\left(\left(u^{2N+5}v\right)^{-N-1}w^{-2}a_{k}q^{m_{N+2}-M}p^{l_{N+2}-L}\right)\,.$
(B.5)

To get the $A\Delta O$ acting on $\mathrm{SU}(N+1)_{x}$ puncture, we should S
glue this tube to the index of an arbitrary theory ${\mathcal{I}}(\tilde{x})$
with $SU(N+1)_{\tilde{x}}$ puncture and sum over all partitions,

$\displaystyle{\mathcal{I}}_{1}(x)=\kappa_{N}\sum_{\\{m_{i}\\},\\{l_{i}\\}}\oint\prod\limits_{i=1}^{N}\frac{d\tilde{x}_{i}}{2\pi
i\tilde{x}_{i}}\prod\limits_{i\neq
j}^{N+1}\frac{1}{\Gamma_{e}\left(\frac{\tilde{x}_{i}}{\tilde{x}_{j}}\right)}K_{\left(2;2N+6;\vec{m},\vec{l}\right)}^{A_{N}}(\tilde{x},x){\mathcal{I}}_{0}(\tilde{x})\,,$
(B.6)

Note that there is a zero in (B.5) coming from $\Gamma_{e}\left(pq\right)$ but
this is canceled against the pinching of $\tilde{x}$ at the following values,

$\displaystyle\tilde{x}_{i}=x_{i}q^{m_{i}-s_{i}}p^{l_{i}-r_{i}}\,,\quad\sum\limits_{i=1}^{N+1}s_{i}=M-m_{N+2}\,,\quad\sum\limits_{i=1}^{N+1}r_{i}=L-l_{N+2},$
(B.7)

up to permutations of $x_{i}$ which give the same result due to the Weyl
symmetry of $A_{N}$ root system. If we specify $\tilde{x}_{i}$ to the values
written above we obtain double pole due to the contour pinching of (B.6) but
as we mentioned one of them is cancelled by the zero of
$\Gamma_{e}\left(pq\right)$. Computing the residue of the remaining pole we
obtain the following contribution of each individual partition:

$\displaystyle{\mathcal{I}}^{\boldsymbol{m,l,s,r}}_{1}(x)=\prod_{i=1}^{N+1}\prod_{n=1}^{N+2}\prod\limits_{k=1}^{2N+4}\prod\limits_{j\neq
i}^{N+1}\frac{\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{2(N+2)^{2}}q^{M-m_{N+2}}p^{L-l_{N+2}}x_{i}\right)}{\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{2(N+2)^{2}}q^{M+m_{i}-m_{N+2}}p^{L+l_{i}-l_{N+2}}x_{i}\right)}\times$
$\displaystyle\frac{\prod_{n=1}^{M-m_{N+2}}\theta_{p}\left(q^{-n}\right)\prod_{n=1}^{L-l_{N+2}}\theta_{q}\left(q^{m_{N+2}-M}p^{-n}\right)}{\prod_{n=1}^{m_{j}}\theta_{p}\left(q^{-n}p^{-l_{j}}\right)\prod_{n=1}^{l_{j}}\theta_{q}\left(p^{-n}\right)\prod_{n=1}^{s_{i}}\theta_{p}\left(q^{-n}p^{-r_{i}}\right)\prod_{n=1}^{r_{i}}\theta_{q}\left(p^{-n}\right)}\times$
$\displaystyle\frac{\Gamma_{e}\left((pq)^{-\frac{1}{2}}u^{-2(N+2)^{2}}x_{i}^{-1}q^{-M-m_{i}}p^{-L-l_{i}}\right)}{\Gamma_{e}\left((pq)^{-\frac{1}{2}}u^{-2(N+2)^{2}}x_{i}^{-1}q^{m_{N+2}-M-m_{i}}p^{l_{N+2}-L-l_{i}}\right)}\Gamma_{e}\left(\frac{x_{j}}{x_{i}}q^{m_{j}-s_{j}-m_{i}}p^{l_{j}-r_{j}-l_{i}}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{N+3}v^{-N-1}w^{-2}a_{k}x_{i}q^{m_{i}}p^{l_{i}}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(u^{N}w^{2}\right)^{2N+4}x_{i}^{-1}q^{M-m_{i}}p^{L-l_{i}}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{-2(N+2)^{2}}x_{i}^{-1}q^{-m_{i}}p^{-l_{i}}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{-N-3}v^{N+1}w^{2}a_{k}^{-1}x_{i}^{-1}q^{s_{i}-m_{i}}p^{r_{i}-l_{i}}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(uv^{N+1}\right)^{-2N-4}x_{i}^{-1}q^{s_{i}-m_{i}}p^{r_{i}-l_{i}}\right)\frac{\Gamma_{e}\left(\frac{x_{j}}{x_{i}}q^{-m_{i}}p^{-l_{i}}\right)}{\Gamma_{e}\left(\frac{x_{j}}{x_{i}}q^{m_{j}-m_{i}}p^{l_{j}-l_{i}}\right)}\times$
$\displaystyle\frac{\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(u^{N}w^{2}\right)^{-2N-4}x_{i}q^{m_{i}-s_{i}}p^{l_{i}-r_{i}}\right)}{\Gamma_{e}\left(q^{m_{i}-m_{j}+s_{j}-s_{i}}p^{l_{i}-l_{j}+r_{j}-r_{i}}x_{i}/x_{j}\right)}\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(uv^{N+1}\right)^{2N+4}x_{i}q^{m_{i}}p^{l_{i}}\right)\times$
$\displaystyle\Gamma_{e}\left(pq\left(u^{N+1}w\right)^{4N+8}q^{2M-m_{N+2}}p^{2L-l_{N+2}}\right)\times$
$\displaystyle\Gamma_{e}\left(q^{m_{N+2}-M}p^{l_{N+2}-L}\left(vu^{-1}\right)^{2(N+1)(N+2)}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{2(N+2)^{2}}q^{M-m_{N+2}+m_{i}-s_{i}}p^{L-l_{N+2}+l_{i}-r_{i}}x_{i}\right)\times$
$\displaystyle\Gamma_{e}\left(\left(u^{2N+5}v\right)^{-N-1}w^{-2}a_{k}q^{m_{N+2}-M}p^{l_{N+2}-L}\right){\mathcal{I}}_{0}(q^{m_{i}-s_{i}}p^{l_{i}-r_{i}}x_{i})\,.$
(B.8)

Summing over all partitions and using properties of $\Gamma$ and $\theta$
functions (A.9) we get the difference operator given in (2.10) with the
$\tilde{h}_{k}=\tilde{h}_{10}$. Operators obtained using another ways to close
minimal punctures can be derived in absolutely identical manner and lead to
the same A$\Delta$O (2.10).

##  Appendix C Derivation of $C_{2}$ operator

In this section we give details of the derivations of results summarized in
Section 3 and show in all details how to derive $C_{2}$ type generalization of
the van Diejen model.

Let’s start with the derivation of the trinion that has one maximal
$\mathrm{USp}(2N)$ puncture, one maximal $\mathrm{SU}(N+1)$ puncture and one
minimal $\mathrm{SU}(2)$ puncture. It is important since it will be later used
by us to derive four punctured sphere with two maximal $\mathrm{USp}(2N)$ and
two minimal $\mathrm{SU}(2)$ punctures as specified in (3.7).

To obtain this kind of three punctured sphere theory we start with the trinion
${\mathcal{T}}^{A}_{x,y,z}$ with two maximal $\mathrm{SU}(N+1)_{x,y}$ and one
minimal $\mathrm{SU}(2)_{z}$ punctures. This trinion was derived in
Razamat:2020bix  and is shown on the Figure 1. Corresponding superconformal
index is given by:

$\displaystyle
K_{3}^{A}(x,y,z)=\kappa_{N+1}\oint\prod\limits_{i=1}^{N+1}\frac{dt_{i}}{2\pi
it_{i}}\prod\limits_{i\neq
j}^{N+2}\frac{1}{\Gamma_{e}\left(\frac{t_{i}}{t_{j}}\right)}\prod\limits_{i=1}^{N+2}\prod\limits_{j=1}^{N+1}\Gamma_{e}\left((pq)^{\frac{1}{2(N+2)}}u^{2N+4}t_{i}x_{j}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2(N+2)}}v^{2N+4}t_{i}y_{j}\right)\Gamma_{e}\left((pq)^{\frac{1}{2(N+2)}}w^{2N+4}t_{i}z^{\pm
1}\right)\times$
$\displaystyle\prod\limits_{l=1}^{2N+4}\Gamma_{e}\left((pq)^{\frac{N+1}{2(N+2)}}\left(uv\right)^{-N-1}w^{-2}t_{i}^{-1}a_{l}\right)\,,$
(C.1)

where $u$, $v$, $w$ and $a_{i}$ parametrize Cartans of the $6d$
$\mathrm{SO}(4N+8)$ global symmetry. $\mathrm{SU}(N+2)$ gauge symmetry is
parametrized by $t_{i}$’s with the relation

$\displaystyle\prod\limits_{i=1}^{N+2}t_{i}=1\,.$ (C.2)

Global $\mathrm{SU}(N+1)$ symmetries of the maximal punctures are parametrized
by $x_{i}$ and $y_{i}$ satisfying

$\displaystyle\prod\limits_{j=1}^{N+1}x_{j}=\prod\limits_{j=1}^{N+1}y_{j}=1\,.$
(C.3)

Each puncture has $(2N+6)$ moment map operators with the charges specified in
(2.1).

Now in order to obtain desired trinion we should turn one of the
$\mathrm{SU}(N+1)$ punctures into $\mathrm{USp}(2N)$ type puncture. For this
purpose we use an $A_{N}C_{N}$ tube theory introduced in Kim:2018bpg  and
shown in Figure 2(a). Chirals of these theory correspond to the moment maps of
punctures. For $\mathrm{USp}(2N)$ puncture these are just moment maps
specified in (3.1). For $\mathrm{SU}(N+1)$ puncture as specified in (2.1) but
with the last moment map flipped so that all of them transform in the same
fundamental representation of $\mathrm{SU}(N+1)$:

$\displaystyle M_{u}={\bf N+1}^{x}\otimes\left({\bf
2N+4}_{u^{N+3}v^{-N-1}w^{-2}}\oplus{\bf
1}_{\left(uv^{N+1}\right)^{2N+4}}\oplus{\bf
1}_{\left(u^{N}w^{2}\right)^{-2N-4}}\right)$ (C.4)

The index of this tube theory is given by:

$\displaystyle
K_{2}^{AC}(x,\tilde{x})=\prod\limits_{i=1}^{N+1}\prod\limits_{j=1}^{N}\prod\limits_{l=1}^{2N+4}\Gamma_{e}\left(\left(u^{-1}w\right)^{2N+4}\tilde{x}_{i}^{-1}x_{j}^{\pm
1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{N+3}v^{-N-1}w^{-2}a_{l}\tilde{x}_{i}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(uv^{N+1}\right)^{2N+4}\tilde{x}_{i}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(u^{N}w^{2}\right)^{-2N-4}\tilde{x}_{i}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(uvw^{-2}\right)^{N+1}a_{l}^{-1}x_{j}^{\pm
1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(v^{N+1}w\right)^{-2N-4}x_{j}^{\pm
1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(u^{N+1}w\right)^{2N+4}x_{j}^{\pm
1}\right)\prod\limits_{i<j}^{N+1}\Gamma_{e}\left(pq\left(uw^{-1}\right)^{4(N+2)}\tilde{x}_{i}\tilde{x}_{j}\right)\,,$
(C.5)

Now we take this theory and glue it along the $\mathrm{SU}(N+1)$ puncture to
the $A_{N}$ trinion shown in Figure 1. Consistent gluing in this case is the
mixture of $\Phi$ and S gluings. In particular we $\Phi$ glue all moment maps
except ${\bf 1}_{\left(u^{N}w^{2}\right)^{2N+4}}$ which is S glued. At the
level of the index this operation corresponds to

$\displaystyle
K_{3}^{AC}(x,y,z)=\kappa_{N}\oint\prod\limits_{i=1}^{N}\frac{d\tilde{x}_{i}}{2\pi
i\tilde{x}_{i}}\prod\limits_{i\neq
j}^{N+1}\frac{1}{\Gamma_{e}\left(\frac{\tilde{x}_{i}}{\tilde{x}_{j}}\right)}K_{3}^{A}(\tilde{x},y,z)K_{2}^{AC}(\tilde{x},x)\times$
$\displaystyle\prod\limits_{i=1}^{N+1}\prod\limits_{l=1}^{2N+4}\Gamma_{e}\left((pq)^{\frac{1}{2}}u^{-N-3}v^{N+1}w^{2}a_{l}^{-1}\tilde{x}_{i}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}\left(uv^{N+1}\right)^{-2N-4}\tilde{x}_{i}^{-1}\right)$
(C.6)

Substituting expressions (C.1) for $A_{N}$ trinion and (C.5) for the
$A_{N}C_{N}$ tube we immediately see that the $\mathrm{SU}(N+1)_{\tilde{x}}$
gauge theory appears to be S-confining. Corresponding S-confining theory is
depicted in Figure 8. This duality was first introduced by Spiridonov and
Warnaar MR2264067  as a mathematical identity for superconformal indices of
the corresponding theories:

Figure 8: Spiridonov-Warnaar-Vartanov (SWV) duality. On the picture we also
specify superpotentials, UV on the left and dynamically generated on the right
sides of dualitites. The solid line starting and ending on the same node
denotes multiplet in rank-two antisymmetric representation (AS).

$\displaystyle\kappa_{N}\oint\prod\limits_{i=1}^{N}\frac{d\tilde{x}_{i}}{2\pi
i\tilde{x}_{i}}\prod\limits_{i\neq
j}^{N+1}\frac{1}{\Gamma_{e}\left(\frac{\tilde{x}_{i}}{\tilde{x}_{j}}\right)}\prod\limits_{i<j}^{N+1}\Gamma_{e}\left(S\tilde{x}_{i}^{-1}\tilde{x}_{j}^{-1}\right)\prod\limits_{j=1}^{N+1}\prod\limits_{k=1}^{N}\Gamma_{e}\left(\tilde{x}_{j}t_{k}\right)\Gamma_{e}\left(pqS^{-1}t_{k}^{-1}\tilde{x}_{j}\right)\times$
$\displaystyle\prod\limits_{m=1}^{N+3}\Gamma_{e}\left(s_{m}\tilde{x}_{j}^{-1}\right)=\prod\limits_{k=1}^{N}\prod\limits_{m=1}^{N+3}\frac{\Gamma_{e}\left(t_{k}s_{m}\right)}{\Gamma_{e}\left(St_{k}s_{m}^{-1}\right)}\prod\limits_{l<m}^{N+3}\Gamma_{e}\left(Ss_{l}^{-1}s_{m}^{-1}\right)\,,$
(C.7)

where $S=\prod\limits_{i=1}^{N+3}s_{i}$. Later Spiridonov and Vartanov
Spiridonov:2009za  discussed physical implications of the relation written
above. We refer to this duality as Spiridonov-Warnaar-Vartanov (SWV) duality.
More detailed discussion about it can also been found in the Appendix B.5 of
our previous paper Nazzal:2021tiu . Using this duality for the
$\mathrm{SU}(N+1)_{\tilde{x}}$ node in (C.6) we can finally write down the
index of the desired three-punctured sphere as follows:

$\displaystyle
K_{3}^{AC}(x,y,z)=\kappa_{N+1}\oint\prod\limits_{i=1}^{N+1}\frac{dt_{i}}{2\pi
it_{i}}\prod\limits_{i\neq
j}^{N+2}\frac{1}{\Gamma_{e}\left(\frac{t_{i}}{t_{j}}\right)}\prod\limits_{k=1}^{N}\prod\limits_{i=1}^{N+2}\prod\limits_{j=1}^{N+1}\Gamma_{e}\left((pq)^{\frac{1}{2(N+2)}}w^{2N+4}x_{k}^{\pm
1}t_{i}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{N+1}{2(N+2)}}u^{2(N+1)(N+2)}t_{i}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2(N+2)}}v^{2N+4}y_{j}t_{i}\right)\Gamma_{e}\left((pq)^{\frac{1}{2(N+2)}}w^{2N+4}t_{i}z^{\pm
1}\right)\times$
$\displaystyle\prod\limits_{l=1}^{2N+4}\Gamma_{e}\left((pq)^{\frac{N+1}{2(N+2)}}\left(uv\right)^{-N-1}w^{-2}t_{i}^{-1}a_{l}\right)\prod\limits_{i<j}^{N+2}\Gamma_{e}\left((pq)^{\frac{N+1}{N+2}}w^{-4N-8}t_{i}^{-1}t_{j}^{-1}\right)\,,$
(C.8)

where we have omitted some of the $\mathrm{USp}(2N)$ singlets thus redefining
type of the puncture. Corresponding trinion theory is shown in Figure 9.

Figure 9: Trinion theory with $\mathrm{USp}(2N)_{x}$ maximal puncture (light
orange), $\mathrm{SU}(N+1)_{y}$ maximal puncture (green) and
$\mathrm{SU}(2)_{z}$ minimal puncture (orange). $U(1)$ global symmetries are
parametrized using the parametrization (3.3) more natural for the
$\mathrm{SU}(N+1)$ type punctures. The index of the theory is given in (C.8).

Now we can use this $A_{N}C_{N}$ type trinion theory in order to derive four-
punctured sphere theory with two $\mathrm{USp}(2N)$ maximal punctures. For
this purpose we S glue two copies of this trinion along $A_{N}$ type puncture.
At the level of the index this corresponds to the operation specified in
(3.7). Substituting trinion index (C.8) into this expression we obtain the
following four-punctured sphere index:

$\displaystyle
K_{4}(x,\tilde{x},z,\tilde{z})\equiv\kappa_{N}\oint\prod\limits_{i=1}^{N}\frac{dy}{2\pi
ix}\prod\limits_{i\neq
j}\frac{1}{\Gamma_{e}\left(y_{i}/y_{j}\right)}K_{3}^{AC}(x,y,z)\bar{K}_{3}^{AC}(\tilde{x},y,\tilde{z})=$
$\displaystyle\kappa_{N+1}^{2}\kappa_{N}\oint\prod\limits_{i=1}^{N+1}\frac{dt_{i}}{2\pi
it_{i}}\frac{d\tilde{t}_{i}}{2\pi
i\tilde{t}_{i}}\prod\limits_{i=1}^{N}\frac{dy_{i}}{2\pi
iy_{i}}\prod\limits_{i=1}^{N+2}\frac{1}{\Gamma_{e}\left(t_{i}/t_{j}\right)\Gamma_{e}\left(\tilde{t}_{i}/\tilde{t}_{j}\right)}\prod\limits_{i=1}^{N+1}\frac{1}{\Gamma_{e}\left(y_{i}/y_{j}\right)}\times$
$\displaystyle\prod\limits_{i=1}^{N+2}\prod\limits_{j=1}^{N+1}\prod\limits_{k=1}^{N}\prod\limits_{l=1}^{2N+4}\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{2N+4}x_{k}^{\pm
1}t_{i}\right)\Gamma_{e}\left((pq)^{\frac{N+1}{2N+4}}u^{2(N+1)(N+2)}t_{i}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}v^{2N+4}y_{j}t_{i}\right)\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{2N+4}t_{i}z^{\pm
1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{-2N-4}\tilde{x}_{k}^{\pm
1}\tilde{t}_{i}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{N+1}{2N+4}}\left(uv\right)^{-N-1}w^{-2}t_{i}^{-1}a_{l}\right)\Gamma_{e}\left((pq)^{\frac{N+1}{2N+4}}\left(uv\right)^{N+1}w^{2}\tilde{t}_{i}a_{l}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{N+1}{2N+4}}u^{2(N+1)(N+2)}\tilde{t}_{i}\right)\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}v^{-2N-4}y_{j}^{-1}\tilde{t}_{i}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{-2N-4}\tilde{t}_{i}^{-1}\tilde{z}^{\pm
1}\right)\prod\limits_{i<j}^{N+2}\Gamma_{e}\left((pq)^{\frac{N+1}{N+2}}w^{-4N-8}t_{i}^{-1}t_{j}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{N+1}{N+2}}w^{4N+8}\tilde{t}_{i}\tilde{t}_{j}\right)$
(C.9)

Now if we look on the $\mathrm{SU}(N+1)_{y}$ gauged node we see that it
corresponds to the theory with $N+2$ hypermmultiplets, meaning the node is
S-confining and can be integrated out using the standard Seiberg duality.
Performing this simple operation we land on the theory shown on the Figure 10
with the following superconformal index:

$\displaystyle
K_{4}^{C}(x,\tilde{x},z,\tilde{z})=\kappa^{2}_{N+1}\oint\prod\limits_{i=1}^{N+1}\frac{dt_{i}}{2\pi
it_{i}}\frac{d\tilde{t}_{i}}{2\pi i\tilde{t}_{i}}\prod\limits_{i\neq
j}^{N+2}\frac{1}{\Gamma_{e}\left(t_{i}/t_{j}\right)\Gamma_{e}\left(\tilde{t}_{i}/\tilde{t}_{j}\right)}\times$
$\displaystyle\prod\limits_{i=1}^{N+2}\prod\limits_{k=1}^{N}\prod\limits_{l=1}^{2N+6}\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{2N+4}x_{k}^{\pm
1}t_{i}\right)\Gamma_{e}\left((pq)^{\frac{N+1}{2N+4}}w^{-2\frac{N+2}{N+3}}\tilde{a}_{l}t_{i}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{-2N-4}\tilde{x}_{k}^{\pm
1}\tilde{t}_{i}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{-2N-4}\tilde{t}_{i}^{-1}\tilde{z}^{\pm
1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{2\frac{N+2}{N+3}}\tilde{a}_{l}^{-1}\tilde{t}_{i}\right)\prod\limits_{i,j=1}^{N+2}\Gamma_{e}\left((pq)^{\frac{1}{N+2}}t_{i}\tilde{t}_{j}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{2N+4}t_{i}z^{\pm
1}\right)\times$
$\displaystyle\prod\limits_{i<j}^{N+2}\Gamma_{e}\left((pq)^{\frac{N+1}{N+2}}w^{-4N-8}t_{i}^{-1}t_{j}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{N+1}{N+2}}w^{4N+8}\tilde{t}_{i}\tilde{t}_{j}\right)\,,$
(C.10)

where we have also used dictionary (3.4) in order to express everything in
terms of $C_{N}$ parametrzation (3.2) since now we have only this type of
maximal punctures and hence latter parametrization is more natural.

Figure 10: Four-punctured sphere theory with two maximal $\mathrm{USp}(2N)$
and two minimal $\mathrm{SU}(2)$ punctures. The four-punctured sphere is
derived by gluing two trinion theories shown on the Figure 9 and performing
some duality transformations specified in the text.

Now we finally close one of the two $\mathrm{SU}(2)$ minimal punctures without
introducing the defect. As discussed in Section 3 this amounts to computing
the residue of (C.10) at the point
$\tilde{z}=(pq)^{-\frac{1}{2}}w^{2\frac{(N+2)^{2}}{N+3}}\tilde{a}_{i}$. Indeed
it can be seen that at this value of $\tilde{z}$ there is pole coming from the
contour pinching at

$\displaystyle\tilde{t}_{N+2}=(pq)^{-\frac{N+1}{2N+4}}w^{-2\frac{N+2}{N+3}}\tilde{a}_{i}\,,$
(C.11)

where $\tilde{t}_{N+2}$ variable is chosen without loss of generality. In fact
we should compute such pinchings for each $\tilde{t}_{i}$ variables and sum
the results. But due to the Weyl symmetry of the $\mathrm{SU}(N+1)$ root
system all such contributions are the same and summing them results in an
overall factor of $(N+1)!$. We omit such overall factors since they are
irrelevant for the structure of the A$\Delta$O we get in the end. Computation
of the corresponding residue at the pinching point leads to the following
expression for the three-punctured sphere theory index:

$\displaystyle
K_{(3;i;0)}^{C}(x,\tilde{x},z)=\kappa_{N+1}\kappa_{N}\oint\prod\limits_{j=1}^{N+1}\frac{dt_{j}}{2\pi
it_{j}}\prod\limits_{j=1}^{N}\frac{d\tilde{t}_{j}}{2\pi
i\tilde{t}_{j}}\prod\limits_{k\neq
j}^{N+2}\frac{1}{\Gamma_{e}\left(t_{k}/t_{j}\right)}\prod\limits_{i\neq
j}^{N+1}\frac{1}{\Gamma_{e}\left(\tilde{t}_{k}/\tilde{t}_{j}\right)}\times$
$\displaystyle\prod\limits_{m=1}^{N+2}\prod\limits_{j=1}^{N+1}\prod\limits_{k=1}^{N}\prod\limits_{l\neq
i}^{2N+6}\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{2N+4}x_{k}^{\pm
1}t_{m}\right)\Gamma_{e}\left((pq)^{\frac{N+1}{2N+4}}w^{-2\frac{N+2}{N+3}}\tilde{a}_{l}t_{m}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{2N+4}t_{m}z^{\pm
1}\right)\Gamma_{e}\left(w^{-2\frac{(N+2)^{3}}{(N+1)(N+3)}}\tilde{a}_{i}^{\frac{1}{N+1}}\tilde{x}_{k}^{\pm
1}\tilde{t}_{j}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-2\frac{(N+2)^{2}}{N+3}}\tilde{a}_{i}^{-1}\tilde{z}_{k}^{\pm
1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{2\frac{(N+2)^{2}}{(N+1)(N+3)}}\tilde{a}_{l}^{-1}\tilde{a}_{i}^{-\frac{1}{N+1}}\tilde{t}_{j}\right)\Gamma_{e}\left((pq)^{\frac{1}{2N+4}}w^{-2\frac{N+2}{(N+1)(N+3)}}\tilde{a}_{i}^{\frac{1}{N+1}}t_{i}/\tilde{t}_{j}\right)\times$
$\displaystyle\prod\limits_{i=m<j}^{N+2}\Gamma_{e}\left((pq)^{\frac{N+1}{N+2}}w^{-4N-8}t_{m}^{-1}t_{j}^{-1}\right)\prod\limits_{m<j}^{N+1}\Gamma_{e}\left(pqw^{4\frac{(N+2)^{3}}{(N+1)(N+3)}}\tilde{a}_{i}^{-\frac{2}{N+1}}\tilde{t}_{m}\tilde{t}_{j}\right)\,.$
(C.12)

Corresponding quiver diagram is shown in Figure 3.

In case $N=1$ it can be shown that the theory we derived reduces to the one
shown in Figure 22 (b) in our previous paper Nazzal:2021tiu . There we used it
to derive $C_{1}$ A$\Delta$O which appeared to be van Diejen operator.
Studying general $N$ case appears to be too complicated since there is no
clear way to further simplify theory (C.12). Instead we concentrate here on
$C_{2}$ case. When $N=2$ $\overline{AS}$ representation of $\mathrm{SU}(N+1)$
node becomes anti-fundamental representation and the index of the theory
reads:

$\displaystyle
K_{(3;i,0)}^{C}(x,\tilde{x},z)=\kappa_{3}\kappa_{2}\oint\prod\limits_{m=1}^{3}\frac{dt_{m}}{2\pi
it_{m}}\prod\limits_{i=1}^{2}\frac{d\tilde{t}_{m}}{2\pi
i\tilde{t}_{m}}\prod\limits_{m\neq
j}^{4}\frac{1}{\Gamma_{e}\left(t_{m}/t_{j}\right)}\prod\limits_{m\neq
j}^{3}\frac{1}{\Gamma_{e}\left(\tilde{t}_{m}/\tilde{t}_{j}\right)}\times$
$\displaystyle\prod\limits_{m=1}^{4}\prod\limits_{j=1}^{3}\prod\limits_{k=1}^{2}\prod\limits_{l\neq
i}^{10}\Gamma_{e}\left((pq)^{\frac{1}{8}}w^{8}x_{k}^{\pm
1}t_{m}\right)\Gamma_{e}\left((pq)^{\frac{3}{8}}w^{-\frac{8}{5}}\tilde{a}_{l}t_{m}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{8}}w^{8}t_{m}z^{\pm
1}\right)\times$
$\displaystyle\Gamma_{e}\left(w^{-\frac{128}{15}}\tilde{a}_{i}^{\frac{1}{3}}\tilde{x}_{k}^{\pm
1}\tilde{t}_{j}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{i}^{-1}\tilde{x}_{k}^{\pm
1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{l}^{-1}\tilde{a}_{i}^{-\frac{1}{3}}\tilde{t}_{j}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{8}}w^{-\frac{8}{15}}\tilde{a}_{i}^{\frac{1}{3}}t_{m}/\tilde{t}_{j}\right)\Gamma_{e}\left(pqw^{\frac{256}{15}}\tilde{a}_{i}^{-\frac{2}{3}}\tilde{t}_{j}^{-1}\right)\prod\limits_{m<j}^{N+2}\Gamma_{e}\left((pq)^{\frac{3}{4}}w^{-16}t_{m}^{-1}t_{j}^{-1}\right)\,,$
(C.13)

with the corresponding quiver of the theory shown in Figure 11 (a).

Figure 11: Chain of duality transformations of three-punctured sphere theory
in $C_{2}$ case. We start with the theory derived by gluing two three-
punctured sphere theories ${\mathcal{T}}^{AC}$ along $A$-type puncture,
integrating out $A_{2}$ node and closing $\mathrm{SU}(2)_{\tilde{z}}$ minimal
puncture. This theory is shown on the Figure (a) above and corresponding index
is given by (C.13). Then performing Seiberg duality on the right node of this
theory we obtain the quiver shown on the Figure (b). Finally we notice that
right $\mathrm{SU}(4)$ gauge node of theory (b) is confining due to SWV
duality. Integrating it out we obtain theory shown in the Figure (c).

At the next step we perform Seiberg duality on the $\mathrm{SU}(3)$ gauge node
of the theory above. This node has $9$ falvors so after dualising it we obtain
$\mathrm{SU}(6)$ node with the superconformal index of the resulting theory
given by

$\displaystyle
K_{(3;i,0)}^{C}(x,\tilde{x},z)=\kappa_{3}\kappa_{5}\oint\prod\limits_{m=1}^{3}\frac{dt_{m}}{2\pi
it_{m}}\prod\limits_{m=1}^{5}\frac{d\tilde{t}_{m}}{2\pi
i\tilde{t}_{m}}\prod\limits_{m\neq
j}^{4}\frac{1}{\Gamma_{e}\left(t_{m}/t_{j}\right)}\prod\limits_{m\neq
j}^{6}\frac{1}{\Gamma_{e}\left(\tilde{t}_{m}/\tilde{t}_{j}\right)}\times$
$\displaystyle\prod\limits_{m=1}^{4}\prod\limits_{j=1}^{6}\prod\limits_{k=1}^{2}\prod\limits_{l=2}^{10}\Gamma_{e}\left((pq)^{\frac{1}{8}}w^{8}y_{k}^{\pm
1}t_{m}\right)\Gamma_{e}\left((pq)^{\frac{1}{8}}w^{8}t_{m}z^{\pm
1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{i}^{-1}\tilde{x}_{k}^{\pm
1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{l}^{-1}\tilde{x}_{k}^{\pm
1}\right)\Gamma_{e}\left((pq)^{\frac{1}{4}}w^{\frac{16}{15}}\tilde{a}_{l}\tilde{t}_{j}\right)\Gamma_{e}\left((pq)^{\frac{1}{4}}w^{\frac{16}{3}}\tilde{x}_{k}^{\pm
1}\tilde{t}_{j}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{-\frac{3}{4}}w^{-\frac{304}{15}}\tilde{a}_{i}\tilde{t}_{j}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{8}}w^{-\frac{8}{3}}t_{m}^{-1}\tilde{t}_{j}^{-1}\right)\prod_{m<n}^{4}\Gamma_{e}\left((pq)^{\frac{3}{4}}w^{-16}t_{m}^{-1}t_{n}^{-1}\right)\,,$
(C.14)

and corresponding quiver is shown in Figure 11 (b). We see that
$\mathrm{SU}(4)$ gauge node of this theory appears to be S-confining due to
SWV duality. Hence using SWV index identity (C.7) we arrive to the gauge
theory with the following superconformal index:

$\displaystyle
K_{(3;i,0)}^{C}(x,\tilde{x},z)=\kappa_{5}\oint\prod\limits_{i=1}^{5}\frac{d\tilde{t}_{i}}{2\pi
i\tilde{t}_{i}}\prod\limits_{i\neq
j}^{6}\frac{1}{\Gamma_{e}\left(\tilde{t}_{i}/\tilde{t}_{j}\right)}\prod\limits_{i=1}^{6}\prod\limits_{k=1}^{2}\Gamma_{e}\left((pq)^{\frac{1}{4}}w^{\frac{16}{3}}x_{k}^{\pm
1}\tilde{t}_{i}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{4}}w^{\frac{16}{3}}z^{\pm
1}\tilde{t}_{i}^{-1}\right)\Gamma_{e}\left((pq)^{\frac{1}{4}}w^{\frac{16}{3}}\tilde{x}_{k}^{\pm
1}\tilde{t}_{i}^{-1}\right)\Gamma_{e}\left((pq)^{-\frac{3}{4}}w^{-\frac{304}{15}}\tilde{a}_{1}\tilde{t}_{i}^{-1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{1}^{-1}\tilde{x}_{k}^{\pm
1}\right)\prod\limits_{l=2}^{10}\Gamma_{e}\left((pq)^{\frac{1}{4}}w^{\frac{16}{15}}\tilde{a}_{l}\tilde{t}_{i}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{l}^{-1}\tilde{x}_{k}^{\pm
1}\right)\times$
$\displaystyle\prod\limits_{i<j}^{6}\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{3}}\tilde{t}_{i}\tilde{t}_{j}\right)\,,$
(C.15)

with the quiver shown in Figure 11 (c). Now we are ready to close second
minimal puncture leaving us with the sphere with only two maximal
$\mathrm{USp}(2N)$ punctures left. As discussed in the Section 3 for this
purpose we fix $z$ fugacity to be

$\displaystyle
z=Z^{*}_{i;K,0}=(pq)^{-\frac{1}{2}}w^{-\frac{32}{5}}a_{i}^{-1}q^{-K}\,.$
(C.16)

Carefully studying expression (C.15) we see that at this value of $z$
superconformal index indeed has pole coming from the contour pinching at:

$\displaystyle\tilde{t}_{1,2}=(pq)^{\frac{1}{4}}w^{\frac{16}{3}}y_{1}^{\pm
1}q^{k_{1,\pm}}\,,\quad\tilde{t}_{3,4}=(pq)^{\frac{1}{4}}w^{\frac{16}{3}}y_{1}^{\pm
1}q^{k_{2,\pm}}\,,$
$\displaystyle\tilde{t}_{5}=(pq)^{-\frac{1}{4}}w^{-\frac{16}{5}}a_{i}^{-1}q^{k_{5}-K}\,,\quad\tilde{t}_{6}=(pq)^{-\frac{3}{4}}w^{-\frac{304}{15}}a_{i}q^{k_{6}}\,,$
(C.17)

where $\\{k_{1,\pm},\,k_{2,\pm},\,k_{5},\,k_{6}\\}\equiv\vec{K}$ is partition
of the integer $K$. Of course any permutation of (C.17) would also result in
the same contour pinching and hence the pole of the superconformal index. Due
to the Weyl group symmetry contributions of all these permutations into our
operator is the same and taking them into account just results in an overall
factor which we omit. Substituting values (C.17) into three punctured sphere
index (C.15) we get the following result for the index of the two-punctured
sphere:

$\displaystyle
K_{(2;i;K,0)}(x,\tilde{x})=\sum\limits_{\vec{K}}C_{(\vec{K};i)}\prod\limits_{j=1}^{2}\prod\limits_{l\neq
i}^{10}\frac{\Gamma_{e}\left(x_{j}^{\pm
2}q^{-k_{j,\mp}}\right)}{\Gamma_{e}\left(x_{j}^{\pm
2}q^{\pm(k_{j,+}-k_{j,-})}\right)}\times$
$\displaystyle\frac{\Gamma_{e}\left(x_{1}x_{2}^{\pm
1}q^{-k_{2,\mp}}\right)\Gamma_{e}\left(x_{1}^{-1}x_{2}^{\pm
1}q^{-k_{2,\mp}}\right)}{\Gamma_{e}\left(\left(x_{1}x_{2}^{\pm
1}q^{\left(k_{1,+}-k_{2,\mp}\right)}\right)^{\pm
1}\right)}\frac{\Gamma_{e}\left(x_{1}^{\pm
1}x_{2}q^{-k_{1,\mp}}\right)\Gamma_{e}\left(x_{1}^{\pm
1}x_{2}^{-1}q^{-k_{1,\mp}}\right)}{\Gamma_{e}\left(\left(x_{1}^{-1}x_{2}^{\pm
1}q^{\left(k_{1,-}-k_{2,\mp}\right)}\right)^{\pm 1}\right)}\times$
$\displaystyle\frac{\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{i}x_{j}^{\pm
1}q^{K-k_{j,\mp}}\right)}{\Gamma_{e}\left(\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{i}x_{j}^{\pm
1}q^{K+k_{j,\pm}-k_{5}}\right)^{\pm
1}\right)}\frac{\Gamma_{e}\left(p^{-1}q^{-1}w^{-\frac{128}{5}}\tilde{a}_{i}x_{j}^{\pm
1}q^{-k_{j,\mp}}\right)}{\Gamma_{e}\left(\left(pqw^{\frac{128}{5}}\tilde{a}_{i}^{-1}x_{j}^{\pm
1}q^{k_{j,\pm}-k_{6}}\right)^{\pm 1}\right)}\times$
$\displaystyle\Gamma_{e}\left((pq)^{-\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{i}^{-1}x_{j}^{\pm
1}q^{-K-k_{j,\mp}}\right)\Gamma_{e}\left(w^{-\frac{128}{5}}\tilde{a}_{i}x_{j}^{\pm
1}q^{k_{j,\pm}+k_{6}}\right)\Gamma_{e}\left(pqx_{1}x_{2}^{\pm
1}q^{k_{1,+}+k_{2,\pm}}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{i}^{-1}x_{j}^{\pm
1}q^{k_{j,\pm}+k_{5}-K}\right)\Gamma_{e}\left(pqw^{\frac{128}{5}}\tilde{a}_{i}^{-1}\tilde{x}_{j}^{\pm
1}q^{-k_{6}}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{l}x_{j}^{\pm
1}q^{k_{j,\pm}}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}x_{j}^{\pm
1}\tilde{a}_{i}q^{K-k_{5}}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{i}\tilde{x}_{j}^{\pm
1}q^{K-k_{5}}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{l}^{-1}\tilde{x}_{j}^{\pm
1}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{i}^{-1}\tilde{x}_{j}^{\pm
1}\right)\prod\limits_{k=1}^{2}\Gamma_{e}\left(\tilde{x}_{k}^{\pm
1}x_{j}q^{-k_{j,-}}\right)\Gamma_{e}\left(\tilde{x}_{k}^{\pm
1}x_{j}^{-1}q^{-k_{j,+}}\right)\Gamma_{e}\left(pq^{1+k_{1,+}+k_{1,-}}\right)\times$
$\displaystyle\hskip
56.9055pt\Gamma_{e}\left(pqw^{\frac{128}{5}}\tilde{a}_{i}^{-1}x_{j}^{\pm
1}q^{-k_{6}}\right)\Gamma_{e}\left(pqx_{1}^{-1}x_{2}^{\pm
1}q^{k_{1,-}+k_{2,\pm}}\right)\Gamma_{e}\left(pq^{1+k_{2,+}+k_{2,-}}\right)\,,$
(C.18)

where $C_{(\vec{K};i)}$ is the following constant:

$\displaystyle C_{(\vec{K};i)}$ $\displaystyle=\prod\limits_{l\neq
i}^{10}\Gamma_{e}\left((pq)^{\frac{3}{2}}w^{32}q^{K-k_{6}}\right)\Gamma_{e}\left(\tilde{a}_{l}\tilde{a}_{i}^{-1}q^{k_{5}-K}\right)\Gamma_{e}\left((pq)^{-\frac{1}{2}}w^{-\frac{96}{5}}\tilde{a}_{l}\tilde{a}_{i}q^{k_{6}}\right)\times$
$\displaystyle\frac{\Gamma_{e}\left((pq)^{-\frac{1}{2}}w^{-\frac{96}{5}}\tilde{a}_{i}^{2}q^{K-k_{5}}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{\frac{96}{5}}\tilde{a}_{i}^{-2}q^{-K-k_{6}}\right)}{\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{\frac{96}{5}}\tilde{a}_{i}^{-2}q^{-K+k_{5}-k_{6}}\right)\Gamma_{e}\left((pq)^{-\frac{1}{2}}w^{-\frac{96}{5}}\tilde{a}_{i}^{2}q^{K+k_{6}-k_{5}}\right)}\times$
$\displaystyle\hskip
56.9055pt\Gamma_{e}\left((pq)^{-\frac{1}{2}}w^{-32}q^{k_{5}+k_{6}-K}\right)\Gamma_{e}\left(pqw^{\frac{64}{5}}\tilde{a}_{i}^{2}q^{2K-k_{5}}\right)\,,$
(C.19)

here subscript contains all the information on how the tube was obtained.
Inside this subscript $2$ means that we obtain two-punctured sphere (the tube)
in the end. Index $i$ stands for the choice of the moment map we give vev to
according to (3.8) and (3.10). Finally $K$ denotes the power of holomorphic
derivative of the moment map we give vev to. Summation in (C.18) goes over all
possible partitions of $K$ integer. Due to the presence of the defect this
index expression unlike our previous four- and three-punctured sphere does not
have nice gauge theory interpretation.

Now gluing tube (C.18) to an arbitrary $\mathcal{N}=1$ theory with one maximal
$\mathrm{USp}(2N)$ puncture we can obtain desired A$\Delta$O. At the level of
the superconformal index the gluing is performed as specified in (3.13):

$\displaystyle{\mathcal{O}}^{(C_{2};h_{k};K,0)}_{x}\cdot{\mathcal{I}}(x)=\frac{\kappa_{N}}{2^{N}}\oint\frac{d\tilde{x}_{1,2}}{2\pi
i\tilde{x}_{1,2}}\frac{1}{\Gamma_{e}\left(\tilde{x}_{1,2}^{\pm
2}\right)\Gamma_{e}\left(\tilde{x}_{1}^{\pm 1}\tilde{x}_{2}^{\pm
1}\right)}K_{(2;i;K,0)}^{C}(x,\,\tilde{x}){\mathcal{I}}(\tilde{x})$ (C.20)

Notice that expression (C.18) for the tube index has zeroes coming from the
last two $\Gamma$-functions. Due to these zeroes only the poles of the
integral in (C.20) coming from other $\Gamma$ functions will contribute to the
final expression. In particular these poles are coming from the
$\Gamma_{e}\left(\tilde{x}_{k}^{\pm 1}x_{j}q^{-k_{j,-}}\right)$ and
$\Gamma_{e}\left(\tilde{x}_{k}^{\pm 1}x_{j}^{-1}q^{-k_{j,+}}\right)$ terms and
are located at

$\displaystyle\tilde{x}_{i}=\left(x_{\sigma(i)}q^{-m_{i}}\right)^{\pm
1}\,,\qquad-k_{\sigma(i),+}\leq m_{i}\leq k_{\sigma(i),-}\,.$ (C.21)

where $\sigma(i)$ is permutation of $i$. In the end we should sum over such
permutations as well as over all combinations of $\pm 1$ powers in the
expression above. Since the tube expression (C.18) is symmetric w.r.t.
permutations of $\tilde{x}_{1}\leftrightarrow\tilde{x}_{2}$ and
$\tilde{x}_{i}\to\tilde{x}_{i}^{-1}$ every contribution of the pinchings
specified above is the same and summation just gives an irrelevant overall
factor. So for simplicity we can consider only one of the combinations

$\displaystyle\tilde{x}_{i}=x_{i}q^{-m_{i}}\,,\qquad-k_{i,+}\leq m_{i}\leq
k_{i,-}\,.$ (C.22)

The condition on $m_{i}$ comes from the fact that both types of specified
$\Gamma$ functions should have poles. Half of these terms should lead to
integration contour pinching in (C.20) while other half will cancel zeroes
coming from the last two $\Gamma$ functions in the tube index expression
(C.18).

Now computing the contribution of pinchings (C.22) into gluing (C.20) and
using $\Gamma$ functions identity (A.9) we can directly obtain the full tower
of operators (3.14).

##  Appendix D Proof of the kernel property of $A_{2}C_{2}$ tube theory

Here we give a proof of the kernel property (4.2) where we act with $A_{2}$
operator (2.13) and $C_{2}$ operator (3.17) on the index (C.5) of the tube
with $\mathrm{SU}(3)$ and $\mathrm{USp}(4)$ maximal punctures.

Let’s start the proof of the kernel property by explicitly calculating action
of our A$\Delta$Os on the tube index and write two sides of the equation in
terms of algebraic expressions. First of all we should fix the parametrization
of the Cartans of global $6d$ symmetry. For this it will be more convenient to
use $C_{2}$ parametrization given in (3.2). We should also use the dictionary
(3.4) between $A_{2}$ and $C_{2}$ parametrizations in order to rewrite
everything in terms of the latter one. Fixing $N=2$ in the tube index (C.5)
and writing everything in terms of $C_{2}$ parametrization we obtain:

$\displaystyle
K_{2}^{A_{2}C_{2}}(x,y)=\prod\limits_{i=1}^{3}\prod\limits_{j=1}^{2}\prod\limits_{l=1}^{9}\Gamma_{e}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}y_{i}^{-1}x_{j}^{\pm
1}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{1}{3}}\tilde{a}_{l}y_{i}\right)\times$
$\displaystyle\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{-\frac{224}{15}}\tilde{a}_{10}^{-\frac{2}{3}}y_{i}\right)\Gamma_{e}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{10}x_{j}^{\pm
1}\right)\Gamma_{e}\left(pqw^{-\frac{256}{15}}\tilde{a}_{10}^{\frac{2}{3}}y_{i}^{-1}\right)\,,$
(D.1)

where as previously $x_{1,2}$ are fugacities for the global symmetry of the
$\mathrm{USp}(4)$ puncture, while $y_{1,2,3}$ are fugacities of the
$\mathrm{SU}(3)$ puncture. Latter ones are as usually constrained by the
identity $\prod_{i=1}^{3}y_{i}=1$.

Now we can start with the r.h.s. of (4.2) describing the action of $C_{2}$
operator. Main ingredient we need to find the action of the full operator is
the action of the shift $\Delta_{q}(x_{i})$ on the tube index. Studying this
action on (D.1) we obtain:

$\displaystyle\Delta_{q}(x_{l})K_{2}^{A_{2}C_{2}}(x,y)=D_{l}(x,y)K_{2}^{A_{2}C_{2}}(x,y)\,,$
(D.2)

where the function $D_{l}(x,y)$ is given by:

$\displaystyle
D_{l}(x,y)=\prod\limits_{i=1}^{3}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{a}_{10}w^{\frac{32}{5}}x_{l}\right)}{\theta_{p}\left((pq)^{\frac{1}{2}}q^{-1}\tilde{a}_{10}w^{\frac{32}{5}}x_{l}^{-1}\right)}\frac{\theta_{p}\left(\tilde{a}_{10}^{-\frac{1}{3}}w^{\frac{128}{15}}x_{l}y_{i}^{-1}\right)}{\theta_{p}\left(q^{-1}\tilde{a}_{10}^{-\frac{1}{3}}w^{\frac{128}{15}}x_{l}^{-1}y_{i}^{-1}\right)}$
(D.3)

Then on the r.h.s. of (4.2) we obtain:

$\displaystyle{\mathcal{O}}^{(C_{2};h_{10};1,0)}_{x}K_{2}^{A_{2}C_{2}}(x,y)=\left[F^{(C_{2};h_{10};1,0)}(x,y)+W^{(C_{2};h_{10};1,0)}(x)\right]K_{2}^{A_{2}C_{2}}(x,y)$
$\displaystyle
F^{(C_{2};h_{10};1,0)}(x,y)\equiv\sum_{i=1}^{2}\left(A^{(C_{2};h_{10};1,0)}_{i}(x)D_{i}(x,y)+\left(x_{i}\to
x_{i}^{-1}\right)\right)=$
$\displaystyle\sum_{i=1}^{2}\left(\prod\limits_{j\neq
i}^{2}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{10}x_{j}^{\pm
1}\right)}{\theta_{p}\left(x_{i}\right)\theta_{p}\left(qx_{i}^{2}\right)\theta_{p}\left(x_{i}x_{j}^{\pm
1}\right)}\prod\limits_{k=1}^{10}\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{5}}\tilde{a}_{k}^{-1}x_{i}\right)\right.\times$
$\displaystyle\left.\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{10}x_{i}\right)}{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{10}q^{-1}x_{i}^{-1}\right)}\prod\limits_{l=1}^{3}\frac{\theta_{p}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}y_{l}^{-1}x_{i}\right)}{\theta_{p}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}y_{l}^{-1}q^{-1}x_{i}^{-1}\right)}+\left(x_{i}\to
x_{i}^{-1}\right)\right)\,.$ (D.4)

Now we consider l.h.s. of the equation above. Similarly we need to understand
action of $\Delta_{lm}(y)$ operator on the $A_{2}C_{2}$ tube (4.2):

$\displaystyle\Delta_{lm}(y)K_{2}^{A_{2}C_{2}}(x,y)=D_{lm}(x,y)K_{2}^{A_{2}C_{2}}(x,y)\,,$
(D.5)

where

$\displaystyle
D_{lm}(x,y)=\prod\limits_{j=1}^{2}\prod\limits_{k=1}^{9}\frac{\theta_{p}\left(\tilde{a}_{10}^{-\frac{1}{3}}w^{\frac{128}{15}}x_{j}^{\pm
1}y_{l}^{-1}\right)}{\theta_{p}\left(q^{-1}\tilde{a}_{10}^{-\frac{1}{3}}w^{\frac{128}{15}}x_{j}^{\pm
1}y_{m}^{-1}\right)}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{a}_{10}^{\frac{1}{3}}w^{-\frac{32}{15}}\tilde{a}_{k}y_{m}\right)}{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{a}_{10}^{\frac{1}{3}}w^{-\frac{32}{15}}\tilde{a}_{k}q^{-1}y_{l}\right)}\times$
$\displaystyle\frac{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{a}_{10}^{-\frac{2}{3}}w^{-\frac{224}{15}}y_{m}\right)}{\theta_{p}\left((pq)^{\frac{1}{2}}\tilde{a}_{10}^{-\frac{2}{3}}w^{-\frac{224}{15}}q^{-1}y_{l}\right)}\frac{\theta_{p}\left(\tilde{a}_{10}^{-\frac{2}{3}}w^{\frac{256}{15}}q^{-1}y_{l}\right)}{\theta_{p}\left(\tilde{a}_{10}^{-\frac{2}{3}}w^{\frac{256}{15}}y_{m}\right)}$
(D.6)

So the l.h.s. of (4.2) takes the following form:

$\displaystyle{\mathcal{O}}^{(A_{2};\tilde{h}_{10}^{-1};1,0)}_{y}K_{2}^{A_{2}C_{2}}(x,y)=\left[F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)+W^{(A_{2};\tilde{h}_{10}^{-1};1,0)}(y)\right]K_{2}^{A_{2}C_{2}}(x,y)\,,$
$\displaystyle F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)\equiv\sum\limits_{l\neq
m}^{3}D_{lm}(x,y)A^{(A_{2};\tilde{h}_{10};1,0)}_{lm}(y)=$
$\displaystyle\sum\limits_{l\neq
m}^{3}\prod\limits_{k=1}^{9}\prod\limits_{j=1}^{2}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{1}{3}}\tilde{a}_{k}y_{m}\right)}{\theta_{p}\left(q\frac{y_{m}}{y_{l}}\right)\theta_{p}\left(\frac{y_{m}}{y_{l}}\right)}\frac{\theta_{p}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}y_{l}^{-1}x_{j}^{\pm
1}\right)\theta_{p}\left(w^{\frac{256}{15}}\tilde{a}_{10}^{-\frac{2}{3}}q^{-1}y_{l}\right)}{\theta_{p}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}y_{m}^{-1}q^{-1}x_{j}^{\pm
1}\right)\theta_{p}\left(w^{\frac{256}{15}}\tilde{a}_{10}^{-\frac{2}{3}}y_{m}\right)}\times$
$\displaystyle\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{224}{15}}\tilde{a}_{10}^{-\frac{2}{3}}y_{m}\right)\prod\limits_{i\neq
m\neq
l}^{3}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{224}{15}}\tilde{a}_{10}^{\frac{2}{3}}y_{i}^{-1}\right)\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{4}{3}}y_{i}\right)}{\theta_{p}\left(\frac{y_{i}}{y_{l}}\right)\theta_{p}\left(\frac{y_{m}}{y_{i}}\right)}\,.$
(D.7)

Hence kernel property (4.2) can be reduced to the following algebraic
identity:

$\displaystyle
F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)+W^{(A_{2};\tilde{h}_{10}^{-1};1,0)}(y)=F^{(C_{2};h_{10};1,0)}(x,y)+W^{(C_{2};h_{10};1,0)}(x)$
(D.8)

It can be checked that both $F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)$ and
$F^{(C_{2};h_{10};1,0)}(x,y)$ functions defined above are elliptic w.r.t both
$x_{i}$ and $y_{i}$ variables with periods $1$ and $p$ 444As usually when we
discuss ellipticity w.r.t. $A_{2}$ variables we should keep in mind the
constraint so when we shift one of the variables, say $y_{l}\to py_{l}$ we are
forced to also make another shift $y_{m}\to p^{-1}y_{m}$ for some other
variable. . In order to prove the identity we now need to check poles and
residues in the fundamental domain on two sides of equation 555 We have
complicated functions of many variables. Hence when we speak about poles and
corresponding residues we always consider our functions as functions of one
chosen variable while all other variables are kept fixed. The poles and
residues should be checked in this way for all variables of functions one by
one.. We already know the poles and residues of the constant parts
$W^{(A_{2};\tilde{h}_{10}^{-1};1,0)}(y)$ and $W^{(C_{2};h_{10};1,0)}(x)$
summarized in sections 2 and 3 correspondingly. Now let’s study functions
$F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)$ and $F^{(C_{2};h_{10};1,0)}(x,y)$ coming
from the action of the shift parts.

We start with $F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)$ function given in (D.7).
This elliptic function appears to have poles at the following positions:

$\displaystyle
y_{a}=w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}q^{-1}x_{b}^{s}\,,\quad
y_{a}=qy_{b}\,,\quad y_{a}=q^{-1}y_{b}\,,\quad
y_{a}=sq^{\frac{1}{2}}P_{a}^{-\frac{1}{2}}\,,\quad
y_{a}=sq^{-\frac{1}{2}}P_{a}^{-\frac{1}{2}}\,,$ $\displaystyle
y_{a}=sq^{\frac{1}{2}}p^{\frac{1}{2}}P_{a}^{-\frac{1}{2}}\,,\quad
y_{a}=sq^{-\frac{1}{2}}p^{\frac{1}{2}}P_{a}^{-\frac{1}{2}}\,,\quad
x_{j}=\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}q^{-1}y_{a}^{-1}\right)^{\pm
1}\,,$ (D.9)

where $s=\pm$ and $P_{a}\equiv\prod\limits_{j\neq a}^{3}y_{j}$. Seemingly
there are more poles in expression (D.7) but checking all residues shows that
the only actual poles are the ones listed above.

Residues of poles (D.20) are given by

$\displaystyle\mathrm{Res}_{y_{a}=w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}q^{-1}x_{b}^{s}}F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)=\frac{w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}q^{-1}x_{b}^{s}}{\left(p;p\right)_{\infty}^{2}}\sum\limits_{l\neq
a}^{3}\prod\limits_{k=1}^{9}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{32}{5}}\tilde{a}_{k}q^{-1}x_{b}^{s}\right)}{\theta_{p}\left(w^{\frac{128}{5}}q_{10}^{-1}q^{-1}x_{b}^{s}\right)}\times$
$\displaystyle\frac{\theta_{p}\left(w^{\frac{256}{15}}\tilde{a}_{10}^{-\frac{2}{3}}q^{-1}y_{l}\right)}{\theta_{p}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}q^{-1}y_{l}^{-1}x_{b}^{s}\right)}\prod\limits_{i\neq
l\neq
a}^{3}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{224}{15}}\tilde{a}_{10}^{\frac{2}{3}}y_{i}^{-1}\right)\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{4}{3}}y_{i}\right)}{\theta_{p}\left(\frac{y_{i}}{y_{l}}\right)\theta_{p}\left(w^{\frac{128}{15}}q^{-1}\tilde{a}_{10}^{-\frac{1}{3}}y_{i}^{-1}x_{b}^{s}\right)}\times$
$\displaystyle\prod\limits_{j\neq
b}^{2}\frac{\theta_{p}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}y_{l}^{-1}x_{j}^{\pm
1}\right)\theta_{p}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}y_{l}^{-1}x_{b}^{-s}\right)}{\theta_{p}\left(x_{b}^{-s}x_{j}^{\pm
1}\right)\theta_{p}\left(x_{b}^{-2s}\right)}\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{-1}q^{-1}x_{b}^{s}\right)\,,$
(D.10)
$\displaystyle\mathrm{Res}_{x_{j}=\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}q^{-1}y_{a}^{-1}\right)^{s}}F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)=s\frac{\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}q^{-1}y_{a}^{-1}\right)^{s}}{\left(p;p\right)_{\infty}^{2}}\times$
$\displaystyle\sum_{l\neq a}^{3}\prod\limits_{k=1}^{9}\prod\limits_{i\neq
j}^{2}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{1}{3}}\tilde{a}_{k}y_{a}\right)}{\theta_{p}\left(q\frac{y_{a}}{y_{l}}\right)\theta_{p}\left(\frac{y_{a}}{y_{l}}\right)}\frac{\theta_{p}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}y_{l}^{-1}x_{i}^{\pm
1}\right)}{\theta_{p}\left(w^{\frac{128}{15}}\tilde{a}_{10}^{-\frac{1}{3}}y_{a}^{-1}q^{-1}x_{i}^{\pm
1}\right)}\times$
$\displaystyle\frac{\theta_{p}\left(w^{\frac{256}{15}}\tilde{a}_{10}^{-\frac{2}{3}}q^{-1}y_{l}^{-1}y_{a}^{-1}\right)\theta_{p}\left(w^{\frac{256}{15}}\tilde{a}_{10}^{-\frac{2}{3}}q^{-1}y_{l}\right)}{\theta_{p}\left(w^{\frac{256}{15}}\tilde{a}_{10}^{-\frac{2}{3}}q^{-2}y_{a}^{-2}\right)\theta_{p}\left(w^{\frac{256}{15}}\tilde{a}_{10}^{-\frac{2}{3}}y_{a}\right)}\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{224}{15}}\tilde{a}_{10}^{-\frac{2}{3}}y_{a}\right)\times$
$\displaystyle\theta_{p}\left(qy_{l}^{-1}y_{a}\right)\prod\limits_{k\neq a\neq
l}^{3}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{224}{15}}\tilde{a}_{10}^{\frac{2}{3}}y_{k}^{-1}\right)\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{4}{3}}y_{k}\right)}{\theta_{p}\left(\frac{y_{k}}{y_{l}}\right)\theta_{p}\left(\frac{y_{a}}{y_{k}}\right)}$
(D.11)
$\displaystyle\mathrm{Res}_{y_{a}=qy_{b}}F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)=\frac{qy_{b}}{\theta_{p}\left(q^{-1}\right)\left(p;p\right)_{\infty}^{2}}\prod\limits_{k=1}^{9}\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{1}{3}}\tilde{a}_{k}y_{b}\right)\times$
$\displaystyle\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{224}{15}}\tilde{a}_{10}^{-\frac{2}{3}}y_{b}\right)\prod\limits_{j\neq
a\neq
b}^{3}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{224}{15}}\tilde{a}_{10}^{\frac{2}{3}}y_{j}^{-1}\right)\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{4}{3}}y_{j}\right)}{\theta_{p}\left(q^{-1}\frac{y_{j}}{y_{b}}\right)\theta_{p}\left(\frac{y_{b}}{y_{j}}\right)}\,,$
(D.12)
$\displaystyle\mathrm{Res}_{y_{a}=q^{-1}y_{b}}F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)=-\frac{q^{-1}y_{b}}{\theta_{p}\left(q^{-1}\right)\left(p;p\right)_{\infty}^{2}}\prod\limits_{k=1}^{9}\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{32}{15}}\tilde{a}_{10}^{-\frac{1}{3}}\tilde{a}_{k}^{-1}y_{b}^{-1}\right)\times$
$\displaystyle\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{224}{15}}\tilde{a}_{10}^{\frac{2}{3}}y_{b}^{-1}\right)\prod\limits_{j\neq
a\neq
b}^{3}\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{224}{15}}\tilde{a}_{10}^{\frac{2}{3}}y_{j}^{-1}\right)\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{4}{3}}y_{j}\right)}{\theta_{p}\left(\frac{y_{j}}{y_{b}}\right)\theta_{p}\left(q^{-1}\frac{y_{b}}{y_{j}}\right)}\,,$
(D.13)
$\displaystyle\mathrm{Res}_{y_{a}=sq^{\frac{1}{2}}P_{a}^{-\frac{1}{2}}}F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)=s\frac{q^{\frac{1}{2}}P_{a}^{-\frac{1}{2}}}{2\theta_{p}\left(q^{-1}\right)\left(p;p\right)_{\infty}^{2}}\prod\limits_{k=1}^{9}\theta_{p}\left(sp^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{1}{3}}\tilde{a}_{k}P_{a}^{-\frac{1}{2}}\right)\times$
$\displaystyle\theta_{p}\left(sp^{\frac{1}{2}}w^{-\frac{224}{15}}\tilde{a}_{10}^{-\frac{2}{3}}P_{a}^{-\frac{1}{2}}\right)\frac{\theta_{p}\left((pq)^{\frac{1}{2}}w^{\frac{224}{15}}\tilde{a}_{10}^{\frac{2}{3}}P_{a}^{-1}\right)\theta_{p}\left((pq)^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{4}{3}}P_{a}\right)}{\theta_{p}\left(sq^{-\frac{1}{2}}P_{a}^{\frac{3}{2}}\right)\theta_{p}\left(sq^{-\frac{1}{2}}P_{a}^{-\frac{3}{2}}\right)}\,,$
(D.14)
$\displaystyle\mathrm{Res}_{y_{a}=sq^{-\frac{1}{2}}P_{a}^{-\frac{1}{2}}}F^{(A_{2};\tilde{h}_{10};1,0)}(x,y)=-s\frac{q^{-\frac{1}{2}}P_{a}^{-\frac{1}{2}}}{2\theta_{p}\left(q^{-1}\right)\left(p;p\right)_{\infty}^{2}}\prod\limits_{k=1}^{9}\theta_{p}\left(sp^{\frac{1}{2}}w^{-\frac{32}{15}}\tilde{a}_{10}^{\frac{1}{3}}\tilde{a}_{k}P_{a}^{-\frac{1}{2}}\right)\times$