YITP-20-164

Argyres-Douglas Theories, S-duality and
AGT Correspondence

Takuya Kimura,♢,1 Takahiro Nishinaka,♣,1 Yuji Sugawara,♡,1 and Takahiro
Uetoko♠,2

1 Department of Physical Sciences, College of Science and Engineering
Ritsumeikan University, Shiga 525-8577, Japan

2 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics
Kyoto University, Kyoto 606-8502, Japan

We propose a Nekrasov-type formula for the instanton partition functions of
four-dimensional $\mathcal{N}=2\;\,U(2)$ gauge theories coupled to
$(A_{1},D_{2n})$ Argyres-Douglas theories. This is carried out by extending
the generalized AGT correspondence to the case of $U(2)$ gauge group, which
requires us to define irregular states of the direct sum of Virasoro and
Heisenberg algebras. Using our formula, one can evaluate the contribution of
the $(A_{1},D_{2n})$ theory at each fixed point on the $U(2)$ instanton moduli
space. As an application, we evaluate the instanton partition function of the
$(A_{3},A_{3})$ theory to find it in a peculiar relation to that of $SU(2)$
gauge theory with four fundamental flavors. From this relation, we read off
how the S-duality group acts on the UV gauge coupling of the $(A_{3},A_{3})$
theory.

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###### Contents

  1. 1 Introduction
  2. 2 Generalized AGT correspondence
  3. 3 Irregular states in $Vir\oplus H$ modules
    1. 3.1 $U(2)$ version of the original AGT
    2. 3.2 $U(2)$ version of the generalized AGT
  4. 4 Nekrasov-type formula for AD matter
    1. 4.1 Decomposition
    2. 4.2 Identification of $\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$
    3. 4.3 Identification of $\Lambda$
    4. 4.4 Consistency check
  5. 5 Application to $(A_{3},A_{3})$ theory
    1. 5.1 Prepotential
    2. 5.2 S-duality
    3. 5.3 Peculiarity of $T$
    4. 5.4 Turning on couplings, masses and VEVs
  6. 6 Conclusions and discussions
  7. A Nekrasov’s formula
  8. B Orthogonal Basis
  9. C Prepotential with massive deformations

##  1 Introduction

The Nekrasov partition function of four-dimensional $\mathcal{N}=2$ gauge
theories with Lagrangian description can often be exactly evaluated by
supersymmetric localization [1, 2]. The result is written as a sum over fixed
points of a torus action on the instanton moduli space. For instance, for
$U(2)$ gauge theory with $N_{f}$ fundamental hypermultiplets, the partition
function is evaluated as

$\displaystyle\mathcal{Z}_{U(2)}^{N_{f}}=\mathcal{Z}_{\text{pert}}\sum_{Y_{1},Y_{2}}\Lambda^{b_{0}(|Y_{1}|+|Y_{2}|)}\mathcal{Z}^{\text{vec}}_{Y_{1},Y_{2}}(a)\prod_{i=1}^{N_{f}}\mathcal{Z}^{\text{fund}}_{Y_{1},Y_{2}}(a,m_{i})~{},$
(1.1)

where $b_{0}\equiv 4-N_{f}$, $\Lambda$ is a dynamical scale, $a$ is the vacuum
expectation value (VEV) of a scalar, $Y_{k}$ are Young diagrams, and $|Y_{k}|$
is the number of boxes in $Y_{k}$. The sum over $(Y_{1},Y_{2})$ can be
regarded as a sum over fixed points on the moduli space of $U(2)$ instantons,
and $\mathcal{Z}^{\text{vec}}_{Y_{1},Y_{2}}$ and
$\mathcal{Z}^{\text{fund}}_{Y_{1},Y_{2}}$ are respectively the contributions
of the gauge and matter sectors at these fixed points. The prefactor,
$\mathcal{Z}_{\text{pert}}$, is the perturbative contribution that makes the
power series in $\Lambda$ start with $1$.

$(A_{1},D_{4})$ $2$ $(A_{1},D_{4})$ 1  Figure 1: The quiver diagram of the
$(A_{3},A_{3})$ AD theory. The left and right boxes stand for two copies of
$(A_{1},D_{4})$ theory while the top box stands for a fundamental
hypermultiplet of $SU(2)$. The middle circle stands for $SU(2)$ gauge group
coupled to these “matters.”

Despite the above success for Lagrangian theories, there is a rich class of
four-dimensional $\mathcal{N}=2$ gauge theories whose partition functions are
still to be evaluated. These theories involve strongly-coupled CFTs in their
matter sector, and therefore their partition functions cannot be directly
evaluated by supersymmetric localization. Among other theories, conformal
gauge theories coupled to Argyres-Douglas (AD) theories are of particular
importance in this class, since they provide a new class of $\mathcal{N}=2$
S-dualities [3, 4, 5, 6, 7, 8, 9, 10]. We call such conformal gauge theories
“conformally gauged AD theories.” One of the simplest examples of such
theories is the $(A_{3},A_{3})$ theory described by the quiver diagram in Fig.
1, where the $(A_{1},D_{4})$ theory is a particular AD theory.111This AD
theory is also called $H_{2}$ theory, $D_{2}(SU(3))$ theory, and
$(A_{2},A_{2})$ theory in the literature. The first series of papers on AD
theories are [11, 12, 13]. The beta function of the gauge coupling vanishes
here since the contributions of the gauge and matter sectors are exactly
canceled.

While the partition functions of conformally gauged AD theories have not been
evaluated, there exists a series of non-conformally gauged AD theories whose
partition functions were evaluated via a generalization [14, 15] of the AGT
correspondence [16, 17] (See [18, 19, 20, 21, 22, 23, 24, 25, 26, 27] for
recent developments on this generalization). In particular, for the theory
described by the quiver in Fig. 2, the partition function was evaluated as the
inner product of so-called “irregular states” of Virasoro algebra. The
application of the generalized AGT correspondence is possible here since these
theories can be engineered by compactifying the 6d $(2,0)$ $A_{1}$ theory on a
Riemann surface.

$(A_{1},D_{2n})$ $2$ $(A_{1},D_{2n})$ Figure 2: A typical example of gauge
theories studied in the generalized AGT correspondence, where $n$ is a
positive integer. The circle stands for $SU(2)$ gauge group, and each box
stands for an $(A_{1},D_{2n})$ theory. The $SU(2)$ vector multiplet is
diagonally gauging an $SU(2)$ sub-group of the $(A_{1},D_{2n})$ theories.

The purpose of this paper is to propose a way to compute the partition
function of the conformally gauged AD theory in Fig. 1, using the generalized
AGT correspondence for the non-conformally gauged AD theory in Fig. 2. Since
the former has no known construction from the 6d (2,0) $A_{1}$ theory, one
cannot directly apply the AGT correspondence to it.222The theory shown in Fig.
1 can be constructed by compactifying the 6d (2,0) $A_{2}$ or $A_{3}$ theory
on a Riemann surface [28, 29]. This suggests a possibility of studying its
partition function via the higher-rank generalization of the AGT
correspondence [30]. Going in this direction would, however, be non-trivial
and involved since the 2d side is now a higher-rank Toda theory. In this
paper, we take a different route. Instead, our strategy is to apply the
generalized AGT correspondence to the latter (with a small but crucial
modification discussed below) and decompose the resulting partition function
as a sum over Young diagrams, as in (1.1). While the theory has a strongly-
coupled matter sector, such a decomposition is expected, since the gauge
sector of the theory is still described by Lagrangian and its path integral is
expected to lead to a sum over fixed points on the instanton moduli space.
Once such a decomposition is obtained, one can read off the contribution of
the $(A_{1},D_{2n})$ theory at each fixed point, say
$\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$. It is then straightforward to
compute the partition function of the conformally gauged AD theory shown in
Fig. 1.

One important point in the above discussion is that the decomposition of the
form (1.1) is for $U(2)$ gauge group while the generalized AGT correspondence
is for $SU(2)$ gauge group. Indeed, for fixed points on the instanton moduli
space to be labeled by $(Y_{1},Y_{2})$, the gauge group must be $U(2)$ instead
of $SU(2)$. Therefore, to decompose the resulting partition function as a sum
over $(Y_{1},Y_{2})$, we need a $U(2)$ version of the generalized AGT
correspondence. For the original AGT correspondence, such a $U(2)$ version is
realized by considering the direct sum of Virasoro and Heisenberg algebras
($Vir\oplus H$) on the two-dimensional side [31]. In this paper, we extend it
to the generalized AGT correspondence, by considering irregular states of
$Vir\oplus H$.

Given the $U(2)$ version of the generalized AGT correspondence, one can easily
decompose the partition function of $U(2)$ gauge theory coupled to two
$(A_{1},D_{2n})$ as

$\displaystyle\mathcal{Z}_{U(2)}^{2\times(A_{1},D_{2n})}=\mathcal{Z}_{\text{pert}}\sum_{Y_{1},Y_{2}}\Lambda^{b_{0}(|Y_{1}|+|Y_{2}|)}\mathcal{Z}^{\text{vec}}_{Y_{1},Y_{2}}(a)\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{2n})}(a,m,\boldsymbol{d},\boldsymbol{u})\widetilde{\mathcal{Z}}_{Y_{1},Y_{2}}^{(A_{1},D_{2n})}(a,\widetilde{m},\widetilde{\boldsymbol{d}},\widetilde{\boldsymbol{u}})~{},$
(1.2)

where $m,\boldsymbol{d}=(d_{1},\cdots,d_{n-1})$ and
$\boldsymbol{u}=(u_{1},\cdots,u_{n-1})$ are respectively the mass, relevant
couplings and VEVs of Coulomb branch operators of the $(A_{1},D_{2n})$ theory,
and $b_{0}\equiv 2/n$ is the coefficient of the one-loop $\beta$ function. We
interpret the sum over $(Y_{1},Y_{2})$ as a sum over fixed points on the
moduli space of $U(2)$ instantons, and identify
$\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ and
$\widetilde{\mathcal{Z}}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ as contributions from
the $(A_{1},D_{2n})$ theories at each fixed point. The difference between
$\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ and
$\widetilde{\mathcal{Z}}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ is interpreted as
coming from how the $U(1)$ part of the gauge group is coupled to
$(A_{1},D_{2n})$.

With the above identification of $\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$,
we evaluate the partition function of the $(A_{3},A_{3})$ theory as follows.
We start with the quiver description shown in Fig. 1. When the gauge group in
the quiver is $U(2)$, the partition function is evaluated as

$\displaystyle\mathcal{Z}_{U(2)}=\mathcal{Z}_{\text{pert}}\sum_{Y_{1},Y_{2}}q^{|Y_{1}|+|Y_{2}|}\mathcal{Z}^{\text{vec}}_{Y_{1},Y_{2}}(a)\mathcal{Z}^{\text{fund}}_{Y_{1},Y_{2}}(a,M)\prod_{i=1}^{2}\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{4})}(a,m_{i},d_{i},u_{i})~{},$
(1.3)

where $q$ is the exponential of the marginal gauge coupling, and $M$ is the
mass of the hypermultiplet. The prefactor $\mathcal{Z}_{\text{pert}}$ is again
the perturbative contribution that makes the power series in $q$ start with
$1$. Since $\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ is already read off
from the decomposition (1.2), one can explicitly compute the above partition
function. When the gauge group is $SU(2)$, the partition function differs from
(1.3) by the contribution of the $U(1)$-part of the gauge group. Indeed,
according to a general discussion in [16], the partition function for $SU(2)$
gauge group is expected to be given by

$\displaystyle\mathcal{Z}_{SU(2)}=\frac{\mathcal{Z}_{U(2)}}{\mathcal{Z}_{U(1)}}~{},$
(1.4)

where $\mathcal{Z}_{U(1)}$ is the partition function of the $U(1)$ part, and
called “$U(1)$-factor.” We use (1.3) and (1.4) to show in particular that the
S-duality of the $(A_{3},A_{3})$ theory is in a peculiar relation to that of
$SU(2)$ gauge theory with $4$ fundamental flavors. It is an interesting open
problem to see how this peculiar relation is connected to a similar relation
between the Schur index of the same pair of theories discussed in [32].

The organization of the rest of this paper is the following. In Sec. 2, we
briefly review the AGT correspondence and its generalization to AD theories.
In Sec. 3, we consider a $U(2)$-version of the generalized AGT correspondence,
in terms of irregular states of $Vir\oplus H$. In Sec. 4, we derive a formula
for $\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{2n})}$ corresponding to the gauged
$(A_{1},D_{2n})$ theory, using the $U(2)$-version of the generalized AGT
correspondence discussed in the previous section. In Sec. 5, we evaluate the
partition function of $(A_{3},A_{3})$ theory using our formula for
$\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{4})}$. We particularly discuss the
S-duality of the theory in connection to the S-duality of $SU(2)$ gauge theory
with four flavors. In Sec. 6, we conclude and discuss future directions. There
are several appendices. Sec. A includes Nekrasov’s formulae for Lagrangian
sectors. Sec. B contains the first few examples of states in a special basis
of the highest weight module of $Vir\oplus H$ that we will use in Sec. 4. In
Sec. C, we explain how the prepotential of the $(A_{3},A_{3})$ theory is
constrained by the invariance under (5.45) when all the massive deformations
are turned on.

##  2 Generalized AGT correspondence

In this section, we briefly review the AGT correspondence [16, 17] and its
generalization [14, 15]. Suppose that $\mathcal{T}_{\mathcal{C}}$ is a four-
dimensional $\mathcal{N}=2$ superconformal field theory obtained by
compactifying the 6d $(2,0)\;A_{1}$ theory on a punctured Riemann surface
$\mathcal{C}$. It is known that punctures on $\mathcal{C}$ can be “regular” or
“irregular” depending on whether a six-dimensional BPS scalar operator has a
simple pole or higher order pole at it. When the BPS operator has a pole of
order $(n+1)$ at an irregular puncture, we say the puncture is of rank $n$.
While this rank can be an integer or half-integer in general, we only consider
integer ranks in this paper.

$2$ $2$ $2$ $\cdots$$2$ $2$ Figure 3: The quiver diagram of the class
$\mathcal{S}$ theory $\mathcal{T}_{\mathcal{C}}$ for $\mathcal{C}$ being
sphere with $(n+2)$ regular punctures. There are $(n-1)$ circles, each of
which stands for an $SU(2)$ gauge group.

Let us first focus on the case in which $\mathcal{C}$ has no irregular
puncture. In this case, the AGT correspondence [16] implies that the Nekrasov
partition function of $\mathcal{T}_{\mathcal{C}}$ is identical to the
conformal block of Liouville theory on $\mathcal{C}$, where the Liouville
charge $Q$ is related to the $\Omega$-background parameters by
$Q=(\epsilon_{1}+\epsilon_{2})/\sqrt{\epsilon_{1}\epsilon_{2}}$. As an
example, let us consider the case of $\mathcal{C}$ being a sphere with $(n+2)$
regular punctures. The corresponding $\mathcal{T}_{\mathcal{C}}$ is a linear
quiver $SU(2)$ gauge theory as shown in Fig. 3. The AGT correspondence implies

$\displaystyle\mathcal{Z}_{SU(2)}(\vec{a};m_{1},\cdots,m_{n+2})=\mathcal{F}_{\alpha_{0}}{}^{\alpha_{1}}{}_{\beta_{1}}{}^{\alpha_{2}}{}_{\beta_{2}}{}^{\alpha_{3}}\cdots{}_{\beta_{n-1}}{}^{\alpha_{n}}{}_{\alpha_{n+1}}~{},$
(2.1)

where the LHS is the Nekrasov partition function of $\mathcal{T}_{C}$ with
$\vec{a}\equiv(a_{1},\cdots,a_{n-1})$ being the VEVs of Coulomb branch
operators, and $m_{i}$ being the masses of fundamental and bi-fundamental
hypermultiplets. The subscript, $SU(2)$, is for emphasizing that the gauge
group of $\mathcal{T}_{C}$ is $SU(2)^{n-1}$. The RHS of (2.1) is the
$(n+2)$-point conformal block of Liouville theory with $\beta_{i}$ being
intermediate momenta, and $\alpha_{i}$ being external momenta. The 4d and 2d
parameters are related by

$\displaystyle\frac{a_{i}}{\sqrt{\epsilon_{1}\epsilon_{2}}}$
$\displaystyle=\beta_{i}+\frac{Q}{2}~{},\qquad\frac{m_{i}}{\sqrt{\epsilon_{1}\epsilon_{2}}}=\alpha_{i}+\frac{Q}{2}~{}.$
(2.2)

In the rest of this paper, we rescale dimensionful parameters so that
$\epsilon_{1}\epsilon_{2}=1$, as in [16], and write things in terms of the 2d
parameters. When we need to recover the full $\epsilon_{i}$-dependence, we
rescale the 2d parameters as

$\displaystyle\alpha_{i}\to\frac{\alpha_{i}}{\sqrt{\epsilon_{1}\epsilon_{2}}}~{},\qquad\beta_{i}\to\frac{\beta_{i}}{\sqrt{\epsilon_{1}\epsilon_{2}}}~{},\qquad
Q\to\frac{Q}{\sqrt{\epsilon_{1}\epsilon_{2}}}~{}.$ (2.3)

The AGT correspondence has been generalized to the case in which $\mathcal{C}$
has irregular punctures [17, 14, 15]. The most important difference from the
original AGT is that $\mathcal{T}_{\mathcal{C}}$ generically involves an AD
theory [14, 15]. One simple example is the case in which $\mathcal{C}$ is a
sphere with one irregular puncture of rank $n$ and one regular puncture. In
this case, $\mathcal{T}_{\mathcal{C}}$ is an AD theory called the
$(A_{1},D_{2n})$ theory, and its partition function is identified as the two-
point function of 2d operators corresponding to the punctures on
$\mathcal{C}$. By the state-operator map, this two-point function is mapped to
the inner product of two states

$\displaystyle\mathcal{Z}_{(A_{1},D_{2n})}=\langle a|I^{(n)}\rangle~{},$ (2.4)

where $|a\rangle$ is the Virasoro primary state corresponding to the regular
puncture, while $|I^{(n)}\rangle$ is a state corresponding to the rank-$n$
irregular puncture.

A more interesting situation arises when $\mathcal{C}$ has two irregular
punctures of rank $n$. In this case, $\mathcal{T}_{\mathcal{C}}$ is no longer
conformal, but is an $SU(2)$ gauge theory coupled to two copies of
$(A_{1},D_{2n})$ theories. The quiver diagram of this gauge theory is shown in
Fig. 2. Here, each $(A_{1},D_{2n})$ in the matter sector is associated with an
irregular puncture on $\mathcal{C}$. The $\beta$-function coefficient of this
$SU(2)$ gauge coupling is evaluated as $\beta_{0}=2/n$, and therefore
$\mathcal{T}_{\mathcal{C}}$ is never conformal. The generalized AGT
correspondence then implies that the partition function of this theory is
given by

$\displaystyle\mathcal{Z}_{SU(2)}^{2\times(A_{1},D_{2n})}$
$\displaystyle=\langle I^{(n)}|I^{(n)}\rangle~{},$ (2.5)

where $\langle I^{(n)}|$ and $|I^{(n)}\rangle$ correspond to the two irregular
punctures on $\mathcal{C}$.

In contrast to the state corresponding to a regular puncture, the state
$|I^{(n)}\rangle$ is not a Virasoro primary but a linear combination of a
primary and its descendants. This linear combination is determined to be a
simultaneous solution to a set of equations. There are two characterizations
of this set of equations, and here we follow the characterization proposed in
[15]:333The relation between the characterizations in [3] and [15] was
partially studied in appendix B of [21].

$\displaystyle
L_{k}|I^{(n)}\rangle=\left\\{\begin{array}[]{l}\lambda_{k}|I^{(n)}\rangle\qquad\text{for}\quad
n\leq k\leq 2n\\\\[5.69054pt]
\left(\lambda_{k}+\sum_{\ell=1}^{n-k}\ell\,c_{\ell+k}\frac{\partial}{\partial
c_{\ell}}\right)|I^{(n)}\rangle\qquad\text{for}\quad 0\leq k<n\\\
\end{array}\right.~{},$ (2.8)

where $\lambda_{k}$ are constants fixed by $c_{0},\cdots,c_{n}$ as

$\displaystyle\lambda_{k}=\left\\{\begin{array}[]{l}-\sum_{\ell=k-n}^{n}c_{\ell}c_{k-\ell}\quad\text{for}\quad
n<k\leq 2n\\\\[5.69054pt]
-\sum_{\ell=0}^{k}c_{\ell}c_{k-\ell}+(k+1)Qc_{k}\quad\text{for}\quad k\leq
n\end{array}\right.~{}.$ (2.11)

It was conjectured in [15] that $|I^{(n)}\rangle$ satisfying the above
equations exists in a highest weight module of Virasoro algebra.444Note that
the highest weight of this module is not necessarily equal to $c_{0}$. Indeed,
the highest weight is given by $\beta_{0}$ discussed below.

Since the constraints (2.8) are differential equations, $|I^{(n)}\rangle$ is
not completely fixed by $c_{k}$ but depends on the “boundary condition” or the
“asymptotic behavior.” It was conjectured in [15] that $|I^{(n)}\rangle$ is
uniquely fixed by specifying $n$ extra complex parameters in addition to
$c_{0},\cdots,c_{n-1}$ and $c_{n}$. These extra parameters characterize the
asymptotic behavior of $|I^{(n)}\rangle$ in the small $c_{k}$ limit for
$k=1,\cdots,n$. We denote these extra parameters by
$\beta_{0},\cdots,\beta_{n-1}$.555See section 3 and appendix B of [15] for
more detail. Some explicit examples are also shown in [27]. Then the
generalized AGT correspondence implies that $c_{1},\cdots,c_{n-1}$ are related
to relevant couplings, $\beta_{1},\cdots,\beta_{n-1}$ are related to the VEVs
of Coulomb branch operators, and $c_{0}$ and $\beta_{0}$ are related to mass
parameters of $(A_{1},D_{2n})$ theory. The Liouville charge $Q$ is identified
with $\epsilon_{1}+\epsilon_{2}$.

The irregular puncture of integer rank $n$ can be created by colliding $(n+1)$
regular punctures. As a result, the condition (2.8) is obtained in the
colliding limit of Virasoro primary operators [15], which we briefly review
here for later use. Let us first consider the state

$\displaystyle|\phi_{n}(z_{1},\dots,z_{n})\rangle\equiv\;\;:\\!\left(\prod^{n}_{i=1}V_{\alpha_{i}}^{L}(z_{i})\right)V^{L}_{\alpha_{0}}(0)\\!:|0\rangle~{},$
(2.12)

where $V_{\alpha}^{L}(z)$ is the Virasoro primary vertex operator of conformal
weight $\alpha(Q-\alpha)$, and $:\\!XY\\!:$ is the normal-ordered product of
$X$ and $Y$. The action of $T_{>}(y)\equiv\sum_{k\geq-1}y^{-k-2}L_{k}$ on this
state is expressed as

$\displaystyle\begin{aligned}
T_{>}(y)|\phi_{n}(z_{1},\dots,z_{n})\rangle=\sum_{i=0}^{n}\bigg{(}\frac{\alpha_{i}(Q-\alpha_{i})}{y-z_{i}}+\frac{1}{y-z_{i}}\frac{\partial}{\partial
z_{i}}\bigg{)}|\phi_{n}(z_{1},\dots,z_{n})\rangle~{},\end{aligned}$ (2.13)

where $z_{0}\equiv 0$. The idea is then to consider a limit $z_{i}\to 0$ in
which the above action of $T_{>}(y)$ remains well-defined but gives an
interesting result. If we keep $\alpha_{i}$ finite in the limit,
$|\phi_{n}(z_{1},\cdots,z_{n})\rangle$ just reduces to a single primary vertex
operator acting on the vacuum. A more interesting limit is to take $z_{i}\to
0$ and $\alpha_{i}\to\infty$ with

$\displaystyle
c_{k}\equiv\sum_{i=0}^{n}\alpha_{i}z_{i}^{k}\qquad(k=0,\cdots,n)~{},$ (2.14)

kept finite. We call this latter limit “colliding limit.” It is
straightforward to show that $T_{>}(y)$ acts on the state

$\displaystyle|I^{(n)}\rangle\equiv\lim_{\text{colliding
limit}}|\phi_{n}(z_{1},\cdots,z_{n})\rangle~{},$ (2.15)

as

$\displaystyle
T_{>}(y)|I^{(n)}\rangle=\left(\sum_{k=n}^{2n}\frac{\lambda_{k}}{y^{k+2}}+\sum_{k=0}^{n-1}\frac{\lambda_{k}+\sum_{\ell=1}^{n-k}\ell
c_{\ell+k}\frac{\partial}{\partial
c_{\ell}}}{y^{k+2}}+\frac{L_{-1}}{y}\right)|I^{(n)}\rangle~{},$ (2.16)

where $\lambda_{i}$ are defined by (2.11). This implies that the state (2.15)
satisfies (2.8), and therefore is an irregular state of rank $n$. Note that,
since (2.12) is defined in terms of the normal-ordered product, (2.15) gives a
well-defined state.

As mentioned already, the irregular state $|I^{(n)}\rangle$ generally depends
on $n$ extra parameters, $\beta_{0},\cdots,\beta_{n-1}$, corresponding to the
“boundary condition” of a solution to the differential equations (2.8). These
extra parameters correspond to inserting screening operators in the product
(2.12). Since screening operators commute with the Virasoro algebra, this
insertion does not break the conditions (2.8).

##  3 Irregular states in $Vir\oplus H$ modules

As reviewed in the previous section, the generalized AGT correspondence allows
us to evaluate the partition function of $SU(2)$ gauge theory coupled to two
$(A_{1},D_{2n})$ theories as in (2.5). In this section, we consider an
extension of this generalized AGT correspondence to the case in which the
gauge group is $U(2)$ instead of $SU(2)$. To that end, we start with the
$U(2)$ version of the original AGT correspondence and consider its colliding
limit.

###  3.1 $U(2)$ version of the original AGT

The $U(2)$ version of the original AGT correspondence was studied in the
literature. First, it was found in [16] that the Nekrasov partition function
of a $U(2)$ gauge theory $\mathcal{Z}_{U(2)}$ is generally related to that of
the $SU(2)$ gauge theory with the same matter content, $\mathcal{Z}_{SU(2)}$,
by

$\displaystyle\mathcal{Z}_{U(2)}=\mathcal{Z}_{SU(2)}\mathcal{Z}_{U(1)}~{},$
(3.1)

where $\mathcal{Z}_{U(1)}$ is called “$U(1)$ factor” and regarded as the
partition function of the $U(1)$ part of the gauge theory. The AGT
correspondence implies that $\mathcal{Z}_{SU(2)}$ is identical to a conformal
block of the 2d Liouville CFT.

The 2d interpretation of the $U(1)$ factor was then given in [31];
$\mathcal{Z}_{U(1)}$ is identical to a correlation function of chiral vertex
operators for an extra Heisenberg algebra. To be concrete, let us focus on the
linear quiver gauge theory described by the quiver in Fig. 3. As shown in Eq.
(2.1), $\mathcal{Z}_{SU(2)}$ is identified with the $(n+2)$-point conformal
block of Liouville theory. On the other hand, the $U(1)$ factor is identified
as

$\displaystyle\mathcal{Z}_{U(1)}=\langle V^{H}_{\alpha_{0}}(z_{0})\cdots
V^{H}_{\alpha_{n+1}}(z_{n+1})\rangle~{},$ (3.2)

where
$V^{H}_{\alpha}(z)\equiv\exp\Big{(}{2(\alpha-Q)i\sum_{k<0}\frac{a_{k}}{k}z^{-k}}\Big{)}\exp\Big{(}{2\alpha
i\sum_{k>0}\frac{a_{k}}{k}z^{-k}}\Big{)}$, and the loci $z_{k}$ of the vertex
operators coincide with those of the Liouville vertex operators in Eq. (2.1).
Here our convention for the Heisenberg algebra is such that
$[a_{k},a_{\ell}]=\frac{k}{2}\delta_{k+\ell,0}$.

Combining (2.1),(3.1) and (3.2), we see that $\mathcal{Z}_{U(2)}$ is
identified with the correlator of $(n+2)$ chiral vertex operators of the
form666To be precise, one can also insert screening operators in the Liouville
sector here.

$\displaystyle\widehat{V}_{\alpha}(z)\equiv V^{H}_{\alpha}(z)\otimes
V^{L}_{\alpha}(z)~{},$ (3.3)

where $V^{L}_{\alpha}(z)$ is the Virasoro primary vertex operator of conformal
weight $\alpha(Q-\alpha)$ in the Liouville CFT. Thus, the $U(2)$ version of
the AGT correspondence involves the direct sum of Virasoro and Heisenberg
algebras, which we denote by $Vir\oplus H$. Note that $L_{k}$ and $a_{k}$ are
commutative since we consider the direct sum of the two algebras. The action
of $Vir\oplus H$ on $\widehat{V}_{\alpha}(z)$ is characterized by

$\displaystyle[L_{n},\widehat{V}_{\alpha}(z)]$
$\displaystyle=(z^{n+1}+(n+1)\alpha(Q-\alpha)z^{n})\widehat{V}_{\alpha}(z)~{},$
(3.4) $\displaystyle[a_{n},\widehat{V}_{\alpha}(z)]$
$\displaystyle=\left\\{\begin{array}[]{l}-i\alpha
z^{n}\widehat{V}_{\alpha}(z)\quad(n<0)\\\
i(Q-\alpha)z^{n}\widehat{V}_{\alpha}(z)\quad(n>0)\end{array}\right.~{}.$ (3.7)

###  3.2 $U(2)$ version of the generalized AGT

We now consider the $U(2)$ version of the generalized AGT correspondence. Our
idea is to start with the $U(2)$ version of the original AGT, and take the
same colliding limit as the one reviewed in the latter half of Sec. 2.

To that end, we start with the quiver gauge theory described in Fig. 3 with
$U(2)$ gauge groups. The partition function (3.1) is then identified with the
product of (2.1) and (3.2). Note that the loci of the Heisenberg vertex
operators in (3.2) coincide with those of the Liouville vertex operators,
which reflects the fact that the 4d gauge couplings of $U(1)\subset U(2)$ and
$SU(2)\subset U(2)$ are identical. We now take the limit of parameters in
which the 4d theory flows to the theory described by the quiver in Fig. 2 with
$U(2)$ gauge group. On the 2d side, this corresponds to the colliding limit of
vertex operators (3.3), and gives rise to an irregular state
$|\widehat{I}^{(n)}\rangle$ of $Vir\oplus H$. The precise definition of
$|\widehat{I}^{(n)}\rangle$ will be given below. The same argument as in Sec.
2 then leads us to identifying the inner product

$\displaystyle\mathcal{Z}^{2\times(A_{1},D_{2n})}_{U(2)}=\langle\widehat{I}^{(n)}|\widehat{I}^{(n)}\rangle~{},$
(3.8)

as the partition function of $U(2)$ gauge theory coupled to two
$(A_{1},D_{2n})$ theories.

Our new irregular state $|\widehat{I}^{(n)}\rangle$ is characterized by the
actions of the Virasoro and Heisenberg algebras on it. These actions can be
read off by keeping track of their actions on
$\widehat{V}_{\alpha_{k}}(z)=V^{H}_{\alpha_{k}}(z_{k})\otimes
V^{L}_{\alpha_{k}}(z_{k})$ in the colliding limit. To see this, let us
consider the state

$\displaystyle|\widehat{\phi}_{n}(z_{1},\cdots,z_{n})\rangle\equiv\;\;:\\!\left(\prod^{n}_{i=1}\widehat{V}_{\alpha_{i}}(z_{i})\right)\widehat{V}_{\alpha_{0}}(0)\\!:|0\rangle~{}.$
(3.9)

The action of $T_{>}(y)$ on this state is the same as in Eq. (2.13).
Similarly, the action of $J_{>}(y)\equiv\sum_{k\geq 1}y^{-k-1}a_{k}$ is
written as

$\displaystyle
J_{>}(y)|\widehat{\phi}_{n}(z_{1},\cdots,z_{n})\rangle=-\sum_{i=0}^{n}\frac{i(Q-\alpha_{i})z_{i}}{y(y-z_{i})}|\widehat{\phi}(z_{1},\cdots,z_{n})\rangle~{},$
(3.10)

where $z_{0}\equiv 0$. We now take the colliding limit $z_{i}\to 0$ and
$\alpha_{i}\to\infty$ with (2.14) kept fixed. The irregular state
$|\widehat{I}^{(n)}\rangle$ is now defined by

$\displaystyle|\widehat{I}^{(n)}\rangle\equiv\lim_{\text{colliding
limit}}|\widehat{\phi}_{n}(z_{1},\cdots,z_{n})\rangle~{}.$ (3.11)

It is straightforward to show that $T_{>}(y)$ and $J_{>}(y)$ act on this state
as

$\displaystyle T_{>}(y)|\widehat{I}^{(n)}\rangle$
$\displaystyle=\left(\sum_{k=n}^{2n}\frac{\lambda_{k}}{y^{k+2}}+\sum_{k=0}^{n-1}\frac{\lambda_{k}+\sum_{\ell=1}^{n-k}\ell
c_{\ell+k}\frac{\partial}{\partial
c_{\ell}}}{y^{k+2}}+\frac{L_{-1}}{y}\right)~{},$ (3.12) $\displaystyle
J_{>}(y)|\widehat{I}^{(n)}\rangle$
$\displaystyle=\sum_{k=1}^{n}\frac{-ic_{k}}{y^{k+1}}|\widehat{I}^{(n)}\rangle~{},$
(3.13)

where $\lambda_{k}$ and $c_{k}$ are given by (2.11) and (2.14), respectively.
From the above result, we see that $Vir\oplus H$ acts on
$|\widehat{I}^{(n)}\rangle$ as

$\displaystyle L_{k}|\widehat{I}^{(n)}\rangle$
$\displaystyle=\left\\{\begin{array}[]{l}\lambda_{k}|\widehat{I}^{(n)}\rangle\qquad\text{for}\quad
n\leq k\leq 2n\\\\[5.69054pt]
\left(\lambda_{k}+\sum_{\ell=1}^{n-k}\ell\,c_{\ell+k}\frac{\partial}{\partial
c_{\ell}}\right)|\widehat{I}^{(n)}\rangle\qquad\text{for}\quad 0\leq k<n\\\
\end{array}\right.~{},$ (3.16) $\displaystyle a_{k}|\widehat{I}^{(n)}\rangle$
$\displaystyle=\left\\{\begin{array}[]{l}\;\;\;\;0\qquad\text{for}\quad
n<k\\\\[5.69054pt] -ic_{k}|\widehat{I}^{(n)}\rangle\qquad\text{for}\quad 1\leq
k\leq n\\\ \end{array}\right.~{}.$ (3.19)

Note here that the above characterization of the irregular state does not fix
the overall normalization, as in the case of the $SU(2)$-version of the
generalized AGT correspondence. This means an ambiguity in the computation of
the perturbative part of the partition function (3.8). However, it turns out
that the instanton part of the partition function can be unambiguously
computed, as will be discussed in the following sections.

Note also that $|\widehat{I}^{(n)}\rangle$ is by definition decomposed into
the Virasoro part and the Heisenberg part as
$|\widehat{I}^{(n)}\rangle=|I^{(n)}\rangle\otimes|I^{(n)}_{H}\rangle$, where
$|I^{(n)}\rangle$ is the irregular state of Virasoro algebra that was reviewed
in Sec. 2. While $|I^{(n)}\rangle$ depends on $n$ extra parameters in addition
to $c_{0},\cdots,c_{n-1}$ and $c_{n}$, the state $|I^{(n)}_{H}\rangle$ is
uniquely fixed by $c_{1},\cdots,c_{n-1}$ and $c_{n}$ up to a prefactor. We
here write down its explicit expression:

$\displaystyle|I^{(n)}_{H}\rangle=\exp\left(-2i\sum_{k=1}^{n}\frac{c_{k}}{k}a_{-k}\right)|0\rangle~{},$
(3.20)

where the prefactor is fixed so that $|I^{(n)}_{H}\rangle$ reduces to
$|0\rangle$ when $c_{k}=0$. Since $|I^{(n)}_{H}\rangle$ and $|I^{(n)}\rangle$
are respectively in a highest weight module of $Vir$ and $H$, we see that
$|\widehat{I}^{(n)}\rangle$ is a state in a highest weight module of
$Vir\oplus H$. Note that the Heisenberg sector has no possible insertion of
screening operators, and therefore (3.20) is the unique expression for
$|I^{(n)}_{H}\rangle$ up to a prefactor. Indeed, the constraints (3.19) are
eigenstate equations, whose solution is fixed (up to the normalization) by
$\\{c_{k}\\}$ without specifying a “boundary condition.”

Given the $U(2)$-version of the generalized AGT correspondence (3.8), one can
now study the decomposition of $\mathcal{Z}_{U(2)}^{2\times(A_{1},D_{2n})}$ as
a sum over pairs of Young diagrams as in Eq. (1.2). In the next section, we
explicitly evaluate this decomposition to read off the factor
$\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{2n})}$ in (1.2).

##  4 Nekrasov-type formula for AD matter

Here we read off the factor $\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{2n})}$ in
(1.2) from the $U(2)$-version of the generalized AGT correspondence (3.8).
This factor can be interpreted as the contribution of the $(A_{1},D_{2n})$
theory at the fixed point corresponding to $(Y_{1},Y_{2})$ on the $U(2)$
instanton moduli space.

###  4.1 Decomposition

To read off $\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{2n})}$, we first decompose
(3.8) as a sum over pairs of Young diagrams, using a nice basis of highest
weight modules of $Vir\oplus H$ that was found in [31]. To describe it, let us
denote by

$\displaystyle\mathcal{Z}_{Y_{1},Y_{2};W_{1},W_{2}}^{\text{bifund}}(a,b,\alpha)~{},$
(4.1)

the contribution to the Nekrasov partition function from a bi-fundamental
hypermultiplet of $U(2)\times U(2)$. Here, $(a,b)$ stands for the VEVs of
Coulomb branch operators in the vector multiplet for $SU(2)\times SU(2)\subset
U(2)\times U(2)$, and $\alpha$ is a mass parameter. The explicit expression
for (4.1) is written in Appendix A. It was shown in [31] that there exists an
orthogonal basis, $|a;Y_{1},Y_{2}\rangle$, of the highest weight module of
$Vir\oplus H$ such that

$\displaystyle\frac{\langle
a;Y_{1},Y_{2}|V_{\alpha}(1)|b;W_{1},W_{2}\rangle}{\langle
a|V_{\alpha}(1)|b\rangle}$
$\displaystyle=\mathcal{Z}_{Y_{1},Y_{2};W_{1},W_{2}}^{\text{bifund}}(a,b,\alpha)~{},$
(4.2)

where $|a\rangle$ is the highest weight state satisfying
$L_{0}|a\rangle=\Delta(a)|a\rangle$ and $L_{n}|a\rangle=a_{n}|a\rangle=0$ for
$n>0$, $V_{\alpha}(z)$ is the vertex operator shown in (3.3), and $Y_{k}$ and
$W_{k}$ are arbitrary Young diagrams. Note that the conjugate $\langle
a;Y_{1},Y_{2}|$ is not the usual hermitian conjugate of
$|a;Y_{1},Y_{2}\rangle$; it is obtained by expanding $|a;Y_{1},Y_{2}\rangle$
as a linear combination of $L_{-k_{1}}^{m_{1}}\cdots
L_{-k_{p}}^{m_{p}}a_{-\ell_{1}}^{n_{1}}\cdots a_{-\ell_{q}}^{n_{q}}|a\rangle$,
and then replacing each such state with $\langle a|L_{k_{p}}^{m_{p}}\cdots
L_{k_{1}}^{m_{1}}a_{\ell_{q}}^{n_{q}}\cdots a_{\ell_{1}}^{n_{1}}$ without
changing the coefficients. The first few examples of $|a;Y_{1},Y_{2}\rangle$
are presented in Appendix B. It was also shown in [31] that
$|a;Y_{1},Y_{2}\rangle$ is generally a linear combination of descendants of
$|a\rangle$ at level $(|Y_{1}|+|Y_{2}|)$.777Here, the “level” is defined by
the sum of the levels of the Virasoro and Heisenberg parts. For example, the
level of $L_{-k_{1}}(L_{-k_{2}})^{3}a_{-\ell}|a\rangle$ is
$k_{1}+3k_{2}+\ell$.

As discussed in [31], the condition (4.2) and the fact that
$\mathcal{Z}_{Y_{1},Y_{2}}^{\text{vec}}(a)=1/\mathcal{Z}_{Y_{1},Y_{2};Y_{1},Y_{2}}^{\text{bifund}}(a,a,0)$
imply

$\displaystyle{\bf
1}=\sum_{Y_{1},Y_{2}}\mathcal{Z}_{Y_{1},Y_{2}}^{\text{vec}}(a)\,|a;Y_{1},Y_{2}\rangle\langle
a;Y_{1},Y_{2}|~{},$ (4.3)

on the highest weight $Vir\oplus H$-module associated with $|a\rangle$. Note
again that $\langle a;Y_{1},Y_{2}|$ is not the usual conjugate of
$|a;Y_{1},Y_{2}\rangle$. Let us now take $|a\rangle$ to be the highest weight
state of the $Vir\oplus H$-module that includes $|\widehat{I}^{(n)}\rangle$.
Then, by inserting (4.3) in Eq. (3.8), one obtains the following
decomposition:

$\displaystyle\mathcal{Z}_{U(2)}^{2\times(A_{1},D_{2n})}=\sum_{Y_{1},Y_{2}}\mathcal{Z}_{Y_{1},Y_{2}}^{\text{vec}}(a)\,\langle\widehat{I}^{(n)}|a;Y_{1},Y_{2}\rangle\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle~{}.$ (4.4)

We interpret the above expression as a sum over fixed points on the moduli
space of $U(2)$ instantons, and $\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle$ and
$\langle\widehat{I}^{(n)}|a;Y_{1},Y_{2}\rangle$ as the contributions of the
$(A_{1},D_{2n})$ theories corresponding to $|\widehat{I}^{(n)}\rangle$ and
$\langle\widehat{I}^{(n)}|$, respectively. Note that this particularly implies
that $\langle a;\emptyset,\emptyset|\widehat{I}^{(n)}\rangle$ and
$\langle\widehat{I}^{(n)}|a;\emptyset,\emptyset\rangle$ are the partition
function of $(A_{1},D_{2n})$ theory with its flavor symmetry un-gauged. Since
$|a;\emptyset,\emptyset\rangle=|a\rangle$ [31], this is indeed consistent with
(2.4).

###  4.2 Identification of $\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$

Comparing (4.4) with (1.2), we see that it is natural to interpret

$\displaystyle\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}\sim\frac{\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle}{\langle
a|\widehat{I}^{(n)}\rangle}~{},\qquad\widetilde{\mathcal{Z}}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}\sim\frac{\langle\widehat{I}^{(n)}|a;Y_{1},Y_{2}\rangle}{\langle\widehat{I}^{(n)}|a\rangle}~{},$
(4.5)

with possible proportionality constants. Note that the denominators in (4.5)
are necessary for
$\mathcal{Z}^{(A_{1},D_{2n})}_{\emptyset,\emptyset}=\widetilde{\mathcal{Z}}_{\emptyset,\emptyset}^{(A_{1},D_{2n})}=1$.

Here, $|\widehat{I}^{(n)}\rangle$ and $\langle\widehat{I}^{(n)}|$ correspond
to two different $(A_{1},D_{2n})$ theories. Indeed, as seen from their
colliding-limit derivation, these $(A_{1},D_{2n})$ theories are differently
coupled to the $U(1)$-part of the gauge group. Therefore,
$\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{2n})}$ and
$\widetilde{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ are not identical.

In the rest of this sub-section, we make the relations (4.5) more precise. To
that end, we first focus on the left relation, and read off how the 4d and 2d
parameters are related. Note that $\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle$ on the RHS depends on $(2n+1)$
parameters. Indeed, $|\widehat{I}^{(n)}\rangle$ depends on $n$ extra
parameters $\beta_{0},\cdots,\beta_{n-1}$ in addition to $c_{0},\cdots,c_{n}$,
as reviewed in Sec. 3.2. One of these extra parameters fixes the highest
weight, $a$, of the $Vir\oplus H$-module that includes
$|\widehat{I}^{(n)}\rangle$, and therefore $\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle$ is completely fixed by these $(2n+1)$
parameters. On the other hand,
$\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}(a,m,\boldsymbol{d},\boldsymbol{u})$
on the LHS depends only on $2n$ parameters,
$a,m,\boldsymbol{d}=(d_{1},\cdots,d_{n-1})$ and
$\boldsymbol{u}=(u_{1},\cdots,u_{n-1})$. Here, $a$ and $\boldsymbol{u}$ are
the VEVs of Coulomb branch operators, $\boldsymbol{d}$ are relevant couplings
and $m$ is a mass parameter. Therefore, there is a discrepancy in the number
of parameters between the 2d and 4d sides.

To see this discrepancy more explicitly, let us identify the precise relation
between the 2d and 4d parameters. We start with the Seiberg-Witten (SW) curve
of the $(A_{1},D_{2n})$ theory [14, 28]

$\displaystyle
x^{2}=\frac{a^{2}}{z^{2}}+\sum_{k=1}^{n-1}\frac{u_{k}}{z^{n+2-k}}+\frac{m}{z^{n+2}}+\sum_{k=1}^{n-1}\frac{d_{k}}{z^{2n+2-k}}+\frac{1}{z^{2n+2}}~{},$
(4.6)

where the SW 1-form is given by $xdz$.888Here $a$ is regarded as the mass
parameter corresponding to a flavor $SU(2)$ sub-group of the $(A_{1},D_{2n})$
theory. This curve is identified, on the 2d side, as the classical limit
$\epsilon_{i}\to 0$ of the following [16]

$\displaystyle x^{2}=-\frac{\langle a|T(z)|\widehat{I}^{(n)}\rangle}{\langle
a|\widehat{I}^{(n)}\rangle}=-\frac{\Delta(a)}{z^{2}}+\cdots+\frac{2c_{n}c_{n-1}}{z^{2n+1}}+\frac{c_{n}^{2}}{z^{2n+2}}~{},$
(4.7)

up to a change of variables that preserves the SW 1-form. Note that the RHS of
the above equation can be explicitly evaluated via Eq. (3.16).999Recall here
that, to recover the full $\epsilon_{i}$-dependence, one needs to perform a
replacement corresponding to (2.3). For the irregular state $|I^{(n)}\rangle$,
this replacement implies $c_{k}\to c_{k}/\sqrt{\epsilon_{1}\epsilon_{2}}$ as
seen from its colliding-limit derivation. We here change the variables in
(4.7) as $z\to(c_{n})^{\frac{1}{n}}\,z$ and $x\to(c_{n})^{-\frac{1}{n}}\,x$ so
that the coefficient of $1/z^{2n+2}$ is $1$. Comparing the classical limit of
the resulting equation with (4.6), we obtain the relation between the 2d and
4d parameters.101010Note here that $\Delta(a)$ reduces to $-a^{2}$ in the
classical limit $\epsilon_{1},\epsilon_{2}\to 0$. To make this relation
simple, let us define on the 2d side

$\displaystyle\gamma_{k}\equiv\frac{c_{k}}{(c_{n})^{\frac{k}{n}}}~{},$ (4.8)

for $k=0,\cdots,n-1$. In terms of these variables, the relation between the 2d
and 4d parameters is expressed as

$\displaystyle d_{k}$
$\displaystyle=\sum_{\ell=n-k}^{n}\gamma_{\ell}\gamma_{2n-k-\ell}~{},\quad
m=\sum_{\ell=0}^{n}\gamma_{\ell}\gamma_{n-\ell}~{},\quad
u_{k}=\sum_{\ell=0}^{n-k}\gamma_{\ell}\gamma_{n-k-\ell}-\sum_{\ell=1}^{k}\ell\gamma_{\ell+n-k}\frac{\partial}{\partial\gamma_{\ell}}\mathcal{F}_{(A_{1},D_{2n})}~{},$
(4.9)

where the derivatives $\partial/\partial\gamma_{\ell}$ are defined with
$\vec{\gamma}\equiv(\gamma_{0},\cdots,\gamma_{n-1})$ and $c_{n}$ taken as
independent variables, and $\mathcal{F}_{(A_{1},D_{2n})}$ is the classical
limit of $\log\langle a|\widehat{I}^{(n)}\rangle$. These expressions imply
that, when one takes $\vec{\gamma}$ and $c_{n}$ as independent variables, all
the 4d parameters are independent of $c_{n}$.111111To prove this statement,
one needs to show that
$\frac{\partial}{\partial\gamma_{\ell}}\mathcal{F}_{(A_{1},D_{2n})}$ is
independent of $c_{n}$. This can be shown as follows. As we will show below,
$\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle=(c_{n})^{\frac{\Delta_{a}-\Delta_{c_{0}}+|Y_{1}|+|Y_{2}|}{n}}f_{Y_{1},Y_{2}}(\vec{\gamma})$
for a function $f_{Y_{1},Y_{2}}(\vec{\gamma})$ independent of $c_{n}$. Setting
$Y_{1}=Y_{2}=\emptyset$, we find $\log\langle
a|\widehat{I}^{(n)}\rangle=\frac{\Delta_{a}-\Delta_{c_{0}}}{n}\log c_{n}+\log
f_{\emptyset,\emptyset}(\vec{\gamma})$. This implies that
$\frac{\partial}{\partial\gamma_{\ell}}\log\langle
a|\widehat{I}^{(n)}\rangle=\frac{\partial}{\partial\gamma_{\ell}}\log
f_{\emptyset,\emptyset}(\vec{\gamma})$. Since this is independent of $c_{n}$,
its classical limit
$\frac{\partial}{\partial\gamma_{\ell}}\mathcal{F}_{(A_{1},D_{2n})}$ is also
independent of $c_{n}$ when written in terms of $\vec{\gamma}$ and $c_{n}$.
This reflects the conformal invariance of $(A_{1},D_{2n})$, and explains the
discrepancy in the number of parameters between the 2d and 4d sides.

The above discussion implies that, for the left relation in (4.5) to be an
equality, the $c_{n}$-dependence of the RHS needs to be canceled by a constant
of proportionality. To identify this proportionality constant, let us consider

$\displaystyle n\,c_{n}\left.\frac{\partial}{\partial
c_{n}}\right|_{\vec{\gamma}}\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle~{},$ (4.10)

where $\partial/\partial c_{n}|_{\vec{\gamma}}$ is the derivative with respect
to $c_{n}$ with $\vec{\gamma}=(\gamma_{0},\cdots,\gamma_{n-1})$ kept fixed.
From (3.16), we see that this is identical to

$\displaystyle\langle
a;Y_{1},Y_{2}|(L_{0}-\Delta_{c_{0}})|\widehat{I}^{(n)}\rangle=\left(\Delta_{a}-\Delta_{c_{0}}+|Y_{1}|+|Y_{2}|\right)\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle~{}.$ (4.11)

The equality between the above two implies that $\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle\sim(c_{n})^{\frac{\Delta_{a}-\Delta_{c_{0}}+|Y_{1}|+|Y_{2}|}{n}}$,
and therefore the ratio $(c_{n})^{-\frac{|Y_{1}|+|Y_{2}|}{n}}\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle/\langle a|\widehat{I}^{(n)}\rangle$ is
independent of $c_{n}$ when written in terms of $\vec{\gamma}$ and
$c_{n}$.121212Recall here that $\langle a;\emptyset,\emptyset|=\langle a|$.
This suggests the following identification

$\displaystyle\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}=(\zeta
c_{n})^{-\frac{|Y_{1}|+|Y_{2}|}{n}}\frac{\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle}{\langle
a|\widehat{I}^{(n)}\rangle}~{},$ (4.12)

where $\zeta$ is a possible numerical constant independent of all variables.
Note that $\zeta$ above can be absorbed by rescaling the instanton factor
$\Lambda$.

A parallel discussion shows that the parameters of
$\widetilde{\mathcal{Z}}_{Y_{1},Y_{2}}(a,\widetilde{m},\widetilde{\boldsymbol{d}},\widetilde{\boldsymbol{u}})$
are related to those of $\langle\widehat{I}^{(n)}|a;Y_{1},Y_{2}\rangle$ by a
similar relation to (4.9). As $\langle a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle$
depends on $c_{0},\cdots,c_{n}$ and $\beta_{0},\cdots,\beta_{n-1}$,
$\langle\widehat{I}^{(n)}|a;Y_{1},Y_{2}\rangle$ also depends on $(2n+1)$
parameters, which we denote by $\widetilde{c}_{0},\cdots,\widetilde{c}_{n}$
and $\widetilde{\beta}_{0},\cdots,\widetilde{\beta}_{n-1}$. From the same
argument as above, we see that

$\displaystyle\widetilde{\mathcal{Z}}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}=(-\zeta\widetilde{c}_{n}^{\,*})^{-\frac{|Y_{1}|+|Y_{2}|}{n}}\frac{\langle\widehat{I}^{(n)}|a;Y_{1},Y_{2}\rangle}{\langle\widehat{I}^{(n)}|a\rangle}~{},$
(4.13)

where $\widetilde{c}_{n}^{\,*}$ is the complex conjugate of
$\widetilde{c}_{n}$.131313The opposite sign in the bracket in
$(-\widetilde{c}_{n}^{*})^{-\frac{|Y_{1}|+|Y_{2}|}{n}}$ can be understood as
follows. First recall that $\langle a;Y_{1},Y_{2}|$ in (4.13) is not the usual
conjugate of $|a;Y_{1},Y_{2}\rangle$, as discussed in [31]. Indeed, $\langle
a;Y_{1},Y_{2}|$ is obtained by expanding $|a;Y_{1},Y_{2}\rangle$ as a linear
combination of $L_{-k_{1}}^{m_{1}}\cdots
L_{-k_{p}}^{m_{p}}a_{-\ell_{1}}^{n_{1}}\cdots a_{-\ell_{q}}^{n_{q}}|a\rangle$,
and then replacing each such state with $\langle a|L_{k_{p}}^{m_{p}}\cdots
L_{k_{1}}^{m_{1}}a_{\ell_{q}}^{n_{q}}\cdots a_{\ell_{1}}^{n_{1}}$ without
changing the coefficients. This implies that,
$\langle\widehat{I}^{(n)}|a;Y_{1},Y_{2}\rangle$ is obtained from $\langle
a;Y_{1},Y_{2}|\widehat{I}^{(n)}\rangle$ by replacing $\langle
a|L_{k_{p}}^{m_{p}}\cdots L_{k_{1}}^{m_{1}}a_{\ell_{q}}^{n_{q}}\cdots
a_{\ell_{1}}^{n_{1}}|\widehat{I}^{(n)}\rangle$ with
$\langle\widehat{I}^{(n)}|L_{-k_{p}}^{m_{p}}\cdots
L_{-k_{1}}^{m_{1}}a_{-\ell_{q}}^{n_{q}}\cdots a_{-\ell_{1}}^{n_{1}}|a\rangle$.
From (3.16) and (3.19), we see that this is equivalent to the replacement
$\displaystyle c_{k}\longrightarrow-\widetilde{c}_{k}^{*}~{},\qquad
Q\to-Q^{*}$ (4.14) for $k=0,\cdots,n$. In particular, $c_{n}$ in (4.12) is
replaced by $-\widetilde{c}_{n}^{*}$.

In the above identifications, $\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ and
$\widetilde{\mathcal{Z}}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ are regarded as the
contribution of the $(A_{1},D_{2n})$ theory corresponding to
$|\widehat{I}^{(n)}\rangle$ and $\langle\widehat{I}^{(n)}|$, respectively. As
discussed at the beginning, these two $(A_{1},D_{2n})$ theories have different
couplings to the $U(1)$ part of the gauge group. As we will see in Sec. 4.4,
the difference between $\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ and
$\widetilde{\mathcal{Z}}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ is an AD counterpart
of the difference between the fundamental and anti-fundamental hypermultiplets
of $U(2)$.

###  4.3 Identification of $\Lambda$

The identifications (4.12) and (4.13) imply that (4.4) is re-expressed as

$\displaystyle\mathcal{Z}^{2\times(A_{1},D_{2n})}_{Y_{1},Y_{2}}$
$\displaystyle=\mathcal{Z}_{\text{pert}}\sum_{Y_{1},Y_{2}}(-\zeta^{2}c_{n}\widetilde{c}_{n}^{\,*})^{\frac{|Y_{1}|+|Y_{2}|}{n}}\mathcal{Z}^{\text{vec}}_{Y_{1},Y_{2}}(a)\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{2n})}(a,m,\boldsymbol{d},\boldsymbol{u})\widetilde{\mathcal{Z}}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}(a,\widetilde{m},\widetilde{\boldsymbol{d}},\widetilde{\boldsymbol{u}})~{},$
(4.15)

with the perturbative part
$\mathcal{Z}_{\text{pert}}\equiv\langle\widehat{I}^{(n)}|a\rangle\langle
a|\widehat{I}^{(n)}\rangle$. Comparing (4.15) with (1.2), we identify the 4d
dynamical scale as

$\displaystyle\Lambda^{2}=-\zeta^{2}c_{n}\widetilde{c}_{n}^{\,*}~{}.$ (4.16)

Recall that the $(A_{1},D_{2n})$ sector is independent of $c_{n}$ (and
$\widetilde{c}_{n}^{\,*}$) as a result of its conformal invariance. Here, the
$U(2)$ gauge coupling breaks this conformal invariance through the dynamical
scale, and therefore it is natural that $\Lambda$ depends on $c_{n}$ and
$\widetilde{c}_{n}^{\,*}$.

The identification (4.16) is also consistent with the SW curve. The curve of
the theory shown in Fig. 2 is written as [14]

$\displaystyle
x^{2}=\Lambda_{0}^{2}z^{2n-2}+\cdots+\frac{\Lambda_{0}^{2}}{z^{2n+2}}~{},$
(4.17)

where $\Lambda_{0}$ is a dynamical scale that can differ from $\Lambda$ by a
numerical factor, and the ellipsis stands for a Laurent polynomial of $z$
which is less singular than $z^{2n-2}$ at $z=\infty$ and than $1/z^{2n+2}$ at
$z=0$.141414The SW 1-form is again given by $xdz$. Now, by the same argument
as around Eq. (4.7), this curve is identified as

$\displaystyle
x^{2}=-\frac{\langle\widehat{I}^{(n)}|T(z)|\widehat{I}^{(n)}\rangle}{\langle\widehat{I}^{(n)}|\widehat{I}^{(n)}\rangle}=(\widetilde{c}_{n}^{\,*})^{2}z^{2n-2}+\cdots+\frac{(c_{n})^{2}}{z^{2n+2}}~{},$
(4.18)

up to a change of variables that preserves the SW 1-form. After changing
variables as $z\to z\,(-c_{n}/\widetilde{c}_{n}^{\,*})^{\frac{1}{2n}}$ and
$x\to x\,(-c_{n}/\widetilde{c}_{n}^{\,*})^{-\frac{1}{2n}}$, the curve (4.18)
is re-expressed as
$x^{2}=(-c_{n}\widetilde{c}_{n}^{\,*})z^{2n-2}+\cdots+(-c_{n}\widetilde{c}_{n}^{\,*})/z^{2n+2}$.
Comparing this with (4.17), we find
$\Lambda_{0}^{2}=-c_{n}\widetilde{c}_{n}^{\,*}$, which coincides with (4.16)
up to a numerical factor.

###  4.4 Consistency check

Since the $(A_{1},D_{2})$ theory is a theory of free hypermultiplets in the
doublet of $U(2)$, one can perform a consistency check of our proposals (4.12)
and (4.13) by comparing them with the Nekrasov’s formula for fundamental and
anti-fundamental hypermultiplets.

Let us first consider (4.12). In the case of $n=1$, the irregular state
involved in (4.12) satisfies

$\displaystyle L_{2}|\widehat{I}^{(1)}\rangle$
$\displaystyle=-c_{1}^{2}|\widehat{I}^{(1)}\rangle~{},$ (4.19) $\displaystyle
L_{1}|\widehat{I}^{(1)}\rangle$
$\displaystyle=2(Q-c_{0})c_{1}|\widehat{I}^{(1)}\rangle~{},$ (4.20)
$\displaystyle L_{0}|\widehat{I}^{(1)}\rangle$
$\displaystyle=\left(\Delta_{c_{0}}+c_{1}\frac{\partial}{\partial
c_{1}}\right)|\widehat{I}^{(1)}\rangle~{},$ (4.21) $\displaystyle
a_{1}|\widehat{I}^{(1)}\rangle$
$\displaystyle=-ic_{1}|\widehat{I}^{(1)}\rangle~{},$ (4.22)

together with $a_{k}|\widehat{I}^{(1)}\rangle=0$ for $k\geq 2$. These
equations are enough to compute the ratio of inner products $\langle
a;Y_{1},Y_{2}|\widehat{I}^{(1)}\rangle/\langle a|\widehat{I}^{(1)}\rangle$. We
then find that151515We checked this equality for $|Y_{1}|+|Y_{2}|\leq 6$.

$\displaystyle\left(-\frac{c_{1}}{2}\right)^{-|Y_{1}|-|Y_{2}|}\frac{\langle
a;Y_{1},Y_{2}|\widehat{I}^{(1)}\rangle}{\langle
a|\widehat{I}^{(1)}\rangle}=\mathcal{Z}^{\text{fund}}_{Y_{1},Y_{2}}(a,m)~{},$
(4.23)

where $\mathcal{Z}^{\text{fund}}_{Y_{1},Y_{2}}$ is the contribution from a
fundamental hypermultiplet of $U(2)$ as reviewed in Appendix A. The mass
parameter $m$ is related to $c_{0}$ by

$\displaystyle m=c_{0}-\frac{Q}{2}~{},$ (4.24)

which coincides with (4.9) in the classical limit $\epsilon_{i}\to 0$. We see
that (4.23) is perfectly consistent with our proposal (4.12) for $\zeta=-1/2$.

We also perform the same computation for
$\langle\widehat{I}^{(1)}|a;Y_{1},Y_{2}\rangle/\langle\widehat{I}^{(1)}|a\rangle$
to find that161616We also checked this equality for $|Y_{1}|+|Y_{2}|\leq 6$.

$\displaystyle\left(\frac{\widetilde{c}_{n}^{\,*}}{2}\right)^{-|Y_{1}|-|Y_{2}|}\frac{\langle\widehat{I}^{(1)}|a;Y_{1},Y_{2}\rangle}{\langle\widehat{I}^{(1)}|a\rangle}=\mathcal{Z}^{\text{anti-
fund}}_{Y_{1},Y_{2}}(a,\widetilde{m})~{},$ (4.25)

where $\widetilde{m}$ is similarly identified as

$\displaystyle\widetilde{m}=\left(\widetilde{c}_{0}-\frac{Q}{2}\right)^{*}~{}.$
(4.26)

This is also in perfect agreement with our proposal (4.13) for $\zeta=-1/2$.

The above two checks suggest that the difference between
$\mathcal{Z}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ and
$\widetilde{\mathcal{Z}}^{(A_{1},D_{2n})}_{Y_{1},Y_{2}}$ can be regarded as
the AD counterpart of the difference between the fundamental and anti-
fundamental hypermultiplets.

##  5 Application to $(A_{3},A_{3})$ theory

In this section, we apply our method to the $(A_{3},A_{3})$ theory and compute
its partition function. Recall that the $(A_{3},A_{3})$ theory is described by
the quiver diagram in Fig. 1. When the gauge group is replaced by $U(2)$, the
partition function of the theory is given by

$\displaystyle\mathcal{Z}_{U(2)}=\mathcal{Z}^{U(2)}_{\text{pert}}\sum_{Y_{1},Y_{2}}q^{|Y_{1}|+|Y_{2}|}\mathcal{Z}_{Y_{1},Y_{2}}^{\text{vec}}(a)\mathcal{Z}_{Y_{1},Y_{2}}^{\text{fund}}(a,M)\prod_{i=1}^{2}\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{4})}(a,m_{i},d_{i},u_{i})~{},$
(5.1)

where $\mathcal{Z}^{(A_{1},D_{4})}_{Y_{1},Y_{2}}$ is the contribution of the
$(A_{1},D_{4})$ theory that we have identified in Eq. (4.12), and
$\mathcal{Z}^{U(2)}_{\text{pert}}$ is the perturbative contribution that makes
the series in $q$ start with $1$. The parameters $m_{i},d_{i}$ and $u_{i}$ are
respectively a mass parameter, relevant coupling of dimension $1/2$, and the
VEV of Coulomb branch operator of dimension $3/2$ in the $i$-th
$(A_{1},D_{4})$ theory. Since the $SU(2)$ gauge coupling is exactly marginal,
the above expression includes the exponential of the marginal gauge coupling,
$q$, instead of a dynamical scale.

Note that, depending on how the $U(1)$-part of the gauge group couples to
$(A_{1},D_{4})$, its contribution to the partition function is
$\mathcal{Z}^{(A_{1},D_{4})}_{Y_{1},Y_{2}}$ or
$\widetilde{\mathcal{Z}}^{(A_{1},D_{4})}_{Y_{1},Y_{2}}$. In this section, we
focus on the case in which both of the two $(A_{1},D_{4})$ theories couple to
the $U(1)$ in the way corresponding to
$\mathcal{Z}^{(A_{1},D_{4})}_{Y_{1},Y_{2}}$. Replacing one or two of
$\mathcal{Z}^{(A_{1},D_{4})}_{Y_{1},Y_{2}}$ with
$\widetilde{\mathcal{Z}}^{(A_{1},D_{4})}_{Y_{1},Y_{2}}$, one would obtain to a
different $\mathcal{Z}_{U(2)}$, which is however expected to give the same
$\mathcal{Z}_{SU(2)}$ when the $U(1)$ factor $\mathcal{Z}_{U(1)}$ is removed
as in (1.4).

The factor $\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{4})}$ in Eq. (5.1) is given
by

$\displaystyle\mathcal{Z}_{Y_{1},Y_{2}}^{(A_{1},D_{4})}(a,m,d,u)=\left(-\frac{c_{2}}{2}\right)^{-\frac{|Y_{1}|+|Y_{2}|}{2}}\frac{\langle
a;Y_{1},Y_{2}|\widehat{I}^{(2)}\rangle}{\langle
a|\widehat{I}^{(2)}\rangle}~{},$ (5.2)

where $m,d$ and $u$ are identified as

$\displaystyle m=2c_{0}+\frac{c_{1}^{2}}{c_{2}}~{},\qquad
d=\frac{2c_{1}}{\sqrt{c_{2}}}~{},\qquad
u=\frac{2c_{0}c_{1}}{\sqrt{c_{2}}}-\sqrt{c_{2}}\frac{\partial\mathcal{F}_{(A_{1},D_{4})}}{\partial
c_{1}}~{},$ (5.3)

with $\mathcal{F}_{(A_{1},D_{4})}$ being the classical limit of $\log\langle
a|\widehat{I}^{(2)}\rangle$. Note that we here set $\zeta=-1/2$ in (4.12) to
avoid various numerical factors in the expressions below. This factor can be
generated or absorbed by rescaling $q$ in the expression (5.1). The irregular
state $|\widehat{I}^{(2)}\rangle$ is characterized by

$\displaystyle L_{4}|\widehat{I}^{(2)}\rangle$
$\displaystyle=-c_{2}^{2}|\widehat{I}^{(2)}\rangle~{},$ (5.4) $\displaystyle
L_{3}|\widehat{I}^{(2)}\rangle$
$\displaystyle=-2c_{1}c_{2}|\widehat{I}^{(2)}\rangle~{},$ (5.5) $\displaystyle
L_{2}|\widehat{I}^{(2)}\rangle$
$\displaystyle=-(c_{1}^{2}+c_{2}(2c_{0}-3Q))|\widehat{I}^{(2)}\rangle~{},$
(5.6) $\displaystyle L_{1}|\widehat{I}^{(2)}\rangle$
$\displaystyle=\left(c_{2}\frac{\partial}{\partial
c_{1}}-2c_{1}(c_{0}-Q)\right)|\widehat{I}^{(2)}\rangle~{},$ (5.7)
$\displaystyle L_{0}|\widehat{I}^{(2)}\rangle$
$\displaystyle=\left(\Delta_{c_{0}}+c_{1}\frac{\partial}{\partial
c_{1}}+2c_{2}\frac{\partial}{\partial
c_{2}}\right)|\widehat{I}^{(2)}\rangle~{},$ (5.8)

and $a_{k}|\widehat{I}^{(2)}\rangle=-ic_{k}|\widehat{I}^{(2)}\rangle$ for
$k=1,2$ together with $a_{k}|\widehat{I}^{(2)}\rangle=0$ for $k>2$.

To extract the partition function of the $(A_{3},A_{3})$ theory from (5.1),
one has to remove the contribution of the $U(1)$-part of the gauge group. As
discussed in Sec. 1, this can be done by dividing (5.1) by a $U(1)$ factor,
$\mathcal{Z}_{U(1)}$. Therefore the partition function of the $(A_{3},A_{3})$
theory is identified as

$\displaystyle\mathcal{Z}_{(A_{3},A_{3})}=\frac{\mathcal{Z}_{U(2)}}{\mathcal{Z}_{U(1)}}~{}.$
(5.9)

While it is beyond the scope of this paper to determine the $U(1)$ factor, we
know that $\mathcal{Z}_{U(1)}$ is independent of the parameter $a$, since $a$
is the VEV of a scalar in the $SU(2)$ vector multiplet that is neutral under
the $U(1)$. Below, we use this fact and compute the classical limit of
$\mathcal{Z}_{(A_{3},A_{3})}$.

###  5.1 Prepotential

Here we consider the classical limit $\epsilon_{i}\to 0$, and compute the
prepotential of the $(A_{3},A_{3})$ theory

$\displaystyle\mathcal{F}^{(A_{3},A_{3})}\equiv\lim_{\epsilon_{i}\to
0}\left(-\epsilon_{1}\epsilon_{2}\log\mathcal{Z}_{(A_{3},A_{3})}\right)~{}.$
(5.10)

This prepotential splits into the perturbative and instanton parts as
$\mathcal{F}^{(A_{3},A_{3})}=\mathcal{F}_{\text{pert}}^{(A_{3},A_{3})}+\mathcal{F}^{(A_{3},A_{3})}_{\text{inst}}$,
and we are particularly interested in the instanton part
$\mathcal{F}_{\text{inst}}^{(A_{3},A_{3})}$.171717Note that the perturbative
part contains the prepotential of the $(A_{1},D_{4})$ theories (with their
flavor symmetries ungauged). The instanton part is generally expanded as

$\displaystyle\mathcal{F}^{(A_{3},A_{3})}_{\text{inst}}=\sum_{k=1}^{\infty}\mathcal{F}_{k}q^{k}~{}.$
(5.11)

Below, we will compute the coefficients, $\mathcal{F}_{k}$, in this expansion.

To that end, let us first consider

$\displaystyle\mathcal{F}^{U(2)}\equiv\lim_{\epsilon_{i}\to
0}\left(-\epsilon_{1}\epsilon_{2}\log\mathcal{Z}_{U(2)}\right)~{},$ (5.12)

which is the prepotential of the theory with the gauge group replaced by
$U(2)$. This prepotential also splits into the perturbative part,
${\displaystyle\lim_{\epsilon_{i}\to
0}(-\epsilon_{1}\epsilon_{2}\log\mathcal{Z}_{\text{pert}}^{U(2)})}$, and the
instanton part

$\displaystyle\mathcal{F}^{U(2)}_{\text{inst}}\equiv\lim_{\epsilon_{i}\to
0}\left(-\epsilon_{1}\epsilon_{2}\log\frac{\mathcal{Z}^{U(2)}}{\mathcal{Z}^{U(2)}_{\text{pert}}}\right)~{}.$
(5.13)

The instanton part (5.13) is identical to
$\mathcal{F}^{(A_{3},A_{3})}_{\text{inst}}$ up to the contribution of the
$U(1)$ factor. Our strategy is to compute $\mathcal{F}^{U(2)}_{\text{inst}}$
using the formula (5.1), and then strip off the $U(1)$ factor to obtain
$\mathcal{F}_{\text{inst}}^{(A_{3},A_{3})}$.

Note that the computation of $\mathcal{Z}_{U(2)}$ via (5.1) and (5.2)
eventually reduces to evaluating

$\displaystyle\langle a|L_{k_{p}}^{m_{p}}\cdots
L_{k_{1}}^{m_{1}}a_{\ell_{q}}^{n_{q}}\cdots
a_{\ell_{1}}^{n_{1}}|\widehat{I}^{(2)}\rangle~{},$ (5.14)

for positive integers $k_{i},m_{i},\ell_{j}$ and $n_{j}$. Using (5.4)–(5.6)
and the fact that
$L_{k}|\widehat{I}^{(2)}\rangle=a_{\ell}|\widehat{I}^{(2)}\rangle=0$ for $k>4$
and $\ell>2$, this computation further reduces to evaluating

$\displaystyle\langle
a|L_{1}^{k}|\widehat{I}^{(2)}\rangle=\left(c_{2}\frac{\partial}{\partial
c_{1}}-2c_{1}(c_{0}-Q)\right)^{k}\langle a|\widehat{I}^{(2)}\rangle~{},$
(5.15)

where $\langle a|\widehat{I}^{(2)}\rangle$ is the partition function of
$(A_{1},D_{4})$ theory (with its flavor symmetry ungauged). Therefore, to
compute $\mathcal{Z}_{U(2)}$ for general $\Omega$-background parameters, one
needs to know how $\langle a|\widehat{I}^{(2)}\rangle$ depends on
$c_{1}$.181818The $1/c_{1}$-expansion of $\langle a|I^{(2)}\rangle$ was
carefully studied in [27]. However, in the classical limit $\epsilon_{i}\to
0$, one can skip this procedure. Indeed, recovering the full
$\epsilon_{i}$-dependence by $c_{k}\to c_{k}/\sqrt{\epsilon_{1}\epsilon_{2}}$,
we see that in the classical limit

$\displaystyle\langle
a|L_{1}^{k}|\widehat{I}^{(2)}\rangle=(c_{2})^{\frac{k}{2}}(-u)^{k}\langle
a|\widehat{I}^{(2)}\rangle~{},$ (5.16)

where $u$ is defined by Eq. (5.3). Given (5.4)–(5.6) and (5.16), it is
straightforward to compute the classical limit of $\mathcal{Z}_{U(2)}$, and
therefore $\mathcal{F}_{\text{inst}}^{U(2)}$, order by order in $q$.

We now turn to the $U(1)$ factor. While it is generically non-vanishing, the
contribution from the $U(1)$-factor turns out to vanish when all the
dimensionful parameters in four-dimensions, except for $a$ and $\epsilon_{i}$,
are turned off. Indeed, in the classical limit,
$\mathcal{Z}_{U(1)}\sim\exp\left(-\frac{1}{\epsilon_{1}\epsilon_{2}}\mathcal{F}_{U(1)}\right)$
with $\mathcal{F}_{U(1)}$ being independent of $\epsilon_{i}$. When
dimensionful parameters are turned off except for $a$ and $\epsilon_{i}$ ,
$\mathcal{F}_{U(1)}$ must be proportional to $a^{2}$ for dimensional reasons.
However, as discussed below (5.9), $\mathcal{Z}_{U(1)}$ must be independent of
$a$. This means that the proportionality constant is zero so that
$\mathcal{F}_{U(1)}=0$.

Let us now focus on the case in which $M,m_{i},d_{i}$ and $u_{i}$ in (5.1) are
all turned off. Then the only non-vanishing dimensionful parameters are $a$
and $\epsilon_{i}$. Since the $U(1)$-factor is trivial in this case, one can
identify (5.10) with (5.12), and therefore (5.11) with (5.13). With this
identification, we finally obtain

$\displaystyle\mathcal{F}^{(A_{3},A_{3})}_{\text{inst}}(q;a)=\left(\frac{1}{4}q^{2}+\frac{13}{128}q^{4}+\frac{23}{384}q^{6}+\frac{2701}{65536}q^{8}+\cdots\right)a^{2}~{}.$
(5.17)

Remarkably, this expression is closely related to the instanton part of the
prepotential of the $SU(2)$ gauge theory with four fundamental flavors.
Indeed, when all the mass parameters are turned off, the latter is given by

$\displaystyle\mathcal{F}_{\text{inst}}^{N_{f}=4}(q;a)=\left(\frac{1}{2}q+\frac{13}{64}q^{2}+\frac{23}{192}q^{3}+\frac{2701}{32768}q^{4}+\cdots\right)a^{2}~{},$
(5.18)

as shown in Appendix B.3 of [16]. Comparing (5.17) and (5.18), we see that

$\displaystyle
2\mathcal{F}_{\text{inst}}^{(A_{3},A_{3})}(q;a)=\mathcal{F}_{\text{inst}}^{N_{f}=4}(q^{2},a)~{},$
(5.19)

at least up to $\mathcal{O}(q^{8})$.

###  5.2 S-duality

Here, we show that one can read off the action of the S-duality group on the
$(A_{3},A_{3})$ theory assuming the remarkable identity (5.19) extends to the
full prepotential. To that end, let us first give a quick review of the
S-duality of $SU(2)$ gauge theory with four fundamental flavors. When the mass
parameters are turned off, the full prepotential of this theory is written as

$\displaystyle\mathcal{F}^{N_{f}=4}=(\log q_{\text{IR}})a^{2}~{},$ (5.20)

where $q_{\text{IR}}$ is related to the IR theta angle and electric coupling
by

$\displaystyle
q_{\text{IR}}=e^{i\theta_{\text{IR}}-\frac{8\pi^{2}}{g_{\text{IR}}^{2}}}~{}.$
(5.21)

The full prepotential (5.20) is the sum of the instanton part (5.18) and the
perturbative part $(\log q-\log 16)a^{2}$. This implies that $q$ and
$q_{\text{IR}}$ are related by [33]

$\displaystyle
q=\frac{\theta_{2}(q_{\text{IR}})^{4}}{\theta_{3}(q_{\text{IR}})^{4}}~{},$
(5.22)

where we used the convention that
$\theta_{2}(q)=\sum_{n\in\mathbb{Z}}q^{(n-\frac{1}{2})^{2}}$ and
$\theta_{3}(q)=\sum_{n\in\mathbb{Z}}q^{n^{2}}$. The relation (5.22) implies
that

$\displaystyle\tau_{\text{IR}}\equiv\frac{1}{\pi i}\log
q_{\text{IR}}=\frac{\theta_{\text{IR}}}{\pi}+\frac{8\pi i}{g_{\text{IR}}^{2}}$
(5.23)

is the modulus of the elliptic curve corresponding to the double cover of
$\mathbb{CP}^{1}$ with four branch points whose cross-ratio is $q$. This
elliptic curve is identified as the SW-curve of the theory on the Coulomb
branch. The curve has a natural $PSL(2,\mathbb{Z})$-action generated by

$\displaystyle T:\tau_{\text{IR}}\to\tau_{\text{IR}}+1~{},\qquad
S:\tau_{\text{IR}}\to-\frac{1}{\tau_{\text{IR}}}~{}.$ (5.24)

From (5.22), we see that these $T$ and $S$ transformations are induced by the
following changes of the UV gauge couplings:

$\displaystyle T:q\to\frac{q}{q-1}~{},\qquad S:q\to 1-q~{}.$ (5.25)

Since the SW-curve is invariant under $T$ and $S$, so is the whole BPS
spectrum on the Coulomb branch. It is then natural to expect that the theory
is completely invariant under this $PSL(2,\mathbb{Z})$.191919When mass
parameters are turned on, they are also permuted by the action of
$PSL(2,\mathbb{Z})$. This is the famous S-duality of $SU(2)$ gauge theory with
four flavors.

Let us now turn back to the $(A_{3},A_{3})$ theory. Its quiver description
shown in Fig. 1 has an obvious similarity to the $SU(2)$ gauge theory with
four flavors; it has the same gauge group with the vanishing $\beta$-function.
This similarity has been studied carefully in [3, 4, 5] to show that the IR
physics of the $(A_{3},A_{3})$ theory on its Coulomb branch admits an action
of $PSL(2,\mathbb{Z})$ (see [6, 7, 8, 9, 10] for further studies on this new
class of $\mathcal{N}=2$ S-dualities). The generalization of the
$SO(8)$-triality in this case is carefully discussed in [5]. This
$PSL(2,\mathbb{Z})$-action has only been studied in the IR language such as
the SW curve, the associated Calabi-Yau three-fold, and the spectrum of BPS
states on the Coulomb branch. Here we discuss the action of
$PSL(2,\mathbb{Z})$ on the UV gauge coupling, using the surprising relation
(5.19).

To that end, let us define the IR gauge coupling $q_{\text{IR}}$ of the
$(A_{3},A_{3})$ theory similarly by

$\displaystyle
q_{\text{IR}}=e^{i\theta_{\text{IR}}-\frac{8\pi^{2}}{g_{\text{IR}}^{2}}}$
(5.26)

so that the full prepotential is written as

$\displaystyle\mathcal{F}^{(A_{3},A_{3})}=(\log q_{\text{IR}})a^{2}~{}.$
(5.27)

Assuming the relation (5.19) extends to the full prepotential, one
obtains202020Given the relation (5.19) for the instanton part, this assumption
is rather mild. Indeed, it only requires that the perturbative part
$\mathcal{F}_{\text{pert}}=\mathcal{F}_{\text{cl}}+\mathcal{F}_{\text{1-loop}}$
also satisfies  $\displaystyle
2\mathcal{F}^{(A_{3},A_{3})}_{\text{pert}}(q;a)=\mathcal{F}^{N_{f}=4}_{\text{pert}}(q^{2};a)~{}.$
(5.28) Note that, on both sides of the above relation,
$\mathcal{F}_{\text{pert}}=(\log q+X)a^{2}$ with $X$ being a constant.
Therefore, in proving (5.28), all we need to show is the following equality
between two constants:  $\displaystyle 2X^{(A_{3},A_{3})}=X^{N_{f}=4}~{}.$
(5.29) To prove this, one needs to identify the 1-loop part
$\mathcal{F}_{\text{1-loop}}$ for $(A_{3},A_{3})$, which we leave for future
work.

$\displaystyle
2\mathcal{F}^{(A_{3},A_{3})}(q;a)=\mathcal{F}^{N_{f}=4}(q^{2};a)~{},$ (5.30)

which implies that $q_{\text{IR}}$ is related to $q$ by

$\displaystyle
q^{2}=\frac{\theta_{2}(q_{\text{IR}}^{2})^{4}}{\theta_{3}(q_{\text{IR}}^{2})^{4}}~{}.$
(5.31)

This relation means that

$\displaystyle\tau_{\text{IR}}\equiv\frac{2}{\pi i}\log
q_{\text{IR}}=\frac{2\theta_{\text{IR}}}{\pi}+\frac{16\pi
i}{g_{\text{IR}}^{2}}$ (5.32)

is the modulus of the double cover of $\mathbb{CP}^{1}$ with four branch
points whose cross-ratio is $q^{2}$. The $T$ and $S$ transformations of the
S-duality group are identified as

$\displaystyle T:\tau_{\text{IR}}\to\tau_{\text{IR}}+1~{},\qquad
S:\tau_{\text{IR}}\to-\frac{1}{\tau_{\text{IR}}}~{}.$ (5.33)

Note that (5.31) reveals a non-trivial relation between the IR gauge coupling
$q_{\text{IR}}$ and the UV gauge coupling $q$ of the $(A_{3},A_{3})$ theory.
Compared to (5.22) for $SU(2)$ gauge theory with four flavors, both the UV and
IR gauge couplings are replaced by their squares here. While the replacement
$q\to q^{2}$ can be easily understood from the relation (5.30), the
replacement

$\displaystyle q_{\text{IR}}\to q_{\text{IR}}^{2}$ (5.34)

is a bit more non-trivial. We see that this replacement is a consequence of
the factor $2$ in front of $\mathcal{F}^{(A_{3},A_{3})}$ in (5.30). Note that
(5.34) is crucial to have the correct weak-coupling behavior. Indeed, in the
weak coupling limit, quantum corrections to the IR gauge coupling vanish, and
therefore we expect $q=q_{\text{IR}}$. We see that (5.31) correctly reduces to
$q\sim q_{\text{IR}}$ in the weak coupling limit $q_{\text{IR}}\to 0$ if
(5.34) is simultaneously performed with $q\to q^{2}$.

From the above discussion, we see how $S$ and $T$ act on the UV gauge coupling
of the $(A_{3},A_{3})$ theory. Indeed, combining (5.31) and (5.32), we see
that (5.33) corresponds to

$\displaystyle T:\;q^{2}\to\frac{q^{2}}{q^{2}-1}~{},\qquad S:\;q^{2}\to
1-q^{2}~{}.$ (5.35)

One can explicitly check that (5.17) combined with the classical part
$\mathcal{F}^{(A_{3},A_{3})}_{\text{cl}}=(\log q)a^{2}$ is invariant under
$T$.

###  5.3 Peculiarity of $T$

From (5.32), we see that our $T$-transformation,
$\tau_{\text{IR}}\to\tau_{\text{IR}}+1$, corresponds to

$\displaystyle\theta_{\text{IR}}\to\theta_{\text{IR}}+\frac{\pi}{2}~{}.$
(5.36)

This is remarkable since this means that $T$ maps the monopole of the minimal
magnetic charge to a dyon that has half the electric charge of fundamental
quark. This is not possible if this “electric charge” is a charge associated
with the unbroken $U(1)\subset SU(2)$ gauge group on the Coulomb branch, since
in that case the minimal electric charge is the charge of fundamental quark.
Therefore, the “electric charge” here is not simply associated with the
unbroken $U(1)\subset SU(2)$. As we will argue in Sec. 6, the “electric
charge” here is interpreted as a linear combination of the electric charge
associated with $U(1)\subset SU(2)$ and those arising from the $(A_{1},D_{4})$
theories.

In the rest of this sub-section, we show that the $T$ transformation (5.36) is
also consistent with the SW curve of $(A_{3},A_{3})$ theory. To that end,
first recall that the SW curve of this theory is written as [3]

$\displaystyle 0$
$\displaystyle=x^{4}+\mathtt{q}x^{2}z^{2}+z^{4}+c_{3,0}x^{3}+c_{0,3}z^{3}+c_{2,0}x^{2}+mxz+c_{0,2}z^{2}+c_{1,0}x+c_{0,1}z+c_{0,0}~{},$
(5.37)

where $\mathtt{q}$ is a function of $q_{\text{IR}}$, $c_{3,0}$ and $c_{0,3}$
are relevant couplings of dimension $1/2$, $c_{2,0},\,c_{0,2}$ and $m$ are
mass parameters, and $c_{1,0},\,c_{0,1}$ and $c_{0,0}$ parameterize the
Coulomb branch moduli space. The SW 1-form is given by $xdz$. It was shown in
[3] that the above curve and 1-form are invariant under two transformations
$\widetilde{S}$ and $\widetilde{T}$, which were interpreted as two independent
S-dual transformations. In particular, $\widetilde{S}$ acts on the marginal
gauge coupling as212121The $\widetilde{T}$-transformation acts on the gauge
coupling as $\widetilde{T}:\mathsf{q}\to\frac{12-2\mathsf{q}}{2+\mathsf{q}}$.
Our $T$ and $S$ correspond to $\widetilde{S}$ and $\widetilde{T}$ in [3],
respectively.

$\displaystyle\widetilde{S}:\;\mathsf{q}\to-\mathsf{q}~{}.$ (5.38)

Below, we show that (5.38) is identical to our $T$-transformation (5.36), near
a cusp on the conformal manifold.222222Here, the “conformal manifold” is
defined as the space of values of exactly marginal couplings in the theory. As
shown in [3], $\mathsf{q}\to\infty$ corresponds to a weak-coupling cusp on the
conformal manifold, where $M\equiv m/\mathsf{q}$ and $u\equiv
c_{0,0}/\mathsf{q}$ are respectively identified as the mass of the fundamental
hypermultiplet and the VEV of the Coulomb branch operator in the vector
multiplet, in the quiver description in Fig. 1.232323There are also other
cusps at $\mathsf{q}\to\pm 2$, where the parameters in the SW curve have
different interpretations. In particular, the mass of the fundamental
hypermultiplet is identified with some linear combination of $m,c_{2,0}$ and
$c_{0,2}$. We focus on this cusp, and turn on an infinitely large mass, $M$,
for the fundamental hypermultiplet so that it decouples from the theory in the
infrared. When the hypermultiplet decouples, the theory reduces to a non-
conformally gauged AD theory described in Fig. 2. To realize this limit at the
level of the SW curve, one needs to take $\mathsf{q}\to\infty$ simultaneously
with $M\to\infty$. Indeed, if we take $M\to\infty$ with $\mathsf{q}$ kept
fixed, some periods of the curve are divergent. It turns out that all the
periods of the curve remain finite when one takes $M\to\infty$ and
$\mathsf{q}\to\infty$ with

$\displaystyle\Lambda\equiv\frac{M}{\sqrt{\mathsf{q}}}$ (5.39)

kept fixed. This $\Lambda$ is then identified as a dynamical scale of the
resulting theory. The curve correctly reduces to the curve of the IR theory
(4.17) (for $n=2$) in the limit $M,\mathsf{q}\to\infty$ with (5.39) kept
finite. 242424One can show this explicitly as follows. As show in [3], near
the cusp $\mathsf{q}\to\infty$,
$C_{3,0}\equiv\mathsf{q}^{-\frac{1}{4}}c_{3,0}$ and
$C_{0,3}\equiv\mathsf{q}^{-\frac{1}{4}}c_{0,3}$ are identified with the
(correctly-normalized) relevant couplings of dimension $1/2$,
$C_{2,0}\equiv\mathsf{q}^{-\frac{1}{2}}c_{2,0}$ and
$C_{0,2}\equiv\mathsf{q}^{-\frac{1}{2}}c_{0,2}$ are mass deformation
parameters, and $C_{1,0}\equiv\mathsf{q}^{-\frac{3}{2}}c_{1,0}$ and
$C_{0,1}\equiv\mathsf{q}^{-\frac{3}{4}}c_{0,1}$ are the VEVs of Coulomb branch
operators, in the $(A_{1},D_{4})$ sectors. In terms of these variables, the
curve of the $(A_{3},A_{3})$ theory is written as
$0=x^{4}+\mathsf{q}x^{2}z^{2}+z^{4}+\mathsf{q}^{\frac{1}{4}}C_{3,0}x^{3}+\mathsf{q}^{\frac{1}{4}}C_{0,3}z^{3}+\mathsf{q}^{\frac{1}{2}}C_{2,0}x^{2}+\mathsf{q}Mxz+\mathsf{q}^{\frac{1}{2}}C_{0,2}z^{2}+\mathsf{q}^{\frac{3}{4}}C_{1,0}x+\mathsf{q}^{\frac{3}{4}}C_{0,1}z+\mathsf{q}u$.
We here define $X\equiv
i(\sqrt{z}x^{\frac{3}{2}}+\frac{1}{2}\sqrt{q}\Lambda\sqrt{x/z}),\,Z\equiv\sqrt{z/x}$
and $U\equiv u-\frac{q\Lambda^{2}}{4}$, and take the limit
$M,\mathsf{q}\to\infty$ with $\Lambda,U,C_{i,j}$ and $(X,Y)$ kept finite. Then
the curve reduces to  $\displaystyle X^{2}$
$\displaystyle=\widetilde{\Lambda}^{2}Z^{2}+\widetilde{\Lambda}^{\frac{3}{2}}C_{0,3}Z+\widetilde{\Lambda}C_{0,2}+\frac{\widetilde{\Lambda}^{\frac{1}{2}}C_{0,1}}{Z}+\frac{U}{Z^{2}}+\frac{\widetilde{\Lambda}^{\frac{1}{2}}C_{1,0}}{Z^{3}}+\frac{\widetilde{\Lambda}C_{2,0}}{Z^{4}}+\frac{\widetilde{\Lambda}^{\frac{3}{2}}C_{3,0}}{Z^{5}}+\frac{\widetilde{\Lambda}^{2}}{Z^{6}}~{},$
(5.40) where $\widetilde{\Lambda}\equiv-\Lambda/2$. The SW 1-form is written
as $-\frac{3}{2}iXdZ$ up to exact terms. This curve is precisely identical to
(4.17) for $n=2$. The 1-form is also identical up to a prefactor that can be
absorbed by rescaling dimensionful parameters and $X$. This implies that
(5.39) is the correct identification of the IR dynamical scale.

Note here that, by standard arguments, the dynamical scale $\Lambda$ of the
mass-deformed theory and the gauge coupling
$e^{i\theta_{\text{IR}}-\frac{8\pi^{2}}{g_{\text{IR}}^{2}}}$ of the conformal
theory are related by

$\displaystyle\left(\frac{\Lambda}{M}\right)^{b_{0}}=e^{i\theta_{\text{IR}}-\frac{8\pi^{2}}{g_{\text{IR}}^{2}}}~{},$
(5.41)

where $b_{0}$ is the coefficient of the one-loop $\beta$-function of the IR
theory. Since $b_{0}=1$ in our case, (5.39) and (5.41) imply that

$\displaystyle\mathsf{q}=e^{-2i\theta_{\text{IR}}+\frac{16\pi^{2}}{g_{\text{IR}}^{2}}}~{}.$
(5.42)

From this relation, it is now clear that the $\widetilde{S}$ transformation
(5.38) is precisely identical to our $T$-transformation (5.36). Therefore, our
$T$-transformation (5.36) is perfectly consistent with the earlier analysis of
the S-duality using the SW-curve.

###  5.4 Turning on couplings, masses and VEVs

In the previous sections, we have identified the $PSL(2,\mathbb{Z})$-action in
the case of vanishing dimensionful parameters except for $\epsilon_{i}$ and
$a$. This action has been interpreted as corresponding to the symmetry of the
SW-curve studied in [3]. Since this symmetry of the curve extends to the case
of generic values of dimensionful parameters, we expect that the
$PSL(2,\mathbb{Z})$-action on the partition function has a similar extension
for non-vanishing relevant couplings, masses and VEVs of Coulomb branch
operators. In this sub-section, we discuss such an extension of the
$PSL(2,\mathbb{Z})$-action.

The instanton part of the prepotential for generic values of dimensionful
parameters is expanded as

$\displaystyle\mathcal{F}^{(A_{3},A_{3})}_{\text{inst}}$
$\displaystyle=\sum_{k=-1}^{\infty}\mathcal{F}_{2k}(q,M,\\{m_{i}\\},\\{u_{i}\\},\\{d_{i}\\})a^{-2k}~{}.$
(5.43)

Recall that $T$ and $S$ act on the UV gauge coupling as in (5.35). Since $S$
is non-perturbative in $q$, it is hard to understand how (5.43) behaves under
$S$ using our order-by-order computation. We therefore focus on the
$T$-transformation below.

In the case of $d_{i}=m_{i}=u_{i}=M=0$, (5.43) reduces to

$\displaystyle\mathcal{F}_{-2}(q)a^{2}~{},$ (5.44)

whose explicit expression is shown in Eq. (5.17). When combined with the
perturbative part, this is invariant under $T:q^{2}\to q^{2}/(q^{2}-1)$.

When $d_{i},m_{i},u_{i}$ and $M$ are turned on, (5.44) receives corrections
from $\mathcal{F}_{2k}$ for all $k\geq 0$. We evaluated these corrections
using our formula (5.2) to find that $\mathcal{F}_{-2}$ and
$\mathcal{F}_{k>0}$ are all invariant under

$\displaystyle q$ $\displaystyle\to\frac{iq}{\sqrt{1-q^{2}}}~{},\quad
d_{1}\to\frac{d_{1}+qd_{2}}{\sqrt{1-q^{2}}}~{},\quad d_{2}\to id_{2}~{},\quad
m_{2}\to-m_{2}~{},\quad u_{2}\to-iu_{2}~{},$ (5.45)

with the other parameters kept fixed.252525Note that $\mathcal{F}_{0}$ cannot
be evaluated without identifying the $U(1)$-factor, and therefore we leave the
computation of $\mathcal{F}_{0}$ for future work. The other terms,
$\mathcal{F}_{-2}$ and $\mathcal{F}_{k>0}$, are not affected by the
$U(1)$-factor. We checked this invariance up to a very high order of $q$.
First few terms in the $q$-series of $\mathcal{F}_{-2},\mathcal{F}_{2}$ and
$\mathcal{F}_{4}$ are shown in appendix C. Given this invariance, we interpret
(5.45) as an extension of $q^{2}\to q^{2}/(q^{2}-1)$ to the case of non-
vanishing $d_{i},m_{i},u_{i}$ and $M$.

It turns out that (5.45) is consistent with the symmetry of the SW curve, at
least in the weak coupling limit. To see this, recall that our
$T$-transformation is identified with $\widetilde{S}$ discussed in [3]. This
$\widetilde{S}$ induces $\mathsf{q}\to-\mathsf{q}$, as reviewed already, and
also changes the other parameters in the SW curve as

$\displaystyle c_{3,0}\to-e^{\frac{3\pi i}{4}}c_{3,0}~{},\quad
c_{0,3}\to-e^{-\frac{3\pi i}{4}}c_{0,3}~{},\quad c_{2,0}\to-ic_{2,0}~{},\quad
c_{0,2}\to ic_{0,2}~{},$ $\displaystyle m\to-m~{},\quad
c_{1,0}\to-e^{\frac{\pi i}{4}}c_{1,0}~{},\quad c_{0,1}\to-e^{-\frac{\pi
i}{4}}c_{0,1}~{},\quad c_{0,0}\to-c_{0,0}~{}.$ (5.46)

It is straightforward to show that the curve (5.37) and the SW 1-form is
invariant under these transformations. Since $q^{2}\to q^{2}/(q^{2}-1)$ is
already identified with $\mathsf{q}\to-\mathsf{q}$, (5.46) is expected to be
equivalent to (5.45). One can show this equivalence explicitly at the weak-
coupling cusp $\mathsf{q}=\infty$. As shown in [3], in the limit
$\mathsf{q}\to\infty$, the coefficients $c_{i,j}$ must be appropriately
renormalized so that physical quantities remain finite. In terms of our
$d_{i},m_{i},u_{i},M$ and $u$, such a renormalization is expressed as

$\displaystyle c_{3,0}=\mathtt{q}^{\frac{1}{4}}d_{1}~{},\quad
c_{0,3}=\mathtt{q}^{\frac{1}{4}}d_{2}~{},\quad
c_{2,0}=\mathtt{q}^{\frac{1}{2}}m_{1}~{},\quad
c_{0,2}=\mathtt{q}^{\frac{1}{2}}m_{2}~{},\quad m=\mathtt{q}M~{},$
$\displaystyle c_{1,0}=\mathtt{q}^{\frac{3}{4}}u_{1}~{},\quad
c_{0,1}=\mathtt{q}^{\frac{3}{4}}u_{2}~{},\quad c_{0,0}=\mathtt{q}u~{}.$ (5.47)

From (5.47) and (5.46), we see that $\widetilde{S}$ is equivalent to
$\mathsf{q}\to\exp(-\pi i)\mathsf{q}$ combined with

$\displaystyle d_{2}\to id_{2}~{},\qquad m_{2}\to-m_{2}~{},\qquad u_{2}\to-
iu_{2}~{},$ (5.48)

where $d_{1},m_{1},u_{1},M$ and $u$ are kept fixed. Since
$\mathsf{q}\to\exp(-\pi i)\mathsf{q}$ is identified with $q^{2}\to
q^{2}/(q^{2}-1)$, this is identical to (5.45) at the leading order of $q$. The
reason that we only see the leading order terms is that we are taking the
weak-coupling limit $\mathtt{q}\to\infty$ here, corresponding to $q\to 0$.
Thus, we have shown that our (5.45) is equivalent to (5.46) in the weak-
coupling limit.

Note that the invariance of $\mathcal{F}_{\text{inst}}^{(A_{3},A_{3})}$ under
(5.45) strongly constrains the possible form of $\mathcal{F}_{k>0}$. In
particular, when combined with the trivial symmetry under exchanging two
$(A_{1},D_{4})$ theories in the quiver diagram, imposing this invariance
determines $\mathcal{F}_{k>0}$ up to some $T$-invariant functions for every
$k$. We explicitly show this for $\mathcal{F}_{2}$ in appendix C.

Note also that our check of the invariance under (5.45) is only for the
instanton part of the prepotential. For the full prepotential to be invariant
under (5.45), the perturbative part must also be invariant by itself. One can
show that the perturbative part is invariant under (5.45) when $q=0$, but its
extension to the case of $q\neq 0$ is not straightforward. If (5.45) also
preserves the perturbative part for general $q$, it is natural to identify
(5.45) as the $T$-transformation for generic values of $d_{i},m_{i}$ and
$u_{i}$. If the invariance under (5.45) does not extend to the perturbative
part, (5.45) is a new symmetry that only preserves the instanton part. We
leave a detailed study of these two options for future work.

##  6 Conclusions and discussions

In this paper, we have proposed a Nekrasov-type formula for the instanton
partition function of four-dimensional $\mathcal{N}=2\;\,U(2)$ gauge theories
coupled to $(A_{1},D_{2n})$ Argyres-Douglas theories, by extending the
generalized AGT correspondence to the case of $U(2)$ gauge group. We have
defined irregular states of the direct sum of Virasoro and Heisenberg
algebras, and then identified the contribution of the $(A_{1},D_{2n})$ theory
at each fixed points on the $U(2)$ instanton moduli space, as in (4.12) and
(4.13). Here, 4d and 2d parameters are related by (4.9).

As an application, we evaluate the instanton partition function of the
$(A_{3},A_{3})$ theory. While this partition function cannot be directly
evaluated by the AGT correspondence, we have computed it using our formula
above. Our result shows that, when some parameters are turned off, the
instanton part of the prepotential of $(A_{3},A_{3})$ is in a surprising
relation to that of $SU(2)$ gauge theory with four flavors, as shown in
(5.19). In particular, the two prepotentials are related by the following
replacement of the UV gauge coupling:

$\displaystyle q\to q^{2}~{}.$ (6.1)

From this relation, we have read off the action of the S-duality group on the
UV gauge coupling $q$. We have also discussed its possible extension to
dimensionful parameters.

Here, we give a brief comment on our peculiar $T$-transformation. As shown in
Sec. 5.3, the $T$-transformation, $\tau_{\text{IR}}\to\tau_{\text{IR}}+1$,
corresponds to

$\displaystyle\theta_{\text{IR}}\to\theta_{\text{IR}}+\frac{\pi}{2}~{},$ (6.2)

and therefore maps the monopole of the minimal magnetic charge to a dyon that
has half the electric charge of fundamental quark. This implies that the
“electric charge” here is not simply the electric charge associated with the
$SU(2)$ vector multiplet in Fig. 1. Instead, this electric charge is
interpreted as a linear combination of the electric charge associated with the
$SU(2)$ vector multiplet sector and those associated with the $(A_{1},D_{4})$
sectors. Indeed, it was shown in [5] that $PSL(2,\mathbb{Z})$ naturally acts
on a modified electro-magnetic charge lattice where the “electric charge” is
such a linear combination of charges arising from the three sectors.262626The
“magnetic charge” here is simply associated with the $SU(2)$ vector multiplet
without mixing with those arising from the $(A_{1},D_{4})$ sectors. This
linear combination naturally arises when constructing the $(A_{3},A_{3})$
theory as the IR limit of type IIB string theory on a Calabi-Yau singularity.
With this modified charge lattice, the minimal electric charge is now half the
charge of fundamental quark,272727This can be seen from the discussions around
Eq. (2.64) of [5], where $\mathtt{deg}\,X$ and $\mathtt{rank}\,X$ are
respectively the (modified) electric and magnetic charges naturally acted on
by $PSL(2,\mathbb{Z})$. Their discussions here apply to a larger class of
theories labeled by $p=2,3,4$ and $6$, and the $(A_{3},A_{3})$ theory
corresponds to the $p=4$ case. As the authors explain there, $\mathtt{deg}\,X$
is quantized in the unit of $1/p=1/4$ when it is normalized so that the
W-boson has charge $1$. This means that the minimal value of the modified
electric charge, $\mathtt{deg}\,X$, is half the charge of fundamental
quark.which is consistent with our T-transformation (6.2).

There are obviously many interesting open problems. We list some of them
below.

  * •

While we have focused on the $(A_{1},D_{k})$ theories for even $k$, there are
also those theories for odd $k$. Since they also have $SU(2)$ flavor symmetry
that can be gauged, one can consider the generalization of our work to this
latter class of theories. One difficulty is that the relevant irregular state
in this case cannot be obtained in a colliding limit. Therefore, it is not
straightforward to derive the action of the Heisenberg algebra on the
irregular state.

  * •

It is also interesting to generalize our work to $SU(N)$ gauge theories
coupled to AD theories. For that, we need a $U(N)$-version of the generalized
AGT correspondence. A careful study on the $SU(3)$-version of the generalized
AGT correspondence has been carried out in [21].

  * •

It would be interesting to study the Nekrasov-Shatashvili limit [34] of the
$\Omega$-deformed $(A_{3},A_{3})$ theory. In this limit, combining the results
of [35, 36, 37, 38] with our formula, one can evaluate the deformed
prepotential of the $(A_{3},A_{3})$ theory including both the perturbative and
instanton parts.

  * •

The uplift of the AGT correspondence to five dimensions are known [39, 40, 41,
42, 43, 44, 45, 46, 47, 48]. It would be interesting to search for a 5d uplift
of our results. That would shed some light on the relation between the 4d
$D_{2n}(SU(2))$ theory and the 5d $\widehat{D}_{2n}(SU(2))$ theory [4, 49].
More generally, it would be interesting to study how our results are phrased
in terms of the $W_{1+\infty}$ algebra and the DIM algebra along the lines of
[50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62].

  * •

It was shown in [32] that the Schur indices of the $(A_{3},A_{3})$ theory and
$SU(2)$ gauge theory with four flavors are related by a change of variables
involving

$\displaystyle q\to q^{2}~{},$ (6.3)

where $q$ is now the superconformal fugacity of the index. Although this $q$
is different from our $q$ in (6.1), it would be very interesting to study how
these two peculiar relations are connected.

  * •

Our discussion in the previous bullet suggests that the Schur index and the
prepotential might generally be related. Since the Schur index is identified
as the character of the associated chiral algebra [63], this then suggests a
possible connection between the chiral algebra and the prepotential. Such a
connection has already been suggested in the application of the ODE/IM
correspondence to quantum SW-curves [64]. It would be interesting to study
this connection further.

  * •

A replacement similar to (6.1) arises when considering the Nekrasov partition
functions of $SO/Sp$ gauge theories in connection to $SU$ gauge theories [65,
66]. This seems to suggest that a $\mathbb{Z}_{2}$-orbifolding plays some role
similarly in our case. It would be interesting to study this possibility
further.

## Acknowledgements

We would like to thank Matthew Buican, Katsushi Ito, Kazunobu Maruyoshi,
Hiraku Nakajima, Satoshi Nawata, Takuya Okuda, Jaewon Song, Yuji Tachikawa,
and Wenbin Yan for various illuminating discussions. We would like to
particularly thank Kazunobu Maruyoshi for drawing our attention to the
reference [31] when one of us (T. N.) had a discussion with him during the
international conference “KEK Theory Workshop 2018.” T. N. also thanks KEK
theory group for hosting the wonderful workshop with various opportunities of
discussions. T. N.’s research is partially supported by JSPS Grant-in-Aid for
Early-Career Scientists 18K13547. The work of T. U. is supported by Grant-in-
Aid for JSPS Research Fellows 19J11212.

##  Appendix A Nekrasov’s formula

Here, we list formulae for contributions to the Nekrasov partition function of
$U(2)$ gauge theories. As discussed around Eq. (1.1), the instanton part of
the partition function is written as a sum over pairs of Young diagrams,
$(Y_{1},Y_{2})$. For each $(Y_{1},Y_{2})$, the $U(2)$ vector multiplet
contributes the following factor

$\displaystyle\mathcal{Z}^{\text{vec}}_{Y_{1},Y_{2}}(a)$
$\displaystyle\equiv\prod_{i,j=1}^{2}\prod_{s\in
Y_{i}}\frac{1}{-E_{Y_{i},Y_{j}}(a_{i}-a_{j},s)+\epsilon_{1}+\epsilon_{2}}\prod_{t\in
Y_{j}}\frac{1}{E_{Y_{j},Y_{i}}(a_{j}-a_{i},t)}~{},$ (A.1)

where $a_{1}=-a_{2}=a$ corresponds to the Coulomb branch parameter, and

$\displaystyle E_{Y_{1},Y_{2}}(a,s)$ $\displaystyle\equiv
a-\epsilon_{1}L_{Y_{2}}(s)+\epsilon_{2}(A_{Y_{1}}(s)+1)~{}.$ (A.2)

Here $L_{Y}(s)$ and $A_{Y}(s)$ are defined as follows. For a Young diagram
$Y=\\{\lambda_{1}\geq\lambda_{2}\geq\cdots\\}$, we denote its transpose by
$\\{\lambda_{1}^{\prime}\geq\lambda_{2}^{\prime}\geq\cdots\\}$. We also denote
by $s=(i,j)$ a position in a Young diagram $Y$. Then the leg-length and arm-
length are defined by $L_{Y}(s)\equiv\lambda_{j}^{\prime}-i$ and
$A_{Y}(s)\equiv\lambda_{i}-j$, respectively.

The contribution from a fundamental hypermultiplet of $U(2)$ is given by

$\displaystyle\mathcal{Z}^{\text{fund}}_{Y_{1},Y_{2}}(a,m)$
$\displaystyle\equiv\prod_{i=1}^{2}\prod_{s\in
Y_{i}}\left(\phi(a_{i},s)-m+\epsilon_{1}+\epsilon_{2}\right)~{},$ (A.3)

where

$\displaystyle\phi(a,s)$ $\displaystyle\equiv
a+\epsilon_{1}(i-1)+\epsilon_{2}(j-1)~{},$ (A.4)

and $m$ is the mass of the hypermultiplet. The contribution from a bi-
fundamental hypermultiplet of $U(2)\times U(2)$ is written as

$\displaystyle\mathcal{Z}^{\text{bifund}}_{Y_{1},Y_{2};W_{1},W_{2}}(a,b,\alpha)$
$\displaystyle\equiv\prod_{i,j=1}^{2}\prod_{s\in
Y_{i}}\left(E(a_{i}-b_{j},Y_{i},W_{j},s)-\alpha\right)$
$\displaystyle\qquad\times\prod_{t\in
W_{j}}\left(\epsilon_{1}+\epsilon_{2}-E(b_{j}-a_{i},W_{j},Y_{i},t)-\alpha\right)~{},$
(A.5)

where $(Y_{1},Y_{2})$ and $(W_{1},W_{2})$ correspond to torus fixed points on
the instanton moduli spaces of two $U(2)$ gauge groups, and $a$ and $b$ are
the Coulomb branch parameters of these two $U(2)$. The extra parameter,
$\alpha$, stands for the mass parameter of the bi-fundamental hypermultiplet.

##  Appendix B Orthogonal Basis

Here we list the first few examples of states $|a;Y_{1},Y_{2}\rangle$ in the
basis of the highest weight module of $Vir\oplus H$ that we have used in Sec.
4. These states are defined as solutions to (4.2), and were first found in
[31]. For ease of reference, we here denote the highest weight by $P$ instead
of $a$, as in [31].

To describe these states, we first decompose the space of states by the level
of descendants. Here, the “level” is the sum of the levels of the Virasoro
part and the Heisenberg part, so that $L_{-k_{1}}^{m_{1}}\cdots
L_{-k_{p}}^{m_{p}}a_{-\ell_{1}}^{n_{1}}\cdots a_{-\ell_{q}}^{n_{q}}|P\rangle$
has level $\sum_{i=1}^{p}m_{i}k_{i}+\sum_{j=1}^{q}n_{j}\ell_{j}$. The basis
$\\{|P;Y_{1},Y_{2}\rangle\\}$ found in [31] is such that
$|P;Y_{1},Y_{2}\rangle$ for $|Y_{1}|+|Y_{2}|=k$ span the space of level-$k$
descendants. Recall here that we are rescaling dimensionful parameters so that
$\epsilon_{1}\epsilon_{2}=1$. Therefore we set
$\epsilon_{1}=1/\epsilon_{2}=\mathtt{b}$ when computing the RHS of (4.2). In
this case, the Liouville charge is written as $Q=\mathtt{b}+1/\mathtt{b}$.
Below, we list the first few examples of $|P;Y_{1},Y_{2}\rangle$:

$\displaystyle|P;\emptyset,\emptyset\rangle$ $\displaystyle=|P\rangle~{},$
(B.1) $\displaystyle|P;{\tiny\yng(1)},\emptyset\rangle$
$\displaystyle=\left(-i(\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}-L_{-1}\right)|P\rangle~{},$
(B.2) $\displaystyle|P;{\tiny\yng(1,1)},\emptyset\rangle$
$\displaystyle=\Big{(}-i\mathtt{b}^{-1}(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}+2\mathtt{b}^{-1}+2P)a_{-2}$
$\displaystyle\qquad-\left(\mathtt{b}+\mathtt{b}^{-1}+2P\right)\left(\mathtt{b}+2\mathtt{b}^{-1}+2P\right)a_{-1}^{2}$
$\displaystyle\qquad+2i(\mathtt{b}+2\mathtt{b}^{-1}+2P)a_{-1}L_{-1}-\mathtt{b}^{-1}(\mathtt{b}+\mathtt{b}^{-1}+2P)L_{-2}+L_{-1}^{2}\Big{)}|P\rangle~{},$
(B.3) $\displaystyle|P;{\tiny\yng(2)},\emptyset\rangle$
$\displaystyle=\Big{(}-i\mathtt{b}(\mathtt{b}+\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-2}$
$\displaystyle\qquad-(\mathtt{b}+\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}^{2}$
$\displaystyle\qquad+2i(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}L_{-1}-\mathtt{b}(\mathtt{b}+\mathtt{b}^{-1}+2P)L_{-2}+L_{-1}^{2}\Big{)}|P\rangle~{},$
(B.4) $\displaystyle|P;{\tiny\yng(1)},{\tiny\yng(1)}\rangle$
$\displaystyle=\Big{(}-i(\mathtt{b}+\mathtt{b}^{-1})a_{-2}-(\mathtt{b}^{2}+\mathtt{b}^{-2}+1-4P^{2})a_{-1}^{2}$
$\displaystyle\qquad+2i(\mathtt{b}+\mathtt{b}^{-1})a_{-1}L_{-1}-L_{-2}+L_{-1}^{2}\Big{)}|P\rangle~{},$
(B.5) $\displaystyle|P;{\tiny\yng(1,1,1)},\emptyset\rangle$
$\displaystyle=\Big{(}-2i\mathtt{b}^{-2}(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}+2\mathtt{b}^{-1}+2P)(\mathtt{b}+3\mathtt{b}^{-1}+2P)a_{-3}$
$\displaystyle\qquad-3\mathtt{b}^{-1}(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}+2\mathtt{b}^{-1}+2P)(\mathtt{b}+3\mathtt{b}^{-1}+2P)a_{-2}a_{-1}$
$\displaystyle\qquad+i(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}+2\mathtt{b}^{-1}+2P)(\mathtt{b}+3\mathtt{b}^{-1}+2P)a_{-1}^{3}$
$\displaystyle\qquad+3i\mathtt{b}^{-1}(\mathtt{b}+2\mathtt{b}^{-1}+2P)(\mathtt{b}+3\mathtt{b}^{-1}+2P)a_{-2}L_{-1}$
$\displaystyle\qquad+3(\mathtt{b}+2\mathtt{b}^{-1}+2P)(\mathtt{b}+3\mathtt{b}^{-1}+2P)a_{-1}^{2}L_{-1}$
$\displaystyle\qquad+3i\mathtt{b}^{-1}(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}+3\mathtt{b}^{-1}+2P)a_{-1}L_{-2}$
$\displaystyle\qquad-3i(\mathtt{b}+3\mathtt{b}^{-1}+2P)a_{-1}L_{-1}^{2}-\mathtt{b}^{-2}(\mathtt{b}+4\mathtt{b}^{-1}+4P)(\mathtt{b}+\mathtt{b}^{-1}+2P)L_{-3}$
$\displaystyle\qquad+\mathtt{b}^{-1}(3\mathtt{b}+5\mathtt{b}^{-1}+6P)L_{-2}L_{-1}-L_{-1}^{3}\Big{)}|P\rangle~{},$
(B.6) $\displaystyle|P;{\tiny\yng(2,1)},\emptyset\rangle$
$\displaystyle=\Big{(}-i(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}+2\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-3}$
$\displaystyle\qquad-(\mathtt{b}+\mathtt{b}^{-1})(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}+2\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-2}a_{-1}$
$\displaystyle\qquad+i(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}+2\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}^{3}$
$\displaystyle\qquad+i(\mathtt{b}+\mathtt{b}^{-1})(\mathtt{b}+2\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-2}L_{-1}$
$\displaystyle\qquad+3(\mathtt{b}+2\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}^{2}L_{-1}$
$\displaystyle\qquad+i(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}^{2}+5+\mathtt{b}^{-2}+2(\mathtt{b}+\mathtt{b}^{-1})P)a_{-1}L_{-2}$
$\displaystyle\qquad-i(5\mathtt{b}+5\mathtt{b}^{-1}+6P)a_{-1}L_{-1}^{2}-(\mathtt{b}+\mathtt{b}^{-1}+2P)^{2}L_{-3}$
$\displaystyle\qquad+(\mathtt{b}^{2}+3+\mathtt{b}^{-2}+2(\mathtt{b}+\mathtt{b}^{-1})P)L_{-2}L_{-1}-L_{-1}^{3}\Big{)}|P\rangle~{},$
(B.7) $\displaystyle|P;{\tiny\yng(3)},\emptyset\rangle$
$\displaystyle=\Big{(}-2i\mathtt{b}^{2}(\mathtt{b}+\mathtt{b}^{-1}+2P)(3\mathtt{b}+\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-3}$
$\displaystyle\qquad-3\mathtt{b}(\mathtt{b}+\mathtt{b}^{-1}+2P)(3\mathtt{b}+\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-2}a_{-1}$
$\displaystyle\qquad+i(\mathtt{b}+\mathtt{b}^{-1}+2P)(3\mathtt{b}+\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}^{3}$
$\displaystyle\qquad+3i\mathtt{b}(3\mathtt{b}+\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-2}L_{-1}$
$\displaystyle\qquad+3(3\mathtt{b}+\mathtt{b}^{-1}+2P)(2\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}^{2}L_{-1}$
$\displaystyle\qquad+3i\mathtt{b}(\mathtt{b}+\mathtt{b}^{-1}+2P)(3\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}L_{-2}$
$\displaystyle\qquad-3i(3\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}L_{-1}^{2}-\mathtt{b}^{2}(\mathtt{b}+\mathtt{b}^{-1}+2P)(4\mathtt{b}+\mathtt{b}^{-1}+4P)L_{-3}$
$\displaystyle\qquad+\mathtt{b}(5\mathtt{b}+3\mathtt{b}^{-1}+6P)L_{-2}L_{-1}-L_{-1}^{3}\Big{)}|P\rangle~{},$
(B.8) $\displaystyle|P;{\tiny\yng(1,1)},{\tiny\yng(1)}\rangle$
$\displaystyle=\Big{(}-2i\mathtt{b}^{-1}(\mathtt{b}+\mathtt{b}^{-1})(\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-3}$
$\displaystyle\qquad-\mathtt{b}^{-1}(\mathtt{b}+\mathtt{b}^{-1}+2P)(3\mathtt{b}^{2}+3+2\mathtt{b}^{-2}-2\mathtt{b}^{-1}P-4P^{2})a_{-2}a_{-1}$
$\displaystyle\qquad+i(\mathtt{b}+\mathtt{b}^{-1}+2P)(\mathtt{b}^{2}+1+2\mathtt{b}^{-2}-2\mathtt{b}^{-1}P-4P^{2})a_{-1}^{3}$
$\displaystyle\qquad+i\mathtt{b}^{-1}(3\mathtt{b}^{2}+7+2\mathtt{b}^{-2}+(4\mathtt{b}+6\mathtt{b}^{-1})P+4P^{2})a_{-2}L_{-1}$
$\displaystyle\qquad+(3\mathtt{b}^{2}+7+6\mathtt{b}^{-2}+(4\mathtt{b}+2\mathtt{b}^{-1})P-4P^{2})a_{-1}^{2}L_{-1}$
$\displaystyle\qquad+i\mathtt{b}^{-1}(3\mathtt{b}+\mathtt{b}^{-1}-2P)(\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}L_{-2}$
$\displaystyle\qquad-i(3\mathtt{b}+5\mathtt{b}^{-1}+2P)a_{-1}L_{-1}^{2}-\mathtt{b}^{-1}(\mathtt{b}+\mathtt{b}^{-1}+2P)L_{-3}$
$\displaystyle\qquad+\mathtt{b}^{-1}(3\mathtt{b}+\mathtt{b}^{-1}+2P)L_{-2}L_{-1}-L_{-1}^{3}\Big{)}|P\rangle~{},$
(B.9) $\displaystyle|P;{\tiny\yng(2)},{\tiny\yng(1)}\rangle$
$\displaystyle=\Big{(}-2i\mathtt{b}(\mathtt{b}+\mathtt{b}^{-1})(\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-3}$
$\displaystyle\qquad-\mathtt{b}(\mathtt{b}+\mathtt{b}^{-1}+2P)(2\mathtt{b}^{2}+3+3\mathtt{b}^{-2}-2\mathtt{b}P-4P^{2})a_{-2}a_{-1}$
$\displaystyle\qquad+i(\mathtt{b}+\mathtt{b}^{-1}+2P)(2\mathtt{b}^{2}+1+\mathtt{b}^{-2}-2\mathtt{b}P-4P^{2})a_{-1}^{3}$
$\displaystyle\qquad+i\mathtt{b}(2\mathtt{b}^{2}+7+3\mathtt{b}^{-2}+(6\mathtt{b}+4\mathtt{b}^{-1})P+4P^{2})a_{-2}L_{-1}$
$\displaystyle\qquad+(6\mathtt{b}^{2}+7+3\mathtt{b}^{-2}+(2\mathtt{b}+4\mathtt{b}^{-1})P-4P^{2})a_{-1}^{2}L_{-1}$
$\displaystyle\qquad+i\mathtt{b}(\mathtt{b}+3\mathtt{b}^{-1}-2P)(\mathtt{b}+\mathtt{b}^{-1}+2P)a_{-1}L_{-2}$
$\displaystyle\qquad-i(5\mathtt{b}+3\mathtt{b}^{-1}+2P)a_{-1}L_{-1}^{2}-\mathtt{b}(\mathtt{b}+\mathtt{b}^{-1}+2P)L_{-3}$
$\displaystyle\qquad+\mathtt{b}(\mathtt{b}+3\mathtt{b}^{-1}+2P)L_{-2}L_{-1}-L_{-1}^{3}\Big{)}|P\rangle~{}.$
(B.10)

Note that there are also states obtained by exchanging two Young diagrams in
each of the above, whose expressions are simply obtained via the relation
$|P;Y_{2},Y_{1}\rangle=|-P;Y_{1},Y_{2}\rangle$.

##  Appendix C Prepotential with massive deformations

As discussed in Sec. 5.4, we have checked that the instanton part of the
prepotential of the $(A_{3},A_{3})$ theory is invariant under (5.45), up to
terms affected by the $U(1)$-factor. Indeed, our computation shows that

$\displaystyle\mathcal{F}_{\text{inst}}^{(A_{3},A_{3})}=\sum_{k=-1}^{\infty}\mathcal{F}_{2k}a^{-2k}~{},$
(C.1)

where

$\displaystyle\mathcal{F}_{-2}$
$\displaystyle=q^{2}+\frac{13}{8}q^{4}+\frac{23}{6}q^{6}+\frac{2701}{256}q^{8}+\mathcal{O}(q^{10})~{},$
(C.2) $\displaystyle\mathcal{F}_{2}$
$\displaystyle=\frac{q}{2}Mu_{1}u_{2}+\frac{q^{2}}{16}\big{(}-4d_{1}^{2}d_{2}^{2}M^{2}-8d_{1}^{2}M^{2}m_{2}-d_{1}^{2}u_{2}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad-8d_{2}^{2}M^{2}m_{1}-d_{2}^{2}u_{1}^{2}-16M^{2}m_{1}m_{2}-2m_{1}u_{2}^{2}-2m_{2}u_{1}^{2}\big{)}$
$\displaystyle\quad+\frac{q^{3}}{24}\big{(}12d_{1}^{3}d_{2}M^{2}-6d1^{2}d_{2}Mu_{1}+12d_{1}d_{2}^{3}M^{2}-6d_{1}d_{2}^{2}Mu_{2}+16d_{1}d_{2}M^{3}$
$\displaystyle\qquad\qquad\qquad+24d_{1}d_{2}M^{2}m_{1}+24d_{1}d_{2}M^{2}m_{2}+3d1d_{2}u_{1}^{2}+3d_{1}d_{2}u_{2}^{2}$
$\displaystyle\qquad\qquad\qquad-12d1Mm_{2}u_{2}-12d_{2}Mm_{1}u_{1}+12Mu_{1}u_{2}\big{)}+\mathcal{O}(q^{4})~{},$
(C.3) $\displaystyle\mathcal{F}_{4}$
$\displaystyle=\frac{3q^{2}}{64}\left(4d_{1}^{2}M^{2}u_{2}^{2}+4d_{2}^{2}M^{2}u_{1}^{2}+8M^{2}m_{1}u_{2}^{2}+8M^{2}m_{2}u_{1}^{2}+u_{1}^{2}u_{2}^{2}\right)$
$\displaystyle\qquad+\frac{q^{3}}{48}\Big{(}-12d_{1}^{2}d_{2}^{2}Mu_{1}u_{2}+20d_{1}^{2}d_{2}M^{3}u_{1}-24d_{1}^{2}Mm_{2}u_{1}u_{2}+20d_{1}d_{2}^{2}M^{3}u_{2}$
$\displaystyle\qquad\qquad\qquad-18d_{1}d_{2}M^{2}u_{1}^{2}-18d_{1}d_{2}M^{2}u_{2}^{2}+40d_{1}M^{3}m_{2}u_{2}+5d_{1}Mu_{2}^{3}-24d_{2}^{2}Mm_{1}u_{1}u_{2}$
$\displaystyle\qquad\qquad\qquad+40d_{2}M^{3}m_{1}u_{1}+5d_{2}Mu_{1}^{3}-48Mm_{1}m_{2}u_{1}u_{2}\Big{)}+\mathcal{O}(q^{4})~{}.$
(C.4)

One can explicitly check that these expressions are all invariant under
(5.45).282828Note that $\mathcal{F}_{0}$ depends on the $U(1)$-factor that we
have not identified.

As mentioned at the end of Sec. 5.4, imposing the invariance under (5.45)
constrains the possible form of $\mathcal{F}_{k>0}$ in (5.43). In particular,
when combined with the trivial symmetry of the theory under exchanging two
$(A_{1},D_{4})$ sectors, it determines $\mathcal{F}_{k}$ up to undetermined
$T$-invariant functions. In the rest of this appendix, we explicitly show this
for $\mathcal{F}_{2}$.

To that end, it is useful for us below to re-express (5.45) in terms of the IR
gauge coupling $\tau_{\text{IR}}$ instead of the UV gauge coupling $q$, i.e.,

$\displaystyle\tau_{\text{IR}}\to\tau_{\text{IR}}+1~{},\qquad
d_{1}\to\frac{\theta_{3}^{2}}{\theta_{4}^{2}}d_{1}-\frac{\theta_{2}^{2}}{\theta_{4}^{2}}d_{2}~{},\qquad
d_{2}\to id_{2}~{},\qquad m_{2}\to-m_{2}~{},\qquad u_{2}\to iu_{2}~{},$ (C.5)

where we used the shorthands
$\theta_{2}\equiv\theta_{2}(q_{\text{IR}}^{2}),\theta_{3}\equiv\theta_{3}(q_{\text{IR}}^{2})$
and $\theta_{4}\equiv\theta_{4}(q_{\text{IR}}^{2})$. Note that
$\tau_{\text{IR}}\to\tau_{\text{IR}}+1$ implies
$\theta_{3}(q_{\text{IR}}^{2})\leftrightarrow\theta_{4}(q_{\text{IR}}^{2})$
and $\theta_{2}(q_{\text{IR}}^{2})\to e^{\frac{\pi
i}{4}}\theta_{2}(q_{\text{IR}}^{2})$.

The $(A_{3},A_{3})$ theory also has a trivial symmetry under the permutation
of two $(A_{1},D_{4})$ theories in the quiver description in Fig. 1. This
permutation symmetry implies that $\mathcal{F}_{k>0}$ must also be invariant
under

$\displaystyle d_{1}\longleftrightarrow d_{2}~{},\qquad
m_{1}\longleftrightarrow m_{2}~{},\qquad u_{1}\longleftrightarrow u_{2}~{}.$
(C.6)

Below, we show that demanding the invariance of $\mathcal{F}_{k>0}$ under
(C.5) and (C.6) strongly constrains the possible form of $\mathcal{F}_{k>0}$,
especially focusing on $\mathcal{F}_{2}$.

For that, first note that $\mathcal{F}_{2}$ is a meromorphic function of
$d_{i},m_{i},u_{i},M$ and $\tau_{\text{IR}}$ such that $\mathcal{F}_{2}=0$
when $d_{i}=m_{i}=u_{i}=M=0$. Moreover, since the prepotential is of dimension
two, $\mathcal{F}_{2}$ is of dimension four. These mean that $\mathcal{F}_{2}$
is a polynomial of $d_{i},m_{i},u_{i}$ and $M$ whose coefficients depend on
$\tau_{\text{IR}}$. The invariance under (C.5) and (C.6) implies that these
coefficients are correlated in a highly non-trivial way. For example, let us
consider the possible term

$\displaystyle g(q_{\text{IR}})m_{1}u_{1}u_{2}$ (C.7)

where $g(q_{\text{IR}})$ is an unknown function of $q_{\text{IR}}$. Since it
is of dimension four, this term might appear in $\mathcal{F}_{2}$. The
constraint (C.6) then implies that (C.7) must be accompanied with
$g(q_{\text{IR}})m_{2}u_{1}u_{2}$. Therefore, (C.7) can appear in
$\mathcal{F}_{2}$ only as a part of

$\displaystyle g(q_{\text{IR}})\left(m_{1}+m_{2}\right)u_{1}u_{2}~{}.$ (C.8)

However, the $T$-invariance (C.5) now requires

$\displaystyle
ig(\widetilde{q}_{\text{IR}})(m_{1}-m_{2})u_{1}u_{2}=g(q_{\text{IR}})(m_{1}+m_{2})u_{1}u_{2}~{},$
(C.9)

where $\widetilde{q}_{\text{IR}}$ is the image of $q_{\text{IR}}$ under the
$T$-transformation. For this equality to hold as a function of
$q_{\text{IR}},\,m_{i}$ and $u_{i}$, we need $g(q_{\text{IR}})=0$. Hence, the
invariance under the $T$-transformation together with the permutation symmetry
(C.6) prohibits the term proportional to $m_{1}u_{1}u_{2}$ or
$m_{2}u_{1}u_{2}$ in $\mathcal{F}_{2}$.

The invariance under (C.5) and (C.6) also constrains the coefficients of non-
vanishing terms in $\mathcal{F}_{2}$. Indeed, we find that the most general
expression for $\mathcal{F}_{2}$ that is invariant under (C.5) and (C.6) is
written in terms of only 54 coefficients
$h_{1}(q_{\text{Ir}}),\cdots,h_{54}(q_{\text{IR}})$ as follows.

$\displaystyle\begin{aligned}
&\mathcal{F}_{2}(q,M,\\{m_{i}\\},\\{u_{i}\\},\\{d_{i}\\})\\\
&=h_{1}(q_{\text{IR}})M^{4}+h_{2}(q_{\text{IR}})M^{2}(m_{1}^{2}+m_{2}^{2})+h_{3}(q_{\text{IR}})M^{2}m_{1}m_{2}\frac{\theta_{2}^{4}}{\theta_{3}^{2}\theta_{4}^{2}}+h_{4}(q_{\text{IR}})(m_{1}^{4}+m_{2}^{4})\\\
&\qquad+h_{5}(q_{\text{IR}})m_{1}^{2}m_{2}^{2}+h_{6}(q_{\text{IR}})m_{1}m_{2}(m_{1}^{2}+m_{2}^{2})\frac{\theta_{2}^{4}}{\theta_{3}^{2}\theta_{4}^{2}}\\\
&\quad+h_{7}(q_{\text{IR}})Mu_{1}u_{2}\frac{\theta_{2}^{2}}{\theta_{3}\theta_{4}}+h_{8}(q_{\text{IR}})(m_{1}u_{1}^{2}+m_{2}u_{2}^{2})+h_{9}(q_{\text{IR}})(m_{2}u_{1}^{2}+m_{1}u_{2}^{2})\frac{\theta_{2}^{4}}{\theta_{3}^{2}\theta_{4}^{2}}\\\
&\quad+h_{10}(q_{\text{IR}})M^{2}B_{2}(u_{i},d_{i})+h_{11}(q_{\text{IR}})m_{1}m_{2}B_{2}(u_{i},d_{i})\frac{\theta_{2}^{4}}{\theta_{3}^{2}\theta_{4}^{2}}+h_{12}(q_{\text{IR}})MB_{3}^{(1)}(m_{i},u_{i},d_{i})\\\
&\qquad+h_{13}(q_{\text{IR}})MB_{3}^{(2)}(m_{i},u_{i},d_{i})+h_{14}(q_{\text{IR}})B_{4}^{(1)}(m_{i},u_{i},d_{i})+h_{15}(q_{\text{IR}})B_{4}^{(2)}(m_{i},u_{i},d_{i})\\\
&\quad+h_{16}(q_{\text{IR}})M^{3}D_{1}(d_{i})+h_{17}(q_{\text{IR}})M(m_{1}^{2}+m_{2}^{2})D_{1}(d_{i})+h_{18}(q_{\text{IR}})Mm_{1}m_{2}D_{1}(d_{i})\frac{\theta_{2}^{4}}{\theta_{3}^{2}\theta_{4}^{2}}\\\
&\qquad+h_{19}(q_{\text{IR}})M^{2}E_{2}^{(1)}(m_{i},d_{i})+h_{20}(q_{\text{IR}})M^{2}E_{2}^{(2)}(m_{i},d_{i})+h_{21}(q_{\text{IR}})m_{1}m_{2}E_{2}^{(1)}(m_{i},d_{i})\frac{\theta_{2}^{4}}{\theta_{3}^{2}\theta_{4}^{2}}\\\
&\qquad+h_{22}(q_{\text{IR}})m_{1}m_{2}E_{2}^{(2)}(m_{i},d_{i})\frac{\theta_{2}^{4}}{\theta_{3}^{2}\theta_{4}^{2}}+h_{23}(q_{\text{IR}})E_{4}^{(1)}(m_{i},d_{i})+h_{24}(q_{\text{IR}})E_{4}^{(2)}(m_{i},d_{i})\\\
&\quad+h_{25}(q_{\text{IR}})u_{1}u_{2}D_{1}(d_{i})\frac{\theta_{2}^{2}}{\theta_{3}\theta_{4}}+h_{26}(q_{\text{IR}})F_{4}^{(1)}(u_{i},d_{i})+h_{27}(q_{\text{IR}})F_{4}^{(2)}(u_{i},d_{i})\\\
&\quad+h_{28}(q_{\text{IR}})MG_{3}^{(1)}(u_{i},d_{i})+h_{29}(q_{\text{IR}})MG_{3}^{(2)}(u_{i},d_{i})+h_{30}(q_{\text{IR}})G_{4}^{(1)}(m_{i},u_{i},d_{i})\\\
&\qquad+h_{31}(q_{\text{IR}})G_{4}^{(2)}(m_{i},u_{i},d_{i})+h_{32}(q_{\text{IR}})G_{4}^{(3)}(m_{i},u_{i},d_{i})+h_{33}(q_{\text{IR}})G_{4}^{(4)}(m_{i},u_{i},d_{i})\\\
&\quad+h_{34}(q_{\text{IR}})M^{2}(D_{1}(d_{i}))^{2}+h_{35}(q_{\text{IR}})m_{1}m_{2}(D_{1}(d_{i}))^{2}\frac{\theta_{2}^{4}}{\theta_{3}^{2}\theta_{4}^{2}}+h_{36}(q_{\text{IR}})M^{2}D_{2}(d_{i})\\\
&\qquad+h_{37}(q_{\text{IR}})m_{1}m_{2}D_{2}(d_{i})\frac{\theta_{2}^{4}}{\theta_{3}^{2}\theta_{4}^{2}}+h_{38}(q_{\text{IR}})(m_{1}^{2}+m_{2}^{2})D_{2}(d_{i})+h_{39}(q_{\text{IR}})MH_{3}^{(1)}(m_{i},d_{i})\\\
&\qquad+h_{40}(q_{\text{IR}})MH_{3}^{(2)}(m_{i},d_{i})+h_{41}(q_{\text{IR}})H_{4}^{(1)}(m_{i},d_{i})+h_{42}(q_{\text{IR}})H_{4}^{(2)}(m_{i},d_{i})\\\
&\quad+h_{43}(q_{\text{IR}})I_{4}^{(1)}(u_{i},d_{i})+h_{44}(q_{\text{IR}})I_{4}^{(2)}(u_{i},d_{i})+h_{45}(q_{\text{IR}})I_{4}^{(3)}(u_{i},d_{i})\\\
&\quad+h_{46}(q_{\text{IR}})M(D_{1}(d_{i}))^{3}+h_{47}(q_{\text{IR}})MD_{1}(d_{i})D_{2}(d_{i})+h_{48}(q_{\text{IR}})J_{4}^{(1)}(m_{i},d_{i})\\\
&\qquad+h_{49}(q_{\text{IR}})J_{4}^{(2)}(m_{i},d_{i})+h_{50}(q_{\text{IR}})J_{4}^{(3)}(m_{i},d_{i})+h_{51}(q_{\text{IR}})J_{4}^{(4)}(m_{i},d_{i})\\\
&\quad+h_{52}(q_{\text{IR}})(D_{1}(d_{i}))^{4}+h_{53}(q_{\text{IR}})(D_{1}(d_{i}))^{2}D_{2}(d_{i})+h_{54}(q_{\text{IR}})(D_{2}(d_{i}))^{2}\end{aligned}$
(C.10)

with the following building blocks

$\displaystyle
B_{2}(u_{i},d_{i})=\bigg{(}(u_{1}d_{1}+u_{2}d_{2})+(u_{2}d_{1}+u_{1}d_{2})\frac{\theta_{3}^{2}-\theta_{4}^{2}}{\theta_{2}^{2}}\bigg{)}\frac{\theta_{3}}{\theta_{4}}~{},$
(C.11) $\displaystyle
B_{3}^{(1)}(m_{i},u_{i},d_{i})=\bigg{(}(m_{1}u_{1}d_{1}+m_{2}u_{2}d_{2})+(m_{2}u_{2}d_{1}+m_{1}u_{1}d_{2})\frac{\theta_{3}^{2}+\theta_{4}^{2}}{\theta_{2}^{2}}\bigg{)}\frac{\theta_{3}}{\theta_{4}}~{},$
(C.12) $\displaystyle
B_{3}^{(2)}(m_{i},u_{i},d_{i})=\bigg{(}(m_{2}u_{1}d_{1}+m_{1}u_{2}d_{2})+(m_{1}u_{2}d_{1}+m_{2}u_{1}d_{2})\frac{\theta_{3}^{2}+\theta_{4}^{2}}{\theta_{2}^{2}}\bigg{)}\frac{\theta_{2}^{4}}{\theta_{3}\theta_{4}^{3}}~{},$
(C.13) $\displaystyle
B_{4}^{(1)}(m_{i},u_{i},d_{i})=\bigg{(}(m_{1}^{2}u_{1}d_{1}+m_{2}^{2}u_{2}d_{2})+(m_{2}^{2}u_{2}d_{1}+m_{1}^{2}u_{1}d_{2})\frac{\theta_{3}^{2}-\theta_{4}^{2}}{\theta_{2}^{2}}\bigg{)}\frac{\theta_{3}}{\theta_{4}}~{},$
(C.14) $\displaystyle
B_{4}^{(2)}(m_{i},u_{i},d_{i})=\bigg{(}(m_{2}^{2}u_{1}d_{1}+m_{1}^{2}u_{2}d_{2})+(m_{1}^{2}u_{2}d_{1}+m_{2}^{2}u_{1}d_{2})\frac{\theta_{3}^{2}-\theta_{4}^{2}}{\theta_{2}^{2}}\bigg{)}\frac{\theta_{3}}{\theta_{4}}~{},$
(C.15) $\displaystyle
D_{1}(d_{i})=\bigg{(}(d_{1}^{2}+d_{2}^{2})+2d_{1}d_{2}\frac{\theta_{3}^{2}}{\theta_{2}^{2}}\bigg{)}\frac{\theta_{3}^{2}}{\theta_{4}^{2}}~{},$
(C.16) $\displaystyle
D_{2}(d_{i})=\bigg{(}(d_{1}^{3}d_{2}+d_{1}d_{2}^{3})+d_{1}^{2}d_{2}^{2}\frac{\theta_{3}^{4}+\theta_{2}^{4}}{\theta_{3}^{2}\theta_{2}^{2}}\bigg{)}\frac{\theta_{3}^{4}}{\theta_{4}^{2}\theta_{2}^{2}}~{},$
(C.17) $\displaystyle
E_{2}^{(1)}(m_{i},d_{i})=\bigg{(}(m_{1}d_{1}^{2}+m_{2}d_{2}^{2})+(m_{1}+m_{2})d_{1}d_{2}\frac{\theta_{2}^{2}}{\theta_{3}^{2}}\bigg{)}\frac{\theta_{3}^{2}}{\theta_{4}^{2}}~{},$
(C.18) $\displaystyle
E_{2}^{(2)}(m_{i},d_{i})=\bigg{(}(m_{2}d_{1}^{2}+m_{1}d_{2}^{2})+(m_{1}+m_{2})d_{1}d_{2}\frac{\theta_{2}^{2}}{\theta_{3}^{2}}\bigg{)}\frac{\theta_{2}^{4}}{\theta_{4}^{4}}~{},$
(C.19) $\displaystyle
E_{4}^{(1)}(m_{i},d_{i})=\bigg{(}(m_{1}^{3}d_{1}^{2}+m_{2}^{3}d_{2}^{2})+(m_{1}^{3}+m_{2}^{3})d_{1}d_{2}\frac{\theta_{2}^{2}}{\theta_{3}^{2}}\bigg{)}\frac{\theta_{3}^{2}}{\theta_{4}^{2}}~{},$
(C.20) $\displaystyle
E_{4}^{(2)}(m_{i},d_{i})=\bigg{(}(m_{2}^{3}d_{1}^{2}+m_{1}^{3}d_{2}^{2})+(m_{1}^{3}+m_{2}^{3})d_{1}d_{2}\frac{\theta_{2}^{2}}{\theta_{3}^{2}}\bigg{)}\frac{\theta_{2}^{4}}{\theta_{4}^{4}}~{},$
(C.21) $\displaystyle
F_{4}^{(1)}(u_{i},d_{i})=\bigg{(}(u_{1}^{2}d_{1}^{2}+u_{2}^{2}d_{2}^{2})+d_{1}d_{2}(u_{1}^{2}+u_{2}^{2})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}\bigg{)}\frac{\theta_{3}^{2}}{\theta_{4}^{2}}~{},$
(C.22) $\displaystyle
F_{4}^{(2)}(u_{i},d_{i})=\bigg{(}(u_{2}^{2}d_{1}^{2}+u_{1}^{2}d_{2}^{2})+d_{1}d_{2}(u_{1}^{2}+u_{2}^{2})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}\bigg{)}\frac{\theta_{2}^{4}}{\theta_{4}^{4}}~{},$
(C.23) $\displaystyle
G_{3}^{(1)}(u_{i},d_{i})=\bigg{(}(u_{1}d_{1}^{3}+u_{2}d_{2}^{3})+d_{1}d_{2}\Big{(}(u_{1}d_{1}+u_{2}d_{2})\frac{\theta_{3}^{4}+\theta_{2}^{4}+\theta_{3}^{2}\theta_{4}^{2}}{\theta_{3}^{2}\theta_{2}^{2}}+(u_{2}d_{1}+u_{1}d_{2})\frac{\theta_{3}^{2}+\theta_{4}^{2}}{\theta_{3}^{2}}\Big{)}\bigg{)}\frac{\theta_{3}^{3}}{\theta_{4}^{3}}~{},$
(C.24) $\displaystyle
G_{3}^{(2)}(u_{i},d_{i})=\bigg{(}(u_{2}d_{1}^{3}+u_{1}d_{2}^{3})+d_{1}d_{2}\Big{(}(u_{1}d_{1}+u_{2}d_{2})\frac{\theta_{3}^{2}-\theta_{4}^{2}}{\theta_{3}^{2}}+(u_{2}d_{1}+u_{1}d_{2})\frac{\theta_{3}^{4}+\theta_{2}^{4}-\theta_{3}^{2}\theta_{4}^{2}}{\theta_{3}^{2}\theta_{2}^{2}}\Big{)}\bigg{)}\frac{\theta_{3}^{2}\theta_{2}^{2}}{\theta_{4}^{4}}~{},$
(C.25) $\displaystyle
G_{4}^{(1)}(m_{i},u_{i},d_{i})=\bigg{(}(m_{1}u_{1}d_{1}^{3}+m_{2}u_{2}d_{2}^{3})$
$\displaystyle\qquad\qquad\qquad+d_{1}d_{2}\Big{(}(m_{1}u_{1}d_{1}+m_{2}u_{2}d_{2})\frac{\theta_{3}^{4}+\theta_{2}^{4}-\theta_{3}^{2}\theta_{4}^{2}}{\theta_{3}^{2}\theta_{2}^{2}}+(m_{2}u_{2}d_{1}+m_{1}u_{1}d_{2})\frac{\theta_{3}^{2}-\theta_{4}^{2}}{\theta_{3}^{2}}\Big{)}\bigg{)}\frac{\theta_{3}^{3}}{\theta_{4}^{3}}~{},$
(C.26) $\displaystyle
G_{4}^{(2)}(m_{i},u_{i},d_{i})=\bigg{(}(m_{2}u_{1}d_{1}^{3}+m_{1}u_{2}d_{2}^{3})$
$\displaystyle\qquad\qquad\qquad+d_{1}d_{2}\Big{(}(m_{2}u_{1}d_{1}+m_{1}u_{2}d_{2})\frac{\theta_{3}^{4}+\theta_{2}^{4}-\theta_{3}^{2}\theta_{4}^{2}}{\theta_{3}^{2}\theta_{2}^{2}}+(m_{1}u_{2}d_{1}+m_{2}u_{1}d_{2})\frac{\theta_{3}^{2}-\theta_{4}^{2}}{\theta_{3}^{2}}\Big{)}\bigg{)}\frac{\theta_{3}\theta_{2}^{4}}{\theta_{4}^{5}}~{},$
(C.27) $\displaystyle
G_{4}^{(3)}(m_{i},u_{i},d_{i})=\bigg{(}(m_{1}u_{2}d_{1}^{3}+m_{2}u_{1}d_{2}^{3})$
$\displaystyle\qquad\qquad\qquad+d_{1}d_{2}\Big{(}(m_{2}u_{1}d_{1}+m_{1}u_{2}d_{2})\frac{\theta_{3}^{2}+\theta_{4}^{2}}{\theta_{3}^{2}}+(m_{1}u_{2}d_{1}+m_{2}u_{1}d_{2})\frac{\theta_{3}^{4}+\theta_{2}^{4}+\theta_{3}^{2}\theta_{4}^{2}}{\theta_{3}^{2}\theta_{2}^{2}}\Big{)}\bigg{)}\frac{\theta_{3}^{2}\theta_{2}^{2}}{\theta_{4}^{4}}~{},$
(C.28) $\displaystyle
G_{4}^{(4)}(m_{i},u_{i},d_{i})=\bigg{(}(m_{2}u_{2}d_{1}^{3}+m_{1}u_{1}d_{2}^{3})$
$\displaystyle\qquad\qquad\qquad+d_{1}d_{2}\Big{(}(m_{1}u_{1}d_{1}+m_{2}u_{2}d_{2})\frac{\theta_{3}^{2}+\theta_{4}^{2}}{\theta_{3}^{2}}+(m_{2}u_{2}d_{1}+m_{1}u_{1}d_{2})\frac{\theta_{3}^{4}+\theta_{2}^{4}+\theta_{3}^{2}\theta_{4}^{2}}{\theta_{3}^{2}\theta_{2}^{2}}\Big{)}\bigg{)}\frac{\theta_{3}^{4}}{\theta_{4}^{2}\theta_{2}^{2}}~{},$
(C.29) $\displaystyle
H_{3}^{(1)}(m_{i},d_{i})=\bigg{(}(m_{1}d_{1}^{4}+m_{2}d_{2}^{4})$
$\displaystyle\qquad\qquad\qquad+d_{1}d_{2}\Big{(}(m_{1}d_{1}^{2}+m_{2}d_{2}^{2})\frac{2\theta_{3}^{4}+\theta_{2}^{4}}{\theta_{2}^{2}\theta_{3}^{2}}+(m_{2}d_{1}^{2}+m_{1}d_{2}^{2})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}\Big{)}+3(m_{1}+m_{2})d_{1}^{2}d_{2}^{2}\bigg{)}\frac{\theta_{3}^{4}}{\theta_{4}^{4}}~{},$
(C.30) $\displaystyle
H_{3}^{(2)}(m_{i},d_{i})=\bigg{(}(m_{2}d_{1}^{4}+m_{1}d_{2}^{4})$
$\displaystyle\qquad\qquad\qquad+d_{1}d_{2}\Big{(}(m_{1}d_{1}^{2}+m_{2}d_{2}^{2})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}+(m_{2}d_{1}^{2}+m_{1}d_{2}^{2})\frac{2\theta_{3}^{4}+\theta_{2}^{4}}{\theta_{2}^{2}\theta_{3}^{2}}\Big{)}+3(m_{1}+m_{2})d_{1}^{2}d_{2}^{2}\bigg{)}\frac{\theta_{3}^{2}\theta_{2}^{4}}{\theta_{4}^{6}}~{},$
(C.31) $\displaystyle
H_{4}^{(1)}(m_{i},d_{i})=\bigg{(}(m_{1}^{2}d_{1}^{4}+m_{2}^{2}d_{2}^{4})-2d_{1}d_{2}(m_{2}^{2}d_{1}^{2}+m_{1}^{2}d_{2}^{2})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}-(m_{1}^{2}+m_{2}^{2})d_{1}^{2}d_{2}^{2}\frac{2\theta_{3}^{4}+\theta_{2}^{4}}{\theta_{3}^{4}}\bigg{)}\frac{\theta_{3}^{4}}{\theta_{4}^{4}}~{},$
(C.32) $\displaystyle
H_{4}^{(2)}(m_{i},d_{i})=\bigg{(}(m_{2}^{2}d_{1}^{4}+m_{1}^{2}d_{2}^{4})+2d_{1}d_{2}(m_{2}^{2}d_{1}^{2}+m_{1}^{2}d_{2}^{2})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}+(m_{1}^{2}+m_{2}^{2})d_{1}^{2}d_{2}^{2}\frac{\theta_{2}^{4}}{\theta_{3}^{4}}\bigg{)}\frac{\theta_{3}^{4}}{\theta_{4}^{4}}~{},$
(C.33) $\displaystyle
I_{4}^{(1)}(u_{i},d_{i})=\bigg{(}u_{1}d_{1}^{5}+u_{2}d_{2}^{5}+d_{1}d_{2}(u_{2}d_{1}^{3}+u_{1}d_{2}^{3})\frac{-3\theta_{2}^{8}+2\theta_{4}^{6}(\theta_{3}^{2}-\theta_{4}^{2})+\theta_{4}^{2}\theta_{2}^{4}(4\theta_{3}^{2}-5\theta_{4}^{2})}{\theta_{3}^{4}\theta_{2}^{4}}$
$\displaystyle\qquad\qquad\qquad+d_{1}^{2}d_{2}^{2}\Big{(}(u_{2}d_{1}+u_{1}d_{2})\frac{-8\theta_{2}^{8}(\theta_{3}^{2}-\theta_{4}^{2})-2\theta_{4}^{8}(\theta_{3}^{2}-\theta_{4}^{2})-\theta_{4}^{4}\theta_{2}^{4}(8\theta_{3}^{2}-9\theta_{4}^{2})}{\theta_{3}^{4}\theta_{2}^{6}}$
$\displaystyle\qquad\qquad\qquad+(u_{1}d_{1}+u_{2}d_{2})\frac{-6\theta_{2}^{8}+4\theta_{4}^{2}\theta_{2}^{4}(\theta_{3}^{2}-2\theta_{4}^{2})+3\theta_{4}^{6}(\theta_{3}^{2}-\theta_{4}^{2})}{\theta_{3}^{4}\theta_{2}^{4}}\Big{)}\bigg{)}\frac{\theta_{3}^{5}}{\theta_{4}^{5}}~{},$
(C.34) $\displaystyle
I_{4}^{(2)}(u_{i},d_{i})=\bigg{(}u_{2}d_{1}^{5}+u_{1}d_{2}^{5}+d_{1}d_{2}(u_{2}d_{1}^{3}+u_{1}d_{2}^{3})\frac{\theta_{4}^{4}(\theta_{3}^{2}+\theta_{4}^{2})+\theta_{2}^{4}(3\theta_{3}^{2}+\theta_{4}^{2})}{\theta_{3}^{4}\theta_{2}^{2}}$
$\displaystyle\qquad\qquad\qquad+d_{1}^{2}d_{2}^{2}\Big{(}(u_{1}d_{1}+u_{2}d_{2})\frac{\theta_{2}^{2}(\theta_{3}^{2}+\theta_{4}^{2})}{\theta_{3}^{4}}+(u_{2}d_{1}+u_{1}d_{2})\frac{3\theta_{2}^{4}+2\theta_{4}^{2}(\theta_{3}^{2}+\theta_{4}^{2})}{\theta_{3}^{4}}\Big{)}\bigg{)}\frac{\theta_{3}^{4}\theta_{2}^{4}}{\theta_{4}^{6}}~{},$
(C.35) $\displaystyle
I_{4}^{(3)}(u_{i},d_{i})=\bigg{(}d_{1}d_{2}(u_{1}d_{1}^{3}+u_{2}d_{2}^{3})+d_{1}^{2}d_{2}^{2}\Big{(}(u_{1}d_{1}+u_{2}d_{2})\frac{\theta_{4}^{4}(2\theta_{3}^{2}-\theta_{4}^{2})+\theta_{2}^{4}(3\theta_{3}^{2}-\theta_{4}^{2})}{\theta_{3}^{4}\theta_{2}^{2}}$
$\displaystyle\qquad\qquad\qquad+(u_{2}d_{1}+u_{1}d_{2})\frac{3\theta_{2}^{8}-\theta_{4}^{6}(\theta_{3}^{2}-\theta_{4}^{2})-\theta_{4}^{2}\theta_{2}^{4}(2\theta_{3}^{2}+4\theta_{4}^{2})}{\theta_{3}^{4}\theta_{2}^{4}}\Big{)}\bigg{)}\frac{\theta_{3}^{5}}{\theta_{4}^{3}\theta_{2}^{2}}~{},$
(C.36) $\displaystyle
J_{4}^{(1)}(m_{i},d_{i})=\bigg{(}(m_{1}d_{1}^{6}+m_{2}d_{2}^{6})+3d_{1}d_{2}(m_{1}d_{1}^{4}+m_{2}d_{2}^{4})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+3d_{1}^{2}d_{2}^{2}(m_{1}d_{1}^{2}+m_{2}d_{2}^{2})\frac{\theta_{2}^{4}}{\theta_{3}^{4}}+(m_{1}+m_{2})d_{1}^{3}d_{2}^{3}\frac{\theta_{2}^{6}}{\theta_{3}^{6}}\bigg{)}\frac{\theta_{3}^{6}}{\theta_{4}^{6}}~{},$
(C.37) $\displaystyle
J_{4}^{(2)}(m_{i},d_{i})=\bigg{(}(m_{2}d_{1}^{6}+m_{1}d_{2}^{6})+3d_{1}d_{2}(m_{2}d_{1}^{4}+m_{1}d_{2}^{4})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+3d_{1}^{2}d_{2}^{2}(m_{2}d_{1}^{2}+m_{1}d_{2}^{2})\frac{\theta_{2}^{4}}{\theta_{3}^{4}}+(m_{1}+m_{2})d_{1}^{3}d_{2}^{3}\frac{\theta_{2}^{6}}{\theta_{3}^{6}}\bigg{)}\frac{\theta_{3}^{4}\theta_{2}^{4}}{\theta_{4}^{8}}~{},$
(C.38) $\displaystyle
J_{4}^{(3)}(m_{i},d_{i})=\bigg{(}d_{1}d_{2}(m_{1}d_{1}^{4}+m_{2}d_{2}^{4})+d_{1}^{2}d_{2}^{2}\Big{(}(m_{1}d_{1}^{2}+m_{2}d_{2}^{2})\frac{\theta_{3}^{4}+2\theta_{2}^{4}}{\theta_{3}^{2}\theta_{2}^{2}}+(m_{2}d_{1}^{2}+m_{1}d_{2}^{2})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+(m_{1}+m_{2})d_{1}^{3}d_{2}^{3}\frac{\theta_{3}^{8}+\theta_{2}^{8}+\theta_{4}^{4}\theta_{2}^{4}}{\theta_{3}^{8}}\bigg{)}\frac{\theta_{3}^{6}}{\theta_{4}^{4}\theta_{2}^{2}}~{},$
(C.39) $\displaystyle
J_{4}^{(4)}(m_{i},d_{i})=\bigg{(}d_{1}d_{2}(m_{2}d_{1}^{4}+m_{1}d_{2}^{4})+d_{1}^{2}d_{2}^{2}\Big{(}(m_{1}d_{1}^{2}+m_{2}d_{2}^{2})\frac{\theta_{2}^{2}}{\theta_{3}^{2}}+(m_{2}d_{1}^{2}+m_{1}d_{2}^{2})\frac{\theta_{3}^{4}+2\theta_{2}^{4}}{\theta_{3}^{2}\theta_{2}^{2}}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+(m_{1}+m_{2})d_{1}^{3}d_{2}^{3}\frac{\theta_{3}^{8}+\theta_{2}^{8}+\theta_{4}^{4}\theta_{2}^{4}}{\theta_{3}^{8}}\bigg{)}\frac{\theta_{3}^{4}\theta_{2}^{2}}{\theta_{4}^{6}}~{},$
(C.40)

where each block is invariant under the $T$ transformation (5.45) and the
permutation symmetry, and their subscripts coincide with their mass
dimensions.

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