# Four flavours, triality and bimodular forms

Johannes Aspmana, Elias Furrerb, Jan Manschotc
School of Mathematics, Trinity College, Dublin 2, Ireland
Hamilton Mathematical Institute, Trinity College, Dublin 2
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###### Abstract

We consider $\mathcal{N}=2$ supersymmetric SU(2) gauge theory with $N_{f}=4$
massive hypermultiplets. The duality group of this theory contains
transformations acting on the UV-coupling $\tau_{\text{\tiny{UV}}}$ as well as
on the running coupling $\tau$. We establish that subgroups of the duality
group act separately on $\tau_{\text{\tiny{UV}}}$ and $\tau$, while a larger
group acts simultaneously on $\tau_{\text{\tiny{UV}}}$ and $\tau$. For special
choices of the masses, we find that the duality groups can be identified with
congruence subgroups of $\text{SL}(2,\mathbb{Z})$. We demonstrate that in such
cases, the order parameters are instances of bimodular forms with arguments
$\tau$ and $\tau_{\text{\tiny{UV}}}$. Since the UV duality group of the theory
contains the triality group of outer automorphisms of the flavour symmetry
SO(8), the duality action gives rise to an orbit of mass configurations.
Consequently, the corresponding order parameters combine to vector-valued
bimodular forms with $\text{SL}(2,\mathbb{Z})$ acting simultaneously on the
two couplings.

##  1 Introduction

The $\mathcal{N}=2$ supersymmetric Yang-Mills field theory with gauge group
SU(2) and $N_{f}=4$ fundamental hypermultiplets is distinguished for various
reasons Seiberg:1994aj , including:

  * •

The theory is superconformal up to mass terms for the hypermultiplets, and is
a benchmark for four-dimensional SCFT’s with $\mathcal{N}=2$ supersymmetry
Argyres:1995xn ; Argyres:2007tq ; Argyres:2016xua ; Argyres:2020nrr .

  * •

The theory is a building block for other four-dimensional $\mathcal{N}=2$
SCFT’s and the 2d/4d correspondence Gaiotto:2009we ; Alday:2009aq .

  * •

The theory exhibits an intriguing electric-magnetic duality group including
triality Seiberg:1994aj . This duality group acts on the UV coupling
$\tau_{\text{\tiny{UV}}}$ and running coupling constant $\tau$, and contains
elements which act simultaneously on the two couplings as well as separately.

  * •

The theory is a “parent” theory from which the asymptotically free
$\mathcal{N}=2$, SU(2) theories with $N_{f}\leq 3$ hypermultiplets can be
obtained by decoupling one or more hypermultiplets Seiberg:1994aj ;
Seiberg:1994rs .

The focus of the present paper is on the third bullet point. We analyze
duality groups for the couplings $\tau_{\text{\tiny{UV}}}$ and $\tau$ of the
$N_{f}=4$ theory as function of the masses. To this end, explicit expressions
for the order parameter $u=\langle\text{Tr}\,\phi^{2}\rangle$, with $\phi$ the
complex scalar of the vector multiplet, are determined as function of both
$\tau_{\text{\tiny{UV}}}$ and $\tau$. We identify several loci in the space of
masses where $u$ transforms as a modular form for both
$\tau_{\text{\tiny{UV}}}$ and $\tau$. This extends our recent work
aspman2021cutting  on theories with $N_{f}\leq 3$ to $N_{f}=4$. In
aspman2021cutting , we determined fundamental domains for the running coupling
$\tau$ for the asymptotically free theories by analyzing in detail the order
parameter $u$ as function of the effective coupling $\tau\in\mathbb{H}$. We
have demonstrated that for generic masses this function has branch points,
with the consequence that the fundamental domain for $\tau$ is in general not
of the form $\Gamma\backslash\mathbb{H}$ for a congruence subgroup
$\Gamma\subset\text{SL}(2,\mathbb{Z})$. Only for specific values of the
masses, such as those giving rise to Argyres-Douglas (AD) points, the order
parameter is (weakly) holomorphic as function of $\tau$, and the fundamental
domain is that of a congruence subgroup. In this paper, we find that these
features are present as well for the $N_{f}=4$ theory, but with an additional
dependence on $\tau_{\text{\tiny{UV}}}$.

At special modular loci, some properties of the $N_{f}=4$ order parameters are
similar to that of the $\mathcal{N}=2^{*}$ SU(2) theory, i.e. the
superconformal theory with a single adjoint hypermultiplet. The
$\mathcal{N}=2^{*}$ order parameter transforms as a modular form under the
group $\Gamma(2)\times\Gamma(2)$ with the first factor acting on $\tau$ and
the second on $\tau_{\text{\tiny{UV}}}$, while it also transforms as a modular
form under simultaneous $\text{SL}(2,\mathbb{Z})$ transformations of $\tau$
and $\tau_{\text{\tiny{UV}}}$ Ferrari:1997gu ; Labastida:1998sk ; Huang:2011qx
. It was later clarified that $u(\tau,\tau_{\text{\tiny{UV}}})$ is an example
of a meromorphic _bimodular_ form Manschot:2021qqe . Such functions have
appeared, although sporadically, in the mathematical literature
stienstra_zagier_16 ; yang2007differential ; liuquan2020 .

The SU(2) $N_{f}=4$ theory exhibits a richer structure: It has four mass
parameters that give rise to seven singular vacua on the $u$-plane. For
special choices of the masses, the $u$-planes contain any of the three SU(2)
Argyres-Douglas superconformal points $(A_{1},A_{2})$, $(A_{1},A_{3})$ and
$(A_{1},D_{4})$, while in the massless case there is a non-abelian Coulomb
point with a five quaternionic-dimensional Higgs branch Argyres:1995jj ;
Argyres:1995xn . For generic masses, the singularities are roots of a sextic
polynomial, for which there is no known expression. The flavour symmetry
$\mathrm{SO}(8)$ becomes the universal cover $\text{Spin}(8)$ in the quantum
theory. It has a _triality_ group $\mathrm{Out}(\text{Spin}(8))$ of outer
automorphisms, which is isomorphic to the symmetric group $S_{3}$ on three
letters. The full symmetry group of the $N_{f}=4$ curve is then the semidirect
product $\text{Spin}(8)\rtimes_{\varphi}\text{SL}(2,\mathbb{Z})$, which is
induced by the group homomorphism
$\varphi:\text{SL}(2,\mathbb{Z})\to\mathrm{Out}(\text{Spin}(8))$. As the
triality group is of order $|S_{3}|=6$, the orbits of the group action on mass
space $\mathbb{C}^{4}$ generally have six elements. However, there are
specific mass configurations with enhanced global symmetry that are invariant
under subgroups of the triality group, for which the orbits collapse, either
to three elements or to a single element.

We study four such configurations in detail, and show that their order
parameters, periods and discriminants are bimodular forms for subgroups of
$\text{SL}(2,\mathbb{Z})$. For the triality invariant case
$(m_{1},m_{2},m_{3},m_{4})=(m,m,0,0)$ we find that the order parameter is a
bimodular form of weight $(0,2)$ with $\Gamma(2)$ acting on both $\tau$ and
$\tau_{\text{\tiny{UV}}}$ individually, while it is also bimodular for
$\text{SL}(2,\mathbb{Z})$ acting on $\tau$ and $\tau_{\text{\tiny{UV}}}$
simultaneously. If all four hypermultiplets are rather given an equal mass,
the triality orbit has three elements. The $u$-planes for these three mass
configurations are modular curves for the three subgroups of
$\text{SL}(2,\mathbb{Z})$ conjugate to $\Gamma^{0}(4)$. The order parameters,
periods and discriminants are permuted by triality, and can thus be organised
into vectors to form one-parameter families of vector-valued bimodular forms
for $\text{SL}(2,\mathbb{Z})$. We further give some examples of exact
expressions for order parameters of more complicated theories with two
independent mass parameters. These theories then include both AD points and
branch cuts.

The outline of the paper is as follows. In Section 2 we discuss the symmetries
of the $N_{f}=4$ SW curve, and study the action of the triality group on the
mass space. Specific mass configurations with enhanced global symmetry are
then studied in Section 3, where we also provide a definition of bimodular
forms and vector-valued bimodular forms most suited for our analyses. The
Appendix A contains relevant properties of congruence subgroups and modular
forms. In Appendix B, we study the possible singularity spectrum of the
$N_{f}=4$ theory. In Appendix C we obtain similar results for the $N_{f}=4$
curve constructed from the qq-characters of the $N_{f}=4$ theory. Appendix D
finally provides expressions for the limits to the asymptotically free
theories.

##  2 Four flavours and triality

The one-loop beta function of $\mathcal{N}=2$ supersymmetric Yang-Mills theory
with $N_{f}\leq 4$ hypermultiplets in the fundamental representation is
$\beta_{N_{f}}(g_{\text{YM}})=-\frac{g_{\text{YM}}^{3}}{16\pi^{2}}(4-N_{f})$.
The gauge coupling $g_{\text{YM}}$ is combined with the theta angle $\theta$
in the Lagrangian as $\tau=\frac{\theta}{\pi}+\frac{8\pi
i}{g_{\text{YM}}^{2}}$. This complexified gauge coupling can be considered as
the expectation value of a background chiral superfield. In the
renormalisation scheme where the superpotential remains a holomorphic function
of all chiral superfields, the one-loop running coupling at the energy scale
$E$ can be expressed as Intriligator:1995au

$\tau(E)=\tau_{\text{\tiny{UV}}}-\frac{4-N_{f}}{2\pi
i}\log\frac{E}{\Lambda_{\text{UV}}}.$ (1)

It is one-loop exact in the holomorphic scheme, and thus for $N_{f}<4$ the
combination

$\Lambda_{N_{f}}^{4-N_{f}}\coloneqq\Lambda_{\text{UV}}^{4-N_{f}}e^{2\pi
i\tau_{\text{\tiny{UV}}}}$ (2)

of the scale $\Lambda_{\text{UV}}$ and the coupling $\tau_{\text{\tiny{UV}}}$
is invariant to all orders in perturbation theory. This complexified dynamical
scale $\Lambda_{N_{f}}$ sets the overall scale of the theory. For $N_{f}=4$ on
the other hand, there is a distinguished dimensionless parameter
$\tau_{\text{\tiny{UV}}}$, on which the theory depends nontrivially. To
shorten the notation, we will also set $\tau_{0}:=\tau_{\text{\tiny{UV}}}$ and
$q_{0}:=e^{2\pi i\tau_{0}}$ in the following.

###  2.1 The curve

The low-energy physics of $\mathcal{N}=2$ SYM with $N_{f}=4$ massive
hypermultiplets has been determined in Seiberg:1994aj ; Minahan:1996ws ;
Ferrari:1997gu ; Dorey:1996bn ; Argyres:1999ty ; Huang:2011qx . Similar to the
asymptotically free ($N_{f}\leq 3$) cases, the physics is encoded in an
elliptic curve which depends holomorphically on the Coulomb branch parameter
$u\in\mathcal{B}_{4}$. This coordinate $u$ parametrises the Coulomb branch
$\mathcal{B}_{4}$ of the $N_{f}=4$ theory. Let us first define the symmetric
mass combinations

$\displaystyle\left\llbracket
m_{1}^{k}\right\rrbracket=\sum_{i=1}^{4}m_{i}^{k},\qquad\quad\,\,\left\llbracket
m_{1}^{2}m_{2}^{2}\right\rrbracket=\sum_{i<j}m_{i}^{2}m_{j}^{2}$ (3)
$\displaystyle\left\llbracket m_{1}^{4}m_{2}^{2}\right\rrbracket=\sum_{i\neq
j}m_{i}^{4}m_{j}^{2},\quad\left\llbracket
m_{1}^{2}m_{2}^{2}m_{3}^{2}\right\rrbracket=\sum_{i<j<k}m_{i}^{2}m_{j}^{2}m_{k}^{2},$
$\displaystyle\mathrm{Pf}({\boldsymbol{m}})=m_{1}m_{2}m_{3}m_{4}.$

The $N_{f}=4$ curve for generic masses is then Seiberg:1994aj

$y^{2}=W_{1}W_{2}W_{3}+A\left(W_{1}T_{1}\left(e_{2}-e_{3}\right)+W_{2}T_{2}\left(e_{3}-e_{1}\right)+W_{3}T_{3}\left(e_{1}-e_{2}\right)\right)-A^{2}N,$
(4)

where

$\displaystyle W_{i}$ $\displaystyle=$ $\displaystyle x-e_{i}u-e_{i}^{2}R,$
$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(e_{1}-e_{2}\right)\left(e_{2}-e_{3}\right)\left(e_{3}-e_{1}\right),$
$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\llbracket{m}_{1}^{2}\right\rrbracket,$
$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\left\llbracket{m}_{1}^{2}{m}_{2}^{2}\right\rrbracket-\frac{1}{24}\left\llbracket{m}_{1}^{4}\right\rrbracket,$
$\displaystyle T_{2,3}$ $\displaystyle=$
$\displaystyle\mp\frac{1}{2}\mathrm{Pf}({\boldsymbol{m}})-\frac{1}{24}\left\llbracket{m}_{1}^{2}{m}_{2}^{2}\right\rrbracket+\frac{1}{48}\left\llbracket{m}_{1}^{4}\right\rrbracket,$
$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{3}{16}\left\llbracket{m}_{1}^{2}{m}_{2}^{2}{m}_{3}^{2}\right\rrbracket-\frac{1}{96}\left\llbracket{m}_{1}^{4}{m}_{2}^{2}\right\rrbracket+\frac{1}{96}\left\llbracket{m}_{1}^{6}\right\rrbracket,$
(5)

and the half periods

$e_{1}=\frac{1}{3}(\vartheta_{3}^{4}+\vartheta_{4}^{4}),\quad
e_{2}=-\frac{1}{3}(\vartheta_{2}^{4}+\vartheta_{3}^{4})\quad
e_{3}=\frac{1}{3}(\vartheta_{2}^{4}-\vartheta_{4}^{4})$ (6)

are functions of $\tau_{0}=\tau_{\text{\tiny{UV}}}$, with
$e_{1}+e_{2}+e_{3}=0$. The Jacobi theta functions $\vartheta_{i}$ are defined
in Appendix A. Since the rhs of (4) is a cubic polynomial in $x$, it is indeed
an elliptic curve. We obtain the low energy theory with $N_{f}=3$ flavours by
taking the limit $\tau_{0}\to i\infty$ (or, equivalently, $q_{0}\to 0$) and
$m_{4}\to\infty$ while holding $\Lambda_{3}=64q_{0}^{\frac{1}{2}}m_{4}$ fixed.
The order parameters are then related as Seiberg:1994aj

$u_{N_{f}=4}+\frac{1}{4}e_{1}\left\llbracket m_{1}^{2}\right\rrbracket\to
u_{N_{f}=3}.$ (7)

See Appendix D for the corresponding curves.

Let us study the singularity structure of the Coulomb branch. For generic
masses ${\boldsymbol{m}}=(m_{1},m_{2},m_{3},m_{4})$, there are six distinct
strong coupling singularities. By tuning the mass, some of those singularities
can collide. If we weight each singularity by the number of massless
hypermultiplets at that point, the total weighted number of singularities on
the $u$-plane is thus always $6$. Denote by $k_{l}$ the weight of the $l$-th
singularity, and by ${\boldsymbol{k}}({\boldsymbol{m}})=(k_{1},k_{2},\dots)$
the vector of those weights. In Table 1, we list a selection of specifically
symmetric mass configurations. One notices that certain a priori unrelated
cases have the same weight vector ${\boldsymbol{k}}$ and global symmetries,
such as the cases $\\{\text{B, C, D}\\}$ and $\\{\text{E, F, G}\\}$. This will
be explained in the next subsection. It is also clear that
${\boldsymbol{k}}({\boldsymbol{m}})$ gives a partition of $6$, the total
number of singularities on $\mathcal{B}_{4}$. As there are $p(6)=11$ such
partitions, it is a natural question whether all of those 11 partitions are
realised as ${\boldsymbol{k}}({\boldsymbol{m}})$ for a mass
${\boldsymbol{m}}$. We study this question in Appendix B.

Name m k(m) global symmetry A (m,m,0,0) (2,2,2) SU(2)×SU(2)×SU(2)×U(1) B
(m,m,m,m) (4,1,1) SU(4)×U(1) C (2m,0,0,0) (4,1,1) SU(4)×U(1) D (m,m,m,-m)
(4,1,1) SU(4)×U(1) E (m,m,μ,μ) (2,2,1,1) SU(2)×SU(2)×U(1)×U(1) F (m+μ,m-μ,0,0)
(2,2,1,1) SU(2)×SU(2)×U(1)×U(1) G (m,m,μ,-μ) (2,2,1,1) SU(2)×SU(2)×U(1)×U(1)

Table 1: List of some mass cases with enhanced flavour symmetry in $N_{f}=4$,
with $\mu\neq m$. The vector ${\boldsymbol{k}}({\boldsymbol{m}})$ lists the
multiplicities of all singularities on the Coulomb branch $\mathcal{B}_{4}$
with mass ${\boldsymbol{m}}$.

###  2.2 Triality

Let us study the symmetries of the $N_{f}=4$ curve (4) with mass
${\boldsymbol{m}}=(m_{1},m_{2},m_{3},m_{4})$. Scale invariance, the
$\text{U(1)}_{R}$ R-symmetry and the $\text{SL}(2,\mathbb{Z})$ symmetry acting
on $\tau_{0}$ are explicitly broken by the masses. There is a remnant scale
invariance on the Coulomb branch, which manifests itself in the
$\mathcal{J}$-invariant
$\mathcal{B}_{4}\times\mathbb{C}^{4}\times\mathbb{H}\to\mathbb{C}$ of the
curve being a quasi-homogeneous rational function of degree 0 and type
$(2,1,0)$,

$\mathcal{J}(s^{2}u,s\,{\boldsymbol{m}},\tau_{0})=\mathcal{J}(u,{\boldsymbol{m}},\tau_{0}),\quad
s\in\mathbb{C}^{*}.$ (8)

The $N_{f}=4$ theory has an $\mathrm{SO}(8)$ flavour symmetry, which becomes
the universal double cover $\text{Spin}(8)$ in the quantum theory. In
particular, there exists a short exact sequence

$1\to\mathbb{Z}_{2}\to\text{Spin}(8)\to\mathrm{SO}(8)\to 1$ (9)

of Lie groups. The cover $\text{Spin}(8)$ has an order 6 group
$\mathrm{Out}(\text{Spin}(8))$ of outer automorphisms, which is isomorphic to
$S_{3}$ Adams:623418 ; Fulton2004 .111For any Lie group $G$, there are three
associated groups. $\text{Aut}(G)$ is the Lie group consisting of all
automorphisms of $G$ (i.e. group isomorphisms $G\to G$), $\text{Inn}(G)$ is a
normal subgroup of $\text{Aut}(G)$ consisting of inner automorphisms given by
$\alpha_{g}(h)\coloneqq ghg^{-1}$ for any $g\in G$, and
$\text{Out}(G)=\text{Aut}(G)/\text{Inn}(G)$ is the quotient group. The
automorphism group of $\text{Spin}(8)$ is
$\mathrm{Aut}(\mathrm{SO}(8))=\text{PSO}(8)\rtimes S_{3}$ Adams:623418 .

This group of outer automorphisms acts on the $N_{f}=4$ theory as follows. The
states with $(n_{m},n_{e})=(0,1)$ are the elementary hypermultiplets, which
transform in the fundamental vector representation of $\text{Spin}(8)$. The
magnetic monopole $(1,0)$ transforms as one spinor representation, and the
dyon $(1,1)$ transforms as the conjugate spinor representation Seiberg:1994aj
. By an accidental isomorphism, these three representations are all
$8$-dimensional and irreducible, and they are permuted by the outer
automorphism group $\mathrm{Out}(\text{Spin}(8))\cong S_{3}$. It is generated
by

$\mathcal{T}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\end{pmatrix},\qquad\mathcal{S}=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\
1&1&-1&-1\\\ 1&-1&1&-1\\\ 1&-1&-1&1\end{pmatrix},$ (10)

which act on the column vector
${\boldsymbol{m}}\in\mathscr{M}\coloneqq\mathbb{C}^{4}$ from the left
Seiberg:1994aj ; Marino:1998ru ; Tai:2010im . The map $\mathcal{T}$ exchanges
the two spinors keeping the vector fixed, while $\mathcal{S}$ exchanges the
vector with the spinor, keeping the conjugate spinor fixed. This is depicted
in Fig. 1.

Figure 1: Dynkin diagram of $\mathfrak{d}_{4}=\mathrm{Lie}(\text{Spin}(8))$.
The group $\mathscr{T}\cong S_{3}$ of outer isomorphisms acts by permutations
on the three conjugacy classes of irreducible representations $v$, $s$ and
$\bar{s}$ attached to the nodes of the diagram. The 28-dimensional adjoint
representation is left invariant by $\mathscr{T}$.

The generators (10) satisfy the algebra

$\mathcal{T}^{2}=\mathcal{S}^{2}=(\mathcal{S}\mathcal{T})^{3}=\mathcal{S}\mathcal{T}^{2}\mathcal{S}=\mathbbm{1},$
(11)

which is a presentation of the symmetric group $S_{3}$. Since
$\mathcal{T}^{\mathsf{T}}\mathcal{T}=\mathcal{S}^{\mathsf{T}}\mathcal{S}=\mathbbm{1}$
but $\det\mathcal{T}=\det\mathcal{S}=-1$, the matrices $\mathcal{T}$ and
$\mathcal{S}$ generate a subgroup

$\mathscr{T}=\langle\mathcal{T},\mathcal{S}\rangle$ (12)

of the orthogonal group $\text{O}(4,\mathbb{C})$, isomorphic to
$S_{3}$.222They actually form a subgroup of $O(4,\mathbb{Q})$, but act on
${\boldsymbol{m}}\in\mathbb{C}^{4}$. As a consequence, they leave the inner
product $\left\llbracket m_{1}^{2}\right\rrbracket$ (LABEL:bracketnotation)
invariant.

The flavour symmetry mixes with the $\text{SL}(2,\mathbb{Z})$-symmetry acting
on the UV-coupling $\tau_{0}$ in an interesting way. To see this, notice that
the reduction $\mathbb{Z}\to\mathbb{Z}_{2}$ modulo $2$ induces a homomorphism
$\text{SL}(2,\mathbb{Z})\to\text{SL}(2,\mathbb{Z}/2\mathbb{Z})$. Since
$\text{SL}(2,\mathbb{Z}/2\mathbb{Z})\cong S_{3}$ are isomorphic, by
transitivity we have a group homomorphism

$\varphi:\text{SL}(2,\mathbb{Z})\longrightarrow\mathrm{Out}(\text{Spin}(8)).$
(13)

The full symmetry group of the $N_{f}=4$ theory is the semidirect product
Seiberg:1994aj 333Recall that for two groups $G$ and $H$, a group homomorphism
$\varphi:G\to\mathrm{Aut}(H)$ defines a semi-direct product
$H\rtimes_{\varphi}G\subset H\times G$ with the multiplication
$(h_{1},g_{1})(h_{2},g_{2})\coloneqq(h_{1}\varphi(g_{1})(h_{2}),g_{1}g_{2})$.
For $(h,g)\in H\rtimes_{\varphi}G$, the inverse is found as
$(\varphi(g^{-1})(h^{-1}),g^{-1})$.

$\mathfrak{T}\coloneqq\text{Spin}(8)\rtimes_{\varphi}\text{SL}(2,\mathbb{Z}).$
(14)

The group $(\mathfrak{T},\bullet)$ consists of elements
$(A,\gamma)\in\text{Spin}(8)\times\text{SL}(2,\mathbb{Z})$, with group
operation

$(A,\gamma)\bullet(\tilde{A},\tilde{\gamma})\coloneqq(A\,\varphi(\gamma)(\tilde{A}),\gamma\,\circ\,\tilde{\gamma}).$
(15)

The action of (10) is thus accompanied with an action of
$\text{SL}(2,\mathbb{Z})$ on $\tau$ and $\tau_{0}$. From (11) we find that
$\mathcal{T}^{2}$ and $\mathcal{S}\mathcal{T}^{2}\mathcal{S}$ leave any mass
configuration invariant. This implies that the theory should also be invariant
under the simultaneous action of $T^{2}$ and $ST^{2}S$ on the two couplings.
These two matrices in $\text{SL}(2,\mathbb{Z})$ generate the principal
congruence subgroup $\Gamma(2)$. From this it is also clear that

$\text{SL}(2,\mathbb{Z})/\Gamma(2)=\\{I,T,S,TS,ST,TST\\}\cong S_{3},$ (16)

which is another way to see that the group of outer isomorphisms is $S_{3}$
Tai:2010im . This action is depicted in Fig. 2.

Figure 2: Action of $\text{SL}(2,\mathbb{Z})$ on
$\text{SL}(2,\mathbb{Z})/\Gamma(2)\cong S_{3}$

The subgroup $\Gamma(2)$ is the kernel of the above group homomorphism
$\text{SL}(2,\mathbb{Z})\to\text{SL}(2,\mathbb{Z}/2\mathbb{Z})$, such that it
is in fact a normal subgroup $\Gamma(2)\lhd\,\text{SL}(2,\mathbb{Z})$.

The moduli spaces of the cases A–G of Table 1 are related by $\mathscr{T}$ in
the following way. We have that
${\boldsymbol{m}}_{\text{A}},{\boldsymbol{m}}_{\text{C}},{\boldsymbol{m}}_{\text{F}}$
are invariant under $\mathcal{T}$. Case A is invariant under both
$\mathcal{T}$ and $\mathcal{S}$. The $\mathcal{S}$-transformation relates
cases B and C, as well as E and F, while leaving cases D and G invariant. We
depict the relation among cases B, C and D in Fig. 3. For the cases E, F and
G, there is an analogous diagram.

Figure 3: Relation among ${\boldsymbol{m}}_{\text{B}}=(m,m,m,m)$,
${\boldsymbol{m}}_{\text{C}}=(2m,0,0,0)$ and
${\boldsymbol{m}}_{\text{D}}=(m,m,m,-m)$.

An instance of these relations is that the weights of the singular structure
on the Coulomb branch are invariant under those spaces that are related by
triality,

${\boldsymbol{k}}(\mathscr{T}{\boldsymbol{m}})={\boldsymbol{k}}({\boldsymbol{m}}).$
(17)

Using the action of the SO(8) flavour group, a large range of masses with
equivalent duality diagrams can be reached. For example, the mass
${\boldsymbol{m}}=(2m,0,0,0)$ is related to ${\boldsymbol{m}}=(0,0,0,2m)$ by
an SO(8) rotation. The first one is invariant under $\mathcal{T}$ while the
second one is not. The orbit under $\mathcal{T}$ and $\mathcal{S}$ for the
case ${\boldsymbol{m}}={\boldsymbol{m}}_{\text{B}}=(2m,0,0,0)$ is, as we have
just discussed, given by Fig. 3, while that of ${\boldsymbol{m}}=(0,0,0,2m)$
is given in Fig. 4. We see that it is of order six, and includes different
relative signs compared to ${\boldsymbol{m}}_{\text{A}}$ and
${\boldsymbol{m}}_{\text{D}}$. On closer inspection, we note that the mass
vectors come in pairs differing by an overall sign, which is an element of
SO(8). Thus identifying the mass vectors related by SO(8) in diagram 4, we
find that it is equivalent to diagram 3.

Figure 4: Orbit of the mass vector ${\boldsymbol{m}}=(0,0,0,2m)$ under
$\mathcal{T}$ and $\mathcal{S}$.

###  2.3 Group action

The action

$\displaystyle\mathscr{T}\times\mathscr{M}$
$\displaystyle\longrightarrow\mathscr{M}$ (18)
$\displaystyle(g,{\boldsymbol{m}})$ $\displaystyle\longmapsto
g\cdot{\boldsymbol{m}}$

of the triality group $\mathscr{T}$ on mass space $\mathscr{M}$ can be studied
in great detail. It is easy to check that the action is faithful444For every
$g\neq h\in\mathscr{T}$ there exists an ${\boldsymbol{m}}\in\mathscr{M}$ such
that $g\cdot{\boldsymbol{m}}\neq h\cdot{\boldsymbol{m}}$., but neither
free555A group action is free if it has no fixed points, but
${\boldsymbol{m}}=0$ is a fixed point for any $g\in\mathscr{T}$. nor
transitive666For each pair
${\boldsymbol{m}},\tilde{\boldsymbol{m}}\in\mathscr{M}$ there exists
$g\in\mathscr{T}$ such that a $g\cdot{\boldsymbol{m}}=\tilde{\boldsymbol{m}}$.
A counterexample would be ${\boldsymbol{m}}=0$ and $\tilde{\boldsymbol{m}}\neq
0$..

Up to conjugation, $S_{3}\cong\mathscr{T}$ has four subgroups. They are: the
trivial group $\mathbb{Z}_{1}$, the symmetric group
$S_{2}\cong\mathbb{Z}_{2}$, the alternating group $A_{3}\cong\mathbb{Z}_{3}$,
and $S_{3}$ itself. They have order 1, 2, 3, and 6, respectively. All three
proper subgroups are abelian. For a given ${\boldsymbol{m}}$, triality thus
not always acts by the full $S_{3}$ but rather by a subgroup. For every
${\boldsymbol{m}}\in\mathscr{M}$ we can study the orbit
$\mathscr{T}\cdot{\boldsymbol{m}}=\\{g\cdot{\boldsymbol{m}}\,|\,g\in\mathscr{T}\\}$.
The sets of orbits of $\mathscr{M}$ then give a partition of $\mathscr{M}$
under the action (18).

First, notice that since $\mathscr{T}$ is a finite group, all elements have
finite order. In particular,
$\mathcal{T}^{2}=\mathcal{S}^{2}=(\mathcal{T}\mathcal{S}\mathcal{T})^{2}=\mathbbm{1}$
and $(\mathcal{S}\mathcal{T})^{3}=(\mathcal{T}\mathcal{S})^{3}=\mathbbm{1}$.
The stabiliser subgroup of a mass ${\boldsymbol{m}}\in\mathscr{M}$ is defined
as
$\mathscr{T}_{{\boldsymbol{m}}}=\\{g\in\mathscr{T}\,|\,g\cdot{\boldsymbol{m}}={\boldsymbol{m}}\\}$.
By the orbit-stabiliser theorem

$|\mathscr{T}\cdot{\boldsymbol{m}}|=|\mathscr{T}|/|\mathscr{T}_{{\boldsymbol{m}}}|,$
(19)

it suffices to study the fixed point equations in order to identify the
stabiliser subgroups $\\{\mathbb{Z}_{1},S_{2},A_{3},S_{3}\\}$ with the
subgroups of $\mathscr{T}$. It is straightforward to identify the fixed point
loci

$\begin{split}\mathcal{L}_{\mathcal{T}}&=\\{{\boldsymbol{m}}\in\mathscr{M}\,|\,m_{4}=0\\},\\\
\mathcal{L}_{\mathcal{S}}&=\\{{\boldsymbol{m}}\in\mathscr{M}\,|\,m_{1}=m_{2}+m_{3}+m_{4}\\},\\\
\mathcal{L}_{\mathcal{S}\mathcal{T}\mathcal{S}}&=\\{{\boldsymbol{m}}\in\mathscr{M}\,|\,m_{1}=m_{2}+m_{3}-m_{4}\\},\\\
\mathcal{L}_{\mathcal{S}\mathcal{T}}=\mathcal{L}_{\mathcal{T}\mathcal{S}}&=\\{{\boldsymbol{m}}\in\mathscr{M}\,|\,m_{1}=m_{2}+m_{3}\text{
and }m_{4}=0\\},\\\ \end{split}$ (20)

where
$\mathcal{L}_{g}=\\{{\boldsymbol{m}}\in\mathscr{M}\,|\,g\cdot{\boldsymbol{m}}={\boldsymbol{m}}\\}$.
For ${\boldsymbol{m}}$ in precisely one of $\mathcal{L}_{\mathcal{T}}$,
$\mathcal{L}_{\mathcal{S}}$ or
$\mathcal{L}_{\mathcal{S}\mathcal{T}\mathcal{S}}$, one finds that
$|\mathscr{T}\cdot{\boldsymbol{m}}|=3$. From (19) it then follows that
$|\mathscr{T}_{{\boldsymbol{m}}}|=2$, such that
$\mathscr{T}_{{\boldsymbol{m}}}\cong S_{2}$. In fact, since $\mathcal{T}$,
$\mathcal{S}$ and $\mathcal{S}\mathcal{T}\mathcal{S}$ are all order $2$
elements of $\mathscr{T}$, the stabiliser groups
$\mathscr{T}_{{\boldsymbol{m}}}$ for ${\boldsymbol{m}}$ in either of the three
loci are precisely the three order 2 conjugate subgroups of $\mathscr{T}\cong
S_{3}$.777If we represent $S_{3}$ in cycle notation of permutations of
$\\{1,2,3\\}$, the three order 2 conjugate subgroups of $S_{3}$ are
$\\{(),(1,2)\\}$, $\\{(),(1,3)\\}$ and $\\{(),(2,3)\\}$.

The intersection

$\mathcal{L}_{1}=\mathcal{L}_{\mathcal{T}}\cap\mathcal{L}_{\mathcal{S}}=\\{{\boldsymbol{m}}\in\mathscr{M}\,|\,m_{1}=m_{2}+m_{3}\text{
and }m_{4}=0\\}$ (21)

is the locus of triality invariant masses,
$\mathscr{T}\cdot{\boldsymbol{m}}={\boldsymbol{m}}$. Thus, according to (19)
we have $|\mathscr{T}_{{\boldsymbol{m}}}|=6$ for such masses, such that indeed
$\mathscr{T}_{{\boldsymbol{m}}}=\mathscr{T}$. For the last locus in (20), we
see immediately that
$\mathcal{L}_{\mathcal{S}\mathcal{T}}=\mathcal{L}_{\mathcal{T}\mathcal{S}}=\mathcal{L}_{\mathcal{T}}\cap\mathcal{L}_{\mathcal{S}}$
contains precisely the invariant masses. Therefore, if ${\boldsymbol{m}}$ is
kept fixed by either $\mathcal{T}\mathcal{S}$ or $\mathcal{S}\mathcal{T}$ then
it is also fixed by both $\mathcal{T}$ and $\mathcal{S}$ and therefore by all
of $\mathscr{T}$. Since $\mathcal{S}\mathcal{T}$ and $\mathcal{T}\mathcal{S}$
are the only elements of $\mathscr{T}$ of order 3, there is actually no mass
${\boldsymbol{m}}$ such that $\mathscr{T}\cdot{\boldsymbol{m}}$ has 2
elements, and so there is no stabiliser subgroup isomorphic to $A_{3}$. By
case analysis, it is also easy to prove that the set
$\mathscr{T}\cdot{\boldsymbol{m}}$ has $1$, $3$ or $6$ elements.

Let us summarise. If ${\boldsymbol{m}}\in\mathcal{L}_{1}$, it is invariant
under $\mathscr{T}$. If ${\boldsymbol{m}}$ is in any of
$\mathcal{L}_{\mathcal{T}}$, $\mathcal{L}_{\mathcal{S}}$ or
$\mathcal{L}_{\mathcal{S}\mathcal{T}\mathcal{S}}$, it could be in the
intersection of any two of them. These intersections are however all equal to
$\mathcal{L}_{1}$, which is of course because any two elements of
$\\{\mathcal{T},\mathcal{S},\mathcal{S}\mathcal{T}\mathcal{S}\\}$ generate
$\mathscr{T}$. This is depicted in Fig. 5.

Figure 5: The loci (20) with nontrivial stabiliser groups on the subspace
$m_{2}=m_{3}=0$ in $\mathscr{M}$. They all mutually intersect in the locus
$\mathcal{L}_{1}$ of triality invariant masses.

If ${\boldsymbol{m}}$ is then an element of

$\mathcal{L}_{3}=\mathcal{L}_{\mathcal{T}}\cup\mathcal{L}_{\mathcal{S}}\cup\mathcal{L}_{\mathcal{S}\mathcal{T}\mathcal{S}}\setminus\mathcal{L}_{1},$
(22)

then the stabiliser group of ${\boldsymbol{m}}$ is isomorphic to $S_{2}$. If
${\boldsymbol{m}}$ does not lie in either $\mathcal{L}_{1}$ or
$\mathcal{L}_{3}$, then there is no remaining symmetry. It lies in

$\mathcal{L}_{6}=\mathscr{M}\setminus\mathcal{L}_{1}\cup\mathcal{L}_{3},$ (23)

and its stabiliser group is trivial.

##  3 Order parameters and bimodular forms

For the $N_{f}=4$ SW theory, there are other curves than the one introduced by
Seiberg and Witten Huang:2011qx ; Dorey:1996bn ; Nekrasov:2015wsu ;
Jeong:2019fgx ; Manschot:2019pog ; Nekrasov:2012xe ; Gaiotto:2009hg ;
Gaiotto:2009we ; Witten:1997sc . In this paper, we focus on the modularity of
the original SW curve (4). In Appendix C we show that similar results hold for
the curve constructed from the qq-characters of the theory. The $N_{f}=4$ SW
curve however has the advantage over the qq-curve in that it depends
explicitly on $\tau_{0}$, and one can study modular transformations of
$\tau_{0}$. We proceed by studying the mass configurations with the largest
flavour symmetry groups, A, B, C and D. In all these cases, $u$ is a weight
$(0,2)$ bimodular form, which we define below, for a triple of groups related
to the duality group of the decoupling theory where the mass of the
hypermultiplets is infinitely large, and the stabiliser group of the mass
under the triality action. Since case A is triality invariant, $u$ in that
case transforms under the full $\text{SL}(2,\mathbb{Z})$ group. The other
cases B, C and D are permuted by triality, and furnish a vector-valued
bimodular form.

The massless case where ${\boldsymbol{m}}_{0}=(0,0,0,0)$ is very simple, as
$j(\tau)=\mathcal{J}(u,0,\tau_{0})=j(\tau_{0})$ and therefore

$\tau(u)=\tau_{0}$ (24)

is constant over the whole Coulomb branch $\mathcal{B}_{4}\ni u$. In other
words, the coupling $\tau$ is fixed and thus does not run, which is a
consequence of the massless $N_{f}=4$ theory being exactly superconformal.
There are six singularities, which all sit at the origin $u=0$ and form the
non-abelian Coulomb point with a five quaternionic-dimensional Higgs branch
Argyres:1995xn .

Let us recall a method for finding explicit expressions for the Coulomb branch
parameter $u$. This was recently discussed in detail in aspman2021cutting .
The $\mathcal{J}$-invariant of the SW curve (4) is a rational function
$\mathcal{J}(u,{\boldsymbol{m}},\tau_{0})$ in $u$, the masses
${\boldsymbol{m}}=(m_{1},m_{2},m_{3},m_{4})$ and $e_{i}(\tau_{0})$. While in
general it is not possible to solve
$\mathcal{J}(u(\tau),{\boldsymbol{m}},\tau_{0})=j(\tau)$ for $u(\tau)$
analytically, for specific masses we can rather solve
$\mathcal{J}(u,{\boldsymbol{m}},\tau_{0})=\mathcal{R}(\lambda)$ for $u$, where

$\mathcal{R}(p)=2^{8}\frac{(1+(p-1)p)^{3}}{(p-1)^{2}p^{2}}$ (25)

is the unique rational function with the property that
$\mathcal{R}(\lambda)=j$. Here,
$\lambda\coloneqq\frac{\vartheta_{2}^{4}}{\vartheta_{3}^{4}}$ is the modular
lambda function (a Hauptmodul for $\Gamma(2)$) with $\vartheta_{i}$ the Jacobi
theta functions (77), and $j$ is the modular $j$-function (84) . The reason
for this is that in certain mass configurations, the sextic equation
constructed from the rational function
$\mathcal{J}(u,{\boldsymbol{m}},\tau_{0})$ defines a field extension of
$\mathbb{C}(\text{SL}(2,\mathbb{Z}))$ with intermediate field
$\mathbb{C}(\Gamma(2))$, such that the sextic equation factors over
$\mathbb{C}(\Gamma(2))$ into products of lower degree polynomials
aspman2021cutting .888It is true in $N_{f}=0,1,2,3,4$ that $g_{k}$ is a
polynomial in $u$ of degree $k$, such that
$(g_{2}^{3}-27g_{3}^{2})j-12^{3}g_{2}^{3}$ is indeed a sextic polynomial in
$u$.

###  3.1 Case A

For the mass ${\boldsymbol{m}}_{\text{A}}=(m,m,0,0)$, this allows to express
$u$ as a rational function in Jacobi theta functions of $\tau_{0}$ and $\tau$.
There are in fact six solutions to the correspondence
$\mathcal{J}(u(\tau),{\boldsymbol{m}},\tau_{0})=j(\tau)$. A consistent way of
choosing which solution to use, which we will employ throughout, is to take
the one that has the right decoupling limit when decoupling the massive
hypermultiplets, i.e., the one that decouples to the order parameter of
massless $N_{f}=2$, Eq. (110).

In view of the more complicated mass cases, we can further simplify the rather
lengthy expression. The dependence on $\tau$ is in fact only through
$\lambda=\frac{\vartheta_{2}^{4}}{\vartheta_{3}^{4}}$. This is not quite true
for $\tau_{0}$, for which $u$ has weight 2 Huang:2011qx . This weight factor
can be extracted by eliminating $\vartheta_{4}(\tau_{0})$ through the Jacobi
identity (79) and $\vartheta_{2}(\tau_{0})$ through the definition of
$\lambda(\tau_{0})$. This gives

$u_{\text{A}}(\tau,\tau_{0})=-\frac{m^{2}}{3}\vartheta_{3}(\tau_{0})^{4}\frac{\lambda(\tau_{0})^{2}+2\left(\lambda(\tau)-1\right)\lambda(\tau_{0})-\lambda(\tau)}{\lambda(\tau_{0})-\lambda(\tau)}.$
(26)

The simple mass dependence of $u_{\text{A}}$ is a consequence of the scaling
symmetry (8). The second prefactor $\vartheta_{3}(\tau_{0})^{4}$ gives the
weight 2. The remaining quotient is written in a manifestly invariant fashion.
Let us denote by $\Gamma_{\tau}$ ($\Gamma_{\tau_{0}}$) a group acting by
linear fractional transformations on $\tau$ ($\tau_{0}$). As
$\vartheta_{3}(\tau_{0})^{4}$ is a modular form of weight $2$ and
$\lambda(\tau_{0})$ a modular function (of weight 0) for
$\Gamma(2)_{\tau_{0}}$, one can easily see that $u_{\text{A}}(\tau,\tau_{0})$
is a weight $2$ modular form for $\Gamma(2)_{\tau_{0}}$ for fixed $\tau$, and
a modular function for $\Gamma(2)_{\tau}$ for fixed $\tau_{0}$. We thus have
that

$u_{\text{A}}(\gamma_{1}\tau,\gamma_{2}\tau_{0})=(c_{2}\tau_{0}+d_{2})^{2}u_{\text{A}}(\tau,\tau_{0}),\quad\gamma_{i}=\begin{pmatrix}a_{i}&b_{i}\\\
c_{i}&d_{i}\end{pmatrix}\in\Gamma(2)$ (27)

for $i=1,2$. We call $u_{\text{A}}$ modular for
$\Gamma(2)_{\tau}\times\Gamma(2)_{\tau_{0}}$, where the occurrence of two
groups indicates that they act on both variables $\tau$ and $\tau_{0}$
separately.

The mass ${\boldsymbol{m}}_{\text{A}}$ is invariant under the triality group
(12). As triality acts on $\tau$ and $\tau_{0}$ together, this suggests that
$u_{\text{A}}$ transforms under a simultaneous transformation of
$\text{SL}(2,\mathbb{Z})$. Indeed, if one acts simultaneously on $\tau$ and
$\tau_{0}$ with $\text{SL}(2,\mathbb{Z})$, it is easy to check from
$T:\lambda\mapsto\frac{\lambda}{\lambda-1}$ and $S:\lambda\mapsto 1-\lambda$
that $u_{\text{A}}(\tau,\tau_{0})$ transforms as

$u_{\text{A}}(\gamma\tau,\gamma\tau_{0})=(c\tau_{0}+d)^{2}\,u_{\text{A}}(\tau,\tau_{0}),\quad\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z}).$ (28)

We call $u_{\text{A}}$ modular for
$\text{SL}(2,\mathbb{Z})_{(\tau,\tau_{0})}$, where the notation indicates that
the single group $\text{SL}(2,\mathbb{Z})$ acts on both $\tau$ and $\tau_{0}$
simultaneously. The two transformations (27) and (28) are characteristic
properties for functions known as “bimodular forms” stienstra_zagier_16 ;
yang2007differential ; liuquan2020 . For our application to $N_{f}=4$ SQCD, we
will adopt the following definition in this paper:

######  Definition 1 (Bimodular form).

Let $(\Gamma_{1},\Gamma_{2};\Gamma)$ be a triple of subgroups of
$\text{SL}(2,\mathbb{R})$ commensurable with $\text{SL}(2,\mathbb{Z})$.999A
subgroup $\Gamma\subset\text{SL}(2,\mathbb{R})$ is commensurable with
$\text{SL}(2,\mathbb{Z})$ if $\Gamma\cap\text{SL}(2,\mathbb{Z})$ has finite
index in both $\text{SL}(2,\mathbb{Z})$ and $\text{SL}(2,\mathbb{R})$. This
includes in particular all congruence subgroups of $\text{SL}(2,\mathbb{Z})$.
A two-variable meromorphic function
$F:\mathbb{H}\times\mathbb{H}\to\mathbb{C}$ is called a bimodular form of
weight $(k_{1},k_{2})$ for the triple $(\Gamma_{1},\Gamma_{2};\Gamma)$ if it
satisfies both Condition 1 & 2:

  * •

Condition 1: For all $\gamma_{i}=\left(\begin{smallmatrix}a_{i}&&b_{i}\\\
c_{i}&&d_{i}\end{smallmatrix}\right)\in\Gamma_{i}$, $i=1,2$, $F$ transforms as

$F(\gamma_{1}\tau_{1},\gamma_{2}\tau_{2})=\chi(\gamma_{1},\gamma_{2})\,(c_{1}\tau_{1}+d_{1})^{k_{1}}(c_{2}\tau_{2}+d_{2})^{k_{2}}F(\tau_{1},\tau_{2}),$
(29)

for a certain multiplier $\chi:\Gamma_{1}\times\Gamma_{2}\to\mathbb{C}^{*}$.
We call this the separate transformation of $F$ under
$(\Gamma_{1},\Gamma_{2})$, and denote it by
$(\Gamma_{1})_{\tau_{1}}\times(\Gamma_{2})_{\tau_{2}}$.

  * •

Condition 2: For all $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma$, $F$ transforms as

$F(\gamma\tau_{1},\gamma\tau_{2})=\phi(\gamma)\,(c\tau_{1}+d)^{k_{1}}(c\tau_{2}+d)^{k_{2}}F(\tau_{1},\tau_{2}),$
(30)

for a multiplier $\phi:\Gamma\to\mathbb{C}^{*}$. We call this the simultaneous
transformation of $F$ under $\Gamma$, and denote it by
$\Gamma_{(\tau_{1},\tau_{2})}$.

Note that condition 2 follows from condition 1 if $\Gamma$ is the intersection
of $\Gamma_{1}$ and $\Gamma_{2}$, $\Gamma=\Gamma_{1}\cap\Gamma_{2}$ with
$\phi(\gamma)=\chi(\gamma,\gamma)$, $\gamma\in\Gamma$.

This definition contains the main aspects of other definitions of bimodular
forms in the literature stienstra_zagier_16 ; yang2007differential ;
liuquan2020 ; Manschot:2021qqe .

The definition above for the triple
$(\Gamma_{1},\Gamma_{2};\Gamma_{1}\cap\Gamma_{2})$ is equivalent to the
definition in yang2007differential . For the triple
$(\Gamma_{1},\Gamma_{1};\text{SL}(2,\mathbb{Z}))$, our definition is
equivalent to the one of Manschot:2021qqe . Finally, for $k_{1}=k_{2}$ and the
triple $(\Gamma_{1},\Gamma_{1};\Gamma)$, our definition is equivalent with
liuquan2020 . Finally, the definition of Stienstra and Zagier
stienstra_zagier_16 , as cited in yang2007differential , does require
Condition 2 without requiring Condition 1.

From definition 1, we find that
$u_{\text{A}}:\mathbb{H}\times\mathbb{H}\to\mathbb{C}$ in (26) is a bimodular
form of weight $(0,2)$ for the triple

$(\Gamma(2),\Gamma(2);\text{SL}(2,\mathbb{Z})),$ (31)

with trivial multipliers $\chi$ and $\phi$. In fact, $m\mapsto u_{\text{A}}$
is a 1-parameter family of such bimodular forms.

The function (26) can be easily expanded in either $q=e^{2\pi i\tau}$ or
$q_{0}=e^{2\pi i\tau_{0}}$. When expanding $u_{\text{A}}$ around $q_{0}=0$,
every coefficient is a modular function for $\Gamma(2)_{\tau}$. If we denote
the vector space of holomorphic modular forms of weight $k$ for
$\Gamma\subseteq\text{SL}(2,\mathbb{Z})$ by $\mathcal{M}_{k}(\Gamma)$, then
$u_{\text{A}}\in\mathcal{M}_{0}(\Gamma(2))\llbracket
q_{0}^{\frac{1}{4}}\rrbracket$. Conversely, we have that
$u_{\text{A}}\in\mathcal{M}_{2}(\Gamma(2))\llbracket
q^{\frac{1}{4}}\rrbracket$.

Recall that $\Gamma(2)$ is a genus zero congruence subgroup. As such, its
Hauptmodul $\lambda$ is the single transcendental generator of the function
field of $\Gamma(2)\backslash\mathbb{H}^{*}$. Since $u_{\text{A}}$ is modular
in $\tau$ as well as $\tau_{0}$ for $\Gamma(2)$ and no larger subgroup of
$\text{SL}(2,\mathbb{Z})$, the transcendence of $\lambda$ then implies that
(26) cannot be simplified further.

The Coulomb branch $\mathcal{B}_{4}$ for the mass
${\boldsymbol{m}}={\boldsymbol{m}}_{\text{A}}$ has six singularities that come
in three pairs of two. By expanding $\lambda(\tau)$ around the cusps, one
easily finds

$\displaystyle u_{\text{A}}(\tfrac{1}{2},\tau_{0})$
$\displaystyle=-\frac{m^{2}}{3}\vartheta_{3}(\tau_{0})^{4}(\lambda(\tau_{0})-2),$
(32) $\displaystyle u_{\text{A}}(0,\tau_{0})$
$\displaystyle=-\frac{m^{2}}{3}\vartheta_{3}(\tau_{0})^{4}(\lambda(\tau_{0})+1),$
$\displaystyle u_{\text{A}}(1,\tau_{0})$
$\displaystyle=-\frac{m^{2}}{3}\vartheta_{3}(\tau_{0})^{4}(-2\lambda(\tau_{0})+1).$

Notice that the singularities are holomorphic modular forms of weight 2 for
$\Gamma(2)_{\tau_{0}}$, and are permuted by elements of
$(\text{SL}(2,\mathbb{Z}))/\Gamma(2))_{\tau_{0}}$. The reason for
$u_{\text{A}}(\tfrac{1}{2},\tau_{0})=u_{\text{A}}(i\infty,\tau_{0})$ is
explained in Section 3.7. Since $\lambda$ is a Hauptmodul for $\Gamma(2)$, for
given $\tau_{0}\in\Gamma(2)\backslash\mathbb{H}$ there is exactly one
$\tau\in\Gamma(2)\backslash\mathbb{H}$ where $u$ has a pole. It is where
$\tau$ approaches $\tau_{0}$, $u_{\text{A}}(\tau_{0},\tau_{0})=\infty$.

We can furthermore compute the period $\frac{da}{du}$. Actually,
$\frac{da}{du}$ is not invariant under the monodromy around $\infty$, but
multiplied by $-1$. Instead of $\frac{da}{du}$, we may consider
$\left(\frac{da}{du}\right)^{2}$, which is monodromy invariant
Brandhuber:1996ng ; aspman2021cutting . In the pure ($N_{f}=0$) SU(2) case, it
is a modular form of weight $2$ for $\Gamma^{0}(4)$. The weight is the same in
$N_{f}=4$, however it also transforms well under fractional linear
transformations of $\tau_{0}$. More specifically, we find that

$\left(\frac{da}{du}\right)_{\text{A}}^{2}(\tau,\tau_{0})=\frac{1}{8m^{2}}\frac{\vartheta_{3}(\tau)^{4}}{\vartheta_{3}(\tau_{0})^{8}}\frac{\lambda(\tau)-\lambda(\tau_{0})}{\lambda(\tau_{0})(\lambda(\tau_{0})-1)}.$
(33)

The normalisation may be checked from the fact that
$\frac{da}{du}\sim\frac{1}{\sqrt{8u}}$ for $u\to\infty$, which due to (26)
corresponds to $\tau\to\tau_{0}$. Since $\vartheta_{3}^{4}$ is a modular form
of weight $2$ for $\Gamma(2)$, it follows that
$\left(\frac{da}{du}\right)^{2}_{\text{A}}$ satisfies condition 1 of
definition 1 with weight $(2,-4)$ for
$\Gamma(2)_{\tau}\times\Gamma(2)_{\tau_{0}}$. As ${\boldsymbol{m}}_{\text{A}}$
is left invariant by triality, $\left(\frac{da}{du}\right)^{2}_{\text{A}}$
must also be modular for $\text{SL}(2,\mathbb{Z})$. Indeed, one easily finds
that $\left(\frac{da}{du}\right)^{2}_{\text{A}}$ also satisfies condition $2$,
such that it is a bimodular form of weight $(2,-4)$ for the triple
$(\Gamma(2),\Gamma(2);\text{SL}(2,\mathbb{Z}))$.

We can also compute the physical discriminant, which for the case A reads

$\Delta_{\text{A}}=(u-u_{\text{A}}(\tfrac{1}{2},\tau_{0}))^{2}(u-u_{\text{A}}(0,\tau_{0}))^{2}(u-u_{\text{A}}(1,\tau_{0}))^{2}.$
(34)

Since the singularities (32) themselves are modular forms for $\tau_{0}$, it
is again a bimodular form. One easily computes

$\Delta_{\text{A}}(\tau,\tau_{0})=m^{12}\vartheta_{3}(\tau_{0})^{24}\frac{\lambda(\tau)^{2}(\lambda(\tau)-1)^{2}\lambda(\tau_{0})^{4}(\lambda(\tau_{0})-1)^{4}}{(\lambda(\tau)-\lambda(\tau_{0}))^{6}}.$
(35)

As $\vartheta_{3}^{24}$ is a modular form of weight $12$ for $\Gamma(2)$, this
shows that $\Delta_{\text{A}}$ has modular weight $(0,12)$ under
$\Gamma(2)_{\tau}\times\Gamma(2)_{\tau_{0}}$. With the same reasoning as
above, we find that $\Delta_{\text{A}}$ is a bimodular form of weight $(0,12)$
for the same triple (31).

###  3.2 Case B

The equal mass case ${\boldsymbol{m}}_{\text{B}}=(m,m,m,m)$ can be treated
with the same technique as in the previous subsection. Since $N_{f}=4$ with
four equal masses flows to $N_{f}=0$ for $m\to\infty$, we can express the
$\tau$ dependence through the Hauptmodul
$f\coloneqq\frac{\vartheta_{2}^{4}+\vartheta_{3}^{4}}{\vartheta_{2}^{2}\vartheta_{3}^{2}}$
of $\Gamma^{0}(4)$. In fact, the $N_{f}=0$ order parameter (107) is just
$\frac{u}{\Lambda_{0}^{2}}=-\frac{1}{2}f$. The order parameter $u_{\text{B}}$
reads

$u_{\text{B}}(\tau,\tau_{0})=-\frac{m^{2}}{3}\vartheta_{2}(\tau_{0})^{2}\vartheta_{3}(\tau_{0})^{2}\frac{2f(\tau_{0})^{2}+f(\tau)f(\tau_{0})-12}{f(\tau_{0})-f(\tau)},$
(36)

which thus does not involve $\vartheta_{4}$. Since
$\vartheta_{2}(\tau_{0})^{2}\vartheta_{3}(\tau_{0})^{2}$ is a holomorphic
modular form of weight $2$ for $\Gamma^{0}(4)_{\tau_{0}}$, we find that
$u_{\text{B}}(\tau,\tau_{0})$ has bimodular weight $(0,2)$ for the separate
transformations under $\Gamma^{0}(4)_{\tau}\times\Gamma^{0}(4)_{\tau_{0}}$.

As
$\mathcal{T}:{\boldsymbol{m}}_{\text{B}}\mapsto{\boldsymbol{m}}_{\text{D}}$,
there is no simultaneous action of $T$ on $\tau$ and $\tau_{0}$ leaving
$u_{\text{B}}$ invariant. Also, since
$\mathcal{S}:{\boldsymbol{m}}_{\text{B}}\mapsto{\boldsymbol{m}}_{\text{C}}$,
$S$ does not leave $u_{\text{B}}$ invariant. However, a subgroup
$\mathscr{T}_{{\boldsymbol{m}}_{\text{B}}}$ of $\mathscr{T}$ leaves
${\boldsymbol{m}}_{\text{B}}$ invariant: Out of the six elements of
$\mathscr{T}$, ${\boldsymbol{m}}_{\text{B}}$ is left invariant by
$\mathbbm{1}$ and $\mathcal{T}\mathcal{S}\mathcal{T}$. As the action of
$\mathscr{T}$ is combined in (14) with a simultaneous action on $\tau$ and
$\tau_{0}$, we find that $u_{\text{B}}$ is expected to be invariant under a
simultaneous transformation of $TST\in\text{SL}(2,\mathbb{Z})$. However, due
to the algebra (11), the same holds for $T^{2}$. These two matrices generate
the congruence subgroup $\Gamma^{0}(2)$ of $\text{SL}(2,\mathbb{Z})$. It is
straightforward to check from the explicit expression (36) that $u_{\text{B}}$
transforms with weight $(0,2)$ under a simultaneous transformation on $\tau$
and $\tau_{0}$ of $\Gamma^{0}(2)_{(\tau,\tau_{0})}$. This proves that
$u_{\text{B}}$ is an example of a bimodular form of weight $(0,2)$ for the
triple

$(\Gamma^{0}(4),\Gamma^{0}(4);\Gamma^{0}(2)).$ (37)

As classified in Section 2.3, the stabiliser subgroup
$\mathscr{T}_{{\boldsymbol{m}}_{\text{B}}}=\\{\mathbbm{1},\mathcal{T}\mathcal{S}\mathcal{T}\\}$
for the mass ${\boldsymbol{m}}_{\text{B}}$ is isomorphic to the group
$S_{2}\cong\mathbb{Z}_{2}$ of order 2. This agrees with the fact that
$\text{SL}(2,\mathbb{Z})/\Gamma^{0}(2)\cong S_{2}$.

The singularities are

$\displaystyle u_{\text{B}}(1,\tau_{0})$
$\displaystyle=-\frac{m^{2}}{3}\vartheta_{2}(\tau_{0})^{2}\vartheta_{3}(\tau_{0})^{2}(-f(\tau_{0})),$
(38) $\displaystyle u_{\text{B}}(0,\tau_{0})$
$\displaystyle=-\frac{m^{2}}{3}\vartheta_{2}(\tau_{0})^{2}\vartheta_{3}(\tau_{0})^{2}(2f(\tau_{0})+6),$
$\displaystyle u_{\text{B}}(2,\tau_{0})$
$\displaystyle=-\frac{m^{2}}{3}\vartheta_{2}(\tau_{0})^{2}\vartheta_{3}(\tau_{0})^{2}(2f(\tau_{0})-6),$

which again are holomorphic modular forms of weight 2 for
$\Gamma^{0}(4)_{\tau_{0}}$. Due to the duality group $\Gamma^{0}(4)_{\tau}$,
we have that $u_{\text{B}}(1,\tau_{0})=u_{\text{B}}(i\infty,\tau_{0})$. This
singularity has degeneracy 4, and flows to $\infty$ for $m\to\infty$. The
singularity in the interior is $u_{\text{B}}(\tau_{0},\tau_{0})=\infty$. One
can also check that the singularities (38) never collide: The conditions
$u_{\text{B}}(1,\tau_{0})=u_{\text{B}}(0,\tau_{0})$ or
$u_{\text{B}}(1,\tau_{0})=u_{\text{B}}(2,\tau_{0})$ are equivalent to
$f(\tau_{0})=\pm 2$, whose only solutions are the two cusps $\tau_{0}^{+}=0$
and $\tau_{0}^{-}=2$ of $\Gamma(2)$. Since the SW curve is singular for those
values of $\tau_{0}$, the singularities do not merge for any finite masses.

Similarly as before, one finds

$\left(\frac{da}{du}\right)^{2}_{\text{B}}(\tau,\tau_{0})=\frac{1}{8m^{2}}\frac{\vartheta_{2}(\tau)^{2}\vartheta_{3}(\tau)^{2}}{\vartheta_{4}(\tau_{0})^{8}}\left(f(\tau)-f(\tau_{0})\right).$
(39)

Since $f$ is a Hauptmodul, $\vartheta_{2}^{2}\vartheta_{3}^{2}$ a modular form
of weight $2$ and $\vartheta_{4}^{8}$ a modular form of weight 4 for
$\Gamma^{0}(4)$, it follows that $\left(\frac{da}{du}\right)^{2}_{\text{B}}$
is a bimodular form of weight $(2,-4)$ for the triple (37). Finally, the
physical discriminant reads

$\Delta_{\text{B}}(\tau,\tau_{0})=m^{12}\vartheta_{4}(\tau_{0})^{24}\frac{(f(\tau)^{2}-4)(f(\tau_{0})^{2}-4)^{2}}{(f(\tau)-f(\tau_{0}))^{6}},$
(40)

which is a bimodular form of weight $(0,12)$ for the triple (37).

###  3.3 Case C

Let us study the case where only one hypermultiplet is massive,
${\boldsymbol{m}}_{\text{C}}=(2m,0,0,0)$.101010The particular normalisation is
chosen such that the diagram 3 holds without any prefactors. Since in the
limit $m\to\infty$ we get massless $N_{f}=3$, we can express the $\tau$
dependence through the Hauptmodul
$\tilde{f}=\frac{\vartheta_{3}^{2}\vartheta_{4}^{2}}{(\vartheta_{3}^{2}-\vartheta_{4}^{2})^{2}}$
of $\Gamma_{0}(4)$. The order parameter of the massless $N_{f}=3$ theory reads
$\frac{u}{\Lambda_{3}^{2}}=-\frac{1}{64}\tilde{f}$ (112), and the functions
$f$ and $\tilde{f}$ are related by $f(4\tau)=16\tilde{f}(\tau)+2$ (see
Appendix D). One finds for the order parameter $u_{\text{C}}$,

$u_{\text{C}}(\tau,\tau_{0})=-\frac{m^{2}}{3}\vartheta_{3}(\tau_{0})^{2}\vartheta_{4}(\tau_{0})^{2}\frac{2\tilde{f}(\tau_{0})^{2}+(10\tilde{f}(\tau)+1)\tilde{f}(\tau_{0})+2\tilde{f}(\tau)}{\tilde{f}(\tau_{0})(\tilde{f}(\tau_{0})-\tilde{f}(\tau))},$
(41)

which is independent of $\vartheta_{2}(\tau_{0})$. Again, the factor
$\vartheta_{3}(\tau_{0})^{2}\vartheta_{4}(\tau_{0})^{2}$ is a modular form of
weight 2 for $\Gamma_{0}(4)_{\tau_{0}}$, and the quotient is a meromorphic
modular function of $\Gamma_{0}(4)$ for both $\tau$ and $\tau_{0}$. Thus
$u_{\text{C}}$ satisfies condition 1 of definition 1 with weight $(0,2)$ for
$\Gamma_{0}(4)_{\tau}\times\Gamma_{0}(4)_{\tau_{0}}$.

Since
$\mathcal{T}:{\boldsymbol{m}}_{\text{C}}\mapsto{\boldsymbol{m}}_{\text{C}}$,
there is a simultaneous $T$-duality

$\mathcal{T}:u_{\text{C}}(\tau+1,\tau_{0}+1)=u_{\text{C}}(\tau,\tau_{0}),$
(42)

which is straightforward to check from (41). As
$\mathcal{S}:{\boldsymbol{m}}_{\text{C}}\mapsto{\boldsymbol{m}}_{\text{B}}$,
this exchanges the order parameters

$u_{\text{C}}(-\tfrac{1}{\tau},-\tfrac{1}{\tau_{0}})=\tau_{0}^{2}u_{\text{B}}(\tau,\tau_{0}),$
(43)

which we can also explicitly check. We can again study the stabiliser subgroup
of $\mathscr{T}_{{\boldsymbol{m}}_{\text{C}}}$ of $\mathscr{T}$. It is the
group generated by $\mathcal{T}$ and $\mathcal{S}\mathcal{T}^{2}\mathcal{S}$,
such that $u_{\text{C}}$ is expected to transform simultaneously under $T$ and
$ST^{2}S$. These two matrices generate the congruence subgroup $\Gamma_{0}(2)$
of $\text{SL}(2,\mathbb{Z})$, which is conjugate to $\Gamma^{0}(2)$. Thus we
find that $u_{\text{C}}$ is a bimodular form of weight $(0,2)$ for the triple

$(\Gamma_{0}(4),\Gamma_{0}(4);\Gamma_{0}(2)).$ (44)

Lastly, we can also study

$\left(\frac{da}{du}\right)^{2}_{\text{C}}(\tau,\tau_{0})=\frac{1}{8m^{2}}\frac{\vartheta_{3}(\tau)^{2}\vartheta_{4}(\tau)^{2}}{\vartheta_{2}(\tau_{0})^{8}}\frac{\tilde{f}(\tau)-\tilde{f}(\tau_{0})}{\tilde{f}(\tau)\tilde{f}(\tau_{0})}.$
(45)

It is straightforward to check that
$\left(\frac{da}{du}\right)^{2}_{\text{C}}$ is a bimodular form of weight
$(2,-4)$ for the triple (44). For the discriminant $\Delta_{\text{C}}$ there
exists a similar expression to (40), and it is a bimodular form of weight
$(0,12)$ for (44).

###  3.4 Case D

Let us finally also study the case ${\boldsymbol{m}}_{\text{D}}=(m,m,m,-m)$.
It is related to cases B and C as in Fig. 3. We have that
${\boldsymbol{m}}_{\text{D}}\in\mathcal{L}_{\mathcal{S}}$, while
${\boldsymbol{m}}_{\text{B}}\in\mathcal{L}_{\mathcal{S}\mathcal{T}\mathcal{S}}$
and ${\boldsymbol{m}}_{\text{C}}\in\mathcal{L}_{\mathcal{T}}$. From the SW
curve one easily finds

$u_{\text{D}}(\tau,\tau_{0})=-\frac{m^{2}}{3}i\vartheta_{2}(\tau_{0})^{2}\vartheta_{4}(\tau_{0})^{2}\frac{2\hat{f}(\tau_{0})^{2}+\hat{f}(\tau)\hat{f}(\tau_{0})-12}{\hat{f}(\tau_{0})-\hat{f}(\tau)},$
(46)

where

$\hat{f}(\tau)=f(\tau+1)=i\frac{\vartheta_{2}(\tau)^{4}-\vartheta_{4}(\tau)^{4}}{\vartheta_{2}(\tau)^{2}\vartheta_{4}(\tau)^{4}}.$
(47)

Since $f$ is a Hauptmodul for $\Gamma^{0}(4)$, $\hat{f}$ is a Hauptmodul for a
subgroup of $\text{SL}(2,\mathbb{Z})$ conjugate to $\Gamma^{0}(4)$,

$\widetilde{\Gamma^{0}(4)}=T\,\Gamma^{0}(4)\,T^{-1}=\langle
T^{4},ST^{2}\rangle.$ (48)

A fundamental domain for $\widetilde{\Gamma^{0}(4)}$ is given by

$\widetilde{\Gamma^{0}(4)}\backslash\mathbb{H}=\mathcal{F}\cup
T\mathcal{F}\cup T^{2}\mathcal{F}\cup T^{3}\mathcal{F}\cup TS\mathcal{F}\cup
T^{3}S\mathcal{F},$ (49)

with $\mathcal{F}=\text{SL}(2,\mathbb{Z})\backslash\mathbb{H}$. It is
straightforward to check that $u_{\text{D}}(\tau,\tau_{0})$ transforms with
weight $(0,2)$ under
$\widetilde{\Gamma^{0}(4)}_{\tau}\times\widetilde{\Gamma^{0}(4)}_{\tau_{0}}$.

The subgroup $\mathscr{T}_{{\boldsymbol{m}}_{\text{D}}}\subset\mathscr{T}$
leaving invariant ${\boldsymbol{m}}_{\text{D}}$ is generated by $\mathcal{S}$
and $\mathcal{T}^{2}$. The two corresponding $\text{SL}(2,\mathbb{Z})$
transformations $S$ and $T^{2}$ generate the theta group
$\widetilde{\Gamma^{0}(2)}\coloneqq\Gamma_{\theta}$ (74), which is a
congruence subgroup of $\text{SL}(2,\mathbb{Z})$ with index 3, conjugate to
$\Gamma_{0}(2)$ and $\Gamma^{0}(2)$. Thus we find that $u_{\text{D}}$ is a
bimodular form of weight $(0,2)$ for the triple

$(\widetilde{\Gamma^{0}(4)},\widetilde{\Gamma^{0}(4)};\widetilde{\Gamma^{0}(2)}).$
(50)

The three groups
$\\{\Gamma_{0}(2),\Gamma^{0}(2),\widetilde{\Gamma^{0}(2)}\\}\ni\Gamma$ are in
fact the three groups $\text{SL}(2,\mathbb{Z})\supset\Gamma\supset\Gamma(2)$
with index 3 and 2 cusps, and they correspond to the three conjugate order 2
subgroups of $S_{3}$.

###  3.5 Generic mass

The analysis of the A, B, C and D theories may suggest that the order
parameter $u_{\boldsymbol{m}}$ for a generic mass ${\boldsymbol{m}}$
transforms with weight $(0,2)$ under $G_{\tau}\times G_{\tau_{0}}$ for some
subgroup $G\subseteq\text{SL}(2,\mathbb{Z})$. This is however not true in
general, as for generic masses there are branch points and associated branch
cuts, which spoil the modularity aspman2021cutting . The discussion in
aspman2021cutting  for $N_{f}\leq 3$ suggests that for a fixed $\tau$ or fixed
$\tau_{0}$, there is a natural choice of fundamental domain
$\mathcal{F}({\boldsymbol{m}})\subseteq\mathbb{H}$ for $u_{\boldsymbol{m}}$,
such that $u_{\boldsymbol{m}}:\mathcal{F}({\boldsymbol{m}})\to\mathcal{B}_{4}$
is one-to-one. For a generic choice of masses, monodromies on the $u$-plane
give rise to monodromies of $\mathcal{F}({\boldsymbol{m}})$, but these do not
generate a congruence subgroup of $\text{SL}(2,\mathbb{Z})$ for a generic
mass. For special cases however, $\mathcal{F}({\boldsymbol{m}})$ is equal to
$\Gamma\backslash\mathbb{H}$ for some subgroup
$\Gamma\subseteq\text{SL}(2,\mathbb{Z})$, such as when ${\boldsymbol{m}}$ is
equal to ${\boldsymbol{m}}_{\text{A}}$, ${\boldsymbol{m}}_{\text{B}}$,
${\boldsymbol{m}}_{\text{C}}$ or ${\boldsymbol{m}}_{\text{D}}$, for which
$\Gamma$ is $\Gamma(2)$, $\Gamma^{0}(4)$, $\Gamma_{0}(4)$ or
$\widetilde{\Gamma^{0}(4)}$. If the mass ${\boldsymbol{m}}$ is such that
$\mathcal{B}_{4}$ contains a superconformal Argyres-Douglas point,
$\Gamma\subseteq\text{SL}(2,\mathbb{Z})$ can also be a subgroup of index
smaller than $6$ aspman2021cutting ; Closset:2021lhd . An example of this will
be given in Section 3.6.

In the above discussed examples A–D, the duality groups $\Gamma_{1}$ of $\tau$
and $\Gamma_{2}$ of $\tau_{0}$ are identical. We show in Section 3.6 that this
is not generally true, even if $u(\tau,\tau_{0})$ is modular in $\tau$ and
$\tau_{0}$. However, we can demonstrate that $\Gamma_{1}\subset\Gamma_{2}$. A
common non-perturbative definition of the UV coupling constant is the low-
energy effective coupling $\tau$ in the limit where the order parameter is
large,

$\tau_{0}=\lim_{u\to\infty}\tau(u).$ (51)

Since it is not associated with a singularity, it is neither a cusp nor an
elliptic point and therefore an arbitrary interior point in the space of
$\tau\in\mathbb{H}$. If $\Gamma_{1}\subsetneq\Gamma_{2}$ is not a proper
subgroup, then in general $\tau_{0}\in\Gamma_{2}\backslash\mathbb{H}$ is not
an element of a choice of fundamental domain $\Gamma_{1}\backslash\mathbb{H}$.
However, there exists a $\gamma_{1}\in\Gamma_{1}$ with the property that
$\gamma_{1}\tau_{0}\in\Gamma_{1}\backslash\mathbb{H}$. Since
$u(\tau,\tau_{0})$ has weight $0$ in $\tau$, we notice that

$u(\gamma_{1}\tau_{0},\tau_{0})=u(\tau_{0},\tau_{0})=\infty,$ (52)

which is the weak coupling region in $\mathcal{B}_{4}$. If
$\Gamma_{2}\subsetneq\Gamma_{1}$ however, then there exist two points
$\tau_{0}\neq\tilde{\tau}_{0}$ in the fundamental
$\Gamma_{1}\backslash\mathbb{H}$, which are not related by any element
$\gamma_{1}\in\Gamma_{1}$. Then $u(\tau,\tau_{0})$ and
$u(\tau,\tilde{\tau}_{0})$ are two distinct points in $\mathcal{B}_{4}$. This
contradicts the fact that the $N_{f}=4$ Coulomb branch $\mathcal{B}_{4}$ only
contains one such singularity. This shows that indeed
$\Gamma_{1}\subseteq\Gamma_{2}$.

The weight $(0,2)$ of $u$ can be explained as follows. Monodromies on the
$u$-plane act on the low-energy effective coupling $\tau$ and by definition
leave $u$ invariant. Thus $u(\tau,\tau_{0})$ is required to have weight $0$ in
$\tau$. For $\tau_{0}$, recall that the order parameter relates to the
prepotential $F$ of the theory by a logarithmic derivative with respect to the
instanton counting parameter Matone:1995rx ; Minahan:1997if ; Ne ;
Sonnenschein_1996 ; Eguchi:1995jh ; DHoker:1996yyu

$u=4\pi iq_{0}\frac{\partial F}{\partial q_{0}}=2\frac{\partial
F}{\partial\tau_{0}}.$ (53)

As the prepotential $F$ has weight $0$ in $\tau_{0}$, this shows that
$u(\tau,\tau_{0})$, has weight $2$ in $\tau_{0}$.

The other possible modular transformations are those involving the masses,
which is the action of the triality group
$\text{Spin}(8)\rtimes_{\varphi}\text{SL}(2,\mathbb{Z})$. From the above
analysis, we expect that for generic mass ${\boldsymbol{m}}$ the order
parameter $u_{\boldsymbol{m}}$ transforms as

$\displaystyle\mathcal{T}:$ $\displaystyle
u_{\boldsymbol{m}}(\tau+1,\tau_{0}+1)\\!\\!\\!\\!$
$\displaystyle=u_{\mathcal{T}{\boldsymbol{m}}}(\tau,\tau_{0}),$ (54)
$\displaystyle\mathcal{S}:$ $\displaystyle
u_{\boldsymbol{m}}(-\tfrac{1}{\tau},-\tfrac{1}{\tau_{0}})$
$\displaystyle=\tau_{0}^{2}\,u_{\mathcal{S}{\boldsymbol{m}}}(\tau,\tau_{0}).$

Due to the branch points and cuts for generic masses, these transformations
are again very subtle to perform. From (11) and in particular
$\mathcal{T}^{2}=\mathbbm{1}$, (54) implies

$\mathcal{T}^{2}:\quad
u_{\boldsymbol{m}}(\tau+2,\tau_{0}+2)=u_{{\boldsymbol{m}}}(\tau,\tau_{0}).$
(55)

We can check explicitly that it is true for example for case B as in (36),
which is not $\mathcal{T}$-invariant.

As discussed in Section 2.3, the group action
$\mathscr{T}\times\mathscr{M}\to\mathscr{M}$ partitions the mass space
$\mathscr{M}\ni{\boldsymbol{m}}$ into three regions $\mathcal{L}_{1}$,
$\mathcal{L}_{3}$ and $\mathcal{L}_{6}$, where the orbits
$\mathscr{T}\cdot{\boldsymbol{m}}$ have length 1, 3 and 6. The stabiliser
subgroups of ${\boldsymbol{m}}$ are then subgroups of $S_{3}$ of order 6, 2
and 1, i.e. isomorphic to $S_{3}$, $S_{2}$ or $S_{1}=\\{e\\}$. The
homomorphism $\varphi$ (13) between $\text{SL}(2,\mathbb{Z})$ and
$\mathscr{T}=\text{Out}(\text{Spin}(8))$ then dictates the subgroup

$\varphi^{-1}[\mathscr{T}_{{\boldsymbol{m}}}]$ (56)

under which $u_{\boldsymbol{m}}$ is simultaneously invariant. The preimage of
the stabiliser subgroup under $\varphi$ thus constitutes the third component
$\Gamma$ of the triple $(\Gamma_{1},\Gamma_{2};\Gamma)$ in definition 1.

#### The case ${\boldsymbol{m}}\in\mathcal{L}_{1}$

When ${\boldsymbol{m}}\in\mathcal{L}_{1}$, then the stabiliser group of
${\boldsymbol{m}}$ has six elements and the orbit
$\mathscr{T}\cdot{\boldsymbol{m}}$ consists of ${\boldsymbol{m}}$ only. Then
there is only one function in (54), and $u_{\boldsymbol{m}}$ transforms with
weight $(0,2)$ under $\text{SL}(2,\mathbb{Z})_{(\tau,\tau_{0})}$, as in
condition 2 of definition 1. An example is $u_{\text{A}}$ as given in (26),
and the transformation is checked in (28).

#### The case ${\boldsymbol{m}}\in\mathcal{L}_{3}$

The case ${\boldsymbol{m}}\in\mathcal{L}_{3}$ is most interesting, as it is
not trivial (${\boldsymbol{m}}\in\mathcal{L}_{1}$) and not generic
(${\boldsymbol{m}}\in\mathcal{L}_{6}$). Namely, when the orbit
$\mathscr{T}\cdot{\boldsymbol{m}}$ contains three elements, the stabiliser
group is isomorphic to the symmetric group $S_{2}$ with two elements. Then the
three functions associated with the three elements of the orbit
$\mathscr{T}\cdot{\boldsymbol{m}}$ form a vector that transforms under
$\text{SL}(2,\mathbb{Z})$. An example for this are the functions
$u_{\text{B}},u_{\text{C}},u_{\text{D}}$ found in Sections 3.2–3.4. As is
clear from Fig. 3, they are related to each other by triality. If we organise
${\boldsymbol{u}}_{3}=(u_{\text{B}},u_{\text{C}},u_{\text{D}})^{\mathsf{T}}$,
using (36), (41) (46) one can prove that

$\begin{split}{\boldsymbol{u}}_{3}(\tau+1,\tau_{0}+1)&=\begin{pmatrix}0&0&1\\\
0&1&0\\\ 1&0&0\end{pmatrix}{\boldsymbol{u}}_{3}(\tau,\tau_{0}),\\\
{\boldsymbol{u}}_{3}(-1/\tau,-1/\tau_{0})&=\tau_{0}^{2}\begin{pmatrix}0&1&0\\\
1&0&0\\\ 0&0&1\end{pmatrix}{\boldsymbol{u}}_{3}(\tau,\tau_{0}).\end{split}$
(57)

As the matrices are in $\mathrm{GL}(3,\mathbb{C})$, there exists a
3-dimensional representation
$\text{SL}(2,\mathbb{Z})\to\mathrm{GL}(3,\mathbb{C})$. This shows that
${\boldsymbol{u}}_{3}(\tau,\tau_{0})$ furnishes a vector-valued bimodular form
of weight $(0,2)$ for $\text{SL}(2,\mathbb{Z})$, agreeing with the following
definition:111111It is customary to define vector-valued modular forms for
$\text{SL}(2,\mathbb{Z})$, however vector-valued modular forms for proper
subgroups $\Gamma$ of $\text{SL}(2,\mathbb{Z})$ are familiar in rational CFTs
Miyamoto1998 ; krauel2012vertex ; Cheng:2020srs  and so we leave our
definition more generic.

######  Definition 2 (Vector-valued bimodular form).

Let

${\boldsymbol{F}}=\left(\begin{array}[]{c}F_{1}\\\ \vdots\\\
F_{p}\end{array}\right):\mathbb{H}\times\mathbb{H}\to\mathbb{C}^{p}$ (58)

be a $p$-tuple of two-variable meromorphic functions, $p\in\mathbb{N}$. Then
${\boldsymbol{F}}$ is called a vector-valued bimodular form of weight
$(k_{1},k_{2})$ for $\Gamma\subset\text{SL}(2,\mathbb{Z})$, if

  * •

each component $F_{j}$ is a bimodular form of weight $(k_{1},k_{2})$ for some
triple $\allowbreak(\Gamma_{1}^{j},\Gamma_{2}^{j};\Gamma^{j})$, as in
definition 1, and

  * •

there exists a $p$-dimensional complex representation
$\rho:\Gamma\to\text{GL}(p,\mathbb{C})$ such that

${\boldsymbol{F}}(\gamma\tau_{1},\gamma\tau_{2})=(c\tau_{1}+d)^{k_{1}}(c\tau_{2}+d)^{k_{2}}\rho(\gamma){\boldsymbol{F}}(\tau_{1},\tau_{2})$
(59)

for all $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma$ and all
$\tau_{1},\tau_{2}\in\mathbb{H}$.

Since ${\boldsymbol{u}}_{3}$ is parametrised by the mass $m\in\mathbb{C}$,
$m\mapsto{\boldsymbol{u}}_{3}(m,\tau,\tau_{0})$ is in fact a 1-parameter
family of vector-valued bimodular forms of weight $(0,2)$ for
$\text{SL}(2,\mathbb{Z})$. The triality action of $\text{SL}(2,\mathbb{Z})$
permutes the triples $(\Gamma_{1}^{j},\Gamma_{2}^{j};\Gamma^{j})$ in an
interesting way. The action of the $\text{SL}(2,\mathbb{Z})$ generators on $u$
is given by (57). As $\Gamma_{1}^{j}=\Gamma_{2}^{j}$ for the cases B, C, D,
both $\Gamma_{1}^{j}$ and $\Gamma_{2}^{j}$ are conjugated by the corresponding
element of $\text{SL}(2,\mathbb{Z})$. An instance of this is the group
$\widetilde{\Gamma^{0}(4)}$ (48), which is the set of elements of
$\Gamma^{0}(4)$ conjugated by $T$. Similarly, we have that $\Gamma^{0}(4)$ is
conjugate to $\Gamma_{0}(4)$ by conjugation with $S$. The same is true for the
three groups $\Gamma^{0}(2)$, $\Gamma_{0}(2)$ and $\widetilde{\Gamma^{0}(2)}$
that the cases B, C, D simultaneously transform under, these three conjugate
subgroups are permuted under $\text{SL}(2,\mathbb{Z})$ just as
$\Gamma^{0}(4)$, $\Gamma_{0}(4)$ and $\widetilde{\Gamma^{0}(4)}$ are.

#### The case ${\boldsymbol{m}}\in\mathcal{L}_{6}$

The remaining case is that ${\boldsymbol{m}}\in\mathcal{L}_{6}$, where
$\mathscr{T}\cdot{\boldsymbol{m}}$ has six elements. Then we can organise
${\boldsymbol{u}}_{6}=(u_{{\boldsymbol{m}}},u_{\mathcal{T}{\boldsymbol{m}}},u_{\mathcal{S}{\boldsymbol{m}}},u_{\mathcal{T}\mathcal{S}{\boldsymbol{m}}},u_{\mathcal{S}\mathcal{T}{\boldsymbol{m}}},u_{\mathcal{T}\mathcal{S}\mathcal{T}{\boldsymbol{m}}})^{\mathsf{T}}$,
which is a collection of six pairwise distinct functions. By studying the
action of $\mathcal{T}$ and $\mathcal{S}$ on the vector
$({\boldsymbol{m}},\mathcal{T}{\boldsymbol{m}},\mathcal{S}{\boldsymbol{m}},\mathcal{T}\mathcal{S}{\boldsymbol{m}},\mathcal{S}\mathcal{T}{\boldsymbol{m}},\mathcal{T}\mathcal{S}\mathcal{T}{\boldsymbol{m}})^{\mathsf{T}}$,
we find the transformations

$\displaystyle{\boldsymbol{u}}_{6}(\tau+1,\tau_{0}+1)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&1&0&0&0&0\\\ 1&0&0&0&0&0\\\ 0&0&0&1&0&0\\\
0&0&1&0&0&0\\\ 0&0&0&0&0&1\\\ 0&0&0&0&1&0\\\
\end{pmatrix}{\boldsymbol{u}}_{6}(\tau,\tau_{0}),$ (60)
$\displaystyle{\boldsymbol{u}}_{6}(-1/\tau,-1/\tau_{0})$
$\displaystyle=\tau_{0}^{2}\\!\\!\\!\\!\\!$
$\displaystyle\begin{pmatrix}0&0&1&0&0&0\\\ 0&0&0&0&1&0\\\ 1&0&0&0&0&0\\\
0&0&0&0&0&1\\\ 0&1&0&0&0&0\\\ 0&0&0&1&0&0\\\
\end{pmatrix}{\boldsymbol{u}}_{6}(\tau,\tau_{0}).$

The vector ${\boldsymbol{u}}_{6}$ is not a vector-valued bimodular form for
$\text{SL}(2,\mathbb{Z})$, because the components of ${\boldsymbol{u}}_{6}$ do
not transform as modular forms under the separate action of
$\Gamma^{j}_{1,2}$, $j=1,\dots,6$ due to the branch cuts, as discussed for the
$N_{f}\leq 3$ theories in aspman2021cutting . As we demonstrate in Section
3.6, the simultaneous action of $\text{SL}(2,\mathbb{Z})$ on $\tau$ and
$\tau_{0}$ is not obstructed by the branch cuts of $u(\tau,\tau_{0})$.

The matrices in (57) and (60) are not only in $\mathrm{GL}(n,\mathbb{C})$, but
they are in fact permutation matrices: Because triality permutes the
respective moduli spaces, the order parameters are merely permuted and there
are no phases. Thus we have that

${\boldsymbol{u}}(\gamma\tau,\gamma\tau_{0})=(c\tau_{0}+d)^{2}P_{\pi(\gamma)}{\boldsymbol{u}}(\tau,\tau_{0}),$
(61)

where $P_{\pi(\gamma)}$ is the permutation matrix for the permutation
$\pi(\gamma)\in S_{|\mathscr{T}\cdot{\boldsymbol{m}}|}$, which can be found
from the action of $\mathscr{T}$ on ${\boldsymbol{m}}$.

For the period $\left(\frac{da}{du}\right)^{2}$, there are similar results.
For instance, one can check that

$\left(\left(\tfrac{da}{du}\right)^{2}_{\text{B}},\left(\tfrac{da}{du}\right)^{2}_{\text{C}},\left(\tfrac{da}{du}\right)^{2}_{\text{D}}\right)^{\mathsf{T}}$
(62)

is a vector-valued bimodular form of weight $(2,-4)$ for
$\text{SL}(2,\mathbb{Z})$. As $u$ has weight $(0,2)$, it is not obvious how
the discriminant $\Delta$ transforms since it is a polynomial in $u$. However,
because triality acts on the $6$ singularities as well, in general
$\boldsymbol{\Delta}$ is a vector-valued bimodular form of weight $(0,12)$ for
$\text{SL}(2,\mathbb{Z})$. This can be checked explicitly for the cases B, C,
D, where
$\boldsymbol{\Delta}_{3}=(\Delta_{\text{B}},\Delta_{\text{C}},\Delta_{\text{D}})^{\mathsf{T}}$
is a 1-parameter family of vector-valued bimodular forms of weight $(0,12)$
for $\text{SL}(2,\mathbb{Z})$.

###  3.6 Cases E, F and G

To make the analysis in the previous section more explicit, we can study three
cases E, F and G, with ${\boldsymbol{m}}_{\text{E}}=(m,m,\mu,\mu)$,
${\boldsymbol{m}}_{\text{F}}=(m+\mu,m-\mu,0,0)$ and
${\boldsymbol{m}}_{\text{G}}=(m,m,\mu,-\mu)$. These mass vectors share the
same symmetry properties, and diagram, as B, C and D (see Fig. 3), i.e. that
$\mathcal{S}$ interchanges E and F, while leaving G invariant and
$\mathcal{T}$ interchanges E and G while leaving F invariant. They also give
back all cases A, B, C and D in different limits. For example, if we send
$\mu\to 0$ all three cases become case A, while if we send $\mu\to m$ we see
that E becomes B, F becomes C and G becomes D. However, due to the fact that
we now have two masses, the theories become more complicated. New features
such as superconformal fixed points of Argyres-Douglas (AD) type appear, as
well as branch points due to square roots Argyres:1995xn ; aspman2021cutting .

The order parameters are now given by

$\displaystyle u_{\text{E}}$
$\displaystyle=\frac{\vartheta_{3}(\tau_{0})^{4}}{6(\lambda-\lambda_{0})(\lambda\lambda_{0}-1)}\Big{[}(m^{2}+\mu^{2})(1+\lambda_{0})(\lambda_{0}+\lambda(2+\lambda_{0}(\lambda-6+2\lambda_{0})))$
(63)
$\displaystyle\quad+3(\lambda^{2}-1)(\lambda_{0}-1)\lambda_{0}\sqrt{(m^{2}-\mu^{2})^{2}+4m^{2}\mu^{2}\frac{\lambda}{\lambda_{0}}\frac{(\lambda_{0}-1)^{2}}{(\lambda-1)^{2}}}\Big{]},$
$\displaystyle u_{\text{F}}$
$\displaystyle=\frac{\vartheta_{3}(\tau_{0})^{4}}{6(\lambda-\lambda_{0})(\lambda(\lambda_{0}-1)-\lambda_{0})}\Big{[}(m^{2}+\mu^{2})(\lambda_{0}-2)(\lambda^{2}(\lambda_{0}-1)+2\lambda_{0}^{2}(\lambda-1))$
$\displaystyle\quad+3(\lambda-2)(\lambda_{0}-1)\lambda_{0}\sqrt{(m^{2}-\mu^{2})^{2}\lambda^{2}+4m^{2}\mu^{2}\lambda_{0}^{2}\frac{\lambda-1}{\lambda_{0}-1}}\Big{]},$
$\displaystyle u_{\text{G}}$
$\displaystyle=\frac{\vartheta_{3}(\tau_{0})^{4}}{6(\lambda^{2}-\lambda-\lambda_{0}^{2}+\lambda_{0})}\Big{[}(m^{2}+\mu^{2})(2\lambda_{0}-1)((\lambda_{0}-1)\lambda_{0}+2\lambda^{2}-2\lambda)$
$\displaystyle\quad+3(2\lambda-1)(\lambda_{0}-1)\lambda_{0}\sqrt{(m^{2}-\mu^{2})^{2}+4m^{2}\mu^{2}\frac{\lambda}{\lambda_{0}}\frac{\lambda-1}{\lambda_{0}-1}}\Big{]},$

where we have abbreviated $\lambda(\tau)=\lambda$,
$\lambda(\tau_{0})=\lambda_{0}$. Since $\lambda$ is invariant under
$\Gamma(2)$, it naively looks like these order parameters satisfy condition 1
in definition 1 of a bimodular form, namely that they transform as bimodular
forms under the separate transformations of
$\Gamma(2)_{\tau}\times\Gamma(2)_{\tau_{0}}$ (or possibly conjugates thereof)
with weights $(0,2)$, but due to the presence of the square roots the story is
more subtle aspman2021cutting . We do, however, find the expected behaviour
under the simultaneous action of $\mathcal{T}$ and $\mathcal{S}$ for all three
cases,

$\displaystyle\mathcal{T}:\begin{cases}u_{\text{E}}(\tau+1,\tau_{0}+1)=&\\!\\!\\!\\!u_{\text{G}}(\tau,\tau_{0}),\\\
u_{\text{F}}(\tau+1,\tau_{0}+1)=&\\!\\!\\!\\!u_{\text{F}}(\tau,\tau_{0}),\end{cases}$
(64)
$\displaystyle\mathcal{S}:\begin{cases}u_{\text{E}}(-\tfrac{1}{\tau},-\tfrac{1}{\tau_{0}})=&\\!\\!\\!\\!\tau_{0}^{2}u_{\text{F}}(\tau,\tau_{0}),\\\
u_{\text{G}}(-\tfrac{1}{\tau},-\tfrac{1}{\tau_{0}})=&\\!\\!\\!\\!\tau_{0}^{2}u_{\text{G}}(\tau,\tau_{0})\end{cases}$

We thus see that, even though the separate action of the duality group on
$\tau$ and $\tau_{0}$ becomes more subtle in the presence of square roots, the
simultaneous $\text{SL}(2,\mathbb{Z})$ action involving triality is still
preserved for generic masses. We can also check the limits to other theories.
If we send $\mu\to 0$ we indeed find that the order parameter of all three
cases becomes $u_{\text{A}}$ (26), consistent with the limit of the mass
vector. By sending $\mu\to m$ we instead find that $u_{\text{E}}\to
u_{\text{B}}$, $u_{\text{F}}\to u_{\text{C}}$ and $u_{\text{G}}\to
u_{\text{D}}$ as expected.

The cusps are given by

$\displaystyle\text{E}:\begin{cases}u_{m_{i}}\\!\\!\\!\\!&=\frac{\vartheta_{3}(\tau_{0})^{4}}{3}(m_{i}^{2}(2\lambda_{0}-1)+m_{j\neq
i}^{2}(2-\lambda_{0})),\\\
u_{\pm}\\!\\!\\!\\!&=-\frac{\vartheta_{3}(\tau_{0})^{4}}{3}((m^{2}+\mu^{2})(\lambda_{0}+1)\pm
6m\mu\sqrt{\lambda_{0}}),\end{cases}$ (65)
$\displaystyle\text{F}:\begin{cases}u_{m_{i}}\\!\\!\\!\\!&=\frac{\vartheta_{3}(\tau_{0})^{4}}{3}(m_{i}^{2}(2\lambda_{0}-1)-m_{j\neq
i}^{2}(\lambda_{0}+1)),\\\
u_{\pm}\\!\\!\\!\\!&=\frac{\vartheta_{3}(\tau_{0})^{4}}{3}((m^{2}+\mu^{2})(2-\lambda_{0})\pm
6m\mu\sqrt{\lambda_{0}}),\end{cases}$
$\displaystyle\text{G}:\begin{cases}u_{m_{i}}\\!\\!\\!\\!&=\frac{\vartheta_{3}(\tau_{0})^{4}}{3}(m_{i}^{2}(2-\lambda_{0})-m_{j\neq
i}^{2}(1+\lambda_{0})),\\\
u_{\pm}\\!\\!\\!\\!&=\frac{\vartheta_{3}(\tau_{0})^{4}}{3}((m^{2}+\mu^{2})(2\lambda_{0}-1)\pm
6m\mu i\sqrt{\lambda_{0}(1-\lambda_{0})}),\end{cases}$

where $m_{i}=m,\mu$ and $m_{j\neq i}$ then denotes the other mass. The
singularities $u_{m_{i}}$ all have degeneracy two while $u_{\pm}$ have
degeneracy one. There are also singularities in the interior given by
$u(\tau_{0},\tau_{0})=\infty$ for all three cases.

#### Special points

As in the theories with $0<N_{f}\leq 3$ there is a plethora of theories in the
moduli space of generic masses $N_{f}=4$ where the singularity of the fibres
is of a higher type, in the sense of Kodaira, and where mutually non-local
dyons become massless Argyres:1995xn .121212Note however that we should not
expect to find any _new_ types of theories, compared to the ones of $N_{f}\leq
3$ in this moduli space, but only types $II$-$IV$ Argyres:1995xn . This is
because an overall scaling of the masses is not a true parameter of the
theory. These can be classified similarly as in the asymptotically free
theories by finding the values of ${\boldsymbol{m}}$ and $u$ such that
$g_{2}=g_{3}=0$.

As we have just seen, there are also theories where the order parameter has
branch points due to square roots. A possible interpretation of the
corresponding branch points in the asymptotically free theories has been
suggested in aspman2021cutting  as first order phase transitions connected to
the second order transition that is the Argyres-Douglas (AD) theories. This
implies that we might expect to have branch cuts whenever we have an AD
theory. It is straightforward to check that the cases A, B, C and D only have
as superconformal fixed points $m\to 0$, $u\to 0$, so that the lack of branch
points in these theories is consistent with the above claim. For the more
general cases the story changes as we have just seen for cases E, F and G. Let
us therefore study the special points of these theories in more detail.

#### AD points

We define the AD loci as the values of the masses for which there exists an AD
theory. This can then be expressed as the zero loci of the polynomials

$\displaystyle\text{P}^{\text{AD}}_{\text{E}}=$
$\displaystyle\left(m^{2}\lambda_{0}-\mu^{2}\right)\left(\mu^{2}\lambda_{0}-m^{2}\right)$
(66) $\displaystyle\text{P}^{\text{AD}}_{\text{F}}=$
$\displaystyle\left(m^{2}(\lambda_{0}-1)+\mu^{2}\right)\left(\mu^{2}(\lambda_{0}-1)+m^{2}\right),$
$\displaystyle\text{P}^{\text{AD}}_{\text{G}}=$
$\displaystyle\left(m^{2}(\lambda_{0}-1)-\lambda_{0}\mu^{2}\right)\left(\mu^{2}(\lambda_{0}-1)-\lambda_{0}m^{2}\right),$

Since $T:\lambda\mapsto\frac{\lambda}{\lambda-1}$ and $S:\lambda\mapsto
1-\lambda$ we see that the AD loci also satisfy triality, such that if we act
on $\text{P}_{\text{E}}^{\text{AD}}$ with $T$ we get
$\text{P}^{\text{AD}}_{\text{G}}$ (up to an overall non-zero factor which is
not important since we are looking for the roots of the polynomial) and if we
act with $S$ we get $\text{P}^{\text{AD}}_{\text{F}}$.

By tuning the mass to any of the AD values we find that three singularities
merge. Depending on which AD mass is chosen, one of the degeneracy two
singularities $u_{m_{i}}$ merge with one of the degeneracy one singularities
$u_{\pm}$. This gives rise to a singular fibre of type $III$
($\text{ord}(g_{2},g_{3},\Delta)=(1,2,3)$), implying that three mutually non-
local states are becoming massless Argyres:1995xn . It is now easy to find
closed expressions for $u$ for any of the three theories, and the square roots
all disappear.131313Note that for some of the values of the masses the
solution we have picked for general $m$ and $\mu$ will become a constant
function of $\tau$, this is because the chosen solution corresponds to the
solution for $u$ near a singularity that merges with others to become the AD
singularity.

To give an example we take case E and tune the masses such that
$m=\mu\sqrt{\lambda_{0}}$, where
$\sqrt{\lambda}=\frac{\vartheta_{2}^{2}}{\vartheta_{3}^{2}}$ is a holomorphic
modular form. The order parameter becomes

$u_{\text{E}}^{\text{AD}}=\frac{2\mu^{2}\vartheta_{3}(\tau_{0})^{4}}{3}\frac{(\lambda_{0}-1)(\lambda_{0}(\lambda_{0}(f_{2}+8(7+\lambda_{0})))-56)-8}{\lambda_{0}(32+f_{2}-16\lambda_{0})-16},$
(67)

where
$f_{2}=f_{2}(\tau)=16\frac{\vartheta_{4}(\tau)^{8}}{\vartheta_{2}(\tau)^{4}\vartheta_{3}(\tau)^{4}}$
is a Hauptmodul of the index 3 congruence subgroup
$\Gamma^{0}(2)\subset\text{SL}(2,\mathbb{Z})$. It is straightforward to check
that $u_{\text{E}}^{\text{AD}}$ has weight $0$ under separate transformations
for $\Gamma^{0}(2)_{\tau}$, and weight $2$ under separate transformations for
$\Gamma(2)_{\tau_{0}}$. Thus, the group of simultaneous transformations
contains $\Gamma^{0}(2)\cap\Gamma(2)\cong\Gamma(2)$. We therefore find that
$u_{\text{E}}^{\text{AD}}$ is a bimodular form of weight $(0,2)$ for the
triple

$(\Gamma^{0}(2),\Gamma(2);\Gamma(2)).$ (68)

Note that this is our first example of a bimodular form that has two different
modular groups for the two couplings. The fact that the index in
$\text{SL}(2,\mathbb{Z})$ of the modular group of $\tau$ shrinks by the number
of merged non-local singularities, $2+1$ in this case, is the expected
behaviour of AD theories Closset:2021lhd ; aspman2021cutting .

Since the two separate duality groups, $\Gamma^{0}(2)$ and $\Gamma(2)$, are
different, we cannot choose the fundamental domains for $\tau$ and $\tau_{0}$
to coincide as in previous cases. Instead, we can choose the fundamental
domain for $\tau$ as a subset of that for $\tau_{0}$. Equation (67)
demonstrates that $u_{\text{E}}^{\text{AD}}$ has a single pole as a function
of $\tau\in\mathbb{H}/\Gamma^{0}(2)$ for fixed $\tau_{0}$, while it has two
poles as function of $\tau_{0}\in\mathbb{H}/\Gamma(2)$ for fixed $\tau$. The
two points in $\tau_{0}\in\mathbb{H}/\Gamma(2)$ are related by an element in
$\Gamma^{0}(2)/\Gamma(2)$.

We can further note that the AD mass, $m_{\text{AD}}=\mu\sqrt{\lambda_{0}}$,
is not invariant under $\Gamma(2)_{\tau_{0}}$, due to the square root. We
rather have that $m_{\text{AD}}\to-m_{\text{AD}}$ under $T^{2}$, which is of
course another AD point of the theory, and the order parameters of the two
theories are given by the same expression. Furthermore, acting with $S$ and
$T$ on $\tau_{0}$ sends this AD mass to the corresponding AD masses of cases F
and G, respectively.

We also have the possibility of tuning $\tau_{0}$ to a specific value such
that more singularities merge. In the above solution, if we fix
$\tau_{0}=1+i$, or $\lambda_{0}=-1$, we find that the remaining degeneracy two
singularity merge with the degeneracy one singularity such that we get the
weight vector ${\boldsymbol{k}}=(3,3)$. The relation between the masses is now
$m=i\mu$ and the order parameter is actually independent of $\tau$, the curve
is simply given by $\mathcal{J}=j(\tau_{0})=12^{3}$. Therefore, the coupling
$\tau(u)=\tau_{0}=1+i$ is fixed over the whole Coulomb branch. This is
expected from the same argument as before since we merge two sets of 3 non-
local singularities, such that the fundamental domain for $\tau$ just shrinks
to a point $\tau_{0}$.

#### Branch points

As previously mentioned, in the more generic cases there will also be branch
points. For the theories E, F and G these are given by the branch points of
the square roots in 63,

$\displaystyle\text{E}:$ $\displaystyle\quad\frac{\lambda}{(\lambda-1)^{2}}$
$\displaystyle=-\frac{\lambda_{0}}{(\lambda_{0}-1)^{2}}\frac{(m^{2}-\mu^{2})^{2}}{4m^{2}\mu^{2}},$
(69) $\displaystyle\text{F}:$
$\displaystyle\quad\frac{\lambda-1}{\lambda^{2}}$
$\displaystyle=\frac{1-\lambda_{0}}{\lambda_{0}}\frac{(m^{2}-\mu^{2})^{2}}{4m^{2}\mu^{2}},$
$\displaystyle\text{G}:$ $\displaystyle\quad\lambda(\lambda-1)$
$\displaystyle=\lambda_{0}(1-\lambda_{0})\frac{(m^{2}-\mu^{2})^{2}}{4m^{2}\mu^{2}}.$

In the $u$-plane they are given by

$\displaystyle\text{E}:$ $\displaystyle\quad u_{\text{bp}}=$
$\displaystyle-\vartheta_{3}(\tau_{0})^{4}(1+\lambda_{0})\frac{m^{4}-4m^{2}\mu^{2}+\mu^{4}}{3(m^{2}+\mu^{2})},$
(70) $\displaystyle\text{F}:$ $\displaystyle\quad u_{\text{bp}}=$
$\displaystyle-\vartheta_{3}(\tau_{0})^{4}(\lambda_{0}-2)\frac{m^{4}-4m^{2}\mu^{2}+\mu^{4}}{3(m^{2}+\mu^{2})},$
$\displaystyle\text{G}:$ $\displaystyle\quad u_{\text{bp}}=$
$\displaystyle\vartheta_{3}(\tau_{0})^{4}(2\lambda_{0}-1)\frac{m^{4}-4m^{2}\mu^{2}+\mu^{4}}{3(m^{2}+\mu^{2})}.$

It is straightforward to see that also these points satisfy triality.

###  3.7 Fundamental domains

In the asymptotically free theories we argued that the $u$-planes can be
identified with fundamental domains $\mathcal{F}_{N_{f}}({\boldsymbol{m}})$
aspman2021cutting . For this we make the correspondence that the number of
singularities gives the number of rational cusps, the number of BPS states
becoming massless at each singularity gives the width of each cusp, and the
width at $i\infty$ is given by $4-N_{f}$. Then, the sum of all cusps is RG
invariant. By following the RG flow from $N_{f}=3$ to $N_{f}<3$ we find that
gradually a singularity at strong coupling (a rational cusp) moves to infinity
and is identified with the weak coupling region ($i\infty$). Reversing this
argument implies that for $N_{f}=4$ there should be six rational cusps and the
width at infinity vanishes. This is consistent with the fact Seiberg:1994aj
that $u=\infty$ does not correspond to a cusp of the curve anymore. Rather, it
lies in the interior of $\mathbb{H}\ni\tau_{0}$.

It is found in the above subsections that depending on the mass configuration,
the fundamental domain for an order parameter is related to the one of the
underlying theory where all massive hypermultiplets are decoupled. We can
depict those domains in an equivalent way that is more suitable to our
description. For this, one chooses an equivalent fundamental domain with the
property that the width at $i\infty$ is zero and the number of rational cusps
is equal to the number of singularities, with according width. For instance,
in case A where ${\boldsymbol{m}}=(m,m,0,0)$ the duality group is $\Gamma(2)$,
whose cusps in the decoupling limit (with the same duality group) we choose as
$\\{i\infty,0,1\\}$. In $N_{f}=4$ it is more suitable to represent $i\infty$
by a rational number. For this we can use that $\Gamma(2)\ni
ST^{-2}S:i\infty\mapsto\frac{1}{2}$, being a preferable representative of the
third cusp. As it necessarily also has width $2$, both $\mathcal{F}$ and
$T\mathcal{F}$ can be mapped to the region $\tau=\frac{1}{2}$. This is
depicted in Fig. 7. The decoupling to massless $N_{f}=2$ is illustrated in
Fig. 7. The domains in this case are exactly equivalent, Fig. 7 merely allows
to extend the $N_{f}\leq 3$ description of the cusps to $N_{f}=4$.

Figure 6: Fundamental domain of $N_{f}=4$ case A with
${\boldsymbol{m}}=(m,m,0,0)$.

Figure 7: Decoupling the two massive hypermultiplets in $N_{f}=4$ case A gives
the domain of massless $N_{f}=2$ (blue). Two of the differing regions (gray)
are the regions near $\tau=\frac{1}{2}$, which are mapped (orange) to
$i\infty$. The remaining two are merely mapped to $\Gamma(2)$ equivalent
regions near the same cusp such that the resulting domain is connected.
Alternatively, the fundamental domain in this figure is the image of
$ST^{-1}STS$ acting on the domain in Figure 7.

We stress that the decoupling limit for $N_{f}=4$, with the order parameter a
bimodular form as (26), is quite different from the asymptotically free
theories with $N_{f}\leq 3$. In the latter theories, $u(\tau)$ is not
holomorphic and modular except for special points in mass space (complex co-
dimension $N_{f}$) aspman2021cutting . The decoupling of a hypermultiplet is
in these theories accompanied by a branch point moving to infinity. In this
way, a singularity merges with the weak coupling cusp. For cases A, B, C and D
in $N_{f}=4$ on the other hand, there is no branch point for any value of the
mass $m$, and in particular there is also none for $m\to\infty$.

###  3.8 Relation to $\mathcal{N}=2^{*}$

Let us collect some results for the $\mathcal{N}=2^{*}$ theory in order to
point out the analogy to $N_{f}=4$. The $\mathcal{N}=2^{*}$ theory is obtained
by perturbing maximally supersymmetric $\mathcal{N}=4$ gauge theory by an
$\mathcal{N}=2$ invariant mass term. We have to be careful with the
redefinitions. For the $\mathcal{N}=4$ theory we have
$\tau_{\mathrm{0}}=\frac{\theta}{2\pi}+\frac{4\pi i}{g^{2}}$ and thus

$a=\sqrt{2u},\quad a_{D}=\tau a.$ (71)

In this convention we have $u=\frac{a^{2}}{2}$. This redefinition from the
classical formulas of the theories with fundamental matter is necessary for
both representations to feature integral electric-magnetic charges
Seiberg:1994aj . The $\mathcal{N}=2^{*}$ curve with mass $m$ is identical to
the one of $N_{f}=4$ (4) with
$\frac{1}{2}\,{\boldsymbol{m}}_{\text{A}}=(\frac{m}{2},\frac{m}{2},0,0)$
Seiberg:1994aj . A reason for this is that both theories have three
singularities each with monodromy being conjugate to $T^{2}$. This was in fact
the ansatz of Seiberg and Witten to determine the curve with generic masses.

This allows to recycle many results from Section 3.1. The order parameter for
$\mathcal{N}=2^{*}$ is equal to141414The expressions for
$u_{\mathcal{N}=2^{*}}$ in the literature Ferrari:1997gu ; laba1998 ;
Huang:2011qx ; Bonelli:2019boe ; Manschot:2021qqe  are related to
$u_{\text{A}}(\tau,\tau_{0})$ by a transformation in (16), which corresponds
to the choice of a different solution of the sextic equation associated with
the $\mathcal{N}=2^{*}$ theory aspman2021cutting . The different choices can
be absorbed in the double scaling limit. The counting of the number of poles
of $u_{\text{A}}$ is immediate from our expression (26) (see comment on the
transcendence of $\lambda$ in Section 3.1).

$u_{\mathcal{N}=2^{*}}(\tau,\tau_{0})=\frac{1}{4}u_{\text{A}}(\tau,\tau_{0}).$
(72)

In particular, it is a bimodular form of weight $(0,2)$ for
$(\Gamma(2),\Gamma(2);\text{SL}(2,\mathbb{Z}))$. The derivative
$\frac{da}{du}$ only receives an overall normalisation from $N_{f}=4$, due to
(71). In $\mathcal{N}=2^{*}$ the singularities each have degeneracy 1 and not
2 as in $N_{f}=4$ case A. Therefore, we have that
$\Delta_{\mathcal{N}=2^{*}}=\sqrt{\Delta_{\text{A}}}$, which is a polynomial
of degree 3 in $u$ and a bimodular form of weight $(0,6)$ Manschot:2021qqe .

##  4 Conclusion and discussion

In this paper, we have studied in detail the Coulomb branch of the
superconformal $N_{f}=4$ theory with gauge group SU(2), which has remained of
great interest throughout the years Seiberg:1994aj ; Tai:2010im ;
Minahan:1996ws ; Ferrari:1997gu ; Dorey:1996bn ; Argyres:1999ty ; Grimm:2007tm
; Huang:2011qx ; Tachikawa13 ; Marino:1998ru ; Argyres:1995xn ;
Nekrasov:2015wsu ; Jeong:2019fgx ; Manschot:2019pog ; Nekrasov:2012xe ;
Gaiotto:2009we ; Malmendier:2008db ; Gorodentsev:1996cu ; Moore:1997pc ;
Gaiotto:2009hg . For the mass configurations with the largest flavour symmetry
group, such as when one, two and four hypermultiplets have an equal mass, we
show that the Coulomb branch is parametrised by a function $u(\tau,\tau_{0})$
that is not only invariant under separate modular transformations of $\tau$
and $\tau_{0}$, but also exhibits invariance under a simultaneous
transformation under $\tau$ and $\tau_{0}$. By restricting to the stabiliser
subgroup of a given mass under the triality action, such order parameters
constitute nontrivial examples of bimodular forms (see (26) for example).
Furthermore, the moduli spaces are permuted under triality, and the order
parameter, periods, discriminants etc. furnish vector-valued bimodular forms,
which we also introduce (see definition 2).

The analysis of other mass configurations can be done using the techniques
established in aspman2021cutting . As more complicated mass configurations
${\boldsymbol{m}}$ inevitably introduce branch points and cuts, in general
$u_{\boldsymbol{m}}$ is not a bimodular form. A simultaneous transformation of
$\tau$ and $\tau_{0}$ is yet to be expected by triality, while the separate
transformations are induced by monodromies and as such do not in general lie
in $\text{SL}(2,\mathbb{Z})$ Seiberg:1994aj . However, even in such cases the
action of the monodromy group of the $u$-plane can be understood as paths in
the fundamental domain for $\tau$. See Reference Aspman:2021vz  for a
discussion of these aspects for gauge group SU(3).

Our results allow to study the topologically twisted theory on a four-manifold
$X$ Witten:1988ze ; Witten:1995gf ; Moore:1997pc ; LoNeSha ; Laba05 , where
the the path integral can be expressed as an integral over the fundamental
domain for the effective coupling $\tau$. In fact, a closed expression for the
order parameter is enough to define the integrand. The modularity for $\tau$
allows to show that the integral measure is well-defined. The triality action
then gives the S-duality orbit of the $N_{f}=4$ theory on $X$ AFMM:future .

It would also be interesting to apply our results to other theories with an IR
moduli space of vacua as well as a non-trivial conformal manifold. Such
theories may include subsectors with triality symmetry, such as F-theory
Sen:1996vd , quiver gauge theories Gaiotto:2009we , the AGT correspondence
Alday:2009aq , little string theory Duff:1995sm  and string/string/string
triality Bastian:2017ary .

###### Acknowledgements.

We are happy to thank Cyril Closset, Gregory Moore and Xinyu Zhang for
discussions. JA is supported by the Government of Ireland Postgraduate
Scholarship Programme GOIPG/2020/910 of the Irish Research Council. EF is
supported by the TCD Provost’s PhD Project Award. JM is supported by the
Laureate Award 15175 “Modularity in Quantum Field Theory and Gravity” of the
Irish Research Council.

##  Appendix A Modular forms

In this appendix, we collect some properties of modular forms for subgroups of
$\text{SL}(2,\mathbb{Z})$. For further reading, see Bruinier08 ; ono2004 ;
Zagier92 ; Diamond ; schultz2015 ; koblitz1993 .

We make use of modular forms for the congruence subgroups $\Gamma_{0}(n)$ and
$\Gamma^{0}(n)$ of $\text{SL}(2,\mathbb{Z})$. These subgroups are defined as

$\displaystyle\Gamma_{0}(n)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z})\big{|}\,c\equiv 0\;\mod
n\right\\},$ (73) $\displaystyle\Gamma^{0}(n)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z})\big{|}\,b\equiv 0\;\mod
n\right\\},$

and are related by conjugation with the matrix $\text{diag}(n,1)$. We
furthermore define the _principal congruence subgroup_ $\Gamma(n)$ as the
subgroup of $\text{SL}(2,\mathbb{Z})\ni A$ with $A\equiv\mathbbm{1}\mod n$. A
subgroup $\Gamma$ of $\text{SL}(2,\mathbb{Z})$ is called a congruence subgroup
if it contains $\Gamma(n)$ for some $n\in\mathbb{N}$. The smallest such $n$ is
then called the _level_ of $\Gamma$.

We furthermore make use of the theta group Bruggeman1994

$\widetilde{\Gamma^{0}(2)}=\Gamma_{\theta}\coloneqq\langle
T^{2},S\rangle\subseteq\text{SL}(2,\mathbb{Z}).$ (74)

A fundamental domain for $\Gamma_{\theta}$ is

$\Gamma_{\theta}\backslash\mathbb{H}=\mathcal{F}\cup T\mathcal{F}\cup
TS\mathcal{F},$ (75)

with $\mathcal{F}=\text{SL}(2,\mathbb{Z})\backslash\mathbb{H}$. This
demonstrates that $\Gamma_{\theta}$ has index $3$ in
$\text{SL}(2,\mathbb{Z})$. It is a congruence subgroup of
$\text{SL}(2,\mathbb{Z})$, as conrad ; rankin_1977 151515It can also be
written as the group of matrices $\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)$ with $a+b+c+d\equiv 0\mod 2$, or $ab\equiv
cd\equiv 0\mod 2$.

$\Gamma_{\theta}=\left\\{A\in\text{SL}(2,\mathbb{Z})\,|\,A\equiv\mathbbm{1}\,\text{
or }\,S\mod 2\right\\}.$ (76)

The above introduced congruence subgroups host a number of interesting modular
forms. The Jacobi theta functions $\vartheta_{j}:\mathbb{H}\to\mathbb{C}$,
$j=2,3,4$, are defined as

$\begin{split}\vartheta_{2}(\tau)=\sum_{r\in\mathbb{Z}+\frac{1}{2}}q^{r^{2}/2},\quad\vartheta_{3}(\tau)=\sum_{n\in\mathbb{Z}}q^{n^{2}/2},\quad\vartheta_{4}(\tau)=\sum_{n\in\mathbb{Z}}(-1)^{n}q^{n^{2}/2},\end{split}$
(77)

with $q=e^{2\pi i\tau}$. These functions transform under
$T,S\in\text{SL}(2,\mathbb{Z})$ as

$\begin{split}S:\quad&\begin{array}[]{l}\vartheta_{2}(-1/\tau)=\sqrt{-i\tau}\vartheta_{4}(\tau),\\\
\vartheta_{3}(-1/\tau)=\sqrt{-i\tau}\vartheta_{3}(\tau),\\\
\vartheta_{4}(-1/\tau)=\sqrt{-i\tau}\vartheta_{2}(\tau),\end{array}\\\
T:\quad&\begin{array}[]{l}\vartheta_{2}(\tau+1)=e^{\frac{\pi
i}{4}}\vartheta_{2}(\tau),\\\ \vartheta_{3}(\tau+1)=\vartheta_{4}(\tau),\\\
\vartheta_{4}(\tau+1)=\vartheta_{3}(\tau).\end{array}\end{split}$ (78)

They furthermore satisfy the Jacobi abstruse identity

$\vartheta_{2}^{4}+\vartheta_{4}^{4}=\vartheta_{3}^{4}.$ (79)

The modular lambda function
$\lambda=\frac{\vartheta_{2}^{4}}{\vartheta_{3}^{4}}$ is a Hauptmodul for
$\Gamma(2)$. The Dedekind eta function $\eta:\mathbb{H}\to\mathbb{C}$ is
defined as the infinite product

$\eta(\tau)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n}),\quad q=e^{2\pi
i\tau}.$ (80)

It transforms under the generators of $\text{SL}(2,\mathbb{Z})$ as

$\displaystyle S:$ $\displaystyle\eta(-1/\tau)=\sqrt{-i\tau}\,\eta(\tau),$
(81) $\displaystyle T:$ $\displaystyle\eta(\tau+1)=e^{\frac{\pi
i}{12}}\,\eta(\tau),$

and relates to the Jacobi theta series as
$\eta^{3}=\frac{1}{2}\vartheta_{2}\vartheta_{3}\vartheta_{4}$.

#### Eisenstein series

We let $\tau\in\mathbb{H}$ and define $q=e^{2\pi i\tau}$. Then the Eisenstein
series $E_{k}:\mathbb{H}\to\mathbb{C}$ for even $k\geq 2$ are defined as the
$q$-series

$E_{k}(\tau)=1-\frac{2k}{B_{k}}\sum_{n=1}^{\infty}\sigma_{k-1}(n)\,q^{n},$
(82)

with $\sigma_{k}(n)=\sum_{d|n}d^{k}$ the divisor sum. For $k\geq 4$ even,
$E_{k}$ is a modular form of weight $k$ for $\operatorname{SL}(2,\mathbb{Z})$.
Any modular form for $\text{SL}(2,\mathbb{Z})$ can be related to the Jacobi
theta functions (77) by

$E_{4}=\frac{1}{2}(\vartheta_{2}^{8}+\vartheta_{3}^{8}+\vartheta_{4}^{8}),\qquad
E_{6}=\frac{1}{2}(\vartheta_{2}^{4}+\vartheta_{3}^{4})(\vartheta_{3}^{4}+\vartheta_{4}^{4})(\vartheta_{4}^{4}-\vartheta_{2}^{4}).$
(83)

With our normalisation (82) the $j$-invariant can be written as

$j=1728\frac{E_{4}^{3}}{E_{4}^{3}-E_{6}^{2}}=256\frac{(\vartheta_{3}^{8}-\vartheta_{3}^{4}\vartheta_{4}^{4}+\vartheta_{4}^{8})^{3}}{\vartheta_{2}^{8}\vartheta_{3}^{8}\vartheta_{4}^{8}}.$
(84)

##  Appendix B Partitions of six

In Section 2.1 we listed a number of special mass configurations of the
$N_{f}=4$ theory, where multiple singularities have merged. Since the
$\mathcal{S}$ and $\mathcal{T}$ transformations preserve the weight vector of
the singularities ${\boldsymbol{k}}({\boldsymbol{m}})$, we could use this
vector as a partial classification of the loci in mass space with merged
singularities. As there are six singularities on the Coulomb branch
$\mathcal{B}_{4}$, we can consider partitions of six, of which there are
$p(6)=11$. To further simplify the problem, we consider masses which are
either different (but generic), equal or zero. This then means that we should
also consider partitions of four for the mass vector. The results are
collected in Table 2.

${\boldsymbol{k}}({\boldsymbol{m}})$ |  Example |  Singularity Type |  Corresponding
Section
---|---|---|---
$(6)$ |  $(0,0,0,0)$ |  $2I_{0}^{*}$ |  3
$(5,1)$ |  - |  excluded |  -
$(4,2)$ |  $(m,m,m,0)$ with
$\tau_{0}=-\tfrac{1}{2}+\tfrac{i}{2}$ |  $I_{0}^{*}\,\@slowromancap iv@\,II$ |  -
$(4,1,1)$ |  $(m,0,0,0)$ |  $I_{0}^{*}\,I_{4}\,2I_{1}$ |  B 3.2 , C 3.3, D 3.4
$(3,3)$ |  $(m,m,\mu,\mu)$, $\tau_{0}=1+i$, $m=\mu\sqrt{\lambda(\tau_{0})}$, |  $I_{0}^{*}\,2\@slowromancap iii@$ |  3.6
$(3,2,1)$ |  $(m,m,\mu,\mu)$,
$m=\mu\sqrt{\lambda(\tau_{0})}$ |  $\@slowromancap i@_{0}^{*}\,\@slowromancap iii@\,\@slowromancap i@_{2}\,\@slowromancap i@_{1}$ |  3.6
$(3,1,1,1)$ |  $(m,m,m,\mu)$, $(m,m,m,0)$ |  $\@slowromancap i@_{0}^{*}\,\@slowromancap i@_{3}\,3\@slowromancap i@_{1}$ |  -
$(2,2,2)$ |  $(m,m,0,0)$ |  $\@slowromancap i@_{0}^{*}\,3I_{2}$ |  A  3.1
$(2,2,1,1)$ |  $(m,\mu,0,0)$, $(m,m,\mu,\mu)$ |  $\@slowromancap i@_{0}^{*}\,2\@slowromancap i@_{2}\,2I_{1}$ |  E, F, G 3.6
$(2,1,1,1,1)$ |  $(m,m,\mu_{1},\mu_{2})$, $(m,m,\mu,0)$ |  $\@slowromancap i@_{0}^{*}\,\@slowromancap i@_{2}\,4\@slowromancap i@_{1}$ |  -
$(1,1,1,1,1,1)$ |  $(m_{1},m_{2},m_{3},m_{4})$, $(m_{1},m_{2},m_{3},0)$ |  $\@slowromancap i@_{0}^{*}\,6\@slowromancap i@_{1}$ |  -
Table 2: A partial classification of mass loci of the $N_{f}=4$ theory, based
on the weight vector ${\boldsymbol{k}}({\boldsymbol{m}})$. The second column
gives one example of the mass vector, where for some values of
${\boldsymbol{k}}({\boldsymbol{m}})$ it is also necessary to fix $\tau_{0}$.
Since the weight vector is triality invariant, the orbits
$\mathscr{T}\cdot{\boldsymbol{m}}$ of ${\boldsymbol{m}}$ under $\mathscr{T}$
(12) give additional configurations. The third column gives the singularity
type following Persson:1990  for the example in the second column. The last
column lists the examples studied in the main text for a given
${\boldsymbol{k}}({\boldsymbol{m}})$, with a reference to the specific
Section.

More generally, we could consider specific relations among the masses,
including $\tau_{0}$-dependence, such as in the AD theories of Sec. 3.6. For
these cases, we tuned the mass such that a number of non-local singularities
merged and we got a weight vector
${\boldsymbol{k}}({\boldsymbol{m}}_{EFG}^{\text{AD}})=(3,2,1)$. Similar
situations appear in the other theories with several mass parameters. For
example, in the $(m,m,m,\mu)$ theory with
$m=m_{\text{AD}}=\frac{\lambda_{0}}{2-\lambda_{0}}\mu$ we find that one of the
degeneracy one singularities merges with the degeneracy three singularity to
give a theory with weight vector ${\boldsymbol{k}}=(4,1,1)$. We further have
the possibility to tune $\tau_{0}$ to a specific value such that even more
singularities merge. For example, in the AD theory of case E studied in Sec.
3.6 we saw that we could fix $\tau_{0}=1+i$ to get a theory with weight vector
${\boldsymbol{k}}=(3,3)$. Similarly, in the $(m,m,m,\mu)$ theory mentioned
above, with $m=m_{\text{AD}}$ we can set $\tau_{0}=(-1)^{1/3}$ to find a
theory with ${\boldsymbol{k}}=(4,2)$. In the $(m,m,m,0)$ theory there is no
value for $m$ such that more singularities merge (except $m=0$) but we can set
$\tau_{0}=-\frac{1}{2}+\frac{i}{2}$ to find a ${\boldsymbol{k}}=(4,2)$ theory.
This is the only theory with one mass parameter where this is possible. In for
example case A, the value of $\tau_{0}$ needed to merge further singularities
is a singular one, corresponding to $\lambda_{0}=1$, which is not allowed.

As discussed recently in Refs Caorsi:2018ahl ; Closset:2021lhd , the possible
fibration structures of the SW curve over the $u$-plane for fixed masses and
$\tau_{0}$ follow the classification of rational elliptic surfaces by Miranda
and Persson Miranda:1986 ; Persson:1990 ; Miranda:1990 . Using this
classification, each part of the partition
${\boldsymbol{k}}({\boldsymbol{m}})$ can be labelled with the appropriate
singularity type. The singularity for $u\to\infty$ ($\tau\to\tau_{0}$) of the
$N_{f}=4$ theory corresponds to the $I_{0}^{*}$ singularity of Kodaira’s
classification of singular fibres. The cuspidal singularities correspond to
$I_{n}$, with $n$ the width of the cusp, while the AD singularities are of
type $II$, $III$ and $IV$. Note that a part of the partition can be realized
by different singularity types depending on the charges of the corresponding
massless particles. For example, a part “2” can correspond to either $I_{2}$
or $II$. Indeed, there are 19 different configurations with $I_{0}^{*}$
Closset:2021lhd , while there are only 11 different partitions of 6. We have
included in the third column in Table 2 the singularity type for the data in
the second column following Persson’s classification (Persson:1990, , The
list).161616We thank Cyril Closset for pointing out that
${\boldsymbol{k}}({\boldsymbol{m}})=(5,1)$ is excluded based on the
classification by Miranda and Persson.

##  Appendix C The qq-curve

The parameters $u$ and $m_{i}$ in (4) are not immediately related to
$\langle\text{tr}\,\phi^{2}\rangle$ and the masses of the hypermultiplets
Dorey:1996qt ; Dorey:1996zj ; Dorey:1996bn ; Argyres:1999ty ; Dorey_1996 ;
Dorey_1996II ; Manschot:2019pog . In fact, there is discrepancy between $u$
and $\langle\text{tr}\,\phi^{2}\rangle$ for $N_{f}\geq 3$ Dorey:1996bn . Let
us instead consider the curve obtained from the qq-characters of the
$N_{f}\leq 4$ SU(2) theory Nekrasov:2015wsu ; Jeong:2019fgx ; Manschot:2019pog
; Nekrasov:2012xe , which is better suited from the perspective of the
instanton calculus NekOk ; losev1998 . The elliptic curve for $N_{f}\leq 4$ in
the flat space limit is

$(1-\mathtt{q})^{2}y^{2}=T(x)^{2}-4\mathtt{q}\prod_{i=1}^{N_{f}}\left(x+m_{i}\right),$
(85)

where

$T(x)=\left(1+\mathtt{q}\right)x^{2}+\mathtt{q}x\sum_{i=1}^{N_{f}}m_{i}-\left(1-\mathtt{q}\right)\mathtt{u}.$
(86)

In the context of the qq-curve, it is customary to define

$\mathtt{q}\coloneqq\lambda(\tau_{0}),$ (87)

where $\tau_{0}$ is the UV coupling. Expanding out all terms for $N_{f}=4$ and
substituting $(1-\mathtt{q})y$ by $y$, we find the quartic curve

$\displaystyle y^{2}$ $\displaystyle=(1-\mathtt{q})^{2}x^{4}-2\left\llbracket
m_{1}\right\rrbracket(1-\mathtt{q})\mathtt{q}x^{3}+\left((\left\llbracket
m_{1}\right\rrbracket^{2}+2\mathtt{u})\mathtt{q}^{2}-4\left\llbracket
m_{1}m_{2}\right\rrbracket\mathtt{q}-2\mathtt{u}\right)x^{2}$ (88)
$\displaystyle-2\mathtt{q}\left(\left\llbracket
m_{1}\right\rrbracket(1-\mathtt{q})\mathtt{u}+2\left\llbracket
m_{1}m_{2}m_{3}\right\rrbracket\right)x+(1-\mathtt{q})^{2}\mathtt{u}^{2}-4\mathtt{q}\text{Pf}\,{\boldsymbol{m}},$

with the notation (LABEL:bracketnotation) as well as

$\left\llbracket
m_{1}\right\rrbracket=\sum_{i=1}^{4}m_{i},\quad\left\llbracket
m_{1}m_{2}\right\rrbracket=\sum_{i<j}m_{i}m_{j},\quad\left\llbracket
m_{1}m_{2}m_{3}\right\rrbracket=\sum_{i<j<k}m_{i}m_{j}m_{k}.$ (89)

As opposed to (7), the low-energy theory with $N_{f}<4$ flavours and scale
$\Lambda_{N_{f}}$ is obtained from the qq-curve (88) by decoupling $4-N_{f}$
hypermultiplets with masses $m_{j}$ in the scaling limit

$\tau_{\text{\tiny{UV}}}\to i\infty,\quad
m_{j}\to\infty,\quad\Lambda_{N_{f}}^{4-N_{f}}=4\,\mathtt{q}\prod_{j}m_{j}.$
(90)

The order parameter $\mathtt{u}$ in the qq-curve (88) is not identical to the
one of the $N_{f}=4$ SW curve (4) or the one of the $N_{f}\leq 3$ SW family
(105). The relation between the qq-curve and the $N_{f}=4$ SW curve is
difficult to work out explicitly, however one easily finds that by decoupling
hypermultiplets (90) the order parameters are related through the chain

$\mathtt{u}\to u_{N_{f}=3}-\frac{\Lambda_{3}}{8}(m_{1}+m_{2}+m_{3})\to
u_{N_{f}=2}-\frac{\Lambda_{2}^{2}}{8}\to u_{N_{f}=1}\to u_{N_{f}=0}.$ (91)

These constant shifts are likely due to the fact that the instanton partition
function is better suited for U(N) gauge theory rather than SU(N).

###  C.1 Case 0

Let us study the curve (88) for the case ${\boldsymbol{m}}=(0,0,0,0)$. Recall
the rational functional $\mathcal{R}$ (25) which has poles at $p=0,1,\infty$.
It relates the Hauptmoduln $j$ of $\text{SL}(2,\mathbb{Z})$ and
$\lambda\coloneqq\frac{\vartheta_{2}^{4}}{\vartheta_{3}^{4}}$ of $\Gamma(2)$
by $\mathcal{R}(\lambda(\tau))=j(\tau)$. In the massless limit we compute
$\mathcal{J}(\mathtt{u},\mathtt{q},0)=\mathcal{R}(\mathtt{q})$.171717 The
invariant of a quartic curve can be found using the formulas given in
Brandhuber:1996ng , for example.  As we identify $j(\tau)=\mathcal{J}$, it
follows that the UV-coupling is related to the complex structure of the curve
by

$\mathtt{q}=\lambda(\tau),$ (92)

with $\tau$ the low-energy effective coupling and
$\mathtt{q}=\lambda(\tau_{0})$ the UV coupling. This has already been
conjectured in Grimm:2007tm  by matching calculations in the field theory
limit of type IIA string compactification on Enriques Calabi-Yau to the
results from the Nekrasov partition function, and was further explored in
Huang:2011qx . For a more extensive discussion on the non-perturbative finite
renormalisation see (Jaewon:2012wsa, , Sections 3.4–3.5).

In Gaiotto:2009we , the parameters of the moduli space of marginal couplings
are identified with coordinates on Teichmüller spaces of punctured Riemann
surfaces. For the case of SU(2) theory with $N_{f}=4$, the corresponding
surface is a 4-punctured Riemann sphere, and the natural coordinate is a
cross-ratio $\mathtt{q}$ of the location of the punctures. Under conformal
transformations, the punctures are permuted Huang:2011qx

$\mathtt{q}\sim\left\\{\mathtt{q},\frac{1}{\mathtt{q}},\frac{1}{1-\mathtt{q}},1-\mathtt{q},\frac{\mathtt{q}}{\mathtt{q}-1},\frac{\mathtt{q}-1}{\mathtt{q}}\right\\}.$
(93)

We notice that $\mathtt{q}\mapsto\mathcal{R}(\mathtt{q})$ is invariant under
these permutations. The transformations (93) of the modular lambda function
$\lambda$ are generated by $T:\lambda\mapsto\frac{\lambda}{\lambda-1}$ and
$S:\lambda\mapsto 1-\lambda$ chandrasekharan2012elliptic  and form the
anharmonic group. Due to the permutations (93), we can view (92) as an
identification of the equivalence class

$\mathtt{q}\,/\\!\sim\,=\lambda(\tau).$ (94)

It is also clear from $\mathcal{J}=\mathcal{R}(\mathtt{q})$ that (92) is just
one of six solutions (94), since the former equation can be written as a
polynomial equation of degree 6 in $\mathtt{q}$ (see also aspman2021cutting ).
From (92) it is furthermore clear that $\tau=\tau_{0}$ is a solution, such
that $\tau(u)=\tau_{0}$ is constant over the whole Coulomb branch, just as in
the case of the SW curve (24).

###  C.2 Case A

Let us then study the curve for ${\boldsymbol{m}}_{\text{A}}=(m,m,0,0)$. This
theory has global symmetry
$\text{SU(2)}\times\text{SU(2)}\times\text{SU(2)}\times\text{U(1)}$. The
physical discriminant is

$\Delta=(\mathtt{u}-\mathtt{u}_{0,\text{A}})^{2}(\mathtt{u}-\mathtt{u}_{m,\text{A}})^{2}\left(\mathtt{u}-\mathtt{u}_{*,\text{A}}\right)^{2},$
(95)

where $\mathtt{u}_{0,\text{A}}=0$, $\mathtt{u}_{m,\text{A}}=m^{2}$,
$\mathtt{u}_{*,\text{A}}=\frac{\mathtt{q}\,m^{2}}{\mathtt{q}-1}$. The Coulomb
branch parameter $\mathtt{u}$ can be found as described in Section 3,

$\mathtt{u}_{\text{A}}(\tau)=m^{2}\frac{\mathtt{q}}{\mathtt{q}-\lambda(\tau)},$
(96)

where again $\lambda$ is a Hauptmodul for $\Gamma(2)$. This shows that
$\mathtt{u}(\tau,\mathtt{q})$ is a modular function for $\Gamma(2)_{\tau}$.
With the definition (87), $\mathtt{u}_{\text{A}}$ is also a modular function
for $\Gamma(2)_{\tau_{0}}$. However, a simultaneous $\text{SL}(2,\mathbb{Z})$
transformation on $\tau$ and $\tau_{0}$ does not give back the same function.
Therefore, $\mathtt{u}_{\text{A}}$ satisfies only condition 1 of def. 1, and
as such is only invariant under separate transformations
$\Gamma(2)_{\tau}\times\Gamma(2)_{\tau_{0}}$.

The singularities (95) can be easily associated to the cusps of
$\Gamma(2)\backslash\mathbb{H}$ by expanding the Jacobi theta functions. As
opposed to the asymptotically free cases, $\mathtt{u}(\cdot,\mathtt{q})$ is
not weakly holomorphic due to a pole in the interior of $\mathbb{H}$. In other
words, there is a singularity $\tau=\lambda^{-1}(\mathtt{q})$, where
$u\to\infty$ at finite coupling. Just as in (94) and (51), this is essentially
a definition of the UV coupling. We can collect,

$\mathtt{u}_{\text{A}}(\tfrac{1}{2})=\mathtt{u}_{m,\text{A}},\quad\mathtt{u}_{\text{A}}(0)=\mathtt{u}_{*,\text{A}},\quad\mathtt{u}_{\text{A}}(1)=\mathtt{u}_{0,\text{A}},\quad\mathtt{u}_{\text{A}}\left(\lambda^{-1}(\mathtt{q})\right)=\infty.$
(97)

In the decoupling limit (90), we find that $u$ flows to (110) of the massless
$N_{f}=2$ theory, considering the constant shift (91).

While the order parameter (96) corresponding to the qq-curve is significantly
simpler than the one (26) from the SW curve, the action of triality is
obstructed for the qq-curve. Using the relation (87), we can express
$\mathtt{u}_{\text{A}}$ as a two variable function of $\tau$ and $\tau_{0}$.
Through the single dependence of both $u_{\text{A}}$ and
$\mathtt{u}_{\text{A}}$ on $\lambda(\tau)$, we can relate
$\mathtt{u}_{\text{A}}$ and $u_{\text{A}}$,

$\mathtt{u}_{\text{A}}(\tau,\tau_{0})=\frac{1}{\vartheta_{4}(\tau_{0})^{4}}u_{\text{A}}(\tau,\tau_{0})-\frac{m^{2}}{3}\frac{\vartheta_{2}(\tau_{0})^{4}-\vartheta_{4}(\tau_{0})^{4}}{\vartheta_{4}(\tau_{0})^{4}}.$
(98)

It is also immediate from this expression that $\mathtt{u}_{\text{A}}$ is not
bimodular, as it fails to be invariant under a simultaneous
$\text{SL}(2,\mathbb{Z})$ action. However, the obstruction to bimodularity can
be expressed analogously to (98): We have that
$\mathtt{u}_{\text{A}}(\tau,\tau_{0})=h_{1}(\tau_{0})\,u_{\text{A}}(\tau,\tau_{0})+h_{2}(\tau_{0})$
for some meromorphic modular forms $h_{j}$, while also a simultaneous action
of $\text{SL}(2,\mathbb{Z})$ yields

$\displaystyle\mathtt{u}_{\text{A}}(\tau+1,\tau_{0}+1)$
$\displaystyle=\,\frac{\vartheta_{4}(\tau_{0})^{4}}{\vartheta_{3}(\tau_{0})^{4}}\mathtt{u}_{\text{A}}(\tau,\tau_{0})+m^{2}\lambda(\tau_{0}),$
(99)
$\displaystyle\mathtt{u}_{\text{A}}(-\tfrac{1}{\tau},-\tfrac{1}{\tau_{0}})$
$\displaystyle=-\frac{\vartheta_{4}(\tau_{0})^{4}}{\vartheta_{2}(\tau_{0})^{4}}\mathtt{u}_{\text{A}}(\tau,\tau_{0}),$

or equivalently in terms of $\mathtt{q}$,

$\begin{split}\mathtt{u}_{\text{A}}(\tau+1,\tau_{0}+1)&=\,(1-\mathtt{q})\,\mathtt{u}_{\text{A}}(\tau,\tau_{0})+m^{2}\,\mathtt{q},\\\
\mathtt{u}_{\text{A}}(-\tfrac{1}{\tau},-\tfrac{1}{\tau_{0}})&=\frac{\mathtt{q}-1}{\mathtt{q}}\,\mathtt{u}_{\text{A}}(\tau,\tau_{0}).\end{split}$
(100)

We thus find that the action of triality is an affine transformation on
$\mathtt{u}_{\text{A}}$.

###  C.3 The other cases

For the other cases B – G the story is similar. The order parameters of cases
B, C, E and F are listed in Table 3. It is straightforward to show that these
parameters have the correct flows when decoupling hypermultiplets and the
correct limits into each other upon tuning the masses. The singularities are
given in Table 4 and the corresponding degeneracies can be read off from the
physical discriminants, which read

$\displaystyle\Delta_{\text{B}}=$
$\displaystyle(\mathtt{u}-\mathtt{u}_{*,\text{B}})^{4}(\mathtt{u}-\mathtt{u}_{+,\text{B}})(\mathtt{u}-\mathtt{u}_{-,\text{B}}),$
(101) $\displaystyle\Delta_{\text{C}}=$
$\displaystyle\mathtt{u}^{4}(\mathtt{u}-\mathtt{u}_{+,\text{C}})(\mathtt{u}-\mathtt{u}_{-,\text{C}}),$
$\displaystyle\Delta_{\text{E}}=$
$\displaystyle(\mathtt{u}-\mathtt{u}_{m,\text{E}})^{2}(\mathtt{u}-\mathtt{u}_{\mu,\text{E}})^{2}(\mathtt{u}-\mathtt{u}_{+,\text{E}})(\mathtt{u}-\mathtt{u}_{-,\text{E}}),$
$\displaystyle\Delta_{\text{F}}=$
$\displaystyle\mathtt{u}^{2}(\mathtt{u}-\mathtt{u}_{*,\text{F}})^{2}(\mathtt{u}-\mathtt{u}_{+,\text{F}})(\mathtt{u}-\mathtt{u}_{-,\text{F}}).$

Theory | $\mathtt{u}(\tau)$
---|---
B | $m^{2}\left(\frac{1-3\mathtt{q}}{1-\mathtt{q}}-\frac{1-\mathtt{q}}{1-f(\tau)\sqrt{\mathtt{q}}+\mathtt{q}}\right)$
C | $m^{2}\frac{\mathtt{q}^{2}}{(1-\mathtt{q})}\frac{1}{2-\mathtt{q}-\sqrt{1-\mathtt{q}}\left(2+1/\tilde{f}(\tau)\right)}$
E | $-\frac{2\mathtt{q}m\mu}{1-\mathtt{q}}-\frac{\mathtt{q}(-1+2\mathtt{q}\lambda-\lambda^{2})(m^{2}+\mu^{2})+(1+\lambda)\sqrt{\mathtt{q}}\sqrt{\mathtt{q}(\lambda-1)^{2}(m^{2}-\mu^{2})^{2}+4(1-\mathtt{q})^{2}\lambda(m\mu)^{2}}}{2(\lambda-\mathtt{q})(\mathtt{q}\lambda-1)}$
F | $\mathtt{q}\frac{\frac{\vartheta_{2}^{8}-(\vartheta_{3}^{8}+\vartheta_{4}^{8})\mathtt{q}}{1-\mathtt{q}}\mu^{2}-(\vartheta_{2}^{8}+2\vartheta_{3}^{4}\vartheta_{4}^{4}\mathtt{q})m^{2}-(\vartheta_{3}^{4}+\vartheta_{4}^{4})\sqrt{\vartheta_{2}^{8}(m^{2}-\mu^{2})^{2}+\frac{4\mathtt{q}^{2}(m\mu)^{2}\vartheta_{3}^{4}\vartheta_{4}^{4}}{1-\mathtt{q}}}}{2(\vartheta_{2}^{4}+\mathtt{q}\vartheta_{4}^{4})(\vartheta_{2}^{4}-\mathtt{q}\vartheta_{3}^{4})}$
Table 3: Order parameters for the massive $N_{f}=4$ cases B, C, E and F. Theory | Singularities
---|---
B | $\mathtt{u}_{*,\text{B}}=m^{2}\frac{1-3\mathtt{q}}{1-\mathtt{q}};\qquad\mathtt{u}_{\pm,\text{B}}=2m^{2}\frac{-2\mathtt{q}\pm\sqrt{\mathtt{q}}}{1-\mathtt{q}}$
C | $\mathtt{u}_{0,\text{C}}=0;\qquad\mathtt{u}_{\pm,\text{C}}=\pm m^{2}\frac{2(1\pm\sqrt{1-\mathtt{q}})-\mathtt{q}}{1-\mathtt{q}}$
E | $\mathtt{u}_{m,\text{E}}=m^{2}-\frac{2m\mu\mathtt{q}}{1-\mathtt{q}};\qquad\mathtt{u}_{\mu,\text{E}}=\mu^{2}-\frac{2m\mu\mathtt{q}}{1-\mathtt{q}};\qquad\mathtt{u}_{\pm,\text{E}}=\frac{\mathtt{q}(m+\mu)^{2}}{\mathtt{q}-1}\pm 2\sqrt{\mathtt{q}}\frac{m\mu}{\mathtt{q}-1}$
F | $\mathtt{u}_{*,\text{F}}=\mathtt{q}\frac{\mu^{2}-m^{2}}{1-\mathtt{q}};\qquad\mathtt{u}_{\pm,\text{F}}=m^{2}+\frac{\mu^{2}}{1-\mathtt{q}}\mp\frac{2m\mu}{\sqrt{1-\mathtt{q}}}$
Table 4: Singularities for the massive $N_{f}=4$ cases B, C, E and F.

From Table 3 it is clear that $\mathtt{u}_{\text{B}}(\cdot,\mathtt{q})$ is a
modular function for $\Gamma^{0}(4)$ and
$\mathtt{u}_{\text{C}}(\cdot,\mathtt{q})$ for $\Gamma_{0}(4)$, in agreement
with the results from the SW curve. The singularities of the curves correspond
to the cusps of the respective modular curves. The singularities in the
interior can be found as

$\displaystyle\mathtt{u}_{\text{B}}(\tau_{\infty,\text{B}})$
$\displaystyle=\infty,\quad$ $\displaystyle\tau_{\infty,\text{B}}$
$\displaystyle=f^{-1}\left(\tfrac{1+\mathtt{q}}{\sqrt{\mathtt{q}}}\right),$
(102) $\displaystyle\mathtt{u}_{\text{C}}(\tau_{\infty,\text{C}})$
$\displaystyle=\infty,\quad$ $\displaystyle\tau_{\infty,\text{C}}$
$\displaystyle=\tilde{f}^{\\!-1}\left(\tfrac{\sqrt{1-\mathtt{q}}}{2-2\sqrt{1-\mathtt{q}}-\mathtt{q}}\right).$

The order parameter of cases E, F and all of the more general cases contain
branch points. For cases E and F they are given by the argument
$\tau_{\text{bp}}$ for which the radicands vanish. For case E for instance, it
is given by

$f_{2\text{B}}(\tfrac{\tau_{\text{bp,\text{E}}}}{2})=-64\frac{(m\mu)^{2}}{(m^{2}-\mu^{2})^{2}}\frac{(1-\mathtt{q})^{2}}{\mathtt{q}}.$
(103)

The AD loci of the theories E, F and G are now given by

$\displaystyle\text{P}^{\text{AD}}_{\text{E}}=$
$\displaystyle(m^{2}\mathtt{q}-\mu^{2})(\mu^{2}\mathtt{q}-m^{2}),$ (104)
$\displaystyle\text{P}^{\text{AD}}_{\text{F}}=$
$\displaystyle(m^{2}(\mathtt{q}-1)+\mu^{2})(\mu^{2}(\mathtt{q}-1)+m^{2}),$
$\displaystyle\text{P}^{\text{AD}}_{\text{G}}=$
$\displaystyle(m^{2}(1-\mathtt{q})+\mu^{2}\mathtt{q})(\mu^{2}(1-\mathtt{q})+m^{2}\mathtt{q}).$

We can note that these polynomials coincide with (66) upon identifying
$\mathtt{q}\coloneqq\lambda_{0}$. Similarly to (99), the order parameters
$\mathtt{u}_{\text{B}}$, $\mathtt{u}_{\text{C}}$ and $\mathtt{u}_{\text{D}}$
do not form a vector-valued bimodular form, which is due to the fact that they
transform into each other with shifts in $\tau_{0}$.

##  Appendix D The asymptotically free theories

The curves for the asymptotically free theories with $0\leq N_{f}\leq 3$
fundamental hypermultiplets has been determined in Seiberg:1994aj . They read

$\displaystyle N_{f}=0:\quad y^{2}=$ $\displaystyle
x^{3}-ux^{2}+\frac{1}{4}\Lambda_{0}^{4}x,$ (105) $\displaystyle N_{f}=1:\quad
y^{2}=$ $\displaystyle
x^{2}(x-u)+\frac{1}{4}m\Lambda_{1}^{3}x-\frac{1}{64}\Lambda_{1}^{6},$
$\displaystyle N_{f}=2:\quad y^{2}=$
$\displaystyle(x^{2}-\frac{1}{64}\Lambda_{2}^{4})(x-u)+\frac{1}{4}m_{1}m_{2}\Lambda_{2}^{2}x-\frac{1}{64}(m_{1}^{2}+m_{2}^{2})\Lambda_{2}^{4},$
$\displaystyle N_{f}=3:\quad y^{2}=$ $\displaystyle
x^{2}(x-u)-\frac{1}{64}\Lambda_{3}^{2}(x-u)^{2}-\frac{1}{64}(m_{1}^{2}+m_{2}^{2}+m_{3}^{2})\Lambda_{3}^{2}(x-u)$
$\displaystyle+\frac{1}{4}m_{1}m_{2}m_{3}\Lambda_{3}x-\frac{1}{64}(m_{1}^{2}m_{2}^{2}+m_{2}^{2}m_{3}^{2}+m_{1}^{2}m_{3}^{2})\Lambda_{3}^{2}.$

By taking the mass of a hypermultiplet to be infinite while sending the
dynamical scale of the theory to zero, in the double scaling limit Eguchi1999

$m_{i}\to\infty,\quad\Lambda_{N_{f}}\to 0,\quad
m_{i}\Lambda_{N_{f}}^{4-N_{f}}=\Lambda_{N_{f}-1}^{4-(N_{f}-1)}$ (106)

the hypermultiplets decouple. Identifying the $\mathcal{J}$-invariants of the
curves (105) with the modular $j$-invariant $j(\tau)$ allows to find the
modular $u$-parameters. This was done in aspman2021cutting  for the massive
$N_{f}=2,3$ curves. As was discussed in aspman2021cutting  in general there
are six different solutions for the order parameter as a function of $\tau$
from the curve. However, there exists a unique choice of solution such that
the decoupling limits work out directly. These solutions then also determine
the $N_{f}=4$ solutions uniquely in the same way, by demanding that the
decoupling limits work out.

For $N_{f}=0$, the unique choice is

$\displaystyle\frac{u}{\Lambda_{0}^{2}}=$
$\displaystyle-\frac{1}{2}\frac{\vartheta_{2}^{4}+\vartheta_{3}^{4}}{\vartheta_{2}^{2}\vartheta_{3}^{2}}.$
(107)

In massless $N_{f}=1$, the solution is given by

$\displaystyle\frac{u}{\Lambda_{1}^{2}}=$
$\displaystyle-\frac{3}{2^{\frac{7}{3}}}\frac{E_{4}^{\frac{1}{2}}}{(E_{4}^{\frac{3}{2}}-E_{6})^{\frac{1}{3}}}.$
(108)

In equal mass $N_{f}=2$, one finds

$\frac{u}{\Lambda_{2}^{2}}=-\frac{\vartheta_{4}^{8}+\vartheta_{2}^{4}\vartheta_{3}^{4}+(\vartheta_{2}^{4}+\vartheta_{3}^{4})\sqrt{16\frac{m^{2}}{\Lambda_{2}^{2}}\vartheta_{2}^{4}\vartheta_{3}^{4}+\vartheta_{4}^{8}}}{8\vartheta_{2}^{4}\vartheta_{3}^{4}},$
(109)

which becomes

$\frac{u}{\Lambda_{2}^{2}}=-\frac{1}{8}\frac{\vartheta_{3}^{4}+\vartheta_{4}^{4}}{\vartheta_{2}^{4}}$
(110)

in the massless limit. The theory with three hypermultiplets is much more
complicated. For the mass configuration ${\boldsymbol{m}}=(m,0,0)$ the
equations can be solved analogously, and one finds

$\frac{u}{\Lambda_{3}^{2}}=-\frac{2\vartheta_{3}^{4}\vartheta_{4}^{4}+(\vartheta_{3}^{4}+\vartheta_{4}^{4})\sqrt{\frac{64m^{2}}{\Lambda_{3}^{2}}\vartheta_{2}^{8}+\vartheta_{3}^{4}\vartheta_{4}^{4}}}{64\vartheta_{2}^{8}}.$
(111)

In the massless limit this becomes

$\frac{u}{\Lambda_{3}^{2}}=-\frac{1}{64}\frac{\vartheta_{3}^{2}\vartheta_{4}^{2}}{(\vartheta_{3}^{2}-\vartheta_{4}^{2})^{2}}.$
(112)

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