#  Seiberg-Witten curves of $\widehat{D}$-type Little Strings

Baptiste<EMAIL_ADDRESS>, Stefan
<EMAIL_ADDRESS>, and Taro
<EMAIL_ADDRESS>
###### Abstract

Little Strings are a type of non-gravitational quantum theories that contain
extended degrees of freedom, but behave like ordinary Quantum Field Theories
at low energies. A particular class of such theories in six dimensions is
engineered as the world-volume theory of an M5-brane on a circle that probes a
transverse orbifold geometry. Its low energy limit is a supersymmetric gauge
theory that is described by a quiver in the shape of the Dynkin diagram of the
affine extension of an ADE-group. While the so-called $\widehat{A}$-type
Little String Theories (LSTs) are very well studied, much less is known about
the $\widehat{D}$-type, where for example the Seiberg-Witten curve (SWC) is
only known in the case of the $\widehat{D}_{4}$ theory. In this work, we
provide a general construction of this curve for arbitrary $\widehat{D}_{M}$
that respects all symmetries and dualities of the LST and is compatible with
lower-dimensional results in the literature. For $M=4$ our construction
reproduces the same curve as previously obtained by other methods. The form in
which we cast the SWC for generic $\widehat{D}_{M}$ allows to study the
behaviour of the LST under modular transformations and provides insights into
a dual formulation as a circular quiver gauge theory with nodes of $Sp(M-4)$
and $SO(2M)$.

${}^{\,}$11footnotemark: 122footnotemark: 2 Univ Lyon, Univ Claude Bernard
Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France

22footnotemark: 2 Dept. of Physics E. Pancini, Università di Napoli Federico
II, via Cintia, 80126 Napoli, Italy

33footnotemark: 3 Institut de Mathématiques de Bourgogne, Université de
Bourgogne, CNRS, UMR 5584, France

###### Contents

  1. 1 Introduction
  2. 2 Little String Theories and Seiberg-Witten curves
    1. 2.1 $\widehat{ADE}$-type Little String Theories
    2. 2.2 $\widehat{A}$-type Seiberg-Witten curves
  3. 3 $\widehat{D}$-type Seiberg-Witten curves
    1. 3.1 $\widehat{D}$-type LSTs
    2. 3.2 General Ansatz for the Seiberg Witten Curve
  4. 4 Seiberg-Witten Curve of $(\widehat{D}_{4},1)$
    1. 4.1 Construction
    2. 4.2 Modular properties
    3. 4.3 Lower Dimensional Limit
  5. 5 Seiberg-Witten Curve of $(\widehat{D}_{5},1)$
    1. 5.1 Construction
    2. 5.2 Modular properties
    3. 5.3 Lower dimensional limit
  6. 6 Generalisation to the $(\widehat{D}_{M},1)$ LSTs
  7. 7 Conclusion
  8. A Modular functions and properties
    1. A.1 Definitions and properties
    2. A.2 Modular (sub)group
  9. B $\mathbb{M}$ matrices examples

##  1 Introduction

Little String Theories (LSTs) [1, 2, 3, 4, 5] are a class of non-gravitational
quantum theories, which at low energies behave like quantum field theories
with point-like degrees of freedom, but whose UV-completion requires new
extended degrees of freedom. Such theories arise naturally within the
framework of String Theory, through suitable limits in which the gravitational
sector is decoupled [6, 7]. They constitute interesting generalisations of
(conformal) field theories that are obtained through different point-particle
limits and have therefore attracted a lot of attention over the years: for
example, through geometric methods similar to the case of (higher dimensional)
conformal field theories, a classification has been provided in [8, 9, 10].

A very rich class of supersymmetric LSTs (dubbed Little String Orbifolds in
[11]) can be constructed as the world-volume theory of $N$ parallel M5-branes
on a circle, probing a transverse orbifold geometry [12] (see also [13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 27, 28, 30, 31,
32]) of the form $\mathbb{R}_{\perp}^{4}/\Gamma$, where $\Gamma\subset SU(2)$.
At low energies, the world-volume theory on the M5-branes is a six-dimensional
supersymmetric gauge theory, described by a quiver in the shape of the Dynkin
diagram of the affine group $\widehat{\Upsilon}$, which is related to $\Gamma$
through the McKay correspondance. Without additional flavour groups, the rank
of the gauge nodes is fixed through the absence of anomalies and matter
appears in the bifundamental representation.

Among these theories the case with $\widehat{\Upsilon}=\widehat{A}_{M-1}$
(called $\widehat{A}$-type and denoted $(\widehat{A}_{M-1},N)$ theory in the
following) has been studied in quite some detail: indeed, this class of
theories enjoys many different dual descriptions, which allow to calculate
important quantities explicitly. For example, the BPS-(instanton) partition
function can be obtained by counting M2-branes stretched between the M5 branes
[13, 14, 15, 16]. Using a dual description in terms of F-theory compactified
on a toric Calabi-Yau manifold $X_{N,M}$ [33], this counting is captured by
the topological string partition function on this threefold [17, 11, 18, 20],
which in turn can be computed in an algorithmic fashion using the (refined)
topological vertex [34, 35]. Another quantity of the $(\widehat{A}_{M-1},N)$
theory, which can be constructed explicitly using various different approaches
[36, 37], is the Seiberg-Witten curve (SWC). The latter encodes important
information about the non-perturbative structure of the theory, notably its
symmetries. Furthermore, it affords interesting insights into related
algebraic integrable systems [38, 39, 40], where it is identified with their
spectral curve. The SWC depends on $MN+2$ independent parameters and can be
written as a linear combination of a basis of theta functions that are defined
on a genus 2 surface. This surface reflects the doubly periodic nature of the
M5-brane setup (which is for example also visible in the double elliptic
fibration structure of the threefold $X_{N,M}$ mentioned above). Furthermore,
for a generic $(\widehat{A}_{M-1},N)$ theory, this surface has a full period
matrix: the third parameter, besides the two modular parameters, has an
interpretation as the mass of the bifundamental matter from the perspective of
the quiver gauge theory.

In this work, however, we shall be concerned with the case
$\widehat{\Upsilon}=\widehat{D}_{M}$ (for $M\geq 4$) and $N=1$ (_i.e._ a
single M5-brane), which we shall denote as $(\widehat{D}_{M},1)$ theory in the
following. Due to technical difficulties and subtleties in realising dual
descriptions, these so-called $\widehat{D}$-type LSTs are much less studied in
the literature. Explicit computations of the instanton partition function have
been performed in the topological vertex formalism by using so-called
trivalent gluing techniques [41, 42, 43], using the blow-up equations [44, 45]
or using dual realisations in term of two-dimensional $\mathcal{N}=(0,4)$
gauge theories [46]. From the perspective of the SWC, to date only the curve
in the case $M=4$ is known [47], either from the thermodynamic limit [48] of
the instanton partition function or by using the quantum states formalism
[49].

For general $M$, the geometric structure of the SWC is somewhat less
restrictive than in the $(\widehat{A}_{M-1},1)$ case: due to the absence of a
mass parameter [50], a suitable set of basis functions consists of products of
genus 1 theta functions (with two different modular parameters), rather than
genuine genus 2 theta functions as in the $\widehat{A}$-type case. In this
paper, we present a general ansatz for the SWC of the $(\widehat{D}_{M},1)$
theory that is manifestly compatible with all symmetries. To further restrict
this ansatz, we use the fact that the $(\widehat{D}_{M},1)$ theory is dual to
a circular quiver gauge theory with one gauge node of type $Sp(M-4)$ and one
node of type $SO(2M)$. This requires that the SWC can equivalently be re-
written in a way that is compatible with the symmetries of this dual theory.
We argue for low values of $M$ (notably $M=4$ and $M=5$) that consistency of
these dual presentations imposes further conditions on the parameters and
ultimately leaves only a few distinct forms of the curve (two in the case of
$M=4$ and three in the case of $M=5$). By further considering a dimensional
reduction (by assuming a natural behaviour of the remaining parameters in the
decompactification limit) and comparing with known results in five-dimensions
[51], we find further conditions on the parameters leaving a unique form of
the SWC. In particular in the case of $M=4$, the latter agrees with known
results in [47, 49]. By studying further examples up to $M=12$, we find
distinctive patterns, which allow us to conjecture the general form of the
$(\widehat{D}_{M},1)$ SWC for generic $M\geq 4$. The latter could be
identified with the spectral curves of new $\widehat{D}$-type double elliptic
integrable systems. Some steps have been taken in this direction through the
construction of the quantum integrable system of $D$-type conformal matter
(_i.e._ six-dimensional Super Conformal Field Theories (SCFTs)) [52] and
quantum deformations of SWCs [53, 54, 55].

This paper is organised as follows: In Section 2, we provide more details on
LSTs engineered from M5-branes probing transverse orbifold geometries and
their SWCs, recalling in particular the $(\widehat{A}_{M-1},N)$ theories. In
Section 3 we focus on the $(\widehat{D}_{M},1)$ LSTs and present a general
ansatz for the SWC that is compatible with all symmetries of the theory. In
Section 4 we discuss in detail the case $M=4$ and show how this ansatz can be
further restricted by imposing duality of the $(\widehat{D}_{4},1)$ theory to
a quiver gauge theory with one $Sp(0)$ and one $SO(8)$ node and by comparing
to known five-dimensional results. We show that the unique SWC obtained in
this way is equivalent to the result in [47, 49]. In Section 5 we repeat this
construction for the $(\widehat{D}_{5},1)$ theory and show that it again leads
to a unique SWC, which constitutes a novel result in the literature. Based on
further examples up to $M=12$, we present in Section 6 a general form for the
SWC of the $(\widehat{D}_{M},1)$ theory. Finally, Section 7 contains our
conclusions, as well as an outlook on applications of our results. This paper
is accompanied by two Appendices: in Appendix A we compile definitions as well
as useful identities for modular objects that are used throughout the main
body of this text. Appendix B contains the results of the SWC for certain
$M>5$, which corroborate our general conjecture in Section 6.

##  2 Little String Theories and Seiberg-Witten curves

###  2.1 $\widehat{ADE}$-type Little String Theories

Little String Theories (LSTs) arise in string theory (or related dual
descriptions) through different decoupling limits [1, 2, 56, 3, 4, 5] (see
also [6, 7] for reviews). They follow an $\widehat{ADE}$-classification [8]
and can be engineered from numerous setups. For concreteness, we begin with a
construction in M-theory [17, 11], consisting of $N$ M5-branes (along
directions $x^{0,1,\ldots,5}$), separated along a circle $\mathbb{S}_{6}^{1}$
probing a transverse $\mathbb{R}^{4}_{\perp}/\Gamma$ orbifold with a finite
subgroup of $SU(2)$, $\Gamma\subset SU(2)$:

| $\mathbb{S}^{1}_{0}$ | $\mathbb{S}^{1}_{1}$ |  | $\mathbb{R}^{3}_{\parallel}$ |  | $\mathbb{S}_{5}^{1}$ | $\mathbb{S}_{6}^{1}$ |  | $\mathbb{R}^{4}_{\perp}\\!\\!\\!$ | $\\!\\!\\!\\!\\!\\!/\Gamma$ |
---|---|---|---|---|---|---|---|---|---|---|---
| $x^{0}$ | $x^{1}$ | $x^{2}$ | $x^{3}$ | $x^{4}$ | $x^{5}$ | $x^{6}$ | $x^{7}$ | $x^{8}$ | $x^{9}$ | $x^{10}$
$N$ $M5$ | $=$ | $=$ | $=$ | $=$ | $=$ | $=$ | $\times$ |  |  |  |

In the low energy limit, this construction corresponds to a supersymmetric
quiver gauge theory on
$\mathbb{S}_{0}^{1}\times\mathbb{S}_{1}^{1}\times\mathbb{R}^{3}_{\parallel}\times\mathbb{S}_{5}^{1}$.
The McKay correspondence [57] associates the orbifold group $\Gamma$ to an
affine quiver group $\widehat{\Upsilon}$ as follows

Orbifold group $\Gamma$ |  $\mathbb{Z}_{M}$ for $M\geq 1$ |  $\mathbb{BD}_{M-2}$ for $M\geq 4$ | $\mathbb{BT}$ | $\mathbb{BO}$ | $\mathbb{BI}$
---|---|---|---|---|---
Quiver group $\widehat{\Upsilon}$ | $\widehat{A}_{M-1}$ | $\widehat{D}_{M}$ | $\widehat{E}_{6}$ | $\widehat{E}_{7}$ | $\widehat{E}_{8}$

where $\mathbb{BD}_{M}$ is the binary dihedral group of order $4M$ and
$\mathbb{BT},\mathbb{BO}$ and $\mathbb{BI}$ are respectively the binary
isomorphism groups of the tetrahedron, the cube and the dodecahedron. The
quiver takes the form of the Dynkin diagram of $\widehat{\Upsilon}$ (see Fig.
1) and mathematically is described by $(\Upsilon_{0},\Upsilon_{1})$, where
$\Upsilon_{0}$ is the set of nodes and $\Upsilon_{1}$ the set of edges
connecting the latter. Fig. 1 also display an $n$_-coloring_ of the quivers,
_i.e._ a particular assignment of the rank of each gauge node. To explain
this, we introduce the Cartan matrix of the quiver
$(\Upsilon_{0},\Upsilon_{1})$ by

$c_{ij}=2\delta_{ij}-\sum_{e\in\Upsilon_{1}:i\to
j}1-\sum_{e\in\Upsilon_{1}:j\to i}1\,,\quad\quad\forall i,j\in\Upsilon_{0}\,.$
(2.1)

The rank of the gauge group at each node in $\Upsilon_{0}$ is uniquely
fixed444Here we do not assume any additional flavor groups. by demanding the
vanishing of the beta-functions for the gauge couplings [58]:

$\sum_{j\in\Upsilon_{0}}c_{ij}\cdot n_{j}=0\,,\quad\quad\forall
i\in\Upsilon_{0}\,,$ (2.2)

with $n_{j}$ the rank of gauge group associated with the $j$-th node. A
solution of the set of equations (2.2) is called an $n$-coloring of the
quiver. For quivers of simply laced Lie groups, the coloring is fixed by a
single integer $N$ which corresponds to the number of M5-branes probing the
orbifold. Concretely, in Fig. 1, the nodes $N$ of the quiver represent gauge
groups $\widehat{A}_{N-1}$, while lines denote matter in the bifundamental
representation.555Certain quiver gauge theories, such as the $\widehat{A}_{0}$
theory with $M=1$ has matter in the adjoint- rather than the bifundamental
representation [17]. However, we shall not consider this case in the current
paper. We shall denote a LST in this framework as $(\widehat{\Upsilon},N)$.

$N$$N$$N$$N$$N$$N$$N$$\widehat{A}_{M}$

$2N$$2N$$N$$N$$N$$N$$\widehat{D}_{M}$

$N$$2N$$3N$$2N$$N$$2N$$N$$\widehat{E}_{6}$

$N$$2N$$3N$$4N$$3N$$2N$$N$$2N$$\widehat{E}_{7}$$2N$$4N$$5N$$5N$$4N$$3N$$2N$$N$$3N$$\widehat{E}_{8}$

Figure 1: $N$-coloring of $\widehat{A}$, $\widehat{D}$ and $\widehat{E}$-type
quivers.

###  2.2 $\widehat{A}$-type Seiberg-Witten curves

Reviewing first the case of $\Gamma=\mathbb{Z}_{M}$, the so-called A-type LSTs
engineered from the M-brane setup in Table 2.1 can be described by a doubly
elliptic Seiberg-Witten curve [36, 37]. To understand the origin of the
latter, we recall [13, 14, 15, 16, 17, 11] that this M-brane configuration can
be described by $MN+2$ independent parameters: we refer the reader for more
details to [17, 11] and only mention that in the following we denote the
(complexified) radii of $\mathbb{S}_{1}^{1}$ and $\mathbb{S}_{6}^{1}$
(measured in units of the radius of $\mathbb{S}_{0}^{1}$) by $\tau$ and $\rho$
respectively. Furthermore, following [17], we introduce a $U(1)_{S}$
deformation of $\mathbb{R}_{\perp}^{4}/\mathbb{Z}_{M}$ with respect to
$\mathbb{S}_{0}^{1}$

$\displaystyle U(1)_{S}:\hskip
11.38092pt(w_{+},w_{-})\longrightarrow\left(e^{2\pi iS}\,w_{+}\,,e^{-2\pi
iS}\,w_{-}\right)\,,$ with $\displaystyle w_{\pm}=x^{7}\pm ix^{8}\,,$ (2.3)

which is compatible with the $\mathbb{Z}_{M}$ action. The parameters
$(\tau,\rho,S)$ can be arranged into the period matrix $\Omega$ of a genus 2
Riemann surface and the SWC (parametrised by $(z_{1},z_{2})$) can be written
in terms of genus $2$ theta functions $\Theta$ (see (A.1) for the definition)
[36, 37]:

$\sum_{i=0}^{M-1}\sum_{j=0}^{N-1}a_{ij}\,\Theta\\!\begin{bmatrix}i/M&j/N\\\
0&0\end{bmatrix}\\!(z\,|\Omega)=0\,,\quad z=\begin{bmatrix}z_{1}\\\
z_{2}\end{bmatrix},\quad\Omega=\begin{bmatrix}\frac{\tau}{M}&S\\\
S&\frac{\rho}{N}\end{bmatrix}.$ (2.4)

The $\\{a_{ij}\\}$ in (2.4) depend on the remaining parameters of the M-brane
configuration mentioned above.666Indeed, due to an overall scaling symmetry of
(2.4), the moduli space of the SWC is indeed $(NM+2)$-dimensional.  Using a
dual description in terms of F-theory compactified on a Calabi-Yau threefold
$X_{N,M}$ [17, 18, 11] with double elliptic-fibration structure, the
$\\{a_{ij}\\}$ have been characterised in [33] as (the product of) generating
functions of Gromov-Witten invariants.

From the perspective of the $\widehat{A}_{M}$ quiver gauge theory represented
by the leftmost diagram in Fig. 1, the parameter $\tau$ is interpreted as a
complexified gauge coupling (which is common to all $M$ nodes in the quiver),
$\rho$ as the (common) parameter which extends all gauge group nodes to their
affine forms $\widehat{A}_{N-1}$ and $S$ as the mass of the bifundamental
hypermultiplets. The remaining parameters [33] (appearing in the form of
$a_{ij}$ in (2.4)) encode further structure of the gauge- and matter spectrum,
but the details shall not be important in the following. At the classical
level, the theory inherits modular properties from the genus $2$ formulation
(2.4). The functional basis (_i.e._ the particular set
$\Theta\\!\left[\begin{smallmatrix}i/M&j/N\\\
0&0\end{smallmatrix}\\!\right]\\!(z\,|\Omega)$ of genus $2$ theta functions)
over which the curve (2.4) is written, corresponds to theta functions of
polarised Abelian varieties [59, 60]. The modular group acting on the period
matrix $\Omega$ is therefore the paramodular group defined as [60]:

$\Sigma_{M,N}:=Sp(4,\mathbb{Q})\cap\begin{bmatrix}\mathbb{Z}&\mathbb{Z}&\mathbb{Z}/M&\mathbb{Z}/N\\\
\mathbb{Z}&\mathbb{Z}&\mathbb{Z}/M&\mathbb{Z}/N\\\
M\mathbb{Z}&M\mathbb{Z}&\mathbb{Z}&M\mathbb{Z}/N\\\
N\mathbb{Z}&N\mathbb{Z}&N\mathbb{Z}/M&\mathbb{Z}\end{bmatrix}\,,$ (2.5)

with,

$Sp(4,\mathbb{Q})=\left\\{\Lambda=\begin{bmatrix}A&B\\\
C&D\end{bmatrix}\in\mathbb{M}_{4}(\mathbb{Q})\,|\,\Lambda\begin{bmatrix}0&\mathds{1}\\\
\mathds{-1}&0\end{bmatrix}\Lambda^{T}=\begin{bmatrix}0&\mathds{1}\\\
\mathds{-1}&0\end{bmatrix}\right\\},$ (2.6)

where $\mathbb{M}_{4}(\mathbb{Q})$ is the space of $4\times 4$ matrices with
rational entries and non-vanishing determinant. The group $\Sigma_{M,N}$ acts
in the following way on the period matrix $\Omega$:

$\Omega\to(A\Omega+B)(C\Omega+D)^{-1}\,,\quad\forall\begin{bmatrix}A&B\\\
C&D\end{bmatrix}\in\Sigma_{M,N}\,.$ (2.7)

It is interesting to note that the linearly acting subgroup of the paramodular
group (_i.e._ the subgroup with $B=C=0$ in (2.7)) generates the web of
dualities in LSTs that was described at the level of the instanton partition
function in [18, 21, 20]. In this respect, the group $\Sigma_{M,N}$ is a
generalisation of the paramodular group found in [26, 27] for generic $M,N>1$.

The $(\widehat{A}_{M-1},N)$ theory was shown to be dual to
$(\widehat{A}_{M^{\prime}-1},N^{\prime})$ for any
$N^{\prime},M^{\prime}\in\mathbb{N}$ such that $N^{\prime}M^{\prime}=NM$ and
$\mathrm{gcd}(N^{\prime},M^{\prime})=\mathrm{gcd}(N,M)=:p$. The corresponding
duality map at the level of the period matrix $\Omega$, was formulated in [18]
in terms of the following $Sp(4,\mathbb{Z})$ transformation

$A=\begin{bmatrix}\frac{M}{p}&-\frac{N}{p}\\\
1&-1\end{bmatrix}\,,\quad\Omega\to(A^{T})^{-1}\Omega
A^{-1}=\begin{bmatrix}\frac{\tau^{\prime}}{M^{\prime}}&S^{\prime}\\\
S^{\prime}&\frac{\rho^{\prime}}{N^{\prime}}\end{bmatrix}\,.$ (2.8)

This transformation acts naturally on the basis of genus 2 theta functions
$\Theta\\!\left[\begin{smallmatrix}i/M&j/N\\\
0&0\end{smallmatrix}\\!\right]\\!(z\,|\Omega)$ (for $i\in\\{0\ldots M-1\\}$
and $j\in\\{0\ldots N-1$) over which (2.4) is formulated: using (A.2) and
noticing the equivalence of the following sets

$\displaystyle\Bigg{\\{}A\cdot\left(\frac{i}{M},\frac{j}{N}\right),$
$\displaystyle\,0\leq i\leq M-1\,,\,0\leq j\leq N-1\Bigg{\\}}$
$\displaystyle\cong\Bigg{\\{}\left(\frac{k}{p}\\!\\!\\!\\!\mod
1,\frac{lp}{MN}\\!\\!\\!\\!\mod 1\right),\,0\leq k\leq p-1\,,\,0\leq
l\leq\frac{MN}{p}-1\Bigg{\\}}\,,$ (2.9)

it indeed follows that (2.8) (along with a suitable duality map of the
remaining parameters $\\{a_{ij}\\}$) leaves the SWC (2.4) invariant.

##  3 $\widehat{D}$-type Seiberg-Witten curves

In this Section, we review some of the key properties of $(\widehat{D}_{M},1)$
LSTs, which allow us to formulate a general ansatz for their SWCs. In the
following Sections 4 and 5 we shall use this ansatz to provide the unique form
of the curves in the cases $M=4$ and $M=5$ respectively, before generalising
our results in Section 6 to generic $M\geq 4$.

###  3.1 $\widehat{D}$-type LSTs

First of all, contrary to $\widehat{A}$-type, $\widehat{D}$-type LSTs do not
allow for a mass-deformation parameter [50], _i.e._ there is no equivalent to
the parameter $S$ in the period matrix in (2.4). Therefore, also the genus 2
theta functions $\Theta$ are effectively reduced to two types of genus $1$
(_i.e._ Jacobi) theta functions. Secondly, the transverse $\widehat{D}$-type
orbifold itself has two important effects, which are of importance to the SWC:

  * _(i)_

the orbifold gives rise to a $\mathbb{Z}_{2}$ symmetry, under which the SWC is
invariant [47]

  * _(ii)_

$M5$-brane fractionalisation: as explained in [61] due to the orbifold, a
single $M5$-brane splits into two ‘half-branes’ on the $\mathbb{S}_{6}^{1}$
circle of Table 2.1. At the level of the quantum states introduced in [49],
each $M5$ state can be described by the product of two $1/2$ $M5$-brane
states.

Finally, there is another point of view, which provides crucial information
for the SWCs, namely a dual description of the ($\widehat{D}_{M},1)$ LSTs in
terms of an $Sp(M-4)$–$SO(2M)$ quiver gauge theory, which is schematically
shown in Figure 2. Indeed, for $M=4$ this duality was first argued in [46] at
the level of the partition function and in [47] at the level of the SWC using
fiber-base duality, while for general $M\geq 4$, it was established at the
level of the partition function in [46].

$SU(2)$$SU(2)$$SU(2)$$SU(1)$$SU(1)$$SU(1)$$SU(1)$

$Sp(M-4)$$SO(2M)$

Figure 2: Quiver diagram of $(\widehat{D}_{M},1)$ LST with $M-3$ $SU(2)$ nodes
(which are indicated explicitly) and its dual $\widehat{A}_{1}$ base
$Sp(M-4)-SO(2M)$ quiver theory

To understand the implications of such a duality at the level of the SWC, we
remark that, in the absence of a mass-deformation, curves with
$\widehat{A}_{1}$ base can generally be written in the following form [47,
49]:

$0=\theta_{3}^{2}(0;\rho)\theta_{2}^{2}(0;\rho)\theta_{1}^{2}(z_{2};\rho)c_{1}(\tau)F_{1}(z_{1};\tau)-\theta_{4}^{2}(z_{2};\rho)F_{2}(z_{1};\tau),$
(3.1)

where the Jacobi theta functions $\theta_{2,3}$ are defined in (A.1),
$c_{1}(\tau)$ is a modular constant and $F_{1,2}$ correspond to elliptic gauge
polynomials777The gauge polynomial is the polynomial whose roots are the
Coulomb branch moduli. Elliptic gauge polynomials are their natural elliptic
uplift. of each of the two nodes of the (right) quiver in Figure 2. For
example, in [47], these polynomials have been found for $SO(8)$ and $Sp(0)$
nodes to be respectively

$\displaystyle
F_{1}(z_{1};\tau)=\prod_{i=1}^{4}\frac{\theta_{1}(z_{1}-a_{i}^{SO};\tau)\theta_{1}(z_{1}+a_{i}^{SO};\tau)}{\theta_{1}^{2}(z_{1};\tau)}\,,$
and $\displaystyle
F_{2}(z_{1};\tau)=\frac{\theta_{1}^{2}(2z_{1};\tau)}{\theta_{1}^{8}(z_{1};\tau)}\,.$
(3.2)

The existence of the dual description of the $(\widehat{D}_{M},1)$ theory
requires that their SWC need to admit a presentation of the form (3.1). As we
shall see in the following, by starting from a generic ansatz (based on a
finite dimensional basis of theta-function building blocks), this requirement
leads to non-trivial constraints.

###  3.2 General Ansatz for the Seiberg Witten Curve

We are now ready to discuss the general form of the SWC of the
$(\widehat{D}_{M},1)$-LST. This curve is doubly elliptic and we shall denote
the corresponding parameters by $\tau$ and $\rho$, while the coordinates shall
be called $z_{1}$ (the coordinate on the $\tau$-parameterised elliptic curve)
and $\mathbb{Z}_{2}$ even in $z_{2}$ (the coordinate on the
$\rho$-parameterised elliptic curve). Following the discussion of the previous
Subsection, we shall now provide a general ansatz for the
$(\widehat{D}_{M},1)$-LST, which is based on the following principles:

  1. 1.

The SWC is independently $\mathbb{Z}_{2}$ even in $z_{1}$ and $z_{2}$. This
symmetry is due to the $\mathbb{Z}_{2}$ symmetry of the $\widehat{D}$-type
orbifold, explained in point _(i)_ of the previous Subsection. The fact that
it (a priori) can act independently on $z_{1}$ and $z_{2}$ is a consequence of
the absence of a mass deformation in the $\widehat{D}$-type LSTs.

  2. 2.

Using (pairs of) genus 1 theta functions as building blocks, the general form
of the $\widehat{D}$-type-LST SWC can be cast in the following form

$\sum_{0\leq i,j\leq
M}\widetilde{T}_{i}(z_{2},\rho,\tau)\,\mathbb{M}_{ij}(\tau)\,\Phi_{j}(z_{1},\tau)=0\,.$
(3.3)

Indeed, here $\\{\Phi_{j}\\}_{j\in\\{0\ldots M\\}}$ is a basis of
$\mathbb{Z}_{2}$ even theta functions of degree $2M$ (defined in (A.7)) and
$\widetilde{T}_{i}$ are $\mathbb{Z}_{2}$ even combinations of elliptic Weyl
characters of $SU(1)$ and $SU(2)$ [50, 62], concretely

$\widetilde{T}_{i}(z_{2},\rho)=\begin{cases}\alpha_{i}\theta_{1}^{2}(z_{2};\rho),&\text{for
}i\in\\{0,1,M-1,M\\}\,,\\\
\alpha_{i}\theta_{1}^{2}(z_{2};\rho)+\beta_{i}\theta_{4}^{2}(z_{2};\rho),&\text{for
}i\in\\{2\ldots M-2\\}\,,\end{cases}$ (3.4)

with $\theta_{1}$ and $\theta_{4}$ defined eq. (A.1). $\alpha_{i},\beta_{i}$
are $2M-2=h^{\vee}(D_{M})$, which are independent of $z_{1,2}$, but may depend
on $\tau$ and $\rho$ (as well as other Coulomb moduli of the
$\widehat{D}_{M}$-LST).

  3. 3.

The form (3.3) of the SWC must permit an alternative decomposition either on
sections of line bundles with polarisation $(2,2M)$ or on products of sections
of line bundles with polarisation $(1,M)$. As detailed in [49], this is
interpreted in term of quantum states as a consequence of the $M5$-brane
fractionalisation discussed in point _(ii)_ of the previous Subsection. This
can alternatively be motivated by analysing the 4d SWC of the nodes involved
in the dual construction (corresponding to the quiver on the right of Fig. 2).
The gauge polynomial of $SO(2M)$ and $Sp(M-4)$ gauge theories can be
decomposed into the product of two degree $M$ polynomials [63]. We assume that
this structure stems from a decomposition in term of sections of line bundle
as explained above in the SWC of the LST.

The most general form of the SWC that is compatible with points 1. and 3. and
permits a presentation of the form (3.3) in point 2., with the correct
structure for the (elliptic) Weyl characters $\widetilde{T}_{i}$ takes the
following form

$\begin{split}\theta_{1}^{2}(z_{2};\rho)&\left[\sum_{0\leq i\leq
j\leq\lfloor\frac{M}{2}\rfloor}a_{ij}^{(+)}X_{i}(z_{1};\tau)X_{j}(z_{1};\tau)+\sum_{1\leq
i\leq
j\leq\lfloor\frac{M-1}{2}\rfloor}a_{ij}^{(-)}Y_{i}(z_{1};\tau)Y_{j}(z_{1};\tau)\right]\\\
&+\theta_{4}^{2}(z_{2};\rho)\left[\sum_{0\leq i\leq
j\leq\lfloor\frac{M}{2}\rfloor}b_{ij}^{(+)}X_{i}(z_{1};\tau)X_{j}(z_{1};\tau)+\sum_{1\leq
i\leq
j\leq\lfloor\frac{M-1}{2}\rfloor}b_{ij}^{(-)}Y_{i}(z_{1};\tau)Y_{j}(z_{1};\tau)\right]=0\,,\end{split}$
(3.5)

where $X_{i}$ and $Y_{i}$ (with $i\in\\{0,\ldots,\lfloor\frac{M-1}{2}\rfloor$)
are defined in (A.6) in Appendix A.1. Here $a_{ij}^{\pm}$ and $b_{ij}^{\pm}$
are a total of $(M+1)^{2}/2+3/2$ parameters for $M$ even and $(M+1)^{2}/2$
parameters for $M$ odd, which exceeds the expected number of $2M-2$. This
implies that these parameters are not all independent and the ansatz (3.5) is
too general. We therefore require further conditions to uniquely determine the
SWC: such conditions are provided by demanding an alternative presentation of
the curve (3.5) in the form of (3.1), compatible with a dual description as a
quiver theory such as in the right part of Figure 2. Finally, any remaining
ambiguities can be fixed by comparison with known results of the 5-dimensional
compactification of the LSTs. To explain this reasoning concretely, we shall
explain in detail in the following Section 4 the case of the
$(\widehat{D}_{4},1)$ theory (for which the SWC was previously constructed in
[47, 49]) and in Section 5 the case of the $(\widehat{D}_{5},1)$ theory (for
which the SWC is not known in the literature). Finally, in Section 6, we
present the generalisation of our results to theories $(\widehat{D}_{M},1)$,
with generic $M\geq 4$.

##  4 Seiberg-Witten Curve of $(\widehat{D}_{4},1)$

In this Section, we focus on the theory $(\widehat{D}_{4},1)$ and explain how
the general ansatz (3.5) can be further restricted to obtain the unique
Seiberg-Witten curve. As we shall explain, our results agree with [47, 49],
which acts as a verification of the general ansatz presented in the previous
Subsection 3.2. We shall furthermore discuss modular properties of our
solution and connect the result to known $5$d SCFTs.

###  4.1 Construction

We start from the general ansatz (3.5) for $M=4$

$\displaystyle\theta_{1}^{2}(z_{2};\rho)\left[a_{00}^{(+)}X_{0}^{2}+a_{11}^{(+)}X_{1}^{2}+a_{02}^{(+)}X_{0}X_{2}+a_{11}^{(-)}Y_{1}^{2}+a_{22}^{(+)}X_{2}^{2}+a_{01}^{(+)}X_{0}X_{1}+a_{12}^{(+)}X_{1}X_{2}\right]$
$\displaystyle+\theta_{4}^{2}(z_{2};\rho)\left[b_{00}^{(+)}X_{0}^{2}+b_{11}^{(+)}X_{1}^{2}+b_{02}^{(+)}X_{0}X_{2}+b_{11}^{(-)}Y_{1}^{2}+b_{22}^{(+)}X_{2}^{2}+b_{01}^{(+)}X_{0}X_{1}+b_{12}^{(+)}X_{1}X_{2}\right]=0\,.$
(4.1)

In order to rearrange this expression into the form (3.3), we systematically
use Riemann’s addition formula (A.8) in Appendix A.1 for genus $1$ theta
functions [60]. This allows us to systematically replace the basis functions
$X_{i}X_{j}$ and $Y_{i}Y_{j}$ by the degree 8 theta functions $\Phi_{k}$
(defined in (A.7)), as is explicitly shown in eq. (A.1) of Appendix A.1.
Indeed, using the shorthand notation:

$\phi_{i}:=\Phi_{i}(0,\tau),\quad\Phi_{i}:=\Phi_{i}(z_{1},\tau),\quad\theta_{1}:=\theta_{1}(z_{2};\rho),\quad\theta_{4}:=\theta_{4}(z_{2};\rho),$
(4.2)

we find that (4.1) can equivalently be written as

$\begin{split}&\left[(a_{00}^{(+)}\theta_{1}^{2}+b_{00}^{(+)}\theta_{4}^{2})\phi_{0}+\left((a_{11}^{(+)}-a_{11}^{(-)})\theta_{1}^{2}+(b_{11}^{(+)}-b_{11}^{(-)})\theta_{4}^{2}\right)\frac{\phi_{2}}{2}+(a_{22}^{(+)}\theta_{1}^{2}+b_{22}^{(+)}\theta_{4}^{2})\phi_{4}\right]\Phi_{0}\\\
&+\left[(a_{01}^{(+)}\theta_{1}^{2}+b_{01}^{(+)}\theta_{4}^{2})\phi_{1}+(a_{12}^{(+)}\theta_{1}^{2}+b_{12}^{(+)}\theta_{4}^{2})\phi_{3}\right]\Phi_{1}\\\
&+\left[\left((a_{11}^{(+)}+a_{11}^{(-)})\theta_{1}^{2}+(b_{11}^{(+)}+b_{11}^{(-)})\theta_{4}^{2}\right)\frac{\phi_{0}+\phi_{4}}{2}+(a_{02}^{(+)}\theta_{1}^{2}+b_{02}^{(+)}\theta_{4}^{2})\phi_{2}\right]\Phi_{2}\\\
&+\left[(a_{01}^{(+)}\theta_{1}^{2}+b_{01}^{(+)}\theta_{4}^{2})\phi_{3}+(a_{12}^{(+)}\theta_{1}^{2}+b_{12}^{(+)}\theta_{4}^{2})\phi_{1}\right]\Phi_{3}\\\
&+\left[(a_{00}^{(+)}\theta_{1}^{2}+b_{00}^{(+)}\theta_{4}^{2})\phi_{4}+\left((a_{11}^{(+)}-a_{11}^{(-)})\theta_{1}^{2}+(b_{11}^{(+)}-b_{11}^{(-)})\theta_{4}^{2}\right)\frac{\phi_{2}}{2}+(a_{22}^{(+)}\theta_{1}^{2}+b_{22}^{(+)}\theta_{4}^{2})\phi_{0}\right]\Phi_{4}=0\,.\end{split}$
(4.3)

In order to further write (4.3) in the form of (3.3), we need to identify the
coefficients of $\Phi_{i}$ (for $i\in\\{0,\ldots,4\\}$) with quantities
$\sum_{0\leq i,j\leq
M}\widetilde{T}_{i}(z_{2},\rho,\tau)\,\mathbb{M}_{ij}(\tau)$, where in
particular the $z_{2}$ dependent part needs to respect the correct assignment
of $SU(2)$ and $SU(1)$ characters: by identifying (some of) the coefficients
$a_{ij}^{\pm}$ and $b_{ij}^{(\pm)}$ we have to create combinations of the
$\theta_{1,4}^{2}$, which can be identified with the characters (3.4). A
priori, there are multiple possibilities to do this, however, not all of them
lead to a consistent structure (3.3) with one $SU(2)$ and 4 $SU(1)$
characters. However, the number of possibilities can be further reduced by
assuming additional symmetries:888In fact, by fixing some of the
$a_{ij}^{\pm}$ and $b_{ij}^{(\pm)}$ we have found $7$ independent candidates
of the form $\alpha\theta_{1}^{2}+\beta\theta_{4}^{2}$ for the $5$
$\widetilde{T}_{i}$ entering (3.3). We have carefully analysed all of these
possible solutions and found that the only consistent ones are compatible with
this assumption. Furthermore, we shall motivate this assumption by modular
considerations in the following Subsection 4.2. we assume that the modular
matrix $\mathbb{M}$ defined by (3.3) remains invariant under the exchanges

$\displaystyle\widetilde{T}_{0}\leftrightarrow\widetilde{T}_{4},\hskip
5.69046pt\Phi_{0}\leftrightarrow\Phi_{4}\,,$ and
$\displaystyle\widetilde{T}_{1}\leftrightarrow\widetilde{T}_{3},\hskip
5.69046pt\Phi_{1}\leftrightarrow\Phi_{3}\,.$ (4.4)

This is equivalent to stating that none of the Jacobi theta functions
multiplied by $a_{ij}^{\pm}$ or $b_{ij}^{\pm}$ with
$(ij)\in\\{(00)\,,(22)\,,(01)\,,(12)\\}$ can be formed into $SU(2)$ characters
such that

$b_{00}^{(+)}=b_{22}^{(+)}=b_{01}^{(+)}=b_{12}^{(+)}=0\,,$ (4.5)

and (4.3) takes the form

$\displaystyle\left[a_{00}^{(+)}\theta_{1}^{2}\,\phi_{0}+\left(\text{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{$(a_{11}^{(+)}-a_{11}^{(-)})\theta_{1}^{2}+(b_{11}^{(+)}-b_{11}^{(-)})\theta_{4}^{2}$}}}\right)\frac{\phi_{2}}{2}+a_{22}^{(+)}\theta_{1}^{2}\,\phi_{4}\right]\Phi_{0}+\left[a_{01}^{(+)}\,\phi_{1}+a_{12}^{(+)}\,\phi_{3}\right]\,\theta_{1}^{2}\,\Phi_{1}$
$\displaystyle+\left[\left(\text{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{$(a_{11}^{(+)}+a_{11}^{(-)})\theta_{1}^{2}+(b_{11}^{(+)}+b_{11}^{(-)})\theta_{4}^{2}$}}}\right)\frac{\phi_{0}+\phi_{4}}{2}+\left(\text{{\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}{$a_{02}^{(+)}\theta_{1}^{2}+b_{02}^{(+)}\theta_{4}^{2}$}}}\right)\phi_{2}\right]\Phi_{2}+\left[a_{01}^{(+)}\,\phi_{3}+a_{12}^{(+)}\,\phi_{1}\right]\theta_{1}^{2}\Phi_{3}$
$\displaystyle+\left[a_{00}^{(+)}\theta_{1}^{2}\,\phi_{4}+\left(\text{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{$(a_{11}^{(+)}-a_{11}^{(-)})\theta_{1}^{2}+(b_{11}^{(+)}-b_{11}^{(-)})\theta_{4}^{2}$}}}\right)\frac{\phi_{2}}{2}+a_{22}^{(+)}\theta_{1}^{2}\,\phi_{0}\right]\Phi_{4}=0\,.$
(4.6)

Among the terms highlighted in colour, one (combination) needs to assume the
role of the $SU(2)$ character. This can be achieved in two inequivalent and
consistent ways

  1. I

) Setting $a_{11}^{(+)}=b_{11}^{(+)}=a_{02}^{(+)}=b_{02}^{(+)}=0$ such that
the red and blue terms in (4.6) are identified and play the role of the
$SU(2)$ character, while the green term is eliminated:

$\displaystyle\left[a_{00}^{(+)}\theta_{1}^{2}\,\phi_{0}-\left(a_{11}^{(-)}\theta_{1}^{2}+b_{11}^{(-)}\theta_{4}^{2}\right)\frac{\phi_{2}}{2}+a_{22}^{(+)}\theta_{1}^{2}\,\phi_{4}\right]\Phi_{0}+\left[a_{01}^{(+)}\,\phi_{1}+a_{12}^{(+)}\,\phi_{3}\right]\theta_{1}^{2}\Phi_{1}$
$\displaystyle+\left[\left(a_{11}^{(-)}\theta_{1}^{2}+b_{11}^{(-)}\theta_{4}^{2}\right)\frac{\phi_{0}+\phi_{4}}{2}\right]\Phi_{2}+\left[a_{01}^{(+)}\,\phi_{3}+a_{12}^{(+)}\,\phi_{1}\right]\theta_{1}^{2}\Phi_{3}$
$\displaystyle+\left[a_{00}^{(+)}\theta_{1}^{2}\,\phi_{4}-\left(a_{11}^{(-)}\theta_{1}^{2}+b_{11}^{(-)}\theta_{4}^{2}\right)\frac{\phi_{2}}{2}+a_{22}^{(+)}\theta_{1}^{2}\,\phi_{0}\right]\Phi_{4}=0\,.$
(4.7)

The same form can also be achieved by identifying
$a_{02}^{(+)}=-a_{11}^{(+)}\frac{\phi_{0}+\phi_{4}}{\phi_{2}}$ and
$b_{02}^{(+)}=-b_{11}^{(+)}\frac{\phi_{0}+\phi_{4}}{\phi_{2}}$). The curve
(4.7) can indeed be cast into the form (3.3) with

$\widetilde{\mathbf{T}}^{T}=\begin{bmatrix}a_{00}^{(+)}\theta_{1}^{2}\\\
a_{01}^{(+)}\theta_{1}^{2}\\\
\tfrac{1}{2}(a_{11}^{(-)}\theta_{1}^{2}+b_{11}^{(-)}\theta_{4}^{2})\\\
a_{12}^{(+)}\theta_{1}^{2}\\\
a_{22}^{(+)}\theta_{1}^{2}\end{bmatrix},\quad\mathbb{M}_{\widehat{D}_{4},\rm
I}(\tau)=\begin{bmatrix}\phi_{0}&0&0&0&\phi_{4}\\\ 0&\phi_{1}&0&\phi_{3}&0\\\
-\phi_{2}&0&\phi_{0}+\phi_{4}&0&-\phi_{2}\\\ 0&\phi_{3}&0&\phi_{1}&0\\\
\phi_{4}&0&0&0&\phi_{0}\\\ \end{bmatrix}.$ (4.8)

In fact, this curve is the result obtained in [47, 49] since the terms
proportional to $\theta_{4}^{2}$ are
$\beta_{2}\left(\phi_{2}\Phi_{0}-(\phi_{0}+\phi_{4})\Phi_{2}+\phi_{2}\Phi_{4}\right)=2\beta_{2}Y_{1}^{2}$
which is indeed compatible with [49, eq. (3.86)].

  2. II

) Setting $a_{02}^{(+)}=b_{02}^{(+)}=0$ and identifying
$a_{11}^{(-)}=a_{11}^{(+)}$ and $b_{11}^{(-)}=b_{11}^{(+)}$, such that the red
and green terms are eliminated and the curve becomes

$\displaystyle\left[a_{00}^{(+)}\,\phi_{0}+a_{22}^{(+)}\,\phi_{4}\right]\theta_{1}^{2}\,\Phi_{0}+\left[a_{01}^{(+)}\,\phi_{1}+a_{12}^{(+)}\,\phi_{3}\right]\,\theta_{1}^{2}\,\Phi_{1}+\left[a_{11}^{(+)}\theta_{1}^{2}+b_{11}^{(+)}\theta_{4}^{2}\right](\phi_{0}+\phi_{4})\,\Phi_{2}$
$\displaystyle+\left[a_{01}^{(+)}\,\phi_{3}+a_{12}^{(+)}\,\phi_{1}\right]\theta_{1}^{2}\Phi_{3}+\left[a_{00}^{(+)}\,\phi_{4}+a_{22}^{(+)}\,\phi_{0}\right]\,\theta_{1}^{2}\,\Phi_{4}=0\,.$
(4.9)

This curve can be cast into the form (3.3) with

$\widetilde{\mathbf{T}}^{T}=\begin{bmatrix}a_{00}^{(+)}\theta_{1}^{2}\\\
a_{01}^{(+)}\theta_{1}^{2}\\\
a_{11}^{(+)}\theta_{1}^{2}+b_{11}^{(+)}\theta_{4}^{2}\\\
a_{12}^{(+)}\theta_{1}^{2}\\\
a_{22}^{(+)}\theta_{1}^{2}\end{bmatrix},\quad\mathbb{M}_{\widehat{D}_{4},\rm
II}(\tau)=\begin{bmatrix}\phi_{0}&0&0&0&\phi_{4}\\\ 0&\phi_{1}&0&\phi_{3}&0\\\
0&0&\phi_{0}+\phi_{4}&0&0\\\ 0&\phi_{3}&0&\phi_{1}&0\\\
\phi_{4}&0&0&0&\phi_{0}\\\ \end{bmatrix}.$ (4.10)

So far, both solutions are compatible with all symmetries of the
$(\widehat{D}_{4},1)$ theory and cannot be transformed into one another
through any obvious reparametrisations or symmetries. We shall see, however,
in Subsection 4.3 that case I) is compatible with known results of the 5
dimensional theory and thus is the correct SWC to describe the
$(\widehat{D}_{4},1)$ LST. Concerning case II), it is not clear whether it
describes a consistent LST (or any 6 dimensional gauge theory).

###  4.2 Modular properties

Before discussing the 5 dimensional limit of the $(\widehat{D}_{4},1)$ LST, we
first discuss modular properties of the two potential SWC characterised by
(4.8) and (4.10). Since modular transformations acting on the parameter $\rho$
(and coordinate $z_{2}$) only act on the elliptic Weyl characters
$\widetilde{T}_{i}$ (and are thus relatively straight-forward), we shall focus
on modular transformations acting on $\tau$.

From the perspective of the dual theory (_i.e._ the right quiver in Figure 2),
the gauge polynomials (3.2) transform covariantly under an $SL(2,\mathbb{Z})$
symmetry acting on the modular parameter $\tau$ (and the coordinate $z_{1}$),
owing to the transformations (A.14) of the Jacobi theta functions. However,
the picture is more complicated from the perspective of the
($\widehat{D}_{4},1)$ LST form in eq. (3.3), where in fact only a subgroup of
this $SL(2,\mathbb{Z})$ is manifestly realised. In the following we shall
therefore discuss only transformations under the congruence subgroup
$\Gamma_{0}(4)$,999This group also corresponds to the modular subgroup acting
on the coefficients in the change of basis between Weierstrass functions and
theta functions given in [62]. which is the largest subgroup we have found
that preserves the general structure (3.3) of the SWC. $\Gamma_{0}(4)$ is
generated by

$\displaystyle T=\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}\,,$ $\displaystyle
C=\begin{bmatrix}-1&0\\\ 0&-1\end{bmatrix}\,,$
$\displaystyle\widetilde{S}_{4}=C\cdot S\cdot T^{4}\cdot
S=\begin{bmatrix}1&0\\\ -4&1\end{bmatrix}\,,$ (4.11)

where the action of $S$ and $T$ on Jacobi theta functions is explained in
Appendix A.2 and the set $\mathbf{\Phi}:=\\{\Phi_{1},\ldots,\Phi_{4}\\}^{T}$
is invariant under $C$ by construction (which effectively acts as $z_{1}\to-
z_{1}$). We start by discussing the action of $T$ on $\mathbb{M}$ and
$\mathbf{\Phi}$. We can treat (4.8) and (4.10) simultaneously by introducing

$\displaystyle\epsilon=\left\\{\begin{array}[]{ll}-1&\text{for
I)}\,,\\\\[4.0pt] 0&\text{for II)}\,,\end{array}\right.$
$\displaystyle(\alpha_{2},\beta_{2})=\left\\{\begin{array}[]{ll}\left(\tfrac{a_{11}^{(-)}}{2},\tfrac{b_{11}^{(-)}}{2}\right)&\text{for
I)}\,,\\\\[8.0pt] \left(a_{11}^{(+)},b_{11}^{(+)}\right)&\text{for
II)}\,,\end{array}\right.$
$\displaystyle\mathbb{M}_{\widehat{D}_{4}}(\epsilon)=\left[\begin{smallmatrix}\phi_{0}&0&0&0&\phi_{4}\\\
0&\phi_{1}&0&\phi_{3}&0\\\
\epsilon\phi_{2}&0&\phi_{0}+\phi_{4}&0&\epsilon\phi_{2}\\\
0&\phi_{3}&0&\phi_{1}&0\\\ \phi_{4}&0&0&0&\phi_{0}\\\
\end{smallmatrix}\right]\,,$ (4.16)

and
$(\alpha_{0},\alpha_{1},\alpha_{3},\alpha_{4})=\left(a_{00}^{(+)},a_{01}^{(+)},a_{12}^{(+)},a_{22}^{(+)}\right)$,
such that the SWC can be written in the form

$\theta_{1}^{2}(z_{2};\rho)\begin{bmatrix}\alpha_{0}&\alpha_{1}&\alpha_{2}&\alpha_{3}&\alpha_{4}\end{bmatrix}\cdot\mathbb{M}_{\widehat{D}_{4}}(\epsilon)\cdot{\bf\Phi}+\theta_{4}^{2}(z_{2};\rho)\beta_{2}(\epsilon\phi_{2}(\Phi_{0}+\Phi_{4})+(\phi_{0}+\phi_{4})\Phi_{2})=0\,,$
(4.17)

covering both cases. Indeed, under the action of $T$, the matrix
$\mathbb{M}_{\widehat{D}_{4}}(\epsilon)$ transform as

$\mathbb{M}_{\widehat{D}_{4}}(\epsilon)\underset{T}{\longrightarrow}\begin{bmatrix}\phi_{0}&0&0&0&\phi_{4}\\\
0&e^{i\pi/4}\phi_{1}&0&e^{i\pi/4}\phi_{3}&0\\\ \epsilon
i\phi_{2}&0&i(\phi_{0}+\phi_{4})&0&\epsilon i\phi_{2}\\\
0&-e^{i\pi/4}\phi_{3}&0&-e^{i\pi/4}\phi_{1}&0\\\
\phi_{4}&0&0&0&\phi_{0}\end{bmatrix},$ (4.18)

which can be compensated by the transformations

$\displaystyle\alpha_{2}\to-i\alpha_{2}\,,$
$\displaystyle\beta_{2}\to-i\beta_{2}\,,$ $\displaystyle\alpha_{1}\to
e^{-i\pi/4}\alpha_{1}\,,$
$\displaystyle\alpha_{3}\to-e^{-i\pi/4}\alpha_{3}\,,.$ (4.19)

Therefore $T$ leaves the overall form of the SWC invariant for both cases I)
and II).

The generator $\widetilde{S}_{4}$ acts in the following way on
$\mathbb{M}_{\widehat{D}_{4}}(\epsilon)$

$\mathbb{M}_{\widehat{D}_{4}}(\epsilon)\underset{\widetilde{S}_{4}}{\longrightarrow}\begin{bmatrix}\phi_{4}&0&0&0&\phi_{0}\\\
0&\phi_{3}&0&\phi_{1}&0\\\
\epsilon\phi_{2}&0&\phi_{0}+\phi_{4}&0&\epsilon\phi_{2}\\\
0&\phi_{1}&0&\phi_{3}&0\\\ \phi_{0}&0&0&0&\phi_{4}\end{bmatrix}\,,$ (4.20)

which can be compensated by the permutations

$\displaystyle\alpha_{0}\longleftrightarrow\alpha_{4}\,,$ and
$\displaystyle\alpha_{1}\longleftrightarrow\alpha_{3}\,.$ (4.21)

Therefore also $\widetilde{S}_{4}$ leaves the overall form of the SWC
invariant for both cases I) and II), which are therefore both manifestly
invariant under $\Gamma_{0}(4)$.

We remark, that invariance under $\widetilde{S}_{4}$ (as explained above) only
applies to matrices $\mathbb{M}$ that remain invariant under the exchanges
(4.4). This property therefore serves as additional motivation for imposing
this condition when searching for viable forms of the SWC in the previous
Subsection.

###  4.3 Lower Dimensional Limit

So far, both solutions I) and II) have been compatible with all symmetries of
the $(\widehat{D}_{4},1)$ LST and are therefore completely equivalent.
However, we shall now demonstrate that only solution I) correctly reproduces
known results in the literature upon decompactification to 5 dimensions.

To describe the 5 dimensional limit, we shall use the same notation as in the
previous Subsection to treat the two cases I) and II) in parallel. First, we
take the limit $Q_{\rho}=e^{2\pi i\rho}\to 0$ (in which the
$(\widehat{D}_{4},1)$ LST is described in terms of a conformal field theory):
let

$\displaystyle w=e^{2\pi iz_{1}}\,,$ and $\displaystyle t=e^{2\pi iz_{2}}\,,$
(4.22)

and assume that101010Here we assume a scaling of all parameters of the theory,
which preserves as much structure as possible of the 6 dimensional theory.
Indeed, a weaker scaling of $\beta_{2}$ would imply that the lower dimensional
theory has a different gauge group structure.
$\beta_{2}=-Q_{\rho}^{-1/4}\widetilde{\beta_{2}}+\mathfrak{o}(Q_{\rho}^{-1/4})$
(while $\alpha_{i}=\widetilde{\alpha}_{i}+\mathfrak{o}(Q_{\rho}^{0})$), then
the leading order contribution of (4.17) takes the form

$(t-1)^{2}\begin{bmatrix}\widetilde{\alpha}_{0}&\widetilde{\alpha}_{1}&\widetilde{\alpha}_{2}&\widetilde{\alpha}_{3}&\widetilde{\alpha}_{4}\end{bmatrix}\cdot\mathbb{M}_{\widehat{D}_{4}}\cdot{\bf\Phi}+t\widetilde{\beta}_{2}(\epsilon\phi_{2}(\Phi_{0}+\Phi_{4})+(\phi_{0}+\phi_{4})\Phi_{2})=0\,.$
(4.23)

We next extract the leading order in the limit $Q_{\tau}=e^{2\pi i\tau}\to 0$,
which we shall perform in two steps: first we expand the
$\\{\Phi_{i}\\}_{i\in\\{0\dots M\\}}$ in powers of $Q_{\tau}$

$\displaystyle 0$
$\displaystyle=4\widetilde{\alpha}_{0}\left[1+\mathcal{O}(Q_{\tau})\right]+\widetilde{\alpha}_{1}\left[2(w+w^{-1})Q_{\tau}^{\frac{1}{8}}+\mathcal{O}(Q_{\tau}^{9/8})\right]+\widetilde{\alpha}_{2}\left[2(w^{2}+2\epsilon+w^{-2})Q_{\tau}^{\frac{1}{4}}+\mathcal{O}(Q_{\tau}^{5/4})\right]$
$\displaystyle\hskip
14.22636pt+\widetilde{\alpha}_{3}\left[2(w^{3}+w^{-3})Q_{\tau}^{\frac{5}{8}}+\mathcal{O}(Q_{\tau}^{13/8})\right]+\widetilde{\alpha}_{4}\left[4(w^{4}+w^{-4})Q_{\tau}+\mathcal{O}(Q_{\tau}^{2})\right]$
$\displaystyle\hskip
14.22636pt+\widetilde{\beta}_{2}\left[2(w^{2}+2\epsilon+w^{-2})Q_{\tau}^{\frac{1}{4}}+\mathcal{O}(Q_{\tau}^{5/4})\right]\,.$
(4.24)

In order to perform the full limit, we need to make certain assumptions
regarding the $Q_{\tau}$ dependence of the parameters
$\widetilde{\alpha}_{0,\ldots,4}$ and $\widetilde{\beta}_{2}$. As in the case
of the $Q_{\rho}\to 0$ limit, we shall assume a behaviour that preserves as
much of the 6 dimensional symmetries as possible. Concretely, we maintain the
exchange symmetries

$\displaystyle\widetilde{\alpha}_{0}\longleftrightarrow\widetilde{\alpha}_{4}\,,$
and
$\displaystyle\widetilde{\alpha}_{1}\longleftrightarrow\widetilde{\alpha}_{3}\,,$
(4.25)

which enforces that $\widetilde{\alpha}_{0}$ and $\widetilde{\alpha}_{4}$ (as
well as $\widetilde{\alpha}_{1}$ and $\widetilde{\alpha}_{3}$) have the same
leading behaviour in $Q_{\tau}$. However, this implies that the contribution
with $\widetilde{\alpha}_{4}$ in (4.24) (as well as the contribution with
$\widetilde{\alpha}_{3}$) is subleading and therefore drops out in the limit
$Q_{\tau}\to 0$. In order to ensure that none of the remaining terms drop out
in the limit (which we would interpret as a signal of a modification of the
symmetries in the lower dimensional theory), we assume the following scaling
behaviour

$\displaystyle\widetilde{\alpha}_{0}=\widetilde{\alpha}^{\prime}_{0}\,Q_{\tau}^{s}+\mathfrak{o}(Q_{\tau}^{s})\,,$
$\displaystyle\widetilde{\alpha}_{1}=\widetilde{\alpha}^{\prime}_{1}\,Q_{\tau}^{s-\frac{1}{8}}+\mathfrak{o}(Q_{\tau}^{s-1/8})\,,$
$\displaystyle\widetilde{\alpha}_{2}=\widetilde{\alpha}^{\prime}_{2}\,Q_{\tau}^{s-\frac{1}{4}}+\mathfrak{o}(Q_{\tau}^{s-1/4})\,,$
$\displaystyle\widetilde{\beta}_{2}=\widetilde{\beta}^{\prime}_{2}\,Q_{\tau}^{s-\frac{1}{4}}+\mathfrak{o}(Q_{\tau}^{s-1/4})\,.$
(4.26)

where $s\in\mathbb{R}$ is undetermined and shall not be important in the
following. We thus obtain the following limit for the SWC

$(t-1)^{2}\left(w^{2}+w^{-2}+\frac{\widetilde{\alpha}^{\prime}_{1}}{\widetilde{\alpha}^{\prime}_{2}}(w+w^{-1})+2\frac{\widetilde{\alpha}^{\prime}_{0}+\epsilon\widetilde{\alpha}^{\prime}_{2}}{\widetilde{\alpha}^{\prime}_{2}}\right)+t\frac{\widetilde{\beta}^{\prime}_{2}}{\widetilde{\alpha}^{\prime}_{2}}\left(w^{2}+2\epsilon+w^{-2}\right)=0.$
(4.27)

This curve can be directly compared with the SWC of the $Sp(N)$ with $N_{f}$
fundamentals discussed in [51]: let

$\displaystyle
M_{1}+M_{1}^{-1}+M_{2}+M_{2}^{-1}=-\frac{\widetilde{\alpha}^{\prime}_{1}}{\widetilde{\alpha}^{\prime}_{2}}\,,$
$\displaystyle(M_{1}+M_{1}^{-1})(M_{2}+M_{2}^{-1})+2=\frac{2\widetilde{\alpha}^{\prime}_{0}}{\widetilde{\alpha}^{\prime}_{2}}+2\epsilon,$
(4.28)

such that we can rewrite (4.27) in the language of [51]:

$\displaystyle
t^{2}p_{2}(w)+\Bigg{(}\frac{\widetilde{\beta}^{\prime}_{2}}{\widetilde{\alpha}^{\prime}_{2}}$
$\displaystyle(w^{2}+2\epsilon+w^{-2})+2\left(M_{1}+M_{1}^{-1}+M_{2}+M_{2}^{-1}\right)(w+w^{-1})$
$\displaystyle-2(M_{1}+M_{1}^{-1})(M_{2}+M_{2}^{-1})-4-2(w^{2}+w^{-2})\Bigg{)}t+p_{2}(w^{-1})=0\,,$
(4.29)

where we have defined

$p_{2}(w):=w^{-2}\prod_{i=1}^{2}(w-M_{i})(w-M_{i}^{-1})=p_{2}(w^{-1})\,.$
(4.30)

In order to match the contribution of order $t$ in (4.29) (with
$(1-M_{i}^{\pm})=(1-M_{i})(1-M_{i}^{-1})$)

$p_{1}(w)=-\frac{(w+1)^{2}\prod_{i=1}^{2}(1-M_{i}^{\pm})}{2w}+\frac{(w-1)^{2}\prod_{i=1}^{2}(1+M_{i}^{\pm})}{2w}+\frac{\beta_{2}}{\alpha_{2}}(w^{2}+2\epsilon+w^{-2})-2(w^{2}-2+w^{-2}),$
(4.31)

to [51] requires $\epsilon=-1$ and $\beta_{2}/\alpha_{2}=2+q^{-1}$, where $q$
is the exponentiated complex gauge coupling of $Sp(0)$. In this way, we can
indeed interpret the resulting SWC as that of a $Sp(0)$ with $SO(4)$ flavor
symmetry $5$d SCFT [64, 65] since $p_{2}(w)$ has the structure of the
corresponding $SO(4)$ gauge polynomial. This is indeed a consistent lower-
dimensional limit of the 6 dimensional $(\widehat{D}_{4},1)$ LST. Since this
fixes $\epsilon=-1$, this also suggests that the correct SWC of the latter
theory is in fact described by case I) of Subsection 4.1, _i.e._ eq. (4.8).
This is also compatible with the result obtained in [47, 49].

##  5 Seiberg-Witten Curve of $(\widehat{D}_{5},1)$

After having discussed in detail the construction of the SWC of the
$(\widehat{D}_{4},1)$ LST in the previous Section, we now focus on the
$(\widehat{D}_{5},1)$ theory, for which the result is not known in the
literature. Following the same steps as in the previous case, we show that we
find again a unique solution for the SWC.

###  5.1 Construction

We explain the construction of the $(\widehat{D}_{5},1)$ LST SWC following the
same general ansatz (3.5) outlined in Section 3.2. Using the change of basis
(A.1), we can reformulate this curve in the basis
$\\{\Phi_{i}\\}_{i\in\\{0\ldots 5\\}}$ of degree 10 theta functions

$\begin{split}&\left[(a_{00}^{(+)}\theta_{1}^{2}+b_{00}^{(+)}\theta_{4}^{2})\phi_{0}+\Xi_{11}^{-}\phi_{2}+\Xi_{22}^{-}\phi_{4}\right]\Phi_{0}+\left[(a_{01}^{(+)}\theta_{1}^{2}+b_{01}^{(+)}\theta_{4}^{2})\phi_{1}+\Xi_{12}^{-}\phi_{3}+\Xi_{22}^{+}\phi_{5}\right]\Phi_{1}+\\\
&\left[(a_{02}^{(+)}\theta_{1}^{2}+b_{02}^{(+)}\theta_{4}^{2})\phi_{2}+\Xi_{12}^{+}\phi_{4}+\Xi_{11}^{+}\phi_{5}\right]\Phi_{2}+\left[(a_{02}^{(+)}\theta_{1}^{2}+b_{02}^{(+)}\theta_{4}^{2})\phi_{3}+\Xi_{12}^{+}\phi_{1}+\Xi_{11}^{+}\phi_{0}\right]\Phi_{3}+\\\
&\left[(a_{01}^{(+)}\theta_{1}^{2}+b_{01}^{(+)}\theta_{4}^{2})\phi_{4}+\Xi_{12}^{-}\phi_{2}+\Xi_{22}^{+}\phi_{0}\right]\Phi_{4}+\left[(a_{00}^{(+)}\theta_{1}^{2}+b_{00}^{(+)}\theta_{4}^{2})\phi_{5}+\Xi_{11}^{-}\phi_{3}+\Xi_{22}^{-}\phi_{1}\right]\Phi_{5}=0\,,\end{split}$
(5.1)

where we have defined

$\Xi_{ij}^{\pm}:=\frac{a_{ij}^{(+)}\pm
a_{ij}^{(-)}}{2}\theta_{1}^{2}+\frac{b_{ij}^{(+)}\pm
b_{ij}^{(-)}}{2}\theta_{4}^{2}.$ (5.2)

To simplify the assignment of $SU(2)$ and $SU(1)$ Weyl characters, we shall
assume the symmetries111111We motivate this symmetry by the natural
$\mathbb{Z}_{2}$ action of the $SU(1)$ nodes of the quiver which are labelled
by $\widetilde{T}_{0,1,4,5}$.

$\displaystyle\widetilde{T}_{i}\leftrightarrow\widetilde{T}_{j},\hskip
5.69046pt\Phi_{i}\leftrightarrow\Phi_{j}\,,$ for
$\displaystyle(i,j)\in\\{(0,5)\,,(1,4)\\}\,,$ (5.3)

analogous to (4.4) in the case of the $(\widehat{D}_{4},1)$ LST. These
symmetries impose

$b_{00}^{(+)}=b_{01}^{(+)}=a_{02}^{(+)}=b_{02}^{(+)}=0.$ (5.4)

and we are left with three candidates ($\Xi_{11}^{\pm},\Xi_{12}^{\pm}$ and
$\Xi_{22}^{\pm}$) for two $SU(2)$ characters and two $SU(1)$ characters. The
natural choice that provides a symmetry among the $SU(1)$ nodes is to break
one of the $\Xi_{ij}^{\pm}$ (for $(ij)\in\\{(11),(12),(22)\\}$) into two
$SU(1)$ characters. There is a priori three possibilities, schematically:

$\displaystyle\Xi_{11}^{\pm}$ $\displaystyle\to SU(1)^{2}$
$\displaystyle\Xi_{11}^{\pm}$ $\displaystyle\to SU(2)$
$\displaystyle\Xi_{11}^{\pm}$ $\displaystyle\to SU(2)$ $\displaystyle
1)\quad\Xi_{12}^{\pm}$ $\displaystyle\to SU(2),$ $\displaystyle
2)\quad\Xi_{12}^{\pm}$ $\displaystyle\to SU(1)^{2},$ $\displaystyle
3)\quad\Xi_{12}^{\pm}$ $\displaystyle\to SU(2).$ $\displaystyle\Xi_{22}^{\pm}$
$\displaystyle\to SU(2)$ $\displaystyle\Xi_{22}^{\pm}$ $\displaystyle\to
SU(2)$ $\displaystyle\Xi_{22}^{\pm}$ $\displaystyle\to SU(1)^{2}$ (5.5)

All three choices lead to SWC that are compatible with all symmetries of the
$(\widehat{D}_{5},1)$ LST and (as we shall discuss in the next Subsection),
have interesting modular properties: we shall in particular see that there
exist modular transformations that transform the solutions 2) and 3) into one
another. Furthermore, we shall see in Subsection 5.3, that choice 3) provides
a (natural) $Q_{\rho},Q_{\tau}\to 0$ limit that is compatible with known 5
dimensional results in the literature. Since this is a strong indication that
3) is the correct 6 dimensional SWC of the $(\widehat{D}_{5},1)$ LST, we
provide here the explicit details of this solution:

$a_{11}^{(+)}=b_{11}^{(+)}=a_{22}^{(+)}=b_{22}^{(+)}=b_{12}^{(+)}=b_{12}^{(-)}=0,$
(5.6)

such that

$\widetilde{\mathbf{T}}^{T}=\begin{bmatrix}a_{00}^{(+)}\theta_{1}^{2}\\\
(a_{22}^{(+)}+a_{22}^{(-)})\theta_{1}^{2}\\\
a_{11}^{(-)}\theta_{1}^{2}+b_{11}^{(-)}\theta_{4}^{2}\\\
a_{12}^{(-)}\theta_{1}^{2}+b_{12}^{(-)}\theta_{4}^{2}\\\
a_{01}^{(+)}\theta_{1}^{2}\\\
(a_{22}^{(+)}-a_{22}^{(-)})\theta_{1}^{2}\end{bmatrix},\quad\mathbb{M}_{\widehat{D}_{5}}(\tau)=\begin{bmatrix}\phi_{0}&0&0&0&0&\phi_{5}\\\
0&\phi_{5}&0&0&\phi_{0}&0\\\ -\phi_{2}&0&\phi_{0}&\phi_{5}&0&-\phi_{3}\\\
0&-\phi_{3}&\phi_{4}&\phi_{1}&-\phi_{2}&0\\\ 0&\phi_{1}&0&0&\phi_{4}&0\\\
\phi_{4}&0&0&0&0&\phi_{1}\end{bmatrix}.$ (5.7)

###  5.2 Modular properties

In the same way as in the $(\widehat{D}_{4},1)$ LST in Subsection 4.2, we
indicate the action of the generators of a $\Gamma_{0}(5)$ modular group
(acting on the modular parameter $\tau$ and coordinate $z_{1}$) on the
solutions (5.5). The congruence subgroup $\Gamma_{0}(5)\subset
SL(2,\mathbb{Z})$ is generated by:

$T=\begin{bmatrix}1&1\\\
0&1\end{bmatrix},\quad\widetilde{S}_{5,1}=ST^{2}ST^{3}ST^{2}=\begin{bmatrix}-3&-5\\\
5&8\end{bmatrix},\quad\widetilde{S}_{5,2}=ST^{5}S=\begin{bmatrix}-1&0\\\
5&-1\end{bmatrix}.$ (5.8)

The overall form of $\mathbb{M}$ is left untouched by the action of $T$ and
$\widetilde{S}_{5,2}$. However, $\widetilde{S}_{5,2}\circ\widetilde{S}_{5,1}$
acts non-trivially on $\bf\Phi$ and $\mathbb{M}_{\widehat{D}_{5}}$ given in
(5.7) (see (A.13)), in particular

$\begin{bmatrix}\phi_{0}&0&0&0&0&\phi_{5}\\\ 0&\phi_{5}&0&0&\phi_{0}&0\\\
-\phi_{2}&0&\phi_{0}&\phi_{5}&0&-\phi_{3}\\\
0&-\phi_{3}&\phi_{4}&\phi_{1}&-\phi_{2}&0\\\ 0&\phi_{1}&0&0&\phi_{4}&0\\\
\phi_{4}&0&0&0&0&\phi_{1}\end{bmatrix}\underset{\widetilde{S}_{5,2}\circ\widetilde{S}_{5,1}}{\longrightarrow}\begin{bmatrix}\phi_{0}&0&0&0&0&\phi_{5}\\\
0&0&\phi_{0}&\phi_{5}&0&0\\\ -\phi_{4}&\phi_{5}&0&0&\phi_{0}&-\phi_{1}\\\
0&\phi_{3}&-\phi_{4}&-\phi_{1}&\phi_{2}&0\\\ 0&0&\phi_{2}&\phi_{3}&0&0\\\
\phi_{2}&0&0&0&0&\phi_{3}\end{bmatrix}$ (5.9)

This matrix, however, corresponds precisely to the solution called in 2) in
(5.5), indicating that the two are not independent, but are related through a
non-trivial modular transformation (and are thus simply equivalent
presentations of the same curve.)

###  5.3 Lower dimensional limit

In order to compute the 5 dimensional limit of the SWC characterised by (5.7),
we follow the same double limit $Q_{\rho}\to 0$ and $Q_{\tau}\to 0$ as
explained in Section 4.3 in the case of the $(\widehat{D}_{4},1)$ LST, which
preserves as much of the higher dimensional symmetries as possible. In the
interest of brevity, we shall only display the final result, which we can cast
in the form

$\begin{split}(t-1)^{2}&\left[(w^{3}+w^{-3})+\frac{\alpha_{2}}{\alpha_{3}}(w^{2}+w^{-2})+\left(\frac{\alpha_{1}}{\alpha_{3}}-1\right)(w+w^{-1})+2\frac{\alpha_{0}}{\alpha_{3}}-2\frac{\alpha_{2}}{\alpha_{3}}\right]\\\
&\quad\quad+\left[\frac{\beta_{3}}{\alpha_{3}}(w^{3}-(w+w^{-1})+w^{-3})+\frac{\beta_{2}}{\alpha_{3}}(w^{2}-2+w^{-2})\right]t=0,\end{split}$
(5.10)

In order to compare to the results of [51], we define

$\begin{split}&\frac{\alpha_{2}}{\alpha_{3}}=-(M_{1}+M_{1}^{-1}+M_{2}+M_{2}^{-1}+M_{3}+M_{3}^{-1}),\\\
&\frac{\alpha_{1}}{\alpha_{3}}=(1+M_{1}M_{2}+M_{1}M_{3}+M_{2}M_{3})(1+(M_{1}M_{2})^{-1}+(M_{1}M_{3})^{-1}+(M_{2}M_{3})^{-1}),\\\
&\frac{\alpha_{0}}{\alpha_{3}}=-\frac{(1+M_{1}^{2})(1+M_{2}^{2})(1+M_{3}^{2})}{2M_{1}M_{2}M_{3}}.\end{split}$
(5.11)

such that the lower dimensional curve can be written in the form

$t^{2}p_{2}(w)+\left[\frac{\beta_{3}}{\alpha_{3}}(w^{3}-(w+w^{-1})+w^{-3})+\frac{\beta_{2}}{\alpha_{3}}(w^{2}-2+w^{-2})-2p_{2}(w)\right]t+p_{2}(w)=0,$
(5.12)

with

$p_{2}(w)=\prod_{i=1}^{3}(w-(M_{i}+M_{i}^{-1})+w^{-1}).$ (5.13)

Following [51], we redefine $t\to p_{2}(w)^{-1}t$, the SWC becomes:

$\begin{split}t^{2}+\Big{[}\left(\frac{\beta_{3}}{\alpha_{3}}-2\right)&(w^{3}+w^{-3})+\left(\frac{\beta_{2}-2\alpha_{2}}{\alpha_{3}}\right)(w^{2}+w^{-2})\\\
&+\left(-\chi_{c}+2-\frac{\beta_{3}}{\alpha_{3}}\right)(w+w^{-1})+2\left(\chi_{s}-\frac{\beta_{2}-2\alpha_{2}}{\alpha_{3}}\right)\Big{]}t+p_{2}(w)^{2}=0,\end{split}$
(5.14)

where we used the definition of $SO(2N_{f})$ characters for spinor and
conjugate spinor representation [65]:

$\chi_{s}:=\frac{1}{2}\left(\prod_{i=1}^{N_{f}}\left(M_{i}^{-\frac{1}{2}}+M_{i}^{\frac{1}{2}}\right)+\prod_{i=1}^{N_{f}}\left(M_{i}^{-\frac{1}{2}}-M_{i}^{\frac{1}{2}}\right)\right),$
(5.15a)
$\chi_{c}:=\frac{1}{2}\left(\prod_{i=1}^{N_{f}}\left(M_{i}^{-\frac{1}{2}}+M_{i}^{\frac{1}{2}}\right)-\prod_{i=1}^{N_{f}}\left(M_{i}^{-\frac{1}{2}}-M_{i}^{\frac{1}{2}}\right)\right).$
(5.15b)

We can match the result of [51, eq. (4.31)] by identifying
$U_{2}=q(\beta_{2}-2\alpha_{2})/\alpha_{3}$ and
$\beta_{3}/\alpha_{3}=2+q^{-1}$. Furthermore, we impose $M_{3}=M_{3}^{-1}=-1$,
which has the interpretation of an exponentiated complex mass parameter, which
leads to $\chi_{c}=-\chi_{s}$. This is a natural choice since in the
$Q_{\tau}\to 0$ limit the original $SO(10)$ group is broken down to $SO(5)$
which involves two Coulomb branch moduli, there is therefore a spurious
parameter in (5.11). Thus we obtain:

$t^{2}+q^{-1}\left((w^{3}+w^{-3})+U_{2}(w^{2}+w^{-2})+(q\chi_{s}-1)(w+w^{-1})-2(q\chi_{c}+U_{2})\right)t+p_{2}(w)^{2}=0.$
(5.16)

This result matches the SWC of the $Sp(1)$ with $SO(5)$ flavor symmetry $5$d
SCFT [64, 65] as a specialisation of $Sp(1)$ with $6$ flavours. This therefore
serves as a strong consistency check, that (5.7) provides the correct SWC of
the $(\widehat{D}_{5},1)$ LST.

##  6 Generalisation to the $(\widehat{D}_{M},1)$ LSTs

In the same manner as for the $(\widehat{D}_{4},1)$ and $(\widehat{D}_{5},1)$
theories, we can analyse $(\widehat{D}_{M},1)$ LSTs for generic $M\geq 4$.
Here we only indicate a general pattern we have found by studying explicit
cases up to $M=12$: In the following we simply state the form of the modular
matrix $\mathbb{M}$ and $\widetilde{\mathbf{T}}$.

For $M$ odd, we distinguish the cases

  * •

$M=4M^{\prime}+1$

$\widetilde{\mathbf{T}}^{T}=\begin{bmatrix}\alpha_{0}\theta_{1}^{2}&\alpha_{1}\theta_{1}^{2}&\alpha_{2}\theta_{1}^{2}+\beta_{2}\theta_{4}^{2}&\cdots&\alpha_{4M^{\prime}-1}\theta_{1}^{2}+\beta_{4M^{\prime}-1}\theta_{4}^{2}&\alpha_{4M^{\prime}}\theta_{1}^{2}&\alpha_{4M^{\prime}+1}\theta_{1}^{2}\end{bmatrix}\\!,$
(6.1)
$\mathbb{M}_{\widehat{D}_{4M^{\prime}+1}}=\left[\\!\\!\begin{smallmatrix}\phi_{0}&0&0&\cdots&0&0&\cdots&0&0&\phi_{4M^{\prime}+1}\\\
0&\phi_{4M^{\prime}+1}&0&\cdots&0&0&\cdots&0&\phi_{0}&0\\\
-\phi_{2}&0&\phi_{0}&\cdots&0&0&\cdots&\phi_{4M^{\prime}+3}&0&-\phi_{4M^{\prime}-1}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots\\\
-\phi_{2M^{\prime}+1}&0&0&\cdots&\phi_{0}&\phi_{4M^{\prime}+1}&\cdots&0&0&-\phi_{2M^{\prime}}\\\
0&-\phi_{2M^{\prime}}&0&\cdots&\phi_{4M^{\prime}}&\phi_{1}&\cdots&0&-\phi_{2M^{\prime}+1}&0\\\
\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&-\phi_{4M^{\prime}-1}&\phi_{4M^{\prime}+2}&\cdots&0&0&\cdots&\phi_{1}&-\phi_{2}&0\\\
0&\phi_{1}&0&\cdots&0&0&\cdots&0&\phi_{4M^{\prime}}&0\\\
\phi_{4M^{\prime}}&0&0&\cdots&0&0&\cdots&0&0&\phi_{1}\end{smallmatrix}\\!\\!\right].$
(6.2)

  * •

$M=4M^{\prime}+3$:

$\widetilde{\mathbf{T}}^{T}=\begin{bmatrix}\alpha_{0}\theta_{1}^{2}&\alpha_{1}\theta_{1}^{2}&\alpha_{2}\theta_{1}^{2}+\beta_{2}\theta_{4}^{2}&\cdots&\alpha_{4M^{\prime}+1}\theta_{1}^{2}+\beta_{4M^{\prime}+1}\theta_{4}^{2}&\alpha_{4M^{\prime}+2}\theta_{1}^{2}&\alpha_{4M^{\prime}+3}\theta_{1}^{2}\end{bmatrix}\\!,$
(6.3)
$\mathbb{M}_{\widehat{D}_{4M^{\prime}+3}}=\left[\\!\\!\begin{smallmatrix}\phi_{0}&0&0&\cdots&0&0&\cdots&0&0&\phi_{4M^{\prime}+3}\\\
0&\phi_{4M^{\prime}+3}&0&\cdots&0&0&\cdots&0&\phi_{0}&0\\\
-\phi_{2}&0&\phi_{0}&\cdots&0&0&\cdots&\phi_{4M^{\prime}+3}&0&-\phi_{4M^{\prime}+1}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\iddots&\vdots&\vdots&\vdots\\\
-\phi_{2M^{\prime}+2}&0&0&\cdots&\phi_{4M^{\prime}+3}&\phi_{0}&\cdots&0&0&-\phi_{2M^{\prime}+1}\\\
0&-\phi_{2M^{\prime}+1}&0&\cdots&\phi_{1}&\phi_{4M^{\prime}+2}&\cdots&0&-\phi_{2M^{\prime}+2}&0\\\
\vdots&\vdots&\vdots&\iddots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&-\phi_{4M^{\prime}+1}&\phi_{4M^{\prime}+2}&\cdots&0&0&\cdots&\phi_{1}&-\phi_{2}&0\\\
0&\phi_{1}&0&\cdots&0&0&\cdots&0&\phi_{4M^{\prime}+2}&0\\\
\phi_{4M^{\prime}+2}&0&0&\cdots&0&0&\cdots&0&0&\phi_{1}\end{smallmatrix}\\!\\!\right].$
(6.4)

while for $M$ even we distinguish

  * •

$M=4M^{\prime}$

$\widetilde{\mathbf{T}}=\begin{bmatrix}\alpha_{0}\theta_{1}^{2}&\alpha_{1}\theta_{1}^{2}&\alpha_{2}\theta_{1}^{2}+\beta_{2}\theta_{4}^{2}&\cdots&\alpha_{4M^{\prime}-2}\theta_{1}^{2}+\beta_{4M^{\prime}-2}\theta_{4}^{2}&\alpha_{4M^{\prime}-1}\theta_{1}^{2}&\alpha_{4M^{\prime}}\theta_{1}^{2}\end{bmatrix}\\!,$
(6.5)
$\mathbb{M}_{\widehat{D}_{4M^{\prime}}}=\left[\\!\\!\begin{smallmatrix}\phi_{0}&0&0&\cdots&0&0&\cdots&0&\phi_{4M^{\prime}}\\\
0&\phi_{1}&0&\cdots&0&\cdots&0&\phi_{4M^{\prime}-1}&0\\\
-\phi_{2}&0&\phi_{2}&\cdots&0&\cdots&\phi_{4M^{\prime}}&0&-\phi_{4M^{\prime}-2}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\iddots&\vdots&\vdots&\vdots\\\
-\phi_{2M^{\prime}}&0&0&\cdots&\phi_{0}+\phi_{4M^{\prime}}&\cdots&0&0&-\phi_{2M^{\prime}}\\\
\vdots&\vdots&\vdots&\iddots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
-\phi_{4M^{\prime}-2}&0&\phi_{4M^{\prime}}&\cdots&0&\cdots&\phi_{0}&0&-\phi_{2}\\\
0&\phi_{4M^{\prime}-1}&0&\cdots&0&\cdots&0&\phi_{1}&0\\\
\phi_{4M^{\prime}}&0&0&\cdots&0&\cdots&0&0&\phi_{0}\end{smallmatrix}\\!\\!\right],$
(6.6)

  * •

$M=4M^{\prime}+2$

$\widetilde{\mathbf{T}}=\begin{bmatrix}\alpha_{0}\theta_{1}^{2}&\alpha_{1}\theta_{1}^{2}&\alpha_{2}\theta_{1}^{2}+\beta_{2}\theta_{4}^{2}&\cdots&\alpha_{4M^{\prime}}\theta_{1}^{2}+\beta_{4M^{\prime}}\theta_{4}^{2}&\alpha_{4M^{\prime}+1}\theta_{1}^{2}&\alpha_{4M^{\prime}+2}\theta_{1}^{2}\end{bmatrix}\\!,$
(6.7)
$\mathbb{M}_{\widehat{D}_{4M^{\prime}+2}}=\left[\\!\\!\begin{smallmatrix}\phi_{0}&0&0&\cdots&0&0&\cdots&0&\phi_{4M^{\prime}}\\\
0&\phi_{1}&0&\cdots&0&\cdots&0&\phi_{4M^{\prime}-1}&0\\\
-\phi_{2}&0&\phi_{2}&\cdots&0&\cdots&\phi_{4M^{\prime}}&0&-\phi_{4M^{\prime}-2}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\iddots&\vdots&\vdots&\vdots\\\
0&-\phi_{2M^{\prime}+1}&0&\cdots&\phi_{1}+\phi_{4M^{\prime}+1}&\cdots&0&-\phi_{2M^{\prime}+1}&0\\\
\vdots&\vdots&\vdots&\iddots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
-\phi_{4M^{\prime}-2}&0&\phi_{4M^{\prime}}&\cdots&0&\cdots&\phi_{0}&0&-\phi_{2}\\\
0&\phi_{4M^{\prime}-1}&0&\cdots&0&\cdots&0&\phi_{1}&0\\\
\phi_{4M^{\prime}}&0&0&\cdots&0&\cdots&0&0&\phi_{0}\end{smallmatrix}\\!\\!\right].$
(6.8)

Explicit examples for the matrices $\mathbb{M}$ for $M\in\\{6\ldots 9\\}$ can
be found in Appendix B.

####

These four classes of result can be expressed is a unifying manner from the
dual perspective of $\widehat{A}_{1}$ base theory (_i.e._ the right quiver in
Figure 2). The results translate to the elliptic gauge polynomials of $SO(2M)$
and $Sp(M^{\prime})$ with $M^{\prime}=M-4$ and can be understood as a direct
generalisation of the case $M=4$ (3.1). Elliptic gauge polynomials are finite
products of Jacobi theta functions and can be systematically understood as a
Higgsing of a $SU(N)$ elliptic gauge polynomial [37]:

$\displaystyle
s_{SU(N)}(z_{1};\tau)=\prod_{i=1}^{N}\frac{\theta_{1}(z_{1}-a_{i};\tau)}{\theta_{1}(z_{1};\tau)},\quad\text{and}\quad\langle\varphi\rangle=\left(a_{1},\ldots,a_{N}\right),$
(6.9)

$\langle\varphi\rangle$ corresponds to the gauge holonomies of $SU(N)$. For
$SO(2M)$, we obtain:

$F_{1}(z_{1};\tau)=s_{SO(2M)}(z_{1},\tau)=\prod_{i=1}^{M}\frac{\theta_{1}(z_{1}-a_{i}^{SO};\tau)\theta_{1}(z_{1}+a_{i}^{SO};\tau)}{\theta_{1}^{2}(z_{1};\tau)}\,.$
(6.10)

which corresponds to a Higgsing of $SU(2M)$ gauge polynomial given by:

$\langle\varphi\rangle=\left(\pm a_{1}^{SO},\ldots,\pm a_{M}^{SO}\right).$
(6.11)

For $Sp(M^{\prime})$, we have:

$F_{2}(z_{1};\tau)=s_{Sp(M^{\prime})}(z_{1},\tau)=\frac{\theta_{1}^{2}(2z_{1};\tau)}{\theta_{1}^{8}(z_{1};\tau)}\prod_{i=1}^{M^{\prime}}\frac{\theta_{1}(z_{1}-a_{i}^{Sp};\tau)\theta_{1}(z_{1}+a_{i}^{Sp};\tau)}{\theta_{1}^{2}(z_{1};\tau)}\,,$
(6.12)

which corresponds to a Higgsing of $SU(2M^{\prime}+8)$ gauge polynomial given
by:

$\langle\varphi\rangle=\left(0,0,\pm\frac{1}{2},\pm\frac{\tau}{2},\pm\frac{1+\tau}{2},\pm
a_{1}^{Sp},\ldots,\pm a_{M^{\prime}}^{Sp}\right).$ (6.13)

We remark that $F_{2}$ can be seen as a specialisation of $F_{1}$ by using
(A.5) and identifying
$\left(a_{1}^{SO},\ldots,a_{4}^{SO}\right)=\left(0,\frac{1}{2},\frac{\tau}{2},\frac{\tau+1}{2}\right)$
(which correspond to the so-called discrete holonomies). This result is in
line with observations in lower dimensions. In $5$ dimensions, $SO$ or $Sp$
gauge theories with flavors can be engineered from Type IIB String Theory
through $5$-brane webs and $O7^{\pm}$ orientifold planes [51]. Typically, $5$d
$\mathcal{N}=1$ $SO$ theories are constructed using $O7^{+}$-planes while $Sp$
theories arise from constructions involving $O7^{-}$. In [51], it was pointed
out that an $O7^{+}$ plane admits an equivalent description as an $O7^{-}$
along with $8D7\big{|}_{\rm frozen}$ i.e. with no moduli attached to them. We
interpret the relation between $F_{1}$ and $F_{2}$ as a $6$d manifestation of
this observation.

##  7 Conclusion

In this paper we have constructed the Seiberg-Witten Curves (SWCs) for a
family of Little String Theories (LSTs) engineered by a single M5-brane
probing a transverse (binary) dihedral orbifold geometry
$\mathbb{R}_{\perp}^{4}/\mathbb{BD}_{M-2}$ (with $M\geq 4$). The six-
dimensional world-volume theory on the M5-brane can be described by a quiver
gauge theory, as shown in the left part of Figure 2: the quiver is in the form
of the Dynkin diagram of $\widehat{D}_{M}$, with nodes of type $SU(2)$ and
$SU(1)$ respectively and matter in the bifundamental representations. This
theory, denoted $(\widehat{D}_{M},1)$ in this work, was shown to be dual [46]
(see also [47] for the case $M=4$) to a circular quiver gauge theory (_i.e._
with quiver group $\widehat{A}_{0}$) with one node of type $Sp(M-4)$ and one
node of type $SO(2M)$ and bifundamental matter, as shown in the right part of
Figure 2. While a form for the SWC of the theory $(\widehat{D}_{M},1)$ with
$M=4$ was written down in [47, 49], a construction for general $M$ is not
known in the literature.

In (3.5) we provide a general ansatz for the SWC for generic $M\geq 4$ that is
compatible with important symmetries and dualities, notably a decomposition in
the form of equation (3.3) in terms of (elliptic) Weyl characters
$\widetilde{\mathbf{T}}$ associated with the different gauge nodes, a basis
$\\{\Phi_{j}\\}_{j\in\\{0,\ldots,M\\}}$ of $\mathbb{Z}_{2}$-even theta
functions of degree $2M$ (see (A.7)) and a modular matrix $\mathbb{M}_{ij}$.
The ansatz (3.5) depends on more than the $2M-2$ expected parameters, such
that further conditions need to be imposed to determine the SWC: indeed,
further parameters are fixed by demanding that the curve can also be cast in
the form (3.1), thus reflecting the dual description of the
$(\widehat{D}_{M},1)$ theory in terms of the $Sp(M-4)$–$SO(2M)$ quiver gauge
theory explained above. Finally, by assuming a natural dimensional reduction
to 5 dimensions, and comparison with known results of the lower dimensional
SWC [51] further puts restrictions on the possible solutions. In this way, we
have found in Sections 4 and 5 respectively, a unique solution for the SWC of
the $(\widehat{D}_{4},1)$ and $(\widehat{D}_{5},1)$ respectively. In the
former case, the solution precisely agrees with the curve presented in [47,
49], while the latter constitutes a novel result.

We have systematically analysed in a similar fashion all $(\widehat{D}_{M},1)$
LSTs up to $M=12$, which present a number of clear structured patterns (see
Appendix B for the modular matrices up to $M=10$). These have lead us to
conjecture the general form of the $(\widehat{D}_{M},1)$ SWC, as explained in
Section 6. Furthermore, using the dual form of the SWC, we also find the form
(6.10) and (6.12) respectively, of the gauge potentials $F_{1,2}$ entering
into (3.1).

By determining the form of the SWC of the $(\widehat{D}_{M},1)$ theory for
generic $M$, our work casts further light on the $\widehat{D}$-type LSTs,
which are far less studied than their $\widehat{A}$-type counterparts [37,
36]. For example, the relation between the two (equivalent) presentations
(3.3) and (3.1) encodes details on the duality between the
$(\widehat{D}_{M},1)$ and the $Sp(M-4)$–$SO(2M)$ theory.

Our results may also hint to further $\widehat{D}$-type orbifold theories in 6
dimensions. Indeed, in the case of $M=4$ (Section 4) and $M=5$ (Section 5),
the final form of the SWC could only be determined after taking a (natural)
decompactification limit and comparison with lower dimensional results. In
both cases, we found alternative forms of the curve ((4.10) and (5.5)
respectively) that are compatible with all symmetries of the problem, and only
whose natural dimensional reduction leads to a different curve than expected
in the lower-dimensional theory [51]. It is an interesting question, whether
these curves nevertheless correspond to viable SWC of – possibly deformed –
theories in 6 dimensions and whether they are dual to the
$(\widehat{D}_{M},1)$ LSTs. If this is the case, there may exist more
intricate limits than the ones considered in Sections 4.3 and 5.3,
respectively, that makes them compatible with the known lower dimensional
results. We have seen one example of such a scenario being realised, namely
the case entitled 2) in (5.5), which we showed to be related to the case 3) by
a modular transformation (an element of the congruence subgroup
$\Gamma_{0}(5)$). Since the latter SWC in turn reduces by a natural limit to a
known 5 dimensional SWC, this implies that there exists also a (transformed)
limit, that reduces the case 2) to a viable 5 dimensional result. Similar
mechanisms can a priori not be ruled out for all other 6 dimensional SWCs that
we found to be compatible with all higher dimensional symmetries, thus leaving
the question whether they indeed realise viable gauge theories. We leave this
question for future research.

Let us remark another potential application of our results. It has been
established that the Coulomb branch of the supersymmetric vacua of eight
supercharge theory is identified with the phase space of the algebraic
integrable system [38, 39]. In this context, the SWC is identified with the
spectral curve of the corresponding integrable system. Since the integrable
system associated with the $\widehat{A}$-type LST is known to be the double
elliptic system [66, 67, 68], the SWC discussed in this paper would be
identified with a spectral curve of new $\widehat{D}$-type double elliptic
integrable systems.

A further interesting question concerns the generalisation of our work to more
general LSTs. Indeed, here we have only considered theories engineered by a
single M5 brane probing a dihedral orbifold geometry. It would be interesting
to study the generalisation to multiple M5-branes, _i.e._ LSTs of the type
$(\widehat{D}_{M},N)$ for $N\geq 1$. For $N>1$, the existence of a duality
generalising Figure 2 is not evident and it will be interesting to analyse the
structure of the SWC in this case. Finally, instead of considering a
transverse dihedral orbifold, it would be interesting to consider other
transverse group actions, which lead to the $\widehat{E}$-type LSTs.

## Acknowledgements

We are very grateful to B. Haghighat for useful correspondence on the SWC of
the $(\widehat{D}_{4},1)$ LST. BF and SH are grateful to the Quantum Theory
Center ($\hbar$QTC, D-IAS, University of Southern Denmark) for kind
hospitality when this work was being finalised. BF and TK also acknowledge the
Galileo Galilei Institute for Theoretical Physics (GGI) for support and
hospitality. The work of TK was supported by CNRS through MITI
interdisciplinary programs, EIPHI Graduate School (No. ANR-17-EURE-0002) and
Bourgogne-Franche-Comté region.

##  Appendix A Modular functions and properties

In this Appendix, we define modular functions and give some of their
properties that are useful in the computation of SWC.

###  A.1 Definitions and properties

We use the notation $\Theta$ for genus $2$ theta functions and $\theta$ for
genus $1$ theta functions. Genus $2$ theta functions are defined as:

$\Theta\\!\begin{bmatrix}{\delta}\\\
{\epsilon}\end{bmatrix}\\!(z|\Omega)=\sum_{m\in\mathbb{Z}^{2}}\exp\left(i\pi(m+\delta)^{T}\cdot\Omega\cdot(m+\delta)+2i\pi(z+\epsilon)^{T}(m+\delta)\right),$
(A.1)

and verify:

$\begin{split}\Theta\\!\begin{bmatrix}{\delta}\\\
\mathbf{0}\end{bmatrix}\\!(z|A\cdot\Omega\cdot
A^{T})=\Theta\\!\begin{bmatrix}{A\cdot\delta}\\\
\mathbf{0}\end{bmatrix}\\!(z\cdot A^{-1}|\Omega)\,.\end{split}$ (A.2)

We define the standard Jacobi theta function as:

$\displaystyle\theta_{1}(z;\tau)=\theta\\!\begin{bmatrix}1/2\\\
1/2\end{bmatrix}\\!(z|\tau),$
$\displaystyle\theta_{2}(z;\tau)=\theta\\!\begin{bmatrix}1/2\\\
0\end{bmatrix}\\!(z|\tau),$
$\displaystyle\theta_{3}(z;\tau)=\theta\\!\begin{bmatrix}0\\\
0\end{bmatrix}\\!(z|\tau),$
$\displaystyle\theta_{4}(z;\tau)=\theta\\!\begin{bmatrix}0\\\
1/2\end{bmatrix}\\!(z|\tau),$ (A.3)

that are related by shifts in $\mathbb{Z}/2\oplus\tau\mathbb{Z}/2$:

$\displaystyle\theta_{2}(z;\tau)=\theta_{1}\left(z+\frac{1}{2};\tau\right),$
$\displaystyle\theta_{3}(z;\tau)=\theta_{4}\left(z+\frac{1}{2};\tau\right),$
$\displaystyle\theta_{1}(z;\tau)=-ie^{i\pi
z}e^{i\pi\frac{\tau}{4}}\theta_{4}\left(z+\frac{\tau}{2};\tau\right),$ (A.4)

and verify:

$\theta_{1}(z;\tau)\theta_{2}(z;\tau)\theta_{3}(z;\tau)\theta_{4}(z;\tau)=\frac{\theta_{1}(2z;\tau)}{2\theta_{2}(0;\tau)\theta_{3}(0;\tau)\theta_{4}(0;\tau)}\,.$
(A.5)

In this work, we are interested in basis of modular forms that arise as
sections of a degree $r$ line bundle. Elements of such basis
$\\{\varphi_{i}(z;\tau)\\}_{i\in I}$ will be denoted as degree $r$ modular
forms and have the property that each
$\varphi_{i}(z;\tau)/\theta_{1}(z;\tau)^{r}$ has a pole of maximal order $r$
in $z$ in the $\tau\to 0$ limit, for all $i\in I$. For example, we define
$X_{i}$ $(Y_{i})$, the elements of $\mathbb{Z}_{2}$ even (odd) basis of degree
$M$ modular forms:

$\begin{split}&X_{i}(z_{1},\tau)=\theta\\!\begin{bmatrix}\frac{i}{M}\\\
0\end{bmatrix}\\!(Mz_{1}|M\tau)+\theta\\!\begin{bmatrix}\frac{M-i}{M}\\\
0\end{bmatrix}\\!(Mz_{1}|M\tau)\,,\quad i\in\\{0\ldots\lfloor
M/2\rfloor\\}\,,\\\ &Y_{i}(z_{1},\tau)=\theta\\!\begin{bmatrix}\frac{i}{M}\\\
0\end{bmatrix}\\!(Mz_{1}|M\tau)-\theta\\!\begin{bmatrix}\frac{M-i}{M}\\\
0\end{bmatrix}\\!(Mz_{1}|M\tau)\,,\quad
i\in\\{1\ldots\lfloor(M-1)/2\rfloor\\}\,,\end{split}$ (A.6)

and $\Phi_{i}$ the elements of $\mathbb{Z}_{2}$ even basis of degree $2M$
modular forms:

$\Phi_{j}(z_{1},\tau)=\theta\\!\begin{bmatrix}\frac{j}{2M}\\\
0\end{bmatrix}\\!(2Mz_{1}|2M\tau)+\theta\\!\begin{bmatrix}\frac{2M-j}{2M}\\\
0\end{bmatrix}\\!(2Mz_{1}|2M\tau)\,,\quad j\in\\{0\ldots M\\}\,,$ (A.7)

the two basis are related by Riemann’s addition formula

$\theta\\!\begin{bmatrix}\frac{i}{M}\\\
0\end{bmatrix}\\!(z_{1}|\tau)\,\theta\\!\begin{bmatrix}\frac{j}{M}\\\
0\end{bmatrix}\\!(z_{1}|\tau)=\sum_{k\in\\{0,\frac{1}{2}\\}}\theta\\!\begin{bmatrix}k+\frac{i-j}{2M}\\\
0\end{bmatrix}\\!(0|2\tau)\,\theta\\!\begin{bmatrix}k+\frac{i+j}{2M}\\\
0\end{bmatrix}\\!(2z_{1}|2\tau)\,.$ (A.8)

In particular, we have the following product rules for $\widehat{D}_{4}$:

$\displaystyle X_{0}^{2}=\phi_{0}\Phi_{0}+\phi_{4}\Phi_{4},$ $\displaystyle
X_{2}^{2}=\phi_{0}\Phi_{4}+\phi_{4}\Phi_{0},$ $\displaystyle
X_{0}X_{2}=2\phi_{2}\Phi_{2},$ $\displaystyle
X_{1}X_{2}=\phi_{1}\Phi_{3}+\phi_{3}\Phi_{1},$ $\displaystyle
X_{1}^{2}=\frac{\phi_{0}+\phi_{4}}{2}\Phi_{2}+\frac{\phi_{2}}{2}(\Phi_{0}+\Phi_{4}),$
$\displaystyle
Y_{1}^{2}=\frac{\phi_{0}+\phi_{4}}{2}\Phi_{2}-\frac{\phi_{2}}{2}(\Phi_{0}+\Phi_{4}),$
$\displaystyle X_{0}X_{1}=\phi_{1}\Phi_{1}+\phi_{3}\Phi_{3}.$ (A.9)

and product rules for $\widehat{D}_{5}$:

$\displaystyle X_{0}^{2}=\phi_{0}\Phi_{0}+\phi_{5}\Phi_{5},$ $\displaystyle
X_{0}X_{1}=\phi_{1}\Phi_{1}+\phi_{4}\Phi_{4},$ $\displaystyle
X_{1}^{2}=\frac{\phi_{0}}{2}\Phi_{2}+\frac{\phi_{5}}{2}\Phi_{3}+\frac{\phi_{4}}{2}\Phi_{5}+\frac{\phi_{1}}{2}\Phi_{0},$
$\displaystyle
Y_{1}^{2}=\frac{\phi_{0}}{2}\Phi_{2}+\frac{\phi_{5}}{2}\Phi_{3}-\frac{\phi_{4}}{2}\Phi_{5}-\frac{\phi_{1}}{2}\Phi_{0},$
$\displaystyle
X_{1}X_{2}=\frac{\phi_{1}}{2}\Phi_{3}+\frac{\phi_{4}}{2}\Phi_{2}+\frac{\phi_{2}}{2}\Phi_{4}+\frac{\phi_{3}}{2}\Phi_{1},$
$\displaystyle
Y_{1}Y_{2}=\frac{\phi_{1}}{2}\Phi_{3}+\frac{\phi_{4}}{2}\Phi_{2}-\frac{\phi_{2}}{2}\Phi_{4}-\frac{\phi_{3}}{2}\Phi_{1},$
$\displaystyle
X_{2}^{2}=\frac{\phi_{0}}{2}\Phi_{4}+\frac{\phi_{5}}{2}\Phi_{1}+\frac{\phi_{1}}{2}\Phi_{5}+\frac{\phi_{4}}{2}\Phi_{0},$
$\displaystyle
Y_{2}^{2}=\frac{\phi_{0}}{2}\Phi_{4}+\frac{\phi_{5}}{2}\Phi_{1}-\frac{\phi_{1}}{2}\Phi_{5}-\frac{\phi_{4}}{2}\Phi_{0},$
$\displaystyle X_{0}X_{2}=\phi_{2}\Phi_{2}+\phi_{3}\Phi_{3}.$ (A.10)

###  A.2 Modular (sub)group

Defining the action of elements of a modular subgroup $\Gamma\subseteq
SL(2,\mathbb{Z})$ on $(z,\tau)$:

$\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\Gamma:(z,\tau)\longmapsto\left(\frac{z}{c\tau+d},\frac{a\tau+b}{c\tau+d}\right),$
(A.11)

the set of theta functions used to formulate the SWC (3.3) is vector-valued
under such modular transformations [69]:

$\theta\\!\begin{bmatrix}\frac{j}{2M}\\\
0\end{bmatrix}\\!(2Mz_{1}|2M(\tau+1))\underset{T:\tau\to\tau+1}{\longrightarrow}e^{i\pi\frac{j^{2}}{2M}}\theta\\!\begin{bmatrix}\frac{j}{2M}\\\
0\end{bmatrix}\\!(2Mz_{1}|2M\tau),$ (A.12a)
$\theta\\!\begin{bmatrix}\frac{j}{2M}\\\
0\end{bmatrix}\\!(2Mz_{1}|2M\tau)\underset{S:z_{1},\tau\to\frac{z_{1}}{\tau},\frac{-1}{\tau}}{\longrightarrow}\omega(z_{1}|\tau)\sum_{k=0}^{2M-1}e^{-i\pi\frac{kj}{M}}\theta\\!\begin{bmatrix}\frac{k}{2M}\\\
0\end{bmatrix}\\!(2Mz_{1}|2M\tau),$ (A.12b)

with
$\omega(z_{1}|\tau)=\sqrt{\frac{\tau}{2iM}}\exp\left(\frac{2iMz_{1}^{2}}{\tau}\right)$,
$S=\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}$ and $T=\begin{bmatrix}1&1\\\
0&1\end{bmatrix}$. This generates the following transformation on the base of
even modular forms of degree $2M$:

$\mathbf{\Phi}:=\begin{bmatrix}\Phi_{0}\\\ \vdots\\\
\Phi_{M}\end{bmatrix}\underset{T:\tau\to\tau+1}{\longrightarrow}\left[\exp\left(i\pi\frac{j^{2}}{2M}\right)\right]_{jj}\cdot\mathbf{\Phi},\quad\mathbf{\Phi}\underset{S:z_{1},\tau\to\frac{z_{1}}{\tau},\frac{-1}{\tau}}{\longrightarrow}\frac{\omega(z_{1}|\tau)}{\sqrt{2M}}\left[2\cos\left(\frac{ij\pi}{M}\right)\right]_{ij}\cdot{\bf\Phi}.$
(A.13)

Standard Jacobi theta functions are re-scaled by modular transformations, in
particular $\theta_{1}$ verifies:

$\theta_{1}(z_{1};\tau)\underset{T:\tau\to\tau+1}{\longrightarrow}e^{i\frac{\pi}{4}}\theta_{1}(z_{1};\tau),\quad\theta_{1}(z_{1};\tau)\underset{S:z_{1},\tau\to\frac{z_{1}}{\tau},\frac{-1}{\tau}}{\longrightarrow}-i\sqrt{\frac{\tau}{i}}e^{i\frac{\pi
z_{1}^{2}}{\tau}}\theta_{1}(z_{1};\tau).$ (A.14)

We define the modular congruence subgroup $\Gamma_{0}(M)$ as:

$\Gamma_{0}(M)=\left\\{\begin{bmatrix}a&b\\\ c&d\end{bmatrix}\in
SL(2,\mathbb{Z}),\,\,c\equiv 0\,\,\text{mod}\,\,M\right\\}.$ (A.15)

##  Appendix B $\mathbb{M}$ matrices examples

In this appendix, we give explicit examples of $\mathbb{M}$ matrices entering
(3.3) of the SWC for $\widehat{D}_{6\ldots 9}$. For $(\widehat{D}_{6},1)$ LST,
we have:

$\mathbb{M}_{\widehat{D}_{6}}(\tau)=\begin{bmatrix}\phi_{0}&0&0&0&0&0&\phi_{6}\\\
0&\phi_{1}&0&0&0&\phi_{5}&0\\\ -\phi_{2}&0&\phi_{0}&0&\phi_{6}&0&-\phi_{4}\\\
0&-\phi_{3}&0&\phi_{1}+\phi_{5}&0&-\phi_{3}&0\\\
-\phi_{4}&0&\phi_{6}&0&\phi_{0}&0&-\phi_{2}\\\ 0&\phi_{5}&0&0&0&\phi_{1}&0\\\
\phi_{6}&0&0&0&0&0&\phi_{0}\end{bmatrix},$ (B.1)

for $(\widehat{D}_{7},1)$ LST:

$\mathbb{M}_{\widehat{D}_{7}}(\tau)=\begin{bmatrix}\phi_{0}&0&0&0&0&0&0&\phi_{7}\\\
0&\phi_{7}&0&0&0&0&\phi_{0}&0\\\
-\phi_{2}&0&\phi_{0}&0&0&\phi_{7}&0&-\phi_{5}\\\
-\phi_{4}&0&0&\phi_{7}&\phi_{0}&0&0&-\phi_{3}\\\
0&-\phi_{3}&0&\phi_{1}&\phi_{6}&0&-\phi_{4}&0\\\
0&-\phi_{5}&\phi_{6}&0&0&\phi_{1}&-\phi_{2}&0\\\
0&\phi_{1}&0&0&0&0&\phi_{6}&0\\\ \phi_{6}&0&0&0&0&0&0&\phi_{1}\end{bmatrix},$
(B.2)

for $(\widehat{D}_{8},1)$ LST:

$\mathbb{M}_{\widehat{D}_{8}}(\tau)=\begin{bmatrix}\phi_{0}&0&0&0&0&0&0&0&\phi_{8}\\\
0&\phi_{1}&0&0&0&0&0&\phi_{7}&0\\\
-\phi_{2}&0&\phi_{0}&0&0&0&\phi_{8}&0&-\phi_{6}\\\
0&-\phi_{3}&0&\phi_{1}&0&\phi_{7}&0&-\phi_{5}&0\\\
-\phi_{4}&0&0&0&\phi_{0}+\phi_{8}&0&0&0&-\phi_{4}\\\
0&-\phi_{5}&0&\phi_{7}&0&\phi_{1}&0&-\phi_{3}&0\\\
-\phi_{6}&0&\phi_{8}&0&0&0&\phi_{0}&0&-\phi_{2}\\\
0&\phi_{7}&0&0&0&0&0&\phi_{1}&0\\\
\phi_{8}&0&0&0&0&0&0&0&\phi_{0}\end{bmatrix},$ (B.3)

for $(\widehat{D}_{9},1)$ LST:

$\mathbb{M}_{\widehat{D}_{9}}(\tau)=\begin{bmatrix}\phi_{0}&0&0&0&0&0&0&0&0&\phi_{9}\\\
0&\phi_{9}&0&0&0&0&0&0&\phi_{0}&0\\\
-\phi_{2}&0&\phi_{0}&0&0&0&0&\phi_{9}&0&-\phi_{7}\\\
-\phi_{4}&0&0&\phi_{9}&0&0&\phi_{0}&0&0&-\phi_{5}\\\
-\phi_{6}&0&0&0&\phi_{0}&\phi_{9}&0&0&0&-\phi_{3}\\\
0&-\phi_{3}&0&0&\phi_{8}&\phi_{1}&0&0&-\phi_{6}&0\\\
0&-\phi_{5}&0&\phi_{1}&0&0&\phi_{8}&0&-\phi_{4}&0\\\
0&-\phi_{7}&\phi_{8}&0&0&0&0&\phi_{1}&-\phi_{2}&0\\\
0&\phi_{1}&0&0&0&0&0&0&\phi_{8}&0\\\
\phi_{8}&0&0&0&0&0&0&0&0&\phi_{1}\end{bmatrix}.$ (B.4)

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