††institutetext: Department of Mathematics, Tsinghua University, Beijing,
100084, China

# Pseudo-periodic map and classification of theories with eight supercharges

Dan Xie

###### Abstract

The classification of one parameter local Coulomb branch solution of theories
with eight supercharges is given by assuming that it is given by a genus $g$
fiberation of Riemann surfaces. The crucial point is the fact that certain
conjugacy class (so-called pseudo-periodic map of negative type) in mapping
class group determines the topological type of the degeneration. The
classification of conjugacy class has a simple combinatorial description. Each
such conjugacy class gives rise to a dual graph and a 3d mirror quiver gauge
theory can be derived, which is then used to identify the low energy theory
(assuming generic deformation). Some global Seiberg-Witten geometries are
given by using the topological data of the degeneration. The geometric setup
unifies 4d $\mathcal{N}=2$ SCFTs (such as $T_{n}$ theory and Argyres-Douglas
theory), 5d $\mathcal{N}=1$ SCFTs, 6d $(1,0)$ SCFTs, 4d IR free theories, and
4d asymptotical free theories in a single combinatorial framework.

##  1 Introduction

One of most important ingredient in solving the Coulomb branch of four
dimensional $\mathcal{N}=2$ theory is the electric-magnetic duality of low
energy abelian gauge theory Seiberg:1994rs , namely, when one go along the
special vacua (where there are extra massless particles), the effective photon
coupling changes by an element of electric-magnetic duality group 111In the
rank one case, this group is $SL(2,Z)$.. By combining the duality group
element around various special vacua (solving a Riemann-Hilbert problem),
Seiberg-Witten (SW) successfully solved Coulomb branch of $\mathcal{N}=2$
$SU(2)$ gauge theory by finding a SW curve Seiberg:1994rs ; Seiberg:1994aj : a
family of genus one curves $F(x,y,u)=0$, here $u$ is the Coulomb branch
parameter. The physical data are extracted as follows: a): The low energy
photon coupling $\tau(u)$ is given by the complex structure of smooth curve
$F(x,y,u)$; b): The special vacua is given by the degeneration of SW curve;
c): The electric-magnetic duality group around it is given by the monodromy
group which acts on homology class of the SW curve, see figure. 1.

$M_{1}$$M_{2}$$M_{3}$$C_{1}$$M(C_{1})$ Figure 1: Left: The electric-magnetic
duality group is given by the monodromy group $M$ which acts on homology group
of SW curve; Right: there is one monodromy group element around each special
vacua (including the $\infty$ point on the Coulomb branch), and the global
constraint is $M_{1}M_{2}M_{3}=I$.

In practice, one usually solve the Coulomb branch by finding a SW geometry
(often using string theory construction), and then try to analyze the IR
physics from SW geometry. While the photon couplings at generic vacua can be
computed using period integral, the IR physics at special vacua is much more
complicated to determine and one usually need to use extra mathematical
structure like mixed Hodge structure Xie:2021hxd . One of the findings in
Xie:2021hxd ; Xie:2022aad  is that the electric-magnetic duality (monodromy)
group around the special vacua is not enough to find the IR theory. For
example, in rank two case namikawa1973complete , besides the monodromy group,
one also need to specify two extra set of data ( an integer $m$ and the
degeneration type at the singularity) so that the low energy theory can be
determined (by assuming generic deformation).

While a large class of SW geometries were already found Gaiotto:2009we ;
Xie:2012hs ; Wang:2015mra ; Wang:2018gvb ; Xie:2015rpa ; Wang:2016yha , to
perform complete classification one need to follow the original approach of
SW: namely first classify the special IR vacua and then try to solve the
Riemann-Hilbert problem to get the global solution Seiberg:1994rs ;
Seiberg:1994aj . Certainly, the first and crucial step is to classify the
local behavior around the special vacua, which have been finished for rank one
case Argyres:2015ffa ; Argyres:2016xmc ; Argyres:2016xua  and rank two case
Xie:2022aad .

The purpose of this paper is to classify the local behavior of SW geometry for
theory with arbitrary rank $g$. Several assumptions on SW geometry are made:
a): The local solution is described by a fiberation of principally polarized
abelian varieties Donagi:1995cf  which can be identified with Jacobian of
genus $g$ Riemann surfaces, and so the special vacua is given by the
degeneration of Riemann surfaces; b): We take a one parameter slice of Coulomb
branch and so there is a one parameter family of Riemann surfaces; c) The
theory is associated with the generic deformation of the degeneration.
Therefore, the classification of local theory is reduced to the problem of
classification of one parameter degeneration of Riemann surface.

This task seems quite difficult given the fact that the number of degeneration
type increases dramatically with the genus (140 for rank two
namikawa1973complete , 1600 for rank three ashikaga2002classification ), and
they seem to be hard to organize even for rank two case. Luckily, Matsumoto-
Montesinos matsumoto2011pseudo  (MM) gave a remarkable combinatorial
description for such classification. The crucial idea is to use the conjugacy
class of mapping class group action on the Riemann surface in going around the
special vacua, see figure. 2. The mapping class group action certainly
determines the action on homology group, and therefore gives the electric-
magnetic duality group, but it contains more information, and in particular
the degeneration can be completely specified once the conjugacy class in
mapping class group is specified matsumoto2011pseudo !

$re^{i\theta}$$r$$x$$h_{\theta}(x)$$h_{2\pi}(x)$ Figure 2: There is a
homeomorphism action $h$ associated with the loop around special vacua. The
action acts on any point on Riemann surface, and in particular it induces an
action on homology group.

In the context of degeneration of Riemann surface, the conjugacy class of
mapping class group is of special type: it is the so-called pseudo-periodic
map of negative type. Those maps are classified by following data:

  1. 1.

An admissible system of cut curves ${\cal C}=\cup C_{i}$. Admissible means
that the irreducible component of $B=\Sigma_{g}/{\cal C}$ has non-negative
Euler number $\chi_{i}=2-2g_{i}+n_{i}\geq 0$, here $n_{i}$ is the number of
boundary curves for an irreducible component of $B$, and $g_{i}$ is the genus.

  2. 2.

The action of $f$ on the oriented graph $G_{\cal C}$ induced by ${\cal C}$.

  3. 3.

The screw numbers of $f$ around each annulus $C_{i}$. Here the screw number is
required to be negative.

  4. 4.

The action $f$ restricted on each irreducible component of $B$ is a periodic
map, which is in turn determined by the so-called valency data:
$(n,g^{{}^{\prime}},{\sigma_{1}\over\lambda_{1}}+\frac{\sigma_{2}}{\lambda_{2}}+\ldots+\frac{\sigma_{s}}{\lambda_{s}})$,
here $n$ is the order of the map ($f^{n}=id$), and $g^{{}^{\prime}}$ is the
genus of the base defined by the covering map
$f:\Sigma\to\Sigma^{{}^{\prime}}$, and $\sigma_{i},\lambda_{i}$ are the
integral value, see section 2.1.

In summary, one get a weighted graph for first two steps, See figure. 3 for an
example. There is also an integer $K\geq-1$ for every annulus (this number is
determined by screw number and the boundary data on the annulus); finally,
there is a periodic map for each irreducible component in $B$. Therefore, the
classification can be done in a combinatorial way!

$C_{1}$$C_{2}$$1$$1$$r=1$$r=0$$f_{1}$$f_{2}$$S(C_{2})$$S(C_{1})$ Figure 3: Up:
An admissible cut system for a genus two Riemann surface, and there are two
irreducible components after the cutting; there is a screw number associated
with each cutting curve, and a periodic map on each irreducible component.
Bottom: A weighted graph for the above cut system: here one draw an edge for
every separating curve (which will cut the Riemann surface into two separate
components), and one write the number of non-separating cutting curves for a
vertex in the graph. $\left(n,g^{{}^{\prime}}=0,\
\frac{1}{n}+\frac{1}{n}+\frac{n-1}{n}+\frac{n-1}{n}\right)$$(1)$$n$$1$$1$$n-1,..,2,1$$1,2,..n-1$$(2)$$0$$0$$f=\frac{1}{n}+\frac{n-1}{n}+\bm{1}$$n$$f=\frac{1}{n}+\frac{n-1}{n}+\bm{1}$$SU(n)$$n$$n$$K$
Figure 4: (1): The data for a periodic map of a genus $n-1$ Riemann surface,
and the dual graph for the degeneration is drawn; the dual graph is just the
3d mirror for $SU(n)$ theory coupled with $2n$ fundamental matter; (2): A
weighted graph for a genus $n-1$ curve, here $K\geq 1$; the IR theory is
$SU(n)$ gauge theory coupled with $2n+K$ hypermultiplets in fundamental
representation.

Once the degeneration is classified, the next question is to find the low
energy theory associated with it, which is in general a quite difficult
question. It turns out that the dual graph in MM’s theory will help us solve
this problem. The crucial observation is the link between the dual graph and
the 3d mirror of the low energy theory Xie:2022aad . Notice that one can not
always find a 3d mirror from the dual graph, which might give a criteria to
determine whether a degeneration can appear in the SW geometry or not. Using
the dual graph (and the associated 3d mirror) for the degeneration, the low
energy theory can be roughly described as follows:

  1. 1.

One find a four dimensional $\mathcal{N}=2$ SCFT for periodic map, and many
familiar theories such as $(A_{n-1},A_{k-1})$ theory cecotti2010r ; Xie:2012hs
, $T_{n}$ theory Gaiotto:2009we , and free bi-fundamental matter can be found.
See table. 1 for more examples.

  2. 2.

The gluing of different components in the weighted graph are interpreted as
gauging the flavor symmetry of the matter system.

See figure. 4 for a couple of examples. This type of description of low energy
theory is very much like what was used in class ${\cal S}$ theory, see figure.
24. In our case, the Argyres-Douglas theory, $T_{n}$ theory, and IR free gauge
theory are described in a unified framework.

Once the local studies are finished, the global SW geometry can be studied by
adding a point $\infty$ on the physical Coulomb branch Xie:2021hxd . A
singular fiber $F_{\infty}$ would also be added at $\infty$ which fits into
previous local study. However, the property of $F_{\infty}$ determines the UV
property. We give simple topological constraints on $F_{\infty}$ so that it
can defines a UV complete 4d, 5d and 6d theory. The results are: a): 4d
theory: the dual graph is a tree of rational curves; b): 5d theory: the dual
graph is a chain of rational curves with just one loop; c): 6d theory: the
dual graph is a chain of rational curves with two loops! It is interesting to
note the number of loops on the dual graph matches the number of loops in
doing the compactification. Some global SW geometries for many familiar
theories in 4d, 5d and 6d are discussed, and thorough studies would be left
for separate publications.

This paper is organized as follows: section two reviews the pseudo-periodic
map; section three discusses how to attach a dual graph for a pseudo-periodic
map; section four discusses how to find the low energy theory from the dual
graph and related 3d mirror; section five discusses topological constraints on
UV singular fiber, and several classes of global SW geometries for 4d, 5d, and
6d theories are given; finally a conclusion is given in section six.

##  2 Pseudo-periodic map

Let’s first recall the definition of mapping class group, for more details,
see farb2011primer . Let’s assume that $S$ is the connect sum of $g\geq 0$
tori with $b\geq 0$ open disks removed and $n\geq 0$ points removed from the
interior. Let $Homeo^{+}(S,\partial S)$ denote the group of orientation-
preserving homeomorphisms of $S$ that restrict to the identity of $\partial
S$. The mapping class group of $S$, denoted as $Mod(S)$, is the group

$Mod(S)=Hemeo^{+}(S,\partial S)/Homeo_{0}(S,\partial S)$

Here $Homeo_{0}(S,\partial S)$ denotes the connected component of the identity
in $Hemeo^{+}(S,\partial S)$. For example, the mapping class group of a closed
genus one Riemann surface is $SL(2,Z)$. There are also other equivalent
definition of mapping class group, i.e. the isotopy class of
$Hemeo^{+}(S,\partial S)$.

We also need to consider the generalization of above mapping class group,
namely, the homeomorphism $f$ might just keep the boundary $\partial S$ as a
set, and more generally, $f$ could also permute the elements in $\partial S$.

As discussed in the introduction, a local SW solution of the particular type
222If the associated abelian variety is given by the Jacobian of a Riemann
surface. gives rise to a one parameter family of genus $g$ fiberation. If
there is a special vacua, then one can associate an element of mapping class
group, see figure. 2. Actually, one associate a conjugacy class by changing
the base point of the loop around the singularity.

Instead of starting with known SW solution and try to compute the mapping
class group element around the degeneration, we go in the opposite direction.
We first try to classify all possible topological type of one parameter
degeneration of genus $g$ fiberation 333Let’s emphasize that unlike rank one
case, it appears that some degenerations can not appear in the Coulomb branch
solution., and then try to determine the IR theory for each degeneration.

The one parameter degeneration can be described as follows. Let
$\phi:S\to\Delta$ be a proper surjective holomorphic map to the unit disk
$\Delta=\\{|t|<1\\}$ such that $\phi^{-1}(t)$ is a smooth Riemann surface of
genus $g\geq 2$ for any $t\in\Delta^{*}=\Delta-\\{0\\}$. We call $\phi$ a
degeneration of genus $g$, and call $F=\phi^{-1}(0)$ the singular fiber. Now
assuming the $\phi$ is normally minimal, i.e. the reduced scheme of $F$ has
normal crossing and any $-1$ curve in $F$ intersects the other components at
at least three components. For two degenerations $\phi:S\to\Delta$ and
$\phi^{{}^{\prime}}:S^{{}^{\prime}}\to\Delta$, if there is an orientation-
preserving homeomorphism $\psi:S\to S^{{}^{\prime}}$ which satisfies
$\phi^{{}^{\prime}}\cdot\psi=\phi$, then we say $\phi$ and
$\phi^{{}^{\prime}}$ are topologically equivalent. Notice that for the purpose
of classifying local theories, only the topological type of the degeneration
is needed. Other physical quantities such the low energy couplings are given
by the holomorphic data.

By fixing $t_{0}\in\Delta^{*}$ as a base point, the canonical generator of the
fundamental group of the punctured disc acts naturally on the Riemann surface
$\phi^{-1}(t_{0})$ as an orientation-preserving homeomorphism modulo isotopy.
Since a change of the base point corresponds to conjugation in mapping class
group, therefore one get a conjugacy class from the degeneration. This element
is called topological monodromy of the degeneration. One should be careful
about that the monodromy here refers to the conjugacy class of mapping class
group, instead of conjugacy class of group $Sp(2g,Z)$.

Let’s now state the most important fact between the conjugacy class of mapping
class group and one parameter degeneration of Riemann surface
matsumoto1994pseudo ; matsumoto2011pseudo . First, the conjugacy class
realized as the topological monodromy of a degeneration of curves (with genus
$g\geq 2$) is represented by a pseudo-periodic map of negative type; Secondly,
any conjugacy class of pseudo-periodic map of negative type is realized as the
topological monodromy of a certain degeneration of curves.

The conjugacy class of a pseudo-periodic map $f:\Sigma_{g}\to\Sigma_{g}$ of
negative type of a smooth curve $\Sigma_{g}$ is determined by following data:

  1. 1.

An admissible system of cut curves ${\cal C}=\cup C_{i}$. Admissible means
that the irreducible component of $B=\Sigma_{g}/{\cal C}$ has non-negative
Euler number $\chi_{i}=2-2g_{i}+n_{i}\geq 0$, here $n_{i}$ is the number of
boundary curves for an irreducible component of $B$.

  2. 2.

The action of $f$ on the oriented graph $G_{\cal C}$ induced by ${\cal C}$.

  3. 3.

The screw numbers of $f$ around each annulus $C_{i}$. Here the screw number is
required to be negative.

  4. 4.

The action $f$ restricted on each component of $B$ is a periodic map, which is
in turn determined by the so-called valency data.

Let’s now explain above ingredients in more details:

Cut system of curves: The classification of an admissible system of cut curves
${\cal C}$ is the same as classification of the stable curves introduced by
Deligne-Mumford. Such cut systems can be represented by a weighted graph as
follows: a) The vertex $v$ represents an irreducible component, and two
integers $(g(v),\rho(v))$: $g(v)$ the genus of $v$, $\rho(v)$ the number of
cut curves which only belongs to $v$. b) The edge $e$, and the multiplicities
$\epsilon(e)$ is the number of double points between two components, see
figure. 5 for an example.

$1$$1$$e_{1}$$v_{1}$$v_{2}$ Figure 5: Up: There is only one cut curve for a
genus two Riemann surface; There are two components after the cut; Bottom: The
weighted graph for the cut system.

Automorphism: Given a cut system and so a weighted graph $X$. By an
automorphism $\sigma:X\to X$, we mean an automorphism of the graph such that
the weight $(g(B_{v}),\rho(v))$ is the same as
$(g(B_{\sigma(v)}),\rho(\sigma(v)))$ for each vertex $v$ of $X$. For example,
for $X$ listed in figure. 5, there is a $\mathbb{Z}_{2}$ action exchanging
$v_{1}$ and $v_{2}$, and fixing the edge $e_{1}$.

Screw number: For a curve $\vec{C}_{i}$ of ${\cal C}$, there exists a minimal
integer $\alpha_{i}$ such that $f^{\alpha_{i}}(\vec{C}_{i})=\vec{C}_{i}$ (they
are equal as a set). There also exists a minimal integer $L_{i}$ such that
$f^{L_{i}}|_{C_{i}}$ is a Dehn twist of $e_{i}$ times ($e_{i}$ could be
positive or negative integers). The screw number is defined as

$s(C_{i})={e_{i}\alpha_{i}/L_{i}}.$

The cut curves are classified into two types. A curve $C_{i}$ is called
amphidrome if $\alpha_{i}$ is even and
$f^{\alpha_{i}/2}(\vec{C}_{i})=-\vec{C}_{i}$, which means the orientation is
changed. Otherwise it is non-amphidrome.

Dehn twist$C$ Figure 6: Dehn twist along curve $C$, and this action is taken
to be negative around the indicated direction.

Valency data: The valency data for periodic map is more complicated to define.
Let $\Sigma$ be a Riemann surface, and $f:\Sigma\to\Sigma$ be a periodic map
(namely, $f^{n}$ is an identity map). For any point $P$ on $\Sigma$, there
exists an integer $\alpha(P)$ such that $P,f(P),\ldots,f^{\alpha(P)-1}(P)$ are
distinct points, and $f^{\alpha(P)}(P)=P$. If $\alpha(P)=n$, $P$ is called
simple points, and P is called multiple point if $\alpha(P)<n$.

Let $\vec{C}$ be an oriented simple closed curve, and let $m$ be the smallest
positive integer such that $f^{m}(\vec{C})=\vec{C}$ as a set, and
$n=m\lambda$. The restriction of $f$ on a $\vec{C}$ is then a periodic map.
Let $Q$ be any point on $C$, the orbit of $Q$ under the iteration of the
action $f^{m}$, and the points are ordered as
$(Q,f^{m\sigma}(Q),f^{2m\sigma}(Q),\ldots,f^{(\lambda-1)m\sigma}(Q))$, here
$\sigma$ is an integer which satisfies the condition
$0\leq\sigma\leq\lambda-1$, and $gcd(\sigma,\lambda)=1$. Finally, one
introduce another integer $\delta$ which satisfies the condition

$\sigma\delta=1(mod\lambda),~{}~{}0\leq\delta\leq\lambda-1.$

This means that the action $f^{m}$ on $\vec{C}$ is rotation with angle
$2\pi\delta\lambda$. The data $(m,\lambda,\sigma)$ is the valency data
associated with a curve $\vec{C}$. The valency data for a multiple point $P$
is defined as that of the closed curve around $P$.

Given a periodic map $f$, one has a $n$-fold cyclic covering map
$\Pi:\Sigma\to\Sigma^{{}^{\prime}}$ associated with $f$, and
$\Sigma^{{}^{\prime}}$ is the quotient space with respect to $f$ (the space of
orbits). The multiple points of $f$ gives rise to ramification points of
$\Pi$. The valency data of the ramification point is defined as that of the
multiple points. So finally, a periodic map is specified by the genus of
$\Sigma^{{}^{\prime}}$ and the valency data for the ramification points.

Induced action on Homology group: One of the fundamental aspects of
$Mod(S_{g})$ is its action on $H_{1}(S_{g},Z)$, and the representation
$\Psi:Mod(S_{g})\to Aut(H_{1}(S_{g},Z)$ is the first approximation to
$Mod(S_{g})$. Any $\phi\in Homeo^{+}(S_{g})$ induces an automorphism
$\phi_{*}:H_{1}(S_{g},Z)\to H_{1}(S_{g},Z)$, and it follows that the map

$\Psi_{0}:Mod(S_{g})\to Aut(H_{1}(S_{g};Z))$

has the property that

$\Psi_{0}(Mod(S_{g}))\subset Sp(2g,Z)$

Let’s now describe the action of Dehn twist on the homology class. Let $a$ and
$b$ be isotopy classes of oriented simple closed curves in $S_{g}$. For any
$k\geq 0$, we have

$\Psi_{0}(T_{b}^{k})([a])=[a]+k\hat{i}(a,b)[b].$

Here $\hat{i}(a,b)$ are the intersection number of $a,b$. Using the above
formula, one can see that the Dehn twist acts trivially on separating curves,
as the homology class of those curves is trivial. Those mapping class elements
which act trivially on homology groups are called Torreli groups.

###  2.1 Periodic map

Let $\Sigma_{g}$ be a closed Riemann surface of genus $g\geq 2$, and
$\Gamma_{g}$ be its mapping class group. In the isotopy class of a periodic
map of $\Sigma_{g}$, one can choose a representative which is an analytic
automorphism under a certain complex structure on $\Sigma_{g}$. Now let
$f:\Sigma_{g}\to\Sigma_{g}$ be a cyclic analytic automorphism of order $n$,
and $\Pi:\Sigma\to\Sigma^{{}^{\prime}}$ be the $n$-fold cyclic covering. Let
$g^{{}^{\prime}}$ be the genus of $\Sigma^{{}^{\prime}}$, and
$\lambda_{1},\ldots,\lambda_{l}$ be the ramification indices of $\Pi$ and
$(n/\lambda_{i},\lambda_{i},\sigma_{i})$ be the valency data. The valency data
satisfies following conditions:

  1. 1.

The Hurwitz formula:
$2g-2=n[2g^{{}^{\prime}}-2+\sum_{i=1}^{l}(1-{1\over\lambda_{i}})]$.

  2. 2.

Nielson theorem: $\sum\sigma_{i}/\lambda_{i}$ is an integer.

  3. 3.

Wiman: $n\leq 4g+2$.

  4. 4.

Havey: Set $M=lcm(\lambda_{1},\ldots,\lambda_{l})$ 444Here $lcm$ denotes the
least common multiple of the numbers within bracket..

    1. (a)

$lcm(\lambda_{1},\ldots,\hat{\lambda}_{i},\ldots,\lambda_{l})=M$ for all $i$,
here $\hat{\lambda}_{i}$ denotes the omission of $\lambda_{i}$.

    2. (b)

$M$ divides $n$, and if $g^{{}^{\prime}}=0$, then $M=n$.

    3. (c)

$l\neq 1$, and if $g^{{}^{\prime}}=0$, then $l\geq 3$.

    4. (d)

If $M$ is an even number and so $M=2^{\delta}M_{1}$ with $M_{1}$ an odd
number, then the number of $\lambda_{i}$ which are divisible by $2^{\delta}$
is even.

Usually, one label the periodic map by the data
$(n,g^{{}^{\prime}},{\sigma_{1}\over\lambda_{1}}+{\sigma_{2}\over\lambda_{2}}+\ldots+{\sigma_{l}\over\lambda_{l}})$.
On the other hand, given a data satisfying above condition, one can find a
period map of the Riemann surface $\Sigma_{g}$.

Example 1: The periodic map is given by
$(n,g^{{}^{\prime}}=0,\underbrace{{1\over n},\ldots,{1\over n}}_{a},{n-a\over
n})$, $a\geq 2$. If the common divisor of $n$ and $a$ is $b$, the genus of
$\Sigma_{g}$ is

$2g-2=(a-1)n-a-b\to g={(a-1)n-a-b+2\over 2}.$

In particular, if $b=(n,a)=1$, the genus $g={(a-1)(n-1)\over 2}$.

Example 2: The periodic map is given by
$(n,g^{{}^{\prime}}=0,\underbrace{{n-1\over n},\ldots,{n-1\over n}}_{n})$. The
genus of $\Sigma_{g}$ is

$2g-2=n(n-3)\to g={(n-1)(n-2)\over 2}.$

Duality for periodic map: Given a period map, one can change the valency data
${\sigma_{i}\over\lambda_{i}}\to\frac{\lambda_{i}-\sigma_{i}}{\lambda_{i}}$ to
define another periodic map. So the following two periodic maps are dual to
each other:

$(n,g^{{}^{\prime}},{\sigma_{1}\over\lambda_{1}}+{\sigma_{2}\over\lambda_{2}}+\ldots+{\sigma_{l}\over\lambda_{l}})\to(n,g^{{}^{\prime}},{\lambda_{1}-\sigma_{1}\over\lambda_{1}}+\ldots+{\lambda_{l}-\sigma_{l}\over\lambda_{l}}).$

Periodic map for surface with boundaries: For a genus $g$ surface with $k$
boundaries, the period map is defined as the one associated with closed genus
$g$ surface by adding disks around $k$ boundaries. One often need to specify
the transformation properties of the boundary curves. Here is an example,
let’s take the order of $f$ to be $4$, and $f$ permute the boundaries as
$(\partial_{1},\ldots,\partial_{4})$ (so $m=4$ for these points), and so under
the coving map, the center of these boundaries are mapped to a single point on
$\Sigma^{{}^{\prime}}$, and the associated valency data is
$(m,\lambda,\sigma)=(4,1,1)$, with $\frac{\lambda}{\sigma}=1$.

###  2.2 Weighted graph

Given a cut system $X$ and an automorphism $G$, one can define weighted graph
$Y$ from $X$ as follows. We need to enlarge the action by adding an
orientation for each edge. Then an automorphism has to preserve the
orientation.

The weighted graph $Y$ is derived as follows. Let $h:X\to Y$ be a map, a): let
$\bar{v}$ be a vertex of $Y$, then $h^{-1}(\bar{v})$ consists of finite number
of vertices, say $l(\bar{v})$, such that their weights
$(g(v_{i}),\rho(v_{i}))$ coincide with each other; so vertex $\bar{v}$ has the
triple weight $(l(\bar{v}),g(\bar{v}),\rho(\bar{v}))$; b) let $\bar{e}$ be an
edge of $Y$; Then $h^{-1}(\bar{e})$ consists of a finite number
$\zeta(\bar{e})$, of edges of $X$, so there is a weight $\zeta(\bar{e})$ for
$\bar{e}$. Notice that if there is an edge $e$ in $X$ connecting $v$ and
$v^{{}^{\prime}}$, and such that $h(v)=h(v^{{}^{\prime}})$, then there would
be a loop in $Y$.

Finally one need to define the resolution graph $\tilde{Y}$ of $Y$. Suppose
that there is an edge of $X$ and a positive integer $m$ such that
$\sigma^{m}(e)=e,\sigma^{m}(v)=v^{{}^{\prime}},\sigma^{m}(v^{{}^{\prime}})=v$
555This condition do not include the orientation of the edges., and $e$
connects $v$ and $v^{{}^{\prime}}$. The above condition means that
$\sigma^{m}$ fixes $e$ and exchanges the vertex $v$ and $v^{{}^{\prime}}$. Let
$m$ be smallest integer satisfies above condition, then one replace the edge
$h(e)$ by a line with weight $2m$ where the top part is branched into two
lines with weight $m$.

Example: Let’s consider a cut system for genus two Riemann surface, and it is
given by a graph with two vertex with genus $0$ and three edges between them.
There are following automorphism for it:

  1. 1.

$\sigma$ acts trivially.

  2. 2.

$\sigma$ fixes $v_{1}$ and $v_{2}$, and exchanges $e_{1}$ and $e_{2}$.

  3. 3.

$\sigma$ fixes $v_{1}$ and $v_{2}$, and cyclic permute $e_{1},e_{2},e_{3}$.

  4. 4.

$\sigma$ exchanges $v_{1}$ and $v_{2}$, fixes $e_{1},e_{2},e_{3}$ as sets, but
with their orientations reversed.

  5. 5.

$\sigma$ exchanges $v_{1}$ and $v_{2}$, and exchanges $e_{1}$ to $-e_{2}$, and
$e_{2}$ to $-e_{1}$, $e_{3}$ to $-e_{3}$.

  6. 6.

$\sigma$ exchanges $v_{1}$ and $v_{2}$, and exchanges $e_{1}$ to $-e_{2}$, and
$e_{2}$ to $-e_{3}$, $e_{3}$ to $-e_{2}$.

So there are a total of six types of weighted graph, see the bottom of figure.
7. The interested reader might try to match the weighted graph with above
automorphism action.

$2$$1$$1$$e_{1}$$0$$0$$2$$2$$e_{1}$$e_{2}$$e_{3}$$r=1$$r=2$$1$$1$$e_{1}$$r=0$$r=1$$1$$1$$e_{1}$$1$$r=1$$r=1$$2$$1$$2$$l=2$$0$$l=2$$2$$l=2$$0$$2$$l=2$$2$$2$$2$$0$$l=2$$0$$0$$1$$2$$0$$0$$3$$6$$3$$3$
Figure 7: All possible weighted graph for genus two Riemann surface.

##  3 Dual graph

The pseudo-periodic map of negative type have been described in last section.
One can attach a dual graph for each conjugacy class which will play a crucial
role in our later study of physical theories.

###  3.1 Dual graph for period map

First, one can define a dual graph for a period map as follows. A periodic map
is given as
$(n,g^{{}^{\prime}},{\sigma_{1}\over\lambda_{1}}+\ldots+{\sigma_{l}\over\lambda_{l}})$,
and the dual graph is constructed as follows:

  1. 1.

First given the valency data ${\sigma\over\lambda}$ ($m\lambda=n$), one attach
a linear chain of spheres with following nonzero multiplicities
$a_{0}>a_{1}>a_{2}\ldots>a_{s}=1$:

$a_{0}=\lambda,~{}~{}a_{1}=\sigma,~{}~{}{a_{i+1}+a_{i-1}\over
a_{i}}=\lambda_{i}\in Z$

Given $a_{i}$ and $a_{i-1}$, the above formula uniquely determines the integer
$a_{i+1}$. Since $n=\lambda m$, the final chain of spheres for the valency
data ${\sigma\over\lambda}$ are

$ma_{0}-ma_{1}-ma_{2}-\ldots-ma_{s-1}-m$

Example 1: If the data is $n=7,{\sigma\over\lambda}={5\over 7}$, the tail is
$7-5-3-1$; On the other hand, for the data $n=6,{\sigma\over\lambda}={2\over
3}$, the tail is $6-4-2$.

Example 2: If the data is ${n-1\over n}$. The tail would be

$n-(n-1)-(n-2)-\ldots-(2)-1$

Notice that this is the quiver tail corresponding to the so-called full
regular puncture of $A_{n-1}$ type Benini:2010uu .

Example 3: If the data is ${1\over n}$. The tail would be

$n-1$

This is the quiver tail corresponding to the so-called simple regular puncture
of $A_{n-1}$ type Benini:2010uu .

  2. 2.

The final graph is formed as follows: one has a core component with genus
$g^{{}^{\prime}}$, and the multiplicities $n$, then attach the linear tail
from each multiple point. The gluing is simple as the first component of each
tail has the common multiplicities $n$. At the end, one get a star-shaped
graph!

Example 1: The data for the periodic map is $(n=6,g^{{}^{\prime}}=0,{1\over
6}+{1\over 3}+{1\over 2})$, and the dual graph is shown in figure. 8.

Example 2: The data for the periodic map is $(n=3,g^{{}^{\prime}}=0,{1\over
3}+{1\over 3}+{1\over 3}+\frac{2}{3})$, and the graph is shown in figure. 8.
Notice that the dual graph is the 3d mirror for $SU(3)$ gauge theory with
$n_{f}=6$.

$6$$3$$2$$1$$\left(n=6,g^{{}^{\prime}}=0,\frac{1}{6}+\frac{1}{3}+\frac{1}{2}\right)$$3$$2$$1$$2$$\left(n=3,g^{{}^{\prime}}=0,\frac{1}{3}+\frac{1}{3}+\frac{2}{3}+\frac{2}{3}\right)$$1$$1$$1$
Figure 8: The dual graph for periodic maps.

###  3.2 Gluing

Let’s now construct dual graph for general pseudo-periodic map. The first data
is a weighted graph $X$, and the irreducible components in $X$ are given by
genus $g$ Riemann surface with $k$ boundary curves (Let’s use $\partial_{i}$
to denote them.), and $r$ internal non-separating cutting curves (Let’s use
$C_{i}$ to denote them.). There is an associated periodic map on each
irreducible component. More explanation on the action of $f$ on the boundary
curves is needed (Here $n$ is the order of $f$):

  1. 1.

First, one need to specify the action of $f$ on boundary curves
$\partial_{i}$, for example $f$ could permute
$(\partial_{1},\partial_{2},\partial_{3})$, this implies that branch points of
$\Sigma^{{}^{\prime}}$ associated with three boundary curves has the data
$m=3$, and so $\lambda={n\over 3}$.

  2. 2.

Secondly, one need to specify the action of $f$ on the cutting curve $C_{i}$.
There are two types of action according to whether the action is amphidrome or
not. Let’s take two curves as an example: a): if $C_{1},C_{2}$ is non-
amphidrome, and the boundary curves are
$C_{1}^{{}^{\prime}},C_{1}^{{}^{\prime\prime}},C_{2}^{{}^{\prime}},C_{2}^{{}^{\prime\prime}}$,
then $f$ permutes $C_{1}^{{}^{\prime}},C_{2}^{{}^{\prime}}$ and so the branch
point $P_{1}$ corresponding to them has $m=2$, similarly $f$ also permute
$C_{1}^{{}^{\prime\prime}},C_{2}^{{}^{\prime\prime}}$ (with $m=2$), and branch
point is $P_{2}$ with $m=2$. One then specify the valency data for $P_{1}$,
$P_{2}$, and in particular $\lambda=\frac{n}{2}$. In general, if $\langle
C_{1},\ldots,C_{s}\rangle$ is permuted by $f$, then there are two branch
points on $\Sigma^{{}^{\prime}}$, and the order of them is $m=s$.

If the action on $C_{1}$ is amphidrome, then $f$ permutes two boundary curves
$C_{1}^{{}^{\prime}},C_{1}^{{}^{\prime\prime}}$ and so $m=2$. Similarly, if
the action on $C_{1},C_{2}$ is amphidrome, namely,
$f(\vec{C}_{1})=-\vec{C}_{2},f^{2}(\vec{C}_{1})=-\vec{C}_{1}$, so the boundary
curves
$(C_{1}^{{}^{\prime}},C_{1}^{{}^{\prime\prime}},C_{2}^{{}^{\prime}},C_{2}^{{}^{\prime\prime}})$
is permuted by $f$, then $m=4$ for these boundary curves. In general, if
$\langle C_{1},\ldots,C_{s}\rangle$ is amphidrome under $f$, then there is
just one branch point on $\Sigma^{{}^{\prime}}$, with order $m=2s$.

Remark: We will use bold number for valency data for the boundary curves, and
use bracket for the valency data of the internal cutting curves (two for non-
amphidrome action, and one for amphidrome action). For exmaple, if
$g=0,k=4,r=0$, we have following possibilities: a): $f=id$; b):
$\langle\partial_{1},\partial_{2}\rangle,f=\bm{1}+\bm{\frac{1}{2}}+\bm{\frac{1}{2}}$;
c):
$\langle\partial_{1},\partial_{2}\rangle,\langle\partial_{3},\partial_{4}\rangle,f=\bm{1}+\bm{1}+\frac{1}{2}+\frac{1}{2}$;
d):
$\langle\partial_{1},\partial_{2},\partial_{3}\rangle,f=\bm{1}+\bm{\frac{1}{3}}+\frac{2}{3}~{}or~{}\bm{1}+\bm{\frac{2}{3}}+\frac{1}{3}$;
e):
$\langle\partial_{1},\partial_{2},\partial_{3},\partial_{4}\rangle,f=\bm{1}+\frac{1}{4}+\frac{3}{4}$.

In last subsection, one associate a dual graph for each periodic map. Now one
can get a dual graph for any pseudo-periodic map by gluing the graphs for
period maps together. Let $A_{i}$ be an annular neighborhood of $C_{i}$
(possibly many curves, see the discussion above). Let’s also use
$(m^{(1)},\lambda^{(1)},\sigma^{(1)})$ and
$(m^{(2)},\lambda^{(2)},\sigma^{(2)})$ to be the valencies of the boundary
curves $C_{i}^{{}^{\prime}}$ and $C_{i}^{{}^{\prime\prime}}$.

Let’s first assume $C_{i}$ to be non-amphidrome. Then $m^{(1)}=m^{(2)}=m$, and
one obtain two sequences of integers $a_{0}>a_{1}>\ldots>a_{u}=1$ and
$b_{0}>b_{1}>\ldots>b_{v}=1$. Graphically, one get two quiver tails from above
sequence of integers. Define an integer

$K=-s(C_{i})-\delta^{(1)}/\lambda^{(1)}-\delta^{(2)}/\lambda^{(2)}$ (1)

where $\delta^{(j)}$ are integers such that
$\sigma^{(j)}\delta^{(j)}=1(mod\lambda^{(j)}$),
$0\leq\delta^{(j)}<\lambda^{(j)}-1$. If $\lambda^{(j)}=1$, one set
$\delta^{(j)}=0$. $K$ satisfies condition $K\geq-1$, as
$s(C_{i})<0,0\leq\delta^{(j)}/\lambda^{(j)}<1$. The gluing for the two quiver
tails is defined as follows

  1. 1.

If $K\geq 1$, the the glued tail looks as follows

$(ma_{0},ma_{1},\ldots,ma_{u},\underbrace{m,m,\ldots,m}_{K-1},mb_{v},\ldots,mb_{1},mb_{0})$

  2. 2.

If $K=0$, the the glued tail looks as follows

$(ma_{0},ma_{1},\ldots,ma_{u-1},m,mb_{v-1},\ldots,mb_{1},mb_{0})$

  3. 3.

Finally, if $K=-1$, then one can find $u_{0}<u$ and $v_{0}<v$ so that
$a_{u_{0}}=b_{v_{0}}$, and $(a_{u_{0}-1}+b_{v_{0}-1})/a_{u_{0}}$ is an integer
greater than one. Then the quiver tail looks like

$(ma_{0},ma_{1},\ldots,ma_{u_{0}},mb_{v_{0}-1},\ldots,mb_{1},mb_{0})$

Let’s now assume $C_{i}$ to be amphidrome, then
$C_{i}^{{}^{\prime}},C_{i}^{{}^{\prime\prime}}$ has valency data
$(2m,\lambda,\sigma)$. Similarly, one has a sequence of integers
$a_{0}>a_{1}>\ldots>a_{u}=1$, from which one can get a quiver tail. Then
$K=-s(C_{i})/2-\delta/\lambda$ is a non-negative integer where
$\delta\sigma=1(mod\lambda)$. The glued quiver tail now has $u+K+2$ spheres,
and it is a Dynkin diagram of $D$ type

$(2ma_{0},2ma_{1},\ldots,2ma_{u},\underbrace{2m,\ldots,2m}_{K}\text{(the tree
part)},~{}m,m~{}\text{(the terminal part}))$

Example1: Let’s take $C_{i}$ to be non-amphidrome, and the valency data for
both of them is $(m,\lambda,\sigma)=(1,3,2)$, so $\delta^{1}=\delta^{2}=2$;
the integer $K=-S(C_{i})-\frac{2}{3}-\frac{2}{3}\geq-1$. The glued tail is
shown in figure. 9.

Example2: Let’s take $C_{i}$ to be amphidrome, and the valency data is
$(2,2n,n-1)$ (${\sigma\over\lambda}={n-1\over n},~{}m=1$), and the integer
$K=-S(C_{i})-\frac{n-1}{n}\geq 0$. The glued tail is shown in figure. 9.

$K\geq
1$$K=0$$K=-1$$3$$2$$3$$2$$1$$3$$2$$3$$2$$1$$1$$2n$$(2n-2)$$4$$2$$(2n-4)$$3$$2$$3$$2$$1$$1$$1$$1$$K-1$$1$$2$$2$$K$$1$
Figure 9: Up: The glued tail for a cylinder with same valency data
$(m,\lambda,\sigma)=(1,3,2)$, and $K=-S(C)-\frac{2}{3}-\frac{2}{3}\geq-1$.
Bottom: The glued tail for the amphidrome cut, here the valency data is
$(m,\lambda,\sigma)=(2,n,n-1)$.

###  3.3 Dual graph for arbitrary pseudo-periodic map

$\zeta(e_{2})$$\zeta(e_{1})$$l$$g,\rho$$\zeta(e_{2})/l$$\zeta(e_{1})/l$$g,\rho$$\zeta(e_{2})/l$
Figure 10:  For a component in weighted graph, one get an associated genus $g$
curve with $k$ boundaries, and the multiplicity of the boundary edges is
divided by $l$.

As discussed in last section, the conjugacy class of pseudo-periodic map is
classified by weighted graphs. There is an associated periodic map for each
vertex $v$ in the graph $\tilde{Y}$, and $e_{1},\ldots,e_{s}$ are the edges on
$v$, $e_{s+1},\ldots e_{s+s^{{}^{\prime}}}$ are the loops. So a vertex of
$\tilde{Y}$ has data $(l(v),g(v),\rho(v))$ and the edges $1\leq i\leq
s+s^{{}^{\prime}}$ with weight $\zeta(e_{i})$, and the edge
$e_{i}^{{}^{\prime\prime}}$ ($s+1\leq i\leq s+s^{{}^{\prime}})$ has weights
$\zeta(e_{i})$. One associate a curve with following data:

$g=g(v),~{}~{}r=\rho(v),~{}~{}k={\sum_{i=1}^{s+s^{{}^{\prime}}}\zeta(e_{i}^{{}^{\prime}})+\sum_{i=s+1}^{s+s^{{}^{\prime}}}\zeta(e_{i}^{{}^{\prime\prime}})\over
l(v)}$

Namely there are a total of $s+2s^{{}^{\prime}}$ tails. Write the dual graph
for the component (with data $(g(v),\rho(v),k(v))$ as follows:

$S_{f}=\sum_{j}m_{j}E_{j}+\sum_{i}n_{i}\vec{F}_{i}$

Then the dual graph is found by replacing the component $\sum m_{j}E_{j}$ by
$\sum l(v)m_{j}E_{j}$. Essentially, one need to multiply $l$ for every
component in the dual graph of the vertex $(g,r,k)$.

Example: Consider weighted graph shown in figure. 11, and the data for
associated curve has $g=1,r=0,k=2$, and $f$ keeps $\partial_{1},\partial_{2}$
separately (so $m=1$ for them). The data for periodic map is taken to be
$\bm{\frac{3}{4}}+\bm{\frac{3}{4}}+\frac{1}{2}$. See the dual graph in figure.
11.

$l=2$$1$$2$$1$$1$$1$$f=\bm{\frac{3}{4}}+\bm{\frac{3}{4}}+\frac{1}{2}$$4$$3,2,1$$2$$3,2,1$$1,..,1$$8$$6,4,2$$4$$6,4,2$$2,..,2$
Figure 11: Up: A weighted graph with $l=2$ and a loop with weight 2. The
associated curve with boundaries are shown on the right, and the associated
periodic map is given too; Bottom: We first write the dual graph for the
periodic map, and then get the dual graph of original theory by multiplying
two on the multiplicity of all the nodes.

##  4 IR theory

Let’s determine the 4d IR theory associated with a conjugacy class of pseudo-
periodic map. Specifying only the conjugacy class is not enough to determine
the IR theory, as there would be more than one theory associated with a single
degeneration, see Argyres:2015ffa  for rank one example. The ambiguity can be
fixed if one further assume the IR theory is associated with the generic
deformation of the singularity. To identify the IR theory, one need to use the
interesting relation between the 3d mirror of the IR theory and the dual graph
studied in last section Xie:2022aad . By using the dual graph and the related
3d mirror Benini:2010uu ; Nanopoulos:2010bv ; Xie:2012hs ; Xie:2021ewm , one
can determine the IR theory effectively.

###  4.1 Dual graph and 3d mirror

Given a conjugacy class of pseudo-periodic map, one can get a dual graph:

$F=\sum n_{i}C_{i};$

Here $C_{i}$ is an irreducible curve and $n_{i}$ is the multiplicity. The
intersection number $C_{i}\cdot C_{j}$ and the genus $g_{i}$ are also
specified. The self-intersection number for a component $C_{i}$ can be
computed by requiring:

$C_{i}\cdot F=0.$

The intersection number $k_{i}=C_{i}\cdot K$ 666Here $K$ is the canonical
class of the surface, and only play a formal role here. can be computed from
the genus $g_{i}$ and self-intersection number $C_{i}^{2}$:

$1+{1\over 2}(C_{i}^{2}+k_{i})=g_{i}\to k_{i}=2(g_{i}-1)-C_{i}^{2}.$ (2)

The genus of the configuration $F$ is expressed by the data $k_{i}$ as
follows:

$1+{1\over 2}\sum n_{i}k_{i}=g.$

See artin1971degenerate  for more details on configuration of curves.

$1$$1$$n$$n-2$$n-4$$1$$1$$1$$n-2$$n-4$$1$$1$$1$$1$$\frac{n-1}{2}$ Figure 12:
Left: the dual graph for a periodic map with data
$(n,g^{{}^{\prime}}=0,\frac{1}{n}+\frac{1}{n}+\frac{n-2}{n})$, and node $n$ is
a -1 curve. Middle: Contracting node $n$, and there would be one edge between
the three nodes connecting node $n$; After contraction, the node with
multiplicity $n-1$ becomes a $-1$ curve; right: Continuing the contracting
process, one finally get a model without any $-1$ curve.

Contracting $-1$ curve: The dual graph studied in last section is called
normally minimal model, since the intersections of components are all double
points. There could be $-1$ rational curves (self-intersection number is $-1$)
in these models. To get information for the low energy theory, one need to
contract $-1$ curve to get a so-called relatively minimal model (no $-1$ curve
in the model). The contraction formula is given as follows. Assuming that a
$-1$ curve $C_{n}$ is contracted, and the new intersection number and genus
are given as

$\displaystyle C_{i}^{{}^{\prime}}\cdot C_{j}^{{}^{\prime}}=C_{i}\cdot
C_{j}+(C_{i}\cdot C_{n})(C_{j}\cdot C_{n}),~{}$
$\displaystyle~{}g(C_{i}^{{}^{\prime}})=g(C_{i})+{1\over 2}((C_{i}\cdot
C_{n})^{2}-C_{i}\cdot C_{n}).$ (3)

Example: Let’s consider a periodic map with data
$(n,g^{{}^{\prime}}=0,\frac{1}{n}+\frac{1}{n}+\frac{n-2}{n})$, and $n$ is odd.
The dual graph is shown in figure. 12, and all the components are rational
curves. The self-intersection number of node $n$ is computed as:

$C_{i}\cdot F=0\to C_{i}\cdot(nC_{i}+\delta)=0\to
nC_{i}^{2}+C_{i}\cdot\delta=0\to C_{i}^{2}=-1.$

Here $\delta=F-nC_{i}$ (which are the components of $F$ minus the central
node), and $C_{i}\cdot\delta=n$ is used. So the central node is a $-1$ curve,
which can be contracted and one get a triangle connecting nodes with
multiplicity $1,1,n-2$. After contraction, the node with multiplicity $n-2$
becomes a $-1$ curve; One continue doing the contraction until getting a model
without any $-1$ curve, see figure. 12. For our later purpose, we draw the
contracting process using the curves, see figure. 13.

$n$$1$$1$$n-2$$n-4,..,1$$n-2$$n-6,..,1$$1$$1$$1$$1$$1$ Figure 13: The
contracting process. At the end, the intersection number between two red lines
is $\frac{n-1}{2}$. The intersection number of red and black lines is just
one.

3d Mirror from relative minimal model: Once the relative minimal model $F=\sum
n_{i}C_{i}$ is found, a 3d $\mathcal{N}=4$ quiver gauge theory can be found as
follows Xie:2022aad : a): Associate a quiver node $Q_{i}$ with gauge group
$U(n_{i})$ for a component $C_{i}$; b): The number of bi-fundamental fields
between $Q_{i}$ and $Q_{j}$ is given as $n_{ij}=C_{i}\cdot C_{j}$; c): Add
$g_{i}$ adjoint hypermultiplets for a node $Q_{i}$. The Higgs branch dimension
of the quiver gauge theory is

$\displaystyle r_{H}=\sum_{i<j}n_{i}n_{j}C_{i}\cdot C_{j}+\sum
g_{i}n_{i}^{2}-\sum n_{i}^{2}+1$ $\displaystyle={1\over 2}(\sum
C_{i}n_{i})^{2}+\sum g_{i}n_{i}^{2}-{1\over 2}C_{i}^{2}n_{i}^{2}-\sum
n_{i}^{2}+1$ $\displaystyle=\sum n_{i}^{2}(g_{i}-{1\over 2}C_{i}^{2}-1)+1$
$\displaystyle=\boxed{\sum{(n_{i}^{2}-n_{i})\over 2}k_{i}}+g.$ (4)

Here $F^{2}=(\sum n_{i}C_{i})^{2}=0$ is used, and an extra $+1$ is added
because an overall $U(1)$ gauge group is decoupled.

Let’s extract a 3d mirror for the IR theory from above quiver gauge theory.
The basic match is that the above dimension formula $r_{H}$ should be equal to
$g$ (which is the Coulomb branch dimension of the 4d IR theory), but in
general the above formula is larger than $g$. It was noticed in Xie:2022aad
that one can match the known 3d mirror by modifying the proposed quiver gauge
theory, and it is possible to generalize the construction in Xie:2022aad  to
theory of arbitrary rank (again by matching the known 3d mirror for 4d
theories). Here is the summary for the modification procedure:

  1. 1.

If a quiver node has $n_{i}=1$ or $k_{i}=0$ (this is only possible for $-2$
rational curve, see formula. 2 by using $C_{i}^{2}<0$), then no modification
is needed, as these nodes do not contribute to the extra terms in the Higgs
branch dimension formula. 4.

  2. 2.

For other nodes, one need to peel off $None$ number of following quiver tails

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}n_{i}}-(n_{i}-1)-(n_{i}-2)-\ldots-2-1$

Each such tail contributes Higgs branch dimension ${(n_{i}^{2}-n_{i})\over
2}$, and so would cancel the contribution of the $i$th node in the formula. 4.
Notice that this is not always possible.

Example: Let’s consider the periodic map given by the data
$(n=4,g^{{}^{\prime}}=0,\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4})$. The
quiver gauge theory derived from it is a star-shaped quiver with four maximal
quiver tails. The central node has self-intersection number $C_{i}^{2}=-3$,
and has $k_{i}=1$, so one need to peel off one maximal quiver tail. The final
quiver is the 3d mirror for $T_{4}$ theory Benini:2010uu . More evidence for
the identification would be given later.

The rule listed above is based on the match between the 3d mirrors for known
SCFTs. It would be interesting to find a deeper reason for it, and perhaps
those bad ones do not appear as the singular fiber for the SW geometry.
Indeed, one can not find physical interpretation if above constraints are not
imposed. Here let’s illustrate this point by an interesting example. Consider
a genus two example, and the weighted graph is $(g=2,r=1)$, and the cut is
amphidrome. The valency data is $(\frac{1}{2})+\frac{3}{4}+\frac{3}{4}$ (the
one in the bracket is for the cutting curve) (this is the $III^{*}-II_{n}^{*}$
singularity in Xie:2022aad ). Now the conjugacy class has an integral
parameter $n$, whose physical interpretation should be a gauge theory coupled
with free hypermultiplets (see section. 4.4). However, according to our
dimension formula (see table. 4 in Xie:2022aad ), there are only two possible
scaling dimensions $(2,\frac{3}{2})$ and $(4,3)$ for this degeneration. The
scaling dimension $(2,\frac{3}{2})$ gives the gauge theory description for the
fiber $(III-II_{n}^{*})$, so the scaling dimension for fiber
$III^{*}-II_{n}^{*}$ should be $(4,3)$ which implies that no gauge theory is
available (there has to be a dimension two Coulomb branch operator so that a
gauge theory description is possible). This seemingly contradiction is saved
by our constraints from finding a 3d mirror. In fact, one only find a good 3d
mirror by setting $n=0$, and interestingly the scaling dimension read from the
3d mirror is indeed $(4,3)$. The above example gives certain justification to
our procedure, and more evidence would be given later, see section. 4.4.

Bad tail: Let’s now analyze a general tail determined by the fraction ${a\over
n}$ with the order $mn$, here $(a,n)=1$. Let’s assume following decomposition
of $n$:

$n=(n-a)x+b,~{}~{}0<b\leq(n-a)$

The quiver tail takes the following form (for $a\geq\frac{n}{2}$):

$n-(n-(n-a))-(n-2(n-a))-\ldots-(n-(x-1)(n-a))-\boxed{b}-\ldots$

The node with multiplicity $b$ is a bad node (its self-intersection number is
not $-2$). For $a<\frac{n}{2}$, the quiver tail is

$n-\boxed{a}-c-\ldots$

Here the node with multiplicity $a$ is a bad node.

Let’s now analyze the constraint on the data $(a,n)$ so that a good tail can
be defined using our modification procedure.

  * •

For $m=1$: Let’s first assume $a\geq\frac{n}{2}$: 1): the tail has to take the
form $\ldots-(n-(x-1)(n-a))-\boxed{b}-(b-1)-\ldots-1$. To implement our
modification procedure, the self-intersection number of node $b$ has to be
$-3$:

$3b=(b-1)+(n-(x-1)(n-a))\to b=n-a-1.$

2): If $b=1$, then no problem arises. Similarly, if $a<\frac{n}{2}$, the tail
has to take the form $n-a-(a-1)-\ldots$ and $a$ should have self-intersection
number $-3$, which gives $a=\frac{n-1}{2}$. The above constraints on $(n,a)$
can be put in following uniform way:

$\displaystyle n=(n-a)x+(n-a-1)~{}~{}~{}or~{}$ $\displaystyle n=(n-a)x+1$ (5)

One can associate a quiver tail from a Young Tableaux Benini:2010uu . The
Young Tableaux for above cases are shown in figure. 14.

  * •

For $m>1$, one can not use the modification procedure to cure the bad node,
and so all the node has to be good. This has only following two solution:
$a=n-1$ or $n=1$. The associated Young Tableaux is listed in figure. 14.

  * •

Finally, let’s point out that one must be careful if the dual graph has $-1$
curve as the core, and one need to first contract the $-1$ curves and then
decide whether the dual graph is good or not. A case by case study is needed.

$n-a$$n-a-1$$n-a$$n-a$$m=1$$m>1$$m$ Figure 14: Up: The Young Tableaux for
$m=1$ which would give the good tail, and the flavor symmetry is $SU(x)\times
U(1)$ (after modifying the quiver tail), here $n=(n-a)x+(n-a-1)$ or
$n=(n-a)x+1$; Bottom: The Young Tableaux for $m>1$, and the flavor symmetry is
$SU(\lambda)$, with $\lambda m=n$.

###  4.2 Periodic map and 4d SCFTs

It is natural to associate a 4d $\mathcal{N}=2$ SCFT for the periodic map, see
Argyres:2015ffa  for rank one and Xie:2022aad  for rank two cases. Here let’s
do it for theory with arbitrary rank, see table. 1 for many interesting class
of examples, and the main check is the relation between dual graph and 3d
mirror. Let’s now point out some interesting relations between geometric data
and the physical data.

Maximal scaling dimension and order of period map: There is a relation between
the order of periodic map and the maximal scaling dimension in Coulomb branch
spectrum: Assume the common divisor of the scaling dimension is $u$, and the
maximal scaling dimension takes the form $u_{max}={n\over u}$, then the order
of periodic map is $n$. The reason is the relation between the scaling
dimension and singularity spectrum Xie:2015xva . In particular, the maximal
scaling dimension is given as

$u_{max}=\frac{1}{\alpha_{0}};$

Here $\lambda_{0}=\exp^{2\pi i\alpha_{0}}$ with $\lambda_{0}$ the eigenvalue
of the monodromy group, whose order is the same as that of the periodic map,
i.e. $\lambda_{0}^{n}=1$. This implies $\alpha_{0}n=1(mod~{}\mathbb{Z})$,
which gives the desired relation.

Example: Let’s consider the possible periodic map associated with the
$(A_{n-1},A_{k-1})$ theory. The maximal scaling dimension of this theory is
${nk\over n+k}$, and so the order of the periodic map should be $nk$, and
indeed this is the case, see table. 1 for examples.

Generic deformation and number of $A_{1}$ singularities: Let’s verify our
proposal by looking at the $T_{n}$ theory. An interesting number is the number
of $A_{1}$ singularities under the generic deformation. These numbers can be
computed as follows: first let’s compute central charge $c$, and then use the
following formulas shapere2008central :

$c=\frac{R(B)}{3}+\frac{r}{6},~{}~{}R(B)={\alpha_{max}\mu\over
4},~{}\to~{}\mu=\frac{12c-2r}{\alpha_{max}}.$

Here $\mu$ is the number of $A_{1}$ singularities, $\alpha_{max}$ is the
maximal scaling dimension in the Coulomb branch spectrum, $r$ is the rank of
theory. Since
$c=\frac{n^{3}}{6}-\frac{n^{2}}{4}-\frac{n}{12}+\frac{1}{6},r={(n-1)(n-2)\over
2},~{}\alpha_{max}=n$ for $T_{n}$ theory Xie:2013jc , one get:

$\mu=2(n-1)^{2}.$

Now for the periodic map which is used to engineer $T_{n}$ theory, it admits
deformation so that all the singularities are the $A_{1}$ singularities
takamura2004towards , and the number is:

$\\#A_{1}=n_{t}+2g-1;$

Here $n_{t}$ is the number of components in the dual graph of the periodic
map, and is equal to $n_{t}=n(n-1)+1$. Since $g=\frac{(n-1)(n-2)}{2}$, and
finally $\\#A_{1}=2(n-1)^{2}$. This is in agreement with the computation using
central charge. For the theories engineered using isolated three dimensional
singularities, the number of $A_{1}$ singularities is equal to $2r+f$ where
$f$ is the number of mass parameters. This relation does not hold in general,
as the example with $T_{n}$ theory shows.

Theory | $Data$
---|---
$SU(n)$ with $n_{f}=2n$ |  $g=n-1$, $(n,g^{{}^{\prime}}=0,{1\over n}+{1\over n}+{n-1\over n}+{n-1\over n})$
$SU(2)^{k}~{}linear~{}quiver$ |  $g=k$, $(n=2,g^{{}^{\prime}}=0,({1\over 2})^{2k+2})$
$T_{n}$ |  $g={(n-2)(n-1)\over 2}$, $(n,g^{{}^{\prime}}=0,\underbrace{{n-1\over n}+{n-1\over n}+\ldots+{n-1\over n}}_{n})$
$(A_{1},A_{n-1}),~{}n~{}even$ |  $g={n-2\over 2}$, $(n,g^{{}^{\prime}}=0,{1\over n}+{1\over n}+{n-2\over n})$
$(A_{1},A_{n-1}),~{}n~{}odd$ |  $g={n-1\over 2}$, $(n,g^{{}^{\prime}}=0,{1\over 2n}+{1\over 2}+{n-1\over 2n})$
$(A_{1},D_{n+1}),~{}n~{}even$ |  $g={n\over 2}$, $(n,g^{{}^{\prime}}=0,{1\over 2n}+{1\over 2}+{n-1\over 2n})$
$(A_{1},D_{n+1}),~{}n~{}odd$ |  $g={n-1\over 2}$, $(n,g^{{}^{\prime}}=0,{1\over n}+{1\over n}+{n-2\over n})$
$(A_{2},A_{3n-1})$ |  $g=3n-2$, $(3n,g^{{}^{\prime}}=0,{1\over 3n}+{1\over 3n}+{1\over 3n}+{n-1\over n})$
$(A_{k-1},A_{nk-1})$ |  $g=\frac{(k-1)(nk-2)}{2}$, $(nk,g^{{}^{\prime}}=0,\underbrace{{1\over nk}+\ldots+{1\over nk}}_{k}+{n-1\over n})$
$(A_{n-1},A_{k-1}),~{}(n,k)=1$ |  $g={(n-1)(k-1)\over 2}$, $(nk,g^{{}^{\prime}}=0,{1\over nk}+{a\over n}+{b\over k})$
| $(n=3,k=3x+1\to a=2,b=x)$
| $(n=3,k=3x+2\to a=1,b=2x+1)$
$D_{2}SU(2n+1)$ |  $g=n$, $(2n+1,g^{{}^{\prime}}=0,{1\over 2n+1}+{n\over 2n+1}+{n\over 2n+1})$
$D_{n+k}(SU(n)),~{}~{}(n,k)=1$ |  $g={(n-1)(n+k-1)\over 2}$, $(n(n+k-1),g^{{}^{\prime}}=0,{1\over n(n+k-1)}+{a\over n}+{b\over n(n+k-1)})$
| $n=3,n+k=3x+1\to a=2,b=n+k-2,~{}~{}$
| $n=3,n+k=3x+2\to a=1,b=2(n+k)-3~{}$
Table 1: Periodic maps and the associated SCFTs.

Inverse engineering: Since the 3d mirror for a large class of 4d
$\mathcal{N}=2$ theories are known Benini:2010uu ; Xie:2012hs ; Xie:2021ewm ,
one can try to get the periodic map by doing inverse enginneering. The idea is
following: a) if the 3d mirror is a star-shaped quiver with central node $n$,
and the quiver tail is constrained so that it can be derived from a ratio
$\frac{a_{i}}{\lambda_{i}}$ ($m_{i}\lambda_{i}=n$), see section. 3.1; then one
need to add $x$ maximal tails so that the new quiver can be described by the
dual graph of a periodic map. $x$ is determined by following equations:

$\displaystyle N={x(n-1)+\sum m_{i}a_{i}\over n}\in\mathbb{Z},~{}~{}$
$\displaystyle N-2=x.$

The first equation ensures the valency data satisfies integral equation for
the periodic map, and the second equation is the condition for peeling off $x$
extra maximal tails. In conclusion, one solve $x$ to be

$x=\sum m_{i}a_{i}-2n.$ (6)

b): The 3d mirror for Argyres-Douglas is more complicated, and it is often
given by a quiver core (which consists more than one quiver nodes, such as a
triangle, see Xie:2012hs  for details), and there are also tails attached to
the core quiver nodes; To find the periodic map, one need to do the inverse
operation of contraction to get a star-shaped quiver.

Example 1: Let’s consider $A_{2}$ class ${\cal S}$ theory which is engineered
by a sphere with $a$ simple punctures and $b$ full punctures Gaiotto:2009we .
The 3d mirror is a star-shaped quiver with $a$ simple tail $3-1$, and $b$
maximal tail $3-2-1$ Benini:2010uu . According to our formula 6, $x=a+2b-6$,
and the valency data is

$(\frac{1}{3})^{a}+(\frac{2}{3})^{b}+(\frac{2}{3})^{a+2b-6}.$

A simple check is following: the rank of the SCFT is $2a+3b-8$. The genus
formula from the valency data is

$2g-2=3(-2+\frac{2}{3}(2a+3b-6))\to g=2a+3b-8,$

which is consistent. The above computation can be done similarly for $A_{n-1}$
theory defined by a sphere with $a$ simple and $b$ full punctures.

Example 2: Consider an AD theory whose 3d mirror has a triangle core with
ranks $n_{1},n_{2},n_{3}$. To get a star-shaped quiver whose contraction will
give rise to the 3d mirror for AD theory, one add a core node with rank
$N=n_{1}+n_{2}+n_{3}$, see figure. 15. The valency data is

$\frac{n_{1}}{N}+\frac{n_{2}}{N}+\frac{n_{3}}{N}$

$n_{1},n_{2},n_{3}$ is chosen that the above data is legitimate to define a
periodic map, and moreover they are constrained so that one can use the
modification procedure to get a 3d mirror. Some solutions are
$n_{1}=1,n_{2}=n,n_{3}=n$ (one get 3d mirror for $D_{2}SU(2n+1)$ theory), and
$n_{1}=n+1,n_{2}=n,n_{3}=n$ (here $n\leq 4$).

$n_{1}$$n_{2}$$n_{3}$$n_{1}$$n_{2}$$n_{3}$$N$ Figure 15: Left: the 3d mirror
for an AD theory whose core has three nodes and are connected by a single
edge; Right: One add a node with rank $N=n_{1}+n_{2}+n_{3}$ so it becomes a
star-shaped quiver.

###  4.3 Other 4d SCFTs

It was noticed in Xie:2022aad  that many familiar rank two 4d $\mathcal{N}=2$
SCFTs are not given by the periodic maps. The reason is following: our study
is based on one parameter family of genus $g$ fiberation, but one can not find
a one parameter scale invariant geometry for many SCFTs, therefore a periodic
map can not be associated. However, it seems possible to find a non-scale
invariant one parameter genus $g$ family associated with such family, which
fits in our framework. So it should be possible to discover those SCFTs using
our methods. More importantly, one always need to put a genus $g$ degeneration
at $\infty$ Xie:2021hxd , and so it is important to identify the conjugacy
classes of these SCFTs. Here are some possibilities:

  1. 1.

First, let’s consider rank $l$ version of $E_{n}$ Minahan:1996cj  and $H_{n}$
Argyres:1995xn  type theories. The corresponding weighted graph is shown in
figure. 16. One might wonder whether it is possible to find other similar rank
$l$ generalizations, namely, the weighted graph is similar to that shown in
figure. 16, and with the replacement of genus one component by a genus $g$
component. The search turns out to be difficult, as there are strong
constraints on original theory: a) no modification is needed for original dual
graph; b) all the valency data should be of the form $\frac{n-1}{n}$. Most
likely, one can only consider $E_{n}$ and $H_{n}$ theories.

$1$$1$$1$$1$$1$$l$$l$$ln$$l$$T_{1}$$1$$2l$$ln$$l$$1$$2l$ Figure 16: Up: The
weighted graph corresponding to rank $l$ version of $H_{n}$ and $E_{n}$
theories, and the automorphism permutes the genus one nodes. The periodic map
on genus zero component is $\bm{1}+\frac{1}{l}+\frac{l-1}{l}$, and on the
genus one component is $\bm{\frac{3}{4}}+\frac{3}{4}+\frac{1}{2}$ (we take
$E_{7}$ for an example, and other cases are similar). Bottom: the dual graph
for above configuration, here $K$ (the integer associated with the cut curves)
is take to be zero; After modification on the bad node, one get the 3d mirror
for rank $l$ theory.

  2. 2.

One might also get SCFT from the geometry $(g,r=1)$ with the cut curve
amphidrome. The valency data for the cut curve $C_{1},C_{1}^{{}^{\prime}}$ is
${n-1}\over n$, and the order is $2n$. The tail for the amphidrome curve is
then

$2n-(2n-2)-\ldots-4-2-(1,1)(terminal)$

One can take off one terminal node with rank one to get a good 3d mirror. So
the valency data is

$(\frac{n-1}{n})+\sum{a_{i}\over 2n}$

Here $a_{i}$ is chosen so that one has a good tail. This will produce the 3d
mirror for many known SCFT.

  3. 3.

There are some other possibilities: the weighted graph has one component and
the edges are all of the $D$ type. We leave the detailed analysis to
interested reader. Here we give a class of examples.

$l=2$$2$$2$$1$$\bm{\frac{3}{4}}+\bm{\frac{3}{4}}+\frac{1}{2}$$\bm{\frac{1}{4}}+\bm{\frac{1}{4}}+\frac{1}{2}$$\bm{\frac{2}{3}}+\bm{\frac{2}{3}}+\frac{2}{3}$$\bm{\frac{1}{3}}+\bm{\frac{1}{3}}+\frac{1}{3}$$\bm{\frac{1}{2}}+\bm{\frac{1}{2}}+\frac{1}{2}+\frac{1}{2}$$8$$6,4,2,1$$6,4,2,1$$4$$1$$2$$2$$1$$6$$4,2,1$$4,2,1$$4$$2$$4$$2,1$$2,1$$2$$2$$2$$2$$2$$1$
Figure 17: Top: Weighted graph which might give a SCFT. The valency data for
periodic map is given on the left, and the 3d mirror from the dual graph is
shown on the right.

Example: The weighted graph is given in figure. 17. It has just one node with
$l=2$, and there are two D type tails. The corresponding data is
$(g=1,r=0,k=2)$, and the periodic map fixes each boundary map separately.
There are following choices for the periodic map

$\bm{\frac{3}{4}}+\bm{\frac{3}{4}}+\frac{1}{2},~{}~{}\bm{\frac{1}{4}}+\bm{\frac{1}{4}}+\frac{1}{2},~{}~{}\bm{\frac{2}{3}}+\bm{\frac{2}{3}}+\frac{2}{3},~{}\bm{\frac{1}{3}}+\bm{\frac{1}{3}}+\frac{1}{3},~{}~{}\bm{\frac{1}{3}}+\bm{\frac{1}{3}}+\frac{1}{3}$

The corresponding dual graph (after the modification and so is the 3d mirror)
is shown in figure. 17. It is easy to identify them as the rank three theories
listed in table 10 of Li:2022njl .

  4. 4.

Finally, it might happen that the gluing parameter $K$ takes value $K\geq 0$,
and only $K=0$ can be consistently to find a 3d mirror (this happens for
$m>1$). This often implies that one can associate a SCFT for it.

Example: Let’s give a family of examples to illustrate this class of examples.
The weighted graph is given by two components (with genus $g_{1}$ and $g_{2}$
separately) connected by an edge with weight 2, see figure. 18. The valency
data are

$\bm{\frac{1}{3n+4}}+\frac{1}{6n+8}+\frac{6n+5}{6n+8},~{}~{}~{}\bm{1}+(\frac{1}{2})^{2n+2}$

The dual graph gives (after modification) a good 3d mirror if $K=0$. In fact,
the 3d mirror agrees with the theory found in Xie:2012hs : it is engineered by
an irregular singularity with data $A_{2}$ theory on a sphere with one
irregular singularity with data $\Phi={T_{2n+4}\over
z^{2n+4}}+\ldots+{T_{2}\over z^{2}}+\frac{T_{1}}{z}$, with
$T_{2n+4},\ldots,T_{2}$ are of type $[2,1]$, and $T_{1}$ are type $[1,1,1]$,
there is also a maximal regular singularity.

$6n+8$$6n+5,6n+2,..,2,1$$1$$2$$2n+2$$1$$2$$2n+2$$2$$1$$1$$2$$2$$1$$Contraction$$Modification$$1$$2n+2$$2n+2$$1$$1$
Figure 18: Up: the dual graph for a pseudo-periodic map, and the weighted
graph has one genus $g_{1}$ component connected by a genus zero component.
Bottom: the corresponding 3d mirror.

###  4.4 IR free theory

Let’s now consider IR theory associated with general pseudo-periodic map. It
is expected that the IR theory is given by gauge theory coupled with matter.
Roughly, the matter is represented by irreducible components in the cut
system, and the gauge group is represented by edges (the rank of the gauge
group is given by the weights of the edge), and the number $K$ for the edges
gives the hypermultiplets coupled with gauge group.

To begin with, let’s analyze the case where the weighted graph has only one
node, so the data is $(g,r)$, here $r$ denotes the non-separating cutting
curves. One can find the IR theory (with arbitrary $K$) as follows:

  1. 1.

Let’s take $r=1$ and the cut non-amphidrome, and $f$ be the identity map on
$\Sigma_{g}/C$, the low energy theory is $U(1)$ gauge theory with $K$
fundamental hypermultiplets and $g-1$ free vector multiplets. This can be seen
from the dual graph (and 3d mirror) as follows: the 3d mirror consists $g-1$
free hypermultiplets 777The adjoint hypermultiplets on $U(1)$ node decouple.
which gives $g-1$ free vector multiplets of the original theory, and a cyclic
quiver with $K$ $U(1)$ gauge nodes whose mirror is just $U(1)$ with $K$
hypermultiplets Intriligator:1996ex , see figure. 19.

  2. 2.

Again, let’s take $r=1$ and the cut curve non-amphidrome, but $f$ is now a
general periodic map. Now the low energy theory can be found as follows: Let’s
assume the boundary curves are $C_{1}$ and $C_{2}$, and the theory associated
with $\Sigma_{g-1}$ is $T$, which gives a rank $g-1$ theory. The dual graph
has tail $T_{1}$ and $T_{2}$ corresponding to boundary curves $C_{1},C_{2}$
(The valency data for them has $m=1$). The low energy theory is then described
as follows: it is given by diagonally gauging $U(1)$ flavor symmetry of the
tail $T_{1}$ and $T_{2}$, and there is also $K$ extra free hypermultiplets
charged over $U(1)$ gauge group. Since the modification procedure is no longer
available (as the two tails have to be glued together), one has following
constraint on the valency data $(n,a)$: $n=(n-a)x+1$, and examples are
$\frac{n-1}{n}$ or $\frac{1}{n}$.

  3. 3.

Let’s take $r=1$, but this time the action is amphidrome. The boundary curve
$C_{1}^{{}^{\prime}}$ and $C_{1}^{{}^{\prime\prime}}$ now are related by the
periodic map (the valency data for them has $m=2$). And the dual graph has a
$D$ type tail. To get a valid 3d mirror with arbitrary link number $K$, the
order of periodic map should be two, and the valency data for the cutting
curves are $\bm{1}$, see figure. 19. We interpret it as a $SU(2)$ gauge theory
coupled with $K+2$ fundamental flavors and a theory $T$ represented by the
periodic map on $\Sigma_{g-1}$. One of the evidence is that by increasing $K$
by $1$, the Coulomb branch dimension of the 3d mirror is increased by two, and
so the Higgs branch of the original theory is increased by two. This is
consistent with the interpretation that $K$ is the number of fundamental
hypermultiplets coupled with $SU(2)$ gauge group.

  4. 4.

The situation with general $r$ is now clear: a): if the action is non-
amphidrome, and one of the orbit of $f$ is $\langle
C_{1},\ldots,C_{s}\rangle$, the boundary curves after the cut are
$(C_{1}^{{}^{\prime}},\ldots,C_{s}^{{}^{\prime}}),(C_{1}^{{}^{\prime\prime}},\ldots,C_{s}^{{}^{\prime\prime}})$,
then there are two tails for the cutting curves
$(C_{1}^{{}^{\prime}},\ldots,C_{s}^{{}^{\prime}})$,
$(C_{1}^{{}^{\prime\prime}},\ldots,C_{s}^{{}^{\prime\prime}})$: the low energy
theory is just gauging diagonally $U(s)$ flavor symmetry of $T_{1}$ and
$T_{2}$, and the valency data for them is $\bm{1}$ ; b): if the action is
amphidrome, and the orbit of $f$ is $\langle C_{1},\ldots,C_{s}\rangle$, the
boundary curves after the cut are
$(C_{1}^{{}^{\prime}},\ldots,C_{s}^{{}^{\prime}}),(C_{1}^{{}^{\prime\prime}},\ldots,C_{s}^{{}^{\prime\prime}})$,
then
$(C_{1}^{{}^{\prime}},\ldots,C_{s}^{{}^{\prime}},C_{1}^{{}^{\prime\prime}},\ldots,C_{s}^{{}^{\prime\prime}})$
are under the same orbit of $f$ (the valency data is $\bm{1}$), and the gauge
group is conjectured to be $Sp(2s)$. An example is shown in figure. 19.

$1$$g-1$$1)$$2)$$T_{1}$$T_{2}$$3)$$2$$2$$1$$1$$1$$1$$K-1$$1$$T$$1$$K-1$$1$$1$$2$$K$
Figure 19: (1): Dual graph for $(g,r=1)$: the cut curve is non-amphidrome, and
the corresponding periodic map is trivial; The IR theory is $U(1)$ coupled
with $K$ free hypermultiplets and $g-1$ free vector multiplets; (2): Dual
graph for $(g,r=1)$: the cut curve is non-amphidrome and the corresponding
periodic map is generic; (3): Dual graph for $(g,r=1)$, the cut curve is
amphidrome and the corresponding periodic map has order two. The IR theory is
$SU(2)$ gauge theory coupled with $K+2$ free hypermultiplets and an
interacting theory.

Let’s now discuss the situation where weighted graph has several components.
Now each component has the label $(g_{i},r_{i},k_{i})$ with $k_{i}$ the
boundary curves connecting with other components. The physical interpretation
for the non-separating curves $r_{i}$ are gauging, which is also true for
separating curves. The difference for the separating curves is that the gauge
group is $SU(m)$ (instead of $U(m)$ ). This is due to the fact the
contribution of the edge with multiplicities $m$ to the genus is $m-1$. There
are also $K$ fundamental hypermultiplets charged over $SU(m)$ gauge group. In
particular, if $m=1$, there are decoupled massless hypermultiplets (as the
gauge group is $SU(1)$).

So the IR theory for a general pseudo-periodic map is given by gauging the
matter systems represented by periodic maps. Some familiar matter systems are
given:

  1. 1.

Free matter: A set of free matter is represented by a sphere with $n$ boundary
curves, which are permuted by the action $f$. The periodic map is represented
by the data $\textbf{1}+{1\over n}+{n-1\over n}$, which gives rise to bi-
fundamental matter.

More generally, one can consider a sphere with $2n$ boundary curves, and $f$
permutes $n$ components separately. The periodic map is given by the data
$\textbf{1}+\textbf{1}+{1\over n}+{n-1\over n}$. The dual graph is shown in
figure. 20. The self-intersection number for the core component is $-3$, and
so one need to take off a maximal quiver tail, and the 3d mirror is shown in
figure. 20. It is then easy to see that this corresponds to bi-fundamental
matter.

  2. 2.

$T_{n}$ matter: It is represented by a genus ${(n-1)(n-2)\over 2}$ curve with
$3n$ boundary components, with $f$ permuting $n$ components each. The periodic
map is given by the data
$\textbf{1}+\textbf{1}+\textbf{1}+\underbrace{{n-1\over n}+\ldots+{n-1\over
n}}_{n}$. The dual graph and the corresponding 3d mirror are shown in figure.
20. It is then natural to identify it as the $T_{n}$ matter, with three
$SU(n)$ flavor symmetry groups gauged.

  3. 3.

General matter: It is now clear how to get general matter system. Let’s start
with a periodic map with order $n$, so that it can be interpreted as a SCFT.
Now assume that there is still maximal tail left (with the valency data
$\frac{n-1}{n}$) in the 3d mirror, then one can consider the following
configuration: a genus $g$ curve with $n$ boundary curves which are permuted
by periodic map. Then the data for the periodic map of this configuration is
$\textbf{1}+\ldots$, where 1 represents the $n$ boundary curves. To get the
good dual graph, one simply remove a maximal tail for the modified dual graph,
and replace it with a tail with one node $\boxed{n}$. The gluing of this
matter system with other parts can be interpreted physically as gauging the
$SU(n)$ flavor symmetry of the original matter system.

  4. 4.

Special case: The following matter system deserves special treatment: The
periodic map is given by $\sum{\sigma_{i}\over\lambda_{i}}+\bf{1}$, here
$\sum{\sigma_{i}\over\lambda_{i}}=1$. The order is $n$, and there are $n$
boundary curves which are permuted by the period map. The matter system is
interpreted as the class $S$ theory with regular punctured labeled by
${\sigma_{i}\over\lambda_{i}}$, and a full regular puncture. The gluing is
interpreted as gauging the $SU(n)$ flavor symmetry.

Using above matter system, one can easily construct familiar IR free theory
and the associated pseudo-periodic map, see figure. 21. The 3d mirror for such
IR free theory has been found in Nanopoulos:2010bv .

$n$$n$$n$$n$$T_{1}$$T_{n}$$n$$n$$n$$n$$n$$n$$n$$T_{1}$$n$$n$$n$$1$$1$ Figure
20: Up: the dual graph for bi-fundamental matter, and after modification it
becomes a good 3d mirror (part of a larger quiver); Bottom: the dual graph for
$T_{n}$ matter, and after modification it becomes a good 3d mirror.
$1)$$2)$$g-SU(g)-SU(g)-...-SU(g)-g$$n_{1}$$n_{2}$$n_{s}$$3)$$T_{g}$SU(g)SU(g)SU(g)$n_{1}$$n_{2}$$n_{3}$$g$$g$$g$$g$$g$$2g$$2g$$Sp(2g)$$SU(2g)$$K+2$
Figure 21: Some IR free gauge theories.

Constraints on gluing: Important rule on gluing is that the dual graph can be
interpreted as the 3d mirror for the low energy theory (with possible
modification). This puts constraints on possible gluing pattern, and a
detailed analysis will be provided.

Let’s look at the formula in section. 3.2 for gluing dual graphs, and one can
make following observations. If the curves $C_{i}$ are non-amphidrome, then

  1. 1.

If $m=1$, since it is no longer possible to do the surgery to modify the
quiver tails, the two tails $T_{1},T_{2}$ have to be good by itself (all the
quiver nodes on the tail has self-intersection number $-2$ and no modification
is needed). This puts the constraint on valency data: it can only take the
form $\frac{a}{n}$, with $n=(n-a)x+1$. The glued tail would be

$\displaystyle(n_{1}-\ldots-1-\underbrace{1-\ldots-1}_{K-1}-1-\ldots-
n_{2}),~{}~{}K\geq 1$ $\displaystyle(n_{1}-\ldots-1-1-\ldots-n_{2}),~{}~{}K=0$

For $K=-1$, one need to verify the condition case by case.

  2. 2.

If $m>1$, the only solution would be

$(m-\underbrace{m-\ldots-m}_{K-1}-m)$

then there would be no bad component; the boundary valency data is actually
$(\textbf{1},\textbf{1})$, and $K\geq 1$;

If the curve $C_{i}$ is amphidrome (the valency data for the cutting curves
are $\frac{a}{\lambda}$ with $n=2m\lambda$), by looking at the glued quiver,
one find that:

  1. 1.

$2m=n$ ($n$ the order of the periodic map, and the valency data is $(1)$),
then there would be no bad node as the tail takes the form

$2m-\underbrace{2m-\ldots-2m}_{K}-(m,m)(\text{terminal node})$

  2. 2.

$m=1$, $K=0$, and $a_{u-1}=~{}2~{}or~{}3$. Since the tail takes the form

$\ldots-2a_{u-1}-\textbf{2}-(1,1),$

and the bad node 2 has $k=a_{u-1}-1$, so if $a_{u-1}=2,3$, one can peel off
one or two maximal tails to get a good tail.

The above constraints then significantly reduce the possibilities for the
gluing:

  * •

If two components in weighted graph are connected by an edge with weight $m$,
then physically it is interpreted as gauging a $SU(m)$ flavor symmetry of the
matter. There are also $K$ fundamental hypermultiplet charged with $SU(m)$. In
particular, if $m=1$, there is no gauging, and the IR theory is the direct sum
of the theory associated with each component; there are also $K$ free
hypermultiplets which do not carry any electric-magnetic charge.

  * •

For the cuts corresponding to non-separating curves, the gluing is interpreted
as a $U(m)$ gauge group, or $Sp(2m)$ gauge group depending whether the cut is
amphidrome or not. For $m>1$, there is only $SU(m)$ flavor symmetry for the
matter associated with theory defined using period map, but there is an extra
$U(1)$ gauge theory, so that total rank of the IR theory is the same as the
genus of the Riemann surface.

  * •

When the periodic map of one component is special (namely the sum of valency
data is one), one should be careful about the gluing, and a case by case
analysis is needed. The physical interpretation is more complicated, many
gauge theory coupled with two Argyres-Douglas matters can be described in this
way.

  * •

If the vertices form a loop in the weighted graph, then the gauge groups take
the form $\sum SU(m_{i})\oplus U(1)$. The extra $U(1)$ is needed as the
contribution to the genus of the edges in the loop is $\sum(m_{i}-1)+1$.

Example 1: Let’s verify above proposal by looking at an example. The weighted
graph has only one node and the data is $(g,r=g)$, and the periodic map
permutes the curve $\langle C_{1},\ldots,C_{g}\rangle$. The valency data is
$(1)+(1)+\frac{1}{g}+\frac{g-1}{g}$, and the dual graph is shown in figure.
22. It is then easy to recognize that it is the 3d mirror for $U(g)$ gauge
theory with an adjoint and $K$ fundamental deBoer:1996mp  ($K\geq 1$).

Example 2: The weighted graph is given by two genus one curves connected by a
weight two edges, so the genus is three. The valency data on each component is
$\frac{1}{4}+\frac{1}{4}+\bm{\frac{1}{2}}$, and $K=-S(C)-1/2-1/2\geq 0$. The
dual graph is drawn in figure. 23. One recognize that it is the 3d mirror for
a gauge theory: $SU(2)$ gauge theory coupled with two $D_{2}(SU(3)$ matter and
$K+2$ fundamental hypermultiplets. Notice that the limit $K=-1$ is actually a
SCFT, which is represented by a periodic map with data
$(n=4,g^{{}^{\prime}}=0,\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4})$.

$g$$r=g$$f=(1)+(1)+\frac{1}{g}+\frac{g-1}{g}$$g$$g$$g$$g$$1$$g$$g$ Figure 22:
Left: The weighted graph has just one vertex and the data is $(g,r=g)$, and
the periodic map data is also given. Right: The dual graph for the pseudo-
periodic map.
$2$$1$$1$$f=\frac{1}{4}+\frac{1}{4}+\bm{\frac{1}{2}}$$f=\frac{1}{4}+\frac{1}{4}+\bm{\frac{1}{2}}$$4$$1$$1$$2$$4$$2$$K+1$$1$$1$$K=-S(C)-\frac{1}{2}-\frac{1}{2}\geq
0$$1$$1$$2$$2$$K+1$$1$$1$$SU(2)$$D_{2}(SU(3))$$D_{2}(SU(3))$$K+2$ Figure 23:
Up: the data for a pseudo-periodic map which would give the gauge theory
coupled with AD matter; Middle: the dual graph and the corresponding 3d
mirror; Bottom: The corresponding 4d gauge theory which is given by $SU(2)$
gauge group coupled with AD matters.

Relation with class ${\cal S}$ theory: Let’s recall the geometric
representation of a theory in class ${\cal S}$ theory Gaiotto:2009we ;
Xie:2012hs . The theory can be either a SCFT or an asymptotical free gauge
theory. The theory is represented by a Riemann surface $\Sigma$ with regular
punctures and irregular punctures, and the gauge theory is given by finding a
pants decomposition of $\Sigma$. Notice that the AD matter is represented by a
sphere with one irregular and one regular singularity, and the component of AD
theory is living at the boundary of the pants decomposition Xie:2012hs ;
$T_{n}$ matter are represented by a punctured three sphere, see figure. 24.
The picture of the IR theory found above is very similar to class ${\cal S}$
theory: the weighted graph is similar to the pants decomposition of the
Riemann surface $\Sigma_{g}$, and the difference is that: a): We get SCFT and
IR free gauge theory; b): The matter is represented by periodic map on an
irreducible component in the cut system; c): The gauge group is also
represented by edges, but there is an extra integer $K$, which gives rise to
free hypermultiplets coupled with gauge group. In the next section, we will
show how to use our result to describe UV complete theories.

$T_{1}$$G_{1}$$T_{2}$$G_{2}$$T_{3}$$g_{1}$$g_{2}$$g_{3}$$m_{1}$$k_{1}$$m_{2}$$k_{2}$$f_{1}$$f_{2}$$f_{3}$$T_{1}$$G_{1}$$T_{2}$$G_{2}$$T_{3}$$k_{1}$$k_{2}$
Figure 24: Up: The representation of a UV complete theory in class ${\cal S}$
description: it is given by a Riemann surface with regular and irregular
singularities; and the gauge theory is given by the pants decomposition of the
punctured Riemann surface and the irregular singularity stays at the boundary;
Bottom: The representation of a IR free theory by a pseudo-periodic map, here
one need to specify a weighted graph and the corresponding periodic map; there
is an extra integer on the edges of the weighted graph.

##  5 UV theory

In last fewer sections, the IR theory associated with the degeneration is
studied. In this section, we will study the global SW geometry (on a one
parameter slice on the whole Coulomb branch). A point is added at $\infty$ to
make the Coulomb branch compact, i.e. $\mathbb{P}^{1}$. The property of $UV$
theory is then reflected by the singular fiber at $\infty$ Xie:2021hxd , see
figure. 25.

$\infty$ Figure 25: The global structure of SW geometry at Coulomb branch.

Geometrically, we now have a compact surface $S$ fibered over
$\mathbb{P}^{1}$: $f:S\to\mathbb{P}^{1}$, and there are singular fibers at
$\infty$ and bulk. Two further assumptions on the fiberation are made: a)
First the surface $S$ is a rational surface; b): the bulk singularities are
atomic, which means it can no longer be deformed. In our case, the atomic
singular fiber are classified into following kinds:

  1. 1.

There is a single non-separating curve. The weighted graph for it has a single
node with $r=1$ (non-amphidrome and $K=1$), and the periodic map is trivial.
The low energy theory is $U(1)$ gauge theory coupled with one massless
hypermultiplet, and $g-1$ free vector multiplets. This is denoted as $A_{1}$
singularity.

  2. 2.

There is a single separating curve, and it separates a genus $g$ curve into a
genus $h$ and a genus $g-h$ curve, see figure. 26 for the illustration. The
link number is also $K=1$, and this singularity is denoted as $A_{1}^{h}$
singularity. The low energy theory is $g$ free vector multiplets, and one free
hypermultiplet (which does not charged under any $U(1)$ gauge group).

$1)$$2)$ Figure 26: Up: The atomic singularity corresponding to non-separating
curve; Bottom: The atomic singularity corresponding to separating curve.

In the following, we mainly consider the case where the bulk singularities are
all of the $A_{1}$ type, and the study of general atomic singularities is left
for separate publications.

Topological invariants: We’d like to study the global topological constraint
on the singular fiber at $\infty$ for the UV theory in dimension $4,5,6$.
Crucially, one can define two invariants $d_{x},\delta_{x}$ for a singular
fiber. $d_{x}$ is defined using holomorphic data, and $\delta_{x}$ is the
topological invariant. $d_{x}$ is defined for genus one and genus two cases
kenji1988discriminants , and it might be defined for arbitrary genus. However,
$d_{x}$ is in general not easy to compute, and details will be discussed
elsewhere. On the contrary, the topological invariant $\delta_{x}$ can be
computed from the dual graph (the normally minimal model). In general, $d_{x}$
is not equal to $\delta_{x}$ (Though they are equal for a large class of
singular fibers). Given a singular fiber $F$ with its normally minimal model,
first one one get a reduced curve $F_{red}$, and its topological Euler number
$\delta_{x}$ is computed from $F_{red}$ as follows tan1994invariants :

$\delta_{x}=2(g-p_{a}(F_{red}))+\\#intersections$ (7)

here $p_{a}$ denotes the geometric genus, which is given as
$p_{a}(F_{red})=1+\frac{1}{2}(F_{red}^{2}+F_{red}\cdot K)$. Let’s now examine
the behavior of $p_{a}(F_{red})$, which will have interesting implications for
the UV singular fiber. Given $F_{red}=\sum_{i=1}^{k}C_{i}$, then

$\displaystyle p_{a}(F_{red})=1+\frac{1}{2}(F_{red}^{2}+F_{red}\cdot
K)=\sum_{i}[1+\frac{1}{2}(C_{i}^{2}+K_{i})]+\sum_{i<j}C_{i}\cdot C_{j}-k+1$
$\displaystyle=\sum_{i}(p_{a}(C_{i}))+\sum_{i<j}C_{i}\cdot C_{j}-k+1.$

Since $p_{a}(C_{i})\geq 0$ and $\sum_{i<j}C_{i}\cdot C_{j}\geq k-1$, and the
equality holds when: a) all the curves $C_{i}$ is a rational curve; b): all
the curves intersect transversely and they form a tree!

One actually need to compute the topological Euler number for relative minimal
model (no $-1$ rational curve). The way of computing is following: first one
compute $\delta_{x}$ for normal model, and each contraction of $-1$ curve
would reduce the Euler number by $1$. Here are some comments on the Euler
number $\delta_{x}$:

  1. 1.

If the dual graph consists of only spheres, and it has $k$ loops, then the
topological Euler number is given as

$\delta_{x}=n_{t}+2g-1-k;$

Here $n_{t}$ is the number of components in the dual graph. This relation
still holds after contracting $-1$ curve.

  2. 2.

If the dual graph can be regarded as the 3d mirror for a 4d theory (a tree
graph and no modification is needed), the Euler number is related to the
physical data as follows

$\delta_{x}=f+2g;$

Here $f$ is the rank of flavor symmetries. If the modification on dual graph
is needed, the rank of flavor symmetry is smaller than the number
$\delta_{x}-2g$.

Let’s now consider global topological constraints for the fibered surface.
Given a genus $g$ fiberation $f:S\to\mathbb{P}^{1}$, one can define the
following relative invariants

$\displaystyle K_{f}^{2}=c_{1}^{2}(S)+8(g-1),$ $\displaystyle
e_{f}=c_{2}(S)+4(g-1),$ $\displaystyle\chi_{f}=\chi({\cal O}_{S})+(g-1),$
$\displaystyle q_{f}=q(S).$

Here $c_{1}(S)$ ($c_{2}(S)$) are the first (second) Chern class, ${\cal O}$ is
structure sheaf, and $q(S)$ is the irregularity of $S$. The relative
invariants can be expressed by the local data of singular fibers
lu2013inequalities :

$\displaystyle K_{f}^{2}=\kappa(f)+\sum_{i}c_{1}^{2}(F_{i}),$ $\displaystyle
e_{f}=\delta(f)+\sum_{i}c_{2}(F_{i}),$
$\displaystyle\chi_{f}=\lambda(f)+\sum_{i}\chi(F_{i}).$

Here $c_{2}(F)$ and $c_{1}(F)$ are the local Chern numbers, and
$\kappa(f),\delta(f),\lambda(f)$ are modular invariants which are zero for
isotrivial family. We are interested in rational surface which has
$q=p_{g}=0$, and so according to Noether’s theorem:
$c_{2}(S)+c_{1}^{2}(S)=12$. The local data satisfies the following equation:

$K_{f}^{2}+e_{f}=12g\to\kappa(f)+\delta(f)+\sum_{i}(c_{1}^{2}(F_{i})+c_{2}(F_{i}))=12g.$
(8)

The crucial data for the singular fiber is
$\chi(F)=\frac{1}{12}(c_{1}^{2}(F_{i})+c_{2}(F_{i}))$, which can be computed
using the normal minimal model $F=\sum n_{i}C_{i}$:

$\displaystyle\chi(F)=\frac{1}{2}(g-p_{a}(F_{red}))+\sum_{i<j}\chi(n_{i},n_{j})C_{i}\cdot
C_{j},$
$\displaystyle\chi(n_{i},n_{j})=3-\frac{(n_{i},n_{j})^{2}}{n_{i}n_{j}}-\frac{n_{j}}{n_{i}}-\frac{n_{i}}{n_{j}}$
(9)

Here the bracket $(n_{i},n_{j})$ denotes the greatest common divisor for
$n_{i},n_{j}$. Notice that this invariant is not changed by blow-up or
contraction, and so it is the same for relative minimal model.

The above formula 8 has limited use for us as: a): It involves global modular
invariants $\kappa(f),\delta(f),\lambda(f)$ which are not easy to compute; b):
The data $\chi(F)$ is not conserved on deformation. For our purpose, an
invariant which is preserved under splitting of singularity is needed. Indeed,
such invariant exists, and is called local signature arakawa2001local :

$\sigma(F)=\frac{1}{2g+1}(gh_{x}-(g+1)\delta_{x})$

$h_{x}$ 888This number is equal to $d_{x}-\delta_{x}$ for rank one and rank
two case, and we conjecture that it holds for general case. is an invariant
which measures the number of $A_{1}^{h}$ type singularities under deformation.
$h_{x}=0,\delta_{x}=1$ for a $A_{1}$ singularity, and so
$\sigma(F)=-\frac{g+1}{2g+1}$. The signature is preserved for the splitting of
singularity, i.e. if $F$ is split into several singularities
$F_{1},\ldots,F_{s}$, then

$\sigma(F)=\sum_{i}\sigma(F_{i}).$

In particular, if $h_{x}=0$ for a singular fiber $F$, then the number of
$A_{1}$ singularities under the generic splitting is equal to $\delta_{x}$! On
the other hand, if a singular fiber $F$ is split into only $A_{1}$
singularities, then $h_{x}=0$.

Moreover, if the fiberation is hyperelliptic, i.e. the generic fiber is a
hyperelliptic surface, the signature of surface $S$ is the the sum of that of
local singular fibers arakawa2001local :

$\sigma(S)=\sum_{i}\sigma(F_{i}).$

For a rational surface, the Hodge index theorem gives the signature of the
surface as $(1,\rho-1)$ with $\rho$ the Picard number, and so
$\sigma(S)=2-\rho$. Therefore, one get following equation:

$\boxed{2-\rho(S)=\sum_{i}\sigma(F_{i})}.$ (10)

The Picard number for a rational surface is related to relative invariant as
follows (using Noether’s theorem):

$\rho(S)=8g+2-K_{f}^{2}.$

and $K_{f}^{2}$ has a lower bound $4g-4$ which implies an upper bound for
$\rho(S)\leq 4g+6$.

Flavor symmetry and Mordell-Weil lattice: Given a higher genus fiberation of
an algebraic surface, one can define a Mordell-Weil lattice shioda1999mordell
. This lattice is closed related to flavor symmetry of the physical theory,
which is studied in rank one Caorsi:2018ahl ; Argyres:2018taw  and rank two
case Xie:2022aad . Here we summarize some important properties of this
lattice. First, the rank is equal to shioda1999mordell :

$r=\rho-2-\sum(n_{t}-1).$ (11)

Here $\rho$ is the Picard number of $S$. For the rational fibered surface $S$
which is hyperelliptic saito1994mordell , the Picard number is $\rho=4g+6$ for
$K_{f}^{2}=4g-4$, and the MW rank is

$r_{MW}=4g+4-\sum(n_{t}-1).$ (12)

The maximal MW rank is $4g+4$ if all the singular fiber has $n_{t}=1$, and the
corresponding surface is constructed in saito1994mordell , see figure. 27 for
the dual graph for the lattice.

$g+1$$1$$2$$3$$4$$4g+2$$4g+3$$4g+4$$2$$2$$2$$2$$2$$2$ Figure 27: The maximal
Mordell-Weil lattice for a rational surface with genus $g$ fiberation. The
first node has self-intersection number $g+1$, and other node has self-
intersection number $2$.

###  5.1 4d theory

Let’s make following assumptions for the generic deformation of a 4d theory:
a): there is one special singular fiber at $z=\infty$; b) there are only
$A_{1}$ singularities at the bulk. We’d like to determine the topological
constraint on the fiber at $\infty$ so that it can define a sensible UV theory
in dimension 4.

Let’s first assume that the fiberation is hyperelliptic and the Picard number
is $\rho(S)=4g+6$. Following the same argument in Xie:2022aad , the
topological constraint for $F_{\infty}$ so that it defines a 4d theory is

$\boxed{\delta_{\infty}-n_{\infty}=2g-1}.$ (13)

Let’s recall the argument leading to above equation. The basic assumption is
that there is one BPS particle associated with each bulk $A_{1}$ singularity,
and they generate the whole charge lattice 999Notice that in general only a
subset of these BPS particles are needed to generate the full charge lattice.,
and so

$f+2g=\\#A_{1};$ (14)

Here $f$ is the number of mass parameters. Now the number of mass parameters
is given by the rank of Mordell-Weil lattice (see formula. 11):

$f=r_{MW}=\rho-2-(n_{\infty}-1).$

Finally, the number of bulk $A_{1}$ singularities and the topological
invariant $\delta_{\infty}$ (assuming $h_{\infty}=0$) satisfy the equation
(see formula. 10):

$\frac{g+1}{2g+1}(\\#A_{1}+\delta_{\infty})=\rho-2.$

Substituting expression for $f$ and $\\#A_{1}$ into equation. 14, we have
($\rho=4g+6$):

$4g+4-(n_{\infty}-1)+2g=8g+4-\delta_{\infty}\to\delta_{\infty}-n_{\infty}=2g-1.$

In the general case, $f$ is not the rank of physical flavor symmetry, but is
given by rank of MW group, see formula 11. Then one has the same constraint
for the fiber at $\infty$. The above result then implies that the dual graph
for $F_{\infty}$ has to be a tree of rational curves!

Motivated by the above results for hyperelliptic fiberation, we conjecture
that the singular fiber at $\infty$ for 4d theory has to satisfy the condition
13.

Comments: We have used the correspondence of dual graph and 3d mirror to
constrain the possible IR theory. In the UV case, this constraint can be still
imposed on the dual fiber for $F_{\infty}$! The dual fiber is defined for
periodic map, and one can define it for general fiber by taking the dual of
the periodic part of the map.

SCFT: Let’s consider 4d SCFT which is given by a periodic map with singular
fiber $F$. There is a natural candidate for the fiber at $\infty$: the dual
periodic map $F^{{}^{\prime}}$. We first consider a fibered surface with just
two singular fibers $F$ and $F^{{}^{\prime}}$, and the topological constraints
are automatically satisfied, see equation. 8. Then one can deform the fiber
$F$ into $A_{1}$ singularities to get the global SW geometry.

Example: Let’s consider $(A_{1},A_{n-1})$ theory with $n$ even, and the
corresponding periodic map for fiber $F$ is
$(n,g^{{}^{\prime}}=0,\frac{1}{n}+\frac{1}{n}+\frac{n-2}{n}),~{}~{}g=\frac{n-2}{2}$
(the Euler number is $\delta_{x}=n-1$), the dual fiber is given by the data
$(n,g^{{}^{\prime}}=0,\frac{n-1}{n}+\frac{n-1}{n}+\frac{2}{n})$ (the Euler
number is $\delta_{x}=3n-3$). The total Euler number satisfies the condition
$\delta_{x}(F)+\delta_{x}(F^{{}^{\prime}})=4n-4=8g+4$. This implies that: a):
The global curve can be given by the fiberation of hyperelliptic curves; b):
$d_{x}=\delta_{x}$ for $F$ and $F^{{}^{\prime}}$; c): The generic deformation
of the theory has only $A_{1}$ singularities, which agrees with the result
using the singularity theory (one can engineer the SCFT by an isolated
hypersurface singularity Xie:2015rpa , and generic deformation of singularity
has only $A_{1}$ singularities).

One can similarly find the singular fiber at $\infty$ for other SCFTs
discussed in section. 4.3: one simply choose the dual periodic map in the
decomposition of $F$. See figure. 28 for an example.

$F^{{}^{\prime}}$$F$$1)$$F^{{}^{\prime}}$$2)$$F^{{}^{\prime}}$$F=$$F^{{}^{\prime}}=$$2$$1$$l=2$$1$$f=\frac{3}{4}+\frac{1}{2}+\bm{\frac{3}{4}},~{}~{}K=0$$2$$1$$l=2$$1$$f=\frac{1}{4}+\frac{1}{2}+\bm{\frac{1}{4}},~{}~{}K=0$
Figure 28: (1): Left: The global geometry of a SCFT defined by periodic map
$F$, and $F^{{}^{\prime}}$ is the dual map; They can sit together to get an
isotrivial family; Right: $F$ can be deformed into several $A_{1}$
singularities which give rise to global SW geometry for the theory $F$. (2): A
SCFT is represented by a fiber $F$, and the fiber at infinity is given by its
dual $F^{{}^{\prime}}$.

Global curve: The global curves for various SCFTs have been found for many
class of theories, and they are useful to verity our general strategy of
classifying theories. Here we give the global curve for $D_{2}(SU(n+1)$
theory: when $n$ is odd, it is just $SU(\frac{n+1}{2})$ coupled with $n+1$
fundamental flavor; When $n$ is even, it is an interacting AD theory with
flavor symmetry group $SU(n+1)$ Xie:2012hs ; cecotti2013infinitely . The
global curve takes the following simple form

$y^{2}=x^{n+1}+p_{2}x^{n-1}+\ldots+p_{n}x+p_{n+1}+t^{2}.$

Here $t$ is the Coulomb branch operator (the scaling dimension is
$[t]=\frac{n+1}{2}$). The Mordell-Weil lattice is $A_{n}$ ($A_{n}\oplus U(1)$)
for $n$ even ($n$ odd) shioda1999mordell , which is in consistent with the
flavor symmetry of the physical theory.

Asymptotical free theory: Let’s now give some global SW geometries for
asymptotical free theory.

$SU(n)$ gauge theory coupled with $n_{f}$ flavor: The singular fiber at
$\infty$ is given by the data shown in figure. 29. This gives the SW geometry
for $SU(n)$ gauge theory coupled with $n_{f}=2n-K,~{}K\geq 1$ hypermultiplets
in fundamental representation. A simple check is to compute the topological
Euler number $\delta_{x}$ (which is conjectured to be the same as the
holomorphic invariant $d_{x}$), and the answer is

$d_{x}=\delta_{x}=2n+K+2(n-1)=4n+K-2.$

So the number of $A_{1}$ singularities at the bulk are given as (since the
global curve is hyperelliptic):

$8(n-1)+4-(4n+K-2)=2(n-1)+2n-K=2r+f.$

The singular fiber for $Sp(2n)$ gauge theory coupled with $n_{f}=2n+2-K$
hypermultiplets in fundamental representation is also shown in figure. 29.

$n$$n$$n$$n$$n$$1,2,..,n-1$$1$$1$$n-1,..,2,1$K-1$n$$r=n$$f=\bm{1}+\frac{1}{n}+\frac{n-1}{n}$$f=(\mathrm{1})+\frac{1}{2n}+\frac{2n-1}{2n}$$2n$$2n$$2n$$2n$$n$$n$$1$$2n-1,..,2,1$K-1
Figure 29: Upper: The singular fiber at $\infty$ for $SU(n)$ gauge theory with
$2n-K$ flavors; The weighted graph is given by two genus zero curve connected
by an edge with multiplicity $n$. The periodic map on each genus zero
component is $\frac{1}{n}+\frac{n-1}{n}+\bm{1}$. Bottom: The singular fiber at
$\infty$ for $Sp(2n)$ gauge theory with $2n+2-K$ flavor; The weighted graph is
given by a single component with $r=n$ cutting curves, which are amphidrome.
The periodic map on the irreducible components are
$(1)+\frac{1}{2n}+\frac{n-1}{2n}$.

Pure $SU(n)$ gauge theory: Let’s consider pure $SU(n)$ gauge theory in more
detail, and there are a total of $2n-2$ $A_{1}$ singularities at the bulk.
We’d like to determine other special vacua by simple topological constraints:
a) The Euler number is less than $2n-2$; b) There is no flavor symmetry so the
dual graph has just one node. Here are some obvious possibilities:

  1. 1.

The degeneration is given by the weighted graph $(g=n-1,r=g)$, namely there
are $n-1$ non-separating cutting curves, see figure. 30. The periodic map on
the component $\Sigma/C$ is trivial. To have no flavor symmetry, the linking
number $K=1$, so the IR theory is direct sum of $U(1)$ with one
hypermultiplet: $(U(1)\oplus 1)^{n-1}$. The Euler number of this vacua is
$n-1$. This particular vacua is important as it gives the confining vacua for
the corresponding $\mathcal{N}=1$ theory.

  2. 2.

The degeneration gives AD theory and free vector multiplets. Depending $n$,
the AD theory might have a $U(1)$ global symmetry which would be gauged.

$......$$1)$$2)$$g=\frac{n-1}{2}$$n\ odd$$g=\frac{n-1}{2}$$f=\left(n,\
g^{{}^{\prime}}=0,\bm{\frac{1}{2n}}+\frac{1}{2}+\frac{n-1}{2n}\right)$$f=id$$n\
even$ $f=\left(n,\
g^{{}^{\prime}}=0,\bm{\frac{1}{n}}+\bm{\frac{1}{n}}+\frac{n-2}{n}\right)$$f=id$$g=\frac{n-2}{2}$$g=\frac{n-2}{2}$
Figure 30: (1): The special vacua for pure $SU(n)$ theory, and there are $g$
non-separating curves in the cut system and all the associated integers $K=1$.
The IR theory is just g copies of $U(1)$ coupled with one hypermultiplet; (2)
The singular fiber corresponding to AD theory plus free vector multiplets.

$\mathcal{N}=2^{*}$ $SU(n)$ gauge theory: Although we mainly focus on the case
where the bulk singularities are just $A_{1}$ singularities. It is actually
possible to get the SW solution for other theories by considering more general
un-deformable singularities. Notice that the local structure of the
singularities are still classified by the result of this paper: one just do
not deform those singularities into $A_{1}$ type. We leave the general
discussion to a separate publication, and here give the results for
$\mathcal{N}=2^{*}$ $SU(n)$ gauge theory. The fiber at $\infty$ should give a
periodic map, which gives the scaling dimensions $2,3,\ldots,n$ Xie:2021hxd .
The natural candidate is the one for $SU(n)$ with $n_{f}=2n$, namely, it is
given by the period map with data $(n,g^{{}^{\prime}}=0,{1\over n}+{1\over
n}+{n-1\over n}+{n-1\over n})$. For the bulk singularities, there are
following two constraints a): The topological constraints, namely,
$\sum_{bulk}\delta_{x}=4n-2$; b): Dirac quantization Argyres:2015ffa ; c): The
rank of charge lattice is $2n-1$ and so there would be at least $2n-1$
singularities. By using above constraints, one can conclude that there are
precisely $2n-1$ $A_{2}$ singularities. Notice $A_{2}$ singularities are
described by a genus $n-1$ curve with a single cut (the periodic map on the
remaining part is trivial), and the integer along the cut is $K=2$.

$\mathcal{N}=2^{*}$ $Sp(2n)$ gauge theory: The fiber at $\infty$ is given by
periodic map with data $(n,g^{{}^{\prime}}=0,{1\over 2n}+{1\over 2}+{1\over
2}+{2n-1\over 2n})$, and its topological Euler number is $4n+2$. Again we
conjecture that the bulk singularities are of the un-deformable $A_{2}$
singularities, and the number is $2n+1$. This is in perfect agreement with the
dimension of charge lattice (there is only one mass parameter).

Linear quiver: Let’s consider the UV complete linear quiver
$n_{1}-SU(n)-SU(n)-n_{2}$, here $n_{1},n_{2}<n$. It is natural to state that
the degeneration at $\infty$ should be the one shown in figure. 31, the data
is $n_{1}=n-K_{1},~{}~{}n_{2}=n-K_{2}$. Since the SW geometry is no longer
hyperelliptic and the Picard number is not known, and so one do not know the
number of $A_{1}$ singularities at the bulk. We notice that there is a SCFT by
taking $n_{1}=n_{2}=n$, and the total Euler number on the global SW geometry
can be easily computed: the corresponding periodic map is self-dual (with
valency data $(\frac{1}{n})^{3}+(\frac{n-1}{n})^{3}$, and $\delta_{x}=7n-4$),
so the total Euler number is $e_{total}=14n-8$. We assume that the total Euler
number should be still the same, and this gives rise to the number of bulk
$A_{1}$ singularities as

$\displaystyle\\#A_{1}=14n-8-(7n+2n-n_{1}-n_{2}-4)$
$\displaystyle=5n+n_{1}+n_{2}-4=2r+f+(n-1).$

Here $r=2n-2$ is the rank of the theory, $f=n_{1}+n_{2}+1$ are the number of
mass parameters.

$n$$n$$f=\frac{1}{n}+\frac{n-1}{n}+\bm{1}$$f=\frac{1}{n}+\frac{n-1}{n}+\bm{1}+\bm{1}$$f=\frac{1}{n}+\frac{n-1}{n}+\bm{1}$$1$$1,2,..,n-1$$n$$1$$n-1,..,1$$n$$n$$1$$n-1,..,1$$K_{1}-1$$K_{2}-1$
Figure 31: Up: the pseudo-periodic map for a UV complete linear quiver theory;
Bottom: the dual graph and the topological Euler number is
$\delta_{x}=7n+K_{1}+K_{2}-4$.

###  5.2 5d theory

Let’s now turn to the SW geometry for 5d KK theory: the Coulomb branch
solution for 5d $\mathcal{N}=1$ theory compactified on a circle with finite
size. The difference with 4d SW geometry is that the charge lattice has
dimension $\Gamma=2r+f+1$, where the extra one denotes the KK charge.
Following similar argument as 4d theory, the topological constraint for a
singular fiber at $\infty$ of a 5d theory is:

$\delta_{\infty}-n_{\infty}=2g-2.$

Hence the dual graph of $F$ satisfies following condition: a) All the
components are all rational curves; b) It has just one loop; c): The dual
fiber is good in the sense that one can get a good 3d mirror.

Example: Let’s consider several class of possible UV singular fibers for 5d
rank 3 KK theory, see figure. 32. Some of them might be related to 5d theory
engineered using toric diagram (such that there is a $SU(4)$ gauge theory
interpretation along certain special locus.) Xie:2017pfl . A check is that one
can compute the eigenvalues of monodromy around infinity using the method
explained in Xie:2022aad , and there are two eigenvalues $1$, and other
eigenvalues are roots of order $3$; on the other hand, for the pseudo-periodic
map shown in figure. 32, the induced monodromy on homology cycles also has two
eigenvalues $1$, and other eigenvalues are of order 3 (the order of the
periodic map is $3$.).

$3$$r=1$$f=\frac{2}{3}+\frac{2}{3}+\left(\frac{1}{3}\right)+\left(\frac{1}{3}\right)$$f=\frac{1}{3}+\frac{2}{3}+\left(\frac{1}{3}\right)+\left(\frac{2}{3}\right)$$f=\frac{1}{3}+\frac{1}{3}+\left(\frac{2}{3}\right)+\left(\frac{2}{3}\right)$$3$$2,1$$2,1$$1$$1$$1$$1$$K-1$$3$$2,1$$2$$1$$1$$1$$K-1$$3$$1$$1$$1$$1$$1$$1$$K-1$$1$$1$$2$$2$$Dual\
graph$$G$$A_{5}\oplus A_{K-1}\oplus\delta^{2}$$A_{4}\oplus
A_{K-1}\oplus\delta\oplus\delta_{1}^{2}$$A_{3}\oplus
A_{K-1}\oplus\delta_{1}^{4}$ Figure 32: Singular fiber which might give rise
to Coulomb branch geometry for 5d $\mathcal{N}=1$ KK theories. Here $\delta$
has self-intersection number $-4$, and $\delta_{1}$ has self-intersection
number $-3$.

###  5.3 6d theory

Let’s now turn to the SW geometry for 6d KK theory: the Coulomb branch
solution for 6d $(1,0)$ theory compactified on a torus with finite size. The
difference with 4d SW geometry is that the charge lattice has dimension
$\Gamma=2r+f+2$, where the extra two denotes the KK charge along two cycles of
the torus. Following similar argument as 4d theory, the topological constraint
for a 4d theory is

$\delta_{\infty}-n_{\infty}=2g-3.$

Hence the dual graph of $F$ satisfies following condition: a) All the
components are all rational curves; b) It has just two loops; c): The dual
fiber is good in the sense that one can get a good 3d mirror for it.

Example: A class of possible UV singular fiber for 6d rank 3 KK theory are
shown in figure. 33.

$3$$r=2$$2$$f=\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)$$1$$1$$1$$1$$1$$1$$1$$1$$K_{1}-1$$K_{2}-1$$G=A_{1}\oplus
A_{K_{1}-1}\oplus A_{K_{1}-1}\oplus\delta_{1}^{4}$ Figure 33: Singular fiber
that might give rise to the SW geometry for rank three 6d $(1,0)$ KK theories.

##  6 Conclusion

We give a classification of IR and UV behavior for Coulomb branch geometry of
theories with eight supercharges, with the following assumptions: a) the local
SW solution is given by the fiberation of genus $g$ curves; b): The theory is
associated with the generic deformation of the singularity; c): A one
parameter slice of SW geometry is taken. The crucial fact is that such
degeneration is classified by the conjugacy class of mapping class group
matsumoto2011pseudo , which are the so-called pseudo-periodic map of negative
type. The description of such conjugacy class is combinatorial, which makes a
systematical study possible.

To determine the IR theory (associated with the singular fiber at bulk) or the
UV theory (associated with the singular fiber at $\infty$), we find the dual
graph from the pseudo-periodic map play a crucial role: they are closed
related to 3d mirror for these theories. Using the dual graph and the related
3d mirror, we can identify SCFTs (such as Argyres-Douglas theories, $T_{n}$
theories), IR free theories, and UV complete theories with certain conjugacy
class of mapping class group! Given the simplicity of the description, we
believe that many physical questions about theories with eight supercharges
can be solved! In this paper, it helps us explain the number of $A_{1}$
singularities in the generic deformation of $T_{n}$ theories, and confirms the
3d mirror proposal for Argyres-Douglas theories Xie:2021ewm .

There are several further interesting questions that one can study:

  1. 1.

The usual electric-magnetic duality group acts on photon coupling, and it
turns out that this action along is not enough to determine the IR theory. The
duality group is interpreted geometrically as the action on homology groups.
Now if one look at enlarged action and consider the conjugacy class in mapping
class group, then the IR theory can be completely determined. The question is
the physical objects that the mapping class group acts on. Since mapping class
group naturally acts on the simple closed curves on Riemann surface, which is
in turn gives the IR line operators, this suggests that one need to study the
action on the set of IR line operators when one go along the special vacua!

  2. 2.

It would be interesting to give an interpretation for the mysterious relation
between the dual graph and the 3d mirror. We have used the identification of
3d mirror to constrain the possible appearance of singular fiber. It would be
interesting to study the physical meaning of these constraints.

  3. 3.

We have studied one parameter slice of the full Coulomb branch geometry, and
it is certainly useful to study the full parameter space of SW geometry, for
instance, one can not find a one parameter scale-invariant geometry for a
SCFT, but it is possible to find a scale-invariant geometry for a multiple
parameter family Argyres:2022lah . More generally, one might find more
constraints on the appearance of singular fibers by looking at the full
deformation space of the physical theory.

  4. 4.

Certainly, the next major question is the classification of global SW geometry
(see Xie:2022lcm  for rank one case). We hope to report the progress on this
question in the near future.

  5. 5.

In our study, the IR theory is assumed to be associated with generic
deformation. To give a complete classification, one need to classify the
theory associated with the so-called non-generic deformation.

  6. 6.

We also assume the SW geometry is given by a family of Riemann surfaces. This
condition can be relaxed in following way: one can consider a finite group
action on the local family, and study the quotient. In particular, it is
possible to define a so-called “orientifold” action on the conjugacy class to
get quiver theories with orthosymplectic gauge groups. Details will appear in
a separate publication.

## Acklowledgement

DX would like to thank Prof. S. Y. Cheng for constant encouragement.

##  Appendix A Periodic maps for $1\leq g\leq 5$

Non-identical conjugacy classes of periodic maps of closed surfaces of genus
$1\leq g\leq 5$ are listed in table. 2,3,4,5,6.

Order | $Data$
---|---
$n=6$ | $(\frac{1}{6}+\frac{1}{3}+\frac{1}{2},\frac{5}{6}+\frac{2}{3}+\frac{1}{2})$
$n=4$ | $(\frac{1}{4}+\frac{1}{4}+\frac{1}{2},\frac{3}{4}+\frac{3}{4}+\frac{1}{2})$
$n=3$ | $(\frac{1}{3}+\frac{1}{3}+\frac{1}{3},\frac{2}{3}+\frac{2}{3}+\frac{2}{3})$
$n=2$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
Table 2: The periodic maps for genus $g=1$. The genus $g^{{}^{\prime}}=0$ is ignored. Order | $Data$
---|---
$n=10$ |  $(\frac{1}{10}+\frac{2}{5}+\frac{1}{2},\frac{9}{10}+\frac{3}{5}+\frac{1}{2})$, $(\frac{3}{10}+\frac{1}{5}+\frac{1}{2},\frac{7}{10}+\frac{4}{5}+\frac{1}{2})$,
$n=8$ | $(\frac{1}{8}+\frac{3}{8}+\frac{1}{2},\frac{7}{8}+\frac{5}{8}+\frac{1}{2})$
$n=6$ | $(\frac{1}{6}+\frac{1}{6}+\frac{2}{3},\frac{5}{6}+\frac{5}{6}+\frac{1}{3}),~{}~{}(\frac{1}{3}+\frac{2}{3}+\frac{1}{2}+\frac{1}{2})$
$n=5$ | $(\frac{1}{5}+\frac{1}{5}+\frac{3}{5},\frac{4}{5}+\frac{4}{5}+\frac{2}{5}),~{}~{}(\frac{1}{5}+\frac{1}{5}+\frac{2}{5},\frac{4}{5}+\frac{4}{5}+\frac{3}{5})$
$n=4$ | $\frac{1}{4}+\frac{3}{4}+\frac{1}{2}+\frac{1}{2}$
$n=3$ | $\frac{1}{3}+\frac{1}{3}+\frac{2}{3}+\frac{2}{3}$
$n=2$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
$g^{{}^{\prime}}=1,n=2$ | $\frac{1}{2}+\frac{1}{2}$
Table 3: The periodic maps for genus $g=2$. Order | $Data$
---|---
$n=14$ | $(\frac{3}{14}+\frac{2}{7}+\frac{1}{2},\frac{11}{14}+\frac{5}{7}+\frac{1}{2}),~{}~{}(\frac{1}{14}+\frac{3}{7}+\frac{1}{2},\frac{13}{14}+\frac{4}{7}+\frac{1}{2})~{}~{}(\frac{5}{14}+\frac{1}{7}+\frac{1}{2},\frac{9}{14}+\frac{6}{7}+\frac{1}{2})$
$n=12$ | $(\frac{1}{12}+\frac{5}{12}+\frac{1}{2},\frac{11}{12}+\frac{7}{12}+\frac{1}{2}),~{}~{}(\frac{1}{12}+\frac{1}{4}+\frac{2}{3},\frac{11}{12}+\frac{3}{4}+\frac{1}{3})~{}~{}(\frac{5}{12}+\frac{1}{4}+\frac{1}{3},\frac{7}{12}+\frac{3}{4}+\frac{2}{3})$
$n=9$ | $(\frac{1}{9}+\frac{5}{9}+\frac{1}{3},\frac{8}{9}+\frac{5}{7}+\frac{1}{2}),~{}~{}(\frac{1}{9}+\frac{2}{9}+\frac{2}{3},\frac{8}{9}+\frac{7}{9}+\frac{1}{3})~{}~{}(\frac{2}{9}+\frac{4}{9}+\frac{1}{3},\frac{7}{9}+\frac{5}{9}+\frac{2}{3})$
$n=8$ | $(\frac{1}{8}+\frac{5}{8}+\frac{1}{4},\frac{7}{8}+\frac{3}{8}+\frac{3}{4}),~{}~{}(\frac{1}{8}+\frac{1}{8}+\frac{3}{4},\frac{7}{8}+\frac{7}{8}+\frac{1}{4})~{}~{}(\frac{3}{8}+\frac{3}{8}+\frac{1}{4},\frac{5}{8}+\frac{5}{8}+\frac{3}{4})$
$n=7$ | $(\frac{1}{7}+\frac{1}{7}+\frac{5}{7},\frac{6}{7}+\frac{6}{7}+\frac{2}{7}),~{}~{}(\frac{1}{7}+\frac{2}{7}+\frac{4}{7},\frac{6}{7}+\frac{5}{7}+\frac{3}{7})~{}~{}(\frac{1}{7}+\frac{3}{7}+\frac{3}{7},\frac{6}{7}+\frac{4}{7}+\frac{4}{7})$
| $(\frac{2}{7}+\frac{2}{7}+\frac{3}{7},\frac{5}{7}+\frac{5}{7}+\frac{4}{7})$
$n=6$ | $(\frac{5}{6}+\frac{1}{6}+\frac{1}{2}+\frac{1}{2}),(\frac{5}{6}+\frac{1}{3}+\frac{1}{3}+\frac{1}{2},\frac{1}{6}+\frac{2}{3}+\frac{2}{3}+\frac{1}{2})$
$n=4$ | $(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4},\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}),~{}(\frac{1}{4}+\frac{1}{4}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2},\frac{3}{4}+\frac{3}{4}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}),(\frac{3}{4}+\frac{3}{4}+\frac{1}{2}+\frac{1}{2})$
$n=3$ | $(\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{1}{3},\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{2}{3})$
$n=2$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
$g^{{}^{\prime}}=1,n=4$ | $\frac{1}{2}+\frac{1}{2}$
$g^{{}^{\prime}}=1,n=3$ | $\frac{1}{3}+\frac{2}{3}$
$g^{{}^{\prime}}=1,n=2$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
$g^{{}^{\prime}}=2,n=2$ |  $f:\Pi\to\Pi^{{}^{\prime}}$ is an umramified covering.
Table 4: The periodic maps for genus $g=3$. Order | $Data$
---|---
$n=18$ | $(\frac{1}{2}+\frac{1}{9}+\frac{7}{18},\frac{1}{2}+\frac{8}{9}+\frac{11}{18}),~{}~{}(\frac{1}{2}+\frac{2}{9}+\frac{5}{18},\frac{1}{2}+\frac{7}{9}+\frac{13}{18})~{}~{}(\frac{1}{2}+\frac{4}{9}+\frac{1}{18},\frac{1}{2}+\frac{5}{9}+\frac{17}{18})$
$n=16$ | $(\frac{1}{2}+\frac{3}{16}+\frac{5}{16},\frac{1}{2}+\frac{13}{16}+\frac{11}{16}),~{}~{}(\frac{1}{2}+\frac{7}{16}+\frac{1}{16},\frac{1}{2}+\frac{9}{16}+\frac{15}{16})~{}~{}$
$n=15$ | $(\frac{1}{3}+\frac{1}{5}+\frac{7}{15},\frac{2}{3}+\frac{4}{5}+\frac{8}{15}),~{}~{}(\frac{1}{3}+\frac{2}{5}+\frac{4}{15},\frac{2}{3}+\frac{3}{5}+\frac{11}{15})~{}~{}(\frac{1}{5}+\frac{2}{3}+\frac{2}{15},\frac{4}{5}+\frac{1}{3}+\frac{13}{15})$
|
$(\frac{3}{5}+\frac{1}{3}+\frac{1}{15},\frac{2}{5}+\frac{2}{3}+\frac{14}{15})$
$n=10$ | $(\frac{1}{2}+\frac{1}{2}+\frac{2}{5}+\frac{3}{5}),~{}(\frac{2}{5}+\frac{3}{10}+\frac{3}{10},\frac{3}{5}+\frac{7}{10}+\frac{7}{10})~{}~{}(\frac{1}{10}+\frac{3}{10}+\frac{3}{5},\frac{9}{10}+\frac{7}{10}+\frac{2}{5})$
|
$(\frac{1}{10}+\frac{7}{10}+\frac{1}{5},\frac{9}{10}+\frac{3}{10}+\frac{4}{5})~{}~{}(\frac{1}{10}+\frac{1}{10}+\frac{4}{5},\frac{9}{10}+\frac{9}{10}+\frac{1}{5})$
$n=9$ | $(\frac{1}{9}+\frac{4}{9}+\frac{4}{9},\frac{8}{9}+\frac{5}{9}+\frac{5}{9}),~{}~{}(\frac{1}{9}+\frac{1}{9}+\frac{7}{9},\frac{8}{9}+\frac{8}{9}+\frac{2}{9})~{}~{}(\frac{2}{9}+\frac{2}{9}+\frac{5}{9},\frac{7}{9}+\frac{7}{9}+\frac{4}{9})$
$n=8$ | $(\frac{1}{2}+\frac{1}{2}+\frac{1}{8}+\frac{7}{8}),(\frac{1}{2}+\frac{1}{2}+\frac{3}{8}+\frac{5}{8})$
$n=6$ | $(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{3}+\frac{1}{6},\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{2}{3}+\frac{5}{6}),(\frac{1}{6}+\frac{1}{6}+\frac{1}{3}+\frac{1}{3},\frac{5}{6}+\frac{5}{6}+\frac{2}{3}+\frac{2}{3}),~{}~{}(\frac{5}{6}+\frac{1}{6}+\frac{2}{3}+\frac{1}{3})$
|
$(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{2},\frac{5}{6}+\frac{5}{6}+\frac{5}{6}+\frac{1}{2}),~{}(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{2}+\frac{1}{2},\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{1}{2}+\frac{1}{2})$
$n=5$ | $(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{2}{5},\frac{4}{5}+\frac{4}{5}+\frac{4}{5}+\frac{1}{5}),~{}~{}(\frac{2}{5}+\frac{2}{5}+\frac{2}{5}+\frac{4}{5},\frac{3}{5}+\frac{3}{5}+\frac{3}{5}+\frac{1}{5})~{}~{}$
|
$(\frac{2}{5}+\frac{2}{5}+\frac{3}{5}+\frac{3}{5})),(\frac{4}{5}+\frac{4}{5}+\frac{1}{5}+\frac{1}{5})$
$n=4$ | $(\frac{1}{4}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{3}{4}),~{}~{}(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{2}+\frac{3}{4},\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{1}{2}+\frac{1}{4})$
$n=3$ | $(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3},\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}),~{}(\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3})$
$n=2$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
$g^{{}^{\prime}}=1,n=6$ | $\frac{1}{2}+\frac{1}{2}$
$g^{{}^{\prime}}=1,n=4$ | $\frac{1}{4}+\frac{3}{4}$
$g^{{}^{\prime}}=1,n=3$ | $(\frac{1}{3}+\frac{1}{3}+\frac{1}{3},\frac{2}{3}+\frac{2}{3}+\frac{2}{3})$
$g^{{}^{\prime}}=1,n=2$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
$g^{{}^{\prime}}=2,n=3$ | Unramified covering
$g^{{}^{\prime}}=2,n=2$ | $\frac{1}{2}+\frac{1}{2}$
Table 5: The periodic maps for genus $g=4$. Order | $Data$
---|---
$n=22$ | $(\frac{1}{2}+\frac{1}{11}+\frac{9}{22},\frac{1}{2}+\frac{10}{11}+\frac{13}{22}),~{}~{}(\frac{1}{2}+\frac{2}{11}+\frac{7}{22},\frac{1}{2}+\frac{9}{11}+\frac{15}{22})~{}~{}(\frac{1}{2}+\frac{3}{11}+\frac{5}{22},\frac{1}{2}+\frac{8}{11}+\frac{17}{22})$
|
$(\frac{1}{2}+\frac{4}{11}+\frac{3}{22},\frac{1}{2}+\frac{7}{11}+\frac{19}{22}),~{}~{}(\frac{1}{2}+\frac{5}{11}+\frac{1}{22},\frac{1}{2}+\frac{6}{11}+\frac{21}{22})$
$n=20$ | $(\frac{1}{2}+\frac{1}{20}+\frac{9}{20},\frac{1}{2}+\frac{11}{20}+\frac{19}{20}),~{}~{}(\frac{1}{2}+\frac{3}{20}+\frac{7}{20},\frac{1}{2}+\frac{17}{20}+\frac{13}{20})$
$n=15$ | $(\frac{1}{3}+\frac{2}{15}+\frac{8}{15},\frac{2}{3}+\frac{13}{15}+\frac{7}{15}),~{}~{}(\frac{2}{3}+\frac{1}{15}+\frac{4}{15},\frac{1}{3}+\frac{14}{15}+\frac{11}{15})$
$n=12$ | $(\frac{1}{6}+\frac{5}{12}+\frac{5}{12},\frac{5}{6}+\frac{7}{12}+\frac{7}{12}),~{}~{}(\frac{1}{12}+\frac{1}{12}+\frac{5}{6},\frac{11}{12}+\frac{11}{12}+\frac{1}{6})$
$n=11$ | $(\frac{1}{11}+\frac{1}{11}+\frac{9}{11},\frac{10}{11}+\frac{10}{11}+\frac{2}{11}),~{}~{}(\frac{1}{11}+\frac{2}{11}+\frac{8}{11},\frac{10}{11}+\frac{9}{11}+\frac{3}{11})~{}~{}(\frac{1}{11}+\frac{3}{11}+\frac{7}{11},\frac{10}{11}+\frac{8}{11}+\frac{4}{11})$
|
$(\frac{1}{11}+\frac{4}{11}+\frac{6}{11},\frac{10}{11}+\frac{7}{11}+\frac{5}{11}),~{}~{}(\frac{1}{11}+\frac{5}{11}+\frac{5}{11},\frac{10}{11}+\frac{6}{11}+\frac{6}{11})~{}~{}(\frac{2}{11}+\frac{2}{11}+\frac{7}{11},\frac{9}{11}+\frac{9}{11}+\frac{4}{11})$
|
$(\frac{2}{11}+\frac{4}{11}+\frac{5}{11},\frac{9}{11}+\frac{7}{11}+\frac{6}{11}),~{}~{}(\frac{3}{11}+\frac{3}{11}+\frac{5}{11},\frac{8}{11}+\frac{8}{11}+\frac{6}{11})~{}~{}(\frac{2}{11}+\frac{3}{11}+\frac{6}{11},\frac{9}{11}+\frac{8}{11}+\frac{5}{11})$
|
$(\frac{3}{11}+\frac{4}{11}+\frac{4}{11},\frac{8}{11}+\frac{7}{11}+\frac{7}{11})$
$n=10$ | $(\frac{1}{2}+\frac{1}{2}+\frac{1}{10}+\frac{9}{10}),(\frac{1}{2}+\frac{1}{2}+\frac{3}{10}+\frac{7}{10})$
$n=8$ | $(\frac{1}{8}+\frac{1}{8}+\frac{1}{2}+\frac{1}{4},\frac{7}{8}+\frac{7}{8}+\frac{1}{2}+\frac{3}{4}),~{}(\frac{1}{2}+\frac{5}{8}+\frac{1}{8}+\frac{3}{4},\frac{1}{2}+\frac{3}{8}+\frac{7}{8}+\frac{1}{4})$
|
$(\frac{1}{2}+\frac{3}{8}+\frac{3}{8}+\frac{3}{4},\frac{1}{2}+\frac{5}{8}+\frac{5}{8}+\frac{1}{4})$
$n=6$ | $(\frac{1}{6}+\frac{1}{3}+\frac{1}{3}+\frac{2}{3}+\frac{1}{2},\frac{5}{6}+\frac{2}{3}+\frac{2}{3}+\frac{1}{3}+\frac{1}{2}),~{}(\frac{1}{6}+\frac{1}{6}+\frac{1}{2}+\frac{1}{2}+\frac{2}{3},\frac{5}{6}+\frac{5}{6}+\frac{1}{2}+\frac{1}{2}+\frac{1}{3})$
|
$(\frac{1}{3}+\frac{2}{3}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}),~{}(\frac{1}{6}+\frac{1}{6}+\frac{5}{6}+\frac{5}{6})$
$n=4$ | $(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{2}+\frac{1}{2},\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{1}{2}+\frac{1}{2}),~{}(\frac{1}{4}+\frac{1}{4}+\frac{1}{2}+\frac{1}{2}+\frac{3}{4}+\frac{3}{4})$
|
$(\frac{1}{4}+\frac{1}{4}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2},\frac{3}{4}+\frac{3}{4}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2})$
$n=3$ | $(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{2}{3}+\frac{2}{3},\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{1}{3}+\frac{1}{3})$
$n=2$ | $(\frac{1}{2})^{12}$
$g^{{}^{\prime}}=1,n=8$ | $\frac{1}{2}+\frac{1}{2}$
$g^{{}^{\prime}}=1,n=6$ | $\frac{1}{3}+\frac{2}{3}$
$g^{{}^{\prime}}=1,n=5$ | $\frac{1}{5}+\frac{4}{5}$
$g^{{}^{\prime}}=1,n=4$ | $(\frac{1}{4}+\frac{1}{4}+\frac{1}{2},\frac{3}{4}+\frac{3}{4}+\frac{1}{2}),~{}(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2})$
$g^{{}^{\prime}}=1,n=3$ | $\frac{1}{3}+\frac{2}{3}+\frac{1}{3}+\frac{2}{3}$
$g^{{}^{\prime}}=1,n=2$ | $(\frac{1}{2})^{8}$
$g^{{}^{\prime}}=2,n=4$ | Unramified covering
$g^{{}^{\prime}}=2,n=2$ | $(\frac{1}{2})^{4}$
$g^{{}^{\prime}}=3,n=2$ | Unramified covering
Table 6: The periodic maps for genus $g=5$.

##  Appendix B Rank 4 and rank 5 4d SCFTs from periodic map

We list the SCFTs associated with the periodic maps for genus 4 case (figure.
34) and genus 5 case (figure. 35).

$\frac{1}{2}+\frac{4}{9}+\frac{1}{18}$$1$$4$$(A_{1},A_{8})$$\frac{1}{2}+\frac{1}{16}+\frac{9}{16}$$1$$3$$(A_{1},D_{9})$$\frac{3}{5}+\frac{1}{15}+\frac{1}{3}$$(A_{2},A_{4})$$1$$2$$\frac{1}{10}+\frac{3}{10}+\frac{3}{5}$$\frac{2}{5}+\frac{3}{10}+\frac{3}{10}$$3$$3$$2$$\left(5,\frac{9}{2},3,\frac{3}{2}\right)$$4$$\frac{1}{10}+\frac{7}{10}+\frac{1}{5}$$1$$2$$1$$3$$\frac{1}{10}+\frac{1}{10}+\frac{4}{5}$$1$$1$$5$$\frac{1}{9}+\frac{4}{9}+\frac{4}{9}$$1$$4,3,2,1$$1,2,3,4$$\frac{2}{9}+\frac{2}{9}+\frac{5}{9}$$1$$2$$2$$\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{2}{5}$$1$$1$$1,2$$1$$4$$1$$3,2,1$$2$$\left(\frac{10}{3},\frac{7}{3},\frac{5}{3},\frac{4}{3}\right)$$\left(\frac{5}{2},\frac{7}{4},\frac{6}{4},\frac{5}{4}\right)$$(A_{1},A_{9})$$D_{2}(SU(9))$$\frac{1}{9}+\frac{1}{9}+\frac{7}{9}$$1$$1$$1$$4$$(A_{1},D_{10})$$2$$\left(3,\frac{8}{3},\frac{5}{3},\frac{4}{3}\right)$$1$$\left(2,\frac{7}{2},\frac{5}{2},\frac{3}{2}\right)$
Figure 34: Rank 4 SCFTs and periodic map. Left: the data for periodic map;
Middle: 3d mirror from dual graph; Right: the name for known theories or the
scaling dimensions.
$\frac{1}{2}+\frac{5}{11}+\frac{1}{22}$$1$$5$$(A_{1},A_{10})$$\frac{1}{2}+\frac{1}{20}+\frac{9}{20}$$1$$4$$(A_{1},D_{11})$$\frac{2}{3}+\frac{1}{15}+\frac{4}{15}$$1$$3$$(J_{3}^{4},S)$$1$$2$$1$$3$$\frac{1}{12}+\frac{1}{12}+\frac{5}{6}$$1$$1$$6$$(A_{1},A_{11})$$\frac{1}{11}+\frac{1}{11}+\frac{9}{11}$$1$$1$$5$$(A_{1},A_{11})$$1$$\frac{1}{11}+\frac{2}{11}+\frac{8}{11}$$1$$2$$2$$1$$3$$\frac{11}{4},\frac{10}{4},\frac{9}{4},\
\frac{7}{4},\frac{6}{4}$$\frac{1}{11}+\frac{3}{11}+\frac{7}{11}$$1$$3$$3$$2$$2$$1$$1$$\frac{11}{3},\frac{8}{3},\frac{7}{3},\
\frac{5}{3},\frac{4}{3}$$\frac{1}{11}+\frac{4}{11}+\frac{6}{11}$$1$$2$$1$$1$$3$$1$$\frac{11}{5},\frac{9}{5},\frac{8}{5},\
\frac{7}{5},\frac{6}{5}$$\frac{1}{11}+\frac{5}{11}+\frac{5}{11}$$1$$5,4,3,2,1$$1,2,3,4,5$$1$$D_{2}(SU(11))$$\frac{1}{8}+\frac{1}{8}+\frac{1}{4}+\frac{1}{2}$$1$$1$$2$$\frac{8}{3},\frac{6}{3},\frac{5}{3},\
\frac{4}{3},\frac{4}{3}$ Figure 35: Rank 5 SCFTs and periodic maps. Left: the
data for periodic map; Middle: 3d mirror from dual graph; Right: the name for
known theories or the scaling dimensions.

## References

  * (1) N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B426 (1994) 19–52, [hep-th/9407087]. [Erratum: Nucl. Phys.B430,485(1994)].
  * (2) N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B431 (1994) 484–550, [hep-th/9408099].
  * (3) D. Xie and D. Zhang, Mixed Hodge structure and $\mathcal{N}=2$ Coulomb branch solution, arXiv:2107.1118.
  * (4) D. Xie, On rank two theories with eight supercharges part I: local singularities, arXiv:2212.0247.
  * (5) Y. Namikawa and K. Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta mathematica 9 (1973), no. 2 143–186.
  * (6) D. Gaiotto, N=2 dualities, JHEP 08 (2012) 034, [arXiv:0904.2715].
  * (7) D. Xie, General Argyres-Douglas Theory, JHEP 1301 (2013) 100, [arXiv:1204.2270].
  * (8) Y. Wang and D. Xie, Classification of Argyres-Douglas theories from M5 branes, arXiv:1509.0084.
  * (9) Y. Wang and D. Xie, Codimension-two defects and Argyres-Douglas theories from outer-automorphism twist in 6d $(2,0)$ theories, arXiv:1805.0883.
  * (10) D. Xie and S.-T. Yau, 4d N=2 SCFT and singularity theory Part I: Classification, arXiv:1510.0132.
  * (11) Y. Wang, D. Xie, S. S. T. Yau, and S.-T. Yau, $4d$ $\mathcal{N}=2$ SCFT from complete intersection singularity, Adv. Theor. Math. Phys. 21 (2017) 801–855, [arXiv:1606.0630].
  * (12) P. Argyres, M. Lotito, Y. Lü, and M. Martone, Geometric constraints on the space of $\mathcal{N}$ = 2 SCFTs. Part I: physical constraints on relevant deformations, JHEP 02 (2018) 001, [arXiv:1505.0481].
  * (13) P. Argyres, M. Lotito, Y. Lü, and M. Martone, Geometric constraints on the space of $\mathcal{N}$ = 2 SCFTs. Part III: enhanced Coulomb branches and central charges, JHEP 02 (2018) 003, [arXiv:1609.0440].
  * (14) P. C. Argyres, M. Lotito, Y. Lü, and M. Martone, Expanding the landscape of $\mathcal{N}$ = 2 rank 1 SCFTs, JHEP 05 (2016) 088, [arXiv:1602.0276].
  * (15) R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299–334, [hep-th/9510101].
  * (16) T. Ashikaga and M. Ishizaka, Classification of degenerations of curves of genus three via matsumoto-montesinos’ theorem, Tohoku Mathematical Journal, Second Series 54 (2002), no. 2 195–226.
  * (17) Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic maps and degeneration of Riemann surfaces, vol. 2030.  Springer Science & Business Media, 2011.
  * (18) S. Cecotti, A. Neitzke, and C. Vafa, R-twisting and 4d/2d correspondences, arXiv preprint arXiv:1006.3435 (2010).
  * (19) B. Farb and D. Margalit, A primer on mapping class groups (pms-49), vol. 41.  Princeton university press, 2011.
  * (20) Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic homeomorphisms and degeneration of riemann surfaces, Bulletin of the American Mathematical Society 30 (1994), no. 1 70–75.
  * (21) F. Benini, Y. Tachikawa, and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063, [arXiv:1007.0992].
  * (22) D. Nanopoulos and D. Xie, More Three Dimensional Mirror Pairs, JHEP 05 (2011) 071, [arXiv:1011.1911].
  * (23) D. Xie, 3d mirror for Argyres-Douglas theories, arXiv:2107.0525.
  * (24) M. Artin and G. Winters, Degenerate fibres and stable reduction of curves, Topology 10 (1971), no. 4 373–383.
  * (25) D. Xie and S.-T. Yau, Semicontinuity of 4d N=2 spectrum under renormalization group flow, JHEP 03 (2016) 094, [arXiv:1510.0603].
  * (26) A. D. Shapere and Y. Tachikawa, Central charges of n= 2 superconformal field theories in four dimensions, Journal of High Energy Physics 2008 (2008), no. 09 109.
  * (27) D. Xie and P. Zhao, Central charges and RG flow of strongly-coupled N=2 theory, JHEP 03 (2013) 006, [arXiv:1301.0210].
  * (28) S. Takamura, Towards the classification of atoms of degenerations, i-splitting criteria via configurations of singular fibers dedicated to professor yukio matsumoto on the occasion of his sixtieth birthday, Journal of the Mathematical Society of Japan 56 (2004), no. 1 115–145.
  * (29) J. A. Minahan and D. Nemeschansky, Superconformal fixed points with E(n) global symmetry, Nucl. Phys. B 489 (1997) 24–46, [hep-th/9610076].
  * (30) P. C. Argyres, M. R. Plesser, N. Seiberg, and E. Witten, New N=2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71–84, [hep-th/9511154].
  * (31) B. Li, D. Xie, and W. Yan, On low rank 4d $\mathcal{N}=2$ SCFTs, arXiv:2212.0308.
  * (32) K. A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys.Lett. B387 (1996) 513–519, [hep-th/9607207].
  * (33) J. de Boer, K. Hori, H. Ooguri, and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101–147, [hep-th/9611063].
  * (34) K. Ueno, Discriminants of curves of genus 2 and arithmetic surfaces, in Algebraic geometry and commutative algebra, pp. 749–770.  Elsevier, 1988.
  * (35) S.-L. Tan, On the invariants of base changes of pencils of curves, i, Manuscripta mathematica 84 (1994), no. 1 225–244.
  * (36) J. Lu and S.-L. Tan, Inequalities between the chern numbers of a singular fiber in a family of algebraic curves, Transactions of the American Mathematical Society 365 (2013), no. 7 3373–3396.
  * (37) T. Arakawa and T. Ashikaga, Local splitting families of hyperelliptic pencils, i, Tohoku Mathematical Journal, Second Series 53 (2001), no. 3 369–394.
  * (38) T. Shioda, Mordell-weil lattices for higher genus fibration over a curve, London Mathematical Society Lecture Note Series (1999) 359–374.
  * (39) M. Caorsi and S. Cecotti, Special Arithmetic of Flavor, JHEP 08 (2018) 057, [arXiv:1803.0053].
  * (40) P. C. Argyres and M. Lotito, Flavor symmetries and the topology of special Kähler structures at rank 1, JHEP 02 (2019) 026, [arXiv:1811.0001].
  * (41) M.-H. Saito and K.-I. Sakakibara, On mordell-weil lattices of higher genus fibrations on rational surfaces, J. Math. Kyoto Univ 34 (1994), no. 4 859–871.
  * (42) S. Cecotti and M. Del Zotto, Infinitely many $n=2$ scft with ade flavor symmetry, Journal of High Energy Physics 2013 (2013), no. 1 1–22.
  * (43) D. Xie and S.-T. Yau, Three dimensional canonical singularity and five dimensional $\mathcal{N}$ = 1 SCFT, JHEP 06 (2017) 134, [arXiv:1704.0079].
  * (44) P. C. Argyres and M. Martone, The rank 2 classification problem I: scale invariant geometries, arXiv:2209.0924.
  * (45) D. Xie, Classification of rank one 5d $\mathcal{N}=1$ and 6d $(1,0)$ SCFTs, arXiv:2210.1732.