aainstitutetext: Yau Mathematics Science Center, Tsinghua University, Beijing,
100084, Chinabbinstitutetext: Department of Mathematics, Tsinghua University,
Beijing, 100084, China

# On rank two theories with eight supercharges part I: local singularities

Dan Xie

###### Abstract

A complete study of local singularities of rank two $\mathcal{N}=2$ Coulomb
branch geometry is given. Low energy theory associated with the local
singularity is identified: it can be superconformal field theory (SCFT), or IR
free gauge theory, or the combination of them. Various invariants for local
singularity are also listed which are essential for the study of global
Coulomb branch. As a first application, global Coulomb branch with only
simplest local singularities in the bulk are given for 4d theories (including
SCFTs and asymptotical free theories), 5d KK theories, and 6d KK theories;
those examples appear to cover all the findings in the literature and suggest
there are more possibilities. More general global Coulomb branch geometry
would be discussed in the sequel of this paper.

##  1 Introduction

There are lots of interests in studying supersymmetric theory with eight
supercharges in various dimensions, such as 3d $\mathcal{N}=4$ theory
Intriligator:1996ex , 4d $\mathcal{N}=2$ theory Seiberg:1994rs ;
Seiberg:1994aj , 5d $\mathcal{N}=1$ theory Seiberg:1996bd , and 6d $(1,0)$
theory Seiberg:1996qx . These theories have many physical applications, i.e.
the studies of them help us understand strongly coupled quantum field theory,
confinement, strong-weak duality, etc; They also have many applications in
modern mathematics, such as geometric representation theory Braverman:2016pwk
and invariants for four manifolds Witten:1994cg ,etc.

Most of those theories are strongly coupled and so conventional field theory
tools are of little use, but the powerful geometric methods make it possible
to study various deep properties of those theories. In particular, one can
construct a large class of 3d $\mathcal{N}=4$ theory using type IIB branes
Hanany:1996ie , 4d $\mathcal{N}=2$ theory using M5 branes Witten:1997sc ;
Gaiotto:2009we ; Xie:2012hs  and 3-fold canonical singularities Shapere:1999xr
; Xie:2015rpa , 5d $\mathcal{N}=1$ theory using 5 brane webs Aharony:1997bh
and 3-fold canonical singularities Intriligator:1997pq ; Xie:2017pfl , 6d
$(1,0)$ theory using F theory on local elliptic fibered 3-folds
Heckman:2013pva .

Although there are now a quite large space of theories, it is desirable to
have a classification of theories with 8 supercharges 111Theories with 16
supercharges have a fairly simple classification and are closed related to the
classification of simple Lie algebra Seiberg:1997ax .. While the geometric
constructions listed above are quite powerful, it is never clear whether they
would give a classification (in fact most of times they would miss some
theories).

Argyres and his collaborators Argyres:2015ffa ; Argyres:2015gha ;
Argyres:2016xmc ; Argyres:2016xua ; Caorsi:2018ahl  have took a different
approach in classifying 4d $\mathcal{N}=2$ SCFTs by classifying the rank one
Coulomb branch solution found in Seiberg:1994rs ; Seiberg:1994aj . The
classification scheme is a lot more complicated than the geometric approach,
but it is more complete in the sense that they would generate theories which
are not found using geometric tools. Their approach is implemented by the
author in classifying rank one 5d $\mathcal{N}=1$ and 6d $(1,0)$ SCFTs
Xie:2022lcm . The link of Coulomb branch geometry with the rational elliptic
surface schutt2019elliptic  plays a crucial role in the classification of rank
one theory.

The purpose of this paper and the follow-ups Xie:ranktwob ; Xie:ranktwoc  is
to give a complete classification for rank two theories with eight
supercharges in dimension $D\geq 4$ (see Argyres:2018zay ; Bourget:2021csg ;
Kaidi:2021tgr ; Martone:2021ixp ; Argyres:2022lah ; Argyres:2022puv ;
Argyres:2022fwy  for the attempt in classifying 4d rank two SCFTs). Now the
general structure of rank two Coulomb branch and its underlying mathematical
structure (genus two fibered rational surface) have not been thoroughly
studied, and our work will fill the gap.

The basic idea of the classification for theories in $D\geq 4$ dimension has
been described in Xie:2022lcm . Namely, one put 6d theory on $T^{2}$ and 5d
theory on $S^{1}$ so that one get effective 4d $\mathcal{N}=2$ theory in the
low energy (the resulting 4d effective theory is called KK theory). Then one
use the general structure of 4d $\mathcal{N}=2$ Coulomb branch advocated in
Xie:2021hxd  to do the classification, see figure, 1 for the illustration of
Coulomb branch geometry.

The crucial new ingredient in figure. 1 is the object attached to the special
points (We all them singularities.) at Coulomb branch. Those special points
are the place where new massless particles appear. The new object attached at
singularity is defined by the limiting behavior of the structure attached to
the nearby points. These limiting objects are crucial in finding the low
energy theory at special vacua.

Another crucial new ingredient is to compactify the Coulomb branch and so a
limiting object can also be defined at the $\infty$ point of Coulomb branch.
The limiting structure at $\infty$ can tell us the information of UV theory,
such as which space-time dimension the theory lives.

$\infty$$H\left(F^{\bullet},W^{\bullet},Q(\cdot,\cdot)\right)$$H_{s\ \ \ \
}\left(F^{\bullet},W^{\bullet}\right)$$H_{lim}\left(F^{\bullet},W^{\bullet}(N)\right)$$H_{van}\left(F^{\bullet},W^{\bullet}(N)\right)$$T$$H_{\infty}\left(F^{\bullet},W^{\bullet}(N_{\infty})\right)$$u$
Figure 1: The structure of $\mathcal{N}=2$ Coulomb branch: 1): at generic
point, one has a vector space with a mixed Hodge structure; 2): at special
point, there are three vector spaces and all of them carry mixed Hodge
structure; and there is a monodromy group $T$ acting on these vector spaces;
3): One can also have a vector space at $\infty$ of moduli space.

So the local behavior near the singularities (including $\infty$) is crucial
in our classification program. For rank one theory, the local singularity is
classified by the conjugacy class of $SL(2,Z)$ group satisfying
$(T^{k}-I)^{2}=0$, and there are 8 types which coincide with the Kodaira’s
list of singular fibers of elliptic surface. The associated low energy theory
222Assuming the generic condition, namely, the singularities can always be
split into a set of simplest $I_{1}$ singularities. has been well studied
Argyres:1995xn ; Minahan:1996cj .

The local singularities of rank two Coulomb branch is a lot more complicated.
Firstly, there are many more possibilities for the choices of conjugacy
classes of monodromy group; Secondly, unlike rank one case, the monodromy
group can not by itself decide the singularity. Now if one focus on a one
dimensional slice of Coulomb branch, the local classification of singularities
can be reduced to the study of degeneration of genus two pencils over the
disk, which has been classified in namikawa1973complete . The classification
is given by data $(M,\tau_{ij}(0),m)$, here $M$ is the monodromy group,
$\tau_{ij}(0)$ is the fixed point of $M$, and $m$ is an invariant which is
subtle to define. There are now a total of 120 classes.

There are several quantities attach to local singularities: a) invariants
$d_{x},\delta_{x},n_{t}$; b): a dual graph; c): monodromy group. All the data
for these local singularities are listed in table. [6,7,8,9,10], which are the
main results of this paper. We then determine the low energy theory by using
following two set of data: a): the possible scaling dimension inferred from
the monodromy group Xie:2015xva ; Caorsi:2018zsq ; b): the dual graph attached
to the singularity, which can be used to find the 3d mirror for the low energy
theory. There are a couple of new features for rank two local singularities:

  1. 1.

One can define two invariants $d_{x},\delta_{x}$ for the local singularity.
$\delta_{x}$ is called topological Euler number, and $d_{x}$ is related to the
minimal holomorphic family of fibration. In the rank one case
$d_{x}=\delta_{x}$. This is no longer the case for rank two cases: there are a
few examples with $d_{x}\neq\delta_{x}$.

  2. 2.

In the rank one case, the local singularities can be split into several
$I_{1}$ singularity. In the rank two case, it is also possible to split the
singularity into simpler ones, but now there are two kinds of atomic
singularities (those which can not be further splitted): $I_{1}$ and
$\tilde{I}_{1}$, see figure. 8 for the illustration. The number of $I_{1}$ and
$\tilde{I}_{1}$ singularities are determined by the number $d_{x}$ and
$\delta_{x}$.

  3. 3.

In the rank one case, the component $n_{t}$ in the dual graph is related to
the rank of flavor symmetry $f$ as $f=n_{t}-1$. Although the formula
$f=n_{t}-1$ is still true for most of cases of rank two case, there are
exceptions.

As a first application of our approach, we construct some candidate global
Coulomb branch geometries for 4d SCFTs (table. 12 and 13) and asymptotical
free theories (table. 14), 5d $\mathcal{N}=1$ (table. 17) and 6d $(1,0)$
theories (table. 19). Here we only use simplest singularities in the bulk of
the Coulomb branch. Our lists seem to be able to covers almost all the related
results in the literature and actually suggest more possibilities.

Our approach is powerful in that it can deal with theories in various
dimensions simultaneously, and it can determine the low energy theory at every
vacua of Coulomb branch. The results of this paper would be useful in studying
the field theory dynamics, which we will explore elsewhere.

This paper is organized as follows: section 2 describes the local
singularities of rank two theory, such as various invariants and the low
energy theory attached to it; section 3 describes some global Coulomb branch
geometries; section 4 gives a conclusion.

##  2 Local singularities of rank two Coulomb branch geometry

The basic properties of Coulomb branch of a 4d $\mathcal{N}=2$ theory are:

  1. 1.

At a generic point, the low energy theory is described by $U(1)^{r}$ abelian
gauge theory, free hypermultiplets, and possibly interacting SCFT whose
Coulomb branch deformation is trivial. An important goal is to determine the
effective coupling for the $U(1)^{r}$ gauge theory, here $r$ is called the
rank of the theory.

  2. 2.

At a special point, new massless degrees of freedom appear. The low energy
theory could be IR free gauge theory, or SCFT, or the direct sum of them.

  3. 3.

The new massless degrees of freedom at special point come from massive BPS
particles at the generic point.

The Coulomb branch solution for 4d $SU(2)$ gauge theory was solved in an
elegant way by Seiberg and Witten Seiberg:1994rs ; Seiberg:1994aj . They
solved the theory by finding a family of algebraic curves
$F(x,y,u,m,\Lambda)=0$ (Here $u$ parameterizes the Coulomb branch, $m$ the
mass parameters, and $\Lambda$ the dynamical generated scale.), and a SW
differential $\lambda$ is also needed.

More generally, the Coulomb branch solution could be represented as a mixed
Hodge module over the generalized Coulomb branch (parameter space including
Coulomb branch operators, masses, relevant and marginal couplings). Let’s
assume the rank of the theory is $r$ and the flavor symmetry has rank $f$
333This flavor number could be modified for 5d KK and 6d KK theories..

At generic point: There is a flat 444The flat structure gives an integrable
connection which is required for the definition of the mixed Hodge module.
holomorphic vector bundle whose rank is $2r+f$. To get the information of the
low energy theory, two extra structures are needed on the fiber $H$:

  * •

A mixed Hodge structure, namely a Hodge filtration and a weight filtration;
The weight filtration is an increasing filtration which takes the following
form 555If the SW geometry is given by a three dimensional variety Xie:2015rpa
, then the maximal weight is 4. These MHS could be brought to the form
presented here by doing a Tate twist.:

$\\{0\\}=W^{0}\subset W^{1}\subset W^{2}=H;$ (1)

so we have two quotient spaces
$Gr_{1}^{W}=W^{1}/W^{0},~{}~{}Gr_{2}^{W}=W^{2}/W^{1}$, with dimension
$dim(Gr_{1}^{W})=2r,~{}dim(Gr_{2}^{W})=f$. The weight filtration is needed so
that we can separate the electric-magnetic part and the flavor part of the
central charge: $Gr_{1}^{W}$ gives the electric-magnetic charge, and
$Gr_{2}^{W}$ gives the flavor charge. The Hodge filtration is a decreasing
filtration and takes the following form

$H=F^{0}\supset F^{1};$ (2)

So in our case, two holomorphic sub-bundles $W^{1}$ and $F^{1}$ are needed.
The weight filtration and Hodge filtration together defines a so-called Mixed
Hodge structure, and Hodge decomposition takes the form
$Gr_{1}^{W}=H^{1,0}\oplus H^{0,1}$ and $Gr_{2}^{W}=H^{1,1}$, with dimension
$h^{1,0}=h^{0,1}=r$ and $h^{1,1}=f$;

  * •

A polarization $Q(\cdot,\cdot)$ (which satisfies Riemann-Hodge bilinear
relations on $Gr_{1}^{W}$ and acts trivially on $Gr_{2}^{W}$) on $H$ so that
positive definite coupling constants can be defined. In fact, a period matrix
$Z_{ij},~{}i,j=1,\ldots,r$ which is symmetric and satisfies the condition
$Im(Z)>0$ can be defined using the polarization.

At singular point 666The Coulomb branch is not singular, but the physics is
different from that of the generic point of the Coulomb branch.: There is also
a vector space $H_{s}$ whose dimension is smaller than $H$, so the
mathematical structure is not the vector bundle which is more familiar to
physicists. The physics of the abelian gauge theory at singular point is
described by $H_{s}$. The crucial point of the mixed Hodge module is that one
can define two more vector spaces at the singular point. The first is the so-
called nearby cycle $H_{lim}$ which can be thought of as the limiting objects
for the nearby vector spaces. There is a mixed Hodge structure on $H_{lim}$
which is quite different from that of the generic fiber described earlier: The
weight filtration is now determined by the nilpotent part $N$ of the monodromy
group $T$ around the singularity.

For the known solution, the monodromy group $T$ satisfies the following
condition

$(T^{k}-1)^{2}=1.$ (3)

Namely the maximal size of the Jordan block is two, and the eigenvalue
satisfies $\lambda^{k}=1$. We conjecture that this is true for the Coulomb
branch solution of any $\mathcal{N}=2$ field theory. Furthermore, if we
restrict the monodromy on the weight two part of the generic fiber, its action
is trivial

$T|_{Gr_{2}^{W}}=I.$ (4)

What this implies is that the monodromy matrix takes the form

$T=\left[\begin{array}[]{cc}I&0\\\ &M\\\ \end{array}\right].$

and $M$ acts on weight one part. Finally, $H_{s}$ and $H_{lim}$ can be used to
define a third vector space called vanishing cycle $H_{van}$. All of these
three spaces carry mixed Hodge structure, and they form an exact sequence of
mixed Hodge structure.

Using the limit mixed Hodge structure (let’s assume $H_{s}=0$), one can define
a set of rational numbers $(\alpha_{1},\alpha_{2},\ldots,\alpha_{s})$ called
spectrum kulikov1998mixed , and its relation to the eigenvalue of the
monodromy group $T$ is given as

$\lambda_{i}=\exp(2\pi i\alpha_{i}).$

The monodromy group acts on the vector space $H_{lim}$, and so it has the
decomposition $H_{lim}=\oplus_{\lambda}H^{lim}_{\lambda}$. The limit Hodge
filtration defines a filtration on $H_{\lambda}^{lim}$:
$F^{0}(H_{\lambda}^{lim})\supset F^{1}(H_{\lambda}^{lim})$. Now for a basis
element $e_{i}$ in $H_{\lambda}^{lim}$, a spectral number $\alpha$ is defined
as

$\begin{cases}&e_{i}\in
F^{1}(H_{\lambda}^{lim}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-1<\alpha\leq 0\\\
&e_{i}\in
F^{0}(H_{\lambda}^{lim})/F^{1}(H_{\lambda}^{lim}),~{}~{}~{}~{}0<\alpha\leq
1\end{cases}$

here $\exp(2\pi\alpha)=\lambda$. An important consistent condition is that the
spectral numbers are in pair

$\alpha_{i}+\alpha_{j}=0.$

One can find the Coulomb branch spectrum from the spectral numbers as follows.
Let’s denote the minimal spectrum number as $\alpha_{min}$, then one associate
a Coulomb branch scaling dimension for a spectral number $\alpha_{i}$ as
follows Xie:2015xva :

$[u_{i}]={1+\alpha_{min}-\alpha_{i}\over 1+\alpha_{min}}.$ (5)

So if $\alpha_{i}=0$, then $[u_{i}]=1$ which gives a mass parameter. The
maximal scaling dimension is given as ${1\over 1+\alpha_{min}}$.

At Infinity: While the above analysis is carried for the finite points of the
Coulomb branch, it is possible to do the similar computation for the $\infty$
point on the moduli space, and the MHS at $\infty$ is useful to extract
information for UV theory. The general structure of the Coulomb branch
solution is summarized in figure. 1.

###  2.1 Rank two Coulomb branch geometry and genus two pencils

Let’s now look at rank two Coulomb branch geometry: namely the dimension of
$Gr_{1}^{W}$ is $4$. The generalized Coulomb branch parameters include
expectation values of two Coulomb branch operators $u,v$ (here one assume the
scaling dimension of $v$ is no less than $u$), the mass parameters, and
possibly relevant deformations for 4d theory. There would be also dynamical
generated scale or exact marginal deformation for 4d theory, but those
parameters would be regarded as parameterizing the UV theory, and are not
regarded as the Coulomb branch parameters of a given UV theory.

We now make a couple of simplifying assumption for the Coulomb branch
geometry. First of all, since the monodromy group acts trivially on weight two
part $Gr_{2}^{W}$, one might focus on weight one part, and now there is a rank
four flat holomorphic bundle at generic points. One must be careful that lots
of information such as the flavor group is lost in this assumption, and the
hope is to recover those information by looking at extra structure (such as
the Mordell-Weil lattice) on the weight one part of Coulomb branch geometry.

Secondly, one fix all the parameters but one (typically the parameter $v$, but
it is not always possible to do it, then one need to look at a different
family defined by the hypersurface $f(u,v)=0$ in the $u,v$ plane), and a one
parameter family of rank four flat bundles would be considered. See figure. 2.

$v$$\infty$$\tau_{ij}(v)$$v$$\infty$ Figure 2: Left: The weight one part of
Coulomb branch solution of a rank two theory: A multivalued holomorphic
function $\tau_{ij}(v)$ is defined over the one dimensional slice of Coulomb
branch parameterized by $v$; Right: One get a genus two fibration over
$\mathbb{P}^{1}$ by identifying the complex structure of the genus two curve
with $\tau_{ij}(v)$.

Now one has a multi-valued holomorphic function $\tau_{ij}(v)$ at a generic
point of the one family Coulomb branch geometry. The matrix $\tau_{ij}$ is a
$2\times 2$ symmetric matrix and satisfies following condition

$Im(\tau_{ij})>0;$

We now attach a genus two curve at each base point on $v$ plane by identifying
its complex structure with $\tau_{ij}$. Therefore one has a family of genus
two curves over $\mathbb{P}^{1}$, and the total space is a surface $X$ with
the map $f:X\to\mathbb{P}^{1}$. At singularity, the genus two curves would
degenerate, and the study of local singularities of rank two Coulomb branch
(weight one part) is then reduced to the study of the one parameter
degeneration of genus two curves!

###  2.2 Pencils of genus two curves

The result of last subsection links the study of rank two Coulomb branch
geometry to that of genus two pencils, and the degeneration of those pencils
are completely classified in namikawa1973complete . In this subsection, we
will first review their classifications, and the local invariants, and finally
the low energy theory would be given by physical input.

####  2.2.1 Numerical type of genus singular fibers

In this subsection, the numerical type of a genus two pencils will be
reviewed. Here the singular fiber $X_{0}$ is regarded as a divisor in $X$, and
$X_{0}$ does not have a component with self-intersection number $-1$.

Let’s consider a family of genus two fibration $\pi:X\to\Delta$, and assume
the generic fiber on $b\in\Delta$ is a smooth genus two curve. Only the fiber
$X_{0}$ at $b=0$ is a singular fiber. $X_{0}$ is a reducible curve $X_{0}=\sum
n_{i}C_{i}$, which satisfies following conditions:

  1. 1.

$X_{0}$ is connected, and $n_{i}>0$ for all $i$.

  2. 2.

$C_{i}\cdot C_{j}=C_{j}\cdot C_{i}\geq 0$ if $i\neq j$, and $X\cdot C_{i}=0$.

  3. 3.

$p(C_{i})=\frac{1}{2}[C_{i}^{2}+C_{i}\cdot K)+1\geq 0$ for all $i$. Here
$p(C_{i})$ is the genus of curve $C_{i}$.

$K$ is the canonical class of the compact surface $X$, which only plays a
formal role here (all we need are numbers $k_{i}=C_{i}\cdot K\geq 0$).
Therefore a configuration of curves is specified by the data

$(n_{i},k_{i},C_{ij}),~{}~{}~{}i=1,\ldots,n.$ (6)

The genus of $X_{0}$ is given by following formula:

$g(X_{0})=1+\frac{1}{2}(X_{0}\cdot K)=1+{1\over 2}\sum(n_{i}k_{i}),$ (7)

Which is determined by the data $n_{i},k_{i}$ only. Since there is no $-1$
component (self-intersection number is $-1$) in $X_{0}$, there is only a
finite number of possibilities for a given genus artin1971degenerate . There
is a complete classification of genus one pencils by Kodaira, and the
configurations are represented by the dual graphs, see figure. 3.

$II$$III$$IV$$n$$2$$2$$2$$2$$I_{n}^{*}$$I_{n}$$n-1$$3$$2$$2$$2$$IV^{*}$$III^{*}$$II^{*}$$2$$3$$3$$2$$2$$6$$4$$5$$4$$3$$2$$4$$2$
Figure 3: The dual graph for genus one degeneration, which is classified by
Kodaira, see barth2015compact . The data $(n_{i},k_{i},C_{ij})$ are read from
the dual graph as follows: the number $n_{i}$ is listed for each component
$C_{i}$ in the graph (if $n_{i}=1$, the number is omitted); $C_{ij}$ is read
from the intersection number of component $C_{i}$ and $C_{j}$ in the graph; In
genus one case, all the components are rational curve with $-2$ self-
intersection number, and so $k_{i}=0$ for all the components.

For genus two, the individual component of the singular fiber $\Gamma$ is one
of the five types listed in table. 1. The proof is following: By using the
formula 7 for the genus, one has following solution for the components with
non-zero $k_{i}$: a): there is one component with $k=2$ and $n=1$; and this
component has genus $p(C)=\frac{1}{2}(C^{2}+2)+1$; Either $C^{2}=-4,p(C)=0$
(type D), or $C^{2}=-2,p(C)=1$ ( type $C$). b): Two components with $k=1$; The
genus of each component is given as $p(C)=\frac{1}{2}(C^{2}+1)+1$, and so
$C^{2}=-3$ (type B) or $C^{2}=-1$ (type A). The genus formula does not put
constraints on type $E$ component $(k_{i}=0)$, and they are always allowed.

Type | $k_{i}$ | $\Gamma^{2}$ | $p(\Gamma)$
---|---|---|---
A: | $\Gamma\cdot K=1$ | $\Gamma^{2}=-1$ | $p(\Gamma)=1$
B: | $\Gamma\cdot K=1$ | $\Gamma^{2}=-3$ | $p(\Gamma)=0$
C: | $\Gamma\cdot K=2$ | $\Gamma^{2}=-2$ | $p(\Gamma)=1$
D: | $\Gamma\cdot K=2$ | $\Gamma^{2}=-4$ | $p(\Gamma)=0$
E: | $\Gamma\cdot K=0$ | $\Gamma^{2}=-2$ | $p(\Gamma)=0$
Table 1: The possible components for the genus two singular fiber.

Given the data in table. 1, there are following choices for components with
$k_{i}>0$:

$C,~{}2A,~{}~{}(A,A),~{}~{}2B,~{}~{}(A,B),~{}~{}(B,B),~{}~{}~{}D.$

The dual graphs for all genus two singular fibers have been given in
ogg1966pencils , and there are a total of 44 types.

Example: Here we’d like to give an example to illustrate the idea of
classification. Let’s assume that there is a type $D$ component, and so the
singular fiber can be written as $X_{0}=D+\delta$. Now the condition $D\cdot
X_{0}=0$ implies that $D^{2}+D\cdot\delta=0$, therefore the following equation
holds

$D\cdot\delta=4.$

The above equation implies that $D$ intersects with the rest of $X_{0}$ (minus
D) four times. The decomposition of $4$ is $4,3+1,2+2$. Let’s look at the
decomposition $4$, and so $X_{0}=D+4C_{1}+\delta_{1}$, where $C_{1}$ is a $-2$
curve with $C_{1}\cdot D=4$ . We then use the condition $C_{1}\cdot X_{0}=0$,
which then gives the equation

$C_{1}\cdot(D+4C_{1}+\delta_{1})=0\to C_{1}\cdot\delta_{1}=-C_{1}\cdot
D-4C_{1}^{2}=7.$

So the total intersection number of $C_{1}$ with $\delta_{1}$ is 7. The
legitimate decomposition of $7$ is $7=2+2+3,3+4,2+5,7$, and one of the
configurations (with decomposition $7=2+2+3$) is shown in figure. 4. One can
do the similar computation to find all the other configurations listed in
ogg1966pencils .

$D$$4$$2$$2$$2$$1$$3$ Figure 4: A genus two singular fibers $X_{0}=\sum
n_{i}C_{i}$: there is a component $D$, and the multiplicities $n_{i}$ are
indicated. The type E component is not explicitly stated.

####  2.2.2 Classification of genus two singular fiber

$X_{0}$$X_{t}$ Figure 5: A family of genus two curves over the disc. The fiber
is smooth for $t\neq 0$, and the fiber degenerates at $t=0$

The numerical type does not completely determine the degeneration of the
family of genus two curves. Mathematically, one consider a family of genus two
curves over the disc $f:X\to\Delta$. One further assume that the fiber
$X_{t}=f^{-1}(t)$ is smooth for $t\neq 0$. The only possible singular fiber is
the fiber $X_{0}$ at $t=0$, see figure. 5. Once the above family is given, one
can get a period mapping by looking at the first cohomology group of $X_{t}$:
the first cohomology group $H^{1}(X_{t})$ has a Hodge decomposition
$H^{1}(X_{t})=H^{1,0}\otimes H^{0,1}$, and $H^{1,0}$ is spanned by the
holomorphic differentials $w_{1},w_{2}$. By choosing a symplectic basis of
homology cycles $A_{1},A_{2},B_{1},B_{2}$, one can form the period integrals
and compute period matrix $\Omega$ whose entries are

$\int_{A_{j}}\omega_{i},~{}~{}\int_{B_{j}}\omega_{i},$

The period matrix takes the form $\Omega=[I,\tau_{ij}]$ by choosing the basis
of holomorphic differentials property. Here $\tau_{ij}$ is a two by two matrix
satisfies following condition

$\tau=\tau^{T},~{}~{}Im(\tau)>0.$

The entry of $\tau_{ij}(t)$ is moreover a multivalued holomorphic function of
$t$, and so it has the following monodromy property

$\tau(e^{2\pi i}t)=M\tau(t).$

Here $M$ is the monodromy group. In fact one has a conjugacy class of
$Sp(4,Z)$, as if the symplectic basis of the homology group is changed, the
monodromy group is also transformed by conjugating a $Sp(4,Z)$ matrix. The
monodromy group satisfies the relation

$(M^{k}-I)^{2}=0.$

This means that $M$ is a quasi-unipotent matrix, namely, the eigenvalue is the
roots of unity, and the maximal size of its Jordan block is two. $M$ is called
elliptic if it is semi-simple (finite order), otherwise it is called parabolic
(infinite order).

It was shown in namikawa1973complete  that the isomorphic family of genus two
pencil are classified by the following three types of data: 1) The conjugacy
class of monodromy group $M$; 2) The limiting point $\tau(0)$ which can be
represented by stable curves; 3): A third invariant $m$ which is more subtle
to define. In summary, a degeneration of genus two family is classified by the
data

$\boxed{(M,\tau(0),m)}.$ (8)

There are a total of around 120 types of degenerations for genus two curves
(comparing 8 classes of genus one degenerations).

The study of the conjugacy class of monodromy groups is left for section. 2.3.
Here let’s explain the data $\tau(0)$ which is the fixed point of monodromy
group. $\tau(0)$ is represented by the genus two stable curves. A stable curve
is a curve whose only singularities are nodes and whose smooth rational
components contain at least three singular points of the curve. For genus two,
there are following classes of stable curves: M: an irreducible non-singular
hypelliptic curve; N: two elliptic curves meeting at one point transversally;
Ba): an elliptic curve with one ordinary double point; Bb): an elliptic curve
and a rational curve with ordinary double point, the above two components
intersect transversely; Ca): A rational curve with two ordinary double points;
Cb): Two rational curves with one ordinary double points intersecting
transversely; D): Two non-singular rational curves meeting at three points
transversally. See figure. 6 for the configuration.

$M$$N$$Ba$$Bb$$Ca$$Cb$$D$ Figure 6: Genus two stable curves. The red points
are node where the local equation is $xy=0$ for the singularity.

There are a total of five classes of genus two degenerations, see table. 2.
The meaning of the classification is following: a) First the conjugacy class
of monodromy group is classified as elliptic if it is semi-simple (finite
order, namely $M^{k}=I$ for some k), and parabolic if it is of infinite order;
b): The limiting point of the period matrix $\tau_{ij}(0)$ correspond to a
type of stable curve, see figure. 6; c): the invariant $m$ is the most subtle
one which can be defined using the limiting behavior of the period map and its
intersection with the divisor $\bar{N}$ ($=N\cup Bb\cup Cb$ 777These three
types have separating node: namely one get two disconnected components by
taking off the node.) in the space of stable curves, see namikawa1973complete
for further explanation of this number.

Type | Monodromy group | Fixed point | degree
---|---|---|---
Elliptic [1] | Elliptic | M | $m=0$
Elliptic [2] | Elliptic | N | $m\geq 0$
Parabolic [3] | Infinite | B | $m\geq 0$
Parabolic [4] | Infinite | C | $m\geq 0$
Parabolic [5] | Infinite | D | $m=0$
Table 2: Five classes of genus two singular fibers.

The invariant $m$ deserves further explanation. First of all, let’s look the
degeneration of type $[2]-I_{0}-I_{0}-m$ (We use the label used in
namikawa1973complete ). The period mapping has the following form

$\tau=\left(\begin{array}[]{cc}a(t)&t^{m}\\\ t^{m}&b(t)\\\
\end{array}\right),~{}~{}Im(a(0))>0,~{}~{}Im(b(0))>0.$

Here the number $m$ is just the degree for the degeneration. We notice that
the number $m$ does not affect the limiting value of the period matrix, which
is determined by $a(0)$ and $b(0)$ only. The dual graph of this singular fiber
is shown in figure. 7, and the number $m$ is related to the number of $-2$
rational curves in between two elliptic curves (they form a chain of $A_{m-1}$
Dynkin diagram). $a(0)$ and $b(0)$ gives the moduli of two elliptic curves.

$A$$A$$m-1$ Figure 7: The dual graph for $[2]-I_{0}-I_{0}-m$ singular fiber.
The invariant $m$ for the singular fiber is given by the $A_{m-1}$ chain of
rational curves with self-intersection number $-2$.

###  2.3 Monodromy group and Coulomb branch spectrum

We’d like to derive the possible scaling dimension from the data of monodromy
group of a singular fiber. The conjugacy class of $Sp(4,Z)$ has been
classified, see eie1984dimensions . Here we’d like to focus on the so-called
elliptic conjugacy class, i.e. the conjugacy class with the property
$M^{k}=I$; they are also called conjugacy class with finite order. The
eigenvalues for these conjugacy classes are roots of unity. The characteristic
polynomial and the minimal polynomial for them are listed in table. 3.

Characteristic polynomial $\Psi(x)$ | minimal polynomial $P(x)$
---|---
$(x\pm 1)^{4}$ | $(x\pm 1)$
$(x-1)^{2}(x+1)^{2}$ | $x^{2}-1$
$(x^{2}+1)^{2}$ | $x^{2}+1$
$(x^{2}\pm x+1)^{2}$ | $x^{2}\pm x+1$
$(x^{2}+1)(x\pm 1)^{2}$ | $(x^{2}+1)(x\pm 1)$
$(x\pm 1)^{2}(x^{2}+x+1)$ | $(x\pm 1)(x^{2}+x+1)$
$(x\pm 1)^{2}(x^{2}-x+1)$ | $(x\pm 1)(x^{2}-x+1)$
$x^{4}+1,~{}x^{4}\pm x^{2}+1,~{}(x^{2}+1)(x^{2}\pm x+1),~{}~{}x^{4}\pm x^{3}+x^{2}\pm x+1$ |
Table 3: Characteristic polynomial and its minimal polynomial for elliptic
conjugacy class of $Sp(4,Z)$ group.

The Coulomb branch spectrum are related to singularity spectrum by the formula
5. Using the limit mixed Hodge structure, one can define a set of rational
numbers $(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$ called spectrum
kulikov1998mixed , and its relation to the eigenvalue of the monodromy group
is

$\lambda_{i}=\exp(2\pi i\alpha_{i}).$ (9)

An important consistent condition is that the spectral numbers are in pair

$\alpha_{i}+\alpha_{j}=0.$ (10)

Even without knowing the exact limit mixed Hodge structure, one can get
possible scaling dimension by using the information of eigenvalues and the two
constraints 9, 10. The data is listed in table. 4 (We assume that there is no
eigenvalue one.), where the exponent $e_{i}=\alpha_{i}+1$ is shown. The
pairing for the exponent takes the form $e_{i}+e_{j}=2$. The formula for
scaling dimension is changed

$[u_{i}]={1+e_{min}-e_{i}\over e_{min}}.$

and the pairing for $[u_{i}]$ is $[u_{i}]+[u_{j}]=2$. Only the scaling
dimension with $[u_{i}]>1$ is listed in table. 4, and others can be found by
using the pairing symmetry.

Eigenvalues |  Possible exponents |  Scaling dimension | Characteristic polynomial
---|---|---|---
$(\exp(\pi i),\exp(\pi i),\exp(\pi i),\exp(\pi i))$ |  $[\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{3}{2}]$ |  $[2,2]$ | $(x+1)^{4}$
$(\exp(\pi i/2),\exp(\pi i/2),\exp(3\pi i/2),\exp(3\pi i/2))$ |  $[\frac{1}{4},\frac{1}{4},\frac{7}{4},\frac{7}{4}]$
$[\frac{1}{4},\frac{3}{4},\frac{5}{4},\frac{7}{4}]$
$[\frac{3}{4},\frac{3}{4},\frac{5}{4},\frac{5}{4}]$ |  $[4,4]$
$[4,2]$
$[\frac{4}{3},\frac{4}{3}]$ | $(x^{2}+1)^{2}$
$(\exp(2\pi i/3),\exp(2\pi i/3),\exp(4\pi i/3),\exp(4\pi i/3))$ |  $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3}]$
$[\frac{1}{3},\frac{2}{3},\frac{4}{3},\frac{5}{3}]$
$[\frac{2}{3},\frac{2}{3},\frac{4}{3},\frac{4}{3}]$ |  $[3,3]$
$[3,2]$
$[\frac{3}{2},\frac{3}{2}]$ | $(x^{2}+x+1)^{2}$
$(\exp(2\pi i/6),\exp(2\pi i/6),\exp(10\pi i/6),\exp(10\pi i/6))$ |  $[\frac{1}{6},\frac{1}{6},\frac{11}{6},\frac{11}{6}]$
$[\frac{1}{6},\frac{5}{6},\frac{7}{6},\frac{11}{6}]$
$[\frac{5}{6},\frac{5}{6},\frac{7}{6},\frac{7}{6}]$ |  $[6,6]$
$[6,2]$
$[\frac{6}{5},\frac{6}{5}]$ | $(x^{2}-x+1)^{2}$
$(\exp(2\pi i/4),\exp(6\pi i/4),\exp(\pi i),\exp(\pi i))$ |  $[\frac{1}{4},\frac{1}{2},\frac{3}{2},\frac{7}{4}]$
$[\frac{1}{2},\frac{3}{4},\frac{5}{4},\frac{3}{2}]$ |  $[4,3]$
$[2,\frac{3}{2}]$ | $(x+1)^{2}(x^{2}+1)$
$(\exp(2\pi i/3),\exp(4\pi i/3),\exp(\pi i),\exp(\pi i))$ |  $[\frac{1}{3},\frac{1}{2},\frac{3}{2},\frac{5}{3}]$
$[\frac{1}{2},\frac{2}{3},\frac{4}{3},\frac{3}{2}]$ |  $[3,\frac{5}{2}]$
$[2,\frac{5}{3}]$ | $(x+1)^{2}(x^{2}+x+1)$
$(\exp(2\pi i/6),\exp(10\pi i/6),\exp(\pi i),\exp(\pi i))$ |  $[\frac{1}{6},\frac{1}{2},\frac{3}{2},\frac{11}{6}]$
$[\frac{1}{2},\frac{5}{6},\frac{7}{6},\frac{3}{2}]$ |  $[6,4]$
$[2,\frac{4}{3}]$ | $(x+1)^{2}(x^{2}-x+1)$
$(\exp(2\pi i/8),\exp(6\pi i/8),\exp(10\pi i/8),\exp(14\pi i/8))$ |  $[\frac{1}{8},\frac{3}{8},\frac{13}{8},\frac{15}{8}]$
$[\frac{1}{8},\frac{5}{8},\frac{11}{8},\frac{15}{8}]$
$[\frac{3}{8},\frac{7}{8},\frac{9}{8},\frac{13}{8}]$
$[\frac{5}{8},\frac{7}{8},\frac{9}{8},\frac{11}{8}]$ |  $[8,6]$
$[8,4]$
$[\frac{8}{3},\frac{4}{3}]$
$[\frac{8}{5},\frac{6}{5}]$ | $x^{4}+1$
$(\exp(2\pi i/12),\exp(10\pi i/12),\exp(22\pi i/12),\exp(14\pi i/12))$ |  $[\frac{1}{12},\frac{5}{12},\frac{19}{12},\frac{23}{12}]$
$[\frac{1}{12},\frac{7}{12},\frac{17}{12},\frac{23}{12}]$
$[\frac{5}{12},\frac{11}{12},\frac{13}{12},\frac{19}{12}]$
$[\frac{7}{12},\frac{11}{12},\frac{13}{12},\frac{17}{12}]$ |  $[12,8]$
$[12,6]$
$[\frac{12}{5},\frac{6}{5}]$
$[\frac{12}{7},\frac{8}{7}]$ | $x^{4}-x^{2}+1$
$(\exp(2\pi i/3),\exp(4\pi i/3),\exp(2\pi i/6),\exp(10\pi i/6))$ |  $[\frac{1}{6},\frac{1}{3},\frac{5}{3},\frac{11}{6}]$
$[\frac{1}{6},\frac{2}{3},\frac{4}{3},\frac{11}{6}]$
$[\frac{1}{3},\frac{5}{6},\frac{7}{6},\frac{5}{3}]$
$[\frac{2}{3},\frac{5}{6},\frac{7}{6},\frac{4}{3}]$ |  $[6,5]$
$[6,3]$
$[3,\frac{3}{2}]$
$[\frac{3}{2},\frac{5}{4}]$ | $x^{4}+x^{2}+1$
$(\exp(2\pi i/4),\exp(6\pi i/4),\exp(2\pi i/3),\exp(4\pi i/3))$ |  $[\frac{1}{4},\frac{1}{3},\frac{5}{3},\frac{7}{4}]$
$[\frac{1}{4},\frac{2}{3},\frac{4}{3},\frac{7}{4}]$
$[\frac{1}{3},\frac{3}{4},\frac{5}{4},\frac{5}{3}]$
$[\frac{2}{3},\frac{3}{4},\frac{5}{4},\frac{4}{3}]$ |  $[4,\frac{11}{3}]$
$[4,\frac{7}{3}]$
$[3,\frac{7}{4}]$
$[\frac{3}{2},\frac{23}{8}]$ | $(x^{2}+1)(x^{2}+x+1)$
$(\exp(2\pi i/6),\exp(10\pi i/6),\exp(2\pi i/4),\exp(6\pi i/4))$ |  $[\frac{1}{6},\frac{1}{4},\frac{7}{4},\frac{11}{6}]$
$[\frac{1}{6},\frac{3}{4},\frac{5}{4},\frac{11}{6}]$
$[\frac{1}{4},\frac{5}{6},\frac{7}{6},\frac{7}{4}]$
$[\frac{3}{4},\frac{5}{6},\frac{7}{6},\frac{5}{4}]$ |  $[6,\frac{11}{2}]$
$[6,\frac{5}{2}]$
$[4,\frac{5}{3}]$
$[\frac{4}{3},\frac{11}{9}]$ | $(x^{2}+1)(x^{2}-x+1)$
$(\exp(2\pi i/5),\exp(4\pi i/5),\exp(8\pi i/5),\exp(6\pi i/5))$ |  $[\frac{1}{5},\frac{2}{5},\frac{8}{5},\frac{9}{5}]$
$[\frac{1}{5},\frac{3}{5},\frac{7}{5},\frac{9}{5}]$
$[\frac{2}{5},\frac{4}{5},\frac{6}{5},\frac{8}{5}]$
$[\frac{3}{5},\frac{4}{5},\frac{6}{5},\frac{7}{5}]$ |  $[5,4]$
$[5,3]$
$[\frac{5}{2},\frac{3}{2}]$
$[\frac{5}{3},\frac{4}{3}]$ | $x^{4}+x^{3}+x^{2}+x+1$
$(\exp(2\pi i/10),\exp(6\pi i/10),\exp(14\pi i/10),\exp(18\pi i/10))$ |  $[\frac{1}{10},\frac{3}{10},\frac{17}{10},\frac{19}{10}]$
$[\frac{1}{10},\frac{7}{10},\frac{13}{10},\frac{19}{10}]$
$[\frac{3}{10},\frac{9}{10},\frac{11}{10},\frac{17}{10}]$
$[\frac{7}{10},\frac{9}{10},\frac{11}{10},\frac{13}{10}]$ |  $[10,8]$
$[10,4]$
$[\frac{10}{3},\frac{4}{3}]$
$[\frac{10}{7},\frac{8}{7}]$ | $x^{4}-x^{3}+x^{2}-x+1$

Table 4: The possible scaling dimension given the characteristic polynomial of
the elliptic conjugacy class.

In the above computations, we assume that the limit mixed Hodge structure is
irreducible. For some cases, the limit is reducible and so the monodromy group
is the direct sum of two $SL(2,Z)$ groups. The characteristic polynomial is
then the product of two degree two polynomials. One can get the scaling
dimension by simply using the result of rank one case, see Xie:2022lcm .

For the parabolic conjugacy class, one can also find the possible scaling
dimension by looking at the eigenvalues. The method is completely same. The
dual graph is useful in finding the precise scaling dimension, see the
discussion in the section. 2.5.

Isolated fixed points of monodromy group: A conjugacy class of $Sp(4,Z)$ is
called regular elliptic if it has isolated fixed points. There are only six
type of fixed points:

$\displaystyle Z_{1}=\mathop{\rm diag}\nolimits[i,i],~{}~{}Z_{2}=\mathop{\rm
diag}\nolimits[\rho,\rho],~{}~{}\rho=\exp(\pi i/3)$ $\displaystyle
Z_{3}=\mathop{\rm
diag}\nolimits[i,\rho],~{}~{}Z_{4}={i\over\sqrt{3}}\left(\begin{array}[]{cc}2&1\\\
1&2\end{array}\right)$ $\displaystyle
Z_{5}=\left(\begin{array}[]{cc}\eta&(\eta-1)/2\\\
(\eta-1)/2&\eta\end{array}\right),~{}~{}\eta=\frac{1}{3}+{2\sqrt{2}i\over 3}$
$\displaystyle Z_{6}=\left(\begin{array}[]{cc}w&w+w^{-2}\\\
w+w^{-2}&-w^{-1}\end{array}\right),~{}~{}w=\exp(2\pi i/5)$

The isotropy subgroups at $Z_{i}$ are groups of order $16,36,12,12,24,5$.
These subgroups are useful for the studies of discrete gauging of 4d
$\mathcal{N}=2$ SCFTs.

###  2.4 Local invariants

One can define a set of local invariant $n_{t},\delta_{x},d_{x}$ for a
singular fiber. These invariants are very important for extracting the low
energy theory associated with the singular fibers. They are also crucial in
constructing global Coulomb branch geometry.

The number $n_{t}$: This number is given by the number of irreducible
components in the dual graph, and it can be easily computed from the dual
graph.

The number $\delta_{x}$: This number $\delta_{x}$ is defined as

$\delta_{x}=2+\epsilon(F).$

here $\epsilon(F)$ is the topological Euler number of the dual graph
$\Gamma=\sum n_{i}C_{i}$. It can be computed as follows: first, the Euler
number of a component $C_{i}$ is given as $2-2g$ ($g$ is the genus), and there
is a minus one contribution for every double point. The total Euler number is
the sum of the contributions from each component and the double points.

Example: Let’s compute $\delta_{x}$ for the dual graph in figure. 4. There are
seven component with genus zero and each contributes $2$ to the total Euler
number, and there are six intersection points, so $\epsilon(F)=14-6=8$, and
$\delta_{x}=10$.

The number $d_{x}$: This number is the most complicated one to define, see
kenji1988discriminants  for details. To begin with, we notice that an
arbitrary one parameter genus two curves can be represented by an equation

$y^{2}=f(x,t).$

with $f(x,t)$ a polynomial of degree $5$ or $6$ in $x$. The discriminant for a
sextic $g(x)=u_{0}\prod_{i=1}^{6}(x-a_{i})$ is defined as

$\Delta=u_{0}^{10}\prod_{i<j}(a_{i}-a_{j})^{2}.$ (11)

Notice that there are many equations which would give the same surface. Now
the basis for the holomorphic differential for a genus two hyper-elliptic
curve is taken as $w_{1}=dx/y$ and $w_{2}=xdx/y$, and $\Delta(w_{1}\wedge
w_{2})^{10}$ is actually independent of the defining equation for the given
surface 888At a fixed point $t$ on the base $\mathbb{P}^{1}$, there is fiber
of a rank two holomorphic vector bundle $V$ spanned by the differential
$w_{1}$ and $w_{2}$. The antisymmetric tensor product $\Lambda^{2}V$ (the
basis is $w_{1}\wedge w_{2}$) gives a line bundle over $\mathbb{P}^{1}$.. Now
the invariant $d_{x}$ is defined as

$d_{x}=ord_{t}\Delta(w_{1}\wedge w_{2})^{10}.$

Here $ord_{t}$ means the order of the zero of $\Delta(w_{1}\wedge w_{2})^{10}$
at $t=0$. A hypelliptic equation is called minimal equation if
$ord_{t}\Delta=d_{x}$, here $\Delta$ is the discriminant of the defining
equation.

The invariants $d_{x}$ and $\delta_{x}$ can also be similarly defined for
genus one fibration, and it can be shown $d_{x}=\delta_{x}$. The equality does
not hold for genus two case. The difference of $d_{x}$ and $\delta_{x}$ is
given by the following equation (for all the cases except eight types):

$d_{x}=\delta_{x}+l+1.$

here $l$ is a simple function of $m$ (in most cases $l=m-1$). Let’s recall
that, $m$ is the degree needed for the classification of degenerations. For
the following eight types:
$[1]-II,~{}[1]-IV,~{}[1]-VIII-3,~{}[3]-II_{n-0}^{*},~{}[3]-IV^{*}-II_{0},~{}[3]-III^{*}-II_{0},~{}[5]-II_{n-p}^{*},~{}[5]-III_{n}^{*}$,
the formula is changed to

$d_{x}=\delta_{x}+l+2.$

Gauge algebra: One can also associate a gauge algebra on the singular fiber,
which can be read from the dual graph $F$. The crucial idea is the balanced
chain in the dual graph. Let’s first review the notion of a balanced chain.
Given a dual graph $F=\sum n_{i}C_{i}$, a node $C_{i}$ is called balanced if
$\sum_{j\neq i}C_{ij}n_{j}=2n_{i}$, and one can verify that only type $E$
component is balanced. Proof: Let’s write the dual graph as
$F=n_{i}C_{i}+\delta$, and since $C_{i}\cdot F=0$, we have

$\displaystyle C_{i}\cdot(n_{i}C_{i}+\delta)=0\to(C_{i}\cdot
C_{i})n_{i}+C_{i}\cdot\delta=0,$ $\displaystyle\to\sum_{j\neq
i}C_{ij}n_{j}=-(C_{i}\cdot C_{i})n_{i}.$

so the balance condition only holds for type $E$ component (with $C_{i}\cdot
C_{i}=-2$).

A connected subgraph in $F$ is called balanced chain if all of the components
in this subgraph is balanced, and it can be proven that the balanced chain
form a $ADE$ or affine $ADE$ Dynkin diagram. Now the gauge algebra is read
from the dual graph as follows : each balanced ADE chain (or affine ADE chain)
gives a corresponding ADE algebra.

Example: Let’s look at the example in figure. 4. There is a balanced chain
forming a $D_{6}$ Dynkin diagram (only D component is not balanced), and so
the gauge algebra of this singular fiber is $D_{6}$.

Physically, the gauge algebra gives the flavor symmetry of low energy theory.

####  2.4.1 Local splitting of singularities

It is possible to deform a local singularity $F$ by changing the defining
family. In fact, it was shown in ashikaga2000global  that each local
singularity can be deformed into some number of $a_{x}$ $I_{1}$ (type
$I_{[1-0-0]}$ in cite) singularity and $t_{x}$ $\tilde{I}_{1}$ (type
$I_{[0-0-1]}$ in namikawa1973complete ) singularity, see figure.8. Those
numbers $a_{x},t_{x}$ are related to the local invariants $d_{x},\delta_{x}$
as follows:

$\boxed{t_{x}=d_{x}-\delta_{x},~{}~{}a_{x}=2\delta_{x}-d_{x}}.$ (12)

The dual graph for $I_{1}$ and $\tilde{I_{1}}$ singularity are shown in
figure. 8, and they are called atomic singular fiber takamura2004towards . The
local data $(d_{x},\delta_{x})$ for $I_{1}$ ($\tilde{I}_{1}$) singularities
are $I_{1}=(1,1)$ ($\tilde{I}_{1}=(2,1)$), and it can be easily verified the
data for the original singularity $F$ is equal to the sum of the local data of
atomic singularity ($I_{1}$ and $\tilde{I}_{1}$ type).

$\widetilde{I_{1}}$$I_{1}$Splitting$I_{1}$$\widetilde{I_{1}}$ Figure 8: Up:
Local splitting of a singularity: the complicated singularity is split into
several $I_{1}$ and $\tilde{I}_{1}$ singularities. Bottom: The geometric
representation of $I_{1}$ and $\tilde{I}_{1}$ singularity; $I_{1}$ singularity
has a non-separating node while $\tilde{I}_{1}$ singularity has a separating
node.

###  2.5 Low energy theory

The important data for genus two singular fibers are summarized in following
tables. 6, 7,8,9,10 . The invariants $d_{x},\delta_{x},n_{t}$ are included,
and so the invariant $t_{x}=d_{x}-\delta_{x}$ can be computed directly.
$n_{t}$ and $\delta_{x}$ can be found from the dual graph listed in
namikawa1973complete . $d_{x}$ has been computed in kenji1988discriminants .

The gauge algebra is also computed using the dual graph, which should be part
of the flavor symmetry of the low energy theory. The scaling dimension of the
Coulomb branch operator is constrained by the eigenvalues of the monodromy
group, and there are actually several choices for a given set of eigenvalues,
see table. 4. The dual graph plays an essential role for us to find the exact
scaling dimension and the low energy theory. First of all, if the dual graph
is formed by two Kodaira’s graphs intersected with a type $D$ or $C$ component
(see figure. 9), it is natural to guess the low energy theory is just the
direct sum of two rank one theories. The map between Kodaira’s singularity and
low energy theory is listed in table. 5. More generally, one can find a 3d
mirror from the dual graph, which is then used to find the low energy theory.

$Kod_{1}$$Kod_{2}$$D$$Kod$$C$ Figure 9: The dual graph for a genus two
singular fiber: it is given by two Kodaira dual graphs intersecting with a
type $D$ component or type $C$ component. Kodaira’s dual graph always has a
component with multiplicity one, which is replaced by $C$ or $D$ component in
the figure.

Kodaira’s list | Theory
---|---
$II^{*}$ |  $E_{8}$ SCFT
$III^{*}$ |  $E_{7}$ SCFT
$IV^{*}$ |  $E_{6}$ SCFT
$II$ |  $H_{0}$ SCFT
$III$ |  $H_{1}$ SCFT
$IV$ |  $H_{2}$ SCFT
$I_{n}$ | $U(1)-n$
$I_{n}^{*}$ | $SU(2)-(n+4)$

Table 5: The theory associated with Kodaira’s list of rank one singular fiber.
The $E_{n}$ type theories were found in Minahan:1996cj , and the $H_{i}$ type
theories were found in Argyres:1995xn .

####  2.5.1 3d mirror from dual graph

Let’s first review 3d mirror symmetry relevant for our context. One can
compactify 4d $\mathcal{N}=2$ theory on a circle to get a 3d $\mathcal{N}=4$
theory $A$. 3d $\mathcal{N}=4$ theory has interesting mirror symmetry
property: there is a mirror theory $B$ such that the Higgs branch of $B$ is
identified with the Coulomb branch of $A$ and vice versa. In our case, the
hyperkahler dimension of the theory $A$ is two, and so the Higgs branch of the
mirror theory $B$ should have dimension 2.

The way of attaching a quiver gauge theory for a singular fiber $F=\sum
n_{i}C_{i}$ is following: a): attach a quiver node for each component $C_{i}$,
and the gauge group is $U(n_{i})$; b): the number of bi-fundamental
hypermultiplets between two quiver nodes $i,j$ is $C_{ij}$; c): There is one
adjoint hypermultiplet for genus one component. Let’s now compute the Higgs
branch of the constructed quiver (there is an added one because the overall
$U(1)$ gauge group is decoupled; we also leave the extra adjoint
hypermultiplet aside for a while.):

$\displaystyle dimHiggs={1\over 2}\sum_{i\neq j}n_{i}n_{j}C_{ij}-\sum
n_{i}^{2}+1$ $\displaystyle={1\over 2}(\sum n_{i}C_{i})\cdot(\sum
n_{i}C_{i})+{1\over 2}\sum(-C_{i}^{2}n_{i}^{2}-2n_{i}^{2})+1$
$\displaystyle={1\over 2}\sum_{C_{i}\neq
E}(-C_{i}^{2}n_{i}^{2}-2n_{i}^{2})+1.$

Here the fact $C_{i}\cdot F=0$ is used. By looking at the data listed in
table. 1, one find (one need to add the contribution of adjoint hypermultiplet
for the $2A$ case):

$\displaystyle dimHiggs=3,~{}~{}\text{for type}~{}~{}2B$ $\displaystyle
dimHiggs=2,~{}~{}\text{for type}~{}~{}D,~{}(B,B),$ $\displaystyle
dimHiggs=1,~{}~{}\text{for type}~{}~{}C,~{}(A,B)$ $\displaystyle
dimHiggs=0,~{}~{}\text{for type}~{}~{}(A,A)$ $\displaystyle
dimHiggs=1,~{}~{}\text{for type}~{}~{}2A$

After some experiments, the following rules for finding the 3d mirror are
found:

  1. 1.

For type $D$ and $(B,B)$, the 3d mirror is just the quiver gauge theory found
from the dual graph.

  2. 2.

For type $2B$, if there is a type $E$ component with multiplicity one which
intersects $2B$ component, then the 3d mirror is found by removing the $U(1)$
node corresponding to $E$. One can easily see that the Higgs branch of the new
quiver has dimension two. One can do similar thing for type $2A$.

  3. 3.

For type $C$, the 3d mirror is just that of a rank one theory represented by
Kodaira dual graph.

We do not know how to extract 3d mirror for other type of dual graphs.

Example 2: The dual graph for $III_{n}$ type singular fiber is shown in
figure. 10. We recognize that it is the 3d mirror for $SU(3)$ gauge theory
coupled with $n+6$ fundamental flavors Nanopoulos:2010bv , which is therefore
the low energy theory associated with the singularity $III_{n}$.

$B$$B$$2$$2$$3$$3$$n+1$$2$$2$$1$$1$$3$$n+1$$1$$1$$3$ Figure 10: Left: the dual
graph for singularity $III_{n}$; right: the 3d mirror for the low energy
theory associated with the singular fiber $III_{n}$.

####  2.5.2 List of tables

The monodromy group, invariants $d_{x},\delta_{x},n_{t}$ for every genus two
singular fibers are included. The scaling dimension, gauge algebra, and the
theory are included. The fixed points and dual graph for each singular fiber
can be found in namikawa1973complete . We only show data for singular fiber
with $m=0$, with the only exception $I_{0}-I_{0}-1$ ($\tilde{I}_{1}$) fiber.

Table 6: Elliptic fiber [1].

Type | Monodromy | $d_{x}$ | $\delta_{x}$ | $l$ | Theory | Scaling dimension | Gauge algebra |  $n_{t}$ (components)
---|---|---|---|---|---|---|---|---
$I_{0-0-0}$ | $\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right)$ | 0 | 0 | -1 | $U(1)^{2}$ | $(1,1)$ | $\emptyset$ | 1
$I^{*}_{0-0-0}$ | $\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $10$ | $10$ | $-1$ | $2-SU(2)-SU(2)-2$ | $(2,2)$ | $A_{1}^{6}$ | 7
$II$ | $\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{array}\right)$ | $5$ | $4$ | $-1$ | $?$ | $(2,1)$ | $A_{1}^{2}$ | 3
$III$ | $\left(\begin{array}[]{cccc}0&-1&0&0\\\ 1&-1&0&0\\\ 0&0&-1&-1\\\ 0&0&1&0\end{array}\right)$ | $10$ | $10$ | $-1$ | $SU(3)-6$ | $(3,2)$ | $A_{5}$ | 7
$IV$ | $\left(\begin{array}[]{cccc}0&1&0&0\\\ -1&1&0&0\\\ 0&0&1&1\\\ 0&0&-1&0\end{array}\right)$ | $10$ | $9$ | $-1$ | $?$ | $(6,2)$ | $D_{5}$ | 6
$V$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&1&1\\\ -1&1&0&0\\\ 0&-1&0&0\end{array}\right)$ | $5$ | $5$ | $-1$ | SCFT | $(\frac{6}{4},\frac{5}{4})$ |  | 2
$V^{*}$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&-1&-1\\\ 1&-1&0&0\\\ 0&1&0&0\end{array}\right)$ | $15$ | $15$ | $-1$ | SCFT | $(6,5)$ | $A_{11}$ | 12
$VI$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&-1&-1\\\ 1&-1&0&0\\\ 0&1&0&0\end{array}\right)$ | $10$ | $10$ | $-1$ | $Sp(4)-6(\textbf{F})$ | $(4,2)$ | $D_{6}$ | 7
$VII$ | $\left(\begin{array}[]{cccc}0&1&1&0\\\ 1&0&0&-1\\\ 0&0&0&-1\\\ 0&0&1&0\end{array}\right)$ | $5$ | $5$ | $-1$ | $SCFT$ | $(\frac{8}{5},\frac{6}{5})$ | $A_{1}$ | 2
$VII^{*}$ | $\left(\begin{array}[]{cccc}0&-1&-1&0\\\ 1&0&0&1\\\ 0&0&0&1\\\ 0&0&-1&0\end{array}\right)$ | $15$ | $15$ | $-1$ | $SCFT$ | $(8,6)$ | $D_{10}\oplus A_{1}$ | 12
$VIII-1$ | $\left(\begin{array}[]{cccc}0&1&1&0\\\ 1&0&0&1\\\ -1&1&1&0\\\ 0&-1&0&0\end{array}\right)$ | $4$ | $4$ | $-1$ | $SCFT$ | $(\frac{10}{7},\frac{8}{7})$ | $\emptyset$ | 1
$VIII-2$ | $\left(\begin{array}[]{cccc}0&0&1&1\\\ -1&1&1&0\\\ 0&-1&0&0\\\ 0&0&-1&0\end{array}\right)$ | $12$ | $12$ | $-1$ | $SCFT$ | $(10,4)$ | $E_{8}$ | 9
$VIII-3$ | $\left(\begin{array}[]{cccc}0&-1&-1&-1\\\ 0&0&-1&0\\\ 0&0&0&-1\\\ 1&0&0&1\end{array}\right)$ | $8$ | $7$ | $-1$ | $SCFT$ | $(\frac{10}{3},\frac{4}{3})$ | $A_{3}$ | 4
$VIII-4$ | $\left(\begin{array}[]{cccc}1&0&-1&0\\\ 0&0&0&-1\\\ 1&0&0&1\\\ -1&1&1&0\end{array}\right)$ | $16$ | $16$ | $-1$ | $SCFT$ | $(10,8)$ | $D_{12}$ | $13$
$I_{X}-1$ | $\left(\begin{array}[]{cccc}0&1&1&1\\\ 0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&-1\end{array}\right)$ | $8$ | $8$ | $-1$ | $D_{2}(SU(5))$ | $(\frac{5}{2},\frac{3}{2})$ | $A_{4}$ | 5
$I_{X}-2$ | $\left(\begin{array}[]{cccc}-1&0&1&0\\\ 0&0&0&1\\\ -1&0&0&-1\\\ 1&-1&-1&0\end{array}\right)$ | $6$ | $6$ | $-1$ | $SCFT$ | $(\frac{5}{3},\frac{4}{3})$ | $A_{1}$ | 3
$I_{X}-3$ | $\left(\begin{array}[]{cccc}0&-1&-1&0\\\ -1&0&0&-1\\\ 1&-1&-1&0\\\ 0&1&0&0\end{array}\right)$ | $14$ | $14$ | $-1$ | $SCFT$ | $(5,4)$ | $A_{9}\oplus A_{1}$ | 11
$I_{X}-4$ | $\left(\begin{array}[]{cccc}0&0&-1&-1\\\ 1&-1&-1&0\\\ 0&1&0&0\\\ 0&0&1&0\end{array}\right)$ | $12$ | $12$ | $-1$ | $SCFT$ | $(5,3)$ | $D_{7}\oplus A_{1}$ | 9

Table 7: Elliptic fiber [2].

Type | Monodromy | $d_{x}$ | $\delta_{x}$ | $l$ | Theory | Scaling dimension | Gauge algebra |  $n_{t}$ (components)
---|---|---|---|---|---|---|---|---
$I_{0}-I_{0}-1$ | $\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right)$ | 2 | 1 | 0 | $U(1)^{2}$ | $(1,1)$ | $\emptyset$ | 2
$I^{*}_{0}-I_{0}^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $12$ | $12$ | $-1$ | $D_{4}\oplus D_{4}$ | $(2,2)$ | $D_{4}\oplus D_{4}$ | 9
$I_{0}-I_{0}^{*}-0$ | $\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&1&0\\\ 0&0&0&-1\end{array}\right)$ | $6$ | $6$ | $-1$ | $D_{4}\oplus U(1)$ | $(2,1)$ | $D_{4}$ | 5
$2I_{0}-0$ | $\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{array}\right)$ | $7$ | $5$ | $1$ | $?$ | $(2,1)$ | $D_{3}$ | 4
$2I_{0}^{*}-0$ | $\left(\begin{array}[]{cccc}0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\end{array}\right)$ | $11$ | $10$ | $0$ | $SCFT$ | $(4,2)$ | $D_{4}\oplus A_{1}^{2}$ | 7
$I_{0}-II-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $2$ | $2$ | $-1$ | $H_{0}\oplus U(1)$ | $(\frac{6}{5},1)$ | $\emptyset$ | 1
$I_{0}-II^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&1&0&0\\\ 1&0&1&0\\\ 0&0&0&1\end{array}\right)$ | $10$ | $10$ | $-1$ | $E_{8}\oplus U(1)$ | $(6,1)$ | $E_{8}$ | 10
$I_{0}-IV-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&-1&-1\\\ 1&-1&0&0\\\ 0&1&0&0\end{array}\right)$ | $4$ | $4$ | $-1$ | $H_{2}\oplus U(1)$ | $(\frac{3}{2},1)$ | $A_{2}$ | 4
$I_{0}-IV^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&1&0&0\\\ -1&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $8$ | $8$ | $-1$ | $E_{6}\oplus U(1)$ | $(3,1)$ | $E_{6}$ | 8
$I_{0}^{*}-II-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&-1&0&0\\\ -1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $8$ | $8$ | $-1$ | $D_{4}\oplus H_{0}$ | $(2,\frac{6}{5})$ | $D_{4}$ | 5
$I_{0}^{*}-II^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&-1&0&0\\\ 1&0&1&0\\\ 0&0&0&-1\end{array}\right)$ | $16$ | $16$ | $-1$ | $D_{4}\oplus E_{8}$ | $(6,2)$ | $E_{8}\oplus D_{4}$ | 14
$I_{0}^{*}-II^{*}-\alpha$ |  | $14$ | $14$ | $-1$ | $?$ |  $(6,2)$? | $E_{7}\oplus A_{1}^{3}$ | 11
$I_{0}^{*}-IV-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&-1&0&0\\\ -1&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $10$ | $10$ | $-1$ | $D_{4}\oplus H_{2}$ | $(2,\frac{3}{2})$ | $D_{4}\oplus A_{2}$ | 7
$I_{0}^{*}-IV^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $14$ | $14$ | $-1$ | $D_{4}\oplus E_{6}$ | $(2,3)$ | $D_{4}\oplus E_{6}$ | 12
$I_{0}^{*}-IV^{*}-\alpha$ |  | $12$ | $12$ | $-1$ | $?$ |  $(3,2)$? | $A_{5}\oplus A_{1}^{3}$ | 9
$I_{0}-III-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $3$ | $3$ | $-1$ | $U(1)\oplus H_{1}$ | $(\frac{4}{3},1)$ | $A_{1}$ | 3
$I_{0}-III^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $9$ | $9$ | $-1$ | $U(1)\oplus E_{7}$ | $(4,1)$ | $E_{7}$ | 9
$I_{0}^{*}-III-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&-1&0&0\\\ -1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $9$ | $9$ | $-1$ | $D_{4}\oplus H_{1}$ | $(2,\frac{4}{3})$ | $D_{4}\oplus A_{1}$ | 6
$I_{0}^{*}-III^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $15$ | $15$ | $-1$ | $D_{4}\oplus E_{7}$ | $(4,2)$ | $D_{4}\oplus E_{7}$ | 12
$I_{0}^{*}-III^{*}-\alpha$ |  | $13$ | $13$ | $-1$ | $?$ |  $(4,2)$? | $D_{6}\oplus A_{1}^{3}$ | 10
$2II-0$ | $\left(\begin{array}[]{cccc}0&1&0&1\\\ 1&0&0&0\\\ 0&-1&0&0\\\ 0&0&1&0\end{array}\right)$ | 7 | 6 | 0 | $SCFT$ | $(\frac{12}{5},\frac{6}{5})$ | $A_{1}^{2}$ | 3
$2II^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&0&-1\\\ 1&0&0&0\\\ 0&1&0&1\\\ 0&0&1&0\end{array}\right)$ | $15$ | $14$ | $0$ | $SCFT$ | $(12,6)$ | $E_{8}\oplus A_{1}^{2}$ | 11
$II-II-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&1&0&1\\\ -1&0&0&0\\\ 0&-1&0&0\end{array}\right)$ | 4 | 4 | $-1$ | $H_{0}\oplus H_{0}$ | $(\frac{6}{5},\frac{6}{5})$ | $\emptyset$ | 1
$II-II^{*}-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&0&0&-1\\\ -1&0&0&0\\\ 0&1&0&1\end{array}\right)$ | $12$ | $12$ | $-1$ | $H_{0}\oplus E_{8}$ | $(6,\frac{6}{5})$ | $E_{8}$ | 12
$II^{*}-II^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&-1\\\ 1&0&1&0\\\ 0&1&0&1\end{array}\right)$ | $20$ | $20$ | $-1$ | $E_{8}\oplus E_{8}$ | $(6,6)$ | $E_{8}\oplus E_{8}$ | 19
$II^{*}-II^{*}-\alpha$ |  | $18$ | $18$ | $-1$ | $?$ | $?$ | $E_{7}\oplus E_{7}$ | 15
$II-IV-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&1\\\ -1&0&-1&0\\\ 0&-1&0&0\end{array}\right)$ | $6$ | $6$ | $-1$ | $H_{0}\oplus H_{2}$ | $(\frac{3}{2},\frac{6}{5})$ | $A_{2}$ | 3

Table 8: Elliptic fiber [2]-continued.

Type | Monodromy | $d_{x}$ | $\delta_{x}$ | $l$ | Theory | Scaling dimension | Gauge algebra |  $n_{t}$ (components)
---|---|---|---|---|---|---|---|---
$II-IV^{*}-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&-1&0&-1\\\ -1&0&0&0\\\ 0&1&0&0\end{array}\right)$ | $10$ | $10$ | $-1$ | $H_{0}\oplus E_{6}$ | $(3,\frac{6}{5})$ | $E_{6}$ | 8
$II^{*}-IV-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&1&0&0\\\ -1&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $14$ | $14$ | $-1$ | $H_{2}\oplus E_{8}$ | $(6,\frac{3}{2})$ | $E_{8}\oplus A_{2}$ | 11
$II^{*}-IV-\alpha$ |  | $12$ | $12$ | $-1$ | $?$ | $?$ | $E_{7}$ | 9
$II^{*}-IV^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&-1&0&-1\\\ 1&0&1&0\\\ 0&1&0&0\end{array}\right)$ | $18$ | $18$ | $-1$ | $E_{6}\oplus E_{8}$ | $(6,3)$ | $E_{8}\oplus E_{6}$ | 17
$II^{*}-IV^{*}-\alpha$ |  | $16$ | $16$ | $-1$ | $?$ | $?$ | $E_{7}\oplus A_{5}$ | 13
$2IV-0$ | $\left(\begin{array}[]{cccc}0&0&0&1\\\ 1&0&0&0\\\ 0&-1&0&-1\\\ 0&0&1&0\end{array}\right)$ | $9$ | $8$ | $0$ | $SCFT$ | $(3,\frac{3}{2})$ | $A_{2}\oplus A_{1}^{2}$ | 5
$2IV^{*}-0$ | $\left(\begin{array}[]{cccc}0&-1&0&-1\\\ 1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\end{array}\right)$ | $13$ | $12$ | $0$ | $SCFT$ | $(6,3)$ | $E_{6}\oplus A_{1}^{2}$ | 9
$IV-IV-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\ -1&0&-1&0\\\ 0&-1&0&-1\end{array}\right)$ | $8$ | $8$ | $-1$ | $H_{2}\oplus H_{2}$ | $(\frac{3}{2},\frac{3}{2})$ | $A_{2}\oplus A_{2}$ | 5
$IV-IV^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&-1&0&-1\\\ -1&0&-1&0\\\ 0&1&0&0\end{array}\right)$ | $12$ | $12$ | $-1$ | $E_{6}\oplus H_{2}$ | $(3,\frac{3}{2})$ | $A_{1}$ | 10
$IV^{*}-IV^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&-1&0&-1\\\ 1&0&0&0\\\ 0&1&0&0\end{array}\right)$ | $16$ | $16$ | $-1$ | $E_{6}\oplus E_{6}$ | $(3,3)$ | $E_{6}\oplus E_{6}$ | 15
$IV^{*}-IV^{*}-\alpha$ |  | $14$ | $14$ | $-1$ | $?$ | $?$ | $A_{5}\oplus A_{5}$ | 11
$II-III-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&0&0&-1\\\ -1&0&0&0\\\ 0&1&0&0\end{array}\right)$ | $5$ | $5$ | $-1$ | $H_{0}\oplus H_{1}$ | $(\frac{4}{3},\frac{6}{5})$ | $A_{1}$ | 2
$II-III^{*}-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\ 0&-1&0&0\end{array}\right)$ | $11$ | $11$ | $-1$ | $H_{0}\oplus E_{7}$ | $(4,\frac{6}{5})$ | $E_{7}$ | 9
$II^{*}-III-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&1\\\ 1&0&1&0\\\ 0&-1&0&0\end{array}\right)$ | $13$ | $13$ | $-1$ | $E_{8}\times H_{1}$ | $(6,\frac{4}{3})$ | $E_{8}\oplus A_{1}$ | 11
$II^{*}-III-\alpha$ |  | $11$ | $11$ | $-1$ | $?$ | $?$ | $E_{7}$ | 8
$II^{*}-III^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&-1\\\ 1&0&1&0\\\ 0&1&0&0\end{array}\right)$ | 19 | 19 | -1 | $E_{8}\oplus E_{7}$ | $(6,4)$ | $E_{8}\oplus E_{7}$ | 17
$II^{*}-III^{*}-\alpha$ |  | $17$ | $17$ | $-1$ | $?$ |  $(6,4)$? | $E_{7}\oplus D_{6}$ | 14
$IV-III-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\ -1&0&-1&0\\\ 0&-1&0&0\end{array}\right)$ | $7$ | $7$ | $-1$ | $H_{1}\oplus H_{2}$ | $(\frac{3}{2},\frac{4}{3})$ | $A_{2}\oplus A_{1}$ | 4
$IV-III^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\ -1&0&-1&0\\\ 0&1&0&0\end{array}\right)$ | $13$ | $13$ | $1$ | $H_{1}\oplus E_{7}$ | $(\frac{3}{2},4)$ | $A_{2}\times E_{7}$ | 11
$IV-III^{*}-\alpha$ |  | $11$ | $11$ | $0$ | $?$ |  $(4,3)$? | $D_{6}$ | 8
$IV^{*}-III-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\ 0&-1&0&0\end{array}\right)$ | $11$ | $11$ | $-1$ | $H_{1}\oplus E_{6}$ | $(3,\frac{4}{3})$ | $E_{6}\oplus A_{1}$ | 9
$IV^{*}-III^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&0&0&-1\\\ 1&0&0&0\\\ 0&1&0&0\end{array}\right)$ | $17$ | $17$ | $-1$ | $E_{7}\oplus E_{6}$ | $(4,3)$ | $E_{7}\oplus E_{6}$ | 16
$IV^{*}-III^{*}-\alpha$ |  | $15$ | $15$ | $-1$ | $?$ | $?$ | $A_{5}\oplus D_{6}$ | 12
$2III-0$ | $\left(\begin{array}[]{cccc}0&0&0&1\\\ 1&0&0&0\\\ 0&-1&0&0\\\ 0&0&1&0\end{array}\right)$ | $8$ | $7$ | $0$ | $SCFT$ | $(\frac{8}{3},\frac{4}{3})$ | $A_{1}\oplus A_{1}^{2}$ | 4
$2III^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&0&-1\\\ 1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\end{array}\right)$ | $14$ | $13$ | $0$ | $SCFT$ | $(8,4)$ | $E_{7}\oplus A_{1}^{2}$ | 10
$III-III-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\ 0&-1&0&0\end{array}\right)$ | $6$ | $6$ | $-1$ | $H_{1}\oplus H_{1}$ | $(\frac{4}{3},\frac{4}{3})$ | $A_{1}\oplus A_{1}$ | 3
$III-III^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\ -1&0&0&0\\\ 0&1&0&0\end{array}\right)$ | $12$ | $12$ | $-1$ | $E_{7}\oplus H_{1}$ | $(4,\frac{4}{3})$ | $E_{7}\oplus A_{1}$ | 10
$III^{*}-III^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&-1\\\ 1&0&0&0\\\ 0&1&0&0\end{array}\right)$ | $18$ | $18$ | $-1$ | $E_{7}\oplus E_{7}$ | $(4,4)$ | $E_{7}\oplus E_{7}$ | 17
$III^{*}-III^{*}-\alpha$ |  | $16$ | $16$ | $-1$ | $?$ | $?$ | $D_{6}\oplus D_{6}$ | 13

Table 9: Parabolic fiber [3].

Type | Monodromy | $d_{x}$ | $\delta_{x}$ | $l$ | Theory | Scaling dimension | Gauge algebra |  $n_{t}$(components)
---|---|---|---|---|---|---|---|---
$I_{n-0-0}$ | $\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&n\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right)$ | $n$ | $n$ | -1 | $U(1)-n\oplus U(1)$ | $(1,1)$ | $A_{n-1}$ | $n$
$I_{0}-I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&-n\\\ 0&0&1&0\\\ 0&0&0&-1\end{array}\right)$ | $n+6$ | $n+6$ | $-1$ | $SU(2)-(n+4)\oplus U(1)$ | $(2,1)$ | $D_{n+4}$ | $n+5$
$I_{n}-I_{0}^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&1&0&n\\\ 0&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $n+6$ | $n+6$ | $-1$ | $SU(2)-(4)\oplus U(1)-n$ | $(2,1)$ | $D_{4}\oplus A_{n-1}$ | $n+4$
$I_{n-0-0}^{*}$ | $\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&-1&0&-n\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $n+10$ | $n+10$ | $-1$ | $(n+2)-SU(2)-SU(2)-2$ | $(2,2)$ | $D_{n+2}\times A_{1}^{3}$ | $n+7$
$I_{0}^{*}-I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&-1&0&-n\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $n+12$ | $n+12$ | $-1$ | $SU(2)-4\oplus SU(2)-(n+4)$ | $(2,2)$ | $D_{4}\times D_{n+4}$ | $n+9$
$II_{n-0}$ | $\left(\begin{array}[]{cccc}-1&0&0&0\\\ -1&1&0&n\\\ 0&0&-1&-1\\\ 0&0&0&1\end{array}\right)$ | $n+5$ | $n+5$ | $-1$ | $(n-1)-U(1)-SU(2)-3$ | $(2,1)$ | $A_{n-2}\oplus D_{3}$ | $n+3$
$II_{n-0}^{*}$ | $\left(\begin{array}[]{cccc}1&0&0&0\\\ 1&-1&0&-n\\\ 0&0&1&1\\\ 0&0&0&-1\end{array}\right)$ | $n+5$ | $n+4$ | $-1$ | $?$ | $(2,1)$ | $D_{n+2}$ | $n+3$
$II-I_{n}-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&1&0&n\\\ -1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $n+2$ | $n+2$ | $-1$ | $H_{0}\oplus I_{n}$ | $(\frac{6}{5},1)$ | $A_{n-1}$ | $n$
$II^{*}-I_{n}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&1&0&n\\\ 1&0&1&0\\\ 0&0&0&1\end{array}\right)$ | $n+10$ | $n+10$ | $-1$ | $E_{8}\oplus I_{n}$ | $(6,1)$ | $E_{8}\oplus A_{n-1}$ | $n+9$
$IV-I_{n}-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&n\\\ -1&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $n+4$ | $n+4$ | $-1$ | $H_{2}\oplus I_{n}$ | $(\frac{3}{2},1)$ | $A_{2}\oplus A_{n-1}$ | $n+2$
$IV^{*}-I_{n}-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&1&0&n\\\ 1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $n+8$ | $n+8$ | $-1$ | $E_{6}\oplus I_{n}$ | $(3,1)$ | $E_{6}\oplus A_{n-1}$ | $n+7$
$II-I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&-1&0&-n\\\ -1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $n+8$ | $n+8$ | $-1$ | $H_{0}\oplus I_{n}^{*}$ | $(2,\frac{6}{5})$ | $D_{n+4}$ | $n+5$
$II^{*}-I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&-1&0&-n\\\ 1&0&1&0\\\ 0&0&0&-1\end{array}\right)$ | $n+16$ | $n+16$ | $-1$ | $E_{8}\oplus I_{n}^{*}$ | $(6,2)$ | $E_{8}\oplus D_{n+4}$ | $n+14$
$II^{*}-I_{n}^{*}-\alpha$ |  | $n+14$ | $n+14$ | $-1$ | $(n+2)-SU(2)-E_{8}$ | $(6,2)$ | $D_{n+2}\oplus E_{7}\oplus A_{1}$ | $n+11$
$IV-I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&-1&0&-n\\\ -1&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $n+10$ | $n+10$ | $-1$ | $H_{2}\oplus I_{n}^{*}$ | $(\frac{3}{2},2)$ | $E_{6}\oplus A_{2}$ | $n+7$
$IV^{*}-I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&-1&0&-n\\\ 1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $n+14$ | $n+14$ | $-1$ | $E_{6}\oplus I_{n}^{*}$ | $(3,2)$ | $E_{6}\oplus D_{n+4}$ | $n+12$
$IV^{*}-I_{n}^{*}-\alpha$ |  | $n+12$ | $n+12$ | $-1$ | $(n+2)-SU(2)-E_{6}$ | $(3,2)$ | $A_{5}\oplus D_{n+2}\oplus A_{1}$ | $n+9$
$IV-II_{n}$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&-1&n+1\\\ -1&0&-1&1\\\ 0&0&0&1\end{array}\right)$ | $n+4$ | $n+4$ | $-1$ | $n-U(1)-H_{2}$ | $(\frac{3}{2},1)$ | $A_{n-1}$ | $n+2$
$IV^{*}-II_{0}$ |  | $7$ | $6$ | $-1$ | $U(1)\phantom{.}^{\frown}_{\smile}\phantom{.}E_{6}$ | $(3,1)$ | $A_{3}$ | $4$
$IV^{*}-II_{n}$ | $\left(\begin{array}[]{cccc}-1&0&-1&-1\\\ -1&1&0&n\\\ 1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $n+7$ | $n+7$ | $-1$ | $(n-1)-U(1)-E_{6}$ | $(3,1)$ | $D_{5}\oplus A_{n-2}$ | $n+5$
$II-II^{*}_{n}$ | $\left(\begin{array}[]{cccc}1&0&1&1\\\ 1&-1&0&-n\\\ -1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $n+7$ | $n+7$ | $-1$ | $(n+3)-SU(2)-H_{1}$ | $(2,4/3)$ | $D_{n+3}$ | $n+4$
$II^{*}-II_{n}^{*}$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&-1&1&-n\\\ 1&0&1&-1\\\ 0&0&0&1\end{array}\right)$ | $n+14$ | $n+14$ | $-1$ |  $(n+2)-SU(2)-E_{8}$ $(E_{8}=D_{7}\oplus A_{1})$ |  $(6,4)$? | $A_{1}\oplus D_{7}\oplus D_{n+2}$ | $n+11$
$III-I_{n}-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&n\\\ -1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $n+3$ | $n+3$ | $-1$ | $H_{1}\oplus I_{n}$ | $(\frac{4}{3},1)$ | $A_{1}\oplus A_{n-1}$ | $n+1$
$III^{*}-I_{n}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&1&0&n\\\ 1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $n+9$ | $n+9$ | $-1$ | $E_{7}\oplus I_{n}$ | $(4,1)$ | $E_{7}\oplus A_{n-1}$ | $n+8$
$III-I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&-1&0&-n\\\ -1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $n+9$ | $n+9$ | $-1$ | $H_{1}\oplus I_{n}^{*}$ | $(\frac{4}{3},2)$ | $A_{1}\oplus A_{n+4}$ | $n+6$

Table 10: Parabolic fiber [3] [4] [5].

Type | Monodromy | $d_{x}$ | $\delta_{x}$ | $l$ | Theory | Scaling dimension | Gauge algebra |  $n_{t}$ (components)
---|---|---|---|---|---|---|---|---
$III^{*}-I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&-1&0&-n\\\ 1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $n+15$ | $n+15$ | $-1$ | $E_{7}\oplus I_{n}^{*}$ | $(4,2)$ | $E_{7}\oplus D_{n+4}$ | $n+13$
$III^{*}-I_{n}^{*}-\alpha$ |  | $n+13$ | $n+13$ | $-1$ | $(n+2)-SU(2)-E_{7}$ | $(4,2)$ | $D_{6}\oplus D_{n+2}\oplus A_{1}$ | $n+9$
$III-II_{n}$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&1&n+1\\\ -1&0&0&-1\\\ 0&0&0&1\end{array}\right)$ | $n+3$ | $n+3$ | $-1$ | $n-U(1)-H_{1}$ | $(\frac{4}{3},1)$ | $A_{n-1}$ | $n+1$
$III^{*}-II_{n}$ | $\left(\begin{array}[]{cccc}0&0&-1&1\\\ 1&1&0&n\\\ 1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $n+8$ | $n+8$ | -1 | $(n-1)-U(1)-E_{7}$ | $(4,1)$ | $A_{n-2}\oplus E_{6}$ | $n+6$
$III-II_{n}^{*}$ | $\left(\begin{array}[]{cccc}0&0&1&-1\\\ -1&-1&0&-n\\\ -1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $n+8$ | $n+8$ | $-1$ | $(n+3)-SU(2)-H_{2}$ | $(\frac{3}{2},2)$ | $D_{n+3}$ | $n+5$
$III^{*}-II_{n}^{*}$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&-1&-1&-n-1\\\ 1&0&0&1\\\ 0&0&0&-1\end{array}\right)$ | $n+13$ | $n+13$ | $-1$ |  $(n+2)-SU(2)-E_{7}$ $(E_{7}=SU(2)\oplus A_{6})$ |  $(4,2)$? | $D_{n+2}\oplus A_{6}\oplus A_{1}$ | $n+10$
Parabolic[4] |  |  |  |  |  |  |  |
$I_{n-p-0}$ | $\left(\begin{array}[]{cccc}1&0&p&0\\\ 0&1&0&n\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right)$ | $n+p$ | $n+p$ | $1$ | $I_{n}\oplus I_{p}$ | $(1,1)$ | $A_{n-1}\oplus A_{p-1}$ | $n+p-1$
$I_{n-p-0}^{*}$ | $\left(\begin{array}[]{cccc}-1&0&-p&0\\\ 0&-1&0&-n\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $n+p+10$ | $n+p+10$ | $0$ | $(n+2)-SU(2)-SU(2)-(p+2)$ | $(2,2)$ | $D_{n+2}\oplus A_{1}^{2}\oplus D_{p+2}$ | $n+p+7$
$I_{n}^{*}-I_{p}^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&-p&0\\\ 0&-1&0&-n\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $n+p+12$ | $n+p+12$ | $-1$ | $I_{n}^{*}\oplus I_{p}^{*}$ | $(2,2)$ | $D_{n+4}\oplus D_{p+4}$ | $n+p+9$
$I_{n}-I_{p}^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&-p&0\\\ 0&1&0&n\\\ 0&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $n+p+6$ | $n+p+6$ | $-1$ | $I_{n}\oplus I_{p}^{*}$ | $(2,1)$ | $A_{n-1}\oplus D_{p+4}$ | $n+p+4$
$2I_{n}-0$ | $\left(\begin{array}[]{cccc}0&1&0&n\\\ 1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{array}\right)$ | $n+5$ | $n+4$ | $0$ | $?$ | $(2,1)$ | $A_{n-1}\oplus A_{1}^{2}$ | $n+2$
$2I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}0&-1&0&-n\\\ 1&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\end{array}\right)$ | $n+11$ | $n+10$ | $0$ | $V-Sp(4)-(n+4)(\textbf{4})$ | $(4,2)$ | $D_{n+4}\oplus A_{1}^{2}$ | $n+7$
$II_{n-p}$ | $\left(\begin{array}[]{cccc}-1&0&-p&-1\\\ 0&1&1&n\\\ 0&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $n+p+5$ | $n+p+5$ | $-1$ |  $(n-1)-U(1)-SU(2)-(p+3)$? | $(2,1)$ | $A_{n-2}\oplus D_{p+3}$ | $n+p+3$
$\tilde{III}_{n}$ | $\left(\begin{array}[]{cccc}0&-1&1&0\\\ 1&0&n&-1\\\ 0&0&0&-1\\\ 0&0&1&0\end{array}\right)$ | $n+10$ | $n+10$ | $-1$ | $Sp(4)-(n+6)(\textbf{4})$ | $(4,2)$ | $D_{n+6}$ | $n+7$
Parabolic[5] |  |  |  |  |  |  |  |
$I_{n-p-q}$ | $\left(\begin{array}[]{cccc}1&0&p+q&-q\\\ 0&1&-q&n+q\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right)$ | $n+p+q$ | $n+p+q$ | $-1$ |  $n-U(1)-q-U(1)-p$? | $(1,1)$ | $A_{n-1}\oplus A_{p-1}\oplus A_{q-1}$ | $n+p+q-1$
$I_{n-p-q}^{*}$ | $\left(\begin{array}[]{cccc}-1&0&-p-q&q\\\ 0&-1&q&-n-q\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $n+p+q+10$ | $n+p+q+10$ | $-1$ |  $(n+1)-SU(2)-q+1-SU(2)-(p+1)$? | $(2,2)$ | $D_{n+2}\oplus D_{p+2}\oplus D_{q+2}$ | $n+p+q+7$
$\tilde{II}_{n-p}$ | $\left(\begin{array}[]{cccc}-1&0&-p&0\\\ 1&1&p&n\\\ 0&0&-1&1\\\ 0&0&0&1\end{array}\right)$ | $n+p+5$ | $n+p+5$ | $-1$ | $?$ | $(2,1)$ | $A_{n-2}\oplus A_{p+3}$ | $n+p+3$
$II_{n-p}^{*}$ | $\left(\begin{array}[]{cccc}1&0&-p&0\\\ -1&-1&p&-n\\\ 0&0&1&-1\\\ 0&0&0&-1\end{array}\right)$ | $n+p+5$ | $n+p+4$ | $-1$ | $?$ | $(2,1)$ | $A_{p-1}\oplus D_{n+2}$ | $n+p+2$
$III_{n}$ | $\left(\begin{array}[]{cccc}0&1&-n&n\\\ -1&-1&0&0\\\ 0&0&-1&1\\\ 0&0&-1&0\end{array}\right)$ | $n+10$ | $n+10$ | $-1$ | $SU(3)-(n+6)$ | $(3,2)$ | $A_{n+5}$ | $n+7$
$III_{n}^{*}$ | $\left(\begin{array}[]{cccc}0&-1&n&-n\\\ 1&1&0&0\\\ 0&0&1&-1\\\ 0&0&1&0\end{array}\right)$ | $n+10$ | $n+9$ | $-1$ | $G_{2}-(n+4)\textbf{7}$ | $(6,2)$ | $D_{n+5}$ | $n+6$

##  3 Global coulomb branch geometries

Let’s now construct some global Coulomb branch geometries, where all the
fibers at the bulk are the $I_{1}$ $(I_{1-0-0}$ in table. 9) or
$\tilde{I}_{1}$ ($I_{0}-I_{0}-1$ in table. 7) type fibers. These would
describe the singular fiber configuration of generic deformation on Coulomb
branch. The general structure is shown in figure. 11: there is a singular
fiber at infinity which encodes the UV behavior of the theory, i.e. which
space time dimension the UV theory lives.

$\infty$$I_{1}$$\tilde{I}_{1}$ Figure 11: Global Coulomb branch geometry with
$I_{1}$ and $\tilde{I}_{1}$ type singular fibers at the bulk. The fiber at
infinity gives the information for UV theory, i.e. which space-time dimension
the UV theory lives.

There are several global constraints that a singular fiber configuration has
to satisfy, for example, the total monodromy around the singularities should
be trivial, and the holomorphic quantities (the photon couplings) should be
consistently defined. The study of those constraints would be left in
Xie:ranktwob . Here we’d like to discuss some topological constraints which
are related to the local invariants of the singularity.

The first constraint comes from the geometry of the total space $X$ formed by
genus two fibration over $\mathbb{P}^{1}$: the total space $X$ should be a
rational surface so that it can describe the Coulomb branch configuration for
a local field theory. The rational condition would put constraint on the
choice of singular fibers.

Let’s first review some facts about complex surface. For a compact rational
surface, the irregularity and geometric genus satisfies $q=p_{g}=0$ 999The
irregularity of a compact complex surface $X$ is defined as
$q=dimH^{1,0}(X,C)$, and the geometric genus is given as $p_{g}=dimH^{2,0}$.
The only nontrivial Hodge numbers for a connected rational surface are
$h^{0,0}=h^{2,2}=1$, and $h^{1,1}$., and the Euler number for the rational
surface is $\chi(X)=1$. For a compact complex surface, Norther’s theorem
relates the Euler number of the surface to the first Chern class $c_{1}$ and
second Chern class $c_{2}$ as follows:

$12\chi=c_{1}^{2}+c_{2}.$ (13)

Given a genus two fibered surface, $c_{1}^{2}$ and $c_{2}$ can be computed
from the local data associated with singular fiber kenji1988discriminants .
The local contribution for a singular fiber $F$ is given by the local data
$d_{x},\delta_{x}$ as follows kenji1988discriminants :

$\displaystyle c_{1}^{2}(F)=\frac{6}{5}d_{x}-\delta_{x},$ $\displaystyle
c_{2}(F)=\delta_{x}.$

The total Chern class of the surface $X$ are computed from the local data as:

$\displaystyle c_{1}^{2}(X)=-8+\sum
c_{1}^{2}(F)=-8+\sum(\frac{6}{5}d_{x}-\delta_{x})$ $\displaystyle
c_{2}(X)=-4+\sum c_{2}(F)=-4+\sum\delta_{x}$ (14)

Using Noether’s theorem (see equation 13), the local data for the singular
fibers should satisfy following equation:

$12\chi=-12+\sum(c_{1}^{2}(F)+c_{2}(F))=-12+\sum(\frac{6}{5}d_{x}).$

Since the Euler number $\chi$ of a rational surface $X$ is one, one get from
above equation

$\boxed{24=\sum(\frac{6}{5}d_{x})\to\sum d_{x}=20}.$ (15)

This is the first important equation that a configuration of singular fibers
relevant for the Coulomb branch should satisfy. Notice only the assumption
that the total space $X$ should be rational is used in deriving above
equation.

The second constraint comes from the relation between flavor symmetry and the
Mordell-Weil lattice. One can define a Mordell-Weil lattice for higher genus
fibration over $\mathbb{P}^{1}$, whose rank is given as shioda1992mordell :

$rank(MW(X))=\rho(X)-2-\sum(n_{t}-1).$ (16)

Here $n_{t}$ is the number of components in the dual graph of the singular
fiber; and $\rho(X)$ is the Picard number of $X$. $11\leq\rho(X)\leq 14$ for a
genus two fibration $X$.

Let’s now relate $\rho(X)$ to the local data of the singular fibers. First of
all, there is the following formula for the Picard number:

$\rho(X)=c_{2}(X)-2=\sum\delta_{x}-6.$

Formula 14 for $c_{2}(X)$ is used in deriving above equation. Let’s now recall
that the number $n_{x}=2\delta_{x}-d_{x},t_{x}=d_{x}-\delta_{x}$ ($n_{x}$
($t_{x}$) is the number of $I_{1}$ ($\tilde{I}_{1}$) singularities in the
deformation of the singularity): and the Picard number becomes

$\rho(X)=\sum(d_{x}-t_{x})-6=14-\sum t_{x}.$

Here we used the constraint 15. The maximal value for Picard number is 14 when
all the local data $t_{x}$ is zero!

So finally, the rank of MW group is given by following formula:

$\boxed{rank(MW(X))=12-\sum t_{x}-\sum(n_{t}-1)}.$ (17)

Here the sum is over all the singular fibers. The Mordell-Weil group forms a
lattice by introducing a pairing shioda1992mordell . The maximal Mordell-Weil
lattice for genus two fibration is given in figure. 12, which is a 12
dimensional lattice, see khac2002mordell . This lattice is called $D_{12}^{+}$
in conway2013sphere  and is the unique unimodular lattice in dimension 12.

$3$ Figure 12: The maximal Mordell-Weil lattice for a genus two fibered
rational surface. There is one basis section with self-intersection number
$-3$, and all the other basis sections have self-intersection number $-2$.

It was argued in Caorsi:2018ahl  that the rank of the Mordell-Weil lattice is
the same as that of the flavor symmetry of rank one theory. In rank two case,
the above identification no longer holds. However, there is a more general
formula relating the rank of Mordell-Weil group and the number of bulk
singularities $\Gamma$:

$\Gamma=(MW(X)+4)+D-4=MW(X)+D.$ (18)

Here $D$ is the space-time dimension the UV theory lives. The difference comes
from the fact there is one (two) extra KK charges for 5d (6d) theories.

Now assuming the local data of the singular fiber at $\infty$ is
$d_{\infty},\delta_{\infty},n_{\infty}$, we have following two equations
(using 15, 17 and 18):

$\displaystyle d_{\infty}+\Gamma+\sum_{F/\infty}t_{x}=20,$ $\displaystyle
rank(MW)+D=[12-\sum_{F/\infty}t_{x}-t_{\infty}-(n_{\infty}-1)]+D=\Gamma.$

Here $\Gamma$ is the total number of singular fibers at the bulk. Solving
above equations, one find:

$\boxed{d_{\infty}-n_{\infty}=7+t_{\infty}-D}.$ (19)

This equation would tell us which kind of singular fiber can be put at
infinity for a theory with given space-time dimension $D$.

Remark 1: We’d like to clarify an important subtly here. The rank of flavor
symmetry is the same as the rank of Mordell-Weil lattice for rank one theory
Caorsi:2018ahl ; this is no longer true for rank two theories. For example,
consider 4d theory defined by quiver $2-SU(2)-SU(2)-2$, its Coulomb branch
geometry is given by the configuration $(I_{0-0-0}^{*},I_{1}^{10})$; the rank
of MW lattice of above configuration is 6, but the rank of physical flavor
symmetry is only $5$. So the real meaning of 18 is that the $MW(X)+D=\Gamma$,
with $\Gamma$ the number of $I_{1}$ and $\tilde{I}_{1}$ singularities in the
bulk.

Remark 2: For a large class of 4d $\mathcal{N}=2$ theories defined by
hypersurface singularity, the number of $I_{1}$ singularities in the generic
deformation is given by the formula $\mu=2r+f$. This is not true in general,
as also illustrated by the theory $2-SU(2)-SU(2)-2$, here $2r+f=9$, but there
are 10 $I_{1}$ singularities in the bulk (the Coulomb branch geometry is
$(I_{0-0-0}^{*},I_{1}^{10})$).

###  3.1 4d theories

Let’s now consider Coulomb branch configurations for 4d theories. The
constraint on the topological data of the fiber at $\infty$ is (see formula.
19):

$\boxed{d_{\infty}-n_{\infty}=3+t_{\infty}}$ (20)

One can search from tables of section. 2.5 to find candidate singular fiber at
$\infty$ for a 4d theory.

####  3.1.1 Isotrivial family and SCFTs

Let’s first consider the so-called iso-trivial family over $\mathcal{P}^{1}$,
which has only two singular fibers. Those families are already classified in
gong2016families , see table. 11. Interestingly, all the singular fibers in
table. 11 satisfies the condition 20 and so they would give the 4d
$\mathcal{N}=2$ SCFT.

The configuration in table. 11 should give the SW geometry for a rank two 4d
SCFTs with only $t$ deformation (without the $u$ deformation and mass
deformation). Here one use $t,u$ to denote the expectation values for two
Coulomb branch operators (the scaling dimension satisfies $[v]\geq[u]$)
101010See Argyres:2022lah  for the curves with two Coulomb branch deformations
(one can easily check that if one set the $u$ parameters of their curves to be
zero, one can get the same curve as listed in table. 11. Notice that all the
singular fibers are of the elliptic type [1], and the dual graphs for them are
listed in figure. 13.

No | $F_{0},F_{\infty}$ | Families
---|---|---
$1$ | $(I_{0-0-0}^{*},I_{0-0-0}^{*})$ | $y^{2}=t(x^{5}+ax^{4}+bx^{3}+cx^{2}+x)$
$2$ | $II,II$ | $y^{2}=t^{6}+dx^{4}t+ex^{2}t^{2}+t^{3}$
$3$ | $III,III$ | $y^{2}=t^{6}+dx^{3}t+t^{2}$
$2$ | $IV,IV$ | $y^{2}=(t^{6}+dx^{3}t+t^{2})t$
$2$ | $V,V^{*}$ | $y^{2}=t^{6}+t$
$2$ | $VI,VI$ | $y^{2}=t^{5}+dtx^{3}+t^{2}x$
$2$ | $VII,VII^{*}$ | $y^{2}=x^{5}+xt$
$2$ | $VIII-1,VIII-4$ | $y^{2}=x^{5}+t$
$2$ | $VIII-2,VIII-3$ | $y^{2}=x^{5}+t^{3}$
$2$ | $IX-1,IX-4$ | $y^{2}=t^{5}+t^{2}$
$2$ | $IX-2,IX-3$ | $y^{2}=t(t+x^{5})$
Table 11: The isotrivial families of genus two fibration with two singular
fibers.
$2B$$8$$5$$4$$7$$6$$5$$4$$3$$2$$1$$1$$(8,6)$$5$$4$$4$$3$$2$$1$$3$$2$$1$$2B$$1$$(5,4)$$(5,3)$$5$$4$$3$$B$$3$$2$$1$$3$$B$$(4,2)$$2$$4$$6$$8$$5$$10$$7$$4$$D$$(10,4)$$4$$3$$2$$2$$1$$D$$2$$(3,2)$$3$$2$$2$$1$$1$$B$$B$$(5/2,3/2)$$2$$2$$(5/3,4/3)$$D$$B$$B$$(6/4,5/4)$$B$$B$$triple$$(6/5,8/5)$$(10/7,8/7)$$C$$cusp$$cusp$$VIII-2$$VII^{*}$$IX-3$$IX-4$$VI$$III$$IX-1$$IX-2$$V$$VII$$VIII-1$
Figure 13: The dual graph for the singular fibers relevant for 4d rank two
SCFTs admitting an isotrivial family.

Scaling dimension from the curve: The eigenvalues of monodromy group for the
singular fibers can be used to constrain the possible scaling dimension, see
table. 4. We also use the relation between the dual graph and the 3d mirror to
get the precise scaling dimension in section. 2.5. On the other hand, one can
find the scaling dimension of coordinate $t$ by using following formula
satisfied by SW differential:

$\partial_{t}\lambda\subset\Omega^{1,0}.$

Here $\lambda$ is the SW differential, and $\Omega^{1,0}$ is the space of
holomorphic differential for the genus two curve. $\Omega$ is generated by
${dx\over y},{xdx\over y}$ for genus two hypelliptic curve defined by the
equation $y^{2}=f(x)$. Now $\lambda$ has scaling dimension one, so the
following equations for scaling dimensions hold (using above equation, so
$[\partial_{t}\lambda]=[\omega_{i}]$ (with $\omega$ one of the basis of the
holomorphic differential):

$1-[t]=[x]-[y],~{}~{}~{}or~{}~{}~{}1-t=2[x]-[y].$

On the other hand, the SW curve is weighted homogenous, and from which one can
get two more equations relating the scaling dimensions of $t,x,y$. Using above
two three equations, one can find out the scaling dimension of $t$. The result
is consistent with the data listed in section. 2.5.

Example: Let’s take the configuration $(V,V^{*})$, and the curve takes the
form (the singular fiber of $V$ is put at $t=0$):

$y^{2}=x^{6}+t.$

There are following equations for the scaling dimensions:

$1-[t]=[x]-[y],~{}~{}2[y]=6[x],~{}~{}2[y]=[t],$

and the solution is $[x]=\frac{1}{4}~{},[y]=\frac{3}{4},~{}[t]=\frac{3}{2}$.
Here one need to assume $\partial_{t}\Omega\propto{dx\over y}$.

Let’s now instead put singular fiber $V^{*}$ at $t=0$, and the curve changes
as $y^{2}=x^{6}+t^{5}$ (since $y$ ($x$) is a section of $O(-3)$ ($O(-1)$)
bundle on $\mathbb{P}^{1}$), and the equations for the scaling dimension
change as

$1-[t]=2[x]-[y],~{}~{}2[y]=6[x],~{}~{}2[y]=5[t]$

Notice that here we have to use $\partial_{t}\lambda\subset{xdx\over y}$ to
get a consistent solution. The solution for above equation is
$[x]=5,[y]=15,[t]=6$.

Singular configuration for generic deformation: Using the result for the
deformation of singularity, one can easily write down the generic deformation
patter for the SCFT. See table. 12 for details. We get the result by keeping
the singular fiber at infinity intact, but split the singular fiber at $t=0$
into $I_{1}$ and $\tilde{I}_{1}$ fibers, whose number is given by the local
invariant $d_{x},\delta_{x}$. We notice that several of them do not correspond
to known theories.

Singular configuration | Scaling dimension | Flavor
---|---|---
$(II,I_{1}^{10})$ | $(2,2)$ | $SU(2)^{5}$
$(III,I_{1}^{10})$ | $(3,2)$ | $U(6)$
$(V^{*},I_{1}^{5})$ | $(\frac{6}{4},\frac{5}{4})$ | $U(1)$
$(VI,I_{1}^{10})$ | $(4,2)$ | $SO(12)$
$(VII^{*},I_{1}^{5})$ | $(\frac{8}{5},\frac{6}{5})$ | $SU(2)$
$(VII,I_{1}^{15})$ | $(8,6)$ | $SO(20)$
$(VIII-4,I_{1}^{4})$ | $(\frac{10}{7},\frac{8}{7})$ | $\emptyset$
$(VIII-3,I_{1}^{12})$ | $(10,4)$ | $E_{8}$
$(IX-4,I_{1}^{8})$ | $(\frac{5}{2},\frac{3}{2})$ | $SU(5)$
$(IX-3,I_{1}^{6})$ | $(\frac{5}{3},\frac{4}{3})$ | $SU(2)\times U(1)$
$(IX-2,I_{1}^{14})$ | $(5,4)$ | $SU(10)$
$(IX-1,I_{1}^{12})$ | $(5,3)$ | $SO(14)\times U(1)$

Singular configuration | Scaling dimension | Theory
---|---|---
$(IV,I_{1}^{8}\tilde{I}_{1})$ | $(6,2)$ |  $G_{2}-4(\textbf{7})$?
$(V,I_{1}^{15})$ | $(6,5)$ | $?$
$(VIII-1,I_{1}^{16})$ | $(10,8)$ | $?$
$(VIII-2,I_{1}^{6}\tilde{I}_{1})$ | $(\frac{10}{3},\frac{4}{3})$ | $?$

Table 12: The generic deformations for 4d rank two SCFTs which admit an
isotrivial family. The status for the entries in the second table is not
clear.

3d Mirror: If the 4d theory is put on a circle, then one can get a 3d
$\mathcal{N}=4$ SCFT in the IR. 3d $\mathcal{N}=4$ theory has interesting
mirror symmetry. In our case, the 3d mirror theory could be read from the dual
graph of the singular fiber $X_{0}$, see figure. 14. The rule is the
following: Let’s take $X_{0}=\sum n_{i}C_{i}$, the 3d mirror is found as
follows:

  1. 1.

If $X_{0}$ is of $D$ or $(B,B)$ type, then the mirror quiver is formed as
follows: there is a $U(n_{i})$ quiver node, and the number of bi-fundamental
hypers between them is $C_{i}\cdot C_{j}$.

  2. 2.

If $X_{0}$ is $2B$ type, one need to first modify the dual graph as follows:
if there are two $-2$ curve intersecting with the $2B$ curve, and one need to
remove one of them. The new dual graph is denoted as $X_{0}^{{}^{\prime}}$,
and one can find a quiver by the same rule as above.

One can check that the mirror quiver is the same as found in the literature,
see xie:2022lowrank .

2B54321432111443321252$(5,4)$$(5/2,3/2)$$D$22$B$$B$$(3/2,5/4)$12121$triple$11
Figure 14: 3d mirror for 4d SCFT defined using the genus two singular fiber.
The 3d mirror is read from the dual graph of the singular fiber.

####  3.1.2 Other rank two SCFTs

There are other rank two SCFTs which does not admit an isotrivial family. The
reason is: if only Coulomb branch operator $v$ with maximal scaling dimension
is turned on, the genus two curve would be degenerated into a singular genus
one curve along the $v$ direction. To use the genus two fibration studied in
this paper, one have to turn on other deformations (such as the other Coulomb
branch operator $u$ or the mass deformation), so that a genus two fibration
over $\mathbb{P}^{1}$ can be formed, see figure. 15. Now the singular fiber at
$\infty$ will still reflect the UV properties, but one can not find a genus
two fibration with just two singular fibers. One might still define these
SCFTs by specifying the singular fiber at $\infty$ and the singular fibers for
generic deformations, see table. 12. The dual graph for the singularity at
infinity is shown in figure. 16, from which one can also find the 3d mirror.
Those 3d mirrors match the known results.

Example: Consider configuration $(2II-0,I_{1}^{13})$, which should give rank
two $E_{8}$ theories. Notice that $2II-0$ fiber has $t_{\infty}=1$, and so
(see formula. 17): $rank(MW)=11-n_{t}(2II-0)+1=9$, which is the same as the
rank of flavor symmetry.

$v$$u$$f(u,v)=0$ Figure 15: The two dimensional Coulomb branch for a rank two
theory. On the $u=0$ plane ($v$ flane), the genus two fibration is degenerated
into a (singular) genus one fibration. To get a genus two fibration, one need
to find a slice defined by an equation $f(u,v)=0$.

Singular configuration | Scaling dimension | Flavor
---|---|---
$(2I_{0}^{*}-0,I_{1}^{9})$ | $(4,2)$ | $SO(8)\times SU(2)$
$(2II-0,I_{1}^{13})$ | $(12,6)$ | $E_{8}\times SU(2)$
$(2II^{*}-0,I_{1}^{5})$ | $(\frac{12}{5},\frac{6}{5})$ | $SU(2)$
$(2IV-0,I_{1}^{11})$ | $(6,3)$ | $E_{6}\times SU(2)$
$(2IV^{*}-0,I_{1}^{7})$ | $(3,\frac{3}{2})$ | $SU(3)\times SU(2)$
$(2III-0,I_{1}^{12})$ | $(8,4)$ | $E_{7}\times SU(2)$
$(2III^{*}-0,I_{1}^{6})$ | $(\frac{8}{3},\frac{4}{3})$ | $SU(2)\times SU(2)$

Singular configuration | Scaling dimension | Flavor
---|---|---
$(II-II^{*}_{0},I_{1}^{13})$ | $(6,4)$ | $SO(16)\times SU(2)$
$(III-II_{0}^{*},I_{1}^{12})$ | $(4,3)$ | $SU(8)\times SU(2)$

Table 13: The generic deformations for 4d rank two SCFTs which do not have an
isotrivial limit. The entries in first table are rank two version of rank one
$E_{n}$ and $H_{n}$ type theories.
$2B$$4$$4$$2$$4$$6$$2$$(6,3)$$6$$4$$4$$4$$8$$2$$2B$$(8,4)$$(12,6)$$6$$12$$8$$2B$$8$$4$$10$$6$$(3,3/2)$$2$$2B$$4$$(4,2)$$(8/3,4/3)$$(12/5,6/5)$$2$$2$$6$$4$$2B$$2$$2A$$2$$2B$$2$$2$$(6,4)$$5$$2$$3$$2B$$4$$3$$6$$4$$1$$(4,3)$$1$$2$$3$$2B$$4$$3$$2$$1$$(2,2)$$2B$
Figure 16: The dual graph for singularities relevant for 4d $\mathcal{N}=2$
SCFT. Notice that for the Coulomb branch configuration $(F,I_{1}^{b})$, the
property of the UV theory should be derived from the dual fiber $F^{*}$. The
dual pairs in table.13 are
$(2II-0,2II^{*}-0),(2III-0,2III^{*}-0),(2IV-0,2IV^{*}-0)$. $2I_{0}^{*}-0$ is
self-dual.

####  3.1.3 Asymptotic free theories

The rank two gauge groups are $A_{2},B_{2},C_{2},G_{2},A_{1}\times A_{1}$. One
can couple the gauge group with matter fields in various kinds of
representations, so that the theory is asymptotic free. The typical example is
$SU(3)$ gauge group coupled with $N_{f}\leq 6$ fundamental hypermultiplets. In
our framework, the SW solutions are represented by a configuration of genus
two singular fibers on $\mathbb{P}_{1}$: Given the classification of singular
fiber in last section, it is easy to find the SW geometry for various
asymptotic free theories. See table .14.

Another possibility is to consider $SU(2)$ gauge group coupled with rank one
SCFT and various free matter. The possibilities are: a)
$n-SU(2)-H_{1},~{}n=0,1,2$, b) $n-SU(2)-H_{2},n=0,1,2$. The SW geometries for
these theories are also listed in table. 14.

Configuration | Theory
---|---
$(III_{n},I_{1}^{10-n}),~{}~{}n=0,1\ldots,6$ | $SU(3)-(6-n)$
$(\tilde{III}_{n},I_{1}^{10-n}),~{}~{}n=0,1\ldots,6$ | $Sp(4)-(6-n)$
$(2I_{n}^{*}-0,I_{1}^{9-n}),~{}~{}n=0,1,\ldots,4$ | $V-Sp(4)-(4-n)$
$(III_{n}^{*},I_{1}^{8-n}\tilde{I}_{1}),~{}~{}n=0,1\ldots,4$ | $G_{2}-(4-n)(\textbf{7})$
$(I^{*}_{n-p-0},I_{1}^{10-n-p}),~{}~{}n=0,1,2,~{}~{}p=0,1,2$ | $(2-n)-SU(2)-SU(2)-(2-p)$
$(II^{*}-II_{n}^{*},I_{1}^{6-n})$ | $(2-n)-SU(2)-H_{1},~{}n=0,1,2$
$(III^{*}-II_{n}^{*},I_{1}^{7-n})$ | $(2-n)-SU(2)-H_{2},~{}n=0,1,2$
Table 14: The singular configuration for 4d asymptotic free theories.

Example: Let’s consider $SU(3)$ gauge theory coupled with $6-n$
hypermultiplets in fundamental representation. The fiber at $\infty$ is
$III_{n}$, and the bulk has $(10-n)$ $I_{1-0-0}$ (which will be denoted as
$I_{1}$ singularity). This would give the SW geometry for $SU(3)$ with $6-n$
fundamental flavors.

####  3.1.4 Other possibilities

There are other singular fibers satisfying the condition 20. Potentially, they
would define 4d $\mathcal{N}=2$ UV complete theories. The basic data for these
singular fibers are listed in table. 15, here one assume that all the
eigenvalues are different from one. Whether those theories give rise to SCFT
or just asymptotical free theories would be discussed in Xie:ranktwob .

Type | Monodromy | $d_{x}$ | $\delta_{x}$ | $l$ | Eigenvalue |  $n_{t}$ (components)
---|---|---|---|---|---|---
$I^{*}_{0}-I_{0}^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $12$ | $12$ | -1 | $(-1,-1,-1,-1)$ | 9
$I_{0}^{*}-II-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&-1&0&0\\\ -1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $8$ | $8$ | $-1$ | $(-1,-1,\exp(2\pi i/6),\exp(10\pi/6))$ | 5
$I_{0}^{*}-II^{*}-\alpha$ |  | $14$ | $14$ | $-1$ | $(-1,-1,\exp(2\pi i/6),\exp(10\pi/6))$ | 11
$I_{0}^{*}-IV-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&-1&0&0\\\ -1&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ | $10$ | $10$ | $-1$ | $(-1,-1,\exp(2\pi i/3),\exp(4\pi/3))$ | 7
$I_{0}^{*}-IV^{*}-\alpha$ |  | $12$ | $12$ | $-1$ | $(-1,-1,\exp(2\pi i/3),\exp(4\pi/3))$ | 9
$I_{0}^{*}-III-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&-1&0&0\\\ -1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $9$ | $9$ | $-1$ | $(-1,-1,\exp(2\pi i/4),\exp(6\pi/4))$ | 6
$I_{0}^{*}-III^{*}-0$ | $\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\end{array}\right)$ | $15$ | $15$ | $-1$ | $(-1,-1,\exp(2\pi i/4),\exp(6\pi/4))$ | 12
$I_{0}^{*}-III^{*}-\alpha$ |  | $13$ | $13$ | $-1$ | $(-1,-1,\exp(2\pi i/4),\exp(6\pi/4))$ | 10
$II-IV-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&1\\\ -1&0&-1&0\\\ 0&-1&0&0\end{array}\right)$ | $6$ | $6$ | $-1$ | $(\exp(2\pi i/6),\exp(10\pi/6),\exp(2\pi i/3),\exp(4\pi/3))$ | 3
$II^{*}-IV-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&1&0&0\\\ -1&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $14$ | $14$ | $-1$ | $(\exp(2\pi i/6),\exp(10\pi/6),\exp(2\pi i/3),\exp(4\pi/3))$ | 11
$II^{*}-IV^{*}-\alpha$ |  | $16$ | $16$ | $-1$ | $(\exp(2\pi i/6),\exp(10\pi/6),\exp(2\pi i/3),\exp(4\pi/3))$ | 13
$IV^{*}-IV^{*}-\alpha$ |  | $14$ | $14$ | $-1$ | $(\exp(2\pi i/6),\exp(10\pi/6),\exp(2\pi i/3),\exp(4\pi/3))$ | 11
$II-III-0$ | $\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&0&0&-1\\\ -1&0&0&0\\\ 0&1&0&0\end{array}\right)$ | $5$ | $5$ | $-1$ | $(\exp(2\pi i/6),\exp(10\pi/6),\exp(2\pi i/4),\exp(6\pi/4))$ | 2
$II^{*}-III-\alpha$ |  | $11$ | $11$ | $-1$ | $(\exp(2\pi i/6),\exp(10\pi/6),\exp(2\pi i/4),\exp(6\pi/4))$ | 8
$IV-III-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\ -1&0&-1&0\\\ 0&-1&0&0\end{array}\right)$ | $7$ | $7$ | $-1$ | $(\exp(2\pi i/6),\exp(10\pi/6),\exp(2\pi i/3),\exp(4\pi/3))$ | 4
$IV-III^{*}-\alpha$ |  | $11$ | $11$ | $0$ | $(\exp(2\pi i/4),\exp(6\pi/4),\exp(2\pi i/3),\exp(4\pi/3))$ | 8
$IV^{*}-III^{*}-\alpha$ |  | $15$ | $15$ | $-1$ | $(\exp(2\pi i/4),\exp(6\pi/4),\exp(2\pi i/3),\exp(4\pi/3))$ | 12
$III-III-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\ 0&-1&0&0\end{array}\right)$ | $6$ | $6$ | $-1$ | $(\exp(2\pi i/4),\exp(6\pi/4),\exp(2\pi i/4),\exp(6\pi/4))$ | 3
$III^{*}-III^{*}-\alpha$ |  | $16$ | $16$ | $-1$ | $(\exp(2\pi i/4),\exp(6\pi/4),\exp(2\pi i/4),\exp(6\pi/4))$ | 13

Table 15: Singular fibers which may give rise to UV complete 4d
$\mathcal{N}=2$ theories.

###  3.2 5d KK theories

Let’s now discuss the singular fiber configuration for 5d KK theories. The
constraint on the topological data of the fiber at $\infty$ is

$\boxed{d_{\infty}-n_{\infty}=2+t_{\infty}}.$ (21)

One can search from tables listed in section. 2.5 for the singular fiber
satisfying above condition, and there are many candidates. Of course, one
would like to find more constraints, i.e. the eigenvalues for the monodromy
group at $\infty$ of the SW curve of 5d theory. In the following, we will use
the known results of 5d rank two theory to get some insights.

In fact, one can engineer 5d $\mathcal{N}=1$ theory by putting M theory on a
3d canonical singularity Xie:2017pfl . A particular interesting class is the
toric singularity, and the rank two cases are classified in Xie:2017pfl . The
SW geometry for those 5d KK theories are easy to find Aharony:1997bh : they
are just given by the monomials associated with the toric diagram. The
eigenvalues at infinity can be computed by the combinatorial formula found in
libgober1995zeta .

Let’s give a review for how to compute the eigenvalues at infinity. A toric
diagram is given as a convex polygon $P$, see an example in figure. 17. The SW
geometry is given as

$f=\sum_{(a,b)\in\Gamma}\lambda_{a,b}x^{a}y^{b}.$

Here $\Gamma$ contains lattice points inside or at the boundary of the convex
polygon $P$, $x$ and $y$ are $\mathbb{C}^{*}$ variable. We further assume that
one of the interior point of $P$ is at the origin. So there is a one parameter
family of maps $f:\mathbb{C}^{*}\to C$, and the generic fiber
$X_{t}=f^{-1}(t)$ is a smooth curve. When $t\to\infty$, there is monodromy
group $M$ action acting on the homology group $H_{i}(X_{t})$. One define a
Zeta function by using the monodromy group action:

$\zeta(s)=\prod det((I-sM)|H_{i}(X_{t}))^{(-1)^{i}}.$

In our particular case, only $H_{0}$ and $H_{1}$ are non-vanishing, and so the
$\zeta$ function takes the form

$\zeta(s)={(1-s)\over(1-s\lambda_{1})\ldots(1-s\lambda_{n})}.$ (22)

Here $n$ is the dimension of $H_{1}$, and $\lambda_{1},\ldots,\lambda_{n}$ are
the eigenvalues of the monodromy group acting on $H_{1}$. On the other hand,
$\zeta(s)$ has the following combinatorial description

$\zeta(s)=\prod(1-s^{m_{\sigma}})^{-Vol(\sigma)}.$

Here the product is over the one dimensional boundary $\sigma$ of the convex
polygon, and $\sigma$ has the equation $\sum
a_{i}^{\sigma}x_{i}=m_{\sigma},m_{\sigma}>0$. $Vol(\sigma)$ is equal to
interior lattice points of $\sigma$ plus one.

Example: Let’s look at the convex polygon in figure. 17. The SW geometry is
given by:

$f=y^{2}+x+{1\over xy}+y,$

and there are three one dimensional boundaries with equations:
$y+2x=2,~{}~{}y-3x=2,~{}~{}x-2y=1$, and there is no interior lattice points
for those boundaries. So we have

$\zeta(s)={1\over(1-s^{2})(1-s^{2})(1-s)}={1\over(1-s)(1-s)(1-s)(1+s)(1+s)}$

Comparing with formula $\zeta(s)$ in 22, one can see that the eigenvalue for
$M$ on $H_{1}$ are $(1,1,1,1,-1,-1)$. The weight one part of $H_{1}$ is just
four dimensional, and the eigenvalues of the monodromy acting on weight one
part is $(1,1,-1,-1)$.

One quick comment is that there are always two eigenvalues $1$ if one put an
interior point of the convex polygon at the origin, and the other pair of
eigenvalues could be $(-1,-1)$ for most cases. However, it is also possible
that the other two eigenvalues are $(\exp(i2\pi/3),\exp(4\pi/3))$.

$y$$x$ Figure 17: Left:The toric diagram for a 3d canonical singularity. M
theory on it gives rise to a 5d $\mathcal{N}=1$ SCFT without any flavor
symmetry; Right: The Coulomb branch geometry for the 5d KK theory defined
using the toric diagram on the left. There are a total of five $I_{1}$
singularities at the bulk.

All possible singular fibers satisfying equation 21 are listed in table. 16,
and the eigenvalues are $(1,1,-1,-1)$ or $(1,1,\exp(i2\pi/3),\exp(4\pi/3))$.

Type | Monodromy | $d_{x}$ | $\delta_{x}$ | $l$ | Eigenvalue |  $n_{t}$ (components) | Gauge algebra
---|---|---|---|---|---|---|---
$IV-I_{n}-0$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&n\\\ -1&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $n+4$ | $n+4$ | $-1$ | $(1,1,\exp(2i\pi/3),\exp(4i\pi/3))$ | $n+2$ | $A_{2}\oplus A_{n-1}$
$IV-II_{n}$ | $\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&-1&n+1\\\ -1&0&-1&1\\\ 0&0&0&1\end{array}\right)$ | $n+4$ | $n+4$ | $-1$ | $(1,1,\exp(2i\pi/3),\exp(4i\pi/3))$ | $n+2$ | $A_{1}\oplus A_{n-1}$
$IV^{*}-II_{0}$ |  | $7$ | $6$ | $-1$ | $(1,1,\exp(2i\pi/3),\exp(4i\pi/3))$ | $4$ | $D_{5}$
$IV^{*}-II_{n}$ | $\left(\begin{array}[]{cccc}-1&0&-1&-1\\\ -1&1&0&n\\\ 1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $n+7$ | $n+7$ | $-1$ | $(1,1,\exp(2i\pi/3),\exp(4i\pi/3))$ | $n+5$ | $D_{5}\oplus A_{n-2}$
$I_{n}-I_{p}^{*}-0$ | $\left(\begin{array}[]{cccc}-1&0&-p&0\\\ 0&1&0&n\\\ 0&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $n+p+6$ | $n+p+6$ | $-1$ | $(1,1,-1,-1)$ | $n+p+4$ | $A_{n-1}\oplus D_{p+4}$
$II_{n-p}~{}(n>0,~{}p\geq 0)$ | $\left(\begin{array}[]{cccc}-1&0&-p&-1\\\ 0&1&1&n\\\ 0&0&-1&0\\\ 0&0&0&1\end{array}\right)$ | $n+p+5$ | $n+p+5$ | $-1$ | $(1,1,-1,-1)$ | $n+p+3$ | $A_{n-2}\oplus D_{p+3}$
$\tilde{II}_{n-p},~{}(n>0,p\geq 0)$ | $\left(\begin{array}[]{cccc}-1&0&-p&0\\\ 1&1&p&n\\\ 0&0&-1&1\\\ 0&0&0&1\end{array}\right)$ | $n+p+5$ | $n+p+5$ | $-1$ | $(1,1,-1,-1)$ | $n+p+3$ | $A_{n-2}\oplus A_{p+3}$
$II_{n-p}^{*},~{}~{}p>0$ | $\left(\begin{array}[]{cccc}1&0&-p&0\\\ -1&-1&p&-n\\\ 0&0&1&-1\\\ 0&0&0&-1\end{array}\right)$ | $n+p+5$ | $n+p+4$ | $-1$ | $(1,1,-1,-1)$ | $n+p+2$ | $A_{p-1}\oplus D_{n+2}$

Table 16: List of singular fibers which would give the Coulomb branch geometry
for the 5d KK theories.

Once the singular fiber at $\infty$ is given, it is possible to construct some
candidate Coulomb branch geometry of 5d KK theories (see also Martone:2021drm
for related studies.), the results are listed in table. 17. One interesting
observation is that there are many possibilities for a given rank of flavor
symmetry, one can compare our findings with the geometric classification in
Xie:2017pfl ; Jefferson:2018irk ; Hayashi:2018lyv ; Bhardwaj:2018yhy ;
Apruzzi:2019opn ; Apruzzi:2019enx ; Bhardwaj:2019jtr ; Saxena:2020ltf , and it
seems that our theory can cover all of them (comparing our table with that in
Jefferson:2018irk ; Apruzzi:2019opn ). Moreover, it suggests that there are
more. A detailed studies of those theories will be left in Xie:ranktwob .

Configuration | Flavor
---|---
$(IV-II_{0},I_{1}^{16})$ | $SO(20)$
$(II_{0-1}^{*},I_{1}^{14})$ | $E_{8}\times SU(2)$
$(II_{n-p},I_{1}^{15-(n+p)}),~{}~{}1\leq(n+p)\leq 10$ | $rank=10-(n+p)$
$(\tilde{II}_{n-p},I_{1}^{15-(n+p)}),~{}~{}1\leq(n+p)\leq 10$ | $rank=10-(n+p)$
$(II_{n-p}^{*},I_{1}^{15-(n+p)}),~{}~{}1\leq(n+p)\leq 10$ | $rank=10-(n+p)$
Table 17: Possible singular configurations for 5d $\mathcal{N}=1$ KK theories.

###  3.3 6d KK theories

Let’s now give the singular fiber configuration for 6d KK theories. The
constraint on the topological data of the fiber at $\infty$ is

$\boxed{d_{\infty}-n_{\infty}=1+t_{\infty}}$ (23)

The list of singularities satisfying the above equation is given in table. 18.
Here we further assume that at least two eigenvalues of the monodromy group is
identity. Some global Coulomb branch geometries are listed in table. 19.

Type | Monodromy | $d_{x}$ | $\delta_{x}$ | $l$ | Eigenvalue |  $n_{t}$ (components) | Gauge algebra
---|---|---|---|---|---|---|---
$I_{0}-I_{n}^{*}-0$ | $\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&-n\\\ 0&0&1&0\\\ 0&0&0&-1\end{array}\right)$ | $n+6$ | $n+6$ | $-1$ | $(1,1,-1,-1)$ | $n+5$ | $D_{n+4}$
$II_{n-0}^{*}$ | $\left(\begin{array}[]{cccc}1&0&0&0\\\ 1&-1&0&-n\\\ 0&0&1&1\\\ 0&0&0&-1\end{array}\right)$ | $n+5$ | $n+4$ | $-1$ | $(1,1,-1,-1)$ | $n+3$ | $D_{n+2}$
$IV^{*}-I_{n}-0$ | $\left(\begin{array}[]{cccc}-1&0&-1&0\\\ 0&1&0&n\\\ 1&0&0&0\\\ 0&0&0&1\end{array}\right)$ | $n+8$ | $n+8$ | $-1$ | $(1,1,\exp(2\pi/3),\exp(4\pi/3))$ | $n+7$ | $E_{6}\oplus A_{n-1}$
$I_{n-p-q},~{}n>0,p>0$ | $\left(\begin{array}[]{cccc}1&0&p+q&-q\\\ 0&1&-q&n+q\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right)$ | $n+p+q$ | $n+p+q$ | $-1$ | $(1,1,1,1)$ | $n+p+q-1$ | $A_{n-1}\oplus A_{p-1}\oplus A_{q-1}$

Table 18: List of singular fibers which would give the Coulomb branch geometry for the 6d KK theories. Configuration | Flavor
---|---
$(I_{1-1-1},I_{1}^{17})$ | $SO(20)$
$(II_{0-0}^{*},I_{1}^{15})$ | $E_{8}\times SU(2)$
$(I_{n-p-q},I_{1}^{20-n-p-q}),~{}n>0,p>0,q>0$ | Rank=$14-(n+p+q)$
$(II_{n-0}^{*},I_{1}^{15-n})$ | rank= $9-n$
Table 19: Possible singular configuration for 6d $(1,0)$ KK theories.

##  4 Conclusion

We have studied the local singularities for rank two $\mathcal{N}=2$ Coulomb
branch geometries, and the local invariants $d_{x},\delta_{x},n_{t}$, gauge
algebra are listed in table. 6,7,8,9, 10. The low energy theories for most of
them are also included there, which could be SCFT, IR free gauge theory, or
the sum of the above components, etc.

We then construct some potential global Coulomb branch geometries for 4d SCFTs
(table. 12 and 13) and asymptotical free theories (table. 14), 5d
$\mathcal{N}=1$ KK (table. 17) and 6d $(1,0)$ KK theories (table. 19). Here we
assume that the bulk singularities are just $I_{1}$ type (the low energy
theory is $U(1)$ coupled with one massless hypermultiplet plus another free
$U(1)$ vector-multiplet). The condition on the singular fiber at $\infty$ is
given for a UV theory in space time dimension $D\geq 4$. It is quite a
remarkable fact that our approach can essentially cover all the known results
in the literature.

The next question is the systematical study of global genus two fiberation,
and the details will appear in Xie:ranktwob , where one can construct other
type of undeformable singularities, such as $I_{n-0-0}$ singularities. To
study general undeformable singularities, one need to study the base change of
genus two fiberation, and the details will appear in Xie:ranktwoc .

It is possible to generalize the study of the local singularities to rank
$g\geq 3$ theory. Similarly, locally one has a one dimensional family of
abelian varieties whose complex structure is determined by $\tau_{ij}(v)$ (the
photon couplings). One might first study the case where there is an associate
genus $g$ fibration and so one should study a genus $g$ pencils. A complete
classification for genus three degeneration is given in
ashikaga2002classification , and it seems not difficult to get the low energy
theory by following the method proposed in this paper. A even small subset
would be the study of genus $g$ hypelliptic pencils, which would give some of
the well-known theories (such as $SU(N)$ gauge theory with $N_{f}$ fundamental
flavors). The details will appear elsewhere.

## Acklowledgement

DX would like to thank D.X Zhang for helpful discussions. DX is supported by
Yau mathematical science center at Tsinghua University.

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