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

# On rank two theories with eight supercharges part II: Lefschetz pencils

Dan Xie

###### Abstract

The global Seiberg-Witten (SW) geometries for rank two theories with eight
supercharges are studied. The theory is deformed generically so that there are
only simplest $I_{1}$ or $\tilde{I}_{1}$ singularities on the Coulomb branch,
which geometrically gives the so-called Lefschetz pencils. The local
singularity was shown to be determined by the conjugacy class of mapping class
group (MCG); The global study is then reduced to the questions about MCG: a)
Find the factorization of the MCG element of the singular fiber into positive
products of Dehn twists (which gives the $I_{1}$ singularity or
$\tilde{I}_{1}$ singularity); b) Find the factorization of identity element in
terms of Dehn twists. We solved above two MCG problems for most rank two
theories.The results are very helpful in determining IR physics for all vacua
of 4d SCFTs. Our approach is combinatorial and many aspects can be
straightforwardly generalized to the study of higher rank theory.

##  1 Introduction

This is the second one of a series of papers in trying to classify rank two
theories with eight supercharges. The basic idea is to classify consistent
Seiberg-Witten (SW) solution on the effective 4d Coulomb branch 111See results
in classifying 4d rank one SCFTs Argyres:2015ffa  and rank two SCFTs in
Argyres:2022lah ; Argyres:2022puv ; Argyres:2022fwy .. In previous paper
Xie:2022aad , the local of local singularities of SW solution are classified
and the corresponding IR physical theories are identified by using the
associated dual graph.

The purpose of this paper is to study the global SW geometry for rank two
theories, namely glue the local singularities consistently. There are a couple
of topological constraints: a) the product of local monodromy should be
identity; b) there are simple constraints on the sum of local invariants (see
discussion in Xie:2022aad ) by assuming the geometry of the total space of the
SW fiberation. However, the above constraints are not very strong and a
systematical study seems quite difficult.

It is certainly very helpful if one can write down the full SW families which
encode the prescribed local singularities (See Argyres:2022lah ), which we
will pursue elsewhere. We do not take that approach in this paper. Instead a
topological approach is taken so that the construction of global SW geometry
can be carried out in a combinatorial way. The following two topological facts
are important for us:

  1. 1.

We will use Mastumoto-Montesinos’s (MM) theory matsumoto2011pseudo  for local
singularity: the local degeneration is given by the conjugacy class of mapping
class group. So there is a mapping class group element around each
degeneration, which completely characterizes the local singularity.

  2. 2.

We consider the generic deformation of the theory so that only the simplest
possible singularity exists at the bulk: they are called $I_{1}$ or
$\tilde{I_{1}}$ singularities, see figure. 1 for the geometric illustration.
The physical interpretation for them is that there is an extra massless
particle associated with the vanishing cycle Seiberg:1994rs . One also need to
add a singularity at $\infty$ which is also assumed to be split into $I_{1}$
and $\tilde{I_{1}}$ singularities. Therefore, one has a genus two SW
fiberation with just $I_{1}$ or $\tilde{I}_{1}$ singularities, see figure. 2,
and such fiberation is called Lefschetz pencils matsumoto1996lefschetz .

Now let’s fixed a generic point at Coulomb branch, and one has an element in
mapping class group by following a path around each singularity, see figure.
2. The mapping class group element along $I_{1}$ or $\tilde{I}_{1}$
singularities is rather simple: it is given by the so-called Dehn twist along
the vanishing cycle farb2011primer .

$I_{1}$$\tilde{I}_{1}{}$ Figure 1: The simplest local degenerations. $I_{1}$
singularity: there is a non-separating vanishing cycle; $\tilde{I}_{1}$
singularity: there is a separating vanishing cycle: the Riemann surface is
split into two parts after the degeneration.

Once we have the topological picture of global SW geometry shown in figure. 2,
the classification is achieved by solving following problems in genus two
mapping class group $M_{2}$, which are generated by five Dehn twists, see
figure. 14:

  1. 1.

The singular fiber at $\infty$ determines the UV theory, and so there is an
associated mapping class group element. Since it is now split into $I_{1}$ and
$\tilde{I}_{1}$ singular fibers, which means that the corresponding mapping
class group is given by the product of positive Dehn twists. So the group
theory question is to find out the positive factorization of the mapping class
group element associated with the UV theory, and this has been solved for 4d
SCFTs, see table. [4,5,6,7]. The candidate factorization for 4d asymptotical
free theories, 5d and 6d KK theories are also found.

  2. 2.

For the global SW geometry, since the base space is now compact, the product
of Dehn twists should satisfy the topological condition

$\tau_{i_{1}}\tau_{i_{2}}\ldots\tau_{i_{s}}=1.$

This amounts to find the factorization of the identity element in terms of
Dehn twist. Another topological constraint is the assumption of the total
space being a rational surface Xie:2022aad , which put the constraint on the
number of $I_{1}$ and $\tilde{I}_{1}$ singularities 222The choice of
$(I_{1},\tilde{I}_{1})=(n,m)$ singularities are $n+2m=20$, and so the choices
are $(n,m)=(20,0),(18,1)$, $(16,2)$, etc. on the Coulomb branch. Such
factorization of identity element was found in this paper.

  3. 3.

The final step of finding global SW geometry is then to rearrange the
factorization of identity so that one can get a desirable UV configuration at
infinity (see step one), and this problem has been solved for most of 4d UV
complete theories in this paper, see table. 8 and 9.

The factorization of mapping class group element associated with 4d SCFT is
extremely useful physically: One can use the braid move and Hurwitz move to
get configuration involving more complicated singularities, which would then
determine all the IR configuration of the theory, so that one can solve this
theory completely.

This paper is organized as follows: section two revisited the local
singularities by using the classification of pseudo-periodic map; section
three discusses the factorization of mapping class group elements in terms of
Dehn twists, and the factorization for local singularity and global SW
geometry for 4d SCFTs are given; section four gives several representation of
mapping class group of genus two curve which would be useful for the further
study, such as the UV singular fiber of 5d and 6d KK theory; finally a
conclusion is given in section five.

$\infty$$\tau_{1}$$\tau_{2}$$\sigma$$\tau_{3}$$\tau_{4}$ Figure 2: Up: there
are only $I_{1}$ or $\tilde{I}_{1}$ singularities on the one dimensional slice
of Coulomb branch; Notice that we should not mix the singularities at $\infty$
and the bulk ones; Bottom: there is an associated mapping class group element
called Dehn twist associated with each $I_{1}$ or $\tilde{I}_{1}$ singularity.

##  2 Mapping class group and genus two degeneration

###  2.1 Genus two degeneration revisited

In the context of 4-dimensional $\mathcal{N}=2$ supersymmetric field theories,
the Coulomb branch is a critical part of the moduli space Seiberg:1994rs ;
Seiberg:1994aj . This branch is associated with the vacuum expectation values
(VEVs) of scalar fields known as Coulomb branch operators.

The Seiberg-Witten solution for Coulomb branch can be described as a bundle of
abelian varieties over the moduli space. In the majority of cases along the
Coulomb branch (they are called generic vacua), the low-energy dynamics of the
theory can be effectively modeled as a $U(1)^{r}$ abelian gauge theory, here
$r$ represents the rank of the theory, and it corresponds to the number of
massless photon fields in the theory. The complex structure of the abelian
variety is identified with coupling constants of photons.

In contrast to the generic vacua, at certain special points along the Coulomb
branch, the abelian variety undergoes a degeneration (i,e. it becomes
singular). This results in additional degrees of freedom in the low-energy
theory. Depending on the specifics of this degeneration, the low-energy
behavior can become much more intricate and may exhibit features like
interacting superconformal field theory (SCFT), infrared (IR) free abelian
theories, or non-abelian gauge theories. The exact nature of the low-energy
theory depends on the details of the degeneration of the abelian variety at
these special points.

When the family of abelian varieties associated with a Coulomb branch solution
can be described using the Jacobian of Riemann surfaces, it implies that for
each point on the Coulomb branch moduli space, one can associate a Riemann
surfaces. This correspondence simplifies the study of the Coulomb branch, as
it allows us to focus on the geometric properties of these curves, which is a
more well-developed area in algebraic geometry.

In the special scenario where the rank of the theory is two, it is possible to
represent all abelian varieties in terms of the Jacobian of genus two curves.
This reduction is particularly significant because it streamlines the
investigation of the Coulomb branch, effectively reducing it to the study of
the properties of genus two curves and the associated Jacobians. It’s useful
to note that every genus two curve is indeed hyperelliptic 333The equation for
a genus two hyperelliptic curve is $y^{2}=x^{5}+\ldots$ or
$y^{2}=x^{6}+\ldots$..

The first key step in analyzing rank two ($r=2$) Coulomb branch solutions
involves the classification of local degenerations of genus two curves. This
classification is fundamental for comprehending how the theory behaves at
specific points along the Coulomb branch. It’s noteworthy that this
classification has already been completed, with detailed information available
in namikawa1973complete . To accomplish this classification, they make use of
Hodge theory, a powerful mathematical framework within algebraic geometry.
Distinguishing the various degenerations in the rank two case requires
considering three essential components: Monodromy Group: This group captures
the transformations on homology groups that occur as one traverses loops
around singular points in the moduli space; Type of Modulus Point: Specifying
the type of modulus point is critical, as it signifies the nature of the
singularity where the curve degenerates; Additional Discrete Parameter $m$:
The inclusion of the discrete parameter $m$ serves to fine-tune the
characterization of the degenerations, providing further details that refine
the classification.

In the context of classifying degenerations of Riemann surfaces, the
topological approach put forward by Matsumoto-Montesinios (MM) is highly
valuable. Their theory, detailed in matsumoto2011pseudo , is particularly
useful for our purposes. The key insight in MM’s theory is that the conjugacy
class of the mapping class group action serves as a complete determinant of
the degeneration type, see figure. 3 for the description of mapping class
group. This approach, aside from being systematic, is also combinatorial in
nature, which greatly facilitates a comprehensive physical investigation
Xie:2022aad . We will apply MM’s theory specifically to the degeneration of
genus two curves, aiming to recover the classification results presented in
namikawa1973complete ; Xie:2022aad .

$re^{i\theta}$$r$$x$$h_{\theta}(x)$$h_{2\pi}(x)$ Figure 3: There is a
homeomorphism action $h$ associated with the loop around special vacua. The
action acts on any point on Riemann surface, and in particular it induces an
action on homology group.

In the study of Riemann surface degenerations, the conjugacy class of the
mapping class group has a distinctive character known as a pseudo-periodic map
of negative type matsumoto2011pseudo . This map, denoted as $f$, can be
classified using specific combinatorial data:

  1. 1.

Admissible System of Cut Curves ${\cal C}=\cup C_{i}$: The classification
begins with an admissible system of cut curves, denoted as ${\cal C}=\cup
C_{i}$. An admissible system is one where the irreducible component
$B=\Sigma_{g}/{\cal C}$ satisfies certain conditions. Each component $B_{i}$
should have a non-negative Euler number $\chi_{i}$, which is calculated as
$\chi_{i}=2-2g_{i}+n_{i}\geq 0$. Here, $n_{i}$ represents the number of
boundary curves for an irreducible component $C_{i}$ , and $g_{i}$ is the
genus of that component.

  2. 2.

Finite Group Action on Oriented Graph $G_{\cal C}$: The map $f$ induces a
finite group action on an oriented graph $G_{\cal C}$ defined in last step.

  3. 3.

Screw Numbers for Annuli $C_{i}$: For each annulus $C_{i}$ in the system, the
screw number is given. It’s worth noting that these screw numbers must be
negative in accordance with the classification.

  4. 4.

Periodic Map Action: The action of $f$ on each irreducible component of $B$ is
a periodic map. This periodicity is, in turn, determined by the valency data,
denoted as
$(n,g^{{}^{\prime}},{\sigma_{1}\over\lambda_{1}}+\frac{\sigma_{2}}{\lambda_{2}}+\ldots+\frac{\sigma_{s}}{\lambda_{s}})$.
Here: $n$ is the order of the map (i.e., $f^{n}=id$). $g^{{}^{\prime}}$
represents the genus of the base, defined by the covering map
$f:\Sigma\to\Sigma^{{}^{\prime}}$, $\sigma_{i},\lambda_{i}$ are integral
values that further specify the characteristics of the periodic map action.

In summary, the classification of pseudo-periodic maps of negative type for
Riemann surface degenerations relies on a systematic consideration of
admissible cut curves, group actions on oriented graphs, screw numbers for
annuli, and the valency data determining periodic map actions. This detailed
combinatorial data provides a comprehensive understanding of the degeneration
types. The first two step gives rise to a weighted graph: each node has label
representing genus and the number of internal cut curves; each edge represents
a separating cut curve and a multiplicity from the finite group action. The
third step gives an integer $K\geq-1$ along each weighted curve in weighted
graph. Finally, one has a periodic map for each component in the cut system.
See figure. 6 for an example.

The three set of data of genus two degeneration given in namikawa1973complete
are recovered from MM’s theory as follows: a) The monodromy group action is
induced from the mapping group action farb2011primer ; b) The modulus point is
given by the cut system (before the finite group action); c) The integral
value $m$ is given by the screw number and the periodic map data on the
boundaries of the annulus.

$C_{1}$$C_{2}$$1$$1$$r=1$$r=0$$f_{1}$$f_{2}$$S(C_{2})$$S(C_{1})$ Figure 4: Up:
An admissible cut system for a genus two Riemann surface, and there are two
irreducible components after the cutting; there is a screw number associated
with each cutting curve, and a periodic map on each irreducible component.
Bottom: A weighted graph for the above cut system: here one draw an edge for
every separating curve (which will cut the Riemann surface into two separate
components), and one write the number of non-separating cutting curves for a
vertex in the graph. $Elliptic\ 1$$2$$a)$$Elliptic\
2$$1$$a)$$1$$1$$b)$$l=2$$2$$Parabolic\ 3$$Parabolic\ 4$$Parabolic\
5$$1$$b)$$1$$r=1$$r=1$$2$$a)$$r=2$$2$$a)$$1$$b)$$1$$r=1$$r=1$$g=1$$c)$$l=2$$2$$r=1$$0$$a)$$0$$0$$b)$$l=2$$2$$0$$c)$$0$$0$$e)$$0$$f)$$0$$0$$d)$$l=2$$6$$2$$1$$3$$3$$3$$l=2$$2$$2$$2$$2$
Figure 5: All possible weighted graphs for genus two Riemann surface. The
number in the vertex represents the genus, and the number $r$ denotes the
number of internal cut.

Let’s now revisit the classification of genus two degeneration by using MM’s
theory. In fact, there are some missing items in namikawa1973complete , but
will be found here. There are a total of five classes, and the combinatorial
data is described as follows.

Remark: The weighted graph 4a) with the periodic data
[$(g=2,k=0,r=2):~{}ord(f)=2,~{}(C_{1},C_{2}),~{}~{}\frac{1}{2}+\frac{1}{2}+(1)+(1)$]
seems missing in namikawa1973complete .

Weighted graph: The weighted graph for genus two degeneration is described in
figure. 5, and we identify them with the names in namikawa1973complete ;
Xie:2023lko . They are found by first imposing the non-negative Euler number
constraint (This is just the constraint in Deligne-Mumford theory), and then
find the finite group actions.

Periodic Maps: The list of periodic maps for curves with genus one and two is
given in table. 1. Additionally, we’ll explore periodic maps on curves that
include boundaries. The fundamental component in this context is labeled as
$(g,k,r)$, where $k$ represents the count of boundary edges, and $r$ signifies
the number of internal cutting curves. We employ bold letters to indicate data
associated with the boundary curves, while data enclosed within brackets
pertains to the internal curves. See table. 2 for the full data.

Order | $Data$
---|---
$n=6$ | $(\frac{1}{6}+\frac{1}{3}+\frac{1}{2},\frac{5}{6}+\frac{2}{3}+\frac{1}{2})$
$n=4$ | $(\frac{1}{4}+\frac{1}{4}+\frac{1}{2},\frac{3}{4}+\frac{3}{4}+\frac{1}{2})$
$n=3$ | $(\frac{1}{3}+\frac{1}{3}+\frac{1}{3},\frac{2}{3}+\frac{2}{3}+\frac{2}{3})$
$n=2$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$

Order | $Data$
---|---
$n=10$ |  $(\frac{1}{10}+\frac{2}{5}+\frac{1}{2},\frac{9}{10}+\frac{3}{5}+\frac{1}{2})$, $(\frac{3}{10}+\frac{1}{5}+\frac{1}{2},\frac{7}{10}+\frac{4}{5}+\frac{1}{2})$,
$n=8$ | $(\frac{1}{8}+\frac{3}{8}+\frac{1}{2},\frac{7}{8}+\frac{5}{8}+\frac{1}{2})$
$n=6$ | $(\frac{1}{6}+\frac{1}{6}+\frac{2}{3},\frac{5}{6}+\frac{5}{6}+\frac{1}{3}),~{}~{}(\frac{1}{3}+\frac{2}{3}+\frac{1}{2}+\frac{1}{2})$
$n=5$ | $(\frac{1}{5}+\frac{1}{5}+\frac{3}{5},\frac{4}{5}+\frac{4}{5}+\frac{2}{5}),~{}~{}(\frac{1}{5}+\frac{1}{5}+\frac{2}{5},\frac{4}{5}+\frac{4}{5}+\frac{3}{5})$
$n=4$ | $\frac{1}{4}+\frac{3}{4}+\frac{1}{2}+\frac{1}{2}$
$n=3$ | $\frac{1}{3}+\frac{1}{3}+\frac{2}{3}+\frac{2}{3}$
$n=2$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
$g^{{}^{\prime}}=1,n=2$ | $\frac{1}{2}+\frac{1}{2}$

Table 1: The periodic maps for genus $g=1$ an $g=2$. The genus $g^{{}^{\prime}}=0$ is ignored. Type | $Data$
---|---
$(g=1,k=1,r=0)$ |  $(1):~{}f=id,~{}~{}~{}~{}~{}~{}(2):~{}\bm{\frac{5}{6}}+\frac{2}{3}+\frac{1}{2},~{}~{}(3):~{}\bm{\frac{1}{6}}+\frac{1}{3}+\frac{1}{2}$,
|
$(4):~{}\bm{\frac{1}{4}}+\frac{1}{4}+\frac{1}{2},~{}~{}(5):~{}\bm{\frac{3}{4}}+\frac{3}{4}+\frac{1}{2},~{}~{}(6):~{}\bm{\frac{1}{3}}+\frac{1}{3}+\frac{1}{3},$
|
$(7):~{}\bm{\frac{2}{3}}+\frac{2}{3}+\frac{2}{3},~{}~{}(8):~{}\bm{\frac{1}{2}}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
$(g=1,k=1,r=1)$ | $(1):f=id$
|  $(2):~{}ord(f)=2:~{}Amp(C_{1}),~{}~{}\bm{\frac{1}{2}}+\frac{1}{2}+(1)$,
$(g=2,k=0,r=1)$ | $(1):~{}f=id,~{}~{}~{}~{}~{}~{}~{}(2):~{}(\frac{1}{4})+(\frac{1}{4})+\frac{1}{2},~{}~{}~{}~{}~{}~{}(3):~{}(\frac{3}{4})+(\frac{3}{4})+\frac{1}{2}$
$f(C_{1})=C_{1}$ | $(4):~{}(\frac{1}{3})+(\frac{1}{3})+\frac{1}{3},~{}~{}(5):~{}(\frac{2}{3})+(\frac{2}{3})+\frac{2}{3},~{}~{}~{}(6):~{}(\frac{1}{2})+(\frac{1}{2})+\frac{1}{2}+\frac{1}{2}$
$Amp(C_{1})$ | $(1):~{}\frac{1}{4}+\frac{1}{4}+(\frac{1}{2}),~{}~{}(2):~{}\frac{3}{4}+\frac{3}{4}+(\frac{1}{2}),~{}(3):~{}\frac{5}{6}+(\frac{2}{3})+\frac{1}{2},$
|
$(4):~{}\frac{1}{6}+(\frac{1}{3})+\frac{1}{2},~{}(5):~{}\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+(1)$
$(g=2,k=0,r=2)$ | $(1):~{}f=id,~{}~{}~{}~{}~{}~{}~{}$
| $(2):~{}ord(f)=2,~{}(C_{1},C_{2}),~{}~{}\frac{1}{2}+\frac{1}{2}+(1)+(1)$
|
$(3):~{}ord(f)=2,~{}Amp(C_{1}),~{}f(C_{2})=C_{2},~{}~{}(\frac{1}{2})+(\frac{1}{2})+(1)$
|
$(4):~{}ord(f)=2,~{}Amp(C_{1}),~{}Amp(C_{2}),~{}~{}\frac{1}{2}+\frac{1}{2}+(1)+(1)$,
| $(5):~{}ord(f)=4,~{}Amp(C_{1},C_{2}),~{}~{}\frac{3}{4}+\frac{1}{4}+(1)$
$(g=0,k=3,r=0)$ | $(1):~{}f=id,~{}~{}~{}~{}~{}~{}~{}$
|
$(2):~{}ord(f)=2,~{}(\partial_{1},\partial_{2}),~{}f(\partial_{3})=\partial_{3},~{}\frac{1}{2}+\bm{\frac{1}{2}}+\bm{1}$
|
$(3):~{}ord(f)=3,~{}(\partial_{1},\partial_{2},\partial_{3}),~{}~{}\frac{1}{3}+\frac{2}{3}+\bm{1}$
Table 2: The periodic maps for curves with boundaries and internal cuts.

Dual graph: One can attach a dual graph (star-shaped) for the periodic maps,
and these sub-graphs are glued together to form a full dual graph for each
degeneration. The dual graph is closely related to 3d mirror of the IR theory,
and is used in an essential way in Xie:2023lko  to determine the IR theory.
Here we give a short review of the results in Xie:2023lko .

First, one can define a dual graph for a period map as follows. The data for a
periodic map is
$(n,g^{{}^{\prime}},{\sigma_{1}\over\lambda_{1}}+\ldots+{\sigma_{l}\over\lambda_{l}})$,
here $n$ is the order of the map, $g^{{}^{\prime}}$ is the genus of the base
defined by the covering map $f:g\to g^{{}^{\prime}}$, and
${\sigma_{i}\over\lambda_{i}}$ gives the data for the ramification point of
the covering map. These data are constrained by Hurwitz formula

$2g-2=n[(2g^{{}^{\prime}}-2)+\sum_{i}(1-\frac{1}{\lambda_{i}})]$

The dual graph is constructed from the combinatorial data
$(n,g^{{}^{\prime}},{\sigma_{1}\over\lambda_{1}}+\ldots+{\sigma_{l}\over\lambda_{l}})$
as follows:

  1. 1.

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

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

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

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

So one get a star-shaped dual graph, with the central node having genus
$g^{{}^{\prime}}$ and all the other node having genus zero.

  2. 2.

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

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

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

    1. (a)

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

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

    2. (b)

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

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

    3. (c)

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

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

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

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

One can get a 3d quiver gauge theory from the dual graph: the multiplicities
gave the gauge group $U(n_{i})$ and the edges give the bi-fundamental matter.
The quiver gauge theory determines wether the degeneration is allowed or not:
it is allowed if the Higgs branch dimension of it is equal to two (after
contracting $-1$ curve and peeling off the quiver tail), see details in
Xie:2023lko , and this is not always possible. If it is an allowed
degeneration, the (modified) quiver gives the 3d mirror for the IR theory,
which can then be used to determine the IR theory.

Example: Let’s now give an example showing how to find the classification for
a given weighted graph. The weighted graph is taken to be $(g=2,r=2)$ (4(a) in
figure. 5), which means that there are two non-separating cutting curves for
the genus two Riemann surface. There are following situations that one need to
consider (see table. 2):

  1. 1.

The action of $f$ on two cutting curves $C_{1},C_{2}$ is non-amphidrome, and
$f(C_{1})=C_{1},f(C_{2})=C_{2}$. After the cut, one has a sphere with four
marked points representing four boundaries, see figure, 6. The periodic map
fixed all the boundary curves, i.e.
$f(C_{1}^{{}^{\prime}})=C_{1}^{{}^{\prime}}$, etc, and so the order of
periodic map is one.

  2. 2.

The action of $f$ is $f(C_{1})=C_{2},f(C_{2})=C_{1}$. The periodic map on the
fourth punctured sphere now acts as
$f(C_{1}^{{}^{\prime}})=C_{2}^{{}^{\prime}},f(C_{1}^{{}^{\prime\prime}})=C_{2}^{{}^{\prime\prime}}$.
So the periodic map has order two, and
$(C_{1}^{{}^{\prime}},C_{2}^{{}^{\prime}})$,
$(C_{1}^{{}^{\prime\prime}},C_{2}^{{}^{\prime\prime}})$ are in $Z_{2}$ orbits.
The periodic map is then $f:\Sigma\to\Sigma^{{}^{\prime}}$ with both $\Sigma$
and $\Sigma^{{}^{\prime}}$ genus zero curve. To satisfy the Hurwitz formula,
there must be two fixed points on $\Sigma$ whose valency data is
$\frac{1}{2}$. Therefore the valency data on the punctured sphere is
$(1)+(1)+\frac{1}{2}+\frac{1}{2}$, with the $1$ in the bracket indicating that
the boundaries of two cutting curves are in the $Z_{2}$ orbit.

  3. 3.

The action of $f$ on $C_{1}$ is amphidrome: $Amp(C_{1}),f(C_{2})=C_{2}$. The
periodic map on the fourth punctured sphere now acts as
$f(C_{1}^{{}^{\prime}})=C_{1}^{{}^{\prime\prime}},f(C_{2}^{{}^{\prime}})=C_{2}^{{}^{\prime}},f(C_{2}^{{}^{\prime\prime}})=C_{2}^{{}^{\prime\prime}}$.
This means that $C_{1},C_{1}^{{}^{\prime}}$ are in a $Z_{2}$ orbit, while
$C_{2}^{{}^{\prime\prime}},C_{2}^{{}^{\prime}}$ are fixed points. So the
periodic map on the fourth punctured sphere has order two, and the valency
data is $(\textbf{1})+(\frac{1}{2})+(\frac{1}{2})$.

  4. 4.

The action of $f$ on $C_{1}$ and $C_{2}$ is amphidrome:
$Amp(C_{1}),Amp(C_{2})$. The periodic map on the fourth punctured sphere now
acts as
$f(C_{1}^{{}^{\prime}})=C_{1}^{{}^{\prime\prime}},f(C_{2}^{{}^{\prime}})=C_{2}^{{}^{\prime\prime}}$.
This means that $C_{1}^{{}^{\prime}},C_{1}^{{}^{\prime\prime}}$ are in a
$Z_{2}$ orbit, while $C_{2}^{{}^{\prime}},C_{2}^{{}^{\prime\prime}}$ are
another $Z_{2}$ orbit. So the periodic map on the fourth punctured sphere has
order two, and the valency data is
$(\textbf{1})+(\textbf{1})+\frac{1}{2}+\frac{1}{2}$.

  5. 5.

The action of $f$ on two cutting curves $C_{1},C_{2}$ is non-amphidrome, i.e.
$Am(C_{1},C_{2})$. The boundaries
$C_{1}^{{}^{\prime}},C_{1}^{{}^{\prime\prime}},C_{2}^{{}^{\prime}},C_{2}^{{}^{\prime\prime}}$
are in a $Z_{4}$ orbit. So the periodic map on the fourth punctured sphere has
order four, and the valency data on the punctured sphere is
$(\textbf{1})+(\textbf{1})+\frac{3}{4}+\frac{1}{4}$.

$C_{1}$$C_{2}$$C_{1}^{{}^{\prime}}$$C_{2}^{{}^{\prime}}$$C_{1}^{{}^{\prime\prime}}$$C_{2}^{{}^{\prime\prime}}$
Figure 6: Genus two curve has two non-separating cutting curves, and one get a
fourth punctured sphere after the cutting.

###  2.2 IR theory

Let’s now use the result of last subsection to classify the possible
degeneration for rank two theories. The subtly is that one can not get
sensible physical interpretation for all the degenerations. To determine the
IR theory, we use the link between the dual graph and 3d mirror, and the basic
assumption is that one should get a consistent 3d mirror for a physical
sensible degeneration, otherwise we will not consider it.

####  2.2.1 4d $\mathcal{N}=2$ SCFTs

The basic assumption to get a SCFT is following: a): one get consistent 3d
mirror from the dual graph ; b): there is no variable link number $K$: the
link number is truncated dual to the consistency of 3d mirror; This excludes
the weighted graph with separating curves, since if the gluing is possible,
the link number is never truncated; for the internal cutting curve, one has to
consider the amphidrome cutting.

One find the following possibilities by looking at the weighted graph in
figure.5:

  * •

The weighted graph 1a) in figure. 5 has no cut curve and so there is no
variable link number K. The only constraint would be that the modification of
the mirror quiver is possible. By looking at the list of genus two periodic
map, we find that the degeneration
$(\frac{9}{10}+\bm{\frac{3}{5}}+\frac{1}{2}),~{}(\frac{3}{10}+\bm{\frac{1}{5}}+\frac{1}{2})$,
$(\frac{5}{6}+\frac{5}{6}+\bm{\frac{1}{3}}),~{}(\frac{2}{3}+\bm{\frac{1}{3}}+\frac{1}{2}+\frac{1}{2})$
are not good. The bad valency data is in put in bold letter. Others give SCFT,
see Xie:2022aad .

  * •

The weighted graph 2b) in figure. 5 would give SCFT. The basic data for the
periodic map is $(g=1,r=0,k=1)$. The corresponding dual graph are listed in
figure. 7. Here we need to first do the contraction of $-1$ curve Xie:2023lko
, and then use the modification procedure to get the 3d mirror.

  * •

The weighted graph is 3a) in figure. 5, and the cut curve is taken to be
amphidrome. The basic building block of the periodic map is $(g=2,r=1)$, See
figure. 7 for the dual graphs.

$1$$l=2$$2$$1$$1$$\frac{5}{6}+\frac{2}{3}+\frac{1}{2}$12$4,8$$6$$10,8,6,4,2$$1$$1$$\frac{1}{6}+\frac{1}{3}+\frac{1}{2}$12$4$$6$$2$$1$$1$$\frac{3}{4}+\frac{3}{4}+\frac{1}{2}$8$2,4,6$$4$$6,4,2$$1$$1$$2$$1$$1$$\frac{1}{4}+\frac{1}{4}+\frac{1}{2}$8$2$$4$$2$$1$$1$$2$$1$$1$$2$$\frac{2}{3}+\frac{2}{3}+\frac{2}{3}$6$2,4$$4,2$$1$$1$$2,4$$\frac{1}{3}+\frac{1}{3}+\frac{1}{3}$6$2$$2$$2$$1$$1$$2$$1$$1$$2$$2$$\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$4$2$$2$$2$$1$$1$$2$

$2$$r=1$$\frac{3}{4}+\frac{3}{4}+\left(\frac{1}{2}\right)$$4$$1,2,3$$1,2,3$$2$$1$$1$$\frac{5}{6}+\frac{1}{2}+\left(\frac{2}{3}\right)$$6$$1,2,3,4,5$$3$$4,2$$1$$1$

Figure 7: Upper: dual graph for SCFTs from weighted graph 2b). Here one first
contract $-1$ curve and then peel off the red tail. These give the rank two
$H_{i}$ type theories. Bottom: dual graph for SCFTs from weighted graph 3a),
and one also peel off the red tail. The 3d mirror for those theories were
found in Xie:2012hs .

####  2.2.2 Non-abelian gauge theory

Let’s now consider the degeneration whose low energy theory is IR free non-
abelian gauge theory. The rank two gauge group could be $A_{2},A_{1}\times
A_{1},B_{2}(=C_{2}),G_{2}$. One can find non-abelian gauge groups from
weighted graph as follows Xie:2023lko : 1) if there is a weight $n$ edges
connecting two components in the weighted graph, the gauge group would be
$SU(n)$; 2) The internal cut is amphidrome with weight $m$ ($m$ cutting curves
will be mapped to each other by the mapping group action), the gauge group
would be $Sp(2m)$.

By looking at all the weighted graphs in figure. 5, one find the following
possibilities: 1) weighted graph 3a), and the cut is taken to be non-
amphidrome; the gauge group is $SU(2)$ and is coupled with a rank one SCFT.
2): weighted graph 4a), and the gauge group is ($Sp(4)$ (weight 2), and the
gauge group is $SU(2)\times SU(2)$ (for weight $(1,1)$). 3) weighted graph
5e), and the gauge group is $SU(3)$, see figure. 8.

Example: For the weighted graph 4a in figure. 5, one can get non-abelian gauge
theory for following two situations. Assuming the cut curves are
$C_{1},C_{2}$: a): the action of the mapping class group is
$Am(C_{1}),Am(C_{2})$, and the gauge group is $SU(2)\times SU(2)$, and there
is one $SU(2)$ associated with cutting; b) the action acts as
$(Am(C_{1},C_{2}))$, and the gauge group is $Sp(4)$.

$2$$r=1$$\frac{1}{4}+\frac{1}{4}+\left(\frac{1}{2}\right)$$4$$1$$1$$2...2$$1$$1$$\frac{1}{6}+\frac{1}{2}+\left(\frac{1}{3}\right)$$6$$1$$3$$2...2$$1$$1$$\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+(1)$$2$$1$$1$$2...2$$1$$1$$1$$1$$1$$2...2$$1$$1$$K+1$$1$$2...2$$1$$1$$K+1$$K$$2$$r=2$$\frac{3}{4}+\frac{1}{4}+(1)$$4$$1,2,3$$1$$4...4$$2$$2$$\frac{1}{2}+\frac{1}{2}+(1)+(1)$$2$$1$$2...2$$1$$1$$1$$2...2$$1$$1$SU(2)$K+3$$\
(A_{1},\ D_{4})$SU(2)$K+3$$\ (A_{1},\
A_{3})$SU(2)$-K+2$$2-SU(2)-$Sp(4)$6+K$$K$SU(2)$K_{2}+2$$K_{1}$$K_{2}$SU(2)$K_{1}+2$$0$$0$$3$$f_{1}=f_{2}=\frac{1}{3}+\frac{2}{3}+1$$3$$1,2$$1$$3...3$$2,1$$1$$K+1$SU(3)$6+K$
Figure 8: The dual graph for non-abelian gauge theory. The lower bound for $K$
is determined so that the theory is IR free. $1$$1$$Two\ decoupled\ rank\ one\
SCFT$$Elliptic\ 2a)$$2$$a)$$r=1$$Parabolic\ 3$$n\oplus U(1)\oplus\ rank\ one\
SCFT$$1$$b)$$1$$r=1$$\ U(1)\oplus n\ +\ a\ rank\ one\ SCFT$$\ SU(2)\oplus n\
+\ a\ rank\ one\ SCFT$$Parabolic\
4$$2$$a)$$r=2$$(C_{1},C_{2}):U(2)+adj+nF$$Amp(C_{1}),f(C_{2})=C_{2}$$Parabolic\
5(c)$$1$$b)$$1$$r=1$$r=1$$g=1$$c)$$2$$l=2$$\ U(1)\oplus n\ +\ U(1)\oplus m$$\
U(1)\oplus n\ +\ SU(2)\oplus m$$\ SU(2)\oplus n\ +\ SU(2)\oplus m$$f=id:\
U(2)+adj+nF$$r=1$$order(2):\
Sp(4)+(n+4)F+V$$n-U(1)-SU(2)-m$$0$$0$$2$$1$$n-U(1)\ \ \ \ \ \ \ \ SU(2)-m$
Figure 9: Weighted graph and periodic map for IR free theories involving
abelian gauge group.

####  2.2.3 Other cases: abelian gauge theory

The IR theory for other cases involves decoupled systems: if the edge in the
weighted graph has multiplicity one, the gauge group is $SU(1)$ and so the two
adjacent system is decoupled. The appearance of abelian gauge group: this
happens if the internal cut is non-amphidrome: for an internal cut with
multiplicity $n$, the gauge group is $U(n)$. The full list is:

  * •

Weighted graph 2a): two decoupled rank one SCFTs, plus possible uncharged
hypermultiplets;

  * •

Weighted graph 3a) and the cut is taken to be non-amphidrome. The IR theory is
$U(1)$ gauge group coupled with $n$ free hypermultiplets plus a rank one SCFT.

  * •

Weighted graph 3b). If the cut is amphidrome, the IR theory is $SU(2)$ gauge
group coupled with $n$ fundamental hypermultiplets plus a rank one SCFT; If
the cut is non-amphidrome, the IR theory is $U(1)$ gauge group coupled with
$n$ free hypermultipelts plus a rank one SCFT.

  * •

The IR theory for weighted graph 4a), 4b), 4c) and 5c) are listed in figure.
9.

Example: Let’s consider the last item in figure. 9. The periodic map on each
genus zero component should be $(\frac{1}{2})+(1)+\frac{1}{2}$, and the dual
graph is shown in figure. 10. To find the IR theory, we try to find the 3d
mirror of the dual graph, which can be achieved using S duality of type IIB
string theory.

$2$$2$$1$$1$$2$$2$$1$$1$$NS5$$NS5$$D5$$NS5$$D5$$D5$$S\ duality$$D5$$NS5$
Figure 10: Left: the dual graph of a genus two degeneration determined by
weighted graph 5c) in figure. 5. Right: we represent the quiver on the left
side by a type IIB configuration Hanany:1996ie , notice here one of the $U(1)$
gauge group is ungauged; We then do the S duality to find the 3d mirror, and
the gauge group is $U(2)\times U(1)$. To get the mirror of the left side
quiver, one need to change the $U(2)$ gauge group to $SU(2)$ gauge group.

##  3 Global SW geometry

Let’s now study the global SW geometry for rank one and rank two theory from
the perspective of mapping class group. A one dimensional slice of the Coulomb
branch is taken, and there are singular fibers where the low energy physics is
different from the generic point. We add a point at $\infty$ to get a compact
base, and this introduces a singular fiber at $\infty$ which determines the UV
theory. We first study the generic one dimensional slice, so that the bulk
singularity is either $I_{1}$ or $\tilde{I}_{1}$ type, see figure. 1 for the
illustration. The physical understanding of these singularities are: a) The
vanishing cycle associated with $I_{1}$ singularity is non-separating and its
homology class is non-trivial; the low energy theory is a massless
hypermultiplet charged with a $U(1)$ gauge group (in proper duality frame),
plus other free vector multiplets; b) There is a separating vanishing cycle
associated with $\tilde{I}_{1}$ singularity, and there is an extra massless
hypermultiplet which is not charged under the low energy gauge groups.

The SW fiberation with just $I_{1}$ or $\tilde{I}_{1}$ singular fibers are
called Lefschetz fiberation. Notice that in our case the singular fiber at
$\infty$ could be special, however, we assume that it could also be deformed
into $I_{1}$ and $\tilde{I}_{1}$ singularities, see the discussion below. The
Coulomb branch structure is shown in figure. 11.

$\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 following topological constraints that the global SW geometry has to
satisfy:

  * •

One can define two topological invariants for a singular fiber: $d_{x}$ and
$\delta_{x}$. $\delta_{x}$ can be easily computed from the dual graph and
$d_{x}$ can be computed from holomorphic data, see Xie:2022aad  for those
numbers. The difference of $d_{x}$ and $\delta_{x}$ reflects the number of
$\tilde{I}_{1}$ singularity in the generic deformation. The first topological
constraint is

$\sum d_{x}=2,$

which reflects the fact the total space of SW fiberation is a rational
surface. The topological data for $I_{1}$ singularity is $d_{x}=\delta_{x}=1$,
while the data for $\tilde{I}_{1}$ singularity is $d_{x}=2,\delta_{x}=1$. The
above topological constraint comes from the assumption that the total space of
the genus two fiberation is rational. The above constraint implies that the
choice of $I_{1},\tilde{I}_{1}$ singularities are $(20,0),(18,1),(16,2)$, etc.

  * •

The second topological constraint is due to the compactness of the base of the
fiberation. By choosing a generic point on the moduli space, there is one
mapping class group element $M_{i}$ for each singular fiber by choosing a
path, and the product of them should be trivial:

$M_{1}\ldots M_{s}=I.$

Therefore, the global SW geometry has a simple topological meaning: a
factorization of identity element in terms of mapping class group elements
associated with $I_{1}$ and $\tilde{I}_{1}$ singularity.

Therefore we need to solve two problems in mapping class group: a) Find the
factorization of the UV MCG element in terms of the MCG element of $I_{1}$ and
$\tilde{I}_{1}$ singularity; b) Find the factorization of the identity element
in terms of suitable number MCG element of $I_{1}$ and $\tilde{I}_{1}$
singularity.

###  3.1 Dehn twist

As we discussed at the beginning of this section, the global SW geometry has a
meaning in terms of factorization of mapping class group (MCG) element into
$I_{1}$ or $\tilde{I}_{1}$ singularity. There is a vanishing cycle for each
$I_{1}$ or $\tilde{I}_{1}$ singularity, and one can have an associated MCG
element called Dehn twist, see figure. 12.

The following are the basic relations for Dehn twist associated with two
cycles $a,b$:

  1. 1.

If $a,b$ is disjoint, then $T_{a}T_{b}=T_{b}T_{a}$.

  2. 2.

If $b=h(a)$ with $h$ an element of mapping class group, then
$T_{b}=hT_{a}h^{-1}$.

  3. 3.

If $(a,b)=1$, then $T_{a}T_{b}T_{a}=T_{b}T_{a}T_{b}$.

  4. 4.

If $(a,b)>1$, then there are no relations between $T_{a}$ and $T_{b}$.

The first and third relations are called braid relations, which would play
crucial roles later.

Dehn twist$C$ Figure 12: Dehn twist along curve $C$, and this action is taken
to be negative around the indicated direction.

###  3.2 Rank one theory

Let’s now apply our classification strategy to rank one theory, and we’d like
to recover the classification of rank one theory.

The genus one MCG is just $SL(2,Z)$ group, and it is generated by the Dehn
twist around two cycles $a_{1},b_{1}$, see figure. 13. The generators are
denoted as $\tau_{1},\tau_{2}$, and there is just one non-trivial relation:

$(\tau_{1}\tau_{2})^{6}=1.$

We also have the braid relation
$\tau_{1}\tau_{2}\tau_{1}=\tau_{2}\tau_{1}\tau_{2}$. The intersection form is
$(a_{1},b_{1})=1$, and so the action of the Dehn twist on homology classes
$[a_{1}],[b_{1}]$ is

$\displaystyle
T_{1}(a_{1})=a_{1}+(a_{1},a_{1})a_{1}=a_{1},~{}~{}~{}~{}~{}~{}~{}~{}T_{1}(b_{1})=b_{1}+(b_{1},a_{1})a_{1}=-a_{1}+b_{1}$
$\displaystyle
T_{2}(a_{1})=a_{1}+(a_{1},b_{1})b_{1}=a_{1}+b_{1},~{}~{}T_{2}(b_{1})=b_{1}+(b_{1},b_{1})b_{1}=b_{1}$

So the representation matrix on the basis of homology groups
$([a_{1}],[b_{1}])$ is given as

$\tau_{1}=\left(\begin{array}[]{cc}1&-1\\\
0&1\end{array}\right),~{}~{}\tau_{2}=\left(\begin{array}[]{cc}1&0\\\
1&1\end{array}\right)$

which gives the standard representation for the generators of $SL(2,Z)$ group.

$a_{1}$$b_{1}$ Figure 13: The two cycles generate the homology group of genus
one curve.

Factorization for MCG element of singular fibers: The classification of rank
one IR theory is given by the degeneration of elliptic curve, and it is the
same as the classification of the conjugacy class of $M_{1}$ whose homology
representation $M$ satisfying $Tr(M)\leq 2$. We will find the factorization of
these conjugacy class by requiring:

  1. 1.

It is given by a product of positive Dehn twist, namely it involves only the
generators $\tau_{1},\tau_{2}$, but not the inverse of them.

  2. 2.

The number of elements in the factorization is the same as the Euler number,
which is also equal to the number of $I_{1}$ singularities under the generic
deformation of the singularity.

The factorizations for genus one degeneration is shown in table. 3.

Mapping class group element | Data | Name | Euler number
---|---|---|---
$\zeta_{1}=\tau_{1}\tau_{2}$ | $\frac{1}{6}+\frac{1}{3}+\frac{1}{2}$ | II | 2
$\zeta_{1}^{2}$ | $\frac{1}{3}+\frac{1}{3}+\frac{1}{3}$ | IV | 4
$\zeta_{1}^{3}$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$ | $I_{0}^{*}$ | 6
$\zeta_{1}^{4}$ | $\frac{2}{3}+\frac{2}{3}+\frac{2}{3}$ | $IV^{*}$ | 8
$\zeta_{1}^{5}$ | $\frac{5}{6}+\frac{2}{3}+\frac{1}{2}$ | $II^{*}$ | 10
$\zeta_{1}^{6}=1$ | $1$ | $I_{0}$ |

Mapping class group element | Data | Name | Euler number
---|---|---|---
$\zeta_{2}=\tau_{1}\tau_{2}\tau_{1}$ | $\frac{1}{4}+\frac{1}{4}+\frac{1}{2}$ | $III$ | 3
$\zeta_{2}^{2}$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$ | $I_{0}^{*}$ | 6
$\zeta_{2}^{3}$ | $\frac{3}{4}+\frac{3}{4}+\frac{1}{2}$ | $III^{*}$ | 9
$\zeta_{2}^{4}$ | $1$ | $I_{0}$ |

Mapping class group element | Name | Euler number
---|---|---
$\tau_{2}^{k}(\tau_{1}\tau_{2})^{3}$ | $I_{k}^{*}$ | $k+6$
$\tau_{1}^{k}$ | $I_{k}$ | $k$

Table 3: The factorization of mapping class group element associated with
genus one degeneration.

Given the representation of MCG element in table. 3, one can then easily find
the possible IR configurations: one simply rearrange the word by using the
braid relation and the Hurwitz move

$\displaystyle\tau_{1}\tau_{2}\tau_{1}=\tau_{2}\tau_{1}\tau_{2},$
$\displaystyle(\tau_{1}\ldots\underbrace{\tau_{i}\tau_{i+1}}\ldots\tau_{r})=(\tau_{1}\ldots\underbrace{(\tau_{i}\tau_{i+1}\tau_{i}^{-1})\tau_{i}}\ldots\tau_{r})$
(2)

Since $\tau_{i}\tau_{i+1}\tau_{i}^{-1}=\tau_{i(i+1)}$ is also a positive Dehn
twist around the curve $i(i+1)$ (which is the resulting curve of the Dehn
twist $\tau_{i}$ acting on the curve $(i+1)$.). The above two moves would also
give the positive factorization. One can find the IR configuration by looking
at the possible collapsing of the $I_{1}$ singularities.

Example 1: Let’s look at $E_{8}$ SCFT which is represented by the word
$(\tau_{1}\tau_{2})^{5}$, and it is easy to find various singular fiber
combinations. Here we just give several simple examples. The simplest one is
the configuration with five type II singularities:

$(\tau_{1}\tau_{2})(\tau_{1}\tau_{2})(\tau_{1}\tau_{2})(\tau_{1}\tau_{2})(\tau_{1}\tau_{2})$

We use bracket to indicate that the singularities inside it is collapsed. The
next one involves a $I_{0}^{*}$ singularity

$(\tau_{1}\tau_{2}\tau_{1}\tau_{2}\tau_{1}\tau_{2})\tau_{1}\tau_{2}\tau_{1}\tau_{2}$

Finally one can find a configuration with a $I_{3}^{*}$ singularity

$(\tau_{1}\tau_{2}\tau_{1})\tau_{2}\tau_{1}\tau_{2}\tau_{1}\tau_{2}\tau_{1}\tau_{2}\to(\tau_{2}\tau_{1})\tau_{2}\tau_{2}\tau_{1}\tau_{2}\tau_{1}\tau_{2}\tau_{1}\tau_{2}\to\tau_{2(1)}(\tau_{2}^{3}(\tau_{1}\tau_{2})^{3})$

Here one use the braid relation in the first step, and use the Hurwitz move in
the second step, and finally one get a $I_{3}^{*}$ singularity. The interested
reader can work out all the other configurations listed in
persson1990configurations ; Xie:2022lcm .

Global SW geometry: The total mapping class group around all the singularities
on the compactified Coulomb branch should be trivial, which implies that the
ordered product of Dehn twists should be trivial. We conjecture that the
corresponding elliptic fibered surface should be a rational surface, which
implies that the total number of $I_{1}$ singularities should be 12
persson1990configurations ; Xie:2022lcm . So one should find a positive
factorization of identity element with length 12, and the only choice is

$(\tau_{1}\tau_{2})^{6}=(\tau_{1}\tau_{2}\tau_{1})^{4}=1.$

We can then find the global SW geometry by using braid move and Hurwitz move
to rearrange the above letters, and get the configuration for the
factorization of the UV fiber, see table. 3.

4d SCFT: The configuration for SCFT is quite simple, we have

$\displaystyle((\tau_{1}\tau_{2})^{i},(\tau_{1}\tau_{2})^{6-i}),$
$\displaystyle((\tau_{1}\tau_{2}\tau_{1})^{i},(\tau_{1}\tau_{2}\tau)^{4-i}).$
(3)

So one can get a pair of SCFT by putting one configuration at $\infty$, and
others are bulk singularities which can be moved freely.

4d asymptotical free theory: The configuration for asymptotical free theory is

$(I_{bulk},I_{\infty})=(\tau_{1}\tau_{22(1)}\tau_{2}^{4-k},\tau_{2}^{k}(\tau_{1}\tau_{2})^{3}).$
(4)

This is the SW geometry for $SU(2)$ with $4-k$ fundamental hypermultiplets.
The configuration is found by doing the braid move and Hurwitz move on the
fundamental factorization of identity, so that one can form a $I_{k}^{*}$
singularity:

$1=\tau_{1}\tau_{2}(\tau_{1}\tau_{2}\tau_{1})\tau_{2}(\tau_{1}\tau_{2})^{3}=\tau_{1}(\tau_{2}\tau_{2}\tau_{1})\tau_{2}\tau_{2}(\tau_{1}\tau_{2})^{3}=\tau_{1}\tau_{22(1)}\tau_{2}^{4}(\tau_{1}\tau_{2})^{3}.$

One do the braid move for the first step, and do the Hurwitz move for the
second step. The notation $\tau_{22(1)}$ means the positive Dehn twist along
the curve $22(1)$, which is the curve derived by first doing Dehn twist along
curve 2 on curve $1$ to get a curve $2(1)$, and then do the Dehn twist along
curve 2 on curve $2(1)$. Using the action of Dehn twist on homology, one has

$22([1])=2([1]+(1,2)[2])=[1]+(1,2)[2]+(1,2)[2]=[1]+2[2].$

Here one use the intersection number $(1,2)=1$. For the pure $SU(2)$ theory
$(k=4)$ in 4, the bulk singularity is $\tau_{1}\tau_{22(1)}$; So there is one
vanishing cycle with homology $[1]$, and another vanishing cycle with homology
$[1]+2[2]$. The intersection pair of these two vanishing cycles are $2$, which
indeed gives the BPS quiver of this theory.

Notice that the singular fiber at $\infty$ is $I_{k}^{*}$ singular fiber,
which is represented by $\tau_{2}^{k}(\tau_{1}\tau_{2})^{3}$, and is the
singular fiber at $\infty$ for 4d asymptotical free theory.

5d KK theory: One simply use the braid move and Hurwitz move to get a $I_{k}$
singularity on the right:

$\displaystyle\tau_{1}(\tau_{2}\tau_{1}\tau_{2})\tau_{1}(\tau_{2}\tau_{1}\tau_{2})\tau_{1}(\tau_{2}\tau_{1}\tau_{2})\to\tau_{1}\tau_{1}\tau_{2}\tau_{1}\tau_{1}\tau_{1}\tau_{2}\tau_{1}\tau_{1}\tau_{1}\tau_{2}\tau_{1}$
$\displaystyle\to\tau_{1^{2}(2)}\tau_{1^{5}(2)}\tau_{1^{8}(2)}\tau_{1}^{9}.$

and one can get a $I_{9}$ singularity. Different type of 5d theory is found by
putting $I_{k}$ singularity at $\infty$.

6d KK theory: This one is the simplest one as there is no singularity at
$\infty$, so the bulk singularity is just $(\tau_{1}\tau_{2})^{6}$.

Non-deformable singularity: To get theory with non-deformable singularity, one
simply use braid move and Hurwitz move on the word for a SCFT to get the non-
deformable singularity type (such as $I_{n},I_{k}^{*},II^{*},III^{*},IV^{*}$)
on the bulk.

$\mathcal{N}=2^{*}$ theory: SW geometry of $\mathcal{N}=2^{*}$ theory is found
by $SU(2)$ with $N_{f}=4$ word, which is a $I_{0}^{*}$ singularity. It can be
written as $I_{4}I_{1}^{2}$ configuration, which can be found as follows:

$\tau_{1}(\tau_{2}\tau_{1}\tau_{2})\tau_{1}\tau_{2}=(\tau_{1}^{2}\tau_{2})\tau_{1}^{2}\tau_{2}=\tau_{1^{2}(2)}(\tau_{1}^{4})\tau_{2}.$

Here we used braid move and the Hurwitz move to get a $I_{4}$ singularity.
Another realization is given by $I_{2}I_{2}I_{2}$ configuration, which can be
found from $I_{0}^{*}$ word as follows:

$\tau_{1}(\tau_{2}\tau_{1}\tau_{2})\tau_{1}\tau_{2}=\tau_{1}^{2}(\tau_{2}\tau_{1}\tau_{1})\tau_{2}=(\tau_{1}^{2})(\tau_{2(1)}^{2})(\tau_{2}^{2}).$

###  3.3 Rank two theory

The mapping class group $M_{2}$ of genus two curve is generated by Dehn twists
associated with the five curves $\delta_{i}$ shown in figure. 14. The
generators are labeled as

$\tau_{1},\tau_{2},\tau_{3},\tau_{4},\tau_{5}$ (5)

and the relations are

$\displaystyle\tau_{j}\tau_{j+1}\tau_{j}=\tau_{j+1}\tau_{j}\tau_{j+1}$
$\displaystyle\tau_{i}\tau_{j}=\tau_{j}\tau_{i},~{}~{}if~{}~{}|i-j|>1$
$\displaystyle I\tau_{j}=\tau_{j}I$ $\displaystyle I^{2}=1$
$\displaystyle(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{6}=1$ (6)

Here
$I=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}$,
and is the hyperelliptic involution. The first two relations are called braid
relations.

$\delta_{1}$$\delta_{2}$$\delta_{3}$$\delta_{4}$$\delta_{5}$ Figure 14: The
Dehn twists along curves $\delta_{i}$ generate the genus two mapping class
group.

An element in $M_{2}$ can be represented by a positive product of generators;
However, the representation is far from unique due to the braid relations and
other relations in $M_{2}$. Furthermore, we also need to impose following two
equivalence relations:

  * •

The Hurwitz equivalence:

$(\tau_{1}\ldots\underbrace{\tau_{i}\tau_{i+1}}\ldots\tau_{r})=(\tau_{1}\ldots\underbrace{(\tau_{i}\tau_{i+1}\tau_{i}^{-1})\tau_{i}}\ldots\tau_{r})=(\tau_{1}\ldots\tau_{i(i+1)}\tau_{i}\ldots\tau_{r}).$

  * •

Global conjugacy:

$\phi(\tau_{1}\ldots\tau_{r})\phi^{-1}=(\phi\tau_{1}\phi^{-1}\ldots\phi\tau_{r}\phi^{-1}).$

This is due to the fact the degeneration is given by the conjugacy class of
mapping class group.

The goal for finding a special factorization for the degeneration is
following: a): It should be given by a positive factorization, namely, the
word consists of only Dehn twist, but not its inverse; b): The number of
generators are determined by the local invariant $d_{x}$ and $\delta_{x}$:

$\displaystyle\\#\tau=2\delta_{x}-d_{x},$
$\displaystyle\\#\sigma=d_{x}-\delta_{x}.$ (7)

Here $\\#\tau$ is the number of Dehn twist along non-separating curve, and
$\\#\sigma$ is the Dehn twist along the separating curve $\sigma$. The task of
finding above factorization is a difficult one, and we will solve it for the
SCFT and many other IR theories in this paper.

Let’s now summarize some important relations regarding group $M_{2}$, which
will be quite useful for our later studies.

  1. 1.

Let $\zeta_{a,b}=\prod_{i=a}^{b}\tau_{i}$, then

$\tau_{i}\zeta_{a,b}=\zeta_{a,b}\tau_{i-1}$ (8)

Here $a<i\leq b$. This equation can be proven using the braid relation:

Proof:

$\displaystyle\tau_{i}(\tau_{a}\tau_{a+1}\ldots\tau_{b})=\tau_{a}\ldots(\tau_{i}\tau_{i-1}\tau_{i})\tau_{i+1}\ldots\tau_{b}=\tau_{a}\ldots\tau_{i-1}\tau_{i}(\tau_{i-1}\tau_{i+1})\ldots\tau_{b}$
$\displaystyle=(\tau_{a}\ldots\tau_{b})\tau_{i-1}$

  2. 2.

Let
$\zeta=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5},\eta=\tau_{1}\tau_{2}\tau_{3}\tau_{4}$,
we have the relation

$\displaystyle\tau_{i}\zeta^{j}=\zeta^{j}\tau_{i-j},~{}~{}~{}~{}~{}(i\neq
j)~{}~{}$
$\displaystyle\tau_{i}\zeta^{i}=\zeta^{i+1}\eta^{-1},~{}~{}(\eta=\zeta^{-i}\tau_{i}^{-1}\zeta^{i+1})$
(9)

These two equations are derived using the relation 8. From the second
relation, one find that the generators can be expressed in terms of
$\zeta,\eta$:

$\tau_{i}=\zeta^{i+1}\eta^{-1}\zeta^{-i}$ (10)

So the mapping class group $M_{2}$ is generated by $\zeta,\eta$ subject to
relation $\zeta^{6}=\eta^{10}=1$.

  3. 3.

Let
$\epsilon=\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4},\eta=\tau_{1}\tau_{2}\tau_{3}\tau_{4}$.
We have the relation $\epsilon^{4}=\eta^{5}$, and $\eta^{5}$ is conjugate with
$I$.

Proof: Let’s first do the computation for $\eta^{5}$. Using formula. 9, we
have

$\displaystyle\eta^{5}=(\zeta^{-1}\tau_{1}^{-1}\zeta^{2})(\zeta^{-2}\tau_{2}^{-1}\zeta^{3})\ldots(\zeta^{-5}\tau_{5}^{-1}\zeta^{6})=$
$\displaystyle=\zeta^{-1}\tau_{1}^{-1}\tau_{2}^{-1}\tau_{3}^{-1}\tau_{4}^{-1}\tau_{5}^{-1}=$
$\displaystyle=\tau_{5}^{-1}\tau_{4}^{-1}\tau_{3}^{-1}\tau_{2}^{-1}\tau_{1}^{-1}\tau_{1}^{-1}\tau_{2}^{-1}\tau_{3}^{-1}\tau_{4}^{-1}\tau_{5}^{-1}=$
$\displaystyle=\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}=\tilde{I}$

The last expression is named $\tilde{I}$ which is conjugate to $I$, and the
corresponding conjugate element is
$\phi=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}$. On the other hand, we have
$\epsilon=\tau_{1}\eta$ (definition), and $\tau_{i+1}\eta=\eta\tau_{i}$ (see
8), we have

$\displaystyle\epsilon^{4}=\tau_{1}(\eta\tau_{1})\eta\tau_{1}\eta\tau_{1}\eta=\tau_{1}\tau_{2}\eta(\eta\tau_{1})\eta\tau_{1}\eta=\tau_{1}\tau_{2}(\eta\tau_{2})\eta\eta\tau_{1}\eta$
$\displaystyle=\tau_{1}\tau_{2}\tau_{3}\eta\eta(\eta\tau_{1})\eta=\tau_{1}\tau_{2}\tau_{3}\eta\eta\tau_{2}\eta\eta=\ldots=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\eta^{4}=\eta^{5}.$

In the following, we’d like to list some useful conjugation transformation.
Let $\zeta_{a,b}=\prod_{i=a}^{b}\tau_{i}$, then we have

  1. 1.

For any $a,b$ such that $1\leq a<b\leq 5$, we have

$C(\zeta_{a,b})(\tau_{i})=\begin{cases}\tau_{i},~{}~{}~{}~{}if~{}i<a-1\\\
\tau_{i+1},~{}~{}if~{}a\leq i<b,\\\ \tau_{i},~{}~{}if~{}i>b+1\end{cases}$

  2. 2.

For any $a,b$ such that $1\leq a<b\leq 5$, let
$\eta_{a,b}=\prod_{c=0}^{b-a}\tau_{a,b-c}$, then

$C(\eta_{a,b})(\zeta_{i})=\begin{cases}\tau_{i},~{}~{}~{}~{}if~{}i<a-1\\\
\tau_{a+b-i},~{}~{}if~{}a\leq i\leq b,\\\
\tau_{i},~{}~{}if~{}i>b+1\end{cases}$ (11)

In particular, one can take $a=1,b=5$, and so its action on the index would be
$5\to 1,4\to 2,3\to 3$.

  3. 3.

$C(\zeta_{a,b}^{2})(\tau_{b})=\tau_{a}$.

  4. 4.

There is following conjugacy condition

$\zeta_{1}^{t_{1}}\zeta_{2}^{t_{2}}\ldots\zeta_{5}^{t_{5}}\sim\zeta_{s(1)}^{t_{s(1)}}\zeta_{s(1)}^{t_{s(1)}}\ldots\zeta_{s(5)}^{s(t_{5})}$

Here $s$ is a permutation.

One can prove above conjugacy equation by using braid relations.

####  3.3.1 Mapping class elements for 4d SCFT

We are going to list the mapping class elements for rank two $\mathcal{N}=2$
SCFT realized as the periodic map of genus two curve. We’d like to find a
factorization of the mapping class element so that the number of $I_{1}$ and
$\tilde{I}_{1}$ singularities are given as (see next section for the
derivation):

$\displaystyle\\#I_{1}=2\delta_{x}-d_{x}$
$\displaystyle\\#\tilde{I}_{1}=d_{x}-\delta_{x}$ (12)

Here the $I_{1}$ singularity is represented by the Dehn twist along any closed
curve whose homology class is nontrivial, and $\tilde{I}_{1}$ singularity is
given by the Dehn twist along the cycle with trivial homology class. This
problem has been studied in hirose2010presentations ; nakamura2018generation ;
dhanwani2023factoring ; sakalli2023singular  and the results are listed in
table. [4, 5, 6, 7]. One of the crucial relation is the following
representation of the Dehn twist along the separating curve:

$\boxed{(\tau_{1}\tau_{2})^{6}=\sigma}$ (13)

Here $\sigma$ is the Dehn twist along the non-separating curve, see figure.
15.

$\sigma$ Figure 15: The separating curve on genus two curve. Monodromy | Data | Singularity | Scaling dimension | $(d_{x},\delta_{x})$
---|---|---|---|---
$\eta=\tau_{1}\tau_{2}\tau_{3}\tau_{4}$ | $\frac{1}{10}+\frac{2}{5}+\frac{1}{2}$ | $y^{2}=x^{5}+t$ | $(\frac{10}{7},\frac{8}{7})$ | $(4,4)$
$\eta^{2}$ | $\frac{1}{5}+\frac{2}{5}+\frac{2}{5}$ | $y^{2}=x^{5}+t^{2}$ | $(\frac{5}{2},\frac{3}{2})$ | $(8,8)$
$\eta^{3}$ | $\frac{4}{5}+\frac{7}{10}+\frac{1}{2}$ | $y^{2}=x^{5}+t^{3}$ | $(10,4)$ | $(12,12)$
$\phi_{\tilde{4}}=\eta^{4}=\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}$ | $\frac{1}{5}+\frac{1}{5}+\frac{3}{5}$ |  | $(\frac{5}{3},\frac{4}{3})$ | $(6,6)$
$\eta^{5}=\tilde{\text{I}}$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$ |  | $(2,2)$ | $(10,10)$
$\eta\tilde{\text{I}}$ | $\frac{4}{5}+\frac{4}{5}+\frac{2}{5}$ |  | $(5,4)$ | $(14,14)$
$\eta^{7}$ | $\frac{1}{5}+\frac{3}{10}+\frac{1}{2}$ |  | $(\frac{10}{4},\frac{4}{3})$ | $(8,7)$
$\phi_{\tilde{4}}\phi_{\tilde{4}}$ | $\frac{4}{5}+\frac{3}{5}+\frac{3}{5}$ |  | $(5,3)$ | $(12,12)$
$\tilde{\text{I}}\phi_{\tilde{4}}$ | $\frac{9}{10}+\frac{3}{5}+\frac{1}{2}$ |  | $(10,8)$ | $(16,16)$
$\eta^{10}=1$ |  |  |  |
Table 4: Factorization for periodic maps. This family is generated by the factorization $\eta$. Monodromy | Data | Singularity | Scaling dimension |  $(d_{x},\delta_{x})$)
---|---|---|---|---
$\epsilon=\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4}$ | $\frac{1}{8}+\frac{3}{8}+\frac{1}{2}$ | $y^{2}=x(x^{4}+t)$ | $(\frac{8}{5},\frac{6}{5})$ | $(5,5)$
$\epsilon^{2}$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{4}+\frac{3}{4}$ | $y^{2}=x(x^{4}+t^{2})$ | $(4,2)$ | $(10,10)$
$\epsilon^{3}=\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}$ | $\frac{1}{8}+\frac{3}{8}+\frac{1}{2}$ | $y^{2}=x(x^{4}+t^{3})$ | $(\frac{8}{5},\frac{6}{5})$ | $(5,5)$
$\epsilon^{4}=\tilde{I}$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$ |  | $(2,2)$ | $(10,10)$
$\epsilon\tilde{I}$ | $\frac{7}{8}+\frac{5}{8}+\frac{1}{2}$ |  | $(8,6)$ | $(15,15)$
$(\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2})^{2}$ | $\frac{1}{2}+\frac{1}{2}+\frac{3}{4}+\frac{1}{4}$ |  | $(4,2)$ | $(10,10)$
$\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tilde{I}$ | $\frac{7}{8}+\frac{5}{8}+\frac{1}{2}$ |  | $(8,6)$ | $(15,15)$
$\epsilon^{8}=1$ |  |  |  |
Table 5: Factorization for periodic maps. This family is generated by the factorization $\epsilon$. Monodromy | Data | Singularity | Scaling dimension | $(d_{x},\delta_{x})$
---|---|---|---|---
$\zeta=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}$ | $\frac{1}{6}+\frac{1}{6}+\frac{2}{3}$ | $y^{2}=x^{6}+t$ | $(\frac{3}{2},\frac{5}{4})$ | $(5,5)$
$\zeta^{2}$ | $\frac{1}{3}+\frac{2}{3}+\frac{2}{3}+\frac{1}{3}$ | $y^{2}=x^{6}+t^{2}$ | $(3,2)$ | $(10,10)$
$\zeta^{3}=\sigma 1_{32}2_{43}3_{54}$ | $(n=2,g^{{}^{\prime}}=1,\frac{1}{2}+\frac{1}{2})$ | $y^{2}=x^{6}+t^{3}$ | $(2,1)$ | $(5,4)$
$\zeta^{4}=\phi_{A}=(\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1})^{2}$ | $\frac{1}{3}+\frac{2}{3}+\frac{2}{3}+\frac{1}{3}$ |  | $(3,2)$ | $(10,10)$
$\phi_{A}\zeta$ | $\frac{5}{6}+\frac{5}{6}+\frac{1}{3}$ |  | $(6,5)$ | $(15,15)$
$\zeta^{6}=1$ |  |  |  |
Table 6: Factorization for periodic maps. This family is generated by the factorization $\zeta$. Monodromy | Data | Scaling dimension | $(d_{x},\delta_{x})$
---|---|---|---
$\phi_{4}=\tau_{1}\tau_{2}\tau_{4}^{-1}\tau_{5}^{-1}$ | $\frac{1}{2}+\frac{1}{2}+\frac{2}{3}+\frac{1}{3}$ | $(6,2)$ | $(10,9)$
$\phi_{4}^{2}$ | $\frac{1}{3}+\frac{1}{3}+\frac{2}{3}+\frac{2}{3}$ | $(3,2)$ | $(10,10)$
$\phi_{4}^{3}=I$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$ | $(2,2)$ | $(10,10)$
$\phi_{4}^{4}\sim\phi_{4}^{2}$ | $\frac{1}{3}+\frac{1}{3}+\frac{2}{3}+\frac{2}{3}$ | $(3,2)$ | $(10,10)$
$\phi_{4}^{5}\sim\phi_{4}$ | $\frac{1}{2}+\frac{1}{2}+\frac{2}{3}+\frac{1}{3}$ | $(6,2)$ | $(10,9)$
$\phi_{4}^{6}=1$ |  |  |
Table 7: Factorization for periodic maps. This family is generated by the
factorization $\phi_{4}$.

Let’s verify some of the results in table. [4, 5, 6, 7]:

  1. 1.

The first one would be the factorization of $\eta^{4}$:

$\displaystyle\eta^{4}=(\zeta^{-1}\tau_{1}^{-1}\zeta^{2})(\zeta^{-2}\tau_{2}^{-1}\zeta^{3})\ldots(\zeta^{-4}\tau_{4}^{-1}\zeta^{5})=\zeta^{-1}\tau_{1}^{-1}\tau_{2}^{-1}\tau_{3}^{-1}\tau_{4}^{-1}\zeta^{-1}$

Here we used equation $\eta=\zeta^{-i}\tau_{i}^{-1}\zeta^{i+1}$, and
$\zeta^{5}=\zeta^{-1}$. We then use the relation $I^{2}=1$ to express above
expression in terms of positive product of generators:

$\displaystyle
1=I^{2}=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}(\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\tau_{5}}\tau_{4}\tau_{3}\tau_{2}\tau_{1}$
$\displaystyle=(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})\tau_{5}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}$
$\displaystyle=\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})$
$\displaystyle=\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}(\zeta\tau_{4}\tau_{3}\tau_{2}\tau_{1}\zeta)\to$
$\displaystyle\zeta^{-1}\tau_{1}^{-1}\tau_{2}^{-1}\tau_{3}^{-1}\tau_{4}^{-1}\zeta^{-1}=\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}$

Here in the first step we used the fact
$\tilde{I}=\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}$
commutes with all the generators (in particular $\tau_{5}$), and in the second
step one used the cyclic equivalence by moving letters
$\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}$ to the end of the word.

  2. 2.

Second, let’s compute $\epsilon^{3}$. Since $\epsilon^{4}=\tilde{I}$, and so

$\epsilon^{3}=\epsilon^{-1}\tilde{I}^{-1}$

Next, we’d like to use the relation

$\displaystyle
1=\tilde{I}^{2}=\tilde{I}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}=\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tilde{I}\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4}=\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tilde{I}\epsilon$
$\displaystyle\to\epsilon^{-1}\tilde{I}^{-1}=\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}$

Here in the first step we used the fact that $\tilde{I}$ commutes with any
generators, and in the second step one used conjugation to move $\tau_{5}$
from the end to the beginning (cyclic relation).

  3. 3.

Thirdly, we’d like to compute the word $\zeta^{3}$.

$\displaystyle\zeta^{3}=12(345123)4512345=1213243(545)12345=12132(434)5412345$
$\displaystyle=121(323)435412345=12123243(5412)345=(12)^{2}3243125(434)5=(12)^{2}3243125(343)5$
$\displaystyle=(12)^{2}3243123(545)3=(12)^{2}3241(323)4543=(12)^{2}3(212)(434)2543=(12)^{2}1(323)1432543$
$\displaystyle=(12)^{2}12321432543=(12)^{3}321432543=(12)^{6}\overline{12}^{3}321432543$

We then need to compute following word:

$\displaystyle\overline{12}^{3}321432543=\bar{2}\bar{1}\bar{2}\bar{1}\bar{2}\bar{1}321432543=\bar{2}\bar{1}\bar{2}\bar{1}\bar{2}3(\bar{1}21)432543=\bar{2}\bar{1}\bar{2}\bar{1}(\bar{2}32)14(\bar{2}32)543$
$\displaystyle=\bar{2}\bar{1}\bar{2}3(\bar{1}21)(\bar{3}43)25(\bar{3}43)=\bar{2}\bar{1}\bar{2}3(21\bar{2})(43\bar{4})25(43\bar{4})=\bar{2}\bar{1}(\bar{2}32)14(\bar{2}32)(\bar{4}54)3\bar{4}$
$\displaystyle=\bar{2}\bar{1}(32\bar{3})14(32\bar{3})(54\bar{5})3\bar{4}=\bar{2}3(\bar{1}21)\bar{3}4325(\bar{3}43)\bar{5}\bar{4}=(\bar{2}32)1\bar{2}\bar{3}432543(\bar{4}\bar{5}\bar{4})$
$\displaystyle=321(\bar{3}\bar{2}\bar{3})432(54\bar{5})3\bar{4}\bar{5}=321\bar{2}\bar{3}(\bar{2}432\bar{4})543\bar{4}\bar{5}=1_{32}\cdot(\bar{2}432\bar{4})\cdot
3_{54}$ $\displaystyle=1_{32}\cdot(432\bar{3}\bar{4})\cdot 1_{45}=1_{32}\cdot
2_{43}\cdot 3_{54}$

In the process of the computation, we used the following braid relation:

$\bar{\tau}_{i+1}\tau_{i}\tau_{i+1}=\tau_{i}\tau_{i+1}\bar{\tau}_{i}$

which is easily derived from the standard braid relation. So finally one has
the following important result

$\zeta^{3}=\sigma\cdot 1_{32}\cdot 2_{43}\cdot 3_{54}$ (14)

  4. 4.

We’d like to compute $\eta^{7}$:

$\displaystyle\eta^{7}=\tilde{I}\eta^{2}=\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{1}\tau_{2}\tau_{3}\tau_{4}=$
$\displaystyle\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\tau_{1}\tau_{2}\tau_{3}\tau_{4}(\tau_{5}\bar{\tau}_{5})\tau_{1}\tau_{2}\tau_{3}\tau_{4}(\tau_{5}\bar{\tau}_{5})=$
$\displaystyle\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{2}\tau_{1}\tau_{2}\tau_{3}(\bar{\tau}_{5}\tau_{4}\tau_{5})\bar{\tau}_{5}=$
$\displaystyle\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{2}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\bar{\tau}_{4}\bar{\tau}_{5}=$
$\displaystyle\tau_{5}\tau_{4}[\tau_{3}\tau_{2}\tau_{1}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{3}]\bar{\tau}_{4}\bar{\tau}_{5}$

So $\eta^{7}$ is conjugate to the element
$\tau_{3}\tau_{2}\tau_{1}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{3}=\tau_{3}\tau_{2}\tau_{1}\zeta^{3}$,
and so it has a factorization with $6$ $I_{1}$ singularities and one
$\tilde{I}_{1}$ singularity by using the factorization of $\zeta^{3}$. This is
also consistent with the local invariant of degeneration
$(\frac{3}{10}+\frac{1}{5}+\frac{1}{2})$: $(d_{x}=8,\delta_{x}=7)$.

  5. 5.

Let’s now verify $\phi_{4}^{3}=I$. First, we have trivial relation

$\tau_{1}^{-1}\tau_{2}^{-1}\tau_{3}^{-1}\tau_{4}^{-1}\tau_{5}^{-1}=\zeta
I\to\tau_{4}^{-1}\tau_{5}^{-1}=\tau_{3}\tau_{2}\tau_{1}\zeta I$

and so

$\displaystyle\phi_{4}=(\tau_{1}\tau_{2})(\tau_{4}^{-1}\tau^{-1}_{5})=(\zeta\tau_{5}^{-1}\tau_{4}^{-1}\tau_{3}^{-1})(\tau_{3}\tau_{2}\tau_{1}\zeta
I)=\zeta\tau_{5}^{-1}\tau_{4}^{-1}\tau_{2}\tau_{1}\zeta I$
$\displaystyle=(\zeta\tau_{2})\tau_{5}^{-1}\tau_{1}(\tau_{4}^{-1}\zeta)I=\tau_{3}(\zeta\tau_{5}^{-1})(\tau_{1}\zeta)\tau_{3}^{-1}I=\tau_{3}\eta\zeta^{2}\eta^{-1}\tau_{3}^{-1}I$
$\displaystyle=(\tau_{3}\eta)\zeta^{2}(\tau_{3}\eta)^{-1}I$

here we used the relation $\tau_{1}\zeta=\zeta^{2}\eta^{-1}$. So
$\phi_{4}^{3}=I$ (notice that $I$ is the central element of the mapping class
group), by using $\zeta^{6}=1,I^{2}=1$. On the other hand, using the above
formula, we find that

$\displaystyle\phi_{4}=(\tau_{3}\eta)\zeta^{2}(\tau_{3}\eta)^{-1}I=(\tau_{3}\eta)\zeta^{3}\zeta^{-1}I(\tau_{3}\eta)^{-1}$
$\displaystyle=(\tau_{3}\eta)\zeta^{3}(\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1})(\tau_{3}\eta)^{-1}$
(15)

Now use the fact that $\zeta^{3}$ can be factorized into a $\tilde{I}_{1}$
singularity and three $I_{1}$ singularity, one see from above formula that
$\phi_{4}$ can be factorized into a $\tilde{I}_{1}$ singularity and eight
$I_{1}$ singularity. This agrees with the result from the local invariant
($d_{x}=10,\delta_{x}=9$).

Singular configurations for SCFT: Once we find out the desired factorization
for mapping class group elements of SCFT, one can find various singular
configurations of them by doing braid moves and Hurwitz moves. Here let’s just
give several simple examples.

Example 1: Consider the theory whose word is
$\zeta=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}$, one can have following
singular configuration: a): $(\tau_{1}\tau_{2}\tau_{3}\tau_{4})\tau_{5}$,
namely there is a AD theory represented by
$\eta=(\tau_{1}\tau_{2}\tau_{3}\tau_{4})$, and a $I_{1}$ singularity; b):
$(\tau_{1}\tau_{2})\tau_{3}\tau_{4}\tau_{5}$, namely there is a rank one AD
theory represented by $\tau_{1}\tau_{2}$, and three $I_{1}$ singularities.

Example 2: Consider the theory whose word is
$(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{2}$, one can have following
singular configuration: a):
$(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})$,
namely there are two rank two AD theories represented by the word $\zeta$; b):
$(\tau_{1}\tau_{2}\tau_{3}\tau_{4})\tau_{5}(\tau_{1}\tau_{2}\tau_{3}\tau_{4})\tau_{5}$,
namely there are two AD theory represented by the word $\eta$, and two extra
$I_{1}$ singularities.

It is possible to use the braid move and Hurwitz move to get undeformable
singularities, see following examples for $I_{n}$ type. Here we use braid
moves to give the configuration with four identical letter, and then one can
use Hurwitz moves to move those letters together (moving the letters from left
to right). The $I_{4}$ series corresponding to scaling dimension $(5,3)$ and
$(4,2)$ were discussed in Xie:2023zxn .

$\displaystyle(8,6):~{}~{}\bm{\tau_{1}^{2}}\tau_{2}\tau_{3}\tau_{4}(\tau_{5}\tau_{4}\tau_{3}\tau_{2}\bm{\tau_{1}^{2}}\tau_{2}\tau_{3}\tau_{4}\tau_{5})$
$\displaystyle(10,4):~{}~{}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{1}\tau_{2}\tau_{3}(\tau_{4}\tau_{1}\tau_{2}\tau_{3}\tau_{4})=\tau_{1}\tau_{2}\bm{\tau_{3}}\tau_{4}\tau_{1}\tau_{2}\bm{\tau_{3}}\tau_{1}\tau_{2}\bm{\tau_{3}}\tau_{4}\bm{\tau_{3}}$
$\displaystyle(5,3):~{}~{}(\bm{\tau_{1}^{2}}\tau_{2}\tau_{3}\tau_{4}\tau_{5})(\bm{\tau_{1}^{2}}\tau_{2}\tau_{3}\tau_{4}\tau_{5})$
$\displaystyle(4,2):~{}~{}~{}(\bm{\tau_{1}^{2}}\tau_{2}\tau_{3}\tau_{4})(\bm{\tau_{1}^{2}}\tau_{2}\tau_{3}\tau_{4})$
$\displaystyle(2,2):~{}~{}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\sim\tau_{2}\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}=\tau_{2(1)}^{2}\tau_{2}^{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}$
$\displaystyle=\tau_{2(1)}^{2}\tau_{2}^{2}\tau_{3}\tau_{4(5)}^{2}\tau_{4}^{2}\tau_{3}=\tau_{2(1)}^{2}\tau_{2}^{2}\tau_{34(5)}^{2}\tau_{3(4)}^{2}\tau_{3}^{2}$

There are also underformable singularities of $Z_{2}$ type Xie:2023out , which
is given by mapping class group element $I$. By looking at table. [4, 5, 6,
7], i.e. theory with scaling dimension $(10,4)$, $(8,6)$ and $(5,4)$.

####  3.3.2 Global SW geometry

The global SW geometry is given by the positive factorization of the identity
element in mapping class group. Since $I^{2}=1$, the first choice is

$(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1})^{2}=1.$
(16)

This is the $(20,0)$ type as there are a total of 20 $I_{1}$ singularities.
The second choice would be $\eta^{10}=\zeta^{6}=1$, and since one need to have
the topological constraint on the number of $I_{1}$ and $\tilde{I_{1}}$
singularities, we use the equivalent factorization

$\eta^{7}\eta^{3}=[\tau_{5}\tau_{4}[\tau_{3}\tau_{2}\tau_{1}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{3}]\bar{\tau}_{4}\bar{\tau}_{5}]\eta^{3}=[\tau_{5}\tau_{4}[\tau_{3}\tau_{2}\tau_{1}\sigma\cdot
1_{32}\cdot 2_{43}\cdot 3_{54}]\bar{\tau}_{4}\bar{\tau}_{5}]\eta^{3}=1.$ (17)

This is the $(18,1)$ type. Finally, we have the relation $\phi_{4}^{6}=1$, and
so $\phi_{4}^{5}=\phi_{4}^{-1}=\tau_{5}\tau_{4}\tau_{2}^{-1}\tau_{1}^{-1}$,
and one can find an element in mapping class group so that its action would
change the index as $5\to 1,~{}4\to 2$ (see 11, and take $a+b=6$), so
$\phi_{4}^{-1}$ is conjugate with $\phi_{4}$, and its factorization involves a
Dehn twist along $\sigma$ and eight Dehn twists along non-separating curves.
So the factorization is just

$\phi_{4}\phi_{4}^{-1}=1.$ (18)

There are now two $\sigma$ Dehn twists in above factorization, and so it gives
the $(16,2)$ type.

To find the global configuration for the SCFT, one need to rearrange the above
configuration to get a sensible singular fiber at $\infty$. Using the data in
table. [4,5,6,7], one can easily find the results, see table. 8.

Factorization | Theory
---|---
$(\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4},\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4}(\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2})^{2})$ | $((\frac{8}{5},\frac{6}{5}),(8,6))$
$(\tau_{1}\tau_{2}\tau_{3}\tau_{4},\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}I)$ | $((\frac{8}{7},\frac{10}{7}),(10,8))$
$(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5},\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}I)$ | $((\frac{3}{2},\frac{5}{4}),(6,5))$
$(\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1},\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4})$ | $((\frac{3}{2},\frac{5}{4}),(5,4))$
$(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1},\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1})$ | $((2,2),(2,2))$
$((\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{2},(\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1})^{2})$ | $((3,2),(3,2))$
$((\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4})^{2},(\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2})^{2})$ | $((4,2),(4,2))$
$((\tau_{1}\tau_{2}\tau_{3}\tau_{4})^{2},(\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1})^{2})$ | $((\frac{5}{2},\frac{3}{2}),(5,3))$
$(\eta^{7},(\tau_{1}\tau_{2}\tau_{3}\tau_{4})^{3})$ | $((\frac{10}{3},\frac{4}{3}),(10,4))$
$(\phi_{4},\phi_{4}^{-1})$ | $((6,2),(6,2))$
Table 8: The global SW geometry for a pair of rank two SCFTs. We use scaling
dimension to denote the theory.

Let’s show some moves to derive the equivalent factorization of identity which
is used to derive the result in table. 8.

  1. 1.

First, we have the following equivalent factorization:

$1=I^{2}=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}I=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}I\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}=(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{2}(\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1})^{2}.$

The fact $I$ commuting with all the generators is used.

  2. 2.

Secondly, we have the following factorization (see table. 5):

$1=\epsilon^{8}=\epsilon^{2}(\epsilon^{3})^{2}=(\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4})^{2}(\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2})^{2}.$

Other 4d $\mathcal{N}=2$ SCFTs: Let’s now give the global SW geometry of all
rank two $\mathcal{N}=2$ SCFT discussed in section 2. We have found the
configuration for theories engineered using periodic maps. Here we’d list the
global SW geometry for other choices, see table. 9. The important difference
is that the fiber at $\infty$ is no longer given by the periodic map. We first
use the following factorization of identity

$\zeta^{6}=1,$

here $\zeta=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}$. We then use the
factorization $\zeta^{3}=\sigma\cdot 1_{32}\cdot 2_{43}\cdot 3_{54}$, and
finally the fact $\sigma=(\tau_{1}\tau_{2})^{6}$, and so the factorization of
identity becomes

$\zeta^{3}(\zeta)^{3}=\sigma\cdot 1_{32}\cdot 2_{43}\cdot 3_{54}\cdot\sigma
1_{32}2_{43}3_{54}=\sigma\cdot 1_{32}\cdot 2_{43}\cdot
3_{54}\cdot(\tau_{1}\tau_{2})^{6}\cdot 1_{32}\cdot 2_{43}\cdot 3_{54}$

We then split the middle factor $(\tau_{1}\tau_{2})^{6}$ to the bulk and
infinity.

Scaling dimension | $[I_{\infty},bulk]$
---|---
$(4,3)$ | $[\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\tau_{1}\tau_{2}\tau_{3},~{}\tau_{4}\tau_{5}(\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1})^{2}]$
$(6,4)$ | $[\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{1}\tau_{2}\tau_{3},~{}\tau_{5}\tau_{4}\tau_{5}(\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1})^{2}$
$(12,6)$ | $[\zeta^{3}\tau_{1}\tau_{2},~{}(\tau_{1}\tau_{2})^{5}\zeta_{5}]$
$(8,4)$ | $[\zeta^{3}\tau_{1}\tau_{2}\tau_{1},~{}\tau_{2}(\tau_{1}\tau_{2})^{4}\zeta_{5}]$
$(6,3)$ | $[\zeta^{3}(\tau_{1}\tau_{2})^{2},~{}(\tau_{1}\tau_{2})^{4}\zeta_{5}]$
$(4,2)$ | $[\zeta^{3}(\tau_{1}\tau_{2})^{3},~{}(\tau_{1}\tau_{2})^{3}\zeta_{5}]$
$(3,\frac{3}{2})$ | $[\zeta^{3}(\tau_{1}\tau_{2})^{4},~{}(\tau_{1}\tau_{2})^{2}\zeta_{5}]$
$(\frac{8}{3},\frac{4}{3})$ | $[\zeta^{3}(\tau_{1}\tau_{2}\tau_{1})^{3},~{}\tau_{2}(\tau_{1}\tau_{2})\zeta_{5}]$
$(\frac{12}{5},\frac{6}{5})$ | $[\zeta^{3}(\tau_{1}\tau_{2})^{5},~{}(\tau_{1}\tau_{2})\zeta_{5}]$
Table 9: Here $\zeta_{5}=1_{32}2_{43}3_{54}$, and one use the factorization of
$\zeta^{3}=\sigma\cdot 1_{32}\cdot 2_{43}\cdot 3_{54}$ so that the type would
be $(18,1)$. We used the fact $\sigma=(\tau_{1}\tau_{2})^{6}$.

##  4 Representation of mapping class group of genus two

To have a complete understanding of the mapping class group elements for rank
two theory, one need to have several useful representation for $M_{2}$. In
this section, we are going to discuss three important representation: a): the
action on homology groups; b): Jones representation; c): the signature
function. We then use these representations to discuss the candidate
configuration for 4d asymptotical free theories, 5d and 6d KK theories.

###  4.1 Symplectic representation

The Dehn twist action on homology represented by oriented curves is given as:

$(T_{b})([\vec{a}])=[\vec{a}]+(\vec{a},\vec{b})[\vec{b}].$

Here $(\vec{a},\vec{b}))$ is the intersection number, see figure. 16 for the
illustration.

$\vec{a}$$\vec{b}$$T_{a}\ \vec{b}$$\vec{a}\ +\vec{b}$ Figure 16: The action of
Dehn twist on oriented curve.

For genus two, the basis for homology group is $a_{1},b_{1},a_{2},b_{2}$ and
the intersection form is
$(a_{1},b_{1})=1,(a_{2},b_{2})=1,~{}\delta_{3}=a_{1}+b_{2}$, see figure. 17.
we have the matrix representation for those generators $\tau_{i}$:

$\displaystyle\tau_{1}=\left(\begin{array}[]{cccc}1&-1&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\
0&0&0&1\end{array}\right),~{}\tau_{2}=\left(\begin{array}[]{cccc}1&0&0&0\\\
1&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right),\tau_{3}=\left(\begin{array}[]{cccc}1&-1&1&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&-1&1&1\end{array}\right),$
$\displaystyle\tau_{4}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&-1\\\
0&0&0&1\end{array}\right),~{}\tau_{5}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&1&1\end{array}\right).$

The representation matrix for some mapping class groups are then:

$\displaystyle I=\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&-1&0&0\\\
0&0&-1&0\\\
0&0&0&-1\end{array}\right),~{}\zeta=\left(\begin{array}[]{cccc}0&-1&0&0\\\
1&0&0&-1\\\ 0&0&0&-1\\\
0&-1&1&0\end{array}\right),\phi_{4}=\left(\begin{array}[]{cccc}0&-1&0&0\\\
1&1&0&0\\\ 0&0&0&1\\\ 0&0&-1&1\end{array}\right),$
$\displaystyle\epsilon=\left(\begin{array}[]{cccc}-1&-1&-1&1\\\ 1&0&1&-1\\\
0&0&1&-1\\\
0&-1&1&0\end{array}\right),~{}\eta=\left(\begin{array}[]{cccc}0&-1&0&0\\\
1&0&1&-1\\\ 0&0&1&-1\\\ 0&-1&1&0\end{array}\right).$

$a_{2}$$b_{2}$$b_{1}$$a_{1}$$a_{1}+b_{2}$ Figure 17: The oriented curves on
genus two Riemann surface.

Given an element $\alpha$ if mapping class group $M_{2}$, the characteristic
polynomial of its symplectic representation $Sp(\alpha)$ is

$Det(yI_{4}-Sp(\alpha))=y^{4}+i_{1}(\alpha)y^{3}+i_{2}(\alpha)y^{2}+i_{1}(\alpha)y+1.$

###  4.2 Jones representation

Jones jones1987hecke  gave another representation for $M_{2}$ which is given
by $M_{5\times 5}[q]$, namely the representation is $5\times 5$ matrix with
the entries in polynomial of $q$, and the explicit form for the generators are
shown here:

$\displaystyle\tau_{1}=\left(\begin{array}[]{ccccc}-1&0&0&0&q\\\ 0&-1&1&0&0\\\
0&0&q&0&0\\\ 0&0&1&-1&0\\\
0&0&0&0&q\end{array}\right),~{}\tau_{2}=\left(\begin{array}[]{ccccc}-1&0&0&0&q\\\
0&-1&1&0&0\\\ 0&0&q&0&0\\\ 0&0&1&-1&0\\\
0&0&0&0&q\end{array}\right),~{}\tau_{3}=\left(\begin{array}[]{ccccc}-1&0&0&q&0\\\
0&-1&1&0&0\\\ 0&0&0&q&0\\\ 0&0&1&-1&0\\\ 0&0&1&0&-1\end{array}\right)$
$\displaystyle\tau_{4}=\left(\begin{array}[]{ccccc}q&0&0&0&0\\\ 1&-1&0&0&0\\\
0&0&-1&0&q\\\ 1&0&0&-1&0\\\
0&0&0&0&q\end{array}\right),~{}\tau_{5}=\left(\begin{array}[]{ccccc}-1&q&0&0&0\\\
0&q&0&0&0\\\ 0&0&q&0&0\\\ 0&0&1&-1&0\\\ 0&0&1&0&-1\end{array}\right).$

Unlike homology representation where different mapping class group would give
the same monodromy group, Jones’s presentation is more unique.

###  4.3 Signature funcation

One can define a Meyer’s function endo2000meyer  for a mapping class group as
follows. First, given two element $\alpha,\beta\in M_{2}$, we have two
$4\times 4$ matrices $A=Sp(\alpha)$ and $B=Sp(\beta)$ by using the homology
representation. Now define a real vector space by following equation

$V_{A,B}=\\{(x,y)\in R^{4}\times R^{4}|(I_{4}-A^{-1})x+(I_{4}-B)y=0\\};$

The above equation defines a linear subspace inside $R^{4}\times R^{4}$, since
one can solve some components of $x$ and $y$ coordinates in terms of other
components linearly. The dimension actually depends on the specific form of
$A$ and $B$. Then define the following quadratic form on $V_{A,B}$ as follows

$\psi_{A,B}((x_{1},y_{1}),(x_{2},y_{2}))=(x_{1}+y_{1})^{t}J(I_{4}-B)y_{2};$

Here $J=\left(\begin{array}[]{cc}0&I_{2}\\\ -I_{2}&0\end{array}\right)$. One
then define a cocycle as follows

$\tau_{2}(\alpha,\beta):=sgn\psi_{A,B};$ (19)

Here $sgn$ counts the difference of the positive and negative eigenvalues of
the quadratic form $\psi_{A,B}$.

For a word, one can define a Meyer’s function for it as follows: First for the
generators: $\phi_{2}(\tau_{i})=\frac{3}{5}$, and secondly for a word
$(\tau_{1}\tau_{2}\ldots\tau_{r})$, its Meyer’s function is

$\phi_{2}(\tau_{1}\tau_{2}\ldots\tau_{r})=\frac{3}{5}r-\sum_{j=1}^{r-1}\tau_{2}(\tau_{1}\ldots\tau_{j},\tau_{j+1});$

Here $\phi_{2}$ is the function defined in 19. In particular, the Meyer’s
function for $I_{1}$ and $\tilde{I}_{1}$ singularity are
$\phi_{2}(I_{1})=\frac{3}{5},~{}~{}\phi_{2}(\tilde{I}_{1})=\frac{4}{5}$. The
Meyer’s function satisfies the equation:

$\displaystyle\phi_{2}(1)=0,$
$\displaystyle\phi_{2}(\alpha^{-1})=-\phi_{2}(\alpha),$
$\displaystyle\phi_{2}(\beta\alpha\beta^{-1})=\phi_{2}(\alpha).$

In particular, the value of the signature depends only on the conjugacy class.
The Meyer’s function also satisfies the equation

$\phi_{2}(B)-\phi_{2}(AB)+\phi_{2}(A)=\tau_{2}(A,B).$

One can then define a signature function for a singular fiber $F$ as follows:

$\sigma(F)=-\phi_{2}(F)+sgn(f^{-1}(D)),$

and $D$ is the small disk around the singularity. On the other hand, the local
signature can also be computed using the local invariant $d_{x},\delta_{x}$ as
follows:

$\sigma(F)=\frac{2}{5}d_{x}-\delta_{x}=-\frac{3}{5}(2\delta_{x}-d_{x})-\frac{1}{5}(d_{x}-\delta_{x}).$
(20)

So one can compute the signature for a local singularity from local
invariants. In particular, The signature value for the $I_{1}$ and
$\tilde{I}_{1}$ singularity is

$\sigma(I_{1})=-\frac{3}{5},~{}~{}\sigma(\tilde{I}_{1})=-\frac{1}{5}.$

The signature function is conserved under the deformation of a degeneration.
if a degeneration $F$ is split into several singularities $F_{i}$, then the
signatures of the corresponding mapping class group satisfies the important
condition matsumoto2011pseudo :

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

The above fact is useful to find the number of terms in the factorization of
the mapping class group element (see formula 20):

$\displaystyle\\#I_{1}=2\delta_{x}-d_{x},$
$\displaystyle\\#\tilde{I}_{1}=d_{x}-\delta_{x}.$ (21)

which is derived by the conservation of the signature function.

###  4.4 Asymptotical free theory

To find the configuration for asymptotical free theory, one might use the
method similar to rank one theory: namely one start with the conformal theory
and then move the bulk $I_{1}$ singularities to the fiber at $\infty$.

$SU(3)$ gauge theory: The factorization for $SU(3)$ with $N_{f}=6$ fundamental
flavor is

$(I_{bulk},I_{\infty})=((\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1})^{2},(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{2}).$

To get the configuration for $SU(3)$ gauge theory with $N_{f}<6$, we’d like to
move some letter from the bulk to $\infty$. One of the constraint is that the
characteristic polynomial of the corresponding monodromy group at $\infty$ is

$y^{4}+2x^{3}+3x^{2}+2x+1=0.$

One possible choice is to move the letter $\tau_{1},\tau_{3},\tau_{5}$ for the
conformal case to $\infty$ and the end result is

$(\tau_{5(4)}\tau_{3(2)}\tau_{35^{2}(4)}\tau_{13^{2}(2)},\tau_{3}\tau_{5}\tau_{1}\tau_{3}\tau_{5}\tau_{1}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{2}).$
(22)

We’d like to conjecture that the SW geometry for the SU(3) with $N_{f}=k$
fundamental flavor is just

$(\tau_{5(4)}\tau_{3(2)}\tau_{35^{2}(4)}\tau_{13^{2}(2)}I_{k},I_{6-k}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5})^{2}).$

Here $I_{k}$ involves $k$ letter of $1,3,5$. Since $1,3,5$ commute with each
other, the ordering in $I_{k}$ is not important. An interesting check is that
when $k=0$, the BPS quiver defined from the BPS particle associated with the
vanishing cycle is given in figure. 18, which is exactly the same as found
earlier. We have bulk word $\tau_{a}\tau_{b}\tau_{c}\tau_{d}$, and the
symplectic pairing as

$(a,b)=-1,~{}~{}(a,c)=2,~{}~{}(a,d)=-2,~{}~{}(b,c)=0,~{}~{}(b,d)=2,~{}~{}(c,d)=-1.$

Here

$\displaystyle a=5(4)=[4]+[5],$ $\displaystyle b=3(2)=[2]-[3]=[2]-[1]-[5],$
$\displaystyle c=35^{2}(4)=[4]+[3]+2[5]=[4]+[1]+3[5],$ $\displaystyle
d=13^{2}(2)=[2]-[1]-2[3]=[2]-3[1]-2[5].$ (23)

We used the relation $[3]=[1]+[5]$. The basis for BPS quiver is $(-a,-b,c,d)$,
which gives the BPS quiver.

$a$$d$$c$$b$ Figure 18: The BPS quiver for the pure $SU(3)$ theory. The
charges are $(-a,-b,c,d)$, see 23 for the charge vectors.

$SU(2)\times SU(2)$ gauge theory: The factorization for the conformal theory
is

$(I_{bulk},I_{\infty})=(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1},\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}).$
(24)

This time one should move the letter $\tau_{2},\tau_{4}$ of the bulk letter to
$\infty$, and there are just four possibilities, which agree with the field
theory expectation. The above choice might be understood from the cut system
for the corresponding mapping class group element.

###  4.5 5d and 6d KK theory

Let’s now discuss the global SW geometry for 5d $\mathcal{N}=1$ KK theory. The
main difficulty here is to determine the singular fiber at $\infty$. There are
several clues about the monodromy group that would be useful: firstly the
eigenvalues of the monodromy group at $\infty$ for many 5d theory engineered
using toric singularity are $(1,1,-1,-1)$ 444More generally, one pair of
eigenvalues can be one, and the other pair is $(\exp(2\pi
i/3),\exp(4\pi/3))$.; secondly the monodromy group is not periodic; thirdly
the number of Dehn twist is determined by the local invariants
$d_{x},\delta_{x}$.

We’d like to find the SW geometry for 5d theory whose flavor symmetry is
$SO(20)$ Jefferson:2018irk ; Apruzzi:2019opn , and other cases should be found
in a similar way. Our basic idea is following: one can find the UV singular
fiber of the 5d theory by using the singular fiber of the corresponding 4d
theory with same flavor symmetry, namely one need to move a $I_{1}$ factor of
the 4d UV fiber to the bulk. The reason is following: the dimension of the
charge lattice of the 5d theory is one bigger than that of the 4d theory, as
the BPS particle of the KK theory could carry the winding mode charge. Now the
$I_{\infty}$ for the 4d theory with $SO(20)$ theory is
$\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4}$, and by moving $\tau_{4}$ to the bulk,
we get the word:

$\tau_{1}\tau_{1}\tau_{2}\tau_{3},$

which gives the required eigenvalues: $(1,1,-(-1)^{1/3},(-1)^{2/3})$, and
certainly this is not a periodic map. The global SW geometry is then

$(I_{\infty},I_{bulk})=(\tau_{1}\tau_{1}\tau_{2}\tau_{3},~{}\tau_{4}\tau_{1}^{2}\tau_{2}\tau_{3}\tau_{4}(\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2})^{2}).$

For the fiber at $\infty$ for 6d $(1,0)$ KK theory, one of the possible choice
is the so-called $I_{n-p-q}$ singularity: the eigenvalues of the monodromy
group is $(1,1,1,1)$. In the MM’s theory, the cut curves are shown in figure.
19. The periodic map on two genus zero component is just trivial. Since the
mapping group action associated with each cut curve is a Dehn twist, it is
natural to identify the factorization as

$\tau_{1}^{n}\tau_{3}^{p}\tau_{5}^{q}.$

When $n=p=q=1$, this should give the $F_{\infty}$ so that the theory has
$SO(20)$ flavor symmetry. The global SW geometry $I_{1-1-1}$ case can be found
as follows:

$\displaystyle
1=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}=$
$\displaystyle\tau_{1(2)}\tau_{1}\tau_{3}\tau_{4}\tau_{5}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}=$
$\displaystyle\tau_{1(2)}\tau_{3(4)}(\tau_{1}\tau_{3}\tau_{5})\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1}=$
$\displaystyle(\tau_{1}\tau_{3}\tau_{5})\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1})\tau_{1(2)}\tau_{3(4)}.$

In the first two steps, we used Hurwitz move to move the letter $1,3,5$
together, and finally we use cyclic equivalence. So finally, we have

$(I_{\infty},I_{bulk})=(\tau_{1}\tau_{3}\tau_{5},~{}\tau_{5}\tau_{4}\tau_{3}\tau_{2}\tau_{1}(\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{5}^{2}\tau_{4}\tau_{3}\tau_{2}\tau_{1})\tau_{1(2)}\tau_{3(4)}).$

Alternatively, one can move the one letter of UV fiber of 5d KK theory with
flavor symmetry $SO(20)$ to the bulk, as the charge lattice of 6d theory is
one dimensional higher, since there is one more winding charge for 6d KK
theory. To satisfy the eigenvalue condition, the choice is
$\tau_{1}^{2}\tau_{3}$, which might be conjugate to the simple choice
$\tau_{1}\tau_{3}\tau_{5}$.

Figure 19: The cutting system for the $I_{n-p-q}$ singularity.

##  5 Conclusion

We studies the global SW geometry for rank two theories with eight
supercharges by using a topological approach. The main task is to find a
factorization of a mapping class group element for the degeneration in terms
of positive Dehn twist (up to conjugation, braid move, and Hurwitz move), and
a factorization of identify element. We achieved this for most local
singularities, and find the global SW geometry for most 4d theories with
generic deformations. The rank one case studied in Argyres:2015ffa  can now
easily recovered using the topological approach, while the rank two case seems
significantly more complicated, although it is still manageable.

We take a topological approach in the classification, and one need to find the
SW geometries to solve the Coulomb branch solution, i.e. the photon couplings.
The genus two fiberations are all hyperelliptic which makes the study easier
and the study of those curves are under progress. The algebraic curves and the
choice of SW differential can give us further constraints (for example, the
theory with scaling dimension $(10,8)$ seem not exist although topologically
we do not find any obstruction, and one may explain it using holomorphic
constraint, i.e. no such hyperelliptic family can be written down Xie:2023zxn
).

The approach taken in this paper can be straightforwardly generalized to
higher rank theory whose SW geometry can be given by the fiberation of genus
$g$ curves. The reason is that the mapping class group can also be generated
by the Dehn twists. One can easily find the factorization for some familiar
theories (such as $(A_{1},A_{n})$ or $(A_{1},D_{n})$ theories). A thorough
study takes more efforts, and we hope to report the progress in the future.

One of the interesting result in this paper is that the mapping class group
$M_{2}$ is generated by that of the simple $(A_{1},A_{n})$ and $(A_{1},D_{n})$
type SCFTs. This suggests that one may start with simple SCFTs and generate
other SCFTs.

In this paper, we studied the theory with generic deformation (with only
$I_{1}$ or $\tilde{I}_{1}$ singularities). One can find more general theories
by allowing so-called un-deformable singularities. We have already discussed
how to form those un-deformable singularities by moving the letters of the
Dehn twist around, and a thorough understanding needs the study of the
automorphism of the genus two fiberation, which will be left for part III of
the series.

## References

  * (1) P. Argyres, M. Lotito, Y. Lü, and M. Martone, Geometric constraints on the space of $\mathcal{N}$ = 2 SCFTs. Part I: physical constraints on relevant deformations, JHEP 02 (2018) 001, [arXiv:1505.0481].
  * (2) P. C. Argyres and M. Martone, The rank 2 classification problem I: scale invariant geometries, arXiv:2209.0924.
  * (3) P. C. Argyres and M. Martone, The rank 2 classification problem II: mapping scale-invariant solutions to SCFTs, arXiv:2209.0991.
  * (4) P. C. Argyres and M. Martone, The rank-2 classification problem III: curves with additional automorphisms, arXiv:2209.1055.
  * (5) D. Xie, On rank two theories with eight supercharges part I: local singularities, arXiv:2212.0247.
  * (6) Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic maps and degeneration of Riemann surfaces, vol. 2030.  Springer Science & Business Media, 2011.
  * (7) N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B426 (1994) 19–52, [hep-th/9407087]. [Erratum: Nucl. Phys.B430,485(1994)].
  * (8) Y. Matsumoto, Lefschetz fibrations of genus two-a topological approach, in Topology and Teichmüller spaces, pp. 123–148.  World Scientific, 1996.
  * (9) B. Farb and D. Margalit, A primer on mapping class groups (pms-49), vol. 41.  Princeton university press, 2011.
  * (10) N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B431 (1994) 484–550, [hep-th/9408099].
  * (11) Y. Namikawa and K. Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta mathematica 9 (1973), no. 2 143–186.
  * (12) D. Xie, Pseudo-periodic map and classification of theories with eight supercharges, arXiv:2304.1366.
  * (13) D. Xie, General Argyres-Douglas Theory, JHEP 1301 (2013) 100, [arXiv:1204.2270].
  * (14) A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152–190, [hep-th/9611230].
  * (15) U. Persson, Configurations of kodaira fibers on rational elliptic surfaces, Mathematische Zeitschrift 205 (1990), no. 1 1–47.
  * (16) D. Xie, Classification of rank one 5d $\mathcal{N}=1$ and 6d $(1,0)$ SCFTs, arXiv:2210.1732.
  * (17) S. Hirose, Presentations of periodic maps on oriented closed surfaces of genera up to 4, Osaka J. Math 47 (2010) 385–421.
  * (18) G. Nakamura and T. Nakanishi, Generation of finite subgroups of the mapping class group of genus 2 surface by dehn twists, Journal of Pure and Applied Algebra 222 (2018), no. 11 3585–3594.
  * (19) N. K. Dhanwani, A. K. Nair, and K. Rajeevsarathy, Factoring periodic maps into dehn twists, Journal of Pure and Applied Algebra 227 (2023), no. 1 107159.
  * (20) S. Sakallı, V. Horn-Morris, et al., Singular fibers in algebraic fibrations of genus 2 and their monodromy factorizations, arXiv preprint arXiv:2303.01554 (2023).
  * (21) D. Xie and Z. Yu, Hyperelliptic families and 4d $\mathcal{N}=2$ SCFT, arXiv:2310.0279.
  * (22) D. Xie, Discrete gauging and automorphism of Seiberg-Witten geometry, to appear, .
  * (23) V. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987) 335–388.
  * (24) H. Endo, Meyer’s signature cocycle and hyperelliptic fibrations, Mathematische Annalen 316 (2000), no. 2 237–257.
  * (25) P. Jefferson, S. Katz, H.-C. Kim, and C. Vafa, On Geometric Classification of 5d SCFTs, JHEP 04 (2018) 103, [arXiv:1801.0403].
  * (26) F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki, and Y.-N. Wang, Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States, JHEP 11 (2019) 068, [arXiv:1907.0540].