aainstitutetext: Department of Mathematics, Tsinghua University, Beijing,
100084, Chinabbinstitutetext: Qiuzhen College, Tsinghua University, Beijing,
100084, China

# Hyperelliptic families and 4d $\mathcal{N}=2$ SCFT

Dan Xie b , Zekai Yu

###### Abstract

We classify four dimensional $\mathcal{N}=2$ SCFTs whose Seiberg-Witten (SW)
geometries can be written as hyperelliptic families. By using special Kähler
condition of SW geometry, we reduce the problem to one parameter quasi-
homogeneous hyperelliptic families $y^{2}=f(x,t)$. The classification is given
by further demanding that the complex algebraic surface defined by
$y^{2}=f(x,t)$ has an isolated singularity. We then write down the full SW
geometry by looking at mini-versal deformations of the one parameter family,
and the SW differential is also written down. The detailed physical data for
these theories are found by matching the theory with other known construction.
Our solutions recover the known rank one and rank two results, and give some
infinite sequences valid at arbitrary ranks. We also studied $Z_{2}$ quotient
of above hyperelliptic families which give rise to $B$ type and $D$ type
conformal gauge theory, and further generalizations.

##  1 Introduction

The Coulomb branch solution of a general $\mathcal{N}=2$ theory can be
described by finding a family of algebraic varieties
$F(z_{i},u_{i},\lambda_{i},m_{i})=0$ fibered over the space of generalized
Coulomb branch parameters involving expectation value $u_{i}$ of Coulomb
branch operators, mass parameters $m_{i}$, and coupling constants
$\lambda_{i}$ (including relevant couplings and exactly marginal
deformations). One also need to find a SW differential $\lambda_{sw}$ to
describe the central charge of BPS particles. The combination of varieties $F$
and $\lambda_{sw}$ are called SW geometry, see Seiberg:1994rs ; Seiberg:1994aj
for the SW curves 111The SW geometries for some 4d $\mathcal{N}=2$ theories
must be described by a high dimensional algebraic varieties. of $SU(2)$ gauge
theory.

The SW geometry is usually found through string/M theory-motivated methods
Witten:1997sc ; gaiotton2  and the connection of SW solution to integrable
system DonagiWitten ; Martinec:1995by . Most SW geometries were discovered by
following two methods: a) The spectral curve of an integral system, and the
most well studied ones are the so-called Hitchin system Hitchin:1987mz ;
DonagiWitten ; Nanopoulos:2009uw ; Xie:2012hs ; Wang:2015mra ; Wang:2018gvb ;
b) The mini-versal deformation of an isolated three dimensional canonical
singularities xie20154d ; Wang:2016yha .

While a large number of SW geometries can be found using the methods above,
one should not forget that the resulting geometries are often written in a
complicated way which makes it hard to perform a detailed study of the low
energy physics. This is in particular true for some SW geometries engineered
using Hitchin system. Often an equivalent but simpler SW geometry would make
the physical exploration much more viable.

Holding this in mind, here we try to classify SW geometry which can be
expressed in terms of the simplest algebraic curves, i.e. the hyperelliptic
curves:

$y^{2}=f(x,u_{i},\lambda_{i},m_{i});$

Here $f$ is a degree $2g+1$ or $2g+2$ polynomial in $x$, and
$u_{i},i=1,\ldots,g$ denotes the expectation value of Coulomb branch
operators. We focus on the SW geometry for the SCFT, and perform the
classification through the following steps:

  1. 1.

In the first step, we turn off the mass parameters and coupling constants, so
that the geometry $y^{2}=f(x,u_{i})$ have a $\mathbb{C}^{*}$ action to reflect
the $U(1)_{R}$ symmetry of a SCFT.

  2. 2.

We then impose the special Kähler condition which implies that the derivative
of SW differential $\lambda_{sw}$ should be in the space of holormorphic
differential of the hyperelliptic curve Xie:2021hxd .

${\partial\lambda_{sw}\over\partial u_{i}}\in H^{1,0}.$

The simplest solution would indicate that the SW curve and SW differential are
written in the form 5 and 6.

  3. 3.

The above two considerations reduce the classification into finding out
possible one-parameter quasi-homogeneous families $y^{2}=f(x,t)$. We impose a
further condition, namely the surface singularity defined by the equation is
isolated, and so a complete classification of one-parameter geometries is
possible, see 8 for the resulting 8 types of geometries.

  4. 4.

We then find the full SW geometry by looking at the versal deformations of one
parameter geometry, and then replace the parameter $t$ by a polynomial in $x$
which involves VEV of Coulomb branch operators and possibly other coupling
constants.

The detailed studies of those theories are listed in table.
[1,2,3,4,5,6,7,8,9]. Our list recovers the rank one Argyres:1995xn ;
Minahan:1996cj  and rank two results Argyres:20221 , and generalizes them to
arbitrary rank.

Two further generalizations are studied: first we study the $\mathbb{Z}_{2}$
quotient of the hyperelliptic families and get three classes of theories, see
Table. 10 and 11. These can be thought of as the arbitrary rank-generalization
of the rank one $I_{4}$ series studied in Argyres:2015gha , and include the
$B$ type conformal gauge theory. We then also study the case where the square
root of the holomorphic differential is used, which would describe $D$ type
conformal gauge theory, and generalizations.

This paper is organized as follows: section 2 gave a detailed illustration of
our classification strategy and also the detailed physical properties of those
theories, by matching them with known constructions; $\mathbb{Z}_{2}$
quotients of the curves are studied in section 3; section 4 discusses the SW
differential and the $D$ type gauge theory; A conclusion is given in section
5.

##  2 Hyperelliptic families and SCFTs

First we assume that the SW curve of a 4d $\mathcal{N}=2$ SCFT is given by a
family of genus $g$ hyperelliptic curves:

$y^{2}=f(x,u_{i})$ (1)

where $f(x,u_{i})$ is a polynomial in $x$ of degree $2g+1$ or $2g+2$. Here
$u_{i},~{}i=1,\ldots,g$ denote the Coulomb branch parameters. To get the SW
geometry of a SCFT, the above curve needs to have a $\mathbb{C}^{*}$ action
acting on $(x,y,u_{i})$ which is the incarnation of the (complexified)
$U(1)_{R}$ symmetry of the SCFT.

The SW differential is given by $\lambda_{sw}=R(x,y,u_{i})dx$ with $R$ a
rational function of $(x,y,u_{i})$, and the derivative of $\lambda_{sw}$ with
respect to Coulomb branch operator is brieskorn2012plane :

${\partial\over\partial
u_{i}}\int_{\lambda}\lambda_{sw}={\partial\over\partial
u_{i}}\int_{\lambda}Rdx=\int({\partial R\over\partial u_{i}}-{\partial
R\over\partial y}({\partial F\over\partial y})^{-1}{\partial F\over\partial
u_{i}})dx;$ (2)

Here $\lambda$ is the cycle on Riemann surface, and $F$ is the equation
$F=y^{2}-f(x,u_{i})=0$. The special Kähler condition requires that Xie:2021hxd
:

${\partial\lambda_{sw}\over\partial u_{i}}\subset H^{1,0}(C);$ (3)

Here $H^{1,0}(C)$ denotes the space of holomorphic differential of the
hyperelliptic curve $C$, which has a canonical basis

$\omega_{i}=\frac{x^{i}dx}{y},~{}~{}i=0,\ldots,g-1.$

Let $t$ denote the Coulomb branch operator with maximal scaling dimension, and
assume that the derivative of SW differential with respect of $t$ is
proportional to $\frac{dx}{y}$, i.e.

${\partial\lambda_{sw}\over\partial t}=({\partial R\over\partial t}-{\partial
R\over\partial y}({\partial F\over\partial y})^{-1}{\partial F\over\partial
t})dx=c{dx\over y};$ (4)

Here $c$ is a nonzero constant. Now if the SW curve takes the following
special form

$\boxed{y^{2}=f(x,u_{g-1}x^{g-1}+u_{g-2}x^{g-2}+\ldots+u_{1}x+t).}$ (5)

and similarly the SW differential takes the form

$\boxed{\lambda_{sw}=R(x,y,u_{g-1}x^{g-1}+u_{g-2}x^{g-2}+\ldots+u_{1}x+t)dx.}$
(6)

Then we have

${\partial\lambda_{sw}\over\partial u_{i}}=({\partial R\over\partial
u_{i}}-{\partial R\over\partial y}({\partial F\over\partial y})^{-1}{\partial
F\over\partial u_{i}})dx=x^{i}{\partial\lambda_{sw}\over\partial
t}=c{x^{i}dx\over y};$ (7)

Here we used the fact ${\partial R\over\partial u_{i}}=x^{i}{\partial
R\over\partial t}$, and ${\partial F\over\partial u_{i}}=x^{i}{\partial
F\over\partial t}$. So the special Kähler condition is satisfied automatically
for the SW curve 5 and SW differential 6, as long as the equation 4 has a
solution (we will discuss this issue in section 4).

One parameter curve: By looking at the curve 5 and 6, one can see that the SW
family is determined by the curve with just one parameter, which corresponds
to the Coulomb branch operator of maximal scaling dimension. The
classification of theories (or SW geometries) is now reduced to the
classification of the one parameter hyperelliptic curve $y^{2}=f(x,t)$ with a
$\mathbb{C}^{*}$ action. We impose a further constraint on the one parameter
family: There is an isolated singularity (of the complex surface defined by
$y^{2}=f(x,t)$) at the origin $x=y=t=0$. Such one parameter hyperelliptic
families can be easily classified as follows:

$\displaystyle I:y^{2}=x^{2g+1}+t^{a},~{}~{}$ $\displaystyle
II:y^{2}=x^{2g+2}+t^{a},~{}~{}$ $\displaystyle
III:y^{2}=x(x^{2g}+t^{a}),~{}~{}$ $\displaystyle
IV:y^{2}=x(x^{2g+1}+t^{a}),~{}~{}$ $\displaystyle
V:y^{2}=t(x^{g+2}+t^{a}),~{}~{}$ $\displaystyle
VI:y^{2}=t(x^{g+3}+t^{a}),~{}~{}$ $\displaystyle
VII:y^{2}=xt(x^{g+1}+t^{a}),~{}~{}$ $\displaystyle
VIII:y^{2}=xt(x^{g+2}+t^{a}).~{}~{}$ (8)

Proof: The isolated quasi-homogeneous singularity of two variables $(x,t)$ is
classified as $x^{a}+t^{b},x^{a}+xt^{b},~{}~{}x^{a}t+y^{b}t$
arnold1998singularity , so one has the above 8 types.

Coulomb branch spectrum: Now one can proceed to the calculation of Coulomb
branch spectrum of the theory. The basic idea is that the SW differential
$\lambda_{sw}$ gives the mass of BPS particle and so it has scaling dimension
one. The equation 7 gives a relation for the scaling dimension

$1-[u_{i}]=i[x]-[y];$

We find that

$[u_{i-1}]-[u_{i}]=[x];$

So if the scaling dimension of $x$ is positive, the maximal scaling dimension
of Coulomb branch operators is $[t]$ and the gap of the Coulomb branch
spectrum is given by $[x]$. The $\mathbb{C}^{*}$ action of the curve
$y^{2}=f(x,t)$ yields two equations relating the scaling dimensions, which
would help us to completely determine the spectrum.

Example: Let’s consider one parameter curve $y^{2}=x^{2g+1}+t^{2}$. We have
the following three equations

$\displaystyle 2[y]=(2g+1)[x],$ $\displaystyle 2[y]=2[t],$ $\displaystyle
1-[t]=[x]-[y].$

and so we find $[t]=\frac{g+1}{2},~{}~{}[x]=1$, so the Coulomb branch spectrum
is $\frac{3}{2},\frac{5}{2},\ldots,\frac{2g+1}{2}$.

Full SW curve: One can turn on exactly marginal deformations, relevant
deformations, and mass deformations without lifting the Coulomb branch. These
are encoded in the deformations of above SW curves. They can be derived as
follows:

  1. 1.

To get the SW curve with general Coulomb branch deformations, we replace the
parameter $t$ in the one parameter curve by a degree $g-1$ polynomial in $x$.
More generally, substitution by a polynomial in higher degree in $x$ is
possible, as long as the scaling dimensions of the parameters that come with
the higher order terms are non-negative:

$t\to\ldots+\lambda_{2}x^{g+1}+\lambda_{1}x^{g}+u_{g-1}x^{g-1}+\ldots+t.$

The parameters $\lambda_{1},\lambda_{2},\ldots$ includes mass parameters,
relevant deformations, and exactly marginal deformations. Extra care must be
taken to ensure that the above replacement do not affect the genus of the
curve.

  2. 2.

Mass deformations can be found by finding the mini-versal deformation of the
one parameter curve. The mini-versal deformation of an isolated singularity of
our type is defined as follows

$y^{2}=f(x,t)+\sum_{\alpha}\lambda_{\alpha}\phi_{\alpha};$ (9)

Here $\phi_{\alpha}$ is the monomial basis of the Jacobi algebra defined as
follows

$J_{f}={C[x,t]\over\\{{\partial f\over\partial x},{\partial f\over\partial
t}\\}}.$

The scaling dimension of $\lambda_{\alpha}$ could be larger than or equal to
one. Parameters with scaling dimension integral and larger than one should be
interpreted as the Casimiers of the non-abelian flavor symmetry.

Example: Let’s consider the one parameter curve $y^{2}=x(x^{2g}+t^{2})$, and
the scaling dimensions are $[x]=2,[t]=2g$. The basis of Jacobi algebra $J_{f}$
is

$J_{f}=\\{t,x^{2g},x^{2g-1},\ldots,1\\};$

and so there are a total of $2g+2$ mass parameters. The deformed one parameter
curve is

$y^{2}=x^{2g+1}+xt^{2}+[T_{2}x^{2g}+T_{4}x^{2g-1}+\ldots+T_{4g+2}]+T_{2g+2}^{{}^{\prime}}t.$

Now the replacement of $t$ (with maximal number of deformations) is

$t\to\tau x^{g}+u_{g-1}x^{g-1}+\ldots+t;$

We see that there is an exactly marginal deformation represented by . The full
SW curve is

$\displaystyle y^{2}=x^{2g+1}+x(\tau
x^{g}+u_{g}x^{g-1}+\ldots+t)^{2}+[T_{2}x^{2g}+T_{4}x^{2g-1}+\ldots+T_{4g+2}]$
$\displaystyle+T_{2g+2}^{{}^{\prime}}(\tau x^{g}+u_{g-1}x^{g-1}+\ldots+t).$

and the scaling dimension of the mass deformations
$(T_{2},T_{4},\ldots,T_{4g+2},T_{2g+2}^{{}^{\prime}})$ is
$(2,4,\ldots,4g+2,2g+2)$, so the flavor symmetry is $SO(4g+4)$.

###  2.1 Detailed data of SCFTs: type I-IV

Now we study those SCFTs whose SW curve is given by the form 5. The one
parameter family 8 is $I,II,III,IV$. These families share the property that
the one parameter family already has genus $g$.

We list the Coulomb branch spectrum, the central charge, flavor symmetry, and
the number of $A_{1}$ singularities for the generic deformations. The
corresponding periodic map xie2023pseudoperiodic . Factorization of the
related mapping class group element is also listed (see ishizakapresent ),
which would be important to understand the low energy physics Xie:2023fact .
The $(A_{1},G)$ sequence was first found in Eguchi:1996vu , and the class $S$
construction was first found in gaiotton2 .

Table 1: Type I: $y^{2}=x^{2g+1}+t^{a}$.

Theory | $(A_{1},A_{2g})$ | $D_{2}(SU(2g+1)$ |
---|---|---|---
curve | $y^{2}=x^{2g+1}+t$ | $y^{2}=x^{2g+1}+t^{2}$ | $y^{2}=x^{2g+1}+t^{a},a<\frac{4g+2}{2g-1}$
CB spectrum | $\frac{2g+4}{2g+3},\frac{2g+6}{2g+3},\ldots,\frac{4g+2}{2g+3}$ | $\frac{3}{2},\frac{5}{2},\ldots,\frac{2g+1}{2}$ | $a\leq 5,~{}~{}g=1$
periodic map | $\frac{1}{2(2g+1)}+\frac{1}{2}+\frac{g}{2g+1}$ | $\frac{g}{2g+1}+\frac{g}{2g+1}+\frac{1}{2g+1}$ | $a\leq 3,~{}~{}g=2$
factorization | $\phi_{1}=\tau_{1}\tau_{2}\ldots\tau_{2g}$ | $\phi_{1}^{2}$ | $a\leq 2,~{}~{}g\geq 3$
central charges | $a=\frac{g(24g+19)}{24(2g+3)},c=\frac{g(6g+5)}{6(2g+3)}$ | $a=\frac{7(g^{2}+g)}{24},c=\frac{g^{2}+g}{3}$ |
flavor symmetry | None | $su(2g+1)$ |
# $A_{1}$ singularity | $2g$ | $4g$ |

Full SW curve $a=1$ | $y^{2}=x^{2g+1}+(\lambda_{1}x^{2g-1}+\lambda_{2}x^{2g-2}+\ldots+u_{g-1}x^{g-1}+\ldots+t)$
---|---
Full SW curve $a=2$ | $y^{2}=x^{2g+1}+(T_{2}x^{2g-1}+\ldots+T_{2g+1})+(\lambda_{\frac{1}{2}}x^{g}+u_{g-1}x^{g-1}+\ldots+t)^{2}$
Full SW curve $a=3,g=2$ | $y^{2}=x^{5}+(u_{4}x+t)^{3}+(T_{12}x^{3}+T_{18}x^{2}+T_{24}x+T_{30})$
flavor $e_{8}$ | $+(T_{2}x^{3}+T_{8}x^{2}+T_{14}x+T_{20})(u_{4}x+t)$

Table 2: Type II: $y^{2}=x^{2g+2}+t^{a}$.

Theory | $(A_{1},A_{2g+1})$ | $SU(g+1)-(2g+2)$ |
---|---|---|---
curve | $y^{2}=x^{2g+2}+t$ | $y^{2}=x^{2g+2}+t^{2}$ | $y^{2}=x^{2g+2}+t^{a}$
CB spectrum | $\frac{g+3}{g+2},\ldots,\frac{2g+2}{g+2}$ | $2,3,\ldots,g+1$ | $a<\frac{2g+2}{g}$
periodic map | $\frac{1}{2g+2}+\frac{1}{2g+2}+\frac{g}{g+1}$ | $\frac{1}{g+1}+\frac{1}{g+1}+\frac{g}{g+1}+\frac{g}{g+1}$ | $a\leq 3,g=1$
factorization | $\phi_{2}=\tau_{1}\tau_{2}\ldots\tau_{2g}\tau_{2g}$ | $\phi_{2}^{2}$ | $a\leq 2,g\geq 2$
central charges | $a=\frac{12g^{2}+19g+2}{24(g+2)},c=\frac{3g^{2}+5g+1}{6(g+2)}$ | $a=\frac{7g^{2}+14g+2}{24},c=\frac{2g^{2}+4g+1}{6}$ |
flavor symmetry | $u(1)$ | $su(2g+2)\times u(1)$ |
# $A_{1}$ singularity | $2g+1$ | $4g+2$ |

Full SW curve $a=1$ | $y^{2}=x^{2g+2}+(\lambda_{1}x^{2g}+\lambda_{2}x^{2g-1}+\ldots+mx^{g}+u_{g-1}x^{g-1}+\ldots+t)$
---|---
Full SW curve $a=2$ | $y^{2}=x^{2g+2}+(T_{2}x^{2g}+\ldots+T_{2g+2})+(\tau x^{g+1}+mx^{g}+u_{g-1}x^{g-1}+\ldots+t)^{2}$

Table 3: Type III: $y^{2}=x(x^{2g}+t^{a})$.

Theory | $(A_{1},D_{2g+1})$ | $USp(2g)-(2g+2)$ |
---|---|---|---
curve | $y^{2}=x(x^{2g}+t)$ | $y^{2}=x(x^{2g}+t^{2})$ | $y^{2}=x(x^{2g}+t^{a}),~{}~{}a<\frac{4g}{2g-1}$
CB spectrum | $\frac{2g+2}{2g+1},\ldots,\frac{4g}{2g+1}$ | $2,4,\ldots,2g$ | $a\leq 3,g=1$
periodic map | $\frac{1}{4g}+\frac{2g-1}{4g}+\frac{1}{2}$ | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2g}+\frac{2g-1}{2g}$ | $a\leq 2,g\geq 2$
factorization | $\phi_{3}=\tau_{1}\tau_{2}\ldots\tau_{2g}\tau_{2g+1}$ | $\phi_{3}^{2}$ | $\phi_{3}^{a}$
central charges | $a=\frac{g(8g+3)}{8(2g+1)},c=\frac{g}{2}$ | $a=\frac{14g^{2}+9g}{24},c=\frac{4g^{2}+3g}{6}$ |
flavor symmetry | $su(2)$ | $so(4g+4)$ |
# $A_{1}$ singularity | $2g+1$ | $4g+2$ |

Full SW curve $a=1$ | $y^{2}=x^{2g+1}+x(\lambda_{1}x^{2g-1}+\ldots+u_{g-1}x^{g-1}+\ldots+t)+T_{2}$
---|---
Full SW curve $a=2$ | $y^{2}=x^{2g+1}+x(\tau x^{g}+u_{g-1}x^{g-1}+\ldots+t)^{2}+[T_{2}x^{2g}+T_{4}x^{2g-1}+\ldots+T_{4g+2}]$
| $+T_{2g+2}^{{}^{\prime}}(\tau x^{g}+u_{g-1}x^{g-1}+\ldots+t)$

Table 4: Type IV: $y^{2}=x(x^{2g+1}+t^{a})$.

Theory | $(A_{1},D_{2g+2})$ | Class S |
---|---|---|---
curve | $y^{2}=x(x^{2g+1}+t)$ | $y^{2}=x(x^{2g+1}+t^{2})$ |  $y^{2}=x(x^{2g+1}+t^{a})$, $a<\frac{2g+1}{g}$
CB spectrum | $\frac{g+2}{g+1},\ldots,\frac{2g+1}{g+1}$ | $3,5,\ldots,2g+1$ | $a\leq 2$
periodic map | $\frac{1}{2g+1}+\frac{1}{2g+1}+\frac{2g-1}{2g+1}$ | $\frac{2g}{2g+1}+\frac{g+1}{2g+1}+\frac{g+1}{2g+1}$ |
factorization | $\phi_{4}$ | $\phi_{4}^{2}$ | $\phi_{4}^{a}$
central charges | $a=\frac{g}{2}+\frac{1}{12},c=\frac{g}{2}+\frac{1}{6}$ | $a=\frac{14g^{2}+23g+4}{24},c=\frac{4g^{2}+7g+2}{6}$ |
flavor symmetry | $su(2)\times u(1)$ |  $so(4g+6)\times u(1)(E_{6}$ if $g=1)$ |
# $A_{1}$ singularity | $2g+2$ | $4g+4$ |

Full SW curve $a=1$ | $y^{2}=x^{2g+2}+x(\lambda_{1}x^{2g}+\ldots+mx^{g}+u_{g-1}x^{g-1}+\ldots+t)+T_{2}$
---|---
Full SW curve $a=2$ | $y^{2}=x^{2g+2}+x(mx^{g}+u_{g-1}x^{g-1}+\ldots+t)^{2}+[T_{2}x^{2g+1}+T_{4}x^{2g-1}+\ldots+T_{4g+4}]$
| $+T_{2g+3}^{{}^{\prime}}(mx^{g}+u_{g-1}x^{g-1}+\ldots+t)$

###  2.2 Detailed data of SCFTs: type V-VIII

The one parameter families of those theories have genus less than $g$, so that
substituting $t$ by a degree $g-1$ polynomial in $x$ would give us a genus $g$
hyperelliptic family. The periodic map is for the one parameter curve.

Table 5: Type V: $y^{2}=t(x^{g+2}+t^{a})$.

Theory |  |
---|---|---
curve | $y^{2}=t(x^{g+2}+t)$ |  $y^{2}=t(x^{g+2}+t^{a})$, $a<\frac{g+2}{g}$
CB spectrum | $3,4,\ldots,g+2$ | $a\leq 1$
Class S | $([g,1^{2}],[1^{g+2}],[1^{g+2}])$ |
Periodic map |  $\frac{g+1}{g+2}+\frac{g+1}{g+2}+\frac{2}{g+2}$ $g~{}odd$ |
|  $\frac{g+1}{g+2}+\frac{g+1}{g+2}+\frac{1}{(g+2)/2}$ $g~{}even$ |
central charges | $a=\frac{7g^{2}+28g+6}{24},c=\frac{2g^{2}+8g+3}{6}$ |
flavor |  $su(2g+4)\times su(2)(E_{6}$ if $g=1)$ |
# $A_{1}$ singularities | $4g+4$ |

Full SW curve $a=1$ | $y^{2}=(T_{2}^{\prime}x^{g}+u_{g-1}x^{g-1}+\ldots+t)[x^{g+2}+(T_{2}^{\prime}x^{g}+u_{g-1}x^{g-1}+\ldots+t)]$
---|---
| $+[T_{2}x^{2g+2}+T_{3}x^{2g+1}+\ldots+T_{2g+4}]$

Table 6: Type VI: $y^{2}=t(x^{g+3}+t^{a})$.

Theory |  |
---|---|---
curve | $y^{2}=t(x^{g+3}+t)$ |  $y^{2}=t(x^{g+3}+t^{a})$, $a<\frac{g+3}{g+1}$
CB spectrum | $4,5,\ldots,g+3$ | $a\leq 1$
Class S | $[(g+1),2],[1^{g+3}],[1^{g+3}]$ |
Periodic map |  $\frac{g+2}{g+3}+\frac{g+2}{g+3}+\frac{2}{g+3}$ $g~{}even$ |
|  $\frac{g+2}{g+3}+\frac{g+2}{g+3}+\frac{1}{(g+3)/2}$ $g~{}odd$ |
central charges | $a=\frac{7g^{2}+42g+10}{24},c=\frac{2g^{2}+12g+5}{6}$ |
flavor | $su(2g+6)$ |
# $A_{1}$ singularities | $4g+5$ |

Full SW curve $a=1$ | $y^{2}=(u_{g-1}x^{g-1}+\ldots+t)[x^{g+3}+(u_{g-1}x^{g-1}+\ldots+t)]$
---|---
| $+[T_{2}x^{2g+4}+T_{3}x^{2g-1}+\ldots+T_{2g+6}]$

Table 7: Type VII: $y^{2}=xt(x^{g+1}+t^{a})$.

Theory |  |
---|---|---
curve | $y^{2}=xt(x^{g+1}+t)$ |  $y^{2}=xt(x^{g+1}+t^{a})$, $a<\frac{g+1}{g}$
CB spectrum | $4,6,\ldots,2g+2$ | $a\leq 1$
Class S | $[(g+1)^{2}],[g^{2},1^{2}],[1^{2g+2}]$ |
Periodic map |  $\frac{2g+1}{2g+2}+\frac{g+2}{2g+2}+\frac{1}{2}$ $g~{}odd$ |
|  $\frac{2g+1}{2g+2}+\frac{(g+2)/2}{g+1}+\frac{1}{2}$ $g~{}even$ |
central charges | $a=\frac{14g^{2}+37g+8}{24},c=\frac{4g^{2}+11g+4}{6}$ |
flavor | $so(4g+8)\times su(2)$ |
# $A_{1}$ singularities | $4g+5$ |

Full SW curve $a=1$ | $y^{2}=x(T_{2}^{\prime}x^{g}+u_{g-1}x^{g-1}+\ldots+t)[x^{g+1}+(T_{2}^{\prime}x^{g}+u_{g-1}x^{g-1}+\ldots+t)]$
---|---
|
$+[T_{2}x^{2g+2}+T_{4}x^{2g+1}+\ldots+T_{4g+6}]+T_{2g+4}^{{}^{\prime}}(T_{2}^{\prime}x^{g}+u_{g-1}x^{g-1}+\ldots+t)$

Table 8: Type VIII: $y^{2}=xt(x^{g+2}+t^{a})$.

Theory |  |
---|---|---
curve | $y^{2}=xt(x^{g+2}+t)$ |  $y^{2}=xt(x^{g+2}+t^{a})$, $a<\frac{g+2}{g+1}$
CB spectrum | $6,8,\ldots,2g+4$ | $a\leq 1$
Class S | $[(g+2)^{2}],[(g+1)^{2},2],[1^{2g+4}]$ |
Periodic map |  $\frac{2g+3}{2g+4}+\frac{g+3}{2g+4}+\frac{1}{2}$ $g~{}even$ |
|  $\frac{2g+3}{2g+4}+\frac{(g+3)/2}{g+2}+\frac{1}{2}$ $g~{}odd$ |
central charges | $a=\frac{14g^{2}+65g+16}{24},c=\frac{4g^{2}+19g+8}{6}$ |
flavor | $so(4g+12)$ |
# $A_{1}$ singularities | $4g+6$ |

Full SW curve $a=1$ | $y^{2}=x(u_{g-1}x^{g-1}+\ldots+t)[x^{g+2}+(u_{g-1}x^{g-1}+\ldots+t)]+$
---|---
|
$[T_{2}x^{2g+4}+T_{4}x^{2g+3}+\ldots+T_{4g+10}]+T_{2g+10}^{{}^{\prime}}(u_{g-1}x^{g-1}+\ldots+t)$

Table 9: The case when $[x]=0$.

Theory | $2-\underbrace{SU(2)-\ldots-SU(2)}_{g}-2$ |
---|---|---
curve | $y^{2}=t(x^{g+3}+\tau_{1}x^{g+2}+\ldots+\tau_{g+2})$ |
CB spectrum | $2,\ldots,2$ |
Class S | $[1^{2}]\times(g+3)$ |
central charges | $a=\frac{19g+4}{24},c=\frac{5g+2}{6}$ |
Periodic map | $(\frac{1}{2})^{g+4}$ |
flavor | $su(2)^{g+3}$ |
# $A_{1}$ singularities | $4g+2$ |

Full SW curve | $y^{2}=(u_{g-1}x^{g-1}+\ldots+t)[x^{g+3}+\tau_{1}x^{g+2}+\ldots+\tau_{g+2})]$
---|---

##  3 $\mathbb{Z}_{2}$ quotient

The Coulomb branch solution is given by the fibration of abelian varieties
over the generalized Coulomb branch DonagiWitten . In above studies, the
abelian varieties are given by the Jacobian of a hyperellliptic curves. Some
abelian varities are given by the finite quotient of the Jacobian variety, and
are called Prym variety. Given above hyperelliptic families, we would like to
study the finite quotient to get the SW geometries for other SCFTs.

Let’s consider the curves which admit a $\mathbb{Z}_{2}$ action $x\to-x$, then
the anti-invariant holomorphic differentials are

${x^{2i}dx\over y},~{}i=0,1\ldots,$

So the number would be $\frac{g}{2}$ for even $g$, and $\frac{g+1}{2}$ for $g$
odd. Now the special Kähler condition is

${\partial\lambda_{sw}\over\partial t}=c\frac{dx}{y};$

So $\lambda_{sw}$ is required to be odd under $x\to-x$ if we’d like to
preserve the $t$ in the spectrum. Once we keep $t$ in the spectrum, the
operator $u_{i}$ with $i$ even would be preserved. The Dirac quantization
would put further constraint on the $t$ transformation of the curve (we would
like to discuss this subtle issue elsewhere).

Example: Let’s now consider the curve defined as $y^{2}=x^{2g+2}+t^{2}$, with
$g$ odd. The full SW curve would be

$y^{2}=x^{2g+2}+(\tau
x^{g+1}+u_{g-1}x^{g-1}+u_{g-3}x^{g-3}+\ldots+t)^{2}+(T_{2}x^{2g}+T_{4}x^{2g-2}+\ldots+T_{2g}x^{2}).$

Terms of odd powers of $x$ are eliminated in the last bracket since the curve
is required to be invariant under $x\rightarrow-x$. This curve should describe
the conformal gauge theory $SO(g+2)$ coupled with $2g$ half-hypermultiplets,
so the flavor symmetry is $usp(2g)$. Notice that the constant deformation term
is not allowed (see Danielsson:1995is ).

We search all such theories and list them in table. 10 and 11. Notice the rank
one theories are the $I_{4}$ series found in Argyres:2015gha .

Table 10: $\mathbb{Z}_{2}$ quotient: $y^{2}=x^{2g+2}+t^{2}$.

Theory | $SO(g+2)-2g(half-hyper)$ |
---|---|---
curve | $y^{2}=x^{2g+2}+t^{2},~{}g~{}odd$ | $y^{2}=x^{2g+2}+t^{2},~{}g~{}even$
CB spectrum | $2,4,\ldots,g+1$ | $3,5\ldots,g+1$
central charges | $a=\frac{1}{48}(g+2)(7g+5)$ | $a=\frac{1}{48}(g(7g+19)+2)$
| $c=\frac{1}{12}(g+2)(2g+1)$ | $c=\frac{1}{12}(g(2g+5)+1)$
Singularities | $I_{4}I_{1}^{2g}$ | $I_{4}I_{1}^{2g}$
free hyper on CB | $g$ | $g$
flavor | $usp(2g)$ | $usp(2g)\times u(1)$

Full SW curve $g~{}odd$ | $y^{2}=x^{2g+2}+(\tau x^{g+1}+u_{g-1}x^{g-1}+u_{g-3}x^{g-3}+\ldots+t)^{2}$
---|---
| $+(T_{2}x^{2g}+T_{4}x^{2g-2}+\ldots+T_{2g}x^{2})$
Full SW curve $g~{}even$ | $y^{2}=x^{2g+2}+(mx^{g}+u_{g-2}x^{g-2}+u_{g-4}x^{g-4}+\ldots+t)^{2}$
| $+(T_{2}x^{2g}+T_{4}x^{2g-2}+\ldots+T_{2g}x^{2})$

Table 11: $\mathbb{Z}_{2}$ quotient: $y^{2}=t(x^{g+2}+t)$. Theory |
---|---
curve | $y^{2}=t(x^{g+2}+t),~{}g~{}even$
CB spectrum | $4,6,\ldots,g+2$
central charges | $a=\frac{1}{48}g(7g+36)$
| $c=\frac{1}{24}g(4g+21)$
Singularities | $I_{4}I_{1}^{2g+1}$
free hyper | $\frac{7g}{2}-4$
flavor | $usp(2g+2)\times su(2)$
Full SW curve $g~{}even$ | $y^{2}=(T_{2}^{\prime}x^{g}+u_{g-2}x^{g-2}+\ldots+t)[x^{g+2}+(T_{2}^{\prime}x^{g}+u_{g-2}x^{g-2}+\ldots+t)]$
| $+[T_{2}x^{2g+2}+T_{4}x^{2g}+\ldots+T_{2g+2}x^{2}]$

##  4 SW differential and other possibilities

In our previous studies, we have assumed that the derivative of $\lambda_{sw}$
with respect to $t$ is given as ${\partial\lambda_{sw}\over\partial
t}=c\frac{dx}{y}$. We would like to solve $\lambda_{sw}$ explicitly. Let’s
assume the SW curve and SW differential take the following form

$y^{2}=f(x,P(x,u_{i})),~{}~{}\lambda_{sw}=R(x,y,\sum_{i}P(x,u_{i}));$

and the polynomial which contains the Coulomb branch operators could take the
following general form

$P(x,u_{i})=\sum_{i}u_{i}x^{i}.$

Now let’s consider the constraints on SW differential
$\lambda_{sw}=R(P,x,y)\frac{dx}{y}$, and the derivative reads

${\partial\lambda_{sw}\over\partial u_{i}}=[\frac{\partial R}{\partial
u_{i}}+\frac{R}{y^{2}}{\partial f\over\partial P}x^{i}]\frac{dx}{y}.$ (10)

We’d like to write it in the form of ${\partial\lambda_{sw}\over\partial
u_{i}}=a{x^{i}dx\over y}+bd({x^{i+1}\over y})$. Using

$d(\frac{x^{i+1}}{y})={(i+1)x^{i}dx\over y}-\frac{1}{y^{2}}x^{i+1}({\partial
f\over\partial P}P^{{}^{\prime}}+{\partial f\over\partial x})\frac{dx}{y}.$
(11)

So if we’d like to derivative is given by a holomorphic differential, one has

$\boxed{-x({\partial f\over\partial P}P^{{}^{\prime}}+{\partial f\over\partial
x})=R{\partial f\over\partial P}+cf.}$ (12)

Here $c$ is a constant. The final result would be

${\partial\lambda_{sw}\over\partial u_{i}}={\partial R\over\partial
u_{i}}dx+[c-(i+1)]{x^{i}dx\over y}+d({x^{i+1}\over y}).$

and the constant in equation 12 is $c=-k$.

Example 1: If $f$ is linear in $P$ (Type I and type II), the solution of SW
differential is simple: $R=y^{2}$ an $\lambda_{sw}=ydx$. A total derivative
term is not needed. Similarly for type III and type IV, one may need to take
$R=\frac{y^{2}}{x}$, and $\lambda_{sw}=y\frac{dx}{x}$.

Example 2: If $f$ takes the form $f=P^{2}+x^{k}$, the solution would be

$R=\frac{k}{2}P-xP^{{}^{\prime}}.$

There exists an interesting possibility that involves the $D_{n}$ type
conformal gauge theory. In this case, the polynomial encoding the Coulomb
branch operator is written as the form

$P(x,u_{i})=\sum u_{i}x^{i}+t^{2}.$

and the derivative of $\lambda_{sw}$ with respect to $t$ is now proportional
the ”square root” of the differential $\frac{dx}{y}$.

Example: The curve is $y^{2}=f(x,t^{2})=x^{4g}+(t^{2})^{2}$, We impose a
$\mathbb{Z}_{2}$ invariance $x\to-x$, and so the full SW curve is written as

$y^{2}=x^{4g}+(\tau
x^{2g}+u_{2}x^{2g-2}+u_{2g-2}x^{2}+t^{2})^{2}+T_{2}x^{4g-2}+\ldots+T_{4g-4}x^{4}.$

This gives the SW curve of $SO(2g)$ coupled with $2g-2$ fundamental
hypermultiplets. Notice that the mass deformations from $x^{2}$ term and
constant are not allowed Brandhuber:1995zp . We might generalized above
construction to the following curve ($g$ is odd)

$y^{2}=x^{2g}+(mx^{g-1}+\ldots+u_{g-2}x^{2}+t^{2})^{2}+T_{2}x^{2g-2}+\ldots+T_{2g-4}x^{4}.$

The Coulomb branch spectrum is $(3,5,\ldots,g-2,\frac{g}{2})$, and the flavor
symmetry is $usp(2g-4)\times U(1)$. When $g=7$, a theory with same Coulomb
branch spectrum and flavor symmetry was indeed found in Li:2022njl .

##  5 Conclusion

We classify 4d $\mathcal{N}=2$ SCFTs whose SW geometry can be written as
hyperelliptic family. The classification is made possible by assuming 5 and 6,
and reduce the problem to one parameter family. It would be interesting to try
to find other solutions to the special Kähler conditions.

Once the full SW family is given, one can further study the low energy physics
of any vacuum Xie:2021hxd  by using the period integral. Geometrically, one
can also compute the flavor symmetry by computing the Mordell-Weil lattice
shioda1999mordell . One might also compute the torsion part of Mordell-Weil
lattice which could be identified as higher form symmetry.

We focus on the SCFT in this paper, and the method can be applied to non
conformal theory. The SW geometry can still be written as the form 5 and 6,
and the only difference is that there is no $\mathbb{C}^{*}$ action acting on
the one parameter family.

Another interesting generalization is to consider other algebraic curves such
as order three algebraic curves $x^{3}=f(y,u_{i})$. We hope to report the
progress in this direction in the near future.

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