aainstitutetext: Yau Mathematics Sciences center, Tsinghua University,
Beijing, 100084, Chinabbinstitutetext: Department of Mathematics, Tsinghua
University, Beijing, 100084, China

# Mixed Hodge structure and $\mathcal{N}=2$ Coulomb branch solution

Dan Xie a Dingxin Zhang

###### Abstract

The Coulomb branch of four dimensional $\mathcal{N}=2$ theories can be solved
by finding a Seiberg–Witten (SW) geometry and a SW differential. While lots of
SW geometries are found, the extraction of low energy theory out of it is
limited due to following reasons: (a) the difficulty of distinguishing
electric-magnetic and flavor charges; (b) the difficulty of determining the
low energy theory at singular point, (c) the lack of SW differential. We show
that the mixed Hodge structure (MHS) can be used to fully solve the low energy
physics of Coulomb branch at every vacuum. The MHS can be used to solve above
three problems as follows: (a) The smooth fiber of SW geometry carries a Mixed
Hodge Structure structure with two weights: one weight describes electric-
magnetic charge and the other for the flavor charge; (b) for the singular
fiber, there are three MHS which can used to determine the IR theory. (c) the
MHS at $\infty$ is used to find the SW differential.

##  1 Introduction

The low energy effective theory at the Coulomb branch111It is called Coulomb
branch because the low energy theory of generic vacua on this branch is
described by abelian gauge theory. of a four dimensional $\mathcal{N}=2$
$SU(2)$ gauge theory was solved by Seiberg–Witten Seiberg:1994rs ;
Seiberg:1994aj . The solution was found by using a number of deep physical
insights: electric-magnetic duality, holomorphy, and the existence of
singularities on the moduli space where extra massless BPS particles appear.

In the theory of Seiberg and Witten, the physical information is encoded in an
elegant way: they introduced a family of algebraic curves (the Seiberg–Witten
(SW) curve) $f(x,y,u,\Lambda)=0$ defined over the Coulomb branch moduli space,
and specified a family of differential forms $\lambda_{SW}$ (the SW
differential). From these geometric objects they were able to extract various
interesting physical properties:

  1. (1)

the low energy photon couplings are found from periods of the SW differential;

  2. (2)

the central charges of massive BPS particles are also given by the periods of
$\lambda_{SW}$;

  3. (3)

the singularities on the moduli space are identified with the degenerating
curve in the SW family.

Therefore, the Coulomb branch solution of a 4d $\mathcal{N}=2$ theory is given
by SW curve and SW differential, and many solutions are henceforth found for
simple gauge theories Argyres:1994xh ; Klemm:1994qs ; Argyres:1995wt ;
Hanany:1995na  via this method. Nevertheless, the original method
Seiberg:1994rs ; Seiberg:1994aj  is difficult to implement in a more general
situation. Instead, a number of solutions were found with the help of other
tools. For example by string theory constructions Kachru:1995fv ;
Witten:1997sc ; Katz:1997eq ; Gaiotto:2009we ; Xie:2012hs ; Xie:2015rpa , and
the relation to integrable system Donagi:1995cf ; Martinec:1995by . The
construction of Seiberg and Witten may be generalized since one can try to
define higher dimensional algebraic varieties fibered over the Coulomb branch
moduli space. We call it the associated SW geometry.

While there are a number of SW geometries available, it is not straightforward
to obtain physical information form them. Let us explain what are the
obstacles.

To begin with, recall that the Coulomb branch vacua are classified by their
low energy behavior: a _generic vacuum_ is a vacuum whose the low energy
theory is an abelian gauge theory. If a vacuum is not generic, it is called
_special_.

Generic vacua. Given a generic vacuum with Coulomb branch modulus $t$, we have
an $n$-dimensional smooth algebraic variety from the SW geometry (we shall
consider the cases when $n=1$ or $n=3$). Its singular cohomology group
$H^{n}(X_{t},\mathbb{C})$222If the SW geometries are curves, the only non-
vanishing reduced cohomology group is $H^{1}$; If the SW geometries are three-
folds, the only non-vanishing reduced cohomology groups are $H^{3}$. We can
also take the coefficient of the cohomology group to be $\mathbb{Z}$ or
$\mathbb{Q}$, which are useful for the electric-magnetic pairing. is the
crucial data for us in order to extract physical information. We face
following challenges:

  1. (1)

The central charges should be given by the periods of the SW differential
$\lambda_{SW}$. If a theory has global symmetry, then there would be periods
for flavor central charges. How can we distinguish those corresponding to
global symmetries from the one that give rise the electric-magnetic charges?

  2. (2)

The constraint of unitarity requires that there is a family of abelian
varieties fibered over the Coulomb branch moduli space. However, in general,
the SW geometries are non-compact algebraic varieties, and
$H^{n}(X_{t},\mathbb{C})$ does not give an abelian variety. How can we find
these abelian varieties?

  3. (3)

Finally, the Coulomb branch solution carries a special Kähler geometry. But
defining the SKG requires the presence of the SW differential $\lambda_{SW}$.
However how to determine the SW differential is rather difficult, even if the
SW geometry is given.

Special vacua. A special vacuum is a place on the moduli space where the SW
geometry $X_{0}$ degenerates333A special vacuum does not necessarily means
that the algebraic variety given by the SW geometry is singular, we will
describe exactly what this means in section 3.. The low energy theory at these
vacua are much more complicated. It could be a superconformal field theory, an
IR free non-abelian gauge theory, or an interacting theory coupled with an
abelian gauge theory. From the SW geometry we could obtain the cohomology
groups $H^{n}(X_{0},\mathbb{C})$, and a monodromy action on the cohomology
groups of fibers near $X_{0}$. These data, however, are not enough to
determine the IR theory of the vacuum.

Special vacua are the most interesting and important vacua. Some of the
special vacua of pure $SU(N)$ gauge theory give the confining vacua of the
resulting $\mathcal{N}=1$ theory formed by soft-breaking of $\mathcal{N}=2$
supersymmetry Seiberg:1994rs .

UV theory. When we are given just a SW geometry, how do we determine the
properties of UV theory from it?

The purpose of this paper is to propose that the mixed Hodge structure (MHS)
can be used to solve above problems. The reader should probably be familiar
with the notion of a Hodge structure, which is modeled on the Hodge
decomposition of a nonsingular projective variety. By definition, a Hodge
structure of weight $n$ consists of a finite rank free abelian group
$H_{\mathbb{Z}}$ together with a decomposition:

$H_{\mathbb{C}}:=H_{\mathbb{Z}}\otimes\mathbb{C}=\bigoplus_{p+q=n}H^{p,q},$

such that the complex conjugation of $H^{p,q}$ equals $\overline{H^{q,p}}$.

A Hodge structure of weight $n$ can be reformulated as a pair
$(H_{\mathbb{Z}},F^{\bullet})$, where $F^{\bullet}$ is a decreasing filtration

$H_{\mathbb{C}}=F^{0}\supset F^{1}\supset\cdots\supset F^{n-1}\supset F^{n}$

satisfying the condition $H_{\mathbb{C}}=F^{p}\oplus\overline{F^{n-p+1}}$ for
every $p$. In this formulation, the $H^{p,q}$ are recovered by
$H^{p,q}=F^{p}\cap\overline{F^{q}}$.

A mixed Hodge structure is a generalization of a HS and was defined by Deligne
deligne:hodge2 ; deligne:hodge3 . It consists of a triple
$H=(H_{\mathbb{Z}},F^{\bullet},W_{\bullet})$, where

  * •

$H_{\mathbb{Z}}$ is a finite rank free abelian group,

  * •

$F^{\bullet}$ is an increasing filtration on $H_{\mathbb{C}}$ called Hodge
filtration, and

  * •

$W_{\bullet}$ is a decreasing filtration on
$H_{\mathbb{Q}}=H_{\mathbb{Z}}\otimes\mathbb{Q}$ called the weight filtration.

They are required to satisfy the following property: the induced filtration of
$F^{\bullet}$ on the quotient space
$\mathrm{Gr}^{W}_{k}H_{\mathbb{Q}}=W_{k}/W_{k-1}$ defines a weight $k$ Hodge
structure. In particular $\mathrm{Gr}^{W}_{k}H_{\mathbb{Q}}\otimes\mathbb{C}$
has a Hodge decomposition:

$\mathrm{Gr}^{W}_{k}H_{\mathbb{Q}}\otimes\mathbb{C}=\bigoplus H^{p,k-p}.$

We now explain how the MHS can help us overcome the difficulties listed above.

MHS and generic vacua. For a generic vacuum described by a smooth algebraic
variety $X_{t}$, its cohomology group $H^{n}(X_{t},\mathbb{Q})$ carries a MHS
as shown by Deligne deligne:hodge2 . For the SW geometries studied in the
literature, this MHS is quite special. It consists of only weight $n$ and
weight $n+1$ part, and the Hodge decomposition has the following form

$\mathrm{Gr}^{W}_{n}H\otimes\mathbb{C}=H^{{n+1\over 2},{n-1\over 2}}\oplus
H^{{n-1\over 2},{n+1\over
2}},\quad\mathrm{Gr}^{W}_{n+1}H\otimes\mathbb{C}=H^{{n+1\over 2},{n+1\over
2}}.$

This special MHS can help us solve the previous problems for the generic
vacua.

  1. (a)

The weight $n$ part of the cohomology gives the electric-magnetic charges, and
weight $n+1$ part gives the flavor charge.

  2. (b)

The Hodge structure of weight $n$ part gives an abelian variety. The
consistent condition is then $h^{{n+1\over 2},{n-1\over 2}}=r,h^{{n+1\over
2},{n+1\over 2}}=f$, here $r$ is the rank of the theory, and $f$ is the rank
of the flavor symmetry group.

  3. (c)

We will explain in §7 how to find a SW differential from the MHS of the SW
geometry.

Example. Let us consider a geometry $X:x^{3}+y^{3}+1=0$, which is a generic
vacuum of an $(A_{2},A_{2})$ theory. The dimension of the cohomology group
$H^{1}(X,\mathbb{Q})$ is 4. Its MHS has is of the form

$\mathrm{Gr}^{W}_{1}H=H^{1,0}\oplus
H^{0,1},\quad\mathrm{Gr}_{2}^{W}H=H^{1,1},$

The Hodge numbers are $h^{1,0}=h^{0,1}=1$, and $h^{1,1}=2$. The
$(A_{2},A_{2})$ theory has a rank one Coulomb branch, and a rank two flavor
symmetry group, agreeing the MHS computation.

MHS and special vacua. Consider a special vaccum corresponding to a variety
$X_{0}$ given by the SW geometry. In general the variety $X_{0}$ is singular
(or has “hidden singularities” as will be explained in §4). According to
Deligne [loc. cit.], its cohomology group $H^{n}(X_{0},\mathbb{Q})$ also
carries a MHS. This MHS alone is not sufficient to describe the physics at the
special vacuum. In Hodge theory, one needs two extra MHS in order to fully
describe the degeneration, they are called

  * •

the nearby cycle, and

  * •

the vanishing cycle.

The cohomology of $X_{0}$, the nearby cycle, and the vanishing cycle form an
exact sequence of MHS. It turns out we can extract the low energy physics from
these MHS in a very satisfying manner:

  1. (a)

The interacting theory is described by the vanishing cycle, and the abelian
gauge theory is described by the cohomology group of $X_{0}$. These two parts
are coupled together through the exact sequence with the nearby cycle.

  2. (b)

Due to non-trivial monodromy action on vanishing cycle and nearby cycle, the
MHS on the vanishing cycle and nearby cycle have finer structures, so that we
can get a set of rational numbers called spectral numbers from these MHS.
These numbers can give us important information of the Coulomb branch spectrum
of the interacting theory.

Example. Consider the geometry $X_{0}:x^{3}+y^{4}+y^{3}=0$, which is a special
vacuum for the $(A_{2},A_{3})$ theory. $X_{0}$ has a singularity at $x=y=0$,
which is equivalent to the singularity of that of $x^{3}+y^{3}=0$. The MHS for
the singular fiber has the form $H^{1}(X_{0},\mathbb{C})=H^{0,0}$ with
$h^{0,0}=2$. The MHS for the vanishing cycle can be computed from the local
singularity $x^{3}+y^{3}=0$, and its Hodge numbers are
$h^{1,0}=h^{0,1}=1,h^{1,1}=2$.

The physical interpretation of the Hodge numbers are: each Hodge number
$h^{1,1}$ of the vanishing cycles gives the mass parameter, and it pairs with
the $h^{0,0}$ part of the singular variety; physically it means that the
$U(1)$ flavor groups of the interacting theory is gauged. So the low energy
theory is given in Figure 1.

T11 Figure 1: The low energy theory for the vacua described by
$X_{0}:x^{3}+y^{4}+y^{3}=0$. Here $T$ is the $(A_{2},A_{2})$ SCFT.

MHS and UV theory. To extract the information of the UV theory, it is natural
to take the Coulomb branch parameter to be $\infty$. An important type of
deformation takes the form $f(x,y,\lambda_{i})=t$, here $t$ has the largest
scaling dimension, and $\lambda_{i}$ are other parameters. The UV limit is
recovered by taking $t$ to infinity. In this case, one can also define a limit
MHS when $t\to\infty$, from which one can extract information of the UV
theory. The MHS only depends the UV theory and is independent of the
parameters $\lambda_{i}$ in the geometry.

Example. Consider the SW geometry $x^{3}+y^{3}+ax+by+cxy=t$. This is the
generic deformations for the $(A_{2},A_{2})$ theory. The MHS for the
$t=\infty$ limit (with arbitrary $a,b,c$) is the same as the MHS for the
singularity $x^{3}+y^{3}=0$, which gives the correct UV theory.

SW differential. The SW differential plays a pivotal role in defining the
special Kähler geometry on the Coulomb branch moduli space. It turns out that
one can indeed define a special section on the Coulomb branch (primitive form,
see the discussion in Li:2018rdd ), and this special section can be identified
with the Seiberg-Witten differential. The MHS in the $t\rightarrow\infty$
limit will be crucial for finding this section.

The great advantage of the approach advocated in this paper is that many
computations are algebraic, and can be carried out explicitly. This is because
for each 4d $\mathcal{N}=2$ theory, one can attach a combinatorial object
called Newton polyhedron and various MHSs can be computed from the
combinatorial data of the polyhedron.

This paper is organized as follows. Section 2 reviews the basic structure of
Coulomb branch solution of 4d $\mathcal{N}=2$ theories. Section 3 describes
how to use MHSs to study generic vacua. Section 4 describes how to find MHS to
study special vacua. Section 5 describe the MHS associated with UV theory.
While previous section focuses on one particular slice of the Coulomb branch,
Section 6 describes MHS on the general Coulomb branch space. Section 7
describes how to find SW differential from the MHS of the UV theory. Finally a
conclusion is given in section 8.

##  2 Coulomb branch of 4d $\mathcal{N}=2$ theories

###  2.1 $\mathcal{N}=2$ theory and its Coulomb branch solution

Basics of $\mathcal{N}=2$ theory. We review some basic facts of 4d
$\mathcal{N}=2$ supersymmetric field theories, for more details, see
bagger1990supersymmetry .

The 4d $\mathcal{N}=2$ superalgebra consists of the generators $M_{\mu\nu}$
for four dimensional Lorentz group, eight supercharges $Q_{\alpha}^{A},$,
$A=1,2$ and $J_{i}$ generating a $SU(2)_{R}$ symmetry. It is also possible to
have generators for other global symmetries $G_{F}$. The representation theory
of the 4d $\mathcal{N}=2$ superalgebra can be found in bagger1990supersymmetry
.

One can construct 4d $\mathcal{N}=2$ supersymmetric field theory using the
superspace method. These models have Lagrangians and admit gauge theory
description. Examples are $\mathcal{N}=2$ SQCD with $SU(N)$ gauge group
coupled with $N_{f}$ fundamental hypermultiplets (We take $N_{f}\leq 2N$ so
that the theory is asymptotically free).

4d $\mathcal{N}=2$ theories have two class of half-BPS operators: Coulomb
branch operators $\mathcal{O}$ (singlet under $SU(2)_{R}$ symmetry), and Higgs
branch operator $\widehat{\mathcal{B}}_{R}$ which transforms under $SU(2)_{R}$
symmetries ($R$ denotes the representation type under $SU(2)_{R}$ symmetry).
These operators are important as they determine the $\mathcal{N}=2$
supersymmetry preserving deformation of the theory. The deformations of these
theories are (we consider $\mathcal{N}=2$ theories which can be defined for
arbitrarily high energy, this includes the asymptotically free gauge theory
and conformal field theory) classified as follows.

  1. (1)

Relevant or marginal deformation using Coulomb branch operators $\mathcal{O}$:

$\delta S=\lambda\int\mathrm{d}^{4}x\;\mathrm{d}^{4}\theta\;\mathcal{O}+c.c$

This type of deformation exists only for a subset of Coulomb branch
operators444In the UV limit, we can assign a scaling dimension to
$\mathcal{O}$, and we require $[\mathcal{O}]\leq 2$ so that the deformation is
relevant or marginal. .

  2. (2)

If our theory has global symmetries, then the theory consists of Higgs branch
operators $\widehat{\mathcal{B}}_{1}$, and the following relevant mass
deformation is also possible:

$\delta
S=m\int\mathrm{d}^{4}x\;\mathrm{d}^{2}\theta\;\widehat{{\mathcal{B}}}_{1}+c.c$

  3. (3)

We can have Coulomb branch deformation by turning on expectation values of
Coulomb branch operators $u_{i}=\langle\mathcal{O}_{i}\rangle$. The number of
Coulomb branch operators is called the rank of the theory. The Coulomb branch
is not lifted if we turn on deformations of first and second kind, but the low
energy physics is typically changed. So the Coulomb branch is parameterized by
$u_{i},\lambda,m$.

  4. (4)

We can have Higgs branch deformation by turning on expectation values of Higgs
branch operators $\widehat{\mathcal{B}}_{R}$. These vacua would be lifted by
turning on deformations of first and second kind.

  5. (5)

It is also possible to have mixed branch where we turn on expectation values
of both Coulomb branch and Higgs branch operators.

The deformations listed in 3, 4, 5 are called moduli space of vacua of
$\mathcal{N}=2$ theories, and the typical structure of the moduli space is
shown in Figure 2.

Figure 2: The structure of moduli space of 4d $\mathcal{N}=2$ theories. Here
Coulomb branch is parameterized by expectation value of Coulomb branch
operator ${\cal O}$; the Higgs branch is parameterized by expectation value of
Higgs branch operator $\hat{{\cal B}}_{R}$; the mixed branch is parameterized
by both the Coulomb and Higgs branch operators.

Superconformal field theory. Some $\mathcal{N}=2$ theories have larger
symmetry group called $\mathcal{N}=2$ superconformal symmetry. These theories
are called superconformal field theories (SCFT), and play fundamental roles in
studying $\mathcal{N}=2$ theories. This is because general $\mathcal{N}=2$
theories are defined as turning on relevant deformation of SCFT. SFCTs have an
extra $U(1)_{R}$ symmetry (besides the $SU(2)_{R}$ symmetry for a general
$\mathcal{N}=2$ theory). The bosonic symmetry group of a general
$\mathcal{N}=2$ SCFT is $SO(2,4)\times SU(2)_{R}\times U(1)_{R}\times G_{F}$,
here $SO(2,4)$ is the conformal group, $SU(2)_{R}\times U(1)_{R}$ is the $R$
symmetry group which exist for every $\mathcal{N}=2$ SCFT, and $G_{F}$ are
other global symmetry groups which could be absent for some theories.

The representation theory for 4d $\mathcal{N}=2$ superconformal algebra is
studied in Dolan:2002zh . A highest weight representation is labeled as
$|\Delta,R,r,j_{1},j_{2}\rangle$, here $\Delta$ is the scaling dimension, $R$
is $SU(2)_{R}$ charge, $r$ is $U(1)_{R}$ charge, $j_{1}$ and $j_{2}$ are left
and right spin. These states could also carry quantum numbers of flavor
symmetry group $G_{F}$. Representation theory of $\mathcal{N}=2$ superalgebra
has been studied in Dolan:2002zh , and its short representations are
completely classified in Dolan:2002zh . Three short (Half-BPS) representations
that we are interested in are Coulomb branch operators, Higgs branch
operators, and supercurrent multiplet:

$\begin{split}&\text{Coulomb~{}branch operators}:~{}~{}~{}~{}{\cal
E}_{r,(0,0)},~{}~{}R=0,~{}~{}\Delta=r,\\\ &\text{Higgs~{}branch
operators}:~{}~{}~{}~{}~{}~{}~{}~{}\hat{{\cal
B}}_{R},~{}~{}~{}~{}~{}~{}~{}r=j_{1}=j_{2}=0,~{}~{}\Delta=2R,\\\
&\text{Supercurrent}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{{\cal
C}}_{0,(0,0)},~{}~{}r=R=0,~{}~{}\Delta=2.\\\ \end{split}$

$\widehat{\mathcal{B}}_{1}$ is a multiplet which contains conserved current
for the flavor symmetry group $G_{F}$, and transforms in adjoint
representation of $G_{F}$. Here we list the quantum numbers for the bottom
component of the multiplet. ${\cal E}_{r,(0,0)}$ play the same role as the
Coulomb branch operator ${\cal O}$ for the general $\mathcal{N}=2$ theory,
here it just carries an extra $U(1)_{R}$ quantum number. and $\hat{{\cal
B}}_{R}$ is just the usual Higgs branch operator we discussed for a general
$\mathcal{N}=2$ theory.

For most of theories, Coulomb branch operators form a freely generated ring.
Determining the Coulomb branch chiral ring is equivalent to determining the
set of $U(1)_{R}$ charges $(r_{1},\ldots,r_{s})$ (unitarity bound requires
these numbers to be larger than 1). The Higgs branch operators also form a
ring, which is in general quite complicated, see Xie:2019vzr  for recent
development.

An important question for SCFT is to determine its Half-BPS spectrum.
Supercurrent multiplet exists for every $\mathcal{N}=2$ SCFT. There are also a
large class of $\mathcal{N}=2$ SCFTs whose full Coulomb branch spectrum and
Higgs branch spectrum are known Gaiotto:2009we ; Xie:2012hs ; Wang:2015mra ;
Wang:2018gvb ; Xie:2015rpa .

One can also define central charges $a_{\mathcal{N}=2}$ and
$c_{\mathcal{N}=2}$ for a 4d conformal field theories. The central charges
$a_{\mathcal{N}=2}$ and $c_{\mathcal{N}=2}$ are related to the anomalies of
$R$ symmetries as follows Shapere:2008zf :

$\begin{split}&\text{Tr}(R_{\mathcal{N}=2}^{3})=6(a_{\mathcal{N}=2}-c_{\mathcal{N}=2}),~{}~{}\text{Tr}(R_{\mathcal{N}=2})=24(a_{\mathcal{N}=2}-c_{\mathcal{N}=2}),\\\
&\text{Tr}(R_{\mathcal{N}=2}R^{2}_{SU(2)})=(2a_{\mathcal{N}=2}-c_{\mathcal{N}=2}).\end{split}$

Here $R_{\mathcal{N}=2}$ is the generator of $U(1)_{R}$ symmetry, and
$R_{SU(2)}$ is the Cartan subalgebra of $SU(2)_{R}$ algebra. Central charges
are computed for a large class of SCFTs in Xie:2015rpa .

Examples of 4d $\mathcal{N}=2$ SCFT are $SU(N)$ gauge theory with $N_{f}=2N$
fundamental matter, which has conventional Lagrangian description. There are
also non-Lagrangian $\mathcal{N}=2$ SCFTs such as $T_{N}$ theory
Gaiotto:2009we , and Argyres-Douglas theories Argyres:1995jj .

Coulomb branch solution. In this paper, we focus on the structure of Coulomb
branch, which is defined as the moduli space parameterized by Coulomb branch
operators. Since the Coulomb branch is not lifted by the relevant/marginal and
mass deformation, we also consider turning on these deformations. So we have a
family of Coulomb branch parameterized by the relevant/marginal and mass
deformations. One of the basic question of understanding Coulomb branch is to
determine the low energy physics at every point of the moduli space. The
Coulomb branch has following structure (see Figure 3):

  1. (1)

The low energy theory at a generic vacuum of Coulomb branch is described by an
abelian $U(1)^{r}$gauge theory555$r$ denotes the number of Coulomb branch
operators ${\cal O}_{i}$, and is also called the “rank” of the theory.. The
important thing is to determine the low energy effective action, which has a
rigid special Kähler structure (RSK).

At a generic point, there are stable massive BPS particles carrying electric-
magnetic charge, and flavor symmetry charge. If the rank of the theory is $r$
and the rank of the flavor symmetry is $f$, then the dimension of the charge
lattice of BPS particle is:

$\mu=2r+f.$

The central charge of a BPS particles with charge $(n_{i},n_{i}^{D},n_{f})$ is
given by

$Z=n_{i}a^{i}+n_{i}^{D}a_{D}^{i}+n_{f}m_{f}.$

here $(a^{i},a_{D}^{i},m_{f})$ are holomorphic functions on the moduli space.
An important question is to determine those functions so that the central
charge of BPS particles can be computed, and the mass of the BPS particle is
given by $M=|Z|$.

  2. (2)

At special points of Coulomb branch, besides the abelian gauge theory, new
massless degrees of freedoms appear. The low energy theory would consist of
several parts: (1) abelian gauge theory; (2) IR free gauge theories: abelian
or non-abelian gauge theory coupled with enough matter; (3) superconformal
field theory. In general, these parts are coupled together. The extra massless
degree of freedoms come from massive BPS particles. The physical question is
to determine the low energy theory and their physical properties at these
special points.

Figure 3: The Coulomb branch is separated into generic points and special
points: the low energy physics at a generic point is described by a $U(1)^{r}$
abelian gauge theory; there are extra massless particles at the special point
and the low energy physics would be much more complicated.

It is very difficult to answer above physical questions of Coulomb branch by
using conventional physical tools, as the study often involves complicated
instanton computations in the weakly coupled region or strongly coupled
dynamics of gauge theory. The remarkable discovery of Seiberg and Witten is to
encode the low energy physics into a family of curves fibered over the Coulomb
branch, and such a curve is called the Seiberg-Witten (SW) curve. Seiberg-
Witten curve is denoted as $f(x,y,u,\lambda,m)=0$, and various physical data
are read from it as follows.

  * •

At generic point of Coulomb branch, the SW curve is smooth and the low energy
photon couplings are identified with the complex structure of the curve. There
is a SW differential $\lambda_{SW}$ whose periods of integral over cycles of
the curve give the central charge of the BPS particles. The derivative of
these periods along the Coulomb branch coordinate would give the complex
structure of the curve, and thereby determines the low energy physics.

  * •

The special point of the Coulomb branch is identified with the degenerating
curve. The nature of the low energy physics at these special points are often
complicated and has to be determined carefully, see Seiberg:1994rs ;
Seiberg:1994aj .

See Figure 4 for the illustration of SW curves fibered over the Coulomb
branch. The SW curve is found based on various deep physical inputs such as
holomorphy of superpotential, BPS particles, electric-magnetic duality, etc.

Figure 4: There is an auxiliary curve (SW curve) at each point of Coulomb
branch. At a generic point, the SW curve is smooth; At a special point, the
curve degenerates and often develops singularity.

Let’s now describe the SW solution of the pure $SU(2)$ gauge theory due to
Seiberg and Witten Seiberg:1994rs ; Seiberg:1994aj . The solution is given by
the family of curves:

$f=x+{\Lambda^{2}\over x}+y^{2}+2u=0;$

Here $\Lambda$ is the dynamical generated scale, and $u$ is the Coulomb branch
moduli. $x$ is a $\mathbb{C}^{*}$ variable and $y$ is a $\mathbb{C}$
variable666This is not the original curve given in Seiberg:1994rs . It is not
difficult to prove that the curve given here is equivalent to that of
Seiberg:1994rs .. Various physical quantities are computed from the curve as
follows:

  * •

At a point $u$ where the corresponding curve is smooth, the photon coupling is
identified with the complex structure of the curve. This automatically
captures the electric-magnetic duality of the low energy $U(1)$ theory.

  * •

The curve degenerates at three points $u=\Lambda,-\Lambda,\infty$, see Figure
5. The physics at $\infty$ is the non-abelian $SU(2)$ gauge theory. At
$u=\pm\Lambda$, there is an extra massless hypermultiplet which is due to BPS
particle becoming massless. The low energy theory is a $U(1)$ gauge theory
coupled with a hypermultiplet.

  * •

One has a SW differential $\lambda_{SW}=y{\mathrm{d}x\over x}$, which gives us
the SKG required by $\mathcal{N}=2$ supersymmetry. Moreover, the periods of
integral of $\lambda_{SW}$ gives us the central charges for BPS particles: the
two periods are $a=\int_{A}\lambda$, $a_{D}=\int_{B}\lambda$ (here $A$ and $B$
are two cyles on the curve), and $Z=na+n_{D}a^{D}$ for a BPS particle with
charge $(n,n_{D})$.

$u$$\Lambda$$-\Lambda$$\infty$ Figure 5: Coulomb branch moduli space for 4d
pure $SU(2)$ SYM theory. Here we use $u$ to parameterize Coulomb branch. At
generic point of moduli space, the low energy theory is an abelian $U(1)$
gauge theory. There are three singularities: $u=\pm\Lambda$ where there is one
extra massless hypermultiplet, and $u=\infty$ where the theory is the non-
abelian $SU(2)$ gauge theory.

###  2.2 Seiberg-Witten geometry and Newton polyhedron

As we discussed in last subsection, the Coulomb branch solution of a 4d
$\mathcal{N}=2$ theory is encoded by a SW curve fibered over the Coulomb
branch777It was found in Katz:1997eq  that one need to consider higher
dimensional variety for more general $\mathcal{N}=2$ theories., together with
a Seiberg-Witten differential. While this approach of encoding Coulomb branch
is very powerful, there is no systematical way to find SW solution given a 4d
$\mathcal{N}=2$ theory. Usually, one can find SW solution by using the the
connection to integrable systems Donagi:1995cf , and string theory
construction Kachru:1995fv ; Witten:1997sc . The two most powerful approaches
are the Hitchin system Nanopoulos:2009uw ; Xie:2012hs ; Wang:2015mra ;
Wang:2018gvb  and complex singularities in one dimension or three dimension
Chen:2016bzh ; Xie:2015rpa ; Wang:2016yha ; Chen:2017wkw .

Most of the Seiberg-Witten geometry found in the literature takes the
following form: it is extracted from a function defined on a smooth affine
variety $f:U\rightarrow\mathbb{C}$, $U$ being $\mathbb{C}^{2}$,
$\mathbb{C}^{*}\times\mathbb{C}$, $\mathbb{C}^{4}$, or
$\mathbb{C}^{*}\times\mathbb{C}^{3}$.888A $\mathbb{C}^{*}$ variable can be
thought of as removing the origin of the complex plane. It is an affine
variety as its coordinate ring is given by $\mathbb{C}[x,y]/(xy-1)$. If we
solve $y$ using the defining ideal $xy=1$, the coordinate ring of
$\mathbb{C}^{*}$ is then isomorphic to $\mathbb{C}[x,x^{-1}]$, which is the
ring of Laurent polynomial of variable $x$. If the function has a
$\mathbb{C}^{*}$-action with positive weights, it is believed that the
geometry should describe a SCFT. Here we list several class of examples (the
corresponding 4d theories are SCFTs) that we would like to study in more
detail.

  * •

Take a polynomial $f:\mathbb{C}^{4}\rightarrow\mathbb{C}$ with an isolated
critical point at origin. We require that $f$ has a $\mathbb{C}^{*}$ action
satisfying

$f(\lambda^{q_{i}}x_{i})=\lambda f(z_{i}),\quad\sum q_{i}>1.$

This type of polynomial has been studied in Xie:2015rpa . The corresponding 4d
theory is an Argyes-Douglas type theory. Many properties of those theories
have been studied in Xie:2016evu ; Xie:2017vaf .

  * •

Take $f:\mathbb{C}^{*}\times\mathbb{C}^{3}\rightarrow\mathbb{C}$ of following
form

$f=f_{ADE}(x_{1},x_{2},x_{3})+zg(x_{1},x_{2},x_{3},z);$

Here, $f_{ADE}(x_{1},x_{2},x_{3})$ is a polynomial defining a standard two
dimensional ADE singularity, and $z$ is a $\mathbb{C}^{*}$ variable; the
polynomial $g$ is taken so that the polynomial has a $\mathbb{C}^{*}$ action.
The corresponding 4d theory can have non-abelian global symmetries. This class
of theory can be engineered using the 6d $(2,0)$ theory Xie:2012hs ;
Wang:2015mra ; Wang:2018gvb , and is important to understand $S$ duality of
Argyres-Douglas theories.

  * •

Take $f:\mathbb{C}^{*}\times\mathbb{C}\rightarrow\mathbb{C}$ of the form

$f=v^{N}g(z)$

Here $z$ is a $\mathbb{C}^{*}$ variable, and $v$ is a $\mathbb{C}$ variable.
$g(z)$ is a degree $(n+1)$ polynomial with distinct roots. This type of theory
can be engineered by using the 6d $(2,0)$ theory on a sphere with regular
singularities Gaiotto:2009we . The physical theories and the above curve were
found using the M-theory construction Witten:1997sc

One can also engineer non-conformal theory which is not the deformation of
above quasi-homogeneous polynomials, and some of the examples are

  * •

$f=v^{n_{1}}z^{s+1}+v^{n_{2}}z^{s}+\ldots+v^{n_{s-1}}z+v^{n_{s}}$. Here $z$ is
a $\mathbb{C}^{*}$ variable, and $v$ is a $\mathbb{C}$ variable. To ensure
that the 4d theory is UV complete, the constraint on $n_{i}$ is following

$2n_{i}<n_{i-1}+n_{i+1},\quad i=2,\ldots,n_{s-1}$

This class of curves describe the Coulomb branch solution of linear quiver of
$SU$ type, and is derived using brane construction Witten:1997sc .

  * •

$f=f_{ADE}(x_{1},x_{2},x_{3})+z^{a}+{1\over z^{b}}$. This curve describes the
Coulomb branch solution of a single ADE gauge theory coupled with Argyres-
Douglas matters Xie:2012hs ; Cecotti:2012jx . Here $z$ is a $\mathbb{C}^{*}$
variable.

Newton polyhedron. Some properties of most of the SW geometries listed above
can be described by the so-called Newton polyhedron. Here we give a review
about how to associate a Newton polyhedron to a polynomial function.

Let $f:(\mathbb{C}^{*})^{n}\times\mathbb{C}^{m}\rightarrow\mathbb{C}$ be given
by a Laurent polynomial

$f=\sum\prod_{i=1}^{n+m}x_{i}^{\omega_{i}}.$

For a $\mathbb{C}$ variable in a monomial of above sum, its exponent must be
non-negative. There is no such constraint for $\mathbb{C}^{*}$ variables. For
each monomial $\prod_{i}x_{i}^{\omega_{i}}$ appearing in $f$, one draws a dot
with coordinate $(\omega_{1},\ldots,\omega_{n+m})$ in $\mathbb{R}^{n+m}$; the
Newton polyhedron $\Delta(f)$ (at infinity) is the convex hull of the origin
and these dots.

For example, the Newton polyhedron of
$f(x_{1},x_{2})=x_{2}^{2}-(x_{1}+x_{1}^{-1})$ is given by Figure 6.

Figure 6: The Newton polyhedron for polynomial
$f(x_{1},x_{2})=x_{2}^{2}-(x_{1}+x_{1}^{-1})$. Here $x_{1}$ is a
$\mathbb{C}^{*}$ variable, and $x_{2}$ is a $\mathbb{C}$ variable.

We say $f$ is non-degenerate with respect to $\Delta(f)$ if for every face
$\sigma$ of $\Delta(f)$ not containing the origin, the equations

$\frac{\partial f_{\sigma}}{\partial x_{1}}=\frac{\partial
f_{\sigma}}{\partial x_{2}}=\ldots=\frac{\partial f_{\sigma}}{\partial
x_{n+m}}=0$

have no common zeros in $(\mathbb{C}^{\ast})^{n+m}$. Here
$f_{\sigma}=\sum_{(\omega_{1},\ldots,\omega_{n+m})\in\sigma}\prod_{i=1}^{n+m}x_{i}^{\omega_{i}}$
is the Laurent polynomial defined using the lattice points on the face
$\sigma$.

If $f(x_{1},x_{2})=x_{2}^{2}-(x_{1}+x_{1}^{-1})$, there are two faces $\sigma$
and $\tau$ which do not contain the origin, as indicated in Figure 6. We have
$f_{\sigma}(x_{1},x_{2})=x_{2}^{2}-x_{1}^{-1}$ and $f_{\tau}=x_{2}^{2}-x_{1}$.
A simple computation shows that this Laurent polynomial is nondegenerate with
respect to its Newton polyhedron.

Now let us define the concept of a convenient polynomial. If all the variables
in defining $f$ are $\mathbb{C}$ variables, $f$ is called convenient if and
only if $f$ contains a term of the form $ax^{j_{i}}_{i}$ for all $i$, namely
its lattice points intersect with every coordinate axis. For the
$\mathbb{C}^{*}$ variable, the convenient condition is that origin is in the
interior (we set $\mathbb{C}$ variable to be zero). For example, the Laurent
polynomial $f(x_{1},x_{2})=x_{2}^{2}-(x_{1}+x_{1}^{-1})$ is convenient. On the
other hand, $f(x_{1},x_{2})=1+x_{1}x_{2}^{2}+x_{1}^{2}x_{2}$ is not
convenient, see Figure 7.

Figure 7: The Newton polyhedron of a non-convenient polynomial
$f(x_{1},x_{2})=1+x_{1}x_{2}^{2}+x_{1}^{2}x_{2}$.

As we will show later, Newton non-degenerate and convenient polynomial have
many nice properties. Some more examples are shown in Figure 8, and the
properties for the underlying physical theories are:

  1. (1)

The theories described by polynomial $f=x^{N}+y^{k}$ are the
$(A_{N-1},A_{k-1})$ theories Cecotti:2010fi . Here $x$ and $y$ are both
$\mathbb{C}$ variables. Such theories can be engineered by the 6d $A_{N-1}$
$(2,0)$ theory on a sphere with one irregular singularity Xie:2012hs . Some
physical data of these theories are: (a) The dimension of charge lattice is
$\mu=2r+f=(N-1)(k-1)$, here $r$ is the dimension of Coulomb branch and $f$ is
the number of mass parameters; (b) the number of mass parameters $f$ is given
by the interior lattice points on the red boundary.

  2. (2)

The theories described by the polynomial $f=x^{N}+z^{k}$, where $x$ is a
$\mathbb{C}$ variable and $z$ is a $\mathbb{C}^{*}$ variable, have a $SU(N)$
flavor symmetry, and can be engineered by the 6d $A_{N-1}$ $(2,0)$ theory on a
sphere with one irregular singularity and one full type regular singularity
Xie:2012hs . The physical data is summarized as follows: (a) The dimension of
charge lattice is $\mu=2r+f=(N-1)(k-1)$; (b) the number of mass parameters $f$
is given by the interior lattice points on the red boundary. In the example
described in the figure, we have two red boundaries.

  3. (3)

The theories described by polynomial $f=x^{N}+x^{n_{1}}z^{b}$, where $x$ is a
$\mathbb{C}$ variable and $z$ is a $\mathbb{C}^{*}$ variable, have
$SU(N)\times U(n_{1})$ flavor symmetry, which are found in Xie:2017vaf  using
a 6d $(2,0)$ construction. This polynomial is not Newton non-degenerate, but
the physical spectrum found in Xie:2017vaf  suggests that we should consider
Newton polyhedron shown in Figure 8 instead.

Figure 8: The Newton polyhedron for three polynomial functions. Left: Here
$f=x^{N}+y^{k}$ and both $x$ and $y$ are $\mathbb{C}$ variable; Middle: Here
$f=x^{N}+z^{k}$, and $x$ is a $\mathbb{C}$ variable and $z$ is a
$\mathbb{C}^{*}$ variable; Right: Here $f=x^{N}+x^{n_{1}}z^{b}+z^{b}$, and $x$
is a $\mathbb{C}$ variable and $z$ is a $\mathbb{C}^{*}$ variable.

The Newton polyhedrons of the second and third example are not convenient. In
practice, we often perform a coordinate transformation of the form
$x=x^{{}^{\prime}}z$, and consider a different Laurent polynomial (take case 2
as an example):

$f=x^{{}^{\prime}N}+z^{k-N}+{a^{N}\over z^{N}}$

Notice that here we always need to turn on a deformation parameterized by $a$,
so that the polynomial is convenient. The New polyhedron for the case two and
three are shown in figure. 9.

$x$$z$$x$$z$ Figure 9: The Newton polyhedron for the Argyres-Douglas matter.
It is convenient and non-degenerate if we turn on generic deformation.

Constraints on polynomials. What kind of polynomials would give us SW geometry
of a four dimensional $\mathcal{N}=2$ theories? If $f$ is quasi-homogeneous,
we might use the 4d/2d correspondence to give a constraint Shapere:1999xr :
i.e., the corresponding 2d Landau-Ginzburg model based on $f$ should have a
central charge $\hat{c}\leq 2$. For a $\mathbb{C}^{*}$ variable, its
contribution to the 2d central charge would be $1$, and a $\mathbb{C}$
variable would contribute to $1-2q$ with $q$ the weight of the variable. If
our polynomial is given by
$f:(\mathbb{C}^{*})^{n}\times\mathbb{C}^{m}\rightarrow\mathbb{C}$, the central
charge of 2d model would be

$\hat{c}=n+\sum_{i=1}^{m}(1-2q_{i}).$

If we require $\hat{c}<2$, we have the following.

  * •

There is at most one $\mathbb{C}^{*}$ variable. If there is a $\mathbb{C}^{*}$
variable, the contribution to central charge of other $\mathbb{C}$ variables
should be less than one, which implies that they form a $ADE$ singularity.

  * •

If all the variables are $\mathbb{C}$ variables ($n=0$), the constraint is

$\sum_{i}^{m}(1-2q_{i})<2\Longleftrightarrow\sum_{i=1}^{m}q_{i}-1>{m-4\over
2}.$

Here we take $m$ to be even, so that the middle homology has a symplectic
pairing (which would give the Dirac pairing for the BPS particles). For $m=2$,
there is no constraint; For $m=4$, the constraint is $\sum_{i=1}^{m}q_{i}>1$
and the list of such polynomials can be found in Xie:2015rpa . For $m\geq 6$,
there are very fewer possibilities, see DelZotto:2015rca  for some interesting
examples.

We now give a general criterion for whether a polynomial (actually its Newton
polyhedron with respect to infinity) gives a SW geometry for a 4d
$\mathcal{N}=2$ theory.

  * •

If $n+m=2$, the only constraint is $n\leq 1$.

  * •

If $n+m=4$. There are two constraints. (a) $n\leq 1$; (b) there is no interior
point in the Newton polyhedron of $\Delta(f)$; (c) if $n=1$, the $\mathbb{C}$
variable part of the polynomial should be of the ADE type.

Later on, we will give an explanation of this fact using Hodge structure of
the variety defined by $f$. A generalization to arbitrary $n,m$ will be given
there.

Deformations. Let us start with a Newton non-degenerate and convenient Laurent
polynomial $f$ which gives SW geometry for 4d $\mathcal{N}=2$ theory. The
deformations of the polynomial is captured by the Jacobian algebra of $f$:

$J_{f}=\frac{\mathbb{C}[x_{1},\ldots,x_{n},x_{n+1},\ldots,x_{n+m}]}{\\{x_{1}{\partial
f\over\partial x_{1}},\ldots,x_{n}{\partial f\over\partial x_{n}},{\partial
f\over\partial x_{n+1}},\ldots,{\partial f\over\partial x_{n+m}}\\}};$ (1)

The deformed polynomials are

$F_{\lambda}=f+\sum_{i=1}^{\mu}\lambda_{i}\phi_{i}$ (2)

where $\mu$ is the dimension of Jacobi algebra, the images of $\phi_{i}$ form
a basis of $J_{f}$. When $f$ is quasi-homogeneous, we can assign a scaling
dimension to $\lambda_{i}$ as follows Shapere:1999xr :

$[\lambda_{i}]={2(1-q_{i})\over 2-\hat{c}};$ (3)

Here $q_{i}$ are the weights of $\phi_{i}$ under $\mathbb{C}^{*}$ action. The
parameters $\lambda_{i}$ are identified with the parameters of Coulomb branch,
and are classified as follows:

  1. (1)

Marginal deformations: if $[\lambda_{i}]=0$.

  2. (2)

Relevant deformations: if $0<[\lambda_{i}]<1$.

  3. (3)

Mass deformations: if $[\lambda_{i}]=1$.

  4. (4)

Coulomb branch moduli: if $[\lambda_{i}]>1$.

  5. (5)

Irrelevant deformation: if $[\lambda_{i}]<0$.

We are particularly interested in the relevant deformations, mass
deformations, and the Coulomb branch moduli, as they would not change the UV
theory. These deformations have a nice geometric descriptions: they are
associated with the so-called under-diagram deformations, namely, the
monomials $\phi_{\alpha}$ are under the boundary of the Newton polyhedron in
the region $x_{i}>0$ (here $x_{i}$ is the $\mathbb{C}$ variable). The marginal
deformation is associated with the lattice points on the boundary of Newton
polyhedron.

The interesting deformation gives the following SW geometry:

$F_{\lambda}=f+\sum_{i=1}^{\mu_{1}}\lambda_{\alpha}\phi_{\alpha};$ (4)

Here $\mu_{1}$ is the total number of relevant/marginal, mass, and Coulomb
branch moduli. The full SW geometry could be regarded as giving a polynomial

$F:U\times\mathbb{C}^{\mu_{1}}\to\mathbb{C}^{\mu_{1}}\times\mathbb{C},$ (5)

and the map is given by

$F:(z_{i},\lambda_{\alpha})=(\lambda_{\alpha},F(z_{i},\lambda_{\alpha})).$ (6)

The above polynomials will be the fundamental objects of this paper.

Polynomial or hypersurface. In the usual study of Seiberg-Witten solution, one
has a hypersurface fibered over Coulomb branch. In our study, we find that it
is better to attach a polynomial map $f_{\lambda}:U\to\mathbb{C}$ for each
vacuum, and the usual hypersurface can be recovered by considering the fiber
at point zero $f_{\lambda}^{-1}(0)$.

Complete intersection. Up to this point, we only consider the SW geometry
defined by a polynomial over an affine space. However, there are lots of
$\mathcal{N}=2$ theories whose SW geometry can not be described by a
polynomial: their SW geometries are given by more general geometry such as
complete intersections. Namely, we have map
$f:\mathbb{C}^{*n}\times\mathbb{C}^{m}\rightarrow\mathbb{C}^{k}$. The case
$f:\mathbb{C}^{5}\rightarrow\mathbb{C}^{2}$ with a $\mathbb{C}^{*}$ action is
studied in Chen:2016bzh ; Wang:2016yha . One can similarly attach a Newton
polyhedron for the complete intersection, etc.

##  3 IR theory: generic vacua

The SW geometries of a large class of $\mathcal{N}=2$ theories can be
described by polynomial maps
$F:U\times\mathbb{C}^{\mu_{1}}\to\mathbb{C}^{\mu_{1}}\times\mathbb{C}$. In
most examples, $U$ is the affine variety $\mathbb{C}^{\ast
n}\times\mathbb{C}^{m}$. We shall consider this case exclusively.

The low energy theory on the Coulomb branch is then determined by a polynomial

$F(x;\lambda_{\alpha})=f(x)+\sum_{\alpha=1}^{\mu_{1}}\lambda_{\alpha}\phi_{\alpha}(x);$
(7)

where $f(x)$ is a polynomial (which determines the Newton polyhedron),
$\lambda_{\alpha}$ are parameters of the relevant deformations, marginal
deformations, mass deformations, and deformations by Coulomb branch moduli.
The map is then given by

$(x,\lambda_{\alpha})\mapsto(\lambda_{\alpha},F(x,\lambda_{\alpha})).$

We would like to extract low energy physics from topologies of these
polynomial maps. This is related to the fiber at $t=0$ and its nearby fibers.
This is one of the crucial point of this paper: the low energy physics is not
completely determined by the hypersurface at a particular vacuum but by both
the singular fiber and the nearby fibers.

First, for given physical parameters $\lambda_{\alpha}$, we would like to
determine whether it is a generic vacuum or special vacuum. This question is
related to how the topology of the fibers $F^{-1}(\lambda,t)$ change with the
parameters $\lambda$ and $t$. Vaguely, if the topology changes at a particular
parameters $(\lambda_{0},0)$, it is a special vacuum, otherwise, it is a
generic vacuum. We shall make this precise in §3.1.

In this section, we would like to discuss the physics associated with generic
vacua, where the low energy physics is given by abelian gauge theories. As we
discussed in the introduction, the cohomology group of the fiber
$F^{-1}(\lambda,t)$ itself is not enough to determine the low energy physics,
one needs to know the mixed Hodge structure defined on the cohomology group.
The MHS is useful in that: (1) It can help us distinguish the cycles relevant
for the flavor central charge from that of the electric-magnetic central
charge; (2) One can define an abelian variety from the cohomology group, which
is necessary from the unitarity requirement of the low energy interaction.

###  3.1 Topology of polynomial maps and vacua type

Fix the deformation parameters $\lambda$. We have a polynomial map
$F:U\to\mathbb{C}$. Physically, this means that we study a particular
deformation of the theory parameterized by the coordinate $t$ (in many cases,
this parameter has largest scaling dimension and is the most important
deformation).

Let us review some basic facts about the topology of polynomial maps
broughton1988milnor . Let $F:\mathbb{C}^{\ast
n}\times\mathbb{C}^{m}\to\mathbb{C}$ be a polynomial map. We shall be
concerned with the topology of the fibers of $F$. It is known that there is a
finite subset $\alpha_{F}=\\{b_{1},\ldots,b_{s}\\}$ of $\mathbb{C}$ such that

$F:F^{-1}(\mathbb{C}-\alpha_{F})\to\mathbb{C}-\alpha_{F}$

is a (topologically) locally trivial fibration. We shall assume that
$\alpha_{F}$ is the smallest possible such subset, and call it the set of
_atypical values_. A point $c\in\mathbb{C}-\alpha_{F}$ is thereby called a
_typical value_ of the function $F$. See Figure 10.

Figure 10: A family of hypersurfaces associated with the polynomial map $F$.
Here the typical value $c$ on the $\mathbb{C}$ plane means that locally around
$c$, the fibration is locally trivial.

The notion of an atypical value should not be confused with the notion of a
“critical value”. By definition, a _critical point_ of $F$ is a point in
$\mathbb{C}^{*n}\times\mathbb{C}^{m}$ where the differential $\mathrm{d}F$
equals zero. If $P$ is a critical point of $F$, $c=F(P)$ is called a _critical
value_. A critical value is necessarily an atypical value. But the converse is
not true.

Example broughton1988milnor . Consider the polynomial $F(X,Y)=X^{2}Y-X$. Then
$F$ has no critical points at all, but the topological type of the fiber
$F_{t}=\\{(x,y,t):x^{2}y-x-t=0\\}$ could change when $t$ changes. When $t=0$,
$F_{t}$ is the union of two curves $(x=0)$ and $(xy-1=0)$; whereas if $t\neq
0$, $F_{t}$ is isomorphic to $\mathbb{C}-\\{0\\}$.

The cohomology of the typical fibers can be computed in terms of the so-called
vanishing cycles. Consider for instance a polynomial map
$F:\mathbb{C}^{n}\to\mathbb{C}$. We are interested in the cohomology of
$X_{c}=F^{-1}(c)$ for a typical value $c$ of $F$. This will be described in
terms of “vanishing cohomology groups”. Let $b$ be an _atypical value_ of $f$.
Then there is a small disk $\Delta_{b}$ centered at $b$ such that
$\Delta_{b}-\\{b\\}$ consists of only typical values. For any
$\eta\in\Delta_{b}-\\{b\\}$, there is a natural “specialization map”

$H^{q}(X_{b})\to H^{q}(X_{\eta})$

The cokernel is denoted by $V^{q}_{b}$, and is called the _vanishing
cohomology_ group near $b$. Now let $\alpha_{F}=\\{b_{1},\ldots,b_{m}\\}$ be
the set of atypical values of $F$. Let $c\in\mathbb{C}\setminus\alpha_{F}$ be
a typical value, see Figure 11. Then we can draw paths
$\gamma_{1},\ldots,\gamma_{m}$ joining $c$ and $b_{i}$. If we choose
$\gamma_{i}$ and $\Delta_{b_{i}}$ suitably, we can ensure that the path
$\gamma_{i}$ is the only path in the above collection that intersect
$\Delta_{b_{i}}$.

Figure 11: The point $b_{1}$ and $b_{2}$ are atypical values. For typical
value $c$, one has a path connecting $c$ and $b_{1}$, and along the path, some
of the cycles on the typical fiber vanishes at $b_{1}$. The cohomology group
of the typical fiber can be computed from the vanishing cohomology along these
paths.

Since $F$ is a locally trivial fibration away from $\alpha_{F}$, we know that
parallel transport along $\gamma_{i}$ gives rise to an isomorphism between the
cohomology group $H^{q}(X_{c})$ and $H^{q}(X_{\eta_{i}})$ for any
$\eta_{i}\in\Delta_{b_{i}}$. Thus, we have a surjective map $H^{q}(X_{c})\to
V_{b_{i}}^{q}$. Summing over all $i$, we get a surjective map

$H^{q}(X_{c})\to\bigoplus_{i=1}^{m}V_{b_{i}}^{q}.$

which is in fact an isomorphism. In other words, the cohomology of a typical
fiber is the direct sum of the parallel transports of all vanishing cohomology
groups. This fact links the cohomology of a generic fiber with that of
vanishing cohomology.

We assumed the variety $U$ equals $\mathbb{C}^{n}$ to fix ideas. In case there
exist $\mathbb{C}^{\ast}$-variables, then the formulation should be changed
slightly. For a subvariety $Z$ of $U$, the primitive cohomology of $Z$ is the
cokernel of

$H^{q}(U)\to H^{q}(Z).$

In the case when $U=\mathbb{C}^{n}\times\mathbb{C}^{\ast m}$, the correct
formulation is then the primitive cohomology of $Z$ equals the direct sum of
the vanishing cohomologies.

Another useful fact for the cohomology of typical fiber is the relation to the
so-called local Milnor fiber. Assume that $P$ is an isolated critical point of
$F$, then Milnor’s Fibration Theorem asserts that there is a small ball $B$
centered around $P$, such that for all sufficiently small discs $\Delta^{*}$,
the map $F:B\cap F^{-1}(\Delta^{*})\rightarrow\Delta^{*}$ is a locally trivial
fibration. Moreover, the fibers $F^{-1}(c)\cap B$ have the homotopy type of a
bouquet of $\mu_{p}$ spheres, where $\mu_{p}$ is called the local Milnor
number. The middle cohomology group has the following structure:

> Let $F:\mathbb{C}^{n+1}\to\mathbb{C}$ be a polynomial. Let $\bar{\mu}$ be
> the sum of all the Milnor numbers of $F$ at isolated critical points of $F$.
> Then for all typical fiber, we have
>
> $H_{q}(F^{-1}(c))\approx\mathbb{Z}^{\bar{\mu}}\oplus A$ (8)
>
> where $A$ is some finitely generated abelian group.

If $F$ has only isolated critical points, then the subgroup $A$ comes from the
contribution of “vanishing cycles at infinity”. Again, with the presence of
$\mathbb{C}^{\ast}$ variables, one should use “primitive homology”, i.e., the
kernel of $H_{q}(F^{-1}(c))\to H_{q}(U)$, instead of the usual homology.

If the parameter $\lambda$ is varying, we then get the map
$F:\mathbb{C}^{*n}\times\mathbb{C}^{m}\times\mathbb{C}^{\mu_{1}}\rightarrow\mathbb{C}^{\mu_{1}}\times\mathbb{C}$,
where the coordinate of $\mathbb{C}^{\mu_{1}}$ is given by $\lambda_{\alpha}$,
and that of $\mathbb{C}$ is given by $t$. We are then interested in the fibers
$F^{-1}(\lambda,t)$. Similarly, the points on
$\mathbb{C}^{\mu_{1}}\times\mathbb{C}$ could be classified as typical and
atypical values. Let
$F:\mathbb{C}^{*n}\times\mathbb{C}^{m}\times\mathbb{C}^{\mu_{1}}\to
B=\mathbb{C}^{\mu_{1}}\times\mathbb{C}$ be a polynomial map. Let
$X_{\lambda,t}$ be the fiber $F^{-1}(\lambda,t)$. Then the sheaf
$R^{n+m-1}F_{\ast}\mathbb{Q}$ (sheaf associated with the presheaf $V\mapsto
H^{n+m-1}(F^{-1}(V),\mathbb{Q})$) is an algebraically constructible sheaf.
This means that there is a collection of locally closed subvarieties $B_{i}$
of $B$ such that

  1. (1)

$B=\coprod_{i=0}^{m}B_{i}$,

  2. (2)

$B_{i}\subset\overline{B_{j}}$ if $i>j$

(when these two conditions hold, we say $\\{B_{i}\\}$ is a stratification of
$B$), such that $R^{n+m-1}F_{\ast}\mathbb{Q}|_{B_{j}}$ is locally constant. In
particular, $B_{0}$ is a Zariski open subvariety of $B$. Points in $B_{0}$ are
called _typical values_ of the function $F$. Points in $B\setminus B_{0}$ are
called _atypical values_ of $F$. In particular, atypical values form a proper
Zariski closed subset of $\mathbb{C}^{\mu_{1}}\times\mathbb{C}$.

Again, a critical value is necessarily an atypical value, but a regular value
could equally be atypical. A useful fact for us is: since
$R^{n-1}F_{\ast}\mathbb{Q}|_{B_{0}}$ is locally constant, its stalk at a
typical value $c=(\lambda,t)\in B_{0}$ is equal to the cohomology of the
fiber: $H^{n+m-1}(X_{\lambda,t})$. The above fact is in general not true for
atypical fibers.

Let us now relate the above discussion of topology of the polynomial map to
the IR physics of the corresponding $\mathcal{N}=2$ theory:

  * •

The typical fiber at $(\lambda,0)$ describes the generic vacua of
$\mathcal{N}=2$ theory, while the atypical fiber at $(\lambda,0)$ describes
special vacua.

Next, we introduce a special type of polynomial called “tame” polynomial, and
for these polynomials the atypical value is the same as the critical value,
which will simplify our computations significantly. This definition is quite
involved broughton1988milnor , and in practice not easy to check. But in our
situation, Newton non-degenerate and convenient polynomials will turn out to
be tame broughton1988milnor ; sabbah2006hypergeometric . As we saw in the
examples given in last section, these polynomials give a large class of SW
geometries.

We summarize some of the useful properties of topologies of tame polynomials.

  1. (1)

The atypical value of a tame polynomial is the same as the critical value.

  2. (2)

For tame polynomial, the only non-trivial cohomology of a tame polynomial is
the middle cohomology group.

  3. (3)

For the tame polynomials, the stalk of $R^{n+m-1}F_{\ast}\mathbb{Q}$ at an
arbitrary point is equal to the cohomology of the fiber.

In Sections 3, 4, 5, we shall consider exclusively Seiberg–Witten geometries
associated with tame polynomials.

Consider a typical fiber associated with a Newton non-degenerate and
convenient polynomial. We would like to extract the physical information from
the variety defined by $F_{\lambda}(x)=0$. We face the following challenges:

  1. (1)

The central charge of the BPS particles should be given by the periods of
integral of SW differential. The central charge contains the electric-magnetic
charge for the $U(1)^{r}$ gauge groups, and flavor charges. The question is:
how can we distinguish the electric-magnetic charge and flavor central charge
in the cohomology group? How can we find a SW differential whose periods would
give us the central charge?

  2. (2)

The low energy photon couplings are identified with the complex structure of
the variety and should have a positivity property. In fact, there is an
abelian variety fibration over the Coulomb branch moduli space Donagi:1995cf .
In the compact curve case, this abelian variety is given by the Jacobian of
the curve. How to define an abelian variety from the cohomology groups of
fibers which are non-compact?

  3. (3)

Finally, as we vary the parameters of the Coulomb branch moduli, we should get
a special Kahler geometry, a strong physical constraint. The key question is
to find a Seiberg-Witten differential.

In this section, we will answer the first two questions by using the mixed
Hodge structure defined on cohomology groups of the open smooth variety. The
construction of Seiberg-Witten differential will be given in 7.

###  3.2 Mixed Hodge structure

In this subsection, we recall the concept of a mixed Hodge structure. The
reader with physics background might already be familiar with the Hodge
decomposition of the cohomology group of a compact Kähler manifold, see
griffiths1978principles .

> On a compact Kähler manifold (e.g., a _projective_ algebraic variety) $M$,
> there exists a canonical decomposition (called the _Hodge decomposition_) on
> the cohomology group:
>
> $H^{n}(M,\mathbb{Q})\otimes\mathbb{C}\cong\bigoplus_{p+q=n}H^{p,q}(M).$
>
> Under the complex conjugation induced on the second tensor factor, the
> conjugate of the subspace $H^{p,q}(M)$ is $H^{q,p}(M)$.

The Hodge decomposition is already crucial for understanding compact Riemann
surfaces. For a connected compact Riemann surface $C$, the above theorem
implies that the rank $b_{1}$ of $H^{1}(C,\mathbb{C})$ is necessarily even,
the space $H^{1}(C,\mathbb{C})$ splits into a direct sum $H^{1,0}(C)\oplus
H^{0,1}(C)$ in a way that the two summands are conjugate to each other. It can
be shown that $H^{1,0}$ is the space of holomorphic one forms. So we can write
$b_{1}=2g$. The number $g$, called the _genus_ of $C$, is the number of
linearly independent holomorphic 1-forms on $C$. A classical theorem of
Torelli says that the abelian group $H^{1}(C,\mathbb{Z})$, the above Hodge
decomposition, and the intersection pairing on $H^{1}(C,\mathbb{Z})$
_completely_ determines the complex structure of the Riemann surface $C$.

Pure Hodge structure. A pure Hodge structure is an abstraction of above
decomposition of compact Kähler manifold. By definition, a pure Hodge
structure of weight $n$ consists of a lattice $H_{\mathbb{Z}}$ together with a
decomposition:

$H_{\mathbb{C}}:=H_{\mathbb{Z}}\otimes\mathbb{C}=\bigoplus_{p+q=n}H^{p,q},$

such that the complex conjugation of $H^{p,q}$ equals $\overline{H^{q,p}}$.

Hodge structure can be equivalently given in terms of its _Hodge filtration_.
The $p$th Hodge filtration is
$F^{p}=\bigoplus_{p^{\prime}>p}H^{p^{\prime},n-p^{\prime}}$, thus
$F^{0}=H,\ldots,F^{n-1}=H^{n,0}\oplus H^{n-1,1},~{}F^{n}=H^{n,0}$, and we have

$H_{\mathbb{C}}=F^{0}\supset\cdots\supset F^{n},$

for each $p$, we have

$H=F^{p}\oplus\overline{F^{n-p+1}},$ (9)

and finally the Hodge decomposition can be recovered by

$H^{p,q}=F^{p}\cap\overline{F^{q}}.$ (10)

Example. A trivial example is the “Tate structure” $\mathbb{Z}(m)$, where $m$
is an integer. This is a pure Hodge structure of weight $-2m$ of one
dimensional lattice. The ambient abelian group is
$(2\pi\sqrt{-1})^{m}\mathbb{Z}$, and upon tensoring with $\mathbb{C}$ all the
$(p,q)$-components are zero unless $(p,q)=(-m,-m)$. So

$H=H^{-m,-m}.$ (11)

For example, the pure Hodge structure associated with $H^{2}(\mathbb{P}^{1})$
(one dimensional vector space) is of weight 2, but it has no $(2,0)$ and
$(0,2)$ parts, and it has only $(1,1)$ part. Thus $H^{2}(\mathbb{P}^{1})$, as
a Hodge structure, is isomorphic to the Tate structure $\mathbb{Q}(-1)$.

In general, if $H$ is a Hodge structure of weight $n$, we use $H(m)$ to denote
the tensor product $H\otimes_{\mathbb{Q}}\mathbb{Q}(m)$. This is a Hodge
structure of weight $n-2m$. This is called _$m$ th Tate twist_ of $H$.

Polarized Hodge structure. A polarized Hodge structure of weight $n$ is an HS
$\\{H_{\mathbb{Z}},F^{\bullet}\\}$ together with a non-degenerate integral
bilinear form

$S:H_{\mathbb{Z}}\times H_{\mathbb{Z}}\rightarrow\mathbb{Z}$ (12)

This bilinear form is symmetric for $n$ even, and antisymmetric for $n$ odd,
and satisfying following Riemann bilinear relations (first and second):

$\displaystyle S(H^{p,q},H^{p^{\prime},q^{\prime}})=0~{}\text{if
$(p^{\prime},q^{\prime})\neq(q,p)$)}$
$\displaystyle\sqrt{-1}^{p-q}S(\psi,\bar{\psi})>0~{}~{}\text{for}~{}\psi\in
H^{p,q},~{}\psi\neq 0$ (13)

In terms of Hodge filtration, the bilinear relations are:

$\displaystyle S(F^{p},F^{n-p+1})=0,~{}~{}$ $\displaystyle
S(C\psi,\bar{\psi})>0~{}~{}\text{for}~{}~{}\psi\neq 0$ (14)

Here $C:H\rightarrow H$ is the Weil operator $C|_{H^{p,q}}=i^{p-q}$.

Geometrically, a polarization is induced by an integral Kähler form $\omega$
on a projective manifold $M$ and the usual cup product on the cohomology: for
differential $n$ forms $v$ and $w$, define

$S(v,w)=(-1)^{n(n-1)/2}\int_{M}\omega^{\dim M-n}\wedge v\wedge w.$

This induces a polarization on the Hodge structure of the so-called “primitive
cohomology” (griffiths1978principles, , p.122). This is known as the first and
second _Hodge–Riemann bilinear relations_.

For example, on a compact Riemann surface, there is no $(2,0)$\- or
$(0,2)$-forms, so the only non-zero cup product exists between a $(1,0)$-form
and $(0,1)$-form, this gives the first bilinear relation. The second bilinear
relation is: $\sqrt{-1}\int\eta\wedge\overline{\eta}>0$ for a holomorphic
1-form $\eta$, this is true because the integral gives the volume of the
Riemann surface.

Mixed Hodge structure. The above story works well for smooth projective
varieties. It is natural to ask what extra structure of the same flavor can be
put on a singular or open variety. In his dissertation “Théorie de Hodge, II
et III”, Pierre Deligne introduced a new structure called _mixed Hodge
structure_ , see deligne:hodge2 . A mixed Hodge structure (MHS) is a triple
$(H_{\mathbb{Z}},W_{\bullet},F^{\bullet})$, where $W_{\bullet}$ is a finite
increasing filtration on $H_{\mathbb{Q}}=H_{\mathbb{Z}}\otimes\mathbb{Q}$, and
$F^{\bullet}$ is a finite decreasing filtration on
$H=H_{\mathbb{Z}}\otimes\mathbb{C}$, so that the induced filtration on the
quotient $\mathrm{Gr}_{k}^{W}=W_{k}/W_{k-1}$ defines a HS of weight $k$. Here
$F^{p}\mathrm{Gr}_{k}^{W}H$ is the image of $F^{p}\cap W_{k}H$ in
$\mathrm{Gr}_{k}^{W}H$:

$F^{p}\mathrm{Gr}_{k}^{W}H=(F^{p}\cap W_{k}+W_{k-1})/W_{k-1}.$ (15)

Given a MHS $(H_{\mathbb{Z}},W_{\bullet},F^{\bullet})$, $F^{\bullet}$ is
called the Hodge filtration, and $W^{\bullet}$ is called weight filtration.
Let $H^{p,q}=\mathrm{Gr}_{F}^{p}(\mathrm{Gr}_{p+q}^{W}H\otimes\mathbb{C})$,
then $\mathrm{Gr}_{k}^{W}H\otimes\mathbb{C}=\oplus_{p+q=k}H^{p,q}$. The number
$h^{p,q}=\dim H^{p,q}$ is called Hodge numbers.

If there are two MHS $H$ and $H^{\prime}$, then a map $\phi:H\rightarrow
H^{{}^{\prime}}$ is called morphism of type $(r,r)$, if
$\phi(H_{\mathbb{Z}})\subset H_{\mathbb{Z}}^{\prime}$, $\phi(W_{k}H)\subset
W_{k+2r}H^{\prime}$, and $\phi(F^{p}H)\subset F^{p+r}H^{\prime}$. The map
$\phi$ is called a morphism of mixed Hodge structures if it is of type
$(0,0)$.

Example. Let $P$ be a smooth projective variety. Let $D\subset P$ be a
nonsingular hypersurface. Set $U=P-D$. Denote by $j$ the inclusion map

$j\colon U=P-D\rightarrow P.$ (16)

Then there is a long exact sequence of cohomology groups

$\cdots\to H^{i}(P)\to H^{i}(U)\to H^{i+1}(P,U)\to\cdots.$

By excision, the cohomology of the pair $(P,U)$ is the same as the pair
$(N,N-D)$, where $N$ is the normal bundle of $D$ inside $P$. The “Thom
isomorphism” in algebraic topology implies that

$H^{i+1}(P,U)\cong H^{i+1}(N,N-D)\cong H^{i-1}(D)\otimes\mathbb{Z}(-1)$

Here we include a Tate twist so that it is an isomorphism of Mixed Hodge
Structure. We get a Gysin sequence

$\cdots\to
H^{k-1}(U)\xrightarrow{r}H^{k-2}(D)\xrightarrow{i_{*}}H^{k}(P)\xrightarrow{j^{*}}H^{k}(U)\to\cdots.$

Here $r$ is the topological residue map; in rational cohomology it is the
transpose of the “tube over cycle map” $\tau:H_{k-2}(D;\mathbb{Q})\rightarrow
H_{k-1}(U,\mathbb{Q})$, which associates to the class of a cycle $c$ in $D$
the class of the boundary in $U$ of a suitable small tubular neighborhood of
$c$. The map $i_{*}$ is the Gysin map associated to $i$, namely, we have the
restriction map $H^{i}(P)\xrightarrow{i}H^{i}(D)$, and the Poincare dual gives
the map $H^{2n-i}(P)\xleftarrow{i_{*}}H^{2n-2-i}(D)$. Here we used the fact
that the real dimension of $D$ is $2n-2$, and that of $P$ is $2n$.

From the displayed exact sequence we see that $H^{i}(U)$ has two pieces of
different weights: the piece $W_{i}=\mathrm{Im}(H^{i}(P)\to H^{i}(U))$, which
should inherit the weight of $H^{i}(P)$, namely $i$. Set $W_{i+1}=H^{i}(U)$.
Then the quotient $W_{i+1}/W_{i}$ is then a subspace of $H^{i-1}(D)(-1)$,
which has weight $i+1$. This suggests that in general on a nonsingular
algebraic variety $U$ there should be more than one weight appearing in its
cohomology.

In general, Deligne proved that the cohomology group of any open smooth
algebraic variety carries a MHS, and the nontrivial weights on $H^{m}(U)$ are
in the range $[m,2m]$.

Example. Consider $H^{1}$ of an open curve $\tilde{C}$ which is described by a
genus $g$ curve with $m$ points removed, its MHS is given in table. 1.

Weight | $\mathrm{Gr}_{1}^{W}H$ | $\mathrm{Gr}_{2}^{W}H$
---|---|---
HS | $H^{1,0}\oplus H^{0,1}$ | $H^{1,1}$
dimension |  $2g$ dim. |  $m-1$ dim.
Table 1: The MHS of an open curve derived by removing $m$ points of a genus
$g$ compact curve. Figure 12: An open curve derived by removing $m$ points of
genus $g$ compact curve. The weight two part of $H^{1}$ is associated with the
cycles around the removed points, and the weight one part is associated with
the cycles $\gamma_{1}$ and $\gamma_{2}$.

The above result can be computed using the exact sequence (3.2) (here we take
$P=C$ a compact genus $g$ curve, and $D=m~{}\text{points}$, so
$U=P-D=\tilde{C}$).

$0\to H^{1}(C)\to H^{1}(U)\to H^{0}(m~{}\text{pts})(-1)\to H^{2}(C)\to 0$

Notice that there is a Tate twist for the cohomology group of
$H^{0}(m~{}\text{points})$, so it gives a weight 2 piece of $H^{1}(U)$; the
weight $1$ piece of $H^{1}(U)$ is then given by $H^{1}(C)$ of compact curve
$C$, see figure.12.

###  3.3 Computations

For a four dimensional $\mathcal{N}=2$ theories, the charge lattice has rank

$d=2r+f$ (17)

Here $r$ is the dimension of Coulomb branch moduli (or the rank of abelian
gauge groups at a generic vacuum) and $f$ is the number of flavor symmetries.
This rank should be equal to the middle (primitive) cohomology of the variety
$X$ in the SW geometry associated with the vacuum

$d=H^{n}(X)$ (18)

In this subsection, we shall explain how to determine the Hodge numbers of the
MHS on $H^{n}(X)$. For curves, we can compute MHS using resolution of
singularity and the result of last subsection. More generally, we can compute
MHS from the combinatorial data of the Newton polyhedron associated with the
polynomial.

#### Direct computation

One can compute the MHS for an arbitrary smooth curve $C$ defined by
$f(x,y)=0$ using following strategy (here $x$ and $y$ are $\mathbb{C}$
variables):

  1. (1)

First we consider a projective completion $\overline{C}$ of $C$ which is given
by a homogenous polynomial $F(x,y,z)$ ($F(x,y,z)$ is the homogenization of
$f(x,y)$). This procedure will typically add some extra points called points
at infinity.

  2. (2)

The points at infinity might be singular points of $\overline{C}$, and we
resolve them (using blow-up, see appendix) to get a smooth curve $\tilde{C}$.
The Hodge numbers of $H^{1}(C)$ are

$h^{1,0}(H^{1}(C))=h^{0,1}(H^{1}(C))=g({\tilde{C}}),\quad
h^{1,1}(H^{1}(C))=m-1$ (19)

Here $m$ are the number of points removed from $\widetilde{C}$. The details of
the computation of these numbers are given in appendix.

Example. Consider the affine curve $C$ given by $f=x^{n}+y^{n}+1$. The
corresponding projective curve $\overline{C}$ is cut out by
$F(x,y,z)=x^{n}+y^{n}+z^{n}$, and is smooth. Its genus is $g={(n-1)(n-2)\over
2}$. The points added at infinity is given by the solution of the equation
$F(x,y,0)=x^{n}+y^{n}=0$, and there are $n$ points
$(x,y)=(1,y_{i}),i=1,\ldots,n$ with $y_{i}^{n}=-1$. So the Hodge numbers are

$h^{1,0}=h^{0,1}={(n-1)(n-2)\over 2},\quad h^{1,1}=n-1$ (20)

The number of Coulomb branch moduli is $r={(n-1)(n-2)\over 2}$, and the rank
of flavor symmetries is given by $f=n-1$. This agrees with the physical result
derived in Xie:2012hs .

#### Combinatorial formulas

The MHS of a smooth hypersurface can be computed using the combinatorial data
of Newton polyhedron, see danilov1987newton . We apply their results to the
polynomials relevant to 4d $\mathcal{N}=2$ theories.

We begin by reviewing their formulas. Generalizing the notion of Euler
characteristic, for each pair of integers $(p,q)$, we introduce the following
invariant of algebraic variety $X$:

$e^{p,q}(X)=\sum_{k}(-1)^{k}h^{p,q}(H_{c}^{k}(X))$ (21)

Here $H_{c}^{k}(X)$ is the $k$th compactly supported cohomolology (if $X$ is
nonsingular, then as a Hode structure, the compactly supported cohomology is
isomorphic to $H^{n-k}(X)^{\ast}\otimes\mathbb{Q}(-2n)$), and $h^{p,q}$ is the
Hodge number defined using MHS. In our case, we only need to consider middle
cohomology group or maximal cohomology group, so $k=n-1$ (the complex
dimension of the fiber is $n-1$). Thus (for $p+q\leq n-1$):

$e^{p,q}(X)=(-1)^{n-1}h^{p,q}(H_{c}^{n-1}(X))$ (22)

Hence, knowing the information of $e^{p,q}(X)$ is the same as knowing the
Hodge numbers of the compacted support middle cohomology group, and the
information of the usual middle cohomology group is found using Poincare
duality.

Consider a smooth hypersurface $Z$ in $\mathbb{C}^{n}$, and its associated
Newton polyhedron is $\Delta$. We have

$e^{p,q}(Z)=\sum_{I}e^{p,q}(Z_{I}).$ (23)

Here we sum over the index set $I\subset\\{1,\ldots,n\\}$, and $Z_{I}$ is the
hypersurface defined by the equation $f_{I}=0$ in the algebraic torus
$(\mathbb{C}\setminus\\{0\\})^{|I|}$. So the Hodge number of the hypersurface
in $\mathbb{C}^{n}$ is known once we know the formula for the hypersurface in
$\mathbb{C}^{*n}$, which is computed in danilov1987newton .

The generalization to a hypersurface in $\mathbb{C}^{*n}\times\mathbb{C}^{m}$
is the following: for the formula 23, the set $I$ is $(1,\ldots,m)$, namely we
consider hypersurfaces $f_{I}$ in $\mathbb{C}^{*n}\times\mathbb{C}^{*I}$.
instead. In the sum of 23, only $\mathbb{C}$ variable in the polynomial $f$
can be set to zero.

The Hodge numbers for the middle cohomology $H_{c}^{n-1}(Z)$ (Here $Z$ is a
hypersurface in $\mathbb{C}^{*n}$) can be computed using the combinatorial
data of the Newton polytope, see table. 2, 3 and 4. These tables can be used
to compute MHS for arbitrary polynomials in dimension less or equal four.

$h^{0,1}=l^{*}(\Delta)$ |
---|---
$h^{0,0}=\Pi-1$ | $h^{1,0}=l^{*}(\Delta)$
Table 2: Here $\Delta$ is a two dimensional polytope, and $\Pi$ denotes the number of integral points in the 1-skeleton of $\Delta$. $h^{0,2}=l^{*}(\Delta)$ |  |
---|---|---
$h^{0,1}=\sum l^{*}(\Gamma)$ | $h^{1,1}$ |
$h^{0,0}=\Pi-1$ | $h^{1,0}=\sum l^{*}(\Gamma)$ | $h^{2,0}=l^{*}(\Delta)$
Table 3: Here $\Delta$ is a three dimensional polytope, $\Gamma$ is the two dimensional face of $\Delta$, and $\Pi$ denotes the number of integral points in the 1-skeleton of $\Delta$. Here $h^{1,1}=l^{*}(2\Delta)-4l^{*}(\Delta)$. $h^{0,3}=\ell^{\ast}(\Delta)$ |  |  |
---|---|---|---
$h^{0,2}=\sum_{\Gamma}\ell^{\ast}(\Gamma)$ | $h^{1,2}$ |  |
$h^{0,1}=\sum_{F}\ell^{\ast}(F)$ | $h^{1,1}$ | $h^{2,1}$ |
$h^{0,0}=\Pi-1$ | $h^{1,0}$ | $h^{2,0}$ | $h^{3,0}=l^{*}(\Delta)$
Table 4: Here $\Delta$ is a four dimensional polytope, $\Gamma$ is the three
dimensional face of $\Delta$, and $F$ is the two dimensional face, while $\Pi$
is the one dimensional face. We have
$h^{2,1}=l^{*}(2\Delta)-5l^{*}(\Delta)-\sum l^{*}(\Gamma)$, and
$h^{1,1}=l(2\Delta)-l^{*}(2\Delta)-5l(\Delta)+5l^{*}(\Delta)+10+\sum_{\Gamma}l^{*}(\Gamma)-\sum_{F}l^{*}(F)$.

Curve case. Let $f:\mathbb{C}^{2}\to\mathbb{C}$ be a polynomial. Let $\Delta$
be the Newton polyhedron of $f$ at infinity. Then the zero locus $Z$ of $f$
defines a curve in $\mathbb{C}^{2}$. Assume that $Z$ is smooth. The MHS for
the cohomology $H^{1}(Z)$ is given in the table 5.

$h^{1,0}=\ell^{\ast}(\Delta)$ | $h^{1,1}=l^{*}(F)$
---|---
| $h^{0,1}=\ell^{\ast}(\Delta)$
Table 5: Here $\Delta$ is the two dimensional polytope, and $F$ denotes the
one-dimensional boundary where no coordinate axis vanishes identically. Here
$\ell^{\ast}(\Delta)$ denotes the interior points of $\Delta$.

Let us explain formulas in Table 5. According to formula 23, we have

$e^{1,0}(Z)=e^{1,0}(Z_{1,2})+e^{1,0}(Z_{1})+e^{1,0}(Z_{0})=e^{1,0}(Z_{1,2})\rightarrow
h^{1,0}(X)=h^{1,0}(Z_{1,2})=l^{*}(\Delta).$ (24)

We also have

$e^{0,0}(Z)=e^{0,0}(Z_{1,2})+e^{0,0}(Z_{1})+e^{0,0}(Z_{2})$ (25)

and

$\displaystyle
h^{0,0}(Z)=h^{0,0}(Z_{1,2})-h^{0,0}(Z_{1})-h^{0,0}(Z_{2})=\Pi-1-(\Pi_{x}-1)-(\Pi_{y}-1)=$
$\displaystyle\Pi-\Pi_{x}-\Pi_{y}+1=l^{*}(F)$ (26)

Here $Z_{1}$ is the hypersurface defined by the equation $f(x_{1},0)$ and
$x_{1}\neq 0$, and $Z_{2}$ is defined by $f(0,x_{2})$ with $x_{2}\neq 0$.
$\Pi_{x}$ is the number of lattice points of $\Delta$ at $x$ axis, and
similarly for $\Pi_{y}$. $F$ is the one dimensional boundary within first
quadrant. The above formula gives rise to the number listed in table. 5.

Notice that what the table computes are the Hodge number of the compact
support cohomology, the MHS for ordinary cohomology is found using the
Poincare duality.

Example. Consider the polynomial map $f=x^{N}+y^{k}$ (here $x,y$ are both
$\mathbb{C}$ variables); We include the deformations so that the zero set of
$f$ is smooth, see the Newton polyhedron in Figure 14. Assume the maximal
common divisor for $(N,k)$ is $a$. The Hodge numbers are (using table. 5)

$h^{1,0}=h^{0,1}={(N-1)(k-1)-a+1\over 2},~{}~{}h^{1,1}=a-1.$ (27)

Here $h^{1,0}$ is the same as rank of the corresponding field theory, and
$h^{1,1}$ is equal to the number of mass parameters.

$x$$y$ Figure 13: The Newton polyhedron for $f=x^{4}+y^{6}+t$ (here $x,y$ are
both $\mathbb{C}$ variable and $t$ is chosen so that $f=0$ is smooth). The
number of interior points is $7$, and the number of point on the boundary in
the first quadrant is $1$. The Hodge numbers are $h^{1,0}=h^{0,1}=7$, and
$h^{1,1}=1$.

Now we consider the polynomial map
$f:\mathbb{C}^{\ast}\times\mathbb{C}\to\mathbb{C}$. We only consider the
_primitive_ cohomology of $Z=\\{f=0\\}$. (in this case, we need to subtract
the contribution from the ambient space). The primitive cohomology carries a
MHS, which has a weight one part and a weight two part. The Hodge numbers are
summarized in the table. 6.

$h^{1,0}=\ell^{\ast}(\Delta)$ | $h^{1,1}=l^{*}(F)+l^{*}(F_{x})$
---|---
| $h^{0,1}=\ell^{\ast}(\Delta)$
Table 6: $\ell^{\ast}(\Delta)$ is the interior points of $\Delta$, $F$ is the
one dimensional boundary where no coordinate vanishes identically, and $F_{x}$
is the boundary on the $x$ axis (here $x$ is a $\mathbb{C}$ variable).

We only need to explain $h^{0,0}$ for the compact support cohomology
$H_{c}^{1}(Z)$ which is given by (assuming $y$ is the $\mathbb{C}^{*}$
variable). First, we have (using formula 23):

$e^{0,0}(Z)=e^{0,0}(Z_{1,2})+e^{0,0}(Z_{2})=-(\Pi-1-(\Pi_{y}-1))=-(l^{*}(F)+l^{*}(F_{x})+1)$
(28)

Here $Z_{2}$ is defined by the equation $f(0,x_{2})$, and $F$ is the one
dimensional boundary inside first quadrant, and $F_{x}$ is the boundary on the
$x$ axis. We find that

$h^{0,0}(H^{1}_{c}(Z))=(l^{*}(F)+l^{*}(F_{x})+1)$ (29)

The primitive part has one less dimension:
$h^{0,0}(H^{1}_{c}(Z))_{prim}=l^{*}(F)+l^{*}(F_{x})$, which gives the desired
answer.

Example. Consider the polynomial map $f=x^{N}+z^{k}+\ldots$, here $x$ is a
$\mathbb{C}$ variable and $z$ is a $\mathbb{C}^{\ast}$ variable; We include
the deformations so that the zero set of $f$ is smooth, see the Newton
polyhedron in Figure 14. Assume the maximal common divisor for $(N,k)$ is $a$.
The Hodge numbers are (using formula in table. 6):

$h^{1,0}=h^{1,0}={(N-1)k-(a-1+N-1)\over 2},~{}~{}h^{1,1}=a-1+N-1.$ (30)

Here $h^{1,0}$ is the same as rank of the corresponding field theory, and
$h^{1,1}$ is equal to the number of mass parameters.

$x$$z$ Figure 14: The Newton polyhedron for $f=x^{4}+z^{6}+t$ (here $t$ is
chosen so that $f=0$ is smooth. The number of interior lattice points is $7$,
and the number of interior lattice points on two boundaries are $4$ (not
including the boundary of $z$ axis).

Threefold case. Let $f:\mathbb{C}^{4}\to\mathbb{C}$ be a polynomial map. Let
$\Delta$ be the Newton polyhedron of $f$ at infinity which is Newton non-
degenerate. Then the zero locus $Z$ of $f$ defines a 3-fold in
$\mathbb{C}^{4}$. Assume that $Z$ is smooth. Then the cohomology $H^{3}(Z)$ is
a mixed Hodge structure which possibly has weights $3,4,5,6$. These numbers
are summarized in the table. 7.

$h^{0,3}=\ell^{\ast}(\Delta)$ | $h^{1,3}=\sum_{\Gamma^{{}^{\prime}}}\ell^{\ast}(\Gamma^{{}^{\prime}})$ | $h^{2,3}=\sum_{F^{{}^{\prime}}}\ell^{\ast}(F^{{}^{\prime}})$ | $h^{3,3}=0$
---|---|---|---
| $h^{1,2}$ | $h^{2,2}$ | $h^{3,2}$
|  | $h^{2,1}$ | $h^{3,1}$
|  |  | $h^{3,0}=l^{*}(\Delta)$
Table 7: The Hodge numbers for $H^{3}(Z)$ of a smooth hypersurface $Z$ in
$\mathbb{C}^{4}$. Here $l^{*}(\sigma)$ denotes the internal lattice points of
a face $\sigma$. Here $\Gamma^{{}^{\prime}}$ denotes the three dimensional
face where no coordinates vanish identically, similarly $F^{{}^{\prime}}$
denotes two dimensional face where no coordinates vanish identically.

Let us explain the entries in Table 7. We take $h^{1,0}(H_{c}^{3}(Z))$999This
is equal to $h^{2,3}(H^{3}(Z))$ due to Poincaré duality. as an example. Using
formula. 23, we have

$\displaystyle
h^{1,0}(Z)=h^{1,0}(Z_{1234})-h^{1,0}(Z_{123})-h^{1,0}(Z_{124})-h^{1,0}(Z_{134})-h^{1,0}(Z_{234})$
$\displaystyle+h^{1,0}(Z_{12})+h^{1,0}(Z_{13})+h^{1,0}(Z_{14})+h^{1,0}(Z_{23})+h^{1,0}(Z_{24})+h^{1,0}(Z_{34})=$
$\displaystyle\sum_{2d~{}F~{}\in\Delta}l^{*}(F)-\sum_{2d~{}F\in\Delta_{ijk}}l^{*}(\Gamma)+\sum_{2d~{}F~{}\in~{}\Delta_{ij}}l^{*}(F)=\sum
l^{*}(F^{{}^{\prime}})$ (31)

In the last row, the first term consists of sum of the face of polytope
$\Delta$, and the second term consists of sum of the face of the three
dimensional polytope $\Delta_{ijk}$ (defined by setting one of the coordinate
in $f$ to be zero), and finally the last term consists of sum of the two
dimensional polytope $\Delta_{ij}$. Finally $F^{{}^{\prime}}$ is the two
dimensional face of three dimensional face $\Gamma^{{}^{\prime}}$ where no
coordinate vanishes identically. One can similarly get the result for
$h^{2,0}(Z)$ (for the compactly supported cohomology).

Let us now impose the condition that $f$ should describe the SW geometry of a
4d $\mathcal{N}=2$ theory. Then there is no internal points in $\Delta$
($l^{*}(\Delta)=0$), and also no internal points on $\Gamma^{{}^{\prime}}$ and
$F^{{}^{\prime}}$ (This is true for quasi-homogeneous case with $\sum q_{i}>1$
and we have to impose this for the general case); So $h^{3,0}=h^{0,3}=0$,
$h^{1,3}=h^{3,1}=0$, and $h^{2,3}=h^{3,2}=0$. The only non-trivial Hodge
numbers are:

$\displaystyle
h^{1,2}=h^{2,1}=l^{*}(2\Delta)-\sum_{\Gamma}l^{*}(\Gamma),~{}~{}$
$\displaystyle
h^{2,2}=-E+1-2h^{1,2}=\sum_{I}(-1)^{|I|-1}|I|!V_{|I|}(\Delta_{I})-2h^{1,2}+1$
(32)

So we need to know the lattice points of polytope in order to compute
$h^{1,2}$. Let us review some basic facts about lattice points of polytope.
For a $d$ dimensional polytope ${\cal P}$, we have following polynomials

$\displaystyle i({\cal P},n)=\\#\\{n{\cal P}\cap\mathbb{Z}^{d}\\}$
$\displaystyle\bar{i}({\cal P},n)=\\#\\{n{\cal P}^{o}\cap\mathbb{Z}^{d}\\}$
(33)

Here $i({\cal P},n)$ counts the lattice points of polytope $n{\cal P}$, and
$\bar{i}({\cal P},n)$ counts the interior lattice points of $n{\cal P}$.
$i({\cal P},n)$ is a polynomial in $n$ and is called Ehrhart polynomial.
Moreover, we have

$\bar{i}({\cal P},n)=(-1)^{d}i({\cal P},-n)$ (34)

The Ehrhart polynomial takes the following form

$i({\cal P},n)=\mathrm{Vol}(P)n^{d}+\ldots$ (35)

In practice, once we know the polytope ${\cal P}$ (One can use Macaulay2 mac
to compute it), we can compute $h^{1,2}$ and $h^{2,2}$ explicitly.

Example. Let us consider polynomial map given by $f=x^{3}+y^{3}+z^{3}+w^{3}$.
The corresponding Ehrhart polynomial is given by

$i({\cal P},n)={27\over 8}n^{4}+{45\over 4}n^{3}+{105\over 8}n^{2}+{25\over
4}n+1.$ (36)

and so

$\bar{i}({\cal P},n)={27\over 8}n^{4}-{45\over 4}n^{3}+{105\over
8}n^{2}-{25\over 4}n+1.$ (37)

Now we we use Ehrhart polynomial and formula (39) to compute the Hodge
numbers: We have $h^{2,1}=h^{1,2}=\bar{i}({\cal P},2)=5$ (There is no
contribution from the interior points of five faces of the polytope). To
compute $h^{2,2}$, we need to compute the volume of various polytopes, and we
have

$E=-4!{27\over 8}+4\times 3!{9\over 2}-6\times 2!{9\over 2}+4\times{3}=-15$
(38)

So we find $h^{2,2}=6$. The Coulomb branch spectrum of this theory can be
found in Xie:2015rpa , and the rank of Coulomb branch is 5, the number of mass
parameter is $6$. This gives the identification $r=h^{1,2}$ and $f=h^{2,2}$.

Now, let us consider a polynomial map
$f:\mathbb{C}^{\ast}\times\mathbb{C}^{3}\to\mathbb{C}$ which is nondegenerate
with respect to its Newton polyhedron. If we impose the condition that it
describes the SW geometry of a 4d $\mathcal{N}=2$ theory, then the non-trivial
Hodge numbers are again $h^{1,2}=h^{2,1}$ and $h^{2,2}$, see formula (39). The
formula for $h^{1,1}$ and $h^{1,2}$ is the same. One thing that we need to be
careful is the formula for $h^{2,2}$

$\displaystyle
h^{1,2}=h^{2,1}=l^{*}(2\Delta)-\sum_{\Gamma}l^{*}(\Gamma),~{}~{}$
$\displaystyle
h^{2,2}=-E-2h^{1,2}=\sum_{I}(-1)^{|I|-1}|I|!V_{|I|}(\Delta_{I})-2h^{1,2}$ (39)

We used the fact that $h^{3}(X)=-E$ for the current case, and the sum in the
second equation includes the set of $\mathbb{C}$ variable. So we see that the
formula for $h^{1,2}$ is not changed, but the formula for $h^{2,2}$ is
changed, as we only consider the set $I\subset\\{1,2,3\\}$ with
$x_{1},x_{2},x_{3}$ $\mathbb{C}$ variables, and there is a difference one
because we only consider primitive cohomology.

Example. Consider the polynomial map given by
$f=x_{1}^{2}+x_{2}^{2}+x_{3}^{3}+z^{9}$, where $z$ is a $\mathbb{C}^{*}$
variable. The polytope is generated by four lattice points

$(0,0,0,0),(2,0,0,0),(0,2,0,0),(0,0,3,0),(0,0,0,9)$ (40)

and the Ehrhart polynomial is given by

$i({\cal P},n)={9\over 2}n^{4}+{27\over 2}n^{3}+{14}n^{2}+{6}n+1.$ (41)

So $l^{*}(2\Delta)=\bar{i}(P,2)=i(P,-2)=9$ (no contribution), and the
following two faces would also contribute to the computation:

$\displaystyle\Gamma_{1}:(0,0,0,0),(2,0,0,0),(0,0,3,0),(0,0,0,9)$
$\displaystyle\Gamma_{2}:(0,0,0,0),(0,2,0,0),(0,0,3,0),(0,0,0,9)$ (42)

The corresponding Ehrhart polynomial is $i(\Gamma,n)=9n^{3}+{27\over
2}n^{2}+{13\over 2}n+1$, so we find
$l^{*}(\Gamma_{1})=l^{*}(\Gamma_{2})=i(P,-1)=1$. This gives that
$h^{1,2}=9-2=7$. The Euler number of the hypersurface is

$E=-4!{9\over 2}+3!(6+9\times 2)-1\times 2!\times{27\over 2}-2\times 2!\times
9+9=-18$ (43)

so we find $h^{2,2}=4$. This equals the number of mass parameters of the
corresponding four dimensional $\mathcal{N}=2$ model, see Xie:2012hs .

Summary. From explicit computation, we see that the weight filtration of
middle cohomology $H_{prim}^{n-1}(X)$ have two weights $n-1$ and $n$ ($n=2$ or
$n=4$) 101010Here the embedding dimension of hypersurface $X$ is $n$.. The
weight $n-1$ part has the Hodge decomposition
$\mathrm{Gr}^{W}_{n-1}H^{n-1}(X)=H^{{n\over 2},{n\over 2}-1}\oplus H^{{n\over
2}-1,{n\over 2}}$, while the weight $n$ piece has the decomposition
$\mathrm{Gr}^{W}_{n}H^{n-1}(X)=H^{{n\over 2},{n\over 2}}$. Based on this fact,
we would make the following statement:

######  Statement.

The weight $n-1$ piece of the primitive middle cohomology $H^{n-1}_{prim}(X)$
gives the electric-magnetic charge, and the weight $n$ part gives the flavor
charge.

This result implies that dimension of weight $n$ part of MHS is equal to $2r$
where $r$ is the rank of the theory, and dimension of weight $n+1$ part of
Mixed Hodge Structure is equal to $f$ which is the number of mass parameters.
So the MHS would solve the problem regarding separating the electric-magnetic
and flavor charges.

Constraints on the Newton polyhedra. If we require a hypersurface to have
above type of MHS, then the Newton polyhedron for the polynomial
$f:U\to\mathbb{C}$ is constrained. The constraints are as follows.

  * •

If the hypersurface is one dimensional, there is no constraint on $U$ and $f$.
$U$ can be either $\mathbb{C}\times\mathbb{C}$,
$\mathbb{C}\times\mathbb{C}^{*}$, or $\mathbb{C}^{*}\times\mathbb{C}^{*}$. To
describe the Coulomb branch of 4d $\mathcal{N}=2$ theories, we require $U$ to
be $\mathbb{C}\times\mathbb{C}$ or $\mathbb{C}\times\mathbb{C}^{*}$. The case
$\mathbb{C}^{*}\times\mathbb{C}^{*}$ is relevant for the study of circle
compactification of 5d $\mathcal{N}=1$ SCFT.

  * •

If the hypersurface is three dimensional, there are strong constraints on $U$
and $f$.

    1. (1)

If $U=\mathbb{C}^{4}$, the polytope $\Delta$ has to satisfy following
condition: $l^{*}(\Delta)=0,l^{*}(F^{\prime})=0,l^{*}(\Gamma^{\prime})=0$.
Here $\Gamma^{\prime}$ and $F^{\prime}$ are the three dimensional face and two
dimensional face, where no coordinate vanishes identically. In the quasi-
homogeneous case, this implies that $\sum q_{i}>1$, where $q_{i}$ are the
weights of the coordinate $x_{i}$.

    2. (2)

Now let us consider $U$ consists of one $\mathbb{C}^{*}$ variable. To ensure
$h^{3,0}$ to be zero, the polytope $\Delta$ should have no interior point. We
now also have new constraint from vanishing of $h^{1,0}$ and $h^{2,0}$. Now
$h^{1,0}$ and $h^{2,0}$ (of compactly supported cohomology) takes following
form (see formula 31):

$h^{1,0}(Z)=\sum l^{*}(F^{{}^{\prime}})+\sum_{F\in\Delta_{123}}l^{*}(F),\quad
h^{2,0}=\sum l^{*}(\Gamma^{{}^{\prime}})+l^{*}(\Delta_{123})$ (44)

Here $\Delta_{123}$ is the polytope formed from $f$ by setting the
$\mathbb{C}^{*}$ variable to be zero. The vanishing of $h^{1,0}(Z)$ then
implies $l^{*}(F)=0,~{}F\in\Delta_{123}$, which is a new constraint.
Similarly, the vanishing of $h^{2,0}(Z)$ implies $l^{*}(\Delta_{123})=0$. For
a quasi-homogeneous case, this implies that the polynomial $f_{123}$ (formed
from $f$ by setting $\mathbb{C}^{*}$ variable to be zero):

$\sum_{i=1}^{3}q_{i}>1$ (45)

If the sum of the weights are bigger than one, the polynomial $f_{123}$ is
constrained to be ADE singularity.

We get above constraint by requiring the middle cohomology has only weight 3,
and weight 4 pieces. We might further relax the constraints by requiring that
each weight has only one type of Hodge numbers, then we have:

$\displaystyle h^{3,0}=0,\quad\rightarrow~{}l^{*}(\Delta)=0$ $\displaystyle
h^{2,0}=0,\quad\rightarrow~{}l^{*}(\Delta_{123})=0$ (46)

and for the quasi-homogeneous case, this implies that the weights of
polynomials $f_{123}$ should satisfy the following condition

$\sum_{i=1}^{3}q_{i}\geq 1.$ (47)

The case $\sum_{i=1}^{3}q_{i}=1$ is new and they are useful to study the torus
compactification of 6d $(1,0)$ theory.

    3. (3)

$U$ consists of maximal two $\mathbb{C}^{*}$ variables, and the polynomial
takes the form $f(z_{1},z_{2})+x^{2}+y^{2}$, here $x,y$ are $\mathbb{C}$
variables, and $z_{1},z_{2}$ are $\mathbb{C}^{*}$ variables. This case does
not add anything new.

###  3.4 Abelian variety and mixed Hodge structure

As we discussed in section 2, the low energy photon couplings should be
identified with the complex structure of the abelian variety, and so we should
get an abelian variety from the SW geometry Donagi:1995cf . In the compact
curve case, the abelian variety is found from the Jacobian variety of the
curve. Now that we have the open variety, it is not easy to get an abelian
variety from the cohomology of the variety. We now show that we did get an
abelian variety from MHS for the polynomials used to describe a
$\mathcal{N}=2$ theory.

Let us briefly review basics about abelian variety. An $n$ dimensional complex
torus is defined as $\mathbb{C}^{n}/\Lambda$, Here $\Lambda$ is a $2n$
dimensional lattice. The lattice is generated by $2n$ linearly independent
vectors in the vector space $\mathbb{R}^{2n}$. Using the standard complex
structure, we can use a $n\times 2n$ complex matrix $A$ to encode the lattice.
By choosing a suitable basis of the lattice, the matrix for the lattice can be
taken to be of form $[I_{n}|B]$. If the imaginary part of $B$ is positive
definite, the corresponding complex torus is called an abelian variety.

Abelian varieties arise naturally from the polarized Hodge structure of weight
$1$. Let $H=(H_{\mathbb{Z}},H^{1,0},H^{0,1})$ be a Hodge structure of weight
$1$. Let $2g$ be the rank of $H_{\mathbb{Z}}$. Then
$A=H^{1,0}/H_{\mathbb{Z}}\cong\mathbb{C}^{g}/\mathbb{Z}^{2g}$ is a a complex
torus.

A complex torus $A$ is an abelian variety if and only if it can be
holomorphically embedded into a complex projective space. In general, the
complex torus $A$ constructed from a plain Hodge structure of weight $1$ is
not an abelian variety. A sufficient and necessary condition for $A$ to be an
abelian variety is that $H$ is polarizable. If $S$ is a polarization on $H$,
it gives rise to a line bundle on $A$, and the Hodge–Riemann bilinear relation
translates to the positivity of this line bundle.

The weight one part of the MHS of an open curve is polarizable (the weight one
part comes from compact Riemann surface which carries a polarizable weight one
Hodge structure), we can obtain an abelian variety from it.

Now consider a Hodge structure
$H=(H_{\mathbb{Z}},H^{3,0},H^{2,1},H^{1,2},H^{0,3})$ of weight $3$. Let again
$2g$ be the rank of $H_{\mathbb{Z}}$. Then we can mimic the previous
construction and define a complex torus $\mathrm{IJ}=(H^{3,0}\oplus
H^{2,1})/H_{\mathbb{Z}}$ of dimension $g$.

This time, even if $H$ admits a polarization, the Hodge–Riemann bilinear
relation does not ensure that $\mathrm{IJ}$ is an abelian variety. Because it
only says that $\sqrt{-1}S(v,\overline{v})$ is positive definite on $H^{2,1}$,
but is negative definite on $H^{3,0}$. So the line bundle defined by $S$ is
not positive.

Fortunately, for the open variety we considered, $H^{3,0}=0$. See the remark
at the end of §3.3 The positivity provided by the Hodge–Riemann bilinear
relation tells us that $\mathrm{IJ}=H^{2,1}/H_{\mathbb{Z}}$ is indeed an
abelian variety.

##  4 IR theory: special vacua

In last section, we studied the MHS defined on the cohomology of a typical
fiber of the SW geometry. The typical fiber describes the generic vacuum, and
the MHS is crucial for defining an abelian variety, and useful in separating
the electric-magnetic and flavor charges. In this section, we study special
vacua, which are described by atypical fibers.

Related to the cohomology of atypical fibers are three mixed Hodge structures:

  1. (1)

In “Théorie de Hodge III” deligne:hodge3 , Deligne proves that the cohomology
groups of the singular fiber $F_{\lambda}=0$ carries a functorial mixed Hodge
structure.

  2. (2)

As we discussed in last section, the topology of the atypical fiber is related
to the behavior of the nearby fibers. We have shown that the every nearby
fiber carries a MHS. When we vary the parameters $\lambda$, this MHS changes
accordingly and we get a “variations of MHS”. For this variation, there is a
limit MHS as the parameters approach to atypical fiber, due to Schmid
schmid:variation-hodge-structure  and Steenbrink–Zucker steenbrink-
zucker:vmhs-1 . The Hodge filtration is given by the limit Hodge filtration of
the smooth fibers, and the weight filtration now is given by the nilpotent
part of the monodromy matrix.

  3. (3)

When we pushing to the atypical fiber, some elements of the cohomology group
become zero. These vanishing cycles also carry a limit MHS, due to Steenbrink
steenbrink:mixed-hodge-on-vanishing-cohomology .

These three MHS are not independent, they form an exact sequence. We refer the
reader to kulikov1998mixed  for a comprehensive review. We shall explain how
can one determine the low energy physics at a special vacuum using these three
MHS.

###  4.1 MHS for the singular variety

The middle cohomology group of an open smooth variety carries a MHS as we
reviewed in last section. The weights of the MHS on middle cohomology group
takes the range $n\leq w\leq 2n$ (Here $n$ is the complex dimension of the
variety). For an open singular variety, it is still possible to define a MHS
on it deligne:hodge3 . This time, the range of weights $w$ on the middle
cohomology satisfy $0\leq w\leq 2n$. We do not attempt to give a general
description of MHS of singular variety, but content to giving an example to
show how the computation can be reduced to the that of smooth variety reviewed
in last section.

Consider a singular variety $X$, and assume that the singular locus $S$ of $X$
is smooth. Then we can find a resolution of singularity:

${E}$${\widetilde{X}}$${S}$${X}$

where $E$ is a normal crossing divisor in $\widetilde{X}$. Then the cohomology
of $X$ is determined by those of $\widetilde{X}$, $S$, and $E$, via an exact
sequence

$\cdots\to H^{i}(X)\to H^{i}(\widetilde{X})\oplus H^{i}(S)\to H^{i}(E)\to
H^{i+1}(X)\to\cdots$

As we have explained, the cohomology of a smooth variety always has a MHS,
therefore the mixed Hodge structure of $H^{i}(X)$ can be computed by above
sequences.

In general, the singular locus $S$ of $X$ may again be singular. Nevertheless,
we know the singular locus has lower dimension. Thus we could inductively
determine the MHS of $S$ using the procedure above, and then proceed to
determine that of $X$.

Example 1. Let $C$ be a possibly singular Riemann surface with $n$ ordinary
double points. Let $\widetilde{C}$ be its normalization (resolution). Assume
that $\widetilde{C}$ is obtained from a _compact_ Riemann surface of genus $g$
with $m$ points removed. Then the mixed Hodge structure of $C$ is summarized
as follows ($H=H^{1}(C)$):

$\mathrm{Gr}_{0}^{W}H$ | $\mathrm{Gr}_{1}^{W}H$ | $\mathrm{Gr}_{2}^{W}H$
---|---|---
$H^{0,0}$ | $H^{1,0}\oplus H^{0,1}$ | $H^{1,1}$
$n$ dim. |  $2g$ dim. |  $m-1$ dim.
Table 8: The MHS of a genus $g$ curve with $n$ double points, and $m$ points
removed.

The weight $2$ terms of the mixed Hodge structure are from the two punctures,
whereas the weight zero part of the cohomology are from the nodal points.

Figure 15: The MHS on a $H^{1}(C)$, here $C$ is a curve with derived from
removing $m$ points of a genus $g$ compact Riemann surface with a double
point. The weight one part of $H^{1}(C)$ comes from the cycles of the original
compact Riemann surface, and weight two comes from the cycles around the
removed points. Weight zero part $(\gamma_{1}^{*})$ comes from the cycles
around the double point.

Example 2. Let $C$ be a singular curve defined by the affine equation
$x^{3}+y^{3}+y^{4}=0$. This curve has a singular point at the origin
$S=(0,0)$. The singularity of $S$ is equivalent to that of $x^{3}+y^{3}=0$ at
the origin. The singularity can be resolved by blowing up the $(x,y)$-plane at
the point $S$. Then $E$ consists of $3$ points. The MHS of $\tilde{C}$ can be
computed using the method given in last section (one first compute the genus
of $\tilde{C}$ using the Pluker formua, the weight two part is found by doing
blow-up of the singular point at infinity), and we have $g=0$. We have
following exact sequence (see formula 4.1):

$\displaystyle 0\to H^{0}(C)\to H^{0}(\widetilde{C})\oplus H^{0}(S)\to
H^{0}(E)\to\boxed{H^{1}(C)}\to H^{1}(\tilde{C})\oplus H^{1}(S)\to H^{1}(E)$
$\displaystyle\to H^{2}(C)\to H^{2}(\tilde{C})\oplus H^{2}(S)\to 0$

From which we find that $h^{1,0}=h^{0,1}=0$, and $h^{0,0}=2$.

Physical interpretation. Now we explain how to extract the low energy physics
from the MHS associated with the singular variety. Let us assume the SW
geometries are curves, and the general cases are similar. The MHS for
$H^{1}(C)$ has weight zero, weight one, and weight two parts. The weight one
part gives an abelian gauge theory; the weight two part gives the flavor
central charge. The weight zero part is new: they are interpreted as gauging
the $U(1)$ flavor symmetry for the interacting theory associated with the
singular points. In next subsection, we will discuss how to find the
interacting theory from the so-called limit MHS.

###  4.2 Limit Mixed Hodge Structure

We have discussed the MHS on the cohomology of the singular fiber. Its weight
one piece was interpreted as the abelian gauge theory part of the special
vacuum. However, the singular points of the variety should be able to describe
a more complicated part of the theory, i.e., a SCFT or an IR free non-abelian
gauge theory. We argue that the limit mixed Hodge structure will allow us to
extract this more intricate piece of information.

Variation of (mixed) Hodge structure: Let us first describe briefly variation
of (mixed) Hodge structures. Let $f:U\to B$ be a map of algebraic varieties
which is a locally topologically trivial fibration. This is the case for
example when $f$ is proper and is a submersion of manifolds (Ehresmann’s
lemma). In this case, the cohomology $H^{m}(U_{b})$ of the fibers of $f$ form
a local system $\mathbb{V}$ on $B$. Let
$\mathcal{V}=\mathbb{V}\otimes_{\mathbb{Q}}\mathcal{O}_{B}$ be the associated
vector bundle on $B$. Thus $\mathcal{V}$ is equipped with a flat connection
$\nabla$ known as the Gauss–Manin connection. The following are known.

  * •

The fiber-by-fiber weight filtrations on $\mathbb{V}$ are horizontal, i.e.,
they form a sub-local system $W_{\bullet}$ of $\mathbb{V}$.

  * •

The Hodge filtration $F^{\bullet}\mathcal{V}$ are holomorphic subbundles of
$\mathcal{V}$, but they are not horizontal (i.e., $\nabla$ does not respect
$F^{\bullet}\mathcal{V}$ in general).

  * •

Instead, the so-called “infinitesimal period relation” holds: for any tangent
vector field $v$ on $B$

$\nabla_{v}F^{i}\mathcal{V}\subset F^{i-1}\mathcal{V}.$

It is therefore not difficult to make an abstract version of the above and
define a variation of MHS. A variation of mixed Hodge structure on a complex
manifold $B$ then consists of

  * •

a local system $\mathbb{V}$ of $\mathbb{Q}$-vector spaces, whose associated
vector bundle with connection is denoted by $(\mathcal{V},\nabla)$;

  * •

sub-local systems $\mathbb{W}_{\bullet}$ such that
$\mathbb{W}_{i}\subset\mathbb{W}_{i+1}$; and

  * •

holomorphic subbundles $F^{\bullet}\mathcal{V}\subset\mathcal{V}$.

These data satisfy the following requirements:

  * •

the infinitesimal period relation holds: $\nabla_{v}F^{i}\mathcal{V}\subset
F^{i-1}\mathcal{V}$;

  * •

for each $b\in B$, the triple $(\mathbb{V}_{b},\mathbb{W}_{k,b},F_{k,b})$ is a
mixed Hodge structure.

If in a variation of mixed Hodge structure the fiberwise mixed Hodge
structures are pure, we then say this is a variation of pure Hodge structure,
or simply variation of Hodge structure. A _polarization_ of a variation of
Hodge structure $(\mathbb{V},F^{\bullet})$ on a complex manifold $B$ is a
horizontal map

$S:\mathbb{V}\otimes\mathbb{V}\to\mathbb{C}_{B}$

that induces a polarization on each fiber. If $(\mathbb{V},\mathbb{W},F)$ is a
variation of mixed Hodge structure, such that $(\mathrm{Gr}^{W}_{k},F)$ admits
a polarization, then we say $(\mathbb{V},\mathbb{W},F)$ is _graded
polarizable_.

For example, if $f:X\to B$ is a smooth projective map of algebraic varieties
the cohomology bundle $R^{m}f_{\ast}\mathbb{Q}$ then underlies a variation of
_pure_ Hodge structure which admits a polarization.

If $f:X\to S$ is a morphism of complex algebraic varieties, then there exists
a Zariski open subset $U\subset S$ such that $R^{m}f_{\ast}\mathbb{Q}$
underlies a variation of mixed Hodge structure. If $f$ is quasi-projective,
then $R^{m}f_{\ast}\mathbb{Q}$ underlies a graded polarizable variation of
mixed Hodge structure.

If we have a variation of polarized Hodge structure on a punctured disk
$\Delta^{*}$, then a theorem of Schmid schmid1973variation  says that the
Hodge filtration (holomorphic subbundles) can be extended to the origin. This
limit filtration, together with a monodromy weight filtration (see below),
form a MHS called the limit MHS. The limit Hodge filtration comes from the
Hodge structure of the nearby fibers, while the limit weight filtration comes
from the monodromy operator. This result is extended to a nice class of
variations of mixed Hodge structure by Kashiwara kashiwara:a-study-vmhs  and
Steenbrink–Zucker steenbrink-zucker:vmhs-1 . In the situation that interests
us, we should really use the theory of Kashiwara–Steenbrink–Zucker. However in
this paper we only need to perform computations about Hodge numbers, so we
confine ourselves to review the result of Schmid only.

Monodromy theorem. In order to define the monodromy weight filtration of the
limit MHS, we need some knowledge about the monodromy action. Suppose we have
a continuous map of topological spaces

$\pi:X\to\Delta.$

Denote by $X_{t}$ the fiber $\pi^{-1}(t)$. Assume that
$\pi|_{X-X_{0}}:X\setminus X_{0}\to\Delta\setminus\\{0\\}$ is a locally
topologically trivial fibration. The cohomology groups $H^{i}(X_{t})$, with
varying $t$, patch together and form a “cohomology bundle” $\mathbf{V}$ on
$\Delta\setminus\\{0\\}$ (thus the fiber of $\mathbf{V}$ at $t$ is
$H^{i}(X_{t})$).

For a generic element $\eta\in\Delta\setminus\\{0\\}$, we can draw a circle
$\xi$ based on $\eta$ and contains $0\in\Delta$ in its interior. Then parallel
transport along the path $\eta$ defines an automorphism

$h:H^{i}(X_{\eta})\to H^{i}(X_{\eta})$

called the _monodromy transform_ or _Picard–Lefschtz transform_. Thus, to
describe the cohomology bundle we only need two pieces of information: a
vector space $V$ (which is the vector space that underlies the various
$H^{i}(X_{t})$) and a linear operator $h$ (which prescribes the monodromy
operator).

A typical example is demonstrated in the following diagram. In this diagram,
we have a family of tori fibered over the punctured disk. When approaching to
zero, a cycle $\delta$ is shrinking to zero. Going around the circle, the
cycle $\gamma$ is transformed via the monodromy transform to $\gamma+\delta$.
This picture is known in topology as the “Dehn twist”.

In this case, the monodromy group action is expressed in terms of famous
Picard-Lefschtz formula:

$h(\sigma)=\sigma+(\delta,\sigma)\delta;$ (48)

Here $\sigma$ is an arbitrary cycle, $\delta$ is the vanishing cycle, and
$(\sigma,\delta)$ is the intersection number of these two cycles.

If $\pi\colon X\to\Delta$ is a degenerating family of algebraic varieties,
then we have the famous monodromy theorem: the monodromy operator $h$ is
quasi-unipotent:

$(h^{q}-1)^{n+1}=0$ (49)

Here $n$ is the dimension of the variety. The monodromy theorem implies that

  1. (1)

the eigenvalues of the monodromy operator are roots of unity,

  2. (2)

the size of the Jordan block of $T$ is less than $n+1$.

In the above setting, we discussed the monodromy action $h$ acting on the
homology group. If we look at the cohomology group, the monodromy group is
given by $h^{*}=(h^{T})^{-1}$.

### The limit Hodge filtration

Now assume that we have a variation of polarized Hodge structure (with weight
$\omega$) over a punctured disk $\Delta^{*}$ with coordinate $t$. So we have a
local system $\mathbf{V}$ together with a monodromy operator $h$ (satisfying
monodromy theorem), holomorphic sub-bundles $F^{\bullet}(t)$, and polarization
$S(t)$. Denote by

$\displaystyle p:\mathfrak{h}$
$\displaystyle=\\{z\in\mathbb{C}:\mathrm{Im}(z)>0\\}\to\Delta^{*},$
$\displaystyle z$ $\displaystyle\mapsto t=\exp(2\pi\mathrm{i}z)$

the universal covering of $\Delta^{*}$ by the upper half plane. Since
$\mathfrak{h}$ is simply connected, the pullback bundle $p^{-1}\mathbf{V}$ is
a trivial local system on $\mathfrak{h}$. Fix a trivialization of
$p^{-1}\mathbf{V}$ on $\mathfrak{h}$ and denote by $V$ the complex vector
space represents a typical fiber of $\mathbf{V}$. The space $V$ has the
following structures.

  1. (1)

Polarization. The polarization $S$ induces a sesquilinear form on the complex
vector space $V$, which we still denote by $S$.

  2. (2)

Monodromy. Since the space $\mathfrak{h}$ endows an automorphism $z\mapsto
z+1$ which generates the group of deck transformations of $p$, the monodromy
action defines an endomorphism $T$, called the _monodromy operator_ , on $V$.
The operator $T$ preserves the polarization.

  3. (3)

Hodge Filtrations. The Hodge filtration $F^{\bullet}$ on $\mathcal{V}$ induces
a holomorphically varying family of flags on $V$ which we shall denote by
$F^{\bullet}(z)$. Then by the definition of $T$ we have
$hF^{\bullet}(z)=F^{\bullet}(z+1)$.

The monodromy theorem implies that $h$ can be written as

$h=T_{s}\cdot\exp(2\pi\mathrm{i}N)$ (50)

of a semisimple matrix whose entries are all of length $1$, and the
exponential of a nilpotent matrix, in a way that $[T_{s},N]=0$. We can
decompose the space $V$ into the sum of the eigenspaces of $T_{s}$:

$V=\bigoplus_{|\lambda|=1}E_{\lambda}(T_{s})$ (51)

where the nilpotent operator $N$ preserves this decomposition. In particular,
the monodromy invariant part (invariant vector) of $V$ equals
$\mathrm{Ker}(N|_{E_{1}(T_{s})})$, and the monodromy coinvariant part of $V$
equals $\mathrm{Coker}(N|_{E_{1}(T_{s})})$. It can be shown that the
decomposition (51) is orthogonal with respect to the polarization form $S$.

Schmid studied the behavior of the the filtrations $F^{\bullet}(z)$ when
$\mathrm{Im}(z)$ turns to $\infty$. He proved the the limit of a suitable
twist (see (52) below) of $F^{\bullet}(z)$ exists. We write

$F^{\bullet}=\lim_{\mathrm{Im}(z)\to\infty}\exp(-2\pi\mathrm{i}zN)\cdot
F^{\bullet}(z).$ (52)

for the limit of the Hodge filtrations. This is called the limit Hodge
filtration. Note that one can easily verify, using the fact
$hF^{\bullet}(z)=F^{\bullet}(z+1)$, that we have

$\exp(2\pi\mathrm{i}(z+1)N)F^{\bullet}(z+1)=T_{s}\exp(2\pi\mathrm{i}zN)F^{\bullet}(z).$
(53)

Hence the limit Hodge filtration is thus preserved by the semisimple part
$T_{s}$ of the monodromy operator:

$h\cdot F=F,$ (54)

hence the limit Hodge filtration is compatible with the decomposition (51).

The vector space $V$ now has been endowed with

(a) a polarization $S$,

(b) a monodromy operator $T=T_{s}\exp(2\pi\mathrm{i}N)$,

(c) an $S$-orthogonal decomposition (51), and

(d) a limit Hodge filtration $F^{\bullet}$.

Deligne observed that the presence of the nilpotent operator $N$ also puts
another increasing filtration $M_{\bullet}$, the _monodromy weight filtration_
, on the space $V$. Assume $N^{k+1}=0$, then this filtration satisfies the
following properties:

  1. (1)

$N(M_{\ell})\subset M_{\ell+2}$,

  2. (2)

$M_{\ell}=0$ if $|\ell|>k$,

  3. (3)

$N^{\ell}:\mathrm{gr}^{M}_{\ell}(V)\to\mathrm{gr}^{M}_{-\ell}(V)$ is an
isomorphism.

One defines the _primitive part_ of $\mathrm{gr}^{M}_{\ell}(V)$ to be the
kernel of $N^{\ell+1}$. Then the usual $\mathfrak{sl}_{2}$-argument decomposes
the spaces $\mathrm{gr}^{M}_{\ell}(V)$ into the sum of primitive parts.

Deligne conjectured that the filtration $M$, plus the limit Hodge filtration
$F$, defines a mixed Hodge structure on the vector space $V$. This is proved
by Schmid schmid1973variation . Precisely, we have

Schmid’s theorem. The data $(V,M_{\bullet},F^{\bullet},N,S)$ is a graded
polarized mixed Hodge structure on $V$, in the sense that

$(\mathrm{gr}^{M}_{\ell}(V){}_{\textup{prim}},F,S(\cdot,N^{\ell}\cdot))$

is a polarized Hodge structure of weight $w+\ell$.

This mixed Hodge structure is called the _limit mixed Hodge structure_ of
$\mathcal{V}$. A very useful interpretation is following: Deligne defines a
canonical extension of the holomorphic vector bundle on $\Delta^{*}$ to the
whole disk $\Delta$, and the limit MHS can be thought of as a natural mixed
Hodge structure on the fiber of canonical extension at the origin of the disk.

Example. Let us give an example of limit MHS coming from degeneration of
Riemann surfaces of genus $2$. For instance, such a degeneration is given by
the projective completion of

$X_{t}:y^{2}=x(x-1)(x-2)(x-3)(x-t)$

where $t$ is a number sufficiently close to $0$. In this example, the
monodromy operation on $X_{t}$ is already unipotent. To determine the Hodge
filtration on cohomology of nearby cycles, we first fix a homology basis
$\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}$ of the curve $X_{t}$. Then the
Hodge filtration on $F_{t}$ is determined by the period matrix $Z(t)$ (i.e.,
the matrix whose column vectors are the integration of the two linearly
independent abelian differentials on $\gamma_{i}$). The multivalued matrix
function $Z(t)$ should be thought of as a single valued function from
$\mathfrak{h}$ into period domain. A standard trick of “untwisting” gives rise
to a different matrix valued function:

$\widehat{Z}(t)=\exp(-2\pi\sqrt{-1}tN)Z(t)$

which does not land in the period domain, but is single valued. At any rate,
Schmid’s nilpotent orbit theorem states that $\widehat{Z}(0)$ exists in a
suitable flag space containing the period domain. This gives the limit Hodge
filtration.

The monodromy weight filtration is a purely topological quantity. In the
present case, the degeneration is caused by the vanishing cycle $\gamma_{1}$,
see figure. 16 (the intersection number is given as
$(\gamma_{1},\gamma_{2})=-1$). And the monodromy action on the cohomology
group is given by the Picard–Lefschetz formula:

$T=\left[\begin{array}[]{cccc}1&0&0&0\\\ 1&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
\end{array}\right]$ (55)

Write $T=(\exp(2\pi\sqrt{-1}N)$, and $N$ takes the following form

$N=\left[\begin{array}[]{cccc}0&0&0&0\\\ 1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}\right]$ (56)

The weight filtration is then given by (here $\gamma_{i}^{*}$ is the dual
basis in the cohomology groups):

$W_{0}=\\{\gamma_{2}^{*}\\},\quad
W_{1}=\mathrm{Ker}(N)=\\{\gamma_{2}^{*},\gamma_{3}^{*},\gamma_{4}^{*}\\},\quad
W_{2}=\\{\gamma_{1}^{*},\gamma_{2}^{*},\gamma_{3}^{*},\gamma_{4}^{*}\\}$

The monodromy invariant part of the cohomology is generated by the basis
$\\{\gamma_{2}^{*},\gamma_{3}^{*},\gamma_{4}^{*}\\}$, and is the same as the
cohomology of the singular variety.

$\gamma_{2}$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$$\gamma_{4}$$\gamma_{3}$$\gamma_{4}$
Figure 16: The degeneration of a genus two Riemann surface. Here $\gamma_{1}$
is a vanishing cycle.

###  4.3 Limit MHS on vanishing cohomology

We have recalled two MHS associated with singular fiber: the MHS associated
with singular variety itself, and the limit MHS using the variation of Hodge
structures of nearby fibers. Here we will introduce a third MHS associated
with singular fiber: the MHS on vanishing cohomology. We refer the reader to
kulikov1998mixed  for a more detailed introduction.

We assume that the atypical fiber has at worst isolated singularities. A
feature of the vanishing cohomology is that it is concerns only the local
behavior of the polynomial map near its critical points. Thus we only need to
consider the function germ modeling the local behavior. Let
$f:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ be a germ of a holomorphic function.
Let $\partial_{i}f=\partial f/\partial x_{i}$. Define the Jacobian algebra to
be

$J_{f}=\frac{\mathbb{C}[\\![x_{0},\ldots,x_{n}]\\!]}{(\partial_{0}f,\cdots,\partial_{n}f)}=\frac{\mathbb{C}\\{x_{0},\ldots,x_{n}\\}}{(\partial_{0}f,\cdots,\partial_{n}f)}.$

We say $0$ is an isolated critical point of $f$ if $J_{f}$ is a finite
dimensional $\mathbb{C}$-vector space.

It is known milnor:singular-points-complex-hypersurfaces  that there is a
neighborhood $X$ of $0$ in $\mathbb{C}^{n}$ and a small neighborhood
$S=\\{t\in\mathbb{C}:|t|<\epsilon\\}$ of $0\in\mathbb{C}$ such that $f$
defines a holomorphic map $f:X\to S$ such that

  * •

$X-f^{-1}(0)\to S-\\{0\\}$ is a locally topologically trivial fibration,

  * •

$f^{-1}(0)$ is a strong deformation retract of $X$, and

  * •

$X$ is homeomorphic to a cone over $f^{-1}(t)$ .

We shall refer to $f:X\to S$ as a _good representative_ of $f$. Moreover, when
$f$ has an isolated critical point at $0$, Milnor proved that for $t\in
S-\\{0\\}$, $f^{-1}(t)$ is homotopy equivalent to a bouquet of $n$-dimensional
spheres. Therefore,

$H^{n}(f^{-1}(t),\mathbb{Z})\cong\mathbb{Z}^{\mu},\quad\forall t\in
S-\\{0\\},$

and we shall refer to this cohomology group as the _vanishing cohomology_ of
the germ $f$. It is a theorem of Milnor that $\mu=\dim_{\mathbb{C}}J_{f}$.

One can define a MHS on the vanishing cohomology of a germ of a holomorphic
function with isolated critical point, see steenbrink1976mixed . In the
following, we will give an algebraic way of computing MHS for the vanishing
cohomology, due to Varchenko, Morihiko Saito, Scherk and Steenbrink
scherk1985mixed .

Gauss–Manin system. First, we would like to describe cohomological bundles.
Given a good representative $f:X\to S$, the sheaf $R^{n}f_{\ast}\mathbb{C}$
capturing the cohomology of fibers of $f$ admits an equivalent description in
terms of D-modules. D-modules refer to the (sheaves of) modules of the ring of
differential operators defined on $S$. Denote the local coordinate on $S$ as
$t$, and the derivative as $\partial_{t}$, the ring of differential operators
is generated by $t$ and $\partial_{t}$ with relation $[\partial_{t},t]=1$.

The vanishing cohomology groups $H^{n}(f^{-1}(t),\mathbb{C})$ can be described
by a D-module, known as the “Gauss–Manin system” of $f$, denoted by $M$. It is
defined as follows. First we consider the “twisted” de Rham complex

$(\Omega^{\bullet}_{X}[\tau],\mathrm{d}+\tau\mathrm{d}f\wedge),$

where $\tau$ is a formal parameter, and $\Omega^{m}_{X}$ is the sheaf of
holomorphic $m$-forms on $X$. We define an action of $\partial_{t}$ on complex
$\Omega^{\bullet}_{X}[\tau]$ by

$\partial_{t}(\tau^{m}\omega)=\tau^{m+1}\omega,$

and define an action of $\mathcal{O}_{S}$ by

$t\cdot(\tau^{m}\omega)=\tau^{m}f\omega-m\tau^{m-1}\omega.$

One checks that the actions of $t$ and $\partial_{t}$ commutes with the
differential $d+\tau\mathrm{d}f\wedge$. Let $M$ be the $(n+1)$th cohomology of
this complex. Then $M$ has an action of $\partial_{t}$ and $t$, thereby
obtaining a structure of D-module on $S$. We call $M$ the _Gauss–Manin system_
of the $f$. An important property of $\partial_{t}$ action in the current case
is

Theorem (Brieskorn brieskorn:monodromy ). The map $\partial_{t}:M\to M$ is an
isomorphism (as a map between sheaves of abelian groups).

This implies that $\partial_{t}$ is invertible on $M$, and the inverse is
denoted as $\partial_{t}^{-1}$. The upshot of this construction is that the
D-module $M$ gives an “analytic” way to visualize the cohomology bundle. For
instance, the information of the monodromy action can be decoded from $M$ via
the actions of $t$ and $\partial_{t}$. Malgrange malgrange:asymptotic-
integrals-and-monodromy  proves that $M$ has $0\in S$ as a regular singular
point. More or less equivalent to this statement is that $M$ admits a “root
decomposition”

$M=\mathop{\widehat{\bigoplus}}_{\alpha\in\mathbb{Q}}M^{\alpha}$

where

$\boxed{M^{\alpha}=\bigcup_{i=1}^{\infty}\mathrm{Ker}(t\partial_{t}-\alpha)^{i}}$

and the hat above the direct sum sign means that $M$ is contained in
$\prod_{\alpha\in\mathbb{Q}}M^{\alpha}$ is generated by the direct sum
$\bigoplus M^{\alpha}$ as a $\mathbb{C}\\{t\\}$-submodule. Moreover, for each
$\alpha\in\mathbb{Q}$, the root space $M^{\alpha}$ is isomorphic to the
generalized eigenspace of $T$ on the Milnor cohomology with eigenvalue
$\exp(-2\pi\sqrt{-1}\alpha)$. Thus, that this decomposition is indexed by
rational numbers is a consequence of the monodromy theorem.

The root decomposition defines a (decreasing) _Kashiwara–Malgrange_ filtration
on $M$ defined by

$V^{\alpha}M=\mathop{\widehat{\bigoplus}}_{\beta\geq\alpha}M^{\beta},\quad
V^{>\alpha}M=\mathop{\widehat{\bigoplus}}_{\beta>\alpha}M^{\beta}$

and it satisfies the following conditions:

  * •

$V^{\alpha}M$ are finite $\mathbb{C}\\{t\\}$-submodules of $M$ and
$M=\bigcup_{\alpha}M^{\alpha}$.

  * •

$tV^{\alpha}M\subset V^{\alpha+1}M$, $\partial_{t}V^{\alpha}M\subset
V^{\alpha-1}M$, and for $\alpha>0$, $tV^{\alpha}M=V^{\alpha+1}M$.

  * •

$t\partial_{t}-\alpha$ is a nilpotent operator on
$\boxed{\mathrm{Gr}_{V}^{\alpha}M=V^{\alpha}M/V^{>\alpha}M}$.

The idea of the Kashiwara–Malgrange filtration comes from “asymptotic
expansions”. On the punctured disk $S^{\ast}=S\setminus\\{0\\}$ the
$\mathcal{D}$-module $M$ determines a local system $\mathbf{H}$ (i.e., the
cohomology bundle of the Milnor fibration) and we can form its dual local
system $\mathbf{H}^{\vee}$ on $S^{\ast}$ (the homology bundle). For an element
$\omega\in M$ defines a holomorphic section (i.e., a family of differential
forms on the fibers of the Milnor fibration) of
$\mathbf{H}\otimes_{\mathbb{C}}\mathcal{O}_{S^{\ast}}$ (upon shrinking $S$, if
necessary). Thus (as $M$ is a $\mathcal{D}$-module with regular singularity at
$0$) for each multivalued horizontal section $\gamma$ of $\mathbf{H}^{\vee}$,
the pairing $s[\omega,\gamma]=\langle\gamma,\omega\rangle$ (i.e., period
integrals $\int_{\gamma}\omega$) is a multivalued holomorphic function on
$S^{\ast}$ with logarithmic singularity at $0$. Whence we have an asymptotic
expansion

$s[\omega,\gamma]=\sum_{\alpha\in\mathbb{Q}}\sum_{m=0}^{\infty}a_{\alpha,m,\gamma}t^{\alpha-1}(\log
t)^{m}$

with $\inf\\{\alpha:a_{\alpha,m,\gamma}\neq 0\\}>-\infty$, where the infimum
runs through all multivalued sections $\gamma$ of $\mathbf{H}^{\vee}$, all
integers $m$, and all rational numbers $\alpha$. This number is then called
the _order_ of $\omega$.

Theorem. Let $\omega$ be a section of $M$. Then $\omega\in
V^{\alpha}M/V^{>\alpha}M$ if and only if the order of $\omega$ is $\alpha$.

The _canonical lattice_ is the $\mathbb{C}\\{t\\}$-module

$\mathcal{L}=V^{>-1}M.$

The _vanishing cohomology of the canonical Milnor fiber_ is

$\boxed{H^{n}(X_{\infty})=\mathcal{L}/t\mathcal{L}=\bigoplus_{-1<\alpha\leq
0}M^{\alpha}.}$

On the vector space $H^{n}(X_{\infty})$ one has a “monodromy action”
$T:H^{n}(X_{\infty})\to H^{n}(X_{\infty})$ defined by

$T|_{M^{\alpha}}=\exp\left\\{-2\pi\sqrt{-1}\alpha\right\\}\mathrm{Id}\cdot\exp\left\\{2\pi\sqrt{-1}(t\partial_{t}-\alpha)\right\\}.$

In other words, $M^{\alpha}$ is the root space of the operator $T$ associated
with the eigenvalue $\lambda=\exp\left\\{-2\pi\sqrt{-1}\alpha\right\\}$. We
would like to define a MHS on vector space $H^{n}(X_{\infty})$, which could be
thought of the zero fiber of the Deligne extension of the cohomological
bundle.

The Brieskorn lattice. To define the Hodge filtration on vanishing cohomology,
we will need a filtration on the Gauss-Manin system. The Gauss–Manin system
$M$ comes with an exhaustive filtration $F^{\bullet}M$, with
$F^{n-i}=\partial_{t}F^{i}$, and

$\boxed{F^{n}M=M_{0}=\frac{\Omega^{n+1}_{X}}{\mathrm{d}f\wedge\mathrm{d}\Omega^{n-1}_{X}}}$

called the _Brieskorn lattice_. The filtration looks like

$F^{n}M=M_{0},~{}~{}F^{n-1}M=\partial_{t}M_{0},~{}~{}\ldots,~{}~{}~{},F^{n-k}M=\partial_{t}^{k}M_{0},~{}\ldots,$
(57)

By a theorem of Sebastiani, $M_{0}$ is a finite free
$\mathbb{C}\\{t\\}$-module. This lattice plays an important role in defining
Mixed Hodge Structure on vanishing cohomology.

The Brieskorn lattice is not stable under the action of $\partial_{t}$, rather
it is stable under $\partial_{t}^{-1}$, indeed, if $\omega$ is an $(n+1)$-form
representing an element in the Brieskorn lattice, then

$\boxed{\partial_{t}^{-1}\omega=\mathrm{d}f\wedge\eta},\quad\text{ where
}\omega=\mathrm{d}\eta.$

In particular,

$\mathop{\mathrm{Im}}\partial_{t}^{-1}=\frac{\mathrm{d}f\wedge\Omega_{X}^{n}}{\mathrm{d}f\wedge\mathrm{d}\Omega^{n-1}_{X}}.$

It follows that $M_{0}/\partial^{-1}_{t}M_{0}=\Omega_{f}$, where

$\boxed{\Omega_{f}=\frac{\Omega^{n+1}_{X}}{\mathrm{d}f\wedge\Omega^{n}_{X}}}.$

An useful property of Brieskorn lattice is that it is contained in the
canonical lattice $M_{0}\subset V^{>-1}M$, and in general $M_{0}$ is smaller
than $V^{>-1}M$.

Hodge filtration after Saito–Scherk–Steenbrink. We now explain how to define
the Hodge filtration on vanishing cohomology according to Morihiko Saito and
Scherk–Steenbrink saito1982gauss ; scherk1985mixed .

We start with a germ of a holomorphic function with an isolated critical
point, and we then consider its Gauss-Mannin system and Brieskorn lattice
$(M,M_{0})$. The vanishing cohomology can be seen from the $V$-filtration by
$\phi_{t}M=\bigoplus_{-1<\alpha\leq 0}M^{\alpha}$, where $M^{\alpha}$ is the
root space in the Kashiwara–Malgrange filtration. To put a Hodge filtration on
each individual root space $M^{\alpha}$ in $\phi_{t}M$, Morihiko Saito and
Scherk–Steenbrink use the formula

$\boxed{F^{p}M^{\alpha}=\mathrm{Gr}_{V}^{\alpha}F^{p}M=\frac{V^{\alpha}M\cap
F^{p}M+V^{>\alpha}M}{V^{>\alpha}M}},\quad-1<\alpha\leq 0.$

The formula for the Hodge filtration then reduces to

$\displaystyle F^{p}M^{\alpha}$
$\displaystyle=\frac{V^{\alpha}M\cap\partial_{t}^{n-p}M_{0}+V^{>\alpha}M}{V^{>\alpha}M}$
(58)

Notice that $\partial_{t}$ action does not preserve $M_{0}$. The above formula
shows that one can compute the Hodge filtration on vanishing cohomology by
using Brieskorn lattice $M_{0}$:

  1. (1)

$F^{n}M^{\alpha}$ consists of elements of $M_{0}$ whose order is exactly
$\alpha$.

  2. (2)

$F^{n-1}M^{\alpha}$ consists of elements of $M_{0}$ whose order is $\alpha+1$.
These can be separated into two kinds: one kind takes the form
$\partial_{t}^{-1}\omega$ with $\omega\in F^{n}M^{\alpha}$, and others are
new. See Figure 17.

$\alpha$$\alpha+1$$F^{n}$$F^{n-1}$$\alpha+2$0-1$F^{n-2}$ Figure 17: The black
block with order $\alpha+p$ are elements of $M_{0}$ which come from the action
of $\partial_{t}^{-1}$ on elements of $M_{0}$ with order $\alpha+p-1$.

Weight filtration. The weight filtration comes from the nilpotent part of the
monodromy operator $h$ (this is the monodromy operator acting on vanishing
cohomology associated with the singularity). It is given by
$N=\partial_{t}t-\alpha$ on $M^{\alpha}$. Given a nilpotent operator $N$
acting on a space $H$, one can define a filtration as follows. Choose a Jordan
basis $u_{1},\ldots,u_{q}$, so that $N$ takes the form

$N=\left(\begin{array}[]{cccc}0&{}\hfil&{}\hfil&{}\hfil\\\
1&0&{}\hfil&{}\hfil\\\ {}\hfil&1&0&{}\hfil\\\ {}\hfil&{}\hfil&1&0\\\
\end{array}\right),~{}~{}Nu_{1}=u_{2},\ldots,Nu_{q-1}=u_{q},~{}~{}Nu_{q}=0$
(59)

We renumber the indices of the basis so that the action of $N$ is
$N(u_{i})=u_{i-2}$, so we have

  1. (1)

If $q=2k+1$ is odd, the basis is numbered as
$u_{2k},\ldots,u_{2},u_{0},u_{-2},\ldots,u_{-2k}$.

  2. (2)

if $q=2k$ is even, the basis is numbered as
$u_{2k-1},\ldots,u_{2},u_{0},u_{-2},\ldots,u_{-(2k-1)}$.

In this way, each basis has a weight corresponding its index, and the
nilpotent operator induces a filtration on $H$: $W_{k}H$ consists of the
subspace generated by the vectors of weight $\leq k$.

Definition. The weight filtration of a nilpotent operator $N$ with center $n$
is a filtration ${}^{n}W$ which is defined as ${}^{n}W_{k}=W_{-n+k}$.

Let’s go back to the root space $M^{\alpha}$, if $\alpha\neq 0$, the weight
filtration is given by ${}^{-n}W$, and if $\alpha=0$, the weight filtration is
given by ${}^{-n-1}W$.

Spectral pairs. We have explained how to define a Hodge and weight filtration
on each root space $M^{\alpha}$ of vanishing cohomology. The discrete
invariant of MHS on vanishing cohomology is recorded by the so-called spectral
pairs. Let $f$ be a function germ with isolated singularity
$f:\mathbb{C}^{n+1}\to\mathbb{C}$, and its vanishing cohomology admits a limit
mixed Hodge structure. Thus the vanishing cohomology admits a decomposition

$\bigoplus H_{\lambda}$

according to the eigenvalues of the monodromy operator. Each $H_{\lambda}$ has
two filtrations: the Hodge filtration $F^{\bullet}H_{\lambda}$ and the weight
filtration $W_{\bullet}H_{\lambda}$.

Thus for each integer $w$, the associated graded vector space

$H_{\lambda,w}=\mathrm{Gr}_{w}^{W}H_{\lambda}$

is a pure Hodge structure of weight $w$, the Hodge filtration being the one
induced by the limit Hodge filtration which we still denote by $F^{\bullet}$.
For an element in $F^{p}\mathrm{Gr}_{w}^{W}H_{\lambda}$, we define the
spectral pairs $(\alpha,l)$ as follows:

$\alpha=-{1\over 2\pi i}\log\lambda,\quad n-p-1<\alpha\leq n-p$ (60)

and

$l=\begin{cases}w&\text{if $\lambda\neq 1$ }\\\
w-1,&\text{$\lambda=1$}\end{cases}$ (61)

We say a pair of numbers $(\alpha,l)$ is a _spectral pair of the vanishing
cohomology of multiplicity $m=n_{\alpha,l}$_ if

$m=\dim F^{p}H_{\lambda,w}/F^{p+1}H_{\lambda,w},$

These spectral numbers play a very important role in determining the physical
theory associated with the local singularity, see Xie:2015rpa . The spectral
numbers have following important properties:

  1. (1)

The spectral number is paired: $\alpha_{i}+\alpha_{j}=n-1$, namely, the
spectral number is symmetric with respect to ${n-1\over 2}$. The corresponding
weight number is also paired $w_{1}+w_{2}=2(n+1)$.

  2. (2)

If we have two singularities $f$ and $g$ whose spectral numbers are
$\alpha_{i},\beta_{j}$ respectively, then the singularity $f+g$ has spectra
$\alpha_{i}+\beta_{j}$.

  3. (3)

The multiplicity of the minimal spectra $\alpha_{1}$ is one.

Example. Tthe spectral pairs with nonzero multiplicities of the function germ
$f(x,y)=x^{5}+x^{2}y^{2}+y^{5}$ are given by the following table

spectral pair | multiplicity
---|---
$(-\frac{1}{2},2)$ | 1
$(-\frac{3}{10},1)$ | 2
$(-\frac{1}{10},1)$ | 2
$(0,1)$ | 1
$(\frac{1}{10},1)$ | 2
$(\frac{3}{10},1)$ | 2
$(\frac{1}{2},0)$ | 1

One can extract the physical information from the spectral pairs as follows
(We shall only discuss 3-fold case, so we consider
$f:\mathbb{C}^{4}\to\mathbb{C}$. This is because the curve singularity case
can be lifted to a three-fold one by adding two extra quadratic terms):

  1. (1)

The spectral number $1$ is identified with the mass parameters.

  2. (2)

For a three dimensional quasi-homogeneous singularity, assume the spectra is
ordered as $(\alpha_{1},\ldots,\alpha_{\mu})$ with
$\alpha_{1}\leq\ldots\leq\alpha_{\mu}$, the scaling dimension of Coulomb
branch spectrum associated with a spectral number $\alpha_{i}$ is

$\Delta_{i}={1\over\alpha_{1}}+{\alpha_{1}-\alpha_{i}\over\alpha_{1}}$ (62)

Summary. Let $f:(\mathbb{C}^{n+1},0)\to(S,0)$ be a germ of a holomorphic
function, and $f=0$ has an isolated singularity at the origin. We associate a
Gauss-Manin system $M$ which is a D-module over $S$. The MHS on vanishing
cohomology can be found as follows:

  1. (1)

There is a Kashiwara-Magrange filtration on $M$ so that
$M=\mathop{\widehat{\bigoplus}}_{\alpha\in\mathbb{Q}}M^{\alpha}$. Here
$\exp(-2\pi\sqrt{-1}\alpha)$ is the eigenvalue of the monodromy operator.

  2. (2)

The vanishing cohomology is given by $\bigoplus_{-1<\alpha\leq 0}M^{\alpha}$.

  3. (3)

In defining Hodge filtration on vanishing cohomology, we can use Brieskorn
module $M_{0}$ which induces a Hodge filtration $F^{\bullet}$ on $M$.

  4. (4)

The weight filtration is given by the operator $t\partial_{t}-\alpha$ on
$M^{\alpha}$.

Example 1. Let’s consider an isolated quasi-homogeneous singularity $f$, and
the weights of the variables are $q_{1},\ldots,q_{n}$ so that the weight of
$f$ is one. The Bireskorn lattice is generated by the differential forms
$\omega_{m}=x^{m}dx^{m}$, here $x^{m}$ denotes the monomial basis of the
Jacobi algebra $J_{f}$. The $\partial_{t}$ action is kulikov1998mixed :

$t\partial_{t}\omega_{m}=(a(m)-1)\omega_{m}$ (63)

here $a(m)=\sum m_{i}q_{i}+\sum q_{i}$, here $m_{i}$ is the exponents of
$x^{m}dx^{m}=x_{1}^{m_{1}}\ldots x_{n}^{m_{n}}dx_{1}\wedge\ldots\wedge