# Spectral Data For Parabolic Projective Symplectic/Orthogonal Higgs Bundles

Sumit Roy  Center for Geometry and Physics, Institute for Basic Science (IBS),
Pohang 37673, Korea<EMAIL_ADDRESS>
###### Abstract.

Hitchin in [Duke Math. J. 54 (1), 91-114 (1987)] introduced a proper morphism
from the moduli space of stable $G$-Higgs bundles
($G=\mathrm{GL}(n,\mathbb{C}),\mathrm{Sp}(2m,\mathbb{C})$ and
$\mathrm{SO}(n,\mathbb{C})$) over a curve to a vector space of invariant
polynomials and he described the generic fibers of that morphism. In this
paper, we first describe the generic Hitchin fibers for the moduli space of
stable parabolic projective symplectic/orthogonal Higgs bundles without fixing
the determinant. We also describe the generic fibers when the determinant is
trivial.

###### Key words and phrases:

Integrable systems; Moduli space; Parabolic bundle; Higgs bundle

###### 2020 Mathematics Subject Classification:

14H70, 14D22, 14H60

E-mail<EMAIL_ADDRESS>
Affiliation: Center for Geometry and Physics, Institute for Basic Science
(IBS), Pohang 37673, Korea

##  1\. Introduction

Let $X$ be a compact (closed and connected) Riemann surface $X$ of genus
$g\geq 2$. Higgs bundles over $X$ were introduced by Hitchin in [2]. A Higgs
bundle over $X$ is a pair $(E,\phi)$ consisting of a a holomorphic vector
bundle $E$ and a Higgs field $\phi:E\to E\otimes K$, where $K$ is the
canonical bundle over $X$. The coefficients of the characteristic polynomial
of $\phi$ defines a morphism
$h:\mathcal{M}_{\mathrm{Higgs}}(r,d)\longrightarrow\mathcal{A}\coloneqq\bigoplus_{i=1}^{r}H^{0}(X,K^{i})$,
from the moduli space of stable Higgs bundles over $X$ of fixed rank $r$ and
degree $d$ to a vector space $\mathcal{A}$, called the Hitchin map (see [3]).
Hitchin in [3], showed that the generic fibers of $h$ are abelian varieties
and this map gives the Higgs bundles moduli space a structure of an
algebraically completely integrable system. Later in [6], Markman generalized
this result for the moduli space of $L$-twisted Higgs bundles $(E,\phi_{L})$,
where $L$ is a line bundle over $X$ and $\phi_{L}:E\to E\otimes L$.

Let $D\subset X$ be a fixed finite subset. The notion of parabolic bundles
over a curve and their moduli spaces were constructed by Mehta and Seshadri in
[1]. Their motivation was to extend the Narasimhan-Seshadri correspondence in
the case of irreducible unitary representations of $\pi_{1}(X-D)$. A parabolic
bundle is a holomorphic vector bundle together with a weighted flag over each
parabolic point $p\in D$. A parabolic Higgs bundle on $X$ is a parabolic
bundle $E$ on $X$ together with a parabolic Higgs field $\phi:E\to E\otimes
K(D)$. The moduli space of parabolic Higgs bundles was constructed by Yokogawa
[11].

Symplectic (resp. orthogonal) parabolic bundles are parabolic bundles with a
suitably defined nondegenerate anti-symmetric (resp. symmetric) form taking
values in a line bundle $L$ (see [13] for more details). In [5], Bhosle and
Ramanathan described the notion of parabolic principal $G$-bundles, where $G$
is a connected reductive group, and also constructed its moduli space. When
all weights are rational, the notion of symplectic (resp. orthogonal)
parabolic bundles coincides with the notion of parabolic principal $G$-bundles
where $G$ is a symplectic (resp. orthogonal) complex group (see [13]). A
symplectic (resp. orthogonal) parabolic Higgs bundle is a symplectic (resp.
orthogonal) parabolic bundle together with a parabolic Higgs field which is
compatible (in a suitable sense) with the symplectic (resp. orthogonal)
structures.

In [3], Hitchin also showed that the moduli space of stable
symplectic/orthogonal Higgs bundles also forms an algebraically completely
integrable system, fibered over a vector space, either by a Jacobian or a Prym
variety of so-called spectral curves. In [15], Roy generalized this result for
the moduli space of stable parabolic symplectic/orthogonal Higgs bundles.

In this paper, we consider the moduli space of parabolic projective
symplectic/orthogonal Higgs bundles with fixed rank and degre and fixed
parabolic structure. We know that for odd rank the projective orthogonal group
is same as the odd orthogonal group, i.e.
$\mathrm{PSO}(2m+1,\mathbb{C})=\mathrm{SO}(2m+1,\mathbb{C})$. Therefore, we
only consider the projective symplectic group $\mathrm{PSp}(2m,\mathbb{C})$
and projective even orthogonal group $\mathrm{PSO}(2m,\mathbb{C})$. A
parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs (resp.
$\mathrm{PSO}(2m,\mathbb{C})$-Higgs) bundle lifts to a parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs (resp.
$\mathrm{GSO}(2m,\mathbb{C})$-Higgs) bundle. We gave an alternative
description of the Prym varieties and we consider an action of the Jacobian
group $\mathrm{Jac}(X)$ on the Prym varieties. We showed that the generic
Hitchin fibers are isomorpic to the quotient variety (see Theorem 3.1 and
Theorem 4.1).

Finally we consider the case when the symplectic/orthogonal form takes values
in the trivial line bundle $\mathcal{O}_{X}$ and we fix this line bundle for
the respective moduli spaces. In this case, the $2$-torsion subgroup
$\mathrm{Jac}_{2}(X)\subset\mathrm{Jac}(X)$ acts on the Prym varieties and the
generic fibers of the Hitchin map are isomorphic to the quotient variety (see
Theorem 5.1 and Theorem 5.2).

##  2\. Preliminaries

###  2.1. Parabolic bundles

Let $X$ be a compact Riemann surface of genus of genus $g\geq 2$. Fix a subset
$D=\\{p_{1},\dots,p_{n}\\}\subset X$ of $n$ distinct marked points.

######  Definition 1.

A parabolic bundle $E_{*}$ of rank $r$ over $X$ is a vector bundle $E$ of rank
$r$ over $X$ with a parabolic structure over the subset $D$, i.e. for each
point $p\in D$

  1. (1)

every fiber $E_{p}$ has a filtration of subspaces, i.e.

$E_{p}\eqqcolon E_{p,1}\supsetneq E_{p,2}\supsetneq\dots\supsetneq
E_{p,r_{p}}\supsetneq E_{p,r_{p}+1}=\\{0\\},$

  2. (2)

an increasing sequence of real numbers (parabolic weights) satisfying

$0\leq\alpha_{1}(p)<\alpha_{2}(p)<\dots<\alpha_{r_{p}}(p)<1,$

where $1\leq r_{p}\leq r$ is an integer.

We denote the collection of all parabolic weights by
$\alpha=\\{(\alpha_{1}(p),\alpha_{2}(p),\dots,\alpha_{r_{p}}(p))\\}_{p\in D}$
corresponding to a fixed parabolic structure. The parabolic structure $\alpha$
is said to have full flags if

$\mathrm{dim}(E_{p,i}/E_{p,i+1})=1$

for all $i\in\\{1,\dots,r_{p}\\}$ and for each $p\in D$, or equivalently
$r_{p}=r$ for each $p\in D$. In this paper, we will assume that the parabolic
structure have full flag at every parabolic point $p\in D$.

The parabolic degree of $E_{*}$ is defined by

$\operatorname{pardeg}(E_{*})\coloneqq\deg(E)+\sum\limits_{p\in
D}\sum\limits_{i=1}^{r_{p}}\alpha_{i}(p)\cdot\dim(E_{p,i}/E_{p,i+1})$

and the parabolic slope of $E_{*}$ is defined by

$\mu_{\mathrm{par}}(E_{*})\coloneqq\frac{\text{pardeg}(E_{*})}{r}.$

######  Definition 2.

A parabolic homomorphism $\phi:E_{*}\to E^{\prime}_{*}$ between two parabolic
bundles is a homomorphism between underlying vector bundles such that at each
parabolic point $p\in D$ we have

$\alpha_{i}(p)>\alpha_{j}^{\prime}(p)\implies\phi(E_{p,i})\subseteq
E_{p,j+1}^{\prime}.$

Furthermore, we call such a homomorphism strongly parabolic if

$\alpha_{i}(p)\geq\alpha_{j}^{\prime}(p)\implies\phi(E_{p,i})\subseteq
E_{p,j+1}^{\prime}$

for every $p\in D$.

We denote by $\mathrm{PEnd}(E_{*})$ and $\mathrm{SPEnd}(E_{*})$ the parabolic
and strongly parabolic endomorphisms of $E_{*}$ respectively.

The dual and tensor product of parabolic bundles can be defined in a natural
way (see [12]).

######  Definition 3.

A parabolic subbundle $F_{*}$ of a parabolic bundle $E_{*}$ is a subbundle
$F\subset E$ of the underlying vector bundle endowed with an induced parabolic
structure. An induced parabolic structure on $F$ is defined as follows. For
every parabolic point $p\in D$, the quasi-parabolic structure on $F$, i.e. the
flag in $F_{p}$ is given by

$F_{p}\eqqcolon F_{p,1}\supsetneq F_{p,2}\supsetneq\dots\supsetneq
F_{p,r^{\prime}_{p}}\supsetneq\\{0\\},$

where $F_{p,i}=F_{p}\cap E_{p,i}$, i.e. we are considering the intersection
with the already given flag in $E_{p}$, and also scrapping all the repetitions
of subspaces in the filtration. The weights
$0\leq\alpha^{\prime}_{1}(p)<\alpha^{\prime}_{2}(p)<\dots<\alpha^{\prime}_{r^{\prime}_{p}}(p)<1$
are taken to be the largest possible among the given weights which are allowed
after the intersections, i.e.

$\alpha^{\prime}_{i}(p)=\mathrm{max}_{j}\\{\alpha_{j}(p)|F_{p}\cap
E_{p,j}=F_{p,i}\\}=\mathrm{max}_{j}\\{\alpha_{j}(p)|F_{p,i}\subseteq
E_{p,j}\\}$

That is to say, the weight associated to $F_{p,i}$ is the weight
$\alpha_{j}(p)$ such that $F_{p,i}\subseteq E_{p,j}$ but $F_{p,i}\nsubseteq
E_{p,j+1}$.

######  Definition 4.

A parabolic bundle $E_{*}$ is called semistable (resp. stable) if every
nonzero proper subbundle $F_{*}\subset E_{*}$ satisfies

$\mu_{\mathrm{par}}(F_{*})\leq\mu_{\mathrm{par}}(E_{*})\hskip
5.69046pt(\mathrm{resp.}\hskip 5.69046pt<).$

The moduli space $\mathcal{M}(\alpha,r,d)$ of stable parabolic bundles over
$X$ of fixed rank $r$ and degree $d$ and parabolic structure $\alpha$ was
constructed by Mehta and Seshadri in [1]. They also showed that
$\mathcal{M}(\alpha,r,d)$ is a normal projective variety of dimension

$\dim\mathcal{M}(\alpha,r,d)=r^{2}(g-1)+1+\dfrac{n(r^{2}-r)}{2},$

where $n$ is the number of marked points and the last summand comes from the
fact that the parabolic structure have full flags over each parabolic point.

###  2.2. Parabolic Higgs bundles

Let $K$ be the canonical bundle over $X$. We write $K(D)\coloneqq
K\otimes\mathcal{O}(D)$.

######  Definition 5.

A (strongly) parabolic Higgs bundle over $X$ is a parabolic bundle $E_{*}$
over $X$ together with Higgs field $\Phi:E_{*}\to E_{*}\otimes K(D)$, such
that $\Phi$ is strongly parabolic i.e. $\Phi(E_{p,i})\subset
E_{p,i+1}\otimes\left.K(D)\right|_{p}$ for all $p\in D$.

There is also a notion of parabolic Higgs bundle where the Higgs field $\Phi$
is only assumed to be parabolic, i.e. $\Phi(E_{p,i})\subset
E_{p,i}\otimes\left.K(D)\right|_{p}$ for all $p\in D$. However in this paper
we will always assume that the Higgs field is strongly parabolic.

######  Definition 6.

A subbundle $F_{*}\subset E_{*}$ is called $\Phi$-invariant if
$\Phi(F_{*})\subset F_{*}\otimes K(D)$.

######  Definition 7.

A parabolic Higgs bundle $(E_{*},\Phi)$ is called semistable (resp. stable) if
every nonzero proper $\Phi$-invariant subbundle $F_{*}\subset E_{*}$ satisfies

$\mu_{\mathrm{par}}(F_{*})\leq\mu_{\mathrm{par}}(E_{*})\hskip
5.69046pt(\mathrm{resp.}\hskip 5.69046pt<).$

The moduli space $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ of stable
parabolic Higgs bundles of fixed rank $r$, degree $d$ and parabolic structure
$\alpha$ was constructed by Yokogawa in [11] (see [7] for more details). It is
a normal quasi-projective complex variety of dimension

$\dim\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)=2\dim\mathcal{M}(\alpha,r,d).$

The parabolic version of the Serre duality (see [7], [12]) says that

$H^{1}(\mathrm{PEnd}(E_{*}))\cong H^{0}(\mathrm{SPEnd}(E_{*})\otimes
K(D))^{*}.$

Therefore, there is an open embedding
$T^{*}\mathcal{M}(\alpha,r,d)\xhookrightarrow{}\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$
and that is why the dimension of $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ is
twice the dimension of $\mathcal{M}(\alpha,r,d)$. Thus the moduli space of
parabolic Higgs bundles $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ has a
symplectic structure induced from the natural symplectic structure of the
cotangent space.

Let $\mathrm{Jac}^{d}(X)$ denote the space of degree $d$ line bundles over
$X$. Consider the determinant map

$\displaystyle\mathrm{det}:\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$
$\displaystyle\longrightarrow\mathrm{Jac}^{d}(X)\times H^{0}(X,K)$
$\displaystyle(E_{*},\Phi)$
$\displaystyle\longmapsto(\wedge^{r}E_{*},\mathrm{trace}(\Phi)).$

Since $\Phi$ is strongly parabolic, $\mathrm{trace}(\Phi)\in H^{0}(X,K)\subset
H^{0}(X,K(D))$. The moduli space
$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$ of stable parabolic Higgs
bundles with fixed determinant $\xi$ is defined by the fiber
$\mathrm{det}^{-1}(\xi,0)$, i.e.

$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)\coloneqq\mathrm{det}^{-1}(\xi,0).$

If the Higgs field is zero, then the dimension of the moduli space
$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$ is given by

$\dim\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)=2(g-1)(r^{2}-1)+nr(r-1).$

###  2.3. Spectral correspondence

We will give a description of the spectral correspondence for the moduli space
$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$ (although a similar
description can be given for the moduli space
$\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$).

Let $p:\mathrm{Tot}(K(D))\to X$ be the natural projection from the total space
of $K(D)$ to $X$ and let $x\in H^{0}(\mathrm{Tot}(K(D)),p^{*}K)$ denote the
tautological section. Since the Higgs field $\Phi$ is strongly parabolic, the
residue at every point of $D$ is nilpotent. Therefore the trace of the map

$\wedge^{i}\Phi:\wedge^{i}E_{*}\to\wedge^{i}E_{*}\otimes K(D)^{i}$

lies in $K^{i}(D^{i-1})$ for each $2\leq i\leq r$, where $K^{i}(D^{j})$ denote
the tensor product of the $i$-th power of $K$ and the $j$-th power of
$\mathcal{O}(D)$. The coefficients of the characteristic polynomial of $\Phi$
are precisely given by $s_{i}=\mathrm{trace}(\wedge^{i}\Phi)$. Therefore, we
have the Hitchin map

$\displaystyle
h_{\mathrm{par}}:\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$
$\displaystyle\longrightarrow\mathcal{H}_{\mathrm{par}}\coloneqq\bigoplus_{i=2}^{r}H^{0}(X,K^{i}(D^{i-1}))$
$\displaystyle(E_{*},\Phi)$ $\displaystyle\longmapsto(s_{2},\dots,s_{r}).$

By Riemann-Roch theorem, the dimension of the base
$\mathcal{H}_{\mathrm{par}}$ is

$r^{2}(g-1)+\dfrac{nr(r-1)}{2},$

which is same as the half the dimension of the moduli space
$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$.

Given $s=(s_{2},s_{3}\dots,s_{r})\in\mathcal{H}_{\mathrm{par}}$ with

$s_{i}\in H^{0}(X,K^{i}(D^{i-1}))\subset H^{0}(X,K(D)^{i}),$

the spectral curve $X_{s}$ in $\mathrm{Tot}(K(D))$ is defined by

$x^{r}+\tilde{s}_{2}x^{r-2}+\tilde{s}_{3}x^{r-3}\cdots+\tilde{s}_{r}=0$

where $\tilde{s}_{i}=p^{*}(s_{i})$ and $x\in H^{0}(\mathrm{Tot}(K(D)),p^{*}K)$
is the tautological section. Let

$\pi:X_{s}\to X$

be the restriction of the projection $p$. For a generic point
$s\in\mathcal{H}_{\mathrm{par}}$, the spectral curve $X_{s}$ is smooth and by
[10] the fiber $h_{\mathrm{par}}^{-1}(s)$ of the Hitchin map is isomorphic to

$\mathrm{Prym}(X_{s}/X)=\\{L\in\mathrm{Pic}(X):\det\pi_{*}L\cong\xi\\}.$

###  2.4. Parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundles

Let us consider the standard symplectic form

$J=\begin{bmatrix}0&I_{m}\\\ -I_{m}&0\end{bmatrix}$

on $\mathbb{C}^{2m}$. Then the general symplectic group defined by

$\mathrm{GSp}(2m,\mathbb{C})=\\{A\in\mathrm{GL}(2m,\mathbb{C}):AJA^{t}=\lambda_{A}J\text{
for some }\lambda_{A}\in\mathbb{C}^{*}\\}$

is an extension of $\mathbb{C}^{*}$ by the symplectic group
$\mathrm{Sp}(2m,\mathbb{C})$, i.e. there exist an exact sequence

$1\to\mathrm{Sp}(2m,\mathbb{C})\xrightarrow{}\mathrm{GSp}(2m,\mathbb{C})\xrightarrow[]{p}\mathbb{C}^{*}\xrightarrow[]{}1,$

where $p(A)=\lambda_{A}$. Therefore, $\det(A)=p(A)^{m}=\lambda_{A}^{m}$ for
every $A\in\mathrm{GSp}(2m,\mathbb{C})$.
The Lie algebra of $\mathrm{GSp}(2m,\mathbb{C})$ is given by

$\mathfrak{gsp}(2m,\mathbb{C})=\\{M\in\mathfrak{gl}(2m,\mathbb{C}):MJ+JA^{t}=\frac{\mathrm{tr}(M)}{m}J\\}\cong\mathfrak{sp}(2m,\mathbb{C})\oplus\mathbb{C}.$

The decomposition $M=N+\frac{\mathrm{tr}(M)}{m}I_{2m}$, with
$N\in\mathfrak{sp}(2m,\mathbb{C})$ produces the above isomorphism.

Let $L$ be a line bundle over $X$ of degree $l$. Let $E_{*}$ be a parabolic
bundle and let

$\varphi:E_{*}\otimes E_{*}\to L$ (1)

be a homomorphism of parabolic bundles. The trivial line bundle
$\mathcal{O}_{X}$ equipped with the trivial parabolic structure is realized as
a parabolic subbundle of $E_{*}\otimes E^{\vee}_{*}$ by sending a locally
defined function $f$ to the locally defined endomorphism of $E$ given by
pointwise multiplication with $f$. Let

$\tilde{\varphi}:E_{*}\to L\otimes E^{\vee}_{*}$ (2)

be the homomorphism defined by the composition

$E_{*}=E_{*}\otimes\mathcal{O}_{X}\xhookrightarrow{}E_{*}\otimes(E_{*}\otimes
E^{\vee}_{*})=(E_{*}\otimes E_{*})\otimes
E^{\vee}_{*}\xrightarrow{\varphi\otimes Id}L\otimes E^{\vee}_{*}.$

######  Definition 8.

A symplectic parabolic bundle is a pair $(E_{*},\varphi)$ of the above form
such that $\varphi$ is anti-symmetric and the homomorphism $\tilde{\varphi}$
is an isomorphism.

Suppose $E$ is the underlying vector bundle of a symplectic parabolic bundle
$(E_{*},\varphi)$. The tensor product $E\otimes E$ is a coherent subsheaf of
the vector bundle underlying the parabolic bundle $E_{*}\otimes E_{*}$.
Therefore, $\varphi$ induces a homomorphism

$\hat{\varphi}:E\otimes E\to L$ (3)

of vector bundles.

######  Definition 9.

A subbundle $F\subset E$ of the underlying bundle of $(E_{*},\varphi)$ is
called isotropic if $\hat{\varphi}(F\otimes F)=0$.

A parabolic Higgs field $\Phi$ on a symplectic parabolic bundle
$(E_{*},\varphi)$ is said to be compatible with $\varphi$ if $\tilde{\varphi}$
takes $\Phi$ to the induced parabolic Higgs field on $L\otimes E^{\vee}_{*}$
(we are considering the zero section as the Higgs field on $L$). We can
describe this compatibility condition locally. A strongly parabolic Higgs
field $\Phi$ on $E_{*}$ can be viewed as a holomorphic section of
$\mathrm{SPEnd}(E_{*})\otimes K(D)$. Let $s$ and $t$ be any holomorphic
sections of $E_{*}$ defined over an open subset $U\subset X$. Consider

$\hat{\varphi}_{\Phi}(s,t)\coloneqq\hat{\varphi}(\Phi(s)\otimes
t)+\hat{\varphi}(s\otimes\Phi(t))\in\Gamma(U,L\otimes K(D)),$

where $\hat{\varphi}$ is the pairing defined in 3. The Higgs field $\Phi$ is
said to be compatible with $\phi$ if and only if $\hat{\varphi}_{\Phi}(s,t)=0$
for all sections $s$ and $t$.

######  Definition 10.

A symplectic parabolic Higgs bundle $(E_{*},\varphi,\Phi)$ is a symplectic
parabolic bundle $(E_{*},\varphi)$ equipped with a parabolic Higgs field
$\Phi$ on $E_{*}$ which is compatible with $\varphi$.

When the parabolic weights are all rational, the notion of symplectic
parabolic Higgs bundle is equivalent to the notion of parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundles (see [13]).

###  2.5. Parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundles

Let $B$ be a non-degenerate symmetric bilinear form on $\mathbb{C}^{2m}$. Also
for notational convenience, we denote the corresponding symmetric matrix by
$B$. Then the general (even) orthogonal group

$\mathrm{GO}(2m,\mathbb{C})=\\{A\in\mathrm{GL}(2m,\mathbb{C}):ABA^{t}=\lambda_{A}J\text{
for some }\lambda_{A}\in\mathbb{C}^{*}\\}.$

therefore,, we have $(\det(A))^{2}=\lambda_{A}^{2m}$ for every
$A\in\mathrm{GO}(2m,\mathbb{C})$.

In this case, there is a sgn morphism

$\textit{sgn}:\mathrm{GO}(2m,\mathbb{C})\longrightarrow\\{\pm 1\\}$

sending $A$ to $\det(A)/\lambda_{A}^{m}$. The general special orthogonal group
is the kernel of this sgn morphism and it is denoted by
$\mathrm{GSO}(2m,\mathbb{C})=\mathrm{ker}(\textit{sgn})$. So we have a short
exact sequence

$1\to\mathrm{GSO}(2m,\mathbb{C})\xrightarrow{}\mathrm{GO}(2m,\mathbb{C})\xrightarrow[]{\textit{sgn}}\\{\pm
1\\}\xrightarrow[]{}1.$

######  Definition 11.

An orthogonal parabolic bundle is a pair $(E_{*},\varphi)$, where $\varphi$
(as in 1) is symmetric and the homomorphism $\tilde{\varphi}$ (as in 2) is an
isomorphism.

######  Definition 12.

An orthogonal parabolic Higgs bundle $(E_{*},\varphi,\Phi)$ is an orthogonal
parabolic bundle $(E_{*},\varphi)$ equipped with a parabolic Higgs field
$\Phi$ on $E_{*}$ which is compatible with $\varphi$.

As in the $\mathrm{GSp}$-case, for rational weights the notions of (even)
orthogonal parabolic Higgs bundles and parabolic
$\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundles coincide.

###  2.6. Moduli space

######  Definition 13.

A symplectic/orthogonal parabolic Higgs bundle $(E_{*},\varphi,\Phi)$ is said
to be semistable (resp. stable) if every nonzero isotropic subbundle $F\subset
E$ such that $\Phi(F_{*})\subset F_{*}\otimes K(D)$ satisfies

$\mu_{par}(F_{*})\leq\mu_{par}(E_{*})\hskip 11.38092pt(\text{resp.}\hskip
4.26773pt<)$

holds, where $F_{*}\subset E_{*}$ has the induced parabolic structure.

The moduli space $\mathcal{M}_{G}(\alpha)$ of stable parabolic $G$-bundles of
a fixed topological type and with a fixed parabolic structure $\alpha$ is a
normal quasi-projective variety (see [8], [5]) of dimension

$\dim\mathcal{M}_{G}(\alpha)=\dim Z(G)+(g-1)\dim(G)+n\dim(G/B),$

where $Z(G)$ denotes the the center of $G$ and $n$ is the number of parabolic
points. The last summand comes from the fact that the flags we are considering
over each point of $D$ are full flags and $B$ is the Borel subgroup of $G$
determined by $\alpha$. The moduli space
$\mathcal{M}_{G\mathrm{-Higgs}}(\alpha)$ of stable parabolic $G$-Higgs bundles
(see [14]) is a normal quasi-projective variety of dimension

$\dim\mathcal{M}_{G\mathrm{-Higgs}}(\alpha)=2\dim\mathcal{M}_{G}(\alpha).$

In particular when $G=\mathrm{GSp}(2m,\mathbb{C})$, the moduli space
$\mathcal{M}_{\mathrm{GSp-Higgs}}(\alpha,2m,d)$ of stable parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle of fixed degree $d$ has dimension

$\dim\mathcal{M}_{\mathrm{GSp-Higgs}}(\alpha,2m,d)=2m(2m+1)(g-1)+2m^{2}n.$

Similarly, the moduli space $\mathcal{M}_{\mathrm{GSO-Higgs}}(\alpha,2m,d)$ of
stable parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundle of fixed degree
$d$ has dimension

$\dim\mathcal{M}_{\mathrm{GSO-Higgs}}(\alpha,2m,d)=2m(2m-1)(g-1)+2mn(m-1).$

### Notation:

From now on, for notational convenience we shall denote a parabolic bundle
$E_{*}$ by $E$.

##  3\. Parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles

In this section we will discuss the Hitchin fibration for the moduli space of
parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles. The projective
symplectic group $\mathrm{PSp}(2m,\mathbb{C})$ is given by the following exact
sequence:

$1\xrightarrow{}\mathbb{C}^{*}\xrightarrow{c\to
cI_{2m}}\mathrm{GSp}(2m,\mathbb{C})\xrightarrow{}\mathrm{PSp}(2m,\mathbb{C})\xrightarrow{}1.$

The sheaf version of this sequence induces the following exact sequence in
homology :

$H^{1}(X,\mathcal{O}_{X}^{*})\xrightarrow{}H^{1}(X,\mathrm{GSp}(2m,\mathcal{O}_{X}))\xrightarrow{q}H^{1}(X,\mathrm{PSp}(2m,\mathcal{O}_{X}))\to
0$ (4)

The surjectivity of the map $q$ implies that there is a bijective
correspondence between the parabolic $\mathrm{PSp}(2m,\mathbb{C})$-bundles and
the equivalence classes of parabolic $\mathrm{GSp}(2m,\mathbb{C})$-bundles
with respect to the action given by the tensor product of line bundles on the
associated bundles. If $V$ is a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-bundle
and $\tilde{V}$ is a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-bundle such that
$q(\tilde{V})=V$, then we call that $\tilde{V}$ is a lifting of $V$ to a
parabolic $\mathrm{GSp}(2m,\mathbb{C})$-bundle.

The group $\mathrm{PSp}(2m,\mathbb{C})$ can also be defined by the quotient of
the symplectic group $\mathrm{Sp}(2m,\mathbb{C})$ by the action of a finite
group by the following exact sequence :

$1\xrightarrow{}\\{\pm 1\\}\xrightarrow{1\to
I_{2m}}\mathrm{Sp}(2m,\mathbb{C})\xrightarrow{}\mathrm{PSp}(2m,\mathbb{C})\xrightarrow{}1.$

Since it is a quotient by a finite group, the Lie algebras are equal, i.e.

$\mathfrak{sp}(2m,\mathbb{C})=\mathfrak{psp}(2m,\mathbb{C}).$ (5)

Consider a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle $(V,\eta)$
which lifts to a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle
$(\tilde{V},\tilde{\eta})$, i.e. $q(\tilde{V},\tilde{\eta})=(V,\eta)$. Let
$(E,\Phi,\varphi,L)$ be the bundle corresponding to
$(\tilde{V},\tilde{\eta})$. Then $(V,\eta)$ corresponds to the equivalence
class $[(E,\Phi,\varphi,L)]$ where the equivalence relation
$\sim_{\mathrm{Jac}(X)}$ is given by :

$(E,\Phi,\varphi,L)\sim_{\mathrm{Jac}(X)}(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},L\otimes M^{2})\hskip
28.45274pt\mathrm{for}\hskip 2.84544pt\mathrm{any}\hskip
5.69046ptM\in{\mathrm{Jac}(X)}$

where

$\varphi_{M}:(E\otimes M)\otimes(E\otimes M)\to L\otimes M^{2}$

is the induced symplectic form on $E\otimes M$ taking values in $L\otimes
M^{2}$.

Let $M_{\mathrm{GSp-
Higgs}}(\alpha,2m)=\coprod_{d\in\mathbb{Z}}\mathcal{M}_{\mathrm{GSp-
Higgs}}(\alpha,2m,d)$ denote the moduli stack of parabolic
$\mathrm{GSp}$-Higgs bundles of fixed rank $2m$ with any degree. Then the
$\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ of $(E,\Phi,\varphi,L)$ under
the action of $\mathrm{Jac}(X)$ on the stack $M_{\mathrm{GSp-
Higgs}}(\alpha,2m)$ is defined by:

$\displaystyle M_{\mathrm{GSp-Higgs}}(\alpha,2m)\times\mathrm{Jac}(X)$
$\displaystyle\longrightarrow M_{\mathrm{GSp-Higgs}}(\alpha,2m)$
$\displaystyle((E,\Phi,\varphi,L),M)$ $\displaystyle\mapsto(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},L\otimes M^{2}).$

Note that $\deg(L\otimes M^{2})=\deg(L)+2\deg(M)$, i.e. it changes the degree
of $L$ by a multiple of $2$. For a parabolic $\mathrm{PSp}(2m,\mathbb{C})$\-
Higgs bundle $[(E,\Phi,\varphi,L)]$, the isomorphism $E\cong E^{\vee}\otimes
L$ implies that $\mathrm{pardeg}(E)=m\deg(L)$. So the parabolic degree of $E$
is determined by the degree of $L$. Therefore,

$\displaystyle\mathrm{pardeg}(E\otimes M)$ $\displaystyle=m\deg(L\otimes
M^{2})$ $\displaystyle=m\deg(L)+2m\deg(M)$
$\displaystyle=\mathrm{pardeg}(E)+2m\deg(M).$

Therefore, the parabolic degree of a parabolic
$\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle can be defined as follows :

######  Definition 14.

Let $(V,\eta)$ be a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle and
let $(\tilde{V},\tilde{\eta})$ be a lifting to a parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle. Let $(E,\Phi,\varphi,L)$ be the
datum corresponding to $(\tilde{V},\tilde{\eta})$ with parabolic degree $ml$,
where $l=\deg(L)$. Then the parabolic degree of $(V,\eta)$ is given by the
class $\overline{ml}\in\mathbb{Z}/m\mathbb{Z}$.

Threfore, we will consider the moduli space $\mathcal{M}_{\mathrm{PSp-
Higgs}}(\alpha,2m,\overline{ml})$ of stable parabolic
$\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles of fixed rank $2m$ and parabolic
degree $\overline{ml}$.

Consider a basis $\\{s_{2i}\\}_{i=1,\dots,m}$ of invariant polynomials of the
lie algebra $\mathfrak{sp}(2m,\mathbb{C})=\mathfrak{psp}(2m,\mathbb{C})$,
where $s_{2i}=\mathrm{tr}(\wedge^{2i}\eta)$. Therefore the Hitchin morphism
for the moduli space of parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles
is of the form

$\displaystyle h_{\mathrm{PSp-par}}:\mathcal{M}_{\mathrm{PSp-
Higgs}}(\alpha,2m,\overline{ml})$
$\displaystyle\longrightarrow\mathcal{H}_{\mathrm{PSp-
par}}\coloneqq\bigoplus_{i=1}^{m}H^{0}(X,K^{2i}(D^{2i-1}))$
$\displaystyle(V,\eta)$ $\displaystyle\longmapsto(s_{2},\dots,s_{2m}).$

Observe that if $(\tilde{V},\tilde{\eta})$ is a lifting of $(V,\eta)$ to a
parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle then

$h_{\mathrm{GSp-par}}(\tilde{V},\tilde{\eta})=h_{\mathrm{PSp-par}}(V,\eta).$

Let $s=(s_{2},s_{4},\dots,s_{2m})\in\mathcal{H}_{\mathrm{PSp-par}}$ be a
generic point of the Hitchin base. The spectral curve

$\pi:X_{s}\to X$

is defined by the equation

$x^{2m}+s_{2}x^{2m-2}+s_{4}x^{2m-4}+\cdots+s_{2m}=0.$

For a generic $s\in\mathcal{H}_{\mathrm{PSp-par}}$, the corresponding spectral
curve $X_{s}$ is smooth (see [9]). Since all odd coefficients of the above
equation are zero, the spectral curve $X_{s}$ possesses an involution
$\sigma:X_{s}\to X_{s}$ defined by $\sigma(\lambda)=-\lambda$. Therefore, we
can define a $2$-fold covering map

$q:X_{s}\to X_{s}/\sigma.$

Since $\sigma$ sends a degree zero line bundle on $X_{s}$ to a degree zero
line bundle, it acts on the Jacobian $\text{Jac}(X_{s})$. The Prym variety
$P_{s,\sigma}\coloneqq\mathrm{Prym}(X_{s},X_{s}/\sigma)$ is given by

$P_{s,\sigma}\coloneqq\mathrm{Prym}(X_{s},X_{s}/\sigma)=\\{N\in\text{Jac}(X_{s}):\sigma^{*}N\cong
N^{\vee}\\}$

and it is of dimension

$\dim P_{s,\sigma}=g(X_{s})-g(X_{s}/\sigma).$

Following [15, Theorem 4.1], we can give a different description of the Prym
variety. Let $J\in P_{s,\sigma}$ be an element in the Prym variety. Consider
the line bundle

$U=J\otimes R^{\vee},$

where $R=(K_{X_{s}}\otimes\pi^{*}K^{\vee}\otimes\pi^{*}L^{\vee})^{1/2}$ is a
holomorphic square root. Then it follows that $U$ satisfies the isomorphism

$\sigma^{*}U\cong
U^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L.$ (6)

Similarly, let $U\in\text{Jac}(X_{s})$ be a line bundle satisfying the
equation (6). Then the line bundle $J=U\otimes R\in P_{s,\sigma}$ is an
element in the Prym variety. Therefore, there is a bijective correspondence
between the Prym variety $P_{s,\sigma}$ and

$\Omega_{s,\sigma}\coloneqq\\{(U,L,\tau)\hskip 2.84544pt|\hskip
2.84544ptU\in\text{Jac}(X_{s}),\tau:\sigma^{*}U\cong
U^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L\\}.$ (7)

From now on, we will refer an element in the Prym variety as an element of
$\Omega_{s,\sigma}$.

The Jacobian $\mathrm{Jac}(X)$ acts on $\Omega_{s,\sigma}$ as follows:

$\displaystyle\Omega_{s,\sigma}\times\mathrm{Jac}(X)$
$\displaystyle\longrightarrow\Omega_{s,\sigma}$ $\displaystyle((U,L,\tau),M)$
$\displaystyle\longmapsto(U\otimes\pi^{*}M,L\otimes M^{2},\tau_{M})$

where $\tau_{M}=\tau\otimes\text{Id}_{\pi^{*}M}$ is the following isomorphism

$\displaystyle\sigma^{*}(U\otimes\pi^{*}M)$
$\displaystyle\cong\sigma^{*}U\otimes\pi^{*}M$ $\displaystyle\cong
U^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L\otimes\pi^{*}M$
$\displaystyle\cong
U^{\vee}\otimes\pi^{*}M^{\vee}\otimes\pi^{*}M\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L\otimes\pi^{*}M$
$\displaystyle\cong(U\otimes\pi^{*}M)^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}(L\otimes
M^{2}).$

######  Theorem 3.1.

For a generic point $s\in\mathcal{H}_{\mathrm{PSp-par}}$, the fiber
$h_{\mathrm{PSp-par}}^{-1}(s)$ is isomorphic to the quotient
$\Omega_{s,\sigma}/\mathrm{Jac}(X)$.

###### Proof.

Let $(V,\eta)\in h_{\mathrm{PSp-par}}^{-1}(s)$ be a parabolic
$\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle in the fiber of the Hitchin
morphism and let $(\tilde{V},\tilde{\eta})$ be a lifting to a parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle which corresponds to the datum of
$(E,\Phi,\varphi,L)$. Then $(V,\eta)$ corresponds to the datum of the
$\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ uniquely. By [15, Theorem 4.1],
the datum of $(E,\Phi,\varphi,L)$ corresponds to an element of the Pyrm
variety $\Omega_{s,\sigma}$ via the spectral correspondence. Consider an
element $M\in\mathrm{Jac}(X)$, i.e. a degree zero line bundle over $X$. Then
by the projection formula the datum of $(E\otimes
M,\Phi\otimes\text{Id}_{M},\varphi_{M},L\otimes M^{2})$ corresponds uniquely
to an element of $\Omega_{s,\sigma}/\mathrm{Jac}(X)$. Therefore, we conclude
that the datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$
corresponds uniquely to the datum of the $\mathrm{Jac}(X)$-orbit of an element
of $\Omega_{s,\sigma}$ via the spectral correspondence. ∎

##  4\. Parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundles

As in the previous section, there is a bijective correspondence between the
parabolic $\mathrm{PSO}(2m,\mathbb{C})$-bundles and the equivalence classes of
parabolic $\mathrm{GSO}(2m,\mathbb{C})$-bundles. So, for every parabolic
$\mathrm{PSO}(2m,\mathbb{C})$-bundle $V$ there is a lifting $\tilde{V}$ of $V$
to a parabolic $\mathrm{GSO}(2m,\mathbb{C})$-bundle.

The group $\mathrm{PSO}(2m,\mathbb{C})$ can be defined by the quotient of
$\mathrm{SO}(2m,\mathbb{C})$ by the action of a finite group by the following
exact sequence :

$1\xrightarrow{}\\{\pm 1\\}\xrightarrow{1\to
I_{2m}}\mathrm{SO}(2m,\mathbb{C})\xrightarrow{}\mathrm{PSO}(2m,\mathbb{C})\xrightarrow{}1.$

Therefore, we have

$\mathfrak{so}(2m,\mathbb{C})=\mathfrak{pso}(2m,\mathbb{C}).$ (8)

Let $(V,\eta)$ be a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle
lifting to the parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundle
$(\tilde{V},\tilde{\eta})$ and let $(E,\Phi,\varphi,L)$ be the bundle
corresponding to $(\tilde{V},\tilde{\eta})$. Then $(V,\eta)$ corresponds to
the equivalence class $[(E,\Phi,\varphi,L)]$ where the equivalence relation
$\sim_{\mathrm{Jac}(X)}$ is given by :

$(E,\Phi,\varphi,L)\sim_{\mathrm{Jac}(X)}(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},L\otimes M^{2})\hskip
28.45274pt\mathrm{for}\hskip 2.84544pt\mathrm{any}\hskip
5.69046ptM\in{\mathrm{Jac}(X)}$

where

$\varphi_{M}:(E\otimes M)\otimes(E\otimes M)\to L\otimes M^{2}$

is the induced symmetric bilinear nondegenerate form on $E\otimes M$ taking
values in $L\otimes M^{2}$.

As in the previous case, the parabolic degree of a parabolic
$\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle can be defined as follows:

######  Definition 15.

Let $(V,\eta)$ be a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle and
let $(\tilde{V},\tilde{\eta})$ be a lifting to a parabolic
$\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundle. Let $(E,\Phi,\varphi,L)$ be the
datum corresponding to $(\tilde{V},\tilde{\eta})$ with parabolic degree $ml$,
where $l=\deg(L)$. Then the parabolic degree of $(V,\eta)$ is given by the
class $\overline{ml}\in\mathbb{Z}/m\mathbb{Z}$.

Let $\\{s_{2i}\\}_{i=1,\dots,m}$ be a basis of invariant polynomials of the
lie algebra $\mathfrak{so}(2m,\mathbb{C})=\mathfrak{pso}(2m,\mathbb{C})$ In
this case, the coefficient $s_{2m}$ is a square of a polynomial $p_{m}\in
H^{0}(X,K(D)^{m})$, the Pfaffian, of degree $m$. A basis for the invariant
polynomials on the Lie algebra
$\mathfrak{so}(2m,\mathbb{C})=\mathfrak{pso}(2m,\mathbb{C})$ is given by the
coefficients $\\{s_{2},...,s_{2m-2},p_{m}\\}$. Therefore, the Hitchin map for
the moduli space of parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundles is
given by

$\displaystyle h_{\mathrm{PSO-par}}:\mathcal{M}_{\mathrm{PSO-
Higgs}}(\alpha,2m,\overline{ml})$
$\displaystyle\longrightarrow\mathcal{H}_{\mathrm{PSO-
par}}\coloneqq\bigoplus_{i=1}^{m-1}H^{0}(X,K^{2i}(D^{2i-1}))\oplus
H^{0}(X,K(D)^{m})$ $\displaystyle(V,\eta)$
$\displaystyle\longmapsto(s_{2},\dots,s_{2m-2},p_{m}).$

For $s=(s_{2},\dots,s_{2m-2},p_{m})\in\mathcal{H}_{\mathrm{PSO-par}}$, the
corresponding spectral curve $X_{s}$ is given by the equation

$x^{2m}+s_{2}x^{2m-2}+\cdots+s_{2m-2}x^{2}+p_{m}^{2}=0.$

The zeroes of $p_{m}$ are singularities of $X_{s}$ and these are the only
singularities. Since $p_{m}\in H^{0}(X,K(D)^{m})$, there are
$K(D)^{m}=m(2g-2+n)$ many singularities. As in the previous case, $X_{s}$
possesses an involution $\sigma(\eta)=-\eta$. Then the fixed points of this
involution $\sigma$ are exactly the singularities of $X_{s}$. Let
$\hat{X_{s}}$ denote the desingularisation of $X_{s}$ with genus

$g(\hat{X_{s}})=g(X_{s})-\mathrm{number\hskip 2.84544ptof\hskip
2.84544ptsingularities}.$

Since the singularities of $X_{s}$ are double points, the involution $\sigma$
on $X_{s}$ extends to an involution $\hat{\sigma}$ on $\hat{X_{s}}$. Therefore
the Prym variety
$P_{s,\hat{\sigma}}\coloneqq\mathrm{Prym}(\hat{X_{s}},\hat{X_{s}}/\hat{\sigma})$
is given by

$P_{s,\hat{\sigma}}\coloneqq\mathrm{Prym}(\hat{X_{s}},\hat{X_{s}}/\hat{\sigma})=\\{N\in\text{Jac}(\hat{X_{s}}):\hat{\sigma}^{*}N\cong
N^{\vee}\\}$ (9)

As in the symplectic case, there is a bijective correspondence between the
Prym variety $P_{s,\hat{\sigma}}$ and

$\Omega_{s,\hat{\sigma}}\coloneqq\\{(U,L,\tau)\hskip 2.84544pt|\hskip
2.84544ptU\in\text{Jac}(\hat{X_{s}}),\tau:\hat{\sigma}^{*}U\cong
U^{\vee}\otimes(K_{\hat{X_{s}}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L\\}.$

Again the action of the group $\mathrm{Jac}(X)$ on $\Omega_{s,\hat{\sigma}}$
is given by

$\displaystyle\Omega_{s,\hat{\sigma}}\times\mathrm{Jac}(X)$
$\displaystyle\longrightarrow\Omega_{s,\hat{\sigma}}$
$\displaystyle((U,L,\tau),M)$
$\displaystyle\longmapsto(U\otimes\pi^{*}M,L\otimes M^{2},\tau_{M})$

where $\tau_{M}=\tau\otimes\text{Id}_{\pi^{*}M}$ is the isomorphism

$\hat{\sigma}^{*}(U\otimes\pi^{*}M)\cong(U\otimes\pi^{*}M)^{\vee}\otimes(K_{\hat{X_{s}}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}(L\otimes
M^{2}).$

######  Theorem 4.1.

For a generic point $s\in\mathcal{H}_{\mathrm{PSO-par}}$, the fiber
$h_{\mathrm{PSO-par}}^{-1}(s)$ is isomorphic to
$\Omega_{s,\hat{\sigma}}/\mathrm{Jac}(X)$.

###### Proof.

Let $(V,\eta)\in h_{\mathrm{PSp-par}}^{-1}(s)$ lifts to a parabolic
$\mathrm{GSO(2m,\mathbb{C})}$-Higgs bundle $(\tilde{V},\tilde{\eta})$ whose
corresponding datum is $(E,\Phi,\varphi,L)$. Then $(V,\eta)$ corresponds to
the datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ uniquely. By
[15, Theorem 4.2], the datum of $(E,\Phi,\varphi,L)$ corresponds to an element
of the Pyrm variety $\Omega_{s,\hat{\sigma}}$. Let $M\in\mathrm{Jac}(X)$. Then
the datum of $(E\otimes M,\Phi\otimes\text{Id}_{M},\varphi_{M},L\otimes
M^{2})$ corresponds uniquely to an element of
$\Omega_{s,\hat{\sigma}}/\mathrm{Jac}(X)$. Therefore, we conclude that the
datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ corresponds
uniquely to the datum of the $\mathrm{Jac}(X)$-orbit of an element of
$\Omega_{s,\hat{\sigma}}$.

∎

##  5\. Fixed line bundle : $\mathcal{O}_{X}$

In this section, we will assume that the symplectic/orthogonal form in 1 takes
values in the trivial line bundle $\mathcal{O}_{X}$. In particular, we
consider the moduli space of parabolic symplectic/orthogonal Higgs bundles
with fixed rank, degree and fixed line bundle $\mathcal{O}_{X}$. In other
words, we are considering the moduli spaces with trivial determinant.

###  5.1. Parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles with fixed
line bundle $\mathcal{O}_{X}$

Let $\mathrm{Jac}_{2}(X)$ be the subgorup of the Jacobian $\mathrm{Jac}(X)$
which contains the $2$-torsion elements of $\mathrm{Jac}(X)$, i.e.

$\mathrm{Jac}_{2}(X)\coloneqq\\{M\in\mathrm{Jac}(X)\hskip 2.84544pt|\hskip
2.84544ptM^{2}\cong\mathcal{O}_{X}\\}.$

Let $(E,\Phi,\varphi,\mathcal{O}_{X})$ be a parabolic symplectic Higgs bundle
with the symplectic form $\varphi:E\otimes E\to\mathcal{O}_{X}$ taking values
in $\mathcal{O}_{X}$ and let $M\in\mathrm{Jac}_{2}(X)$. Then

$\varphi_{M}:(E\otimes M)\otimes(E\otimes
M)\longrightarrow\mathcal{O}_{X}\otimes
M^{2}\cong\mathcal{O}_{X}\otimes\mathcal{O}_{X}\cong\mathcal{O}_{X}$

defines a symplectic form on $E\otimes M$ with values in $\mathcal{O}_{X}$.

Since the symplectic form takes values in $\mathcal{O}_{X}$, a parabolic
$\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle $(V,\eta)$ lifts to a parabolic
$\mathrm{Sp}(2m,\mathbb{C})$-Higgs bundle $(\tilde{V},\tilde{\eta})$. Let
$(E,\Phi,\varphi,\mathcal{O}_{X})$ be the parabolic symplectic Higgs bundle
corresponding to $(\tilde{V},\tilde{\eta})$. Then $(V,\eta)$ corresponds to
the equivalence class $[(E,\Phi,\varphi,\mathcal{O}_{X})]$ where the
equivalence relation $\sim_{\mathrm{Jac}_{2}(X)}$ is given by :

$(E,\Phi,\varphi,\mathcal{O}_{X})\sim_{\mathrm{Jac}_{2}(X)}(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},\mathcal{O}_{X})\hskip
28.45274pt\mathrm{for}\hskip 2.84544pt\mathrm{any}\hskip
5.69046ptM\in{\mathrm{Jac}_{2}(X)}.$

As in 7, the Prym variety is given by

$\Omega_{s,\sigma}\coloneqq\\{(U,\mathcal{O}_{X},\tau)\hskip 2.84544pt|\hskip
2.84544ptU\in\text{Jac}(X_{s}),\tau:\sigma^{*}U\cong
U^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}\mathcal{O}_{X}\\}.$

Also, the group $\mathrm{Jac}_{2}(X)$ acts on $\Omega_{s,\sigma}$ by

$\displaystyle\Omega_{s,\sigma}\times\mathrm{Jac}_{2}(X)$
$\displaystyle\longrightarrow\Omega_{s,\sigma}$
$\displaystyle((U,\mathcal{O}_{X},\tau),M)$
$\displaystyle\longmapsto(U\otimes\pi^{*}M,\mathcal{O}_{X}\otimes
M^{2},\tau_{M})\cong(U\otimes\pi^{*}M,\mathcal{O}_{X},\tau_{M})$

where $\tau_{M}=\tau\otimes\text{Id}_{\pi^{*}M}$ is the isomorphism

$\sigma^{*}(U\otimes\pi^{*}M)\cong(U\otimes\pi^{*}M)^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}\mathcal{O}_{X}.$

######  Theorem 5.1.

For a generic point $s\in\mathcal{H}_{\mathrm{PSp-par},\mathcal{O}_{X}}$ of
the parabolic $\mathrm{PSp}$-Hitchin map with the fixed line bundle
$\mathcal{O}_{X}$, the fiber $h_{\mathrm{PSp-par},\mathcal{O}_{X}}^{-1}(s)$ is
isomorphic to the quotient $\Omega_{s,\sigma}/\mathrm{Jac}_{2}(X)$.

###### Proof.

The proof is similar to the proof of the Theorem 3.1. ∎

###  5.2. Parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundles with fixed
line bundle $\mathcal{O}_{X}$

As in the symplectic case, a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs
bundle $(V,\eta)$ with the orthogonal form taking values in $\mathcal{O}_{X}$
lifts to a parabolic $\mathrm{SO}(2m,\mathbb{C})$-Higgs bundle
$(\tilde{V},\tilde{\eta})$. Let $(E,\Phi,\varphi,\mathcal{O}_{X})$ be the
parabolic (even) orthogonal Higgs bundle corresponding to
$(\tilde{V},\tilde{\eta})$. Then $(V,\eta)$ corresponds to the equivalence
class $[(E,\Phi,\varphi,\mathcal{O}_{X})]$ where the equivalence relation
$\sim_{\mathrm{Jac}_{2}(X)}$ is given by:

$(E,\Phi,\varphi,\mathcal{O}_{X})\sim_{\mathrm{Jac}_{2}(X)}(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},\mathcal{O}_{X})\hskip
28.45274pt\mathrm{for}\hskip 2.84544pt\mathrm{any}\hskip
5.69046ptM\in{\mathrm{Jac}_{2}(X)}.$

Following the above section 4, the alternative description of the Prym variety
(9) is given by

$\Omega_{s,\hat{\sigma}}\coloneqq\\{(U,\mathcal{O}_{X},\tau)\hskip
2.84544pt|\hskip
2.84544ptU\in\text{Jac}(\hat{X_{s}}),\tau:\hat{\sigma}^{*}U\cong
U^{\vee}\otimes(K_{\hat{X_{s}}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}\mathcal{O}_{X}\\}.$

Also, the action of the subgroup $\mathrm{Jac}_{2}(X)$ on
$\Omega_{s,\hat{\sigma}}$ is given by

$\displaystyle\Omega_{s,\hat{\sigma}}\times\mathrm{Jac}_{2}(X)$
$\displaystyle\longrightarrow\Omega_{s,\hat{\sigma}}$
$\displaystyle((U,\mathcal{O}_{X},\tau),M)$
$\displaystyle\longmapsto(U\otimes\pi^{*}M,\mathcal{O}_{X},\tau_{M})$

where $\tau_{M}=\tau\otimes\text{Id}_{\pi^{*}M}$ is the isomorphism

$\hat{\sigma}^{*}(U\otimes\pi^{*}M)\cong(U\otimes\pi^{*}M)^{\vee}\otimes(K_{\hat{X_{s}}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}\mathcal{O}_{X}.$

######  Theorem 5.2.

For a generic point $s\in\mathcal{H}_{\mathrm{PSO-par},\mathcal{O}_{X}}$ of
the parabolic $\mathrm{PSO}$-Hitchin map with the fixed line bundle
$\mathcal{O}_{X}$, the fiber $h_{\mathrm{PSO-par},\mathcal{O}_{X}}^{-1}(s)$ is
isomorphic to the quotient $\Omega_{s,\hat{\sigma}}/\mathrm{Jac}_{2}(X)$.

###### Proof.

The proof is similar to the proof of the Theorem 4.1. ∎

######  Remark 5.3.

We can actually consider any degree zero line bundle in place of
$\mathcal{O}_{X}$.

## Acknowledgement

This work was supported by the Institute for Basic Science (IBS-R003-D1).

## Data Availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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###### Abstract

We study primordial black holes (PBHs) formation in the excursion set theory
(EST) in a vast range of PBHs masses with and without confirmed constraints on
their abundance. In this work, we introduce a new concept of the first touch
in the context of EST for PBHs formation. This new framework takes into
account the earlier horizon reentry of smaller masses. Our study shows that in
the EST, it is possible to produce PBHs in different mass-range, with enhanced
power spectrum, which could make up all dark matter. We also show that in a
broad blue-tilted power spectrum, the production of PBHs is dominated by
smaller masses. Our analysis put an upper limit $\sim\,$0.1 on the amplitude
of the curvature power spectrum at length scales relevant for PBHs formation.

Primordial Black Holes in the Excursion Set Theory

Encieh Erfani${}^{1,\,*}$, Hamed Kameli2 and Shant Baghram${}^{2,\,{\dagger}}$

1Department of Physics, Institute for Advanced Studies in Basic Sciences
(IASBS), Zanjan 45137-66731, Iran
2Department of Physics, Sharif University of Technology, Tehran 11155-9161,
Iran

000∗erfani<EMAIL_ADDRESS>
##  1 Introduction

####

Primordial black holes (PBHs) can form from the collapse of large density
fluctuations in the early Universe [1, 2, 3]. When the density fluctuations
larger than the threshold reenters the horizon after inflation, the region
will collapse to form a PBH with a mass roughly equal to the mass of the
horizon. According to the Hawking radiation [4], PBHs with mass larger than
$\sim 10^{15}$ g have a lifetime longer than the age of the Universe.
Moreover, since they have formed before matter-radiation equality, they are
non-baryonic. Thus they could be a candidate for dark matter (DM).

After the discovery of gravitational waves (GWs) in 2016 by LIGO/Virgo from
mergers of tens of Solar mass black holes [5], the possibility that these BHs
could be primordial rather than astrophysical [6, 7], has led to a great
interest in PBH version of DM. In addition, the abundance of PBHs in our
observable Universe can give us some clues of small-scale density fluctuations
which are not accessible via observations of the cosmic microwave background
(CMB) and the large scale structure (LSS) (i.e. constraints on the primordial
power spectrum for scales between $k\sim(10^{-3}-1)$ Mpc-1) [8, 9, 10]. (For
recent reviews on PBHs, see also refs. [11, 12].) The power spectrum at small
scales relevant for PBHs formation must become orders of magnitude larger than
$\mathcal{O}(10^{-9})$ detected on large scales via the CMB [13] which is a
matter of active research. Since the amplitude of the power spectrum is only
logarithmically sensitive to the PBHs abundance during the radiation
domination, this variation has little to do with the different PBH masses [14,
15]. However, the PBH mass distribution differs when using different
theoretical techniques for their formation [16]111For instance, primordial
non-Gaussianity can have an important effect on the required power spectrum
amplitude [17]. However, we will not consider this issue in this paper.. PBH’s
formation has been studied with the use of Press-Schechter (PS) formalism
[18], peaks theory [19, 20] and recently by excursion set theory (EST) [21,
22]. In this paper, we make the first detailed numerical study of the mass
distribution of PBH in the EST [23] by counting the number of trajectories
that reenter the horizon corresponding to a specific mass. We study the
formation of PBHs when a broad spectrum characterizes the scalar
perturbations, and the goal of the paper is to answer the question of the mass
distribution of PBHs for broad spectra in the EST. We will show that despite
the broadness of the blue-tilted power spectrum, the production of PBHs
enhances towards small masses.

There are various constraints on PBHs abundance which have recently been
compiled in [12, 14]. In our study, we will consider these observational
constraints in mass ranges if they exist. Otherwise, we will assume that PBHs
can comprise all of the DM. We will find the required blue-tilled spectral
indexes for these mass ranges in the EST formalism. Finally, we will translate
these results into upper limit on the amplitude of the curvature power
spectrum at length scales relevant for PBHs formation.

The paper is laid out as follows: in section 2 we elaborate PBH formation in
PS formalism, and we review current constraints on their density being
associated with a variety of gravitational lensing [24, 25, 26, 27] and GW
[28] effects. In section 3 we briefly explain the EST method and we develop
the PBHs formation in this context. Finally, section 4 is devoted to our
conclusions.

##  2 Primordial Black Holes

####

The most common mechanism for PBHs formation is the collapse of large density
perturbations generated by inflation in the very early Universe. These density
perturbations that reenter the Hubble horizon in the radiation dominated (RD)
era would gravitationally collapse into a BH if their amplitude is larger than
a critical value. Since PBHs are not formed by stellar core collapse, they may
be of any size from the Planck mass $\sim 10^{-5}\,$g to $\sim 10^{50}\,$g
[12]. Mass of PBH, $M_{\rm PBH}$ produced at a given time is limited by the
horizon mass at that time

$M_{\rm PBH}=\gamma\,M_{\rm PH}\sim 10^{15}\left(\dfrac{t}{10^{-23}\,{\rm
s}}\right){\rm g}\,,$ (2.1)

where $M_{\rm PH}$ is the particle horizon mass and $\gamma$ is the fraction
of the total energy that ends up inside the PBH222For simplicity we assume
that the PBH mass is a fixed fraction of the horizon mass corresponding to the
smoothing scale; $\gamma\simeq w^{3/2}\simeq 0.2$ during the RD era ($w=1/3$)
[29].. Since PBHs radiate thermally [4], they evaporate on a time scale

$\tau_{{}_{\rm PBH}}(M)\sim
10^{64}\,\left(\frac{M}{M_{\odot}}\right)^{3}\,\rm{yr}\,,$ (2.2)

therefore, the ones with mass greater than $10^{15}$ g survived until now and
would be plausible DM candidates. In this paper, we will focus on DM PBHs
formed in the RD era.

In order to investigate the abundance of formed PBHs, we define a parameter
which represents the mass fraction (the energy density fraction) of PBHs,

$\beta\equiv\frac{\rho_{\rm PBH}(t_{\rm i})}{\rho(t_{\rm i})}\,,$ (2.3)

where the subscript “i” indicates values at the epoch of PBH formation. It is
straightforward to show that this fraction can be related to present day
abundance of PBHs, $f_{\rm PBH}$ [30]

$\beta\simeq 3.7\times
10^{-9}\,\gamma^{-1/2}\,\left(\frac{g_{*,\,i}}{10.75}\right)^{1/4}\left(\frac{M_{\rm
PBH}}{M_{\odot}}\right)^{1/2}\,f_{\rm PBH}\,,$ (2.4)

where $g_{*,\,i}$ is a number of relativistic degree of freedom at the time of
formation. Note that $f_{\rm PBH}\equiv\Omega_{\rm PBH}/\Omega_{\rm DM}$ is a
fraction of PBHs against the total DM component, $\Omega_{\rm DM}$. Thus, for
each mass of PBHs, the observational constraint on $f_{\rm PBH}$ can be
interpreted as that on $\beta$.
The fraction of the energy density of the Universe contained in regions
overdense enough to form PBHs is usually calculated in PS theory [18] as

$\beta=\gamma\int_{\delta_{c}}^{\infty}P(\delta;\,R)\,d\delta\,.$ (2.5)

Here $P(\delta;\,R)$ is the probability distribution function of the linear
density field $\delta$ smoothed on a scale $R$, and $\delta_{c}$ is the
critical threshold for PBHs formation. The estimation of $\delta_{c}$ is under
discussion by different researches both analytically [31] and numerically [32,
33]. In general, the value of $\delta_{c}$ is not unique and depends on the
shape of the primordial curvature power spectrum [34]. The value of
$\delta_{c}$ is varying between 0.4 and 2/3. In this paper, we consider
$\delta_{c}=0.47$ which is the upper bound of the threshold calculated in [34]
for a Gaussian curvature profile 333Note that in general, the value of
$\delta_{c}$ is a function of the equation of state parameter [32]. For
example, a change in the relativistic degrees of freedom at the electroweak
and QCD phase transitions induces a change in $\delta_{c}$. This will have
unavoidable features in the mass function [35, 36].. For the Gaussian
fluctuations, $\beta$ is given by [37]

$\displaystyle\beta(M_{\rm PBH})$
$\displaystyle=\gamma\int_{\delta_{c}}^{\infty}\frac{d\delta}{\sqrt{2\pi\,\sigma^{2}_{\delta}(R)}}\,\exp\left(-\frac{\delta^{2}}{2\,\sigma^{2}_{\delta}(R)}\right)$
$\displaystyle=\frac{\gamma}{2}\,\operatorname{erfc}\left(\frac{\delta_{c}}{\sqrt{2\sigma^{2}}}\right)\,,$
(2.6)

where $\operatorname{erfc}(x)=1-\operatorname{erf}(x)$ is the complementary
error function and, the variance of $\delta$ is given by

$\sigma^{2}_{\delta}(R)=\int\dfrac{dk}{k}\,\mathcal{P}_{\delta}(k)\,\widetilde{W}^{2}(k,\,R)=\int\dfrac{dk}{k}\,\dfrac{16}{81}\left(\dfrac{k}{aH}\right)^{4}\,\mathcal{P_{R}}(k)\,\widetilde{W}^{2}(k,\,R)\,,$
(2.7)

where $\widetilde{W}(k,\,R)$ is the Fourier transform of the window function
used to smooth the density contrast on a comoving scale $R$. In this paper, we
will consider the $k$-space top-hat window function. Recall that
$\mathcal{P}_{\delta}(k)$ is the power spectrum of $\delta$ which is related
to the power spectrum of curvature perturbations on comoving hypersurfaces in
RD era as follows [38]

$\mathcal{P}_{\delta}(k)=\dfrac{16}{81}\left(\dfrac{k}{aH}\right)^{4}\mathcal{P_{R}}(k)\,,$
(2.8)

and we parameterize the curvature power spectrum as

$\mathcal{P_{R}}(k)=A_{0}\,\left(\frac{k}{k_{0}}\right)^{n_{s}(k)-1}\,,$ (2.9)

where $\ln(10^{10}A_{0})=3.044\pm 0.014$ and the spectral index,
$n_{s}(k_{0})=0.9649\pm 0.0042$ are known by the CMB observation at
$k_{0}=0.05$ Mpc-1 [13].
As already mentioned, the mass of PBH, $M_{\rm PBH}$ is related to scale of
horizon at the time of formation when the perturbation reenters the horizon
($k=aH$). It is straightforward to obtain the relation between the mass of
PBHs and the comoving wavenumbers as [39],

$M_{\rm PBH}(k)\simeq
30\,M_{\odot}\,\left(\dfrac{\gamma}{0.2}\right)\,\left(\dfrac{g_{*,\,i}}{10.75}\right)^{-1/6}\left(\dfrac{k_{\rm
PBH}}{2.9\times 10^{5}\,{\rm Mpc}^{-1}}\right)^{-2}\,.$ (2.10)

If the power spectrum of curvature perturbations were scale invariant, using
Eq. (2), and assuming $\delta_{c}\simeq 0.47$ and $\sigma_{\delta}^{2}\propto
A_{0}$, then the initial mass fraction of PBHs formed in the RD era would be
completely negligible. On the other hand, if we assume that all of the DM is
in PBHs with $M_{\rm PBH}\sim M_{\odot}$ (i.e. $f_{\rm PBH}=1$) then, from Eq.
(2.4), the initial PBH mass fraction must be $\beta\sim 10^{-9}$. This means
that the required power spectrum on this scale (see Eq. (2.10)) must be
$\mathcal{P_{R}}({k_{\rm PBH}})\sim 10^{-2}$ which is 7 orders of magnitude
larger than the measured value on cosmological scales $(A_{0}\sim 10^{-9})$.
Thus for DM PBHs formation, the amplitude of power must be enhanced
exponentially in specific wavenumbers corresponding to PBHs mass. One way to
enhance the power is to have a blue-tilted spectral index in the scale of PBHs
formation, $n_{s(b)}(k_{\rm PBH})$. Hence the abundance of PBHs with different
masses (formed in different scales) could put constraints on the power
spectrum beyond the range probed by cosmological observations.

There are various constraints on PBHs abundance. In this paper, we are
interested in PBHs with a mass range from $\sim(10^{16}-10^{46})$ g. Part of
this mass range is probed by the gravitational lensing and the GWs
observations. However, there remain mass windows where PBHs may constitute the
whole DM. We review the potential constraints as follows:

  * •

Gravitational Lensing: Long-duration microlensing is a significant probe to
find the contribution of compact objects as DM. These constraints have been
acquired by a) Subaru HSC (Hyper Suprime-Cam) survey observing the
microlensing of stars in M31 (Andromeda galaxy) [24], b) OGLE (Optical
Gravitational Lensing Experiment) microlensing survey of the Galactic bulge
[25], and c) EROS/MACHO [26, 27] which monitors millions of stars in
Magellanic Cloud, respectively.
\- $10^{-11}\,M_{\odot}\lesssim M_{\rm PBH}\lesssim 10^{-6}\,M_{\odot}$ with
$f_{\rm PBH}\sim 10^{-3}\,,$
\- $10^{-6}\,M_{\odot}\lesssim M_{\rm PBH}\lesssim 10^{-3}\,M_{\odot}$ with
$f_{\rm PBH}\sim 10^{-2}\,,$
\- $10^{-3}\,M_{\odot}\lesssim M_{\rm PBH}\lesssim 10^{-1}\,M_{\odot}$ with
$f_{\rm PBH}<0.04$ .

Note that if $f_{\rm PBH}$ is not negligible and $M_{\rm PBH}\sim$ stellar
mass, the Poissonian noise due to the discrete nature of PBHs implies a
boosted formation of PBH at high redshift, compared to the standard model
[40]. In turn, when most PBHs are in massive halos, they can evade the
microlensing limits due to the additional lensing effect of the cluster, as
shown by [36]. However, in this work, we evade these complications and use the
constraints mentioned by [12].

  * •

Gravitational Waves: The merger of sub-Solar PBH binaries emits GWs like those
observed by LIGO/Virgo [28]. The observed merger rates of the second
LIGO/Virgo run impose constraints on PBH fraction in the following mass range
\- $0.2\,M_{\odot}\lesssim M_{\rm PBH}\lesssim 1.0\,M_{\odot}$, corresponding
to at most $0.16$ or $0.02$ of the DM, respectively 444For the Poisson noise,
the limits from GWs and BH merging rates still allow $f_{\rm PBH}>0.1$ [41]..
The main constraints on solar mass PBH are GW and lensing, which both are
model-dependent. For example, one can evade the lensing constraints by
clustering [36]. Therefore, we consider the optimistic fraction
$f_{\text{PBH}}=1$ as well.
\- intermediate mass range, $10\,M_{\odot}\lesssim M_{\rm PBH}\lesssim
10^{3}\,M_{\odot}$: In this mass range there are several limits such as ultra-
faint dwarf galaxies, X-rays and radio binaries, wide binaries, and CMB
distortion limits [12]. Thus we choose the average limit of $f_{\rm PBH}\sim
10^{-4}$.

  * •

For the following mass ranges there is no (model independent) confirmed
constraints. Therefore, PBHs could be the whole DM, i.e. $f_{\rm PBH}\sim 1$
[12].
\- asteroid mass range $10^{16}\,{\rm g}\lesssim M_{\rm PBH}\lesssim 10^{17}$
g ,
\- sublunar mass range $10^{20}\,{\rm g}\lesssim M_{\rm PBH}\lesssim 10^{24}$
g ,
\- stupendously large black holes (SLABs) $M_{\rm PBH}\geq 10^{11}\,M_{\odot}$
[42].

Thus, in this paper, we will consider eight mass ranges. It is worth to
mention that we only consider the constraints without any assumption about the
distribution of PBHs at their formation, their clustering and the environment
since these conditions can have a significant effect in their abundances.

##  3 Abundance of PBH in the Context of EST

As already mentioned in section 2, the PS formalism can determine the
abundance of PBHs. In this section, we will consider this issue in EST as a
sophisticated development of PS formalism. In subsection 3.1, we will review
the main idea of EST, and in subsection 3.2, we will have our main results on
studying the abundance of PBHs in this context.

###  3.1 Theoretical Background

According to the EST, the spherical/ellipsoidal collapse of a compact object
of mass $M$ forms where the corresponding smoothed linear density contrast,
$\delta$ is larger than critical value, $\delta_{c}$ [43]. In this framework,
the computation of the mass function of a compact object is formulated in
terms of a stochastic process. In other words, if we smooth the density
contrast in an initial box (higher redshifts where perturbations are almost
linear and Gaussian) and plot the density contrast in terms of variance of
perturbations $S$, it executes a stochastic time series. Since $\delta$ is a
function of variance, its evolution is governed by Gaussian white noise which
implies the random walk with respect to $S$. The trajectories start from an
initial value $\delta=0$ at $S=0$ (see Fig. 1).

Figure 1: The Markov trajectories were performed by the smoothed density
contrast. The trajectories which pass the threshold, $\delta_{c}$ for the
first time are called a) First Up-Crossing (FU): at lower $S$ (larger mass) b)
First Touch (FT): at higher $S$ (smaller mass). The blue inverted coordinates
show the comoving scale versus cosmic time. The dashed inclined lines show the
dependence of the comoving horizon $\left(R_{H}=(aH)^{-1}\right)$ to scale and
time. The dash-dotted and dotted blue vertical lines correspond to the
perturbation modes reentering the horizon in radiation dominated (RD) era,
corresponding to FT and FU concepts, respectively.

In this context, the first up-crossing (FU) trajectories that pass
$\delta_{c}$ will give the mass of the collapsed object. Accordingly, the
statistics of FU trajectories will give the distribution of collapsed objects.
If we use the $k$-space top-hat window for smoothing, the random walks become
Markovian and the distribution of FU trajectories is given by

$f_{\rm
FU}(S,\,\delta_{c})=\dfrac{1}{\sqrt{2\pi}}\,\dfrac{\delta_{c}}{S^{3/2}}\exp\left(-\frac{\delta^{2}_{c}}{2S}\right)\,,$
(3.1)

where the variance of density perturbations, $S$ is given by Eq. (2.7).
We will produce Markov trajectories by the method developed in our previous
works [44, 45, 46] where the distribution of DM halos in the (extended)
standard model are addressed in more detail.
It is worth mentioning that there is a fundamental difference between the
statistics of DM halos and PBHs. PBHs form when the scale corresponding to
their mass reenters the horizon. In Fig. 1, the vertical lines represent the
modes that reenter the horizon on a specific scale. We should consider the
trajectories which touch the barrier in larger variance for the first time,
hereafter first touch (FT). This approach is reasonable since the scale of PBH
with lower mass (larger variance) reenters the horizon sooner than the massive
one. The concept of FT is clear from Fig. 1, where the FT trajectories touch
the threshold in larger variance; i.e. smaller mass PBHs form first. For more
clarification, compare the vertical dashed-dotted lines for small mass PBH
corresponding to FT with vertical dotted lines for larger masses corresponding
to FU. Accordingly, the number of FT trajectories at each variance will give
the abundance of PBHs with corresponding mass. Note that according to
correspondence between PS and EST, the cumulative summation of FU distribution
(Eq. (3.1)) is equal to the fraction of PBHs at their formation, $\beta$ (see
Eq. (2.5)). We show that by knowing the FU distribution for any given power
spectrum, one can find $\beta$ and $f_{\rm PBH}$ (see Eq. (2.4)). The fraction
of PBHs is given by counting the FT trajectories. However, statistically, the
FT distribution is a correction to the FU ones with almost similar behaviour.
To clarify this issue, in the schematic Fig. 2, we plot the FU and FT
distribution (top plot) and their cumulative distribution $\beta$ (bottom
plot) for an arbitrary mass range. FT distribution is larger than FU at
smaller masses and vice versa. One could anticipate this result as the FT is
related to the crossings in the smaller mass ranges in comparison to FU. The
cumulative distribution, $\beta$ of both FT and FU is almost the same for the
lower bound of the mass range.

Figure 2: Top panel: The Schematic illustration of both FU (solid line) and FT
(dotted line) for $f$ (fraction) versus mass is plotted. Bottom panel: The
corresponding cumulative fraction, $\beta$ versus mass for both FU (solid
line) and FT (dotted line) is shown.

###  3.2 PBH in the Context of EST

To calculate the abundance of PBHs at different scales, the distribution is
confined with a window function to smooth out modes smaller than the horizon.
In the EST formalism for halo formation, the variance could be calculated at
redshift $z=0$ and density threshold, $\delta_{c}$ changes with growth
function, $D(z)$. However in the case of PBH, we use a constant
$\delta_{c}=0.47$ in RD era [34] and the power spectrum is identified in
various horizon crossing scales, $R_{H}=(aH)^{-1}$. This means the redshift
dependence is encapsulated in power spectrum instead of $\delta_{c}$. So we
have a unique power spectrum for each horizon scale $R_{H}$ which is related
to a specific mass. Therefore, Eq. (2.8) at horizon crossing scale $R_{H}$ is
given by

$\mathcal{P}_{\delta}(k,\,R_{H})\equiv\mathcal{P}_{\delta,R_{H}}(k)=\dfrac{16}{81}A_{0}\,\left(kR_{H}\right)^{4}\left(\dfrac{k}{k_{0}}\right)^{n_{s}(k)-1}\,.$
(3.2)

We use a variable spectral index to apply a blue-tilted power spectrum in a
broad interval $k\sim[k_{s},\,k_{l}]$ where $k_{s}$ and $k_{l}$ are short and
long wavenumbers. This means

$n_{s}(k)=\begin{cases}n_{s}(k_{0})\,,&k<k_{s}\\\
n_{s(b)}>1\,,&k_{s}<k<k_{l}\\\ n_{s}(k_{0})\,,&k>k_{l}\\\ \end{cases}$ (3.3)

Figure 3: Top panel: The enhanced initial power spectrum is plotted versus
wavenumber. The enhancement is for blue-tilted (dashed and dotted lines) and
red-tilted with $A=10^{11}A_{0}$ (solid line) models in wavenumber range of
$k\sim[k_{s},\,k_{l}]$. Bottom panel: The variance, $S$ and the fraction, $f$
versus PBH mass is plotted in left and right panels, respectively.

where $n_{s}(k_{0})=0.9649\pm 0.0042$ and $n_{s(b)}$ are the observed spectral
index and the blue-tilted one, respectively. In Fig. 3, top panel, we show the
blue-tilted power enhancement in the wavenumber interval
$k\sim[k_{s},\,k_{l}]$. Since we have a whole sequence of $S$ for different
scales, $R_{i}$ at any specific horizon crossing scale $R_{H}$, we rewrite Eq.
(2.7) as follows

$S_{R_{H}}(R_{i})=A\,R_{H}^{4}\int_{0}^{\infty}dk\,k^{n_{s}(k)+2}\,\widetilde{W}^{2}(k,\,R_{i})\,,$
(3.4)

where $A\equiv\dfrac{16}{81}\dfrac{A_{0}}{k_{0}^{n_{s}(k)-1}}$. The above
equation means that using each sequence of $S$, we form a whole realization of
trajectories in $(\delta-S)$ plane for each $R_{H}$. Hence, we could count the
FT numerically, i.e. $f_{\rm FT,\,R_{H}}$. Therefore, for PBHs which form at
horizon crossing, $f_{\rm FT,\,R_{H}}$ is only meaningful where the FT is
counted at $R_{i}=R_{H}$.
Note that since PBHs formation at horizon reentry is independent of other
scales, therefore their formation mechanism is history independent. In the
EST, one can get memory independent trajectories by using the $k$-space top-
hat window function which is known as Markov model555Note that the non-Markov
extension of EST has great importance in halo formation due to more physical
smoothing window functions which lead to correction of statistics of FUs,
especially in small scales [44, 45, 47].. For Markov model Eq. (3.4) is given
by

$\displaystyle\frac{S_{R_{H}}(R_{i})}{A\,R_{H}^{4}}$
$\displaystyle=\frac{1}{R_{i}^{n_{s}+3}}\int_{0}^{min(\frac{R_{i}}{R_{s}},\,1)}{dk\,k^{n_{s}+2}}$
(3.5)
$\displaystyle\,+\,\frac{1}{R_{i}^{n_{s(b)}+3}}\int_{min(\frac{R_{i}}{R_{s}},\,1)}^{min(\frac{R_{i}}{R_{l}},\,1)}{dk\,k^{n_{s(b)}+2}}$
$\displaystyle\,+\,\frac{1}{R_{i}^{n_{s}+3}}\int_{min(\frac{R_{i}}{R_{l}},\,1)}^{\infty}{dk\,k^{n_{s}+2}}\,.$

The $k$-space window function applies sharp conditions in upper/lower limits
of the integral.
Note that due to random nature of these trajectories, we only need to apply
one sequence of $S$ and substitute $S(R_{H})\equiv S_{R_{H}}(R_{i}=R_{H})$. We
can analytically calculate $f_{\rm FU}$ and count the first touch of
trajectories to evaluate $f_{\rm FT}$. In counting process, we will exclude
any trajectory which is already collapsed to a PBH.

Figure 4: Different trajectories are shown for different mass ranges. Higher
masses require larger spectral index for their formation. In any mass range,
the number of trajectories that pass the threshold is higher towards the
smaller mass. The FU distribution (Eq. (3.1)) is shown in the inset figure.

PBH formation needs amplification in primordial power spectrum. We modify the
initial power spectrum introduced by early universe physics in a small
wavenumber, where we do not have the constraints of CMB and LSS. We model this
amplification by phenomenological enhancement. In Fig. 3, we show the red and
blue-tilted enhanced curvature power spectrum in wavenumber range of
$k\sim[k_{s},\,k_{l}]$ for a specific case. Note that for the red-tilted power
enhancement, one needs an amplitude of an order $10^{11}A_{0}$. We also show
that the variance and FU statistics change due to these two types of
modifications are almost in the same order of magnitude (bottom panels for $S$
and $f$ versus mass) because the first crossing statistics depend only on the
variance. The variance is a monotonic decreasing function of mass (scale),
which is obtained by an integral over the power spectrum (see Eqs. (3.3) and
(3.4)). In what follows, we use the blue-tilted power spectrum for PBHs
formation, where the modification in spectral index requires the desired
amplification in the mass range more naturally with the same amplitude of
$A_{0}$. Therefore, we need to enhance the power spectrum on a specific scale
(see Eq. (3.5)). In Fig. 4, we show three schematic realizations of
trajectories corresponding to three different blue-tilted spectral indexes at
different mass ranges (different variances). Although all realizations of
trajectories have the same $f_{\rm FU}$ distribution versus $S$ (see Eq. (3.1)
and inset Fig. 4), however, the difference between realizations is due to the
variation of $S$ at different masses. These realizations also show that a
larger spectral index leads to a higher PBH mass. In each mass range, the
number of trajectories that pass the threshold is higher in the smaller mass
range. Note that although $f_{\rm FU}$ has a maximum in variance ($S_{max}$),
however, the maximum fraction of PBHs are formed in declining part of the
$f_{\rm FU}$ which justified by the $f_{\rm PBH}$ constraints proposed in
table 1.

Hence, the obtained value of $S$ from Eq. (3.5) will be larger than variance,
which makes the fraction maximum. This calculated $S$ corresponds to a
specific blue-tilted spectral index required for PBH formation (see Fig. 4).
We should note that $S$ is highly sensitive to spectral index, therefore a
small increase (decrease) in $n_{s(b)}$ leads to a larger (smaller) mass PBH.
For stupendously large masses [42] the obtained value of $S$ could be close to
$S_{max}$.

We report the results of our studies for eight desired mass ranges in table 1.
In each mass range, we consider intervals with different lower limits because
the statistic of FU/FT is higher for the lower mass of the interval. In this
table, $f_{\rm PBH}$ is based on observational constraint if there exist,
otherwise, we suppose that PBHs comprise the whole of DM. We report the
required value of blue-tilted spectral indexes for both EST and PS formalism
for comparison. One can see that these values are close to each other as
expected.

Figure 5: From left for right: $f_{\rm FU/FT}$, $\beta$, and $f_{\rm PBH}$
with respect to the mass of DM PBHs. From top to bottom: OGLE, intermediate
and SLAB mass ranges. Note that $f_{\rm PBH}$ is the required constraint for
each mass range reported in table 1.

| mass range | $f_{\rm PBH}$ | lower bound | spectral index
---|---|---|---|---
| of mass range | EST | PS
asteroid mass range | $\left(10^{16}-10^{17}\right)\,{\rm g}$ | 1 |  $10^{16}$ g | 1.490 | 1.616
sublunar mass range | $\left(10^{20}-10^{24}\right)\,{\rm g}$ | 1 |  $10^{20}$ g | 1.540 | 1.703
$10^{22}$ g | 1.600 | 1.756
Subaru HSC | $\left(10^{-11}-10^{-6}\right)\,M_{\odot}$ | $10^{-3}$ | $10^{-10}\,M_{\odot}$ | 1.604 | 1.560
$10^{-8}\,M_{\odot}$ | 1.666 | 1.605
OGLE | $\left(10^{-6}-10^{-3}\right)\,M_{\odot}$ | $10^{-2}$ | $10^{-6}\,M_{\odot}$ | 1.729 | 1.757
$10^{-4}\,M_{\odot}$ | 1.845 | 1.835
EROS/MACHO | $\left(10^{-3}-10^{-1}\right)\,M_{\odot}$ | $0.04$ | $10^{-3}\,M_{\odot}$ | 1.862 | 1.947
$10^{-2}\,M_{\odot}$ | 1.942 | 1.970
sub-Solar mass range* | $\left(0.2-1.0\right)\,M_{\odot}$ | $0.02$ | $0.2\,M_{\odot}$ | 2.018 | 2.046
$0.6\,M_{\odot}$ | 2.115 | 2.078
$1$ | $0.2\,M_{\odot}$ | 2.028 | 2.258
$0.6\,M_{\odot}$ | 2.126 | 2.297
intermediate mass range | $\left(10^{1}-10^{3}\right)\,M_{\odot}$ | $10^{-4}$ | $10\,M_{\odot}$ | 2.103 | 1.848
$10^{2}\,M_{\odot}$ | 2.204 | 1.911
SLABs | $\geq 10^{11}\,M_{\odot}$ | 1 | $10^{11}\,M_{\odot}$ | 5.220 | 5.598
$10^{12}\,M_{\odot}$ | 6.660 | 6.891

Table 1: Reported spectral indexes in EST and PS for eight mass ranges based
on $f_{\rm PBH}$ constraints. * Note that the sub-Solar mass range is studied
with conservative limit of $f_{\rm PBH}=0.02$ and optimistic limit $f_{\rm
PBH}=1$.

In Fig. 5, we show the abundance of PBHs by counting both the FT and FU for
OGLE, intermediate, and SLAB mass ranges from top to bottom, respectively. In
each mass range, we show $f_{\rm FU/FT}$, $\beta$, and $f_{\rm PBH}$ from left
to right, respectively. Our numerical results show that despite considering a
broad spectrum characterized by a range of scales running from $1/k_{l}$ to
$1/k_{s}$, production of PBHs will be dominated by smaller masses (i.e.
smaller scales, $1/k_{l}$). For SLAB mass range we show both FU and FT,
however for smaller mass ranges due to computational limits, we only report
FU. According to Eq. (2.4), $f_{\rm PBH}$ is much larger than $\beta$ for
small mass range, which is clear from the middle and the right plots of Fig.
5. This means that to get the desired $f_{\rm PBH}$ for smaller mass, we need
very low FU/FT. And since we calculate FU analytically and FT numerically,
small FT abundance requires enormous computational effort to produce enough FT
trajectories. Since we have already shown in Fig. 2 that there is no
considerable difference between FT and FU, therefore it is convenient to
report only the results of FU for small mass ranges.

The constraints on the spectral indexes (see table 1) can be translated to the
upper limit on the power spectrum of curvature perturbations,
$\mathcal{P_{R}}$ by using Eq. (2.10) to find wavenumbers corresponding to PBH
masses. We obtain $\mathcal{P_{R}}(k_{\rm PBH})\lesssim 10^{-1}$ with some
scale dependence in the range of $k_{\rm
PBH}\sim(10^{5}-10^{15})\,\text{Mpc}^{-1}$. The power spectrum corresponding
to the scale of SLABs ($k\sim(1-5)\,\text{Mpc}^{-1}$) is tightly constrained
by Lyman-$\alpha$ [48]. Hence, the whole DM can not be made of SLAB PBHs.

##  4 Conclusions

Although the Primordial black holes were first theorized decades ago, the
first gravitational wave detection by LIGO/Virgo, the great interest towards
PBHs as a DM candidate has revived. Consequently, there have been significant
improvements in the theoretical calculations of PBH formation and the
observational constraints on their abundance. Traditionally, the Press-
Schechter (PS) formalism has been used for PBHs formation. In this work, we
use the Excursion Set Theory (EST) as a reasonable and sophisticated extension
of PS. In the EST formalism, the trajectories which first up-cross (FU) the
density threshold, $\delta_{c}$ (which changes with growth function), form a
dark matter halo. However, for PBHs, we implement the EST with a constant
density threshold and encapsulate the redshift dependence in the power
spectrum. We also introduce a new concept of first touch (FT) instead of FU
for the first time (see Fig. 1 and related discussion in section 3.1). Since
PBHs form when the scale corresponding to their mass reenters the horizon,
this leads to the use of the Markov trajectories as a memory-independent
method. We showed in any mass range, the small PBHs form first and dominate
the mass profile of PBHs. For this reason, we count FT at smaller masses
instead of FU (see Fig. 2).
Since PBHs formation is only possible by enhancing the power spectrum at small
scales, we use a blue-tilted power spectrum in our analysis. We also show that
a red-tilted power with a larger value of amplitude leads to almost similar
results in PBH formation. Our numerical results showed that in a broad
spectrum, the production of PBHs is dominated by smaller masses. We report the
results of required blue-tilted spectral indexes in table 1 for eight mass
ranges in the EST formalism. As a by-product, we compared EST with PS, and we
showed that they are in good agreement in the considered masses. The
constraints on the spectral indexes were translated to the upper bound on
curvature power spectrum, $\mathcal{P_{R}}(k_{\rm PBH})\lesssim 10^{-1}$ with
some scale dependence in the range of $k_{\rm
PBH}\sim(10^{5}-10^{15})\,\text{Mpc}^{-1}$.
It is worth mentioning that our approach could be applied for any
redshift/scale dependence of the density threshold. For future work, we will
apply the EST to study the merger history of PBHs and their redshift
evolution. Also, it worths extending our work to non-Markov and non-Gaussian
perturbations.

## Acknowledgments

We are grateful to Sohrab Rahvar for his insightful comments on the
manuscript. EE and SB are partially supported by Abdus Salam International
Center of Theoretical Physics (ICTP) under the junior associateship scheme.
They thanks the hospitality of ICTP where this work is initiated. We thank
Nasim Derakhshanian for her collaboration in the early stage of this work. SB
is supported by Sharif University of Technology Office of Vice President for
Research under Grant No. G960202.

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