# Regular singular differential equations and free proalgebraic groups

Michael Wibmer  Michael Wibmer, Institute of Analysis and Number Therory, Graz
University of Technology, Kopernikusgasse 24, 8010 Graz, Austria,
https://sites.google.com/view/wibmer<EMAIL_ADDRESS>
###### Abstract.

We determine the differential Galois group of the family of all regular
singular differential equations on the Riemann sphere. It is the free
proalgebraic group on a set of cardinality $|\mathbb{C}|$.

###### Key words and phrases:

Regular singular differential equation, differential Galois theory, free
proalgebraic group

###### 2020 Mathematics Subject Classification:

14L15, 34M50

This work was supported by the NSF grants DMS-1760212, DMS-1760413,
DMS-1760448 and the Lise Meitner grant M 2582-N32 of the Austrian Science Fund
FWF

##  1\. Introduction

The following table depicts the beautiful analogy between classical Galois
theory and differential Galois theory over the rational function field
$\mathbb{C}(x)$. Excellent introductions to these topics can be found in
[Sza09] and [vdPS03].

|  Classical Galois theory |  Differential Galois theory
---|---|---
1 |  Univariate polynomials over $\mathbb{C}(x)$ |  Linear differential equations over $\mathbb{C}(x)$
2 |  Galois extension of $\mathbb{C}(x)$ and their Galois groups |  Picard-Vessiot extensions of $\mathbb{C}(x)$ and their differential Galois groups
3 |  A finite Galois extensions $L$ of $\mathbb{C}(x)$ corresponds to a ramified cover $p\colon X\to\mathbb{P}^{1}(\mathbb{C})$. If $S$ is the finite set of branch points and $x_{0}\in\mathbb{C}$ is not in $S$, then the Galois group of $L/\mathbb{C}(x)$ can be identified with the image of $\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\smallsetminus S,x_{0})$ under its action on $p^{-1}(x_{0})$. |  Schlesinger’s density theorem: Let $S$ be a finite subset of $\mathbb{P}^{1}(\mathbb{C})$ and $x_{0}\in\mathbb{C}$ not in $S$. The differential Galois group of a regular singular linear differential equation with singularities in $S$, can be identified with the Zariski closure of the image of $\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\smallsetminus S,x_{0})$ under its action on the local solution space at $x_{0}$.
4 |  For $x_{0}\in\mathbb{C}$ and $S$ a finite subset of $\mathbb{P}^{1}(\mathbb{C})$ not containing $x_{0}$, the Galois $G$ group of the maximal algebraic extension of $\mathbb{C}(x)$ with ramification only over $S$, is the profinite completion of $\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\smallsetminus S,x_{0})$, i.e., $G$ is the free profinite group on a set of cardinality $|S|-1$. |  For $x_{0}\in\mathbb{C}$ and $S$ a finite subset of $\mathbb{P}^{1}(\mathbb{C})$ not containing $x_{0}$, the differential Galois group $G$ of the family of all regular singular differential equations with singularities inside $S$, is the proalgebraic completion of $\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\smallsetminus S,x_{0})$, i.e., $G$ is the free proalgebraic group on a set of cardinality $|S|-1$.
5 |  Douady’s Theorem: The absolute Galois group of $\mathbb{C}(x)$ is the free profinite group on a set of cardinality $|\mathbb{C}|$. |  ?

The main goal of this article is to fill in the above question mark. On the
face of it, it may seem that the appropriate differential analog of Douady’s
theorem is “The absolute differential Galois group of $\mathbb{C}(x)$ is the
free proalgebraic group on a set of cardinality $|\mathbb{C}|$”. This is in
fact a true statement ([BHHW21a]). However, Douady arrived at 5 via 3 and 4.
In this sense, an appropriate differential analog of Douady’s theorem should
only be concerned with regular singular differential equations. Our main
result is the following differential analog of Douady’s theorem.

######  Theorem A (Theorem 4.4).

The differential Galois group of the family of all regular singular
differential equations over $\mathbb{C}(x)$ is the free proalgebraic group on
a set of cardinality $|\mathbb{C}|$.

At first glance, it might seem that 5 should follow from 4 rather immediately.
However, in general, the projective limit of free profinite groups need not be
a free profinite group ([RZ10, Ex. 9.1.14]). The question, when a projective
limit of free profinite groups is itself free has attracted some attention but
does not seem to be fully understood ([RZ10, Thm. 3.5.15 and Open Question
9.5.2]).

To get from 4 to 5 in the proof of Douady’s theorem (see [Sza09, Sec. 3.4] or
[Dou64] for the original reference) one uses a compactness argument and that
$\pi_{1}(\mathbb{P}^{1}(\mathbb{C})\smallsetminus S,x_{0})$ has more or less
canonical generators. Another ingredient of the proof is that in the free
profinite group on a set with $r$ elements, any subset of $r$ topological
generators is a basis. As we will show (Example 2.3), the corresponding
statement fails for free proalgebraic groups. Therefore, genuinely new ideas
are needed in the differential case. In particular, we will use a
characterization of free proalgebraic groups in terms of embedding problems.

For a finite subset $X$ of $\mathbb{C}$, it is an immediate consequence of the
Riemann-Hilbert correspondence, that the differential Galois group of the
family of all regular singular differential equations with singularities in
$X\cup\\{\infty\\}$, is the free proalgebraic group on a set of cardinality
$|X|$. We generalize this result from finite subsets of $\mathbb{C}$ to
arbitrary subsets of $\mathbb{C}$. Indeed, the case $X=\mathbb{C}$ is exactly
Theorem A.

We note that there is also an analogy between differential Galois theory over
$\mathbb{C}(x)$ and classical Galois theory over $k(x)$, with $k$ an
algebraically closed field of characteristic $p>0$, such that regular singular
points correspond to tamely ramified points, while irregular singular points
correspond to wildly ramified points ([vdPS03, Section 11.6]). Based on this
analogy, our result may seem more surprising, because the Galois group of the
maximal tamely ramified extension of $k(x)$ with branch locus in a fixed
subset $S$ of $\mathbb{P}^{1}(k)$ is not a free profinite group.

We conclude the introduction with an outline of the article. In Section 2 we
recall the definition of free proalgebraic groups and the required results
concerning differential Galois theory and the Riemann-Hilbert correspondence.
We then study projetive systems of abstract free groups in Section 3. Finally,
in the last section the previous results are applied to prove Theorem A.

The author is grateful to David Harbater and Michael Singer for helpful
comments.

##  2\. Preliminaries and Notation

In this preparatory section we recall the basic definitions and results
concerning regular singular differential equations and differential Galois
theory. We also review the definition of free proalgebraic groups.

We use “$=$” or “$\simeq$” to denote canonical isomorphisms and “$\cong$” to
denote isomorphisms. (As the implied isomorphism in Theorem A is not
canonical, it seems worthwhile to make this distinction.)

###  2.1. Free proalgebraic groups

Throughout this article we work over the field $\mathbb{C}$ of complex
numbers. We use the term “algebraic group” in lieu of “affine group scheme of
finite type over $\mathbb{C}$”. Similarly, a “proalgebraic group” is an
“affine group scheme over $\mathbb{C}$”. By a _closed subgroup_ of a
proalgebraic group, we mean a closed subgroup scheme. Following [Mil17, Def.
5.5] a morphism $G\to H$ of proalgebraic groups is called a _quotient map_ if
it is faithfully flat. We use $G\twoheadrightarrow H$ to indicate quotient
maps.

We begin by recalling the definition of free proalgebraic groups from [Wib20].
Let $\Gamma$ be a proalgebraic group and let $X$ be a set. A map
$\varphi\colon X\to\Gamma(\mathbb{C})$ _converges to $1$_ if almost all
elements of $X$ map to $1$ in any algebraic quotient of $\Gamma$, i.e., for
every algebraic group $G$ and every quotient map
$\phi\colon\Gamma\twoheadrightarrow G$, all but finitely many elements of $X$
map to $1$ under
$X\xrightarrow{\varphi}\Gamma(\mathbb{C})\xrightarrow{\phi_{\mathbb{C}}}G(\mathbb{C})$.

The following definition is the special case of [Wib20, Def. 2.18], where
$\mathcal{C}$ is the formation of all algebraic groups and $R=k=\mathbb{C}$.

######  Definition 2.1.

Let $X$ be a set. A proalgebraic group $\Gamma(X)$ together with a map
$\iota\colon X\to\Gamma(X)(\mathbb{C})$ converging to one is called a _free
proalgebraic group_ on $X$ if $\iota$ satisfies the following universal
property. For every proalgebraic group $G$ and every map $\varphi\colon X\to
G$ converging to $1$, there exists a unique morphism $\phi\colon\Gamma(X)\to
G$ of proalgebraic groups such that

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{\varphi}$$\textstyle{\Gamma(X)(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\mathbb{C}}}$$\textstyle{G(\mathbb{C})}$

commutes.

As the pair $(\iota,\Gamma(X))$ is unique up to a unique isomorphism, we will
usually speak of _the_ free proalgebraic group $\Gamma(X)$ on $X$.

The map $\iota\colon X\to\Gamma(X)(\mathbb{C})$ is injective. In fact, the
induced map $F(X)\to\Gamma(X)(\mathbb{C})$ from the (abstract) free group
$F(X)$ on $X$ to $\Gamma(X)(\mathbb{C})$ is injective ([Wib21, Lem. 1.6]). We
will therefore in the sequel identify $X$ with a subset of
$\Gamma(X)(\mathbb{C})$ via $\iota$.

######  Remark 2.2.

To verify that $\Gamma(X)$ is the free proalgebraic group on $X$, it suffices
to verify the universal property of Definition 2.1 for $G$ algebraic ([Wib20,
Rem. 2.19]).

For a proalgebraic group $G$ and a subset $X$ of $G(\mathbb{C})$, we denote
the smallest closed subgroup $H$ of $G$ such that $X\subseteq H(\mathbb{C})$
with $\langle X\rangle$. In other words, $\langle X\rangle$ is the closed
subgroup of $G$ generated by $X$. By [Wib20, Thm. 2.17] we have
$\Gamma(X)=\langle X\rangle$.

We next explain why the proof of Douady’s theorem (as presented in [Dou64] or
[Sza09, Sec. 3.4]) does not have a direct differential analog. This proof uses
a result ([Dou64, Prop. 1] or [Sza09, Lem. 3.4.11]) attributed to Serre by
Douady, stating that, any set of $n$ elements that topologically generates a
free profinite group of rank $n$ is a basis. Equivalently, a surjective
endomorphism of a free profinite group of finite rank is an isomorphism. The
proof of this result uses a counting argument and therefore does not apply in
our context, where finite groups are replaced by algebraic groups. In fact, as
shown in the following example, Serre’s result does not hold in our context.

######  Example 2.3.

Let $X=\\{*\\}$ be a set with one element $*$. We will show that not every
generator of $\Gamma(X)$ is a basis, i.e., there exists a quotient map
$\Gamma(X)\to\Gamma(X)$ that is not an isomorphism.

The free proalgebraic on one element is of the form
$\Gamma(X)=\mathbb{G}_{a}\times D(\mathbb{C}^{\times})$. See [Wib20, Ex. 2.22]
or [Sau16, Cor. 16.26]. Here $\mathbb{G}_{a}$ is the additive group and, as in
[DG70, Ch. IV, §1, Sec. 1], for any abelian group $M$, $D(M)$ denotes the
diagonalizable proalgebraic group with character group $M$, i.e.,
$D(M)(T)=\operatorname{Hom}(M,T^{\times})$ for any $\mathbb{C}$-algebra $T$.
The map $\iota\colon X\to\Gamma(X)(\mathbb{C})$ is given by
$\iota(*)=(1,\operatorname{id})\in\mathbb{C}\times\operatorname{Hom}(\mathbb{C}^{\times},\mathbb{C}^{\times})=\mathbb{G}_{a}(\mathbb{C})\times
D(\mathbb{C}^{\times})(\mathbb{C})$.

As an abelian group, $\mathbb{C}^{\times}$ is isomorphic to
$(\mathbb{Q}/\mathbb{Z})\oplus V$, where $V$ is a $\mathbb{Q}$-vector space of
dimension $|\mathbb{C}|$. In particular, there exists an injective
endomorphism $\psi\colon\mathbb{C}^{\times}\to\mathbb{C}^{\times}$ that is not
an isomorphism. Dualizing $\psi$, we find a quotient map
$D(\mathbb{C}^{\times})\to D(\mathbb{C}^{\times})$ that is not an isomorphism.
This trivially extends to a quotient map $\Gamma(X)\to\Gamma(X)$ that is not
an isomorphism.

We will need the notion of _proalgebraic completion_ of an abstract group. See
e.g., [BLMM02]. Note that the proalgebraic completion is sometimes also
referred to as the _proalgebraic hull_ (e.g., in [Sau16]) or as the
_Hochschild-Mostow group_ (in honor of [HM57]).

######  Definition 2.4.

Let $F$ be an (abstract) group. The _proalgebraic completion_
$F^{\operatorname{alg}}$ of $F$ is a proalgebraic group equipped with a
morphism $F\to F^{\operatorname{alg}}(\mathbb{C})$ of groups satisfying the
following universal property: If $G$ is a proalgebraic and $F\to
G(\mathbb{C})$ is a morphism of groups, then there exists a unique morphism
$\phi\colon F^{\operatorname{alg}}\to G$ of proalgebraic groups such that

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\operatorname{alg}}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\mathbb{C}}}$$\textstyle{G(\mathbb{C})}$

commutes.

For $X$ a finite (!) set and $F(X)$ the (abstract) free group on $X$, it
follows from the universal properties that
$F(X)^{\operatorname{alg}}\simeq\Gamma(X)$. The proalgebraic completion
$F^{\operatorname{alg}}$ of $F$ can be constructed as the fundamental group of
the neutral tannakian category of all finite dimensional $\mathbb{C}$-linear
representations of $F$.

###  2.2. Differential Galois theory

Introductions to this topic can be found in [Mag94, vdPS03],[CH11] and
[Sau16]. We recall the basic definitions and results, introducing our notation
for the subsequent sections along the way.

We fix a differential field $K$ with derivation $\delta\colon K\to K$. We
assume that the field of constants $K^{\delta}=\\{a\in K|\ \delta(a)=a\\}$ of
$K$ is the field $\mathbb{C}$ of complex numbers. We are mainly interested in
the case when $K=\mathbb{C}(x)$ is the rational function field in one variable
$x$ and $\delta=\frac{d}{dx}$. We consider a family
$\mathcal{F}=(\delta(y)=A_{i}y)_{i\in I}$ of linear differential equations
indexed by some set $I$, where $A_{i}\in K^{n_{i}\times n_{i}}$ is a square
matrix for every $i\in I$.

######  Definition 2.5.

A differential field extension $L/K$ with $L^{\delta}=\mathbb{C}$ is a
_Picard-Vessiot extension_ for $\mathcal{F}$ if there exist matrices
$Y_{i}\in\operatorname{GL}_{n_{i}}(L)$ such that $\delta(Y_{i})=A_{i}Y_{i}$
for $i\in I$ and $L$ is generated as a field extension of $K$ be all entries
of all $Y_{i}$’s.

The $K$-subalgebra $R$ of $L$ generated by all of entries of all $Y_{i}$’s and
$\frac{1}{\det(Y_{i})}$’s is a $K$-$\delta$-subalgebra of $L$ and called a
_Picard-Vessiot ring_ for $\mathcal{F}$.

For a given family $\mathcal{F}$, a Picard-Vessiot extension exists and is
unique up to a $K$-$\delta$-isomorphism. The _differential Galois group_
$G(L/K)$ of the Picard-Vessiot extension $L/K$, or of the family
$\mathcal{F}$, is the functor
$T\rightsquigarrow\operatorname{Aut}(R\otimes_{\mathbb{C}}T/K\otimes_{\mathbb{C}}T)$,
from the category of $\mathbb{C}$-algebras to the category groups, where $T$
is considered as a constant differential ring and the automorphisms are
required to commute with the derivation. The functor $G(L/K)$ can be
represented by a $\mathbb{C}$-algebra, i.e., $G(L/K)$ is a proalgebraic group.

A Picard-Vessiot extension $L/K$ is _of finite type_ if it is the Picard-
Vessiot extension for a single differential equation. This is the case if and
only of $G(L/K)$ is algebraic.

Since $L$ is the field of fractions of $R$, any $g\in G(L/K)(\mathbb{C})$
extends uniquely to a $K$-$\delta$-automorphism of $L$. For a closed subgroup
$H$ of $G=G(L/K)$ we set

$L^{H}=\\{a\in L|\ h(a)=a\ \forall\ h\in H(\mathbb{C})\\}.$

######  Theorem 2.6 (The differential Galois correspondence).

Let $L/K$ be a Picard-Vessiot extension. The assignment $M\mapsto G(L/M)$
defines an inclusion reversing bijection between the set of intermediate
differential fields of $L/K$ and the set of closed subgroups of $G(L/K)$. The
inverse is given by $H\mapsto L^{H}$.

If $M$ corresponds to $H$ under this bijection, then $M/K$ is Picard-Vessiot
if and only if $H$ is normal in $G(L/K)$. Moreover, if this is the case, the
restriction morphism $G(L/K)\to G(M/K)$ is a quotient map with kernel
$G(L/M)$. In particular, $G(M/K)\simeq G(L/K)/G(L/M)$.

An alternative definition of the differential Galois group of a family of
linear differential equations can be given via the tannakian formalism ([DM82,
Del90]). As a first step, one has to define the “category of differential
equations”. This is formalized through the notion of a differential module. A
_differential module_ over $K$ is a finite dimensional $K$-vector space $M$
equipped with an additive map $\partial\colon M\to M$ such that
$\partial(am)=\delta(a)m+a\partial(m)$ for all $a\in K$ and $m\in M$. A
morphism of differential modules over $K$ is a $K$-linear map that commutes
with $\partial$.

To a linear differential equation $\delta(y)=Ay$ with $A\in K^{n\times n}$,
one associates the differential module $M_{A}$ by setting $M_{A}=K^{n}$ and
$\partial(\xi)=\delta(\xi)-A\xi$ for all $\xi\in K^{n}$. Conversely, if
$(M,\partial)$ is a differential module with basis
$\underline{e}=(e_{1},\ldots,e_{n})$, we can write
$\partial(\underline{e})=\underline{e}(-A)$ for a unique matrix
$A=A_{M,\underline{e}}\in K^{n\times n}$. Then
$\partial(\underline{e}\xi)=\underline{e}\delta(\xi)+\partial(\underline{e})\xi=\underline{e}(\delta(\xi)-A\xi)$
so that $M$ is isomorphic to $M_{A}$. Via the choice of the basis
$\underline{e}$, we can thus associate to $M$ the differential equation
$\delta(y)=Ay$. A different choice of basis leads to a Gauge equivalent
differential equation.

For a differential module $(M,\partial)$ over $K$ one sets
$M^{\partial}=\\{m\in M|\ \partial(m)=0\\}$. This is a $\mathbb{C}$/̄subspace
of $M$. If $\underline{e}$ is a $K$-basis of $M$ and $A\in K^{n\times n}$ is
such that $\partial(\underline{e}\xi)=\underline{e}(\delta(\xi)-A\xi)$ for all
$\xi\in K^{n}$, then $M^{\partial}$ can be identified with the
$\mathbb{C}$-space of all solutions of $\delta(y)=Ay$ in $K^{n}$. More
generally, if $L$ is a differential field extension of $K$, then
$M\otimes_{K}L$ is a differential module over $L$ and $M\otimes_{K}L$
identifies with the space of all solutions of $\delta(y)=Ay$ in $L^{n}$.

The category of all differential modules over $K$ is a neutral Tannakian
category over $\mathbb{C}$. If $\mathcal{F}=(\delta(y)=A_{i}y)_{i\in I}$ is a
family of differential equations over $K$ and $\mathcal{M}=(M_{A_{i}})_{i\in
I}$ is the corresponding family of differential modules, then the differential
Galois group of $\mathcal{F}$ is isomorphic to the fundamental group of the
neutral Tannakian category $\langle\langle\mathcal{M}\rangle\rangle$ generated
by $\mathcal{M}$.

###  2.3. Regular singular differential equations

For background on regular singular differential equations and the Riemann-
Hilbert correspondence see [Sau16, Part 3], [vdPS03, Chapters 5 and 6], [MS16,
Part I] and [ABC20, Part III].

We first treat the local definitions. Consider a differential module
$(M,\partial)$ over the field $\mathbb{C}((t))$ of formal Laurent series in
$t$ equipped with the usual derivation $\delta=\frac{d}{dt}$. Let
$\mathbb{C}[[t]]\subseteq\mathbb{C}((t))$ be the differential subring of
formal power series. A $\mathbb{C}[[t]]$/̄lattice in $M$ is a
$\mathbb{C}[[t]]$/̄submodule $N$ of $M$ such that there exists a
$\mathbb{C}[[t]]$-basis of $N$ that is also a $\mathbb{C}((t))$-basis of $M$.

The differential module $M$ is called _regular_ if there exists a
$\mathbb{C}[[t]]$/̄lattice $N$ in $M$ such that $\partial(N)\subseteq N$. The
differential module $M$ is called _regular singular_ if there exists a
$\mathbb{C}[[t]]$/̄lattice $N$ in $M$ such that $t\partial(N)\subseteq N$. So
a regular differential module is regular singular.

We now consider the global picture. Let $(M,\partial)$ be differential module
over $\mathbb{C}(x)$. Here, as throughout the paper, the rational function
field $\mathbb{C}(x)$ is considered as a differential field via the derivation
$\delta=\frac{d}{dx}$.

Let $\mathbb{P}^{1}(\mathbb{C})=\mathbb{C}\cup\\{\infty\\}$ denote the Riemann
sphere. For every point $p\in\mathbb{P}^{1}(\mathbb{C})$ we have a “local”
differential field $\mathbb{C}(x)_{p}=\mathbb{C}((t))$. For $p\in\mathbb{C}$
this is $\mathbb{C}(x)_{p}=\mathbb{C}((x-p))=\mathbb{C}((t))$, the field of
formal Laurent series in $t=x-p$ with derivation
$\frac{d}{dt}=\frac{d}{d(x-p)}$. For $p=\infty$ this is
$\mathbb{C}(x)_{p}=\mathbb{C}((x^{-1}))=\mathbb{C}((t))$, the field of formal
Laurent series in $t=x^{-1}$ with derivation $\frac{d}{dt}=\frac{d}{dx^{-1}}$.

Note that $(\mathbb{C}(x),\frac{d}{dx})$ is a differential subfield of
$(\mathbb{C}((x-p)),\frac{d}{d(x-p)})$ for $p\in\mathbb{C}$. Thus, we obtain a
differential module $M_{p}=M\otimes_{\mathbb{C}(x)}\mathbb{C}(x)_{p}$ over
$\mathbb{C}(x)_{p}$ for every $p\in\mathbb{C}$. For $p=\infty$ it is not true
that $(\mathbb{C}(x),\frac{d}{dx})$ is a differential subfield of
$(\mathbb{C}((x^{-1})),\frac{d}{dx^{-1}})$. However,
$(\mathbb{C}(x),-x^{2}\frac{d}{dx})$ is a differential subfield of
$(\mathbb{C}((x^{-1})),\frac{d}{dx^{-1}})$. So we can base change the
differential module $(M,{-x^{2}}\partial)$ over
$(\mathbb{C}(x),-x^{2}\frac{d}{dx})$ to a differential module $M_{\infty}$
over $\mathbb{C}(x)_{\infty}$. This awkwardness at infinity is one of the
reasons why some authors prefer to work with connections, rather than
differential modules. This way one can avoid the a priori choice of a
derivation on $\mathbb{C}(x)$. However, since the module
$\Omega_{\mathbb{C}(x)/\mathbb{C}}$ of differentials of $\mathbb{C}(x)$ over
$\mathbb{C}$ is one dimensional, these two approaches are equivalent.

A point $p\in\mathbb{P}^{1}(\mathbb{C})$ is a _singularity_ of $M$ if the
differential module $M_{p}$ over $\mathbb{C}((t))$ is not regular. A point
$p\in\mathbb{P}^{1}(\mathbb{C})$ is _regular singular_ for $M$ if the
differential module $M_{p}$ over $\mathbb{C}((t))$ is regular singular.
Finally, $M$ is called _regular singular_ if every point
$p\in\mathbb{P}^{1}(\mathbb{C})$ is regular singular for $M$.

Consider a differential equation $\delta(y)=Ay$ with
$A\in\mathbb{C}(x)^{n\times n}$. A point $p\in\mathbb{P}^{1}(\mathbb{C})$ is a
_singularity_ of $\delta(y)=Ay$ if it is a singularity of the associated
differential module $M_{A}$. Note that this definition is at odds with the
common terminology, referring to the poles of $A$ as the singularities of
$\delta(y)=Ay$. If $p\in\mathbb{C}$ is a singularity of $\delta(y)=Ay$, then
$p$ must be a pole of $A$. However, the converse is not true. A pole of $A$
that is not a singularity of $\delta(y)=Ay$ is sometimes called an apparent
singularity. The differential equation $\delta(y)=Ay$ is called _regular
singular_ if the associated differential module $M_{A}$ is regular singular.

Fix a proper subset $X$ of $\mathbb{C}\subseteq\mathbb{P}^{1}(\mathbb{C})$. As
$X$ is assumed to be a proper subset of $\mathbb{C}$, we can choose a “base
point” $x_{0}\in\mathbb{C}$ with $x_{0}\notin X$. Let
$\operatorname{RegSing}(\mathbb{C}(x),X)$ denote the category of all regular
singular differential modules over $\mathbb{C}(x)$ with singularities
contained in $X\cup\\{\infty\\}$. This is a Tannakian category over
$\mathbb{C}$. A fibre functor
$\omega_{X,x_{0}}\colon\operatorname{RegSing}(\mathbb{C}(x),X)\to\operatorname{Vec}_{\mathbb{C}}$,
with values in the category $\operatorname{Vec}_{\mathbb{C}}$ of finite
dimensional $\mathbb{C}$-vector spaces, is given by

$\omega_{X,x_{0}}((M,\partial))=(M\otimes_{\mathbb{C}(x)}\mathcal{M}_{x_{0}})^{\partial},$

where $\mathcal{M}_{x_{0}}$ is the differential field of germs of meromorphic
functions at $x_{0}$. We denote with
$\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$ the proalgebraic
group of tensor automorphisms of $\omega_{X,x_{0}}$.

Now assume that $X$ is finite. Let $\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})$
be the topological fundamental group of the Riemann sphere with the points
$X\cup\\{\infty\\}$ removed, with base point $x_{0}$. Then the local solution
space $(M\otimes_{\mathbb{C}(x)}\mathcal{M}_{x_{0}})^{\partial}$ is naturally
equipped with the monodromy action of $\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0})$. We denote with $\operatorname{Rep}(\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0}))$ the category of finite dimensional $\mathbb{C}$-linear
representations of $\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})$. The following
theorem is sometimes referred to as the (global) Riemann-Hilbert
correspondence. The essential surjectivity of the functor in the theorem is
also known as the solution of the (weak form of the) Riemann-Hilbert problem.

######  Theorem 2.7 (Riemann-Hilbert correspondence).

Let $X\subseteq\mathbb{C}$ be finite. Then the functor

$\operatorname{RegSing}(\mathbb{C}(x),X)\to\operatorname{Rep}(\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0})),\quad(M,\partial)\rightsquigarrow(M\otimes_{\mathbb{C}(x)}\mathcal{M}_{x_{0}})^{\partial}$

is an equivalence of Tannakian categories.

From Theorem 2.7 we immediately obtain:

######  Corollary 2.8.

For $X\subseteq\mathbb{C}$ finite we have
$\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})\simeq\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0})^{\operatorname{alg}}$. In particular, the differential Galois group
of the family of all regular singular differential equations over
$\mathbb{C}(x)$ with singularities in $X\cup\\{\infty\\}$ is isomorphic to the
free proalgebraic group on $X$.

###### Proof.

As $\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})^{\operatorname{alg}}$ is the
proalgebraic group of tensor automorphisms of the forgetful functor
$\operatorname{Rep}(\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0}))\to\operatorname{Vec}_{\mathbb{C}}$, Theorem 2.7 yields an
isomorphism $\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0})^{\operatorname{alg}}\to\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$.
The last statement follows because the differential Galois group of the family
of all regular singular differential equations with singularities in
$X\cup\\{\infty\\}$ is isomorphic to
$\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$ and the group
$\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})$ is free on $|X|$ generators. ∎

We stress the fact that the morphism of groups
$\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0})\to\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})(\mathbb{C})$
is canonical. If $X^{\prime}$ is a subset of the finite set
$X\subseteq\mathbb{C}$, then
$\operatorname{RegSing}(\mathbb{C}(x),X^{\prime})$ is a subcategory of
$\operatorname{RegSing}(\mathbb{C}(x),X)$ and $\omega_{X^{\prime},x_{0}}$ is
the restriction of $\omega_{X,x_{0}}$ to
$\operatorname{RegSing}(\mathbb{C}(x),X^{\prime})$. We thus have a morphism
$\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})\to\underline{\operatorname{Aut}}^{\otimes}(\omega_{X^{\prime},x_{0}})$
of proalgebraic groups. As $\mathbb{C}\smallsetminus
X\subseteq\mathbb{C}\smallsetminus X^{\prime}$, we also have a morphism
$\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})\to\pi_{1}(\mathbb{C}\smallsetminus
X^{\prime},x_{0})$ of groups. The diagram

$\textstyle{\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(\mathbb{C}\smallsetminus
X^{\prime},x_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underline{\operatorname{Aut}}^{\otimes}(\omega_{X^{\prime},x_{0}})(\mathbb{C})}$

commutes and so also the diagram

$\textstyle{\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0})^{\operatorname{alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(\mathbb{C}\smallsetminus
X^{\prime},x_{0})^{\operatorname{alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\underline{\operatorname{Aut}}^{\otimes}(\omega_{X^{\prime},x_{0}})}$
(1)

commutes.

The family of isomorphisms $\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0})^{\operatorname{alg}}\simeq\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$,
one for every finite subset $X$ of $\mathbb{C}$ not containing $x_{0}$, can
thus be seen as defining an isomorphism between two projective systems of
proalgebraic groups. The projective limit on the right hand side is
$\varprojlim_{X}\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})=\underline{\operatorname{Aut}}^{\otimes}(\omega_{\mathbb{C}\smallsetminus\\{x_{0}\\},x_{0}})$,
whereas the projective limit $\varprojlim_{X}\pi_{1}(\mathbb{C}\smallsetminus
X,x_{0})^{\operatorname{alg}}$ on the left hand side is a projective limit of
free proalgebraic groups. The following section provides the necessary tools
to show that this limit itself is free.

##  3\. Projective systems of free groups

Let $X$ be a set and consider the directed set $\mathcal{Y}$ of all finite
subsets of $X$ ordered by inclusion. For $Y\in\mathcal{Y}$ let $F(Y)$ denote
the (abstract) free group on $Y$ and for $Y\subseteq Y^{\prime}$, define a map
$\varphi_{Y,Y^{\prime}}\colon F(Y^{\prime})\to F(Y)$ by

$\varphi_{Y,Y^{\prime}}(y^{\prime})=\begin{cases}y^{\prime}&\text{ if
}y^{\prime}\in Y,\\\ 1&\text{ if }y^{\prime}\notin Y.\end{cases}$

The projective limit $\varprojlim_{Y\in\mathcal{Y}}F(Y)$ (in the category of
groups) of the projective system
$(F(Y)_{Y\in\mathcal{Y}},(\varphi_{Y,Y^{\prime}})_{Y^{\prime}\supseteq Y})$ is
in general not a free group: We have a map $\varphi\colon
X\to\varprojlim_{Y\in\mathcal{Y}}F(Y),\
x\mapsto(\varphi(x)_{Y})_{Y\in\mathcal{Y}}$ given by

$\varphi(x)_{Y}=\begin{cases}x&\text{ if }x\in Y,\\\ 1&\text{ if }x\notin
Y\end{cases}$

and the induced map $F(X)\to\varprojlim_{Y\in\mathcal{Y}}F(Y)$ is injective.
However, this map need not be surjective. Intuitively, surjectivity fails
because $F(X)$ only contains words of finite length, while
$\varprojlim_{Y\in\mathcal{Y}}F(Y)$ may contain words of infinite length. For
example, if $X=\\{x_{1},x_{2},\ldots\\}$ is countably infinite, write
$Y=\\{x_{i_{Y,1}},x_{i_{Y,2}},\ldots,x_{i_{Y,|Y|}}\\}$ with
$i_{Y,1}<i_{Y,2},\ldots<i_{Y,|Y|}$. Then $(x_{i_{Y,1}}x_{i_{Y,2}}\ldots
x_{i_{Y,|Y|}})_{Y\in\mathcal{Y}}$ lies in $\varprojlim_{Y\in\mathcal{Y}}F(Y)$
but not in the image of $F(X)$. In the nomenclature of [Hig52] this limit is
an _unrestricted free product_. For more on projective limits of abstracts
free groups see [CK12] and [EN13].

However, when working with free profinite groups instead of abstract free
groups, the above construction leads to a free profinite group. See [RZ10,
Cor. 3.3.10 b)] or [Sza09, Lem. 3.4.10]. As we shall now explain, also in the
case of free proalgebraic groups, the above construction leads to a free
proalgebraic group.

As above, let $X$ be a set and let $\mathcal{Y}$ be the directed set of all
finite subsets of $X$. For $Y\subseteq Y^{\prime}$, by the universal property
of $\Gamma(Y^{\prime})$ (Definition 2.1), the map
$\varphi_{Y,Y^{\prime}}\colon Y^{\prime}\to\Gamma(Y)(\mathbb{C})$ defined by

$\varphi_{Y,Y^{\prime}}(y^{\prime})=\begin{cases}y^{\prime}&\text{ if
}y^{\prime}\in Y,\\\ 1&\text{ if }y^{\prime}\notin Y,\end{cases}$

extends to a morphism
$\varphi_{Y,Y^{\prime}}\colon\Gamma(Y^{\prime})\to\Gamma(Y)$. Then
$((\Gamma(Y))_{Y\in\mathcal{Y}},(\varphi_{Y,Y^{\prime}})_{Y^{\prime}\supseteq
Y})$ is a projective system of free proalgebraic groups on finite sets. The
projective limit is $\Gamma(X)$.

######  Lemma 3.1.

$\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)=\Gamma(X)$.

###### Proof.

For $Y\in\mathcal{Y}$ we define $\iota_{Y}\colon X\to\Gamma(Y)(\mathbb{C})$ by
$\iota_{Y}(x)=\begin{cases}x\text{ if }x\in Y,\\\ 1\text{ if }x\notin
Y.\end{cases}$ Then

$\iota\colon
X\to\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)(\mathbb{C})=\Big{(}\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)\Big{)}(\mathbb{C}),\quad
x\mapsto(\iota_{Y}(x))_{Y\in\mathcal{Y}}$

converges to $1$ because every quotient map
$\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)\twoheadrightarrow G$ to an algebraic
group $G$ factors through some $\Gamma(Y)$.

To verify the universal property of $\iota$, according to Remark 2.2, it
suffices to consider a map $\varphi\colon X\to G(\mathbb{C})$ converging to
$1$, with $G$ an algebraic group. Then $Y_{0}=\\{x\in X|\ \varphi(x)\neq 1\\}$
is finite. Define a morphism $\psi\colon\Gamma(Y_{0})\to G$ by
$\psi(y)=\varphi(y)$ for $y\in Y_{0}$ and let $\phi$ be the composition
$\phi\colon\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)\to\Gamma(Y_{0})\xrightarrow{\psi}G$.
Then $\phi(\iota(x))=\varphi(x)$ for every $x\in X$.

If $\phi^{\prime}\colon\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)\to G$ is another
morphism such that $\phi^{\prime}(\iota(x))=\varphi(x)$ for all $x\in X$, then
$T\rightsquigarrow\\{g\in(\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y))(T)|\
\phi(g)=\phi^{\prime}(g)\\}$ is a closed subgroup of
$\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)$. It thus suffices to show that
$\langle\iota(X)\rangle=\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)$. The
projections $\langle\iota(X)\rangle\to\Gamma(Y)$ are quotient maps because
$\Gamma(Y)=\langle Y\rangle$. But then also
$\langle\iota(X)\rangle\to\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)$ is a
quotient map. Therefore
$\langle\iota(X)\rangle=\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)$. ∎

We now specialize to the case that $X$ is a proper subset of $\mathbb{C}$. In
this case, besides the projective system
$((F(Y))_{Y\in\mathcal{Y}},(\varphi_{Y^{\prime},Y})_{Y^{\prime}\supseteq Y})$
from the beginning of this section, we can associate another projective system
of finite rank (abstract) free groups to $X$ as follows. Fix
$x_{0}\in\mathbb{C}$ with $x_{0}\notin X$. For
$Y=\\{y_{1},\ldots,y_{n}\\}\in\mathcal{Y}$, it is well known (and follows from
Van Kampen’s theorem) that the fundamental group
$\pi_{1}(\mathbb{C}\smallsetminus Y,x_{0})$ of $\mathbb{C}\smallsetminus Y$
with base point $x_{0}$ is isomorphic to the free group on $n$ generators. A
free set of generators is given by choosing, for each $i=1,\ldots,n$, a loop
based at $x_{0}$ that passes once around $y_{i}$ counterclockwise and does not
enclose any other points of $Y$.

For $Y,Y^{\prime}\in\mathcal{Y}$ with $Y\subseteq Y^{\prime}$, the inclusion
$\mathbb{C}\smallsetminus Y^{\prime}\subseteq\mathbb{C}\smallsetminus Y$ gives
rise to a morphism

$\psi_{Y,Y^{\prime}}\colon\pi_{1}(\mathbb{C}\smallsetminus
Y^{\prime},x_{0})\to\pi_{1}(\mathbb{C}\smallsetminus Y,x_{0})$

of groups. In fact, $((\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0}))_{Y\in\mathcal{Y}},(\psi_{Y,Y^{\prime}})_{Y^{\prime}\supseteq Y})$ is
a projective system of groups.

######  Lemma 3.2.

The projective systems $\big{(}(\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0}))_{Y\in\mathcal{Y}},(\psi_{Y,Y^{\prime}})_{Y^{\prime}\supseteq
Y}\big{)}$ and
$\big{(}(F(Y))_{Y\in\mathcal{Y}},(\varphi_{Y,Y^{\prime}})_{Y^{\prime}\supseteq
Y}\big{)}$ are isomorphic, i.e., there exists a family
$(\alpha_{Y})_{Y\in\mathcal{Y}}$ of isomorphisms $\alpha_{Y}\colon
F(Y)\to\pi_{1}(\mathbb{C}\smallsetminus Y,x_{0})$ such that

$\textstyle{F(Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{Y^{\prime}}}$$\scriptstyle{\varphi_{Y^{\prime},Y}}$$\textstyle{\pi_{1}(\mathbb{C}\smallsetminus
Y^{\prime},x_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{Y,Y^{\prime}}}$$\textstyle{F(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{Y}}$$\textstyle{\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0})}$ (2)

commutes for every $Y^{\prime}\supseteq Y$.

###### Proof.

For $Y\in\mathcal{Y}$ and $y\in Y$, we define a _canonical generator at $y$_
to be an element of $\pi_{1}(\mathbb{C}\smallsetminus Y,x_{0})$ that is the
homotopy class of a loop based at $x_{0}$ that passes once counterclockwise
around $y$, not enclosing any other points of $Y$. The following graphics
depict two canonical generators at $y_{1}$.

Note that $\pi_{1}(\mathbb{C}\smallsetminus Y,x_{0})$ contains only finitely
many canonical generators at $y$. Thus the set

$B_{Y}=\\{(g_{y})_{y\in Y}|\ g_{y}\in\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0})\text{ is a canonical generator at }y\text{ for every }y\in Y\\}$

is finite. The map $\psi_{Y,Y^{\prime}}\colon\pi_{1}(\mathbb{C}\smallsetminus
Y^{\prime},x_{0})\to\pi_{1}(\mathbb{C}\smallsetminus Y,x_{0})$ maps a
canonical generator at $y^{\prime}\in Y^{\prime}\supseteq Y$ either to a
canonical generator at $y$ (if $y^{\prime}\in Y$) or to $1$ if
$y^{\prime}\notin Y$.

For $Y,Y^{\prime}\in\mathcal{Y}$ with $Y\subseteq Y^{\prime}$, we can define a
map $\Psi_{Y,Y^{\prime}}\colon B_{Y^{\prime}}\to B_{Y}$, by
$\Psi_{Y,Y^{\prime}}((g_{y^{\prime}})_{y^{\prime}\in
Y^{\prime}})=(\psi_{Y,Y^{\prime}}(g_{y}))_{y\in Y}$.

Then
$\big{(}(B_{Y})_{Y\in\mathcal{Y}},(\Psi_{Y,Y^{\prime}})_{Y^{\prime}\supseteq
Y}\big{)}$ is a projective system of finite sets. Thus the corresponding
projective limit is non-empty ([RZ10, Prop. 1.1.4]). Let $((g_{y})_{y\in
Y})_{Y\in\mathcal{Y}}$ be an element of $\varprojlim_{Y\in\mathcal{Y}}B_{Y}$.
For $Y\in\mathcal{Y}$ define $\alpha_{Y}\colon
F(Y)\to\pi_{1}(\mathbb{C}\smallsetminus Y,x_{0})$ by $\alpha_{Y}(y)=g_{y}$ for
$y\in Y$. Then $\alpha_{Y}$ is an isomorphism and by construction diagram (2)
commutes. ∎

##  4\. Main result

Throughout Section 4 we will use the following notation. We fix a Picard-
Vessiot extension $L/\mathbb{C}(x)$ for the family all regular singular
differential equations over $\mathbb{C}(x)$. For a subset $X$ of $\mathbb{C}$
we denote with $L_{X}\subseteq L$ the Picard-Vessiot extension of
$\mathbb{C}(x)$ for the family of all regular singular differential equations
over $\mathbb{C}(x)$ with singularities in $X\cup\\{\infty\\}$. So
$L=L_{\mathbb{C}}$. We also set $\Gamma_{X}=G(L_{X}/\mathbb{C}(x))$. With this
notation, our goal is to show that $\Gamma_{X}\cong\Gamma(X)$ for every subset
$X$ of $\mathbb{C}$.

We first tackle the case of proper subsets of $\mathbb{C}$. The following
proposition generalizes Corollary 2.8 from finite subsets of $\mathbb{C}$ to
arbitrary proper subsets of $\mathbb{C}$.

######  Proposition 4.1.

Let $X$ be a proper subset of $\mathbb{C}$. Then the differential Galois group
$\Gamma_{X}$ of the family of all regular singular differential equations over
$\mathbb{C}(x)$ with singularities contained in $X\cup\\{\infty\\}$, is
isomorphic to the free proalgebraic group $\Gamma(X)$ on $X$.

###### Proof.

Fix a base point $x_{0}\in\mathbb{C}$ with $x_{0}\notin X$. As $\Gamma_{X}$ is
isomorphic to $\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$, it
suffices to show that
$\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$ is isomorphic to
$\Gamma(X)$. (Note that the isomorphism between $\Gamma_{X}$ and
$\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$ is arguably not
canonical. It depends on an isomorphism between the fibre functor
$\omega_{X,x_{0}}$ and the fibre functor defined by $L_{X}$.)

As in Section 3, we consider the directed set $\mathcal{Y}$ of all finite
subsets of $X$. Because the category $\operatorname{RegSing}(\mathbb{C}(x),X)$
is the union of the subcategories $\operatorname{RegSing}(\mathbb{C}(x),Y)$,
where $Y$ runs through all elements of $\mathcal{Y}$, it is clear that
$\varprojlim_{Y\in\mathcal{Y}}\underline{\operatorname{Aut}}^{\otimes}(\omega_{Y,x_{0}})\simeq\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$.
The projective systems
$(\underline{\operatorname{Aut}}^{\otimes}(\omega_{Y,x_{0}}))_{Y\in\mathcal{Y}}$
and $(\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0})^{\operatorname{alg}})_{Y\in\mathcal{Y}}$ are canonically isomorphic
by (1). So also
$\varprojlim_{Y\in\mathcal{Y}}\underline{\operatorname{Aut}}^{\otimes}(\omega_{Y,x_{0}})\simeq\varprojlim_{Y\in\mathcal{Y}}\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0})^{\operatorname{alg}}$.

By Lemma 3.2, there exists an isomorphism between the projective systems (of
abstract groups) $\big{(}(\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0}))_{Y\in\mathcal{Y}},(\varphi_{Y^{\prime},Y})_{Y\subseteq
Y^{\prime}}\big{)}$ and
$\big{(}(F(Y))_{Y\in\mathcal{Y}},(\psi_{Y,Y^{\prime}})_{Y\subseteq
Y^{\prime}}\big{)}$. As the proalgebraic completion defines a functor from the
category of groups to the category of proalgebraic groups, we obtain an
isomorphism between the projective systems (of proalgebraic groups)
$\big{(}(\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0})^{\operatorname{alg}})_{Y\in\mathcal{Y}},(\varphi_{Y^{\prime},Y}^{\operatorname{alg}})_{Y\subseteq
Y^{\prime}}\big{)}$ and
$\big{(}(F(Y)^{\operatorname{alg}})_{Y\in\mathcal{Y}},(\psi^{\operatorname{alg}}_{Y,Y^{\prime}})_{Y\subseteq
Y^{\prime}}\big{)}$. This isomorphism in turn yields an isomorphism
$\varprojlim_{Y\in\mathcal{Y}}\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0})^{\operatorname{alg}}\cong\varprojlim_{Y\in\mathcal{Y}}F(Y)^{\operatorname{alg}}$
between the corresponding projective limits. The latter limit is, by Lemma
3.1, isomorphic to $\Gamma(X)$.

In summary, we have

$\Gamma_{X}\cong\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})\simeq\varprojlim_{Y\in\mathcal{Y}}\underline{\operatorname{Aut}}^{\otimes}(\omega_{Y,x_{0}})\simeq\varprojlim_{Y\in\mathcal{Y}}\pi_{1}(\mathbb{C}\smallsetminus
Y,x_{0})^{\operatorname{alg}}\cong\varprojlim_{Y\in\mathcal{Y}}F(Y)^{\operatorname{alg}}\simeq\varprojlim_{Y\in\mathcal{Y}}\Gamma(Y)\simeq\Gamma(X).$

∎

Note that Proposition 4.1 and its proof does not apply to the case
$X=\mathbb{C}$ of prime interest, because for $X=\mathbb{C}$ we cannot choose
a base point $x_{0}\notin\mathbb{C}\smallsetminus X$. To accomplish the case
$X=\mathbb{C}$, we will use a characterization of free proalgebraic groups in
terms embedding problems.

To this end, we need to recall some definitions from [Wib20]. Let $\Gamma$ be
a proalgebraic group. An _embedding problem_ for $\Gamma$ consists of two
quotient maps $\alpha\colon G\twoheadrightarrow H$ and
$\beta\colon\Gamma\twoheadrightarrow H$ of proalgebraic groups. The embedding
problem is _algebraic_ if $G$ (and therefore also $H$) is an algebraic group.
The embedding problem is trivial if $\alpha$ is an isomorphism. A _solution_
of the embedding problem is a quotient map $\phi\colon\Gamma\twoheadrightarrow
G$ such that

$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\phi}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{H}$

commutes. A family $(\phi_{i})_{i\in I}$ of solutions is _independent_ if the
induced map

$\prod_{i\in I}\phi_{i}\colon\Gamma\to\prod_{i\in I}(G\twoheadrightarrow H)$

is a quotient map. Here $\prod_{i\in I}(G\twoheadrightarrow H)$ denotes the
fibre product of $|I|$-copies of $G$ over $H$.

The _rank_ $\operatorname{rank}(\Gamma)$ of a proalgebraic group $\Gamma$ is
defined as the smallest cardinal $\kappa$ such that $\Gamma$ can be written as
a projective limit of algebraic groups over an directed set of cardinality
$\kappa$. See [Wib20, Prop. 3.1] for other characterizations of the rank.

The following theorem provides a characterization of free proalgebraic groups
in terms of algebraic embedding problems.

######  Theorem 4.2.

Let $\Gamma$ be a proalgebraic group with
$\operatorname{rank}(\Gamma)\leq|\mathbb{C}|$. Then $\Gamma$ is isomorphic to
the free proalgebraic group on a set of cardinality $|\mathbb{C}|$ if and only
if every non-trivial algebraic embedding problem for $\Gamma$ has
$|\mathbb{C}|$ independent solutions.

###### Proof.

First assume that $\Gamma$ is isomorphic to $\Gamma(X)$, where $X$ is a set of
cardinality $|\mathbb{C}|$. Since
$\operatorname{rank}(\Gamma(X))=|X|=|\mathbb{C}|$ by [Wib20, Cor. 3.12], we
find $\operatorname{rank}(\Gamma)=|\mathbb{C}|$. Thus the claim follows from
Theorem 3.42 (applied with $\mathcal{C}$ the formation of all algebraic
groups) paired with Definition 3.25 of [Wib20].

Conversely, assume that every non-trivial algebraic embedding problem for
$\Gamma$ has $|\mathbb{C}|$ independent solutions. We claim that
$\operatorname{rank}(\Gamma)=|\mathbb{C}|$. Consider the embedding problem
$\alpha\colon\mathbb{G}_{a}\to 1$, $\beta\colon\Gamma\to 1$ for $\Gamma$,
where $\mathbb{G}_{a}$ is the additive group. By assumption, there exist
solutions $(\phi_{x})_{x\in\mathbb{C}}$ such that the induced morphism
$\Gamma\to\mathbb{G}_{a}^{|\mathbb{C}|}$ is a quotient map. The rank of
$\mathbb{G}_{a}^{|\mathbb{C}|}$ is $|\mathbb{C}|$ ([Wib20, Ex. 3.3]) and the
rank can only decrease when passing to a quotient ([Wib20, Lem. 3.5]). So
$\operatorname{rank}(\Gamma)\geq|\mathbb{C}|$ and consequently
$\operatorname{rank}(\Gamma)=|\mathbb{C}|$. Our assumption on $\Gamma$
therefore implies that $\Gamma$ satisfies condition (vii) of [Wib20, Thm.
3.24]. Thus the claim follows again from [Wib20, Thm. 3.42]. ∎

The following lemma is our crutch to go from proper subsets of $\mathbb{C}$ to
all of $\mathbb{C}$.

######  Lemma 4.3.

Let $\Gamma$ be a proalgebraic group with
$\operatorname{rank}(\Gamma)\leq|\mathbb{C}|$ and let $X$ be a set of
cardinality $|\mathbb{C}|$. Assume that every quotient map
$\Gamma\twoheadrightarrow H$ to an algebraic group $H$ can be factored as
$\Gamma\twoheadrightarrow\Gamma(X)\twoheadrightarrow H$. Then $\Gamma$ is
isomorphic to $\Gamma(X)$.

###### Proof.

Let

$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{H}$
(3)

be an non-trivial algebraic embedding problem for $\Gamma$. By assumption,
$\beta\colon\Gamma\twoheadrightarrow H$ factors as
$\beta\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.125pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.14531pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\beta_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.125pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.125pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Gamma(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
62.19948pt\raise 6.66666pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30556pt\hbox{$\scriptstyle{\beta^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.22223pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.22223pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{H}$}}}}}}}\ignorespaces}}}}\ignorespaces$.
The embedding problem

$\textstyle{\Gamma(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{\prime}}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{H}$

has $|\mathbb{C}|$ independent solutions $(\phi^{\prime}_{x})_{x\in X}$
(Theorem 4.2). Then $\phi_{x}\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.125pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.14531pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\beta_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.125pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.125pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Gamma(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
61.18507pt\raise 6.66666pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30556pt\hbox{$\scriptstyle{\phi_{x}^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.22223pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.22223pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{G}$}}}}}}}\ignorespaces}}}}\ignorespaces$
is a solution of (3). Moreover, as $\prod_{x\in
X}\phi_{x}\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.125pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.14531pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\beta_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.125pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.125pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Gamma(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
60.85489pt\raise 9.48616pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.51393pt\hbox{$\scriptstyle{\prod\phi_{x}^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.22223pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.22223pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\prod_{x\in X}(G\twoheadrightarrow
H)}$}}}}}}}\ignorespaces}}}}\ignorespaces$ is a quotient map, the family
$(\phi_{x})_{x\in X}$ is independent. Thus $\Gamma$ is isomorphic to
$\Gamma(X)$ by Theorem 4.2. ∎

We are now prepared to prove our main result.

######  Theorem 4.4.

The differential Galois group of the family of all regular singular
differential equations over $\mathbb{C}(x)$ is isomorphic to the free
proalgebraic group on a set of cardinality $|\mathbb{C}|$.

###### Proof.

Let $\Gamma=\Gamma_{\mathbb{C}}=G(L/\mathbb{C}(x))$ be the differential Galois
group of the family of all regular singular differential equations. By
[BHHW21b, Lem. 3.3], the rank of $\Gamma$ is the smallest cardinal number
$\kappa$ such that $L/\mathbb{C}(x)$ is a Picard-Vessiot extension for a
family of differential equations of cardinality $\kappa$. Since the family of
all differential equations over $\mathbb{C}(x)$ has cardinality
$|\mathbb{C}|$, we have $\operatorname{rank}(\Gamma)\leq|\mathbb{C}|$.

Let $H$ be an algebraic group and $\beta\colon\Gamma\twoheadrightarrow H$ a
quotient map. By the differential Galois correspondence (Theorem 2.6) the
extension $L^{\ker(\beta)}/\mathbb{C}(x)$ is a Picard-Vessiot extensions with
differential Galois group $H$. Since $H$ is algebraic,
$L^{\ker(\beta)}/\mathbb{C}(x)$ is a Picard-Vessiot extension of finite type.
Therefore, there exists a finite family $\mathcal{F}$ of regular singular
differential equations over $\mathbb{C}(x)$ such that the Picard-Vessiot
extension $L_{\mathcal{F}}$ of $\mathbb{C}(x)$ for $\mathcal{F}$ (inside $L$)
contains $L^{\ker(\beta)}$. Let $Y\subseteq\mathbb{C}$ be the finite set of
(finite) singularities of the differential equations contained in
$\mathcal{F}$. Let $X$ be a proper subset of $\mathbb{C}$ with
$|X|=\mathbb{C}$ and $Y\subseteq X$ (e.g.,
$X=\mathbb{C}\smallsetminus\\{x_{0}\\}$ with $x_{0}\notin Y$). Then

$\mathbb{C}(x)\subseteq L^{\ker(\beta)}\subseteq L_{\mathcal{F}}\subseteq
L_{Y}\subseteq L_{X}$

and so, the quotient map $\beta\colon\Gamma\twoheadrightarrow H$ factors as
$\beta\colon\Gamma\twoheadrightarrow\Gamma_{X}\twoheadrightarrow H$. As
$\Gamma_{X}\cong\Gamma(X)$ by Proposition 4.1, we see that the condition of
Lemma 4.3 is satisfied. ∎

Combining Proposition 4.1 and Theorem 4.4 we obtain:

######  Corollary 4.5.

Let $X$ be a subset of $\mathbb{C}$. Then the differential Galois group of the
family of all regular singular differential equations with singularities in
$X\cup\\{\infty\\}$ is isomorphic to the free proalgebraic group on $X$.

###  4.1. Open Questions

It seems natural to wonder if Corollary 4.5 remains valid when $\mathbb{C}$ is
replaced with an algebraically closed field of characteristic zero. Our proof
uses transcendental tools, such as the fundamental group, and therefore does
not generalize. Even when $X$ is finite, it seems to be unknown if Corollary
4.5 is true for an algebraically closed field of characteristic zero in place
of $\mathbb{C}$.

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# Hidden-charm Hexaquarks from Lattice QCD

Hang Liu  INPAC, Key Laboratory for Particle Astrophysics and Cosmology (MOE),
Shanghai Key Laboratory for Particle Physics and Cosmology, School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Jinchen
He  INPAC, Key Laboratory for Particle Astrophysics and Cosmology (MOE),
Shanghai Key Laboratory for Particle Physics and Cosmology, School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Liuming
Liu  Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu
Province 730000, China  School of Physical Sciences, University of Chinese
Academy of Sciences, Beijing 100049, China  Peng Sun  Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou, Gansu Province 730000, China
Department of Physics and Institute of Theoretical Physics, Nanjing Normal
University, Nanjing, Jiangsu 210023, China  Wei Wang  INPAC, Key Laboratory
for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for
Particle Physics and Cosmology, School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai 200240, China  Yi-Bo Yang  School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China  School of Fundamental
Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study,
UCAS, Hangzhou 310024, China  International Centre for Theoretical Physics
Asia-Pacific, Beijing/Hangzhou, China  Qi-An Zhang  School of Physics, Beihang
University, Beijing 102206, China

###### Abstract

We present a lattice QCD study of hidden-charm hexaquarks with quark content
$usc\bar{d}\bar{s}\bar{c}$ based on four ensembles of gauge configurations
generated by CLQCD Collaboration with pion mass in the range of 220-300MeV.
Four operators with quantum numbers $0^{++},0^{-+},1^{++}$ and $1^{--}$
respectively are constructed to interpolate the hexaquarks. After validating
the spectrum and the dispersion relation for ordinary hadrons, we calculate
the masses of the hexaquarks and extrapolate the results to the physical pion
mass and the continuum limit. We find that the masses of the four hexaquarks
are all below the $\Xi_{c}\bar{\Xi}_{c}$ threshold, while the $0^{-+}$
hexaquark lies around the $\eta_{c}K^{+}K^{-}$ threshold. These results will
be helpful for experimental searches in future and for a deep understanding of
the nature of multiquark states.

Introduction: The spectrum of hadron excitations discovered at experimental
facilities around the world manifests the fundamental interactions of
elementary quarks and gluons, governed by quantum gauge field theory of QCD.
Understanding the complex emergent phenomena of this field theory has
captivated the attention of theoretical physicists in the last decades. To
date one of the perennial problems in hadron physics is to establish the
existence of exotic hadrons that defy the the quark-antiquark interpretation
for mesons and three-quark assignment for baryons Gell-Mann:2015moa ;
Zweig:1964jf ; Jaffe:1976ig ; Jaffe:1977cv ; Lipkin:1987sk . Candidates of
such exotic hadrons, including tetraquark and pentaquark states, have been
recently discovered and confirmed in various experimental measurements
Belle:2003nnu ; CDF:2003cab ; D0:2004zmu ; LHCb:2014zfx ; BESIII:2016adj ;
BESIII:2016bnd ; BESIII:2017tqk ; BaBar:2005hhc ; CLEO:2006tct ; Belle:2007dxy
; Belle:2013yex ; BaBar:2006ait ; BaBar:2012hpr ; Belle:2007umv ;
Belle:2014wyt ; BESIII:2020qkh ; LHCb:2015yax ; LHCb:2016ztz ; LHCb:2019kea .
These experimental progresses give us a strong confidence for the existence of
hexaquark states.

The proposition of six quarks as a single hadron structure was firstly
presented in 1964 Dyson:1964xwa , and a renowned realization is deuteron. In
addition, the $d^{*}(2380)$ resonance, reported by CELSIUS/WASA and WASA-at-
COSY collaborations Faldt:2011zv ; WASA-at-COSY:2011bjg ; WASA-at-COSY:2012seb
; Kim:2020rwn ; Dong:2018emq , is widely believed to be a dibaryon. Until now,
a lot of explorations of hexaquarks with different quark constituents have
been conducted in theory, such as heavy dibaryons (qqqqqQ) Oka:2019mrd ;
Pepin:1998ih , doubly-heavy dibaryons (qqqqQQ) Vijande:2016nzk ; Meng:2017fwb
, fully light dibaryons Zhang:1997ny ; Gerasyuta:2010hn ; Park:2015nha ;
Chen:2019vdh , and fully heavy dibaryons Huang:2020bmb .

Hadrons composed of three quarks and three anti-quarks belong to another
category of hexaquarks. Other than light ones, hidden-charm and hidden-bottom
hexaquarks are of special interest since heavy quarks have much larger masses
and thus are more easily distinguished from ordinary mesons. In this work, we
focus on the hadronic structures containing three quarks and three antiquarks,
i.e. $usc\bar{d}\bar{s}\bar{c}$. We consider four hexaquarks with
$I(J^{PC})=1(0^{++}),1(0^{-+}),1(1^{++}),1(1^{--})$ respectively and calculate
their masses from ab-initio framework–lattice QCD. After making the chiral and
continuum extrapolations, we find that the four hexaquarks are all below the
$\Xi_{c}\bar{\Xi}_{c}$ threshold, while the lightest $0^{-+}$ hexaquark lies
around the $\eta_{c}K^{+}K^{-}$ threshold. This interesting observation can be
further examined by more theoretical analyses and validated by future
experimental measurements.

Theoretical Framework: One of the most powerful method to systematically
tackle the nonperturbative strong interaction is lattice QCD Wilson:1974sk ,
in which the quark and gluon fields are discretized on a space-time grid of
finite size, allowing numerical computations by averaging over large numbers
of possible field configurations generated by Monte-Carlo methods. In
particular, from the euclidean time dependence of correlation functions
calculated in this way, one can determine a discrete spectrum of various
hadrons. Thus Lattice QCD provides a first-principles technique for
explorations of quantities of interest, such as spectrum, scattering phase and
radiative transitions Dudek:2007wv ; Dudek:2006ej ; Dudek:2009kk .

To determine the mass spectrum, one firstly needs to construct appropriate
interpolating operators with the desired quantum numbers. For the hexaquarks
we are interested in this work, the interpolating operators with quantum
numbers $0^{++},0^{-+},1^{++}$ and $1^{--}$ are constructed as

$\displaystyle 0^{++}:\mathcal{O}_{1}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}\epsilon^{def}[{u^{T}_{a}}C\gamma_{5}{s_{b}}][{\bar{d}_{d}}C\gamma_{5}{\bar{s}_{e}}^{T}]\times[{\bar{c}_{f}}c_{c}]$
$\displaystyle 0^{-+}:\mathcal{O}_{2}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}\epsilon^{def}[{u^{T}_{a}}C\gamma_{5}{s_{b}}][{\bar{c}_{d}}C\gamma_{5}{\bar{s}_{e}}^{T}]\times[{\bar{c}_{f}}\gamma_{5}c_{c}]$
$\displaystyle 1^{++}:\mathcal{O}_{3}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}\epsilon^{def}[{u^{T}_{a}}C\gamma_{5}{s_{b}}][{\bar{d}_{d}}C\gamma_{5}{\bar{s}_{e}}^{T}]\times[{\bar{c}_{f}}\gamma_{i}\gamma_{5}q_{c}]$
$\displaystyle 1^{--}:\mathcal{O}_{4}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}\epsilon^{def}[{u^{T}_{a}}C\gamma_{5}{s_{b}}][{\bar{d}_{d}}C\gamma_{5}{\bar{s}_{e}}^{T}]\times[{\bar{c}_{f}}\gamma_{i}c_{c}].$
(1)

Here $C=i\gamma_{2}\gamma_{4}$ is the charge conjugation matrix, and $a,...,f$
are color indices.

Then the determination of the mass spectrum proceeds from the calculation of
the correlation functions between these operator and their hermitian
conjugates at Euclidean times $t$ and $0$ of the form

$\displaystyle C_{2,i}(t)=\langle
0|\mathcal{O}_{i}(t)\mathcal{O}_{i}^{\dagger}(0)|0\rangle,$ (2)

where $i$ labels the four operators.

Lattice Simulation: We employ four (2+1)-flavor gauge ensembles with Wilson
clover fermion and Symanzik gauge action at three lattice spacings
$a=0.054~{}\mathrm{fm},0.081~{}\mathrm{fm},0.108~{}\mathrm{fm}$. The
parameters of the four ensembles are listed in Table 1. A first analysis of
$\Xi_{c}\to\Xi$ form factors using two of the ensembles (C08P30S and C11P29S)
has been conducted in Ref. Zhang:2021oja , and predictions on partial widths
for semi-leptonic $\Xi_{c}$ decays were used in the experimental background
simulation by Belle collaboration Belle:2021crz . The bare strange quark mass
is determined such that the mass of $\eta_{s}$ is around $700$ MeV, and the
bare charm quark mass is tuned to accommodate the spin-average value of the
$J/\psi$ and $\eta_{c}$ masses. The quark propagators are computed using the
Coulomb gauge fixed wall source at one source time slice for the ensembles
C11P29S, C11P22M and C08P30S, while two source time slices are used for the
ensemble C06P30S.

| $\beta$ | $L^{3}\times T$ | $a$ | $\kappa_{l}$ | $m_{\pi}$ | $\kappa_{s}$ | $\kappa_{c}$ |
---|---|---|---|---|---|---|---|---
C11P29S | $6.2$ | $24^{3}\times 72$ | $0.108$ | $0.1343$ | $285(2)$ | $0.1327$ | $0.1117$ |
C11P22M | $6.2$ | $32^{3}\times 64$ | $0.108$ | $0.1344$ | $220(2)$ | $0.1326$ | $0.1116$ |
C08P30S | $6.41$ | $32^{3}\times 96$ | $0.080$ | $0.1326$ | $301(3)$ | $0.1316$ | $0.1181$ |
C06P30S | $6.72$ | $48^{3}\times 144$ | $0.054$ | $0.1311$ | $311(6)$ | $0.1305$ | $0.1227$ |
Table 1: Parameters of the 2+1 flavor Wilson clover fermion ensembles used in
this calculation. The pion masses and the lattice spacings are given in units
of MeV, and fm, respectively.

We first analyze the ordinary mesons including $K$, $D$, $D_{s}$, $\eta_{c}$
and $J/\psi$, and also baryons $\Lambda$, $\Xi$, $\Omega$, $\Lambda_{c}$, and
$\Xi_{c}$. We calculate the two-point correlation functions with the pertinent
interpolating operators. The masses of the ground state hadrons are obtained
from the two-state fits of the corresponding correlation functions with
reasonable $\chi^{2}$/d.o.f.. We simultaneously extrapolate all of the ground
state masses to the physical pion mass and continuum limit using the following
parametrization for all of the hadrons except kaon,

$\displaystyle m_{H}(m_{\pi},a)=m_{H,{\rm
phys}}+g^{H}_{1}(m_{\pi}^{2}{-m_{\pi,{\rm phys}}^{2}})+g^{H}_{2}a^{2},$ (3)

and the parameterization inspired by the chiral symmetry is used in the kaon
case:

$\displaystyle m_{K}^{2}(m_{\pi},a)=m_{K,{\rm
phys}}^{2}+g^{K}_{1}(m_{\pi}^{2}{-m_{\pi,{\rm phys}}^{2}})+g^{K}_{2}a^{2},$
(4)

where $m_{\pi,{\rm phys}}=135$ MeV is the physical pion mass without iso-spin
symmetry breaking and QED effect. As shown in Fig. 1, all of our predictions
after the continuum and chiral extrapolations agree with experimental values
within 2$\sigma$ deviation, expect the kaon mass which requires a better
control on the chiral extrapolation and strange quark mass determination.

Figure 1:  The ordinary hadron masses from this work compared with their
experimental values Zyla:2020zbs . The QED effects and iso-spin symmetry
breaking effects have been removed for $m_{K}$, $m_{D}$ and $m_{D_{s}}$
Giusti:2017dmp . $\Delta_{\rm HFS}=m_{J/\psi}-m_{\eta_{c}}$ is the hyperfine
splitting of the charmonia.

In addition, we explore the mass-energy dispersion relations for the ordinary
mesons to ensure that all the lattice discretization errors are under control.
We have generated the two-point correlation functions for the $\pi$ and
$J/\psi$ mesons at different momenta, and obtained the energy from two-state
fits of the correlations. Then the dispersion relation is parametrized as

$\displaystyle E^{2}=m^{2}+c_{2}p^{2}+c_{3}p^{4}a^{2},$ (5)

where $c_{2}$ and $c_{3}$ are dimensionless parameters to be determined from
the fits. The deviations of $c_{2}$ from unity (the speed of light) and
$c_{3}$ from zero characterize the discretization errors. The data and fit of
pion with the ensembles C11P29S, C08P30S and C06P30S having $m_{\pi}\simeq
300$ MeV are shown in the upper panel of Fig. 2, and the $J/\psi$ case is
shown in the lower panel. The joint fit with three lattice spacings using the
parameterization Eq. 5 (with individual $m$ at different lattice spacing)
provide good $\chi^{2}$/d.o.f. for both cases, with

$\displaystyle c_{2,\pi}$ $\displaystyle=$ $\displaystyle 1.078(26),\
1.037(41),\ 1.093(53),$ $\displaystyle c_{3,\pi}$ $\displaystyle=$
$\displaystyle 0.07(06),\ -0.54(46),\ \ 0.3(1.1),$ $\displaystyle
c_{2,J/\psi}$ $\displaystyle=$ $\displaystyle 0.730(18),\ 0.852(38),\
0.954(45),$ $\displaystyle c_{3,J/\psi}$ $\displaystyle=$
$\displaystyle-0.04(03),-0.07(18),\ 0.41(49),$ (6)

at $a$=0.108, 0.080, 0.054 fm, respectively. Thus the dispersion relation for
the ordinary mesons with both the light and heavy quarks can be well-respected
after the continuum extrapolation.

Figure 2: The dispersion relation for $\pi$ and $J/\psi$ at three lattice
spacings.

Results and discussions: The focus of this work is the hexaquark states with
$I=1,J^{PC}=0^{-+},0^{++},1^{--},1^{++}$. To extract the masses of the
hexaquark states, we adopt the two-state parametrization for the two-point
correlation functions:

$\displaystyle C_{2}(t)=c_{0}e^{-M_{H}t}[1+\Delta c\times e^{-\Delta Et}].$
(7)

where $c_{0},\Delta c,M_{H}$ and $\Delta E$ are parameters to be determined
through a correlated fit of the lattice data. $M_{H}$ is the lowest-lying
hexaquark state and $\Delta E$ corresponds to the relative mass of the excited
state. $c_{0}$ denotes the contributions of the lowest-lying hexaquark to the
two-point correlation functions, and $\Delta c$ reflects the excited state
contamination. We perform two independent fits by fitting the correlation
functions and the effective masses $M_{\rm eff}(t)=\ln C_{2}(t+a)/C_{2}(t)$,
and find consistent results for the $M_{H}$ within $1\sigma$. As an
illustration, the effective masses and the fitted energy plateaus for the four
emsembles are plotted in Fig. 3. Panel (a,b,c,d) corresponds to the four
ensembles, respectively. From the figures, we find that all the lattice data
can be well described with $\chi^{2}/d.o.f.$ smaller than 1.

(a) C11P22M

(b) C11P29S

(c) C08P30S

(d) C06P30S

Figure 3: Effective mass for hexaquarks on the four ensembles. Panel (a), (b),
(c), and (d) corresponds to the C11P22M, C11P29S, C08P30S, and C06P30S
ensembles, respectively.

(a) $0^{-+}$

(b) $0^{++}$

(c) $1^{--}$

(d) $1^{++}$

Figure 4: The continuum and physical pion mass extrapolation for the mass of hexaquark states. $I(J^{PC})$ | $1(0^{-+})$ | $1(1^{--})$ | $1(0^{++})$ | $1(1^{++})$
---|---|---|---|---
mass(GeV) | $3.945(32)$ | $4.144(35)$ | $4.15(20)$ | $4.22(23)$
Table 2:  The obtained hexaquark masses (in units of GeV) after the chiral and
continuum extrapolation Figure 5: The effective energy shift between
hexaquarks and three mesons $K,K,\eta_{c}$ on three different ensembles.

The continuum and chiral extrapolations of the masses of the hexaquark states
with quantum numbers $0^{-+},1^{--},0^{++},1^{++}$ are illustrated in Fig. 4
and the extrapolated values are collected in Tab. 2. It is interesting to
notice that the threshold of $\Xi^{+}_{c}-\bar{\Xi}^{0}_{c}$ is about
$4.938{\rm GeV}$ Zyla:2020zbs , while the masses of the hexaquark states are
below this threshold by around 700 to 950 MeV. This feature is partially
consistent with the result from chromo-magnetic interaction model which also
finds large binding energies for the $0^{-+}$ and $1^{--}$ hexaquark states
with $m_{0^{-+}}=3.815$ GeV and $m_{1^{--}}=4.005$ GeV Liu:2021gva . However,
it should be noticed that the hexaquarks we study in this work can mix with
three-meson states. For instance, the $0^{-+}$ hexaquark can mix with
$K^{+}K^{-}\eta_{c}$ and the hexaquark mass we obtained is indeed very close
to the threshold of $K^{+}K^{-}\eta_{c}$, which is $3.971$ GeV Zyla:2020zbs .
To have an illustration, we show the mass difference

$\displaystyle\Delta
m\equiv\mathrm{\log}\frac{C_{2,O_{2}}}{C^{2}_{2,K}C_{2,\eta_{c}}}$ (8)

in Fig. 5, and one can see that $\Delta m$ is consistent with zero within 1-2
sigma at all the lattice spacings.

More extensive studies on the hexaqaurk spectra by including more
interpolating operators are required to furthur clarify the properties of the
hexaquarks. We defer these studies to the future.

Conclusions: Based on the newly-generated (2+1)-flavor configurations with the
pion mass around $m_{\pi}{\simeq}220,300$ MeV, we have presented a first
lattice simulation of hidden-charm hexaquark states with the quark content
$usc\bar{d}\bar{s}\bar{c}$. Four different quantum numbers are assigned for
the hexaquarks and the corresponding mass spectrum are derived. We have also
extrapolated the results both to the physical pion mass and continuum limit,
and find that the ground state we obtained for all the hexaquark interpolation
operators are around 700-950 MeV below the $\Xi_{c}\bar{\Xi}_{c}$ threshold,
while that of $I(J^{PC})=1(0^{-+})$ channel is consistent with the
$\eta_{c}K^{+}K^{-}$ threshold. These results are helpful towards the search
for such types of the exotic states in the future experiments.

Acknowledgments: We thank Min-Huan Chu and Jun Hua for useful discussions.
This work is supported in part by Natural Science Foundation of China under
grant No. 11735010, U2032102, and 12125503. HL, JCH, and WW are also supported
by Natural Science Foundation of Shanghai under grant No. 15DZ2272100. LL
thanks the support from the CAS Interdisciplinary Innovation Team program. P.
Sun is supported by Natural Science Foundation of China under grant No.
11975127, and Jiangsu Specially Appointed Professor Program. YBY is also
supported by the Strategic Priority Research Program of Chinese Academy of
Sciences, Grant No. XDB34030303, XDPB15. PS, WW, and YBY are also supported by
the NSFC-DFG joint grant under grant No. 12061131006 and SCHA 458/22. The
calculation was supported by National Supercomputing Center in Zhengzhou, and
the Siyuan Mark 1 cluster at Center for High Performance Computing, Shanghai
Jiao Tong University.

##  Appendix A Hadron Masses

In this appendix, we have collected the masses for the
$K,D,D_{s},\Lambda,\Xi,\Omega,\Lambda_{c},\Xi_{c},\eta_{c}$ and $J/\psi$ on
the four ensembles. After the continuum and physical pion extrapolation, the
results for these hadrons are collected in Fig. 1.

(a) $K$

(b) $D$

(c) $D_{s}$

(d) $\Lambda$

(e) $\Xi$

(f) $\Omega$

(g) $\lambda_{c}$

(h) $\Xi_{c}$

(i) $\eta_{c}$

(j) $J/\psi$

Figure 6: Extrapolation for the mass for the ordinary hadrons: $K$, $D$,
$D_{s}$, $\Lambda$, $\Xi$, $\Omega$, $\Lambda_{c}$,$\Xi_{c}$,$\eta_{c}$ and
$J/\psi$.

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