\section*{Introduction}

The monodromy group is one of the most important invariants of a Fuchsian differential equation. If $\mathcal{P}$ is a Picard-Fuchs operator of a one-parameter family of Calabi-Yau threefolds, the monodromy group $Mon(\mathcal{P})$ can be realized as a subgroup of $\mathrm{Sp}(4,\mathbb{Z})$. A natural question is whether $Mon(\mathcal{P})$ is Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$. If it is, one can further ask whether it is \emph{arithmetic}, i.e. of finite index, or \emph{thin}.

The problem of arithmeticity is complicated even for hypergeometric operators \cite{Behr, Singh, Singh-Venkataramana}. They form the simplest class of Picard-Fuchs operators. Their monodromy group is generated by symplectic reflection and a maximally unipotent matrix. Furthermore, they have a point of maximal unipotent monodromy which allows one to construct a symplectic basis using the Doran-Morgan construction.

Some progress towards constructing symplectic realizations and establishing their indexes has been made for more complicated operators possessing a maximally unipotent singularity \cite{Hofmann-van Straten, Chmiel2}. Existence of a point of maximal unipotent monodromy, or a \emph{MUM point}, is also crucial in the context of mirror symmetry.

In this paper we focus on operators which do not have a MUM singularity. For this reason they are called \emph{orphan operators}. For this class of operators there is no general construction of a rational basis, neither a distinguished choice of a symplectic realization. Since every finite index subgroup of $\mathrm{Sp}(4,\mathbb{Z})$ contains a maximally unipotent matrix, one could expect that monodromy groups of orphan operators are small.

We study 18 order four orphan operators which are Picard-Fuchs operators of one-parameter families of double octic Calabi-Yau threefolds defined over $\mathbb{Q}$ (see \cite{Cynk-van Straten}). We construct rational bases for all of these operators, using an approach which is also easily applicable in general. Using these rational realizations we study the properties of the corresponding monodromy groups as linear groups generated by the local monodromy operators. In particular, we obtain symplectic realizations for (subgroups of) orphan double octic monodromy groups.

The main result of this paper can be formulated as the following theorem, which combines Theorem \ref{th:dense}, Theorem \ref{th:MUM} and Theorem \ref{th:arithmetic} below:

\begin{theorem*}\ 

\begin{enumerate}
	\item Monodromy groups of all orphan double octic operators are Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$.
	\item Monodromy groups of orphan double octic operators contain maximally unipotent elements (with the~possible exception of operator \textbf{35}).
	\item Monodromy group of the orphan double octic operator \textbf{250} is of finite index in $\mathrm{Sp}(4,\mathbb{Z})$.
\end{enumerate}
\end{theorem*}
\noindent These results show that the na\"ive expectation that the monodromy groups of Picard-Fuchs operators lacking a MUM point are small is false.

The structure of this paper is as follows. Section \ref{sec:general} is of introductory nature and contains basic facts about the monodromy group and Picard-Fuchs operator. Section \ref{sec:ratbas} explains our method of finding rational basis for the monodromy action. In section \ref{sec:mongr} we present results obtained using this method when applied to Picard-Fuchs operators of orphan families of double octic Calabi-Yau threefolds. Section \ref{sec:MUMor} concerns maximally unipotent elements in the considered monodromy groups. In section \ref{sec:symplectic} we construct symplectic bases for orphan monodromy groups and show that they are Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$. The last section contains a more detailed analysis of three interesting orphan operators, particularly of operator \textbf{250} for which we prove that its monodromy group is arithmetic.

\section{Monodromy group of a Picard-Fuchs equation}\label{sec:general}

A Calabi-Yau threefold is a smooth complex projective variety $X$ of dimension $3$ such that $\omega_X\simeq\mathcal{O}_X$ and $H^1(X,\mathcal{O}_X)=0$. Arguably the most fundamental invariants of a Calabi-Yau threefold are its Hodge numbers. Of particular importance for us is the Hodge number $h^{2,1}(X)$, which by the Bogomolov-Tian-Todorov unobstructedness theorem is the dimension of a smooth deformation space of the manifold $X$.

Let $X$ be a Calabi-Yau threefold satisfying $h^{2,1}(X)=1$, i.e. whose universal deformation space is one-dimensional. This means that there exists a (germ of a) smooth curve $\mathcal{S}$, a distinguished point $s_0\in\mathcal{S}$ and a family of smooth Calabi-Yau threefolds $(X_t)_{t\in\mathcal{S}}=\mathcal{X}\rightarrow\mathcal{S}$ such that $X\simeq X_{s_0}$ under a fixed isomorphism. Furthermore, the morphism $\mathcal{X}\rightarrow\mathcal{S}$ is universal with respect to this property. Such a manifold $X$ is said to \emph{deform in a one-parameter family}.

Let $X\simeq X_{s_0}$ be a member of a one-dimensional family $(X_t)_{t\in\mathcal{S}}=\mathcal{X}$. Choose a family of complex volume elements $\omega_t\in H^{3,0}(X_t)$ depending holomorphically on $t$ and a locally constant family of $3$-cycles $\gamma_t\in H_3(X_t,\mathbb{Z})$. Now we can introduce the \emph{period function}
$$y(t):=\displaystyle\int_{\gamma_t}\omega_t,$$
defined in a some neighbourhood $U$ of $s_0$. Since by the Hodge decomposition $b_3(X_t)=2(h^{3,0}(X_t)+h^{2,1}(X_t))=4$ for all $t\in U$, the spaces $H^3(X_t,\mathbb{C})$ are four-dimensional. Thus for all $t_0\in U$ the elements
$$\left(\nabla_{\frac{\partial}{\partial t}}^4\omega_{t}\right)\Big\rvert_{t=t_0},\quad
\left(\nabla_{\frac{\partial}{\partial t}}^3\omega_{t}\right)\Big\rvert_{t=t_0},\quad
\left(\nabla_{\frac{\partial}{\partial t}}^2\omega_{t}\right)\Big\rvert_{t=t_0},\quad
\left(\nabla_{\frac{\partial}{\partial t}}\omega_t,\right)\Big\rvert_{t=t_0},\quad
\omega_{t_0}$$
must be linearly dependent over $\mathbb{C}$.

As a consequence the period function $y(t)$ satisfies a differential equation $\mathcal{P}y=0$ for some order four differential operator $\mathcal{P}$ with coefficients in $\mathbb{C}(U)$. The operator $\mathcal{P}$ is called the \textit{Picard-Fuchs operator} of the family $\mathcal{X}$.

The main interest of this paper lies in the study of monodromy groups of Picard-Fuchs operators. Let $\mathcal{P}$ be a Picard-Fuchs operator of a one-parameter family $\mathcal{X}\rightarrow\mathcal{S}$, choose a base point $t_0\in\mathcal{S}$ and let $\mathcal{S}ol(\mathcal{P},t_0)\simeq H^3(X_{t_0},\mathbb{C})$ denote the space of solutions of the equation $\mathcal{P}=0$ in a small neighbourhood of the point $t_0$.

Take a loop $\gamma:[0,1]\rightarrow\mathcal{S}$ with $\gamma(0)=\gamma(1)=t_0$. Any solution $f\in\mathcal{S}ol(\mathcal{P},t_0)$ can be continued analytically along $\gamma$ and the result depends only on the equivalence class $[\gamma]\in\pi_1(\mathcal{S},t_0)$. This defines the \emph{monodromy representation}, which is a homomorphism $Mon:\pi_1(\mathcal{S},t_0)\rightarrow\mathrm{Aut}\left(\mathcal{S}ol(\mathcal{P},t_0)\right)$. After choosing a basis $\mathcal{B}$ of $\mathcal{S}ol(\mathcal{P},s_0)$ we obtain a linear representation $Mon^\mathcal{B}: \pi_1(\mathcal{S},t_0)\rightarrow\mathrm{GL}(4,\mathbb{C})$. We denote by $Mon(\mathcal{P})$ and $Mon^\mathcal{B}(\mathcal{P})$ the images of these homomorphisms respectively.

In our further considerations we will deal with families of Calabi-Yau threefolds over curves of the form $\mathbb{P}^1\setminus\Sigma$, where $\Sigma$ is a finite set of points. The fundamental group $\pi_1(\mathbb{P}^1\setminus\Sigma)$ is generated by small loops $\gamma_\sigma$ encircling a single point $\sigma\in\Sigma$. Consequently, operators $Mon(\gamma_\sigma)$ for $\sigma\in\Sigma$ generate the entire monodromy group. Element of the form $Mon(\gamma_\sigma)$ is called the \emph{local monodromy operator} at a singular point $\sigma$.

Picard-Fuchs operators are of Fuchsian type, i.e. they possess only regular singular points. It is a classical result that local monodromy operators $M_\sigma:=Mon(\gamma_\sigma)$ are quasi-unipotent, i.e. there exist integers $k,n\in\mathbb{N}$ such that $(M_\sigma^k-\operatorname{Id})^n=0$ (see \cite{Landman}). Of course in general not all elements of the monodromy group $Mon(\mathcal{P})$ are quasi-unipotent.

For Picard-Fuchs operators of families of Calabi-Yau threefolds four types of singular points appear. They are classified by the Jordan forms of their associated local monodromies. More precisely, after normalization we have the following possibilities:
\begin{center}
	\begin{tabular}{|c|c|}
		\hline
		Type of singularity&Jordan form of local monodromy\\
		\hline
		MUM&
		$\begin{pmatrix}
		1&1&0&0\\0&1&1&0\\0&0&1&1\\0&0&0&1\\
		\end{pmatrix}$\\
		\hline
		$\tfrac{1}{n}$K&
		$\begin{pmatrix}
		1&1&0&0\\0&1&0&0\\0&0&\zeta_n&1\\0&0&0&\zeta_n\\
		\end{pmatrix}$\\
		\hline
		$\tfrac{1}{n}$C&
		$\begin{pmatrix}
		1&0&0&0\\0&\zeta_n&1&0\\0&0&\zeta_n&0\\0&0&0&\zeta_n^2\\
		\end{pmatrix}$\\
		\hline
		$F$&
		$\begin{pmatrix}
		1&0&0&0\\0&\zeta_{n_1}&0&0\\0&0&\zeta_{n_2}&0\\0&0&0&\zeta_{n_1}\zeta_{n_2}\\
		\end{pmatrix}$\\
		\hline
	\end{tabular}
\end{center}
Here $\zeta_n$ denotes some primitive root of unity of order $n$.

Singular points of type MUM are also called \emph{points of maximal unipotent monodromy}, since they have maximal possible index of unipotency equal to $4$. Singular points of type $\tfrac{1}{n}K$ are connected with $K3$ surfaces, while singular points of type $\tfrac{1}{n}C$ are connected to (nodal models of) rigid Calabi-Yau threefolds and are also called \emph{conifold points}. Singular points of type $F$ are called \emph{finite singularities}, since in this case the local monodromy action is of finite order. It is important that for each conifold point $\sigma$ of type $\tfrac{1}{n}C$ there exists a unique solution $f_c$, called the \emph{conifold period}, which is defined as the generator of $\operatorname{im}(M_\sigma^n-\operatorname{Id})$.

In what follows we will be interested in the study of monodromy groups of \emph{orphan} Picard-Fuchs operators, i.e. those that do not possess a MUM singularity. There is no general method for computing the monodromy group of a Picard-Fuchs equation and one usually resorts to numerical computation. Analytic continuation of any given solution can be computed numerically (see \cite{Hofmann}). Thus if we choose a basis $\mathcal{B}$ of the space of solutions $\mathcal{S}ol(\mathcal{P})$, we can compute arbitrary precision approximations of the coefficients of the local monodromy matrices $Mon^\mathcal{B}(M_\sigma)$ generating the monodromy group $Mon^\mathcal{B}(\mathcal{P})$.

The problem is that for an arbitrary choice of the basis $\mathcal{B}$ we only have $Mon^\mathcal{B}(\mathcal{P})\subset\mathrm{GL}(4,\mathbb{C})$. Hence the coefficients of the monodromy matrices $Mon^\mathcal{B}(M_\sigma)$ are \emph{a priori} arbitrary complex numbers. If $\mathcal{B}_{m}$ is the Frobenius basis at a point of maximal unipotent monodromy, it is conjectured that $Mon^{\mathcal{B}_{m}}(\mathcal{P})\subset\mathrm{GL}(4,\mathbb{Q}(\tfrac{\zeta(3)}{(2\pi i)^3}))$. For the Frobenius basis at a half-conifold point we have observed in \cite{Chmiel1} that coefficients of the monodromy matrices include special values $L(f,1)$ and $\tfrac{L(f,2)}{2\pi i}$ of the $L$-function of a certain modular form but there also appears one yet unidentified constant.

Therefore in general it is hard to identify the coefficients of the monodromy matrices, unless we can find a \emph{rational} basis $\mathcal{B}$ such that $Mon^\mathcal{B}(\mathcal{P})\subset\mathrm{GL}(4,\mathbb{Q})$. Using the isomorphism $\mathcal{S}ol(\mathcal{P},t)\simeq H^3(X_t,\mathbb{C})\simeq H^3(X_t,\mathbb{Q})\otimes\mathbb{C}$ one can conclude that a rational basis always exists. A generalization of the construction of Doran and Morgan (see \cite{Doran-Morgan}, \cite{Chmiel1}) provides a rational basis for operators which have at least one MUM point. However, for orphan operators no such general construction is known. In the next section we describe our approach for finding rational bases for these operators.

\section{Rational basis of solutions}\label{sec:ratbas}

As explained in the previous section to numerically identify the monodromy group it is convenient, if not crucial, to isolate linearly independent solutions $f_1,\ldots,f_4\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t)\simeq H^3(X_t,\mathbb{Q})$. In this section we explain how this can be accomplished. A natural point to start is the following lemma:

\begin{lemma}\label{lemma:mon-basis}
Assume that the monodromy group $Mon(\mathcal{P})$ acts irreducibly on $\mathcal{S}ol(\mathcal{P},t_0)$. Let $0\neq f\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$. Then there exist monodromy operators $M_1,\ldots,M_4\in Mon(\mathcal{P})$ such that
$$\mathcal{B}:=\Big\{M_1(f),M_2(f),M_3(f),M_4(f)\Big\}$$
is a a basis of $\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$.
\end{lemma}

\begin{proof}
The subspace $\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$ is monodromy-invariant, hence we always have $\mathcal{B}\subset\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$. It is thus enough to show that for some $M_1,\ldots,M_4\in Mon(\mathcal{P})$ elements of $\mathcal{B}$ are linearly independent. Assume that for each choice of $M_i$'s elements of $\mathcal{B}$ span inside $\mathcal{S}ol(\mathcal{P},t_0)$ a subspace of dimension $\leq 3$. Let $S$ be the $\mathbb{C}$-linear span of the elements $M(f)$ for all $M\in Mon(\mathcal{P})$. Clearly $S$ is monodromy-invariant but by our assumption $\dim S\leq 3$. Thus $S\neq\mathcal{S}ol(\mathcal{P},t_0)$, which is a contradiction because $Mon(\mathcal{P})$ acts irreducibly.
\end{proof}

Given Lemma \ref{lemma:mon-basis}, there are two problems remaining:
\begin{enumerate}
\item\label{a} find some solution $f\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$;
\item\label{b} find $M_1\ldots,M_4\in Mon(\mathcal{P})$ such that $\mathcal{B}$ is linearly independent.
\end{enumerate}

\bigskip

We begin by addressing (\ref{a}). First of all, assume that a basis $\mathcal{B}$ consists of elements of the form $M_1(f),M_2(f),M_3(f),M_4(f)$ for some $f\in\mathcal{S}ol(\mathcal{P},t_0)$, $M_i\in Mon(\mathcal{P})$. Let $\mathcal{B}':=\alpha\cdot\mathcal{B}$ for some $\alpha\in\mathbb{C}^*$. Then clearly the realizations $Mon^{\mathcal{B}}(\mathcal{P})$ and $Mon^{\mathcal{B}'}(\mathcal{P})$ are exactly the same: $M^{\mathcal{B}}=M^{\mathcal{B}'}$ for all $M\in Mon(\mathcal{P})$. Furthermore, $\mathcal{B}'$ is the basis $M_1(\alpha f),M_2(\alpha f),M_3(\alpha f),M_4(\alpha f)$. Hence to get a realization in $\mathrm{GL}(4,\mathbb{Q})$ as in Lemma \ref{lemma:mon-basis}, it is not really necessary that $f\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$ but only that $\alpha f\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$ for some $\alpha\in\mathbb{C}^*$. This motivates the following lemma:

\begin{lemma}\label{lemma:unique}
Let $A=\sum_{i=1}^{n} q_iM_i$ for some $q_i\in\mathbb{Q},\ M_i\in Mon(\mathcal{P})$ and treat it as an endomorphism of $\mathcal{S}ol(\mathcal{P},t_0)$. Assume that $\operatorname{rank}(A)=1$, resp. $\operatorname{corank}(A)=1$, and take $f\in\mathcal{S}ol(\mathcal{P},t_0)$ such that $\operatorname{im}(A)=\mathbb{C}\cdot f$, resp. $\operatorname{ker}(A)=\mathbb{C}\cdot f$. Then there exists $\alpha\in\mathbb{C}^*$ such that $\alpha f\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$.
\end{lemma}	

\begin{proof}
Clearly $A$ acts as an endomorphism of $\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$. Thus there exists $g\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$ such that $\operatorname{im}\left(A|_{\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)}\right)=\mathbb{Q}\cdot g$, resp. $\operatorname{ker}\left(A|_{\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)}\right)=\mathbb{Q}\cdot g$. Hence $g\in\mathbb{C}\cdot f$ and the claim follows.
\end{proof}	

The elements $M_i\in Mon(\mathcal{P})$ in the statement of the lemma can be arbitrary. However, in practice they will always be powers of a fixed local monodromy $M_s$ for a singular point $s\in\mathbb{P}^1$, and consequently $A$ will be a polynomial in $M_s$. If this is the case, we may assume that $M_s$ is given in its Jordan form and the computation of rank and corank of $A$ is elementary. Otherwise, we have \textit{a priori} no control over the behaviour of products of local monodromies.

In the construction of the Doran-Morgan basis (and its generalization from \cite{Chmiel1}) one takes $A:=\left(M_0-\Id\right)^3$, where $M_0$ is a local monodromy at a MUM point at $0$. Then $\operatorname{rank}(A)=1$ and the generator of $\operatorname{im}(A)$ is the unique holomorphic solution $v$ in a neighbourhood of $0$. It is of course equivalent to taking $A:=M_0-\Id$, as then $\operatorname{corank}(A)=1$ and the kernel of $A$ is spanned by the same solution $v$.

In this paper we consider operators which do not have a point of maximal unipotent monodromy. However, if we assume that an operator has a singular point of type $\tfrac{1}{n}C$ at $s$, we can also easily find a matrix $A$ satisfying the assumptions of Lemma \ref{lemma:unique}. Namely, let $A:=M_s^n-\Id$. The Jordan form of $M_s^n$ is
$$
\begin{pmatrix}
1&0&0&0\\
0&1&1&0\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix}
$$
and thus $\operatorname{rank}(A)=1$. The space $\operatorname{im}(A)$ is spanned by the conifold period $f_c$. It follows that, up to scaling, we have $f_c\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$.

Note that the conifold period is also a preferable solution from the computational point of view. First of all, it is an element of the basis constructed using the Frobenius method and thus can always be computed locally. Secondly, if instead we are given any other solution $f\in\mathcal{S}ol(\mathcal{P})$, we can always recover the conifold period by analytic continuation as $f_c=(M_s^n-\Id)(f)$, provided it is non-zero.

Unfortunately, singular points of of type $\tfrac{1}{n}K$ do not provide an appropriate matrix $A$ for $n=1$. If $M_s$ is a local monodromy around a singular point of type $K$ at $s$, then a polynomial in $M_s$ can never distinguish between two eigenvectors of $M_s$ and thus will not provide a unique $f$ as in the statement of the lemma. On the other hand, if $s$ is a $\tfrac{1}{n}K$ singularity with $n>1$ and consequently has Jordan form
$$
\begin{pmatrix}
1&1&0&0\\
0&1&0&0\\
0&0&\zeta_n&1\\
0&0&0&\zeta_n\\
\end{pmatrix}
$$
then $A:=M_s-\Id$ satisfies $\operatorname{corank}(A)=1$, as its kernel is generated by the \textit{unique} eigenvector with eigenvalue~$1$. If $n=2$, we can take $A:=M_s+\Id$ and then $\operatorname{corank}(A)=1$ as well. As for the conifold period, all of the solutions obtained this way are easily seen to arise from the Frobenius method (but \textit{not} from the monodromy). 

In any case, we see that it is possible to use Lemma \ref{lemma:unique} to find $f\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$, up to rescaling. Thus we are left with problem (\ref{b}): given such $f$, find matrices $M_1,\ldots,M_4$ for which $M_1(f),\ldots,M_4(f)$ are linearly independent. Of course we may always take $M_1=\Id$. Assuming $Mon(\mathcal{P})$ acts irreducibly, we know that appropriate $M_2,M_3,M_4$ exist. As can be seen from the proof of Lemma \ref{lemma:mon-basis}, in fact a weaker condition suffices: the solution $f$ generates $\mathcal{S}ol(\mathcal{P},t_0)$ under the action of monodromy. If the last condition does not hold, the solution $f$ would have to satisfy a differential equation of order $\leq 3$ and this is usually easy to check provided enough coefficients of its power series expansion are known.

Let $f$ be some solution and let $M_2,M_3,M_4\in Mon(\mathcal{P})$. Put $f_1:=f,f_i:=M_i(f)$ for $i=2,3,4$. To check numerically whether $f_1,\ldots,f_4$ forms a basis is straightforward. For a fixed non-singular base point $t_0$ consider a matrix:
$$
R:=\left(f_i^{(j)}(t_0)\right)_{i,j=1\ldots,4}
$$
The Frobenius method implies that the map
$$
\mathcal{S}ol(\mathcal{P},t_0)\ni g\mapsto \begin{pmatrix}g(t_0)\\g'(t_0)\\g''(t_0)\\g^{(3)}(t_0)\end{pmatrix}\in\mathbb{C}^4
$$
is an isomorphism. Thus if $\det R\neq 0$, the elements $f_1,\ldots,f_4$ are linearly independent. Note that even though we can never proof that $\det R=0$ using numerical computations, in practice the condition $\det R\neq 0$ can be verified provided sufficient accuracy.

To sum up: we know that appropriate operators $M_2,M_3,M_4$ exist and if they are given, we can verify that they indeed satisfy the desired criterion. This still leaves a problem of actually \textit{finding} these monodromy operators. In general there seems to be no easy procedure to do that. For example, the Doran-Morgan basis can be seen as a specific solution to this problem, where one takes $f=f_c$ to be the conifold period, assumes the existence of a MUM point at $0$ and then puts $M_i:=(M_0-\Id)^{i-1}$ for $i=2,3,4$. But this does not work in general: there are numerous examples of operators with a conifold point and a MUM point at $0$ such that $(M_0-\Id)^3(f_c)=0$ (see \cite{Chmiel1}, \cite{Chmiel2}). Of course in the absence of a MUM point the situation is even more complicated.

For this reason we take an \textit{ad hoc} approach and for each considered operator find appropriate $M_2,M_3,M_4$ case-by-case. As for the initial solution, we always take $f$ to be the conifold period $f_c$ associated with some singular point $s$ of type $\tfrac{1}{n}C$. This choice is preferable for several reasons: as already mentioned, $f_c$ is determined both by the Frobenius method and the monodromy. Furthermore, it is possible to compute it for any $n$, unlike the solutions near singular points of type $\tfrac{1}{n}K$ or $F$. Another important point is that for every choice of $f$, we have $M^n_s(f)=f+\alpha f_c$ for some $\alpha\in\mathbb{C}$. Furthermore, if $f,f_c\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$, then $\alpha\in\mathbb{Q}$. Thus even if our initial solution was different than $f_c$, as soon as the local monodromy $M_s^n$ is a factor of some $M_i$, the conifold period appears anyway.

Results of this approach when applied to \emph{double octic orphan operators} are described in the next section.

\section{Monodromy groups of orphan operators}\label{sec:mongr}

In this section we present rational realizations of certain orphan monodromy groups, which were found using the method described in the previous section. These groups are the monodromy groups of Picard-Fuchs operators of one-parameter families of double octic Calabi-Yau threefolds.

A \textit{double octic} is a smooth Calabi-Yau threefold $\hat X$ obtained as a resolution of singularities of a double cover $X\rightarrow\mathbb{P}^3$ branched along a union of eight planes. A \textit{double octic operator} is a Picard-Fuchs operator of a one-parameter family of double octics $\hat X_t$ defined over $\mathbb{Q}$ and satisfying $h^{2,1}(\hat X_t)=1$. Double octics $\hat X$ defined over $\mathbb{Q}$ such that $h^{2,1}(\hat X)\leq 1$ were classified in \cite{Cynk-Kocel} and the Picard-Fuchs operators for those with $h^{2,1}(\hat X)=1$ are known.

Rational realizations of the monodromy groups for double octic operators possessing a MUM point were found in \cite{Chmiel2} using a generalization of the Doran-Morgan basis (see also \cite{Chmiel1}). The focus of the present paper is on double octic \textit{orphan} operators which do not have a point of maximal unipotent monodromy.

All of the double octic orphan operators can be found in \cite{Cynk-van Straten}. In total there are 25 examples. Among them there are 7 examples of order 2; thus for them the monodromy action is not irreducible and the method described above does not apply. In any case, these examples are less interesting from our point of view since for them the local system of solutions $\mathcal{S}ol(\mathcal{P})$ decomposes into a tensor product of a constant system and a well-understood system of rank $2$ \cite{GvG, Rohde}.

Thus there are 18 double octic orphan operators of order 4. As explained in \cite{Cynk-van Straten}, they correspond to the following families of arrangements of eight planes:
$$
\textbf{33,\ 35,\ 70,\ 71,\ 97,\ 98,\ 152,\ 153,\ 197,\ 198,\ 243,\ 247,\ 248,\ 250,\ 252,\ 258,\ 266,\ 273}
$$
We will label the Picard-Fuchs operators of these families with the same numbering.

It is natural to only consider operators up to some mild equivalence relation. The most obvious one is when an operator $\mathcal{P}$ can be transformed into $\mathcal{P}'$ using the M\"obius transformation of the parameter space. The other one is a \emph{shift of the local exponents} at a singular point. More precisely, let
$$
\mathcal{P}=\mathcal{P}(\Theta,t)=\Theta^4+p_3(t)\Theta^3+p_2(t)\Theta^2+p_1(t)\Theta+p_0(t),\quad p_i\in\mathbb{Q}(t)
$$
be a differential operator written using the logarithmic derivative $\Theta:=t\cdot\tfrac{\textnormal{d}}{\textnormal{d}t}$. Under the M\"obius transformation $t\mapsto\tfrac{at+b}{ct+d}=y(t)$ this derivative transforms as $\Theta_t\mapsto \Theta_{y(t)}$, and hence the operators $\mathcal{P}(\Theta,t)$ and $\mathcal{P}(\Theta_{y},y)$ are equivalent. Shifting of the exponents may be assumed to happen at $0$ and then it is simply the operation $\mathcal{P}(\Theta,t)\mapsto\mathcal{P}(\Theta+\sigma,t)$.

Note that these transformations do not affect the monodromy, or at least its image in $\mathrm{PGL}(4,\mathbb{C})$. Indeed, fix a basis of solutions $\{f_1\ldots,f_4\}$ of $\mathcal{P}$. If $\mathcal{Q}$ is related to $\mathcal{P}$ by the M\"obius transformation $t\mapsto y(t)$,  $\{f_1\circ y,\ldots,f_4\circ y\}$ is a basis of solutions of $\mathcal{Q}$ and clearly the local monodromies are the same. Similarly, if we shift exponents at $0$, the new basis is $\{t^\sigma\cdot f_1,\ldots,t^\sigma\cdot f_4\}$ and the only change in the generators of the monodromy group is $M_0\mapsto e^{2\pi i\sigma}M_0$. Hence for a classification of the monodromy groups it is enough to consider only one operator from each equivalence class of the described relation.

For 18 double octic orphan operators we have the following equivalence classes, with a fixed representative typesetted in boldface:
$$
\begin{tabular}{l}
\textbf{33},\ 70 \\
\textbf{35},\ 71,\ 247,\ 252 \\
\textbf{97},\ 98 \\
\textbf{152},\ 198 \\
\textbf{153},\ 197 \\
\textbf{243} \\
\textbf{248} \\
\textbf{250},\ 258 \\
\textbf{266},\ 273 \\
\end{tabular}
$$
Hence from now on we will only consider the nine operators from the first column. We begin by listing these operators together with their \textit{Riemann symbols}. The Riemann symbol of an operator consists of the list of its singular points together with their \textit{local exponents}. The local exponents determine the type of the singular point, in particular the Jordan form of the local monodromy. For Picard-Fuchs operators of families of Calabi-Yau threefolds we have the following possibilities with $\alpha,\beta,\gamma,\delta\in\mathbb{Q}$:

\begin{center}
	\begin{tabular}{|c|c|}
		\hline
		Local exponents&Jordan form of local monodromy\\
		\hline
		$(\alpha,\alpha,\alpha,\alpha)$&
		$\begin{pmatrix}
		e^{2\pi i \alpha}&1&0&0\\0&e^{2\pi i \alpha}&1&0\\0&0&e^{2\pi i \alpha}&1\\0&0&0&e^{2\pi i \alpha}\\
		\end{pmatrix}$\\
		\hline
		$(\alpha,\alpha,\beta,\beta)\quad \alpha<\beta$&
		$\begin{pmatrix}
		e^{2\pi i \alpha}&1&0&0\\0&e^{2\pi i \alpha}&0&0\\0&0&e^{2\pi i \beta}&1\\0&0&0&e^{2\pi i \beta}\\
		\end{pmatrix}$\\
		\hline
		$(\alpha,\beta,\beta,\gamma)\quad\alpha<\beta<\gamma,\ \alpha+\gamma=2\beta$&
		$\begin{pmatrix}
		e^{2\pi i \alpha}&0&0&0\\0&e^{2\pi i \beta}&1&0\\0&0&e^{2\pi i \beta}&0\\0&0&0&e^{2\pi i \gamma}\\
		\end{pmatrix}$\\
		\hline
		$(\alpha,\beta,\gamma,\delta)\quad\alpha<\beta<\gamma<\delta,\ \alpha+\delta=\beta+\delta$&
		$\begin{pmatrix}
		e^{2\pi i \alpha}&0&0&0\\0&e^{2\pi i \beta}&0&0\\0&0&e^{2\pi i \gamma}&0\\0&0&0&e^{2\pi i \delta}\\
		\end{pmatrix}$\\
		\hline
	\end{tabular}
\end{center}
Since the local exponents are rational, local monodromies are always quasi-unipotent, i.e. for all local monodromy matrices $M$ there exist $n\in\mathbb{N}$ such that $(M^n-\operatorname{Id})^4=0$.

Note that by shifting exponents we may assume that $\alpha=0$ for all singular points $s\in\Sigma$ of the operator $\mathcal{P}$, expect possibly one. Indeed, it is enough to normalize all finite singular points $s\in\Sigma\setminus\{\infty\}$ by first moving them to $0$ via a linear map, shifting local exponents and then applying the inverse transformation; this only changes local exponents at $0$ and $\infty$. After performing this procedure for all elements of the set $\Sigma\setminus\{\infty\}$, we end up with an operator whose all finite singular points have the smallest local exponent equal to $0$.

\bigskip

\begin{center}
\textbf{Double octic orphan operators}
\end{center}

For the reader's convenience we now present the list of all considered Picard-Fuchs operators together with their Riemann symbols. 

\begin{small}
\medskip

Operator \textbf{33}

\medskip

\(\displaystyle {\Theta}^{2} \left( \Theta-1 \right) ^{2}
-\frac1{8}\,t{\Theta}^{2} \left( 20\,{\Theta}^{2}+3 \right) 
+\frac1{16}\,{t}^{2} \left( 8\,{\Theta}^{2}+8\,\Theta+3 \right)  \left( 2\,\Theta+1 \right) ^{2}
-\frac{1}{32}\,{t}^{3} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1\right) ^{2}
\)

\begin{center}
\(\displaystyle
\left\{ \begin {array}{cccc}   0 & 1&   2& \infty \\\hline
0&0&0&1/2\\
0&1/2&1&1/2\\
1&1/2&1&3/2\\
1&1&2&3/2
\end {array} \right\} \)
\end{center}

Operator \textbf{35}

\medskip

\(\Theta\, \left( \Theta-1 \right)  \left(\Theta-\frac12 \right) ^{2}
-\frac14\,t{\Theta}^{2} \left( 4\,{\Theta}^{2}+3 \right) 
-\frac14\,{t}^{2} \left( {\Theta}^{2}+\Theta+1 \right)  \left( 2\,\Theta+1 \right) ^{2}
+\frac14\,{t}^{3} \left( 2\,\Theta+3 \right)  \left( 2\,\Theta+1 \right)  \left( \Theta+1 \right) ^{2}
\)

\begin{center}
\(\left\{ \begin {array}{cccc}1&0&-1&\infty  \\
\hline 0&0&0&1/2\\
0&1/2&1&1\\
1&1/2&1&1\\
1&1&2&3/2
\end {array} \right\} \)
\end{center}

Operator \textbf{97}

\medskip

\(\Theta\, \left( \Theta-1 \right)  \left( \Theta -\frac 12\right) ^{2}
+\frac18\,t{\Theta}^{2} \left( 20\,{\Theta}^{2}+7 \right) \
+\frac1{32}\,{t}^{2} \left( 16\,{\Theta}^{2}+16\,\Theta+7 \right)  \left( 2\,\Theta+1 \right) ^{2}
+\frac1{32}\,{t}^{3} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}
\)

\begin{center}
\(\left\{
\begin {array}{cccc}0&-1&-2&\infty  \\
\hline
0&0&0&1/2\\
1/2&0&1&1/2\\
1/2&1&1&3/2\\
1&1&2&3/2
\end {array} \right\} \)
\end{center}

Operator \textbf{152}

\medskip

\(
\Theta\, \left( \Theta-1 \right)  \left( \Theta -\frac 12 \right) ^{2}
+\frac12\,t\Theta\, \left( 2\,{\Theta}^{3}-8\,{\Theta}^{2}+6\,\Theta-1 \right)
+{t}^{2} \left( -2\,{\Theta}^{4}-4\,{\Theta}^{3}-11/4\,{\Theta}^{2}-{\frac {17}{4}}\,\Theta-{\frac {11}{16}} \right)\)
 
\ \ \(+{t}^{3} \left( -2\,{\Theta}^{4}+1/4\,{\Theta}^{2}+7/2\,\Theta+{\frac {9}{8}} \right) 
+\frac1{16}\,{t}^{4} \left( 2\,\Theta+1 \right)  \left( 8\,{\Theta}^{3}+44\,{\Theta}^{2}+62\,\Theta+25 \right) \)

\ \ \ \ \(+\frac14\,{t}^{5} \left( 2\,\Theta+3 \right)  \left( 2\,\Theta+1 \right)  \left( \Theta+1 \right) ^{2}
\)

\begin{center}
\(\left\{
\begin {array}{cccc}
1&0&-1&\infty\\\hline
0&0&0&1/2\\
1/2&1/2&0&1\\
1/2&1/2&2&1\\
1&1&2&3/2
\end {array} \right\} \)
\end{center}

Operator \textbf{153}

\medskip

\(\Theta\, \left( \Theta-1 \right)  \left(\Theta -\frac12\right) ^{2}
+\frac18\,t\Theta\, \left( 28\,{\Theta}^{3}-16\,{\Theta}^{2}+17\,\Theta-2 \right)
+{t}^{2} \left( {\frac {19}{4}}\,{\Theta}^{4}+7/2\,{\Theta}^{3}+{\frac {39}{8}}\,{\Theta}^{2}+{\frac {13}{8}}\,\Theta+{\frac {19}{64}} \right) \)

\ \ \(+{t}^{3} \left( {\frac {25}{8}}\,{\Theta}^{4}+6\,{\Theta}^{3}+{\frac {109}{16}}\,{\Theta}^{2}+7/2\,\Theta+{\frac {89}{128}} \right) 
+{\frac {1}{64}}\,{t}^{4} \left( 2\,\Theta+1 \right)  \left( 32\,{\Theta}^{3}+80\,{\Theta}^{2}+82\,\Theta+29 \right) \)

\ \ \ \ \(+\frac1{32}\,{t}^{5} \left( 2\,\Theta+3 \right)  \left( 2\,\Theta+1 \right)  \left( \Theta+1 \right) ^{2}
\)

\begin{center}
\(\left\{
\begin {array}{cccc}0&-1&-2&\infty  \\\hline
0&0&0&1/2\\
1/2&1/2&1/2&1\\
1/2&1/2&3/2&1\\
1&1&2&3/2
\end {array} \right\} \)
\end{center}

Operator \textbf{243}

\medskip

\(\Theta\, \left( \Theta-2 \right)  \left( \Theta-1 \right) ^{2}
-\frac16\,t\Theta\, \left( \Theta-1 \right)  \left( 19\,{\Theta}^{2}-19\,\Theta+9 \right) 
+\frac13\,{t}^{2}{\Theta}^{2} \left( 11\,{\Theta}^{2}+4 \right)  \)

\ \ \(-\frac1{24}\,{t}^{3} \left( 11\,{\Theta}^{2}+11\,\Theta+5 \right)
\left( 2\,\Theta+1 \right) ^{2}+\frac1{48}\,{t}^{4} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}
\)

\begin{center}
\(\left\{
\begin {array}{ccccc}2&3/2&1&0&\infty\\\hline
0&0&0&0&1/2\\
1&1&1/2&1&1/2\\
1&1&1/2&1&3/2\\
2&2&1&2&3/2
\end {array} \right\} \)
\end{center}

Operator \textbf{248}

\medskip

\(\Theta\, \left( \Theta-2 \right)  \left( \Theta-1 \right) ^{2}
+\frac16\,t\Theta\, \left( \Theta-1 \right)  \left( 37\,{\Theta}^{2}-61\,\Theta+36 \right)
+\frac16\,{t}^{2}\Theta\, \left( 91\,{\Theta}^{3}-124\,{\Theta}^{2}+121\,\Theta-36 \right)  \)


\ \ \(+{t}^{3} \left( {\frac {115}{6}}\,{\Theta}^{4}-\frac53\,{\Theta}^{3}+{\frac {107}{6}}\,{\Theta}^{2}+\frac23\,\Theta+\frac12 \right) 
+{t}^{4} \left( {\frac {79}{6}}\,{\Theta}^{4}+16\,{\Theta}^{3}+{\frac
	{113}{6}}\,{\Theta}^{2}+8\,\Theta+\frac32 \right) \)

\ \ \ \ \(+\frac16\,{t}^{5} \left( 2\,\Theta+1 \right)  \left(
14\,{\Theta}^{3}+29\,{\Theta}^{2}+27\,\Theta+9 \right) 
+\frac16\,{t}^{6} \left( 2\,\Theta+3 \right)  \left( 2\,\Theta+1
\right)  \left( \Theta+1 \right) ^{2} 
\)

\begin{center}
\(\left\{
\begin {array}{cccccc} 0&-1/2&-1&-3/2&-2&\infty\\\hline
0&0&0&0&0&1/2\\
1&1&0&1&1&1\\
1&1&2&1&1&1\\
2&2&2&2&2&3/2
\end {array} \right\} \)
\end{center}

Operator \textbf{250}

\medskip

\(\Theta\, \left( \Theta-1 \right)  \left( \Theta -\frac12\right) ^{2}
+\frac18\,t\Theta\, \left( 44\,{\Theta}^{3}-96\,{\Theta}^{2}+65\,\Theta-12 \right) 
+{t}^{2} \left( \frac{19}2\,{\Theta}^{4}-23\,{\Theta}^{3}+{\frac {131}{8}}\,{\Theta}^{2}-{\frac {47}{8}}\,\Theta-\frac14 \right) \)


\ \ \(+{t}^{3} \left( \frac52\,{\Theta}^{4}-20\,{\Theta}^{3}-{\frac
	{23}{4}}\,\Theta-{\frac {17}{32}} \right)
-\frac1{32}\,{t}^{4} \left( 68\,{\Theta}^{2}+100\,\Theta+53 \right)  \left( 2\,\Theta+1 \right) ^{2}\)

\ \ \ \ \(-\frac14\,{t}^{5} \left( 8\,{\Theta}^{2}+14\,\Theta+9 \right)  \left( 2\,\Theta+1 \right) ^{2}
-\frac18\,{t}^{6} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}
\)


\begin{center}
\(\left\{
\begin {array}{cccccc}1&0&-1/2&-1&-2&\infty  \\\hline
0&0&0&0&0&1/2\\
1&1/2&1&1/2&1&1/2\\
1&1/2&3&1/2&1&3/2\\
2&1&4&1&2&3/2
\end {array} \right\} \)
\end{center}

Operator \textbf{266}

\medskip

\(\Theta\, \left( \Theta-1 \right)  \left(\Theta -\frac12\right) ^{2}
+\frac14\,t\Theta\, \left( 44\,{\Theta}^{3}-48\,{\Theta}^{2}+37\,\Theta-6 \right) 
+{t}^{2} \left( 50\,{\Theta}^{4}-56\,{\Theta}^{3}+40\,{\Theta}^{2}-\frac52\,\Theta+\frac38 \right)\)

\ \ \(+{t}^{3} \left( 120\,{\Theta}^{4}-288\,{\Theta}^{3}-75\,{\Theta}^{2}-105\,\Theta-21 \right) 
+{t}^{4} \left( 112\,{\Theta}^{4}-1008\,{\Theta}^{3}-718\,{\Theta}^{2}-720\,\Theta-{\frac {303}{2}} \right)\)
 
\ \ \ \ \(+{t}^{5} \left( -224\,{\Theta}^{4}-2464\,{\Theta}^{3}-1924\,{\Theta}^{2}-1628\,\Theta-324 \right) 
+{t}^{6} \left( -960\,{\Theta}^{4}-4224\,{\Theta}^{3}-4296\,{\Theta}^{2}-2448\,\Theta-450 \right)\)
 
\ \ \ \ \ \ \(+{t}^{7} \left( -1600\,{\Theta}^{4}-4992\,{\Theta}^{3}-6368\,{\Theta}^{2}-3504\,\Theta-696 \right)
-32\,{t}^{8} \left( 2\,\Theta+1 \right)  \left( 22\,{\Theta}^{3}+57\,{\Theta}^{2}+59\,\Theta+21 \right) \)

\ \ \ \ \ \ \ \ \(-128\,{t}^{9} \left( 2\,\Theta+3 \right)  \left( 2\,\Theta+1 \right)  \left( \Theta+1 \right) ^{2}
\)

\begin{center}
\(\left\{
\begin {array}{cccccccc}
1/2&0&-1/4&-1/2&-1&(-1+\sqrt{-3})/4&(-1-\sqrt{-3})/4&\infty\\\hline
0&0&0&0&0&0&0&1/2\\
1&1/2&1&1/2&1&1&1&1\\
1&1/2&1&1/2&1&3&3&1\\
2&1&2&1&2&4&4&3/2
\end {array} \right\} \)
\end{center}

\end{small}

\bigskip

\begin{center}
\textbf{Monodromy groups of double octic orphan operators}
\end{center}

We now present rational realizations of the monodromy groups for all the double octic orphan operators listed above. Rational basis were found using the procedure from the previous section. More precisely, except for the operator \textbf{266}, the basis is always of the form
$$\big\{f_c,N_1(f_c),N_2(N_1(f_c)),N_3(N_2(N_1(f_c)))\big\},$$
where $N_i:=M_i-\operatorname{Id}$ for certain monodromy operators $M_i$ and $f_c$ is the conifold period associated with some singular point $s_0$ of type $\tfrac{1}{n}$C, i.e. $\operatorname{im}(M_{s_0}^n-\operatorname{Id})=\mathbb{C}\cdot f_c$.

Let $\mathcal{P}$ be a double octic orphan operator. For every finite singular point $s$ of the operator $\mathcal{P}$ the list contains the local monodromy generator $M_s$ of $Mon(\mathcal{P})$ written in certain rational basis $\mathcal{B}$. The list also presents the basis $\mathcal{B}$ written in the format $[s_0;M_1,M_2,M_3]$, where $s_0$ is a singular point of type $\tfrac{1}{n}C$ and $M_i\in Mon(\mathcal{P})$ is given as a product of the listed generators.

\medskip

\textbf{Operator 33}

\medskip

$[1;M_0,M_2,M_0]$

\medskip

$
M_{1}=
\begin{pmatrix}
-1&-4&-24&16\\
0&2&2&4\\
0&-1&-1&-4\\
0&-\tfrac{1}{4}&-\tfrac{1}{2}&0\\
\end{pmatrix}
$
$
M_{0}=
\begin{pmatrix}
1&0&0&0\\
1&1&0&0\\
0&0&1&0\\
0&0&1&1\\
\end{pmatrix}
$
$
M_{2}=
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
-\tfrac{1}{2}&1&1&12\\
0&0&0&1\\
\end{pmatrix}
$

\medskip

\textbf{Operator 35}

\medskip

$[0;M_1,M_{-1},M_1]$

\medskip

$
M_{0}=
\begin{pmatrix}
-1&-24&-32&-512\\
0&3&-8&32\\
0&\tfrac{1}{2}&-1&8\\
0&-\tfrac{1}{8}&\tfrac{1}{2}&-1\\
\end{pmatrix}
$
$
M_{1}=
\begin{pmatrix}
1&0&0&0\\
1&1&0&0\\
0&0&1&0\\
0&0&1&1\\
\end{pmatrix}
$
$
M_{-1}=
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
\tfrac{1}{4}&1&1&32\\
0&0&0&1\\
\end{pmatrix}
$

\medskip

\textbf{Operator 97}

\medskip

$[-2;M_{-1},M_{0},M_{-1}]$

\medskip

$
M_{0}=
\begin{pmatrix}
-3&0&0&-32\\
2&1&0&16\\
-\tfrac{1}{2}&1&-1&-8\\
\tfrac{1}{4}&0&0&3\\
\end{pmatrix}
$
$
M_{-2}=
\begin{pmatrix}
1&4&-16&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix}
$
$
M_{-1}=
\begin{pmatrix}
1&0&0&0\\
1&1&0&0\\
0&0&1&0\\
0&0&1&1\\
\end{pmatrix}
$

\medskip

\textbf{Operator 152}

\medskip

$[0;M_{-1},M_{1},M_{-1}]$

\medskip

$
M_{0}=
\begin{pmatrix}
-1&-4&32&-64\\
0&1&0&0\\
0&\tfrac{1}{2}&-1&0\\
0&\tfrac{1}{8}&-\tfrac{1}{2}&1\\
\end{pmatrix}
$
$
M_{1}=
\begin{pmatrix}
-3&0&0&-128\\
1&1&0&32\\
-\tfrac{1}{4}&1&-1&-16\\
\tfrac{1}{16}&0&0&3\\
\end{pmatrix}
$
$
M_{-1}=
\begin{pmatrix}
1&0&0&0\\
1&1&0&0\\
0&0&1&0\\
0&0&1&1\\
\end{pmatrix}
$

\medskip

\textbf{Operator 153}

\medskip

$[-1;M_{0},M_{-2},M_{0}]$

\medskip

$
M_{-1}=
\begin{pmatrix}
1&0&0&0\\
1&1&0&32\\
0&0&1&0\\
0&-\tfrac{1}{8}&1&-3\\
\end{pmatrix}
$
$
M_{0}=
\begin{pmatrix}
-1&-68&448&-2368\\
0&-\tfrac{91}{3}&\tfrac{470}{3}&-\tfrac{3008}{3}\\
0&\tfrac{4}{3}&-\tfrac{17}{3}&\tfrac{128}{3}\\
0&\tfrac{9}{8}&-\tfrac{45}{3}&37\\
\end{pmatrix}
$
$
M_{-2}=
\begin{pmatrix}
-9&0&0&-512\\
-\tfrac{25}{6}&1&0&-\tfrac{640}{3}\\
\tfrac{1}{6}&1&-1&\tfrac{112}{3}\\
\tfrac{5}{32}&0&0&9\\
\end{pmatrix}
$

\medskip

\textbf{Operator 243}

\medskip

$[1;M_{\infty}^2,M_{\frac{3}{2}},M_{\infty}^2]$

\medskip

$
M_{1}=
\begin{pmatrix}
-1&-204&360&-14880\\
0&\tfrac{43}{4}&-\tfrac{65}{2}&650\\
0&\tfrac{9}{10}&-2&60\\
0&-\tfrac{21}{160}&\tfrac{7}{16}&-\tfrac{31}{4}\\
\end{pmatrix}
$
$
M_{2}=
\begin{pmatrix}
-2&-12&-120&-1440\\
-\tfrac{9}{16}&-\tfrac{5}{4}&-\tfrac{45}{2}&-270\\
\tfrac{3}{49}&\tfrac{3}{10}&4&36\\
\tfrac{3}{640}&\tfrac{3}{160}&\tfrac{3}{16}&\tfrac{13}{4}\\
\end{pmatrix}
$
$
M_{\frac{3}{2}}=
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
\tfrac{1}{20}&1&1&88\\
0&0&0&1\\
\end{pmatrix}
$
$
M_{\infty}=
\begin{pmatrix}
-1&0&0&0\\
-\tfrac{1}{2}&-1&0&0\\
0&0&-1&0\\
0&0&-\tfrac{1}{2}&-1\\
\end{pmatrix}
$

\medskip

\textbf{Operator 248}

\medskip

$[-\frac{1}{2};M_{-1},M_{0},M_{-1}]$

\medskip

$
M_{0}=
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
-\tfrac{1}{2}&1&1&16\\
0&0&0&1\\
\end{pmatrix}
$
$
M_{-\tfrac{1}{2}}=
\begin{pmatrix}
1&4&16&32\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix}
$
$
M_{-1}=
\begin{pmatrix}
1&0&0&0\\
1&1&0&0\\
0&0&1&0\\
0&0&1&1\\
\end{pmatrix}
$
$
M_{-\tfrac{3}{2}}=
\begin{pmatrix}
3&4&32&32\\
-1&-1&-16&-16\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix}
$
$
M_{-2}=
\begin{pmatrix}
17&16&128&0\\
-8&-7&-64&0\\
-1&-1&-7&0\\
\tfrac{1}{2}&\tfrac{1}{2}&4&1\\
\end{pmatrix}
$

\medskip

\textbf{Operator 250}

\medskip

$[1;M_{\infty},M_{-2},M_{\infty}]$

\medskip

$
M_{1}=
\begin{pmatrix}
1&-8&64&-128\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix}
$
$
M_{\infty}=
\begin{pmatrix}
1&-4&0&0\\
1&-3&0&0\\
0&0&1&-4\\
0&0&1&-3\\
\end{pmatrix}
$
$
M_{-1}=
\begin{pmatrix}
-1&24&-80&224\\
0&7&-24&80\\
\tfrac{3}{8}&\tfrac{3}{4}&-5&28\\
\tfrac{1}{16}&-\tfrac{3}{8}&1&-1\\
\end{pmatrix}
$
$
M_{-2}=
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
-\tfrac{1}{2}&1&1&-8\\
0&0&0&1\\
\end{pmatrix}
$

\medskip

\textbf{Operator 266}

\medskip

The rational basis is $\{f_c,(M_{-\frac{1}{2}}-\operatorname{Id})(f_c),(M_{-\frac{1}{4}}-\operatorname{Id})(f_c),(M_{0}-\operatorname{Id})(f_c)\}$; $f_c$ is the conifold period at $-1$.

\medskip

$
M_{-\tfrac{1}{4}}=
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
1&-6&1&-3\\
0&0&0&1\\
\end{pmatrix}
$
$
M_{-\tfrac{1}{2}}=
\begin{pmatrix}
1&-\tfrac{12}{5}&\tfrac{24}{5}&\tfrac{12}{5}\\
1&-\tfrac{11}{5}&-\tfrac{8}{5}&-\tfrac{24}{5}\\
0&\tfrac{3}{5}&\tfrac{1}{5}&-\tfrac{3}{5}\\
0&-\tfrac{2}{5}&\tfrac{4}{5}&\tfrac{7}{5}\\
\end{pmatrix}
$
$
M_{-1}=
\begin{pmatrix}
1&12&-16&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix}
$
$
M_{0}=
\begin{pmatrix}
1&\tfrac{96}{5}&-\tfrac{48}{5}&0\\
0&\tfrac{13}{5}&-\tfrac{4}{5}&0\\
0&\tfrac{36}{5}&-\tfrac{13}{5}&0\\
1&\tfrac{16}{5}&-\tfrac{28}{5}&-1\\
\end{pmatrix}
$
$
M_{\tfrac{1}{2}}=
\begin{pmatrix}
5&-12&-16&-12\\
0&1&0&0\\
1&-3&-3&-3\\
0&0&0&1\\
\end{pmatrix}
$

\bigskip

\section{Maximal unipotent monodromy for orphans}\label{sec:MUMor}

The results presented in the previous section realize monodromy groups of orphans as subgroups of $\mathrm{GL}(4,\mathbb{Q})$. However, for Picard-Fuchs operators of families of Calabi-Yau threefolds, one can do much better. Recall that there is an isomorphism $\mathcal{S}ol(\mathcal{P},t_0)\simeq H^3(X_{t_0},\mathbb{C})$. We have previously used this isomorphism to infer the existence of a rational, monodromy-invariant subspace $\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)\subset\mathcal{S}ol(\mathcal{P},t_0)$ such that $\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)\otimes\mathbb{C}=\mathcal{S}ol(\mathcal{P},t_0)$ but of course much more is true. The inclusion $H^3(X_{t_0},\mathbb{Z})\subset H^3(X_{t_0},\mathbb{C})$ implies the existence of a monodromy-invariant lattice $\mathcal{S}ol_\mathbb{Z}(\mathcal{P},t_0)\subset\mathcal{S}ol(\mathcal{P},t_0)$. Furthermore, the cup product
$$
\cup:H^3(X_{t_0},\mathbb{Z})\times H^3(X_{t_0},\mathbb{Z})\rightarrow H^6(X_{t_0},\mathbb{Z})\simeq\mathbb{Z}
$$
provides a skew-symmetric, bilinear and non-degenerate form on $H^3(X_{t_0},\mathbb{Z})$ and hence on $\mathcal{S}ol_\mathbb{Z}(\mathcal{P},t_0)$. It follows that there exists a \textit{symplectic} basis of  $\mathcal{S}ol(\mathcal{P},t_0)$, i.e. a basis $\mathcal{B}$ such that $Mon^\mathcal{B}(\mathcal{P})\subset \mathrm{Sp}(4,\mathbb{Z})$.

In the case of hypergeometric operators the group $Mon(\mathcal{P})$ is generated by local monodromies around a MUM point and a conifold point. The matrices of these monodromies in the Doran-Morgan basis are respectively
$$
\begin{pmatrix}
1&0&0&0\\
1&1&0&0\\
0&d&1&0\\
0&0&1&1\\
\end{pmatrix}\quad\textnormal{ and }\quad
\begin{pmatrix}
1&-k&-1&-1\\
0&1&0&0&\\
0&0&1&0&\\
0&0&0&1&\\
\end{pmatrix}
$$
for some $d,k\in\mathbb{Z}$. They can be brought to the symplectic group $\mathrm{Sp}(4,\mathbb{Z})$ by a change of basis given by the matrix
$$
\begin{pmatrix}
0&-1&0&0\\
-1&0&0&0&\\
0&-d&-1&0&\\
0&k&1&1&\\
\end{pmatrix}
$$
Then the corresponding generators of the monodromy group are
$$
\begin{pmatrix}
1&1&0&0\\
0&1&0&0\\
d&d&1&0\\
0&-k&-1&1\\
\end{pmatrix}\quad\textnormal{ and }\quad
\begin{pmatrix}
1&0&0&0\\
0&1&0&1\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix}
$$
Note that in this basis monodromy around a conifold singularity is a symplectic reflection.

A similar procedure can be applied to any Picard-Fuchs operator with a MUM point and a singular point of type $\tfrac{1}{n}C$, although it usually realizes the monodromy group only as a subgroup of $\mathrm{Sp}(4,\mathbb{Q})$. This is simply because a generalized Doran-Morgan basis spans $\mathcal{S}ol_\mathbb{Q}(\mathcal{P},t_0)$ but not $\mathcal{S}ol_\mathbb{Z}(\mathcal{P},t_0)$ in general. However, in many examples one can obtain a symplectic basis after a simple modification of the generating set (see \cite{Chmiel2}).

Once the symplectic basis $\mathcal{B}$ is found, one can ask arithmetical questions about the monodromy group $Mon^\mathcal{B}(\mathcal{P})$ as a subgroup of $\mathrm{Sp}(4,\mathbb{Z})$. The most natural one is whether the index $[\mathrm{Sp}(4,\mathbb{Z}):Mon^\mathcal{B}(\mathcal{P})]$ is finite, or in other words whether $Mon^\mathcal{B}(\mathcal{P})$ is \textit{arithmetic}.

\begin{definition}
Let $G$ be a finitely generated, Zariski-dense subgroup of $\mathrm{Sp}(4,\mathbb{Z})$. Then $G$ is called \emph{arithmetic}, if the index $[\mathrm{Sp}(4,\mathbb{Z}):G]$ is finite. Otherwise it is called \emph{thin}.
\end{definition}

\begin{remark}
Note that if $G\subset\mathrm{Sp}(4,\mathbb{Z})$ is a subgroup such that $[\mathrm{Sp}(4,\mathbb{Z}):\overline G]<+\infty$, then $\overline{G}=\mathrm{Sp}(4,\mathbb{Z})$. Indeed, in this case the closure $\overline{G}$ is a Zariski-clopen subgroup of $\mathrm{Sp}(4,\mathbb{Z})$. In particular, the group $G$ is Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$. Consequently, if $G$ is of finite index, it is also arithmetic.
\end{remark}

Arithmeticity question is very difficult even for hypergeometric operators. There are $14$ hypergeometric operators and the monodromy group of each of them, in the basis described above, depends only on two parameters $(d,k)\in\mathbb{N}^2$. It turns out that the monodromy groups of hypergeometric operators are arithmetic for $(d,k)=$ (1,2), (1,3), (2,3), (3,4), (4,4), (6,5), (9,6) but thin for the remaining pairs $(d,k)=$ (1,4), (2,4), (4,5), (5,5), (8,6), (12,7), (16,8) (see \cite{Brav-Thomas,Singh,Singh-Venkataramana}). Hence already in this simplest possible case there is no easy answer for the question of arithmeticity.

One could expect that the monodromy groups of \textit{orphan} operators always have infinite index. A very basic motivation for such a claim could be the following observation:

\begin{observation}\label{ob:index_MUM}
Let $G\subset \mathrm{Sp}(4,\mathbb{Z})$ be a subgroup which does not contain a maximally unipotent matrix. Then the index $[\mathrm{Sp}(4,\mathbb{Z}):G]$ is infinite.
\end{observation} 

\begin{proof}
Let
$$
M:=\begin{pmatrix}
1&1&0&0\\
0&1&1&0\\
0&0&1&1\\
0&0&0&1\\
\end{pmatrix}
$$
be the Jordan form of a maximally unipotent matrix. Then for all $k\in\mathbb{N}$ we have
$$
M^k=\begin{pmatrix}
1&k&\binom{k}{2}&\binom{k}{3}\\
0&1&k&\binom{k}{2}\\
0&0&1&k\\
0&0&0&1\\
\end{pmatrix}
$$
It follows that a power of a maximally unipotent matrix is again maximally unipotent. If $A$ is any maximally unipotent element of $\mathrm{Sp}(4,\mathbb{Z})$, we conclude that $A^k\not\in G$ for all $k\in\mathbb{N}$. Thus the cosets $A^kG$ are all different.
\end{proof}	

\noindent Since by definition orphan operators do not have a MUM point, one could expect that their monodromy groups do not contain \textit{any} maximally unipotent elements. If that was the case, one could deduce that their index in $\mathrm{Sp}(4,\mathbb{Z})$ is infinite. Note that this approach does not refer to any fixed integral symplectic realization of the monodromy group and hence does not require finding a basis $\mathcal{B}$ such that $Mon^\mathcal{B}(\mathcal{P})\subset\mathrm{Sp}(4,\mathbb{Z})$.

Contrary to this expectation we have the following surprising result:

\begin{theorem}\label{th:MUM}
Let $\mathcal{P}$ be a double octic orphan operator, other than operator \textbf{35}. The monodromy group $Mon(\mathcal{P})$ contains a maximally unipotent element.
\end{theorem} 

\begin{proof}
Using MAPLE we have carried out a random search through products of (at most ten) generators of the monodromy groups and their inverses. In every case expect \textbf{35} we have discovered that one of these products has the minimal polynomial $(X-\Id)^4$ and hence is maximally unipotent. In the case of operator \textbf{35} even a more extensive search did not produce a maximally unipotent matrix.

Keeping the notation from the previous section, we have found the following maximally unipotent elements in the orphan monodromy groups:

\medskip

\textbf{Operator 33}

$\left(M_2\cdot M_0\cdot M_1^{-1}\right)^{-2}=
\begin{pmatrix}
81&-184&240&-2848\\
34&-77&92&-1128\\
-23&52&-63&768\\
-2&\tfrac{9}{2}&-5&63\\
\end{pmatrix}
$

\textbf{Operator 97}

$
\left(M_{-2}\cdot M_{0}^{-1}\cdot M_{-2}^{-1}\cdot M_{-1}^{-1}\cdot M_{-2}\cdot M_{-2}\cdot M_0\right)^{-1}=
\begin{pmatrix}
157&-772&2704&1920\\
-36&181&-624&-448\\
-10&52&-175&-128\\
-13&64&-224&-159\\
\end{pmatrix}
$

\textbf{Operator 152}

$
\left(M_{1}^{-1} \cdot M_{-1}^{1} \cdot M_{-1}^{-1} \cdot M_{1} \cdot M_{1} \cdot M_{0} \cdot M_{0} \cdot M_{-1}\right)^{-1}=
\begin{pmatrix}
-1031&-8&-992&-32768\\
1032&9&992&32768\\
-32&0&-31&-1024\\
33&0&32&1057\\
\end{pmatrix}
$

\textbf{Operator 153}

$
\left(M_{-1}^{-1} \cdot M_{0} \cdot M_{-2} \cdot M_{0}^{-1} \cdot M_{0}^{-1} \cdot M_{-1}^{-1} \cdot M_{0}^{-1} \cdot M_{-2}\right)^{-1}=
\begin{pmatrix}
2673&-14016&75072&-315392\\
\tfrac{3496}{3}&-5759&32576&-129024\\
-\tfrac{196}{3}&352&-1839&7936\\
-\tfrac{89}{2}&220&-1244&4929\\
\end{pmatrix}
$

\textbf{Operator 243}

$
M_{\infty} \cdot M_{2}^{-1} \cdot M_{1} \cdot M_{1} \cdot M_{\infty}^{-1} \cdot M_{1}^{-1} \cdot M_{1}^{-1}=
\begin{pmatrix}
-578&167628&-597720&11175840\\
-\tfrac{5937}{16}&\tfrac{430453}{4}&-\tfrac{767385}{2}&7174590\\
\tfrac{87}{8}&-\tfrac{31539}{10}&11248&-210276\\
\tfrac{783}{128}&-\tfrac{283851}{160}&\tfrac{101223}{16}&-\tfrac{473117}{4}\\
\end{pmatrix}
$

\textbf{Operator 248}

$
\left(M_{-\frac{3}{2}}^{-1} \cdot M_{0} \cdot M_{-\frac{1}{2}} \cdot M_{-\frac{3}{2}}^{-1}\right)^{-1}=
\begin{pmatrix}
-23&-28&-304&-96\\
-6&-7&-80&-64\\
\tfrac{5}{2}&3&33&16\\
0&0&0&1\\
\end{pmatrix}
$

\textbf{Operator 250}

$
M_{-1}^{-1} \cdot M_{-2}^{-1} \cdot M_{-2}^{-1} \cdot M_{-1}^{-1} \cdot M_{1}=
\begin{pmatrix}
-175&1592&-10560&16256\\
-28&257&-1696&2624\\
\tfrac{9}{2}&-40&269&-408\\
\tfrac{15}{4}&-34&226&-347\\
\end{pmatrix}
$

\textbf{Operator 266}

$
M_{-\frac{1}{4}} \cdot M_{0} \cdot M_{\frac{1}{2}} \cdot M_{-\frac{1}{4}} \cdot M_{-\frac{1}{2}}=
\begin{pmatrix}
-\tfrac{2927}{5}&624&\tfrac{19208}{5}&\tfrac{23544}{5}\\
-\tfrac{61}{5}&13&\tfrac{404}{5}&\tfrac{492}{5}\\
-\tfrac{197}{5}&42&\tfrac{1293}{5}&\tfrac{1584}{5}\\
-\tfrac{197}{5}&42&\tfrac{1288}{5}&\tfrac{1589}{5}\\
\end{pmatrix}
$
\end{proof}

\begin{remark}
The usual construction of the Doran-Morgan basis can be applied to operators with a point of maximal unipotent monodromy and a conifold point. However, a straightforward generalization of this construction from \cite{Chmiel1} requires only \emph{some} maximally unipotent monodromy operator, not necessarily a local monodromy one. Therefore Theorem \ref{th:MUM} shows that for double octic orphan operators, other than \textbf{35}, one can construct a generalized Doran-Morgan basis, despite the lack of a MUM singularity.
\end{remark}

Theorem \ref{th:MUM} shows that we cannot (with a single possible exception of operator \textbf{35}) prove that the index of an orphan monodromy group is infinite using Observation \ref{ob:index_MUM}. As already mentioned, this approach would be convenient, since it does not require the construction of an actual symplectic realization $Mon^\mathcal{B}(\mathcal{P})\subset\mathrm{Sp}(4,\mathbb{Z})$. Due to its failure, to proceed we must find a (rational) symplectic basis for orphan monodromy groups. This is accomplished in the next section.

\section{Symplectic bases for monodromy subgroups}\label{sec:symplectic}

The first step towards establishing whether monodromy groups of orphan operators are arithmetic or thin is to check whether the Zariski-closure of $Mon(\mathcal{P})$ is the entire symplectic group $\mathrm{Sp}(4,\mathbb{Z})$. If the answer is positive, the second step is to study the index $[\mathrm{Sp}(4,\mathbb{Z}):Mon(\mathcal{P})]$. Both of these questions implicitly assume that we are given a fixed symplectic basis $\mathcal{B}$ such that $Mon^\mathcal{B}(\mathcal{P})\subset\mathrm{Sp}(4,\mathbb{Z})$. Thus we must begin by finding such a basis.

Let $G:=Mon^\mathcal{B}(\mathcal{P})\subset\mathrm{GL}(4,\mathbb{Q})$ be a rational realization of the monodromy group with local monodromy generators $M_1:=M_{\sigma_1}^\mathcal{B},\ldots,M_n:=M_{\sigma_n}^\mathcal{B}$. Let $S=\left(s_{i,j}\right)_{i,j=1,\ldots,4}$ be a generic non-singular matrix such that $s_{i,j}=-s_{j,i}$. First we find a $G$-invariant symplectic form $\omega=\omega(\cdot,\cdot)$ on the four-dimensional rational vector space $\mathbb{Q}^4$. This is accomplished by solving the system of $16n$ linear equations in the variables $s_{i,j}$ given by
\begin{equation}\label{eq:system}
\big(M_k\cdot S\cdot M_k^T-S\big)_{i,j}=0, \quad k=1,\ldots,n,\quad i,j=1,\ldots,4.
\end{equation}
Let $\mathrm{S}=(\mathrm{s}_{i,j})_{i,j=1,\ldots,4}\in\mathrm{GL}(4,\mathbb{Q})$ be a solution of this system.  The symplectic form defined by $\omega(x,y):=x^T\cdot\mathrm{S}\cdot y$ is a $G$-invariant skew-symmetric bilinear form on $\mathbb{Q}^4$.

For operators with a MUM point and a conifold point, where we have access to the generalized Doran-Morgan basis, it is easy to see that the solution $\mathrm{S}$ is unique, up to a scalar. Local monodromy generators $M_1,\ldots,M_n$ for double octics orphan operators for certain rational realizations were given in Section \ref{sec:mongr}. Solving the system (\ref{eq:system}) for each of these sets of generators one can directly verify that in all cases the solution $\mathrm{S}$ is also unique (up to scaling).

Let
$$
\Omega:=
\begin{pmatrix}
0&\operatorname{Id}_2\\
-\operatorname{Id}_2&0
\end{pmatrix}
$$
be a matrix of the standard symplectic form invariant under $\mathrm{Sp}(4,\mathbb{Z})$. 
To find a rational symplectic basis for the action of $G$ it is now enough to find an invertible matrix $T=\left(t_{i,j}\right)_{i,j=1,\ldots,4}\in\mathrm{GL}(4,\mathbb{Q})$ such that
\begin{equation}\label{eq:form}
T^T\cdot \mathrm{S}\cdot T=\Omega
\end{equation}
Again, treating coefficients of $T$ as indeterminates we can solve this system of non-linear equations. If $\mathrm{T}\in\mathrm{GL}(4,\mathbb{Q})$ is any solution of (\ref{eq:form}), then the monodromy realization with respect to the basis $\mathcal{B}_s:=\mathrm{T}\cdot\mathcal{B}$ is rational and symplectic: $\mathrm{T}\cdot G\cdot \mathrm{T}^{-1}=Mon^{\mathcal{B}_s}(\mathcal{P})\subset\mathrm{Sp}(4,\mathbb{Q})$.

\begin{remark}
Unlike $\mathrm{S}$, the solution $\mathrm{T}$ is highly non-unique. Different choices of solutions will lead to different symplectic realizations of $Mon(\mathcal{P})$. All of these realizations will be of course conjugated in $\mathrm{GL}(4,\mathbb{Q})$ but not in $\mathrm{Sp}(4,\mathbb{Z})$. Thus it is important to note that arithmeticity of $G$ as a subgroup of $\mathrm{Sp}(4,\mathbb{Z})$ does not depend on the choice of the symplectic realization. Indeed, if $G_1,G_2\subset\mathrm{Sp}(4,\mathbb{Z})$ are two subgroups conjugated in $\mathrm{GL}(4,\mathbb{Q})$ and $[\mathrm{Sp}(4,\mathbb{Z}):G_1]<+\infty$, then $[\mathrm{Sp}(4,\mathbb{Z}):G_2]<+\infty$ as well. Thus for the question of arithmeticity the exact choice of the symplectic realization is irrelevant.
\end{remark}

Somewhat problematic is the fact that an arbitrary solution $\mathrm{T}$ of the system of equations (\ref{eq:form}) will only realize the monodromy group $G$ as a subgroup of $\mathrm{Sp}(4,\mathbb{Q})$, not $\mathrm{Sp}(4,\mathbb{Z})$. Solutions $\mathrm{T}$ providing integral bases can easily be found in several cases but in general this problem is open even for Picard-Fuchs operators with a MUM point and a conifold point.

It is of course easier to find an invariant lattice for the action of some proper subgroup $H\subset Mon(\mathcal{P})$. It then follows from the remark above that if $H$ is arithmetic, so is $Mon(\mathcal{P})$. Due to the lack of a natural symplectic basis for the entire monodromy action, this is the approach we have taken.

For each double octic orphan operator $\mathcal{P}$ the following list contains generators of a certain subgroup $H:=H(\mathcal{P})\subset Mon(\mathcal{P})$, written as powers of local monodromy generators $M_\sigma$. Furthermore, it lists a transition matrix $T:=T(\mathcal{P})$ between the rational basis from section \ref{sec:mongr} and a symplectic basis $\mathcal{B}_s$ for which $H^{\mathcal{B}_s}\subset\mathrm{Sp}(4,\mathbb{Z})$. Except for operators \textbf{153} and \textbf{250}, the subgroup $H$ is generated by unipotent powers of the local monodromy operators.

We have always normalized our choice of the symplectic basis so that the local monodromy around one of the conifold points has the form of a symplectic reflection $v\mapsto v+d\omega(f_c,v)f_c$, where $f_c$ is the conifold period and $0\neq d\in\mathbb{Z}$.

\begin{center}
\textbf{Integral symplectic bases for orphan operators}
\end{center}

\textbf{Operator 33}

$H:=\langle M_{1}^{2},M_{0}^{2^3},M_{2}\rangle$;\quad
$
T:=\begin{pmatrix}
	-4&0&192&-16\\
	-2&0&0&0\\
	1&8&-16&0\\
	0&0&8&0\\
\end{pmatrix}
$

\textbf{Operator 35}

$H:=\langle M_{0}^{2},M_{1},M_{-1}\rangle$;\quad
$
T:=\begin{pmatrix}
4&0&20&-4\\
-4&0&12&0\\
-1&-4&-1&0\\
0&0&-1&0\\
\end{pmatrix}
$

\textbf{Operator 97}

$H:=\langle M_{0}^{2},M_{-2},M_{-1}\rangle$;\quad
$
T:=\begin{pmatrix}
	0&16&0&0\\
	8&0&0&4\\
	2&0&0&0\\
	0&0&1&0\\
\end{pmatrix}
$


\textbf{Operator 152}

$H:=\langle M_{0}^{2},M_{1}^{2},M_{-1}\rangle$;\quad
$
T:=\begin{pmatrix}
	0&32&0&0\\
	4&0&0&4\\
	1&0&0&0\\
	0&0&1&0\\
\end{pmatrix}
$


\textbf{Operator 153}

$H:=\langle M_{-1},M_{0},M_{-2}\rangle$;\quad
$
T:=\begin{pmatrix}
	0&96&-96&0\\
	40&0&-96&24\\
	8&0&0&0\\
	0&0&3&0\\
\end{pmatrix}
$


\textbf{Operator 243}

$H:=\langle M_{1}^{2^3},M_{2},M_{\frac{3}{2}}^{3},M_{\infty}^{2^5\cdot 3}\rangle$;\quad
$
T:=\begin{pmatrix}
-5120&-1920&-480&3200\\
0&-360&120&-160\\
128&48&0&-48\\
0&3&0&0\\
\end{pmatrix}
$


\textbf{Operator 248}

$H:=\langle M_{0},M_{-\frac{1}{2}},M_{-1}^{2^3},M_{-\frac{3}{2}}^{2},M_{-2}^{2^2}\rangle$;\quad
$
T:=\begin{pmatrix}
-8&0&256&-16\\
-4&0&0&0\\
1&8&-16&0\\
0&0&8&0\\
\end{pmatrix}
$


\textbf{Operator 250}

$H:=\langle M_{1},M_{\infty},M_{-1},M_{-2}\rangle$;\quad
$
T:=\begin{pmatrix}
32&-16&0&0\\
16&0&0&16\\
0&0&-2&0\\
-1&0&-1&0\\
\end{pmatrix}
$


\textbf{Operator 266}

$H:=\langle M_{-\frac{1}{4}},M_{-\frac{1}{2}}^{2^2\cdot3\cdot5^2},M_{-1},M_{0}^{2\cdot 3},M_{\frac{1}{2}}\rangle$;\quad
$
T:=\begin{pmatrix}
45&0&-24&15\\
0&0&-4&0\\
0&15&-3&0\\
15&0&0&0\\
\end{pmatrix}
$

\medskip

Thus for all orphan operators we have obtained a symplectic realization, at least for some subgroup. To determine whether the monodromy groups of orphan operators with respect to the presented symplectic bases are dense in $\mathrm{Sp}(4,\mathbb{Z})$ we use the following very convenient criterion (see \cite[Theorem 2.4]{Rivin}):

\begin{theorem}\label{th:criterium}
Let $G\subset\mathrm{Sp}(4,\mathbb{Z})$ be a subgroup. Assume that there exists a prime $p\geq 5$ such that the reduction $\mod$ $p$ map $\pi_p:G\rightarrow\mathrm{Sp}(4,\mathbb{Z}/p\mathbb{Z})$ is surjective. Then $G$ is Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$.
\end{theorem}

\noindent Given a set of generators of a subgroup $\widetilde{H}\subset\mathrm{Sp}(4,\mathbb{Z}/p\mathbb{Z})$, it is a standard functionality in GAP to check whether $\widetilde{H}=\mathrm{Sp}(4,\mathbb{Z}/p\mathbb{Z})$. Using integral symplectic realizations of (subgroups of) double octic orphan monodromy groups listed above, we were able to verify the following:

\begin{lemma}\label{ob:surjective}
Let $\mathcal{P}$ be a double octic orphan operator. Let $H^{\mathcal{B}_s}\subset\mathrm{Sp}(4,\mathbb{Z})$ be one of the symplectic realizations of the subgroup $H\subset Mon(\mathcal{P})$ listed above. Let $\pi_p:\mathrm{Sp}(4,\mathbb{Z})\rightarrow\mathrm{Sp}(4,\mathbb{Z}/p\mathbb{Z})$ be the reduction $\mod$ $p$ map. Then the map $\pi_p\big\rvert _{H^{\mathcal{B}_s}}:H^{\mathcal{B}_s}\rightarrow\mathrm{Sp}(4,\mathbb{Z}/p\mathbb{Z})$ is surjective for $p=7$.
\end{lemma}

\noindent Using Theorem \ref{th:criterium} and Lemma \ref{ob:surjective} we immediately obtain the following:

\begin{theorem}\label{th:dense}
The monodromy groups of double octic orphan operators are Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$.
\end{theorem}

\section{Arithmetic monodromy. Some further remarks}\label{sec:35-153}

\subsection{Orphan family with arithmetic monodromy group}\label{ss:250}

\ 

The final theorem of the previous section shows that for all double octic orphan monodromy groups the question of their arithmeticity is well-posed, i.e. they are Zariski-dense in the symplectic group $\mathrm{Sp}(4,\mathbb{Z})$. We will now describe our results concerning operator \textbf{250} for which we verified that its monodromy group $Mon(\mathcal{P}_\textbf{250})$ is in fact arithmetic.

Note that in the list from the previous section  for the operator $\textbf{250}$ we have $H(\mathcal{P}_\textbf{250})=Mon(\mathcal{P}_\textbf{250})$. Hence with respect to the symplectic basis $\mathcal{B}_s$ the entire monodromy group is realized as a subgroup of $\mathrm{Sp}(4,\mathbb{Z})$. More precisely, it is generated by the following four symplectic matrices corresponding to the standard local monodromy generators:

$$
\begin{pmatrix}
	1&0&0&0\\
	0&1&0&8\\
	0&0&1&0\\
	0&0&0&1\\
\end{pmatrix},\quad
\begin{pmatrix}
	-1&0&0&0\\
	0&1&0&4\\
	-2&0&-1&0\\
	0&-1&0&-3\\
\end{pmatrix},\quad
\begin{pmatrix}
	1&-2&-8&12\\
	-6&-5&-12&0\\
	2&3&9&-6\\
	1&2&6&-5\\
\end{pmatrix},\quad
\begin{pmatrix}
	5&4&4&8\\
	8&9&8&16\\
	-4&-4&-3&-8\\
	-4&-4&-4&-7\\
\end{pmatrix}.
$$

The symplectic group $\mathrm{Sp}(4,\mathbb{Z})$ is finitely presented with known generators and relations (see \cite{Behr}). Using an easy brute force algorithm one can write any element of the symplectic group as a product of these distinguished generators. In particular, we can find such presentation for our symplectic generators of the monodromy group $Mon^{\mathcal{B}_s}(\mathcal{P}_\textbf{250})$.

Now the most straightforward approach for determining the index $[\mathrm{Sp}(4,\mathbb{Z}):Mon^{\mathcal{B}_s}(\mathcal{P}_\textbf{250})]$ is to apply the Todd-Coxeter algorithm for a finitely generated subgroup $Mon^{\mathcal{B}_s}(\mathcal{P}_\textbf{250})$ of a finitely presented group $\mathrm{Sp}(4,\mathbb{Z})$. This functionality is implemented in most computational algebra systems e.g., Magma or GAP.

Of course the Todd-Coxeter algorithm stops only when the considered subgroup is of finite index. Thus it can only be used to prove that the monodromy group is arithmetic, not thin. Furthermore, even if the index $[\mathrm{Sp}(4,\mathbb{Z}):Mon(\mathcal{P})]$ is finite, it may be practically impossible for the Todd-Coxeter algorithm to finish, due to the enormous amount of RAM required and a very long running time.

Nevertheless, for the operator \textbf{250} this approach yields satisfying results. Thus we have been able to verify that monodromy groups of orphan operators can be "large" not only in the sense of containing a maximally unipotent matrix (as in Theorem \ref{th:MUM}) but also in a much stronger sense of being of finite index. More precisely, we have the following:

\begin{theorem}\label{th:arithmetic}
The monodromy group $Mon(\mathcal{P}_{\textbf{250}})$ of the operator \textbf{250} is arithmetic and its index in $\mathrm{Sp}(4,\mathbb{Z})$ equals $23592960=2^{19}\cdot3^2\cdot5$.
\end{theorem}

\bigskip
\bigskip

We end this paper with a more detailed analysis of two particularity interesting orphan operators. The first one is operator \textbf{35}, the only one for which we were unable to identify a maximally unipotent element in the monodromy group. The second one is operator \textbf{153} for which the \emph{unipotently generated subgroup} of the monodromy group is very small, even though the entire monodromy group is dense in $\mathrm{Sp}(4,\mathbb{Z})$.

\subsection{Operator 35}\label{ss:35}


\ 

Operator \textbf{35} is the only one for which a random search for a maximally unipotent matrix in $Mon(\mathcal{P}_\textbf{35})$ did not yield any results. We were also unable to use the Todd-Coxeter algorithm to establish its index in $\mathrm{Sp}(4,\mathbb{Z})$. We present now a slightly modified approach to both of these questions, which perhaps is more promising.

For any natural number $n\in\mathbb{N}_{\geq 1}$ let us define
$$
A:=
\begin{pmatrix}
0&12&-17&4\\
0&-3&4&-1\\
1&4&2&0\\
4&32&-12&5\\\end{pmatrix}\quad
B:=
\begin{pmatrix}
1&0&0&0\\
0&1&0&1\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix},\quad
C_n:=
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&n&0&1\\
\end{pmatrix}
$$
Note that $A,B,C_n\in\mathrm{Sp}(4,\mathbb{Z})$. Let $G_n:=\langle A,B,C_n\rangle\subset\mathrm{Sp}(4,\mathbb{Z})$ be the subgroup they generate. Note that for any $a,b\in\mathbb{N}$ we have $C_a^b=C_{ab}$, hence there is an inclusion $G_{ab}\subset G_a$. In particular, $G_1$ is maximal among all the subgroups $G_n$.

The crucial observation is the following:

\begin{observation}\label{ob:mon}
	There is a basis $\mathcal{B}$ such that $G_{32}\subset Mon^\mathcal{B}(\mathcal{P}_\textbf{35})\cap\mathrm{Sp}(4,\mathbb{Z})$.
\end{observation}

\begin{proof}
	Conjugating the matrices in the rational realization of $Mon(\mathcal{P}_\textbf{35})$ from section \ref{sec:mongr} by the matrix
	$$
	\begin{pmatrix}
	16&0&80&-16\\
	-4&0&12&0\\
	-1&-4&-1&0\\
	0&0&-1&0\\
	\end{pmatrix}
	$$
	we obtain a new realization with generators
	$$
	M_0:=
	\begin{pmatrix}
	1&-2&0&0\\
	0&-1&0&0\\
	0&2&1&0\\
	-2&16&-2&-1\\\end{pmatrix},\quad
	M_1:=
	\begin{pmatrix}
	0&12&-17&4\\
	0&-3&4&-1\\
	1&4&2&0\\
	4&32&-12&5\\\end{pmatrix},\quad
	M_{-1}:=
	\begin{pmatrix}
	1&0&0&0\\
	0&1&0&1\\
	0&0&1&0\\
	0&0&0&1\\\end{pmatrix}
	$$
	Since $A=M_1,B=M_{-1}$ and $C_{32}=M_0^{-2}$, we obtain the claim.
\end{proof}

Thus we hope that to study the size of the group $Mon(\mathcal{P}_\textbf{35})$ it is enough to understand the size of certain auxiliary groups $G_{n}$, which seem to be better suited for computations. In particular, using the Todd-Coxeter algorithm we were able to obtain:

\begin{observation}\label{ob:g1}
The group $G_4$ is of finite index in $\mathrm{Sp}(4,\mathbb{Z})$.
\end{observation} 
\noindent More precisely, $[\mathrm{Sp}(4,\mathbb{Z}):G_1]=122880=2^{13}\cdot3\cdot5$, $[\mathrm{Sp}(4,\mathbb{Z}):G_2]=11796480=2^{18}\cdot3^2\cdot5$ and $[\mathrm{Sp}(4,\mathbb{Z}):G_4]=1509949440=2^{25}\cdot3^2\cdot5$. In particular, together with Observation \ref{ob:index_MUM} this implies that there is a maximally unipotent matrix in $G_4$. Using the Todd-Coxeter algorithm we were not able to compute the index $[\mathrm{Sp}(4,\mathbb{Z}):G_n]$ for $n\geq 5$. Nevertheless, as in Theorem \ref{th:MUM}, a random search for maximally unipotent matrices yielded:
\begin{observation}
	The group $G_n$ contains maximally unipotent elements for $n\in\{1,\ldots,9\}$.
\end{observation}
\noindent Again we were unable to apply the same approach to identify maximally unipotent elements in $G_n$ for $n\geq 10$.

In any case we see that these two observations have the following corollary:

\begin{corollary}
	\ 
	\begin{itemize}
		\item If the index $[G_4:G_{32}]$ is finite, then $Mon(\mathcal{P}_\textbf{35})$ is arithmetic.
		\item If the index $[G_8:G_{32}]$ is finite, then $Mon(\mathcal{P}_\textbf{35})$ contains a maximally unipotent matrix.
	\end{itemize}
\end{corollary}
\noindent It may of course happen that $[G_1:G_{32}]=[G_8:G_{32}]=+\infty$, yet $Mon(\mathcal{P}_\textbf{35})$ is still of finite index.

\subsection{Operator 153}\label{ss:153}

\ 

Let $\mathcal{P}$ be a Fuchsian operator, let $n\in\mathbb{N}_{\geq 2}$ and assume that $0$ is a singular point of type $\tfrac{1}{n}C$, $\tfrac{1}{n}K$ or $F$. For the singularity of finite type we assume that the local monodromy is of order $n$. Pull-back by the map $z\mapsto z^n$ transforms the operator $\mathcal{P}$ into a different operator $\mathcal{P}'$ and near the origin this corresponds to changing the local monodromy around $0$ by $M_0\mapsto M_0^n$. The monodromy operator $M_0^n$ has all eigenvalues equal to $1$ and hence is unipotent. Furthermore, the singular point at $0$ is now of type $C$, $K$ or is an \emph{apparent singularity}, i.e. has trivial local monodromy.

Of course pull-back by $z\mapsto z^n$ introduces new singular points and thus globally changes the monodromy group in a much less clear way. Nevertheless, the subgroup generated by local monodromies under (local) pull-backs is particularity convenient for investigation. In particular, finding an integral symplectic basis seems to be much easier once one restricts to this subgroup. This motivates the following definition:

\begin{definition}
Let $Mon(\mathcal{P})$ be a monodromy group of a Picard-Fuchs operator $\mathcal{P}$ with the set of singular points $\Sigma\subset\mathbb{P}^1$. Recall that it is generated by the local monodromy operators $M_\sigma$, $\sigma\in\Sigma$, all of which are quasi-unipotent. For every singular point $\sigma$ let $k_\sigma$ denote the unipotency index of the local monodromy operator $M_\sigma$, i.e. the smallest positive integer such that $(M_\sigma^{k_\sigma}-\operatorname{Id})^4=0$. Then the \emph{unipotently generated subgroup} $Mon(\mathcal{P})^{un}$ is the subgroup of $Mon(\mathcal{P})$ generated by $M_\sigma^{k_\sigma}$.
\end{definition}

For almost all double octic orphan operators $\mathcal{P}$ the unipotently generated subgroup of the monodromy group is already dense in $\mathrm{Sp}(4,\mathbb{Z})$. Indeed, it can easily be checked using Lemma \ref{ob:surjective} and in fact subgroups $H=H(\mathcal{P})$ from the previous section are usually contained in the unipotently generated subgroup.

For operator \textbf{153} th situation is different. Even though the monodromy group $Mon(\mathcal{P}_{\textbf{153}})$ is Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$ (as for all double octic orphan operators), the unipotently generated subgroup $Mon(\mathcal{P}_{\textbf{153}})^{un}$ is not.

\begin{lemma}
Let $\langle A_1,A_2,A_3 \rangle=G\subset\mathrm{Sp}(4,\mathbb{Z})$ be a subgroup generated by three matrices with $\operatorname{rank}(A_i-\operatorname{Id})=1$ for $i=1,2,3$. Then $G$ is not Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$.
\end{lemma}

\begin{proof}
Assume that $G$ is Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$. Since $\operatorname{rank}(A_i-\operatorname{Id})=1$, we also have $\dim\operatorname{ker}(A_i-\operatorname{Id})=3$. Letting $K_i:=\operatorname{ker}(A_i-\operatorname{Id})$ and $K:=K_1\cap K_2\cap K_3$, we see that $\dim K\geq 1$ by dimension count. By definition $A_i(v)=v$ for all $v\in K$ and $i=1,2,3$. Hence the subspace $K$ is $G$-invariant. Since any Zariski-dense subgroup of $\mathrm{Sp}(4,\mathbb{Z})$ must act irreducibly on $\mathbb{C}^4$, we conclude that $K=\mathbb{C}^4$. But this implies $A_1=A_2=A_3=\operatorname{Id}_4$, a contradiction.
\end{proof}

As can be seen from its Riemann scheme in Section \ref{sec:mongr}, the operator \textbf{153} has four singular points: three of type $\tfrac{1}{2}C$ and one of type $F$ with local monodromy of order $2$. Hence its unipotently generated subgroup $Mon(\mathcal{P}_\textbf{153})^{un}$ is generated by three monodromies of type $C$ (and an identity transformation). By definition any local monodromy operator $A$ of type $C$ satisfies $\operatorname{rank}(A-\operatorname{Id})=1$ and we conclude that:

\begin{theorem}
The unipotently generated subgroup $Mon(\mathcal{P}_{\textbf{153}})^{un}$ is not Zariski-dense in $\mathrm{Sp}(4,\mathbb{Z})$.
\end{theorem}

Thus there is a big difference between the monodromy group and the group generated by monodromies under natural pull-backs, witnessed already by their different Zariski-closures. If we restrict ourselves to an even smaller subgroup $H\subset Mon(\mathcal{P}_{\textbf{153}})^{un}$ generated by unipotent powers of local monodromies at \emph{finite} singular points, a precise description of the group $H$ can be obtained and the Zariski-closure $\overline H$ can be computed as follows.

The subgroup $H$ is generated by two monodromy operators $M_1,M_2$ such that $\operatorname{rank}(M_i-\operatorname{Id})=1$ for $i=1,2$. Let $f_i\in\mathcal{S}ol_\mathbb{Q}(\mathcal{P}_\textbf{153})$ be the conifold period spanning the image of $M_i-\operatorname{Id}$, $i=1,2$. For any solution $v\in\mathcal{S}ol(\mathcal{P}_{\textbf{153}})$ we have $M_i(v)=v+\alpha_i(v)\cdot f_i$ for some $\alpha_i(v)\in\mathbb{C}$. The existence of a monodromy-invariant lattice implies that (after appropriate scaling) we have $\alpha_1(f_2),\ \alpha_2(f_1)\in\mathbb{Z}$. It easy to verify numerically that $\alpha_1(f_2),\ \alpha_2(f_1)\neq 0$. In particular, $f_1,f_2$ are linearly independent. Since the conifold period $f_i$ is an eigenvector of $M_i$ with eigenvalue $1$, we also conclude that $\dim\big( \operatorname{ker}(M_1-\operatorname{Id})\cap\operatorname{ker}(M_2-\operatorname{Id})\big)=2$. Let $f_3,f_4\in\mathcal{S}ol(\mathcal{P}_\textbf{153})$ be any basis of $\operatorname{ker}(M_1-\operatorname{Id})\cap\operatorname{ker}(M_2-\operatorname{Id})$.

With respect to the basis $\mathcal{B}:=\{f_1,\ldots,f_4\}$ the subgroup $H$ is generated by matrices
$$
M_1^\mathcal{B}=\begin{pmatrix}
1&a&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\
\end{pmatrix},\quad
M_2^\mathcal{B}=\begin{pmatrix}
1&0&0&0\\b&1&0&0\\0&0&1&0\\0&0&0&1\\
\end{pmatrix}\quad \textnormal{for some }0\neq a,b\in\mathbb{Z}.
$$
\noindent Numerically computing monodromies of the conifold periods one can check that $(a,b)=(1,128)$. Thus
$$
H^\mathcal{B}\subset\left\{
\begin{pmatrix}
	A&\operatorname{0}\\\operatorname{0}&\operatorname{Id}_2\\
\end{pmatrix}\in\mathrm{SL}(4,\mathbb{Z}):\quad A\in\Gamma_0(128)
\right\}
$$
\noindent Note that with respect to this realization the Zariski-closure of $H$ is
$$
\overline{H^\mathcal{B}}=\left\{
\begin{pmatrix}
	A&\operatorname{0}\\\operatorname{0}&\operatorname{Id}_2\\
\end{pmatrix}\in\mathrm{SL}(4,\mathbb{Z}):\quad A\in\mathrm{SL}(2,\mathbb{Z})
\right\}\simeq\mathrm{SL}(2,\mathbb{Z})
$$
