For any hypergeometric differential equation of order $n$ on a Riemann sphere with three punctures $\mathbb{P}^1(\mathbb{C}) / \{0,1,\infty \}$, one has linearly $n$ independent solutions which are called hypergeometric functions. For the $\alpha, \beta$ parameters, we write the local monodromies using the (local) space of solutions through the monodromy representation $\rho: \pi_1 \to GL(V)$. The subgroup $\rho(\pi_1)$ of $GL(V)$ is called the monodromy group associated with our hypergeometric equation and parameters $\alpha, \beta$. Accoriding to Levelt's theorem (and later worked by Beukers–Heckman), there exists a (local) basis of solution space of hypergeometric equation such that the monodromy group associated to it (corresponding to $\alpha, \beta$) is generated by the companion matrices $A$ and $B$ of polynomials $f,g$ such that the polynomials are cyclotomic (then one can conjugate the monodromy groups) which means their roots are roots of unity, primitive, self-reciprocal, and thus have no common root. The monodromies are defined locally using these companion matrices. This monodromy group $\Gamma(f,g)$ is a subgroup of $GL_n(\mathbb{C})$.

Now, when we restrict to the polynomials $f,g$ with integer coefficients, then the monodromy group becomes a subgroup of $GL_n(\mathbb{Z})$. We are very much interested in this special representation. However, there is a very large gap between the monodromy group $\Gamma$ and $GL_n(\mathbb{C})$, hence we look for something in between, which is the Zariski closure of $\Gamma$. For the order $n=1$, we see that $\Gamma$ sits in $GL_1(\mathbb{Z})$; however, with order $2$ and higher, we have non-trivial things going on like finite and infinite index, and so on. For order $n=2$, the Zariski closure (here it is the smallest algebraic group that contains $\Gamma$ in $GL_n(\mathbb{Z})$) of $\Gamma$ is $SL_2(\mathbb{Z})$. Then a nice question is to ask if $\Gamma$ is of finite index in its Zariski closure, then it is called arithmetic, and if it infinite index, then it is called thin.

Let us denote for $\Gamma$ its Zariski closure $\Gamma^{Zar}$ as $G$, so $\Gamma \subset G(\mathbb{Z})$. Typically, one has $G=Sp_\Omega(\mathbb{Z})$ where $\Omega$ is a symplectic form on $\mathbb{Z}^n$ or $G=O_Q$ where $Q$ is a quadratic form. The details about when $G$ is symplectic and orthogonal are here. Anyway, the important question is to ask for any given order, $\Gamma(f,g)$ has a finite (arithmetic) or infinite (thin) index in $G(\mathbb{Z})$. For the number theoretic motivation behind the arithmeticity and thinness, see these notes by Sarnak. In order $n=2,3$, there are arithmetic monodromy groups. However, for $n=4$, the situation is more non-trivial and it contains both arithmetic and thin groups. There are $14$ cases in which the monodromy groups are associated with Calabi-Yau 3-folds. The periods of the three-form $\omega$ on the moduli of a CY 3-fold solve a fourth-order differential equation. (I do not understand much connection more than this for these 14 special cases connected to CY right now.) Anyway, again, one is asking how many of them are arithmetic and how many are thin. They were studied in this paper. As it turned out, 7 of the 14 cases are thin and the other 7 are arithmetic, and Kontsevich was also involved in the problem.