#### A-incarnation

If a Calabi-Yau operator $L$ arises as a regularisation of a
quantum differential equation of a symplectic variety $Z$, we
call $Z$ a (strong) A-incarnation of $L$. The regularisation defines
a complete intersection Calabi-Yau manifold inside $Z$ and the
instanton numbers of the differential operator are supposed to be
equal to the number of rational curves ($g=0$ GV-invariants) of $X$..
#### Apparent singularity

An apparent singularity is a singular point of a differential equation with trivial local
monodromy. The local exponents are all integeral.
For a Calabi-Yau operator, apparent singularities with exponents $0,1,3,4$ are very common,
although also other exponents do occur. They are quite mysterious and I do not know what
exactly is happening to the geometrical model of a $B$-incarnation at such an apparent
singularity.
#### B-incarnation

If a Calabi-Yau operator $L$ appears as (factor of) the Picard-Fuchs operator
a one-parameter family $\mathcal{X} \longrightarrow P^1$ of varieties, we call this
family a $B$-incarnation of $L$. One then says that $L$ is of geometrical origin.
If at appears in a family of Calabi-Yau varieties, we call it
a strong $B$-incarnation. Many more condition may be put here. We may require $h^{12}=1$,
we may ask for a projectice family, etc. For a given operator $L$ there are usually a very
large number of different $B$-incarnations. One may start wondering about possible correspondences
between different incarnations which conjecturally always should exist.
If an operator is derived from a binomial sum, it is the diagonal of a rational function
and hence has a $B$-incarnation. A constant term series of Laurent-series is a special
diagonal, and obviously define $B$-incarnation.
#### Calabi-Yau operator

Under a Calabi-Yau operator $L \in C[x,\Theta]$ we understand a differential operator $L$ with the
following properties:
1. $L$ is fuchsian of order $4$.
2. It has $0$ as MUM-point.
3. $L$ is self-dual.
4. $L$ is strongly arithmetic.
Here {\em strongly arithmetic} means:
a) The solution $y_0(x)$ is $N$-integral.
b) The q-coordinate $q(x)$ is $N$-integral.
c) The instanton numbers are $N$-integral.
Conjecturally, any operator with an $N$-integral power-series solution comes from geometry.
There are many operators that satify a) but not b) or c).
By shifting of exponents we may always assume the the exponents at $0$ are all $0$, so the
operator in $\Theta$-form reads
$L=\Theta^4+xP_1(\Theta)+\ldots+x^rP_r(\Theta)$
#### C-point

A singular point of a Calabi-Yau operator is called a {\em C}-point, if the local
monodromy has a single Jordan block of size $2$. The most common C-point is the
{\em conifold singularity}, which can be recognised by the local exponents $0,1,1,2$.
The local monodromy around a conifold singularity is a symplectic reflection. If the
operator has a $B$-incarnation, such conifold points appear where the variety aquires
one or more $A_1$-points (also called conifold points). At a C-point, the limiting
mixed Hodge structure $Gr^W_3H^3$ has Hodge numbers $1,0,0,1$ and both $Gr^W_4$ and
$Gr^W_2H^3$ are one-dimensional and identified via $N$, the monodromy logarithm.
The two dimensional Hodge structure looks like that of a rigid Calabi-Yau threefold.
#### Degree

For a differential operator $L$ in $\Theta$-form
$$L=P_0(\Theta)+xP_1(\Theta)+t^2P_2(\Theta)+\ldots+x^rP_r(\Theta)$$
with $P_k(\Theta)$ polynomials in $\Theta$, $P_r(\Theta) \neq 0$.
we call the number $r$ the {\em degree} of $L$.
Hypergeometric operators are operators of degree $1$, and the degree serves
as a simple measure of complexity of the operator. We always tried to transform
the operator into one with lowest possible degree, but sometimes this makes the
operator uglier.
For Calabi-Yau operators the degree is equal to the {\em sum the weights of all
singular points $p \neq 0, \neq \infty$}.
For a series $\phi(x)=\sum_n a_n x^n$ the equation $L\phi(x)=0$ translates in a
recursion relation of length $r$
$$ P_0(n) a_n + P_1(n-1) a_n-1+\ldots P_r(n-r)a_{n-r}=0$$
on the coefficients $a_n$.
#### Diagonal

If $$f(x_1,x_2,\ldots,x_n)=\sum_{k_1,k_2,\ldots,k_n} a_{k_1, k_2 \ldots, k_n}x_1^{k_1}x_2^{k_2}\ldots x_n^{k_n}$$ is a power series in $n$-variables one puts
$$\Delta_n(f):=\sum_{k}a_{k,k,\ldots,k} x^k \in \mathbb{C}[[x]]$$
A power series of the form $\Delta_n\left(\frac{P}{Q}\right)$, where $P$ and $Q$ are polynomials with $Q(0) \neq 0$
is called an *$n-diagonal*. Such $n$-diagonals always satisfy a Fuchsian differential equation of
geometrical origin. If $W$ is a Laurent series in $x_1,x_2,\ldots, x_n$, then its constant term series is
$$\sum_n [W^n]_0 x^n =\Delta_{n+1} \frac{1}{1-x_0x_1\ldots x_n W(x_1,x_2,\ldots,x_n)},$$
so constant term series are special diagonals. If a Calabi-Yau operator has a $B$-incarnation, it has a
representation as an $n$-diagonal with $n \le 8$.
#### Differential operator

By a {\em differential operator} $L$ we mean here a linear differential operator
in a single variable $x$, in other words an element of the ring
$C[x, \frac{d}{dx}]$.
We can write an operator $L$ in so-called {\em $\frac{d}{dx}$-form} as
$$L =a_0(x) \frac{d^n}{dx^n}+a_1(x) \frac{d^{n-1}}{dx^{n-1}}+\ldots +a_n(x)$$
where the $a_i(x)$ are polynomials. By dividing by the greatest common factor
of the polynomials $a_0,a_1,\ldots,a_n$ we obtain the associated {\em reduced}
operator.
We work often with operators $L \in C[x,\Theta]$, $\Theta=x\frac{d}{dx}$.
Such an operator can be written in so-called {\em $\Theta$-form} as
$$L=P_0(\Theta)+xP_1(\Theta)+x^2P_2(\Theta)+\ldots+x^rP_r(\Theta)$$
with $P_k(\Theta)$ polynomials in $\Theta$, $P_r(\Theta) \neq 0$.
#### Discriminant

For a reduced differential operator in $\frac{d}{dx}$-form
$$ L =a_0(x) \frac{d^n}{dt^n}+a_1(x) \frac{d^{n-1}}{dt^{n-1}}+\ldots +a_n(x)$$
the coefficient $a_0(x)$ is called the discriminant. The roots of the
polynomial $a_0(t)$ are singularities of the operator.
#### Exponents

For an operator $L$ in $\Theta$-form
$L=P_0(\Theta)+xP_1(\Theta)+x^2P_2(\Theta)+\ldots+x^rP_r(\Theta)$$
with $P_k(\Theta)$ polynomials in $\Theta$, $P_r(\Theta) \neq 0$,
the roots of the polynomial $P_0(Theta)$ are called the {\em exponents} of $L$ at $0$.
The exponents at a general point $p$ are obtained by translating $p$ to the origin
and compute $P_0$ for the translated operator. The exponents at infinity are the roots of
$P_r(-\Theta)$.
There is a single relation between the exponents of a differential operator, called the {\em Fuchs relation}. For a Calabi-Yau operator that relation reads
$\sum_{p} (6-\textup{exponent sum at}\; p)=12$
#### F-point

A singular point of a Calabi-Yau operator is called an $F$-point, if the local
monodromy around it has finite order. It means that the limiting mixed Hodge structure
stays pure here. A base change of finite order converts this to a point with trivial
local monodromy. It then can be a non-singular point, or a so called {\em apparent singularity}.
#### Frobenius basis

A Calabi-Yau operator has a preferred basis of solutions on a slit-disc
neigbourhood of a MUM-point.
They can be written as
$$y_0(x)=f_0(x)$$
$$y_1(x)=\log(x) f_0(x)+f_1(x)$$
$$y_2(x)=\frac{1}{2}\log(x)^2 f_0(x)+\log(x) f_1(x)+f_2(x)$$
$$y_3(x)=\frac{1}{6}\log(x)^3 f_0(x)+\frac{1}{2}\log(x)^2 f_1(x)+\log(x) f_2(x)+f_3(x)$$
Here $f_0(x)$ is a power-series with integral coefficients, $f_1,f_2,f_3 \in x\Q[[x]]$$
#### Instanton numbers

The normalised instanton numbers
$n_0:=1, n_1, n_2, \ldots $
are defined by expanding the Yukawa-coupling $K(q)$ into a Lambert series
of the form
$K(q)=1+\sum_{d=^}^{infinity} n_d \frac{d^3q^d}{1-q^d}$
These numbers are usually rational and in most cases there is a least common
denominator that we call the instanton denominator.
If the operator has characteristic invariants $(H^3,c_2H,c_3)$, we
can multiply through by the degree $D:=H^3$ and obtain the scaled
instanton numbers
$n_0^*:=D, n_1^*=D n_1, n_2^* =D n_2, \ldots $
and the number $n_d^*$ is supposed to coincide with the number of
rational degree $d$ curves (or $g=0$ Gopakumar-Vafa invariants) on
an $A$-model incarnation.
#### K-point

A singular point of a Calabi-Yau operator is called a {\em K}-point, if the local
monodromy has a two Jordan blocks of size $2$. Such points are recognised by having
two pairs of equal exponents. If the operator has a $B$-incarnation, such K-points
appear where the variety aquires more complicated singularities. At a K-point, the limiting
mixed Hodge structure $Gr^W_3H^3 =0$, whereas both $Gr^W_4$ and
$Gr^W_2H^3$ are two dimensional with Hodge numbers $1,0,1$ which are identified via $N$.
the monodromy logarithm. The Hodge structure $1,0,1$ looks like the trancendental lattice
of a K3-surface with Picard number $20$, and in the semi-stable reducion such a K3-surface
should appear, hence the name $K$-point for such singularities.
#### MUM-point

If the local monodromy around a singularity has a Jordan-block
of maximal size, it is called a MUM-point. By definition, a
Calabi-Yau operator has a MUM-point at the origin as defining
property. At a MUM-point all exponents have to be equal. For
Calabi-Yau operators MUM-points are characterised by having all
equal exponents, which can be seen from the Riemann symbol.
#### $q$-coordinate

The $q$-coordinate of a Calabi-Yau operator $L$ is defined
as $q=e^{y_1(x)/y_0(x)}$
where $y_0(x)$ and $y_1(x)$ are the first two solutions of $L$
mear the MUM-point.
As $y_1(x)=\log(x) y_0(x)+f_1(x)$, $f_1(0)=0$, we in fact have
$ q(x)=x.e^{f_1(x)}=x+ \alpha_2 x^2+\alpha_3 x^3+\ldots$
so $q$ can indeed seen as a coordinate near $0 \in P^1$, and
the above series rather defines the $q$-coordinate in terms
of the algebraic coordinate $x$. In a way it plays a role similar
to the $q$ in the theory of the Tate-elliptic curve.
#### Quantum differential equation

Gromov-Witten invariants of a symplectic manifold $Z$ can be used to
define its so-called quantum $D$-module/Dubrovin-Givental connection via
$\nabla_{A} S = A \ast S$
Here $S$ is a cohomology-valued function and $\ast$ denotes quantum product.
If $Z$ is a Fano manifold with $H^2(Z)$ is one-dimensional and $A=H$ is the
ample generator of $H^2(Z)$, the horizontal sections satisfy the ordinary
differential equation $ \Theta S(x)= H \ast S(x)$, where $\Theta=x\frac{d}{dx}$
and $S(x)=\sum_{a \in H^*(Z) } s_a(x) a$ the cohomology valued function.
The differential equation satisfied by $s_{pt}(x)$, ($pt \in H^{*}(Z)$ the
class of a point in $Z$) is called the {\em quantum differential equation}
of $Z$; it will have an irregular singularity at $x=\infinity$.
#### Regularisation

The quantum differential equation of a Fano-manifold with $Pic(Z)= \Z H$
has an irregular singularity at infinity.
It can be converted into a fuchsian differential equation using a {\em Laplace
transformation.}
If $\psi(t)=\sum a_n t^n$ solves the quantum differential equation of $Z$,
and $K=-rH$, then its Laplace transform
$L\psi(s) =\frac{1}{x}\int_0^{infty} \psi(t^r) e^{-t/s}bdt$
expands as
$L\ psi(s)=\sum (r n)! a_n s^{r n}$
When we put $x=s^r$ we obtain the function
$\psi(x):=\sum_n (rn)! a_n x^n$ that satisfies a differential equation that
we call the {\em regularised quantum differential equation}. It is supposed
to be the Picard-Fuchs equation of a Calabi-Yau manifold that is mirror dual
to the anti-canonical hyperplane section of $Z$.
In fact, to any nef-partition of the canonical bundle $K$ of $Z$ we can associate
a regularisation of the quantum differential equation. If $r=r_1+r_2+\ldots+r_k$
is a partition of $r$ into $k$ parts, we can regularise $\psi(t)=\sum_n a_n t^n$
to the series $\phi(x)=\sum_n (r_1 n)! (r_2 n)! \ldots (r_k n)! a_n x^n$
which now satisfies a fuchsian differential equation. It should be Picard-Fuchs
operator of a family of Calabi-Yaus mirror dual to the Calabi-Yau that appears
as complete intersection in $Z$ of $k$ general hypersurfaces of degree
$r_1,r_2, \ldots r_k$.
#### Riemann symbol

The Riemann symbol of an operator contains the exponents of the
operator at all singular points, including the point infinity. Under
each singular point of the operator we write a column containing the
exponents of the operator at that point. Usually, curly brackets are
written around the columns of exponents and a horizontal bar is written
to separate the singular points from the corresponding exponents.
The Rieman symbol is a convenient representation of the ramification
properties of the operator at the singular points. For Calabi-Yau operators
one can usually read off the local monodromy around the singularities from the
Riemann symbol.
#### Shifting

Multiplication of the solutions of a differential equation by an algebraic function of $x$
lead to a shift of exponents at the singularities of the algebraic function. For
example, multiplication by $x^{\alpha}$ lead to a shift of the exponents at $0$ and $\infty$
by $\alpha$ and $-\alpha$. By appropriate shifting, we can reduce the {\em weight} of the
singularities of the operator as much as possible, which leads to an operator of lowest
possible degree.
#### Singular point

A singular point of a differential operator $L$ of order $n$ is a point where
the exponents are {\em not} $0,1,2,\ldots,n-1$. For a Calabi-Yau operator
a great variety of different exponents may occur.
It is useful to use a rough classification into MUM-points, C-point, K-points and
F-points that reflect the Jordan structure of the local monodromies.
#### Weight

The {\em weight} of a singular point $p$ of a differential operator $L$ of order $n$ is
defined as the number $n-k$, where $k$ is zero if $0$ is not an exponent of $L$ at $p$
and else $k$ is the largest number such that all numbers
$0,1,\ldots,k-1$ are exponents of $L$ at $p$.
So $w(0,0,0,0)=3$, $w(0,1/2,1/2,1)$, $w(0,1,1,2)=1$, $w(1,2,3,4)=4$, $w(0,1,2,3)=0$.
The weights of the singular points determine the degree of a reduced operator:
$\textup{degree}=\sum_{p \neq 0,\neq \infty}$
#### Yukawa coupling

By expressing a Calabi-Yau operator $L$ in its $q$-coordinate, it take the form
$L=\theta^2 \frac{1}{K(q)}\theta^2,\;\;\; \theta=q \frac{d}{dq},$
where $K(q) \in C[[q]] $ is called the Yukawa-coupling of $L$.
The series $K(q)$ is the formal invariant of the operator $L$.
The Yukawa-coupling $\alpha$ of an operator $L$ in the original $x$-coordinate is solution to
the differential equation $\alpha'=\frac{2}{n}a_1(x) \alpha$, where $a_1(x)$ is a first coefficient
of $L$ in $\frac{d}{dx}$-form. (Note however that the Yukawa coupling transforms as a symmetric
$3$-tensor.)