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.


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.

Binomial sum

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 $L=\Theta^4+xP_1(\Theta)+\ldots+x^rP_r(\Theta)$


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.


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 $a_n$.


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 $\Theta$, $P_r(\Theta) \neq 0$.


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.


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 $\sum_{p} (6-\textup{exponent sum at}\; p)=12$


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.


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.



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.


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 $x=\infinity$.


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.


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.


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.)