# Traintrack Calabi-Yaus from Twistor Geometry

Cristian Vergu  Matthias Volk

###### Abstract

We describe the geometry of the leading singularity locus of the traintrack
integral family directly in momentum twistor space. For the two-loop case,
known as the elliptic double box, the leading singularity locus is a genus one
curve, which we obtain as an intersection of two quadrics in $\mathbb{P}^{3}$.
At three loops, we obtain a K3 surface which arises as a branched surface over
two genus-one curves in $\mathbb{P}^{1}\times\mathbb{P}^{1}$. We present an
analysis of its properties. We also discuss the geometry at higher loops and
the supersymmetrization of the construction.

##  1 Introduction

While it was initially hoped that the integrals which appear in computations
in planar $\mathcal{N}=4$ SYM are expressible in terms of generalized
polylogarithms, it has by now become clear that this is not the case.111Work
on the Kontsevich conjecture by Belkale and Brosnan [1] had given good reasons
to be pessimistic. More recently, Brown and Schnetz [2] have given explicit
examples in $\phi^{4}$ theory, which contain K3 geometries. Not only are the
generalized polylogarithms insufficient but, by any reasonable measure, most
of the integrals in $\mathcal{N}=4$ SYM seem to require more complicated
classes of functions, which are as of yet very poorly understood.

One class of integrals which is relatively well-understood is the class of
pure integrals. These integrals have leading singularities (see ref. [3])
which are pure numbers such as $0$ or $\pm 1$. In all known examples they are
computable in terms of generalized polylogarithms.

Recall that to obtain leading singularities one takes residues in the
propagators of the integral. Doing so, Jacobian factors are generated in which
one can often take further residues. If we start with an integral with fewer
propagators than integration variables, two things can happen. Either one can
generate enough Jacobian factors to take residues in, so that the integral
localizes, or not. If the integral does not localize, then the process of
taking residues ends with a holomorphic form. This form may however develop
poles for special kinematics.

The leading singularity locus, when it is not a set of points, turns out to be
an interesting variety of Calabi-Yau type. The discussion above makes it
plausible that one is more likely to find integrals which do not localize if
we consider examples with as few propagators as possible. Since triangles are
not possible in a dual-conformal expansion in planar $\mathcal{N}=4$ SYM, the
examples we consider are box integrals. As it turns out, ladder integrals are
computable in terms of classical polylogarithms (see ref. [4]). The simplest
integral which can not be localized by taking residues is the elliptic double
box integral, studied in refs. [5, 6]. It is part of a family of integrals,
called traintrack integrals (see fig. 1). There are many other examples in the
literature, where Calabi-Yau geometries have been identified in loop
integrals, see e.g. [7, 2, 8, 9, 10, 11, 12, 13, 14].

The traintrack integrals were studied in ref. [15]. This reference studied
three- and four-loop integrals using Feynman parametrization. The leading
singularity loci were defined as hypersurfaces in various weighted projective
spaces, whose coordinates were related to the Feynman parameters of the
original integral. The constructions in ref. [15] were pretty involved, in
that they required complicated changes of variables which did not seem to fit
a pattern that could be generalized to all loops.

In this paper we study the leading singularity locus by using the momentum
twistor description of the traintrack integrals. Momentum twistors were
introduced by Hodges [16] in order to make the dual conformal symmetry [17,
18, 19] more manifest. The translation from momentum space to twistor space
proceeds as follows. Given a planar Feynman integral such as the one in fig.
1, we introduce dual coordinates $x_{\ell_{i}}$ for each loop and $x_{i}$ for
each external region. Under the twistor correspondence, each of these dual
points corresponds to a projective line $\mathbb{P}^{1}$ inside a
$\mathbb{P}^{3}$ space. This $\mathbb{P}^{3}$ is called momentum twistor
space. Under this dictionary, the action of the conformal group on the dual
space with coordinates $x$ becomes the familiar $\operatorname{PSL}(4)$ action
on $\mathbb{P}^{3}$.

Two dual points are light-like separated if their corresponding lines in
momentum twistor space intersect. This simple geometric fact, which is
manifestly invariant under $\operatorname{PSL}(4)$ transformations, will be
central to our discussions below. Indeed, the leading singularity locus is
obtained by imposing a number of light-like conditions between the dual
points. Using the momentum twistor constructions these constraints yield a
configuration of intersecting lines, which is much easier to describe than the
set of quadratic equations which one has to solve in momentum space or dual
space.

Another advantage of the momentum twistor description is that it automatically
picks for us a compactification and complexification of the dual space which
is compatible with the dual conformal symmetry. The complexification is
essential as well since all the varieties we will describe below are complex
varieties.

Our analysis is similar in spirit to the analysis done by Hodges [20] for the
one-loop box integral. The one-loop box example is however much simpler, since
its leading singularity locus is a set of two points.

In this paper we obtain the following results. We describe the leading
singularity locus of the elliptic double box as an intersection of two
quadrics in $\mathbb{P}^{3}$. We compute the $j$-invariant of this
intersection and compare with the answer obtained in ref. [6]. Next, we
analyze the three-loop case and we identify the leading singularity locus with
a K3 surface. The K3 surface is described as a branched surface over the union
of two genus-one curves in $\mathbb{P}^{1}\times\mathbb{P}^{1}$. We compute
its Euler characteristic and the number of moduli. Then, we analyze the
leading singularity locus in the four-loop case. We obtain a Calabi-Yau three-
fold which can be realized as a complete intersection. We analyze its topology
using the methods of Batyrev and Borisov. Finally we end with short
discussions of the higher-loop cases and of the supersymmetrization.

Figure 1: The traintrack integrals

##  2 Two loops: the elliptic double box

###  2.1 Construction

We consider the two-loop traintrack diagram, i.e. the two-loop version of the
class of diagrams depicted in fig. 1. Its leading singularity is determined as
follows. There are three dual points $x_{1}$, $x_{2}$, $x_{3}$ corresponding
to the left loop and three dual points $x_{4}$, $x_{5}$, $x_{6}$ corresponding
to the right loop. The left loop internal dual point $x_{\ell_{1}}$ has to be
light-like separated from the three dual points $x_{1}$, $x_{2}$, $x_{3}$. The
right loop internal dual point $x_{\ell_{2}}$ has to be light-like separated
from the three dual points $x_{4}$, $x_{5}$, $x_{6}$. Finally, the points
$x_{\ell_{1}}$ and $x_{\ell_{2}}$ have to be light-like separated.

In momentum twistor space this can be described as follows. To each dual point
$x_{i}$ we associate a line $A_{i}\wedge B_{i}$ in momentum twistor space
$\mathbb{P}^{3}$. Two dual points are light-like separated if their
corresponding lines in $\mathbb{P}^{3}$ intersect. At first, we assume that
all the lines corresponding to external dual points are skew (do not meet in
$\mathbb{P}^{3}$). When some of these lines intersect, the geometry
simplifies.

Given three skew lines, there is a one-dimensional family of lines which
intersect all of them. This can be seen by using several fundamental results
about quadrics in $\mathbb{P}^{3}$. The first fact is that three skew lines
uniquely determine a non-singular quadric $Q$. The second fact is that a non-
singular quadric $Q$ in $\mathbb{P}^{3}$ contains two families of lines where
the lines in a given family are skew while two lines in different families
always intersect. Finally, through a given point passes a unique line from
each family of lines. Such families of lines on a quadric are called
_rulings_.

More concretely, given three skew lines $A_{i}\wedge B_{i}$ for $i=1,2,3$, the
quadric they determine can be written as

$\displaystyle Q(Z)=\langle ZA_{1}B_{1}A_{3}\rangle\langle
ZA_{2}B_{2}B_{3}\rangle-\langle ZA_{1}B_{1}B_{3}\rangle\langle
ZA_{2}B_{2}A_{3}\rangle.$ (2.1)

Here $Z$, $A_{i}$ and $B_{i}$ are points in $\mathbb{P}^{3}$ and $\langle
ABCD\rangle=\det(A,B,C,D)$ is the usual four-bracket of momentum twistors. The
three lines appear symmetrically, but this is not manifest in the formula
above. Using Plücker relations one can show that the symmetry holds.

Figure 2: Relationship between the endcap of the traintrack and the quadric.

Then, to the dual points $x_{1}$, $x_{2}$, $x_{3}$ neighboring the left loop
we can associate a quadric $Q_{L}$ and to the points $x_{4}$, $x_{5}$, $x_{6}$
neighboring the right loop we can associate a quadric $Q_{R}$; cf. fig. 2.
Next, consider the intersection
$C\text{\makebox[15.5pt][c]{$\displaystyle\text{\makebox[0.0pt][r]{$\raisebox{0.47pt}{\hskip
1.25pt:}$}}=\text{\makebox[0.0pt][l]{$$}}$}}Q_{L}\cap
Q_{R}\subset\mathbb{P}^{3}$ of these two quadrics, which is a curve. To each
point on $C$ we can associate a line in $Q_{L}$ which intersects all the three
lines determining $Q_{L}$. This line corresponds to the interior dual point
$x_{\ell_{1}}$ of the left loop. Similarly, through the same point of $C$ we
can construct a line which intersects all the lines in $Q_{R}$ corresponding
to the interior dual point $x_{\ell_{2}}$. The line in $Q_{L}$ and the one in
$Q_{R}$ intersect in a point in $C$ so their corresponding dual points are
also light-like separated as required for the leading singularity.

Figure 3: Two intersecting quadrics. Their intersection is the genus-one curve
$C$ in the elliptic double box.

The intersection of two quadrics in $\mathbb{P}^{3}$ is a genus-one algebraic
curve, see fig. 3.

We can connect this construction to the more familiar picture of a cubic curve
in $\mathbb{P}^{2}$ as follows: Without loss of generality, we can take the
point $[X_{0}:X_{1}:X_{2}:X_{3}]=[0:0:0:1]$ to belong to both quadrics. Then
the equations for the two quadrics can be written as

$Q_{L}=X_{3}L_{L}+M_{L},\qquad Q_{R}=X_{3}L_{R}+M_{R},$ (2.2)

where $L_{L}$ and $L_{R}$ are of homogeneous of degree one and $M_{L}$ and
$M_{R}$ are homogeneous of degree two in $X_{0}$, $X_{1}$ and $X_{2}$. When
eliminating $X_{3}$, we obtain $L_{L}M_{R}-L_{R}M_{L}=0$, which is a cubic in
$\mathbb{P}^{2}$.

###  2.2 Analysis of the two-loop leading singularity locus

Having constructed a genus-one curve $C$ as the intersection of two quadrics
in $\mathbb{P}^{3}$, we now proceed to analyze its properties.

The holomorphic differential one-form on the curve can be found by taking
Poincaré residues,

$\displaystyle\omega_{C}=\operatorname{Res}_{Q_{L}}\operatorname{Res}_{Q_{R}}\frac{\omega_{\mathbb{P}^{3}}}{Q_{L}Q_{R}}.$
(2.3)

Here $\omega_{\mathbb{P}^{3}}$ is the $\operatorname{PSL}(4)$-invariant,
weight-four holomorphic three-form on $\mathbb{P}^{3}$. The quadrics $Q_{L}$
and $Q_{R}$ both have weight two so that the ratio
$\frac{\omega_{\mathbb{P}^{3}}}{Q_{L}Q_{R}}$ is invariant under rescaling of
the homogeneous coordinates of $\mathbb{P}^{3}$. Then, we take two Poincaré
residues which yields a one-form localized on $C$. This is in fact the unique
holomorphic one-form on $C$ so the curve $C$ is indeed a genus-one curve. A
genus-one curve is characterized by only one modulus, which can be taken to be
its $j$-invariant.

We can also see that there is only one modulus by counting parameters as
follows: There are six dual points, each with four coordinates. From this, we
need to subtract the dimension of the four-dimensional conformal group, which
is $15$. In total we obtain $6\times 4-15=9$, assuming the conformal group
acts effectively. However, there are configurations of the three skew lines in
the left quadric which generate the same quadric. Indeed, consider a line
inside $Q_{L}$ which intersects all the lines which determine $Q_{L}$. We can
displace any of these three lines along the chosen line without changing
$Q_{L}$. Hence, there is a three-dimensional space of three skew lines which
parametrize the same quadric $Q_{L}$. The same holds for $Q_{R}$. Moreover,
the same curve $C$ can be obtained by considering any two members of the so-
called _pencil of quadrics_ generated by $Q_{L}$ and $Q_{R}$.222A pencil is a
set of subvarieties, in this case quadrics, which are parametrized by a line
[21]. In other words, instead of using $Q_{L}$ and $Q_{R}$ we can use linear
combinations of them, $\lambda_{L}Q_{L}+\lambda_{R}Q_{R}$ and
$\mu_{L}Q_{L}+\mu_{R}Q_{R}$, where $[\lambda_{L}:\lambda_{R}]$ and
$[\mu_{L}:\mu_{R}]$ are homogeneous coordinates on a projective line. This
amounts to two extra parameters which do not appear in the moduli of $C$. In
the end, $C$ has $9-3-3-2=1$ moduli.

The pencil of quadrics $\lambda_{L}Q_{L}+\lambda_{R}Q_{R}$ also allows us to
compute the $j$-invariant of the curve $C$. As mentioned above, $C$ is
obtained as the intersection of any two members of the pencil. We now think of
each of the quadrics as a $4\times 4$ symmetric matrix of the coefficients in
the defining equation (2.1) and consider the determinant

$\displaystyle\det(\lambda_{L}Q_{L}+\lambda_{R}Q_{R}).$ (2.4)

This is a polynomial of degree four in the homogeneous coordinates
$[\lambda_{L}:\lambda_{R}]$ of $\mathbb{P}^{1}$. Hence, it vanishes at four
points in $\mathbb{P}^{1}$ and we conclude that there are four singular
members of the pencils.333Note that we assume that the quadrics $Q_{L}$ and
$Q_{R}$ are in general position such that the four roots of (2.4) are
distinct. If they are not, then the intersection degenerates and the integral
can be computed in terms of generalized polylogarithms. The cross-ratio of
these four points is an invariant of the pencil. More concretely, let us
denote the four points where (2.4) vanishes by
$\lambda^{i}\text{\makebox[15.5pt][c]{$\displaystyle\text{\makebox[0.0pt][r]{$\raisebox{0.47pt}{\hskip
1.25pt:}$}}=\text{\makebox[0.0pt][l]{$$}}$}}[\lambda_{L}^{i}:\lambda_{R}^{i}]$.
Then, we can form the cross-ratio $z=\frac{\langle 12\rangle\langle
34\rangle}{\langle 13\rangle\langle 24\rangle}$, where $\langle
ij\rangle=\det(\lambda^{i},\lambda^{j})$, and the $j$-invariant

$\displaystyle j=256\frac{(z^{2}-z+1)^{3}}{z^{2}(z-1)^{2}}.$ (2.5)

As pointed out above, the curve $C$ is obtained as the intersection of _any_
two members of the pencil of quadrics $\lambda_{L}Q_{L}+\lambda_{R}Q_{R}$.
Thus we can characterize isomorphism classes of $C$ by completely
characterizing the pencil. The cross-ratio $z$ formed above classifies the
isomorphism classes of four ordered points on $\mathbb{P}^{1}$ up to
projective equivalence. The $j$-invariant formed in (2.5) has the correct
symmetries for the corresponding elliptic curve: In defining the cross-ratio
$z$, we have the freedom of permuting three of the points $\lambda^{i}$ on
$\mathbb{P}^{1}$ while keeping one fixed without changing $C$. This
permutation acts on $z$ by sending $z\mapsto
z^{\prime}\in\\{z,\frac{1}{z},1-z,1-\frac{1}{z},\frac{1}{1-z},1-\frac{1}{1-z}\\}$.
One can check that the $j$-invariant in (2.5) is invariant under this map.

In [6], the elliptic double box integral was analyzed using the method of
direct integration. Starting from a dual-conformally invariant expression,
Feynman parameters were introduced and as many integrations as possible were
performed in terms of multiple polylogarithms. Eventually, the authors found a
representation of the double box integral of the form

$\displaystyle\int_{0}^{\infty}d\alpha\,\frac{H_{3}(\alpha)}{\sqrt{Q(\alpha)}}.$
(2.6)

Here $H_{3}$ is a combination of weight-three multiple polylogarithms and
$Q(\alpha)$ is a polynomial in $\alpha$ of degree four with coefficients
depending on conformal cross-ratios. The equation $y^{2}=Q(\alpha)$ thus
defines an elliptic curve. We have checked that the $j$-invariant of this
curve matches the $j$-invariant of the curve constructed directly in momentum
twistor space above. This is an encouraging result as it means that the
geometry is not merely an artifact of the chosen parametrization but an
intrinsic property of the leading singularity of the double box integral.

##  3 Three and more loops

###  3.1 K3 surface

####  3.1.1 Construction

The construction of a geometry for the three-loop traintrack integral is
similar to the one for the two-loop case presented in section 2. This time,
however, we have two extra lines in momentum twistor space corresponding to
the two additional external dual points. The geometry in this case is given by
two quadrics $Q_{L}$ and $Q_{R}$, constructed in the same way as at two loops,
together with two lines $\ell_{1}$ and $\ell_{2}$. Given points
$P_{1}\in\ell_{1}$ and $P_{2}\in\ell_{2}$, we can construct a line
$P_{1}\wedge P_{2}$ whose corresponding dual point is light-like separated
from both dual points corresponding to $\ell_{1}$ and $\ell_{2}$. The line
$P_{1}\wedge P_{2}$ corresponds to the middle loop in the the three-loop
traintrack integral. The moduli space of these lines is
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ corresponding to the freedom in choosing
$P_{1}$ and $P_{2}$. We illustrate the construction in fig. 4.

Figure 4: Quadrics and lines defining the K3 surface in the three-loop
traintrack diagram.

The rest of the light-like constraints for the leading singularity can be
imposed as follows. By Bezout’s theorem, the line $P_{1}\wedge P_{2}$
intersects the quadric $Q_{L}$ in two points and the quadric $Q_{R}$ in two
points.444Bezout’s theorem states that $n$ hypersurfaces of degrees
$d_{1},\dotsc,d_{n}$ in complex projective space $\mathbb{P}^{n}$ intersect in
$d_{1}\cdots d_{n}$ points, if the number of intersection points is finite
[21]. In our case, the quadric has degree two, while a line can be seen as the
intersection of two hyperplanes, each of degree one. Hence, the intersection
consists of two points.  Choosing one of these intersections in $Q_{L}$ and
one in $Q_{R}$, we obtain a leading singularity configuration. In total, there
are four choices. The total space of leading singularities is therefore a
four-fold cover of $\mathbb{P}^{1}\times\mathbb{P}^{1}$, branched over the
curves where the line $P_{1}\wedge P_{2}$ is tangent to $Q_{L}$ or $Q_{R}$.

To find out where this branching arises, consider the points
$\alpha_{1}P_{1}+\alpha_{2}P_{2}$ on the line $P_{1}\wedge P_{2}$. The
intersection with $Q_{L}$ is given by the equation

$\displaystyle\alpha_{1}^{2}\,Q_{L}(P_{1},P_{1})+2\alpha_{1}\alpha_{2}\,Q_{L}(P_{1},P_{2})+\alpha_{2}^{2}\,Q_{L}(P_{2},P_{2})=0.$
(3.1)

The line $P_{1}\wedge P_{2}$ is tangent to $Q_{L}$ if this has a double root,
i.e. when the discriminant with respect to $\alpha_{1}$ or $\alpha_{2}$
vanishes,

$\displaystyle\Delta_{L}\text{\makebox[15.5pt][c]{$\displaystyle\text{\makebox[0.0pt][r]{$\raisebox{0.47pt}{\hskip
1.25pt:}$}}=\text{\makebox[0.0pt][l]{$$}}$}}Q_{L}(P_{1},P_{2})^{2}-Q_{L}(P_{1},P_{1})Q_{L}(P_{2},P_{2})=0.$
(3.2)

The polynomial $\Delta_{L}$ is homogeneous of bidegree $(2,2)$ in the
coordinates of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ that parametrize the
points $P_{1}\in\ell_{1}$ and $P_{2}\in\ell_{2}$.

A similar analysis can be done for the right quadric and we obtain another
polynomial $\Delta_{R}$ of bidegree $(2,2)$. The curves determined by
$\Delta_{L}$ and $\Delta_{R}$ intersect in eight points.555To see why,
consider first the intersection of such a genus-one curve with a line in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ which sits at a point in the first or the
second $\mathbb{P}^{1}$. It is easy to see that this intersection consists of
two points. Now, consider a degeneration of the biquadratic into four lines.
Two of the lines sit at a point in the first $\mathbb{P}^{1}$ while the other
two sit at a point in the second $\mathbb{P}^{1}$. Each one of them intersects
the biquadratic in two points. In total, there are eight intersection points.
As we deform from a singular curve consisting of four lines to a non-singular
one, the number of intersections is conserved. This type of argument is often
used in Schubert problems (see ref. [22] for a detailed discussion). At these
eight points, all the branches of the surface meet. Over the remaining points
of the curves determined by $\Delta_{L}$ and $\Delta_{R}$ there are only two
branches, while over the remaining points of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ there are four branches.

The curves in $\mathbb{P}^{1}\times\mathbb{P}^{1}$ defined by the vanishing
locus of $\Delta_{L}$ and $\Delta_{R}$ are themselves genus-one curves as can
be seen as follows. If we choose coordinates $x=[x_{0}:x_{1}]$ and
$y=[y_{0}:y_{1}]$ on $\mathbb{P}^{1}\times\mathbb{P}^{1}$, then we can write
the equation for a biquadratic as

$\displaystyle\Delta(x,y)=\sum_{a,b,a^{\prime},b^{\prime}=0}^{1}A_{ab,a^{\prime}b^{\prime}}\,x_{a}x_{b}\,y_{a^{\prime}}y_{b^{\prime}},$
(3.3)

where $A$ is symmetric in the first and second pair of indices independently
and thus has 9 independent components. We now embed
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ into $\mathbb{P}^{3}$ using the Segre
map. Concretely, we identify the homogeneous coordinates
$[z_{0}:z_{1}:z_{2}:z_{3}]$ on $\mathbb{P}^{3}$ with the coordinates on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ as

$\displaystyle
z_{0}=x_{0}y_{0},\;z_{1}=x_{0}y_{1},\;z_{2}=x_{1}y_{0},\;z_{3}=x_{1}y_{1}.$
(3.4)

The image of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is then a quadric in
$\mathbb{P}^{3}$ given by $z_{0}z_{3}-z_{1}z_{2}=0$. The biquadratic (3.3)
becomes

$\displaystyle\Delta(z)=\sum_{i,j=0}^{3}\tilde{A}_{ij}\,z_{i}z_{j},$ (3.5)

where $\tilde{A}$ is a $4\times 4$ symmetric matrix that depends on the
original coefficients $A_{ab,a^{\prime}b^{\prime}}$. This defines another
quadric in $\mathbb{P}^{3}$. The intersection of these two quadrics is a
genus-one curve with only one modulus, as we have discussed before.

####  3.1.2 Analysis

The holomorphic two-form on the surface is

$\displaystyle\omega_{K3}=\frac{\omega_{\mathbb{P}^{1}}\,\omega_{\mathbb{P}^{1}}}{\sqrt{\Delta_{L}}\sqrt{\Delta_{R}}}.$
(3.6)

Notice that this ratio has the right homogeneity in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$: The first $\omega_{\mathbb{P}^{1}}$ has
bidegree $(2,0$) while the second one has bidegree $(0,2)$. The polynomials
$\Delta_{L}$ and $\Delta_{R}$ both have bidegree $(2,2)$ so that (3.6) has
homogeneity zero as required.

An analogous construction can be done for the simpler case of a genus-one
curve in $\mathbb{P}^{2}$ as a two-fold branched cover over four points in
$\mathbb{P}^{1}$. In that case, we can define a polynomial $P$ whose roots are
the four points and the holomorphic form is
$\frac{\omega_{\mathbb{P}^{2}}}{\sqrt{P}}$.

##### Euler characteristic

It is well-known that the Euler characteristic $\chi$ of a K3 surface is $24$,
but we can directly compute this from the construction in momentum twistor
space. To do so, we will use the basic fact that $\chi$ is additive under
surgery.

According to the branching described above, the K3 surface $S$ has only one
branch on the points $\mathbb{P}^{1}\times\mathbb{P}^{1}$ where the two curves
$\Delta_{L}$ and $\Delta_{R}$ meet, i.e. for the points in
$\Delta_{L}\cap\Delta_{R}$. For the points that lie on either of the two
curves, i.e. for $\Delta_{L}\cup\Delta_{R}\setminus\Delta_{L}\cap\Delta_{R}$,
there are two branches. In the complement of the two curves, i.e. in
$\mathbb{P}^{1}\times\mathbb{P}^{1}\setminus\Delta_{L}\cup\Delta_{R}$, there
are four branches. It follows that

$\displaystyle\begin{split}\chi(S)&=4\left[\chi(\mathbb{P}^{1}\times\mathbb{P}^{1})-\chi(\Delta_{L}\cup\Delta_{R})\right]+2\left[\chi(\Delta_{L}\cup\Delta_{R})-\chi(\Delta_{L}\cap\Delta_{R})\right]\\\
&\quad+\chi(\Delta_{L}\cap\Delta_{R})\\\
&=4\chi(\mathbb{P}^{1}\times\mathbb{P}^{1})-2\chi(\Delta_{L}\cup\Delta_{R})-\chi(\Delta_{L}\cap\Delta_{R}).\end{split}$
(3.7)

Next, we use the fact that
$\chi(\mathbb{P}^{1}\times\mathbb{P}^{1})=\chi(\mathbb{P}^{1})^{2}$,
$\chi(\mathbb{P}^{1})=2$ and
$\chi(\Delta_{L}\cup\Delta_{R})=\chi(\Delta_{L})+\chi(\Delta_{R})-\chi(\Delta_{L}\cap\Delta_{R})$.
The Euler characteristic of a point is one and the intersection
$\Delta_{L}\cap\Delta_{R}$ consists of eight points, thus we get
$\chi(\Delta_{L}\cap\Delta_{R})=8$. Moreover, $\Delta_{L}$ and $\Delta_{R}$
are genus-one curves, thus $\chi(\Delta_{L})=\chi(\Delta_{R})=0$. Finally, we
get

$\displaystyle\chi(S)=4\times 2\times 2-2\times(-8)-8=24.$ (3.8)

This is the expected number for a K3 surface which has Betti numbers
$b_{0}=1$, $b_{2}=22$ and $b_{4}=1$ with the odd Betti numbers vanishing.

##### Counting the number of moduli

We would now like to count the number of moduli of these K3 surfaces. This
amounts to a counting of degrees of freedom of two genus-one curves in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, intersecting in eight points. On top of
that, there are moduli that roughly speaking describe the position of the
quadrics corresponding to the endcaps of the traintrack integrals.

Before solving the first problem, recall the more familiar case of two cubic
curves in the projective plane $\mathbb{P}^{2}$. A cubic curve in the
projective plane is a non-zero linear combination of ten monomials. Hence, the
set of cubic curves forms a $\mathbb{P}^{9}$. The condition that a point
belongs to a cubic curve imposes a linear condition in $\mathbb{P}^{9}$. Given
nine points in _general_ position, there is a single cubic curve which
contains all of them. The condition that the nine points be generic is
essential here. In fact, consider two cubics in the projective plane. By
Bezout’s theorem, they intersect in nine points. In this case, these nine
points can not be generic since they do not uniquely determine a cubic curve.
In fact, they determine a pencil of cubics.

The theorem of Cayley-Bacharach states that if two plane cubics intersect in
nine points, then any other cubic which passes through eight of them
automatically passes through the ninth [21].666The Cayley-Bacharach theorem is
essential in proving the associativity of the group law on a genus-one curve.

Let us now return to genus-one curves in $\mathbb{P}^{1}\times\mathbb{P}^{1}$.
A biquadratic curve in $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is a linear
combination of nine monomials of bidegree $(2,2)$. Hence, these curves form a
$\mathbb{P}^{8}$. As before, the condition that a point belongs to such a
curve is a linear condition in $\mathbb{P}^{8}$. Hence, eight points in
_general_ position uniquely determine a genus-one curve in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Next, consider two such biquadratic curves. They intersect in eight points. If
the equations of the two biquadratics in homogeneous coordinates
$x=[x_{0}:x_{1}]$ and $y=[y_{0}:y_{1}]$ of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ are

$\displaystyle\Delta_{00}(y)x_{0}^{2}+2\Delta_{01}(y)x_{0}x_{1}+\Delta_{11}(y)x_{1}^{2}=0,$
(3.9)
$\displaystyle\Delta_{00}^{\prime}(y)x_{0}^{2}+2\Delta_{01}^{\prime}(y)x_{0}x_{1}+\Delta_{11}^{\prime}(y)x_{1}^{2}=0,$
(3.10)

then the intersection points have $y$ coordinates satisfying

$(\Delta_{00}^{\prime}\Delta_{11}-\Delta_{00}\Delta_{11}^{\prime})^{2}+4(\Delta_{00}^{\prime}\Delta_{01}-\Delta_{00}\Delta_{01}^{\prime})(\Delta_{11}^{\prime}\Delta_{01}-\Delta_{01}^{\prime}\Delta_{11})=0.$
(3.11)

Here $\Delta_{ij}$ and $\Delta_{ij}^{\prime}$ are quadratic in $y$ such that
this is a degree-eight polynomial and that generically there are eight such
intersection points. For each of these values of $y$ the corresponding value
of $x\in\mathbb{P}^{1}$ is given by

$2(\Delta_{00}^{\prime}\Delta_{01}-\Delta_{00}\Delta_{01}^{\prime})x_{0}+(\Delta_{00}^{\prime}\Delta_{11}-\Delta_{00}\Delta_{11}^{\prime})x_{1}=0.$
(3.12)

These eight points can not be in general position, otherwise there would be a
unique biquadratic curve containing them. For this case, we have a variant of
the Cayley-Bacharach theorem, stating that if two biquadratic curves meet in
seven points then they meet in the eighth as well.

Returning to the problem of counting the moduli, we see that we have to
specify seven points in $\mathbb{P}^{1}\times\mathbb{P}^{1}$ which amounts to
14 parameters. From this we have to subtract $2\times 3$ parameters due to
$\operatorname{PSL}(2)$ transformations on each $\mathbb{P}^{1}$. Moreover, we
need to pick two members of the pencil of quadrics
$\lambda_{L}Q_{L}+\lambda_{R}Q_{R}$ which adds two additional moduli. It turns
out that there is one more modulus corresponding to the relative position of
the left and right quadric along the middle line through the points $P_{1}$
and $P_{2}$. In total, the number of moduli is

$\displaystyle 14-2\times 3+2+1=11.$ (3.13)

There is another, more direct way to establish 11 as an upper bound for the
number of moduli: The K3 surface only depends on the left and right quadrics
and the two lines $\ell_{1}$ and $\ell_{2}$. In dual space we have $8\times
4-15=17$, where we subtracted $15$ due to the action of the conformal group.
As discussed in section 2.2, we can move each of the three lines defining a
quadric up and down along a line from the opposite ruling without changing the
quadric. Thus we can subtract $2\times 3=6$ coordinates. In total we get
$8\times 4-15-6=11$ moduli.

For algebraic K3 surfaces, the sum of the dimension of the moduli space and
the generic Picard rank has to equal $20$ (see ref. [23]). Since we found a
moduli space of dimension $11$, then the generic Picard rank should be $9$.
Below, we find the same answer by looking at Nikulin involutions.

In [15], the authors analyzed the three-loop traintrack integral using Feynman
parameters and identified a K3 surface as a hypersurface in a certain weighted
projective space. For a generic hypersurface in this space they found an upper
bound of 18 for the number of moduli which is compatible with the number that
we found above. In the case of the elliptic curve we were able to compare the
momentum twistor construction to the one found in Feynman-parametric
integration using the $j$-invariant of the curve and found that they give the
same geometry. For the K3 surfaces, a more thorough study of their
characteristics is needed to conclude whether or not they are equal.

##### Automorphisms and Nikulin involutions

To further characterize the K3 surface $S$, we study its automorphisms, in
particular those automorphisms that leave the holomorphic two-form on $S$
invariant. Such automorphisms are called symplectic. If $f$ is a symplectic
automorphism of finite order $n$ and $f\neq\operatorname{id}$, then one can
show that the set of fixed points $\operatorname{Fix}(f)\subset S$ is non-
empty and finite. Moreover, the number of fixed points satisfies
$1\leq\left|\operatorname{Fix}(f)\right|\leq 8$ and depends only on the order
$n$ of $f$, see for example ref. [24]. Nikulin [25] also showed that the order
$n$ can at most be eight, i.e. $n\leq 8$, which means that only the
combinations of $n$ and $\left|\operatorname{Fix}(f)\right|$ in table 1 are
possible.

Symplectic automorphisms of order two are called _Nikulin involutions_ and the
corresponding number of fixed points is eight. Such involutions are realized
in our K3 surface as follows.

Consider the left quadric $Q_{L}$ and the line $P_{1}\wedge P_{2}$ transversal
to $\ell_{1}$ and $\ell_{2}$, see also fig. 4. $P_{1}\wedge P_{2}$ intersects
$Q_{L}$ in two points and exchanging these two points constitutes an
involution of the left quadric. Recall that the points of intersection are
given by the two roots of (3.1). Since this in a quadratic equation, the
difference between the two roots is $\sqrt{\Delta_{L}}$. Thus, exchanging the
two points of intersection, sends $\sqrt{\Delta_{L}}$ to $-\sqrt{\Delta_{L}}$.
The fixed points of this involution of the left quadric are the points of
$Q_{L}$ at which $P_{1}\wedge P_{2}$ becomes tangent, i.e. the points
described by the genus-one curve $\Delta_{L}=0$ in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. Since the map we described so far
changes the sign of $\sqrt{\Delta_{L}}$, the holomorphic two-form (3.6) also
changes sign and we only obtain a Nikulin involution of the K3 surface if we
perform the same involution on the right quadric. The fixed points are then
the eight intersection points of the curves $\Delta_{L}$ and $\Delta_{R}$ in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$.

An involution which is not symplectic is the exchange of the two
$\mathbb{P}^{1}$ corresponding to the lines $\ell_{1}$ and $\ell_{2}$. Indeed,
under this transformation the holomorphic two-form in eq. (3.6) picks up a
sign.

The existence of automorphisms implies a lower bound for the Picard number
$\rho(S)$ of the K3 surface [24]. For a Nikulin involution, i.e. a symplectic
automorphism of order two, the bound is $\rho(S)\geq 9$ (see Appendix. A).
Since the Picard number plus the dimension of the moduli space are equal to
20, this bound is consistent with the counting of the moduli above. In fact in
our case the bound is satisfied, i.e. $\rho(S)=9$; for this case a complete
description of the Picard lattice of $S$ can be found in ref. [26].

###  3.2 Three-fold and beyond

In this section, we demonstrate how we can build a Calabi-Yau manifold
embedded in a toric variety for the four- and higher-loop traintrack
integrals. It was shown by Batyrev that mirror families of Calabi-Yau
manifolds can be constructed as anticanonical hypersurfaces in toric varieties
and that their Hodge numbers can be computed combinatorially by counting
points in an associated pair of reflexive polytopes [27]. This construction
was generalized to complete intersection Calabi-Yau (CICY) manifolds by
Batyrev and Borisov using the nef-partitions of a reflexive polytope pair [28,
29]. The Hodge numbers in this case can be computed by means of a recursive
generating function; an implementation of this function is available in PALP
[30].777Note that technically the generating function computes the _stringy_
Hodge numbers introduced in [31].

####  3.2.1 Three-fold

The leading singularity configuration for the four-loop traintrack integral is
depicted in fig. 5. Compared to the three-loop case discussed in section 3.1,
we have two new lines, $\ell_{3}$ and $\ell_{4}$, corresponding to the two
extra external dual points.

Let us introduce coordinates $([\alpha_{1}:\alpha_{2}],[\beta_{1}:\beta_{2}])$
for the $\mathbb{P}^{1}\times\mathbb{P}^{1}$ corresponding to the lines
$\ell_{1}$ and $\ell_{2}$ and similarly
$([\gamma_{1}:\gamma_{2}],[\delta_{1}:\delta_{2}])$ for the lines $\ell_{3}$
and $\ell_{4}$. Then the embedding space is a toric variety defined by the
relations

$\displaystyle\begin{split}(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},y_{L})&\sim(t_{1}\,\alpha_{1},t_{1}\,\alpha_{2},\beta_{1},\beta_{2},t_{1}\,y_{L}),\\\
(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},y_{L})&\sim(\alpha_{1},\alpha_{2},t_{2}\,\beta_{1},t_{2}\,\beta_{2},t_{2}\,y_{L})\end{split}$
(3.14)

for the left part of fig. 5 and

$\displaystyle\begin{split}(\gamma_{1},\gamma_{2},\delta_{1},\delta_{2},y_{R})&\sim(t_{3}\,\gamma_{1},t_{3}\,\gamma_{2},\delta_{1},\delta_{2},t_{3}\,y_{R}),\\\
(\gamma_{1},\gamma_{2},\delta_{1},\delta_{2},y_{R})&\sim(\gamma_{1},\gamma_{2},t_{4}\,\delta_{1},t_{4}\,\delta_{2},t_{4}\,y_{R})\end{split}$
(3.15)

from the right part. Here
$t_{1},t_{2},t_{3},t_{4}\in\mathbb{C}\setminus\\{0\\}$ and the role of $y_{L}$
and $y_{R}$ will be clarified momentarily. Since we have ten coordinates and
four relations, we are left with a six-dimensional space.

Following the same construction as for the three-loop (K3) case, we obtain two
polynomials $\Delta_{L}$ and $\Delta_{R}$ of bidegree $(2,2)$ in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ from the left and right outermost loop of
the traintrack. In the six-dimensional toric variety constructed above, the
Calabi-Yau manifold is defined as a codimension-three subvariety by means of
the constraints

$\displaystyle y_{L}^{2}=\Delta_{L},\quad y_{R}^{2}=\Delta_{R},\quad\langle
P_{1}P_{2}P_{3}P_{4}\rangle=0.$ (3.16)

The last condition forces the two transversals $P_{1}\wedge P_{2}$ and
$P_{3}\wedge P_{4}$ to intersect, see also fig. 5.

The toric variety defined by the relations (3.14) and (3.15) can be described
by a polytope with ten vertices in a six-dimensional integer lattice.
Explicitly, the vertices are given by the columns of the matrix

$\displaystyle\begin{pmatrix}[r]1&0&0&1&-1&0&0&0&0&0\\\
0&1&0&1&-1&0&0&0&0&0\\\ 0&0&1&-1&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&1&-1\\\
0&0&0&0&0&0&1&0&1&-1\\\ 0&0&0&0&0&0&0&1&-1&0\end{pmatrix}.$ (3.17)

The Hodge numbers of a generic codimension-three subvariety in this space can
be obtained by computing the nef-partitions of the polytope defined by (3.17).
Using PALP [30], in particular the component nef.x888Note that we had to set
VERT_Nmax to 96 in Global.h for the computation to succeed., we find that
there are 22 nef partitions. Out of these, we identify three that have
defining equations with degrees compatible with the constraints (3.16). The
Hodge numbers are $h^{11}=12$ and $h^{12}=28$ which gives a Euler
characteristic of $\chi=-32$.

Figure 5: Quadrics and lines defining the CY three-fold in the four-loop
traintrack diagram.

####  3.2.2 General case

The construction used for the three-fold, i.e. the four-loop case of the
traintracks, generalizes to higher loops. For $L\geq 4$, we build a toric
embedding space as follows: There are $2+4(L-2)$ coordinates, $2$ from $y_{L}$
and $y_{R}$ and $2\times 2(L-2)$ from the two external dual points added with
each loop. The number of relations between these coordinates is $2(L-2)$; thus
the dimension of the embedding space is $2+4(L-2)-2(L-2)=2(L-1)$. In this
space, we impose $2$ quadratic constraints, namely $y_{L}^{2}=\Delta_{L}$ and
$y_{R}^{2}=\Delta_{R}$, as well as $L-3$ multilinear constraints. Thus, the
Calabi-Yau manifold is obtained as a subvariety of codimension $L-1$ in a
toric variety of dimension $2(L-1)$. Note that the dimension of the manifold
is also $L-1$.

As above, we can describe the embedding space by a polytope with vertices in
an integer lattice. The dimension of this lattice equals the dimension of the
embedding space, i.e. $2(L-1)$, while the number of vertices is equal to the
number of coordinates, $2+4(L-2)$. The vertices are given in the general case
by the columns of a block-diagonal matrix

$\displaystyle\begin{pmatrix}\mathbf{A}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{0}\\\
\mathbf{0}&\mathbf{A}&\mathbf{0}&&\\\ \mathbf{0}&\mathbf{0}&\mathbf{B}&&\\\
\vdots&&&\ddots&\vdots\\\
\mathbf{0}&&&\cdots&\mathbf{B}\end{pmatrix},\qquad\mathbf{A}=\begin{pmatrix}[r]1&0&0&1&-1\\\
0&1&0&1&-1\\\
0&0&1&-1&0\end{pmatrix},\qquad\mathbf{B}=\begin{pmatrix}[r]1&-1\end{pmatrix}.$
(3.18)

Note that in the case of the threefold (i.e. $L=4$) that was discussed above,
$\mathbf{B}$ does not appear and the matrix reduces to (3.17).

We note that the codimension grows with the loop order and this makes the
analysis of these varieties in terms of complete intersections more
challenging. One may hope for a more “efficient” description of these
varieties, but it remains to be seen if this is possible in way which is
compatible with supersymmetry, as described in sec. 4.

##  4 Supersymmetrization

The constructions presented so far are manifestly dual-conformal invariant.
Indeed, this is one reason why it makes sense to use momentum twistors to
describe their geometry. However, we know that the scattering amplitudes in
$\mathcal{N}=4$ are in fact dual _super_ -conformal invariant. It is then
natural to ask what becomes of the supersymmetry.

In order to describe the supersymmetrization, we will redo the previous
analysis in such a way that the various incidence relations are described in
terms of $\operatorname{PSL}(4)$-invariant delta functions. The basic
ingredient will be the delta function of two points on $\mathbb{P}^{3}$, which
we denote by $\delta_{\mathbb{P}^{3}}^{3}(P_{1};P_{2})$, where
$P_{1},P_{2}\in\mathbb{P}^{3}$.

This quantity can be used to define $\delta^{2}_{\mathbb{P}^{3}}(L;P)$, which
has support when the point $P$ lies on the line $L$. If the line $P$ contains
two points $P_{0}$ and $P_{1}$, then we have

$\delta_{\mathbb{P}^{3}}^{2}(L;P)=\int\omega_{\mathbb{P}^{1}}(\alpha)\delta_{\mathbb{P}^{3}}(\alpha_{0}P_{0}+\alpha_{1}P_{1};P).$
(4.1)

Similarly, we can define $\delta_{\mathbb{P}^{3}}(L_{1};L_{2})$, which has
support when the two lines $L_{1}$ and $L_{2}$ intersect.

To define a delta function with support on a quadric, we use the fact that the
quadric is determined by three skew lines $L_{1}$, $L_{2}$ and $L_{3}$. The
quadric is ruled by a family of lines which intersect $L_{1}$, $L_{2}$ and
$L_{3}$. Moreover, through any point on the quadric passes one line in this
ruling. We can then describe the conditions that a point $P$ belongs to the
quadric $Q$ determined by the skew lines $L_{1}$, $L_{2}$ and $L_{3}$ by the
following integral

$\delta_{\mathbb{P}^{3}}(Q;P)=\int\mu_{\mathbb{P}^{3}}(L)\delta_{\mathbb{P}^{3}}(L;L_{1})\delta_{\mathbb{P}^{3}}(L;L_{2})\delta_{\mathbb{P}^{3}}(L;L_{3})\delta_{\mathbb{P}^{3}}^{2}(L;P),$
(4.2)

where $\mu_{\mathbb{P}^{3}}(L)$ is the integral over the space of lines in
$\mathbb{P}^{3}$. This integral is four-dimensional so, after performing the
integrals, we are left with a single constraint. This is expected since a
quadric is of codimension one in $\mathbb{P}^{3}$.

To obtain the genus-one curve we simply take the product of the two delta
functions corresponding to $Q_{L}$ and $Q_{R}$. This is a distribution which
has support on the intersection of the two quadrics $Q_{L}\cap Q_{R}$. We can
also obtain the holomorphic top form, but instead of taking Poincaré residues,
we proceed as follows. We look for a one-form $\omega_{C}$ such that

$\int_{C}\omega_{C}(Z)f(Z)=\int\omega_{\mathbb{P}^{3}}(Z)\delta_{\mathbb{P}^{3}}(Q_{L};Z)\delta_{\mathbb{P}^{3}}(Q_{R};Z)f(Z),$
(4.3)

for any meromorphic function $f$ on $\mathbb{P}^{3}$ whose poles lie outside
$Q_{L}\cap Q_{R}$.

This construction is rather unnatural when done in $\mathbb{P}^{3}$, but its
advantage lies in the fact that it can be pretty straightforwardly
supersymmetrized to $\mathbb{P}^{3|4}$. Indeed, in $\mathbb{P}^{3|4}$ we have
a delta function
$\delta_{\mathbb{P}^{3|4}}^{3|4}(\mathcal{Z}_{1};\mathcal{Z}_{2})$, and so on.
These supersymmetrizations were introduced in ref. [32]. For the superquadric
we obtain $\delta_{\mathbb{P}^{3|4}}^{1|8}(\mathcal{Q},\mathcal{Z})$. Pursuing
the same strategy as in the $\mathbb{P}^{3}$ case, we finally define
$\omega_{C}^{1|12}$ using

$\int_{C}\omega_{C}^{1|12}(Z)f(Z)=\int\omega_{\mathbb{P}^{3|4}}(\mathcal{Z})\delta_{\mathbb{P}^{3|4}}(\mathcal{Q}_{l};\mathcal{Z})\delta_{\mathbb{P}^{3|4}}(\mathcal{Q}_{r};\mathcal{Z})f(Z),$
(4.4)

where
$\mathcal{Z}=[Z_{0}:Z_{1}:Z_{2}:Z_{3}\,|\,\chi_{1}:\chi_{2}:\chi_{3}:\chi_{4}]$
and
$\omega_{\mathbb{P}^{3|4}}(\mathcal{Z})=\omega_{\mathbb{P}^{3}}(Z)d\chi_{1}d\chi_{2}d\chi_{3}d\chi_{4}$
is the $\operatorname{PSL}(4|4)$-invariant form on $\mathbb{P}^{3|4}$.

This construction can be generalized to higher dimensions.

##  5 Summary and Outlook

We have presented a few examples of Calabi-Yau varieties arising as the
leading singularity loci of the class of traintrack integrals.

For the elliptic double box we have a pretty explicit understanding of the
moduli space and how it relates to the external kinematics of the integral. We
believe this should be a useful ingredient in the computation of these
integrals.

The moduli space of algebraic K3 surfaces has a global description as a double
coset of an orthogonal group (see ref. [23]). This moduli space should be
somehow parametrized by the external kinematics, but this global description
does not seem to arise naturally from the twistor representation of the
kinematics. So, while we have described the topology of these varieties in
some detail, our description of their moduli space has not been as detailed as
we would like. One approach we have sketched is to use a parametrization where
$10$ moduli arise from an intersection of two genus-one curves in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ and an extra modulus arises from the
intersections of transversals to these $\mathbb{P}^{1}$ with the two quadrics
$Q_{L}$ and $Q_{R}$. It remains to be seen if this parametrization will be
useful for expressing the corresponding integral.

One slightly mysterious aspect remains in connection with Calabi-Yau varieties
encountered in non-planar integrals. The twistor methods are well-adapted for
studying planar integrals. How should non-planar integrals be described in
this language? It is not clear yet if the momentum twistor approach is a
useful description for the leading singularity locus of these integrals. We
hope to report on this issue in future work.

We have also discussed supersymmetrization. The approach to
supersymmetrization we have sketched generalizes to other cases as well.
Clearly supersymmetry imposes some restriction on the geometry of these
varieties and it would be interesting to understand this better.

## Acknowledgments

We are grateful to Jacob Bourjaily, Andrew McLeod, Matt von Hippel and
Matthias Wilhelm for discussions, collaboration on related topics and comments
on the draft of this paper. This work was supported in part by the Danish
Independent Research Fund under grant number DFF-4002-00037 (MV), the Danish
National Research Foundation (DNRF91), the research grant 00015369 from Villum
Fonden and a Starting Grant (No. 757978) from the European Research Council
(CV, MV).

##  Appendix A Automorphisms of K3 surfaces

For an account of the automorphisms of K3 surfaces see for example ref. [24,
Chapter 15]. In the following we summarize some of the most important facts.

When studying the group of automorphisms $\operatorname{Aut}(S)$ of a K3
surface $S$, one distinguishes between symplectic and non-symplectic
automorphisms. An automorphism $f:S\rightarrow S$ of a K3 surface $S$ is
symplectic if the induced action on $H^{0}(S,\Omega_{S}^{2})$ is the identity,
i.e. if it leaves the holomorphic two-form on $S$ invariant. One can show that
$\operatorname{Aut}(S)$ is discrete and that the subgroup
$\operatorname{Aut}_{s}(S)\subset\operatorname{Aut}(S)$ of symplectic
automorphisms is of finite index, at least for projective K3 surfaces.

One can moreover show the following result: Let
$f\in\operatorname{Aut}_{s}(S)$ be of finite order $n$ and
$f\neq\operatorname{id}$. Then the set of fixed points $\operatorname{Fix}(f)$
is non-empty and finite and

$\displaystyle\left|\operatorname{Fix}(S)\right|=\frac{24}{n}\prod_{p|n}\left(1+\frac{1}{p}\right)^{-1}.$
(A.1)

Moreover the number of fixed point satisfies
$1\leq\left|\operatorname{Fix}(f)\right|\leq 8$ and only depends on the order
$n$ of $f$.

Nikulin also proved that for $f\in\operatorname{Aut}_{s}(S)$, the order $n$ of
$f$ satisfies $n\leq 8$. This means that only the combinations of $n$ and
$\left|\operatorname{Fix}(S)\right|$ shown in table 1 can occur. For each $n$,
one can also derive a lower bound for the Picard number $\rho(S)$ which is
also shown in table 1. One can see that the Picard number of K3 surfaces with
automorphisms tends to be quite high.

Order $n$ | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---
$\left|\operatorname{Fix}(S)\right|$ | 8 | 6 | 4 | 4 | 2 | 3 | 2
$\rho(S)\geq$ | 9 | 13 | 15 | 17 | 17 | 19 | 19
Table 1: Symplectic automorphism orders and number of fixed points for a
complex K3 surface $S$. Here $\rho(S)$ is the Picard number of $S$. Table from
ref. [24].

Symplectic automorphisms of order two were studied by Nikulin [25] and are
called _Nikulin involutions_. According to table 1, a Nikulin involution of a
complex K3 surface has eight fixed points and Picard number $\rho(S)\geq 9$. A
classification of all algebraic K3 surfaces with Picard number satisfying the
lower bound, i.e. $\rho(S)=9$ can be found in ref. [26].

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