††institutetext: NHETC, Department of Physics and Astronomy
Rutgers University
136 Frelinghuysen Road
Piscataway, NJ 08855
USA

# Hadamard products and BPS networks

Mohamed Elmi<EMAIL_ADDRESS>
###### Abstract

We study examples of fourth-order Picard-Fuchs operators that are Hadamard
products of two second order Picard-Fuchs operators. Each second order Picard-
Fuchs operator is associated with a family of elliptic curves, and the
Hadamard product computes period integrals on the fibred product of the two
elliptic surfaces. We construct $3$-cycles on this geometry as the union of
$2$-cycles in the fibre over contours on the base. We then use the special
Lagrangian condition to constrain the contours on the base. This leads to a
construction reminiscent of spectral networks and exponential networks that
have previously appeared in string theory literature.

##  1 Introduction

Almkvist, Enckevort, van Straten and Zudilin have compiled a list of fourth
order, linear differential operators of Calabi-Yau type that we will refer to
as the AESZ list. 2005math……7430A  By definition, these operators satisfy a
number of conditions that make them candidate Picard-Fuchs operators for a
family of Calabi-Yau threefolds with Hodge number $h^{2,1}=1$. For example,
each operator should have a point of maximal unipotent monodromy at
$\varphi=0$, there should exist a power series solution at $\varphi=0$ with
integer coefficients, etc.

There are many operators in the AESZ list that are the Hadamard product of two
second order Picard-Fuchs equations, each associated with a family of elliptic
curves. They are defined as follows. Let $\mathcal{L}^{(1)}$ and
$\mathcal{L}^{(2)}$ be Picard-Fuchs operators in the variable $s$ such that

$\mathcal{L}^{(i)}\left(\sum_{n=0}^{\infty}a_{n}^{(i)}s^{n}\right)=0.$ (1)

We say that a non-trivial differential operator $\mathcal{L}$ is a Hadamard
product of $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$ if it satisfies

$\mathcal{L}\left(\sum_{n=0}^{\infty}a_{n}^{(1)}a_{n}^{(2)}\varphi^{n}\right)=0.$
(2)

If $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$ are of finite order with
polynomial coefficients, there exists an operator $\mathcal{L}$ that satisfies
(2).andre1989g  However, this operator is not unique, and we know of no
general algorithm for computing it. Moreover, there is no guarantee that the
Hadamard product of two second-order Calabi-Yau operators will be a fourth-
order Calabi-Yau operator. In practice, we simply make the ansatz

$\mathcal{L}=S_{4}(\varphi)\theta^{4}+S_{3}(\varphi)\theta^{3}+S_{2}(\varphi)\theta^{2}+S_{1}(\varphi)\theta^{1}+S_{0}(\varphi),$
(3)

where $\theta=\varphi\frac{d}{d\varphi}$ and $S_{i}(\varphi)$ is a polynomial
in $\varphi$. We then insist that it satisfy (2) and solve for the
coefficients in the polynomials $S_{i}(\varphi)$. In all the examples we study
here, there will be a differential operator $\mathcal{L}$, unique up to some
overall scalar multiple, of minimal degree in the polynomials $S_{i}(\varphi)$
that satisfies (2). We will refer to this operator as the Hadamard product of
$\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$.

Geometrically, the Hadamard products compute the periods of a holomorphic
$3$-form on the fibred product of the two families of elliptic curves over a
common $\mathbb{CP}^{1}$, where the structure of the fibration depends on a
complex parameter $\varphi$ (see, for example, the thesis of Samol
DissertationSamol ). We may construct $3$-cycles on the fibred product as the
union of $S^{1}\times S^{1}$ over some contour on the base $\mathbb{CP}^{1}$.
Here, each $S^{1}$ must lie in one of the elliptic fibres and both the contour
and the homology classes in the fibres must be carefully chosen to ensure that
the $2$-boundary vanishes. We may then integrate the holomorphic $3$-form over
this $3$-cycle. The resulting integral will be a function of $\varphi$ and
will be annihilated by the Hadamard product. When the Hadamard product is a
Picard-Fuchs equation of a family of compact Calabi-Yau threefolds, the same
integrals may be identified with periods of the holomorphic $3$-form on a
compact Calabi-Yau threefold. This gives us a precise map between $3$-cycles
on the fibred product of elliptic curves and $3$-cycles on the Calabi-Yau
threefold. As examples, we will use the Picard-Fuchs operators AESZ 3 and AESZ
101, which are both Hadamard products. In Sections 2 and 3, we compute
solutions of AESZ 3, solutions of AESZ 101 and solutions of the associated
second order Picard-Fuchs equations in appropriate bases. We also explain, for
each example, the precise relationship between the family of smooth Calabi-Yau
threefolds and a fibred product of families of elliptic curves. We then turn
to the subject of special Lagrangian submanifolds.

A D3 brane on a Calabi-Yau threefold at zero string coupling is described by a
special Lagrangian submanifold with a flat $U(1)$ connection.Becker:1995kb
The special Lagrangian condition is a constraint involving the Kähler form and
the holomorphic $3$-form of the Calabi-Yau threefold. We do not have much
control over the Kähler form, but we do have a great deal of control over the
holomorphic $3$-form, so we apply this second condition to the $3$-cycles on
the fibred product of elliptic curves. This leads to a construction that is
very similar to the spectral networks and exponential networks that have
previously appeared in string theory literature.Klemm:1996bj ; Gaiotto:2012rg
; Eager:2016yxd

The networks that appear in this paper are defined as follows. Let $s$ be an
in-homogeneous coordinate on $\mathbb{CP}^{1}$ and let $\gamma_{1}$ and
$\gamma_{2}$ be homology $1$-cycles in the first and second elliptic fibres,
respectively. We integrate the holomorphic $1$-forms of the first and second
elliptic fibres at $s$ over $\gamma_{1}$ and $\gamma_{2}$, respectively. We
will see in following sections that this will result in two periods
$\omega_{\gamma_{1}}(s)$ and
$\widetilde{\omega}\left(\frac{\varphi}{s}\right)$, respectively. Note the
dependence on $\varphi$ and the coordinate $s$. We construct a $3$-cycle by
taking the union of products of circles over contours on the base
$\mathbb{CP}^{1}$. If the circles in the fibre are in the homology classes
$\gamma_{1}$ and $\gamma_{2}$, respectively, the special Lagrangian condition
requires that the contour on the base satisfy

$\omega_{\gamma_{1}}(s)\widetilde{\omega}_{\gamma_{2}}\left(\frac{\varphi}{s}\right)\frac{1}{s}\frac{ds}{dl}=e^{i\theta}$
(4)

for some parametrisation $s=s(l)$ and an angle $\theta$.

We have chosen to focus on Hadamard products and the associated fibred
products of elliptic curves for their simplicity, but we expect that the ideas
in this paper will generalise. We also point out that much of our analysis
only depends on the structure of the associated Picard-Fuchs operators.
Knowledge of the the triple intersection number, second Chern-class and Euler
characteristic of the mirror Calabi-Yau is used to fix a suitable basis of
solutions at the point of maximal unipotent monodromy $\varphi=0$. However,
even this topological data is strongly constrained by the requirement of
integral symplectic monodromy and, given a fourth order Picard-Fuchs equation,
one may find the finitely many choices of topological data consistent with
integral symplectic monodromy.2004math…..12539V

##  2 AESZ 3

###  2.1 Family of Calabi-Yau threefolds

Solutions of the Picard-Fuchs equation AESZ 3 determine periods of the
holomorphic $3$-form on the mirror of the intersection of four quadrics in
$\mathbb{CP}^{7}$ (the precise differential equation may be found in the next
subsection). AESZ 3 is a Hadamard product of the Picard-Fuchs equations of two
families of elliptic curves, so one might ask - how is the mirror of the
intersection of four quadrics related to the fibred product of two elliptic
surfaces? An answer may be found in Bonisch:2022mgw . We summarise their
computation here, with slight modifications.

The mirror of the intersection of four quadrics in $\mathbb{CP}^{7}$ is
constructed via a Green-Plesser like construction.Greene:1990ud  One starts
with an especially symmetric family, defined as the interesections of the four
quadrics

$W_{\psi}:P_{j}=x_{j}^{2}+y_{j}^{2}-2\psi x_{j+1}y_{j+1}=0,$ (5)

where $j\in\\{0,1,2,3\\}$ and $\psi$ is a complex parameter of the family. The
fibre at a given $\psi$ is invariant under the group

$\Gamma=\\{\xi=(\xi_{j})_{j=0,...,3}\in(\mu_{4})^{4}\\}/\mu_{4}^{\text{diag}}$
(6)

where $\mu_{4}$ is the group generated by the primitive fourth roots of unity.
$\Gamma$ is isomorphic to $(\mathbb{Z}/4\mathbb{Z})^{3}$ and acts on the
coordinates of $\mathbb{CP}^{7}$ as

$(x_{j},y_{j})\mapsto(\xi_{j}x_{j},\xi_{j-1}^{2}\xi_{j}^{-1}y_{j}).$ (7)

Away from $\varphi=\frac{1}{(2\psi)^{8}}\in\\{0,\frac{1}{2^{8}},\infty\\}$,
the quotient of $W_{\psi}$ by $\Gamma$ may be resolved to a smooth Calabi-Yau
threefold with Hodge numbers $h^{1,1}=65$ and $h^{2,1}=1$. Now consider the
hypersurface in $(\mathbb{CP}^{1})^{4}$ defined by

$\widetilde{W}_{\phi}:\prod_{i=1}^{4}\left(Y_{i}-\frac{1}{Y_{i}}\right)=\phi.$
(8)

There is a degree $8$ map from $W_{\psi}/\Gamma$ to $W_{\phi}$, given by

$iY_{j}=\frac{x_{j}-y_{j}}{x_{j}+y_{j}}$ (9)

where $j\in\\{0,1,2,3\\}$ and
$\varphi=\left(\frac{\phi}{2^{8}}\right)^{2}=(2\psi)^{-8}$.
$\widetilde{W}_{\phi}$ is a fibred product of two families of elliptic curves.
To see this, note that

$\mathcal{E}_{s}:\left(Y_{1}-\frac{1}{Y_{1}}\right)\left(Y_{2}-\frac{1}{Y_{2}}\right)=s$
(10)

defines a family of elliptic curves with parameter $s$. Thus, if $Y_{1}$ and
$Y_{2}$ are constrained to lie on the elliptic curve $\mathcal{E}_{s}$, then
$Y_{3}$ and $Y_{4}$ are constrained by (8) to lie on the elliptic curve
$\mathcal{E}_{\frac{\phi}{s}}$.

Finally, note that the families of elliptic curves $\mathcal{E}_{s}$ and
threefolds $\widetilde{W}_{\phi}$ both admit the action of an involution.
$\mathcal{E}_{s}$ is isomorphic to the curve $\mathcal{E}_{\frac{2^{4}}{s}}$
via the map

$Y_{i}\mapsto\frac{Y_{i}-1}{Y_{i}+1}.$ (11)

By applying this map to each factor in (8), we see that $\widetilde{W}_{\phi}$
is equivalent to $\widetilde{W}_{\frac{2^{8}}{\phi}}$.

###  2.2 Solutions of AESZ 3 and a double cover

AESZ 3 is the operator

$\mathcal{L}_{\text{AESZ3}}=(1-2^{8}\varphi)\theta_{\varphi}^{4}-2^{9}\varphi\theta_{\varphi}^{3}-(2^{7}\cdot
3)\varphi\theta_{\varphi}^{2}-2^{7}\varphi\theta_{\varphi}-2^{4}\varphi,$ (12)

where $\theta_{\varphi}=\varphi\frac{d}{d\varphi}$. It defines a
hypergeometric differential equation with the Riemann symbol

$\mathcal{P}\left\\{\begin{array}[]{c c c}0&1/2^{8}&\infty\\\ \hline\cr
0&0&1/2\\\ 0&1&1/2\\\ 0&1&1/2\\\ 0&2&1/2\end{array}~{};\varphi\right\\}~{}.$
(13)

A basis of solutions of around the point of maximum unipotent monodromy (MUM)
at $\varphi=0$ is given by

$\varpi(\varphi)=\left(\begin{array}[]{l}F_{0}(\varphi)\\\
F_{0}(\varphi)\log(\varphi)+F_{1}(\varphi)\\\
F_{0}(\varphi)\log^{2}(\varphi)+2F_{1}(\varphi)\log(\varphi)+F_{2}(\varphi)\\\
F_{0}(\varphi)\log^{3}(\varphi)+3F_{1}(\varphi)\log^{2}(\varphi)+3F_{2}(\varphi)\log(\varphi)+F_{3}(\varphi)\end{array}\right)$
(14)

where

$\displaystyle\begin{split}F_{0}(\varphi)&=\sum_{n=0}^{\infty}\binom{2n}{n}^{4}\varphi^{n}\\\
&=1+16\varphi+1296\varphi^{2}+160000\varphi^{3}+24010000\varphi^{4}+4032758016\varphi^{5}+\ldots\\\
F_{1}(\varphi)&=64\varphi+6048\varphi^{2}+\frac{2368000}{3}\varphi^{3}+\frac{365638000}{3}\varphi^{4}+\frac{104147576064}{5}\varphi^{5}+\ldots\\\
F_{2}(\varphi)&=64\varphi+11664\varphi^{2}+\frac{16364800}{9}\varphi^{3}+\frac{2748199300}{9}\varphi^{4}+\frac{1370195072064}{25}\varphi^{5}\ldots\\\
F_{3}(\varphi)&=-384\varphi-25776\varphi^{2}-\frac{20668160}{9}\varphi^{3}-\frac{2221690310}{9}\varphi^{4}\ldots~{}.\end{split}$
(15)

We refer to this basis of solutions as the Frobenius basis. Let $\Omega$ be
the holomorphic $3$-form on the mirror of the intersection of four quadrics in
$\mathbb{CP}^{7}$ and define

$\Pi(\varphi)=\left(\int_{A_{I}}\Omega(\varphi),\int_{B^{I}}\Omega(\varphi)\right)^{T},$
(16)

where $A_{I},B^{J}\in H^{3}(X,\mathbb{Z})$ are generators that satisfy
$A_{I}\cap B^{J}=-B^{J}\cap A_{I}=\delta_{I}^{J}$ with all other intersections
vanishing. Such a homology basis is implicitly determined by the pre-potential
of the complexified Kähler moduli space of a mirror Calabi-Yau
manifold.Candelas:1990rm  The change of basis from Frobenius basis to an
integral symplectic basis is given by

$\Pi(\varphi)=\rho\varpi(\varphi),$ (17)

where

$\rho=(2\pi i)^{3}\begin{pmatrix}\frac{\zeta(3)}{(2\pi
i)^{3}}\chi&\frac{c_{2}\cdot H}{24}&0&\frac{H^{3}}{3!}\\\ \frac{c_{2}\cdot
H}{24}&\frac{\sigma}{2}&-\frac{H^{3}}{2!}&0\\\ 1&0&0&0\\\
0&1&0&0\end{pmatrix}\begin{pmatrix}1&0&0&0\\\ 0&(2\pi i)^{-1}&0&0\\\ 0&0&(2\pi
i)^{-2}&0\\\ 0&0&0&(2\pi i)^{-3}\end{pmatrix}$ (18)

and

$\sigma=\begin{cases}~{}0&\text{if $H^{3}$ is even}\\\ ~{}1&\text{if $H^{3}$
is odd}.\end{cases}$ (19)

See, for example, Braun:2015jdy . In this example, the topological data of the
of the mirror manifold are given by $H^{3}=16$, $c_{2}\cdot H=64$ and
$\chi=-128$. We define the monodromy matrices by choosing a base point in the
upper half $\varphi$-plane and analytically continuing $\Pi$ in an anti-
clockwise fashion around $\varphi_{*}$ so that $\Pi\mapsto
M_{\varphi_{*}}\Pi$. These monodromy matrices are given by111AESZ 3 is
hypergeometric, so the monodromy matrices may be computed via the Mellin-
Barnes integral, as in Candelas:1990rm . In more general examples, as in the
case of AESZ 101, other methods are needed. One may simply evaluate $\Pi$
within the radius of convergence around zero, numerically integrate the
solutions around a closed loop and compare with the original values. Since the
monodromy matrices are integral, only a single decimal place of precision is
required to fix the monodromy matrices exactly. See Braun:2015jdy  for an
example.

$\displaystyle\begin{split}M_{\varphi=0}&=\begin{pmatrix}1&-1&8&8\\\
0&1&-8&-16\\\ 0&0&1&0\\\
0&0&1&1\end{pmatrix},~{}~{}~{}~{}M_{\varphi=\frac{1}{2^{8}}}=\begin{pmatrix}1&0&0&0\\\
0&1&0&0\\\ -1&0&1&0\\\ 0&0&0&1\end{pmatrix}\\\
M_{\varphi=\infty}&=\left(M_{\varphi=0}M_{\varphi=\frac{1}{2^{8}}}\right)^{-1}=\begin{pmatrix}1&1&-8&8\\\
0&1&-8&16\\\ 1&1&-7&8\\\ 0&0&-1&1\end{pmatrix}\end{split}$ (20)

Although $\varphi$ is the most natural variable for AESZ 3 and the mirror of
the intersection of four quadrics in $\mathbb{CP}^{7}$, in order to make
contact with the fibred product of elliptic curves in (8), it is more natural
to use the the variable $\phi$, where
$\varphi=\left(\frac{\phi}{2^{8}}\right)^{2}$. The pullback of AESZ 3 defines
a new operator

$\mathcal{L}_{\text{AESZ3}}^{*}=(\phi^{2}-2^{8})\theta_{\phi}^{4}+2^{2}\phi^{2}\theta_{\phi}^{3}+2\cdot
3\phi^{2}\theta_{\phi}^{2}+2^{2}\phi^{2}\theta_{\phi}+\phi^{2}.$ (21)

with Riemann symbol

$\mathcal{P}\left\\{\begin{array}[]{c c c}0&\pm 4^{2}&\infty\\\ \hline\cr
0&0&1\\\ 0&1&1\\\ 0&1&1\\\ 0&2&1\end{array}~{};\phi\right\\}~{}.$ (22)

We pullback the integral symplectic basis of solutions in (18) and define a
basis of solutions of $\mathcal{L}_{\text{AESZ3}}^{*}\Pi^{*}(\phi)=0$ with
integral symplectic monodromy by

$\Pi^{*}(\phi)=\rho\begin{pmatrix}1&0&0&0\\\ 2\log\frac{1}{2^{8}}&2&0&0\\\
2^{2}\log^{2}\frac{1}{2^{8}}&2^{2}\log\frac{1}{2^{8}}&2^{2}&0\\\
2^{3}\log^{3}\frac{1}{2^{8}}&2^{3}\cdot 3\log^{2}\frac{1}{2^{8}}&2^{3}\cdot
3\log\frac{1}{2^{8}}&2^{3}\end{pmatrix}\varpi^{*}(\phi),$ (23)

where a Frobenius basis of solutions of $\mathcal{L}_{\text{AESZ3}}^{*}$ is
defined as in (14) and $\rho$ is defined in (18). The power series defining
the Frobenius basis of solutions $\varpi^{*}$ are now given by

$\displaystyle\begin{split}F_{0}^{*}(\phi)&=\sum_{n=0}^{\infty}\binom{2n}{n}^{4}\left(\frac{\phi}{2^{8}}\right)^{2n}\\\
&=1+\frac{\phi^{2}}{4096}+\frac{81\phi^{4}}{268435456}+\frac{625\phi^{6}}{1099511627776}+\frac{1500625\phi^{8}}{1152921504606846976}+\ldots\\\
F_{1}^{*}(\phi)&=\frac{\phi^{2}}{2048}+\frac{189\phi^{4}}{268435456}+\frac{4625\phi^{6}}{3298534883328}+\frac{22852375\phi^{8}}{6917529027641081856}+\ldots\\\
F_{2}^{*}(\phi)&=\frac{\phi^{2}}{4096}+\frac{729\phi^{4}}{1073741824}+\frac{63925\phi^{6}}{39582418599936}+\frac{687049825\phi^{8}}{166020696663385964544}+\ldots\\\
F_{3}^{*}(\phi)&=-\frac{3\phi^{2}}{4096}-\frac{1611\phi^{4}}{2147483648}-\frac{80735\phi^{6}}{79164837199872}+\ldots~{}.\end{split}$
(24)

By analytically continuing $\Pi^{*}$ from a base point in the upper half
$\phi$-plane around the various singularities in an anti-clockwise fashion, we
find the following monodromy matrices.

$\displaystyle\begin{split}M_{\phi=-4^{2}}&=\left(\begin{array}[]{cccc}-7&-8&64&-64\\\
8&9&-64&64\\\ -1&-1&9&-8\\\ -1&-1&8&-7\\\
\end{array}\right),M_{\phi=0}=\left(\begin{array}[]{cccc}1&-2&32&32\\\
0&1&-32&-32\\\ 0&0&1&0\\\ 0&0&2&1\\\ \end{array}\right)\\\
M_{\phi=4^{2}}&=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ -1&0&1&0\\\
0&0&0&1\\\ \end{array}\right)\\\
M_{\phi=\infty}&=(M_{\phi=-4^{2}}M_{\phi=0}M_{\phi=4^{2}})^{-1}=\left(\begin{array}[]{cccc}-7&-6&32&-32\\\
-8&-7&32&-32\\\ -6&-5&25&-24\\\ -1&-1&6&-7\\\ \end{array}\right).\end{split}$
(25)

Both $\mathcal{L}_{\text{AESZ3}}$ and its pull-back
$\mathcal{L}_{\text{AESZ3}}^{*}$ are invariant under the action of an
involution. The integral symplectic solutions $\Pi$ of AESZ3 satisfy

$\frac{1}{\sqrt{2^{16}\varphi}}\Pi\left(\frac{1}{2^{16}\varphi}\right)=\left(\begin{array}[]{cccc}\frac{1}{16}&\frac{1}{16}&-\frac{1}{2}&\frac{1}{2}\\\
0&-\frac{1}{16}&\frac{1}{2}&-1\\\ \frac{1}{32}&\frac{1}{64}&-\frac{1}{16}&0\\\
\frac{1}{64}&\frac{1}{64}&-\frac{1}{16}&\frac{1}{16}\\\
\end{array}\right)\Pi(\varphi).$ (26)

This identity was computed for real $\varphi$ close to the fixed point
$-\frac{1}{2^{8}}$ and a path connecting $\varphi$ and
$\frac{1}{2^{16}\varphi}$ along the upper half $\varphi$-plane. A homotopy
inequivalent path would multiply the above matrix by a monodromy matrix.
Similarly, we find that

$\frac{2^{8}}{\phi}\Pi^{*}\left(\frac{2^{8}}{\phi}\right)=\left(\begin{array}[]{cccc}-16&0&0&0\\\
0&16&0&0\\\ 8&4&-16&0\\\ -4&0&0&16\\\ \end{array}\right)\Pi^{*}(\phi),$ (27)

where the above identity was computed near the fixed point $\phi=2^{4}$ and
along a path on the upper half $\phi$-plane connecting $\phi$ and
$\frac{2^{8}}{\phi}$.

Note that it is not possible make the matrices on the right hand side of the
above identities integral symplectic by an overall scaling of the matrices.
This is a reflection of the fact that the MUM points of AESZ 3 at $\varphi=0$
and $\varphi=\infty$ are inequivalent in the sense that they are associated
with different mirror geometries.222We are grateful to Thorsten Schimannek for
clarifying this point.Katz:2023zan  However, at the level of periods, the
Frobenius solutions around $\varphi=0$ and $\varphi=\infty$ are identical up
to multiplication by an algebraic function of $\varphi$. This leads to the
above identities.

###  2.3 Picard-Fuchs equation of an associated family of elliptic curves

In this section, we compute the periods of the holomorphic 1-form for the
family of elliptic curves defined in (10). The family of curves
$\mathcal{E}_{s}$ may be brought into the standard Legendre form

$y^{2}=x(x-1)\left(x-\left(1-\frac{s^{2}}{2^{4}}\right)\right)$ (28)

by the change of variables

$(Y_{1},Y_{2})=\left(\frac{xs+4y}{xs-4y},\frac{x(1-x)}{y}\right).$ (29)

We immediately see that the elliptic curve is singular at $s=0$, $s=\pm 2^{2}$
and $s=\infty$.

The holomorphic 1-form on an elliptic fibre is given by

$\frac{dx}{2y(s)}=-\frac{dY_{1}}{\sqrt{Y_{1}^{4}+\left(\frac{s^{2}}{4}-2\right)Y_{1}^{2}+1}}.$
(30)

Now suppose that $s\in\mathbb{R}$ and $0<s<2^{2}$. We define $A\in
H_{1}(\mathcal{E}_{s},\mathbb{Z})$ as the homology class of the contour that
encircles the branch points

$Y_{1}=-\frac{1}{2}\sqrt{4-\frac{1}{2}s\left(s-\sqrt{s^{2}+16}\right)}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}Y_{1}=-\frac{1}{2}\sqrt{4-\frac{1}{2}s\left(s-\sqrt{s^{2}-16}\right)}$
(31)

in an anti-clockwise fashion. Similarly, $B\in
H_{1}(\mathcal{E}_{s},\mathbb{Z})$ is defined as the homology class of the
contour that encircles the branch points

$Y_{1}=\frac{1}{2}\sqrt{4-\frac{1}{2}s\left(s-\sqrt{s^{2}-16}\right)}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}Y_{1}=\frac{1}{2}\sqrt{4-\frac{1}{2}s\left(s+\sqrt{s^{2}-16}\right)}$
(32)

in a clock-wise fashion. It follows that $A\cap B=-B\cap A=1$ and

$\omega^{*}(s)=\begin{pmatrix}\int_{A}\frac{dx}{2y(s)}\\\
\int_{B}\frac{dx}{2y(s)}\end{pmatrix}=(2\pi
i)\begin{pmatrix}-\frac{1}{2}&~{}0\\\ -\frac{\log 2^{8}}{2\pi
i}&~{}2\end{pmatrix}\begin{pmatrix}1&0\\\ 0&(2\pi
i)^{-1}\end{pmatrix}\left(\begin{array}[]{l}f_{0}^{*}(s)\\\
f_{0}^{*}(s)\log(s)+f_{1}^{*}(s)\end{array}\right),$ (33)

where

$\displaystyle\begin{split}f_{0}^{*}(s)&=\sum_{n=0}^{\infty}\binom{2n}{n}^{2}\left(\frac{s}{2^{4}}\right)^{2n}\\\
&=1+\frac{s^{2}}{64}+\frac{9s^{4}}{16384}+\frac{25s^{6}}{1048576}+\frac{1225s^{8}}{1073741824}+\frac{3969s^{10}}{68719476736}+\ldots\\\
f^{*}_{1}(s)&=\frac{s^{2}}{64}++\frac{21s^{4}}{32768}+\frac{185s^{6}}{6291456}+\frac{18655s^{8}}{12884901888}+\frac{102501s^{10}}{1374389534720}+\ldots~{}.\end{split}$
(34)

The $A$-integral is holomorphic as $s\rightarrow 0$ and may be computed by
first expanding the integrand in $s$, and then evaluating the resulting
integrals by computing residues around $Y_{1}=-1$. Similarly, $B$ is
holomorphic as $s\rightarrow 2^{2}$ and may be computed around this point as a
power series. This solution may then be analytically continued to $s=0$ by
noting that the $A$ and $B$ periods are solutions of the Picard-Fuchs equation

$(s^{2}-4^{2})\theta_{s}^{2}+2s^{2}\theta_{s}+s^{2}$ (35)

with Riemann symbol

$\mathcal{P}\left\\{\begin{array}[]{c c c}0&\pm 4&\infty\\\ \hline\cr 0&0&1\\\
0&0&1\\\ \end{array}~{};s\right\\}~{}.$ (36)

By comparing $f_{0}^{*}$ in (34) with $F_{0}^{*}$ in (24), we see that
$\mathcal{L}_{\text{AESZ3}}^{*}$ is the Hadamard product of the above second
order differential operator with itself. Note that (36) is the pullback of the
hypergeometric differential operator

$(16s^{\prime}-1)\theta_{s^{\prime}}^{2}+16s^{\prime}\theta_{s^{\prime}}+4s^{\prime}$
(37)

where $s^{\prime}=\left(\frac{s}{2^{4}}\right)^{2}$. It follows that AESZ 3 is
a Hadamard product of this operator with itself.

As expected from geometric considerations (see (11)), the solutions
$\omega(s)$ have the symmetry

$\frac{2^{4}}{s}\omega^{*}\left(\frac{2^{4}}{s}\right)=2^{2}\begin{pmatrix}1&1\\\
0&1\end{pmatrix}\omega^{*}(s)~{}.$ (38)

Note that in computing the above matrix, we need to arbitrarily choose a path
between $s$ and $\frac{2^{4}}{s}$. Thus, the above matrix is only defined up
to multiplication by a monodromy matrix. The matrix in (38) was computed in a
small neighbourhood of the fixed point $s=2^{2}$, using a path along the upper
half $s$-plane. Also note that this matrix is not a monodromy matrix of
$\omega(s)$, because it has a $1$ as the upper off-diagonal entry. The
monodromy matrices, for a choice of base point in the upper half $s$-plane and
anti-clockwise loops around the various singularities, are generated by

$m_{s=-4}=\begin{pmatrix}5&2\\\
-8&-3\end{pmatrix},~{}~{}~{}m_{s=0}=\begin{pmatrix}1&0\\\
-4&1\end{pmatrix},~{}~{}~{}\text{and}~{}~{}~{}m_{s=4}=\begin{pmatrix}1&2\\\
0&1\end{pmatrix}~{}.$ (39)

##  3 AESZ 101

###  3.1 Family of Calabi-Yau threefolds

Consider the family of elliptic curves in $\mathbb{CP}^{2}$, defined by the
cubic

$\mathcal{E}_{s}:x(x-z)(y-z)-syz(x-y)=0$ (40)

with parameter $s$. $\mathcal{E}_{s}$ is smooth away from
$s\in\\{0,\frac{1}{2}\left(-11+\pm 5\sqrt{5}\right),\infty\\}$.

It was shown in DissertationSamol , using the results of 2008arXiv0802.3760K ,
that the fibred product of the families $\mathcal{E}_{s}$ and
$\mathcal{E}_{\frac{\varphi}{s}}$ may be resolved to a smooth Calabi-Yau
threefold with Hodge numbers $h^{1,1}=51$ and $h^{2,1}=1$ (and Euler
chracteristic $2\times(51-1)=100$) for generic values of $\varphi$. More
precisely, we consider the singular variety in
$\mathbb{CP}^{2}\times\mathbb{CP}^{2}\times\mathbb{CP}^{1}$, defined by

$\displaystyle\begin{split}x_{1}(x_{1}-z_{1})(y_{1}-z_{1})-sy_{1}z_{1}(x_{1}-y_{1})&=0\\\
x_{2}(x_{2}-z_{2})(y_{2}-z_{2})-\left(\frac{\varphi}{s}\right)y_{2}z_{2}(x_{2}-y_{2})&=0\end{split}$
(41)

where $[x_{i},y_{i},z_{i}]$ are homogeneous coordinates on $\mathbb{CP}^{2}$
and $s$ is an inhomogeneous coordinate on $\mathbb{CP}^{1}$. $\varphi$ is a
free-parameter that determines complex structure of a smooth Calabi-Yau
threefold away from $\varphi\in\\{-1,0,\left(\frac{1}{2}\left(-11\pm
5\sqrt{5}\right)\right)^{2},\infty\\}$. If we consider points on the complex
$s$-plane where an elliptic fibre is singular, these values of $\varphi$ are
the values for which two singular points collide. A projective small
resolution of singularities exists away these values of $\varphi$.

Note that the elliptic curve $\mathcal{E}_{s}$ is equivalent to the curve
$\mathcal{E}_{-\frac{1}{s}}$. This can be seen by making the linear change of
variables

$(x,y,z)\mapsto(-z,~{}x-z,~{}y-z)~{}.$ (42)

By applying this to both elliptic fibres of (41), we see that the Calabi-Yau
variety at $\varphi$ is equivalent to that at $\frac{1}{\varphi}$.

We will need the $A$-model topological data (triple intersection number,
second Chern class and Euler characteristic) in order to fix an integral
symplectic basis of periods around the MUM point $\varphi=0$. One approach is
to construct the mirror of the resolution of the variety defined in (41). A
natural candidate for the mirror appears in 2010arXiv1006.0223K ;
2017arXiv170700534B ; 2017arXiv170609952O ; Knapp:2019cih  that is constructed
as the intersection of two Grassmannians in $\mathbb{CP}^{9}$. More precisely,
the Plücker embedding defines a map from the Grassmannian $Gr(2,5)$ (of two
dimensional linear subspaces of $\mathbb{C}^{5}$) to the projective space
$\mathbb{CP}^{9}$. $GL(10,\mathbb{C})$ acts in the obvious way on the
homogeneous coordinates of $\mathbb{CP}^{9}$ and, by composing the Plucker
embedding with a $GL(10,\mathbb{C})$ transformation, one may generate many
embeddings of $Gr(2,5)$ in $\mathbb{CP}^{9}$. For generic choices of
$GL(10,\mathbb{C})$ transformation, the intersection of the original and a
transformed copy of $Gr(2,5)$ in $\mathbb{CP}^{9}$ will be a smooth Calabi-Yau
threefold with Hodge numbers $h^{1,1}=1$ and $h^{2,1}=51$. This Calabi-Yau
threefold has the triple intersection number $H^{3}=25$, second Chern class
$c_{2}.H=70$ and Euler characteristic $\chi=-100$.2010arXiv1006.0223K

An alternative approach for computing the A-model topological data is to first
compute the Picard-Fuchs equation associated with (41), and then use monodromy
to constrain the topological data of the mirror, as was done in
2004math…..12539V . This approach proceeds as follows. We know that the
Picard-Fuchs equation must be the Hadamard product of two second order Picard-
Fuchs equations associated with families of elliptic curves $\mathcal{E}_{s}$
and $\mathcal{E}_{\frac{\varphi}{s}}$, so it is straightforward to find on a
computer (by writing down the holomorphic solution and searching for a fourth
order differential operator that annihilates it). In this way, we find the
Picard-Fuchs equation AESZ 101 (see following subsection for the equation and
its solutions). We compute the solutions of the Picard-Fuchs equation in a
suitable basis and assume that a matrix like the one in (18) determines a
basis of periods with integral symplectic monodromy. We then numerically
compute the monodromy matrices and require that they be symplectic with
integer coefficients. Since these matrices must have integer coefficients, a
single decimal place of precision is sufficient. We also require that the
monodromy around the nearest conifold take a standard form with a single off-
diagonal component. In this way, we find that $H^{3}=25$ and $c_{2}\cdot H=70$
are the only choices consistent with $\chi=-100$.

###  3.2 Solutions of AESZ 101

AESZ 101 is the operator

$\displaystyle\begin{split}\mathcal{L}_{\text{AESZ101}}=&(\varphi-1)^{2}(\varphi+1)\left(\varphi^{2}-123\varphi+1\right)\theta_{\varphi}^{4}\\\
&+2(\varphi-1)\varphi\left(2\varphi^{3}-125\varphi^{2}+244\varphi+121\right)\theta_{\varphi}^{3}\\\
&+\varphi\left(6\varphi^{4}-205\varphi^{3}+6/89\varphi^{2}-787\varphi-187\right)\theta_{\varphi}^{2}\\\
&+2\varphi\left(2\varphi^{4}-39\varphi^{3}+105\varphi^{2}-277\varphi-33\right)\theta_{\varphi}\\\
&+\varphi\left(\varphi^{4}-12\varphi^{3}+12\varphi^{2}-124\varphi-9\right)\end{split}$
(43)

where $\theta_{\varphi}=\varphi\frac{d}{d\varphi}$. AESZ 101 has the Riemann
symbol

$\mathcal{P}\left\\{\begin{array}[]{c c c c c}-1&0&s_{\pm}^{2}&1&\infty\\\
\hline\cr 0&0&0&0&1\\\ 1&0&1&1&1\\\ 1&0&1&3&1\\\
2&0&2&4&1\end{array}~{};\varphi\right\\}~{},$ (44)

where $s_{+}$ and $s_{-}$ are the positive and negative solutions of
$s_{\pm}^{2}+11s_{\pm}-1=0$ respectively i.e. $s_{\pm}=\frac{1}{2}\left(-11\pm
5\sqrt{5}\right)$. A Frobenius basis of solutions of around the MUM at
$\varphi=0$ is defined as in (14) and the relevant power series are now given
by

$\displaystyle\begin{split}F_{0}(\varphi)&=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}\right)^{2}\varphi^{n}\\\
&=1+9\varphi+361\varphi^{2}+21609\varphi^{3}+1565001\varphi^{4}+126630009\varphi^{5}+\ldots\\\
F_{1}(\varphi)&=30\varphi+1425\varphi^{2}+90895\varphi^{3}+\frac{13604625\varphi^{4}}{2}+\frac{1123000637\varphi^{5}}{2}+\ldots\\\
F_{2}(\varphi)&=26\varphi+\frac{4865\varphi^{2}}{2}+\frac{3369877\varphi^{3}}{18}+\frac{122552385\varphi^{4}}{8}+\frac{2398214876497\varphi^{5}}{1800}+\ldots\\\
F_{3}(\varphi)&=-156\varphi-\frac{10599\varphi^{2}}{2}-\frac{4192853\varphi^{3}}{18}-\frac{195151095\varphi^{4}}{16}+\ldots~{}.\end{split}$
(45)

A basis of periods $\Pi$ with integral symplectic mononodromy is defined as in
(17). We use the matrix $\rho$ in (18) with the topological data $H^{3}=25$,
$c_{2}\cdot H=70$ and $\chi=-100$. We choose a base point in the upper half
$\varphi$-plane and anti-clockwise loops around the various singularities.
Alternatively, this base point may be deformed to the real $\varphi$ axis
(this will be helpful for the discussion in Appendix A). Both choices of base
point lead to the same monodromy matrices, which are given by

$\displaystyle\begin{split}M_{\varphi=-1}&=\left(\begin{array}[]{cccc}-19&-10&50&-20\\\
8&5&-20&8\\\ -8&-4&21&-8\\\ -4&-2&10&-3\\\
\end{array}\right),~{}M_{\varphi=0}=\left(\begin{array}[]{cccc}1&-1&10&13\\\
0&1&-12&-25\\\ 0&0&1&0\\\ 0&0&1&1\\\ \end{array}\right)\\\
M_{\varphi=s_{+}^{2}}&=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
-1&0&1&0\\\ 0&0&0&1\\\
\end{array}\right),M_{\varphi=s_{-}^{2}}=\left(\begin{array}[]{cccc}1&0&0&0\\\
-60&1&0&225\\\ -16&0&1&60\\\ 0&0&0&1\\\ \end{array}\right)\\\
M_{\varphi=\infty}&=(M_{\varphi=-1}M_{\varphi=0}M_{\varphi=s_{+}^{2}}M_{\varphi=s_{-}^{2}})^{-1}=\left(\begin{array}[]{cccc}-19&-9&40&-8\\\
-252&-95&392&208\\\ -75&-29&121&52\\\ -4&-2&9&-3\\\
\end{array}\right)\end{split}$ (46)

As an aside, note that, although $\varphi=1$ is a singularity of the Picard-
Fuchs equation, the underlying Calabi-Yau variety is smooth and the periods
have trivial monodromy around this point.333The determinant of the Wronskian
matrix vanishes at $\varphi=1$. It is an apparent singularity. This follows
from the lack of repeated indices in the Riemann symbol (44) at $\varphi=1$.
This point is quite interesting in its own right. As argued in the previous
subsection, the Calabi-Yau variety at $\varphi$ is equivalent to that at
$\frac{1}{\varphi}$. This should be visible in the periods $\Pi(\varphi)$ and,
indeed, we find that

$\frac{1}{\varphi}\Pi\left(\frac{1}{\varphi}\right)=\left(\begin{array}[]{cccc}-4&0&0&15\\\
7&4&-15&0\\\ 0&1&-4&7\\\ -1&0&0&4\\\ \end{array}\right)\Pi(\varphi)$ (47)

where the matrix on the right hand side has been computed in a small
neighbourhood of the fixed point $\varphi=1$. The matrix on the right hand
side of the above identity is an element of $Sp(4,\mathbb{Z})$ and can be
understood as a map from the cohomology of the underlying smooth Calabi-Yau
$X_{\varphi}$ to that of $X_{\frac{1}{\varphi}}$. It has the eigenvalues
$(+1,+1,-1,-1)$ and, at the smooth fixed point $\varphi=1$, it splits
$H^{3}(X_{1},\mathbb{Q})$ into positive and negative eigenspaces. Elements in
the positive eigenspace will Hodge numbers $(3,0)+(0,3)$ whereas those in the
negative eigenspace will have Hodge numbers $(2,1)+(1,2)$. Such points are
known as attractor points of rank two, which are named as such because the
$H^{3,0}(X_{1})\oplus H^{0,3}(X_{1})$ is generated by an integral lattice of
rank two. A numerical computation reveals that

$\Pi(1)=\eta_{1}\begin{pmatrix}6\\\ 1\\\ 3\\\
2\end{pmatrix}+\eta_{2}\begin{pmatrix}0\\\ 5\\\ 1\\\ 0\end{pmatrix},$ (48)

where

$\displaystyle\begin{split}\eta_{1}=&~{}~{}28.985700535271822856288566710613439081713073465729\ldots\\\
\eta_{2}=&-i89.4161554867742890655454362164259781884639062623\ldots~{}.\end{split}$
(49)

Recall that the covariant derivative of special geometry acts on the
holomorphic $3$-form $\Omega$ (and its periods) as
$(\partial_{\varphi}+\partial_{\varphi}K)\Omega$. As cohomology class,
$D_{\varphi}\Omega\in H^{2,1}(X)$ and the Kähler potential $K$ is defined as

$K=-\log\left(-i\int_{X}\Omega\wedge\overline{\Omega}\right)=-\log\left(-i\Pi^{\dagger}\Sigma_{4}\Pi\right)$
(50)

and

$\Sigma_{4}=\begin{pmatrix}0&\mathds{1}_{2}\\\ -\mathds{1}_{2}&0\end{pmatrix}$
(51)

is the standard $4\times 4$ symplectic form. We find that

$D_{\varphi}\Pi(1)=\eta_{3}\begin{pmatrix}10\\\ 1\\\ 5\\\
2\end{pmatrix}+\eta_{4}\begin{pmatrix}0\\\ 3\\\ 1\\\ 0\end{pmatrix}$ (52)

where

$\displaystyle\begin{split}\eta_{3}=&~{}4.5830075667086921421687455172516021305804602797484\ldots\\\
\eta_{4}=&~{}-i9.96834656548114388389121094203891689421621590045\ldots~{}.\end{split}$
(53)

The transcendental numbers $\eta_{i}$ are known to be critical values of
$L$-functions associated to the Calabi-Yau threefold $X_{\varphi=1}$ (up to
rational multiples and factors of $(2\pi i)$). In other words, they contain
information about the arithmetic geometry of the Calabi-Yau at $\varphi=1$. We
will not have much more to say about such points in this paper. The interested
reader is referred to Candelas:2019llw ; Bonisch:2022mgw .

###  3.3 Picard-Fuchs equation of an associated family of elliptic curves

The elliptic curve in (40) is defined by the polynomial

$P(x,y,z,s)=x(x-z)(y-z)-syz(x-y)$ (54)

where $x$, $y$ and $z$ define homogeneous coordinates on $\mathbb{CP}^{1}$ and
$s$ is a complex parameter. On the $z=1$ coordinate patch, the holomorphic
1-form may be taken to be

$\frac{dx}{\partial_{y}P|_{P=0}}=\frac{dx}{\sqrt{x(-4s+x+s(6+s)x-2(1+s)x^{2}+x^{3})}}~{}.$
(55)

We define a basis of $A$ and $B$ periods as follows. The branch points are the
solutions of

$x(-4s+x+s(6+s)x-2(1+s)x^{2}+x^{3})=0.$ (56)

For $s\in\mathbb{R}$ and $0<s<\frac{1}{2}\left(-11+5\sqrt{5}\right)$, the
branch points are all real and are ordered from left to right on the real
line. We define $A\in H_{1}(\mathcal{E}_{s},\mathbb{Z})$ as the homology class
of a contour encircling the first and second branch points in an anti-
clockwise fashion. Similarly, $B\in H_{1}(\mathcal{E}_{s},\mathbb{Z})$ is
defined as the homology class of a contour encircling the second and third
branch points in an anti-clockwise fashion. This implies that

$\omega(s)=\begin{pmatrix}\int_{A}\frac{dx}{\partial_{y}P|_{P=0}}\\\
\int_{B}\frac{dx}{\partial_{y}P|_{P=0}}\end{pmatrix}=(2\pi
i)\begin{pmatrix}1&0\\\ 0&5\end{pmatrix}\begin{pmatrix}1&0\\\ 0&\left(2\pi
i\right)^{-1}\end{pmatrix}\left(\begin{array}[]{l}f_{0}(s)\\\
f_{0}(s)\log(s)+f_{1}(s)\end{array}\right)$ (57)

where

$\displaystyle\begin{split}f_{0}(s)&=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}\right)s^{n}\\\
&=1+3s+19s^{2}+147s^{3}+1251s^{4}+11253s^{5}+104959s^{6}+1004307s^{7}+\ldots\\\
f_{1}(s)&=5s+\frac{75}{2}s^{2}+\frac{1855}{6}s^{3}+\frac{10875}{4}s^{4}+\frac{299387}{12}s^{5}+\frac{943397}{4}s^{6}+\ldots~{}.\end{split}$
(58)

The $A$ period may be computed by expanding the denominator of the integrand
around $s=0$ and evaluating the resulting integrals by computing residues. The
$B$ integral is holomorphic around the point
$s=\frac{1}{2}\left(-11+5\sqrt{5}\right)$ and may be similarly computed around
this point. We then analytically continue it back to zero by noting that the
$A$ period is a annihilated by the differential operator

$(s^{2}+11s-1)\theta_{s}^{2}+s(2s+11)\theta_{s}+s(s+3)$ (59)

with the Riemann symbol

$\mathcal{P}\left\\{\begin{array}[]{c c c}0&\pm s_{\pm}&\infty\\\ \hline\cr
0&0&1\\\ 0&0&1\\\ \end{array}~{};s\right\\},$ (60)

where $s_{\pm}=\frac{1}{2}\left(-11\pm 5\sqrt{5}\right)$. The $B$ period must
be annihilated by the same operator, so we numerically evaluate the
logarithmic solution of the above differential equation around $s=0$ and
evaluate it for some real $0<s<s_{+}$. By numerically evaluating the $B$
period computed around $s_{+}$ at this same value of $s$, we determine the
constant $5$ in (57).

By comparing the holomorphic solutions in (45) and (58), we see that AESZ 101
is a Hadamard product of (59) with itself.

Recall that the elliptic curve $\mathcal{E}_{s}$ is isomorphic to the elliptic
curve $\mathcal{E}_{-\frac{1}{s}}$. As was the case in (47), we see this as a
symmetry of the elliptic periods

$-\frac{1}{s}\omega\left(-\frac{1}{s}\right)=\begin{pmatrix}-2&1\\\
-5&2\end{pmatrix}\omega(s).$ (61)

The above matrix has been computed in a small neighbourhood of the smooth
fixed point $s=i$ and, as always, is only defined up to multiplication by a
monodromy matrix.444The smooth fixed points at $s=\pm i$ have complex
multiplication as a result of the symmetry of the family. For a choice of base
point in the upper half plane and anti-clockwise loops around the various
singularities, we find that the monodromy matrices are generated by

$m_{s=s_{-}}=\begin{pmatrix}11&-4\\\
25&-9\end{pmatrix},~{}~{}~{}m_{s=0}=\begin{pmatrix}1&0\\\
5&1\end{pmatrix}~{}~{}~{}\text{and}~{}~{}~{}m_{s=s_{+}}=\begin{pmatrix}1&-1\\\
0&1\end{pmatrix}~{}.$ (62)

##  4 $3$-Cycles on Fibred Products of Elliptic Curves

###  4.1 Solutions of Hadamard product

Consider the fibred product of a family of elliptic curves
$\mathcal{E}\rightarrow\mathbb{CP}^{1}$, where the elliptic fibre degenerates
at some finite number of points $\Delta\subset\mathbb{CP}^{1}$. We write
$\mathcal{E}_{s}$ for the fibre at some $s\in\mathbb{CP}^{1}$. In studying
Hadamard products, we have been considering fibred products such that the
fibre at $s$ has the form
$\mathcal{E}_{s}\times\mathcal{E}_{\frac{\varphi}{s}}$ for some parameter
$\varphi$ of the resulting $3$-fold
$\mathcal{E}\times_{\mathbb{CP}^{1}}\mathcal{E}$. If the elliptic curves are
presented in Weierstrass form, and we remove points on the base where an
elliptic fibre degenerates, a holomorphic $3$-form on
$\mathcal{E}\times_{\mathbb{CP}^{1}\backslash\Delta_{1}\cup\Delta_{2}}\mathcal{E}$
is given by

$\frac{dx_{1}}{2y_{1}(s)}\wedge\frac{dx_{2}}{2y_{2}\left(\frac{\varphi}{s}\right)}\wedge\frac{ds}{s}$
(63)

where $y_{i}$ and $x_{i}$ are the usual variables in the Weierstrass equation
of the $i^{th}$ elliptic fibre, $s$ is a coordinate on the the base
$\mathbb{CP}^{1}$ and $\Delta_{i}$ is the finite set of points on the base
where the $i^{th}$ elliptic fibre degenerates. We would like to integrate the
above $3$-form over a 3-chain in
$\mathcal{E}\times_{\mathbb{CP}^{1}\backslash\Delta_{1}\cup\Delta_{2}}\mathcal{E}$
so that the integral does not vanish for generic $\varphi$. A $3$-chain with
this property must necessarily take the form of a product of two circles, one
in each of the factors of
$\mathcal{E}_{s}\times\mathcal{E}_{\frac{\varphi}{s}}$, fibred over a contour
on $\mathbb{CP}^{1}\backslash\Delta_{1}\cup\Delta_{2}$. We may further insist
that the $3$-chain is a $3$-cycle by requiring that the $2$-boundary vanishes.
We may arrange this by choosing a non-trivial contour in
$\mathbb{CP}^{1}\backslash\Delta_{1}\cup\Delta_{2}$ and a choice of
$\gamma_{1}\otimes\gamma_{2}$ in
$H_{1}\left(\mathcal{E}_{s},\mathbb{Z}\right)\otimes
H_{1}\left(\mathcal{E}_{\frac{\varphi}{s}},\mathbb{Z}\right)$ that is
invariant when transported around the contour. This defines a topological
$3$-torus. Alternatively, the contour on the base may be open, with end points
in $\Delta_{1}\cup\Delta_{2}$. The requirement of trivial $2$-boundary is
again a constraint on the choice of homology classes in the fibres. Finally,
we may consider collections of such contours on the base, each with a choice
of homology class in the fibre, such that the boundary contributions cancel
out and the collection of $3$-chains becomes a $3$-cycle. We will see examples
of all three types of $3$-cycles, but first we fix some notation.

We write the periods of the holomorphic $(3,0)$ form on a Calabi-Yau threefold
as

$\Pi(\varphi)=\left(\int_{A_{I}}\Omega(\varphi),\int_{B^{I}}\Omega(\varphi)\right)^{T}=\left(F_{I}(\varphi),X^{I}(\varphi)\right)^{T}$
(64)

where $A_{I}$ and $B^{I}$ generate the torsion-free part of
$H_{3}\left(X,\mathbb{Z}\right)$ and intersect as $A_{I}\cap B^{J}=-B^{J}\cap
A_{I}=\delta_{I}^{J}$, with all other intersections vanishing. The index $I$
runs from $I=1$ to $I=b_{3}(X)=4$. We define a dual basis in cohomology
$\\{\alpha^{I},\beta_{J}\\}$ via the relations

$\int_{A_{J}}\alpha^{I}=-\int_{B^{I}}\beta_{J}=\int_{X}\alpha^{I}\wedge\beta_{J}=\delta_{J}^{I}$
(65)

with all other integrals vanishing. The holomorphic $(3,0)$-form may now be
expressed as

$\Omega(\varphi)=F_{I}(\varphi)\alpha^{I}-X^{I}(\varphi)\beta_{I}~{}.$ (66)

A torsion-free, integral cohomology class

$p_{I}\alpha^{I}-q^{I}\beta_{I}$ (67)

is Poincaré dual to the homology class

$-q^{I}A_{I}+p_{I}B^{I}$ (68)

where $p_{I},q^{J}\in\mathbb{Z}$. We will consistently abuse notation and
represent both by the integral periods

$\Gamma=\begin{pmatrix}p_{I}\\\
q^{I}\end{pmatrix}=\begin{pmatrix}\int_{A_{J}}\left(p_{I}\alpha^{I}-q^{I}\beta_{I}\right)\\\
\int_{B^{I}}\left(p_{I}\alpha^{I}-q^{I}\beta_{I}\right)\end{pmatrix}\in\mathbb{Z}^{4}.$
(69)

We will sometimes write $\Pi_{\Gamma}(\varphi)$ for the integral of the
holomorphic $(3,0)$-form over the homology class represented by $\Gamma$ i.e.

$\Pi_{\Gamma}(\varphi)=\Gamma^{T}\Sigma_{4}\Pi(\varphi)=\int_{-q^{I}A_{I}+p_{I}B^{I}}\Omega=\int_{X}\Omega\wedge\left(p_{I}\alpha^{I}-q^{I}\beta_{I}\right)$
(70)

where

$\Sigma_{n}=\begin{pmatrix}0&\mathds{1}_{\frac{n}{2}}\\\
-\mathds{1}_{\frac{n}{2}}&0\end{pmatrix}$ (71)

is the standard $n\times n$ symplectic form and $n\in 2\mathbb{Z}$.

In precisely the same way, we may compute the integral of the holomorphic
$1$-form on an elliptic curve over a $1$-homology class that is Poincaré dual
to a cohomology class with periods $\gamma\in\mathbb{Z}^{2}$. We denote this
analogously by

$\omega_{\gamma}(s)=\gamma^{T}\Sigma_{2}\omega(s)~{}.$ (72)

Suppose that $N\in\mathbb{Z}$ and consider the network

$\left\\{\left(\ell^{(i)},\gamma_{1}^{(i)},\gamma_{2}^{(i)}\right)\right\\}_{i=1}^{N},$
(73)

defined by a collection of contours $\ell^{(i)}$ between two singularities on
the $s$-plane and $1$-homology classes $\gamma_{1}^{(i)}$ and
$\gamma_{2}^{(i)}$ in the first and second elliptic fibre, respectively. The
tensor product of the two homology classes is fibred over the corresponding
contour $\ell^{(i)}$ in such a way that the sum of the resulting $3$-chains is
a $3$-cycle i.e. the $2$-boundary must vanish. We may integrate (63) over this
$3$-cycle and find that

$\Pi_{\Gamma}(\varphi)=\sum_{i=1}^{N}\int_{\ell^{(i)}}\omega_{\gamma^{(i)}_{1}}(s)\omega_{\gamma^{(i)}_{2}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}$
(74)

for some (co)-homology class, represented by $\Gamma\in\mathbb{Z}^{4}$. This
equation shall be the workhorse of the remainder of this paper. It allows us
to compute the (co)homology class associated to a network like the one in
(73).

When studying integrals like the one in (74), one must worry about monodromy
as $\varphi$ is varied around singular points. Braids provide a useful tool
for keeping track of how the contours on the base must vary as $\varphi$ is
varied. We explain this in Appendix A.

Note that, in the examples described in the previous section, there exists a
smooth Calabi-Yau variety $X_{\varphi}$ (for generic $\varphi$) and a map
$X_{\varphi}\rightarrow\mathcal{E}\times_{\mathbb{CP}^{1}}\mathcal{E}$, so
$\Gamma$ should be understood as representing some class in
$H_{3}(X_{\varphi},\mathbb{Q})$ which is pushed forward onto the collection of
$3$-chains in (73). This is because the basis of periods $\Pi(\varphi)$ is
determined, via mirror symmetry, for $X_{\varphi}$.

###  4.2 Double cover of AESZ 3

In section 2, we studied a double cover of AESZ 3. Recall that both
$\mathcal{L}_{\text{AESZ3}}$ and the pull-back
$\mathcal{L}^{*}_{\text{AESZ3}}$ are Hadamard products. Let $\omega^{*}(s)$ be
the solution of $\mathcal{L}^{*}_{\text{AESZ3}}\omega^{*}(s)=0$ with monodromy
in $SL(2,\mathbb{Z})$, as defined in Section 2, and consider the monodromy of
$\omega^{*}(s)\otimes\omega^{*}\left(\frac{\varphi}{s}\right)$ around some
singularity $s_{*}$. The monodromy matrices are of the form
$m_{*}\otimes\widetilde{m}_{s}^{*}$. We choose some real $0<\phi<2^{4}$ and a
set of generators of
$\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{-4,-\frac{\phi}{4},0,\frac{\phi}{4},4,\infty\\},s_{0}\right)$
in Figure 3. We must also fix the branch that $\omega^{*}(s)$ and
$\omega^{*}\left(\frac{\varphi}{s}\right)$ start on. We do this by choosing
the principal branch of the logarithms in (33) and requiring that
$\omega^{*}(s)$ and $\omega^{*}\left(\frac{\varphi}{s}\right)$ are on the same
branch when evaluated on the interval $\frac{\phi}{2^{2}}<s<2^{2}$. With these
conventions, we find the monodromy matrices in Table 1.

$s=-4$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=-\frac{\phi}{4}$$s=0$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=0$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=\frac{\phi}{4}$$s=4$$s=\infty$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=\infty$$\ell_{-4}$$\ell_{-\frac{\phi}{4}}$$\ell_{0}$$\ell_{\frac{\phi}{4}}$$\ell_{4}$$s_{0}$
Figure 1: Generators of
$\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{-4,-\frac{\phi}{4},0,\frac{\phi}{4},4,\infty\\},s_{0}\right)$
in the complex $s$-plane for real $\phi$ such that $0<\phi<4^{2}$. We have
listed points on the $s$-plane where the first elliptic curve degenerates
below the real axis (in black) and points where the second elliptic curve
degenerates above the real axis (in red). We choose a basepoint $s_{0}$ with
very large imaginary part.

In addition to the monodromy matrices, we also determine the vanishing cycles
associated to the various singularities in Table 1. These are listed as
integer vectors $v_{s_{*}}$ and $\widetilde{v}_{s_{*}}$, which are the periods
of the cohomology class Poincaré dual to the vanishing cycle. They may be read
off from the monodromy matrices in Picard-Lefschetz form

$m_{s_{*}}=\mathds{1}_{2}+c_{s_{*}}v_{s_{*}}\left(\Sigma_{2}v_{s_{*}}\right)^{T}~{}~{}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}~{}~{}\widetilde{m}_{s_{*}}=\mathds{1}_{2}+\widetilde{c}_{s_{*}}\widetilde{v}_{s_{*}}\left(\Sigma_{2}\widetilde{v}_{s_{*}}\right)^{T}.$
(75)

$s_{*}$ | $m_{s_{*}}$ | $c_{s_{*}}$ | $v_{s_{*}}$ | $\widetilde{m}_{s_{*}}$ | $\widetilde{c}_{s_{*}}$ | $\widetilde{v}_{s_{*}}$
---|---|---|---|---|---|---
$-4$ | $\left(\begin{matrix}5&2\\\ -8&-3\end{matrix}\right)$ | $-2$ | $\begin{pmatrix}-1\\\ 2\end{pmatrix}$ | $\mathds{1}_{2}$ | n/a | n/a
$-\frac{\phi}{4}$ | $\mathds{1}_{2}$ | n/a | n/a | $\begin{pmatrix}-3&2\\\ -8&5\end{pmatrix}$ | $-2$ | $\begin{pmatrix}1\\\ 2\end{pmatrix}$
0 | $\begin{pmatrix}1&0\\\ -4&1\end{pmatrix}$ | $-4$ | $\begin{pmatrix}0\\\ 1\end{pmatrix}$ | $\begin{pmatrix}-3&4\\\ -4&5\end{pmatrix}$ | $-4$ | $\begin{pmatrix}1\\\ 1\end{pmatrix}$
$\frac{\phi}{4}$ | $\mathds{1}_{2}$ | n/a | n/a | $\begin{pmatrix}1&2\\\ 0&1\end{pmatrix}$ | $-2$ | $\begin{pmatrix}1\\\ 0\end{pmatrix}$
$4$ | $\begin{pmatrix}1&2\\\ 0&1\end{pmatrix}$ | $-2$ | $\begin{pmatrix}1\\\ 0\end{pmatrix}$ | $\mathds{1}_{2}$ | n/a | n/a
$\infty$ | $\begin{pmatrix}5&4\\\ -4&-3\end{pmatrix}$ | $-4$ | $\begin{pmatrix}-1\\\ 1\end{pmatrix}$ | $\begin{pmatrix}1&0\\\ -4&1\end{pmatrix}$ | $-4$ | $\begin{pmatrix}0\\\ 1\end{pmatrix}$
Table 1: The monodromy matrices of the elliptic periods associated with the
Hadamard product $\mathcal{L}_{\text{AESZ3}}^{*}$.

With the monodromy of
$\omega^{*}(s)\otimes\omega^{*}\left(\frac{\phi}{s}\right)$ in hand, we may
now begin identifying $3$-cycles. The first $3$-cycle we consider is the
$T^{3}$ that is responsible for the Hadamard product structure of the fourth
order Picard-Fuchs equation. We start by re-expressing (24) as

$\frac{1}{4}\left(1,0,0,0\right)\Sigma_{4}\Pi^{*}(\phi)=\frac{(2\pi
i)^{3}}{4}\sum_{n=0}^{\infty}\binom{2n}{n}^{4}\left(\frac{\phi}{2^{8}}\right)^{2n}.$
(76)

Note that $(0,1)^{T}\otimes(0,1)^{T}$ is invariant under monodromy around
$\ell_{-\frac{\phi}{4}}\ell_{0}\ell_{\frac{\phi}{4}}$. The union of an
$S_{1}\times S_{1}$ in the homology class
$\left(0,1\right)^{T}\otimes\left(0,1\right)^{T}$ fibred over a loop in the
homotopy class $\ell_{-\frac{\phi}{4}}\ell_{0}\ell_{\frac{\phi}{4}}$
determines a $3$-cycle. For $0<|\phi|<4^{2}$, we may compute

$\displaystyle\oint_{\ell_{-\frac{\phi}{4}}\ell_{0}\ell_{\frac{\phi}{4}}}\omega^{*}_{\begin{pmatrix}0\\\
1\\\ \end{pmatrix}}(s)\omega^{*}_{\begin{pmatrix}0\\\ 1\\\
\end{pmatrix}}\left(\frac{\phi}{s}\right)\frac{ds}{s}=$
$\displaystyle\frac{(2\pi i)^{2}}{4}\sum_{m,n=0}^{\infty}$
$\displaystyle\Bigg{(}\binom{2m}{m}^{2}\binom{2n}{n}^{2}\left(\frac{1}{2^{8}}\right)^{2m+2n}$
$\displaystyle\times\phi^{2m}\oint_{\ell_{-\frac{\phi}{4}}\ell_{0}\ell_{\frac{\phi}{4}}}s^{2m-2n}\frac{ds}{s}\Bigg{)}$
(77) $\displaystyle=\frac{(2\pi i)^{3}}{4}\sum_{n=0}^{\infty}$
$\displaystyle\binom{2n}{n}^{4}\left(\frac{\phi}{2^{8}}\right)^{2n}$

Both of the power series inside the above integral converge for
$\frac{\phi}{4}<|s|<4$ and we find that

$\frac{1}{4}\left(1,0,0,0\right)\Sigma_{4}\Pi^{*}(\phi)=\oint_{\ell_{-\frac{\phi}{4}}\ell_{0}\ell_{\frac{\phi}{4}}}\omega^{*}_{\begin{pmatrix}0\\\
1\\\ \end{pmatrix}}(s)\omega^{*}_{\begin{pmatrix}0\\\ 1\\\
\end{pmatrix}}\left(\frac{\phi}{s}\right)\frac{ds}{s},$ (78)

which should be interpreted as the statement that the union of
$\left(0,1\right)^{T}\otimes\left(0,1\right)^{T}$ over a loop in the homotopy
class $\ell_{-\frac{\phi}{4}}\ell_{0}\ell_{\frac{\phi}{4}}$ determines a
$3$-cycle in the homology class represented by

$\left(\frac{1}{4},0,0,0\right)^{T}.$ (79)

Note that this is a rational homology class, rather than an integral one. This
is to be expected, because the map (9) from the mirror of four quadrics in
$\mathbb{CP}^{7}$ to the fibred product of elliptic curves is a degree $8$
map. We, therefore, expect coefficients in $\frac{1}{8}\mathbb{Z}$.

As another example, consider the union of
$\left(2,0\right)^{T}\otimes\left(2,0\right)^{T}$ over the real interval
$\frac{\phi}{4}\leq s\leq 4$. We expect that this is a $3$-cycle, because the
class in the fibre is a product of vanishing cycles, with one of them
shrinking to zero volume at either end. A useful picture to keep in mind is
Figure 2. This expectation is confirmed by the identity

$\frac{1}{8}\left(0,0,1,0\right)\Sigma_{4}\Pi^{*}(\phi)=\int_{\frac{\phi}{4}}^{4}\omega^{*}_{\begin{pmatrix}2\\\
0\\\ \end{pmatrix}}(s)\omega^{*}_{\begin{pmatrix}2\\\ 0\\\
\end{pmatrix}}\left(\frac{\phi}{s}\right)\frac{ds}{s},$ (80)

where the above contour is taken to be a straight line in the $s$-plane that
starts at $\frac{\phi}{4}$ and ends at $4$. This identity may be proven by
noting that the periods in the integrand are holomorphic at $s=4$ (as expected
of an integral over a vanishing cycle). We may expand all terms inside the
integral aroudn $(s-4)$ and evaluate the integral term-by-term in powers of
$\left(\phi-4^{2}\right)$ (see the computation of (95)). This just leaves the
problem of analytically continuing the expansion of $\omega^{*}$ around $s=4$
to $s=0$ and the expansion of $\Pi^{*}$ around $\phi=4^{2}$ to $\phi=0$. Both
$\omega^{*}$ and $\Pi^{*}$ are the solutions of the pull-back of
hypergeometric differential operators, so the computation may be carried out
by use of Mellin-Barnes integrals. This approach, while rigorous, is rather
cumbersome when one wants to find many identities like (80) . In practice, a
much more effective approach is to numerically evaluate $\Pi^{*}(\phi)$ and
the right hand side of (80) at some fixed $\phi$. We expect that the
components of $\Pi^{*}(\phi)$ are independent transcendental numbers for
generic $\phi$, so we may use a computer to search for an integer relation
between the numerical components of $\Pi^{*}(\phi)$ and the integral on the
right hand side of (80). Such an integer relation may be quickly found with
the Lenstra–Lenstra–Lovász (LLL) lattice basis reduction
algorithm.lenstra1982factoring

$s=-4$$\begin{pmatrix}-1\\\
2\end{pmatrix}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=-\frac{\phi}{4}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\begin{pmatrix}1\\\
2\end{pmatrix}$$s=0$$\begin{pmatrix}0\\\
1\end{pmatrix}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=0$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\begin{pmatrix}1\\\
1\end{pmatrix}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=\frac{\phi}{4}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\begin{pmatrix}1\\\
0\end{pmatrix}$$s=4$$\begin{pmatrix}1\\\
0\end{pmatrix}$$s=\infty$$\begin{pmatrix}-1\\\
1\end{pmatrix}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=\infty$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\begin{pmatrix}0\\\
1\end{pmatrix}$ Figure 2: The complex $s$-plane and singularities for real
$\phi$ such that $0<\phi<4^{2}$. We also visualise a $3$-cycle that vanishes
at $\phi=4^{2}$. Note that at either end of the interval $\frac{\phi}{4}\leq
s\leq 4$, one of the circles in the fibre shrinks to a point. This is why the
resulting $3$-chain has no $2$-boundary. Points where the first elliptic fibre
degenerates have a label below the real axis (in black) and points where the
second elliptic fibre degenerates have a label above the real axis (in red).
Next to every such point, we list a vanishing cycle, which we compute from the
monodromy matrices. Note that we may always approach the same singularity from
a different branch. When we do this, the associated vanishing cycle should be
multiplied by an appropriate product of the monodromy matrices in Table 1.

Note that the union of $\left(2,0\right)^{T}\otimes\left(2,0\right)^{T}$ over
the real interval $\frac{\phi}{2^{2}}\leq s\leq 2^{2}$ is a vanishing
$3$-cycle, because the two ends of the interval on the $s$-plane approach each
other as $\phi\rightarrow 2^{4}$. This expectation is confirmed by the fact
that $(0,0,1,0)$ is the vanishing cycle at $\phi=2^{4}$, which follows from
the monodromy matrix $M_{\phi=2^{4}}$ in (25), which takes the Picard-Lefshetz
form

$M_{\phi=2^{4}}=\mathds{1}_{2}-\begin{pmatrix}~{}0~{}\\\ 0\\\ 1\\\
0\end{pmatrix}\begin{pmatrix}0,&0,&1,&0\end{pmatrix}\Sigma_{4}^{T}$ (81)

The reader should also note that we have used the fibre
$(2,0)^{T}\otimes(2,0)^{T}$ instead of $(1,0)^{T}\otimes(1,0)^{T}$ in the
integrand in (80). This is because we only expect a denominator as large as
$8$ on the left hand side of (80), and $(1,0)^{T}\otimes(1,0)^{T}$ would give
us a denominator of $32$. We note the extra factors of $2$ agree with
$c_{4}=-2$ and $\widetilde{c}_{\frac{\phi}{4}}=-2$ that we compute from the
monodromy matrices in Table 1. A precise explanation for these extra factors
of $2$ is likely to come from a careful study of the map (9) from the mirror
of the intersection of four quadrics in $\mathbb{CP}^{7}$ to the fibred
product of two families of elliptic curves. We leave this problem for future
investigations.

An identity analogous to (80) for the vanishing $3$-cycle at $\phi=-4^{2}$ is
given by

$\frac{1}{8}\left(-8,8,-1,-1\right)\Sigma_{4}\Pi^{*}(\phi)=\int_{\frac{\phi}{4}}^{-4}\omega^{*}_{\begin{pmatrix}-2\\\
4\\\ \end{pmatrix}}(s)\omega^{*}_{\begin{pmatrix}2\\\ 0\\\
\end{pmatrix}}\left(\frac{\phi}{s}\right)\frac{ds}{s},$ (82)

where the above contour is taken along the upper half $s$-plane for real
$\phi$. Note that the $1$-cycles in the integrand match the vanishing cycles
at either end of the contour in the $s$-plane. As expected, the union of
$\left(-2,4\right)^{T}\otimes\left(2,0\right)^{T}$ over a contour between $-4$
and $\frac{\phi}{4}$ on the upper half $s$-plane determines a vanishing cycle
at $\phi=-4^{2}$. We confirm this by noting that the monodromy matrix
$M_{\phi=-4^{2}}$ in (25) has the Picard-Lefshetz form

$M_{\phi=-4^{2}}=\mathds{1}_{2}-\begin{pmatrix}-8~{}\\\ 8\\\ -1\\\
-1\end{pmatrix}\begin{pmatrix}-8,&8,&-1,&-1\end{pmatrix}\Sigma_{4}^{T}~{}.$
(83)

We consider yet another example, in order to illustrate that, in general, one
must really consider collections of contours on the $s$-plane, each with its
own choice of homology class in the fibre. We numerically confirm that

$\displaystyle\begin{split}\frac{1}{8}\left(0,-1,0,0\right)\Sigma_{4}\Pi^{*}(\phi)=&~{}\int_{0}^{\frac{\phi}{4}}\omega^{*}_{\begin{pmatrix}0\\\
1\end{pmatrix}}(s)\omega^{*}_{\begin{pmatrix}1\\\
1\end{pmatrix}}\left(\frac{\phi}{s}\right)\frac{ds}{s}\\\
&+\int_{\frac{\phi}{4}}^{4}\omega^{*}_{\begin{pmatrix}0\\\
1\end{pmatrix}}(s)\omega^{*}_{\begin{pmatrix}0\\\
1\end{pmatrix}}\left(\frac{\phi}{s}\right)\frac{ds}{s}\\\
&+\int_{4}^{\infty}\omega^{*}_{\begin{pmatrix}-1\\\
~{}1\end{pmatrix}}(s)\omega^{*}_{\begin{pmatrix}0\\\
1\end{pmatrix}}\left(\frac{\phi}{s}\right)\frac{ds}{s}.\end{split}$ (84)

For example, at the regular point $\phi=8$, both the left and right hand side
of (84) evaluate to

$\frac{1}{8}\left(0,-1,0,0\right)\Sigma_{4}\Pi^{*}(8)=-34.44699775873907269740146097820\ldots~{}.$
(85)

Note that each of the above integrals are integrals over a $3$-chain with
boundary, but the sum defines a $3$-cycle. For example,
$(0,1)^{T}\otimes(1,1)^{T}$ does not match the vanishing cycle at
$s=\frac{\phi}{4}$ (see Table 1 and Figure 2). However, the boundary term is
cancelled by the next integral, because

$\begin{pmatrix}1\\\ 1\end{pmatrix}\equiv\begin{pmatrix}1\\\
0\end{pmatrix}~{}\text{mod}~{}\widetilde{v}_{\frac{\phi}{4}}~{}.$ (86)

Geometrically, one should imagine gluing two circles together, one in the
homology class $(1,1)^{T}$ and the other in the homology class $(1,0)^{T}$.
This is only possible at a point where the elliptic curve degenerates and the
two $1$-cycles are equal modulo the vanishing cycle. The same kind of
cancellation takes place between the second and third integrals i.e.

$\begin{pmatrix}0\\\ 1\end{pmatrix}\equiv\begin{pmatrix}-1\\\
1\end{pmatrix}~{}\text{mod}~{}v_{4}~{}.$ (87)

This illustrates the point that the most general identity of the form (74)
involves a collection of contours with different choices of homology classes
in the fibre.

###  4.3 Extensions of AESZ 3 and its double cover

We have seen that, for judicious choices of contours $\ell^{(i)}$ between two
singularities in the $s$-plane and homology classes $\gamma_{1}^{(i)}$ and
$\gamma_{2}^{(i)}$ in the fibres, the integral (74) is a solution of a Picard-
Fuchs equation.555The reader might argue that a closed contour in the
$s$-plane need not start and end at a singularity. This is the case, for
example, in (78). However, we may always deform such a contour so that it
starts and ends at some singularity without changing the associated integral.
Moreover, we will assume that this can be done even when the contours are
constrained by the special Lagrangian condition in Section 5. However, generic
choices of contours and fibres will lead to a $3$-chain with boundaries at the
singularities in the $s$-plane. In other words,

$\mathcal{L}_{\text{AESZ3}}^{*}\left(\sum_{i=1}^{N}\int_{\ell^{(i)}}\omega^{*}_{\gamma^{(i)}_{1}}(s)\omega^{*}_{\gamma^{(i)}_{2}}\left(\frac{\phi}{s}\right)\frac{ds}{s}\right)\neq
0~{}.$ (88)

In the case of AESZ 3 and its pullback, we have been able to find a simple
extension of the differential equation that annihilates the above sum of
integrals. The extension is given by

$\bigg{(}(\phi^{2}-2^{8})\theta_{\phi}^{2}+2(\phi^{2}+2^{9})\theta_{\phi}+(\phi^{2}-2^{10})\bigg{)}\mathcal{L}_{\text{AESZ3}}^{*},$
(89)

and it annihilates the integral

$\int_{\ell}\omega^{*}_{\gamma_{1}}(s)\omega^{*}_{\gamma_{2}}\left(\frac{\phi}{s}\right)\frac{ds}{s}$
(90)

for all possible choices of contours $\ell$ between two singularities and
fibre classes $\gamma_{1}$ and $\gamma_{2}$. The extended Picard-Fuchs
operator has the Riemann symbol

$\mathcal{P}\left\\{\begin{array}[]{c c c}0&\pm 4^{2}&\infty\\\ \hline\cr
0&0&1\\\ 0&1&1\\\ 0&1&1\\\ 0&2&1\\\ 2&2&3\\\ 2&2&3\\\
\end{array}~{};\phi\right\\}~{}.$ (91)

and is also invariant under the $\phi\rightarrow\frac{2^{8}}{\phi}$ symmetry
of $\mathcal{L}_{\text{AESZ3}}^{*}$.

The above extension of $\mathcal{L}_{\text{AESZ3}}^{*}$ may be found as
follows. First, note that $\omega^{*}(s)$ has the following expansion around
$s=4$

$\omega^{*}(s)=\begin{pmatrix}\left(\pi-5i\log 2\right)&i\\\
-\pi&0\end{pmatrix}\left(\begin{array}[]{l}f_{0}^{(4)}(s)\\\
f_{0}^{(4)}(s)\log\left(s-4\right)+f_{1}^{(4)}(s)\end{array}\right)$ (92)

where

$\displaystyle\begin{split}f_{0}^{(4)}(s)&=\sum_{n=0}^{\infty}a_{n}^{(4)}(s-4)^{n}\\\
&=1-\frac{1}{8}(s-4)+\frac{5}{256}(s-4)^{2}-\frac{7}{2048}(s-4)^{3}+\frac{169}{262144}(s-4)^{4}+\ldots\\\
f_{1}^{(4)}(s)&=\sum_{n=0}^{\infty}b_{n}^{(4)}(s-4)^{n}\\\
&=-\frac{1}{8}(s-4)+\frac{7}{256}(s-4)^{2}-\frac{17}{3072}(s-4)^{3}+\frac{1775}{1572864}(s-4)^{4}+\ldots~{}.\end{split}$
(93)

The coefficients $a_{n}^{(4)}$ and $b_{n}^{(4)}$ may be computed recursively
with the Picard-Fuchs equation. We have computed the above transition matrix
numerically to a high precision and identified the terms appearing the matrix.
Alternatively, one may use the fact that $\omega^{*}(s)$ is annihlated by the
pull-back of a hypergeometric differential operator, so the transition
matrices may be computed with the help of Mellin-Barnes integrals. Now
consider the integral

$\int_{\frac{\phi}{4}}^{4}\omega^{*}_{\begin{pmatrix}1\\\
0\end{pmatrix}}(s)\omega^{*}_{\begin{pmatrix}1\\\
0\end{pmatrix}}\left(\frac{\phi}{s}\right)\frac{ds}{s}=\pi^{2}\int_{\frac{\phi}{4}}^{4}f_{0}^{(4)}(s)f^{(4)}_{0}\left(\frac{\phi}{s}\right)\frac{ds}{s}$
(94)

where $0<\phi<4^{2}$ and the contour between $s=\frac{\phi}{4}$ and $s=4$ is
taken to be a straight line. We saw in (80) that this determines a solution of
$\mathcal{L}^{*}_{\text{AESZ3}}$. With this choice of contour, the only way of
matching the vanishing cycles at $s=\frac{\phi}{4}$ and $s=4$ is with a choice
of fibre class proportional to $(1,0)^{T}\otimes(1,0)^{T}$, as in (80). It
follows that

$I(\phi)=\int_{\frac{\phi}{4}}^{4}\left(f_{0}^{(4)}(s)\log(s-4)+f_{1}^{(4)}(s)\right)f^{(4)}_{0}\left(\frac{\phi}{s}\right)\frac{ds}{s}$
(95)

cannot be proportional to an integral over a $3$-cycle, so we expand $I(\phi)$
around the singular point $\phi=4^{2}$ and look for an extension of
$\mathcal{L}^{*}_{\text{AESZ3}}$ that annihilates it. We expand the $f_{j}$ as
power series, swap the order of summation and integration and replace each
occurrence of $s$ with $(s-4)$ e.g.

$\left(\frac{\phi}{s}-4\right)^{n}=\sum_{k=0}^{n}\binom{n}{k}\frac{4^{n-k}}{s^{n}}(-1)^{n-k}(s-4)^{n-k}(\phi-4^{2})^{k}$
(96)

and

$\frac{1}{s^{n+1}}=\sum_{l=0}^{\infty}\binom{n+l}{l}\frac{(-1)^{l}}{4^{n+l+1}}(s-4)^{l}.$
(97)

We then use the identity

$\int x^{n}\log x~{}dx=-\frac{x^{n+1}}{(n+1)^{2}}+\frac{x^{n+1}}{n+1}\log x$
(98)

and find that

$I(\phi)=A(\phi)\log(\phi-4^{2})-\log(4)A(\phi)+B(\phi)+C(\phi)$ (99)

where

$\displaystyle A(\phi)$
$\displaystyle=\sum_{n=0}^{\infty}\Bigg{(}\sum_{i=0}^{n}\sum_{j=0}^{n-i}$
$\displaystyle(-1)^{n-i-j+1}\frac{a_{i}^{(4)}a_{j}^{(4)}}{4^{2n+2-i-j}}\frac{(n-i)!(n-j)!}{(n-i-j)!(n+1)!}\Bigg{)}(\phi-4^{2})^{n+1}$
(100) $\displaystyle B(\phi)$
$\displaystyle=\sum_{n=0}^{\infty}\Bigg{(}\sum_{i=0}^{n}\sum_{j=0}^{n-i}$
$\displaystyle(-1)^{n-i-j}\frac{a_{i}^{(4)}a_{j}^{(4)}}{4^{2n+2-i-j}}\frac{(n-i)!(n-j)!}{(n-i-j)!(n+1)!}~{}$
(101)
$\displaystyle\times\left(H_{n+1}-H_{n-j}\right)\Bigg{)}(\phi-4^{2})^{n+1}$
$\displaystyle C(\phi)$
$\displaystyle=\sum_{n=0}^{\infty}\Bigg{(}\sum_{i=0}^{n}\sum_{j=0}^{n-i}$
$\displaystyle(-1)^{n-i-j+1}\frac{a_{i}^{(4)}b_{j}^{(4)}}{4^{2n+2-i-j}}\frac{(n-i)!(n-j)!}{(n-i-j)!(n+1)!}\Bigg{)}(\phi-4^{2})^{n+1}.$
(102)

and $H_{n}$ is the harmonic sum

$H_{n}=\sum_{k=1}^{n}\frac{1}{k}.$ (103)

In deriving these formulae, we have used the identities

$\displaystyle\begin{split}\sum_{k=0}^{n}\binom{n}{k}\frac{(-1)^{k}}{m+n+l+1-k}&=\frac{(-1)^{n}(m+l)!n!}{(m+n+l+1)!}\\\
\sum_{k=0}^{n}\binom{n}{k}\frac{(-1)^{k}}{(m+n+l+1-k)^{2}}&=\frac{(-1)^{n}(m+l)!n!}{(m+n+l+1)!}\left(H_{m+n+l+1}-H_{m+l}\right)~{}.\end{split}$
(104)

It is now straightforward, with the aid of a computer, to expand $I(\phi)$ to
some finite order in $(\phi-4^{2})$ and search for an extension of
$\mathcal{L}^{*}_{\text{AESZ3}}$ that annihilates it. In this way, we find
(89). For various real values of $\phi$ satisfying $0<|\phi|<2^{4}$, we
numerically confirm that the extension annihilates all integrals of the form
(90) where $\ell$ is some contour between the two singularities and

$\gamma_{1}\otimes\gamma_{2}=e_{i}\otimes
e_{j}~{}~{}~{}~{}~{}~{}\text{where}~{}~{}~{}~{}~{}~{}e_{1}=\begin{pmatrix}1\\\
0\end{pmatrix}~{}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}~{}e_{2}=\begin{pmatrix}0\\\
1\end{pmatrix}.$ (105)

All integrals of the form (90) can be written as a linear combination of such
integrals, so there are only finitely many integrals to check for each $\phi$.
We conjecture that this is true for all $\phi$.

Recall that AESZ 3 and its pull-back are related by
$\varphi=\left(\frac{\phi}{2^{8}}\right)^{2}$. A similar exercise for
$\mathcal{L}_{\text{AESZ3}}$ reveals that

$\bigg{(}(2^{8}\varphi^{2}-1)\theta_{\varphi}^{2}+2(2^{7}\varphi^{2}+1)\theta_{\varphi}+(2^{6}\varphi^{2}-1)\bigg{)}\mathcal{L}_{\text{AESZ3}}$
(106)

annihilates the corresponding integrals on the $s$-plane. It follows that the
extension of $\mathcal{L}^{*}_{\text{AESZ3}}$ is a pull-back of the extension
of $\mathcal{L}_{\text{AESZ3}}$.

###  4.4 AESZ 101

In this section, we compute solutions of AESZ 101 by computing integrals over
$3$-cycles on the fibred product of elliptic curves. The analysis is very
similar to that of AESZ 3, so this section will be rather terse. Let
$s_{\pm}=\frac{1}{2}\left(-11\pm 5\sqrt{5}\right)$. We must fix the branches
that $\omega(s)$ and $\omega\left(\frac{\varphi}{s}\right)$ start on. We do
this by choosing some real $0<\varphi<s_{+}^{2}$ and requiring that
$\omega(s)$ and $\omega\left(\frac{\varphi}{s}\right)$ are on the same branch
when evaluated on the interval $\frac{\varphi}{s_{+}}<s<s_{+}^{2}$. This
defines $\omega(s)\otimes\omega\left(\frac{\varphi}{s}\right)$ by analytic
continuation. We choose generators of the fundamental group as in Figure 3 and
find the monodromy matrices in Table 2. From this, we compute the vanishing
cycles at the various singularities in the $s$-plane, which we also list in
Table 2.

$s=s_{-}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=\frac{\varphi}{s_{-}}$$s=0$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=0$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=\frac{\varphi}{s_{+}}$$s=s_{+}$$s=\infty$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s=\infty$$\ell_{s_{-}}$$\ell_{\frac{\varphi}{s_{-}}}$$\ell_{0}$$\ell_{\frac{\varphi}{s_{+}}}$$\ell_{s_{+}}$$s_{0}$ Figure 3: Generators of $\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{s_{-},\frac{\varphi}{s_{-}},0,\frac{\varphi}{s_{+}},s_{+},\infty\\},s_{0}\right)$ in the complex $s$-plane for real $\varphi$ such that $0<\varphi<s_{+}^{2}$. We have listed points on the $s$-plane where the first elliptic curve degenerates below the real axis (in black) and points where the second elliptic curve degenerates above the real axis (in red). Note that we have chosen a basepoint $s_{0}$ with very large imaginary part. $s_{*}$ | $m_{s_{*}}$ | $c_{s_{*}}$ | $v_{\widetilde{s_{*}}}$ | $\widetilde{m}_{s_{*}}$ | $\widetilde{c}_{s_{*}}$ | $\widetilde{v}_{s_{*}}$
---|---|---|---|---|---|---
$s_{-}=\frac{1}{2}(-11-5\sqrt{5})$ | $\left(\begin{matrix}11&-4\\\ 25&-9\end{matrix}\right)$ | $1$ | $\begin{pmatrix}2\\\ 5\end{pmatrix}$ | $\mathds{1}_{2}$ | n/a | n/a
$\frac{\varphi}{s_{-}}$ | $\mathds{1}_{2}$ | n/a | n/a | $\begin{pmatrix}-9&-4\\\ 25&11\end{pmatrix}$ | 1 | $\begin{pmatrix}-2\\\ 5\end{pmatrix}$
0 | $\begin{pmatrix}1&0\\\ 5&1\end{pmatrix}$ | $5$ | $\begin{pmatrix}0\\\ 1\end{pmatrix}$ | $\begin{pmatrix}-9&-5\\\ 20&11\end{pmatrix}$ | $5$ | $\begin{pmatrix}-1\\\ 2\end{pmatrix}$
$s_{+}=\frac{1}{2}(-11+5\sqrt{5})$ | $\begin{pmatrix}1&-1\\\ 0&1\end{pmatrix}$ | $1$ | $\begin{pmatrix}1\\\ 0\end{pmatrix}$ | $\mathds{1}_{2}$ | n/a | n/a
$\frac{\varphi}{s_{+}}$ | $\mathds{1}_{2}$ | n/a | n/a | $\begin{pmatrix}1&-1\\\ 0&1\end{pmatrix}$ | $1$ | $\begin{pmatrix}1\\\ 0\end{pmatrix}$
$\infty$ | $\begin{pmatrix}11&-5\\\ 20&-9\end{pmatrix}$ | $5$ | $\begin{pmatrix}1\\\ 2\end{pmatrix}$ | $\begin{pmatrix}1&0\\\ 5&1\end{pmatrix}$ | $5$ | $\begin{pmatrix}0\\\ 1\end{pmatrix}$
Table 2: The monodromy matrices of the elliptic periods associated with the
Hadamard product $\mathcal{L}_{\text{AESZ101}}$.

The fibre class $(0,1)^{T}\otimes(0,1)^{T}$ is invariant under monodromy
around a closed contour in the homotopy class
$\ell_{\frac{\varphi}{s_{-}}}\ell_{0}\ell_{\frac{\varphi}{s_{+}}}$. The union
of this fibre class over such a contour defines a topological $T^{3}$. By
integrating over this $T^{3}$, we recover the defining property of a Hadamard
product i.e.

$\left(1,0,0,0\right)\Sigma_{4}\Pi(\varphi)=\oint_{\ell_{\frac{\varphi}{s_{-}}}\ell_{0}\ell_{\frac{\varphi}{s_{+}}}}\omega_{\begin{pmatrix}0\\\
1\\\ \end{pmatrix}}(s)\omega_{\begin{pmatrix}0\\\ 1\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}.$ (107)

Note that, unlike in (78) there are no factors $\frac{1}{4}$ in this
expression. Such factors in (78) and related expression arise from the fact
that the map from the smooth Calabi-Yau threefold (which determines the basis
$\Pi^{*}$) and the fibred product of elliptic curves is a degree $8$ map. For
the geometries associated to AESZ 101, the analogous map is birational.

Let $\varphi$ be real and $0<\varphi<s_{+}^{2}$. The singularities on the
$s$-plane are all real and we may identify the vanishing cycle at the
singularity $\varphi=s_{+}^{2}$ with the union of $(1,0)^{T}\otimes(1,0)^{T}$
over the real interval $\frac{\varphi}{s_{+}^{2}}\leq s\leq s_{+}^{2}$. This
is confirmed by the identity

$\left(0,0,1,0\right)\Sigma_{4}\Pi(\varphi)=\int_{\frac{\varphi}{s_{+}}}^{s_{+}}\omega_{\begin{pmatrix}1\\\
0\\\ \end{pmatrix}}(s)\omega_{\begin{pmatrix}1\\\ 0\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s},$ (108)

which may be checked numerically. Alternatively, one may evaluate the integral
on the right hand side as a power series around $\varphi=s_{+}^{2}$, along the
lines of the calculation in subsection 4.3.

As a final example, we find that

$\displaystyle\begin{split}\left(0,-4,0,0\right)\Sigma_{4}\Pi(\varphi)=&~{}\int_{0}^{\frac{\varphi}{s_{+}}}\omega_{\begin{pmatrix}0\\\
2\end{pmatrix}}(s)\omega_{\begin{pmatrix}-1\\\
2\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}\\\
&+\int_{\frac{\varphi}{s_{+}}}^{s_{+}}\omega_{\begin{pmatrix}0\\\
2\end{pmatrix}}(s)\omega_{\begin{pmatrix}0\\\
2\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}\\\
&+\int_{s_{+}}^{\infty}\omega_{\begin{pmatrix}1\\\
2\end{pmatrix}}(s)\omega_{\begin{pmatrix}0\\\
2\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}.\end{split}$ (109)

This identity may be confirmed numerically for real $0\leq\varphi\leq
s_{+}^{2}$, where the each of the contours on the $s$-plane are taken along
the positive real axis. As in the case of (84), none of the above integrals
are themselves annihilated by $\mathcal{L}_{\text{AESZ101}}$. However, all
boundary terms cancel in the sum, which is annihilated by
$\mathcal{L}_{\text{AESZ101}}$.

##  5 BPS networks

###  5.1 Networks from special Lagrangian condition

A D3-brane on a Calabi-Yau threefold $X$ at zero string coupling is determined
by a special Lagrangian submanifold $L$ that supports a flat $U(1)$
connection.Becker:1995kb  The submanifold $L$ has three real dimensions and is
defined by an inclusion map $\iota:L\hookrightarrow X$ such that

$\displaystyle\begin{split}\iota^{*}\omega_{K}&=0\\\
\iota^{*}\text{Im}~{}e^{-i\theta}\Omega&=0\end{split}$ (110)

where $\omega_{K}$ and $\Omega$ are the Kähler form and holomorphic volume
form of the Calabi-Yau threefold, respectively.666 Here we denote the Kähler
form by $\omega_{K}$ to avoid clashing with $\omega$, which we have used to
denote the periods of the holomorphic $1$-form on an elliptic curve. The angle
$\theta$ accounts for the fact that the normalisation of $\Omega$ is arbitrary
and, for a given submanifold $L$, it can be removed by rescaling $\Omega$.
However, it becomes important when one wants to consider the stability of
multiple special Lagrangian submanifolds at the same time. The flat $U(1)$
connections on $L$ are classified by the topology of $L$, so problem of
finding stable D3-branes boils down to the problem of solving (110).
Unfortunately, generic solutions of (110) are extremely difficult to come by.
Most known solutions are constructed as the fixed loci of an anti-holomorphic
involution on $X$. For example, when the polynomials defining $X$ are defined
over the real numbers, $X$ is invariant under complex conjugation and the real
locus is a union of one or more disconnected special Lagrangian submanifolds.
See Walcher:2006rs  for an example.

In previous sections, we have studied the fibred product of two families of
elliptic curves with the holomorphic $3$-form

$\frac{dx_{1}}{2y_{1}(s)}\wedge\frac{dx_{2}}{2y_{2}\left(\frac{\varphi}{s}\right)}\wedge\frac{ds}{s},$
(111)

where $x_{i}$ and $y_{i}$ are the variables used in the Weierstrass equation
of the corresponding elliptic curve. We have been able to identify $3$-cycles
on such geometries with $3$-cycles on a compact Calabi-Yau threefold
$X_{\varphi}$. One naturally asks - can we also impose the special Lagrangian
condition? The Kähler form depends on the Ricci flat Calabi-Yau metric, which
is difficult to compute. However, we have a lot of control over the
holomorphic $3$-form. For generic $\varphi$, the integral of the holomorphic
$3$-form on a $3$-cycle is non-vanishing, so we must consider $3$-cycles
obtained as the union of a circle in each of the elliptic fibres over a
contour on $\mathbb{CP}^{1}$. By integrating (111) over a product of two
circles, one in each of the elliptic fibres, we are led to the constraint

$\omega_{\gamma_{1}}(s)\omega_{\gamma_{2}}\left(\frac{\varphi}{s}\right)\frac{1}{s}\frac{ds}{dl}=e^{i\theta},$
(112)

for the contour $s=s(l)$ on the base. We have chosen to use a parameterisation
by $l$ such that the absolute value of the right hand side is equal to unity.
Here, as in previous sections, $\omega_{\gamma_{i}}$ is the period of the
holmorphic $1$-form on an elliptic fibre over a $1$-cycle in the homology
class $\gamma_{i}$.

Solutions of the constant phase condition (112) define a network of contours
on $\mathbb{CP}^{1}\backslash\Delta_{1}\cup\Delta_{2}$, where $\Delta_{i}$ is
the finite set of points on $\mathbb{CP}^{1}$ where the $i^{\text{th}}$
elliptic curve degenerates. They are strongly reminiscent of the spectral and
exponential networks that have previously appeared in string theory
literature. This is not surprising, because they all arise in essentially the
same way. In the context of the Seiberg-Witten analysis of field theories with
$\mathcal{N}=2$ supersymmetry, BPS states are enumerated by certain critical
spectral networks.Klemm:1996bj ; Gaiotto:2012rg  These are constant phase
contours for a differential on a punctured Riemann surface. In Klemm:1996bj ,
the Seiberg-Witten differential arises from a reduction of a holomorphic
$3$-form on a Calabi-Yau threefold to the base of a fibration, which is quite
similar in spirit to the construction in this paper. Spectral networks have
been generalised to exponential networks.Eager:2016yxd ; Banerjee:2018syt ;
Banerjee:2020moh ; Banerjee:2022oed ; Banerjee:2023zqc  Exponential networks
arise from the special Lagrangian condition on toric Calabi-Yau threefolds.

Two examples of solutions of the constant phase condition (112) are shown in
the top plot of Figure 4. Both of these contours are topological circles with
a monodromy invariant $S^{1}\times S^{1}$ as fibres. In other words, they each
determine a topological $T^{3}$. Between these two contours, there is an
infinite family of solutions of the constant phase condition (112). As
$\varphi\rightarrow 0$, this family of $T^{3}$s defines a SYZ like
fibration.Strominger:1996it  Indeed, as computed in (107), the integral over
any single $T^{3}$ is holomorphic at $\varphi=0$, as expected of the mirror of
a D0-brane at large volume. In this example, the $T^{3}$ fibration only
reaches up to the singularity $s=s_{+}$ on the base $\mathbb{CP}^{1}$.

$\varphi=\frac{s_{+}^{2}}{2}$$s=\frac{\varphi}{s_{-}}$$s=0$$s=\frac{\varphi}{s_{+}}$$s=s_{+}$

$\varphi=\frac{1}{2}(2+i)s_{+}^{2}$$s=\frac{\varphi}{s_{-}}$$s=0$$s=s_{+}$$s=\frac{\varphi}{s_{+}}$

Figure 4: Two plots of constant phase contours with
$\varphi=\frac{s_{+}^{2}}{2}$ (top plot) and
$\varphi=\frac{1}{2}(2+i)s_{+}^{2}$ (bottom plot). All three contours in these
plots have the associated fibre class
$\gamma_{1}\otimes\gamma_{2}=(0,1)^{T}\otimes(0,1)^{T}$ and they all lift to
the homology class $\Gamma=(1,0,0,0)^{T}$. This determines the associated
phases
$\theta=\text{arg}\left(\int_{\Gamma}\Omega(\varphi)\right)=-\frac{\pi}{2}$
for the contours in the top plot and
$\theta=\text{arg}\left(\int_{\Gamma}\Omega(\varphi)\right)\approx-1.49204$
for the contour in the bottom plot. Note that the two contours in the top plot
represent the boundaries of a family of constant phase contours that intersect
the real line at $\frac{\varphi}{s_{+}}\leq s\leq s_{+}$. Also note that the
point at the origin represents the two singularities $s=\frac{\varphi}{s_{-}}$
and $s=0$, which are difficult to distinguish at this scale. The remaining
finite singularity at $s=s_{-}\approx-11.09017$ is not plotted.

The fact that the constant phase contours in Figure 4 come in families is to
be expected. Generally, the embedding of a special Lagrangian submanifold
inside a Calabi-Yau threefold may be deformed while preserving the special
Lagrangian condition. The resulting moduli space of special Lagrangian
submanifolds has real dimension equal to the first Betti number $b_{1}(L)$.
Infinitesimally, such deformations are in one to one correspondence with
harmonic $1$-forms on $L$.1997dg.ga….11002H ; Strominger:1996it

We may also choose
$\theta(\varphi)=\text{arg}\left(\int_{\Gamma}\Omega(\varphi)\right)$ from the
beginning, slightly vary the complex structure parameter $\varphi$ and look
for solutions of (112) while keeping $\Gamma$ and
$\gamma_{1}\otimes\gamma_{2}$ fixed. See, for example, the bottom plot of
Figure 4. The reader may notice that the constant phase contour appears as if
it might collide with the singularity $\frac{\varphi}{s_{+}}$ for a large
enough variation of $\varphi$. This is indeed case and it is expected.
Typically, for a small deformation of complex structure, one can preserve the
special Lagrangian condition by making a compensating small deformation of the
embedding map and the phase $\theta$. However, for a large enough variation,
this may no longer be possible and a special Lagrangian submanifold in a
homology class $\Gamma$ must break up into components in different homology
classes that sum to the original $\Gamma$. This decomposition takes place at a
wall of marginal stability which, for homology classes $\Gamma_{1}$ and
$\Gamma_{2}$, is defined as the region in moduli space where

$\text{arg}\left(\int_{\Gamma_{1}}\Omega(\varphi)\right)=\text{arg}\left(\int_{\Gamma_{2}}\Omega(\varphi)\right).$
(113)

The (dis)-assembling of special Lagrangian submanifolds across such walls may
be organised with split attractor flows.Denef:2001ix  An investigation of how
the networks defined in this paper vary across walls of marginal stability
would be very interesting and we hope to return to this problem in the future.
For now, we plot a simple example of two constant phase contours,
corresponding to different homology classes, that approach their common wall
of marginal stability.

$\varphi=\frac{1}{2}(1+i)s_{+}^{2}$$s=\frac{\varphi}{s_{-}}$$s=0$$s=s_{+}$$s=\frac{\varphi}{s_{+}}$

$\varphi=\frac{1}{2}(3-i)s_{+}^{2}$$s=\frac{\varphi}{s_{-}}$$s=0$$s=s_{+}$$s=\frac{\varphi}{s_{+}}$

Figure 5: Two plots of constant phase contours at
$\varphi=\frac{1}{2}(1+i)s_{+}^{2}$ (top plot) and
$\varphi=\frac{1}{2}(3-i)s_{+}^{2}$ (bottom plot). In both plots, the solid
contour from $s=0$ to $s=\frac{\varphi}{s_{+}}$ has the associated fibre class
$(0,1)^{T}\otimes(1,0)^{T}$ and it lifts to the homology class
$\Gamma_{1}=(-1,0,0,0)^{T}$ when considered together with a constant phase
contour from $s=0$ to $s=\frac{\varphi}{s_{-}}$ with associated fibre class
$(0,1)^{T}\otimes(-4,10)^{T}$ (this constant phase contour is too small to be
seen on these plots). In both plots, the dashed contour from $s=s_{+}$ to
$s=\frac{\varphi}{s_{+}}$ has the associated fibre class
$(-1,0)^{T}\otimes(1,0)$ and lifts to the homology class
$\Gamma_{2}=(0,0,1,0)^{T}$. At $\varphi=\frac{1}{2}(1+i)s_{+}^{2}$, we find
that $\text{arg}\left(\int_{\Gamma_{1}}\Omega(\varphi)\right)\approx 1.61961$
and $\text{arg}\left(\int_{\Gamma_{2}}\Omega(\varphi)\right)\approx-1.36498$.
Similarly, at $\varphi=\frac{1}{2}(3-i)s_{+}^{2}$, we find that
$\text{arg}\left(\int_{\Gamma_{1}}\Omega(\varphi)\right)\approx 1.71821$ and
$\text{arg}\left(\int_{\Gamma_{2}}\Omega(\varphi)\right)\approx 2.62224$. This
determines the associated phase $\theta$ in (112) for all of these contours.

The plots in Figure 5 depict constant phase networks in the homology classes
$\Gamma_{1}=(-1,0,0,0)^{T}$ (solid contour) and $\Gamma_{2}=(0,0,1,0)^{T}$
(dashed contour). Their homology classes are determined by the identities

$\displaystyle\begin{split}\Gamma_{1}^{T}\Sigma_{4}\Pi(\varphi)&=\int_{\frac{\varphi}{s_{-}}}^{0}\omega_{\begin{pmatrix}0\\\
1\\\ \end{pmatrix}}(s)\omega_{\begin{pmatrix}-4\\\ 10\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}+\int_{0}^{\frac{\varphi}{s_{+}}}\omega_{\begin{pmatrix}0\\\
1\\\ \end{pmatrix}}(s)\omega_{\begin{pmatrix}1\\\ 0\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}\\\
\Gamma_{2}^{T}\Sigma_{4}\Pi(\varphi)&=\int_{\frac{\varphi}{s_{-}}}^{0}\omega_{\begin{pmatrix}-1\\\
0\\\ \end{pmatrix}}(s)\omega_{\begin{pmatrix}1\\\ 0\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}~{},\end{split}$ (114)

which we confirm numerically. We see in Figure 5 that the networks in homology
classes $\Gamma_{1}$ and $\Gamma_{2}$ begin to approach each other as we vary
$\varphi$. At the $(\Gamma_{1},\Gamma_{2})$-wall of marginal stability, two
contours from $s_{+}$ to $\frac{\varphi}{s_{+}}$ will overlap and we may
define a new network by replacing their assocaited fibre classes with their
sum i.e.

$\begin{pmatrix}0\\\ 1\end{pmatrix}\otimes\begin{pmatrix}1\\\
0\end{pmatrix}+\begin{pmatrix}-1\\\ 0\end{pmatrix}\otimes\begin{pmatrix}1\\\
0\end{pmatrix}=\begin{pmatrix}-1\\\ 1\end{pmatrix}\otimes\begin{pmatrix}1\\\
0\end{pmatrix}.$ (115)

We can do this because $(1,0)^{T}\otimes\gamma_{2}$ is a vanishing cycle at
the singularity $s_{+}$ for any choice of $\gamma_{2}$. Thus, at the
$(\Gamma_{1},\Gamma_{2})$-wall of marginal stability, we find a new network
associated with the three-term identity

$\displaystyle\begin{split}(\Gamma_{1}+\Gamma_{2})^{T}\Sigma_{4}\Pi(\varphi)=&\int_{\frac{\varphi}{s_{-}}}^{0}\omega_{\begin{pmatrix}0\\\
1\\\ \end{pmatrix}}(s)\omega_{\begin{pmatrix}-4\\\ 10\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}+\int_{0}^{s_{+}}\omega_{\begin{pmatrix}0\\\
1\\\ \end{pmatrix}}(s)\omega_{\begin{pmatrix}1\\\ 0\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}\\\
&+\int_{s_{+}}^{\frac{\varphi}{s_{+}}}\omega_{\begin{pmatrix}-1\\\ 1\\\
\end{pmatrix}}(s)\omega_{\begin{pmatrix}1\\\ 0\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}.\end{split}$ (116)

Finally, it should be noted that the constant phase contours introduced in
this section and the $3$-cycles of the previous section are well-defined when
the homology classes in the fibres are non-seperable. Instead of
$\gamma^{(i)}_{1}\otimes\gamma^{(i)}_{2}\in
H_{1}(\mathcal{E}_{s},\mathbb{Z})\otimes
H_{1}(\mathcal{E}_{\frac{\varphi}{s}},\mathbb{Z})$, we may choose some
generic, non-separable $G^{(i)}\in H_{1}(\mathcal{E}_{s},\mathbb{Z})\otimes
H_{1}\left(\mathcal{E}_{\frac{\varphi}{s}},\mathbb{Z}\right)$ that cannot be
written as a tensor product of two homology classes. We may then ask if the
union of the $G^{(i)}$ over some contours $\ell^{(i)}$ can be identified with
a homology class on a Calabi-Yau threefold. As always, we abuse notation and
identify $G^{(i)}$ with a vector of its components in some suitable basis. We
then check if an expression of the form

$\sum_{i=1}^{N}(G^{(i)})^{T}\Sigma_{2}\otimes\Sigma_{2}\int_{\ell^{(i)}}\omega(s)\otimes\omega\left(\frac{\varphi}{s}\right)\frac{ds}{s}$
(117)

is annihilated by the Hadamard product. The constant phase condition (112) is
also straightforward to generalise for non-separable fibres and is given by

$G^{T}\left(\Sigma_{2}\otimes\Sigma_{2}\right)\omega(s)\otimes\omega_{\gamma_{2}}\left(\frac{\varphi}{s}\right)\frac{1}{s}\frac{ds}{dl}=e^{i\theta}~{}.$
(118)

However, if we want to identify the solutions with actual submanifolds, it is
important that the homology classes in the fibres are separable.

###  5.2 Counting networks?

We now come to the main unresolved question of this paper. We ask - is it
possible to count the number (of families) of solutions of the constant phase
condition (112) that lift to a given (torsion free) homology class $\Gamma\in
H_{3}(X,\mathbb{Z})$ at a given $\varphi$? For example, a family of solutions
like the one in Figure 4 should be counted once. The following heuristic
argument suggests that this is possible. Choose some $\Gamma$ as above and a
non-singular value of $\varphi$. This determines a phase
$\theta=\text{arg}\left(\int_{\Gamma}\Omega(\varphi)\right)$ that we compute
with the fourth order Picard-Fuchs equation (the Hadamard product). We assume
that, when a solution of the constant phase condition (112) belongs to a
family, we may always choose a contour in this family that starts and ends at
singularities on the base $\mathbb{CP}^{1}$, so we restrict our search to such
contours. We analytically continue the elliptic periods to one of the
singularities and try solving the constant phase condition (112) for
$\gamma_{1}\otimes\gamma_{2}$ with small integer coefficients. We may
compensate for different choices of analytic continuation by choosing
$\gamma_{1}\otimes\gamma_{2}$ that are related by known monodromy matrices, so
we make an arbitrary choice. Note that a choice of
$\gamma_{1}\otimes\gamma_{2}$ is a choice of four integers and, if those
integers are too large, the factor

$\omega_{\gamma_{1}}(s)\omega_{\gamma_{2}}\left(\frac{\varphi}{s}\right)\frac{1}{s}$
(119)

will also be large. This means that the absolute value of $\frac{ds}{dl}$ will
be correspondingly small and the constant phase contour will be very slow to
move away from the starting singularity. Since we want a network of constant
phase contours that lift to a given $\Gamma$ at a fixed $\varphi$, the
parameter $l$ is bounded by

$l\leq\left|\int_{\Gamma}\Omega(\varphi)\right|.$ (120)

If the components of $\gamma_{1}\otimes\gamma_{2}$ are too large, it will be
impossible for the constant phase contour to reach another singularity or form
a non-trivial homotopy class in “time” $l$ that satisfies (120).

In principle, one could fix $\Gamma$ and $\varphi$ and then simply perform a
brute force computer search for solutions of $\eqref{eq: constant phase
condition}$ that begin at a given singularity. One would start at some
singularity and iterate over $\gamma_{1}\otimes\gamma_{2}$ with increasingly
large coefficients until it becomes clear that the solutions will not reach
another singularity while satisfying (120). One would then perform this search
for all singularities and then, armed with finitely many contours, find all
combinations of these contours that lift to the given homology class $\Gamma$.
Moreover, since the BPS networks introduced in this section capture
information about the topology of the associated special Lagrangian
submanifold, via their deformations, one would ultimately hope to be able to
count $A$-branes and make contact with known enumerative invariants of the
associated compact Calabi-Yau threefold (and its mirror). An analogous problem
for exponential networks has been studied in Banerjee:2022oed .

###### Acknowledgements.

The author benefited from many helpful conversations with Kilian Bönisch,
Philip Candelas, Xenia de la Ossa, Vasily Golyshev, Amir-Kian Kashani-Poor,
Matthew Kerr, Albrecht Klemm, Gregory Moore, Fernando Rodriguez Villegas and
Mauricio Romo. The author is also grateful for hospitality and support from
the Bethe Centre for Theoretical Physics (Bonn) and The Abdus Salam
International Centre for Theoretical Physics (Trieste). The author is
primarily supported by the US Department of Energy under grant DE-SC0010008.

##  Appendix A $\varphi$-monodromy as a braid

When studying identities of the form

$\Pi_{\Gamma}(\varphi)=\sum_{i=1}^{N}\int_{\ell^{(i)}}\omega_{\gamma^{(i)}_{1}}(s)\omega_{\gamma^{(i)}_{2}}\left(\frac{\varphi}{s}\right)\frac{ds}{s},$
(121)

one must worry about monodromy as $\varphi$ is varied around some singular
points. The left hand side simply transforms as $\Pi(\varphi)\mapsto
M\Pi(\varphi)$ for some monodromy matrix $M$.777Which implies that
$\Pi_{\Gamma}(\varphi)\mapsto\Pi_{{M^{-1}}\Gamma}(\varphi)$, because
$\Sigma_{4}M=(M^{-1})^{T}\Sigma_{4}$ for $M\in Sp(4,\mathbb{Z})$. However, the
analytic continuation of the integrals on the right hand side is more subtle.
As the parameter $\varphi$ is varied, some of the singularities in the
$s$-plane will move around and the contours $\ell^{(i)}$ must be deformed to
avoid these singularities. We will focus on the case of closed contours,
because any open contour is homotopy equivalent to the composition of a small
set of open contours and an arbitrary complicated closed contour. For example,
suppose that $\varphi$ is real and all the singularities in the $s$-plane lie
on the real line. An open contour between singularities $s_{1}$ and $s_{2}$ is
homotopy equivalent to the composition of a path from $s_{1}$ to a base point
$s_{0}$ in the upper half $s$-plane, a loop based at $s_{0}$ and a path from
$s_{0}$ to $s_{2}$ along the upper half plane. The action of
$\varphi$-monodromy on an open contour from the base point $s_{0}$ to a
singularity on the real line is readily computed by hand, and there are
finitely many such contours to consider. However, the closed loop based at
$s_{0}$ can be arbitrarily complicated. Fortunately, it is not too difficult
to quickly compute the action of $\varphi$-monodromy on
$\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{s_{-},\frac{\varphi}{s_{-}},0,\frac{\varphi}{s_{+}},s_{+}\\},s_{0}\right)$.
To be concrete, consider AESZ 101 and suppose that $\varphi_{0}$ is real and
$0\leq\varphi_{0}\leq s_{+}^{2}$. The choice $\varphi=\varphi_{0}$ sets all
the singularities in the $s$-plane to real values and fixes their ordering
from left to right in the $s$-plane. Now fix a set of generators of
$\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{-1,0,s_{+}^{2},s_{-}^{2},\infty\\},\varphi_{0}\right)$
as in Figure 6.

$\varphi=-1$$\varphi=0$$\varphi=s_{+}^{2}$$\varphi=s_{-}^{2}$$\varphi=\infty$$L_{-1}$$L_{0}$$L_{s_{+}^{2}}$$L_{s_{-}^{2}}$$\varphi_{0}$
Figure 6: Generators of
$\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{-1,0,s_{+}^{2},s_{-}^{2},\infty\\},\varphi_{0}\right)$
in the complex $\varphi$-plane. Note that we have chosen a real basepoint
$\varphi_{0}$ with $0<\varphi_{0}<s_{+}^{2}$.

By drawing contours on the complex $s$-plane, one might see that the homotopy
class
$\ell_{\frac{\varphi}{s_{+}}}\in\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{s_{-},\frac{\varphi}{s_{-}},0,\frac{\varphi}{s_{+}},s_{+}\\},s_{0}\right)$
is mapped to
$\ell_{\frac{\varphi}{s_{-}}}\ell_{0}\ell_{\frac{\varphi}{s_{+}}}\ell_{0}^{-1}\ell_{\frac{\varphi}{s_{-}}}^{-1}$
when $\varphi$ is transported along $L_{0}$ (see Figure 3 for the generators
$\ell_{s_{*}}$). However, this quickly becomes impractical. A more systematic
approach is to identify $L_{0}$ with a braid that acts on
$\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{s_{-},\frac{\varphi}{s_{-}},0,\frac{\varphi}{s_{+}},s_{+}\\},s_{0}\right)$.artin1947theory

The $n^{\text{th}}$ braid group $B_{n}$ may be defined in terms of generators
$\sigma_{i}$ as

$B_{n}=\Bigg{\langle}\sigma_{1},\sigma_{2},\ldots,\sigma_{n-1}\begin{array}[]{|l
l}\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{1}&~{}\text{if}~{}2\leq|i-j|\\\
\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}&~{}\text{if}~{}1\leq
i\leq n-2\end{array}\Bigg{\rangle},$ (122)

where each generator $\sigma_{i}$ should be understood as the braid with the
$i^{\text{th}}$ strand moved over the $(i+1)^{\text{th}}$ strand and all other
strands left unchanged. This is represented by the braid diagram in Figure 7.

$i-1$$i+1$$i$$~{}~{}i+2$$\boldsymbol{\cdots}$$\boldsymbol{\cdots}$ Figure 7:
The braid represented by the generator $\sigma_{i}\in B_{n}$.

Our conventions are such that time increases as one moves upwards along a
braid i.e. the braid $\sigma_{i}\sigma_{i+1}$ is represented by the braid in
Figure 8.

$i-1$$i+1$$~{}~{}i+2$$i$$\boldsymbol{\cdots}$$\boldsymbol{\cdots}$ Figure 8:
The braid represented by $\sigma_{i}\sigma_{i+1}\in B_{n}$.

Label the generators of the free group $F_{n}$ as $x_{i}$, where
$i\in\\{1,2,\ldots,n\\}$. The braid group $B_{n}$ acts on $F_{n}$ from the
right. In terms of generators, this action is defined as

$x_{j}\sigma_{i}=\begin{cases}x_{j}x_{j+1}x_{j}^{-1}&\text{if}~{}i=j\\\
x_{j-1}&\text{if}~{}i=j-1\\\ x_{j}&\text{otherwise}\end{cases}$ (123)

One should understand this group action as follows. A braid intersects the
complex plane $\mathbb{C}$ transversely. At the start of the braid, loops
around the points of intersection define an element of the fundamental group
of the $n$-times punctured plane for some choice of base point. This is
isomorphic to the free group $F_{n}$. Now one may transport such a loop
upwards along a braid and express the transported loop in terms of the
original generators of $F_{n}$. In this way, every braid defines an
automorphism of $F_{n}$.

We compute the homomorphism
$\mu:\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{-1,0,s_{+}^{2},s_{-}^{2},\infty\\},\varphi_{0}\right)\rightarrow
B_{5}$ by examining the way singularities in the $s$-plane move (see Figure 9
for an example). We find that $\mu$ maps the generators in Figure 6 to the
braids

$\displaystyle\mu(L_{-1})$
$\displaystyle=(\sigma_{2}\sigma_{3}\sigma_{2})(\sigma_{4}^{2}\sigma_{1}^{2})(\sigma_{2}\sigma_{3}\sigma_{2})^{-1}$
(124) $\displaystyle\mu(L_{0})$
$\displaystyle=(\sigma_{2}\sigma_{3}\sigma_{2})^{2}$
$\displaystyle\mu(L_{s_{+}^{2}})$ $\displaystyle=\sigma_{4}^{2}$
$\displaystyle\mu(L_{s_{-}^{2}})$ $\displaystyle=\sigma_{1}^{2}~{}.$

Note that the image of $\mu$ lies in the subgroup of pure braids i.e. braids
in which the strands start and end in the same order. This, of course, is just
a consequence of the fact that all elements of
$\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{-1,0,s_{+}^{2},s_{-}^{2},\infty\\},\varphi_{0}\right)$
will start and end at the same point.

$s_{-}$$\frac{\varphi}{s_{-}}$$0$$\frac{\varphi}{s_{+}}$$s_{+}$$s_{-}$$\frac{\varphi}{s_{-}}$$0$$\frac{\varphi}{s_{+}}$$s_{+}$
Figure 9: $\mu(L_{0})=(\sigma_{2}\sigma_{3}\sigma_{2})^{2}$ as a braid
diagram.

It is now straightforward to compute the action of $\varphi$-monodromy on an
element of
$\pi_{1}\left(\mathbb{CP}^{1}\backslash\\{s_{-},\frac{\varphi}{s_{-}},0,\frac{\varphi}{s_{+}},s_{+},\infty\\},s_{0}\right)$.
We label the generators in Figure 3 from left to right (i.e.
$x_{1}=\ell_{s_{-}},x_{2}=\ell_{\frac{\varphi}{s_{-}}},\ldots$) and use (123).
This is already implemented in many computer algebra systems e.g.
SageMath.sagemath  As an example, consider the identity (107). Repeated here,
it is given by

$\left(1,0,0,0\right)\Sigma_{4}\Pi(\varphi)=\oint_{\ell_{\frac{\varphi}{s_{-}}}\ell_{0}\ell_{\frac{\varphi}{s_{+}}}}\omega_{\begin{pmatrix}0\\\
1\\\ \end{pmatrix}}(s)\omega_{\begin{pmatrix}0\\\ 1\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s}.$ (125)

Analytically continuing both sides around $L_{s_{-}^{2}}$ leads to the new
identity

$\left(1,0,0,0\right)\left(M_{s_{-}^{2}}^{T}\right)^{-1}\Sigma_{4}\Pi(\varphi)=\oint_{\left(\ell_{\frac{\varphi}{s_{-}}}\ell_{0}\ell_{\frac{\varphi}{s_{+}}}\right)\mu\left(L_{s_{-}^{2}}\right)}\omega_{\begin{pmatrix}0\\\
1\\\ \end{pmatrix}}(s)\omega_{\begin{pmatrix}0\\\ 1\\\
\end{pmatrix}}\left(\frac{\varphi}{s}\right)\frac{ds}{s},$ (126)

where $M_{s_{-}^{2}}$ is the monodromy matrix computed in (46) and

$\left(\ell_{\frac{\varphi}{s_{-}}}\ell_{0}\ell_{\frac{\varphi}{s_{+}}}\right)\mu\left(L_{s_{-}^{2}}\right)=\ell_{s_{-}}\ell_{\frac{\varphi}{s_{-}}}\ell_{s_{-}}^{-1}\ell_{0}\ell_{\frac{\varphi}{s_{+}}}.$
(127)

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