# Gauss-Manin connection in disguise:
Open Gromov-Witten invariants

Felipe Espreafico

###### Abstract

In mirror symmetry, after the work by J. Walcher, the number of holomorphic
disks with boundary on the real quintic lagrangian in a general quintic
threefold is related to the periods of the mirror quintic family with boundary
o two homologous rational curves, known as Deligne conics. Following the
ideias of H. Movasati, we construct a quasi-affine space parametrizing such
objects enhanced with a frame for the relative de Rham cohomology with
boundary at the curves compatible with the mixed Hodge structure. We also
compute a modular vector field attached to such a parametrization.

##  1 Introduction

In the 1980s, physicists working on String Theory discovered a phenomena
called Mirror Symmetry: the idea is that tha same physical theory can be
described by two mathematically different models. Naively, one of this models
(the A-model) is mainly related to the symplectic geometry of Calabi-Yau
manifolds, while the other one (B-model) is related to the complex algebraic
geometry of these spaces. Although many of the tools used by these physicists
were not yet rigorous, this led to some impressing purely mathematical
results. The most famous example is the quintic threefold: mirror symmetry
associates, to a generic quintic threefold in $\mathbb{P}^{4}$, a family of
manifolds known as mirror quintics. In the paper [CDLOGP91], published in
1991, the authors used these techniques to make predictions for the number of
rational curves on a generic quintic threefold in $\mathbb{P}^{4}$. What they
did was to assume that the correlation functions from the A-model and the
B-model were the same.The first was associated to Gromov-Witten invariants and
a prediction to the number of rational curves on a general quintic, and the
second were computed explicitly. This correlation function is usually called
Yukawa coupling.

In the B-model, the Yukawa coupling is related to a generalization of Kaneko
and Zagier’s theory of quasi-modular forms developed by Movasati in [Mov15].
His approach relies on the algebraic de Rham cohomology (see [Gro66]) and on
the Gauss-Manin connection (see [KO68]). The idea is to consider the moduli
space $\sf T_{\rm cl}$ of pairs $(W,[\alpha_{1},\dots,\alpha_{4}])$, where $W$
is a mirror quintic and $[\alpha_{1},\dots,\alpha_{4}]$ is a basis to the
third de Rham cohomology of $W$ that, in some sense, respects the Hodge
filtration. After computing the Gauss-Manin connection matrix, he was able to
define a differential algebra attached to $\sf T_{\rm cl}$ with elements that
behave similarly to modular forms: he found relations between the generators
and their derivatives which generalizes the Ramanujan relations we have for
the Eisenstein series. The procedure described above is part of a more general
program known as Gauss-Manin connection in disguise (GMCD, for short). It is
an attempt to construct a general theory of modular forms via functions on a
moduli space submitted to a differential equation. Some cases have already
been treated, from the elliptic curve case (see [Mov12]) to more general cases
as in [AMSY16] and the more recent paper [AKV22].

In this text we will focus solely on the B-side of mirror symmetry and
consider another enumerative result: the computation of the numbers of disks
with boundary on the real quintic Lagrangian inside the quintic threefold.
These numbers were first predicted in [Wal07] and then fully computed in
[PSW08]. This computation relies on the open Gromov-Witten invariants which
are also computed in these articles. Instead of just considering the mirror
family, we need to fix, in each one of the elements of the family, a pair of
homologous rational curves $C_{\pm}$ called Deligne conics. See Section 2 for
details. We refer to [MW09] for the reasons to consider these two curves. In
this context, we have a new natural period to compute, which is the integral
of a holomorphic three form over the homology connecting the two curves above.
This period satisfy a non homogenous version of the Picard-Fuchs equation,
which can be found on Section 2.4. In order to execute the ideas from the GMCD
program, we consider a relative version of the algebraic de Rham cohomology
$H_{{\rm dR}}^{3}(W,C_{+}\cup C_{-})$ and a relative version of the Gauss-
Manin connection. We define a moduli space $\sf T_{op}$ of triples
$(W,C_{\pm},[\alpha_{0},\dots,\alpha_{4}])$, where $W$ is an element of the
mirror family, $C_{\pm}$ is the pair of curves described above on each element
of the family and $[\alpha_{0},\dots,\alpha_{4}]$ is a basis of the third
relative algebraic de Rham cohomology which respect the mixed Hodge structure
(for details on mixed Hodge structures, we refer to [PS08] and Section 2.5 of
this text). In this case, the Hodge filtration consists of $F^{0}\supset
F^{1}\supset\dots\supset F^{4}=0$ with $\dim F^{0}=5$, $\dim F^{1}=4$, $\dim
F^{2}=2$, $\dim F^{3}=1$ and the weight filtration consists of $0=W_{0}\subset
W_{1}\subset\dots W_{3}$, with $\dim W_{0}=\dim W_{1}=0$, $\dim W_{2}=1$ and
$\dim W_{3}=5$. These dimensions are computed in Section 2.5. The idea of
considering mixed Hodge structures and relative cohomology in the GMCD
framework was examined in the case of elliptic curves with two fixed points in
the paper [CML21], in which the authors recovered the theory of Jacobi forms
of index zero, and in the paper [AKV22], where the authors considered affine
Calabi-Yau varieties.

######  Definition 1.

A relatively enhanced mirror quintic is simply a triple
$(X,C_{\pm},[\alpha_{0},\dots,\alpha_{4}])$, where $X$ is a mirror quintic,
$C_{\pm}$ is the pair of homologous curves cited above and specified in (11),
$\delta_{0}$ is a homology connecting the two curves and
$[\alpha_{0},\dots,\alpha_{4}]$ is a basis of $H^{3}_{dR}(X,C_{+}\cup C_{-})$
satisfying:

  1. i)

$\alpha_{i}\in F^{4-i}\setminus F^{5-i},\quad i=1,2,3,4$;

  2. ii)

$[\langle\alpha_{i},\alpha_{j}\rangle]=\left[\begin{array}[]{c
cccc}0&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&0&1&0\\\ 0&0&-1&0&0\\\
0&-1&0&0&0\end{array}\right]$;

  3. iii)

$\alpha_{0}\in F^{1}\setminus F^{2}$;

  4. iv)

$\alpha_{0}\in W_{2}$;

  5. v)

$\int_{\delta_{0}}\alpha_{0}=1$;

  6. vi)

$\alpha_{i}\in W_{3}\setminus W_{2},\quad i=1,2,3,4$.

Although condition $(v)$ above seems to depend on the choice of $\delta_{0}$,
it is actually an algebraic condition which is not influenced by this choice.
Indeed, $\alpha_{0}$ ends up being a class on the image of the map $H_{{\rm
dR}}^{2}(C_{+}\cup C_{-})\to H_{{\rm dR}}^{3}(W,C_{+}\cup C_{-})$ (see Section
2.5) and, therefore, its integral over any homology class depends only on the
boundary of the homology class.

######  Theorem 1.

Relatively enhanced mirror quintics can be parametrized by the nine
coordinates in affine space given by

${\sf
T_{op}}:=\text{Spec}\left(\mathbb{C}\left[s_{0},s_{1},s_{2},s_{3},s_{4},s_{5},s_{6},s_{7},s_{8},\frac{1}{s_{5}(s_{0}^{10}-s_{4}^{10})s_{0}s_{4}}\right]\right).$
(1)

For an explicit description of the parametrization above, see the proof of
Theorem 1 in Section 3.1. With this moduli space in hands, it is possible to
compute a differential equation relating this generators $s_{i}$ with open
Gromov-Witten invariants and the virtual count of disks.

######  Theorem 2.

Consider the space $\sf T_{op}$ defined above. Let $\sf A$ be the Gauss-Manin
connection matrix in the basis $\alpha$. There is a unique vector field $\sf
R$, for which the connection composed with it is given, in the basis $\alpha$,
by

$\sf A_{\sf R}=\begin{bmatrix}0&0&0&0&0\\\ 0&0&1&0&0\\\ \sf F&0&0&\sf Y&0\\\
0&0&0&0&-1\\\ 0&0&0&0&0\end{bmatrix},$

for regular functions $\sf Y$ and $\sf F$ in $\sf T_{op}$. The expression of
$\sf R$, $\sf F$ and $\sf Y$ in the coordinates from Theorem 1 are

${\sf
Y}=\frac{5^{8}\left(s_{4}^{10}-s_{0}^{10}\right)^{2}}{s_{5}^{3}},\quad{\sf
F}=-s_{7}{\sf Y},$ (2) $\sf R:\left\\{\begin{aligned}
\dot{s}_{0}&=\frac{1}{2s_{0}s_{5}}\left(6\cdot
5^{4}s_{0}^{10}+s_{0}^{2}s_{3}-5^{4}s_{4}^{10}\right)\\\
\dot{s}_{1}&=\frac{1}{s_{5}}\left(-5^{8}s_{0}^{12}+5^{5}s_{0}^{8}s_{1}+5^{8}s_{0}^{2}s_{4}^{10}+s_{1}s_{3}\right)\\\
\dot{s}_{2}&=\frac{1}{s_{5}}\left(-3\cdot
5^{9}s_{0}^{14}-5^{4}s_{0}^{10}s_{1}+2\cdot 5^{5}s_{0}^{8}s_{2}+3\cdot
5^{9}s_{0}^{4}s_{4}^{10}+5^{4}s_{1}s_{4}^{10}+2s_{2}s_{3}\right)\\\
\dot{s}_{3}&=\frac{1}{s_{5}}\left(-5^{10}s_{0}^{16}-5^{4}s_{0}^{10}s_{2}+3\cdot
5^{5}s_{0}^{8}s_{3}+5^{10}s_{0}^{3}s_{4}^{10}+5^{4}s_{2}s_{4}^{10}+3s_{3}^{2}\right)\\\
\dot{s}_{4}&=\frac{1}{10s_{5}}\left(5^{6}s_{0}^{8}s_{4}+5s_{3}s_{4}\right)\\\
\dot{s}_{5}&=\frac{1}{s_{5}}\left(-5^{4}s_{0}^{10}s_{6}+3\cdot
5^{5}s_{0}^{8}s_{5}+2s_{3}s_{5}+5^{4}s_{4}^{10}s_{6}\right)\\\
\dot{s}_{6}&=\frac{1}{s_{5}}\left(3\cdot
5^{5}s_{0}^{8}s_{6}-5^{5}s_{0}^{6}s_{5}-2s_{2}s_{5}+3s_{3}s_{6}\right)\\\
\dot{s}_{7}&=-s_{8}\\\
\dot{s}_{8}&=-\frac{5^{12}\left(s_{0}^{10}-s_{4}^{10}\right)}{s_{5}}\cdot\frac{15}{8}\left(\frac{s_{4}}{s_{0}}\right)^{5}\frac{1}{25\sqrt{5}}\\\
\end{aligned}\right.$ (3)

The proof of Theorem 2 is computational. The interesting part about it is
that, if we consider $s_{i}$ functions on a variable $q$, take the derivation
to be $5q\frac{d}{dq}$, fix initial values $s_{0,0}:=\frac{1}{\sqrt{5}}$,
$s_{0,1}:=12\sqrt{5}$ and $s_{0,4}:=0$ and allow fractional exponents, we get

$-5^{3}{\sf
Y}=5+2875q+4876875q^{2}+8564575000q^{3}+\dots=\sum_{d=0}^{\infty}n_{d}d^{3}\frac{q^{d}}{1-q^{d}}$
$\frac{4}{5^{3}}{\sf
F}(q):=30q^{1/2}+13800q^{3/2}+27206280q^{5/2}+47823842250q^{7/2}+...=$
$=\sum_{d\,{\rm odd}}n_{d}^{\rm disk}d^{2}\frac{q^{d/2}}{1-q^{d}},$

where $n_{d}$ are the virtual counts of rational curves of degree $d$ on a
generic quintic threefold (see [CDLOGP91] and [Mov15]) and $n_{d}^{\rm disk}$
are the virtual counts of disks with boundary on a lagragian of a quintic
threefold (see [Wal07] and [PSW08]). The q-expansions for all functions
$s_{i}$ can be found on the author’s
webpage111www.impa.br/~felipe.espreafico/expansionsi. Notice that
$s_{1},s_{2},s_{3},s_{4},s_{5}$ and $s_{6}$ are the same as the corresponding
$t_{i}$ from [Mov15, Theorem 3] and that $s_{0}^{2}=t_{0}$ and
$s_{4}^{10}=t_{4}$.

Our paper is divided in four parts. Section 2 has the aim of presenting basic
definitions and properties of the objects and concepts we are using. In
Section 3 we present the proofs of the two main results stated in the
introduction. In Section 4 we make some computations using the period matrix
and the generalized period domain to explain, for example, why the disk counts
are appearing in Theorem 2. Moreover, these computations help us understand
why these functions behave as modular forms.

### Acknowledgments

I would like to thank my advisor Hossein Movasati for helping me through the
development of this work and for presenting this problem to me. I also would
like to thank Johannes Walcher, Martin Vogrin, Hans Jockers, Lukas Hanh and
Sebastian Nills for valuable discussions and comments on early versions of
this work. I also would like to thank IMPA for the infrastructure and research
ambient and CNPq for financial support via grant 141069/2020-1. Finally, I
would like to thank the Institute of Mathematics of the Heidelberg University
and the Mainz Institute for Theoretical Physics for the great reception they
provided during my visit.

##  2 Preliminaries

To start, we present some of the necessary concepts to prove our theorems and
fix the notation we are going to use in the rest of the paper.

###  2.1 The case of Elliptic Curves

Before start elaborating on the main topic of the paper, which is to construct
a modular framework for open string invariants for the quintic, we recall how
a geometric framework is construct for the classical quasi-modular forms. This
is an attempt to motivate the constructions in the present text. We refer the
interested reader to [Mov12], where this framework was first developed, for
more details and proofs.

Recall that the algebra of of quasi-modular forms is
$\mathbb{C}[E_{2},E_{4},E_{6}]$, where $E_{k}$ are the Eisenstein series. They
satisfy a system of differential equations known as Ramanujan equations. Our
idea is to interpret these equations as a vector field on a suitable moduli
space of elliptic curves. To do this, we have to deal with elliptic curves and
elliptic integrals. In order to deal with both at the same time we will
consider enhanced elliptic curves.

######  Definition 2.

A triple $(E,\alpha,\omega)$, where $\alpha$ is a holomorphic 1-form (first
piece of the Hodge filtration) and $\omega$ is not holomorphic such that
$\langle\alpha,\omega\rangle=1$ is called enhanced elliptic curve.

Here, $\langle,\rangle$ is the usual intersection product on the algebraic de
Rham cohomology. The definition above allow us to consider, the integrals of
$\alpha$ and $\omega$ over paths in $E$, that is, to study elliptic integrals.

######  Proposition 1 ([Mov12], Prop 5.4).

The moduli space of enhanced elliptic curves is given by

$T=\\{(t_{1},t_{2},t_{3})\in\mathbb{C}^{3}\ |\ 27t_{3}^{2}-t_{2}^{3}\neq
0\\},$ (4)

where the $(t_{1},t_{2},t_{3})$ corresponds to the triple

$E:y^{2}=4(x-t_{1})^{3}+t_{2}(x-t_{1})+t_{3}\qquad\alpha=\frac{dx}{y}\qquad\omega=x\frac{dx}{y}.$

We now have a universal family $X\to T$ of enhanced elliptic curves and a
basis of sections of the de Rham cohomology bundle. If we compute the Gauss-
Manin connection in this basis $(\alpha,\omega)$, we get an explicit matrix in
terms of the differentials $dt_{i}$.

######  Proposition 2 ([Mov12], Prop. 4.1).

Let $R$ be a vector field in $T$ such that $\nabla_{R}\alpha=-\omega$ and
$\nabla_{R}\omega=0$. Then $R$ is unique and it is given by

$\mathrm{R}=\left(t_{1}^{2}-\frac{1}{12}t_{2}\right)\frac{\partial}{\partial
t_{1}}+\left(4t_{1}t_{2}-6t_{3}\right)\frac{\partial}{\partial
t_{2}}+\left(6t_{1}t_{3}-\frac{1}{3}t_{2}^{2}\right)\frac{\partial}{\partial
t_{3}}$ (5)

If $R$ is written as a system of differential equations, after multiplying
$t_{i}$ by some constants, we get exactly the Ramanujan equations. By looking
at the 1-dimensional locus $L$ for which $R$ generates the tangent space, the
maps $t_{1},t_{2},t_{3}$ restricted to $L$ will be the Eisenstein series after
a change of coordinates. This is a sign that we can generalize modular forms
by looking at functions on a suitable moduli space for each case.

######  Remark 1.

The locus $L$ in the last paragraph has an interpretation based on the periods
(integrals) of $\alpha$ and $\omega$ over integral cycles. From this point of
view, $L$ is a fundamental domain for the action of an algebraic group and the
natural coordinate for $L$ is given by a quotient of two such periods. For
more details, see sections 7 and 8 of [Mov12]. In the section 4 of the present
text, we give this interpretation for the case we are considering.

###  2.2 Mirror Quintic, Deligne Conics and Relative Algebraic de Rham
Cohomology

After the motivation, we introduce our main objects, which are the Mirror
quintic threefold and the Deligne conics.

######  Definition 3.

Let $\psi^{5}\neq 1$ and let $G$ be the group given by

$G=\left.\left\\{\left(a_{0},\ldots,a_{4}\right)\in\mathbb{Z}_{5}^{5}:\sum_{i}a_{i}\equiv
0\bmod 5\right\\}\middle/\mathbb{Z}_{5}\right.,$ (6)

where $\mathbb{Z}_{5}$ is embedded diagonally. This group acts on
$\mathbb{P}^{4}$ in the natural way:

$(a_{0},\ldots,a_{4})\bullet[x_{0},\ldots,x_{4}]\mapsto[\mu^{a_{0}}x_{0}:\ldots\mu^{a_{4}}x_{4}],$

where $\mu$ is a primitive fifth root of unit. For us, a mirror quinitic
$X_{\psi}$ is the resolution of singularities of the quotient

$\left\\{\left[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}\right]\in\mathbb{P}^{4}\mid
x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}-5\psi
x_{0}x_{1}x_{2}x_{3}x_{4}=0\right\\}/G.$ (7)

After the quotient and the resolution, one can observe that the varieties
obtained will have the Calabi-Yau property. For details, see [GP90] or [CK99,
Section 2.2]. As we have stated in the introduction, our framework will
include two rational curves, the Deligne concis $C_{\pm}\subset X_{\psi}$.
Besides being rational, these curves are homologous as cycles in
$H_{2}(X,\mathbb{Z})$. They are defined by the equations

$C_{\pm}=\left\\{x_{0}+x_{1}=0,x_{2}+x_{3}=0,{x_{4}}^{2}\pm\sqrt{5\psi}x_{1}x_{3}=0\right\\}.$
(8)

Our goal now is, instead of considering forms on the absolute cohomology, to
consider the relative algebraic de Rham cohomology of the pair
$(X_{\psi},C_{+}\cup C_{-})$. In order to do this, we recall the definition
for the relative algebraic de Rham cohomology in general, which is a
generalization of the original definition from [Gro66].

######  Definition 4.

Let $Y\subset X$ be a closed subvariety of $X$. Define the sheaf
$\Omega^{m}_{X,Y}$ as

$\Omega^{m}_{X,Y}(U)=\Omega^{m}_{X}(U)\oplus\Omega^{m-1}_{Y}(U\cap Y),$

and define a differential operator as

${\rm d}(\omega,\alpha)=({\rm d}\omega,\omega|_{Y}-{\rm d}\alpha),$

where the d’s appearing on the right hand side are the boundary operators of
$X$ and $Y$. We define the the m-th relative de Rham cohomology of X with
boundary on Y, denoted by $H_{\rm dR}^{m}(X,Y)$ as the mth-hypercohomology of
the complex of sheaves

$0\to\Omega^{0}_{X,Y}\mathop{\to}^{d}\Omega_{X,Y}^{1}\mathop{\to}^{d}\dots\mathop{\to}^{d}\Omega^{m}_{X,Y}\to\dots$

With this definition, we get a natural long exact sequence of the pair. In our
case of $C_{+}$ and $C_{-}$, we get $\dim H_{{\rm dR}}^{3}(X_{\psi},C_{+}\cap
C_{-})=5$ and a surjection $H_{{\rm dR}}^{3}(X_{\psi},C_{+}\cap C_{-})\to
H_{{\rm dR}}^{3}(X_{\psi})$, which means that we can come up with a basis for
$H^{3}_{dR}(X_{\psi},C_{+}\cup C_{-})$ by choosing elements in the pre image
of any basis for $H_{{\rm dR}}^{3}(X_{\psi})$ and adding the image of a non
zero element from $H_{{\rm dR}}^{2}(C_{+}\cup C_{-})$. Recall that $\dim
H_{{\rm dR}}^{3}(X_{\psi})=4$.

###  2.3 Moduli Space

After the motivation coming from the Elliptic curve, our goal is to construct
an analogous moduli space, we are going to assign coordinates for the space of
triples $(X_{\psi},C_{\pm},\omega)$, where $\omega$ is a holomorphic
differential 3-form on $X_{\psi}$. First, we recall that, from [Mov15, Section
2.1], there are affine coordinates for the pairs $(X,\omega)$: one can
associate coordinates $(t_{0},t_{4})$ to a mirror quintic $X_{\psi}$, with
$\psi^{-5}=\frac{t_{4}}{t_{0}^{5}}$. If we recall $G$ from Definition 3, we
have:

$\displaystyle X_{t_{0},t_{4}}$ $\displaystyle:=\\{f(x)=0\\}/G$ (9)
$\displaystyle f(x)$
$\displaystyle:=-t_{4}x_{0}^{5}-x_{1}^{5}-x_{2}^{5}-x_{3}^{5}-x_{4}^{5}+5t_{0}x_{0}x_{1}x_{2}x_{3}x_{4}.$
(10)

The 3-form dependent on $(t_{0},t_{4})$ is the form induced on the resolution
of the quotient via the residue form

$\omega_{1}:=\frac{dx_{1}\wedge dx_{2}\wedge dx_{3}\wedge dx_{4}}{df}$

which is clearly invariant after the action of the group $G$. Note that these
coordinates are only defined $t_{0}^{5}\neq t_{4}$ and $t_{4}\neq 0$. The
curves $C_{\pm}$ also depend on this coordinates, but, in order to avoid
taking tenth and square roots, we introduce new coordinates $s_{0}$ and
$s_{4}$ which satisfy $s_{0}^{2}=t_{0}$ and $s_{4}^{10}=t_{4}$. In these
coordinates the curves are the resolution of singularities of the quotient of

$C_{\pm}=\left\\{s_{4}^{2}x_{0}+x_{1}=0,x_{2}+x_{3}=0,s_{4}{x_{4}}^{2}\pm\sqrt{5}s_{0}x_{1}x_{3}=0\right\\}\subset
X_{t_{0},t_{4}},$ (11)

by the group $G$. In the appendix of [MW09], they give an explicit way to
solve this singularities. We can, therefore, associate a pair $(s_{0},s_{4})$
to a triple $(X_{s_{0},s_{4}},C_{\pm},\omega)$. Of course, this association is
only defined when $s_{4}^{10}\neq s_{0}^{10}$ and $s_{4}s_{0}\neq 0$, since
for $s_{0}=0$ both curves $C_{+}$ and $C_{-}$ are equal. We end up with a
quasi affine space ${\sf
S_{op}}:=\mathbb{C}^{2}\setminus\\{s_{0}s_{4}(s_{0}^{10}-s_{4}^{10})=0\\}$
parametrizing the triples $(X_{s_{0},s_{4}},C_{\pm},\omega)$. This
parametrization has an important property which we state below:

######  Proposition 3.

Let $r\in\mathbb{C}^{*}$. If $(s_{0},s_{4})$ is the point corresponding to
$(X,\omega,C)\in M$, then $(rs_{0},rs_{4})$ is the point corresponding to
$(X,r^{-2}\omega,C)$.

###### Proof.

We know that the isomorphism
$(x_{0},x_{1},x_{2},x_{3},x_{4})\mapsto(x_{0},rx_{1},rx_{2},rx_{3},rx_{4})$
between $X_{s_{0},s_{4}}=X_{t_{0},t_{4}}$ and
$X_{rs_{0},rs_{4}}=X_{r^{2}t_{0},r^{10}t_{4}}$ takes $\omega(t_{0},t_{4})$ to
$r^{-2}\omega(r^{2}t_{0},r^{10}t_{4})$ (see [Mov15], sec. 2.1). Using that
$t_{0}=s_{0}^{2}$ and $t_{4}=s_{4}^{10}$, we have our result. We just need to
check that it maps the curve $C$ to its correspondent. Indeed, we have

$C_{rs_{0},rs_{4}}=\left\\{r^{2}s_{4}^{2}x_{0}+x_{1}=0,x_{2}+x_{3}=0,rs_{4}x_{4}^{2}+rs_{0}\sqrt{5}x_{1}x_{3}=0\right\\}\subset
X_{rs_{0},rs_{4}}$

and

$C_{s_{0},s_{4}}=\left\\{s_{4}^{2}x_{0}+x_{1}=0,x_{2}+x_{3}=0,s_{4}x_{4}^{2}+s_{0}\sqrt{5}x_{1}x_{3}=0\right\\}\subset
X_{s_{0},s_{4}}.$

Taking a point of the second curve and applying the isomorphism, we have that
the equations of the first one are satisfied.

$\displaystyle r^{2}s_{4}^{2}x_{0}+(r^{2}x_{1})=r^{2}(s_{4}x_{0}+x_{1})=0,$
$\displaystyle(r^{2}x_{2})+(r^{2}x_{3})=r^{2}(x_{2}+x_{3})=0,$ $\displaystyle
rs_{4}(r^{2}x_{4})^{2}+rs_{0}\sqrt{5}(r^{2}x_{1})(r^{2}x_{3})=r^{5}(s_{4}x_{4}^{2}+s_{0}\sqrt{5}x_{1}x_{3})=0.$

This finishes the proof. ∎

Considering the zero locus of the equation $f$ from (9) in the product
$\mathbb{P}^{4}\times\sf S_{op}$, we get a family $\mathcal{X}\to\sf S_{op}$.
Now, $C_{\pm}$ induce a subfamily $\mathcal{Y}\subset\mathcal{X}$
corresponding to the curves. The next step is to define a structure on this
triple which encodes properties of the relative algebraic de Rham cohomology
of each element on the family.

###  2.4 Relative Gauss-Manin Connection

One of the most important tools in our paper is the algebraic Gauss-Manin
connection. It was first introduced in the paper [KO68], by Katz and Oda. They
defined the Gauss-Manin connection for any smooth morphism $\pi:X\to S$ of
smooth schemes (in other words, a family over $S$). Roughly, it formalizes the
notion of differentiating with respect to the parameters of a family. They
consider the algebraic de Rham cohomology of the family, which is the
hyperderived sheaf

$\mathcal{H}_{dR}^{m}(X/S):=R^{m}\pi_{*}(\Omega^{\bullet}_{X/S})$ (12)

defined over $S$. This is the sheaf associated to the presheaf
$U\mapsto\mathbb{H}^{m}\left(\pi^{-1}(U),\Omega^{\bullet}_{X/S}\right)$, where
$\mathbb{H}$ denotes the hypercomology. They define a connection

$\nabla:\mathcal{H}_{dR}^{m}(X/S)\to\mathcal{H}_{dR}^{m}(X/S)\otimes\Omega^{1}_{S}$
(13)

by considering a filtration on $\Omega^{\bullet}_{X}$ and taking the spectral
sequence associated to this filtration. Here, we need a relative version of
this construction. If we have a smooth closed subvariety $Y\subset X$ for
which the restriction of $\pi$ is smooth, it is not difficult to make the same
construction and get a connection

$\nabla:\mathcal{H}_{dR}^{m}(X,Y/S)\to\mathcal{H}_{dR}^{m}(X,Y/S)\otimes\Omega^{1}_{S},$
(14)

where $\mathcal{H}_{dR}^{m}(X,Y/S)$ is defined as above, by considering
$\Omega^{\bullet}_{X,Y/S}$ and not $\Omega^{\bullet}_{X/S}$ (see Definition 4,
but considering $\Omega^{\bullet}_{X/S}$ and $\Omega^{\bullet}_{Y/S}$ instead
of $\Omega^{\bullet}_{X}$ and $\Omega^{\bullet}_{Y}$). This definition of the
relative version of the Gauss-Manin connection is compatible with the long
exact sequence of the pair and thus many computations from the absolute case
can be transported to the relative case. The most important example would be
the equality

$d\left(\int_{\delta}\omega\right)=\int_{\delta}\nabla\omega,$ (15)

where $\delta$ is any cycle (absolute or relative). In order to compute the
connection for the family $\sf S_{op}$, we will need differential relations
among the periods and use (15). We consider, therefore, the nonhomogenous
version of the Picard-Fuchs equation

$\theta^{4}-z\left(\theta+\frac{1}{5}\right)\left(\theta+\frac{2}{5}\right)\left(\theta+\frac{3}{5}\right)\left(\theta+\frac{4}{5}\right)=15\frac{\sqrt{5^{-5}z}}{8}$
(16)

satisfied by the integral of a holomorphic three form on $X_{\psi}$ over the
homology connecting the two curves $C_{\pm}$. Above, $z=\psi^{-5}$ and
$\theta=z\frac{\partial}{\partial z}$. The equation in the form above is in
[PSW08, page 1170] using coordinates $t$ with $z=5^{5}e^{t}$. Observe that if
we consider the right hand side of (16) to be zero, we get the classical
equation appearing on [CDLOGP91] for which the periods given by integrals of
the holomorphic three form over absolute homology classes are solutions.

###  2.5 Mixed Hodge structure on the relative algebraic de Rham cohomology

Consider the family $\mathcal{X}\to\sf S_{op}$ defined in Section 2.3. We want
to study the structure we have on the relative algebraic de Rham cohomology.
Recall from algebraic topology that we can define the cup product in the
relative cohomology by restricting the cup product from absolute cycles to
relative ones. In the case we are considering, i.e., the cohomology $H_{{\rm
dR}}^{3}(X,Y)$, where $X$ is a mirror quintic and $Y$ is the union of the two
curves $C_{\pm}$, we have that the element coming from $H_{{\rm dR}}^{2}(Y)$
is degenerate for the product. Therefore, we can use the computation from
[Mov15, Section 2.3], since the intersection matrix will be the same after
adding zeros to the first column and row.

The Mixed Hodge structure on the relative cohomology is defined by means of
the mixed cone (see [PS08, Theorem 3.22 and Example 3.24]). The important part
is that the long exact sequence of the pair ends up being a long exact
sequence of mixed Hodge structures and so the Hodge and the weight filtration
may be defined via this sequence. Also, the Gauss-Manin connection satisfies
the Griffths transversality property for the Hodge filtration, that is,
$\nabla(F^{p})\subset F^{p-1}\otimes\Omega^{1}_{S}$. This is easily seen using
that fact that the Gauss-Manin connection commutes with the long exact
sequence and that the long exact sequence is a sequence of mixed Hodge
structure. We have a nice description of the mixed Hodge structure for our
case, given by the proposition below.

######  Proposition 4.

We have that the image of the map $\alpha:H_{\rm dR}^{2}(Y)\to H_{\rm
dR}^{3}(X,Y)$ defined via the long exact sequence is one-dimensional and it is
contained in $F^{1}H^{3}_{\rm dR}(X,Y)\setminus F^{2}H^{3}_{\rm dR}(X,Y)$,
where $F$ represents the Hodge filtration. Also this image is exactly the
second piece of the weight filtration $W_{2}(H^{3}_{\rm dR}(X,Y))$. In
particular, any generator of this image satisfy properties (iii) and (iv) of
Definition 1.

###### Proof.

Recall that, as $Y$ is the union of two $\mathbb{P}^{1}$, it has
$F^{2}H^{2}_{\rm dR}(Y)=0$ and $F^{1}H^{2}_{\rm dR}(Y)=H^{2}_{\rm dR}(Y)$.
Therefore, as the long exact sequence of the pair is a sequence of mixed Hodge
structures, we conclude that the image is contained in $F^{1}$ and not on
$F^{2}$ (since we have $\alpha(F^{p}(H^{2}_{\rm dR}(Y)))={\rm Im}(\alpha)\cap
F^{p}H^{3}_{\rm dR}(X,Y)$). For the part about the weight filtration, we just
need to use that the weight filtration for $H^{2}_{\rm dR}(Y)$ is given by
$W_{0}=0$, $W_{1}=0$ and $W_{k}=H^{2}_{\rm dR}(Y)$ for $k\geq 2$. Therefore,
by again using that the exact sequence is a sequence of mixed Hodge
structures, we conclude that the image of $\alpha$ is $W_{2}H^{3}_{\rm
dR}(X,Y)$. ∎

##  3 Proofs of the main theorems

In this section, our goal is to prove the two main theorems stated in the
introduction. For simplicity, throughout this section, we denote $Y=C_{+}\cup
C_{-}$.

###  3.1 Proof of Theorem 1

###### Proof of Theorem 1.

Consider the basis $W=\\{\omega_{1},\dots,\omega_{4}\\}$ for $H^{3}_{\rm
dR}(X)$, with $\omega_{1}$ a holomorphic 3-form and
$\omega_{i}:=\nabla_{\frac{\partial}{\partial t_{0}}}(\omega_{i-1})$, where
$t_{0}:=s_{0}^{2}$ and $\nabla$ is the Gauss-Manin connection on the absolute
cohomology (see Section 2.3). Notice that we use derivatives with respect to
$t_{0}$ instead of $s_{0}$, since it makes it easier to compare with the
absolute case. Of course, we can go from $t_{0}$ to $s_{0}$ via the relation

$\frac{\partial}{\partial t_{0}}=\frac{1}{2s_{0}}\frac{\partial}{\partial
s_{0}}.$

Using that the map $H_{dR}^{3}(X,Y)\to H_{dR}^{3}(X)$ is surjective, we can
take elements $\omega_{1},\dots,\omega_{4}\in H_{dR}^{3}(X,Y)$ corresponding
to the basis $W$. Then, we choose a generator of ${\rm Im}(H_{{\rm
dR}}^{2}(Y)\to H_{{\rm dR}}^{3}(X,Y))$ and call it $\omega_{0}$. This
generator is chosen to be the image of the of the Poincaré dual of the
difference of the homology classes $[C_{+}]$ and $[C_{-}]$. In this way, we
get that the integral of $\omega_{0}$ over the homology connecting the two
curves is 1. Now, consider the matrix:

$S=\begin{bmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&a&b&0&0\\\
s_{7}&c&s_{6}&s_{5}&0\\\ s_{8}&s_{1}&s_{2}&s_{3}&d\end{bmatrix}$ (17)

and assume it is invertible, which implies $s_{5}\neq 0$. The base
$\alpha=S\omega$ satisfy all properties on Definition 1 execept for (ii).
Indeed, by Proposition 4 above, (iii) and (iv) are satisfied and, as the map
$H^{3}(X,Y)\to H^{3}(X)$ preserves filtrations and the Gauss-Manin connection
sends $F^{i}$ to $F^{i-1}$, we have condition (i). Condition (v) is satisfied
by construction. Demanding $S[\langle\omega_{i},\omega_{j}\rangle]S^{\rm
tr}=\Phi$, that is, condition (ii), we get equations relating $a,b,c,d$ and
the other variables:

$\displaystyle cb-s_{6}a$ $\displaystyle=3125s_{0}^{6}+s_{2},$ (18)
$\displaystyle d$ $\displaystyle=-bs_{5},$ (19) $\displaystyle s_{5}a$
$\displaystyle=-3125s_{0}^{8}-s_{3},$ (20) $\displaystyle d$
$\displaystyle=625\left(s_{4}^{10}-s_{0}^{10}\right).$ (21)

To perform this computation, we make use of the intersection product computed
in [Mov15, Proposition 3] and the fact that $\alpha_{0}$ is degenerate for the
intersection product (see Section 2.5). This relations imply that we can drop
the variables $a$, $b$, $c$ and $d$ and only consider, besides $s_{0}$ and
$s_{4}$, five coordinates $s_{1},s_{2},s_{3},s_{5},s_{6}$, which are the same
as the corresponding $t_{i}$ in [Mov15], and the extra two coordinates $s_{7}$
and $s_{8}$ which only appear in the relative case. Notice that, for each
matrix $S$, we obtain a different basis $\alpha$ and for each basis $\alpha$,
we obtain a matrix by inverting $S$ and solving $\omega=S^{-1}\alpha$. ∎

###  3.2 Proof of Theorem 2

To prove Theorem 2, we need first to compute the Gauss-Manin connection in the
basis $\alpha$. Fix $z=\frac{t_{4}}{t_{0}^{5}}=\frac{s_{4}^{10}}{s_{0}^{10}}$
and consider the non-homogenous Picard-Fuchs differential equation 16. Let
$\eta_{1}=t_{0}\omega_{1}$ and $\eta_{0}=\omega_{0}$. Those forms are the ones
we get by pulling back $\omega_{0}$ and $\omega_{1}$ via the ismorphism
$X_{1,\frac{s_{4}}{s_{0}}}\cong X_{s_{0},s_{4}}$ (see Proposition 3). Define
$\eta_{i}=\nabla_{\frac{\partial}{\partial z}}(\eta_{i-1})$. By the definition
of $z$, we get a relation between $\frac{\partial}{\partial z}$ and
$\frac{\partial}{\partial t_{0}}$ given by

$\frac{\partial}{\partial
z}=\frac{-1}{5}\frac{t_{0}^{6}}{t_{4}}\frac{\partial}{\partial t_{0}}.$ (22)

It is easy therefore to get a relationship between the basis $\eta$ and
$\omega$. We call this matrix $C$. As we observed, the Picard-Fuchs equation
(16) is satisfied by the integral of $\eta_{1}$ over the homology connecting
the curves $C_{+}$ and $C_{-}$. Using that $\int\eta_{0}=\int\omega_{0}=1$,
the fact that $\eta_{1}$ satisfy (16) implies the following equality:

$\int_{\delta_{0}}\nabla_{\frac{\partial}{\partial
z}}\eta_{4}=\frac{-p}{z^{4}(z-1)}\int_{\delta_{0}}\eta_{0}+\frac{-24}{625z^{3}(z-1)}\int_{\delta_{0}}\eta_{1}+\\\
+\frac{-24z+5}{5z^{3}(z-1)}\int_{\delta_{0}}\eta_{2}+\frac{-72z+35}{5z^{2}(z-1)}\int_{\delta_{0}}\eta_{3}+\frac{-8z+6}{z(z-1)}\int_{\delta_{0}}\eta_{4}.$
(23)

By comparing the integrands, we can see that the Gauss-Manin matrix in the
basis $\eta$ is given by:

${\sf B_{1}}=\begin{bmatrix}0&0&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&1&0\\\
0&0&0&0&1\\\ a_{0}&a_{1}&a_{2}&a_{3}&a_{4}\\\ \end{bmatrix}dz,$ (24)

where $a_{i}$ are the coefficients of $\frac{\partial^{i}}{\partial z^{i}}$ in
16. To end, we compute

${\sf B_{2}}:=(dC+C\cdot{\sf B_{1}})C^{-1},$ (25)

which is the Gauss-Manin connection written in basis $\omega$. Observe that
the submatrix formed by lines and columns from 2 to 5 is the Gauss-Manin
connection in $H_{dR}^{3}(X)$ as computed in [Mov15] Section 2.6. To compute
the matrix in the basis $\alpha$ from Theorem 1, we compute:

${\sf A}=(dS+S\cdot{\sf B_{2}})S^{-1},$ (26)

where $S$ is given in (17).

###### Proof of Theorem 2.

We take an unknown vector field $\sf R$ and let its first six coordinates be
equal to the ones in [Mov15, Theorem 3]. As the 4x4 submatrix of the Gauss-
Manin Connection is the same as in the absolute case, by direct computation,
we end up with

$\left[\begin{smallmatrix}0&0&0&0&0\\\ 0&0&1&0&0\\\
\frac{5^{8}\left(s_{0}^{10}-s_{4}^{10}\right)^{2}}{s_{5}^{3}}s_{7}&0&0&\frac{5^{8}\left(s_{4}^{10}-t_{0}^{10}\right)^{2}}{s_{5}^{3}}&0\\\
ds_{7}({\sf R})+s_{8}&0&0&0&-1\\\ ds_{8}({\sf
R})+\frac{5^{12}\left(s_{0}^{10}-s_{4}^{10}\right)}{s_{5}}p&0&0&0&0\\\
\end{smallmatrix}\right]$ (27)

after plugging $\sf R$ in the matrix $\sf A$ from (26). Now, recalling that
${\rm d}s_{i}(\sf R)$ is the i-th coordinate of the vector field $\sf R$ and
that the first column has to have only zeros except for the third line, we can
determine the other coordinates of $\sf R$ uniquely. This gives us the desired
vector field and ends the theorem. ∎

##  4 Relationship with Periods

In this section, our goal is to explain why are the functions $\sf Y$ and $\sf
F$ appearing. For this, we need to look at the period domain of the space $\sf
T_{op}$ from Theorem 1. For us, a period is simply a number obtained by
integration of differential forms over cycles in homology. Here, we are
specially interested in the integration of 3-forms over 3 dimensional cycles.
Consider a symplectic basis of the homology group $H_{3}(X)$ given by
$\\{\delta_{1},\delta_{2},\delta_{3},\delta_{4}\\}$. Also, let $\delta_{0}$ be
the homology connecting the two rational curves $C_{+}$ and $C_{-}$. Of
course, those five homology classes form a basis for $H_{3}(X,Y)$.

######  Definition 5.

The period matrix is defined as

${\sf P}=[p_{ij}]=\left[\int_{\delta_{i}}\alpha_{j}\right]_{ij},$ (28)

where the $\alpha_{j}$ form a basis satisfying the conditions from 1.

Using Poincaré duality, one can easily see that this matrix is related to the
intersection matrix of the $\alpha_{i}$’s by the formula

$\left[\left\langle\alpha_{i},\alpha_{j}\right\rangle\right]=\left[\int_{\delta_{i}}\alpha_{j}\right]^{\mathrm{T}}\Psi^{-\mathrm{T}}\left[\int_{\delta_{i}}\alpha_{j}\right],$
(29)

where $\Psi$ is the intersection matrix of the basis $\delta$, which is given
by

$\Psi:=\left[\begin{array}[]{ccccc}0&0&0&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&1\\\
0&-1&0&0&0\\\ 0&0&-1&0&0\end{array}\right].$

######  Definition 6.

Let $\sf G$ be the group given by:

${\sf G}:=\left\\{g=\left(\begin{array}[]{ccccc}1&0&0&h_{1}&h_{2}\\\
0&g_{11}&g_{12}&g_{13}&g_{14}\\\ 0&0&g_{22}&g_{23}&g_{24}\\\
0&0&0&g_{33}&g_{34}\\\ 0&0&0&0&g_{44}\\\ \end{array}\right),\quad
h_{k},g_{ij}\in\mathbb{C}\quad\begin{array}[]{c}g_{11}g_{44}=1,\\\
g_{22}g_{33}=1,\\\ g_{12}g_{44}+g_{22}g_{34}=0,\\\
g_{13}g_{44}+g_{23}g_{34}-g_{24}g_{33}=0.\end{array}\right\\}$ (30)

The group ${\sf G}$ acts in an element $(X,\alpha)$ in the moduli space by the
right as $(X,\alpha)\bullet g=(X,\alpha g)$ where $\alpha$ is seen as a line
vector. Considering the relations, we can write this group in terms of six
”g”-coordinates and two ”h”-coordinates, as below:

$(g_{1},g_{2},g_{3},g_{4},g_{5},g_{6},h_{1},h_{2})=\left(\begin{array}[]{ccccc}1&0&0&h_{1}&h_{2}\\\
0&g_{1}^{-2}&-g_{3}g_{1}^{-1}&\left(-g_{3}g_{6}+g_{4}\right)g_{1}^{-2}&\left(-g_{3}g_{4}+g_{5}\right)g_{1}^{-2}\\\
0&0&g_{2}^{-1}&g_{6}g_{2}^{-1}&g_{4}g_{2}^{-1}\\\ 0&0&0&g_{2}&g_{2}g_{3}\\\
0&0&0&0&g_{1}^{2}\end{array}\right).$ (31)

Notice that our coordinate $g_{1}$ is different from [Mov15]: ours is the
square root of the one in that article.

######  Proposition 5.

The action of ${\sf G}$ written on the coordinates $s_{i}$ of $\sf T_{op}$ is

$\displaystyle g\bullet s_{0}=s_{0}g_{1}$ $\displaystyle g\bullet
s_{1}=s_{1}g_{1}^{2}+cg_{1}g_{2}g_{3}+ag_{1}g_{2}^{-1}g_{4}-g_{3}g_{4}+g_{5}$
$\displaystyle g\bullet
s_{2}=s_{2}g_{1}^{3}+s_{6}g_{1}^{2}g_{2}g_{3}+bg_{1}^{2}g_{2}^{-1}g_{4}$
$\displaystyle g\bullet s_{3}=s_{3}g_{1}^{4}+s_{5}g_{1}^{3}g_{2}g_{3}$
$\displaystyle g\bullet s_{4}=s_{4}g_{1}$ $\displaystyle g\bullet
s_{5}=s_{5}g_{1}^{3}g_{2}$ $\displaystyle g\bullet
s_{6}=s_{6}g_{1}^{2}g_{2}+bg_{1}^{2}g_{2}^{-1}g_{6}$ $\displaystyle g\bullet
s_{7}=s_{7}g_{2}+h_{1}$ $\displaystyle g\bullet
s_{8}=s_{7}g_{2}g_{3}+s_{8}g_{1}+h_{2}$

where $a,b,c$ are the expressions in given in terms of the coordinates $s_{i}$
from (17).

###### Proof.

We start with a pair $(X_{s_{0},s_{4}},\omega_{1})$. This form $\omega_{1}$,
together with its derivatives and a form $\omega_{0}$ yields a basis of
$H^{3}(X,Y)$. Multiplying by the matrix $S$ from equation (17) we get a basis
satisfying the conditions in Definition 1 depending on the coordinates
$s_{i}$. Now, let $g\in G$ act. By definition, $\alpha_{1}=\omega_{1}$ would
be multiplied by $g_{1}^{-2}$. In order to write the new element of the moduli
space in coordinates, we need to normalize $\omega_{1}$. Consider the form
$g_{1}^{2}\omega_{1}$ in the beginning. After this change, we need to multiply
the basis $\omega$ by the matrix

$K=\begin{pmatrix}1&0&0&0&0\\\ 0&k&0&0&0\\\ 0&0&k^{2}&0&0\\\ 0&0&0&k^{3}&0\\\
0&0&0&0&k^{4}\\\ \end{pmatrix}$

where $k=g_{1}^{2}$. This is because of the other forms of the basis $\omega$
are derivatives of $\omega_{1}$.

Notice that, by doing this, we are considering the point $(ks_{1},ks_{4})$ of
the moduli space of mirror quintics enhanced with a holomorphic 3-form and two
ration curves. Now, the matrix $g^{T}SK$ takes the basis
$\omega_{0},\omega_{1},\dots,\omega_{4}$ to its image by the action of $g$.
The entries of this matrix are the coordinates of the image. For example, the
entry $(5,2)$ should be the coordinate $t_{1}$, etc. After completing this
computation, we get the result. ∎

###  4.1 The $\tau$-matrix

We want to consider the orbits of the action of ${\sf G}$ on $\sf T_{op}$ and
their images by the period map. For this, we notice that ${\sf G}$ acts on the
space of matrices by right-multiplication. This action clearly preserves the
relations (29) and is compatible with the action on ${\sf T_{op}}$, in the
sense that the period matrix relative to a basis $\alpha\bullet g$ is simply
$Ag$, where $A$ is the matrix with respect to $\alpha$.

######  Proposition 6.

For any period matrix $\sf P$ satisfying the relations (29), there exists a
unique $g\in G$ such that ${\sf P}g$ can be written on the form

$\tau=\begin{pmatrix}1&\tau_{4}&\tau_{5}&0&0\\\ 0&\tau_{0}&1&0&0\\\
0&1&0&0&0\\\ 0&\tau_{1}&\tau_{3}&1&0\\\
0&\tau_{2}&-\tau_{0}\tau_{3}+\tau_{1}&-\tau_{0}&1\end{pmatrix}.$ (32)

for some $\tau_{i}$.

###### Proof.

Write $g$ in the form (31). Multiplying $g$ by a general matrix $\sf P$ and
using the relations (29), we get a matrix of the form

${\sf P}g=\left(\begin{array}[]{ccccc}1&*&*&*&*\\\ 0&*&1&0&*\\\ 0&1&0&0&0\\\
0&*&*&*&*\\\ 0&*&*&*&*\end{array}\right)$

Using the computations done in [Mov15, Section 3.3], writing down the
equations for ${\sf P}g=\tau$ when the entries in $\tau$ are independent of
$\tau_{i}$, we get that the first coordinates of $g^{\prime}:=g^{-1}$ are
necessarily given by

$\displaystyle(g^{\prime}_{1})^{2}={\sf P}_{21}^{-1},$ $\displaystyle
g^{\prime}_{2}=\frac{-{\sf P}_{21}}{{\sf P}_{11}{\sf P}_{22}-{\sf P}_{12}{\sf
P}_{21}},$ $\displaystyle g^{\prime}_{3}=\frac{-{\sf P}_{22}}{{\sf P}_{21}},$
$\displaystyle g^{\prime}_{4}=\frac{-{\sf P}_{12}{\sf P}_{23}+{\sf P}_{13}{\sf
P}_{22}}{{\sf P}_{11}{\sf P}_{22}-{\sf P}_{12}{\sf P}_{21}},$ $\displaystyle
g^{\prime}_{5}=\frac{{\sf P}_{11}{\sf P}_{22}{\sf P}_{24}-{\sf P}_{12}{\sf
P}_{21}{\sf P}_{24}+{\sf P}_{12}{\sf P}_{22}{\sf P}_{23}-{\sf P}_{13}{\sf
P}_{22}^{2}}{{\sf P}_{11}{\sf P}_{21}{\sf P}_{22}-{\sf P}_{12}{\sf
P}_{21}^{2}},$ $\displaystyle g^{\prime}_{6}=\frac{{\sf P}_{11}{\sf
P}_{23}-{\sf P}_{13}{\sf P}_{21}}{{\sf P}_{11}{\sf P}_{22}-{\sf P}_{12}{\sf
P}_{21}}$

It suffices to compute $h^{\prime}_{1}$ and $h^{\prime}_{2}$. After computing
${\sf P}g$ = ${\sf P}g^{\prime-1}$, we get:

$\tau=\left(\begin{array}[]{ccccc}1&\frac{{\sf P}_{01}}{{\sf
P}_{21}}&\frac{{\sf P}_{01}{\sf P}_{22}-{\sf P}_{02}{\sf P}_{21}}{{\sf
P}_{11}{\sf P}_{22}-{\sf P}_{12}{\sf P}_{21}}&P&Q\\\ 0&\frac{{\sf
P}_{11}}{{\sf P}_{21}}&1&0&0\\\ 0&1&0&0&0\\\ 0&\frac{{\sf P}_{31}}{{\sf
P}_{21}}&\frac{-{\sf P}_{21}{\sf P}_{32}+{\sf P}_{22}{\sf P}_{31}}{{\sf
P}_{11}{\sf P}_{22}-{\sf P}_{12}{\sf P}_{21}}&1&0\\\ 0&\frac{{\sf
P}_{41}}{{\sf P}_{21}}&\frac{-{\sf P}_{21}{\sf P}_{42}+{\sf P}_{22}{\sf
P}_{41}}{{\sf P}_{11}{\sf P}_{22}-{\sf P}_{12}{\sf P}_{21}}&-\frac{{\sf
P}_{11}}{{\sf P}_{21}}&1\end{array}\right)$ (33)

where $P$ and $Q$ depend linearly on $h^{\prime}_{1}$, $h^{\prime}_{2}$.
Making $P=Q=0$, we find expressions for $h_{1}$ and $h_{2}$. This shows
existence and uniqueness. ∎

Define ${\sf L_{op}}$ as the locus in the moduli space $\sf T_{op}$ from
Theorem 1 for which the period matrix is of the form (32). Our goal is to
express the functions $s_{i}$ restricted to this locus in some coordinate. To
do this, we first consider the points of $\sf T_{op}$ of the form
$(1,0,0,0,y,1,0,0,0)$, where $y^{10}=z$ (the same coordinate used in Section
3.2). Then, we compute the period matrix $\sf P$ for these points and find the
elements $g\in\sf G$ for which $\sf P$ is of the form (32). Then, by computing
the elements $(1,0,0,0,y,1,0,0,0)\bullet g$, we will get expressions for the
coordinates of $\sf L_{op}$. We consider the periods

$x_{ij}=\int_{\delta_{j}}{\tilde{\eta_{i}}},$

where $\tilde{\eta}_{0}=\omega_{0}=\alpha_{0}$, $\tilde{\eta_{1}}$ is the
holomorphic three form associated to the point $(1,\frac{s_{4}}{s_{0}})$ of
the moduli space of triples $(X,\omega,C_{\pm})$ defined in Section 2.3 and
$\tilde{\eta_{i}}=\theta(\tilde{\eta_{i-1}})$ (recall
$\theta=z\frac{\partial}{\partial z}$). These periods are related to the
solutions of the Picard-Fuchs equation via the matrix

$\left(\begin{array}[]{l}x_{01}\\\ x_{11}\\\ x_{21}\\\ x_{31}\\\
x_{41}\end{array}\right)=\left(\begin{array}[]{ccccc}\frac{1}{2\pi^{2}}&0&0&\frac{5^{4}}{2\cdot(2\pi
i)^{2}}&\frac{5^{4}}{4\cdot(2\pi i)^{3}}\\\ 0&0&0&1&0\\\ 0&0&0&0&1\\\
0&0&5&\frac{5}{2}&-\frac{25}{12}\\\
0&-5&0&-\frac{25}{12}&200\frac{\zeta(3)}{(2\pi
i)^{3}}\end{array}\right)\left(\begin{array}[]{c}\varphi\\\
\frac{1}{5^{4}}\psi_{3}\\\ \frac{2\pi i}{5^{4}}\psi_{2}\\\ \frac{(2\pi
i)^{2}}{5^{4}}\psi_{1}\\\ \frac{(2\pi
i)^{3}}{5^{4}}\psi_{0}\end{array}\right),$ (34)

where $\psi_{i}$ are solutions for the homogenous equation as it is presented
in [Mov15, Introduction] and $\varphi$ is the solution for inhomogenous
equation (16) given by the series:

$2\sum_{m\,{\rm odd}}^{\infty}\frac{(5m)!!}{(m!!)^{5}}(5^{-5}z)^{m/2}$

where the double exclamation point mean we multiply all the odd numbers less
or equal to the number (e.g $7!!=1\cdot 3\cdot 5\cdot 7=105$). The expression
for $x_{01}$ is taken from [MW09] and [PSW08]. Notice that the notation for
the series above in [MW09] is different: they take $\tau=\frac{\varphi}{30}$.
The expressions for the other periods are taken from [Mov15, Introduction]. To
find the period matrix in terms of $x_{ij}$, we need to change from
$\tilde{\eta}$ to the basis $\alpha$ from the moduli space. We consider,
therefore, the matrices

$S=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\
0&-5^{5}&-5^{4}(z-1)&0&0\\\ 0&-\frac{5}{z-1}&0&1&0\\\
0&0&0&0&5^{4}(z-1)\end{array}\right)$

and

$T=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&-1&-5&0&0\\\
0&2&15&25&0\\\ 0&-6&-55&-150&-125\end{array}\right),$

where $T$ takes the basis $\eta$ to the basis $\omega$ and $S$ takes $\omega$
to $\alpha$. Notice that the matrices above have already been used in Section
3, but for general $s_{i}$. Therefore, the period matrix is simply ${\sf
P}=[x_{ij}](ST)^{T}$. Now, using Proposition 6 and Proposition 5, it is easy
to find $g\in\sf G$ for which ${\sf P}g$ is of the form (32) and compute the
action of $g$ on $(1,0,0,0,y,1,0,0,0)$ to get the $s_{i}$ restricted to $\sf
L_{op}$. The first 7 were already computed in [Mov15, Theorem 1] and the other
two are given below:

$s_{7}=5^{7}(z-1)\frac{x_{01}x_{12}x_{23}-x_{01}x_{13}x_{22}-x_{02}x_{11}x_{23}+x_{02}x_{13}x_{21}+x_{03}x_{11}x_{22}-x_{03}x_{12}x_{21}}{x_{21}},$
(35)
$s_{8}=5^{7}(z-1)\left(x_{01}x_{24}-x_{02}x_{23}+x_{03}x_{22}-x_{04}x_{21}\right)+\\\
+5^{6}z\left(x_{01}x_{22}+\frac{5}{2}x_{01}x_{23}-x_{02}x_{21}-\frac{5}{2}x_{03}x_{21}\right).$
(36)

######  Proposition 7.

The Gauss-Manin connection restricted to the locus $L$ can be computed in
terms of the coordinate $\tau_{0}$. It is given by:

$\left.{\sf A}\right|_{L}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&1&0&0\\\ \frac{d\tau_{5}}{d\tau_{0}}&0&0&\frac{d\tau_{3}}{d\tau_{0}}&0\\\
0&0&0&0&-1\\\ 0&0&0&0&0\end{array}\right)d\tau_{0},$ (37)

where $\tau$ is given by (33).

###### Proof.

To prove this, use the fact that the Gauss-Manin connection commutes with
integrals, in the following sense:

$d\left(\int_{\delta}\omega\right)=\int_{\delta}\nabla\omega,$

where the integration on the right hand side takes place only on $H_{{\rm
dR}}^{3}(X,Y)$ (recall that $\nabla\omega$ can be written as sum of elements
of the form $\omega^{\prime}\otimes s$, where $s$ a form in $\Omega^{1}_{T}$).
Now, using this, we get:

$d\tau=\left[\int_{\delta_{i}}\nabla\alpha_{j}\right]_{i,j}=\left[\sum_{k}\int_{\delta_{i}}a_{jk}\alpha_{k}\right]_{i,j}=\left[\sum_{k}a_{jk}\tau_{ik}\right]_{i,j}=\left[\sum_{k}\tau_{ik}a^{T}_{kj}\right]_{i,j}=\tau\cdot{\sf
A}^{T}.$

This implies that the Gauss-Manin connection has to be given by

$\left.{\sf
A}\right|_{L}=d\tau^{\mathrm{T}}\cdot\tau^{-\operatorname{T}}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
-\tau_{5}d\tau_{0}+d\tau_{4}&0&d\tau_{0}&-\tau_{3}d\tau_{0}+d\tau_{1}&-\tau_{1}d\tau_{0}+\tau_{0}d\tau_{1}+d\tau_{2}\\\
d\tau_{5}&0&0&d\tau_{3}&-\tau_{3}d\tau_{0}+d\tau_{1}\\\ 0&0&0&0&-d\tau_{0}\\\
0&0&0&0&0\end{array}\right)$

By using Griffths transversality, we conclude that positions (2,1), (2,4) and
(3,5) have to be zero, since our basis respect the Hodge filtration. After
taking $d\tau_{0}$ out, we have the result. This yields the relations

$\displaystyle\tau_{1}$
$\displaystyle=-\frac{d\tau_{2}}{d\tau_{0}}-\tau_{0}\frac{d\tau_{1}}{d\tau_{0}},$
(38) $\displaystyle\tau_{3}$ $\displaystyle=\frac{d\tau_{1}}{d\tau_{0}},$ (39)
$\displaystyle\tau_{5}$ $\displaystyle=\frac{d\tau_{4}}{d\tau_{0}}.$ (40)

Notice the first two had already been computed in [Mov15]. ∎

By the uniqueness statement from Theorem 2, we conclude that
$\frac{\partial}{\partial\tau_{0}}$ is the vector field $\sf R$ restricted to
the locus $\sf L_{op}$. If we consider the coordinate $q=e^{2\pi i\tau_{0}}$,
then $\sf R$ becomes $2\pi iq\frac{\partial}{\partial q}$. Writing the
functions $\sf F$ and $\sf Y$ (or $\frac{d\tau_{5}}{d\tau_{0}}$ and
$\frac{d\tau_{3}}{d\tau_{0}}$, respectively) in the coordinate $q$ gives us
virtual counts of disks with boundary on the real quintic and virtual counts
of rational curves on a quintic threefold. To see this, we just need to look
at the expressions of the $\tau_{i}$ in terms of $x_{ij}$ we get, for example,
that $\tau_{4}$ is given by $\frac{x_{01}}{x_{21}}$, i.e., the quotient of a
solution for the non homogenous Picard-Fuchs equation by the holomorphic
solution to the homogenous equation. This shows that the expressions for $F$
and $Y$ are the expressions in periods that we have for the disk potential and
the Yukawa coupling.

To end, we list some of the possible extensions that can be made to our work.
Firstly, it is natural to consider the case of two moving rational curves or
even a family of divisors in the mirror quintic and see if we get a
generalization of Jacobi forms. A nice starting point would be the paper
[JS09] in which the authors consider families of divisors on the mirror
quintic from a Physical point of view. Secondly, we could consider higher
genus Gromov-Witten invariants and try to carry out a similar program.
Finally, there are still some details missing in the realization of GMCD in
general. A very interesting program would be to try to construct a more
rigorous framework to define the moduli spaces and the functions we are
working with.

## References

  * [AKV22] Murad Alim, Vadym Kurylenko, and Martin Vogrin.  The algebra of derivations of quasi-modular forms from mirror symmetry.  Pure and Applied Mathematics Quartely, in press, 2022.
  * [AMSY16] Murad Alim, Hossein Movasati, Emanuel Scheidegger, and Shing-Tung Yau.  Gauss–Manin Connection in Disguise: Calabi–Yau Threefolds.  Commun. Math. Phys., 344(3):889–914, June 2016.
  * [CDLOGP91] Philip Candelas, Xenia C. De La Ossa, Paul S. Green, and Linda Parkes.  A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory.  Nuclear Physics B, 359(1):21–74, July 1991.
  * [CK99] David A. Cox and Sheldon Katz.  Mirror symmetry and algebraic geometry.  Mathematical surveys and monographs 68. American Mathematical Society, 1999.
  * [CML21] Jin Cao, Hossein Movasati, and Roberto Villaflor Loyola.  Gauss-manin connection in disguise: Quasi jacobi forms of index zero.  arXiv: 2109.00587, September 2021.
  * [GP90] B.R. Greene and M.R. Plesser.  Duality in Calabi-Yau moduli space.  Nuclear Physics B, 338(1):15–37, July 1990.
  * [Gro66] Alexander Grothendieck.  On the de Rham cohomology of algebraic varieties.  Publications Mathématiques de l’IHÉS, 29:95–103, 1966.
  * [JS09] Hans Jockers and Masoud Soroush.  Effective Superpotentials for Compact D5-Brane Calabi-Yau Geometries.  Commun. Math. Phys., 290(1):249–290, August 2009.
  * [KO68] Nicholas M. Katz and Tadao Oda.  On the differentiation of De Rham cohomology classes with respect to parameters.  Journal of Mathematics of Kyoto University, 8(2):199–213, January 1968.  Publisher: Duke University Press.
  * [Mov12] Hossein Movasati.  Quasi-modular forms attached to elliptic curves, I.  Annales Mathématiques Blaise Pascal, 19(2):307–377, 2012.
  * [Mov15] Hossein Movasati.  Modular-type functions attached to mirror quintic Calabi–Yau varieties.  Math. Z., 281(3-4):907–929, December 2015.
  * [MW09] David R. Morrison and Johannes Walcher.  D-branes and normal functions.  Advances in Theoretical and Mathematical Physics, 13(2):553–598, April 2009.  Publisher: International Press of Boston.
  * [PS08] Christiaan Peters and Joseph H. M. Steenbrink.  Mixed Hodge Structures.  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin Heidelberg, 2008.
  * [PSW08] R. Pandharipande, J. Solomon, and J. Walcher.  Disk Enumeration on the Quintic 3-Fold.  Journal of the American Mathematical Society, 21(4):1169–1209, 2008\.
  * [Wal07] Johannes Walcher.  Opening Mirror Symmetry on the Quintic.  Commun. Math. Phys., 276(3):671–689, December 2007.


Some novel considerations about the collective coordinates approximation for
the scattering of $\phi^{4}$ kinks

Carlos F.S. Pereira1, Gabriel Luchini1, Tadeu Tassis2,Clisthenis P.
Constantinidis1
1Departamento de Física, Universidade Federal do Espírito Santo (UFES),

CEP 29075-900, Vitória-ES, Brazil 2Departamento de Física, Universidade
Federal do ABC (UFABC),

CEP 09210-580,- Bangú, Santo André-SP, Brazil

###### Abstract

The collective coordinates approximation for the kink/anti-kink scattering in
the $1+1$ dimensional $\phi^{4}$ model is considered and we discuss how the
results found in the current literature on the topic can be improved by giving
the analytical expression for the lagrangian in the space of parameters. A
comprehensive discussion of the role of the collective coordinates
approximation in this particular situation is given for completeness.

##  1 Introduction

In this paper we revisit the problem of kink/anti-kink scattering in the
$\phi^{4}$ model using collective coordinates. Although this subjet has been
extensively discussed recently in the literature [1, 2, 6] and also models
which are constructed from the $\phi^{4}$ model are also highly considered in
this context [3, 4, 5], our results bring some improvements with respect to
the previous approaches to find the effective theory based on collective
coordinates.

Collective coordinates can be used as an approximation method for studying
scattering processes of solitons in non-integrable field theories, such as for
the $\phi^{4}$ model. It appears as an alternative for the construction of
multisoliton configurations by reducing the degrees of freedom which
characterize the dynamics of these models by essentially making some suitable
choice of a set of relevant dynamical variables (the collective coordinates)
for the problem whose evolution in time will describe the important features
of the scattering.

In the $\phi^{4}$ model kink/anti-kink scattering, the full numerical
evolution of this configuration shows a resonanse behaviour of the system
where the individual solitons will form a bounded system during a certain time
interval which is then undone and the solitons are again free to move[7, 8, 9,
10, 14].

The collective coordinates approximation has been used in attempts to
understand not only the scattering of these solitons as well as a way to shed
some light on this intriguing resonanse phenomenon. Important contributions
were made[11] pointing to the fact that during the collision the excitation
mode of the $\phi^{4}$ solitons becomes important. This is corroborated
essentially by the fact that the collective coordinate approximation is
considerably improved and much closer to the full simuations results when the
possibility of this excitation is considered.

One of the drawbacks of this approximation method is a technical issue: the
construction of the lagrangian in the space of parameters requires the
calculation of many highly non-trivial integrals. In the literature several
mathematical methods have been employed and they are generally quite
complicate. Perhaps, this difficulty has been forcing many authors to find
arguments to ignore some of those integrals in their approximation. The main
goal of this paper is to present the full lagrangian which describes the
dynamics of the kink/anti-kink scattering in the $\phi^{4}$ model for the
collective coordinates which encode information about the translation of the
solitons and their excitation due to mutual interaction.

The first sections of the paper are devoted to an introduction of the system
we are going to study as a way to fix notation and ideas. In section 2 we
discuss the topological aspects of the $\phi^{4}$ model and some
characteristics of the dynamics of the kink and anti-kink solutions.
Collective coordinates are introduced in section 3 where the method is
illustrated for a simple case of a single (anti)-kink solution. Then we
discuss how this approximation is implemented in the case of the scattering of
these objects by looking at was is known in the literature and showing how it
can be improved with the method of integration which is presented in one of
the appendices. Finally, in section 4, we draw our conclusions on the role of
the collective coordinates as employed here and our perspectives concerning
the method.

##  2 The model

The so called $\phi^{4}$ model has a wide range of applications[9], from
condensed matter physics[16] to high energy theories[17], cosmology[18, 19],
biological systems[20] and so on. Our interest here is on the $\phi^{4}$ model
for which the field $\phi$ is a real Lorentz scalar in $1+1$ dimensional
Minkowski space-time with metric $g_{\mu\nu}={\rm diag}(1,-1)$. The action for
this model is defined as

$\mathcal{S}=\int
d^{2}x\;\left(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\mathcal{U}(\phi)\right),$
(1)

where the energy density shown if figure 1 is

$\mathcal{U}(\phi)=\frac{\lambda}{4}\left(\phi^{2}-\eta^{2}\right)^{2},$ (2)

with $\lambda\geq 0$ for boundedness of the potential whose vacua are at
$\phi=\pm\eta\in\mathbb{R}$.

Figure 1: The potential energy density has two degenerate vacua at
$\phi=\pm\eta$.

The dynamical equation of the field

$\partial_{\mu}\partial^{\mu}\phi+\lambda\left(\phi^{2}-\eta^{2}\right)\phi=0,$
(3)

is highly non-linear and can be approached by two different directions. First
there are solutions of the form
$\phi=\pm\eta+\delta\phi+\mathcal{O}\left((\delta\phi)^{2}\right)$, with
$\delta\phi\ll\eta$ a linear perturbation of a vacuum state, which consist of
the trivial solution of the model, i.e., a solution with vanishing energy; the
space of these solutions is known as the perturbative sector. Then, there are
also the so called kink and anti-kink configurations which appear as solutions
that interpolate between the vacua of the potential energy density, definining
the non-perturbative sector of the theory. They are topological solitons[17]:
their existence and stability are related to the behaviour of the field at the
spatial border (the topological data), so defined that the configuration has,
in this case, the least possible finite energy. This interplay between the
topological data and energy reveals, for this model, that the time-independent
(anti-)kink configurations are BPS solutions[21, 22, 23], i.e. static
configurations $\phi(x)$ which render stationary the static energy functional

$M=\int_{-\infty}^{+\infty}\left(\frac{1}{2}\left(\frac{d\phi}{dx}\right)^{2}+\mathcal{U}\right)\;dx$
(4)

with fixed (and constant) field values at spatial infinity,
$\phi(\pm\infty)=\pm\eta$, which satisfy the first order (BPS) equation

$\frac{d\phi}{dx}=\pm\sqrt{2\mathcal{U}}$ (5)

and have energy proportional to the absolute value of the topological degree

$N=\frac{1}{2\eta}\int_{-\infty}^{+\infty}\frac{d\phi}{dx}\;dx=\frac{\phi(+\infty)-\phi(-\infty)}{2\eta}=\pm
1.$ (6)

The solutions of the BPS equation (5) with the potential given in (2) can be
easily obtained by direct integration and read

$\phi_{K}(x)=\eta\tanh{\left(\sqrt{\frac{\lambda}{2}}\;\eta(x-a)\right)},\qquad\phi_{\bar{K}}(x)=-\eta\tanh{\left(\sqrt{\frac{\lambda}{2}}\;\eta(x-a)\right)},$
(7)

respectively the kink with topological degree $N=+1$ and the anti-kink with
$N=-1$. Here, $a$ is an integration constant which defines the position of the
(anti-)kink, as it is interpreted as a relativistic particle-like111But not
pointwise. object with the static energy $M$ regarded as its rest mass. The
kink solution as well as its energy density are shown in figure 2.

Figure 2: The energy density of the kink configuration is a localised function
around the point $x=a$ of the solution, here taken to be zero.

These BPS solutions $\phi_{K,\bar{K}}$ are stable against linear perturbations
(see appendix A)

$\phi_{K,\bar{K}}\rightarrow\phi_{K,\bar{K}}+\epsilon(t,x).$ (8)

At this level the (anti-)kink has a vibrational “zero mode”, which consists of
a spatial translation of the configuration along its tangent direction, thus
costing no energy to the system, and a first excited mode of the form

$\epsilon(t,x)=\frac{1}{2}\operatorname{sech}{(\sigma x)}\tanh{(\sigma
x)}\cos\left(\sqrt{3}\sigma{t}\right),$ (9)

with $\sigma=\sqrt{\frac{\lambda}{2}}\eta$. Such excitation on the kink
solution is shown in figure 3.

Figure 3: The (anti)-kink solition is stable agains perturbations. Here the
effect of the excitation mode on the solution is presented at different times.

The non-linear character of the model is what drives the interaction between
these particle-like objects[24]. Although one cannot have a solution of the
theory consisting of a moving kink/anti-kink pair, such a static configuration
with the kink at $x=-a$ and the anti-kink at $x=a$ can be constructed as shown
if figure 4 where

$\phi(x)=\phi_{K}(\sigma(x+a))+\phi_{\bar{K}}(\sigma(x-a))-\eta.$ (10)

Setting the distance $2a$ to be large enough222Which means larger than the
individual length of the solitons which can be estimated as
$\frac{2\sqrt{2}}{\eta}$. this can be within some approximation considered as
a legit solution.

Figure 4: The kink/anti-kink configuration is defined by sewing both solutions
separated apart by a distance $2a$ where here, $a=15$.

Looking at the well separated kink and anti-kink as individual objects one can
calculate an intrinsic force[17] between them given by

$F=32{\sigma}^{2}{\eta}^{2}e^{-4\sigma{a}},$ (11)

i.e., a weak but non-vanishing attraction.

If left for long enough time this configuration, with a vanishing topological
degree, is unstable against the perturbations provoked by the intrinsic force
between kink and anti-kink and decays after a while, as shown in figure 5.

Figure 5: The energy density of the configuration is shown. We start with the
static configuration (10) with $a=3$ and $\eta=\sigma=1$. The configuration
will evolve numerically in time. While at first the kink and anti-kink where
clearly standing still at their original positions, given by the localised
peaks of energy density, their mutual attraction will lead them to accelerate
towards each other which can be seen after around $t=75$.

The kink/anti-kink pair exhibit interesting behaviour when their scattering is
considered, i.e., when these individual objects acquire a relative velocity
between them. Being a relativistic system, the moving (anti-)kink can be found
by performing a Lorentz boost of the static solution:

$\phi_{K,\bar{K}}(t,x)=\pm\eta\tanh{\left(\sqrt{\frac{\lambda}{2}}\;\eta\frac{(x-a-
vt)}{\sqrt{1-v^{2}}}\right)},$ (12)

the parameter $v\in(-1,1)$ standing for its boost velocity.

For large values of relative velocity the topological solitons will collide
elastically, thus exhibiting a behaviour similar to that of usual point
particles, as seen in figure 6.

(a) An elastic scattering is observed when the kink and anti-kink are thrown
towards each other with $|v|=0.5$ each.

(b) The value of the field $\phi$ at $x=0$ is shown for different times during
the scattering process. A change occurs close to $t=20$, when the collision
takes place.

Figure 6: When the solitonic objects are well separated, the field at $x=0$
has a constant value $\phi=1$. During the collision process this value
abruptly changes and in the present case, being the collision elastic, it
recovers its original value. Nevertheless one observes some small wrinkles
both in the value of the field and in the energy density.

For some lower velocity values one sees what’s called bounce windows[11, 2,
12, 13]: a bound state is formed during some time lapse after which the
solitonic objects recover their unbounded motion. This is shown in figure 7.

(a) A bounce window is observed when the kink and anti-kink are thrown towards
each other with $|v|=0.2249$ each.

(b) The value of the field $\phi$ at $x=0$ is shown for different times during
the scattering process. Two changes occur, indicating two different
collisions: a first one close to $t=40$ and another one close to $t=60$.

Figure 7: For low values of the relative velocity between kink and anti-kink
one observes a resonance phenomenon where the pair of solitonic objects will
form a temporary bound state after which they recover their original “free
motion”.

Although this model is non-integrable and therefore energy is expect to be
lost in the collision process, by performing the numerical calculation of the
total energy while the solitons scatter we observe that the emmition of
radiation is in fact very little, as shown in figure 8.

(a) The energy as a function of time for the collision between kink and anti-
kink with initial velocity of $|v|=0.5$ each. The observed difference between
the energy after the collision and before is $\Delta E=0.00877$

(b) The energy as a function of time for the collision between kink and anti-
kink with initial velocity of $v=0.2249$ each. The observed difference between
the energy after the collision and before is $\Delta E=0.00005$

Figure 8: The energy of the system is calculated in the region between $x=-20$
and $x=20$, from when the solitons start their motion towards each other,
during the collision and after that for a while. It is observed that the
amount of radiation dispersed in this process is very little.

This fact seems to indicate that the ripples observed in the field after the
scattering are not a consequence of this process of radiation emmition only
but perhaps much more related to a reorganisation of the distribution of
energy within the system.

Thus, if from one hand these objects have many particle-like features, from
the other they also exhibit this intriguing resonance phenomenon, which is not
observed for pointwise particles. This is interesting enough to motivate an
attempt to understand which mechanism is responsible to the formation of these
so called bounce windows.

##  3 The collective coordinates approach

One way of studying the dynamics of the soliton collisions, especially when
low values of relative velocity are considered, is through the so called
collective coordinates method[15, 13, 11, 12, 2]: the degrees of freedom of
the field are encoded into time dependent parameters (the collective
coordinates) whose dynamics will determine the relative motion of the solitons
through a geodesic in the space where these parameters are defined as
coordinates.

To study the motion of a free (anti-)kink moving at low velocity the
collective coordinate $a(t)$ can be introduced as the position function of
that object by writing

$\phi=\pm\eta\tanh{\left(\sigma\left(x-a(t)\right)\right)},\qquad\sigma=\sqrt{\frac{\lambda}{2}}\eta.$
(13)

By doing so we have that $\frac{\partial\phi}{\partial
t}=\dot{a}\frac{\partial\phi}{\partial x}$ and all time dependence gets
factorized in terms of $\dot{a}(t)$ and the lagrangian density in (1) can be
integrated over the spatial coordinate giving

$L=\int_{-\infty}^{+\infty}\mathcal{L}\,dx=\frac{1}{2}g(a)\dot{a}^{2}-V(a)$
(14)

where, in this case, $g(a)$ is in fact the constant
$g=\frac{2\eta^{2}}{3}\sigma$ as well as $V(a)$ is the constant function
$V=\frac{2\eta^{2}\sigma}{3}+\frac{\lambda\eta^{4}}{3\sigma}$. The dynamical
equations for the field are now reduced to the Euler-Lagrange equation for the
collective coordinate $a(t)$, which reads $\ddot{a}=0$, whose solution is

$a(t)=a_{0}+vt,$

with $a_{0}$ and $v$ integration constants. As the result we have that the
soliton will move with a constant velocity $v$ and its profile reads

$\phi=\pm\eta\tanh{\left(\sigma\left(x-a_{0}-vt\right)\right)},$ (15)

which approximately describes the time dependent (anti-)kink solution

$\phi=\pm\eta\tanh{\left(\sqrt{\frac{\lambda}{2}}\;\eta\frac{x-a_{0}-vt}{\sqrt{1-v^{2}}}\right)},$
(16)

for $v\ll 1$.

The description of the kink/anti-kink collision dynamics through collective
coordinates is not as straightforward as the case for the free soliton.
Indeed, by setting[11]

$\phi(t,x)=\phi_{K}(x+a(t))+\phi_{\bar{K}}(x-a(t))-\eta$ (17)

the translation coordinate $a(t)$ now describes half of the relative distance
between the kink and the anti-kink and the lagrangian obtained for $a(t)$ in
the form of (14) has the functions $g(a)$ and $V(a)$ given by333The
integration method which is used here is shown in details in appendix B

$\displaystyle g\left(a\right)$ $\displaystyle=$
$\displaystyle{{\eta}^{2}{\sigma}^{2}}\left[\frac{8}{{3}{\sigma}}-\frac{8}{\sigma}\textrm{csch}^{2}\left({2}{\sigma}{a}\right)+{16}{a}{\coth\left({2}{\sigma}{a}\right)}{\textrm{csch}^{2}\left({2}{\sigma}{a}\right)}\right]$
(18) $\displaystyle V\left(a\right)$ $\displaystyle=$ $\displaystyle
8\sqrt{2}{\sigma}^{2}{\eta}^{2}\left[{-\frac{2}{3}}+{2\sigma{a}}+{\frac{3}{\tanh\left(2{\sigma}a\right)}}\right]$
(19) $\displaystyle+$ $\displaystyle
8\sqrt{2}{\eta}^{2}{\sigma}^{2}\left[-{\frac{\left(2+6{\sigma}a\right)}{\tanh^{2}\left(2{\sigma}a\right)}}+{\frac{4{\sigma}a}{\tanh^{3}\left(2{\sigma}a\right)}}\right]$

whose behaviour can be seen in the figure 9 below.

Figure 9: The behavior of the functions $g(a)$ and $V(a)$ with
$\sigma=\eta=1$.

Different initial conditions for $a(t)$ and $\dot{a}(t)$ will imply different
types of predicted motion for the solitons: both bound and unbound motion are
presented in figure 10.

Figure 10: The dynamics of $a(t)$ defines the motion of the solitons in the
scattering process. In both cases presented here, $\sigma=\eta=1$. The figure
on the left shows an elastic scattering where the solitons started at $a(0)=3$
with initial velocity $\dot{a}(0)=0.01$. The figure on the right shows a
bounded motion obtained for initial velocity $\dot{a}(0)=0.005$.

This approximation is well know, so as its problems. We notice in particular
the inconvenient fact that the collective coordinate $a(t)$ assumes negative
values in the collision process, which is something unwanted once this
coordinate stands for the relative distance between kink and anti-kink.

In particular, the dynamics of the solitonic pair described by $a(t)$ does not
present, in any circumstance, the formation of the resonances observed for
certain given initial relative velocities[12, 10, 14]. Such a limitation
indicates that the use of this collective coordinate only is not enough.

The existence of a force between kink and anti-kink, even when these objects
are far apart, lead us to consider the excitation of the internal mode of
vibration (9) in the dynamics through a new collective coordinate $\xi(t)$.
This coordinate will assume the role of the time dependent amplitude of
oscillation of the first excited mode of the (anti-)kink
$\epsilon(t,x)=\xi(t)\chi(x)$, with
$\chi(x)=\frac{1}{2}\operatorname{sech}{(\sigma x)\tanh{(\sigma x)}}$. We thus
consider the possibility of the energy transfer between the translation and
vibration modes through the nonlinearities of the model.

Thus, considering[11]

$\phi=\phi_{K}(x+a(t))+\phi_{\bar{K}}(x-a(t))-\eta+\xi(t)\left(\chi(x+a(t))-\chi(x-a(t))\right)$
(20)

with $\chi(x)=\frac{1}{2}\operatorname{sech}{(\sigma x)}\tanh{(\sigma x)}$, we
can proceed as before and integrate the lagrangian density over the spatial
coordinates thus obtaining a description in terms of the dynamics of the
coordinates $\xi(t)$ and $a(t)$, taken444We have also considered higher orders
in $\xi$ in the lagrangian and no different result than those presented here
was observed. up to quadratic order in $\xi$. Even though this integration is
far from being simple to be performed we have successfully obtained, as
explained in the appendix B, the analytical expressions needed for the
collective coordinates lagrangian555The parameters are fixed as
$\sigma=\eta=1$.

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(M_{0}+I\left(a\right)+{\xi}^{2}K\left(a\right)+{\xi}J\left(a\right)\right){\dot{a}}^{2}+\frac{1}{2}\left(2+Q\left(a\right)\right){\dot{\xi}}^{2}$
(21) $\displaystyle+$
$\displaystyle{\left(C\left(a\right)+{\xi}N\left(a\right)\right)\dot{\xi}}{\dot{a}}-\left(V\left(a\right)-{\xi}F\left(a\right)+{\xi}^{2}W\left(a\right)\right),$

where the coefficients multiplying the canonical variables were found to be

$\displaystyle M_{0}$ $\displaystyle=$ $\displaystyle\frac{8}{3}$
$\displaystyle I\left(a\right)$ $\displaystyle=$ $\displaystyle
16a\coth\left(2a\right)\textrm{csch}^{2}\left(2a\right)-8\textrm{csch}^{2}\left(2a\right)$
$\displaystyle V\left(a\right)$ $\displaystyle=$ $\displaystyle
8\left[-\frac{2}{3}+2a+\frac{3}{\tanh\left(2a\right)}-\frac{2\left(1+3a\right)}{\tanh^{2}\left(2a\right)}+\frac{4a}{\tanh^{3}\left(2a\right)}\right]$
$\displaystyle Q\left(a\right)$ $\displaystyle=$ $\displaystyle
12a\textrm{csch}\left(2a\right)+24a\textrm{csch}^{3}\left(2a\right)-12\coth\left(2a\right)\textrm{csch}\left(2a\right)$
$\displaystyle C\left(a\right)$ $\displaystyle=$
$\displaystyle\pi\sqrt{\frac{3}{2}}\tanh\left(a\right)\operatorname{sech}^{2}\left(a\right)$
$\displaystyle F\left(a\right)$ $\displaystyle=$
$\displaystyle-6\pi\sqrt{\frac{3}{2}}\tanh^{2}\left(a\right)\left[-1+\tanh\left(a\right)\right]^{2}$
$\displaystyle J\left(a\right)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{2}}\pi\left[1+2\operatorname{sech}^{4}\left(a\right)-\operatorname{sech}^{4}\left(a\right)\cosh\left(2a\right)\right]$
$\displaystyle N\left(a\right)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\left[15\textrm{csch}^{3}\left(2a\right)+9\textrm{csch}\left(2a\right)\coth^{2}\left(2a\right)+3\textrm{csch}\left(2a\right)\right]$
$\displaystyle+$
$\displaystyle\frac{3}{2}\left[-6a\textrm{csch}\left(2a\right)\coth\left(2a\right)-46a\coth\left(2a\right)\textrm{csch}^{3}\left(2a\right)\right]$
$\displaystyle+$
$\displaystyle\frac{3}{2}\left[-2a\coth^{3}\left(2a\right)\textrm{csch}\left(2a\right)\right]$
$\displaystyle K\left(a\right)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\left[\frac{28}{15}+160a\textrm{csch}^{3}\left(2a\right)+8a\textrm{csch}\left(2a\right)+192a\textrm{csch}^{5}\left(2a\right)\right]$
$\displaystyle+$
$\displaystyle\frac{3}{2}\left[-96\coth\left(2a\right)\textrm{csch}^{3}\left(2a\right)-16\coth\left(2a\right)\textrm{csch}\left(2a\right)\right].$

From this lagrangian, the dynamical equations for the collective coordinates
$a(t)$ and $\xi(t)$ are

$\displaystyle\left[C(a)+\xi{N(a)}\right]\ddot{a}+\left[2+Q(a)\right]\ddot{\xi}+Q^{\prime}(a)\dot{a}\dot{\xi}+\left[C^{\prime}(a)-\frac{1}{2}J(a)+\xi\left(N^{\prime}(a)-K(a)\right)\right]\dot{a}^{2}$
$\displaystyle+$ $\displaystyle 2\xi{W(a)}-F(a)=0$
$\displaystyle\left[M_{0}+I(a)+\xi{J(a)+{\xi}^{2}K(a)}\right]\ddot{a}+\left[C(a)+\xi{N(a)}\right]\ddot{\xi}+\left[N(a)-\frac{1}{2}Q^{\prime}(a)\right]{\dot{\xi}}^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left[I^{\prime}(a)+{\xi}J^{\prime}(a)+{\xi}^{2}K^{\prime}(a)\right]{\dot{a}}^{2}+\left[J(a)+2{\xi}K(a)\right]{\dot{a}}{\dot{\xi}}+V^{\prime}(a)+{\xi}^{2}W^{\prime}(a)-{\xi}F^{\prime}(a)=0.$
(23)

These equations can be written as a system of the form[12]
$M\mathbf{a}=\mathbf{f}$, where $\mathbf{a}=(\ddot{a},\ddot{\xi})$, with

$M=\left(\begin{array}[]{cc}\left[C(a)+\xi{N(a)}\right]&\left[2+Q(a)\right]\\\
\left[M_{0}+I(a)+\xi{J(a)+{\xi}^{2}K(a)}\right]&\left[C(a)+\xi{N(a)}\right]\end{array}\right)$
(24)

and $\mathbf{f}$ stands for all the other quantities in terms of the
collective coordinates and their derivatives. For nontrivial solutions of this
system it is required that

$\left(M_{0}+I\left(a\right)+{\xi}^{2}K\left(a\right)+{\xi}J\left(a\right)\right)\left(2+Q\left(a\right)\right)-\left(C+\xi
N\right)^{2}\neq 0.$

An analysis of this condition shows that the requirement $C=N=0$ is needed for
its validity for any values of $a(t)$ and $\xi(t)$.

In the recent literature on this subject, an important contribution[2] was
given concerning the correction of the term here labeled as $F(a)$, which was
mistakenly written in [11]. In the appendix B we discuss how the integrals
leading to this term can be done using the method of residues and this
approach confirms the result presented in [2]. Although of remarkable
importance, this correction was not sufficient to produce better results
within the approximation considered in [2], and in general throughout the
literature, which considers, in the lagrangian (21) given above, $Q=K=J=0$ and
$W=3$, besides the already mentioned necessity of taking $C=N=0$.

While the choice of $W=3$ can be justified by the fact that it really makes
the approximation worst if otherwise chosen and thus it is set as the constant
value it assumes in the limit $a\to\infty$, the reason behind the vanishing of
the functions $Q$, $K$ and $J$ given is generally not very clearly justified.

As discussed in [2], even with the correction of the term labeled as $F$, the
approximation considered there and throughout the literature - as far as we
know - not only leads to a physically non-acceptable negative value of $a(t)$
when the solitons collide but also presents some problems in predicting
elastic collisions when they are expected in a situation where high initial
relative velocities are considered. These results are reproduced in figure 11
where for each case we have also given the center of mass of the system
$\langle x\rangle$ calculated from the full numerical simulation as

$\langle x\rangle=\frac{\int dx\;x\mathcal{E}}{\int dx\;\mathcal{E}}$ (25)

where $\mathcal{E}$ stands for the energy density of the field configuration
(20):

$\mathcal{E}=\frac{1}{2}\left(\frac{\partial\phi}{\partial
t}\right)^{2}+\frac{1}{2}\left(\frac{\partial\phi}{\partial
x}\right)^{2}+\mathcal{U}.$ (26)

(a) A resonance is observed for the initial velocity of $0.2249$.

(b) With velocity of $0.5$ the solitons undergo an elastic collision.

Figure 11: The full line gives the dynamics of the coordinate $a(t)$ within
the approximation where $Q=K=J=0$ and the dashed line shows the dynamics of
the system obtained from the full numerical simulation.

###  3.1 The inclusion of more terms in the approximation

Here we are inclined to state that in fact, the problems which appear in the
construction of the collective coordinates approximation presented in [2] and
reproduced in figure 11, no longer exist when most of these terms are taken
into account, i.e., when $Q$, $K$ and $J$ are included. Thus we consider the
space of the parameters $(a,\xi)$ to be defined by a diagonal metric so that
the lagrangian reads

$L=\frac{1}{2}\left[M_{0}+I\left(a\right)+{\xi}^{2}K\left(a\right)+{\xi}J\left(a\right)\right]{\dot{a}}^{2}+\frac{1}{2}\left[2+Q\left(a\right)\right]{\dot{\xi}}^{2}-\left[V\left(a\right)-{\xi}F\left(a\right)+3\,{\xi}^{2}\right].$
(27)

In figure 12 below we show the numerically obtained solution of the Euler-
Lagrange equations for $a(t)$ for some choices of initial relative velocity.

(a) The solution for $a(t)$ with the initial velocity of $0.2249$.

(b) The solution for $a(t)$ with initial velocity of $0.5$

Figure 12: The collective coordinate approximation seems to reproduce some of
the fundamental properties of the dynamics of the kink/anti-kink scattering.
The resonance window for velocity $0.2249$ can be seen in this approximation.

For these velocities, we may say that the collective coordinates approximation
exhibits some good agreement with what is observed in the full numerical
simulation, described by $\langle x\rangle$. Also, we can see that the
resonance phenomenon is at a certain degree, described by the approximation.
This may indicates that indeed, the two degrees of freedom, namely, the
translation and vibration modes, play the crucial role - as expected - in the
scattering process.

One important remark to be made is that the coefficient labeled as $Q(a)$ in
the lagrangian (also found explicitly in appendix B) is different than the one
labeled the same in [11]. While the latter is divergent for $a\rightarrow 0$,
i.e., when the solitons collide, the function $Q$ found here has value $Q(a\to
0)=-2$, which makes the kinetic part associated to the vibrations in the
lagrangian (27) vanishes. This is certainly of great importance for the
results just presented.

In order to get a more detailed picture of the role of the translation and
vibration modes, in figure 13 we show the solution of $\xi(t)$ together with
$a(t)$.

(a) The solutions for $a(t)$ and $\xi(t)$ for the initial velocity of
$0.2249$.

(b) The solution for $a(t)$ with initial velocity of $0.5$

Figure 13: It is interesting to note that the amplitude of the oscillation
$\xi(t)$ increases after the scattering.

We notice that the amplitude of the oscillation $\xi(t)$ is larger after the
scattering. This seems to indicate, together with the previously discussed
fact that no considerable amount of energy is lost in the scattering process,
that part of the initial energy from the translational motion was transfered
to the vibrational mode. In fact, this can be also seen in figures 7(a) and
7(b), from the full numerical simulation results.

The energy transfer process between the two degrees of freedom can be seen by
defining the total energy of the collective coordinate system as the sum of
two functions, $E=E_{a}+E_{\xi}$, where

$\displaystyle E_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(M_{0}+I\left(a\right)\right){\dot{a}}^{2}+V\left(a\right)$
(28) $\displaystyle E_{\xi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(2+Q\left(a\right)\right){\dot{\xi}}^{2}+\left({\xi}^{2}K\left(a\right)+{\xi}J\left(a\right)\right){\dot{a}}^{2}-{\xi}F\left(a\right)+3\,{\xi}^{2}.$
(29)

In figure 14 we show the behaviour of these functions for two cases, with
velocities $v=0.2249$ and $v=0.5$. A qualitative analysis shows that when the
solitons are getting closer, part of the energy stored in the internal mode
will be transfered to the translational motion and the solitons will
accelerate towards each other. Next, the energy is then transfered to the
internal mode and the relative velocity gets lower. This can be lower than the
“escape velocity”, i.e., the minimum velocity the solitons need to perform an
elastic collision and consequently the solitons get trapped. In another
situation, if the initial velocity is enough, the energy can flow back to the
transational mode so that, at some stage, the solitons will recover sufficient
velocity to escape; that is the case of the resonance phenomenon.

(a) The energy of the collective coordinates system for velocity equal to
$0.2249$.

(b) The energy of the collective coordinate system for velocity equal to $0.5$

Figure 14: The translational and vibrational modes become evident with respect
to the exchange of energy if it is separated in terms of $E_{a}$ and
$E_{\xi}$.

In figure 15 we present the behaviour of the velocity of the solitons during
the scattering processes, calculated using the full numerical simulation
results and also the one obtained from the collective coordinates
approximation (which is given in terms of its absolute value). We notice that
indeed, by looking at the full simulation results for the cases shown and for
all other we have seen, the solitons leave the collision with lower velocity
than that with which they entered.

(a) The velocity profile of the soliton during the scattering process with
initial velocity equal to $0.2249$.

(b) The velocity profile of the soliton during the scattering process with
initial velocity equal to $0.5$

Figure 15: The velocity profile of the solitons given from the full numerical
simulation and using the collective coordinates, in which case we present the
absolute value.

As seen from figure 15, the collective coordinates approximation captures the
fact that the velocity gets lower after the collision by increasing the
solitons back and fourth oscillation.

##  4 Conclusions

We have shown that through the technique presented in the appendix B all the
coefficients which appear in the lagrangian of the collective coordinates
system obtained with (20) for the scattering of kink and anti-kink of the
$\phi^{4}$ model can be found analytically. These results enabled us to push
the approximation method to “its best possible performance”, so that no
further inclusion of terms aiming to correct the approximation, as those
described in [2], are needed. The only assumption made are that $W$ is taken
to have the constant value of $W=3$ and $N=C=0$, which are already known in
the literature and well justified.

The collective coordinates approximation as described here allows for the
appearance of the resonance windows which are seen in the full numerical
simulation. Moreover, also elastic scattering can be described through this
approximation. Nevertheless, it must be remarked that the effectiveness of the
approximation is highly dependent on the value of the initial relative
velocity between the solitons. We have seen in our numerical experiments that
for certain values of this parameter, what is observed for $a(t)$ can be very
different from what is found for $\langle x\rangle$.

The relative success of the collective coordinates approximation involving a
translational and a vibrational mode as presented here seems to indicate that
indeed it is the interchange of energy between these modes that results in the
resonances found in the scattering of $\phi^{4}$ solitons. The collective
coordinate system exhibits a recurrence phenomenon where almost all the
energy, after the collision, is distributed back to the translation of the
solitons which allows for them to undone the temporary bound system, when it
is formed. In our numerical experiments we have seen that in all cases
considered, some of the energy absorbed in the vibrational mode during the
collision remains there, so that the solitons have some wiggling after the
collision.

##  Appendix A Perturbation of the kink solution

For completeness we present in this section a detailed calculation concerning
the dynamics of the perturbation of a kink solution of the $\phi^{4}$ model.

Writing $\phi_{0}$ as the (anti-)kink solution we consider a configuration
given by

$\displaystyle\phi\left(x,t\right)=\phi_{0}\left(x\right)+\epsilon\left(x,t\right)$
(30)

where $|\epsilon|\ll\phi_{0}$ is the perturbation field. This configuration
will satisfy the dynamical equation (3) for linear order in the perturbation
if

$-\ddot{\epsilon}+{\epsilon}^{\prime\prime}-\lambda\epsilon{\eta}^{2}\left[3\tanh^{2}\left(\sigma{x}\right)-1\right]=0$
(31)

holds.

This equation has solutions of the form
$\epsilon\left(x,t\right)=\chi\left(x\right)e^{\imath{\omega}t}$ given that
$\chi$ satisfies the eigenvalue equation

$\frac{d^{2}\chi}{dx^{2}}+\left[E+\frac{U_{0}}{\cosh^{2}\left(\sigma{x}\right)}\right]\chi\left(x\right)=0,$
(32)

with $E={\omega}^{2}-4{\sigma}^{2}$ and $U_{0}=6{\sigma}^{2}$.

With the change of variable
$y=\frac{1}{2}\left[1-\tanh\left(\sigma{x}\right)\right]$ and defining
$\frac{E}{{\sigma}^{2}}=-{\varepsilon}^{2}$ e
$\frac{U_{0}}{{\sigma}^{2}}=s\left(s+1\right)$, this equation becomes

$\displaystyle
y\left(1-y\right)\frac{d^{2}\chi}{dy^{2}}+\left(1-2y\right)\frac{d\chi}{dy}+\left[s\left(s+1\right)-\frac{{\varepsilon}^{2}}{4y\left(1-y\right)}\right]\chi=0.$
(33)

In order to obtain a well-behaved solution we take into account the fact that
for $x\to\infty$ we get $y\to 0$ and this equation reads, at this limit666 We
have that $x\to\infty,y\to 0\qquad x\to-{\infty},y\to 1\qquad
x\to{0},y\to\frac{1}{2}.$

$\displaystyle
y^{2}{\chi}^{\prime\prime}+y{\chi}^{\prime}-\frac{{\varepsilon}^{2}}{4}\chi=0,$
(34)

and thus with ${\chi}\;{\sim}\;{y}^{\alpha}$, the asymptotic solution is

$\displaystyle\chi=C_{1}{y}^{\frac{\varepsilon}{2}}+C_{2}{y}^{-\frac{\varepsilon}{2}},$
(35)

and $C_{2}$ is taken to be zero.

Finally, taking

$\displaystyle\chi=\left[y\left(1-y\right)\right]^{\frac{\varepsilon}{2}}W\left(y\right),$
(36)

we end up with the hypergeometric equation[25]

$\displaystyle
y\left(1-y\right)W^{\prime\prime}+\left(\varepsilon+1\right)\left(1-2y\right)W^{\prime}-\left[\left(\varepsilon-s\right)\left(s+\varepsilon+1\right)\right]W=0,$
(37)

whose solution is well known[25] to be

$\displaystyle\chi\left(y\right)=\left[y\left(1-y\right)\right]^{\frac{\varepsilon}{2}}{{}_{2}}F_{1}\left[\varepsilon-s,s+\varepsilon+1,1+\varepsilon,\frac{1}{2}\left(1-y\right)\right].$
(38)

Then, it is required that $\varepsilon-s=-n$, so that the polynomial character
of the solution holds. This gives

$\displaystyle E_{n}=-{\sigma}^{2}\left(2-n\right)^{2}$ (39)

and ${\varepsilon}^{2}=-\frac{E}{{\sigma}^{2}}$ implies $\varepsilon=2-n>0$,
and therefore only two values of $n$ are allowed: $n=0,1$.

###  A.1 The perturbation field

We start by rewriting (38) as

$\displaystyle{\chi}_{n}=\left[y\left(1-y\right)\right]^{\frac{2-n}{2}}\frac{\Gamma\left(3-n\right)\Gamma\left(5\right)}{\Gamma\left(5-n\right)\Gamma\left(3\right)}\left(\frac{1}{2}\left(y-1\right)\right)^{n}.$
(40)

Reintroducing the original variables we find

$\displaystyle{\chi}_{0}=\frac{1}{4}\operatorname{sech}^{2}\left(\sigma{x}\right),$
(41)

and the first excited solution reads

$\epsilon=\frac{1}{4}\operatorname{sech}^{2}\left(\sigma{x}\right),$ (42)

having $E_{0}=-4\sigma^{2}$ and thus a zero frequency of oscillation $\omega$.
This is called a translational mode.

For $n=1$ we find

$\displaystyle{\chi}_{1}=\frac{1}{2}\operatorname{sech}\left(\sigma{x}\right){\tanh\left(\sigma{x}\right)}$
(43)

and with $E_{1}=-\sigma^{2}$ and ${\omega}=\sqrt{3}\sigma$, giving a non-
trivial excitation of the kink:

$\epsilon=\frac{1}{2}\operatorname{sech}\left(\sigma{x}\right){\tanh\left(\sigma{x}\right)}\cos\left(\sqrt{3}{\sigma}t\right).$
(44)

##  Appendix B On the calculation of the integrals appearing in the effective
lagrangian

In this section we show the explicit calculation of two particular terms
appearing in the construction of the effective lagrangian for the collective
coordinates. The method can be used to compute all the integrals involved.

We start by considering the term defined as

$Q(a)=-3\int_{-\infty}^{+\infty}dx\;\tanh{(x+a)}\operatorname{sech}{(x+a)}\tanh{(x-a)}\operatorname{sech}{(x-a)}.$
(45)

With $\omega=x+a$ and $\omega^{\prime}=x-a=\omega-2a$ this is rewritten as

$Q(a)=-3\int_{-\infty}^{+\infty}d\omega\;\tanh{\omega}\operatorname{sech}{\omega}\tanh{(\omega-2a)}\operatorname{sech}{(\omega-2a)}.$
(46)

In order to compute this integral we consider it to be part of the integral
along a closed curve in the complex plane of the function

$f(z)=z\tanh{z}\operatorname{sech}{z}\tanh{(z-2a)}\operatorname{sech}{(z-2a)}$
(47)

where $z=\omega+i\phi\in\mathbf{C}$. This complex integral reads

$\oint_{\gamma}dz\;f(z)=\int_{-R}^{R}d\omega\;f(\omega)+i\int_{0}^{i\pi}d\phi\;f(R+i\phi)-\int_{-R}^{R}d\omega\;f(\omega+i\pi)-i\int_{0}^{i\pi}d\phi\;f(-R+i\phi),$

and the path $\gamma$ is a rectangle from $-R$ to $R$ in the real axis and
from $0$ to $i\pi$ in the imaginary axis.

Next we notice that the integration over the paths with constant $R$ will give
no contribution to the result in the limit $R\to\infty$ and therefore

$\lim_{R\rightarrow\infty}\oint_{\gamma}dz\;f(z)=\int_{-\infty}^{+\infty}d\omega\;f(\omega)-\int_{-\infty}^{+\infty}d\omega\;f(\omega+i\pi)=2i\pi\sum_{k}\textrm{Res}\left(f,z_{k}\right)$

where $z_{k}$ are the poles of the function and
$\textrm{Res}\left(f,z_{k}\right)$ stands for the residue of this function at
that pole. For the function given at (47) we find that the poles are at
$z_{1}=\frac{i\pi}{2}$ and $z_{2}=z_{1}+2a$. Then, we find that

$\lim_{R\rightarrow\infty}\oint_{\gamma}dz\;f(z)=-i\frac{\pi}{3}\pi
Q(a)=2i\pi\sum_{k=1}^{2}\textrm{Res}\left(f,z_{k}\right)$ (48)

with

$\sum_{k=1}^{2}\textrm{Res}\left(f,z_{k}\right)=2a\textrm{csch}{(2a)}-2\textrm{csch}{(2a)}\coth{(2a)}+4a\textrm{csch}^{3}{(2a)}.$
(49)

The final result then reads

$Q(a)=12a\textrm{csch}{(2a)}-12\textrm{csch}{(2a)}\coth{(2a)}+24a\textrm{csch}^{3}{(2a)}.$
(50)

It is important to remark that this result differs from those which are known
in the literature for the same term in the collective coordinates effective
lagrangian (usually also labeled as $Q$). In the seminal papers on this
subject, [12] and also [11], this term is found to be divergent as $a\to 0$
while the above calculation shows that $\lim_{a\to 0}Q(a)=-2$, which makes the
kinetic term of the $\xi$ coordinate in the effective lagrangin vanishes.

In the formulation of the effective lagrangian there is also another type of
integral, for instance, in the case of the term labeled as $C$:

$\displaystyle C(a)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{2}}\int_{-\infty}^{+\infty}dx\;\left(\tanh{(x+a)}\operatorname{sech}{(x+a)}-\tanh{(x-a)}\operatorname{sech}{(x-a)}\right)\left(\operatorname{sech}^{2}(x+a)+\operatorname{sech}^{2}(x-a)\right)$
$\displaystyle=$
$\displaystyle\sqrt{\frac{3}{2}}\int_{-\infty}^{+\infty}dx\;\tanh{(x+a)}\operatorname{sech}{(x+a)}\operatorname{sech}^{2}{(x-a)}$
$\displaystyle-$
$\displaystyle\sqrt{\frac{3}{2}}\int_{-\infty}^{+\infty}dx\;\tanh{(x-a)}\operatorname{sech}{(x-a)}\operatorname{sech}^{2}{(x+a)}$

Here the procedure is essentially the same, i.e., we consider the integral on
the complex plane of the function $p(z)=f(z)-h(z)$ where instead of
considering the complex functions $f$ and $h$ to be the complex extension of
the real ones appearing in the integrands above, multiplied by $z$, we take it
to be simply

$f(z)=\tanh{(z)}\operatorname{sech}{(z)}\operatorname{sech}^{2}{(z-2a)}$ (51)

for the first of the two integrals above and

$h(z)=\tanh{(z-2a)}\operatorname{sech}{(z-2a)}\operatorname{sech}^{2}{(z)}$
(52)

for the second one.

We take the path $\gamma$ to be the same as before once the poles of these
functions are also $\frac{i\pi}{2}$ and $\frac{i\pi}{2}+2a$ and also here in
the limit of $R\to\infty$ the integrals of these functions over the imaginary
coordinate $\phi$ will give no contribution to the result. Then we have that

$\lim_{R\rightarrow\infty}\oint
dz\,p(z)=i\pi\sum_{k=1}^{2}\textrm{Res}\left(p,z_{k}\right)$ (53)

with

$\sum_{k=1}^{2}\textrm{Res}\left(f,z_{k}\right)=-4i\textrm{csch}^{3}(2a)\sinh^{4}(a)\quad\textrm{and}\quad\sum_{k=1}^{2}\textrm{Res}\left(h,z_{k}\right)=4i\textrm{csch}^{3}(2a)\sinh^{4}(a).$

Finally, after simplification, the result reads

$C\left(a\right)=\pi\sqrt{\frac{3}{2}}\tanh\left(a\right)\operatorname{sech}^{2}\left(a\right).$
(54)

Acknowledgements GL would like to thank FAPES for the financial support under
the EDITAL CNPq/FAPES No 22/2018. CFSP would like to thank FAPES for financial
support under grant N∘ 98/2017.

## References

  * [1] Weigel, H. Kink-Antikink Scattering in $\phi^{4}$ and $\phi^{6}$ Models, Proceedings, Physics and Mathematics of Nonlinear Phenomena (PMNP 2013): Gallipoli, Italy, June 22-29, 2013”, J. Phys. Conf. Ser. 482” (2014) 012045 , arXiv:1309.6607.
  * [2] Takyi, I. and Weigel, H., Collective Coordinates in One-Dimensional Soliton Models Revisited, Phys. Rev.D94 ( 2016) 8, 085008, arXiv1609.06833.
  * [3] Adam, C. and Oles, K. and Romanczukiewicz, T. and Wereszczynski, A., Spectral Walls in Soliton Collisions, Physical Review Letters, (2019) v.122, n.24
  * [4] C. Adam and K. Oles and T. Romanczukiewicz and A. Wereszczynski. Kink-antikink scattering in the $\phi^{4}$ model without static intersoliton forces, arXiv hep-th 1909.06901
  * [5] C. Adam and K. Oles and T. Romanczukiewicz and A. Wereszczynski. Thick spectral walls in solitonic collisions, arXiv hep-th 1912.09371
  * [6] Kevrekidis, P.G. and Goodman, R. H., Four Decades of Kink Interactions in Nonlinear Klein-Gordon Models: A Crucial Typo, Recent Developments and the Challenges Ahead” (2019) aXiv:1909.03128.
  * [7] M. Peyrard, M. and D.K. Campbell, Kink Antikink Interactions in a modified sine-Gordon Model, Physica, D, 9, (1983), 33.
  * [8] D.K. Campbell, J.S. Schonfeld and C.A Wingate,Resonance Structure in Kink - Antikink Interactions in $\phi^{4}$ Theory, Physica, D, 9, (1983), 1.
  * [9] D.K. Campbell and M. Peyrard, Solitary Wave Collisions Revisited,Physica, D, 18, (1986), 47.
  * [10] P. Anninos, S. Oliveira and R.A. Matzner, R. A.”, Fractal structure in the scalar lambda $(\phi^{2}-1)^{2}$ theory Phys. Rev. D44 (1991) , 1147.
  * [11] T. Sugiyama, Kink - Antikink Collisions in the Two-Dimensional $\varphi^{4}$ Model, Prog. Theor. Phys. 61, 1550, (1979).
  * [12] T.I. Belova and A.E. Kudryavtsev, Quasi-periodic orbits in the scalar classical $\lambda$$\phi$4 field theory, Physica D 32 (1988) 18.
  * [13] R.H. Goodman and R. Haberman , Kink-Antikink Collisions in the $\phi^{4}$ Equation: The n-Bounce Resonance and the SIAM J. App. Dyn. Sys., 4 (2005) 1195.
  * [14] Mehrdad Moshir Soliton-antisoliton scattering and capture in $\lambda\phi^{4}$ theory, Nuclear Physics B, Volume 185, Issue 2, 1981, Pages 318-332.
  * [15] H. E. Baron, G. Luchini and W. J . Zakrzewski, Collective coordinate approximation to the scattering of solitons in the (1+1) dimensional NLS model, J.Phys (2015) 265201.
  * [16] Bishop A R. and Schneider T (Eds.) 1978.  Solitons in Condensed Matter Physics.  Berlin: Springer Verlag.
  * [17] Manton, N. and Sutcliffe, P. (2004).  Topological Solitons.  Cambridge Monographs on Mathematical Physics. Cambridge University Press.
  * [18] Vilenkin, A. and Shellard, E. P. S. (2000).  Cosmic strings and other topological defects.  Cambridge University Press.
  * [19] Vachaspati T. 2006.  Kinks and domain walls: An introduction to classical and quantum solitons.  Cambridge:Cambridge University Press.
  * [20] Wittkowski, Raphael and Tiribocchi, Adriano and Stenhammar, Joakim and Allen, Rosalind J. and Marenduzzo, Davide and Cates, Michael E. (2014)  Scalar $\phi^{4}$ field theory for active-particle phase separation Nature Communications, 5(1)
  * [21] Bogomolny, E. B. (1976).  Stability of Classical Solutions.  Sov. J. Nucl. Phys., 24:449.  [Yad. Fiz.24,861(1976)].
  * [22] Sutcliffe, P. M. (1997).  BPS monopoles.  Int. J. Mod. Phys., A12:4663–4706, hep-th/9707009.
  * [23] Adam, C., Ferreira, L. A., da Hora, E., Wereszczynski, A., & Zakrzewski, W. J. (2013).  Some aspects of self-duality and generalised BPS theories.  JHEP, 08:062, 1305.7239.
  * [24] Manton, N. S. (2008).  Solitons as elementary particles: a paradigm scrutinized.  Nonlinearity, 21(11):T221–T232.
  * [25] G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, sixth edition (Elsevier Academic Press, New York, 2005)