# A ball quotient parametrizing trigonal genus 4 curves

Eduard Looijenga  University of Chicago (USA) and Universiteit Utrecht
(Nederland)

###### Abstract.

We consider the moduli space of genus 4 curves endowed with a $g^{1}_{3}$
(which maps with degree 2 onto the moduli space of genus 4 curves). We prove
that it defines a degree $\frac{1}{2}(3^{10}-1)$ cover of the 9-dimensional
Deligne-Mostow ball quotient such that the natural divisors that live on that
moduli space become totally geodesic (their normalizations are 8-dimensional
ball quotients).

This isomorphism differs from the one considered by S. Kondō and its
construction is perhaps more elementary, as it does not involve K3 surfaces
and their Torelli theorem: the Deligne-Mostow ball quotient parametrizes
certain cyclic covers of degree 6 of a projective line and we show how a level
structure on such a cover produces a degree 3 cover of that line with the same
discriminant, yielding a genus 4 curve endowed with a $g^{1}_{3}$.

The author was supported by the Jump Trading Mathlab Research Fund.

## Introduction

Shigeyuki Kondō constructed in [5] an isomorphism from a Zariski open subset
of the moduli space of nonsingular complex genus four curves
${\mathcal{M}}_{4}$ onto a Zariski open subset of a ball quotient, where the
latter covers the 9-dimensional ball quotient that appears in the Deligne-
Mostow list. He also noted that this cannot be extended to the full
${\mathcal{M}}_{4}$, but that this is possible if we blow up the hyperelliptic
locus ${\mathcal{H}}_{4}$ (which is of codimension 2 in ${\mathcal{M}}_{4}$).
The exceptional divisor then maps to a hyperball quotient, a property that
also holds for the generic point of the thetanull locus. His approach is based
on the fact that the generic point of ${\mathcal{M}}_{4}$ is represented by a
curve $X$ of bidegree $(3,3)$ on ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. The
$\boldsymbol{\mu}_{3}$-cover of ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ that
totally ramifies along $X$ is a K3 surface with $\boldsymbol{\mu}_{3}$-action
and the Torelli theorem for such surfaces implies that its
$\boldsymbol{\mu}_{3}$-Hodge structure is a complete invariant of that
$\boldsymbol{\mu}_{3}$-surface and hence also of $X$ (as the fixed point set
of the $\boldsymbol{\mu}_{3}$-action). Such $\boldsymbol{\mu}_{3}$-Hodge
structures are parametrized by a ball quotient of the above type.

The goal of this note is twofold. First to observe that there is a modular
interpretation of the hyperelliptic blowup as the moduli space of pairs
consisting of a nonsingular complex genus four curve and a $g^{1}_{3}$ on that
curve, where we note that a generic genus 4 curve has _two_ $g^{1}_{3}$’s and
that this moduli space comes with an involution that exchanges them.

And second, to give this moduli space a complex hyperbolic structure without
resorting to K3 surfaces and their Torelli theorem. For this, we start from
the fact that the the generic point of the Deligne-Mostow ball quotient
parametrizes $\boldsymbol{\mu}_{6}$-covers $C\to{\mathbb{P}}^{1}$ totally
ramified over a 12 element subset $D\subset{\mathbb{P}}^{1}$ and nowhere else.
We then show how a certain level structure on $C$ gives rise to a genus $4$
curve $X$ lying with degree 3 over ${\mathbb{P}}^{1}$ with discriminant $D$
(so $X$ comes with a $g^{1}_{3}$) and a correspondence between $C$ and $X$.

Our approach is in the same spirit as our paper with Heckman [4] on the moduli
space of rational elliptic surfaces (which here appears in the guise of the
thetanull locus). Most of that paper was concerned with the construction and
comparison of certain compactifications of that moduli space (which accounts
for its length) and it is likely that a similar program could be pursued here,
where, of course, a compactification which parametrizes curves of arithmetic
genus 4 endowed with a $g^{1}_{3}$ would be best.

Let us finally point out that our period map is quite different from the one
introduced by Kondō, as the two discriminants that we associate to the two
$g^{1}_{3}$’s on a generic genus $4$ curve will not lie in the same
$\operatorname{PGL}_{2}({\mathbb{C}})$-orbit and hence have different image in
the Deligne-Mostow ball quotient. Presumably the two period maps do not even
coincide on the generic point of the thetanull locus, where the genus 4 curve
has only one $g^{1}_{3}$.

##  1\. The nine-dimensional Deligne-Mostow ball quotient

This section is mostly a review of known results. Its main purpose is to
recall them in a way that suits our purpose and to establish notation.

###  $\boldsymbol{\mu}_{6}$-covers of ${\mathbb{P}}^{1}$

Let $C$ be a nonsingular complex-projective curve endowed with an
$\boldsymbol{\mu}_{6}$-action that has 12 fixed points, is free elsewhere and
is such that its orbit space $P$ is a copy of ${\mathbb{P}}^{1}$. The
discriminant of $C\to P$ is then of the form $5D_{C}$, with $D_{C}$ a
$12$-element subset of $P$, considered as a reduced divisor and Riemann-
Hurwitz shows that $C$ will have genus $25$. Note that the pair $(P,D_{C})$
determines $C$ uniquely, the isomorphism being unique up to a covering
transformation (in $\boldsymbol{\mu}_{6}$). We recall how this gives rise to a
point in a $9$-dimensional ball quotient, which happens to be the one of
highest dimension that appears in the Deligne-Mostow list.

Since we find it convenient to make a distinction between what depends on the
complex structure and what only depends on the underlying topology, we also
fix a closed oriented surface $\Sigma$ for of genus 25 endowed with a
$\boldsymbol{\mu}_{6}$-action of the type above, so with $12$ fixed points and
acting freely elsewhere. We denote by $\pi:\Sigma\to S$ for the formation of
its $\boldsymbol{\mu}_{6}$-orbit space (a smooth $2$-sphere), and by $D\subset
S$ its set of critical values. Then any complex structure on $S$ (which will
make it a copy of the Riemann sphere) determines a unique one on $\Sigma$ for
which the $\boldsymbol{\mu}_{6}$-action and $\pi$ are holomorphic, yielding a
$\boldsymbol{\mu}_{6}$-curve as above with discriminant $5D$. We also note
that a diffeomorphism of $S$ which preserves $D$ lifts to $S$ and does so
almost uniquely: two such lifts will lie in the same
$\boldsymbol{\mu}_{6}$-orbit.

### Hodge structure of a $\boldsymbol{\mu}_{6}$-cover of ${\mathbb{P}}^{1}$

We first note that the standard embedding
$\chi:\boldsymbol{\mu}_{6}\subset{\mathbb{C}}$ and its complex conjugate are
the only two primitive characters of $\boldsymbol{\mu}_{6}$. The group
$\boldsymbol{\mu}_{6}$ acts on $H^{1}(\Sigma;{\mathbb{C}})$ and decomposes the
latter into its character spaces:
$H^{1}(\Sigma;{\mathbb{C}})=\oplus_{i\in{\mathbb{Z}}/6}H^{1}(\Sigma;{\mathbb{C}})_{\chi^{i}}$.
The intersection pairing gives rise to a nondegenerate Hermitian form on
$H^{1}(\Sigma;{\mathbb{C}})$ defined by
$(\alpha,\beta)\mapsto\sqrt{-1}\int_{\Sigma}\alpha\wedge\overline{\beta}$. It
has signature $(25,25)$ and for this form the above decomposition into
character spaces is perpendicular. In the presence of a complex structure it
is also compatible with the Hodge structure. For example, the decomposition

(1.1)
$H^{1}(C;{\mathbb{C}})_{\chi}=H^{1}(C,\Omega_{C})_{\chi}\oplus\overline{H^{1}(C,\Omega_{C})_{\overline{\chi}}}.$

is perpendicular with respect to this hermitian form with the first summand
being positive definite of dimension one and the second summand negative
definite of dimension nine. Since the group of covering transformations
$\boldsymbol{\mu}_{6}$ acts trivially on
${\mathbb{P}}(H^{1}(\Sigma;{\mathbb{C}})_{\chi})$ and so this projective space
only depends on the pair $(S,D)$; we therefore denote it by
${\mathbb{P}}(S,D)$. Its group
$\operatorname{PU}(H^{1}(\Sigma;{\mathbb{C}})_{\chi})$ of projective unitary
transformations, which we denote for a similar reason by
$\operatorname{PU}(S,D)$, is isomorphic to $\operatorname{PU}(1,9)$. The
positive definite complex lines in $H^{1}(\Sigma;{\mathbb{C}})_{\chi}$ make up
an open complex ball ${\mathbb{B}}(S,D)\subset{\mathbb{P}}(S,D)$ on which
$\operatorname{PU}(S,D)$ acts transitively with compact stabilizers; indeed,
this ball is the symmetric domain associated to $\operatorname{PU}(S,D)$. If
there is no risk of confusion, we simply write ${\mathbb{B}}$ for
${\mathbb{B}}(S,D)$.

As we observed above, a complex structure on $S$ determines one on $\Sigma$,
turning it into a compact Riemann surface $C$ with
$\boldsymbol{\mu}_{6}$-action. The Hodge decomposition (1.1) then defines an
element, the _Hodge point_ , of ${\mathbb{B}}$. The main result of Deligne and
Mostow [3], [6] implies that this Hodge point is a complete invariant of the
relative isotopy class of this complex structure.

### Action of a mapping class group

To make this last assertion precise, let $\operatorname{Diff}^{+}(S,D)$ denote
the group of orientation preserving diffeomorphisms of $S$ which preserve $D$.
Since this group $\operatorname{Diff}^{+}(S,D)$ acts on $\Sigma$ up to
covering transformations, it gives rise to a well-defined action on
${\mathbb{P}}(S,D)$ via $\operatorname{PU}(S,D)$ (and so preserves
${\mathbb{B}}$). This action factors through its connected component group
that we shall denote $\operatorname{Mod}(S,D)$ and which we readily recognize
as the spherical braid group on 12 strands. That group has as a distinguished
conjugacy class the set of simple braids. We recall that a _simple braid_ is
given by an (unoriented) arc $\delta$ in $S$ which connects two points of $D$
and has no points of $D$ in its interior: the associated simple braid
$T_{\delta}\in\operatorname{Mod}(S,D)$ has its support in a regular
neighborhood of $\delta$ and lets $\delta$ make a half-turn in a
counterclockwise direction. These simple braids generate
$\operatorname{Mod}(S,D)$. In fact, if we enumerate the points of $D$ by
$\\{p_{i}\\}_{i\in{\mathbb{Z}}/12}$ and let $\delta_{i}$ connect $p_{i}$ with
$p_{i+1}$ in such a way as that their juxtaposition produces an embedded
circle in $S$, then $T_{\delta_{1}},\dots,T_{\delta_{10}}$ already generate
$\operatorname{Mod}(S,D)$. It follows from the work of Deligne-Mostow that the
image of $\operatorname{Mod}(S,D)$ in $\operatorname{PU}(S,D)$ is an
arithmetic group, which therefore acts properly discretely on ${\mathbb{B}}$
with finite covolume. We denote this image group by
$\Gamma\subset\operatorname{PU}(S,D)$.

### A lattice over the Eisenstein ring

In order to make the action of a simple braid on ${\mathbb{B}}$ somewhat more
explicit, we make some general observations first. Write $\tau$ for the
primitive $6$th root of unity with positive imaginary part,
$e^{\pi\sqrt{-1}/3}$, and regard it as a generator of $\boldsymbol{\mu}_{6}$.
As a complex number it satisfies $\tau^{2}=\tau-1$, but this identity is of
course not valid in the group ring ${\mathbb{Z}}[\boldsymbol{\mu}_{6}]$. Given
a ${\mathbb{Z}}[\boldsymbol{\mu}_{6}]$-module $V$, let us denote by
$V^{\circ}$ resp. $V_{\circ}$ the maximal submodule resp. quotient module on
which $\tau$ satisfies the identity $\tau^{2}=\tau-1$. For example,
$V_{\circ}$ is the quotient of $V$ by the subgroup of $v\in V$ with nontrivial
$\boldsymbol{\mu}_{6}$-stabilizer. Then $V^{\circ}$ and $V_{\circ}$ become
modules over the ring of _Eisenstein integers_
${\mathcal{E}}:={\mathbb{Z}}+{\mathbb{Z}}\tau\subset{\mathbb{C}}$, the ring of
integers the cyclotomic field ${\mathbb{Q}}(\sqrt[3]{1})$. Note that
${\mathbb{R}}\otimes V^{\circ}\to{\mathbb{R}}\otimes V_{\circ}$ is an
isomorphism and can be identified with the summand of ${\mathbb{R}}\otimes V$
which decomposes into planes on which $\tau$ acts as a rotation over $2\pi/6$.
Since ${\mathcal{E}}$ is a principal ideal domain, any finitely generated
torsion free ${\mathcal{E}}$-module is in fact free. So if $V$ is a finitely
generated free ${\mathbb{Z}}$-module, then both $V_{\circ}$ and $V^{\circ}$
are finitely generated free ${\mathcal{E}}$-modules. If we are in addition
given a $\boldsymbol{\mu}_{6}$-invariant antisymmetric pairing $a:V\times
V\to{\mathbb{Z}}$, then this gives rise to the
$\frac{1}{2}{\mathcal{E}}$-valued hermitian form on $V^{\circ}$ defined by

$h_{a}(x,y):=\tfrac{1}{2}\tau a(x,y)-\tfrac{1}{2}a(x,\tau y)$

(the justification for the perhaps somewhat unexpected normalization factor
$\tfrac{1}{2}$ will be given below). Note that
$h_{a}(x,x)=-\tfrac{1}{2}a(x,\tau x)$ and
$h_{a}(x,y)-\overline{h_{a}(x,y)}=\tfrac{1}{2}(\tau-\overline{\tau})a(x,y)=\tfrac{1}{2}\theta
a(x,y)$, where

$\theta:=\tau-\overline{\tau}=-1+2\tau=\sqrt{-3}$

is the ‘first’ purely imaginary number in ${\mathcal{E}}$. We apply this to
the ${\mathbb{Z}}[\boldsymbol{\mu}_{6}]$-module $H_{1}(\Sigma)$. We find that
$H_{1}(\Sigma)^{\circ}$ and $H_{1}(\Sigma)_{\circ}$ are free
${\mathcal{E}}$-modules of rank 10. The intersection pairing on
$H_{1}(\Sigma)$ is nondegenerate on $H_{1}(\Sigma)^{\circ}$ and defines via
the receipe above on this lattice the ${\mathcal{E}}$-valued hermitian form

$h(x,y):=\tfrac{1}{2}\tau(x\cdot y)-\tfrac{1}{2}(x\cdot\tau y).$

We will write $L$ for $H_{1}(\Sigma)^{\circ}$ endowed with this form.

Let us see what this induces on the preimage $\pi^{-1}\delta$ of an arc
$\delta$ as above. We now assume $\delta$ oriented and let $\tilde{\delta}$ be
a lift of $\delta$ over $\pi$. Then $c_{\delta}:=(1-\tau^{3})\tilde{\delta}$
is an embedded oriented circle (hence a $1$-cycle) and meets its
$\tau$-translate in 2 points, each with multiplicity $1$. In other words,
$c_{\delta}\cdot\tau c_{\delta}=2$. Now $H_{1}(\pi^{-1}\delta)^{\circ}$ has
the ${\mathcal{E}}$-generator
$a_{\delta}:=(1+\tau)(1-\tau^{3})\tilde{\delta}=(1+\tau)c_{\delta}$ (which is
unique up multiplication by an element of $\boldsymbol{\mu}_{6}$), and the
above formula shows that $h(a_{\delta},a_{\delta})=\tfrac{1}{2}.-6=-3$. A
similar computation shows that
$h(a_{\delta_{i}},a_{\delta_{i+1}})\in\boldsymbol{\mu}_{6}\theta$. So
$h|L\times L$ takes values in $\theta{\mathcal{E}}$, so that $\theta^{-1}h$
defines a skew-hermitian form

$h^{\prime}:=\theta^{-1}h:L\times L\to{\mathcal{E}}$

Since we have the freedom of multiplying each $a_{\delta}$ with an element of
$\boldsymbol{\mu}_{6}$, we can choose $a_{\delta_{1}},\dots,a_{\delta_{10}}$
in such a manner that it is a ${\mathcal{E}}$-basis of $L$ with

$h(a_{\delta_{i}},a_{\delta_{j}})=\begin{cases}-3&\text{ when $j=i$,}\\\
\pm\theta&\text{ when $j=i\pm 1$,}\\\ 0&\text{ otherwise.}\\\ \end{cases}$

This also shows that $L$ is isomorphic to the lattice thus denoted in [2]
(this explains the coefficient $\tfrac{1}{2}$ in our definition of $h$) and if
we $h$ replace by $-h$, it becomes isomorphic to the lattice denoted $\Lambda$
in [4]. In [2], $L$ is in fact constructed from the even
${\mathbb{Z}}$-lattice
$\mathbf{E}_{8}\perp\mathbf{E}_{8}\perp\mathbf{U}\perp\mathbf{U}$. Going in
the opposite direction, this amounts to the statement that we recover this
lattice if take the underlying abelian group and the even symmetric pairing on
it defined by the real part of $h$ multiplied with $-\frac{2}{3}$.

### Mirror hyperballs

The simple braid $T_{\delta}$ acts in $L$ as the unitary transformation
$s_{\delta}$ defined by

$s_{\delta}(x)=x+\tfrac{1}{3}(1-\tau^{2})h(x,a_{\delta})a_{\delta}=x+\tau
h^{\prime}(x,a_{\delta})a_{\delta}$

It is what is called in [2] a _triflection_ in ${\mathcal{E}}a_{\delta}$: it
multiplies $a_{\delta}$ with the third root of unity $\tau^{2}$ and is the
identity on the orthogonal complement of $a_{\delta}$. It indeed preserves
$L$, as follows from the fact that $h^{\prime}$ takes its values in
${\mathcal{E}}$.

Ordinary duality identifies $H^{1}(\Sigma)^{\circ}$ resp.
$H^{1}(\Sigma)_{\circ}$ with the dual of $H^{1}(\Sigma)_{\circ}$ resp.
$H^{1}(\Sigma)^{\circ}$. We use Poincaré duality to identify these with resp.
$L$ resp. $H_{1}(\Sigma)_{\circ}$ and use on these the same hermitian form as
above. This identifies ${\mathbb{P}}(S,D)$ with
${\mathbb{P}}({\mathbb{C}}\otimes_{\mathcal{E}}L)$. In particular,
$T_{\delta}$ acts on ${\mathbb{B}}$ as a triflection. So its fixed point set
${\mathbb{B}}^{T_{\delta}}$ is a hyperball (to which we shall also refer as a
_mirror_ in ${\mathbb{B}}$). Since the $T_{\delta}$ make up a single conjugacy
class in $\operatorname{Mod}(S,D)$, the group $\operatorname{Mod}(S,D)$
permutes the mirror hyperballs transitively.

The group of unitary transformations of the ${\mathcal{E}}$-module $L$ acts
properly discretely on ${\mathbb{B}}$ with kernel its center
$\boldsymbol{\mu}_{6}$. The group $\Gamma$ is a priori contained in the
quotient of this group by its center (which is an arithmetic subgroup of
$\operatorname{PU}(S,D)$), but as Allcock has shown ([2], Thm. 5.1), is in
fact equal to it. So $\Gamma$ acts properly discontinuously on ${\mathbb{B}}$
and the Baily-Borel theory asserts that
${\mathbb{B}}_{\Gamma}:=\Gamma\backslash{\mathbb{B}}$ has the structure of a
quasi-projective orbifold that can be completed to a normal projective variety
by adding a finite set of _cusps_ (in this case, just one). The image of the
Hodge point in this ball quotient ${\mathbb{B}}_{\Gamma}$ is a complete
invariant of the isomorphism type of the $\boldsymbol{\mu}_{6}$-curve $C$, or
equivalently, of the pair $(S,D)$ with the given complex structure.

The $\operatorname{Mod}(S,D)$-centralizer of $T_{\delta}$ acts on the mirror
${\mathbb{B}}^{T_{\delta}}$ with image an arithmetic subgroup and the
resulting ball quotient of ${\mathbb{B}}^{T_{\delta}}$ appears in the Deligne-
Mostow list as the one defined by the sequence
$(\frac{2}{12},\frac{1}{12},\frac{1}{12},\dots,\frac{1}{12})$.

The group $\Gamma$ is as a quotient of $\operatorname{Mod}(S,D)$ obtained by
imposing the relation $T_{\delta}^{3}=1$. The mirrors are locally finite on
${\mathbb{B}}$ and since they are transitively permuted by $\Gamma$, they
define an irreducible totally geodesic hypersurface ${\mathcal{D}}$ in
${\mathbb{B}}_{\Gamma}$ that we shall call the _confluence locus_. We write
${\mathbb{B}}^{\circ}$ for the complement of the union of the mirrors (an open
subset of ${\mathbb{B}}$). So this is the preimage of
${\mathbb{B}}_{\Gamma}^{\circ}\smallsetminus{\mathcal{D}}$.

### The Deligne-Mostow isomorphism

Let ${\mathcal{M}}(12)$ denote the $\operatorname{PGL}(2,{\mathbb{C}})$-orbit
space of the configuration space of $12$-element subsets of ${\mathbb{P}}^{1}$
(this is also equal to the ${\mathfrak{S}}_{12}$-orbit space of
${\mathcal{M}}_{0,12}$). The above construction defines a homomorphic map

${\mathcal{M}}(12)\to{\mathbb{B}}_{\Gamma}.$

If we think of a $12$-element subset of ${\mathbb{P}}^{1}$ as a reduced degree
$12$-divisor, then the above map extends to the locus ${\mathcal{M}}(12)^{\rm
st}$ of Hilbert-Mumford stable positive degree $12$-divisors, that is,
divisors for which the multiplicities are at most $5$. The Deligne-Mostow
theory asserts that this extension is in fact an isomorphism

${\mathcal{M}}(12)^{\rm st}\xrightarrow{\cong}{\mathbb{B}}_{\Gamma}$

that takes ${\mathcal{M}}(12)$ onto ${\mathbb{B}}_{\Gamma}^{\circ}$. It even
extends to an isomorphism of their natural compactifications: the GIT
compactification on the left and the Baily-Borel compactification on the
right. Both are one-point compactifications, with the added point on the left
being represented by a $6$-divisible divisor on ${\mathbb{P}}^{1}$ whose
support consists of two points.

##  2\. Trigonal structures on a genus four curve

###  $\mathbf{g^{1}_{3}}$’s on a genus 4 curve

Let $X$ be a smooth connected complex-projective curve of genus $4$ and let
$P$ be a pencil of degree $3$ on $X$ (in other words, a $g^{1}_{3}$). If $P$
has no base point, then the pencil defines a morphism $X\to P$ of degree $3$
and by Riemann-Hurwitz, the discriminant divisor $D(X,P)$ of this morphism is
of degree $12$. Note that its multiplicities will be $\leq 2$. In case $P$ has
a base point, then the residual base point free pencil $P^{\prime}$ will have
degree $2$ and this makes $X$ hyperelliptic with $P^{\prime}$ being the
hyperelliptic pencil. We will see that it is then natural to define the
discriminant divisor of $P$ to be the degree $12$-divisor $D(X,P)$ that is
(via the identification of $P$ with $P^{\prime}$) the sum of the discriminant
divisor of $X\to P^{\prime}$ (the image of the Weierstraß points, so of degree
10) plus twice the image in $P^{\prime}$ of the fixed point $P$. So its
multiplicities are then $\leq 3$.

Recall that if $X$ is nonhyperelliptic, then a canonical image of $X$ in
${\mathbb{P}}^{3}$ is the transversal intersection of a cubic surface with a
quadric $Q$ surface, the latter being unique and either a smooth (so
isomorphic to ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$) or a quadric cone. In
the first case, the projections on the two factors give us two pencils of
degree $3$. In the second case, projection away from the vertex of the cone
also yields a pencil of degree $3$; the line bundle is question then an
effective (even) theta characteristic. It is well-known that this exhausts all
the $g^{1}_{3}$’s in genus 4. For future reference we state this as a lemma
(and only sketch the proof).

######  Lemma 2.1.

Let $X$ be a smooth connected complex-projective curve of genus $4$. If $X$ is
nonhyperelliptic, then any $g^{1}_{3}$ on $X$ is as above: it comes with a
residual $g^{1}_{3}$ (in the sense that its members supplement the members of
the given $g^{1}_{3}$ to a canonical divisor) and there are no others (so the
two being equal if and only if the $g^{1}_{3}$ is a theta characteristic). If
$X$ is hyperelliptic, then any $g^{1}_{3}$ on $X$ is the sum of an $x\in X$
and its hyperelliptic $g^{1}_{2}$; in that case the pencil is a theta
characteristic if and only if $x$ is a Weierstraß point.

###### Proof.

This is well-known, as it is in a sense a geometric formulation of Clifford’s
theorem in genus $4$. The proof runs as follows. If $P$ is a $g^{1}_{3}$ on
$X$, then Clifford’s theorem implies that $P$ is a complete linear system. By
Riemann-Roch we have a supplementary $g^{1}_{3}$ on $X$, denoted $P^{\prime}$,
whose members supplement the members of $P$ to a canonical divisor. So each
member of $P$ resp. $P^{\prime}$ defines a line in
$\check{\mathbb{P}}(H^{0}(X,\Omega_{X}))\cong{\mathbb{P}}^{3}$ which meets the
canonical image of $X$ in at most three points and $P^{\prime}$ resp. $P$ is
realized by the pencil of planes through that line. We have $P=P^{\prime}$ if
and only if $P$ is a theta characteristic. The rest of the argument is left to
the reader. ∎

### A moduli stack of genus 4 curves with a $\mathbf{g^{1}_{3}}$

We interpret this in terms of moduli spaces. Pairs $(X,P)$ as above define a
Deligne-Mumford stack that we shall denote by ${\mathcal{M}}_{4}(g^{1}_{3})$.
It comes with an involution which assigns to $(X,P)$ the residual pair
$(X,P^{\prime})$ that is characterized by the property that $P+P^{\prime}$
lies in the canonical system. The forgetful morphism
${\mathcal{M}}_{4}(g^{1}_{3})\to{\mathcal{M}}_{4}$ evidently factors through
the orbit space of this involution. Recall that the moduli space
${\mathcal{M}}_{4}$ contains the thetanull locus ${\mathcal{M}}_{4}^{0}$ (the
locus for which the curve admits an even effective theta characteristic) as a
substack of codimension one and that this substack ${\mathcal{M}}_{4}^{0}$ in
turn contains the hyperelliptic locus ${\mathcal{H}}_{4}$ as a substack of
codimension one.

Consider the Stein factorization

${\mathcal{M}}_{4}(g^{1}_{3})\to\hat{\mathcal{M}}_{4}\to{\mathcal{M}}_{4}$

It follows from Lemma 2.1 that $\hat{\mathcal{M}}_{4}\to{\mathcal{M}}_{4}$ is
finite of degree two and ramified along ${\mathcal{M}}_{4}^{0}$, with the
covering transformation being induced by the residual involution. It also
shows that if $\hat{\mathcal{H}}_{4}$ stands for the (isomorphic) preimage of
${\mathcal{H}}_{4}$ in $\hat{\mathcal{M}}_{4}$, then
${\mathcal{M}}_{4}(g^{1}_{3})\to\hat{\mathcal{M}}_{4}$ is an isomorphism over
$\hat{\mathcal{M}}_{4}\smallsetminus\hat{\mathcal{H}}_{4}$ and is over
$\hat{\mathcal{H}}_{4}\cong{\mathcal{H}}_{4}$ the universal hyperelliptic
curve ${\mathcal{H}}_{4}(g^{1}_{3})$ of genus $4$ (so that will be a divisor).
The strict transform of ${\mathcal{M}}_{4}^{0}$ in
${\mathcal{M}}_{4}(g^{1}_{3})$ is another divisor
${\mathcal{M}}_{4}(g^{1}_{3})^{0}$ on ${\mathcal{M}}_{4}(g^{1}_{3})$ which
meets ${\mathcal{H}}_{4}(g^{1}_{3})$ in the locus parametrizing the
Weierstrass points (this locus is geometrically connected, since the monodromy
is transitive on the Weierstrass points).

Our goal is to realize ${\mathcal{M}}_{4}(g^{1}_{3})$ as an open subset of a
quotient of ${\mathbb{B}}$ relative some finite index subgroup of $\Gamma$. We
determine this subgroup in two ways: first in a purely topological manner and
subsequently more in the spirit of algebraic geometry as a level structure on
the $\boldsymbol{\mu}_{6}$-curve $\Sigma$.

### The approach via a classification of coverings

This begins with addressing the question of how many topological types of
smooth connected degree three coverings of the $2$-sphere $S$ exist that have
the $12$-element subset $D\subset S$ as reduced discriminant. Such a covering
is given by its restriction to $S\smallsetminus D$ (which is then unramified).
So if we choose a base point $o\in S\smallsetminus D$, then it can be given by
a group homomorphism $\rho:\pi_{1}(S\smallsetminus D,o)\to{\mathfrak{S}}_{3}$
which assigns to a simple loop around a point of $D$ a transposition. The
image of $\rho$ must be a transitive subgroup of ${\mathfrak{S}}_{3}$ and
hence must be all of ${\mathfrak{S}}_{3}$. Two such epimorphisms define
isomorphic coverings if and only if they differ by an inner automorphism of
${\mathfrak{S}}_{3}$. So if we let
$\operatorname{Epi}^{\prime}(\pi_{1}(S\smallsetminus D,o),{\mathfrak{S}}_{3})$
stand for the set of group of surjective homomorphisms
$\pi_{1}(S\smallsetminus D,o)\to{\mathfrak{S}}_{3}$ that take simple loops to
transpositions, then the set of topological types is naturally identified with

$R(S,D):={\mathfrak{S}}_{3}\backslash\operatorname{Epi}^{\prime}(\pi_{1}(S\smallsetminus
D,o),{\mathfrak{S}}_{3}).$

We determine its number of elements. We choose smooth arcs
$\\{\gamma_{i}\\}_{i\in{\mathbb{Z}}/12}$ from $o$ to the distinct points of
$D$ that only meet at $o$ and depart from there along rays in $T_{o}S$ in a
counterclock wise order. This means that the associated simple loop associated
to $\gamma_{i}$ defines a $c_{i}\in\pi_{1}(S\smallsetminus D,o)$ so that
$c_{11}\cdots c_{0}=1$ and $c_{1},\dots,c_{11}$ are free generators. Then a
$\rho$ as above is given by its values on $c_{1},\dots,c_{11}$. Note that
$\rho(c_{1}),\dots,\rho(c_{11})$ is a sequence in the 3-element set of
transpositions $\\{(12),(23),(31)\\}$ of ${\mathfrak{S}}_{3}$ that cannot be
nonconstant, but can otherwise be arbitrary. So there are $3^{11}-3$ such
$\rho$. The action of ${\mathfrak{S}}_{3}$ on this set by conjugation is free
and so $\\#R(S,D)=(3^{11}-3)/6=(3^{10}-1)/2=3^{9}+3^{8}+\cdots+1$. This number
is also equal to $\\#{\mathbb{P}}^{9}({\mathbb{F}}_{3})$. We shall see that
this is not a coincidence.

Evidently the group of automorphisms of $\pi_{1}(S\smallsetminus D,o)$ which
preserves the simple loops acts on $R(S,D)$. Since we have divided out by
${\mathfrak{S}}_{3}$-conjugation, the inner automorphisms act trivially, so
that we have an outer action of $\operatorname{Mod}(S,D)$ on
$\pi_{1}(S\smallsetminus D,o)$. This gives us a genuine action of
$\operatorname{Mod}(S,D)$ on $R(S,D)$.

######  Lemma 2.2.

The action of $\operatorname{Mod}(S,D)$ on $R(S,D)$ is through $\Gamma$ and is
transitive.

###### Proof.

Recall that an (unoriented) arc $\delta$ in $S$ connecting two distinct points
of $D$ whose relative interior does not meet any point of $D$ represents a
simple braid in $\operatorname{Mod}(S,D)$. If we choose an interior point of
$\delta$ as our base point $o$, then this yields two simple loops and hence
two transpositions of the fiber over $o$. Then $T_{\delta}$ acts trivially on
$\rho$ if these two transpositions are equal and, as a straightforward
verification shows, is of order $3$ otherwise. It follows that this action of
$\operatorname{Mod}(S,D)$ on $R(S,D)$ factors through $\Gamma$.

For the transitivity property it is enough to show that for a given degree 3
covering $\Sigma^{\prime}\to S$ as above there exists a standard generating
set $(c_{0},\dots,c_{11})$ of $\pi_{1}(S\smallsetminus D,o)$ as above and a
numbering of the fiber over $o$ such that $\rho_{\Sigma^{\prime}}(c_{i})$
equals $(12)$ for $i$ even and $(23)$ for $i$ odd. Since
$\operatorname{Mod}(S,D)$ acts transitively on the collection of standard
generating sets, this will indeed suffice.

Suppose that for $i\leq 10$ we have constructed arcs
$\gamma_{1},\dots,\gamma_{i}$ that depart at $o$ along tangent rays in a
counterclockwise order, are otherwise disjoint and are such that
$\rho(c_{1}),\dots\rho(c_{i})$ are as desired, and with the property that the
preimage in $\Sigma^{\prime}$ over
$\Delta_{i}:=S\smallsetminus\\{\gamma_{1}\cup\dots\cup\gamma_{i}\\}$ (a copy
of an open disk) is connected. This means that the monodromy over $\Delta_{i}$
is still the full ${\mathfrak{S}}_{3}$. The reader will then have no trouble
finding an arc $\gamma_{i+1}$ which departs along a ray in the sector spanned
by the rays of departure of $\gamma_{i}$ and $\gamma_{1}$ and which stays in
$\\{0\\}\cup\Delta$ and whose associated monodromy is the prescribed
transposition.

But if $i\leq 9$, extra care is needed to ensure that the preimage in
$\Sigma^{\prime}$ over
$\Delta_{i+1}:=S\smallsetminus\\{\gamma_{1}\cup\dots\cup\gamma_{i+1}\\}$ is
still connected. That this indeed possible follows from the fact that the
preimage over $\Delta_{i}$ is connected and has at least three points of
simple ramification.

On the other hand, if $i=10$ we are done, for then the value of $\rho$ on
$c_{0}:=c_{1}^{-1}\cdots c_{11}^{-1}$ will be
$\big{(}(23)(12)\big{)}^{5}(23)=(12)$. This completes the proof. ∎

Fix a degree 3 covering $\Sigma^{\prime}\to S$ as above and denote by $r\in
R(S,D)$ the associated element and denote by $\Gamma^{\prime}\subset\Gamma$
its stabilizer. It follows from Lemma 2.2 that $\Gamma^{\prime}$ is a subgroup
$\Gamma$ of index $(3^{9}-1)/2$. By construction, the isomorphism type of a
connected smooth degree $3$ covering of a copy of ${\mathbb{P}}^{1}$ with
reduced discriminant of degree $12$ determines a $\Gamma^{\prime}$-orbit in
${\mathbb{B}}^{\circ}$. So if ${\mathcal{M}}_{4}(g^{1}_{3})^{\circ}$ stands
for the open subset of ${\mathcal{M}}_{4}(g^{1}_{3})$ that parametrizes pairs
$(X,P)$ for which $P$ has precisely $12$ nonreduced members (in other words,
for which $X$ is nonhyperelliptic and for which $X\to P$ has reduced
discrimininant), then we find:

######  Corollary 2.3.

The open subset ${\mathcal{M}}_{4}(g^{1}_{3})^{\circ}$ is naturally isomorphic
to the ball quotient ${\mathbb{B}}_{\Gamma^{\prime}}^{\circ}$.

In order to get all of ${\mathcal{M}}_{4}(g^{1}_{3})$ we must allow the
discriminant divisor $D$ to become nonreduced. What we must exclude is that
two points of $D$ lie in a disk $\Delta\subset P$ (which contains no other
points of $D$) over whose boundary the covering $X\to P$ is trivial, for if we
let these points then coalesce inside $\Delta$ to a point $z\in\Delta$, this
will create an ordinary double point of the covering. There is however one
case that we should not throw out: if the covering has in addition a section
over $P\smallsetminus\Delta$, then the degeneration will be the union of a
smooth double covering $X^{\prime}\to P$ (whose discriminant is
$D\cap(P\smallsetminus\Delta)$) and a component $\tilde{P}$ which maps
isomorphically onto $P$ and meets $X^{\prime}$ transversally over $z$. So this
amounts to specifying a hyperelliptic curve of genus $4$ and a point on that
curve; in other words, and element of ${\mathcal{H}}_{4}(g^{1}_{3})$. This
happens for example when
$\rho(c_{0})=\rho(c_{1})\not=\rho(c_{3})=\dots=\rho(c_{11})$ and we take for
$\Delta$ a thin regular neighborhood $\gamma_{0}\cup\gamma_{1}$.

We can state this in terms of the $\Gamma$-action on $R(S,D)$. The confluence
of two points along an arc $\delta$ as above represents a triflection in
$\Gamma$. It may or may not act trivially on $R(S,D)$. If the action is
nontrivial, then this confluence creates a point of total ramification and we
still have defined a genus $4$ covering of $S$. Otherwise this creates an
ordinary double point. These three cases translate into saying that the
stabilizer $\Gamma^{\prime}$ has three orbits in the collection of mirrors.
Equivalently, the preimage of the confluence divisor ${\mathcal{D}}$ in
${\mathbb{B}}_{\Gamma}$ under the natural map

${\mathbb{B}}_{\Gamma^{\prime}}\to{\mathbb{B}}_{\Gamma}$

has three irreducible components ${\mathcal{D}}_{{\rm rm}},{\mathcal{D}}_{{\rm
sg}},{\mathcal{D}}_{{\mathcal{H}}}$ whose generic points are characterized
topologically by the degree 3 covering acquiring respectively (_rm_) a point
of total ramification, (_sg_) an ordinary double point whose complement is
connected and (${\mathcal{H}}$) an ordinary double point which disconnects. We
can can now improve upon Corollary 2.3 as follows.

######  Theorem 2.4.

The above construction gives an identification

${\mathcal{M}}_{4}(g^{1}_{3})\cong{\mathbb{B}}_{\Gamma^{\prime}}\smallsetminus{\mathcal{D}}_{{\rm
sg}},$

thus endowing the left hand side with an (incomplete) complex-hyperbolic
metric. This identifies the preimage of ${\mathcal{D}}_{{\mathcal{H}}}$ with
the universal hyperelliptic curve of genus $4$.

For a different approach to this theorem (that ultimately yields a more
precise statement), we first need to discuss level 3 structures for the
Eisenstein lattice $L$.

### The approach via a level structure

We begin with the observation that ${\mathcal{E}}/\theta{\mathcal{E}}$ is a
${\mathbb{F}}_{3}$-vector space of dimension one on which
$\tau\in\boldsymbol{\mu}_{6}$ acts as minus the identity (because
$\tau+1=\tau^{-1}\theta$). It follows that ${\mathbb{F}}_{3}\otimes L=L/\theta
L$ is a ${\mathbb{F}}_{3}$-vector space of rank 10 on which the skew-hermitian
form $h^{\prime}$ induces a symplectic form
$h^{\prime}_{{\mathbb{F}}_{3}}:={\mathbb{F}}_{3}\otimes h^{\prime}$. The
following lemma generalizes an argument of Allcock (appearing in the proof of
Thm. 5.2 of [2]).

######  Lemma 2.5.

The symplectic form $h^{\prime}_{{\mathbb{F}}_{3}}$ is nondegenerate on
${\mathbb{F}}_{3}\otimes L$ (so that it is isomorphic to
${\mathbb{F}}_{3}^{10}$ equipped with its standard symplectic form) and
$\Gamma$ acts on ${\mathbb{F}}_{3}\otimes L$ with image its full symplectic
group (a copy of $\operatorname{Sp}_{10}({\mathbb{F}}_{3})$). The image of a
triflection is a symplectic transvection.

###### Proof.

The basis $a_{\delta_{1}},\dots,a_{\delta_{10}}$ of $L$ maps to a basis
$\alpha_{1},\dots,\alpha_{10}$ of ${\mathbb{F}}_{3}\otimes L$ with the
property that $h^{\prime}_{{\mathbb{F}}_{3}}(\alpha_{i},\alpha_{j})$ is $\pm
1$ if $j=\pm i$ and is zero otherwise. The triflection defined by
$a_{\delta_{i}}$ becomes $x\mapsto x+\tau
h^{\prime}_{{\mathbb{F}}_{3}}(x,\alpha_{i})\alpha_{i}=x-h^{\prime}_{{\mathbb{F}}_{3}}(x,\alpha_{i})\alpha_{i}$,
which is indeed a symplectic transvection. Let $V$ be the free abelian group
with basis $\tilde{\alpha}_{1},\dots,\tilde{\alpha}_{10}$ endowed with the
symplectic form that assigns to the pair
$(\tilde{\alpha}_{i},\tilde{\alpha}_{j})$ the value $\pm 1$ if $j=\pm i$ and
is zero otherwise, so that its reduction mod $3$ gives
$h^{\prime}_{{\mathbb{F}}_{3}}$. This lattice is well-known in singularity
theory (it is the one for the Milnor fiber of a plane curve singularity of
type $A_{10}$, defined by $z_{1}^{2}+z_{2}^{11}=0$ as given on a standard
basis of vanishing cycles). It was proved by A’Campo [1] that the subgroup of
$\operatorname{Sp}(V)$ generated by these symplectic transvections contains
the principal congruence level 2 subgroup. This implies that it maps onto the
full symplectic group of its mod $3$ reduction. ∎

The covering group $\boldsymbol{\mu}_{6}$ acts on the 10-dimensional
${\mathbb{F}}_{3}$-vector space ${\mathbb{F}}_{3}\otimes L$ by scalars and so
the 9-dimensional projective space ${\mathbb{P}}({\mathbb{F}}_{3}\otimes L)$
only depends on the pair $(S,D)$.

We regard $L$ as a primitive subgroup of $H^{1}(\Sigma)$, so that a one
dimensional subspace $\ell\subset{\mathbb{F}}_{3}\otimes L$ defines (by
restriction) a surjection
$H_{1}(\Sigma)\cong\operatorname{Hom}(H^{1}(\Sigma),{\mathbb{Z}})\to\operatorname{Hom}(\ell,{\mathbb{F}}_{3})=\ell^{\vee}$.
This surjection is $\boldsymbol{\mu}_{6}$-equivariant if we let
$\boldsymbol{\mu}_{6}$ act on $\ell^{\vee}$ via the obvious character
$\boldsymbol{\mu}_{6}\twoheadrightarrow\boldsymbol{\mu}_{2}=\\{\pm 1\\}$. This
also yields an unramified $\ell^{\vee}$-covering $\Sigma_{\ell}\to\Sigma$.

######  Lemma 2.6.

The $\boldsymbol{\mu}_{6}$-action on $\Sigma$ lifts to $\Sigma_{\ell}$. If
$\Sigma^{\prime}_{\ell}$ denotes the orbit space of this lift, then
$\Sigma^{\prime}_{\ell}$ is an oriented surface of genus 4 such that the
obvious map $\Sigma^{\prime}_{\ell}\to S$ is orientation preserving of degree
$3$ with discriminant $D$ (in other words, defines an element of $R(S,D)$).
Moreover, this degree $3$ cover is is independent of the lift of the
$\boldsymbol{\mu}_{6}$-action, in the sense that for two any lifts, the two
degree 3 covers of $S$ are isomorphic by a unique isomorphism.

###### Proof.

We first show that the $\boldsymbol{\mu}_{6}$-action on $\Sigma$ lifts to
$\Sigma_{\ell}$. Recall that if we choose a base point $x\in\Sigma$, then
$\Sigma_{\ell}$ can be obtained as a quotient of the space of paths in
$\Sigma$ that begin at $x$: two such paths $\alpha,\beta$ define the same
point of $\Sigma_{\ell}$ if and only if $\alpha$ and $\beta$ have the same
endpoint and the class of the loop $\beta^{-1}*\alpha$ in $\pi_{1}(\Sigma,x)$
has zero image in $\ell^{\vee}$. If we choose $x$ to be fixed point of the
$\boldsymbol{\mu}_{6}$-action, then $\boldsymbol{\mu}_{6}$ will act on the
space of such of paths. The homomorphism $\pi_{1}(\Sigma,x)\to\ell^{\vee}$ is
$\boldsymbol{\mu}_{6}$-equivariant, in particular, the
$\boldsymbol{\mu}_{6}$-action on $\pi_{1}(\Sigma,x)$ preserves its kernel. So
$\boldsymbol{\mu}_{6}$ preserves the above equivalence relation, meaning that
we have lifted the $\boldsymbol{\mu}_{6}$-action on $\Sigma$ to
$\Sigma_{\ell}$.

This lift is not canonical, but two such lifts differ by an
$\ell^{\vee}$-covering transformation. Hence the isomorphism type of the
$\boldsymbol{\mu}_{6}$-orbit space $\Sigma_{\ell}^{\prime}$ of
$\Sigma_{\ell}$, now viewed as a degree $3$ cover of $S$, is independent of
the lift. Note that $\Sigma_{\ell}^{\prime}$ is a connected oriented surface,
not ramified over $S\smallsetminus D$. To see what happens over $z\in D$, we
observe that the fiber $\Sigma_{\ell}(z)$ of $\Sigma_{\ell}\to S$ over $z$ has
the structure of an affine line over $\ell^{\vee}$ on which
$\boldsymbol{\mu}_{6}$ also acts and in such a manner that the induced action
on $\ell^{\vee}$ is minus the identity. This means that $\boldsymbol{\mu}_{6}$
acts as transposition on $\Sigma_{\ell}(z)$: it fixes a point and exchanges
the remaining two. It follows that $\Sigma^{\prime}_{\ell}\to S$ has no
ramification at the image of the fixed point of the lifted
$\boldsymbol{\mu}_{6}$-action, whereas the common image of the remaining pair
is a point of simple ramification. It follows that $D$ is the discriminant of
$\Sigma^{\prime}_{\ell}\to S$ and that $\Sigma^{\prime}_{\ell}$ has genus 4.
This also shows that $\Sigma^{\prime}_{\ell}$ has no covering transformations.
So $\Sigma^{\prime}_{\ell}$ is unique up to unique isomorphism. ∎

######  Theorem 2.7.

The map which assigns to $[\ell]\in{\mathbb{P}}({\mathbb{F}}_{3}\otimes L)$
the cover $\Sigma^{\prime}_{\ell}\to S$ defines a
$\operatorname{Mod}(S,D)$-equivariant bijection

$[\ell]\in{{\mathbb{P}}}({\mathbb{F}}_{3}\otimes L)\mapsto
r(\Sigma^{\prime}_{\ell}/S)\in R(S,D)$

In particular, $R(S,D)$ thus acquires the structure of the projective space of
a 10-dimensional symplectic space over ${\mathbb{F}}_{3}$ on which $\Gamma$
acts through a symplectic group modulo its center.

###### Proof.

The $\operatorname{Mod}(S,D)$-equivariance of this assignment is clear from
the construction. The theorem then follows from the fact that the two sets
have the same cardinality and that $\operatorname{Mod}(S,D)$ acts transitively
on the target. The rest follows from Lemma 2.5. ∎

### The confluence divisors

Theorem 2.7 also explains why the preimage of
${\mathcal{D}}\subset{\mathbb{B}}_{\Gamma}$ in
${\mathbb{B}}_{\Gamma^{\prime}}$ has three irreducible components, to wit, the
confluence divisors ${\mathcal{D}}_{\mathcal{H}}$, ${\mathcal{D}}_{{\rm rm}}$
and ${\mathcal{D}}_{{\rm sg}}$. Let us first observe that a mirror is given by
the ${\mathcal{E}}$-span of $(-3)$-vector of $L$ and that the mod
$\theta$-reduction of the latter determines a line in ${\mathbb{F}}_{3}\otimes
L$ and hence a point in ${\mathbb{P}}({\mathbb{F}}_{3}\otimes L)$.

######  Proposition 2.8.

Let $\ell$ be a line in ${\mathbb{F}}_{3}\otimes L$. Then the three orbits of
$\Gamma_{\ell}$ in ${\mathbb{P}}({\mathbb{F}}_{3}\otimes L)$ are $[\ell]$,
${\mathbb{P}}(\ell^{\perp})\smallsetminus\\{[\ell]\\}$ and
${\mathbb{P}}({\mathbb{F}}_{3}\otimes
L)\smallsetminus{\mathbb{P}}(\ell^{\perp})$ and represent respectively the
irreducible components ${\mathcal{D}}_{\mathcal{H}}$, ${\mathcal{D}}_{{\rm
rm}}$ and ${\mathcal{D}}_{{\rm sg}}$ of the preimage of the confluence divisor
${\mathcal{D}}$ of ${\mathbb{B}}_{\Gamma}$ in ${\mathbb{B}}_{\Gamma_{\ell}}$.

###### Proof.

By Lemma 2.2 we can choose a standard set of arcs
$\\{\gamma_{i}\\}_{i\in{\mathbb{Z}}/12}$ connecting the base point $o$ with
the points of $D$ such that that the associated collection
$\\{c_{i}\in\pi_{1}(S\smallsetminus D,o)\\}_{i\in{\mathbb{Z}}/12}$ is a
standard set of generators of $\pi_{1}(S\smallsetminus D,o)$ (so this is just
as in the proof of Lemma 2.2) for which the covering $\Sigma^{\prime}\to S$
has the property that
$\rho(c_{0})=\rho(c_{1})\not=\rho(c_{2})=\cdots=\rho(c_{11})$ (so this is
different than in the proof of Lemma 2.2). The juxtaposition of $\gamma_{i+1}$
and the inverse of $\gamma_{i}$ is isotopic to an arc $\delta_{i}$ as in
Section 1, and as explained there, determines a $(-3)$-vector
$a_{\delta_{i}}\in L$ up to a $\boldsymbol{\mu}_{6}$-multiple and hence a line
$\ell_{i}$ in ${\mathbb{F}}_{3}\otimes L$.

Let $\ell$ be the line in ${\mathbb{F}}_{3}\otimes L$ for which
$\Sigma^{\prime}\cong\Sigma^{\prime}_{\ell}$. We claim that $\ell=\ell_{0}$.
Indeed, the covering $\Sigma^{\prime}_{\ell}\to S$ is disconnected over
$S\smallsetminus\delta_{0}$. It is then not hard to see that
$\Sigma_{\ell}\to\Sigma$ must be a trivial $\ell^{\vee}$-covering over the
preimage of $\delta_{0}$. This implies that $\ell$ has its support over
$\delta_{0}$, so that $\ell=\ell_{0}$.

It remains to observe that for $i=0,1,2$, the mirror
${\mathbb{B}}^{T_{\gamma_{i}}}$ parametrizes the points where the covering
$\Sigma^{\prime}_{\ell}\to P$, acquires, after shrinking $\delta_{i}$,
respectively a separating double point, a point of ramification of order $2$
and a nonseparating double point. So $\ell_{0}$, $\ell_{1}$, $\ell_{2}$
represent respectively ${\mathcal{D}}_{\mathcal{H}}$, ${\mathcal{D}}_{{\rm
rm}}$ and ${\mathcal{D}}_{{\rm sg}}$. ∎

### The strict transform of the thetanull locus

The goal of this section is to show that the strict transform
${\mathcal{M}}_{4}(g^{1}_{3})^{0}$ of ${\mathcal{M}}_{4}^{0}$ in
${\mathcal{M}}_{4}(g^{1}_{3})$ is also locally a ball quotient in
${\mathbb{B}}_{\Gamma^{\prime}}$. This was actually established in the paper
with Heckman [4] and the goal of this section is to show how. The generic
point of that locus parametrizes the canonical genus $4$ curves $X$ that lie
on a quadric cone $Q^{0}$ in a 3-dimensional complex projective space and are
transversally cut out on $Q^{0}$ by a cubic hypersurface (so not containing
the vertex). The base of the cone $Q^{0}$ (in other words, the set of _rays_
in $Q^{0}$, lines on $Q^{0}$ that pass through its vertex) is a conic. We
denote it by $P$, since the projection $X\to P$ gives us the unique
$g^{1}_{3}$ on $X$. We assume for now that the discriminant divisor $D$ of
this projection is reduced.

The vertex of $Q^{0}$ is the unique singular point of $Q^{0}$ and is resolved
by a single blowup: $\hat{Q}^{0}\to Q^{0}$. Its exceptional set, which we can
identify with $P$, has self-intersection $-2$. It defines a section of the
evident morphism $\hat{Q}^{0}\to P$ whose fibers are the rays of $Q^{0}$. In
other words, $\hat{Q}^{0}$ is a Hirzebruch surface whose exceptional section
is $P$. Its Picard group is freely generated by $P$ and the class $f$ of
fiber. The curve $X$ on $Q^{0}$ has class $3(P+2f)$ and hence the class of
$X+P$ is $3(P+2f)+P=4P+6f$. Since this is $2$-divisible, it determines admits
a double covering $E_{X}\to\hat{Q}^{0}$ with ramification divisor $X+P$
(111Kondō [5] considers instead the $\boldsymbol{\mu}_{3}$-cover of
$\hat{Q}^{0}$ ramified along $X$, which produces a K3-surface with
$\boldsymbol{\mu}_{3}$-action.). The composite with the projection
$\hat{Q}^{0}\to P$ defines an elliptic fibration for which the preimage of
$P\subset\hat{Q}^{0}$ in $E_{X}$ defines the zero section (self-intersection
number $-1$) for whose natural involution (inversion) is the one defined by
the double cover $E_{X}\to\hat{Q}^{0}$. Note that $D$ now also appears as the
discriminant of the fibration $E_{X}\to P$. Indeed, all fibers are smooth,
except those over $D$: over each point of $D$ we find a nodal curve (a Kodaira
fiber of type $I_{1}$) whose double point lies over the singular point of the
projection $X\to P$.

By taking the $j$-invariant of the fiber, we obtain a morphism

$j:P\to\operatorname{PSL}_{2}({\mathbb{Z}})\backslash{\mathbb{H}}^{*}=\big{(}\operatorname{PSL}_{2}({\mathbb{Z}})\backslash{\mathbb{H}}\big{)}\cup\\{\infty\\}=\overline{\mathcal{M}}_{1,1},$

where ${\mathbb{H}}^{*}:={\mathbb{H}}\cup{\mathbb{P}}^{1}({\mathbb{Q}})$
endowed with the horocyclic topology. This morphism has degree $12$ and
$D=j^{*}(\infty)$. In [4] it was observed that the $D$ that thus appear can be
characterized in ${\mathbb{B}}_{\Gamma}$ as hyperball quotient as follows. The
abelianization of $\operatorname{PSL}_{2}({\mathbb{Z}})$ is naturally
isomorphic with $\boldsymbol{\mu}_{6}$ and hence defines a ramified
$\boldsymbol{\mu}_{6}$-cover
$\operatorname{PSL}_{2}({\mathbb{Z}})^{\prime}\backslash{\mathbb{H}}\to\operatorname{PSL}_{2}({\mathbb{Z}})\backslash{\mathbb{H}}$.
This morphism extends across the cusp $\infty$ with total ramification, so
that we get a $\boldsymbol{\mu}_{6}$-cover

$\overline{\mathcal{M}}{}^{\prime}_{1,1}:=\operatorname{PSL}_{2}({\mathbb{Z}})^{\prime}\backslash{\mathbb{H}}^{*}\to\operatorname{PSL}_{2}({\mathbb{Z}})\backslash{\mathbb{H}}^{*}=\overline{\mathcal{M}}_{1,1}.$

The left hand side can be identified with the (Eisenstein) elliptic curve of
genus zero ${\mathbb{C}}/{\mathcal{E}}$ whose origin is the unique point over
$\infty$ and on which $\boldsymbol{\mu}_{6}$ acts in the standard manner. The
differential on ${\mathbb{C}}/{\mathcal{E}}$ defined by $dz$ transforms under
the $\boldsymbol{\mu}_{6}$-action via the character $\chi^{-1}$ and hence
$d\overline{z}$ transforms with the character $\chi$. So
$H^{1}(\overline{\mathcal{M}}{}^{\prime}_{1,1};{\mathbb{C}})_{\chi}$ is of
Hodge type $(0,1)$.

The $\boldsymbol{\mu}_{6}$-cover $C\to P$ with discriminant $D$ is then the
pull-back of the above $\boldsymbol{\mu}_{6}$-cover under the $j$-morphism:

$\begin{CD}C@>{j^{\prime}}>{}>\overline{\mathcal{M}}{}^{\prime}_{1,1}\\\
@V{\boldsymbol{\mu}_{6}}V{}V@V{\boldsymbol{\mu}_{6}}V{}V\\\
P@>{j}>{}>\overline{\mathcal{M}}_{1,1}\end{CD}$

Since $j$ is of degree $12$, so is $j^{\prime}$.

######  Lemma 2.9.

The map $j^{\prime}{}^{*}$ embeds
$H^{1}(\overline{\mathcal{M}}{}^{\prime}_{1,1})$ in $H^{1}(C)^{\circ}$ as a
primitive rank one ${\mathcal{E}}$-submodule that is generated by a vector
$\varepsilon$ with $h(\varepsilon,\varepsilon)=-6$.

###### Proof.

This is actually proved in [4], but let us give here a simpler argument. First
note that since
$H^{1}(\overline{\mathcal{M}}{}^{\prime}_{1,1})=H^{1}(\overline{\mathcal{M}}{}^{\prime}_{1,1})^{\circ}$,
$j^{\prime}{}^{*}$ takes values in $H^{1}(C)^{\circ}$. If $a\in
H^{1}(\overline{\mathcal{M}}{}^{\prime}_{1,1})$ is primitive, then $a\cdot\tau
a=1$ and hence if we put $\varepsilon:=j^{\prime}{}^{*}a$, then

$h(\varepsilon,\varepsilon)=-\tfrac{1}{2}(\varepsilon,\tau\varepsilon)=-\tfrac{1}{2}.12(a\cdot\tau
a)=-6,$

where we used that $j^{\prime}$ is of degree $12$. This also implies that
$\varepsilon$ is primitive: if it is divisible by $\lambda\in{\mathcal{E}}$:
$\varepsilon=\lambda\varepsilon^{\prime}$ with $\varepsilon^{\prime}\in
H^{1}(C)^{\circ}$, then
$\lambda\overline{\lambda}h(\varepsilon^{\prime},\varepsilon^{\prime})=-6$.
But $\lambda\overline{\lambda}$ is a positive integer that cannot take the
value $2$ and
$h(\varepsilon^{\prime},\varepsilon^{\prime})\in{\mathcal{E}}\theta\cap{\mathbb{Q}}=3{\mathbb{Z}}$.
This implies that $\lambda\overline{\lambda}=1$, meaning that $\lambda$ is a
unit. ∎

Assertions A4 and A6 of [4] state that any $(-6)$-vector in $L$ can be written
as the sum of two $(-3)$-vectors with inner product $\theta$ and that the
$(-6)$-vectors make up a single $\Gamma$-orbit. In particular, the hyperballs
in ${\mathbb{B}}$ defined by such vectors define an irreducible totally
geodesic divisor ${\mathcal{D}}^{0}$ in ${\mathbb{B}}_{\Gamma}$. We can
therefore complete Theorem 2.4 as follows (see also [4], Thm. 8.2):

######  Theorem 2.10.

The isomorphism in Theorem 2.4 takes the thetanull locus
${\mathcal{M}}_{4}(g^{1}_{3})^{0}$ onto
${\mathcal{D}}^{0}\smallsetminus{\mathcal{D}}_{\rm sg}$. In particular,
${\mathcal{M}}_{4}(g^{1}_{3})^{0}$ is a totally geodesic hypersurface in
${\mathcal{M}}_{4}(g^{1}_{3})$.

######  Remark 2.11.

One would expect a stronger assertion, namely that the involution of
${\mathcal{M}}_{4}(g^{1}_{3})$ that assigns to $(X,P)$ the residual pair
$(X,P^{\prime})$ lifts to a reflection in $\Gamma$ in a $(-6)$-vector
$\varepsilon$, in other words, is given by $x\in L\mapsto
x+\frac{1}{3}h(x,\varepsilon)\varepsilon$. But such a reflection will not
preserve $L$, as we must then have $h(x,\varepsilon)\in 3{\mathcal{E}}$.
Indeed, since the $(-6)$-vectors lie in a single
$\operatorname{Aut}(L)$-orbit, it suffices to check this for one such vector,
say $\varepsilon:=a_{\delta_{0}}+a_{\delta_{1}}$. But
$h(\varepsilon,a_{\delta_{2}})=\theta$, which in ${\mathcal{E}}$ is not
divisible by $3$. It is surprising (and still a bit mystifying to us) that
${\mathbb{B}}_{\Gamma^{\prime}}\smallsetminus{\mathcal{D}}_{\rm sg}$ comes
with an involution whose fixed point set is
${\mathcal{D}}^{0}\smallsetminus{\mathcal{D}}_{\rm sg}$, but that is _not_
obtained from a subgroup $\hat{\Gamma}^{\prime}\subset\Gamma$ with
$[\hat{\Gamma}^{\prime}:\Gamma^{\prime}]=2$.

## References

  * [1] N. A’Campo: _Tresses, monodromie et le groupe symplectique,_ Comment. Math. Helv. 54 (1979), 318–327.
  * [2] D. Allcock: The Leech lattice and complex hyperbolic reflections, Invent. Math. 140 (2000), 283–301.
  * [3] P. Deligne, G. D. Mostow: Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89.
  * [4] G. Heckman, E. Looijenga: The moduli space of rational elliptic surfaces, Algebraic geometry 2000, Azumino (Hotaka), 185–248, Adv. Stud. Pure Math., 36, Math. Soc. Japan, Tokyo, 2002
  * [5] S. Kondō: The moduli space of curves of genus 4 and Deligne-Mostow’s complex reflection groups, Algebraic geometry 2000, Azumino (Hotaka), 383–400, Adv. Stud. Pure Math., 36, Math. Soc. Japan, Tokyo, 2002
  * [6] G. D. Mostow: Generalized Picard lattices arising from half-integral conditions, Inst. Hautes Études Sci. Publ. Math. 63 (1986, 91–106.


# Fermions on wobbling kinks: normal versus quasinormal modes

João G. F. Campos
Departamento de Física, Universidade Federal de Pernambuco,
Av. Prof. Moraes Rego, 1235, Recife - PE - 50670-901, Brazil
<EMAIL_ADDRESS>Azadeh Mohammadi
Departamento de Física, Universidade Federal de Pernambuco,
Av. Prof. Moraes Rego, 1235, Recife - PE - 50670-901, Brazil
<EMAIL_ADDRESS>
###### Abstract

The system consisting of a fermion in the background of a wobbling kink is
studied in this paper. To investigate the impact of the wobbling on the
fermion-kink interaction, we employ the time-dependent perturbation theory
formalism in quantum mechanics. To do so, we compute the transition
probabilities between states given in terms of the Bogoliubov coefficients. We
derive Fermi’s golden rule for the model, which allows the transition to the
continuum at a constant rate if the fermion-kink coupling constant is smaller
than the wobbling frequency. Moreover, we study the system replacing the shape
mode with a quasinormal mode. In this case, the transition rate to continuum
decays in time due to the leakage of the mode, and the final transition
probability decreases sharply for large coupling constants in a way that is
analogous to Fermi’s golden rule. Throughout the paper, we compare the
perturbative results with numerical simulations and show that they are in good
agreement.

##  1 Introduction

Solitons and solitary waves are important solutions of field theories that are
topologically stable, propagate without losing the shape, and have localized
energy density [1, 2, 3]. These solutions may couple to bosons and fermions
with fascinating properties such as fractional fermionic charges [4, 5]. In
particular, kinks are solutions of scalar field theories when a discrete
symmetry is broken and degenerate vacua appear. The prototypical models of
kinks are the sine-Gordon, an integrable model, and the $\phi^{4}$ model,
which is non-integrable. The system composed of a fermion interacting with
background $\phi^{4}$ kink is analytically solvable and exhibits bound states
as well as scattering states [6, 7, 8]. The properties of fermion-kink
systems, such as the fermion energy spectrum and Casimir energy, have been
studied for many different models [9, 10, 11, 12, 13, 14]. In these systems,
it is also possible to consider the back-reaction of the fermion on the
soliton, which modifies the kink profile, especially for strong coupling
constants [15, 16, 17, 18]. This problem becomes even more fascinating when
one considers the fermion spectrum in the presence of a static kink-antikink
configuration. It was shown that the spectrum approaches two copies of the one
of a single kink in the limit of large separation for both the $\phi^{4}$
model [6] and the sine-Gordon [19], as expected. A more interesting problem is
the time evolution of the fermion in the presence of a scalar field that is
not static anymore and consists of a kink-antikink collision [20, 21, 22]. In
this case, the fermions may be transferred from one soliton to the other [20,
21] with decreasing probability as the system approaches the Bogomol’nyi-
Prasad-Sommerfield limit, where the scalar field interaction vanishes [22]. An
exciting application to this problem appears in cosmology, where domain walls
can describe branes that live in higher-dimensional universes leading to
fermions localization in the extra dimensions [23, 24, 25]. In this picture,
we could live in a domain wall [26], and wall collisions could cause the big-
bang [27].

The kink has a shape mode in $\phi^{4}$ and many other scalar field models and
can wobble. One of the earliest works to report this property was [28], which
is now well understood in terms of perturbation theory [29, 30, 31]. In one of
the earliest works of kink-antikink collisions, the authors considered an
effective model to describe kink-antikink interactions with two collective
coordinates, the position and the vibrational amplitude [32]. After that, in a
triplet of seminal papers [33, 34, 35], the authors realized that the
vibrational mode existence was essential to explain the resonance phenomenon
where the kink and antikink collide multiple times before separating since the
translational mode exchanges energy with the vibrational mode at each bounce.
The structure of the resonance was shown to have a fractal structure [36] and
was found in several models [37, 38, 39, 40, 41, 42, 43, 44]. Surprisingly,
the $\phi^{6}$, where the isolated kinks do not have a vibrational mode, also
exhibits resonance windows. In this case, it was shown that the energy is
stored in the vibrational mode of the compound kink-antikink configuration
[45]. Interestingly, the resonance exchange mechanism only received a
satisfactorily quantitative confirmation very recently [46, 47].

To understand the role of the shape mode at each bounce, some authors
considered collisions between initially wobbling kinks and showed that the
initial vibrational energy could be converted into translational energy [48,
49]. It is clear that the kink’s vibrational mode is essential and that, even
if the kink is not vibrating initially, this mode will get excited after a
kink-antikink collision. Therefore, if a fermion interacts with a kink, it is
important to ask how this interaction is modified when the kink starts
wobbling.

An interesting modification of the usual scalar field models occurs when the
potential is constructed to turn the vibrational mode into a quasinormal mode.
In fact, many models already possess a tower of QNM for the stability
equations around a kink solution. The quasinormal modes of the Schrödinger and
Klein-Gordon equations can be found analytically for a few potentials, as
described in [50]. See also [51] for a similar analysis. However, we are
interested in the shape mode solution of the stability equation, which is the
most relevant in the kink context. This mode can be turned into a quasinormal
mode if one modifies the scalar potential, which was shown to suppress the
resonance windows, as the decay rate of the quasinormal mode is increased [52,
53].

In this work, we consider a fermion in the presence of wobbling kinks with
both normal and quasinormal modes. In our case, the wobbling kink solution is
considered as a background, that is, without a back-reaction. This is a common
simplifying approximation that allows analytical treatment and is almost
exact, in the limit where the coupling between the fermion and the kink is
small [6, 7]. We will show that the transition probabilities between the
fermion bound and continuum states can be computed by utilizing the time-
dependent perturbation theory formalism in quantum mechanics. We find that the
fermions radiate to infinity if the coupling constant is small. This effect is
quantified by Fermi’s Golden Rule and can be reduced if the vibrational mode
is turned into a quasinormal mode. In the next section, we discuss the
framework we will use to study our system. In section 3 we apply our framework
to the $\phi^{4}$ model and in section 4 to a toy model that exhibits
quasinormal besides normal modes. Finally, in section 5, we summarize our
conclusions.

##  2 Model

Consider the following model representing a scalar field interacting with a
fermion field in $(1+1)$ dimensions

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+V(\phi)+i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-g\phi\bar{\psi}\psi.$
(1)

We ignore the back-reaction of the fermion field on the scalar field, which
simplifies the system considerably and makes it possible to perform some
analytical analysis. This is only a good approximation to the full equations
of motion when $g\ll 1$. However, we will also extrapolate our results for
larger values of $g$. Besides that, it creates the possibility to explore the
effect of fermion excited states, which only appear when $g\geq 1$.

The potential is chosen such that the scalar field has a kink solution
$\phi_{k}$. In this case, the fermion equation of motion in the presence of
the kink is given by

$i\gamma^{\mu}\partial_{\mu}\psi-g\phi_{k}\psi=0.$ (2)

Taking the representation $\gamma^{0}=\sigma_{1}$, $\gamma^{1}=i\sigma_{3}$
for the gamma matrices, where $\sigma_{i}$ are Pauli matrices and multiplying
the above equation by $\gamma^{0}$, it becomes

$i\partial_{t}\psi=H_{0}\psi,$ (3)

with the Hamiltonian operator in the form

$H_{0}=-i\sigma_{2}\partial_{x}+g\phi_{k}\sigma_{1}.$ (4)

This operator can be diagonalized by solving the Schrödinger-like equations

$-\partial_{x}^{2}\psi_{n,\pm}+V_{\pm}\psi_{n,\pm}=E_{n}^{2}\psi_{n,\pm}.$ (5)

with the effective potentials given by
$V_{\pm}=g(g\phi_{k}^{2}\mp\partial_{x}\phi_{k})$ and the eigenfunctions by
$\psi_{n}=(\psi_{n,+},\psi_{n,-})^{T}$. Because they are partner Hamiltonians
[54] it is guaranteed that both have the same eigenvalues, except for the zero
mode ($E=0$), and that the eigenfunctions obey the first order equation

$H_{0}\psi_{n}=E_{n}\psi_{n}.$ (6)

Now, if we perturb the kink with the shape mode, the scalar field becomes
$\phi=\phi_{k}+A\eta_{S}\cos(\omega_{S}t)$ and the Dirac equation gives

$i\partial_{t}\psi=(H_{0}+AH_{1}(t))\psi,$ (7)

where the perturbation is given by
$H_{1}(t)=g\eta_{S}\cos(\omega_{S}t)\sigma_{1}$ and $A$ is the amplitude of
the perturbation. Then, assuming that the perturbation term is small compared
with $H_{0}$, one can write the fermion field as a linear combination of
$H_{0}$ eigenfunctions, which in the Schrödinger picture evolves as

$\psi=\sum_{n}c_{n}(t)e^{-iE_{n}t}\psi_{n}.$ (8)

We expand $c_{n}(t)$ in powers of $A$ in the following form

$c_{n}(t)=c^{(0)}_{n}(t)+Ac^{(1)}_{n}(t)+\cdots.$ (9)

Considering the initial condition $c_{n}(0)=\delta_{ni}$, first-order time-
dependent perturbation theory gives [55]

$c^{(1)}_{n}(t)=-i\int_{0}^{t}e^{i\omega_{ni}t^{\prime}}\langle
n|H_{1}(t^{\prime})|i\rangle dt^{\prime},$ (10)

where $\omega_{ni}=E_{n}-E_{i}$ and the matrix element of any operator
$\mathcal{O}$ is defined as usual

$\langle n|\mathcal{O}|i\rangle=\int\psi_{n}^{T}\mathcal{O}\psi_{i}dx.$ (11)

For a given $i$, the coefficients $c_{n}(t)$ are the Bogoliubov coefficients,
which quantify the probability amplitude of the transitions between states
[21]. Thus, the transition probability for $n\neq i$ is given by

$P_{i\to n}(t)=A^{2}|c_{n}^{(1)}(t)|^{2}.$ (12)

This result is valid as long as $A^{2}|c_{n}^{(1)}(t)|^{2}\ll 1$. We can
perform the time integration in eq. (10) to obtain

$c^{(1)}_{n}(t)=\frac{g}{2}\langle
n|\eta_{S}\sigma_{1}|i\rangle\left(\frac{1-e^{i(\omega_{ni}+\omega_{S})t}}{\omega_{ni}+\omega_{S}}+\frac{1-e^{i(\omega_{ni}-\omega_{S})t}}{\omega_{ni}-\omega_{S}}\right).$
(13)

For large $t$, one can compute the transition rate from the initial to the
final state. It is defined as the large $t$ limit of the ratio between the
transition probability $P_{i\to n}$ and time $t$. According to Fermi’s golden
rule, it is given by

$R_{i\to n}=A^{2}g^{2}\frac{\pi}{2}\left|\langle
n|\eta_{S}\sigma_{1}|i\rangle\right|^{2}\left[\delta\left(\omega_{ni}+\omega_{S}\right)+\delta\left(\omega_{ni}-\omega_{S}\right)\right],$
(14)

for $n\neq i$, which must be integrated over a set of final states.

##  3 $\phi^{4}$ model

Let us start with the perturbative analysis on the fermion field interacting
with the kink of the famous $\phi^{4}$ model where the system is analytically
solvable [6, 7]. We need to set $V(\phi)=\frac{1}{2}(\phi^{2}-1)^{2}$,
$\phi_{k}=\tanh(x)$, $\eta_{S}=\tanh(x)\text{sech}(x)$ and
$\omega_{S}=\sqrt{3}$. In this case, the effective fermionic potentials are in
the Pöschl-Teller form

$V_{\pm}=g^{2}-\frac{g(g\pm 1)}{\cosh^{2}(x)}.$ (15)

They have analytical solutions, which are listed in Appendix A. In the
following analysis, we will focus on the case where the initial fermion state
is the zero mode with $i=0$. Using the analytical expressions for the bound
and scattering states, we can compute the matrix elements $\langle
n|\eta_{S}\sigma_{1}|0\rangle$.

(a)

(b)

Figure 1: Matrix elements for the operator $\eta_{S}\sigma_{1}$ between the
eigenstates of $H_{0}$ with fermion interacting with the $\phi^{4}$ wobbling
kink. We fix $g=1.6$ in (b).

For the $\phi^{4}$ model, a fermion bound state appears whenever $g$ is an
integer. The first excited state appears when $g=1.0$, and the matrix element
between this state and the zero mode is given by

$\langle
1|\eta_{S}\sigma_{1}|0\rangle=\int\left(\psi_{1,-}\eta_{S}\psi_{0,+}+\psi_{1,+}\eta_{S}\psi_{0,-}\right)dx,$
(16)

for example. Because $\psi_{0,+}$ and $\psi_{1,-}$ are even and $\eta_{S}$ is
odd, the first term vanishes and, because $\psi_{0,-}=0$, the second term also
vanishes. In general, the integral vanishes by parity arguments whenever the
difference between the index of the two bound states is an odd number. In
contrast, the transition from the zero mode to the next excited state,
appearing at $g=2.0$, is allowed. In Fig. 1, some of the matrix elements
$\langle n|\eta_{S}\sigma_{1}|i\rangle$ as a function of the coupling between
the fermion and wobbling kink, $g$, are shown in the left panel. In the right
panel, we depict the transition from the zero mode and also the first excited
state to any continuum state111The continuum states are always denoted by the
letter $k$. $|k\rangle$, varying $k$. This shows the amount of fermion
radiating away for the fixed value of the coupling $g=1.6$ and, in general,
these transitions are allowed. We can also see that the radiation to the high
continuum state, i.e., large $k$, is suppressed as the energy gap between the
initial and final states becomes too large, making the transition impossible.
We observe the same behavior in the other models studied here.

The time evolution of the transition probability is shown in Fig. 2. In the
figure, we compare a numerical simulation of the full equations of motion (eq.
(3)) with the approximate result of first-order perturbation theory (eq.
(13)). The numerical methods used in the simulations are described in Appendix
B. In these calculations we take the analytical expression for the scalar
field $\phi=\phi_{k}+A\eta_{S}\cos(\omega_{S}t)$. We set $A=0.1$ because it is
of the same order of magnitude of typical wobbling amplitudes of a kink after
a kink-antikink collision [48]. As one can see, the result of the simulation
and the perturbative one match well.

(a) $g=0.8$

(b) $g=2.1$

Figure 2: Transition probabilities from the zero mode to continuum and bound
states for a perturbation with amplitude $A=0.1$. We compare simulation
results with first-order perturbation theory. The rate from the zero mode to
the continuum (eq. (17)) is also shown in (a) with the dotted line.

For $g=0.8$, we compute the transition probability from the zero mode to the
continuum. This is found semi-analytically by integrating eq. (13) over all
the continuum states. We see that the probability increases at a constant
rate, except for a small oscillation with frequency $\omega_{S}$, the kink
wobbling frequency. This increase is expected for large times whenever Fermi’s
golden rule allows the transition to the continuum. For a final state in the
continuum and the initial state in the zero mode, we have that
$\omega_{ki}=\pm\sqrt{g^{2}+k^{2}}$. Thus, the transition to some state in the
continuum is allowed by Fermi’s golden rule whenever $g<\omega_{S}=\sqrt{3}$.
More explicitly, we integrate eq. (14) over $k$ from $0$ to $\infty$ for the
scattering states given in Appendix A, $\psi_{k,L}$, with incident waves from
the left. Then, including a factor of $4$ to account for the scattering states
with incident waves from the right as well as the ones with negative energy,
it yields

$R_{i\to\text{cont.}}=2A^{2}g^{2}\pi\bigg{|}\bigg{\langle}k=\sqrt{\omega_{S}^{2}-g^{2}}\bigg{|}\eta_{S}\sigma_{1}\bigg{|}i\bigg{\rangle}\bigg{|}^{2}\frac{\omega_{S}}{\sqrt{\omega_{S}^{2}-g^{2}}},$
(17)

for $g<\omega_{S}$ and zero otherwise. The transition rate is shown in Fig.
2(a), where one can see that it gives the correct slope of the transition
probability curve. This is an interesting result because it allows one to
probe whether a kink is wobbling or at least to set an upper bound on the
wobbling amplitude by measuring the fermions escape rate from the kink. For
$g=2.1$, we see that the transition to the continuum is not allowed anymore by
Fermi’s golden rule. Therefore, the transition probability to the continuum
eventually stops increasing returns to zero, and this is repeated many times.
Furthermore, the fermion now has a second excited state, and the transition
probability to this state is finite and does not keep increasing for the same
reason. In Fig. 2(b) we plot the sum of the transition probability for both
second excited states with positive and negative energy. By symmetry, they
both have equal transition probabilities and, therefore, the sum is just twice
the transition probability to the second excited state with positive energy.

##  4 Toy model

Now we repeat the same procedure for a toy model, which allows kinks with
either a normal mode or a quasinormal mode [53]. After rescaling, the toy
model is described by the following potentials

$V(\phi)=\begin{cases}\frac{1}{2}(-\phi^{2}+A^{2}),&0\leq\phi<\phi_{1},\\\
\frac{\gamma}{2}(\phi-1)^{2},&\phi>\phi_{1},\end{cases}$ (18)

and

$V(\phi)=\begin{cases}\frac{1}{2}(-\phi^{2}+A^{2}),&0\leq\phi<\phi_{1},\\\
\frac{\gamma}{2}[(\phi-1+\delta)^{2}+B^{2}],&\phi_{1}<\phi<1-\epsilon,\\\
\frac{\gamma^{\prime}}{2}(\phi-1)^{2},&1-\epsilon<\phi<1+\epsilon,\\\
\frac{\gamma}{2}[(\phi-1-\delta)^{2}+B^{2}],&\phi>1+\epsilon,\end{cases}$ (19)

The latter potential allows for the appearance of quasinormal modes, in
contrast with the former one (18). All quantities in the definition of the
potentials are constants, except for $\phi$. The dependence on $\phi$ for
negative values can be found by assuming that the potentials are even
functions.

The potential (18) is constructed such that the linearized potential of the
stability equation has a square-well shape. Eq. (19) modifies this potential
to the square-well shape with two barriers at the sides, allowing the bound
states of the previous potential to tunnel and escape from the kink if
$\gamma^{\prime}$ is small enough. To ensure that the kinks behave correctly,
we imposed continuity of the potentials and their derivatives. In this case,
it is easy to show that, in the limit that $\epsilon$ goes to zero, eq. (19)
becomes eq. (18). The parameter $\epsilon$ encodes the information about the
energy leak when normal mode turns into the quasinormal mode, and as it
increases, the energy leakage increases. Another consequence of the continuity
relations is that we have only one free parameter in eq. (18), which we fix by
setting $\gamma=3$. This value is interesting because the kink has only a
single vibrational mode and, thus, behaves similarly to the $\phi^{4}$ model.
For eq. (19), the continuity relations imply that there are three free
parameters, which we fix at $\gamma=3.0$, $\gamma^{\prime}=1.0$ and
$\epsilon=0.05$. Choosing these values, we guarantee that the only shape mode
in the square-well stability potential turns into a quasinormal mode.

The kink solution for the normal mode case is given by

$\phi_{K}(x)=\begin{cases}A\sin x,&0<x<L,\\\
1-(1-\phi_{1})e^{-\sqrt{\gamma}(x-L)},&x>L,\end{cases}$ (20)

for positive $x$, where $L$ is the point where
$\phi_{1}=A\sin(\sqrt{\lambda}L)$. For the quasinormal mode case the kink is
given by

$\phi_{K}(x)=\begin{cases}A\sin x,&0<x<L,\\\
1-\delta+B\sinh(C+\sqrt{\gamma}(x-L)),&L<x<L^{\prime},\\\ 1-\epsilon
e^{-\sqrt{\gamma^{\prime}}(x-L^{\prime})},&x>L^{\prime},\end{cases}$ (21)

with the following definitions

$C\equiv\ln(B/D),$ (22)

and

$D\equiv\sqrt{B^{2}+(1-\delta-\phi_{1})^{2}}+1-\delta-\phi_{1}.$ (23)

Similarly, $L$ and $L^{\prime}$ are defined as the points where
$\phi_{K}=\phi_{1}$ and $\phi_{K}=1-\epsilon$, respectively. The fermion
spectra for both kinks, resulting from eq. (5), are shown in Fig. 3. They are
very similar, except that the bound states appear earlier in the quasinormal
mode case. This occurs because, in this case, the kink’s tail decays more
slowly, and the potentials $V_{\pm}$ become wider.

(a) $\epsilon=0.0$

(b) $\epsilon=0.05$

Figure 3: Energy of fermion bound states as a function of coupling parameter
$g$. The scalar field background is the kink of the toy model.

A typical evolution of the fermion field in the presence of a wobbling kink is
shown in Fig. 4. These were obtained by solving Dirac’s equation numerically
and starting the fermion at the zero mode. The plots are similar for all the
models that we considered. The kink stays bound to the kink at the origin but
with noticeable oscillations. As $g$ is increased, the fermion becomes more
localized at the kink because the potentials $V_{\pm}$ become deeper as $g$
increases. However, the effect of the perturbations may increase with $g$ in a
certain range and have the opposite effect. If the transition to the continuum
is allowed according to Fermi’s golden rule, it is possible to see some
radiation as shown in Fig. 4(a) and 4(c). For the quasinormal case, the
oscillations of the fermion field decrease slowly due to a decrease in the
amplitude of quasinormal mode perturbation. This occurs because the
quasinormal mode can tunnel the potential barrier created by the kink and
radiate to infinity. However, the decay is slow and hence difficult to observe
in Fig. 4(c) and 4(d).

(a) $\epsilon=0.0$, $g=0.8$

(b) $\epsilon=0.0$, $g=2.1$

(c) $\epsilon=0.05$, $g=0.8$

(d) $\epsilon=0.05$, $g=2.1$

Figure 4: Evolution of the fermion density by numerically solving the
equations of motion. The fermion starts at the zero mode.

We will discuss the perturbed system for both normal and quasinormal mode
cases in detail and compare both numerical and perturbative solutions in the
following subsections.

###  4.1 Normal mode

As we mentioned before, considering $\gamma=3.0$, the system possesses one
shape mode, which can be found analytically [53]. It has the following form
with arbitrary normalization

$\eta_{S}(x)=\begin{cases}-e^{k_{S}x},&x<-L,\\\
e^{-k_{S}L}\sin(p_{S}x)/\sin(p_{S}L),&-L<x<L,\\\ e^{-k_{S}x},&x>L,\end{cases}$
(24)

where we defined $k_{S}\equiv\sqrt{\gamma-\omega_{S}^{2}}$ and
$p_{S}\equiv\sqrt{\omega_{S}^{2}+1}$ with $\omega_{S}$ being the solution of
the transcedental equation $k_{S}=-p_{S}\cot(p_{S}L)$. For $\gamma=3.0$ we
find $\omega_{S}\simeq 1.568$. The matrix elements of the operator
$\eta_{S}\sigma_{1}$ between the eigenstates of $H_{0}$ are shown in Fig. 5.
It is clear that the result is similar to the $\phi^{4}$ case.

(a)

(b)

Figure 5: Matrix elements for the operator $\eta_{S}\sigma_{1}$ between the
eigenstates of $H_{0}$ for the toy model. The parameters are $\gamma=3.0$ and
$\epsilon=0.0$. We fix $g=1.6$ in (b).

Starting with the fermion on the zero mode we can evolve the system
integrating the equations of motion subject to the perturbation numerically.
These results are shown in Fig. 6, where we compare them with the analytical
expressions. Again we find results very similar to the $\phi^{4}$ model,
including the increase of the transition probability for $g<\omega_{S}$ with
transition rate given by the Fermi’s Golden Rule expression in eq. (17). In
fact, we expect similar results for a large class of models with kink
solutions containing a shape mode.

(a) $g=0.8$

(b) $g=2.1$

Figure 6: Transition probabilities from the zero mode to continuum and bound
states for the toy model. We consider the normal mode case with parameters
$A=0.1$, $\gamma=3.0$. We compare simulation results with first order
perturbation theory. The rate from the zero mode to the continuum (eq. (17))
is also shown in (a) with dotted line.

The main difference between the $\phi^{4}$ and the toy models is that $H_{0}$
has to be diagonalized numerically in the latter. We find the bound states
using the NDEigensystem method in Mathematica and the solve_bvp method of the
Python library SciPy. The continuum states are found as described in [12]. The
method deals with the scattering states instead of the parity eigenstates.
However, a simple linear combination of the scattering states gives the parity
eigenstates. We consider, for instance, the scattering state with an incoming
wave from the left with wavenumber $k$ and energy $E_{k}=\sqrt{g^{2}+k^{2}}$.
In short, the method consists of factoring out the highly oscillatory
transmitted wave dependence as follows

$\psi_{k}=e^{ikx}\chi_{k}(x).$ (25)

Then, we find the initial condition for $\chi_{k}$ by taking the limit
$x\to\infty$ in the eigenvalue equation (6). Using $\phi_{k}(\infty)=1.0$ and
$\chi_{k}^{\prime}(\infty)=0$, this results in

$\chi_{k}(\infty)=\begin{pmatrix}1\\\
\frac{g+ik}{\sqrt{g^{2}+k^{2}}}\end{pmatrix}.$ (26)

Numerically, we set infinity at $x=20.0$, and we integrate $\chi_{k}$ up to
$x=-20.0$ according to the eigenvalue equation (6). After that, we multiply
all eigenfunctions by appropriate normalization constants.

###  4.2 Quasinormal mode

For the toy model with potential (18), there is already a tower of quasinormal
modes of the stability equation. This is also true for the $\phi^{4}$ model
with non-integer $g$. However, physically, the most important solution of the
stability equation is the shape mode for the kink systems. We want this mode
to be turned into a quasinormal mode, and this can be achieved by considering
potential (19). In this case, it is possible to find this mode analytically,
and it could be used as our perturbation to the scalar field if carefully
truncated. This method can work well for large times. However, as we are
interested in the whole transition process and not only the large-time
behavior, a more sensible approach is to use the shape mode of the model with
$\epsilon=0.0$ as the initial condition of the perturbation for the
$\epsilon\neq 0$ case. In this work, we will adopt this approach. The kink’s
shape mode becomes a quasinormal mode because the linearized potential of the
normal mode case has been modified such that this mode can tunnel through the
potential barrier. Therefore, this perturbation describes precisely the quasi-
bound state localized at the kink, which decays in time according to the decay
rate of the quasinormal mode.

Writing $\phi=\phi_{k}+A\eta(x,t)$, the evolution of this perturbation can
then be found numerically using the time-dependent version of the stability
equation

$\frac{\partial^{2}\eta(x,t)}{\partial
t^{2}}-\frac{\partial^{2}\eta(x,t)}{\partial x^{2}}+U(x)\eta(x,t)=0,$ (27)

where $U(x)$ is the linearized potential. In this case the perturbed
Hamiltonian becomes $H_{1}(t)=g\eta(x,t)\sigma_{1}$. If we neglect the
radiation generated by the evolution, we can approximate the perturbation by
$\eta(x,t)\simeq\eta_{S}\cos(\Omega t)e^{-\Gamma t}$, where $\eta_{S}$ is
given by eq. (24) and $\Omega$ and $\Gamma$ are the analytical values of the
frequency and decay rate of the quasinormal mode. For $\gamma=3.0$,
$\gamma^{\prime}=1.0$ and $\epsilon=0.05$, we find $\Gamma\simeq 0.0564$ and
$\Omega\simeq 1.584$ according to the analytical expression in [53]. In this
case, eq. (13) is modified to

$c^{(1)}_{n}(t)=\frac{g}{2}\langle
n|\eta_{S}\sigma_{1}|i\rangle\left(\frac{1-e^{i(\omega_{ni}+\Omega+i\Gamma)t}}{\omega_{ni}+\Omega+i\Gamma}+\frac{1-e^{i(\omega_{ni}-\Omega+i\Gamma)t}}{\omega_{ni}-\Omega+i\Gamma}\right).$
(28)

If we wish to include the effect of the radiation, we should take eq. (10) to
compute $c^{(1)}_{n}$ with the full numerical solution of the scalar field
perturbation as described above.

In Fig. 7 we show the time evolution of the transition probability. It behaves
similarly to the previous cases, but the effect of the perturbation decays
gradually in time due to the leak, which is expected from eq. (28). In the
figure, we compare the numerical simulations with both perturbative results
using eq. 28 and eq. (10) with the inclusion of radiation. The former does not
have a good agreement for small values of $g$, where the effect of the
radiation is more relevant but improves as $g$ is increased.

(a) $g=0.8$

(b) $g=2.1$

Figure 7: Transition probabilities from the zero mode to continuum and bound
states for the toy model. We consider the quasinormal mode case with
parameters $A=0.1$, $\gamma=3.0$, $\gamma^{\prime}=1.0$ and $\epsilon=0.05$.
We compare simulation results with first order perturbation theory. The
perturbative results are given by eq. (10) with the full numerical evolution
of the scalar field (dashed) and by eq. (28) (dotted).

Due to the decrease of the quasinormal mode’s amplitude, the transition
probabilities will reach an asymptotic value if we wait long enough. This can
be used to make a more detailed analysis of the system’s dependence on the
coupling parameter $g$. The result is shown in Fig. 8, where we plot the
asymptotic value of the transition probability as a function of $g$. The
transition probability from the zero mode to the continuum,
$P_{0\to\text{cont.}}$, initially increases with $g$ due to the dependence of
$H_{1}$ on this parameter. After a critical value of $g$, the transition
becomes forbidden because the energy gap between the zero mode and the
continuum becomes too large, and consequently, the transition probability
decreases considerably. This occurs for $g\simeq\Omega$ and is an effect
analogous to Fermi’s golden rule for normal modes. Near $g=1.54$, the second
excited state appears, and there is a finite transition probability to this
state, which has a small peak and then also decreases considerably as $g$
increases due to the large energy gap between the states.

Figure 8: Asymptotic value of the transition probabilities as a function of
$g$. We compare simulation results with first order perturbation theory. We
consider the quasinormal mode case with parameters $A=0.1$, $\gamma=3.0$,
$\gamma^{\prime}=1.0$ and $\epsilon=0.05$.

##  5 Conclusion

In this paper, we considered a system composed of a fermion in the presence of
a wobbling kink and treated the wobbling as a perturbation over the background
kink configuration. We were able to write the Dirac equation for the fermion
field interacting with the static kink background in the Hamiltonian formalism
and diagonalize the unperturbed Hamiltonian. Moreover, we studied the fermion
interaction with the wobbling $\phi^{4}$ kink and also kinks of a toy model
with excited normal and quasinormal modes, which have analytical expressions.
The toy model potential possesses a parameter $\epsilon$, which encodes the
transition from the normal mode of the kink to a quasinormal mode as it
becomes nonzero. As $\epsilon$ approaches zero, the toy model behaves
similarly to the $\phi^{4}$ model [53]. For the $\phi^{4}$ model, we used the
analytical results for the energy and eigenstates, while for the toy model, we
diagonalized the Hamiltonian numerically. In both cases, the threshold energy
increases linearly with $g$, and more bound states with positive and negative
energy appear when $g$ is increased. After diagonalizing, we include the
perturbation and use the first-order perturbation theory to find the
transition probabilities.

The perturbation theory results depend on the matrix elements of the
Hamiltonian perturbation, which can be easily computed in all cases
considered. The matrix elements from the bound states to the continuum are
suppressed when the energy gap becomes too large. Moreover, we found that
these matrix elements obey parity selection rules. The time dependence of the
transition probabilities can be computed analytically, except for the
quasinormal mode case, where the evolution is computed numerically. From the
first-order perturbation theory results, we derived Fermi’s golden rule for
our system, which appears because the perturbation is periodic. It states that
the probability rate to the continuum vanishes if the fermion-kink coupling
$g$ is greater than the wobbling frequency. The transition to the continuum
was obtained by integrating the transition probability and transition rate
over all the final momenta $k$ for both positive and negative energy. For the
transition rate, the integral was performed explicitly.

The perturbative results were compared with numerical simulations of the Dirac
equation, which showed a satisfactory agreement. In the models containing a
normal mode, the transition probability to the continuum shows a steady
increase when $g<\omega_{S}$, in agreement with Fermi’s golden rule. For
larger $g$, the transition probabilities oscillate around an average value,
leading to a vanishing transition rate over a large time interval. The
simulations show the evolution of the fermion density with time. The fermion
is mostly bound to the kink and, ignoring the effect of perturbation, it
becomes more localized around the kink center as $g$ is increased. On the
other hand, the effect of the perturbation can be opposite to that in a
certain range of $g$. Furthermore, for small coupling constants, it is
possible to see more pronounced radiation in the evolution.

The scalar potential was modified to turn the normal mode of the kink into a
quasinormal mode. In this case, the vibrational mode of the toy model with
$\epsilon=0$ was considered as the initial condition of the perturbation.
Then, we evolved the perturbation numerically according to the time-dependent
stability equation. When the normal mode becomes a quasinormal mode, the
perturbation amplitude decays over time because it leaks energy. This causes
the transition rate to also decrease over time and, consequently, the
probability reaches an asymptotic value for large times. Interestingly, the
transition to the continuum is only significant for coupling constants smaller
than the oscillation frequency, which is a reminiscent behavior of Fermi’s
golden rule for the normal mode case. These effects were observed in the
numerical simulations and compared to results from first-order perturbation
theory. We included the full evolution of the scalar field perturbation in the
perturbation theory to have a good agreement between the two. However, we have
also considered the approximation, neglecting the radiation in the scalar
field because, in this case, it is possible to find the time dependence of the
transition probabilities explicitly. The latter approximation gives a
reasonable estimation of the asymptotic value of the transition probability.

Our work clarifies that the fermion decouples from the wobbling kink if the
coupling constant is small and that this effect is attenuated if we have
quasinormal instead of normal modes. This occurs because the perturbation
decreases over time in the former case. Moreover, if it is possible to measure
the fermion escape rate, one could, in principle, probe whether the kink is
wobbling or set a bound on the wobbling parameters, such as the wobbling
amplitude, quasinormal mode decay rate, and the coupling constant $g$. This
could be relevant for electrons in polyacetylene [56, 57] and in the scenario
where we live in a domain wall [26], to name a few examples.

## Acknowledgements

JGFC acknowledges financial support from the Brazilian agency CNPq. AM thanks
financial support from the Brazilian agencies, CAPES and CNPq Grant No.
309368/2020-0, and also Universidade Federal de Pernambuco Edital Qualis A.

##  Appendix A Fermion Bound States of the $\phi^{4}$ model

The bound and scattering states of a fermion coupled to a $\phi^{4}$ kink are
listed in references [6, 7]. The effective fermionic potentials, in this case,
are given by eq. (15), which are Pöschl-Teller potentials. In this case, the
number of bound states is given by the largest integer smaller than $g$. The
bound states are as follows

$\displaystyle\psi_{n,+}$
$\displaystyle=N_{+}^{n}\,\text{sech}^{g-n}(x)\,F\left(-n,2g-n+1,g-n+1;\frac{1}{2}(1-\tanh(x))\right),$
$\displaystyle\psi_{n,-}$
$\displaystyle=N_{-}^{n}\,\text{sech}^{g-n}(x)\,F\left(-n+1,2g-n,g-n+1;\frac{1}{2}(1-\tanh(x))\right),$
(29)

where $N_{\pm}^{n}$ are normalization constants obeying $N_{-}^{0}=0$ and, for
$n\geq 1$, $N_{-}^{n}/N_{+}^{n}=n/E_{n}$. Also $E_{n}=\sqrt{n(2g-n)}$ is the
energy of the state and $F$ is the hypergeometric function. On the other hand,
the scattering states of an incoming incident wave from the left and from the
right are

$\displaystyle\psi^{\pm}_{k,L}$
$\displaystyle=N^{k,L}_{\pm}\cosh^{ik}(x)\,F\left(\frac{1}{2}-ik-\zeta_{\pm},\frac{1}{2}-ik+\zeta_{\pm},1-ik;\frac{1}{1+e^{2x}}\right),$
$\displaystyle\psi^{\pm}_{k,R}$
$\displaystyle=N^{k,R}_{\pm}\cosh^{ik}(x)\,F\left(\frac{1}{2}-ik-\zeta_{\pm},\frac{1}{2}-ik+\zeta_{\pm},1-ik;\frac{1}{1+e^{-2x}}\right),$
(30)

respectively, where $\zeta_{\pm}=g\pm\frac{1}{2}$, and $N^{k,L}_{\pm}$ and
$N^{k,R}_{\pm}$ are normalization factors obeying
$N^{k,L}_{-}/N^{k,L}_{+}=-N^{k,R}_{-}/N^{k,R}_{+}=(ik+g)/\sqrt{k^{2}+g^{2}}$.
Both scattering states have energy $E_{k}=\sqrt{g^{2}+k^{2}}$.

##  Appendix B Numerical technique

To solve the equations of motion numerically for the fermion field, we
discretize space on the interval $-L<x<L$ with spacing $h=0.1$ and $L=100.0$.
We substitute the partial derivative with respect to $x$ by a five-point
stencil approximation

$\frac{\partial\psi}{\partial
x}=\frac{-\psi(x+2h)+8\psi(x+h)-8\psi(x-h)+\psi(x-2h)}{12h},$ (31)

which is fourth order accurate in $h$. At $x=-L+h$, the finite difference
expression is as follows

$\frac{\partial\psi}{\partial
x}=\frac{\psi(-L+4h)-6\psi(-L+3h)+18\psi(-L+2h)-10\psi(-L+h)-3\psi(-L)}{12h}$
(32)

where we impose $\psi(x=-L)=0$. Similar expressions are used at $x=L$. After
discretizing the space, we integrate the resulting ordinary differential
equation using a fifth-order Runge-Kutta method [58] implemented by the
solve_ivp method of Python library SciPy. For the scalar field, we use the
analytical expression for the kink and, whenever possible, for the
perturbation. This way, we ignore the higher-order coupling of the shape mode
with the radiation mode [29]. We cannot use the analytical expression for the
quasinormal modes, and we must integrate the time-dependent stability eq.
(27). In this case, we follow the same procedure to integrate the fermion
field, i.e., the spacetime is discretized the same way, partial derivatives
over $x$ are obtained using a five-point-stencil approximation, and
integration over time is done using a fifth-order Runge-Kutta method.

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