# Constrained instantons and kink-antikink collisions

C. Adam<EMAIL_ADDRESS>Departamento de Física de Partículas, Universidad
de Santiago de Compostela and
Instituto Galego de Física de Altas Enerxias (IGFAE), E-15782 Santiago de
Compostela, Spain  A. García Martín-Caro<EMAIL_ADDRESS>Departamento de Física de Partículas, Universidad de Santiago de Compostela
and
Instituto Galego de Física de Altas Enerxias (IGFAE), E-15782 Santiago de
Compostela, Spain  M. Huidobro<EMAIL_ADDRESS>Departamento de
Física de Partículas, Universidad de Santiago de Compostela and
Instituto Galego de Física de Altas Enerxias (IGFAE), E-15782 Santiago de
Compostela, Spain  K. Oles<EMAIL_ADDRESS>T. Romanczukiewicz
<EMAIL_ADDRESS>A. Wereszczynski
<EMAIL_ADDRESS>Institute of Theoretical Physics, Jagiellonian
University, Lojasiewicza 11, Kraków, Poland

###### Abstract

We show that constrained $CP^{1}$ instantons, combined with the Relativistic
Moduli Space approach, can accurately describe kink-antikink collisions in the
$\phi^{4}$ model.

##  I Introduction

Topological solitons are ubiquitous in many areas of theoretical physics
reaching from particle theory to nuclear physics, condensed matter or
cosmology. A thorough understanding both of their individual properties and
their dynamics in multisoliton processes is, therefore, of utmost importance
both for classical and quantized solitons. On the other hand, topological
solitons typically are solutions of rather complicated, nonlinear field
theories, and exact analytical expressions are available only in rare
occasions, which considerably complicates their analysis. One important tool
in their study is, therefore, a systematic construction of analytical field
configurations which provide an accurate approximation both for static
solitons and for the field configurations which may appear in dynamical
processes like soliton scattering.

It turns out that instantons Inst-1 , that is, localized, finite action
solutions of a field theory in an Euclidean space-time, are particularly
useful in this context. They provide a systematic approach for the
understanding of properties of topological solitons in various physically
important models. One of the best examples is the Skyrme model, which is an
effective theory of quantum chromodynamics in the non-perturbative, low energy
regime. This model admits topological excitations, called Skyrmions, which are
identified with baryons and atomic nuclei. It was shown by Atiyah and Manton
that Skyrmions can be well approximated by the holonomy of the $SU(2)$
instantons in $(4+0)$ space AM . This is important not only because one gets a
relatively simple method to generate and study Skyrmions LM ; SSut , but it is
also a way which defines a truncated, finite dimensional configuration space,
i.e., a moduli space, in a natural manner, provided by the instanton moduli
space (the space of parameters on which a general instanton solution may
depend). This finite dimensional subspace is a crucial ingredient both for a
simplified study of scattering processes and in the quantization of skyrmions
Ho ; Chris-1 ; chris-2 . Furthermore, the instanton approach allowed for the
investigation of dense skyrmionic matter both in the crystal MS-c ; PV-1 ; M-c
and inhomogeneous phases PV-2  as well as nucleon-nucleon forces Chris-3 . It
was also used to construct topological solutions of different types like
vortices Nitta  and sphalerons Sh . More importantly, a similar construction
of skyrmions from instantons becomes realized in some holographic models, such
as the Sakai-Sugimoto SS  or the Sutcliffe Sut-Inst  models, where Skyrme
solitons appear in the boundary theories coupled to an (infinite) tower of
vector mesons.

While applied with success in various areas, the instanton approximation did
not work too well in models with massive fields. This is an obvious
consequence of the power-like localization of instantons, which is typical for
massless fields.

However, it has recently been shown how to extend this framework also for
Skyrmions with massive pions Alberto , where instantons are replaced by the
so-called constrained instantons, namely, topologically non-trivial Euclidean
configurations whose size has been fixed by an additional constraint in the
original action. The effect of such constraint is an exponential decay in the
radial direction, which agrees with the solutions of the massive Skyrme model.

In the present paper, we test the instanton approach for a much simpler model,
which is the $\phi^{4}$ theory in (1+1) dimensions. Therefore we use $CP^{1}$
instantons. However, our aim is to model one of the most difficult and subtle
features of this theory, i.e., the kink-antikink collisions. This goes much
beyond the usual application of instantons to model static properties of
solitons like, e.g., their energies and shapes (geometry).

##  II Instanton approximation

###  II.1 $CP^{1}$ Instanton and kink profile

We begin with a short summary of the instanton approximation to kinks in (1+1)
dimensions proposed by Sutcliffe Sut-1 ; Sut-2 ; Sut-3 . The obvious choice is
to use the $CP^{1}$ model in (2+0) dimensions, which supports self-dual
instantons. They are finite action solutions of the following model

$\mathcal{S}=\int d^{2}x\mbox{Tr}(D_{\mu}Z)^{\dagger}(D^{\mu}Z),$ (1)

defined on the Euclidean space $x^{\mu}=(x^{1},x^{2})$. Here
$D_{\mu}=\partial_{\mu}-A_{\mu}$, $A_{\mu}=Z^{\dagger}\partial_{\mu}Z$ and $Z$
is a two component complex field obeying a constraint $Z^{\dagger}Z=1$. The
constraint can be resolved via the parametrization

$Z=\frac{1}{\sqrt{1+|w|^{2}}}\left(\begin{array}[]{c}1\\\
w\end{array}\right),$ (2)

where $w$ is a complex field. In instanton solutions the complex field $w$ is
a rational function of $z=x_{1}+x_{2}$. Then, the topological charge of the
instanton equals the degree of the rational map $w(z)$. For example, the
charge one instanton is given by

$w=\lambda(z-z_{0})$ (3)

where $z_{0}$ is its position and $\lambda$ its scale, which appear as a
consequence of the translational and conformal symmetry. Anti-instantons can
be obtained by complex conjugation.

Now, we define the $U(1)$ abelian holonomy

$U(x_{1})=\;\mbox{exp}\left(\int_{-\infty}^{\infty}A_{2}(x_{1},x_{2})dx_{2}\right),$
(4)

which can be further related to a static scalar field

$U(x_{1})=e^{i\pi\phi(x_{1})}.$ (5)

Hence, inserting the single instanton solution we arrive at the following
field configuration (from now on, we suppress the lower index, $x\equiv
x_{1}$)

$\Phi(x;\lambda,a)=\frac{\lambda(x-a)}{\sqrt{1+\lambda^{2}(x-a)^{2}}}.$ (6)

So, this profile is independent of the Lagrangian of the scalar field. The
only place where the actual kink model may affect the instanton approximation,
therefore, is the optimal value of $\lambda$.

###  II.2 $\phi^{4}$ theory

In the present work we apply this construction to the $\phi^{4}$ theory in
(1+1) dimensions,

$L=\int_{-\infty}^{\infty}\left(\frac{1}{2}\phi_{t}^{2}-\frac{1}{2}\phi^{2}_{x}-\frac{1}{2}(1-\phi^{2})^{2}\right)dx.$
(7)

This model supports a kink $\Phi_{K}$ and antikink $\Phi_{\bar{K}}$
interpolating between the two vacua $\phi_{v}=\mp 1$

$\Phi^{K}=\tanh(x-a),\;\;\;\Phi^{\bar{K}}=-\tanh(x-a)$ (8)

They are stable, minimal energy solutions in the pertinent topological
sectors. Here $a$ is a moduli parameter denoting the center of the soliton. It
arises due to the translational invariance of the action and is reflected in
the existence of the zero mode.

The (anti)kink possesses also a massive mode, so-called shape mode

$\eta^{sh}(x,t)=\sqrt{\frac{3}{2}}\frac{\sinh(x-a)}{\cosh^{2}(x-a)}e^{i\omega_{sh}t}$
(9)

which has a frequency $\omega_{sh}^{2}=3$. Finally, there are scattering
modes, i.e., radiation for $\omega^{2}>4$.

To approximate the static $\phi^{4}$ solitons, one inserts the single
instanton into the $\phi^{4}$ energy integral and minimizes it w.r.t. the
scale parameter $\lambda$. The optimal value is
$\lambda_{0}=\frac{2}{\sqrt{3}}$ which leads to the energy
$E[\lambda_{0}]=\frac{\sqrt{3}\pi}{4}\approx 1.36035$. This should be compared
with the static energy of the (anti)kink, which is
$E_{K(\bar{K})}=\frac{4}{3}\approx 1.33333$. We remark that the observed $2\%$
discrepancy can be made lower if one uses higher dimensional instantons Sut-2
.

###  II.3 Perturbative Relativistic Moduli Space

Now, we want to go further and model the shape mode of the $\phi^{4}$ kink.
This will be achieved by applying the recently developed perturbative
Relativistic Moduli Space approach AMORW . The is a version of the collective
coordinate method which partially reproduces the Lorentz invariance of the
underlying field theory Rice , AMORW .

In the collective coordinate model, we assume that for a given solitonic
process the infinite dimensional space of all field configurations can be
restricted to a finite dimensional set parametrized by some parameters,
moduli, $X^{i}$

$\mathcal{M}[X^{i}]=\left\\{\Phi(x;X^{i});i=1..N\right\\}.$ (10)

In the next step, we promote the moduli to time dependent variables. Inserting
these configurations into the original Lagrangian and performing the spatial
integration brings us to a mechanical like effective model

$L[{\bf
X}]=\int_{-\infty}^{\infty}\mathcal{L}[\Phi(x;X^{i}(t))]\,dx=\frac{1}{2}g_{ij}({\bf
X})\dot{X}^{i}\dot{X}^{j}-V({\bf X})\,,$ (11)

where

$g_{ij}({\bf X})=\int_{-\infty}^{\infty}\frac{\partial\Phi}{\partial
X^{i}}\frac{\partial\Phi}{\partial X^{j}}\,dx$ (12)

is the metric on $\mathcal{M}$, while

$V({\bf
X})=\int_{-\infty}^{\infty}\left(\frac{1}{2}\left(\frac{\partial\Phi}{\partial
x}\right)^{2}+\frac{1}{2}(1-\Phi^{2})^{2}\right)\,dx$ (13)

is the effective potential which modifies the motion on the moduli space.

The Relativistic Moduli Space framework is based on the inclusion of the scale
(Derrick) deformation, $x\to bx$. Then, we build the moduli space as follows.
We start with energy equivalent set of configurations $\Phi_{0}(x;a)$. They
can be single kink solution, $\phi_{K}(x-a)$ or some other configurations
which are meant to approximate the kink, as, e.g., the instanton motivated
profile, $\Phi(x-a;\lambda_{0})$. Then, we include a deformation $b$, which
rescales the spatial direction, $x\to bx$. Hence, we arrive at the following
set of configurations

$\mathcal{M}[a,b]=\left\\{\Phi_{0}(b(x-a))\right\\}$ (14)

where we assumed the translational invariance of the problem. We remark that
the $\Phi_{0}(x;a,b=1)$ should be an energy minimizer under the Derrick
deformation.

The resulting CCM possesses a stationary solution
$a(t)=vt+a_{0},\;b=\gamma\equiv 1/\sqrt{1-v^{2}}$, which describes a boosted
static solution $\Phi_{0}(x)$, Rice , AMORW .

In our case, the Derrick deformation leads to the following family of
configurations

$\Phi(x;\lambda_{0},a,b)=\frac{\lambda_{0}b(x-a)}{\sqrt{1+\lambda_{0}^{2}b^{2}(x-a)^{2}}}.$
(15)

It is immediately seen that the Derrick deformation can be identified with
deformation of the instanton scale parameter $\lambda$. Therefore,
configurations with $b=1$ are the energy minima also with the scaling
deformation included. In addition we can take $\lambda$ as an equivalent scale
moduli. Thus, the instanton motivated single kink moduli space reads

$\mathcal{M}[a,\lambda]=\left\\{\Phi(x;a,\lambda);\;a\in\mathbb{R},\;\lambda\in\mathbb{R}_{+}\right\\},$
(16)

where the positive sign of $\lambda$ is fixed by the topology, i.e., by the
boundary conditions imposed on $\Phi$. Obviously, contrary to the flat
direction $a$, changes of $\lambda$ correspond to massive deformations which
may cover the internal, vibrational modes. As we have already noticed, the
corresponding collective model has a stationary solution

$a(t)=vt+a_{0},\;\;\;\lambda=\lambda_{0}\gamma\equiv\frac{\lambda_{0}}{\sqrt{1-v^{2}}},$
(17)

which is just the Lorentz contraction of a boosted instanton approximated
kink.

It was shown, however, that to study multi-kink collisions, another version of
the Relativistic Moduli Space approach is much more useful. This follows from
the observation that for kink-antikink collisions a finite dimensional set of
configurations, based on a simple sum of a single kink and antikink, leads to
the appearance of an essential singularity on the moduli space at $a=0$, i.e.,
at the point where solitons are on top of each other and form a vacuum.
(Strictly speaking, this only occurs for solitons related by the symmetry
$\Phi^{\bar{K}}(x)=-\Phi^{K}(x)$, which is the case in the $\phi^{4}$ model.)

To circumvent this issue, one can use the perturbative Relativistic Moduli
Space approach. This means that we expand the profile provided by the
instanton approximation in small perturbations in the scale (Lorentz boost)
parameter around the optimal value, $\lambda=\lambda_{0}+\epsilon$

$\displaystyle\Phi_{0}(x;a,b)=\Phi_{0}(b(x-a))=\Phi_{0}((1+\epsilon)(x-a))$
$\displaystyle\quad\quad=\sum_{k=0}^{n}\frac{\epsilon^{k}}{k!}(x-a)^{k}\Phi_{0}^{(k)}(x-a)+o(\epsilon^{n})\,,$
(18)

Now, we treat each term in the expansion as an independent mode, the so-called
Derrick mode

$\displaystyle\Phi_{0}(x;a,{\bf C})$ $\displaystyle=$
$\displaystyle\Phi_{0}(x-a)$ (19) $\displaystyle+$
$\displaystyle\sum_{k=1}^{n}\frac{C_{k}}{k!}(x-a)^{k}\Phi^{(k)}_{0}(x-a)\,.$

This provides an arbitrarily large set of collective coordinates
$C_{i},\;i=1..n$, which are the amplitudes of the Derrick modes.

In the current work we take the simplest possibility and consider only the
first, lowest Derrick mode. Hence, we arrive at two dimensional instanton
motivated moduli space with configurations of the following form

$\Phi(x;a,C)=\frac{\lambda_{0}(x-a)}{\sqrt{1+\lambda_{0}^{2}(x-a)^{2}}}+C\frac{(x-a)}{(1+\lambda_{0}^{2}(x-a)^{2})^{3/2}}.$
(20)

Figure 1: Kink profiles: $\phi^{4}$ kink (black), the instanton approximation
(blue) and the constrained instanton approximation (orange).

Figure 2: Derrick modes compared to the true shape mode of the $\phi^{4}$ kink
(black): the instanton approximation (blue) and the constrained instanton
approximation (orange).

It is important to notice that the CCM model arising from these configurations
still possesses a stationary solution which approximates the boosted kink.
Indeed, inserting $\Phi(x;a,C)$ into the $\phi^{4}$ Lagrangian we find the
following metric

$g_{aa}=\frac{\sqrt{3}\pi}{4}+\frac{\pi}{8}C+\frac{9\sqrt{3}\pi}{256}C^{2},\;\;\;g_{CC}=\frac{3\sqrt{3}\pi}{64}$
(21)

and the effective potential

$V(C)=\frac{\sqrt{3}\pi}{4}+\frac{39\sqrt{3}\pi}{512}C^{2}+\frac{27\pi}{1024}C^{3}+\frac{27\sqrt{3}\pi}{16384}C^{4}.$
(22)

The resulting equations of motion have a solution

$\dot{a}=v,\;\;\;\tilde{C}=C(v)$ (23)

where $\tilde{C}$ obeys an algebraic equation

$\frac{v^{2}}{2}\partial_{C}g_{aa}=\partial_{C}V.$ (24)

Finally, we look closer at the first instanton motivated Derrick mode which
after normalization reads

$\eta^{inst}=\sqrt{\frac{64}{3\sqrt{3}\pi}}\frac{(x-a)}{(1+\lambda_{0}^{2}(x-a)^{2})^{3/2}},\;\;\;\lambda_{0}=\frac{2}{\sqrt{3}}.$
(25)

It is a massive excitation and therefore gives an approximation to the shape
mode. It is a rather good approximation. Its frequency is
$(\omega_{inst})^{2}=\frac{13}{4}=3.25$. It has also a very good overlap with
the shape mode,
$\langle\eta^{inst}|\eta^{sh}\rangle=\int\eta^{sh}\eta^{inst}dx=0.97565.$

Hence, we can summarize that the instanton approximation provides a quite good
approximation to both static and stationary properties of a kink. Here we mean
the mass of the kink and the properties of its normal mode.

One should also be aware that there is an important property of the kink which
is not correctly reproduced by the instanton approximations, namely, the near
vacuum regime. Indeed, the approximated kinks have very slowly decaying tails.
Instead of the original exponential approach to the vacuum

$\Phi^{K}=\pm 1\mp 2e^{-2x}\;\;\;\mbox{as}\;\;\;x\to\pm\infty,$ (26)

the instanton approximation leads to $x^{-1}$ localized solitons. Hence, the
approximated kinks interact much more stronger and at a much larger distance.
As we will see below, this rather strongly affects the multi-kink dynamics.

##  III Instanton-antiinstanton valley

In order to study multi-soliton solutions, one has to find multi-instanton
configurations in the corresponding topological (instanton) charge sector.
Such $CP^{1}$ multi-instantons are exactly known. They are given by
holomorphic rational maps whose degree is the instanton charge. Then, for
example, a kink-kink scattering may be approximated by a path in the two-
instanton space, where the actual time evolution would arise from the
pertinent collective coordinate model Sut-1 ; Sut-3 .

There is, however, a problem if we want to study kink-antikink collisions.
There are no $CP^{1}$ instantons in trivial charge sector. Technically it
means that there are no nontrivial self-dual solutions with zero topological
charge. They may appear if we couple the $CP^{1}$ model with a background
field, an impurity, in a very specific manner, the so-called BPS impurity
models BPS-imp-1 ; BPS-imp-2 . Here we will not follow this strategy but,
rather, will use a more traditional instanton-antiinstanton ($I\bar{I}$)
valley construction BY .

In this construction, one focuses on trajectories $\Psi_{v}(x^{i};\xi)$ in
functional space which asymptotically, for a parameter $\xi\to\infty$,
represent a pair of infinitely separated instanton and antiinstanton solutions
of the action $S_{0}$. For finite $\xi$, the configurations
$\Psi_{v}(x^{i};\xi)$ are not extrema of the original action. Instead,
$\Psi_{v}(x^{i};\xi)$ are solutions of a reduced theory which is the original
$S_{0}$ under a constraint that only variations orthogonal to the direction
$\xi$ are taken into account

$\frac{\delta
S_{red}}{\delta\Psi}[\Psi_{v}]=\omega_{\Psi}(x^{i};\xi)\frac{\partial\Psi_{v}}{\partial\xi}$
(27)

equipped with above mentioned boundary condition. In other words, we find a
streamline in the functional space and the valley configuration goes along
this streamline.

Examples of such $I\bar{I}$ valleys are know BW ; Dorey ; Elis . For instance,
a valley can have the following simple form Elis

$w(z,\bar{z})=\frac{(\xi-1/\xi)\bar{z}}{1+|z|^{2}}$ (28)

This is a concentric $I\bar{I}$ pair which can be transformed into a non-
concentric one by a conformal transformation.

In the present work, we take the simplest choice relevant for kink-antikink
collisions and assume the valley in a form which leads to the simple kink-
antikink superposition BW . We do not know an exact form of the corresponding
$I\bar{I}$ configurations but this is not needed for our computations.

After this reasoning we propose the instanton motivated kink-antikink moduli
space to be a sum of single soliton cases. Specifically, the relevant
configurations are

$\displaystyle\Phi^{K\bar{K}}(x;a,C)=$ $\displaystyle=$
$\displaystyle\frac{\lambda_{0}(x+a)}{\sqrt{1+\lambda_{0}^{2}(x+a)^{2}}}-\frac{\lambda_{0}(x-a)}{\sqrt{1+\lambda_{0}^{2}(x-a)^{2}}}-1$
$\displaystyle+$ $\displaystyle
C\left(\frac{x+a}{(1+\lambda_{0}^{2}(x+a)^{2})^{3/2}}-\frac{x-a}{(1+\lambda_{0}^{2}(x-a)^{2})^{3/2}}\right),$

where we keep only the first Derrick mode (symmetrically excited). However,
there is still a problem, since the moduli space metric of the resulting CCM
(11) has a singularity at $a=0$. Indeed, $g_{CC}=g_{aC}=0$ at this point.
Fortunately, this is an apparent singularity which can be removed by a
suitable choice of the moduli space coordinates. Here, it is enough to change
$C\to\frac{C}{\tanh(a)}$, see MORW-moduli  and AMORW  for details. This
finally brings us to the following set of configurations

$\displaystyle\Phi^{K\bar{K}}(x;a,C)=$ $\displaystyle=$
$\displaystyle\frac{\lambda_{0}(x+a)}{\sqrt{1+\lambda_{0}^{2}(x+a)^{2}}}-\frac{\lambda_{0}(x-a)}{\sqrt{1+\lambda_{0}^{2}(x-a)^{2}}}-1$
$\displaystyle+$
$\displaystyle\frac{C}{\tanh(a)}\left(\frac{x+a}{(1+\lambda_{0}^{2}(x+a)^{2})^{3/2}}-\frac{x-a}{(1+\lambda_{0}^{2}(x-a)^{2})^{3/2}}\right).$

The final step is to identify the appropriate initial conditions which would
correspond to kink-antikink scattering in the original $\phi^{4}$ theory.
Obviously, for a large initial separation, $a(t=0)=a_{0}\gg 1$ the
configurations represent free solitons. Thus, the initial conditions can be
read of from the single kink sector. Specifically, they read

$a(0)=a_{0},\;\;\;\dot{a}(0)=v,\;\;\;C(0)=\tilde{C},\;\;\;\dot{C}(0)=0.$ (31)

where $\tilde{C}$ obeys equation (24).

Figure 3: Kink-antikink collision in the $\phi^{4}$ model.

Figure 4: Kink-antikink collision in a CCM based on the superposition of
instanton and antiinstanton approximated configurations (III).

Figure 5: Kink-antikink collision in a CCM based on the superposition of
constrained instanton and antiinstanton approximated configurations (42.

We start with presenting the results of kink-antikink scattering in the
$\phi^{4}$ model, see Fig. 5. Here we show the time dependence of the field at
the origin $\phi(x=0,t)$ for different initial velocities of the colliding
solitons $v\in[0.1,0.3]$. One can easily recognize the bounce windows, where
kink and antikink scatter back to infinity after a certain number of
collisions, as well as the bion chimneys where the solitons do not have
sufficient kinetic energy to overcome the attraction and, therefore,
annihilate via the formation of a bion i.e., an oscillating state which slowly
decays to the vacuum via radiation. The actual behaviour strongly depends on
the initial velocity and reveals a fractal like pattern. For
$v>v_{crit}\approx 0.2598$ we see only one-bounce scattering.

In Fig. 5 we show the value of the field at the origin found in the instanton
motivated CCM based on configurations (III). We see bounce windows and bion
chimneys, but the critical velocity is very large, more than twice as big as
in the true collisions, $v_{crit}^{inst}\approx 0.57$. This means that,
although the static and stationary properties of single kink are quite well
reproduced by the instanton approximation, it is too little to cover dynamical
features of the model as e.g., the effects of kink-antikink collisions.

An obvious source of this failure is the wrong asymptotic behaviour of the
approximated kinks. They decay too slowly and feel each other much stronger
than in the original field theory. Therefore, it is not too surprising that
the kinks need significantly bigger energy to overcome this stronger
attraction. This energy can only come from the initial kinetic energy. As a
consequence, we find a much larger critical velocity.

To improve the instanton description of kink-antikink collisions we have to
modify the behaviour of the solitons in the near-vacuum regime and equip the
kink and antikink with exponential tails. This can be done by making use of
the constrained instantons.

##  IV Constrained instanton

In order to impose an exponential decay on the (constrained) instanton
configurations, we may modify the original instanton Lagrangian by including a
mass term in the action

$S_{m}=\frac{1}{2}\int
d^{2}x[\partial_{\mu}n^{a}\partial_{\mu}n^{a}+m^{2}(1-n^{3})].$ (32)

Here we use the $O(3)$ formulation of the model, where instead of the complex
field $w$ we have a unit three component iso-vector
$\vec{n}=(n^{1},n^{2},n^{3})$

$w=\frac{n_{1}+in_{2}}{1-n_{3}}.$ (33)

It will be also useful to introduce polar coordinates in the base manifold,
$\\{r,\theta\\}$, so that an axially symmetric configuration can be easily
written in terms of a radial function $f(r)$ as

$w(r,\theta)=\frac{\sin fe^{i\theta}}{1-\cos f}.$ (34)

For instance, the single-instanton solution centered at the origin is given by
${\cos f(r)=(\lambda^{2}-r^{2})/(\lambda^{2}+r^{2})}$.

The massive $O(3)$ model does not support topologically nontrivial euclidean
action minimizers. However, it can still provide very useful so-called
constrained instanton configurations const-inst1 ; const-inst2 .

The equation of motion in the large $r$ limit simplifies to the modified
Bessel equation,

$f^{\prime\prime}+\frac{1}{r}f^{\prime}-\left(\frac{1}{r^{2}}+m^{2}\right)f=0$
(35)

so that solutions vanishing at infinity must satisfy

$f\sim K_{1}(mr)\qquad\text{for}\quad r\rightarrow\infty.$ (36)

On the other hand, we want our constrained instanton configuration to behave
as the pure $CP^{1}$ instanton for small distances from its center,

$f\sim\arccos{\frac{\lambda^{2}-r^{2}}{\lambda^{2}+r^{2}}}\qquad\text{for}\quad
r\sim 0.$ (37)

A simple function that achieves both conditions is

$f_{c}(r)=\arccos\left({\frac{\lambda^{2}-r^{2}}{\lambda^{2}+r^{2}}}\right)mrK_{1}(mr)$
(38)

This function depends on two parameters $\lambda$ and $m$, which for the
moment we leave as free parameters.

Now, we compute the holonomy of $A_{2}$ generated by the constrained instanton
profile and find

$\Phi^{c}(x;\lambda,m)=\frac{1}{\pi}\int
dt\frac{x}{x^{2}+t^{2}}\cos^{2}\left(\frac{m}{2}\sqrt{x^{2}+t^{2}}K_{1}\left(m\sqrt{x^{2}+t^{2}}\right)\arccos\left(\frac{2\left(x^{2}+t^{2}\right)}{\lambda^{2}+x^{2}+t^{2}}-1\right)\right)$
(39)

Then, we insert the approximated kink profile into the energy integral of the
$\phi^{4}$ model and find the optimal values of the parameters. They are
$\lambda_{0}=1.4076$ and $m_{0}=0.5870$. This guarantees that the scale
perturbation will not provide a lower energy configuration. Note that for the
constrained instanton we have to perform such a two parameter minimization
because for such an ansatz the scaling deformation is not simply equivalent to
a redefinition of $\lambda$. Instead, it affects terms where $\lambda$ is
absent. This is due to the explicit conformal symmetry breaking implied by the
inclusion of the mass term. We also remark that these values of the ansatz
parameters give a very good approximation of the true kink energy. Namely
$E=1.3337$. One can also observe that the optimal value of the mass parameter
$m_{0}$ is not too different from the exponentially decaying tail of the kink,
for which $m=1$.

Of course, the reason why we have to consider scale deformations is clear from
the previous analysis. We need Derrick modes to perturbatively approximate the
Lorentz boost of a static kink configuration and, therefore, to correctly
define the initial conditions in the CCM.

Thus, the simplest perturbative relativistic moduli space describing single-
kink sector is built out of the following configurations

$\Phi^{c}(x-a;\lambda_{0},m_{0})+C(x-a)\partial_{x}\Phi^{c}(x-a;\lambda_{0},m_{0}),$
(40)

where we contain only the first Derrick mode (for simplicity, we show it
without normalizing to one)

$\eta^{c}=(x-a)\partial_{x}\Phi^{c}(x-a;\lambda_{0},m_{0}).$ (41)

Here, $a$ is the position of the kink and $C$ is the amplitude of the first
Derick mode. The constant velocity motion of the kink is found as a stationary
solution of the resulting CCM, $\dot{a}=v$, $C=\tilde{C}(v)$ obeying eq. (24).

The construction of the kink-antikink moduli space is also straightforward. It
is based on the superposition of the kink and antikink configuration with the
previously discussed regularization. Hence,

$\displaystyle\Phi_{K\bar{K}}^{c}$ $\displaystyle=$
$\displaystyle\Phi^{c}(x+a;\lambda_{0},m_{0})-\Phi^{c}(x-a;\lambda_{0},m_{0})$
(42) $\displaystyle+$
$\displaystyle\frac{C}{\tanh(a)}\left((x+a)\partial_{x}\Phi^{c}(x+a;\lambda_{0},m_{0})\right.$
$\displaystyle-$
$\displaystyle\left.(x-a)\partial_{x}\Phi^{c}(x-a;\lambda_{0},m_{0})\right)$

This gives rise to a CCM which we solve with the previously specified initial
conditions (31).

Solving the equations arising from this CCM,

$\ddot{X}^{i}=-\Gamma^{i}_{\,jk}\dot{X}^{j}\dot{X}^{k}-g^{ij}\frac{\partial
V}{\partial X^{j}},$ (43)

requires calculating many integrals numerically, especially because even the
instanton profile is given by a slowly converging integral. Here, the
Christoffel symbols and the gradient are functions of the position on the
moduli space and are given by two dimensional integrals (over $t$ and $x$). In
our earlier attempts, we calculated all the integrals (12), (13) at each time
step. However, this approach resulted in many artefacts and substantial, but
acceptable computation times. Here, calculating integrals at each time step
would be much more time-consuming. Therefore we applied a different strategy.
Prior to solving the effective equation, all elements of
$\Gamma^{i}_{\,jk}(X)$ and $\partial^{i}V(X)$ were calculated on a grid and
two dimensional splines were used to approximate the rhs of the effective
equations. This reduced the computation time of the full scan shown in Fig. 5
down to a couple of minutes (in Julia programming language).

In Fig. 5, we plot the results of kink-antikink collisions in the CCM derived
above. Again we see bounce windows and bion chimneys, but now in very good
agreement with the full field theory. Especially the critical velocity
improves significantly. Now it reads $v_{crit}^{c}=0.2864$, which is only
$10\%$ bigger than the true value. Note that in the usual instanton
approximation the difference is about $120\%$. The improvement is spectacular
and indicates a high importance of the correct localization, that is, the
correct approach to the vacuum of the approximated soliton.

We remark that the results of the constrained instanton based CCM are quite
similar to computations where the exact kink and the shape MORW  or Derrick
mode AMORW  were used. One can verify that the corresponding scans of the
collisions are qualitatively similar, while the critical velocities read
$v_{crit}=0.282$ and $v_{crit}=0.2853$, respectively.

##  V Conclusions

In the present paper we show that the constrained instanton approximation
combined with the instanton-antiinstanton valley construction and the
perturbative relativistic moduli space approach can quite accurately describe
such a delicate phenomenon as kink-antikink collisions in the $\phi^{4}$
model. This is a very nontrivial result, as the kink-antikink collisions lead
to a chaotic system with a fractal structure in the final state formation.

Our results provide clear evidence that the (constrained) instanton
approximation gives a universal tool allowing for the analysis (and/or
approximation) of both individual soliton properties and soliton collisions.
Of course, we cannot expect that this approximation will be competitive with
more specific, taylor-made approaches whenever such a specific treatment can
be found like, e.g., the moduli space approach based on a superposition of
true single soliton solutions combined with single kink normal modes MORW ,
single kink Derrick modes AMORW  or even multi-kink internal modes phi-6 .
This is because the instantons provide a limited set of configurations which
is basically independent of the theory we want to approximate. The only place
where this theory affects the instanton approximation is via the optimal
values of the parameters. However, in many cases as for example the Skyrme
model, a good, i.e., physically motivated choice for a moduli space and the
related CCM is difficult to find, and precisely in these situations the
instanton approximation can be a very useful tool.

Therefore, the main result is the demonstration that the constrained instanton
approximation can be applied to much more complicated phenomena or properties
than the usually considered shapes and masses of static solitons.
Specifically, dynamical properties which involve the internal modes seem to be
quite well captured.

This can also be very important for the application of instantons to the
analysis of the semiclassical version of solitonic models, where normal modes
are the crucial ingredients Q1 . See also Q2 ; Q3 ; Q4  for recent
developments. Especially, in higher dimensions the instanton approach can
provide a simple but trustworthy approach.

## Acknowledgements

The authors acknowledge financial support from the Ministry of Education,
Culture, and Sports, Spain (Grant No. PID2020-119632GB-I00), the Xunta de
Galicia (Grant No. INCITE09.296.035PR and Centro singular de investigación de
Galicia accreditation 2019-2022), the Spanish Consolider-Ingenio 2010
Programme CPAN (CSD2007-00042), and the European Union ERDF. AGMC is grateful
to the Spanish Ministry of Science, Innovation and Universities, and the
European Social Fund for the funding of his predoctoral research activity
(_Ayuda para contratos predoctorales para la formación de doctores_ 2019). MHG
thanks the Xunta de Galicia (Consellería de Cultura, Educación y Universidad)
for the funding of his predoctoral activity through _Programa de ayudas a la
etapa predoctoral_ 2021\. KO was supported by the Polish National Science
Centre (Grant NCN 2021/43/D/ST2/01122).

AW thanks Christopher Halcrow for comments.

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# Period-like polynomials for $L$-series associated with half-integral weight
cusp forms

James Branch  University of Nottingham<EMAIL_ADDRESS>,
Nikolaos Diamantis  University of Nottingham
<EMAIL_ADDRESS>,  Wissam Raji  American University of
Beirut and Number Theory Research Unit at Center for Advanced Mathematical
Sciences<EMAIL_ADDRESS>and  Larry Rolen  Vanderbilt University
<EMAIL_ADDRESS>
###### Abstract.

Given the $L$-series of a half-integral weight cusp form, we construct
polynomials behaving similarly to the classical period polynomial of an
integral weight cusp form. We also define a lift of half-integral weight cusp
forms to integral weight cusp forms that are compatible with the $L$-series of
the respective forms.

##  1\. Introduction

The Dirichlet series associated by Shimura to half-integral weight modular
forms in the last section of his original paper [7] has not received as much
attention as its integral weight counterpart. Partly because of its failure to
possess an Euler product, it has not been extensively studied from arithmetic
and algebraic perspectives that have a long history in the case of integral
weight modular forms.

In view of this, the two main purposes of this note are: (i) to attach Eichler
integrals and period-like polynomials to the $L$-series of a half-integral
weight cusp form. This will lead to cohomology classes with coefficients in a
_finite_ dimensional vector space in a way that parallels the Eichler
cohomology of the integral weight case. (ii) to define a lift of half-integral
weight cusp forms to integral weight cusp forms that are compatible with the
$L$-series of the respective forms.

This section presents special cases of the two main results of the note. The
first one is obtained from Theorem 3.1 in the special case $N=4$, $k$ such
that $4|(k-\frac{5}{2})$ and $a=k-2$:

######  Theorem 1.1.

Let $k\in\frac{1}{2}+\mathbb{Z}$ such that $k-\frac{5}{2}\in 4\mathbb{N}$ and
let $f$ be a cusp form of weight $k$ for $\Gamma_{0}(4)$ such that
$f(-1/(4z))=(-2iz)^{k}f(z).$ For each $z$ in the upper half-plane
$\mathfrak{H}$ define the “Eichler integral”

$F(z)=\Gamma(k-1)\int_{z}^{i\infty}f(w)\left(\sum_{n=0}^{k-\frac{5}{2}}\left(\frac{(4i)^{n}}{n!\Gamma(k-1-n)}+\frac{4^{k-n-\frac{11}{4}}i^{\frac{5}{2}-n}w^{-\frac{1}{2}}}{(k-\frac{5}{2}-n)!\Gamma(n+\frac{3}{2})}\right)z^{n}w^{k-2-n}\right)dw.$

Then,

  1. (1)

For each $z\in\mathfrak{H}$, $P(z):=F(z)-F(-1/(4z))(-2iz)^{k-\frac{5}{2}}$ is
a polynomial of degree at most $k-\frac{5}{2}$ in $z.$

  2. (2)

We have

$P(z)=i^{k-1}\Gamma(k-1)\sum_{n=0}^{k-\frac{5}{2}}\left(\frac{4^{n}\Lambda_{f}(k-1-n)}{n!\Gamma(k-1-n)}+(-1)^{n+1}\frac{4^{k-n-\frac{11}{4}}\Lambda_{f}(k-\frac{3}{2}-n)}{(k-\frac{5}{2}-n)!\Gamma(n+\frac{3}{2})}\right)z^{n},$

where $\Lambda_{f}(s)$ is the $L$-series of $f$ (to be defined precisely in
the next section).

The polynomial $P$ shares some of the defining features of the _period
polynomial_ of integral weight forms: it encodes certain values of
$\Lambda_{f}$ inside the interval $[1,k-1]$ and satisfies one of the period
relations. Indeed, in Prop. 3.2 we will show that $P$ matches exactly the
$(k-\frac{3}{2})$-th partial sum in the Taylor expansion of the (“symmetrised”
version of the) Eichler cocycle, as extended to all real weights in [1]. Since
then the period polynomial equals the value at the Fricke involution of an
Eichler cocycle (based at $i\infty$), we think of $P$ as an analogue of the
period polynomial for half-integral weight cusp forms. The period relation of
$P$ (see Th. 3.1 (i)) is not immediately visible from its expression in part
(2) of Th. 1.1, but it is deduced from its relation with the analogue of the
Eichler integral $F(z)$, as is the case with the classical period polynomial.

A first, to our knowledge, attempt to develop a cohomology for $L$-series of
half-integral cusp forms was made in [3]. Its main construction encodes
$L$-values inside $[1,k-1]$ and satisfies one of the period relations too.
However, the analogue of $P$ in [3] belongs to an infinite dimensional space
whereas $P$ is a polynomial of degree $\leq k-\frac{5}{2}.$ Another difference
is that, as will be seen in the general form of the theorem in the sequel, our
polynomial $P$ can be made to encode values of $\Lambda_{f}(s)$ at a larger
class of finite “arithmetic sequences” inside $[1,k-1]$.

The second main result presented here is a special case of Theorem 4.1:

######  Theorem 1.2.

Let $k\in\frac{1}{2}+\mathbb{Z}$ such that $k-\frac{5}{2}\in 4\mathbb{N}$. For
each cusp form $f$ of weight $k$ for $\Gamma_{0}(4)$ such that
$f(-1/(4z))=(-2iz)^{k}f(z)$ there exists a unique pair $(g,h)$ of cusp forms
of (integral) weight $k-\frac{1}{2}$ and level $4$ such that, for each
$n=0,\dots,k-\frac{5}{2}$, we have

(1.1)
$\left(\frac{2^{2n}}{n!\Gamma(k-1-n)}+(-1)^{n+1}\frac{2^{2k-5-2n}}{(k-\frac{5}{2}-n)!\Gamma(n+\frac{3}{2})}\right)\Lambda_{f}\left(k-\frac{5}{4}-n\right)\\\
=\frac{i^{\frac{3}{4}}}{\Gamma(k-1)}\binom{k-\frac{5}{2}}{n}\left(i^{n+1}\Lambda_{g}\left(k-n-\frac{3}{2}\right)+i^{-n-1}\overline{\Lambda_{h}\left(k-n-\frac{3}{2}\right)}\right)$

The main characteristic of the “lift” induced by Theorem 1.2 is that it is
compatible, in the sense of eq. (1.1), with the $L$-series of the half-
integral weight form and that of the corresponding integral weight forms. On
the other hand, there does not seem to be any compatibility with the Hecke
action, and the “lifted” forms are not explicitly given in terms of Fourier
expansions, as was the case of the Shimura lift. One also notices that the
weight of our “lift” is half the weight of the Shimura lift. Because of those
differences in their behaviour and the different problems they were each
designed to resolve, it does not seem likely that our lift is related to the
Shimura lift.

The lack of compatibility with the Hecke action was expected: The identity of
Theorem 1.2 expresses the “critical” values of $L$-series of a half-integral
weight cusp form directly in terms of $L$-values of _integral_ weight forms.
If our lift were compatible with the Hecke action, then we could immediately
deduce algebraicity results about the $L$-values of some half-integral weight
Hecke eigenforms from the corresponding results in the integral weight case.
However, algebraic properties so similar to those of the integral-weight
$L$-values are not expected for $L$-series of half-integral weight forms.
Therefore, additional input is required to derive algebraic information from
the translation of half-integral weight $L$-values to integral weight
$L$-values provided by Theorem 1.2.

It was also mentioned above that our lift is not given explicitly via Fourier
expansions. Nevertheless, it does have an explicit expression through the
explicit inverse of the Eichler-Shimura map given in [6]. Theorem 5.4 allows
us to obtain the integral weight lift of a given form $f$ of half-integral
weight $k$ from $k-3/2$ values of the $L$-series associated with $f$. To
formulate and prove this result we develop, in Sect. 5, a reformulation of our
lift of Theorem 4.1 which may be of independent interest. This reformulation
(Prop. 5.2), applies only to _odd_ values of $n$ and relies on a version of
the Eichler-Shimura Isomorphism (Th. 5.1) which considers separately the even
and the odd part of a polynomial. As a result, it makes it possible to express
those $L$-values of $f$ that correspond to odd $n$ in terms of a single
integral weight form. By contrast, Th. 4.1 requires two of them, but accounts
for $L$-values responding to both odd and even $n$.

## Data statement

The authors confirm that this manuscript has no associated data.

## Ethics declarations

The authors declare that there are no conflict of interests.

## Acknowledgements

The authors thank V. Pasol and A. Popa, as well as the anonymous referee, for
helpful comments on this work. Part of the work was done while ND was visiting
Max Planck Institute for Mathematics in Bonn and the University of Patras,
both of whose hospitality he acknowledges. Research on this work is partially
supported by the second author’s EPSRC Grant EP/S032460/1. The third named
author would like to thank the Center for Advanced Mathematical Sciences
(CAMS) at the American University of Beirut for the support of the number
theory unit. This work was supported by a grant from the Simons Foundation
(853830, LR). The fourth author is also grateful for support from a Dean’s
Faculty Fellowship from Vanderbilt University, and to the Max Planck Institute
for Mathematics in Bonn for its hospitality and financial support.

##  2\. Terminology and notation

We first fix the terminology and the notation we will be using. They will
mostly be consistent with those of [7] and [3].

Let $k\in\frac{1}{2}+\mathbb{Z}$ and $N\in 4\mathbb{N}.$ We let
$\left(\frac{c}{d}\right)$ be the Kronecker symbol. For an odd integer $d$, we
set

(2.1) $\epsilon_{d}:=\begin{cases}1&\text{ if }d\equiv 1\bmod{4},\\\ i&\text{
if }d\equiv 3\bmod{4},\end{cases}$

so that $\epsilon_{d}^{2}=\left(\frac{-1}{d}\right)$. We set the implied
logarithm to equal its principal branch so that $-\pi<$arg$(z)\leq\pi$. We
define the action $|_{k}$ of $\Gamma_{0}(N)$ on smooth functions $f$ on
$\mathfrak{H}$ as follows:

(2.2)
$(f|_{k}\gamma)(z):=\left(\frac{c}{d}\right)\epsilon_{d}^{2k}(cz+d)^{-k}f(\gamma
z)\qquad\text{ for all }\gamma=\begin{pmatrix}*&*\\\
c&d\end{pmatrix}\in\Gamma_{0}(N).$

Further, let $W_{N}=\begin{pmatrix}0&-1/\sqrt{N}\\\ \sqrt{N}&0\end{pmatrix}$
and $\Gamma_{0}(N)^{*}=\langle W_{N},\Gamma_{0}(N)\rangle.$ We set

(2.3) $(f|_{k}W_{N})(z):=(-i\sqrt{N}z)^{-k}f(-1/(Nz)).$

We extend the action to $\mathbb{C}[\Gamma_{0}(N)^{*}]$ by linearity.

For $n\in\mathbb{Z}$ we let, as usual,

(2.4) $(f|_{n}\gamma)(z):=(cz+d)^{-n}f(\gamma z)\qquad\text{ for all
}\gamma=\begin{pmatrix}*&*\\\
c&d\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{R}).$

Let $k\in\frac{1}{2}\mathbb{Z}$. If $\Gamma$ is either a subgroup of finite
index in $\operatorname{SL}_{2}(\mathbb{Z})$ or $\Gamma^{*}_{0}(N)$ for some
$N$, and $\chi$ a character on $\Gamma$, We set

(2.5) $f|_{k,\chi}\gamma:=\overline{\chi(\gamma)}f|_{k}\gamma\qquad\text{ for
all }\gamma\in\Gamma.$

We will denote the space of cusp forms of weight $k$ and character $\chi$ for
$\Gamma$ by $S_{k}(\Gamma,\chi)$. If $\chi$ is the trivial character, we write
$S_{k}(\Gamma)$.

For $f,g\in S_{k}(\Gamma,\chi)$, we define the Petersson scalar product as

$(f,g)=\int_{\Gamma\backslash\mathfrak{H}}f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^{2}}\quad\mbox{where}\,\,z=x+iy.$

Let $\lambda$ be the width of the cusp $\infty$. For integral or half-integral
weights $k$, we attach to

$f(z)=\sum_{n\geq 1}a_{f}(n)e^{\frac{2\pi inz}{\lambda}}\in
S_{k}(\Gamma,\chi)$

the _$L$ -series_

$L_{f}(s)=\sum_{n\geq 1}\frac{a_{f}(n)}{n^{s}}.$

This is absolutely convergent for $\Re(s)\gg 1$ and can be analytically
continued to the entire complex plane. Its “completed” version is

(2.6) $\Lambda_{f}(s)=\frac{\Gamma(s)\lambda^{s}}{(2\pi)^{s}}\sum_{n\geq
1}\frac{a_{f}(n)}{n^{s}}=\int_{0}^{\infty}f(it)t^{s}\frac{dt}{t}.$

For $\chi$ trivial, it satisfies the functional equation

(2.7)
$\Lambda_{f}(s)=N^{\frac{k}{2}-s}\Lambda_{f|_{k}W_{N}}(k-s),\,\,\,\,\text{if
$k\in\frac{1}{2}+\mathbb{Z}$
and}\,\,\Lambda_{f}(s)=i^{k}N^{\frac{k}{2}-s}\Lambda_{f|_{k}W_{N}}(k-s),\,\,\,\text{if
$k\in\mathbb{Z}$.}$

Finally, we fix the following notation: We set

$T:=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\qquad S:=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\qquad U:=TS=\begin{pmatrix}1&-1\\\ 1&0\end{pmatrix}.$

For $z,w$ not necessarily non-negative integers, set

$\binom{z}{w}:=\frac{\Gamma(z+1)}{\Gamma(w+1)\Gamma(z-w+1)}.$

##  3\. An analogue of the period polynomial

Let $f$ be a cusp form of weight $k\in\frac{1}{2}+\mathbb{Z}$ for
$\Gamma_{0}(N)$ such that $f|_{k}W_{N}=f$. Fix $a\in[0,2k-9/2]$ and set

$P_{a}(z):=\int_{0}^{i\infty}f(w)\Phi_{a}(z,w)dw,$

where

$\Phi_{a}(z,w):=\sum_{n=0}^{k-5/2}\left[\binom{k-2}{n}(iNz)^{n}w^{a-n}+\frac{i^{k}}{\sqrt[4]{N}}\binom{k-2}{n+1/2}(iz)^{n}(-Nw)^{2k-9/2-a-n}\right].$

######  Theorem 3.1.

Let $k\in\frac{1}{2}+\mathbb{Z}$ with $k>5/2$. Suppose that $f\in
S_{k}(\Gamma^{*}_{0}(N))$ and $a\in[0,2k-9/2]$. With the above notation, set

$F_{a}(z):=\int_{z}^{i\infty}f(w)\Phi_{a}(z,w)dw.$

  1. (i)

For $z\in\mathfrak{H}$ we have

$-i^{k-5/2}F_{a}|_{5/2-k}W_{N}+F_{a}=P_{a}.$

Therefore, if $\chi$ is a character on $\Gamma^{*}_{0}(N)$ such that
$\chi(W_{N})=i^{\frac{5}{2}-k}$, then

(3.1) $P_{a}|_{5/2-k,\chi}(W_{N}+1)=0.$

In particular, if $4|(k-5/2)$, then

$P_{a}|_{5/2-k}(W_{N}+1)=0.$

  2. (ii)

For each $z\in\mathbb{C},$

$\displaystyle P_{a}(z)$ $\displaystyle=$ $\displaystyle
i^{a+1}\sum_{n=0}^{k-5/2}\left[\binom{k-2}{n}N^{n}\Lambda_{f}(a+1-n)\right.$
(3.2) $\displaystyle+$
$\displaystyle\left.\binom{k-2}{n+1/2}N^{2k-a-n-\frac{19}{4}}i^{2n-k+\frac{1}{2}}\Lambda_{f}(2k-7/2-a-n)\right]z^{n}.$

###### Proof.

We first show that

(3.3)
$-(i\sqrt{N}z)^{k-5/2}(-i\sqrt{N}w)^{k-2}\Phi_{a}(W_{N}z,W_{N}w)=\Phi_{a}(z,w).$

Indeed, the left-hand side is

$\displaystyle-(i\sqrt{N}z)^{k-5/2}(-i\sqrt{N}w)^{k-2}\sum_{n=0}^{k-5/2}\left[\binom{k-2}{n}(iz)^{-n}(-Nw)^{n-a}\right.$
$\displaystyle\left.+\frac{i^{k}}{\sqrt[4]{N}}\binom{k-2}{n+1/2}(iNz)^{-n}w^{n-2k+9/2+a}\right].$

Observe that

$\binom{k-2}{k-5/2-n}=\binom{k-2}{n+1/2},$

so that the change of variables $n\mapsto k-5/2-n$ yields

(3.4)
$\displaystyle-(i\sqrt{N}z)^{k-5/2}(-i\sqrt{N}w)^{k-2}\sum_{n=0}^{k-5/2}\left[\binom{k-2}{n+1/2}(iz)^{n-k+5/2}(-Nw)^{k-5/2-n-a}\right.$
(3.5)
$\displaystyle\left.+\frac{i^{k}}{\sqrt[4]{N}}\binom{k-2}{n}(iNz)^{n-k+5/2}w^{-k+2-n+a}\right].$

Since $w\in\mathfrak{H}$, we have, for every $t\in\mathbb{R}$,
$(-w)^{t}=e^{-\pi it}w^{t}.$ A routine computation shows

$-(i\sqrt{N}z)^{k-5/2}(-i\sqrt{N}w)^{k-2}(iz)^{n-k+5/2}(-Nw)^{k-5/2-n-a}=\frac{i^{k}}{\sqrt[4]{N}}(iz)^{n}(-Nw)^{2k-9/2-a-n}$

and similarly:

$-(i\sqrt{N}z)^{k-5/2}(-i\sqrt{N}w)^{k-2}\frac{i^{k}}{\sqrt[4]{N}}(iNz)^{n-k+5/2}w^{2-k-n+a}=(iNz)^{n}w^{a-n}.$

Therefore, (3.4) equals $\Phi_{a}(z,w)$, establishing (3.3).

With (3.3) we now have:

(3.6) $\displaystyle i^{k-5/2}(F_{a}|_{5/2-k}W_{N})(z)$ $\displaystyle=$
$\displaystyle(i\sqrt{N}z)^{k-5/2}\int_{W_{N}z}^{i\infty}f(w)\Phi_{a}(W_{N}z,w)dw$
$\displaystyle=$
$\displaystyle(i\sqrt{N}z)^{k-5/2}\int_{z}^{0}f(W_{N}w)\Phi_{a}(W_{N}z,W_{N}w)d(W_{N}w)$
$\displaystyle=$
$\displaystyle-(i\sqrt{N}z)^{k-5/2}\int_{z}^{0}f(w)(-i\sqrt{N}w)^{k}\Phi_{a}(W_{N}z,W_{N}w)\frac{dw}{(-i\sqrt{N}w)^{2}}$
$\displaystyle=$
$\displaystyle\int_{z}^{0}f(w)\Phi_{a}(z,w)dw=F_{a}(z)-P_{a}(z).$

To prove (3.1), we have from (3.6),

$P_{a}|_{\frac{5}{2}-k,\chi}(1+W_{N})=F_{a}|_{\frac{5}{2}-k,\chi}(1-W_{N})|_{\frac{5}{2}-k,\chi}(1+W_{N})=F_{a}|_{\frac{5}{2}-k,\chi}(1-W_{N}^{2})=0.$

To show ((ii)), we expand the defining expression for $P_{a}$ and use the
integral formula for $\Lambda_{f}$ in (2.6). ∎

This theorem and (2.7) show that $P_{a}$ can be thought of as a “period
polynomial” encoding the $L$-values of $f$ at $a+1,a,a-1,\cdots,a-k+7/2$.
Among the various choices of $a$, the most “canonical” is $a=k-2$ because
then, our “period polynomial” $P_{a}$ becomes entirely consistent with the
Eichler cohomology attached to general weight cusp forms, as in [1].
Specifically, the Eichler cocycle on which that cohomology is based is induced
by the assignment (Section 2.2 of [1]) to $f\in S_{k}(\Gamma)$ of the map

$\Gamma\ni\gamma\longrightarrow\psi_{f,\gamma}^{\infty}(z):=\int_{\gamma^{-1}i\infty}^{i\infty}f(w)(w-z)^{k-2}dw$

where $\psi_{f,\gamma}^{\infty}$ is defined on the _lower_ half-plane
$\bar{\mathfrak{H}}$. For $\gamma=W_{N}$, the integral giving
$\psi_{f,W_{N}}^{\infty}\left(x\right)$ is well-defined for $x>0$ as well. We
then have the following relation between our “period polynomial” $P_{k-2}$ and
$\psi_{f,W_{N}}^{\infty}\left(x\right)$.

######  Proposition 3.2.

For each $x>1$, we have

$P_{k-2}(ix)=\psi_{f,W_{N}}^{\infty}\left(Nx\right)-i^{1-2k}\psi_{f,W_{N}}^{\infty}\left(1/x\right)(\sqrt{N}x)^{k-\frac{5}{2}}+O(x^{k-\frac{3}{2}}).$

###### Proof.

We first note that

(3.7)
$\psi_{f,W_{N}}^{\infty}\left(Nx\right)-i^{1-2k}\psi_{f,W_{N}}^{\infty}\left(1/x\right)(\sqrt{N}x)^{k-\frac{5}{2}}\\\
=i^{k-1}\int_{0}^{\infty}f(it)\left((1+iNxt^{-1})^{k-2}-i^{1-2k}(1+ix^{-1}t^{-1})^{k-2}(\sqrt{N}x)^{k-\frac{5}{2}}\right)t^{k-2}dt.$

By Taylor’s formula we have, for each $M\in\mathbb{N}$,

$(1+iNxt^{-1})^{k-2}=\\\
\sum_{n=0}^{M-1}\binom{k-2}{n}(iNxt^{-1})^{n}+O_{M}\left(\int_{0}^{1}(1-y)^{M-1}(1+iNyxt^{-1})^{k-2-M}(iNxt^{-1})^{M}dy\right).$

If $M>k-2$ the error term is $O_{M}((xt^{-1})^{M})$. Likewise, again if
$M>k-2$, the error term of $(1+ix^{-1}t^{-1})^{k-2}$ is
$O_{M}((x^{-1}t^{-1})^{M})$. Therefore for $M=k-\frac{3}{2}$, we deduce, for
$x>1$,

(3.8)
$\psi_{f,W_{N}}^{\infty}\left(Nx\right)-i^{1-2k}\psi_{f,W_{N}}^{\infty}\left(1/x\right)(\sqrt{N}x)^{k-\frac{5}{2}}\\\
=i^{k-1}\sum_{n=0}^{k-\frac{5}{2}}\binom{k-2}{n}\left((iNx)^{n}-i^{1-2k}(ix^{-1})^{n}(\sqrt{N}x)^{k-\frac{5}{2}}\right)\Lambda_{f}(k-1-n)+O(x^{k-\frac{3}{2}})$

where the implied constant is independent of $x$. The change of variables
$n\to k-\frac{5}{2}-n$, followed by (2.7) in the second sum, implies the
result. ∎

###  3.1. Comparison with the period function of [3]

Another function encoding special values of $L_{f}$ is given in [3]. The
result in [3] is stated for cusp forms on Hecke groups, but, thanks to the
embedding of $S_{k}(\Gamma^{*}_{0}(N))$ into a space of cusp forms for Hecke
groups ((8.1) of [3]) it can be formulated for the modular forms studied in
Theorem 3.1:

######  Proposition 3.3.

[3] Let $k\in\frac{1}{2}+\mathbb{Z}$ and $f(z)=\sum a_{f}(n)e^{2\pi inz}\in
S_{k}(\Gamma_{0}(N))$ such that $f|_{k}W_{N}=f$. For each $z\in\mathfrak{H}$
set

$\mathcal{E}^{*}_{f}(z)=\frac{1}{\sqrt{\pi}}\sum_{n\geq
1}\frac{a_{f}(n)}{n^{k-1}}\left(e^{-2\pi inz}\Gamma\left(\frac{1}{2},-2\pi
inz\right)-\frac{1}{\sqrt{-2\pi inz}}\right).$

Then, for all $z\in\mathfrak{H},$

(3.9)
$\left(\mathcal{E}^{*}_{f}|_{2-k}(1-W_{N})\right)(z)=\sum_{n=0}^{k-\frac{3}{2}}\left(\frac{L_{f}(k-n-1)}{\Gamma(n+1)}+\frac{L_{f}(k-n-\frac{1}{2})}{\Gamma(n+\frac{1}{2})}\left(\frac{2\pi
z}{i}\right)^{-\frac{1}{2}}\right)\left(\frac{2\pi z}{i}\right)^{n}.$

The values of $L_{f}(s)$ at $k-1,\dots\frac{3}{2}$ appearing in the right-hand
side of (3.9) are also encoded by $P_{k-2}$. However, the function on the
right-hand side of (3.9) does not belong to a finite-dimensional space closed
under the action of the group. Further, the “Eichler integral”
$\mathcal{E}^{*}_{f}(z)$ is defined as a series. To complete the comparison of
our construction with the “period function” of [3], we show how the main piece
of $\mathcal{E}^{*}_{f}(z)$ can nevertheless be expressed as an integral too.

######  Proposition 3.4.

With the notation of Theorem (3.1), for each $z\in\mathfrak{H}$,

$\mathcal{E}^{*}_{f}(z)=\alpha_{k}(-iz)^{\frac{1}{2}}\int_{z}^{i\infty}F_{f}(w)(w-z)^{k-\frac{5}{2}}dw-\frac{1}{\sqrt{-2\pi^{2}iz}}L_{f}\left(k-\frac{1}{2}\right)$

where

$F_{f}(w):=\int_{0}^{\infty}f\left(xw\right)x^{k-\frac{3}{2}}(x+1)^{-\frac{1}{2}}dx$

and $\alpha_{k}=(-2\pi i)^{k-1}/(\pi^{\frac{1}{2}}(k-\frac{5}{2})!).$

###### Proof.

We first recall ([5], (8.19.1), (8.19.3)) that, for Re$(w)>0,$

$\Gamma\left(\frac{1}{2},w\right)=w^{\frac{1}{2}}\int_{1}^{\infty}e^{-wt}t^{-\frac{1}{2}}dt.$

Therefore, for $z\in\mathfrak{H},$

$\sum_{n\geq 1}\frac{a_{f}(n)}{n^{k-1}}\frac{e^{-2\pi
inz}}{\sqrt{\pi}}\Gamma\left(\frac{1}{2},-2\pi
inz\right)=\frac{1}{\sqrt{\pi}}\sum_{n\geq 1}\frac{a_{f}(n)}{n^{k-1}}e^{-2\pi
inz}\left(-2\pi inz\right)^{\frac{1}{2}}\int_{1}^{\infty}e^{2\pi
inzt}t^{-\frac{1}{2}}dt\\\
=(-2iz)^{\frac{1}{2}}\int_{1}^{\infty}t^{-\frac{1}{2}}\left(\sum_{n\geq
1}\frac{a_{f}(n)}{n^{k-\frac{3}{2}}}e^{2\pi inz(t-1)}\right)dt.$

By the theory of usual (integral weight) Eichler integrals, followed by the
changes of variables $x=t-1$ and $w_{1}=w/x$, this equals

$(-2iz)^{\frac{1}{2}}\int_{1}^{\infty}t^{-\frac{1}{2}}\frac{(-2\pi
i)^{k-\frac{3}{2}}}{\Gamma(k-\frac{3}{2})}\int_{z(t-1)}^{i\infty}f(w)\left(w-z(t-1)\right)^{k-\frac{5}{2}}dwdt\\\
\\\
=\alpha_{k}z^{\frac{1}{2}}\int_{0}^{\infty}x^{k-\frac{3}{2}}(x+1)^{-\frac{1}{2}}\int_{z}^{i\infty}f(xw_{1})\left(w_{1}-z\right)^{k-\frac{5}{2}}dw_{1}dx.$

A change in the order of integration implies the formula. ∎

##  4\. An Eichler cocycle

In this section, we will first use Theorem 3.1 to construct an Eichler
cocycle, with coefficients in the space $\mathbb{C}_{k-5/2}[z]$. We maintain
the notation and assumptions of the last section.

Since $4|N$, $\Gamma_{0}(N)/\\{\pm 1\\}$ is torsion-free and hence free on a
set of generators $\\{\gamma_{j}\\}_{j=1}^{2g+h-1}$, where $\gamma_{1}=T$, $g$
is the genus and $h$ the number of inequivalent cusps of $\Gamma_{0}(N)$.
([2], Prop. 2.4; here by $T,\gamma_{j}$ etc. we mean the images of those
elements of $\Gamma_{0}(N)$ into $\Gamma_{0}(N)/\\{\pm 1\\}$), We also have
$W_{N}\Gamma_{0}(N)=\Gamma_{0}(N)W_{N}$ and thus
$\Gamma^{*}_{0}(N)=\langle\Gamma_{0}(N),W_{N}\rangle$ is generated by
$\\{\gamma_{j}\\}\cup\\{W_{N}\\}$ with only relations $(-1)^{2}=1$ and
$W_{N}^{4}=1$.

From the above, we first deduce that there is always a character $\chi$ on
$\Gamma^{*}_{0}(N)$ such as prescribed in Theorem 3.1, that is such that
$\chi(W_{N})=i^{5/2-k}$. Indeed, since $i^{4(5/2-k)}=1$, the character induced
by the assignment $\chi(-1)=(-1)^{5/2-k}$, $\chi(W_{N})=i^{5/2-k}$ and
$\chi(\gamma_{i})=1$ for $i=1,\dots 2g+h-1$ is well-defined.

Further, let, for $a\in[0,2k-\frac{9}{2}]$, $P_{a}$ be the polynomial defined
in ((ii)). Consider the polynomial

$\hat{P}_{a}(z):=P_{a}(2z/\sqrt{N}).$

It is then easy to deduce from Theorem 3.1 that

(4.1) $\hat{P}_{a}|_{\frac{5}{2}-k,\chi}(W_{4}+1)=0$

for any character $\chi$ on $\Gamma^{*}_{0}(4)$ such that
$\chi(W_{4})=i^{\frac{5}{2}-k}$. (Since there is no risk of confusion, we use
the same notation for both characters). Recall that $\Gamma_{0}(4)/\\{\pm
1\\}$ is freely generated on the generators

$T,\begin{pmatrix}1&0\\\ 4&1\end{pmatrix}$

and thus $\Gamma^{*}_{0}(4)$ is generated by $-1,T,W_{4}$ in
$\Gamma^{*}_{0}(4)$ with only relations $(-1)^{2}=1$ and $W_{4}^{4}=1$. We
consider the map

$\hat{\pi}_{f}:\Gamma^{*}_{0}(4)\to\mathbb{C}_{k-5/2}[z]$

induced by the $1$-cocycle condition from the values

$\hat{\pi}_{f}(W_{4})=\hat{P}_{a}\qquad\mbox{ and
}\qquad\hat{\pi}_{f}(-1)=\hat{\pi}_{f}(T)=0.$

Then $\hat{\pi}_{f}$ is well-defined since by (4.1),

$\hat{\pi}_{f}(W_{4}^{2})=\hat{\pi}_{f}(W_{4})|_{\frac{5}{2}-k,\chi}W_{4}+\hat{\pi}_{f}(W_{4})=\hat{P}_{a}|_{\frac{5}{2}-k}W_{4}+\hat{P}_{a}=0.$

This cocycle induces a non-trivial class in $H_{\text{\rm
par}}^{1}(\Gamma^{*}_{0}(4),\mathbb{C}_{k-5/2}[z])$ where the action of
$\Gamma^{*}_{0}(4)$ on $\mathbb{C}_{k-5/2}[z]$ is $|_{5/2-k,\chi}$.

On the other hand, according to the Eichler-Shimura isomorphism, there is an
isomorphism

$\phi:S_{k-{1/2}}(\Gamma^{*}_{0}(4),\chi)\oplus\overline{S_{k-{1/2}}(\Gamma^{*}_{0}(4),\bar{\chi})}\longrightarrow
H_{\text{\rm par}}^{1}(\Gamma^{*}_{0}(4),\mathbb{C}_{k-5/2}[z])$

induced by the assignment of the following map
$\phi(g,\bar{h}):\Gamma^{*}_{0}(4)\to\mathbb{C}_{k-5/2}[z]$ to $(g,\bar{h})$:

$\phi(g,\bar{h})(\gamma)=\int_{\infty}^{\gamma^{-1}\infty}g(w)(w-z)^{k-{5/2}}dw+\int_{\infty}^{\gamma^{-1}\infty}\overline{h(w)}(\bar{w}-z)^{k-{5/2}}d\bar{w}.$

Therefore, there are unique $g\in S_{k-1/2}(\Gamma^{*}_{0}(4),\chi)$, $h\in
S_{k-1/2}(\Gamma^{*}_{0}(4),\bar{\chi})$ and a polynomial $Q$ in
$\mathbb{C}_{k-5/2}[z]$ such that

$\hat{\pi}_{f}(\gamma)=\phi(g,\bar{h})(\gamma)+Q|_{\frac{5}{2}-k,\chi}(\gamma-1)$

for all $\gamma\in\Gamma^{*}_{0}(4)$. Since
$\hat{\pi}_{f}(T)=\phi(g,\bar{h})(T)=0$, the polynomial $Q$ should vanish too,
because it would otherwise have infinitely many zeros. Therefore, for each
$f\in S_{k}(\Gamma^{*}_{0}(4),\chi),$ there are unique $g\in
S_{k-{1/2}}(\Gamma^{*}_{0}(4),\chi)$, $h\in
S_{k-{1/2}}(\Gamma^{*}_{0}(4),\bar{\chi})$ such that, for all
$\gamma\in\Gamma^{*}_{0}(4),$

(4.2) $\hat{\pi}_{f}(\gamma)=\phi(g,\bar{h})(\gamma).$

An application of the binomial theorem combined with the integral form of
$\Lambda_{g}(s),\Lambda_{h}(s)$ implies that $\phi(g,\bar{h})(W_{4})$ can be
expressed as a polynomial with coefficients involving values of the critical
values of $g,\bar{h}$. In our case this gives

(4.3) $\phi(g,\bar{h})(W_{4})\\\
=-\sum_{n=0}^{k-5/2}\binom{k-5/2}{n}(-z)^{n}\left(i^{k-n-3/2}\Lambda_{g}(k-n-3/2)+i^{3/2+n-k}\overline{\Lambda_{h}(k-n-3/2)}\right).$

Comparing coefficients with the expression for
$\hat{\pi}_{f}(W_{4})=\hat{P}_{a}$ in ((ii)), we deduce

######  Theorem 4.1.

Let $k\in\frac{1}{2}+\mathbb{Z}$ with $k>\frac{5}{2}.$ For each cusp form $f$
of weight $k$ for $\Gamma_{0}(N)$ such that
$f(-1/(Nz))=(-i\sqrt{N}z)^{k}f(z)$, a character $\chi$ on $\Gamma_{0}^{*}(4)$
such that $\chi(W_{4})=i^{\frac{5}{2}-k}$ and for each
$a\in[0,2k-\frac{9}{2}]$, there exists a unique pair $(g,h)$ of cusp forms of
(integral) weight $k-\frac{1}{2}$, level $4$ and character $\chi$ such that,
for each $n=0,\dots,k-\frac{5}{2}$, we have

(4.4)
$i^{a+1}2^{n}\left(\binom{k-2}{n}N^{\frac{n}{2}}\Lambda_{f}(a+1-n)+i^{2n+\frac{1}{2}-k}\binom{k-2}{n+\frac{1}{2}}N^{2k-a-\frac{3n}{2}-\frac{19}{4}}\Lambda_{f}(2k-\frac{7}{2}-a-n)\right)\\\
=\binom{k-\frac{5}{2}}{n}\left(i^{n+\frac{1}{2}+k}\Lambda_{g}(k-n-\frac{3}{2})+i^{-n-\frac{1}{2}-k}\overline{\Lambda_{h}(k-n-\frac{3}{2})}\right).$

Theorem 1.2 is a special case of this for $N=4$, $k$ such that
$4|(k-\frac{5}{2})$ and $a=k-\frac{9}{4}$.

###  4.1. A special case

In low dimensions, Theorem 4.1 can assume a simpler form. For example, we can
specialise to $k$ such that $\mathrm{dim}S_{k-1/2}(\Gamma_{0}^{*}(4))=1$ and
$a=k-9/4$. Then, if $g$ is a normalised eigenform in
$S_{k-1/2}(\Gamma_{0}^{*}(4))$, which, in particular, implies that $g$ has
real Fourier coefficients at infinity, Theorem 4.1 becomes

######  Corollary 4.2.

Let $k\in\frac{5}{2}+4\mathbb{N}$ such that $S_{k-1/2}(\Gamma_{0}^{*}(4))$ is
$1$-dimensional, spanned by a normalised cusp form $g$. For each cusp form $f$
of weight $k$ for $\Gamma_{0}(N)$ such that $f(-1/(Nz))=(-i\sqrt{N}z)^{k}f(z)$
there exists a $\lambda_{f}\in\mathbb{C}$ such that, for each
$n=0,\dots,k-\frac{5}{2}$, we have

(4.5)
$\Lambda_{f}\left(k-\frac{5}{4}-n\right)=C_{k,N,n}\Lambda_{g}\left(k-\frac{3}{2}-n\right),$

where

$C_{k,N,n}=\binom{k-\frac{5}{2}}{n}2^{-n}\left(i^{\frac{7}{4}+n}+\lambda_{f}i^{-n-\frac{1}{4}}\right)\left[\binom{k-2}{n}N^{\frac{n}{2}}+(-1)^{n+1}\binom{k-2}{n+\frac{1}{2}}N^{k-\frac{5}{2}-\frac{3n}{2}}\right]^{-1}.$

Remark. In view of the corollary, it is tempting to try to deduce algebraicity
dependence for $L$-values of $f$ from the corresponding properties for
integral weight forms (e.g. via Manin’s periods’ theorem). However, the number
of $L$-values of $f$ whose algebraic dependence we would like to derive is the
same as the number of independent $L$-values of $g$ in the RHS of (4.5). In
the case of $k=13/2$ for instance, the two values for odd
$n\in\\{0,\dots,4\\}$ are essentially the same because of the functional
equation of $\Lambda_{f}(s)$, leaving us with a single value of
$\Lambda_{f}(s)$, trivially accounted for by the single constant
$\lambda_{f}$.

##  5\. An explicit form of the lift

The lift described in Theorem 4.1 can be made explicit via the explicit
inverse of the Eichler-Shimura map of [6]. To this end, we first reformulate
the construction leading to Theorem 4.1 in a way that parallels the
decomposition of the classical period polynomial into an “even” and “odd”
part. For simplicity, in this section, we will be working with weight $k$ such
that $4|(k-\frac{5}{2}).$

We will first review some cohomological constructions used in [6] and some
notation from [4] which we will then apply to half-integral weight cusp forms.

We first consider the Hecke group $H(2)$ generated by the images of $T^{2}$
and $S$ under the natural projection of $\operatorname{SL}_{2}(\mathbb{Z})$
onto PSL${}_{2}(\mathbb{Z})$. Since $4|(k-\frac{5}{2})$, it will be legitimate
to use the same notation for elements of $\operatorname{SL}_{2}(\mathbb{Z})$
and their images in PSL${}_{2}(\mathbb{Z})$. The group $H(2)$ has only the
relation $S^{2}=1$ (see Sect. 5 of [3] for the summary of some basic
properties of the Hecke groups).

We now recall a construction of [4] that is used to provide an explicit
formula for cocycles on $\operatorname{PSL}_{2}(\mathbb{Z})$ induced by
cocycles on $H(2)$. It is easy to see that a set of representatives of
$H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})$ is

$\\{1,T,U\\},\qquad\mbox{where}\,\,U=TS.$

To be able to keep track of coset representatives, we denote by
$u:\operatorname{PSL}_{2}(\mathbb{Z})\to\\{1,T,U\\}$ the map which sends $x$
to its corresponding coset representative in
$H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})$. Specifically, let
$x\in\operatorname{PSL}_{2}(\mathbb{Z})$. If $H(2)x=H(2)$ (resp.
$H(2)T,H(2)U$), set $u(x)=1$, (resp. $T,U$). Some values of $u$ that will be
repeatedly used below tacitly are: $u(T^{-1})=T,u(TST^{-1})=U.$

For each $x,g\in\operatorname{PSL}_{2}(\mathbb{Z})$, set

(5.1) $\kappa_{x,g}:=u(x)gu(xg)^{-1}\in H(2).$

One notices that, if $x\in H(2)$, $\kappa_{x,g}=gu(g)^{-1}$ and, if $x,g\in
H(2)$, $\kappa_{x,g}=g$. Further, if $H(2)x=H(2)x^{\prime}$, then
$\kappa_{x,g}=\kappa_{x^{\prime},g}$ and thus $\kappa_{x,g}$ is well-defined
as a function of
$\left(H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})\right)\times\operatorname{PSL}_{2}(\mathbb{Z}).$
Finally, $\kappa$ satisfies the relation

(5.2) $\kappa_{x,g_{1}g_{2}}=\kappa_{x,g_{1}}\kappa_{xg_{1},g_{2}}.$

Next, we consider the space

$I_{k}:=\text{Ind}^{\text{PSL}_{2}(\mathbb{Z})}_{H(2)}(\mathbb{C}_{k-5/2}[z])=\\{f:H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})\to\mathbb{C}_{k-5/2}[z]\\}.$

Since $\operatorname{PSL}_{2}(\mathbb{Z})$ acts on $\mathbb{C}_{k-5/2}[z]$,
there is an action of $\operatorname{PSL}_{2}(\mathbb{Z})$ on $I_{k}$, given,
for $v\in I_{k}$, by

$(v||g)(x):=v(xg^{-1})|_{\frac{5}{2}-k}g\quad\mbox{for all}\,\,x\in
H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z}),\,g\in\operatorname{PSL}_{2}(\mathbb{Z}).$

Let $\sigma:H(2)\to I_{k}$ be a $1$-cocycle with values in $I_{k}$. Then, as
in the case of the classical period polynomial, one sees directly by the
cocycle relation that $\sigma(U)$ belongs to the space

$W:=\\{v\in I_{k};v||(S+1)=v||(U^{2}+U+1)=0\\}$

called the space of _period polynomials_ in [6]. (Note that the condition
$v||(-1)=v$ included in the definition of that space in [6] is not needed here
because $k-\frac{5}{2}$ is even.) We also define the following subspace of
$W$:

$C:=\\{P||(1-S);P\in I_{k},P||T=P\\}.$

As in the classical case again, the spaces $W$ and $C$ can be decomposed as a
direct sum of the $\pm$-eigenspaces of a certain involution. Specifically, let
$\epsilon=\left(\begin{smallmatrix}-1&0\\\ 0&1\end{smallmatrix}\right)$ and
let $\epsilon$ act on a $v\in I_{k}$ so that

$(v||\epsilon)(x):=v(\epsilon
x\epsilon)|_{\frac{5}{2}-k}\epsilon\quad\mbox{for all}\,\,x\in
H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z}).$

Then $I_{k}$ decomposes into $\pm$-eigenspaces under the action of $\epsilon$,
denoted $I_{k}^{\pm}$. Further, since $W$ (resp. $C$) is closed under the
action of $\epsilon$, it also decomposes into $\pm$-eigenspaces $W^{\pm}$
(resp. $C^{\pm}$). We denote that $\pm$-component of $\sigma(U)$ by
$\sigma(U)^{\pm}$, i.e.,

$\sigma(U)^{\pm}=\frac{1}{2}\left(\sigma(U)\pm\sigma(U)||\epsilon\right)\in
W^{\pm}.$

The Eichler-Shimura theorem can be formulated as a pair of isomorphisms to
(quotients of) $W^{\pm}$. We present it here in the arrangement of [6]
(Theorem 2.1) as applied to our setting. First, for each cusp form $g$ of
weight $k-\frac{1}{2}$ for $H(2)$, we let $\rho_{g}$ be the element of $I_{k}$
defined by

(5.3)
$\rho_{g}(\gamma)(z)=\int_{0}^{i\infty}\left(g\Big{|}_{k-\frac{1}{2}}\gamma\right)(w)(w-z)^{k-\frac{5}{2}}dw\quad\mbox{for
all}\,\,\gamma\in H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z}).$

Each polynomial $\rho_{g}(\gamma)(z)$ can be expanded as

(5.4)
$\rho_{g}(\gamma)(z)=\sum_{n=0}^{k-\frac{5}{2}}(-1)^{n}\binom{k-\frac{5}{2}}{n}r_{\gamma,n}(g)z^{k-\frac{5}{2}-n},\quad\mbox{for}\,\,r_{\gamma,n}(g)=i^{n+1}\int_{0}^{\infty}\left(g\Big{|}_{k-\frac{1}{2}}\gamma\right)(it)t^{n}dt.$

Further, it can be shown that $\rho_{g}\in W$ and hence that it can be written
as a sum of its $+$ and $-$ components $\rho^{\pm}_{g}\in W^{\pm}$. With this
notation we have

######  Theorem 5.1.

([6], Theorem 2.1 (Eichler-Shimura)) The assignments $g\to\rho^{+}_{g}$ and
$g\to\rho^{-}_{g}$ induce isomorphisms

$\rho^{+}:S_{k-\frac{1}{2}}(H(2))\cong
W^{+}/C^{+}\quad\mbox{and}\,\,\rho^{-}:S_{k-\frac{1}{2}}(H(2))\cong
W^{-}/C^{-}.$

Since $H(2)$ is $\operatorname{SL}_{2}(\mathbb{Z})$-conjugate to
$\Gamma_{0}(2)$, Prop. 4.4 of [6] shows that

$C^{-}\simeq C_{\Gamma_{0}(2)}^{-}=\\{0\\},$

where $C_{\Gamma_{0}(2)}^{-}$ denotes the analogue for $C^{-}$ for
$\Gamma_{0}(2)$, instead of $H(2)$.

Let $k\in\frac{1}{2}+\mathbb{Z}$ with $k>\frac{5}{2}$ and $4|(k-\frac{5}{2})$.
We will construct an element of $W^{-}$ based on a cusp form $f$ of weight $k$
for $\Gamma_{0}(N)$ such that $f(-1/(Nz))=(-i\sqrt{N}z)^{k}f(z)$. In the last
section, we defined the map
$\hat{\pi}_{f}:\Gamma^{*}_{0}(4)\to\mathbb{C}_{k-5/2}[z]$ induced by the
$1$-cocycle condition from the values $\hat{\pi}_{f}(W_{4})=\hat{P}_{a}$ and
$\hat{\pi}_{f}(-1)=\hat{\pi}_{f}(T)=0.$ Since $4|(k-\frac{5}{2})$, the
$1$-cocycle condition of $\hat{\pi}_{f}$ implies that there is a well-defined
$1$-cocycle $\pi^{\prime}_{f}:H(2)\to\mathbb{C}_{k-5/2}[z]$ which is induced
by the $1$-cocycle relation from the values

$\pi^{\prime}_{f}(S)(z)=\hat{P}_{a}(z/2)\qquad\mbox{ and
}\pi^{\prime}_{f}(T^{2})=0.$

This cocycle gives a non-trivial class in $H_{\text{\rm
par}}^{1}(H(2),\mathbb{C}_{k-5/2}[z])$ where the action of $H(2)$ on
$\mathbb{C}_{k-5/2}[z]$ is $|_{5/2-k}$.

We now define a $1$-cocycle of $\operatorname{PSL}_{2}(\mathbb{Z})$ with
coefficients in $I_{k}$ induced by the cocycle $\pi^{\prime}_{f}$. For each
$g\in\operatorname{PSL}_{2}(\mathbb{Z})$ let $\pi_{f}(g)$ be the element of
$I_{k}$ such that

(5.5)
$\pi_{f}(g)(x):=\pi^{\prime}_{f}(\kappa^{-1}_{x,g^{-1}})|_{\frac{5}{2}-k}u(x)\quad\mbox{for
all}\,\,x\in H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z}).$

By Shapiro’s lemma or, directly with (5.1), we can see that it is a
$1$-cocycle. Then, as mentioned when we introduced $W$ earlier in this
section, $\pi_{f}(U)\in W$ and $\pi_{f}(U)=\pi^{+}_{f}(U)+\pi^{-}_{f}(U)$, for

$\pi_{f}(U)^{\pm}=\frac{1}{2}\left(\pi_{f}(U)\pm\pi_{f}(U)||\epsilon\right)\in
W^{\pm}.$

We can now state and prove another version of our lift of half-integral weight
forms.

######  Proposition 5.2.

Let $k\in\frac{1}{2}+\mathbb{Z}$ such that $k>\frac{5}{2}$ and
$4|(k-\frac{5}{2})$. For each $f\in S_{k}(\Gamma_{0}(N))$ such that
$f|_{k}W_{N}=f$ and for each $a\in[0,2k-\frac{9}{2}]$, there is a $g\in
S_{k-\frac{1}{2}}(\Gamma^{*}_{0}(4))$ such that $g|_{k-\frac{1}{2}}W_{4}=g$
and, for all odd $n\in\\{1,\dots,k-\frac{7}{2}\\}$,

(5.6)
$i^{a+1}\left(\binom{k-2}{n}N^{\frac{n}{2}}\Lambda_{f}(a+1-n)+(-1)^{n+1}\binom{k-2}{n+\frac{1}{2}}N^{2k-a-\frac{3n}{2}-\frac{19}{4}}\Lambda_{f}(2k-\frac{7}{2}-a-n)\right)\\\
=\binom{k-\frac{5}{2}}{n}i^{-1-n}2^{k-\frac{3}{2}-n}\Lambda_{g}\left(k-\frac{3}{2}-n\right).$

###### Proof.

By Theorem 5.1, there exists a $g_{1}\in S_{k}(H(2))$ such that

(5.7) $\pi_{f}(U)^{-}=\rho^{-}_{g_{1}}.$

Since $H(2)\epsilon T\epsilon=H(2)T$ and $H(2)\epsilon
U\epsilon=H(2)T^{-1}S^{-1}=H(2)U$, we see that for each $x\in
H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})$, $\pi_{f}(U)^{-}(x)$ (resp.
$\rho^{-}_{g_{1}}(x)$) is the part of the polynomial $\pi_{f}(U)(x)$ (resp.
$\rho_{g_{1}}(x)$) corresponding to its odd powers. By the definition of
$\pi_{f}$ and it $1$-cocycle condition,

$\pi_{f}(U)(1)=\pi^{\prime}_{f}(\kappa^{-1}_{1,U^{-1}})=\pi^{\prime}_{f}(T^{2}S)=\pi^{\prime}_{f}(T^{2})|_{\frac{5}{2}-k}S+\pi^{\prime}_{f}(S)=0+\pi^{\prime}_{f}(S)=P_{a}(z/\sqrt{N}).$

This, together with ((ii)), shows that the $n$-th coefficient of
$\pi_{f}(U)(1)$ equals the left-hand side of (5.6).

By (5.4), the $n$-th coefficient of $\rho_{g_{1}}(1)$ equals

$(-1)^{n}\binom{k-\frac{5}{2}}{n}i^{1-n}\Lambda_{g_{1}}\left(k-\frac{3}{2}-n\right).$

Since, by the analogue of the proposition in Section 8 of [3] for integral
weights, the function $g$ such that $g(z):=g_{1}(2z)$ is a weight
$k-\frac{1}{2}$ cusp form for $\Gamma^{*}_{0}(4)$ and
$\Lambda_{g_{1}}(s)=2^{s}\Lambda_{g}(s)$, we obtain the expression in the
right-hand side of (5.6). With (5.7), we deduce the result. ∎

We will identify explicitly the integral weight cusp form to which a half-
integral weight form is lifted according to Prop 5.2. We will use a theorem of
[6] providing explicit inverses for the Eichler-Shimura maps $\rho^{\pm}$. To
state it, we introduce some additional notation. Firstly, for each $\gamma\in
H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})$ and $n=0,\dots
k-\frac{5}{2}$, we denote by $r^{\pm}_{\gamma,n}(g)$ the constants such that

$\rho^{\pm}_{g}(\gamma)(z)=\sum_{n=0}^{k-\frac{5}{2}}(-1)^{n}\binom{k-\frac{5}{2}}{n}r^{\pm}_{\gamma,n}(g)z^{k-\frac{5}{2}-n}.$

In particular, $r^{+}_{\gamma,n}(g)=0$ (resp. $r^{-}_{\gamma,n}(g)=0$), when
$n$ is odd (resp. even). Then, for each $\gamma\in
H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})$ and $0\leq n\leq
k-\frac{5}{2}$ we denote by $R^{\pm}_{\gamma,n}$ the unique element of
$S_{k-\frac{1}{2}}(H(2))$ such that

$r^{\pm}_{\gamma,n}(h)=(h,R^{\pm}_{\gamma,n})\quad\mbox{for all}\,\,h\in
S_{k-\frac{1}{2}}(H(2))$

and

$s^{\pm}_{\gamma,n}(h)=\sum_{j=0}^{n}\binom{n}{j}(-1)^{n-j}r^{\pm}_{\gamma,j}(h).$

With this notation, Theorem 6.1 of [6] reads, in our case, as

######  Theorem 5.3.

[6] For each $g_{1}\in S_{k-\frac{1}{2}}(H(2))$, we have

$g_{1}=\frac{2}{3}(2i)^{\frac{3}{2}-k}\sum_{\gamma\in
H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})}\sum_{n=0}^{k-\frac{5}{2}}\binom{k-\frac{5}{2}}{n}s^{-}_{\gamma
U^{-1},n}(g_{1})R^{+}_{\gamma,n}.$

To use this theorem, we first compute $\pi_{f}(U)(\gamma)$ for $\gamma\in
H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})$:

(5.8)
$\displaystyle\pi_{f}(U)(1)=\pi^{\prime}_{f}(\kappa^{-1}_{1,S^{-1}})=\pi_{f}^{\prime}(S)=P_{a}(z/\sqrt{N}),$
$\displaystyle\pi_{f}(U)(T)=\pi^{\prime}_{f}(\kappa^{-1}_{T,U^{-1}})|_{k-\frac{5}{2}}T=\pi_{f}^{\prime}(S^{-1})|_{k-\frac{5}{2}}T^{-1}=P_{a}((z-1)/\sqrt{N}),$
$\displaystyle\pi_{f}(U)(U)=\pi^{\prime}_{f}(\kappa^{-1}_{U,U^{-1}})|_{k-\frac{5}{2}}U=\pi^{\prime}_{f}(UU^{-1})|_{k-\frac{5}{2}}U=0.$

Therefore, if $g_{1}$ is the weight $k-\frac{1}{2}$ cusp form for $H(2)$
induced from $f$ by the proof of (5.6), then (5.7) implies that, for $n$ odd,
the $n$-th coefficient of $\rho^{-}_{g_{1}}(1)$ (resp. $\rho^{-}_{g_{1}}(T)$),
equals the $n$-th coefficient of $P_{a}(z/\sqrt{N})$ (resp
$P_{a}((z-1)/\sqrt{N})$). Thus, with (5.7) and ((ii)), we deduce that, for
each $\gamma\in H(2)\backslash\operatorname{PSL}_{2}(\mathbb{Z})$ and odd $n$,

$-\binom{k-\frac{5}{2}}{n}r^{-}_{1,n}(g_{1})=\frac{\alpha_{k-\frac{5}{2}-n}}{N^{\frac{k}{2}-\frac{5}{4}-\frac{n}{2}}}\quad\text{and}\,\,\binom{k-\frac{5}{2}}{n}r^{-}_{T,n}(g_{1})=\sum_{j=k-\frac{5}{2}-n}^{k-\frac{5}{2}}(-1)^{j}\frac{\alpha_{j}}{N^{\frac{j}{2}}}\binom{j}{k-\frac{5}{2}-n}$

where

$\alpha_{j}:=i^{a+1}\left(\binom{k-2}{n}N^{n}\Lambda_{f}(a+1-n)+\binom{k-2}{n+\frac{1}{2}}N^{2k-a-n-\frac{19}{4}}i^{2n-k+\frac{1}{2}}\Lambda_{f}(2k-\frac{7}{2}-a-n)\right).$

This, in turn, implies that, for each $n=0,\dots k-\frac{5}{2}$,

(5.9) $\displaystyle s^{-}_{1,n}:=s^{-}_{1,n}(g_{1})$
$\displaystyle=(-1)^{n}\sum_{\begin{subarray}{c}j=0\\\ \text{$j$
odd}\end{subarray}}^{n}\binom{n}{j}\binom{k-\frac{5}{2}}{j}^{-1}\frac{\alpha_{k-\frac{5}{2}-j}}{N^{\frac{k}{2}-\frac{5}{4}-\frac{j}{2}}}\qquad\text{and}$
$\displaystyle s^{-}_{T,n}:=s^{-}_{T,n}(g_{1})$
$\displaystyle=(-1)^{n+1}\sum_{\begin{subarray}{c}j=0\\\ \text{$j$
odd}\end{subarray}}^{n}\binom{n}{j}\binom{k-\frac{5}{2}}{j}^{-1}\left(\sum_{\ell=k-\frac{5}{2}-j}^{k-\frac{5}{2}}(-1)^{\ell}\frac{\alpha_{\ell}}{N^{\frac{\ell}{2}}}\binom{\ell}{k-\frac{5}{2}-j}\right)$

Recall the cusp form $g(z)=g_{1}(2z)$ of weight $k-\frac{1}{2}$ for
$\Gamma_{0}^{*}(4)$. Then, with Theorem 5.3, we deduce

######  Theorem 5.4.

Let $k\in\frac{1}{2}+\mathbb{Z}$ such that $k>\frac{5}{2}$ and
$4|(k-\frac{5}{2})$. For each $f\in S_{k}(\Gamma_{0}(N))$ such that
$f|_{k}W_{N}=f$ and for each $a\in[0,2k-\frac{9}{2}]$, the “lift” $g\in
S_{k-\frac{1}{2}}(\Gamma^{*}_{0}(4))$ of Prop. 5.2 is given by

$g(z)=\frac{2}{3}(2i)^{\frac{3}{2}-k}\sum_{n=0}^{k-\frac{5}{2}}\binom{k-\frac{5}{2}}{n}\left(s^{-}_{1,n}R^{+}_{U,n}(2z)+s^{-}_{T,n}R^{+}_{1,n}(2z),\right)$

where $s^{-}_{1,n},s^{-}_{T,n}$ are given by (5.9).

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