# Assessing non-Markovian dynamics through moments of the Choi state

Bivas Mallick<EMAIL_ADDRESS>S. N. Bose National Centre for Basic
Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India  Saheli
Mukherjee<EMAIL_ADDRESS>S. N. Bose National Centre for Basic
Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India  Ananda G.
Maity<EMAIL_ADDRESS>Networked Quantum Devices Unit, Okinawa
Institute of Science and Technology Graduate University, Onna-son, Okinawa
904-0495, Japan  A. S. Majumdar<EMAIL_ADDRESS>S. N. Bose National Centre
for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India

###### Abstract

Non-Markovian effects in open quantum system dynamics usually manifest
backflow of information from the environment to the system, indicating
complete-positive divisibility breaking of the dynamics. We provide a
criterion for witnessing such non-Markovian dynamics exhibiting information
backflow, based on partial moments of Choi-matrices. The moment condition
determined by the positive semi-definiteness of a matrix, does not hold for a
Choi-state describing non-Markovian dynamics. We then present some explicit
examples in support of our proposed non-Markovianity detection scheme.
Finally, a moment based measure of non-Markovianity for unital dynamics is
formulated.

##  I Introduction

According to the postulates of quantum mechanics, closed systems evolve
unitarily. However, due to the inevitable interaction with noisy environments,
the system undergoes irreversible phenomena such as dissipation and
decoherence. The theory of open quantum systems provides adequate tools for
studying such dynamics comprising of system-environment interactions (Alicki
and Lendi, 2007; Lindblad, 1976; Gorini  _et al._ , 1976; Breuer and
Petruccione, 2002; Rivas  _et al._ , 2014; Breuer  _et al._ , 2016; de Vega
and Alonso, 2017; Chruściński, 2022). System-environment interactions are
often assumed to be Markovian where the environment does not keep memory of
past interactions with the system and the interaction is considered to be
sufficiently weak. However, in realistic scenarios, when the system-
environment coupling is not sufficiently weak and the environment has some
finite memory, the description of open quantum systems by the Markovian model
may fall short leading to the requirement of the non-Markovian paradigm (Rivas
_et al._ , 2010; Breuer  _et al._ , 2009; Laine  _et al._ , 2010; Bellomo  _et
al._ , 2007; Dijkstra and Tanimura, 2010). Unlike Markovian dynamics (i.e, the
dynamics without memory effects), non-Markovian dynamics usually contains a
backflow of information from the environment to the system providing a unique
signature Breuer  _et al._ (2016); de Vega and Alonso (2017).

In recent times, much effort has been devoted to the study of quantum non-
Markovian dynamics which provides advantages in several quantum information
processing tasks such as perfect teleportation with mixed states (Laine  _et
al._ , 2014), efficient work extraction from Otto cycle (Thomas  _et al._ ,
2018), efficient quantum control (Reich  _et al._ , 2015), entangled state
preparation (Bylicka  _et al._ , 2014; Xiang  _et al._ , 2014), quantum
metrology (Altherr and Yang, 2021), quantum evolution speedup (Deffner, 2017),
and so on. Experimental realization of non-Markovianity has been achieved in
trapped-ion, nuclear magnetic resonance and photonic systems indicating a
potential resource for executing quantum information processing tasks in real
systems Liu  _et al._ (2011); Li  _et al._ (2011); Smirne  _et al._ (2011);
Cialdi  _et al._ (2014); Tang  _et al._ (2015); Sun  _et al._ (2016); Wittemer
_et al._ (2018); Haase  _et al._ (2018); Gessner  _et al._ (2014).

Despite several interesting applications of non-Markovianity, a fundamental
and important question is to assess whether the underlying quantum dynamics is
non-Markovian at all, so that one can utilize it as a resource in legitimate
quantum information processing tasks. Therefore, identifying whether a
dynamics provides non-Markovian traits is a substantial task for advancement
of quantum technologies. Several methods have been proposed till date from
different perspectives and utilizing different properties of non-Markovian
dynamics Rivas  _et al._ (2010); Laine  _et al._ (2010); Bellomo  _et al._
(2007); Dijkstra and Tanimura (2010); Bhattacharya  _et al._ (2017);
Mukhopadhyay  _et al._ (2017); Bhattacharya  _et al._ (2021); Maity  _et al._
(2020); Bhattacharya and Bhattacharya (2021); Maity and Bhattacharya ; Wolf
_et al._ (2008); Jeknić-Dugić  _et al._ (2023). In this work, we provide an
adequate technique to efficiently detect non-Markovian dynamics entailed with
environmental memory. Our approach is based on the determination of partial
moments of the Choi state, and does not require full process tomography,
thereby making it easier to realize in a real experiment.

Our proposal utilizing the moment criterion requires evaluation of simple
functionals which can be efficiently estimated using an experimental technique
called shadow tomography Aaronson (2018); Aaronson and Rothblum (2019); Huang
_et al._ (2020). It is based on a recently proposed methodology for
simultaneous evaluation of several quantities for Noisy Intermediate Scale
Quantum (NISQ) devices and is more efficient than usual tomography. Moreover,
it may be noted here that our criterion is state independent unlike the
witness based detection scheme for which prior information about the quantum
state is necessary. We further provide two explicit examples in support of our
detection scheme for non-Markovian evolution.

In addition to the task of detecting a non-Markovian dynamics, another
important task is to provide a quantitative measure of non-Markovianity.
However, non-Markovianity can be manifested in several ways indicating that
there exists no common or general way of comparing non-Markovian dynamics for
different physical models. Two measures of non-Markovianity proposed earlier,
are based on the concept of divisibility of the dynamical map (RHP measure)
Rivas  _et al._ (2010, 2014); Hakoshima  _et al._ (2021) and
distinguishability of quantum states (BLP measure) Breuer  _et al._ (2009);
Laine  _et al._ (2010); Breuer  _et al._ (2016). In this work, we define a
measure based on partial moments of the Choi-matrix to quantify non-
Markovianity.

The paper is organized as follows. In section II, we provide a brief overview
of the essential mathematical preliminaries concerning the dynamics of open
quantum systems, as well as the moment criteria proposed in earlier works for
entanglement detection. In section III we present our framework for detection
of non-Markovianity along with some explicit examples. A measure of non-
Markovianity is proposed in section IV where we also compare our proposed
measure with the RHP measure for pure dephasing channel with Ohmic spectral
density. Finally, in section V, we summarize our main findings.

##  II Preliminaries

###  II.1 Dynamics of Open Quantum System

Isolated systems undergo unitary evolution. However, a general quantum
evolution (or a quantum channel) can be represented by a completely-positive
trace-reserving (CPTP) map $(\Lambda(t,t_{0}))$ which maps an element
($\rho(t_{0})$) of the set of density operators ($B(\mathcal{H})$) to another
element of the set i.e, $\Lambda(t,t_{0}):\rho(t_{0})\mapsto\rho(t)$. The set
of all such CPTP maps can be represented as $D$. We assume that the inverse
$\Lambda^{-1}(t,t_{0})$ exists for all time from $t_{0}$ to $t$. One can thus
write the dynamical map for any $t\geq s\geq t_{0}$, into a composition

$\Lambda(t,t_{0})=\Lambda(t,s)\circ\Lambda(s,t_{0}).$ (1)

Even though $\Lambda(t,t_{0})$ is always completely positive since it must
correspond to a physically legitimate dynamics (and hence
$\Lambda^{-1}(t,t_{0})$ is well defined) and $\Lambda(s,t_{0})$ is completely
positive, the map $\Lambda(t,s)$ however, need not be completely positive. A
dynamics acting on the system of interest is said to be divisible iff it can
be written as Eq. (1) for any time $t\geq s\geq t_{0},$ where $t_{0}$ is the
initial time of dynamics, and $\circ$ represents the composition between two
maps. The dynamics $\Lambda(t,t_{0})$ is said to be positive divisible (P
divisible) if $\Lambda(t,s)$ is a positive map for every $t\geq s\geq t_{0}$
satisfying the composition law. The dynamics $\Lambda(t,t_{0})$ is said to be
completely positive divisible (CP divisible) if $\Lambda(t,s)$ is a CPTP map
for every $t\geq s\geq t_{0}$ and satisfies the composition law.

The above mathematical characterization of a dynamical map $\Lambda(t,t_{0})$
in terms of ‘divisibility’, describing a memoryless evolution as a composition
of physical maps leads to the definition of quantum Markovianity. According to
the RHP criterion, a dynamics is said to be non-Markovian if it is not CP-
divisible Rivas  _et al._ (2010). Another way of characterizing non-Markovian
dynamics is provided by Breuer et. al. Breuer  _et al._ (2009); Laine  _et
al._ (2010) where distinguishability of quantum states after the action of
dynamical map is considered. Due to the interaction of a quantum system with
the noisy environment, two quantum states lose their state distinguishability
gradually with time. However, if at any instant of time, the
distinguishability increases, then there is backflow of information from the
environment to the system leading to the signature of non-Markovianity. The
former way of representing a dynamics to be non-Markovian is known as RHP-type
non-Markovianity Rivas  _et al._ (2010, 2014), whereas the latter one is known
as BLP-type non-Markovianity Breuer  _et al._ (2009, 2016). A dynamics which
is Markovian in the RHP sense is also Markovian in the BLP sense, but the
converse is not true in general. Therefore, CP divisibility breaking is a
necessary but not sufficient condition for information backflow from the
environment to the system. In this paper we adopt CP divisibility as the sole
property of quantum Markovianity, and any deviation from CP-divisibility
(indivisible) will be considered as the benchmark of non-Markovianity.

Now, for each of the dynamical maps $\Lambda(t,t_{0})\in D$, one can find a
one-to-one correspondence to a state, called the Choi-state
$\mathcal{C}_{\Lambda}(t,t_{0})\in\mathcal{F}$ (where $\mathcal{F}$ is the set
of all Choi-states) via channel-state duality where the Choi state Choi (1975)
is defined as

$\mathcal{C}_{\Lambda}(t,t_{0})=(\mathbb{I}\otimes\Lambda(t,t_{0}))\ket{\phi}\bra{\phi}$
(2)

with $\ket{\phi}\bra{\phi}$ being a maximally entangled bipartite state of
dimension $d\crossproduct d$. According to the Choi–Jamiolkowski isomorphism
Choi (1975); Jamiołkowski (1972), for checking complete-positivity of
$\Lambda(t,t_{0})$, it is sufficient to check the positive semidefiniteness of
the corresponding Choi-state $(\mathcal{C}_{\Lambda}(t,t_{0}))$.

###  II.2 Partial moment criterion

In the bipartite scenario, one of the most well-known detection schemes of
entanglement is based on the PPT criterion which examines whether the partial
transposed state $\rho^{T_{A}}_{AB}$ (where partial transposition is taken
w.r.t subsystem A) is positive semi-definite (all eigenvalues are non-
negative) or not. Violation of this criterion implies that the given state
$\rho_{AB}$ is entangled. This criterion has been shown to be a necessary and
sufficient condition for $2\otimes 2$, $2\otimes 3$ and $3\otimes 2$ systems
and has many applications in theoretical works Castelnovo (2013); Eisler and
Zimborás (2014); Wen  _et al._ (2016); Ruggiero  _et al._ (2016a); Blondeau-
Fournier  _et al._ (2016); Ruggiero  _et al._ (2016b). However, the
transposition map not being a physical one, is impossible to implement exactly
in an experimental scenario. A useful measure using this PPT criterion is the
negativity measure Vidal and Werner (2002), defined as:

$\mathbb{N}=||\rho^{T_{A}}_{AB}||=\sum_{i}{}|\lambda_{i}|$

where $\lambda_{i}$’s are the negative eigenvalues of $\rho^{T_{A}}_{AB}$. But
this again requires an access to the full spectrum of $\rho^{T_{A}}_{AB}$,
which is not obtainable through an experimental setup. To overcome this issue,
the idea of moments of the partially transposed density matrix (PT-moments)
was introduced to study the correlations in many-body systems in relativistic
quantum field theory by Calabrese et al. in 2012 Calabrese  _et al._ (2012).

For a bipartite state $\rho_{AB}$, these PT-moments are given by,

$P_{n}=\text{Tr}[{(\rho_{AB}}^{T_{A}})^{n}]$ (3)

for n=1,2,3,…. One may note that $P_{1}=\text{Tr}[\rho_{AB}^{T_{A}}]=1$ while
$P_{2}=\text{Tr}[(\rho_{AB}^{T_{A}})^{2}]$ is related to the purity of the
state. Therefore, $P_{3}$ is the first non-trivial moment which is necessary
to capture additional information related to the partial transposition. Using
only these first three PT moments, a simple but powerful entanglement
detection criterion was proposed in ref. Elben  _et al._ (2020a). This
suggests that if a state $\rho_{AB}$ is PPT, then ${P_{2}}^{2}\leq
P_{3}P_{1}$. Therefore, from the contrapositivity of this statement, it
follows that if a state $\rho_{AB}$ violates this condition, then it must be
entangled which is the $p_{3}$-PPT criterion. Just like the PPT condition,
this $p_{3}$-PPT condition is also applicable to mixed states and is a state
independent criterion unlike entanglement witness Horodecki  _et al._ (2009);
Gühne and Tóth (2009). While entanglement witness provides a stronger
criterion for entanglement detection, some prior knowledge about the state is
required for the implementation of entanglement witness.

Even though this $p_{3}$-PPT criterion is weaker than the general PPT
criterion, the former involves simple functionals which are easy to realise in
a real experiment by a method called shadow tomography Aaronson (2018);
Aaronson and Rothblum (2019); Huang  _et al._ (2020). For Werner states, the
$p_{3}$-PPT criterion and the full PPT criterion are equivalent, and hence,
the $p_{3}$-PPT criterion is a necessary and sufficient criteria for bipartite
entanglement of Werner states. The PT-moments can be obtained experimentally
with the help of shadow tomography without actually performing full state
tomography, thus making it more efficient in terms of resources consumed. For
a detailed discussion on shadow tomography and its advantage over general
tomography, interested readers are referred to Refs. Aaronson (2018); Aaronson
and Rothblum (2019); Huang  _et al._ (2020); Elben  _et al._ (2020a). The
technique of PT moments offers unparalleled advantage in NISQ and in many-body
systems where a single qubit is used as a control and many distinct PT moments
can be estimated from the same data unlike using random global unitaries for
randomized measurements Huang  _et al._ (2020); Elben  _et al._ (2020a).
Furthermore, the $p_{3}$-moment, in addition to detecting mixed state
entanglement Elben  _et al._ (2020a); Neven  _et al._ (2021); Yu  _et al._
(2021) is also used to study entanglement dynamics in many-body quantum
systems Brydges  _et al._ (2019); Elben  _et al._ (2020b).

It might be noted here that the $p_{3}$-PPT condition provides a necessary
condition for separability. However, each higher order moment ($n\geq 4$)
gives rise to an independent and different entanglement detection criterion,
and evaluating all the higher order moments provides a necessary and
sufficient criterion for NPT entanglement Neven  _et al._ (2021). But this is
very challenging from an experimental point of view and hence, moments up to
third order are used in order to provide an entanglement detection criteria
and this simplifies the task. Motivated by the above considerations, in the
next section we explore whether a moment based detection scheme can be
developed for non-Markovian dynamics, since in realistic scenarios, many
different categories of dynamics consist of non-Markovian memory. We define
$\Lambda$-moments $(r_{n})$, and based on it we develop a formalism for
detection of non-Markovian dynamics characterized via indivisibility.

##  III Detection of non-Markovianity

Definition 1: Let $\Lambda(t,s)$ be a trace preserving, linear map that
satisfies the composition law (1). We define the $n$th order $\Lambda$-moments
$(r_{n})$ as:

$r_{n}=\text{Tr}[(\mathcal{C}_{\Lambda}(t,s))^{n}]$ (4)

with $n$ being an integer and $\mathcal{C}_{\Lambda}(t,s)$ is the Choi state
(defined earlier) corresponding to the dynamics acting on the system between
the time intervals $s$ and $t$, such that $s\leq t$. With the above
definition, we are now ready to propose our criterion for detecting non-
Markovian dynamics.

Theorem 1: If a dynamics is Markovian, then

${r_{2}}^{2}\leq r_{3}$ (5)

where $r_{2}$ and $r_{3}$ are defined in (4).

###### Proof.

If $t_{0}$ be the initial time of a dynamics, then the time evolution of an
open quantum system is governed by a family of completely positive trace
preserving maps $\\{{\Lambda(t,t_{0})\\}}_{t\geq t_{0}}$ satisfying the
composition law (1). Considering the concept of Choi-Jamiolkowski isomorphism
Choi (1975); Jamiołkowski (1972), we shall henceforth use the Choi operator
($\mathcal{C}_{\Lambda}$) corresponding to the map $\Lambda(t,s)$.

Let us now consider the Schatten-$p$ norms for $p\geq 1$, which are defined as

$||X||_{p}=(\sum_{i=1}^{n}{|\chi_{i}|^{p}})^{\frac{1}{p}}=(\text{Tr}[|X|^{p}])^{\frac{1}{p}}$
(6)

where $X$ is a $n\times n$ hermitian matrix having eigenvalue decomposition
$X=\sum_{i=1}^{n}\chi_{i}\ket{x_{i}}\bra{x_{i}}$. Replacing $X$ by the Choi
matrix $\mathcal{C}_{\Lambda}$ in Eq.(6), the Schatten-$p$ norms for Choi
matrix are analogously defined. Further, the $l_{p}$ norm of the vector of
eigenvalues of $\mathcal{C}_{\Lambda}$ corresponding to each Schatten-$p$ norm
is defined by:

$||\lambda||_{l_{p}}:=(\sum_{i=1}^{n}{|\lambda_{i}|^{p}})^{\frac{1}{p}}$ (7)

where ${\\{\lambda_{i}\\}_{{i=1}}^{n}}$ is the spectrum of
$\mathcal{C}_{\Lambda}$. The inner product corresponding to an $n$ vector is
defined as

$\langle u,v\rangle:=\sum_{i=1}^{n}u_{i}v_{i}$ (8)

for $u,v\in\mathbb{R}^{n}$. Now, from Hoelder’s inequality for vector norms,
we know that for $p,q\geq 1$ and $\frac{1}{p}+\frac{1}{q}=1,$ the following
relation holds:

$|\langle
u,v\rangle|\leq\sum_{i=1}^{n}|u_{i}v_{i}|\leq||u||_{l_{p}}||v||_{l_{q}}.$ (9)

Putting $p=3$ and $q=\frac{3}{2}$ in (9), we get

$\text{Tr}[(\mathcal{C}_{\Lambda})^{2}]=\langle\lambda,\lambda\rangle\leq||\lambda||_{l_{3}}||\lambda||_{l_{\frac{3}{2}}}=||\mathcal{C}_{\Lambda}||_{3}||\lambda||_{l_{\frac{3}{2}}}$
(10)

We next apply the Cauchy-Schwarz inequality which is obtained by putting
$p=\frac{1}{2}$ and $q=\frac{1}{2}$ in Hoelder’s inequality. Therefore,

$\displaystyle{||\mathcal{C}_{\Lambda}||_{2}}^{2}=\text{Tr}[(\mathcal{C}_{\Lambda})^{2}]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\leq||\mathcal{C}_{\Lambda}||_{3}||\lambda||_{l_{\frac{3}{2}}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
=||\mathcal{C}_{\Lambda}||_{3}(\sum_{i=1}^{n}{|\lambda_{i}|^{\frac{3}{2}}})^{\frac{2}{3}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
=||\mathcal{C}_{\Lambda}||_{3}(\sum_{i=1}^{n}{{|\lambda_{i}|}{|\lambda_{i}|}^{\frac{1}{2}}})^{\frac{2}{3}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\leq||\mathcal{C}_{\Lambda}||_{3}((\sum_{i=1}^{n}{|\lambda_{i}|^{2}})^{\frac{1}{2}}(\sum_{i=1}^{n}{|\lambda_{i}|})^{\frac{1}{2}})^{\frac{2}{3}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
=||\mathcal{C}_{\Lambda}||_{3}{||\mathcal{C}_{\Lambda}||_{2}}^{\frac{2}{3}}{||\mathcal{C}_{\Lambda}||_{1}}^{\frac{1}{3}}$
(11)

Taking $3$rd power of (11), we get

${||\mathcal{C}_{\Lambda}||_{2}}^{4}\leq{||\mathcal{C}_{\Lambda}||_{3}}^{3}||\mathcal{C}_{\Lambda}||_{1}$
(12)

Since $\Lambda$ is a trace preserving map, $||\mathcal{C}_{\Lambda}||_{1}=1$,
and hence (12) reduces to,

${||\mathcal{C}_{\Lambda}||_{2}}^{4}\leq{||\mathcal{C}_{\Lambda}||_{3}}^{3}$
(13)

i.e.,

$(\text{Tr}[(\mathcal{C}_{\Lambda})^{2}])^{2}\leq\text{Tr}[(\mathcal{C}_{\Lambda})^{3}]$
(14)

which completes the proof. ∎

The above theorem indicates that condition (5) is necessary for a dynamics to
be Markovian. Violation of above theorem is therefore sufficient to conclude
that the underlying dynamics is actually CP-indivisible and hence non-
Markovian. Below, we present some explicit examples of non-Markovian dynamics
which can be detected by the condition mentioned above.

###  III.1 Examples:

We would now like to present two examples in support of Theorem 1. It may be
noted that here we will consider the set of operations which have Lindblad
type generators. For system density matrix $\rho$, the Lindblad master
equation can be written as

$\frac{d\rho}{dt}=-\frac{\iota}{\hbar}[H,\rho]+\sum_{i}\gamma_{i}(L_{i}\rho{L_{i}}^{\dagger}-\frac{1}{2}({L_{i}}^{\dagger}L_{i}\rho+\rho{L_{i}}^{\dagger}L_{i}))$
(15)

where the unitary aspects of the dynamics is described by the Hamiltonian $H$,
$\gamma_{i}$ are the Lindblad coefficients, and $L_{i}$ are the Lindblad
operators which describe the dissipative part of the dynamics Lindblad (1976).

Example 1: We consider a qubit system that interacts with an amplitude damping
environment which is modeled by another qubit system. The non-Markovian
character of this model studied earlier Mukherjee  _et al._ (2015), is
motivated by the experimental realization of such non-Markovian dynamics
through the violation of temporal Bell-like inequalities in a controllable
Nuclear Magnetic Resonance system Souza  _et al._ (2013) .

The master equation is given by

$\frac{d\rho}{dt}=\mathcal{L}(\rho)=\gamma_{1}(\sigma_{x}\rho\sigma_{x}-\rho)+\gamma_{2}(\sigma_{y}\rho\sigma_{y}-\rho)+\gamma_{3}(\sigma_{z}\rho\sigma_{z}-\rho)$
(16)

The Lindblad coefficients are taken as
$\gamma_{1}=\gamma_{2}=\gamma_{3}=e^{-t^{\prime}}\cos{t^{\prime}}$ (for all
$i$) and $t^{\prime}=kt$, with $k$ being a constant having the dimension of
$[T^{-1}]$. The corresponding dynamical map is given by
$\Lambda(\rho)=e^{\mathcal{L}}(\rho)$. For small time approximation (i.e,
$|\epsilon\gamma_{i}|<<1$), the Choi state corresponding to $\Lambda(\rho)$ is
given by

$\mathcal{C}_{\Lambda}=(\mathbb{I}\otimes(\mathbb{I}+\epsilon\mathcal{L}))\ket{\phi^{+}}\bra{\phi^{+}}$
(17)

with $\ket{\phi^{+}}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$. Here, we consider
$\epsilon=0.001$ and $k=1$. It is known that the above dynamics shows its non-
Markovian nature when $\gamma_{i}<0$. Therefore, for $\gamma_{i}<0$, we should
have ${r}_{2}^{2}-{r}_{3}>0$, which is evident from Fig. 1.

Figure 1: Non-Markovian behavior of the dynamics given by Eq.(16),
${r}_{2}^{2}-{r}_{3}>0$, is exhibited for $\gamma<0$.

Example 2: As a second example, we consider a pure dephasing non-Markovian
dynamics. A qubit system interacts with a thermal reservoir which is modeled
by an infinite set of harmonic oscillators in the vacuum state. The
Hamiltonian corresponding to the system-reservoir interaction is

$H=\omega_{\sigma}\sigma_{z}+\sum_{k}\omega_{k}a_{k}^{\dagger}a_{k}+\sum_{k}\alpha_{k}\sigma_{z}[a_{k}exp(i\theta_{k})\\\
+a_{k}^{\dagger}exp(-i\theta_{k})]$

where $\omega_{\sigma},\omega_{k}$ are the energy gap of the system and the
frequency of $k$-th mode of reservoir, respectively, $a_{k}^{\dagger}(a_{k})$
are creation (annihilation) operators of the harmonic oscillator, $\alpha_{k}$
is the coupling constant for the $k$-th mode, and $\theta_{k}$ is the
corresponding phase. The master equation is given by

$\frac{d\rho}{dt}=\mathcal{L}(\rho)=\gamma(\sigma_{z}\rho(t)\sigma_{z}-\rho(t))$
(18)

The time dependent decay rate $\gamma$ corresponding to a Lorentzian spectral
density is (Salimi  _et al._ , 2016; Mukherjee  _et al._ , 2015)

$\gamma=\frac{2\lambda\gamma_{0}\sinh(t^{\prime}g/2)}{g\cosh(t^{\prime}g/2)+\lambda\sinh(t^{\prime}g/2)}$
(19)

with $g=\sqrt{{\lambda}^{2}-2\gamma_{0}\lambda}$, $t^{\prime}=kt$. Here $k$ is
a constant having the dimension of $[T^{-1}]$, $\lambda$ is the spectral width
and $\gamma_{0}$ is the coupling strength.

In the small time approximation (i.e, $|\epsilon\gamma|<<1$), the Choi state
corresponding to the dynamical map, $\Lambda(\rho)=e^{\mathcal{L}}(\rho)$ is
given by
$\mathcal{C}_{\Lambda}=(\mathbb{I}\otimes(\mathbb{I}+\epsilon\mathcal{L}))\ket{\phi^{+}}\bra{\phi^{+}}$
where $\ket{\phi^{+}}$ is the maximally entangled state defined earlier. It is
known that the above dynamics shows its non-Markovian nature when $\gamma<0$
which is possible only when $\gamma_{0}>\lambda/2$ Mukherjee  _et al._ (2015).
So, for $\gamma<0$, we should have ${r}_{2}^{2}-{r}_{3}>0$ which is again
evident from Fig. 2. We consider here $\lambda=1.5$, $\gamma_{0}=1$ and $k=1$
for the figure.

Figure 2: Non-Markovian behavior of the dynamics given by Eq.(18),
${r}_{2}^{2}-{r}_{3}>0$, is exhibited for $\gamma<0$.

##  IV Measure of non-Markovianity

In this section, we would like to define a quantitative measure of non-
Markovianity. Using Schatten-$p$ norms for $p=2$ and $p=3$ we define a measure
of non-Markovianity. Let us first denote:

$f(t)=\lim_{\epsilon\to 0}\frac{M_{\epsilon}}{\epsilon}$ (20)

where,

$\begin{split}M_{\epsilon}&={M_{T}}({t+\epsilon},t)\\\
&=\text{max}\\{0,({||\mathcal{C}_{\Lambda}(t+\epsilon,t)||_{2}}^{2})^{2}-{||\mathcal{C}_{\Lambda}(t+\epsilon,t)||_{3}}^{3}\\}\end{split}$
(21)

We define,

$\mathcal{M}=\int_{0}^{\infty}f(t)\,dt$ (22)

as a measure of non-Markovianity.

Below we will show that $\mathcal{M}$ can be used as a measure of non-
Markovianity for unital dynamics. To show that $\mathcal{M}$ as a measure of
non-Markovianity for unital dynamics, we need to show that $\mathcal{M}=0$ for
all Markovian dynamics and $\mathcal{M}$ is monotone under divisible unital
dynamics. It has been shown earlier that for all unital dynamical maps having
corresponding Lindblad generators, the Lindblad operators are normal
Bhattacharya  _et al._ (2020). Therefore, to prove the monotonicity of our
measure we will consider that the Lindblad operators are normal.

Lemma 1: If the $\alpha$-Renyi entropy defined by

$S_{\alpha}(t):=\frac{1}{1-\alpha}\log_{2}\text{Tr}[X^{\alpha}(t)],\leavevmode\nobreak\
\leavevmode\nobreak\ (\alpha>0)$

evolves under Lindblad type divisible operation having normal Lindblad
operator, then

$\frac{d}{dt}S_{\alpha}(t)\geq 0.$ (23)

###### Proof.

In Refs. Abe (2016); Beigi (2013); Song and Mao (2017); Bhattacharya  _et al._
(2020); Benatti and Narnhofer (1988); Das  _et al._ (2018), it has been shown
that

$\frac{d}{dt}S_{\alpha}(t)=2\sum_{i}\gamma_{i}(t)\chi_{i}(t)$ (24)

where

$\chi_{i}(t)=\frac{\alpha}{1-\alpha}\frac{1}{\text{Tr}[X^{\alpha}(t)]}\text{Tr}[X^{\alpha-1}(t)L_{i}X(t)L_{i}^{\dagger}-X^{\alpha}(t)L_{i}^{\dagger}L_{i}]$

with $L_{i}$ being the Lindblad operator and $\gamma_{i}$ are the Lindblad
coefficients. Further Abe (2016),

$\chi_{i}(t)>\langle[L_{i}^{\dagger},L_{i}]\rangle_{\alpha}$ (25)

where, $\langle
A\rangle_{\alpha}=\text{Tr}[AX^{\alpha}(t)]/\text{Tr}[X^{\alpha}(t)]$. For
normal Lindblad operator $[L_{i}^{\dagger},L_{i}]=0$. Therefore, from (25),
$\chi_{i}(t)>0$. Also, if the dynamics is divisible, i.e., all $\gamma_{i}>0$,
then from Eq. (24) it immediately follows that $\frac{d}{dt}S_{\alpha}(t)\geq
0$. ∎

Lemma 2: If the Schatten-$p$ norms of a positive, Hermitian operator $X$
evolve under divisible operation having Lindblad type generators, and the
Lindblad operators are normal, then

$||X{(t+\delta t)}||_{p}\leq||X(t)||_{p}.$ (26)

where $||X{(t+\delta t)}||_{p},||X(t)||_{p}$ are the Schatten-$p$ norms of $X$
at times $t+\delta t$ and $t$, respectively.

###### Proof.

From Lemma 1 it follows that for Lindblad type divisible operation having
normal Lindblad operator,

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ S_{\alpha}(t+\delta t)-S_{\alpha}(t)\geq 0$
$\displaystyle\implies\frac{1}{1-\alpha}\log_{2}\text{Tr}[X^{\alpha}(t+\delta
t)]\geq\frac{1}{1-\alpha}\log_{2}\text{Tr}[X^{\alpha}(t)].$ Now, if
$\alpha>1$, then $\frac{1}{1-\alpha}<0$ and hence
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\log_{2}\text{Tr}[X^{\alpha}(t+\delta
t)]\geq-\log_{2}\text{Tr}[X^{\alpha}(t)]$
$\displaystyle\implies\alpha\log_{2}\text{Tr}[X^{\alpha}(t+\delta
t)]\leq\alpha\log_{2}\text{Tr}[X^{\alpha}(t)]$
$\displaystyle\implies\text{Tr}[X^{\alpha}(t+\delta
t)]^{\alpha}\leq\text{Tr}[X^{\alpha}(t)]^{\alpha}.$ (27)

Since $\alpha>0$, by taking the ${\alpha}$-th root of both sides, we get

$\displaystyle\text{Tr}[X^{\alpha}(t+\delta t)]\leq\text{Tr}[X^{\alpha}(t)].$
(28)

From definition of Schatten-$p$ norms, it follows

$\displaystyle{||X{(t+\delta
t)}||_{\alpha}}^{\alpha}\leq{||X(t)||_{\alpha}}^{\alpha}.$ (29)

Again, taking the ${\alpha}$-th root of both sides, we get,

$\displaystyle{||X{(t+\delta t)}||_{\alpha}}\leq{||X(t)||_{\alpha}}$ (30)

Now, replacing $\alpha$ by $p$, it follows that

$\displaystyle{||X{(t+\delta t)}||_{p}}\leq{||X(t)||_{p}}$ (31)

∎

The above two lemmas imply the following proposition.

Proposition 1: For unital dynamics having Lindblad type generators,
$\mathcal{M}$ is monotone under divisible operation.

###### Proof.

In order to show $\mathcal{M}$ to be monotone under unital divisible dynamics,
we have to prove that $\mathcal{M}$ satisfies

$\begin{split}(\text{Tr}[({\mathcal{C}_{\Lambda}{(t+\delta
t)})^{2}}])^{2}-\text{Tr}[({\mathcal{C}_{\Lambda}{(t+\delta t)}})^{3}]&\\\
\leq(\text{Tr}[({\mathcal{C}_{\Lambda}{(t)})^{2}}])^{2}-\text{Tr}[({\mathcal{C}_{\Lambda}{(t)}})^{3}]\end{split}$
(32)

It is known that for unital dynamics having Lindblad type generators, the
Lindblad operators are normal Bhattacharya  _et al._ (2020). Therefore,
replacing $X$ by $\mathcal{C}_{\Lambda}$, lemma 2 implies that for $p=2$ (it
follows from (26)),

$||\mathcal{C}_{\Lambda}{(t+\delta
t)}||{{}_{2}}^{4}\leq||\mathcal{C}_{\Lambda}{(t)}||{{}_{2}}^{4}.$ (33)

Using (33), we get

$\begin{split}||\mathcal{C}_{\Lambda}{(t+\delta
t)}||{{}_{2}}^{4}-||\mathcal{C}_{\Lambda}{(t+\delta t)}||{{}_{3}}^{3}&\\\
=(\text{Tr}[({\mathcal{C}_{\Lambda}{(t+\delta
t)})^{2}}])^{2}-\text{Tr}[({\mathcal{C}_{\Lambda}{(t+\delta t)}})^{3}]&\\\
\leq(\text{Tr}[({\mathcal{C}_{\Lambda}{(t)})^{2}}])^{2}-\text{Tr}[({\mathcal{C}_{\Lambda}{(t+\delta
t)}})^{3}]\end{split}$ (34)

For any quantum state $\rho$, evolving under the Lindblad type generators
$\mathcal{L}$,

$\frac{d\rho}{dt}=e^{\mathcal{L}}\rho$ (35)

it follows from (34) that,

$\displaystyle(\text{Tr}[({\mathcal{C}_{\Lambda}{(t+\delta
t)})^{2}}])^{2}-\text{Tr}[({\mathcal{C}_{\Lambda}{(t+\delta t)}})^{3}]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\leq(\text{Tr}[({\mathcal{C}_{\Lambda}{(t)})^{2}}])^{2}-\text{Tr}[{(I+\epsilon\mathcal{L}+{\epsilon}^{2}{\mathcal{L}}^{2}+...)({\mathcal{C}_{\Lambda}(t)})^{3}}]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\leq(\text{Tr}[({\mathcal{C}_{\Lambda}{(t)})^{2}}])^{2}-\text{Tr}[({\mathcal{C}_{\Lambda}{(t)}})^{3}]$
(36)

Using (36), it can be seen that,

$\mathcal{M}(t+\delta t)\leq\mathcal{M}(t)$ (37)

which proves that $\mathcal{M}$ is monotone under divisible unital dynamics.
This completes the proof. ∎

Theorem 2: $\mathcal{M}$ is a measure of unital Lindblad type non-Markovian
dynamics having normal Lindblad operators.

###### Proof.

By the definition of $\mathcal{M}$ and from the proof of Theorem 1, it is
clear that $\mathcal{M}=0$ for all Markovian dynamics. Furthermore, from
Proposition 1 it follows that $\mathcal{M}$ is monotone under divisible
operation for Lindblad type dynamics having normal Lindblad operators.
Therefore, $\mathcal{M}$ satisfies the necessary properties of a measure, and
can be taken to be a measure of non-Markovianity for unital dynamics. ∎

###  IV.1 Comparison between our Measure and the RHP Measure

For a detailed comparison between our proposed measure ($\mathcal{M}$) and the
RHP measure ($\mathcal{I}$), we consider pure dephasing channel as presented
in Example 2. The time dependent dephasing rate is given by Salimi  _et al._
(2016)

$\gamma(t)=\int\frac{J(\omega)\coth({\hbar\omega/2k_{B}T})\sin(\omega
t)}{\omega}d\omega$ (38)

where, $\omega$ and $J(\omega)$ represents the frequency of the reservoir
modes and spectral function of the reservoir respectively. For Ohmic spectral
density, the spectral function is given by

$J(\omega)=\omega\exp{\frac{-\omega}{\omega_{c}}}$ (39)

where, $\omega_{c}$ is the cut-off frequency of the reservoir.

In the small time approximation (i.e, $|\epsilon\gamma|<<1$), the Choi state
corresponding to the dynamical map, $\Lambda(\rho)=e^{\mathcal{L}}(\rho)$ is
given by
$\mathcal{C}_{\Lambda}=(\mathbb{I}\otimes(\mathbb{I}+\epsilon\mathcal{L}))\ket{\phi^{+}}\bra{\phi^{+}}$
where $\ket{\phi^{+}}$ is the maximally entangled state.

RHP Measure: From the Choi–Jamiolkowski isomorphism, we know that the
dynamical map $\Lambda_{\epsilon}=\Lambda(t+\epsilon,t)$ is CP iff
$(\mathbb{I}\otimes\Lambda_{\epsilon})\ket{\phi^{+}}\bra{\phi^{+}}\geq 0$
$\forall\epsilon$. From the trace preserving condition,
$||(\mathbb{I}\otimes\Lambda_{\epsilon})\ket{\phi^{+}}\bra{\phi^{+}}||_{1}=1$
iff $\Lambda_{\epsilon}$ is completely positive and
$||(\mathbb{I}\otimes\Lambda_{\epsilon})\ket{\phi^{+}}\bra{\phi^{+}}||_{1}>1$,
if $\Lambda_{\epsilon}$ is not completely positive. Considering these facts,
one can define

$g(t)=\lim_{\epsilon\to
0}\frac{||(\mathbb{I}\otimes\Lambda_{\epsilon})\ket{\phi}\bra{\phi}||_{1}-1}{\epsilon}$
(40)

where $\Lambda_{\epsilon}=(\mathbb{I}+\epsilon\mathcal{L})$. It is clear that
$g(t)\geq 0$ and $g(t)=0$ iff $\Lambda_{\epsilon}$ is CP. The integral

$\mathcal{I}=\int_{0}^{\infty}g(t)\,dt$ (41)

is the RHP measure Rivas  _et al._ (2014, 2010).

In case of pure dephasing channel with Ohmic spectral density,
$\mathcal{C}_{\Lambda}=(\mathbb{I}\otimes(\mathbb{I}+\epsilon\mathcal{L}))\ket{\phi^{+}}\bra{\phi^{+}}$
has eigen values $\lambda_{1}=0$, $\lambda_{2}=0$,
$\lambda_{3}=\gamma\epsilon$, $\lambda_{4}=1-\gamma\epsilon$.
Therefore,

$\displaystyle g(t)=\begin{cases}0\hskip 28.45274pt\text{when}\hskip
5.69046pt\gamma(t)\geq 0\\\ -2\gamma,\hskip 11.38092pt\text{when}\hskip
5.69046pt\gamma(t)<0.\end{cases}$ (42)

and

$\mathcal{I}=\int_{\gamma<0}-2\gamma\,dt$ (43)

Our Proposed Measure: For the same example, the value of $f(t)$ defined in
Eq.(20) turns out to be

$\displaystyle f(t)=\begin{cases}0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{when}\hskip
5.69046pt\gamma(t)\geq 0\\\ -\gamma,\hskip 22.76228pt\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{when}\hskip
5.69046pt\gamma(t)<0.\end{cases}$ (44)

Therefore, our proposed measure becomes

$\mathcal{M}=\int_{\gamma<0}-\gamma\,dt.$ (45)

Comparing Eqs.(43) and (45), it turns out that the two measures are related by
$\mathcal{I}=2\mathcal{M}$.

It is worth noting that a straightforward calculation for other well-known
channels, such as the depolarizing channel, reveals that the RHP measure is
twice that of our proposed measure. However, establishing a general
relationship between the two measures for arbitrary channels remains a subject
of further investigation.

##  V Conclusions

Non-Markovianity of open quantum system dynamics has already been established
as a useful resource for several information processing tasks Laine  _et al._
(2014); Bylicka  _et al._ (2014); Xiang  _et al._ (2014); Thomas  _et al._
(2018); Reich  _et al._ (2015). However, prior to incorporating it into any
information processing task, it is of foremost importance to detect the
signature of such a resource. In this work, we have developed a methodology to
assess whether a dynamics attributes non-Markovian traits. Our proposal is
based on the moment criterion of Choi matrices, which can be efficiently
demonstrated in an experimental setup. We have presented two explicit examples
in order to illustrate our detection scheme. Further, we have proposed a
measure of non-Markovianity for unital dynamics which is again based on the
partial moment criterion. Interestingly, our proposed measure turns out to be
just half of the RHP measure for some of the well-known channels. However, the
exploration of a general relationship between these two measures for arbitrary
channels requires further analysis.

Our proposed non-Markovianity detection protocol requires computation of
simple functionals without necessitating the evaluation of full spectrum of
the evolution, and hence, is a lot more efficient than full process tomography
Elben  _et al._ (2020a). Moreover, the protocol being state independent, no
prior information about the dynamics is required unlike witness based
detection schemes Huang  _et al._ (2020). Furthermore, since our proposed
measure relies on moments of the Choi state, it is amenable for implementation
in experiments utilizing the tools of shadow tomographyHuang  _et al._ (2020).
The relevance of our proposed detection criterion should be further evident in
the case of non-Markovian dynamics involving multi-qubit systems wherein an
exponentially lesser number of samples would be required. Extension of the
moments based criterion for non-Markovianity detection of multi-qubit systems
is therefore motivated as a natural off-shoot of our present analysis.

##  VI Acknowledgements

BM and SM acknowledge Bihalan Bhattacharya for fruitful discussions. BM
acknowledges DST INSPIRE fellowship program for financial support. ASM
acknowledges support from the Project No. DST/ICPS/QuEST/2018/98 of the
Department of Science and Technology, Government of India.

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# A twisted invariant of a compact Riemann surface ††thanks: Keywords: moduli
of compact Riemann surfaces, the first Mumford-Morita-Miller class, Kawazumi-
Zhang invariant

Nariya Kawazumi

###### Abstract

We introduce a twisted version of the Kawazumi-Zhang invariant
$a_{g}(C)=\varphi(C)$ of a compact Riemann surface $C$ of genus $g\geq 1$, and
discuss how it is related to the first Mumford-Morita-Milller class
$e_{1}=\kappa_{1}$ on the moduli space of compact Riemann surfaces and the
original Kawazumi-Zhang invariant.

##  1 Introduction

A compact Riemann surface is still a fascinating research subject even in the
twenty-first century. In fact, various mathematicians and even physicists are
now studying the Teichmüller space $\mathcal{T}_{g}$ of genus $g$, a universal
covering space of the moduli space ${\mathbb{M}_{g}}$ of compact Riemann
surfaces of genus $g$, from their own various viewpoints. In this paper we
study functions on the moduli space ${\mathbb{M}_{g}}$, i.e., invariant
functions on the Teichmüller space $\mathcal{T}_{g}$ under the mapping class
group action. They are regarded also as analytic invariants of a compact
Riemann surface. There are no non-constant holomorphic functions on
${\mathbb{M}_{g}}$ if $g\geq 3$, while the $3g-3$-dimensional orbifold
${\mathbb{M}_{g}}$ has an enormous amount of $C^{\infty}$ functions. Hence we
would like to find functions which have some geometric and/or arithmetic
meanings. Hyperbolic geometry is a highly effective tool for studying Riemann
surfaces and Teichmüller spaces. But, in the context of hyperbolic geometry,
the author knows no interesting functions on the moduli space
${\mathbb{M}_{g}}$ of compact Riemann surfaces except the injective radius of
a compact hyperbolic surface.

Other effective tools for studying compact Riemann surfaces are the classical
theory of Abelian integrals and its powerful descendant, arithmetic geometry
including the Arakelov geometry. Theta functions provide meaningful functions
on the modulli space ${\mathbb{M}_{g}}$. Moreover, based on his own Arakelov
geometry, Faltings [1] introduced an analytic invariant of a compact Riemann
surface. Hain and Reed [2] introduced another invariant by the Hodge line
bundle and the intermediate Jacobian on the moduli space ${\mathbb{M}_{g}}$.
Zhang [26] and the author [15, 16] introduced the same invariant
$a_{g}=\varphi$ independently. Zhang’s approach comes from the Arakelov
geometry, while the author’s from the Johnson homomorphism similar to that of
Hain and Reed, but based on the classical theory of Abelian integrals. Later
de Jong [8] proved that the Faltings invariant, the Hain-Reed invariant and
the Kawazumi-Zhang invariant are linearly dependent. Moreover he has clarified
various aspects of the invariant $a_{g}$ in papers including [9, 10, 11]. In
particular, de Jong and Shokrieh [13, Theorem 7.1] gave a complete description
of the asymptotic behavior of the invariant $a_{g}$ along $1$-parameter
families of compact Riemann surfaces degenerating into a stable curve. D’Hoker
and Green [4] found some relation of the genus $2$ invariant $a_{2}$ to high
energy physics. D’Hoker, Green, Pioline and Russo [5] proved the function
$a_{2}$ is an eigenfunction with respect to the Laplacian on the moduli space
of principally polarized Abelian varieties $\mathbb{A}_{2}$. In §3.2 we will
show this does not hold in the case $g=3$ by using a twisted version of the
invariant $a_{g}$. The obstruction for $a_{3}$ to be an eigenfunction is the
(extended) first Johnson homomorphism [7, 20].

In this paper, we introduce a twisted version of the invariant $a_{g}$, a map
$A$ from the $4$-fold tensor product of the first cohomology group of the
surface to the complex numbers. One can recover the invariant $a_{g}$ from the
map $A$ by using the intersection pairing on the cohomology group, and can
construct new invariants by a similar way to $a_{g}$. Moreover, from the
invariant $A$, we obtain an explicit $(1,1)$-form on the moduli space
${\mathbb{M}_{g}}$ representing the first Mumford-Morita-Miller class
$e_{1}=\kappa_{1}$ in §3.1.

The invariant $A$ is given as an integral over the $2$-fold product of the
surface. D’Hoker and Schlotterer [6] generalized our construction to integrals
over the $n$-fold product of the surface for $n\geq 3$, and proved highly
nontrivial identities among them. It is amazing that their proof of the
identities is based only on formal natures of the Arakelov Green function
$G(-,-)$.

This paper is organized as follows. The twisted version $A$ of the invariant
$a_{g}$ of a compact Riemann surface $C$ is introduced in §2. In §3 we give
two kinds of interpretation of the map $A$ as tensor fields on the moduli
space ${\mathbb{M}_{g}}$. First it defines a $(1,1)$-form on
${\mathbb{M}_{g}}$. As is proved in §4, it is closed and represents the first
Mumford-Miller-Morita class $e_{1}=\kappa_{1}$. In §3.2, using the second
interpretation, we discuss how the invariant $a_{g}$ behaves in the cases
$g=2,3$. The proof of Proposition 4, which we need for our consideration in
the case $g=3$, is given in §5.

Acknowledgements. First of all the author expresses his sincere thanks and
respect to Athanase Papadopoulos for his friendship and outstanding leadership
in world-wide research networks on Teichmüller theory. As well as his own
research, many books he has edited illustrate his great insight. The author
also thanks M. Uludag for giving him a chance to take part in a tribute to
Athanase Papadopoulos. This paper is based on the author’s talks at OIST,
University of Geneva and University of Strasbourg in 2016 and 2017. The slides
for OIST and Strasbourg are uploaded in [17, 18]. The author thanks S. Hikami,
A. Alekseev and C. Vespa for giving him these opportunities. In particular,
discussions at OIST with Pioline aroused the author’s interest in the genus
$3$ invariant $a_{3}$. He deeply thanks B. Pioline. Furthermore the author
thanks R. de Jong for his helpful comments on the first version of this paper.
The present research is supported in part by the grants JSPS KAKENHI 18KK0071,
18K03283, 19H01784, 20H00115 and 22H01120.

##  2 A twisted invariant of a compact Riemann surface

Throughout this paper, we assume the genus $g$ of a compact Riemann surface
$C$ is not smaller than $1$, $g\geq 1$. Then the vector space
$H^{\prime}:=H^{0}(C;\Omega^{1}_{C})$ of holomorphic $1$-forms on $C$ does not
vanish. Its complex conjugate
$H^{\prime\prime}:=\overline{H^{0}(C;\Omega^{1}_{C})}$ is the vector space of
anti-holomorphic $1$-forms on $C$. The first cohomology group
$H:=H^{1}(C;\mathbb{C})$ is naturally identified with the direct sum
$H^{\prime}\oplus H^{\prime\prime}$, the vector space of harmonic $1$-forms on
the surface $C$. The intersection pairing

$(\phi_{1},\phi_{2})\in H\times
H\mapsto\phi_{1}\cdot\phi_{2}:=\int_{C}\phi_{1}\wedge\phi_{2}\in\mathbb{C},$
(1)

is non-degenerate. Hence we identify $H$ with its dual through the pairing.
Under this identification, the subspaces $H^{\prime}$ and $H^{\prime\prime}$
are dual to each other.

For $\phi^{\prime}_{1},\phi^{\prime}_{2}\in
H^{\prime}=H^{0}(C;\Omega^{1}_{C})$ we can define a positive definite
Hermitian pairing

$(\phi^{\prime}_{1},\phi^{\prime}_{2})\in H^{\prime}\otimes
H^{\prime}\mapsto\frac{\sqrt{-1}}{2}\int_{C}\phi^{\prime}_{1}\wedge\overline{\phi^{\prime}_{2}}\in\mathbb{C}.$
(2)

In fact, any holomorphic $1$-form $\phi^{\prime}$ locally given by
$\phi^{\prime}=f(z)dz$ in terms of a local complex coordinate $z=x+\sqrt{-1}y$
satisfies
$\frac{\sqrt{-1}}{2}\phi^{\prime}\wedge\overline{\phi^{\prime}}=|f(z)|^{2}dx\wedge
dy$. We take an orthonormal basis $\\{\psi_{i}\\}^{g}_{i=1}\subset H^{\prime}$
with respect to the Hermitian pairing

$\frac{\sqrt{-1}}{2}\int_{C}\psi_{i}\wedge\overline{\psi_{j}}=\delta_{ij},\quad(1\leq
i,j\leq g).$

Then the element

$\widehat{\omega}:=\sum^{g}_{i=1}\psi_{i}\otimes\overline{\psi_{i}}\in
H^{\prime}\otimes H^{\prime\prime}(\subset H\otimes H)$ (3)

is independent of the choice of the basis $\\{\psi_{i}\\}^{g}_{i=1}$. In fact,
the set $\\{\psi_{i},\frac{\sqrt{-1}}{2}\overline{\psi_{i}}\\}_{1\leq i\leq
g}$ is a symplectic basis of $H=H^{1}(C;\mathbb{C})$ with respect to the
pairing (1). Hence the symplectic form on $H$, which depends only on the
pairing (1), equals

$\frac{\sqrt{-1}}{2}\sum^{g}_{i=1}\psi_{i}\otimes\overline{\psi_{i}}-\frac{\sqrt{-1}}{2}\sum^{g}_{i=1}\overline{\psi_{i}}\otimes\psi_{i}\in
H^{1}(C;\mathbb{C})^{\otimes 2},$

whose component in $H^{\prime}\otimes H^{\prime\prime}$ equals
$\frac{\sqrt{-1}}{2}\widehat{\omega}$. In particular, a $(1,1)$-form $B$ on
the surface $C$ defined by

$B:=\frac{\sqrt{-1}}{2g}\sum^{g}_{i=1}\psi_{i}\wedge\overline{\psi_{i}},$

is independent of the choice of the basis $\\{\psi_{i}\\}^{g}_{i=1}$. We have
$\int_{C}B=1$. Moreover the form $B$ is a real volume form on the surface $C$,
since the $1$-forms $\psi_{i}$’s have no common zeroes because of the Riemann-
Roch formula.

We denote by $\Omega^{p}(C)$ the vector space of $\mathbb{C}$-valued $p$-forms
on $C$ for $p=0,1,2$. Then the Hodge $*$-operator
$*:\Omega^{1}(C)\to\Omega^{1}(C)$ depends only on the complex structure of the
surface $C$. In fact, in terms of a local complex coordinate $z$, we have
$*dz=-\sqrt{-1}dz$ and $*d\overline{z}=\sqrt{-1}d\overline{z}$. Then, since
the operator $d*d$ is equivalent to the Laplacian by means of the volume form
$B$, we have an exact sequence

$0\to\mathbb{C}\to\Omega^{0}(C)\overset{d*d}{\longrightarrow}\Omega^{2}(C)\overset{\int_{C}}{\longrightarrow}\mathbb{C}\to
0,$

where the left $\mathbb{C}$ means the constant functions. Hence we have a
unique linear map $\widehat{\Phi}:\Omega^{2}(C)\to\Omega^{0}(C)$ such that

$d*d\widehat{\Phi}(\Omega)=\Omega-\left(\int_{C}\Omega\right)B\quad\text{and}\quad\int_{C}\widehat{\Phi}(\Omega)B=0$

for any $\Omega\in\Omega^{2}(C)$. It is nothing but the Green operator
associated with the volume form $B$. The operator $\widehat{\Phi}$ is real,
$\overline{\widehat{\Phi}}=\widehat{\Phi}$, and the induced map

$\widehat{\Phi}|_{\mathrm{Ker}(\int_{C})}:\mathrm{Ker}\left(\int_{C}:\Omega^{2}(C)\to\mathbb{C}\right)\to\Omega^{0}(C)/\mathbb{C}$

depends only on the complex structure on the surface $C$. Moreover we have

$\int_{C}\widehat{\Phi}(\Omega)\Omega^{\prime}=\int_{C}\Omega\widehat{\Phi}(\Omega^{\prime})=\frac{1}{4\pi}\int\\!\\!\\!\\!\int_{(P_{1},P_{2})\in
C\times C}(\Omega)_{P_{1}}(\log G(P_{1},P_{2}))(\Omega^{\prime})_{P_{2}}$ (4)

for any $\Omega,\Omega^{\prime}\in\Omega^{2}(C)$, where we denote by $G(-,-)$
the Arakelov Green function. Here we remark $\widehat{\Phi}(B)=0$. In fact,
the function $\widehat{\Phi}(B)$ is constant since $d*d\widehat{\Phi}(B)=0$,
and so $\widehat{\Phi}(B)=\int_{C}\widehat{\Phi}(B)B=0$.

Now we define a map $A$ by

$A:H^{\prime}\otimes H^{\prime\prime}\otimes H^{\prime}\otimes
H^{\prime\prime}\to\mathbb{C},\quad\phi_{1}\otimes\overline{\phi_{2}}\otimes\phi_{3}\otimes\overline{\phi_{4}}\mapsto\int_{C}\phi_{1}\wedge\overline{\phi_{2}}\,\widehat{\Phi}(\phi_{3}\wedge\overline{\phi_{4}}),$
(5)

which is independent of the choice of the basis $\\{\psi_{i}\\}^{g}_{i=1}$,
and so is a twisted invariant of the compact Riemann surface $C$. One can
extend it to a map $H\otimes H\otimes H\otimes H\to\mathbb{C}$ by the same
formula (5), but with no additional information. In order to compute the map
$A$ explicitly, it is convenient to introduce a complex number
$A_{i\overline{j}k\overline{l}}$ defined by

$\displaystyle A_{i\overline{j}k\overline{l}}$
$\displaystyle:=A(\psi_{i}\otimes\overline{\psi_{j}}\otimes\psi_{k}\otimes\overline{\psi_{l}})=\int_{C}\psi_{i}\wedge\overline{\psi_{j}}\,\widehat{\Phi}(\psi_{k}\wedge\overline{\psi_{l}})$
(6) $\displaystyle=\frac{1}{4\pi}\int\\!\\!\\!\\!\int_{(P_{1},P_{2})\in
C\times C}(\psi_{i}\wedge\overline{\psi_{j}})_{P_{1}}(\log
G(P_{1},P_{2}))(\psi_{k}\wedge\overline{\psi_{l}})_{P_{2}}$

for $1\leq i,j,k,l\leq g$ by using the orthonormal basis
$\\{\psi_{i}\\}^{g}_{i=1}$. Then we have

$\displaystyle\overline{A_{i\overline{j}k\overline{l}}}=\int_{C}\psi_{j}\wedge\overline{\psi_{i}}\,\widehat{\Phi}(\psi_{l}\wedge\overline{\psi_{k}})=A_{j\overline{i}l\overline{k}},$
(7) $\displaystyle
A_{i\overline{j}k\overline{l}}=\int_{C}\widehat{\Phi}(\psi_{i}\wedge\overline{\psi_{j}})\,\psi_{k}\wedge\overline{\psi_{l}}=A_{k\overline{l}i\overline{j}}.$

The latter follows from (4). We can contract the indices $i,k,\dots$ and the
ones $\overline{j},\overline{l},\dots$ to get a new complex number. For
example, the Kawazumi-Zhang invariant111 The author uses the symbol
$a_{g}=a_{g}(C)$, while Zhang and others use the symbol $\varphi=\varphi(C)$.
$a_{g}=a_{g}(C)$ is obtained by

$a_{g}=\sum^{g}_{i,j=1}A_{i\overline{j}j\overline{i}}=A((24)(\widehat{\omega}\otimes\widehat{\omega})),$

which is a real number by the formula (7). Here $(24):H^{\prime}\otimes
H^{\prime\prime}\otimes H^{\prime}\otimes H^{\prime\prime}\to
H^{\prime}\otimes H^{\prime\prime}\otimes H^{\prime}\otimes H^{\prime\prime}$
is the switch map of the 2nd and the 4th components of the tensor product
$H^{\prime}\otimes H^{\prime\prime}\otimes H^{\prime}\otimes
H^{\prime\prime}$.

We can describe the numbers $A_{i\overline{j}k\overline{l}}$ and $a_{g}$ in
the diagrams

$A_{i,\overline{j},k,\overline{l}}=$$i$$\overline{j}$$\overline{l}$$k$$a_{g}=$

.

Here the circle with indices $i$ and $\overline{j}$ means
$\psi_{i}\wedge\overline{\psi_{j}}$, the double line without arrows $\log
G(-,-)$, and a line with an arrow the contraction. For example, since
$\frac{\sqrt{-1}}{2g}\widehat{\Phi}(\sum^{g}_{i=1}\psi_{i}\wedge\overline{\psi_{i}})=\widehat{\Phi}(B)=0$,
we have
$\sum_{i,j,l=1}^{g}A_{i\overline{j}l\overline{l}}=\sum_{j,k,l=1}^{g}A_{j\overline{j}k\overline{l}}=0$,
that is,

$i$$\overline{j}$$=$$\overline{l}$$k$$=0$

Similarly we can introduce complex numbers associated with the Riemann surface
$C$, i.e., functions on the moduli space ${\mathbb{M}_{g}}$ of compact Riemann
surfaces. For example, we can consider a function on ${\mathbb{M}_{g}}$
defined by the diagram

.

The number $A_{i\overline{j}k\overline{l}}$ is given by an integral over the
$2$-fold direct product $C^{\times 2}=C\times C$. We can generalize this
construction to the $3$-fold direct product and more. For example, we can
consider

$\displaystyle\int_{C}\widehat{\Phi}(\psi_{i}\wedge\overline{\psi_{j}})\psi_{m}\wedge\overline{\psi_{n}}\,\widehat{\Phi}(\psi_{k}\wedge\overline{\psi_{l}})=\int_{C}\psi_{i}\wedge\overline{\psi_{j}}\,\widehat{\Phi}((\psi_{m}\wedge\overline{\psi_{n}})\,\widehat{\Phi}(\psi_{k}\wedge\overline{\psi_{l}}))$
$\displaystyle=\frac{1}{(4\pi)^{2}}\int\\!\\!\\!\\!\int\\!\\!\\!\\!\int_{(P_{1},P_{2},P_{3})\in
C\times C\times C}(\psi_{i}\wedge\overline{\psi_{j}})_{P_{1}}(\log
G(P_{1},P_{2}))(\psi_{m}\wedge\overline{\psi_{n}})_{P_{2}}(\log
G(P_{2},P_{3}))(\psi_{k}\wedge\overline{\psi_{l}})_{P_{3}}$

for $1\leq i,j,k,l,m,n\leq g$. This can be described in the diagram

$i$$\overline{j}$$\overline{n}$$m$$\overline{l}$$k$

.

D’Hoker and Schlotterer [6] gave a systematic generalization of
$A_{i\overline{j}k\overline{l}}$ where they constructed modular graph tensors
using integrals on the direct product $C^{\times n}$ for any $n\geq 1$, and
proved highly nontrivial identities among them. It is amazing that their proof
of the identities is based only on formal natures of the Arakelov Green
function $G(-,-)$.

##  3 Tensor fields on the moduli space ${\mathbb{M}_{g}}$

When the Riemann surface $C$ runs over the moduli space ${\mathbb{M}_{g}}$ of
compact Riemann surfaces of genus $g$, the transpose ${}^{t}A$ of the linear
map $A$ in (5) can be regarded as some tensor fields on the moduli space
${\mathbb{M}_{g}}$. Now recall the vector spaces $H^{\prime}$ and
$H^{\prime\prime}$ are dual to each other through the pairing (1). In
particular, the basis $\\{\psi_{i}\\}^{g}_{i=1}$ and
$\\{\frac{\sqrt{-1}}{2}\overline{\psi_{i}}\\}^{g}_{i=1}$ are dual to each
other. Then the transpose ${}^{t}A:\mathbb{C}\to H^{\prime\prime}\otimes
H^{\prime}\otimes H^{\prime\prime}\otimes H^{\prime}$ is determined by its own
value at $1\in\mathbb{C}$

${}^{t}A(1)=\frac{1}{16}\sum_{i,j,k,l=1}^{g}A_{i\overline{j}k\overline{l}}\overline{\psi_{i}}\otimes\psi_{j}\otimes\overline{\psi_{k}}\otimes\psi_{l}\in
H^{\prime\prime}\otimes H^{\prime}\otimes H^{\prime\prime}\otimes H^{\prime}.$
(8)

When the surface $C$ runs over the moduli space ${\mathbb{M}_{g}}$, the
collections of $H^{\prime}$ and $H^{\prime\prime}$ are holomorphic and anti-
holomorphic vector bundles, respectively. Then ${}^{t}A(1)$ is a $C^{\infty}$
section of a tensor product of these vector bundles.

###  3.1 The first Mumford-Morita-Miller class $e_{1}=\kappa_{1}$

Here we recall the holomorphic cotangent space $T^{*}_{[C]}{\mathbb{M}_{g}}$
of the moduli space ${\mathbb{M}_{g}}$ at the equivalence class $[C]$ of the
Riemann surface $C$ is canonically isomorphic to the space of holomorphic
quadratic differentials

$T^{*}_{[C]}{\mathbb{M}_{g}}=H^{0}(C;(\Omega^{1}_{C})^{\otimes 2}).$

For example, we may regard $\psi_{l}\psi_{j}=\psi_{j}\psi_{l}\in
T^{*}_{[C]}{\mathbb{M}_{g}}$ and
$\overline{\psi_{i}}\overline{\psi_{k}}=\overline{\psi_{k}}\overline{\psi_{i}}\in\overline{T^{*}_{[C]}{\mathbb{M}_{g}}}$.
Hence we can define a $(1,1)$-form $e_{1}^{\widehat{\Phi}}$ on the moduli
space ${\mathbb{M}_{g}}$ by222In [17, 18] we denote $e_{1}^{\widehat{\Phi}}$
by $e_{1}^{D}$, whose coefficients was $48\sqrt{-1}$. It is not true. This
mistake comes from computation of the intersection number
$Y_{i}\cdot\overline{Y_{i}}$.

$e_{1}^{\widehat{\Phi}}|_{[C]}:=-3\sqrt{-1}\sum_{i,j,k,l=1}^{g}(\psi_{j}\psi_{l}A_{i\overline{j}k\overline{l}}\overline{\psi_{k}}\overline{\psi_{i}}-\psi_{j}\psi_{l}A_{i\overline{j}k\overline{i}}\overline{\psi_{k}}\overline{\psi_{l}}).$

In fact, from (8), the RHS is independent of the choice of the basis
$\\{\psi_{i}\\}^{g}_{i=1}$. Then we have

######  Theorem 1.

The $(1,1)$-form $e_{1}^{\widehat{\Phi}}$ is closed on the moduli space
${\mathbb{M}_{g}}$, and represents the first Mumford-Morita-Miller class
$e_{1}\in H^{2}({\mathbb{M}_{g}};\mathbb{C})$.

The theorem was stated implicitly in the equation (5.5) of the preprint [15]
without proof. We will give its proof in the next section §4.

Very recently, de Jong and van der Lugt [12, 19] introduced a subalgebra
$\mathcal{R}({\mathbb{M}_{g}})$ of the de Rham complex
$\Omega^{*}({\mathbb{M}_{g}})$, called the ring of tautological forms on
${\mathbb{M}_{g}}$. It consists entirely of closed forms, and its image in the
cohomology algebra $H^{*}({\mathbb{M}_{g}};\mathbb{C})$ is generated by all
the Mumford-Morita-Miller classes, i.e., equals the image of the tautological
ring of ${\mathbb{M}_{g}}$. It is remarkable that the ring
$\mathcal{R}({\mathbb{M}_{g}})$ is finite-dimensional, which suggests us that
study of differential forms on ${\mathbb{M}_{g}}$ could help us to understand
the tautological ring of ${\mathbb{M}_{g}}$. As was shown in [12, 19], the
degree $2$ part of the ring $\mathcal{R}({\mathbb{M}_{g}})$ is closely related
to the invariant $a_{g}$. So it would be very interesting if we could find
further relations of the ring $\mathcal{R}({\mathbb{M}_{g}})$ with the complex
numbers $A_{i\overline{j}k\overline{l}}$’s or generalizations by D’Hoker and
Schlotterer [6].

###  3.2 The Jacobian map $\mathrm{Jac}:{\mathbb{M}_{g}}\to{\mathbb{A}_{g}}$

The transpose ${}^{t}A(1)$ in (8) has another interpretation: it can be
regarded as a $C^{\infty}$ section of the pullback
$\mathrm{Jac}^{*}(T^{*}{\mathbb{A}_{g}}\otimes\overline{T^{*}{\mathbb{A}_{g}}})$
over the moduli space ${\mathbb{M}_{g}}$. Here ${\mathbb{A}_{g}}$ is the
moduli space of principally polarized Abelian varieties of genus $g$, and
$\mathrm{Jac}:{\mathbb{M}_{g}}\to{\mathbb{A}_{g}}$,
$[C]\mapsto[\mathrm{Jac}(C)]$, the Jacobian map. The moduli space
${\mathbb{A}_{g}}$ is the quotient space of the Siegel upper half space
$\mathfrak{H}_{g}$ by the Siegel modular group $Sp_{2g}(\mathbb{Z})$. The
space $\mathfrak{H}_{g}=Sp_{2g}(\mathbb{R})/U_{g}$ is the space of all complex
structures $J$ on the standard symplectic vector space $\mathbb{R}^{2g}$ with
positivity condition. The tautological vector bundle
$E_{g}:=\coprod_{J\in\mathfrak{H}_{g}}(\mathbb{R}^{2g},J)$ over
$\mathfrak{H}_{g}$ is an $Sp_{2g}(\mathbb{R})$-equivariant holomorphic vector
bundle with canonical Hermitian metric, and so descends to the moduli space
${\mathbb{A}_{g}}$. Through a canonical isomorphism
$T{\mathbb{A}_{g}}=\mathrm{Sym}^{2}(E_{g})\subset{E_{g}}^{\otimes 2}$, one can
define a Kähler metric on the moduli space ${\mathbb{A}_{g}}$. See, for
example, [22, Chapter II]. In particular, we may consider the Laplacian
$\Delta$ on the moduli space ${\mathbb{A}_{g}}$. The pullback
$\mathrm{Jac}^{*}E^{*}_{g}$ of the Hermitian vector bundle $E_{g}$ under the
Jacobian map $\mathrm{Jac}$ at the equivalence class $[C]\in{\mathbb{M}_{g}}$
is isometric to the holomorphic $1$-forms $H^{\prime}=H^{0}(C;\Omega^{1}_{C})$
with metric given in (2). Hence the transpose ${}^{t}A(1)$ in (8) can be
regarded as a $C^{\infty}$ section of the pullback
$\mathrm{Jac}^{*}(T^{*}{\mathbb{A}_{g}}\otimes\overline{T^{*}{\mathbb{A}_{g}}}))$.

By Rauch’s formula [24], the differential $d\mathrm{Jac}$ of the Jacobian map
is given by the commutative diagram

$\begin{CD}T^{*}_{[C]}{\mathbb{M}_{g}}@<{(d\mathrm{Jac})^{*}}<{}<T^{*}_{[\mathrm{Jac}(C)]}{\mathbb{A}_{g}}\\\
\Big{\|}\Big{\|}\\\ H^{0}(C;(\Omega^{1}_{C})^{\otimes
2})@<{\text{multiplication}}<{}<\mathrm{Sym}^{2}H^{0}(C;\Omega^{1}_{C}).\end{CD}$
(9)

In particular, from M. Noether’s theorem, the differential
$d\mathrm{Jac}_{[C]}$ is injective if and only if $g=2$ or ($g\geq 3$ and $C$
is non-hyperelliptic). On the other hand,
$\dim{\mathbb{M}_{g}}=\dim{\mathbb{A}_{g}}$ if and only if $g=2$ or $3$. Hence
the Laplacian $\Delta$ on ${\mathbb{A}_{g}}$ acts on functions on
$\mathbb{M}_{2}\overset{\text{open}}{\subset}\mathbb{A}_{2}$ and
$\mathbb{M}_{3}\setminus\mathbb{H}_{3}\overset{\text{open}}{\subset}\mathbb{A}_{3}$.
Here we denote by $\mathbb{H}_{g}\subset{\mathbb{M}_{g}}$ the locus of
hyperelliptic Riemann surfaces of genus $g\geq 2$.

D’Hoker, Green, Pioline and Russo [5] proved the following remarkable theorem.

######  Theorem 2 ([5](4.23)).

$\Delta a_{2}=5a_{2}$. In particular, the invariant $a_{2}$ is an
eigenfunction of the Laplacian $\Delta$.

Based on the theorem, Pioline [23] gave an explicit formula for $a_{2}$ in
terms of theta functions. Later, for any $g\geq 1$, Wilms [25] gave an
explicit formula for $a_{g}$ in terms of a function introduced by de Jong as
well as theta functions. But the methods of Wilms [25] seem to be different
from those discussed in this paper.

There remains the case $g=3$. In this paper we prove

######  Theorem 3.

In the case $g=3$, the invariant $a_{3}$ is not an eigenfunction of the
Laplacian $\Delta$.

The obstruction for $a_{3}$ to be an eigenfunction is the first Johnson
homomorphism [7, 20], which is represented by a twisted $1$-form on
${\mathbb{M}_{g}}$ as follows. We denote by $U=U(C)$ the kernel of the
contraction map $\Lambda^{3}H\to H$,
$\phi_{1}\wedge\phi_{2}\wedge\phi_{3}\mapsto(\phi_{2}\cdot\phi_{3})\phi_{1}+(\phi_{3}\cdot\phi_{1})\phi_{2}+(\phi_{1}\cdot\phi_{2})\phi_{3}$.
It should be remarked that $U$ vanishes if and only if $g\leq 2$. For
$\phi_{i}=\phi^{\prime}_{i}+\phi^{\prime\prime}_{i}\in H=H^{\prime}\oplus
H^{\prime\prime}$, $1\leq i\leq 3$, we define
$Q(\phi_{1},\phi_{2},\phi_{3})\in C^{\infty}(C;(T^{*}C)^{\otimes 2})$ by

$Q(\phi_{1},\phi_{2},\phi_{3}):=-\sqrt{-1}\phi^{\prime}_{1}\partial\widehat{\Phi}(\phi_{2}\wedge\phi_{3})-\sqrt{-1}\phi^{\prime}_{2}\partial\widehat{\Phi}(\phi_{3}\wedge\phi_{1})-\sqrt{-1}\phi^{\prime}_{3}\partial\widehat{\Phi}(\phi_{1}\wedge\phi_{2}).$
(10)

The tensor $Q(\phi_{1},\phi_{2},\phi_{3})$ is alternating in
$\phi_{1},\phi_{2},\phi_{3}$, but is not holomorphic in general. Since
$d*d=-2\sqrt{-1}\overline{\partial}\partial$, we have

$\displaystyle 2\overline{\partial}Q$ $\displaystyle=$
$\displaystyle\phi^{\prime}_{1}(\phi_{2}\wedge\phi_{3}-(\int_{C}\phi_{2}\wedge\phi_{3})B)+\phi^{\prime}_{2}(\phi_{3}\wedge\phi_{1}-(\int_{C}\phi_{3}\wedge\phi_{1})B)$
$\displaystyle\hskip
113.81102pt+\phi^{\prime}_{3}(\phi_{1}\wedge\phi_{2}-(\int_{C}\phi_{1}\wedge\phi_{2})B)$
$\displaystyle=$
$\displaystyle\phi^{\prime}_{1}\phi^{\prime}_{2}\phi^{\prime\prime}_{3}-\phi^{\prime}_{1}\phi^{\prime}_{3}\phi^{\prime\prime}_{2}+\phi^{\prime}_{2}\phi^{\prime}_{3}\phi^{\prime\prime}_{1}-\phi^{\prime}_{2}\phi^{\prime}_{1}\phi^{\prime\prime}_{3}+\phi^{\prime}_{3}\phi^{\prime}_{1}\phi^{\prime\prime}_{2}-\phi^{\prime}_{3}\phi^{\prime}_{2}\phi^{\prime\prime}_{1}$
$\displaystyle\hskip
113.81102pt-(\phi^{\prime}_{1}(\phi_{2}\cdot\phi_{3})+\phi^{\prime}_{2}(\phi_{3}\cdot\phi_{1})+\phi^{\prime}_{3}(\phi_{1}\cdot\phi_{2}))B$
$\displaystyle=$
$\displaystyle-(\phi^{\prime}_{1}(\phi_{2}\cdot\phi_{3})+\phi^{\prime}_{2}(\phi_{3}\cdot\phi_{1})+\phi^{\prime}_{3}(\phi_{1}\cdot\phi_{2}))B.$

This implies the restriction of $Q$ to the subspace $U$ has its value in the
space $H^{0}(C;(\Omega^{1}_{C})^{\otimes 2})$ of holomorphic quadratic
differentials

$Q:U=U(C)\to H^{0}(C;(\Omega^{1}_{C})^{\otimes
2})=T^{*}_{[C]}{\mathbb{M}_{g}}.$

We remark the map $Q$ is equivariant under the action of the holomorphic
automorphism group $\mathrm{Aut}(C)$. In particular, if the Riemann surface
$C$ is hyperelliptic, we have

$Q(U(C))\subset(\text{the $(-1)$-eigenspace of the hyperelliptic involution in
$T^{*}_{[C]}{\mathbb{M}_{g}}$}),$ (11)

since the involution acts on the space $U(C)$ by $-1_{U(C)}$.

We may identify the map $Q$ with its transpose333For simplicity, we use the
same symbol $Q$, not ${}^{t}Q$. $Q:T_{[C]}{\mathbb{M}_{g}}\to(U(C))^{*}$,
which can be regarded as a twisted $(1,0)$ form on the moduli space
${\mathbb{M}_{g}}$. Here the pairing

$\displaystyle\langle-,-\rangle:\Lambda^{3}H\times\Lambda^{3}H$
$\displaystyle\to\mathbb{C},$ (12)
$\displaystyle(\phi^{(1)}_{1}\wedge\phi^{(1)}_{2}\wedge\phi^{(1)}_{3},\phi^{(2)}_{1}\wedge\phi^{(2)}_{2}\wedge\phi^{(2)}_{3})$
$\displaystyle\mapsto\sum_{\sigma,\tau\in\mathfrak{S}_{3}}(\phi^{(1)}_{\sigma(1)}\cdot\phi^{(2)}_{\tau(1)})(\phi^{(1)}_{\sigma(2)}\cdot\phi^{(2)}_{\tau(2)})(\phi^{(1)}_{\sigma(3)}\cdot\phi^{(2)}_{\tau(3)})$

is non-degenerate on $U=U(C)$, and so we identify $U(C)$ with its dual
$(U(C))^{*}$ through the pairing (12). We denote by the same symbol $U$ the
flat vector bundle $\coprod_{[C]\in{\mathbb{M}_{g}}}U(C)$ over the moduli
space ${\mathbb{M}_{g}}$. The tensor $Q$ equals the $(1,0)$-part of the first
variation of the harmonic volumes introduced by Harris [3, Theorem 5.8]. In
particular, the sum $Q+\overline{Q}$ is a twisted closed $1$-form with values
in the flat vector bundle $U$. As was proved in [14, §8], the cohomology class
of $-(Q+\overline{Q})$ equals the extended first Johnson homomorphism
$\tilde{k}$ [20]

$\tilde{k}=-[Q+\overline{Q}]\in H^{1}({\mathbb{M}_{g}};U).$

The cohomology group $H^{1}({\mathbb{M}_{g}};U)$ is canonically isomorphic to
the first group cohomology of the mapping class group of a genus $g$ Riemann
surface with values in the vector space $U=U(C)$. The restriction of
$\tilde{k}$ to the Torelli group equals the first Johnson homomorphism [7]. As
was proved by Morita [21], the first Mumford-Morita-Miller class $e_{1}$
equals the contraction $\langle\tilde{k},\tilde{k}\rangle\in
H^{2}({\mathbb{M}_{g}};\mathbb{C})$ of the cup product
$\tilde{k}\cup\tilde{k}\in H^{2}({\mathbb{M}_{g}};U^{\otimes 2})$ up to non-
zero constant factor. By the same recipe using the pairing in (12), we obtain
a $(1,1)$-form $e_{1}^{J}$ on ${\mathbb{M}_{g}}$ representing the class
$e_{1}$, which equals $\langle Q+\overline{Q},Q+\overline{Q}\rangle$ up to
non-zero constant factor. From [15, Theorem 6.1], we have

$\frac{1}{12}e_{1}^{\widehat{\Phi}}=\frac{-2\sqrt{-1}g}{2(2g+1)}(\partial\overline{\partial}a_{g})+\frac{1}{12}e_{1}^{J}.$
(13)

An alternative proof of the formula (13) using the Deligne pairing is given by
de Jong [11]. If $g=2$, then $U=0$, so that $e_{1}^{J}=0$. But $e_{1}^{J}\neq
0$ for $g\geq 3$.

In [5, Appendices B and C], D’Hoker, Green, Pioline and Russo gave suggestive
computations for any genus $g\geq 2$. In the equations (B.4) and (C.1), they
divide the exact $(1,1)$ form $\overline{\partial}\partial a_{g}$ into two
parts $\psi^{1}_{A}+\psi_{B}$ and $\psi^{2}_{A}+\psi_{C}$. Their formulae
(C.3) and (C.5) say that $\psi^{1}_{A}+\psi_{B}$ and $\psi^{2}_{A}+\psi_{C}$
equal $e_{1}^{\widehat{\Phi}}$ and $e_{1}^{J}$ up to non-zero constant factor,
respectively. These formulae are equivalent to our equation (13). The formula
(C.9) says that $\psi^{2}_{A}+\psi_{C}$ vanishes if $g=2$. In order to
understand the formula (C.4)

$\Delta a_{g}\Bigr{|}_{\psi^{1}_{A}+\psi_{B}}=(2g+1)a_{g},$

we need to introduce a lift of the differential form $e_{1}^{\widehat{\Phi}}$
to an element $\widehat{E_{1}}$ of
$C^{\infty}({\mathbb{M}_{g}};\mathrm{Jac}^{*}(T^{*}{\mathbb{A}_{g}}\otimes\overline{T^{*}{\mathbb{A}_{g}}}))$
defined by

$\widehat{E_{1}}|_{[C]}:=-3\sqrt{-1}\sum_{i,j,k,l=1}^{g}((\psi_{j}\bullet\psi_{l})A_{i\overline{j}k\overline{l}}(\overline{\psi_{k}}\bullet\overline{\psi_{i}})-(\psi_{j}\bullet\psi_{l})A_{i\overline{j}k\overline{i}}(\overline{\psi_{k}}\bullet\overline{\psi_{l}})).$

Here
$\psi_{j}\bullet\psi_{l}:=\frac{1}{2}(\psi_{j}\otimes\psi_{l}+\psi_{l}\otimes\psi_{j})\in\mathrm{Sym}^{2}H^{\prime}$
and so on. Now we recall the Laplacian $\Delta$ on any Kähler manifold $X$ is
given by

$\Delta f=2\sqrt{-1}\Lambda\partial\overline{\partial}f$

for any $f\in C^{\infty}(X;\mathbb{C})$. Here
$\Lambda:T^{*}X\otimes\overline{T^{*}X}\to X\times\mathbb{C}$ is the Kähler
contraction. In our case $X={\mathbb{A}_{g}}$, we have

$\Lambda\widehat{E_{1}}|_{[C]}=6\left(\sum_{j,l=1}^{g}A_{l\overline{j}j\overline{l}}+\sum_{j,l=1}^{g}A_{j\overline{j}l\overline{l}}-g\sum_{i,j=1}^{g}A_{i\overline{j}j\overline{i}}-\sum_{i,l=1}^{g}A_{i\overline{l}l\overline{i}}\right)=-6ga_{g},$

since $\int_{C}\psi_{i}\wedge\overline{\psi_{j}}=-2\sqrt{-1}\delta_{ij}$ and
$\sum_{j,k,l=1}^{g}A_{j\overline{j}k\overline{l}}=0$. Applying the Kähler
contraction to the equation (13), we obtain
$-\frac{g}{2}a_{g}=-\frac{g}{2(2g+1)}\Delta a_{g}+\frac{1}{12}\Lambda
e_{1}^{J}$, and so

$\Delta a_{g}=(2g+1)a_{g}+\frac{g}{6}\Lambda e_{1}^{J},$ (14)

if $g=2,3$. In the case $g=3$, the Kähler contraction $|\Lambda e_{1}^{J}|$
with respect to the metric of $\mathbb{A}_{3}$ satisfies the following.

######  Proposition 4.

There is a hyperelliptic Riemann surface $C_{0}$ of genus $3$ such that

$|\Lambda e_{1}^{J}|\longrightarrow+\infty$

as a non-hyperelliptic Riemann surface $C$ of genus $3$ goes to the surface
$C_{0}$.

The proof will be given in §5, and is based on a theorem of Harris [3, Theorem
6.5].

Now the invariant $a_{3}$ is $C^{\infty}$ near $[C_{0}]\in\mathbb{M}_{3}$, so
is locally bounded near $[C_{0}]$. Proposition 4 together with (14) implies
that $\Delta a_{3}$ is not bounded near $[C_{0}]$. Hence the function $a_{3}$
is not an eigenfunction of the Laplacian $\Delta$. This completes the proof of
Theorem 3 modulo that of Proposition 4. ∎

Since the Jacobian map
$\mathrm{Jac}|_{\mathbb{H}_{g}}:\mathbb{H}_{g}\to{\mathbb{A}_{g}}$ restricted
to the hyperelliptic locus $\mathbb{H}_{g}$ is an embedding, the Kähler metric
on ${\mathbb{A}_{g}}$ induces a Kähler metric on $\mathbb{H}_{g}$ and its
Laplacian $\Delta_{\mathbb{H}_{g}}$. Since the $(1,1)$-form $e_{1}^{J}$
vanishes along $\mathbb{H}_{g}$ from (11), one can clarify
$\Delta_{\mathbb{H}_{g}}(a_{g}|_{\mathbb{H}_{g}})$ if the Kähler contraction
$\Lambda$ on $\mathbb{H}_{g}$ can be computed.

##  4 A proof of Theorem 1

In this section we prove Theorem 1 based on computations given in [15]. There
is introduced a closed $(1,1)$-form $E_{1}^{D}$ on the moduli space
${\mathbb{M}_{g}}$ representing $\frac{1}{12}e_{1}$ in the equation (5.3). We
will prove the equality $E_{1}^{D}=\frac{1}{12}e_{1}^{\widehat{\Phi}}$ to
obtain the theorem.

Our starting point is Lemma 5.5 in [15] which says that

$E^{D}_{1}(\lambda,\overline{\mu})=4M\int_{C}\mathcal{H}(\omega^{\prime}_{(1)}\lambda)\wedge\omega^{\prime}_{(1)}\widehat{\Phi}d*(\omega^{\prime\prime}_{(1)}\overline{\ell^{\mu}}-\overline{\ell^{\mu}}\omega^{\prime\prime}_{(1)})$
(15)

for any Beltrami differentials $\lambda,\mu\in
C^{\infty}(C;TC\otimes\overline{T^{*}C})$.

First of all we need to explain the notation in the equation (15): $M$,
$\omega^{\prime}_{(1)}$, $\omega^{\prime\prime}_{(1)}$, $\mathcal{H}$ and
$\ell^{\mu}$. The first complex homology group $H_{1}(C;\mathbb{C})$ admits
the intersection pairing which we denote by $\cdot:H_{1}(C;\mathbb{C})\times
H_{1}(C;\mathbb{C})\to\mathbb{C}$. The integrand of the RHS is an
$H_{1}(C;\mathbb{C})^{\otimes 4}$-valued $(1,1)$-form. The symbol
$M:H_{1}(C;\mathbb{C})^{\otimes 4}\to\mathbb{C}$ is given by
$M(Z_{1}Z_{2}Z_{3}Z_{4}):=(Z_{2}\cdot Z_{3})(Z_{4}\cdot Z_{1})$ for any
$Z_{i}\in H_{1}(C;\mathbb{C})$. Here and throughout this section we omit the
symbol $\otimes$. The first complex cohomology group $H=H^{1}(C;\mathbb{C})$,
is embedded into the space of $1$-forms $\Omega^{1}(C)$ as the subspace of
harmonic $1$-forms. An $H_{1}(C;\mathbb{C})$-valued real harmonic $1$-form
$\omega_{(1)}\in\Omega^{1}(C)\otimes H_{1}(C;\mathbb{C})$ is defined to be the
embedding
$\in\mathrm{Hom}(H^{1}(C;\mathbb{C}),\Omega^{1}(C))=\Omega^{1}(C)\otimes
H_{1}(C;\mathbb{C})$. We denote by $\omega^{\prime}_{(1)}$ and
$\omega^{\prime\prime}_{(1)}$ its $(1,0)$\- and $(0,1)$\- parts, respectively.
If homology classes $Y_{i}\in H$, $1\leq i\leq g$, satisfy

$\int_{Y_{i}}\psi_{j}=\delta_{i,j},\quad\text{and}\quad\int_{Y_{i}}\overline{\psi_{j}}=0$

for $1\leq i,j\leq g$, in other words,
$\\{Y_{i},-2\sqrt{-1}\overline{Y_{i}}\\}_{1\leq i\leq g}$ is the dual basis of
the symplectic basis
$\\{\psi_{i},\frac{\sqrt{-1}}{2}\overline{\psi_{i}}\\}_{1\leq i\leq g}$, then
we have

$\omega_{(1)}=\omega^{\prime}_{(1)}+\omega^{\prime\prime}_{(1)},\quad\omega^{\prime}_{(1)}=\sum^{g}_{i=1}\psi_{i}Y_{i},\quad\text{and}\quad\omega^{\prime\prime}_{(1)}=\sum^{g}_{i=1}\overline{\psi_{i}}\overline{Y_{i}}.$
(16)

Here we remark $Y_{i}\cdot\overline{Y_{j}}=\frac{\sqrt{-1}}{2}\delta_{ij}$.
The Hodge decomposition on the $1$-forms on $C$ is given by

$\varphi=\mathcal{H}\varphi+*d\widehat{\Phi}d\varphi+d\widehat{\Phi}d*\varphi$

for any $\varphi\in\Omega^{1}(C)$. Here $\mathcal{H}$ is the harmonic
projection, which is written by

$\mathcal{H}(\varphi)=\frac{\sqrt{-1}}{2}\sum^{g}_{j=1}\left(\left(\int_{C}\varphi\wedge\overline{\psi_{j}}\right)\psi_{j}-\left(\int_{C}\varphi\wedge\psi_{j}\right)\overline{\psi_{j}}\right).$

For example, $\omega^{\prime}_{(1)}\lambda$ can be regarded as an
$H_{1}(C;\mathbb{C})$-valued $(0,1)$-form by contracting $T^{*}C$ in
$\omega^{\prime}_{(1)}$ and $TC$ in $\lambda$, so that we have

$\mathcal{H}(\omega^{\prime}_{(1)}\lambda)=\sum^{g}_{l=1}\mathcal{H}(\psi_{l}\lambda)Y_{l}=\frac{\sqrt{-1}}{2}\sum^{g}_{l,j=1}\left(\int_{C}\psi_{j}\psi_{l}\lambda\right)\overline{\psi_{j}}Y_{l},$
(17)

since
$\psi_{l}\lambda\wedge\psi_{j}=-\psi_{j}\wedge\psi_{l}\lambda=-\psi_{j}\psi_{l}\lambda$.
On the other hand, we have

$\varphi^{\prime\prime}-\mathcal{H}(\varphi^{\prime\prime})=*d\widehat{\Phi}d\varphi^{\prime\prime}+d\widehat{\Phi}d*\varphi^{\prime\prime}=2\overline{\partial}\widehat{\Phi}d*\varphi^{\prime\prime}$
(18)

for any $(0,1)$-current $\varphi^{\prime\prime}$.

Finally $\ell^{\mu}\in\Omega^{0}(C)\otimes H_{1}(C;\mathbb{C})$ is defined by

$\ell^{\mu}:=2\widehat{\Phi}d*(\omega^{\prime}_{(1)}\mu).$

From (18) we have

$\displaystyle
d*(\omega^{\prime}_{(1)}\ell^{\mu}-\ell^{\mu}\omega^{\prime}_{(1)})=-\sqrt{-1}\overline{\partial}(\omega^{\prime}_{(1)}\ell^{\mu}-\ell^{\mu}\omega^{\prime}_{(1)})=\sqrt{-1}(\omega^{\prime}_{(1)}\wedge\overline{\partial}\ell^{\mu}-(\overline{\partial}\ell^{\mu})\wedge\omega^{\prime}_{(1)})$
$\displaystyle=2\sqrt{-1}\omega^{\prime}_{(1)}\wedge\overline{\partial}\widehat{\Phi}d*(\omega^{\prime}_{(1)}\mu)+2\sqrt{-1}\overline{\partial}\widehat{\Phi}d*(\omega^{\prime}_{(1)}\mu)\wedge\omega^{\prime}_{(1)}$
$\displaystyle=\sqrt{-1}\omega^{\prime}_{(1)}\wedge(\omega^{\prime}_{(1)}\mu-\mathcal{H}(\omega^{\prime}_{(1)}\mu))+\sqrt{-1}(\omega^{\prime}_{(1)}\mu-\mathcal{H}(\omega^{\prime}_{(1)}\mu))\wedge\omega^{\prime}_{(1)}$
$\displaystyle=-\sqrt{-1}\omega^{\prime}_{(1)}\wedge\mathcal{H}(\omega^{\prime}_{(1)}\mu)-\sqrt{-1}\mathcal{H}(\omega^{\prime}_{(1)}\mu)\wedge\omega^{\prime}_{(1)}.$

The last equality follows from

$\displaystyle\omega^{\prime}_{(1)}\wedge\omega^{\prime}_{(1)}\mu+\omega^{\prime}_{(1)}\mu\wedge\omega^{\prime}_{(1)}=\sum^{g}_{i,j=1}(\psi_{i}\wedge\psi_{j}\mu+\psi_{i}\mu\wedge\psi_{j})Y_{i}Y_{j}$
$\displaystyle=\sum^{g}_{i,j=1}(\psi_{i}\psi_{j}-\psi_{j}\psi_{i})\mu
Y_{i}Y_{j}=0.$

Hence, from (15), we have

$E^{D}_{1}(\lambda,\overline{\mu})=4\sqrt{-1}M\int_{C}\mathcal{H}(\omega^{\prime}_{(1)}\lambda)\wedge\omega^{\prime}_{(1)}\widehat{\Phi}\left(\mathcal{H}(\omega^{\prime\prime}_{(1)}\overline{\mu})\wedge\omega^{\prime\prime}_{(1)}+\omega^{\prime\prime}_{(1)}\wedge\mathcal{H}(\omega^{\prime\prime}_{(1)}\overline{\mu})\right),$

which is exactly the equation (5.5) in [15].

From (17) we have

$\displaystyle
4\sqrt{-1}M\int_{C}\mathcal{H}(\omega^{\prime}_{(1)}\lambda)\wedge\omega^{\prime}_{(1)}\widehat{\Phi}(\mathcal{H}(\omega^{\prime\prime}_{(1)}\overline{\mu})\wedge\omega^{\prime\prime}_{(1)})$
$\displaystyle=\sqrt{-1}\sum_{j,l,i,k,m,n=1}^{g}\left(\int_{C}\psi_{j}\psi_{l}\lambda\right)\left(\int_{C}\overline{\psi_{k}}\overline{\psi_{i}}\overline{\mu}\right)\int_{C}\overline{\psi_{j}}\wedge\psi_{m}\widehat{\Phi}(\psi_{k}\wedge\overline{\psi_{n}})Z(Y_{l}Y_{m}\overline{Y_{i}}\overline{Y_{n}})$
$\displaystyle=-\frac{\sqrt{-1}}{4}\sum_{j,l,i,k,m,n=1}^{g}\left(\int_{C}\psi_{j}\psi_{l}\lambda\right)\left(\int_{C}\overline{\psi_{k}}\overline{\psi_{i}}\overline{\mu}\right)A_{m\overline{j}k\overline{n}}\delta_{mi}\delta_{nl}$
$\displaystyle=-\frac{\sqrt{-1}}{4}\sum_{j,l,i,k=1}^{g}\left(\int_{C}\psi_{j}\psi_{l}\lambda\right)\left(\int_{C}\overline{\psi_{k}}\overline{\psi_{i}}\overline{\mu}\right)A_{i\overline{j}k\overline{l}}.$

On the other hand, we have

$\displaystyle
4\sqrt{-1}M\int_{C}\mathcal{H}(\omega^{\prime}_{(1)}\lambda)\wedge\omega^{\prime}_{(1)}\widehat{\Phi}(\omega^{\prime\prime}_{(1)}\wedge\mathcal{H}(\omega^{\prime\prime}_{(1)}\overline{\mu}))$
$\displaystyle=\sqrt{-1}\sum_{j,l,i,k,m,n=1}^{g}\left(\int_{C}\psi_{j}\psi_{l}\lambda\right)\left(\int_{C}\overline{\psi_{k}}\overline{\psi_{i}}\overline{\mu}\right)\int_{C}\overline{\psi_{j}}\wedge\psi_{m}\widehat{\Phi}(\overline{\psi_{n}}\wedge\psi_{k})Z(Y_{l}Y_{m}\overline{Y_{n}}\overline{Y_{i}})$
$\displaystyle=\frac{\sqrt{-1}}{4}\sum_{j,l,i,k,m,n=1}^{g}\left(\int_{C}\psi_{j}\psi_{l}\lambda\right)\left(\int_{C}\overline{\psi_{k}}\overline{\psi_{i}}\overline{\mu}\right)A_{m\overline{j}k\overline{n}}\delta_{mn}\delta_{il}$
$\displaystyle=\frac{\sqrt{-1}}{4}\sum_{j,l,k,m=1}^{g}\left(\int_{C}\psi_{j}\psi_{l}\lambda\right)\left(\int_{C}\overline{\psi_{k}}\overline{\psi_{l}}\overline{\mu}\right)A_{m\overline{j}k\overline{m}}=\frac{\sqrt{-1}}{4}\sum_{i,j,l,k=1}^{g}\left(\int_{C}\psi_{j}\psi_{l}\lambda\right)\left(\int_{C}\overline{\psi_{k}}\overline{\psi_{l}}\overline{\mu}\right)A_{i\overline{j}k\overline{i}}$

Consequently we obtain

$E^{D}_{1}=-\frac{\sqrt{-1}}{4}\sum_{i,j,k,l=1}^{g}\left(\psi_{i}\psi_{l}A_{i\overline{j}k\overline{l}}\overline{\psi_{k}}\overline{\psi_{i}}-\psi_{j}\psi_{l}A_{i\overline{j}k\overline{i}}\overline{\psi_{k}}\overline{\psi_{l}}\right)=\frac{1}{12}e_{1}^{\widehat{\Phi}},$

as was to be shown. ∎

The present proof is based on direct computations given in [15]. It is
desirable to have a conceptual proof of Theorem 1.

##  5 Proof of Proposition 4

In this section we prove Proposition 4 based on the following theorem of
Harris.

######  Theorem 5 ([3], Theorem 6.5).

For any $g\geq 3$ there exists a hyperelliptic Riemann surface $C_{0}$ such
that the inclusion (11) is the equality

$Q(U(C_{0}))=(\text{the $(-1)$-eigenspace of the hyperelliptic involution in
$T^{*}_{[C_{0}]}{\mathbb{M}_{g}}$}).$

In other words, the extended Johnson homomorphism
$\tilde{k}=-[Q+\overline{Q}]$ at $[C_{0}]$ is non-degenerate in the normal
direction to $\mathbb{H}_{g}$ in ${\mathbb{M}_{g}}$.

The subspace $U(C)\subset\Lambda^{3}(H^{\prime}\oplus
H^{\prime\prime})=\bigoplus_{p+q=3}\Lambda^{p}H^{\prime}\otimes\Lambda^{q}H^{\prime\prime}$
is decomposed into

$U(C)=\bigoplus_{p+q=3}U(C)^{p,q},\quad
U(C)^{p,q}:=U(C)\cap(\Lambda^{p}H^{\prime}\otimes\Lambda^{q}H^{\prime\prime}).$

From the definition of the map $Q$ (10) we have

$Q|_{U(C)^{p,q}}=0$ (19)

if $(p,q)\neq(2,1)$. Through the pairing $\langle-,-\rangle$ in (12),
$U(C)^{p,q}$ and $U(C)^{3-p,3-q}$ are dual to each other. Hence the image of
the transpose $Q:T_{[C]}{\mathbb{M}_{g}}\to U(C)$ is included in
$(U(C)^{2,1})^{*}=U(C)^{1,2}$, i.e., we have
$Q(T_{[C]}{\mathbb{M}_{g}})\subset U(C)^{1,2}$. From the positivity of the
Hermitian pairing (2), we have

$\forall v\in U(C)^{1,2}\setminus\\{0\\},\quad-\sqrt{-1}\langle
v,\overline{v}\rangle>0.$ (20)

In fact, $\\{\psi_{i}\wedge\overline{\psi_{j}}\wedge\overline{\psi_{k}};1\leq
i,j,k\leq g,j<k\\}$ is a orthogonal basis of
$H^{\prime}\otimes\Lambda^{2}H^{\prime\prime}(\supset U(C)^{1,2})$.

Now we go back to the hyperelliptic Riemann surface $[C_{0}]\in\mathbb{M}_{3}$
given by Theorem 5 of Harris. We choose a marking $m_{0}$ of $C_{0}$ to get a
point $[C_{0},m_{0}]$ in the Teichmüller space $\mathcal{T}_{3}$ of genus $3$.
Its Jacobian $\mathrm{Jac}(C_{0})$ with the marking $m_{0}$ defines a point
$[\mathrm{Jac}(C_{0}),m_{0}]\in\mathfrak{H}_{3}$. Then there are complex
coordinates $(z_{1},z_{2},z_{3})$ of $\mathcal{T}_{3}$ centered at
$[C_{0},m_{0}]$, and $(w_{1},w_{2},w_{3})$ of the Siegel upper half space
$\mathfrak{H}_{3}$ centered at $[\mathrm{Jac}(C_{0}),m_{0}]$ such that the
locus $\\{z_{1}=0\\}$ coincides with the inverse image of $\mathbb{H}_{3}$
locally, and that

$\mathrm{Jac}(z_{1},z_{2},z_{3})=({z_{1}}^{2},z_{2},z_{3})=(w_{1},w_{2},w_{3}),$

since the Jacobian map $\mathrm{Jac}$ is a $2$ to $1$ map in the normal
direction of the submanifold $\mathbb{H}_{3}$. There are $U(C)^{1,2}$-valued
functions $v_{1},v_{2},v_{3}$ defined around $[C_{0},m_{0}]\in\mathcal{T}_{3}$
such that $\tilde{k}=v_{1}dz_{1}+v_{2}dz_{2}+v_{3}dz_{3}$ on
$T\mathcal{T}_{3}$. Then we have $v_{1}(0,*,*)\neq 0$ from Theorem 5,
$v_{2}(0,*,*)=v_{3}(0,*,*)=0$ from (11), and the differential form $e_{1}^{J}$
equals

$\sum^{3}_{i,j=1}\langle v_{i},\overline{v_{j}}\rangle dz_{i}\wedge
d\overline{z_{j}}$

up to non-zero constant factor. Since $\frac{\partial}{\partial
w_{1}}=\frac{1}{2z_{1}}\frac{\partial}{\partial z_{1}}$, the amount
$z_{1}\cdot\Lambda e_{1}^{J}$ is a combination of the amounts

$\frac{1}{z_{1}}\langle v_{1},\overline{v_{1}}\rangle,\,\,\langle
v_{1},\overline{v_{j}}\rangle,\,\,\langle
v_{i},\overline{v_{1}}\rangle\,\,\text{and}\,\,z_{1}\langle
v_{i},\overline{v_{j}}\rangle\quad(i,j=2,3)$

with coefficients in non-zero locally bounded functions near $[C_{0},m_{0}]$.
The amounts $\langle v_{1},\overline{v_{j}}\rangle$, $\langle
v_{i},\overline{v_{1}}\rangle$ and $z_{1}\langle
v_{i},\overline{v_{j}}\rangle$ go to $0$ as $z_{1}\to 0$. Together with (20),
this implies

$|z_{1}|\cdot|\Lambda e_{1}^{J}|\geq\frac{(\text{positive
constant})}{|z_{1}|}|\langle
v_{1},\overline{v_{1}}\rangle|\longrightarrow+\infty$

as $z_{1}\to 0$. This completes the proof of Proposition 4 and that of Theorem
3. ∎

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