# Diversity of information pathways drives scaling and sparsity in real-world
networks

Arsham Ghavasieh1,2∗ Manlio De Domenico3,4,5∗

###### Abstract

Empirical complex systems must differentially respond to external
perturbations [1, 2] and, at the same time, internally distribute information
to coordinate their components [3, 4]. While networked backbones help with the
latter [5], they limit the components’ individual degrees of freedom and
reduce their collective dynamical range [6]. Here, we show that real-world
networks are formed to optimize the gain in information flow and loss in
response diversity.

Encoding network states as density matrices [7, 8], we demonstrate that such a
trade-off mathematically resembles the thermodynamic efficiency characterized
by heat and work in physical systems. Our findings explain, analytically and
numerically, the sparsity and the empirical scaling law observed in hundreds
of real-world networks across multiple domains. We show, through numerical
experiments in synthetic and biological networks, that ubiquitous topological
features such as modularity and small-worldness emerge to optimize the above
trade-off for middle- to large-scale information exchange between system’s
units. Our results highlight that the emergence of some of the most prevalent
topological features of real-world networks have a thermodynamic origin.

Fondazione Bruno Kessler, Via Sommarive 18, 38123 Povo, Italy

Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo,
Italy

Department of Physics and Astronomy ”Galileo Galilei”, University of Padua,
Via F. Marzolo 8, 315126 Padova, Italy

Padua Center for Network Medicine, University of Padua, Via F. Marzolo 8,
315126 Padova, Italy

Istituto Nazionale di Fisica Nucleare, Sez. Padova, Italy

$\ast$ Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
Real-world networks are usually sparsely connected [9] – i.e., the number of
existing links between units is much smaller than the one of potentially
available links – and exhibit peculiar topological properties like
heterogeneous connectivity [10, 11, 12], small-worldness [5], modularity [13]
and hierarchy [14] as well as a balance between segregation and integration
[15] or order and disorder [16]. Several network growth mechanisms [17, 18,
19, 20, 21, 22, 23, 24], as well as methods not directly based on growth [25,
26, 27], have been proposed to replicate these features. However, a
theoretical framework to explain why a certain transition from the
disconnected state to a relatively stable wiring configuration is naturally
favored is still lacking. This transition is observed in a variety of complex
biological systems: communication among units – that can be understood in
terms of information exchange of chemical, electric or electrochemical
signals, as well as binary packets or language – allow the system to start
operating and functioning. For instance, human oral bacteria convey
information within multispecies communities via signaling, such as adhesins
and receptors, allowing for adherence and community development [28]. Once a
fungal colony is established the cellular network use communication signals to
regulate colony growth and development [29, 30]. Broadly speaking, information
search and exchange plays an important role, still to be fully uncovered, in
the formation, adaptation and evolution of living, synthetic or engineered
complex systems [31], which have to balance dynamic functions for a rapid
response to internal and external perturbations with the energetic cost of the
intervening actions [32].

In equilibrium and near-equilibrium statistical physics the above questions
are naturally answered by fundamental principles, like the Gibbs entropy
maximization or the free-energy minimization [33, 34, 35]. Conversely, for
complex systems that are open and far from equilibrium, we lack an adequate
theoretical framework to describe, explain and predict state transitions.
Consequently, it is not clear how the aforementioned topological features
observed in several real-world networks might emerge, and to which extent they
are a causal byproduct of fundamental mechanisms related to how information
between units is communicated and at which cost.

In the following, we will firstly introduce a theoretical framework for the
analysis of network formation processes. Successively, we will show how this
framework allows us to make predictions about specific topological features of
complex networks, such as their connectedness and density, the coexistence of
segregation and integration, as well as the coexistence of topological order
and disorder responsible for the emergence of small-worldness. Finally, we
compare our predictions for topological sparsity against hundreds of empirical
networks, finding an excellent agreement between theoretical expectation and
data.

Network formation process. Let $G$ be the structure of a complex system,
modeled as a network of $N$ nodes and $|E|$ connections, respectively. The
links between network units are often encoded into an adjacency matrix
$\mathbf{A}$, where $A_{ij}=1$ if nodes $i$ and $j$ are connected, and
$A_{ij}=0$ otherwise. System’s units can exchange information in several ways,
e.g. to create, destroy or efficiently use links in order to reach a stable
functional regime, which is usually out of equilibrium because the system has
to adapt to a changing environment. In the following we will generally refer
to information exchange to characterize any type of signal that can be used
for unit-unit communication from short- to long-range scales.

A variety of dynamical processes has been used to model the flow of
information between the nodes [36, 37, 38]. Yet, one of the simplest and most
versatile dynamical process that can be used to model the flow of information
through the networks is diffusion [39, 40], which is governed by the Laplacian
matrix $\mathbf{L}=\mathbf{D}-\mathbf{A}$ where $\mathbf{D}$ is a diagonal
matrix, with $D_{ii}=k_{i}$ — i.e., degree— being the connectivity of node
$i$. The solution of the diffusion equation (See Methods) on top of networks
can be understood in terms of a time evolution operator $e^{-\tau\mathbf{L}}$,
where $\left(e^{-\tau\mathbf{L}}\right)_{ij}$ proxies the flow of information
from node $j$ to $i$ and $\tau$ determines the propagation scale. For
instance, when $\tau$ is small signals travel mostly between neighboring
nodes, whereas large values of $\tau$ enable long-range communication.

The state of networks can be suitably encoded into density matrices,
$\boldsymbol{\rho}_{\tau}=e^{-\tau\mathbf{L}}/Z$ [7, 6], with
$Z=\text{Tr}\left(e^{-\tau\mathbf{L}}\right)$ being the partition function.
Network density matrices have been used to tackle a broad range of problems,
from classification of healthy and disease states [41, 42] and robustness
analysis [43, 44] to dimensionality reduction of multilayer networks [45, 46]
and renormalization group [8]. This relatively new framework is versatile
because it is grounded in the physics of linear or non-linear response to the
stochastic perturbations that occur at different locations and propagate
throughout the links. Remarkably, the network Von Neumann entropy [7] defined
by
$\mathcal{S}=-\text{Tr}\left(\boldsymbol{\rho}_{\tau}\log{\boldsymbol{\rho}_{\tau}}\right)$
measures how diverse the system responds to perturbations [6], and the network
free energy, $F=-\log{Z}/\tau$, is a measure of how fast the signal can
transport between the nodes [46]. This framework has been also used to
characterize information cores in empirical networks [47] and multiscale brain
functional connectivity [48].

Here, we characterize the network formation process as a transition between an
initial state $G_{0}$ – a collection of $N$ nodes with no links among them,
$|E|=0$ – and another network $G_{1}$, characterized by topological
connectivity with certain properties. We show (See Methods) that this process
can only lower the response diversity ($\delta\mathcal{S}\leq 0$), whereas it
enhances the transport properties ($\delta F\geq 0$). To compare the two
factors, we define a gain function $W=\delta F$ and a loss function
$Q=\delta\mathcal{S}/\tau$, showing that $W\geq|Q|$ (See Methods), and their
relative trade-off:

$\displaystyle\eta=1-\frac{|Q|}{W}.$ (1)

Note that these quantities characterize the network formation process in terms
of perturbation propagation and the diversity of systems’ response. It is
worth noticing that, from a mathematical perspective, they resemble the
thermodynamic functions of heat $Q$ and work $W$, although such concepts
cannot be directly extended to networks. In Fig. 1, an emblematic example is
shown, where the response diversity and signal flow are compared among four
graphs with different connectivities.

Figure 1: Response diversity and information propagation. $4$ simple graphs
are considered, with different connectivities, and their response to different
environmental perturbations is shown, with the perturbed nodes colored in red.
For convenience, a simple dynamical rule is considered for the propagation of
perturbations, where each perturbed node conveys the perturbation only to its
neighbors. Trivially, connectivity enhances the flow of perturbations captured
by the monotonic increase of $\delta F$ (See the text). For isolated nodes,
the system’s responses to different perturbations have no similarity
(overlap). However, the overlap increases due to connectivity, reducing the
diversity of responses. This is captured by the monotonically decreasing
$\delta\mathcal{S}$ (See the text).

$\eta$ and the signal propagation scale. How deep the perturbations penetrate
through networks depends on many factors, including the underlying topology,
conductance, and the type of signals. Here, we use $\tau$, the temporal
parameter determining the scale of propagation, to characterize such a depth.
For instance, at very small scales $(\tau\ll 1)$, the perturbation can not
propagate through the system, whereas $\tau\gg 1$ let the perturbations flow
between topologically distant nodes.

Firstly, we Taylor expand the generalized heat and work functions while
keeping only the leading terms. We show that at very small scales, where
signals are contained locally $(\tau\approx 0)$, all network formation
processes are indistinguishable, being characterized by $\eta=1$ (See
Methods). Remarkably, the network topology becomes irrelevant to system’s
functionality during this regime. Secondly, we find that networks with fewer
links, in the linear regime $\tau\ll 1$, exhibit higher values of $\eta$.

Finally, a second-order correction reveals the importance of connectivity:
assuming that the number of links in a network scales with the number of nodes
as $|E|=cN^{\gamma}$, where $c$ is a constant, we find that the optimal
exponent which maximizes $\eta$ by $\partial_{\gamma}\eta=0$ is obtained for
$\gamma=1$. Therefore, by requiring that $\eta$ is maximized by a network
formation process, we naturally derive that the corresponding connectivity
distribution must be sparse and scale with a specific exponent.

Figure 2: The effect of topological features on $\eta$. (a) Connectivity
versus disconnectivity, characterized by the emergence of a giant connected
component, modeled by Erdős-Rényi random networks for different wiring
probability $p$. (b) Segregation versus integration, modeled by stochastic
block networks for different mixing parameter $\mu$. (c) Order versus
disorder, modeled by Watts-Strogatz small-world networks with degree $8$ and
rewiring probability $p_{rew}$. In all three cases, different propagation
scales $\tau$ are considered in the heatmap (top panels), while characteristic
line curves have been separately shown for fixed values of $\tau$ (bottom
panels) to better visualize transitions in $\eta$. In all cases, the size of
synthetic networks is $N=10^{3}$. Figure 3: Scaling in empirical networks. The
logarithm of the number of edges $|E|$ and nodes $N$ is represented for $543$
empirical networks, where the best fitting exponent is $\gamma=1.07\pm 0.02$.
Similar analyses have been done for each network domain (informational,
social, biological, technological, economic, and transportation.) separately
and are represented in small blocks. The exponents are reported on top of each
block.

The effect of topological features on $\eta$. Here we study how some of the
most prominent topological features of real-world networks affect $\eta$. More
technically, these features include connectivity, integration versus
segregation, and order versus disorder as captured by synthetic network (See
Fig. 2). First, we use Erdős-Rényi (ER) networks [49, 50], where the existence
of links between every pair of nodes is independent and given by a
connectivity probability $p$, to study the behavior of $\eta$ for varying $p$.
Results indicate that being connected while sparse ensures high $\eta$:
maximum $\eta$ is found between $p\propto N^{-1}$ and $p\propto N^{-1}\log{N}$
(See Fig. 2). It is worth noting that in the thermodynamic limit
$N\rightarrow\infty$, the difference between these two probabilities becomes
negligible ($N^{-1}\approx N^{-1}\log{N}$), and the optimal probability can be
approximated as $p\approx N^{-1}$. Therefore, given that in ER model the
number of links is determined by the connectivity probability ($p$) as
$|E|=p{N\choose 2}$, the numerical result validates our theoretical prediction
of maximum $\eta$ at $|E|\propto N^{\gamma}$, with $\gamma\approx 1$.

Second, we consider a special type of stochastic block networks [51]— i.e.,
models reproducing groups or communities of nodes that are randomly connected
internally with probability $p_{in}$ and externally with probability
$p_{out}$— with $10$ communities and average degree $\bar{k}=10$. This class
of models is useful to investigate the mesoscale organization of complex
networks. We vary the mixing parameter, defined as
$\mu=\frac{k_{out}}{k_{in}+k_{out}}$, where $k_{in}$ is the average number of
connections a node has with the nodes in its community, and $k_{out}$ is the
average number of external connections. Values of $\mu\ll 1$ generate a highly
segregated network with most connections existing only within a community,
while $\mu\approx 1$ gives a network with no particular community structure,
resembling ER models. In this case, at small scales ($\tau\ll 1$), the
integrated communities exhibit the highest $\eta$, whereas at larger scales,
the trade-off between integration and segregation – with $\log{\mu}$ between
$-6$ and $-2$ – is favored. Therefore, our results confirm that in systems
which are either too segregated or too integrated, the trade-off between gain
in information flow and loss in response diversity is not optimal.

Third, we consider small-world networks— i.e., models where nodes are
initially connected in a lattice-like regular pattern, and then the links are
shuffled with a given rewiring probability $p_{rew}$ to introduce topological
shortcuts— with average degree $\bar{k}=8$. Results indicate that at small
scales, $\eta$ is higher in disordered networks ($p_{rew}\approx 1$), while at
larger scales, ordered structures ($p_{rew}\approx 0$) work better than
disordered ones, with the optimal $\eta$ found in a middle regime between
order and disorder exhibiting high small-worldness [5]. Once again, our
framework provides an elegant explanation of why small-worldness is so
ubiquitous in empirical systems.

Scaling and sparsity of empirical networks. We have analytically shown that if
the number of links scales with the number of nodes as $|E|=cN^{\gamma}$, the
optimal exponent which maximizes $\eta$ is given by $\gamma=1$. This
derivation implies that the real-world networks must be sparse in a specific
way: their average degree must scale as
$\bar{k}=\frac{2|E|}{N}=\frac{2cN^{\gamma}}{N}=2c$, and their connectivity
must scale with the number of nodes. Here, we compare this theoretical
expectation with empirical data.

We analyzed $543$ social, biological, informational, economic, transportation
and technological networks from ICON [52], ranging in size from $N\approx
10^{2}$ to $N\approx 10^{8}$ nodes. Assuming a power-law ansatz between
network size and the number of links, $|E|\propto N^{\gamma}$, we perform a
linear regression in the log-log space to obtain the scaling exponent. We find
that the overall best fit is for $\gamma=1.07\pm 0.02$, while the proximity to
$\gamma=1$ result is relatively stable across domains (See Fig. 3), confirming
our analytical and numerical prediction of $|E|\propto N^{1}$, with error
bounds compatible with random topological fluctuations in network formation
and noise in the data set.

Furthermore, we analyze hundreds of biological networks, from fungi to the
human, C Elegans, and Ciona intestinalis connectomes (See Methods, Fig.
Extended Data 1 and Fig. Extended Data 2). A direct comparison between these
empirical networks and different ensembles of null models – namely, Erdős-
Rényi (ER) and Configuration models (CM) – (See Methods), shows that they are
characterized by significantly higher $\eta$ at middle and large propagation
scales, with respect to their randomized null models. This finding provides
further evidence, supporting the numerical experiments of the previous section
(See Fig. 2), that complex systems must exhibit topological correlations to
sustain their flow of information and, simultaneously, their pathway diversity
at the macroscopic scales.

Discussion. While an interconnected structure is essential for information to
flow between systems’ units and for their coordination, it also limits the
units’ independence. This simple fact restricts their collective dynamical
range, posing a significant challenge to complex systems that must diversely
respond, and adapt, to unpredictable environments. Therefore, it is of
paramount importance for real-world networks to exhibit topological features
allowing them to ensure an adequate information flow among units, while
simultaneously minimizing the cost of the diversity of available information
pathways used for communication across multiple time scales.

Here, using network density matrices, we have characterized network formation
as a physical process, quantifying the loss in response diversity, gain in
information flow, and their trade-off. To this aim, we have shown that it is
possible to introduce a new quantity ($\eta$) that, mathematically, resembles
the thermodynamic efficiency related to heat and work. We analytically and
numerically predict that networks must be sparse to optimize the trade-off,
with their connectivity following a scaling law $|E|\propto N^{\gamma}$, with
$\gamma\approx 1$. Our analysis of 543 empirical networks from biological,
social, informational and transportation domains, reveals a scaling with
exponent $\gamma\approx 1.07\pm 0.02$ that is strikingly compatible with our
the theoretical expectation.

Moreover, our numerical experiments show that complex topological features
like modularity and small-worldness can maximize $\eta$ when perturbations
propagate beyond the first neighbors of nodes, enabling middle- to long-range
communications between system’s units. We validated these results with
empirical data from hundreds of fungal and neural systems (See Methods, Fig.
Extended Data 1 and Fig. Extended Data 2), while comparing against suitable
null models that capture distinct topological features of the data. It
demonstrates that topological correlations enable a complex network to
maximize the trade-off and, therefore, play a crucial role in network
formation and function.

Clearly, diffusion dynamics coupled with network topology is only a first step
toward modeling the flow of information in empirical complex systems. This
work opens the doors for future studies engaging nonlinear processes,
especially reaction-diffusion, using suitable generalization of our framework.
Furthermore, higher-order networks including multilayers [53] and hypergraphs
[54], remain to be explored. Most importantly, relaxing some of our
assumptions in the analytical calculation of the optimal exponent
($\gamma=1$), like the mean-field approximation, can lead to even more precise
predictions, spurring further interest for future developments of our
approach.

Overall, our results are compatible with the compelling hypothesis that
complex networks shape their topology to optimize the trade-off between
maximally exchanging information among their units and guaranteeing an
adequate response diversity for middle- to long-range signaling.

Author Contributions AG and MDD designed the study, performed the theoretical
analysis and wrote the manuscript. AG performed the numerical experiments.

Acknowledgements MDD acknowledges financial support from the Human Frontier
Science Program Organization (HFSP Ref. RGY0064/2022), from the University of
Padua (PRD-BIRD 2022) and from the EU funding within the MUR PNRR “National
Center for HPC, BIG DATA AND QUANTUM COMPUTING” (Project no. CN00000013 CN1).

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Methods

Network density matrix. Network density matrices have been recently
generalized to non-linear dynamics [55]. However, here we focus on the
diffusion dynamics governed by the combinatorial Laplacian,
$\mathbf{L}=\mathbf{A}-\mathbf{D}$, where $\mathbf{A}$ and $\mathbf{D}$ are,
respectively, the adjacency matrix ($A_{ij}=1$ indicates the existence of a
link connecting node $i$ and $j$ and $A_{ij}=0$ indicates the absence of a
link between the two) and degree diagonal matrix ($D_{ij}=\delta_{ij}k_{i}$
where $k_{i}$ is the degree of node $i$ and $\delta_{ij}=0$ if $i\neq j$ and
$\delta_{ij}=1$ if $i=j$). The diffusion equation and its solution follows

$\displaystyle\partial_{\tau}|\psi_{\tau}\rangle=-\mathbf{L}|\psi_{\tau}\rangle,\qquad|\psi_{\tau}\rangle=e^{-\tau\mathbf{L}}|\psi_{0}\rangle.$
(2)

From here, it is straightforward to obtain the density matrix
$\boldsymbol{\rho}_{\tau}=e^{-\tau\mathbf{L}}/Z$ [6], with the partition
function $Z=\text{Tr}\left(e^{-\tau\mathbf{L}}\right)$.

Network counterpart of free energy. In diffusion dynamics, the network
partition function $Z$ measures the dynamical trapping [46]— i.e., the
tendency of the network to block the flow of information locally. Let the
eigenvalues of the Laplacian be $\lambda_{\ell},\ell=1,2,...,N$ for a network
with $N$ nodes, the network partition function follows
$Z=\sum\limits_{\ell=1}^{N}e^{-\tau\lambda_{\ell}}$. Since all the eigenvalues
are non-negative, $\lambda_{\ell}\geq 0$, and at least one of the eigenvalues
is $\lambda_{1}=0$, the partition function is bounded $1\leq Z\leq N$. The
lower bound ($Z=1$) is for a connected network at the very large propagation
scales ($\tau\gg 1$). In contrast, the upper bound ($Z=N$) happens at the
smallest scale, $\tau=0$, or for any $\tau$ if the network is composed of
isolated nodes— because all eigenvalues of the Laplacian matrix are zero in
that case.

The network free energy is inversely related to the partition function
$F=-\log{Z_{\tau}}/\tau$. Accordingly, the network of isolated nodes has the
lowest free energy $F_{min}=F_{iso}=-\log{N}/\tau$. Also, the minimum
dynamical trapping ($Z_{\tau}=1$) gives the maximum free energy that is
$F_{max}=0$. Therefore, the network free energy is bounded as $0\geq
F\geq-\log{N}/\tau$.

Network counterpart of Von Neumann entropy. The network Von Neumann entropy is
given by
$\mathcal{S}=-\text{Tr}\left(\boldsymbol{\rho_{\tau}}\log{\boldsymbol{\rho_{\tau}}}\right)$.
Note that the eigenvalues of the density matrix are probabilities of a
Boltzmann distribution ($e^{-\tau\lambda_{\ell}}/Z$) and the microcanonical
ensemble gives the maximum entropy, where all eigenvalues are equal
$\lambda_{\ell}=\lambda_{\ell^{\prime}}$. The only network satisfying this
condition at $\tau>0$ is the network of isolated nodes that has
$\lambda_{\ell}=0,\ell=1,2,...,N$ and its Von Neumann entropy reads
$S_{max}=S_{iso}=\log{N}$.

In contrast, in connected networks, only one eigenvalue of the Laplacian is
$0$, and the rest are positive. Therefore, at large scales, all probabilities
correspond to the non-zero eigenvalues and decay exponentially
($e^{-\tau\lambda_{\ell}}$), except for the first one that does not decay
since $\lambda_{1}=0$. This will lead to the minimum entropy $S_{min}=0$.

Therefore, the network Von Neumann entropy is bounded as
$0\leq\mathcal{S}\leq\log{N}$.

Network counterpart of internal energy. Because the network density matrix, in
case of diffusion dynamics, resembles the Gibbs state, network internal energy
can be obtained from the partition function as follows:

$\displaystyle
U=-\partial_{\tau}\log{Z}=\frac{1}{Z}\sum\limits_{\ell=1}^{N}\lambda_{\ell}e^{-\tau\lambda_{\ell}}.$
(3)

Note that the summation of eigenvalues of the Laplacian matrix is twice the
number of links
$\text{Tr}\left(\mathbf{L}\right)=\sum\limits_{\ell=1}^{N}\lambda_{\ell}=2|E|$,
where $|E|$ is the number of links. Therefore, at $\tau=0$, the network
internal energy reads
$U=\frac{1}{Z_{\tau}}\sum\limits_{\ell=1}^{N}\lambda_{\ell}=\frac{2|E|}{N}$.

Also, it is worth mentioning that for a network of isolated nodes,
$U_{iso}=0$, for all values of $\tau$, since all eigenvalues are $0$.

Finally, the network’s internal energy can be obtained from its free energy
and entropy, as

$\displaystyle U=F+\mathcal{S}/\tau$ (4)

Network formation as a physical process. Network formation comes with an
entropic cost since the entropy of isolated nodes is the maximum
$\mathcal{S}_{iso}=\log{N}$. The cost is given by
$\delta\mathcal{S}=\mathcal{S}-\mathcal{S}_{iso}$. Divided by the propagation
scale $\tau$, it resembles heat dissipation in a thermodynamic process:

$\displaystyle Q=\frac{\mathcal{S}-\log{N}}{\tau}\leq 0.$ (5)

Furthermore, the gain in information flow can be obtained from the alteration
of network free energy $\delta F=F-F_{iso}$, where $F_{iso}=-\log{N}/\tau$ is
the minimum possible network free energy, at the fixed $\tau$, resembling the
work in thermodynamics

$\displaystyle W=F+\log{N}/\tau\geq 0.$ (6)

Since $W+Q=F+\mathcal{S}/\tau=U$ and $U\geq 0$, the summation of gain and loss
is always non-negative in the network formation process— while $Q<0$ the gain
is always larger or equal in magnitude $W\geq|Q|$.

Therefore, it is straightforward to show that $0<W+Q<W$ and
$0<\frac{W+Q}{W}<1$, leading to the relative trade-off between the gain and
loss in the network formation process, bounded between $0$ and $1$, that reads

$\displaystyle\eta=\frac{W+Q}{W}=1-\frac{|Q|}{W}=\frac{U}{W}.$ (7)

##  0.1 Approximation of $\eta$.

Let $\lambda_{\ell}$ be the $\ell$-th eigenvalue of the Laplacian matrix and
$\bar{\lambda^{n}}=\sum\limits_{\ell=1}^{N}\lambda^{n}_{\ell}$. It is
straightforward to write the Taylor series for the partition function
$Z=N\sum\limits_{\ell=1}^{N}\sum\limits_{n=0}^{\infty}\frac{(-\tau)^{n}}{n!}\frac{\lambda_{\ell}^{n}}{N}$,
that we approximate up to the third-order $Z\approx
N(1-\tau\frac{\bar{\lambda}}{N}+\frac{\tau^{2}}{2}\frac{\bar{\lambda^{2}}}{N}-\frac{\tau^{3}}{6}\frac{\bar{\lambda^{3}}}{N})$.
From here, it is straightforward to approximate the logarithm of partition
function,
$-\log{\frac{Z}{N}}\approx\tau\frac{\bar{\lambda}}{N}-\frac{\tau^{2}}{2}\frac{\bar{\lambda^{2}}}{N}+\frac{\tau^{3}}{6}\frac{\bar{\lambda^{3}}}{N}$,
that we use to write the second-order approximation of $U$ and $W$:

$\displaystyle U$
$\displaystyle=-\partial_{\tau}\log{Z}=-\partial_{\tau}\log{\frac{Z}{N}}\approx\frac{\bar{\lambda}}{N}-\tau\frac{\bar{\lambda^{2}}}{N}+\frac{\tau^{2}}{2}\frac{\bar{\lambda^{3}}}{N}$
(8) $\displaystyle W$
$\displaystyle=-\frac{1}{\tau}\log{\frac{Z}{N}}\approx\frac{\bar{\lambda}}{N}-\frac{\tau}{2}\frac{\bar{\lambda^{2}}}{N}+\frac{\tau^{2}}{6}\frac{\bar{\lambda^{3}}}{N}$
(9)

For convenience, we assume that
$\lambda_{\ell}^{2}=(\frac{\bar{\lambda}}{N}-\delta\lambda_{\ell})^{2}$ and a
apply a mean-field approximation
$\bar{\lambda^{2}}\approx\sum\limits_{\ell=1}^{N}(\frac{\bar{\lambda}}{N})^{2}+\delta\lambda_{\ell}^{2}-2\delta\lambda_{\ell}\frac{\bar{\lambda}}{N}\approx\frac{(\bar{\lambda})^{2}}{N}$.
Similarly, we assume that
$\lambda_{\ell}^{3}=(\frac{\bar{\lambda}}{N}-\delta\lambda_{\ell})^{3}$ and,
therefore, a mean-field approximation leads to
$\bar{\lambda^{3}}\approx\frac{(\bar{\lambda})^{3}}{N^{2}}$. Finally, we
assume that the number of links scales with the number of nodes as
$\bar{\lambda}=2|E|=cN^{\delta+1}$. Therefore, we can rewrite $U$ and $W$ as:

$\displaystyle U$ $\displaystyle\approx cN^{\delta}(1-\tau
cN^{\delta}+\frac{\tau^{2}}{2}c^{2}N^{2\delta})$ (10) $\displaystyle W$
$\displaystyle\approx
cN^{\delta}(1-\frac{\tau}{2}cN^{\delta}+\frac{\tau^{2}}{6}c^{2}N^{2\delta}).$
(11)

Finally, we expand the $W$ in the denominator of $\eta$ in Eq. 7 as:

$\displaystyle\frac{1}{W}$
$\displaystyle=\frac{1}{cN^{\delta}}\frac{1}{1-\frac{\tau}{2}cN^{\delta}+\frac{\tau^{2}}{6}c^{2}N^{2\delta}}$
$\displaystyle\approx\frac{1}{cN^{\delta}}(1+\frac{\tau}{2}cN^{\delta}-\frac{\tau^{2}}{6}c^{2}N^{2\delta}+(\frac{\tau}{2}cN^{\delta}-\frac{\tau^{2}}{6}c^{2}N^{2\delta})^{2})$
$\displaystyle\approx\frac{1}{cN^{\delta}}(1+\frac{\tau}{2}cN^{\delta}-\frac{\tau^{2}}{6}c^{2}N^{2\delta}+\frac{\tau^{2}}{4}c^{2}N^{2\delta})$
$\displaystyle=\frac{1}{cN^{\delta}}(1+\frac{\tau}{2}cN^{\delta}+\frac{\tau^{2}}{12}c^{2}N^{2\delta}),$
(12)

to reach the second-order approximation of the $\eta$:

$\displaystyle\eta=\frac{U}{W}$ $\displaystyle\approx(1-\tau
cN^{\delta}+\frac{\tau^{2}}{2}c^{2}N^{2\delta})(1+\frac{\tau}{2}cN^{\delta}+\frac{\tau^{2}}{12}c^{2}N^{2\delta})$
$\displaystyle\approx
1-\frac{\tau}{2}cN^{\delta}+\frac{\tau^{2}}{12}c^{2}N^{2\delta}$ (13)

The first term $\eta\approx 1$ leads the behavior at extremely small
$\tau\approx 0$. In that regime, the relative trade-off is equivalent for all
network formation processes, indicating that all networks are
indistinguishable and the topology is irrelevant to system function.

The first order approximation reads $\eta\approx 1-\frac{\tau
c}{2}N^{\delta}$. Note that the second term is negative ($-\frac{\tau
c}{2}N^{\delta}\leq 0$). Therefore, maximum $\eta$ corresponds to $\delta<0$,
where $\displaystyle{\lim_{N\to\infty}N^{\delta}=0}$. In this regime, a
disconnected network with as few as possible links $|E|$ is favored.

However, taking the derivative of the second-order approximation of $\eta$
with respect to the exponent $\delta$, reveals two optimal points.

$\displaystyle\partial_{\delta}\eta$ $\displaystyle=-\frac{\tau
c}{2}N^{\delta}[1-\frac{\tau c}{3}N^{\delta}]\log{N}=0.$ (14)

One of the roots of Eq. 14 requires $N^{\delta}=0$, favoring disconnected
networks with $\delta<0$, repeating the previously shown first-order
approximation. However, the second root requires $N^{\delta}$ to remain a
constant that does not scale up or down as $N\rightarrow\infty$, leading to
the only possible answer $\delta=0$. Note that in the text, the slope of Fig.
3 is indicated as $\gamma\approx 1$ where $\gamma=\delta+1$.

Empirical $\eta$ profiles. Here, we consider 270 biological systems
representing networks of 6 species of fungi and slime molds: _Physarum
polycephalum_ (Pp), _Phanerochaete velutina_ (Pv), _Resinicium bicolor_ (Rb),
_Agrocybe gibberosa_ (Ag), _Phallus impudicus_ (Pi) and _Strophularia
caerulea_ Sc, introduced by Lee et al. [56], where the weights of the links
are proportional to the cord conductance for pairwise interactions.
Furthermore, we consider the structural (DTI) and functional (fMRI) _Rockland
human connectome_ data for 197 healthy subjects [57]. We consider the
structural (DTI) and functional (fMRI) _Lausanne human connectome_ data at
four resolutions. We consider the structural _Budapest human connectome_ data,
which is obtained from the consensus over 497 healthy subjects at three
resolutions (20k, 200k, 1M), giving four networks, each representing one
single type of connectivity between the brain areas: fiber count, fiber
length, fractional anisotropy, and electrical connectivity [58]. We consider
the connectome of a tadpole larva of _Ciona intestinalis_ [59]. We consider
the representations of _Caenorhabditis elegans_ (C. elegans) connectome, where
links encode Send (S), Send-poly (Sp), Receive (R), Receive-poly (Rp) and
Electric junction (Ej) [60, 61].

We make an ER model and a configuration model (CM) for each empirical network
as null statistical analysis models. Here, the ER null model with adjacency
matrix $A^{(ER)}$ is a network of the same size (N) and the same number of
links $|E|$ as the original network, where links encode a uniform connectivity
probability (or, in case of weighted networks, density):
$A_{ij}^{(ER)}=2|E|/N(N-1)$. Instead, in the configuration model with
adjacency matrix $A^{(CM)}$, the null model preserves the degree sequence, in
addition to the number of links $|ER|$, with $A_{ij}^{(CM)}=k_{i}k_{j}/2|E|$
where $k_{i}=\sum\limits_{j=1}^{N}A_{ij}$ is the degree of node $i$.

See Fig. Extended Data 1 and Fig. Extended Data 2 for a comparison of original
networks with their ER and CM null models in fungal and neural systems.
Consistently, randomness, represented by ER null models, provides the highest
$\eta$ at small scales, $\tau\approx 0$— with the only exception of fMRI
correlation networks that show high similarity to the ER null model. At the
same time, the degree distribution becomes essential for middle- to large-
scale propagations. In some cases, the CM null model can explain the behavior
of $\eta$, exhibiting no significant difference from the original network.
However, topological features arising from degree-degree correlations lead to
deviations from the CM null model in other networks. In conclusion, $\eta$
favors complex topological features at middle- to large-scale propagations,
consistent with the synthetic network analysis in Fig. 2.

Figure Extended Data 1: Fungal networks compared with null models. Six species
of fungi are considered, including Pv, Ag, Rb, Pp, Sc, and Pi (see text for
details). The dashed line shows the average $\eta$ for each species and its
Erdős-Rényi (ER) and Configuration Model (CM) surrogates. Figure Extended Data
2: Neural networks compared with null models. A wide range of connectomes are
considered, including different types of connections in C. elegans, Ciona
intestinalis, and structural and functional maps of the human connectome (See
text for a description of each dataset). The dashed line shows the average
$\eta$ for each species and its ER and CM null models, as in Fig. Extended
Data 1.


Ind}_{\mathfrak{Vir}^{\prime(r_{M}-\delta_{r_{{}_{M}},1})}}^{\mathfrak{Vir}^{\prime}}{\mathbb{C}}v_{0})^{{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$ is a highest weight
$\mathfrak{Vir}^{\prime}$-module. Let $V=\text{\rm
Ind}_{\mathfrak{Vir}^{\prime(r_{M}-\delta_{r_{{}_{M}},1})}}^{\mathfrak{Vir}^{\prime}}{\mathbb{C}}v_{0}$
and $\mathfrak{K}=\text{Ker}(\tau^{\prime})$. It should be noted that

${\mathbb{C}}v_{0}\otimes K^{\prime{\bar{\mathfrak{D}}}^{\prime}}=\\{u\in
V^{{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}\mid d_{0}^{\prime}u=\lambda_{0}u\\}.$

We see that $({\mathbb{C}}v_{0}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}})\cap\mathfrak{K}=0$. Let
$\mathfrak{K}^{\prime}$ be the sum of all $\mathfrak{Vir}^{\prime}$-submodules
$W$ of $V^{{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$ with $({\mathbb{C}}v_{0}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}})\cap W=0$, that is, the unique maximal
(weight) $\mathfrak{Vir}^{\prime}$-submodule of
$V^{{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$ with trivial intersection with
$({\mathbb{C}}v_{0}\otimes K^{\prime{\bar{\mathfrak{D}}}^{\prime}})$. It is
obvious that $\mathfrak{K}\subseteq\mathfrak{K}^{\prime}$. Next we further
show that $\mathfrak{K}=\mathfrak{K}^{\prime}$. For that, take any
$\mathfrak{Vir}^{\prime}$\- submodule $W$ of
$V^{{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$ such that $({\mathbb{C}}v_{0}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}})\cap W=0$. Then for any weight vector
$w=\sum_{{\mathbf{l}}\in\mathbb{M}}d^{\prime{\mathbf{l}}}v_{0}\otimes
u_{{\mathbf{l}}}\in W$, where
$u_{{\mathbf{l}}}\in{K^{\prime}}^{{\bar{\mathfrak{D}}}^{\prime}},d^{\prime{\mathbf{l}}}=\cdots(d^{\prime}_{-2})^{l_{2}}(d^{\prime}_{-1})^{l_{1}}$
if $r_{M}=1,0$, or $d^{\prime{\mathbf{l}}}=\cdots(d^{\prime}_{-2})^{l_{2}}$ if
$r_{M}=-1$, and all ${\rm{w}}({\mathbf{l}})\geq 1$ are equal. Note that
$h_{k}w=\sum_{{\mathbf{l}}\in\mathbb{M}}d^{\prime{\mathbf{l}}}v_{0}\otimes
h_{k}u_{{\mathbf{l}}}$ either equals to $0$ or has the same weight as $w$
under the action of $d_{0}^{\prime}$. So
$\mathcal{U}(\bar{\mathfrak{D}}^{\prime})W\cap({\mathbb{C}}v_{0}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}})=0$, i.e.,
$\mathcal{U}(\bar{\mathfrak{D}}^{\prime})W\subset\mathfrak{K}^{\prime}$. Hence
$\mathcal{U}(\bar{\mathfrak{D}}^{\prime})\mathfrak{K}^{\prime}\cap({\mathbb{C}}v_{0}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}})=0$. The maximality of
$\mathfrak{K}^{\prime}$ forces that
$\mathfrak{K}^{\prime}=\mathcal{U}(\bar{\mathfrak{D}}^{\prime})\mathfrak{K}^{\prime}$
is a proper $\bar{\mathfrak{D}}^{\prime}$-submodule of
$V^{{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$. Since $\mathfrak{K}$ is a maximal
proper $\bar{\mathfrak{D}}^{\prime}$-submodule of
$V^{{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$, it follows that
$\mathfrak{K}=\mathfrak{K}^{\prime}$.

By Lemma 4.8 we know that $\mathfrak{K}$ is generated by
$P_{1}({\mathbb{C}}v_{0}\otimes
K^{{\bar{\mathfrak{D}}}^{\prime}})={\mathbb{C}}P_{1}v_{0}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$ and $P_{2}({\mathbb{C}}v_{0}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}})={\mathbb{C}}P_{2}v_{0}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$. Let $V^{\prime}$ be the maximal
submodule of $V$ generated by $P_{1}v_{0}$ and $P_{2}v_{0}$, then
$\mathfrak{K}=V^{\prime{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$. Therefore,

$M\cong(V^{{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}})/(V^{\prime{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}})\cong(V/V^{\prime})^{{\bar{\mathfrak{D}}}^{\prime}}\otimes
K^{\prime{\bar{\mathfrak{D}}}^{\prime}},$ (5.3)

which forces that $K^{\prime{\bar{\mathfrak{D}}}^{\prime}}$ is a simple
$\bar{\mathfrak{D}}^{\prime}$-module and hence a simple
$\bar{\mathcal{H}}$-module. So $K^{\prime}$ is a simple
$\bar{\mathcal{H}}$-module. By [43, Theorem 12] we know there exists a simple
$\mathfrak{Vir}$-module $U\in{\mathcal{R}}_{\mathfrak{Vir}}$ such that $M\cong
K^{\prime\bar{\mathfrak{D}}}\otimes U^{\bar{\mathfrak{D}}}$. From this
isomorphism and some computations we see that $K_{0}\subseteq
K^{\prime\bar{\mathfrak{D}}}\otimes v_{0}$ where $v_{0}$ is a highest weight
vector. So $K=K^{\prime}$.∎

We are now in a position to present the following main result on
characterization of simple restricted $\bar{\mathfrak{D}}$-modules with
nonzero level.

######  Theorem 5.8.

Let $M$ be a simple restricted $\bar{\mathfrak{D}}$-module with level
$\ell\not=0$. The invariants $n_{M},r_{M}$ of $M$, $K_{0},K$ are defined as
before. Then

$M\cong\begin{cases}K(z)^{\bar{\mathfrak{D}}},&{\text{ if }}r_{M}=-\infty,\cr
K(z)^{\bar{\mathfrak{D}}}\otimes U^{\bar{\mathfrak{D}}},&{\text{ if }}-1\leq
r_{M}\leq 1\text{ or }n_{M}=0,1,\cr\text{\rm
Ind}_{\bar{\mathfrak{D}}^{(0,-(n_{M}-1))}}^{\bar{\mathfrak{D}}}K_{0},&{\text{
otherwise, }}\end{cases}$

for some $U\in\mathcal{R}_{\mathfrak{Vir}}$ and some $z\in{\mathbb{C}}$.

###### Proof.

The assertion follows directly from Proposition 5.2, Proposition 5.4,
Proposition 5.5, Proposition 5.7. ∎

The following result characterizes simple Whittaker modules over the twisted
Heisenberg-Virasoro algebra $\bar{\mathfrak{D}}$.

######  Theorem 5.9.

Let $M$ be a $\bar{\mathfrak{D}}$-module (not necessarily weight) on which the
algebra ${\bar{\mathfrak{D}}}^{+}$ acts locally finitely. Then the following
statements hold.

  1. (i)

The module $M$ contains a nonzero vector $v$ such that
${\bar{\mathfrak{D}}}^{+}\,v\subseteq{\mathbb{C}}v$.

  2. (ii)

If $M$ is simple, then $M$ is a Whittaker module or a highest weight module.

###### Proof.

(i) Let $(M_{1},\rho)$ be a finite dimensional
${\bar{\mathfrak{D}}}^{+}$-submodule of $M$. Then $M_{1}$ is also a finite
dimensional $\mathfrak{Vir}_{\geq 1}$-module. Let
$\mathfrak{a}:=\ker(\rho|_{\mathfrak{Vir}_{\geq 1}})$ be the kernel of the
representation map of $\mathfrak{Vir}_{\geq 1}$ on $M_{1}$. Then
$\mathfrak{a}$ is an ideal of $\mathfrak{Vir}_{\geq 1}$ of finite codimension.
We claim that $d_{n}\in\mathfrak{a}$ for some $n\in{\mathbb{Z}}_{+}$. If this
is not true, then there exists a minimal $m\in{\mathbb{Z}}_{+}$ such that
$\mathfrak{a}$ contains an element of the form
$a_{i_{1}}d_{i_{1}}+a_{i_{2}}d_{i_{2}}+\cdots+a_{i_{m+1}}d_{i_{m+1}}$ for
positive integers $i_{1}<i_{2}<\cdots<i_{m+1}$ and nonzero complex numbers
$a_{i_{1}},a_{i_{2}},\cdots,a_{i_{m+1}}$. We further see that $\mathfrak{a}$
contains

$[d_{i_{1}},a_{i_{1}}d_{i_{1}}+a_{i_{2}}d_{i_{2}}+\cdots+a_{i_{m+1}}d_{i_{m+1}}]=a_{i_{2}}(i_{2}-i_{1})d_{i_{1}+i_{2}}+a_{i_{3}}(i_{3}-i_{1})d_{i_{1}+i_{3}}+\cdots+a_{i_{m+1}}(i_{m+1}-i_{1})d_{i_{1}+i_{m+1}},$

which contradicts with the minimality of $m$. Hence the claim follows.
Consequently,

$\widetilde{\mathfrak{Vir}}_{\geq n}:=\sum_{i\geq n,\,i\neq
2n}{\mathbb{C}}d_{i}={\mathbb{C}}d_{n}+[d_{n},\mathfrak{Vir}_{\geq
1}]\subseteq\mathfrak{a}.$

Then

$\widetilde{\mathfrak{Vir}}_{\geq n}+{\bar{\mathcal{H}}_{\geq
n+1}}=\widetilde{\mathfrak{Vir}}_{\geq n}+[{\bar{\mathcal{H}}_{\geq
1}},\widetilde{\mathfrak{Vir}}_{\geq n}]\subseteq\ker(\rho).$

This implies that $M_{1}$ is a finite dimensional module over a finite
dimensional solvable Lie algebra
$\bar{\mathfrak{D}}^{+}/(\widetilde{\mathfrak{Vir}}_{\geq
n}+{\bar{\mathcal{H}}_{\geq n+1}})$. The desired assertion follows directly
from Lie Theorem.

(ii) follows directly from (i) and [46]. ∎

######  Remark 5.10.

From Theorem 5.9 we know that if $M$ is a simple Whittaker module over
$\bar{\mathfrak{D}}$ with nonzero level, and
$\bar{\mathfrak{D}}^{+}v\subset{\mathbb{C}}v$ for some nonzero vector $v\in
M$, then
$K=\mathcal{U}(\bar{\mathcal{H}})v=\mathcal{U}(\oplus_{r\in-{\mathbb{Z}}_{+}}{\mathbb{C}}h_{r})v$
is a simple Whittaker module over $\bar{\mathcal{H}}$. Therefore, [43, Theorem
12] implies that $M\cong U^{\bar{\mathfrak{D}}}\otimes
K(z)^{\bar{\mathfrak{D}}}$ for some $U\in{\mathcal{R}}_{\mathfrak{Vir}}$.
Clearly, $U$ is a simple Whittaker module or a simple highest weight module
over $\mathfrak{Vir}$.

##  6\. Application one: characterization of simple highest weight modules and
Whittaker modules over the mirror Heisenberg-Virasoro algebra

Based on the results on structure of simple restricted modules over the mirror
Heisenberg-Virasoro algebra $\mathfrak{D}$ given in Theorem 2.11 and Theorem
4.13, we give characterization of simple highest weight $\mathfrak{D}$-modules
and simple Whittaker $\mathfrak{D}$-modules in this section.

We first have the following result characterizing simple highest weight
modules over the mirror Heisenberg-Virasoro algebra.

######  Theorem 6.1.

Let $\mathfrak{D}$ be the mirror Heisenberg-Virasoro algebra with the
triangular decomposition
${{\mathfrak{D}}}={\mathfrak{D}}^{+}\oplus{\mathfrak{D}}^{0}\oplus{\mathfrak{D}}^{-}$.
Let $S$ be a $\mathfrak{D}$-module (not necessarily weight) on which every
element in the algebra ${\mathfrak{D}}^{+}$ acts locally nilpotently. Then the
following statements hold.

  1. (i)

The module $S$ contains a nonzero vector $v$ such that
${\mathfrak{D}}^{+}\,v=0$.

  2. (ii)

If $S$ is simple, then $S$ is a highest weight module.

###### Proof.

(i) It follows from [45, Theorem 1] that there exists a nonzero vector $v\in
S$ such that $d_{i}v=0$ for any $i\in{\mathbb{Z}}_{+}$. If
$h_{\frac{1}{2}}v=0$, then ${\mathfrak{D}}^{+}\,v=0$ as $d_{1},d_{2}$ and
$h_{\frac{1}{2}}$ generate ${\mathfrak{D}}^{+}$. Assume that
$w:=h_{\frac{1}{2}}v\neq 0$. Then

$d_{1}w=d_{1}h_{\frac{1}{2}}v=h_{\frac{1}{2}}d_{1}v+[d_{1},h_{\frac{1}{2}}]v=-\frac{1}{2}h_{\frac{3}{2}}v.$

Similar arguments yield that the element $d_{1}^{j}w=\lambda
h_{j+\frac{1}{2}}v$ for some $\lambda\in{\mathbb{C}}^{*}$ and
$j\in{\mathbb{Z}}_{+}$. As $d_{1}$ acts locally nilpotently on $S$, it follow
that there exists some $n\in{\mathbb{Z}}_{+}$ such that $h_{j+\frac{1}{2}}v=0$
for $j\geq n$.

We now show that for every $m\in{\mathbb{N}}$ there exists some nonzero
element $u\in S$ such that $d_{i}u=h_{k+\frac{1}{2}}u=0$ for
$i\in{\mathbb{Z}}_{+}$ and $k\geq m$ by a backward induction on $m$. The above
arguments imply that the assertion is true for $m\geq n$. Assume that $0\neq
u\in S$ satisfies that $d_{i}u=h_{k+\frac{1}{2}}u=0$ for
$i\in{\mathbb{Z}}_{+}$ and $k\geq m>0$. If $h_{m-\frac{1}{2}}u=0$, then the
induction step is proved. Otherwise, $h_{m-\frac{1}{2}}u\neq 0$, and there
exists some $l\in{\mathbb{N}}$ such that
$u^{\prime}:=h_{m-\frac{1}{2}}^{l}u\neq 0$ and
$h_{m-\frac{1}{2}}u^{\prime}=h_{m-\frac{1}{2}}^{l+1}u=0$. Moreover,
$d_{i}u^{\prime}=h_{k+\frac{1}{2}}u^{\prime}=0$ for $i\in{\mathbb{Z}}_{+}$ and
$k\geq m-1$. The induction step follows.

(ii) By (i), we know that $S$ is a simple restricted $\mathfrak{D}$-module
with $n_{S}=0$ and $m_{S}\leq 1$. From Theorem 2.11 and Case 1 of Theorem 4.13
we know that $S\cong H^{\mathfrak{D}}\otimes U^{\mathfrak{D}}$ as
$\mathfrak{D}$-modules for some simple modules $H\in\mathcal{R}_{\mathcal{H}}$
and $U\in\mathcal{R}_{\mathfrak{Vir}}$. Moreover, $H=\text{\rm
Ind}^{\mathcal{H}}_{\mathcal{H}^{(0)}}({\mathbb{C}}v)$ is a simple highest
weight module over $\mathfrak{D}$. Note that every element in the algebra
$\mathfrak{Vir}^{(1)}$ acts locally nilpotently on ${\mathbb{C}}v\otimes U$ by
the assumption. This implies that the same property also holds on $U$. From
[45, Theorem 1] we know that $U$ is a simple highest weight
$\mathfrak{Vir}$-module. This completes the proof. ∎

As a direct consequence of Theorem 6.1, we have

######  Corollary 6.2.

Let $S$ be an simple restricted $\mathfrak{D}$-module with $m_{S}\leq 1$ and
$n_{S}=0$. Then $S$ is a highest weight module.

###### Proof.

The assumption that $m_{S}\leq 1$ and $n_{S}=0$ implies that there exists a
nonzero vector $v\in M$ such that ${\mathfrak{D}}^{+}v=0$. Then
$M=\mathcal{U}({\mathfrak{D}}^{-}+{\mathfrak{D}}^{0})v$. It follows that each
element in ${\mathfrak{D}}^{+}$ acts locally nilpotently on $M$. Consequently,
the desired assertion follows directly from Theorem 6.1. ∎

The following result characterizes simple Whittaker modules over the mirror
Heisenberg-Virasoro algebra.

######  Theorem 6.3.

Let $M$ be a $\mathfrak{D}$-module (not necessarily weight) on which the
algebra ${\mathfrak{D}}^{+}$ acts locally finitely. Then the following
statements hold.

  1. (i)

The module $M$ contains a nonzero vector $v$ such that
${\mathfrak{D}}^{+}\,v\subseteq{\mathbb{C}}v$.

  2. (ii)

If $M$ is simple, then $M$ is a Whittaker module or a highest weight module.

###### Proof.

(i) Let $(M_{1},\rho)$ be a finite dimensional ${\mathfrak{D}}^{+}$-submodule
of $M$. Then $M_{1}$ is also a finite dimensional $\mathfrak{Vir}_{\geq
1}$-module. Let $\mathfrak{a}:=\ker(\rho|_{\mathfrak{Vir}_{\geq 1}})$ be the
kernel of the representation map of $\mathfrak{Vir}_{\geq 1}$ on $M_{1}$. Then
$\mathfrak{a}$ is an ideal of $\mathfrak{Vir}_{\geq 1}$ of finite codimension.
We claim that $d_{n}\in\mathfrak{a}$ for some $n\in{\mathbb{Z}}_{+}$. If this
is not true, then there exists a minimal $m\in{\mathbb{Z}}_{+}$ such that
$\mathfrak{a}$ contains an element of the form
$a_{i_{1}}d_{i_{1}}+a_{i_{2}}d_{i_{2}}+\cdots+a_{i_{m+1}}d_{i_{m+1}}$ for
positive integers $i_{1}<i_{2}<\cdots<i_{m+1}$ and nonzero complex numbers
$a_{i_{1}},a_{i_{2}},\cdots,a_{i_{m+1}}$. We further see that $\mathfrak{a}$
contains

$[d_{i_{1}},a_{i_{1}}d_{i_{1}}+a_{i_{2}}d_{i_{2}}+\cdots+a_{i_{m+1}}d_{i_{m+1}}]=a_{i_{2}}(i_{1}-i_{2})d_{i_{1}+i_{2}}+a_{i_{3}}(i_{1}-i_{3})d_{i_{1}+i_{3}}+\cdots+a_{i_{m+1}}(i_{1}-i_{m+1})d_{i_{1}+i_{m+1}},$

which contradicts with the minimality of $m$. Hence the claim follows.
Consequently,

$\widetilde{\mathfrak{Vir}}_{\geq n}:=\sum_{i\geq n,\,i\neq
2n}{\mathbb{C}}d_{i}={\mathbb{C}}d_{n}+[d_{n},\mathfrak{Vir}_{\geq
1}]\subseteq\mathfrak{a}.$

Then

$\widetilde{\mathfrak{Vir}}_{\geq n}+\mathcal{H}_{\geq
n}=\widetilde{\mathfrak{Vir}}_{\geq
n}+[{\mathbb{C}}h_{\frac{1}{2}}+{\mathbb{C}}h_{\frac{3}{2}},\widetilde{\mathfrak{Vir}}_{\geq
n}]\subseteq\ker(\rho).$

This implies that $M_{1}$ is a finite dimensional module over a finite
dimensional solvable Lie algebra
${\mathfrak{D}}^{+}/(\widetilde{\mathfrak{Vir}}_{\geq n}+\mathcal{H}_{\geq
n})$. The desired assertion follows directly from Lie Theorem.

(ii) follows directly from (i). ∎

##  7\. Examples

In this section, we will give a few examples of simple restricted
$\bar{\mathfrak{D}}$\- and $\mathfrak{D}$-modules, which are also weak
(simple) untwisted and twisted $\mathcal{V}^{c}$-modules.

######  Example 7.1.

For any $n\in{\mathbb{Z}}_{+}$, let
$\mathcal{W}_{0}={\mathbb{C}}[x_{1},\cdots,x_{n}]$ be the polynomial algebra
in indeterminates $x_{1},\cdots,x_{n}$. Define the $\mathcal{H}^{(-n)}$-module
structure on $\mathcal{W}_{0}$ by

$\displaystyle h_{i-\frac{1}{2}}\cdot
f(x_{1},\cdots,x_{i},\cdots,x_{n})=\lambda_{i}f(x_{1},\cdots,x_{i}-1,\cdots,x_{n}),$
(7.1) $\displaystyle h_{-i+\frac{1}{2}}\cdot
f(x_{1},\cdots,x_{i},\cdots,x_{n})=-\frac{\ell(i-\frac{1}{2})}{\lambda_{i}}(x_{i}+a_{i})f(x_{1},\cdots,x_{i}+1,\cdots,x_{n}),$
$\displaystyle h_{n+j+\frac{1}{2}}\cdot f(x_{1},\cdots,x_{i},\cdots,x_{n})=0,$
$\displaystyle{\bf c}_{2}\cdot f(x_{1},\cdots,x_{n})=\ell
f(x_{1},\cdots,x_{n})$

where
$\ell,\lambda_{i}\in{\mathbb{C}}^{*},a_{i}\in{\mathbb{C}},j\in{\mathbb{N}},1\leq
i\leq n$. It is not hard to check that $\mathcal{W}_{0}$ is a simple
$\mathcal{H}^{(-n)}$-module. Then the induced $\mathcal{H}$-module
$K=\text{\rm Ind}_{\mathcal{H}^{(-n)}}^{\mathcal{H}}\mathcal{W}_{0}$ is a
simple restricted $\mathcal{H}$-module. So $K^{\mathfrak{D}}$ is a simple
restricted $\mathfrak{D}$-module with central charge $1$ and level $\ell$. We
may denote
$K^{\mathfrak{D}}=K^{\mathfrak{D}}(\ell,\Lambda_{n},\mathfrak{a}_{n})$ for any
$\ell\in{\mathbb{C}}^{*},$
$\Lambda_{n}=(\lambda_{1},\cdots,\lambda_{n})\in({\mathbb{C}}^{*})^{n},$
$\mathfrak{a}_{n}=(a_{1},\cdots,a_{n})\in{\mathbb{C}}^{n}$.

Let $U$ be a simple restricted $\mathfrak{Vir}$-module (Theorem 2.9 classified
all simple restricted $\mathfrak{Vir}$-modules). From Corollary 2.14, then
$S=U^{\mathfrak{D}}\otimes
K^{\mathfrak{D}}(\ell,\Lambda_{n},\mathfrak{a}_{n})$ is a simple restricted
$\mathfrak{D}$-module.

If we replace (7.1) by

$\displaystyle h_{i}\cdot
f(x_{1},\cdots,x_{i},\cdots,x_{n})=\lambda_{i}f(x_{1},\cdots,x_{i}-1,\cdots,x_{n}),$
$\displaystyle h_{-i}\cdot f(x_{1},\cdots,x_{i},\cdots,x_{n})=-\frac{\ell
i}{\lambda_{i}}(x_{i}+a_{i})f(x_{1},\cdots,x_{i}+1,\cdots,x_{n}),$
$\displaystyle h_{n+j+1}\cdot f(x_{1},\cdots,x_{i},\cdots,x_{n})=0,$
$\displaystyle\bar{\bf c}_{3}\cdot f(x_{1},\cdots,x_{n})=\ell
f(x_{1},\cdots,x_{n})$

for
$\ell,\lambda_{i}\in{\mathbb{C}}^{*},a_{i}\in{\mathbb{C}},j\in{\mathbb{N}},1\leq
i\leq n$, then $\mathcal{W}_{0}$ is a simple
$\bar{\mathcal{H}}^{(-n)}$-module, and the induced $\bar{\mathcal{H}}$-module
$\bar{K}=\text{\rm
Ind}_{\bar{\mathcal{H}}^{(-n)}}^{\bar{\mathcal{H}}}\mathcal{W}_{0}$ is a
simple restricted $\bar{\mathcal{H}}$-module. Hence, for any
$z\in{\mathbb{C}}$, we have the simple $\bar{\mathfrak{D}}$-module
$\bar{K}(z)^{\bar{\mathfrak{D}}}=\bar{K}(z)^{\bar{\mathfrak{D}}}(\ell,\Lambda_{n},\mathfrak{a}_{n})$
for any $\ell\in{\mathbb{C}}^{*},$
$\Lambda_{n}=(\lambda_{1},\cdots,\lambda_{n})\in({\mathbb{C}}^{*})^{n},$
$\mathfrak{a}_{n}=(a_{1},\cdots,a_{n})\in{\mathbb{C}}^{n}$. For any simple
$\mathfrak{Vir}$-module $U\in\mathcal{R}_{\mathfrak{Vir}}$, the tensor product
$M=U^{\bar{\mathfrak{D}}}\otimes\bar{K}(z)^{\bar{\mathfrak{D}}}(\ell,\Lambda_{n},\mathfrak{a}_{n})$
is a simple restricted $\bar{\mathfrak{D}}$-module.

For characterizing simple induced restricted $\mathfrak{D}$-and
$\bar{\mathfrak{D}}$-module which are not tensor product modules, we need the
following

######  Lemma 7.2.

Let $S=U^{\mathfrak{D}}\otimes V^{\mathfrak{D}}$ be a simple restricted
$\mathfrak{D}$-module with $n_{S}>0$ and nonzero level, where
$U\in{\mathcal{R}}_{\mathfrak{Vir}}$ and $V\in{\mathcal{R}}_{\mathcal{H}}$.
Let $V_{0}=\text{Ann}_{V}(\mathcal{H}^{(n_{S})})$ and
$W_{0}=\text{Ann}_{S}(\mathcal{H}^{(n_{S})})$. Then $V_{0}$ is a simple
$\mathfrak{D}^{(0,-n_{S})}$-module, and $W_{0}=U\otimes V_{0}$. Hence $W_{0}$
contains a simple $\mathcal{H}^{(-n_{S})}$ submodule.

###### Proof.

This is clear. ∎

We also have the $\bar{\mathfrak{D}}$-module version of Lemma 7.2:

######  Lemma 7.3.

Let $M=H(z)^{\bar{\mathfrak{D}}}\otimes U^{\bar{\mathfrak{D}}}$ be a simple
restricted $\bar{\mathfrak{D}}$-module with $n_{M}>1$ and nonzero level, where
$z\in{\mathbb{C}}$, $H\in{\mathcal{R}}_{\bar{\mathcal{H}}}$ and
$U\in{\mathcal{R}}_{\mathfrak{Vir}}$. Let
$H_{0}=\text{Ann}_{H}(\bar{\mathcal{H}}^{(n_{M})})$ and
$M_{0}=\text{Ann}_{M}(\bar{\mathcal{H}}^{(n_{M})})$. Then $H_{0}$ is a simple
$\bar{\mathfrak{D}}^{(0,-n_{M}+1)}$-module, and $M_{0}=H_{0}\otimes U$. Hence
$M_{0}$ contains a simple $\bar{\mathcal{H}}^{(-n_{M}+1)}$ submodule.

Lemma 7.2 (resp. Lemma 7.3) means that if $S\in{\mathcal{R}}_{\mathfrak{D}}$
(resp. $M\in{\mathcal{R}}_{\bar{\mathfrak{D}}}$) is not a tensor product
module, then $W_{0}$ (resp. $M_{0}$) contains no simple
$\mathcal{H}^{(-n_{S})}$-submodule (resp.
$\bar{\mathcal{H}}^{(-n_{M}+1)}$-submodules).

Here we will first consider the case $n_{S}=1$ (resp. $n_{M}=2$). Let
$\mathfrak{b}={\mathbb{C}}h+{\mathbb{C}}e$ be the 2-dimensional solvable Lie
algebra with basis $h,e$ and subject to Lie bracket $[h,e]=e$. The following
concrete example using [40, Example 13] tells us how to construct induced
restricted $\mathfrak{D}$-module (resp. $\bar{\mathfrak{D}}$-module) from a
${\mathbb{C}}[e]$-torsion-free simple $\mathfrak{b}$-module.

######  Example 7.4 (Simple induced restricted module, $n_{S}=1/n_{M}=2$).

Let $c_{1},c_{2}\in{\mathbb{C}}$ with $c_{2}\neq 0$. Let
$W^{\prime}=(t-1)^{-1}{\mathbb{C}}[t,t^{-1}]$. From [40, Example 13] we know
that $W^{\prime}$ is a simple $\mathfrak{b}$-module whose structure is given
by

$h\cdot f(t)=t\frac{d}{dt}(f(t))+\frac{f(t)}{t^{2}(t-1)},\,\,e\cdot
f(t)=tf(t),\forall f(t)\in W^{\prime}.$

We can make $W^{\prime}$ into a $\mathfrak{D}^{(0,0)}$-module by

$\displaystyle{\bf c}_{1}\cdot f(t)$ $\displaystyle=c_{1}f(t),\,\,\,{\bf
c}_{2}\cdot f(t)=c_{2}f(t),$ $\displaystyle d_{0}\cdot f(t)$
$\displaystyle=-\frac{1}{2}h\cdot f(t),\,h_{\frac{1}{2}}\cdot f(t)=e\cdot
f(t),\,d_{i}\cdot f(t)=h_{\frac{1}{2}+i}\cdot
f(t)=0,\,\,\,i\in{\mathbb{Z}}_{+}.$

Then $W^{\prime}$ is a simple $\mathfrak{D}^{(0,0)}$-module. Clearly, the
action of $h_{\frac{1}{2}}$ on $W^{\prime}$ implies that $W^{\prime}$ contains
no simple $\mathcal{H}^{(0)}$-module. Then $W_{0}=\text{\rm
Ind}_{\mathfrak{D}^{(0,0)}}^{\mathfrak{D}^{(0,-1)}}W^{\prime}$ is a simple
$\mathfrak{D}^{(0,-1)}$-module and contains no simple
$\mathcal{H}^{(-1)}$-module. So $W_{0}$ is not a tensor product
$\mathfrak{D}^{(0,-1)}$-module. Let $S=\text{\rm
Ind}_{\mathfrak{D}^{(0,-1)}}^{\mathfrak{D}}W_{0}$. It is easy to see
$n_{S}=1,m_{S}=2=r_{S}$, and $W_{0}=U_{0}=K_{0}$. The proof of Proposition 4.5
implies that $S$ is a simple restricted $\mathfrak{D}$-module. And Lemma 7.2
means that $S$ is not a tensor product $\mathfrak{D}$-module.

For $c,z,z^{\prime}\in{\mathbb{C}},\ell\in{\mathbb{C}}^{*}$, we also can make
$W^{\prime}$ into a $\bar{\mathfrak{D}}^{(0,0)}$-module by

$\displaystyle d_{0}\cdot f(t)=h\cdot f(t),\,h_{1}\cdot f(t)=e\cdot f(t),$
$\displaystyle h_{0}\cdot f(t)=z^{\prime}f(t),h_{1+i}\cdot f(t)=d_{i}\cdot
f(t)=0,\,\,\,i\in{\mathbb{Z}}_{+},$ $\displaystyle\bar{\bf c}_{1}\cdot
f(t)=cf(t),\,\bar{\bf c}_{2}\cdot f(t)=zf(t),\,\bar{\bf c}_{3}\cdot f(t)=\ell
f(t),$

where $f(t)\in W^{\prime}$. Then $W^{\prime}$ is a simple
$\bar{\mathfrak{D}}^{(0,0)}$-module. Clearly, the action of $h_{1}$ on
$W^{\prime}$ implies that $W^{\prime}$ contains no simple
$\bar{\mathcal{H}}^{(0)}$-module. Then $M_{0}=\text{\rm
Ind}_{\bar{\mathfrak{D}}^{(0,0)}}^{\bar{\mathfrak{D}}^{(0,-1)}}W^{\prime}$ is
a simple $\bar{\mathfrak{D}}^{(0,-1)}$-module and contains no simple
$\bar{\mathcal{H}}^{(-1)}$-module. Let $M=\text{\rm
Ind}_{\bar{\mathfrak{D}}^{(0,-1)}}^{\bar{\mathfrak{D}}}M_{0}$. It is easy to
see $n_{M}=2,$$r_{M}=3$. The proof of Proposition 5.5 implies that $M$ is a
simple restricted $\bar{\mathfrak{D}}$-module. And Lemma 7.3 means that $M$ is
not a tensor product $\bar{\mathfrak{D}}$-module.

######  Example 7.5 (Simple induced modules of semi-Whittaker type, $n_{S}\geq
2,n_{M}\geq 3$).

111This example is a modified version of the one provided by Drazen Adamovic.

Take $p,q\in{\mathbb{Z}}_{+},{\bf
a}=(a_{1},\dots,a_{q})\in({{\mathbb{C}}}^{*})^{q}$, ${\bf
b}=(b_{1},\dots,b_{p})\in({{\mathbb{C}}}^{*})^{p}$, $c,\ell\in{\mathbb{C}}$
with $\ell\neq 0$. Define the $1$-dimensional $\mathfrak{D}^{(p,q)}$-module
${{\mathbb{C}}}_{{\bf a},{\bf b}}={{\mathbb{C}}}v_{0}$ with

$\displaystyle{\bf c}_{1}\cdot v_{0}$ $\displaystyle=cv_{0},\,\,\,{\bf
c}_{2}\cdot v_{0}=\ell v_{0},$ (7.2) $\displaystyle d_{p}v_{0}$
$\displaystyle=a_{1}v_{0},\cdots,d_{p+q-1}v_{0}=a_{q}v_{0},\ d_{i}v_{0}=0\
\mbox{for}\ i>p+q-1,$ $\displaystyle h_{q+\tfrac{1}{2}}v_{0}$
$\displaystyle=b_{1}v_{0},\cdots h_{p+q-\tfrac{1}{2}}v_{0}=b_{p}v_{0},\
h_{i-\tfrac{1}{2}}v_{0}=0\ \mbox{for}\ i>p+q.$

It is not hard to show that $U({\bf a},{\bf b}):=\text{\rm
Ind}_{\mathfrak{D}^{(p,q)}}^{\mathfrak{D}^{(0,-1)}}{{\mathbb{C}}}_{{\bf
a},{\bf b}}$ is a simple $\mathfrak{D}^{(0,-1)}$-module. Then in Theorem 3.1
(2) we have $V=U({\bf a},{\bf b}),n=1,k=p+q=l$, and so $S=\widehat{U}({\bf
a},{\bf b}):=\text{\rm Ind}_{\mathfrak{D}^{(0,-1)}}^{\mathfrak{D}}U({\bf
a},{\bf b})$ is a simple restricted $\mathfrak{D}$-module. In Lemma 7.2,
$n_{S}=p+q$, and $W_{0}=\text{\rm
Ind}_{\mathcal{H}^{(q)}}^{\mathcal{H}^{(-(p+q))}}\left(\text{\rm
Ind}_{\mathfrak{D}^{(p,q)}}^{\mathfrak{D}^{(0,q)}}{\mathbb{C}}_{{\bf a},{\bf
b}}\right)$ does not contain any simple $\mathcal{H}^{(-(p+q))}$-module ( for
$h_{\pm 1/2}$ acts freely on $W_{0}$). Hence, by Lemma 7.2, $\widehat{U}({\bf
a},{\bf b})$ is not a tensor product $\mathfrak{D}$-module.

If we, in the above example, replace (7.2) by

$\displaystyle{\bar{\bf c}}_{1}\cdot v_{0}=cv_{0},\,{\bar{\bf c}}_{2}\cdot
v_{0}=zv_{0},\,\bar{\bf c}_{3}v_{0}=\ell v_{0},$ $\displaystyle
d_{p}v_{0}=a_{1}v_{0},\cdots,d_{p+q-1}v_{0}=a_{q}v_{0},\ d_{i}v_{0}=0\
\mbox{for}\ i>p+q-1,$ $\displaystyle h_{q+1}v_{0}=b_{1}v_{0},\cdots
h_{p+q}v_{0}=b_{p}v_{0},\ h_{i}v_{0}=0\ \mbox{for}\ i>p+q,$

where $z\in{\mathbb{C}}$ and leave other parts invariant, then for any
$z^{\prime}\in{\mathbb{C}}$, the induced
$\bar{\mathfrak{D}}^{(0,-(p+q))}$-module

$\bar{V}=\text{\rm
Ind}_{\bar{\mathfrak{D}}^{(p,q+1)}}^{\bar{\mathfrak{D}}^{(0,-(p+q))}}{{\mathbb{C}}}_{\bf
a,\bf
b}/\Big{(}\mathcal{U}(\bar{\mathfrak{D}}^{(0,-(p+q))})(h_{0}-z^{\prime})(1\otimes
v_{0})\Big{)}$

is a simple $\bar{\mathfrak{D}}^{(0,-(p+q))}$-module. Let $M=\text{\rm
Ind}_{\bar{\mathfrak{D}}^{(0,-(p+q))}}^{\bar{\mathfrak{D}}}\bar{V}$. The proof
of Theorem 5.5 implies that $M$ is a simple restricted
$\bar{\mathfrak{D}}$-module where $n_{M}=p+q+1,r_{M}=2(p+q)+1$, and
$K_{0}=\bar{V}=M_{0}$. Since $\bar{V}$ contains no simple
$\bar{\mathcal{H}}^{(-n_{M}+1)}$-module, we see, by Lemma 7.3, that $M$ is not
a tensor product $\bar{\mathfrak{D}}$-module.

######  Remark 7.6.

From Theorem 4.13 (resp. Theorem 5.2) we know that if $n_{S}=0$ (resp.
$n_{M}=0,1$), then simple restricted $\mathfrak{D}$-modules(resp.
$\bar{\mathfrak{D}}$-modules) must be tensor product modules. And Examples
7.4-1 mean that for any $n_{S}>0$ (resp. $n_{M}>1$), there do exist simple
restricted $\mathfrak{D}$-modules (resp. $\bar{\mathfrak{D}}$-modules) which
are not tensor product modules. Clearly, the $\bar{\mathfrak{D}}$-modules here
are simple restricted $\widetilde{\mathfrak{D}}$-modules for $z=0$.

##  Appendix A Application two: simple modules for Heisenberg-Virasoro vertex
operator algebras $\mathcal{V}^{c}$ (by Drazen Adamovic)

A connection between restricted modules over the Heisenberg-Virasoro algebra
and VOA modules in untwisted cases was considered by Guo and Wang in [27]. In
this appendix we extend this correspondence for restricted modules for the
mirror Heisenberg-Virasoro algebra. Restricted modules of nonzero level for
the mirror Heisenberg-Virasoro algebra can be treated as weak twisted modules
for the Heisenberg-Virasoro vertex algebras, and restricted modules of nonzero
level for the twisted Heisenberg-Virasoro algebra can be treated as weak
modules for the Heisenberg-Virasoro vertex algebras. For convenience, in the
Heisenberg-Virasoro algebra $\widetilde{\mathfrak{D}}$, we denote ${\bf
c}_{1}=\bar{\bf c}_{1},{\bf c}_{2}=\bar{\bf c}_{3}$.

Let $\mathcal{P}$ be the subalgebra of $\widetilde{\mathfrak{D}}$ spanned by

${\bf c}_{1},{\bf c}_{2},d_{m},h_{m},\quad(m\in{{\mathbb{Z}}}_{\geq 0}).$

Let $(\ell_{1},\ell_{2})\in{{\mathbb{C}}}^{2}$. Consider the $1$-dimensional
$\mathcal{P}$–module ${{\mathbb{C}}}v_{\ell_{1},\ell_{2}}$ such that

${\bf c}_{1}v_{\ell_{1},\ell_{2}}=\ell_{1}v_{\ell_{1},\ell_{2}},\ {\bf
c}_{2}v_{\ell_{1},\ell_{2}}=\ell_{2}v_{\ell_{1},\ell_{2}},\
d_{m}v_{\ell_{1},\ell_{2}}=h_{m}v_{\ell_{1},\ell_{2}}=0\quad(m\in{{\mathbb{Z}}}_{\geq
0}).$

Let $\mathcal{V}^{\ell_{1},\ell_{2}}$ be the following induced
$\widetilde{\mathfrak{D}}$-module:

$\mathcal{V}^{\ell_{1},\ell_{2}}=U(\widetilde{\mathfrak{D}})\otimes_{U(\mathcal{P})}{{\mathbb{C}}}v_{\ell_{1},\ell_{2}}.$

Then $\mathcal{V}^{\ell_{1},\ell_{2}}$ is a highest weight
$\widetilde{\mathfrak{D}}$-module, with the highest weight vector ${\bf
1}_{\ell_{1},\ell_{2}}=1\otimes v_{\ell_{1},\ell_{2}}$.

Define the following fields acting on $\mathcal{V}^{\ell_{1},\ell_{2}}$:

$h(z)=\sum_{m\in{{\mathbb{Z}}}}h_{m}z^{-m-1},\quad
d(z)=\sum_{m\in{{\mathbb{Z}}}}d_{m}z^{-n-2}.$

  * •

Let $V_{Vir}^{c}$ denotes the universal Virasoro vertex algebra of central
charge $c$ generated by the Virasoro field
$L^{(1)}(z)=\sum_{m\in{\mathbb{Z}}}L^{(1)}_{n}z^{-n-2}$ (cf. [35], [23]).

  * •

Let $M(\ell)$ denotes the Heisenberg vertex algebra of level $\ell$, with the
vertex operator $Y_{M(\ell)}(\cdot,z)$ which is uniquely generated with the
Heisenberg field

$Y_{M(\ell)}(h(-1){\bf 1},z)=h(z)=\sum_{n\in{{\mathcal{Z}}}}h_{n}z^{-n-1}.$

If $\ell\neq 0$, then $M(\ell)$ is simple and always isomorphic to $M(1)$ (cf.
[35]).

  * •

$M(\ell)$ contains a Virasoro vertex subalgebra $V_{Vir}^{c=1}$ generated by
the Virasoro field

$L^{Heis}(z):=\frac{1}{2\ell}:h(z)h(z):=\sum_{n\in{{\mathbb{Z}}}}L^{Heis}_{n}z^{-n-2}.$

Moreover, we have

$L^{Heis}_{n}=\frac{1}{2\ell}\sum_{k\in{\mathbb{Z}}}:h_{k}h_{n-k}:$

For details see [22], [35].

  * •

$M(\ell)$ contains the Virasoro vector

$\omega_{\lambda}=\frac{1}{2\ell}(h_{-1}^{2}){\bf
1}-\frac{\lambda}{\ell}h_{-2}{\bf 1}$

of the central charge $c=1-12{\lambda}^{2}/\ell$. The components of the vertex
operator

$\displaystyle
Y_{M(\ell)}(\omega_{\lambda})=\bar{L}(z)=L^{Heis}(z)-\frac{\lambda}{\ell}\partial_{z}h(z)=\sum_{n\in{{\mathbb{Z}}}}\bar{L}_{n}z^{-n-2}$
(A.1)

represent the operators $\bar{L}_{n}$ defined directly by (2.2).

  * •

Note that the field $\bar{L}(z)$ and operators $\bar{L}_{n}$ acts on any weak
$M(\ell)$-module, and in particular on any restricted module for the
Heisenberg algebra.

Note that $M(\ell_{2})$ is naturally a vertex subalgebra of
$\mathcal{V}^{\ell_{1},\ell_{2}}$.

######  Proposition A.1.

Assume that $\ell_{2}\neq 0$. We have the following isomorphism of vertex
algebras

$V_{Vir}^{c}\otimes M(\ell_{2})\cong\mathcal{V}^{\ell_{1},\ell_{2}},$

such that $c=\ell_{1}-1$ and

$L^{(1)}(z)\mapsto d(z)-L^{Heis}(z),\quad h(z)\mapsto h(z).$

In particular, $\mathcal{V}^{\ell_{1},\ell_{2}}\cong V_{Vir}^{c}\otimes M(1)$.

Since $M(\ell)\cong M(1)$, without loss of generality we can assume that
$\ell=1$. In what follows we set $\mathcal{V}^{c}:=V_{Vir}^{c}\otimes M(1)$.

The following theorem relates restricted $\bar{\mathfrak{D}}$-modules as
(untwisted) modules for the vertex operator algebra $\mathcal{V}^{c}$.

######  Theorem A.2.

(cf. [27]) The following statements holds.

Assume that $W$ is a (simple) restricted $\bar{\mathfrak{D}}$-module of
central charge $\ell_{1}$ and level one. Then $W$ has the unique structure of
a weak (simple) $\mathcal{V}^{c=\ell_{1}-1}$-module generated by the fields:

$h(z)=\sum_{r\in{{\mathbb{Z}}}}h_{r}z^{-r-1},\quad
L^{(1)}(z)=d(z)-L^{Heis}(z)=\sum_{n\in{{\mathbb{Z}}}}L^{(1)}_{n}z^{-n-2}.$

Assume that $W$ is a weak (simple) $\mathcal{V}^{c}$-module. Then $W$ has the
structure of a (simple) restricted $\bar{\mathfrak{D}}$-module of level one
such that

$h_{n}\mapsto h_{n},\quad d_{n}=L^{(1)}_{n}+L^{Heis}_{n}$

The vertex–algebraic interpretation of the restricted $\mathfrak{D}$–modules
is via the twisted $\mathcal{V}^{c}$-modules. There is an automorphism
$\theta_{1}$ of order two of $M(1)$ such that $\theta_{1}(h)=-h$ (cf. [22]).
We extend this automorphism to the automorphism
$\theta=\mbox{Id}\otimes\theta_{1}$ of $\mathcal{V}^{c}$.

We have the following theorem.

######  Theorem A.3.

The following statements holds.

Assume that $W$ is a (simple) restricted $\mathfrak{D}$-module of central
charge $\ell_{1}$ and level one. Then $W$ has the unique structure of a weak
(simple) $\theta$-twisted $\mathcal{V}^{c=\ell_{1}-1}$-module generated by the
fields:

$h^{tw}(z)=\sum_{r\in\tfrac{1}{2}+{{\mathbb{Z}}}}h_{r}z^{-r-1},L^{(1)}(z)=d(z)-L_{tw}^{Heis}(z),$

where

$\displaystyle
L_{tw}^{Heis}(z):=\tfrac{1}{2}:h^{tw}(z)^{2}:+\frac{1}{16}z^{-2}=\sum_{n\in{{\mathbb{Z}}}}L_{n}z^{-n-2}.$
(A.2)

Assume that $W$ is a weak (simple) $\theta$-twisted $\mathcal{V}^{c}$-module.
Then $W$ has the structure of a (simple) restricted $\mathfrak{D}$-module of
level one such that

$h_{r}\mapsto h_{r},\quad d_{n}=L^{(1)}_{n}+L_{n}.$

###### Proof.

As in [22] (see also [54], [29]) we see that the field $h^{tw}(z)$ defines on
$W$ the unique structure of a $\theta_{1}$-twisted $M(1)$-module with the
Virasoro field (A.2). Then we define
$L^{(1)}(z)=d(z)-L_{tw}^{Heis}(z)=\sum_{n\in{{\mathbb{Z}}}}L^{(1)}_{n}z^{-n-2}$.
The field $L^{(1)}(z)$ defines on $W$ the structure of a restricted module for
the Virasoro algebra of central charge $c=\ell_{1}-1$. Since

$[L^{(1)}_{n},h_{r}]=0,\quad\forall n\in{{\mathbb{Z}}},\
r\in\tfrac{1}{2}+{{\mathbb{Z}}},$

we have that the action of $L^{(1)}(z)$ commutes with $h^{tw}(z)$. Therefore
$W$ is a $\theta$-twisted $\mathcal{V}_{Vir}^{c}$-module. Since all components
of the vertex operators are obtained from the action of $d_{n},h_{r}$,
$n\in{{\mathbb{Z}}},r\in\tfrac{1}{2}+{{\mathbb{Z}}}$, we see that $W$ is an
irreducible $\mathfrak{D}$-module if and only if $W$ is an irreducible module
for the vertex algebra $\mathcal{V}^{c}$. This proves the assertion (1).

Assume that $W$ is $\theta$-twisted $\mathcal{V}^{c}$-module with the vertex
operator $Y^{tw}_{W}(v,z)$, $v\in\mathcal{V}^{c}$. Then the twisted Jacobi
identity (cf. [22]) shows that:

  * •

The components of the field

$L^{(1)}(z)=Y_{W}^{tw}(L^{(1)}_{-2}{\bf
1},z)=\sum_{n\in{\mathbb{Z}}}L^{(1)}_{n}z^{-n-2}$

define on $W$ structure of a restricted module for the Virasoro algebra with
central charge $c$;

  * •

The components of the field

$h^{tw}(z)=Y_{W}^{tw}(h_{-1}{\bf
1},z)=\sum_{r\in\tfrac{1}{2}+{\mathbb{Z}}}h_{r}z^{-r-1}$

define on $W$ the structure of a restricted module for the Heisenberg algebra
of level one.

  * •

The fields $h^{tw}(z)$ and $L^{(1)}(z)$ commute.

  * •

The field

$L_{tw}^{Heis}(z)=Y(\tfrac{1}{2}h_{-1}^{2}{\bf
1},z)=\sum_{n\in{\mathbb{Z}}}L_{n}z^{-n-2}$

is a Virasoro field commuting with $L^{(1)}(z)$ such that

$[L_{n},h_{r}]=-rh_{n+r},\quad
n\in{\mathbb{Z}},r\in\tfrac{1}{2}+{\mathbb{Z}}.$

Define
$d(z)=L^{(1)}(z)+L_{tw}^{Heis}(z)=\sum_{n\in{\mathbb{Z}}}d_{n}z^{-n-2}$. Then
the components of the fields $d(z)$ and $h^{tw}(z)$ define on $W$ the
structure of a restricted $\mathfrak{D}$-module of central charge
$\ell_{1}=c+1$ and level one.

Arguments for the irreducibility are the same as in (1). This proves the
assertion (2). ∎

######  Remark A.4.

The simple modules in Examples 7.4-1 show that the vertex operator algebra
$V_{Vir}^{c}\otimes M(1)$ has simple weak (untwisted and twisted) modules
which are not isomorphic to any tensor product $S_{1}\otimes S_{2}$ for any
simple weak $V_{Vir}^{c}$-module $S_{1}$ and any simple weak (untwisted and
twisted) $M(1)$-module $S_{2}$. It would be interesting to find analogs of
these non-tensor product modules for other types of vertex operator algebras.

Acknowledgments

The authors would like to thank Drazen Adamovic to provide Appendix A and
Example 7.5, and help modifying the introduction. The last two authors would
like to thank H. Chen and D. Gao for helpful discussions at the early stage on
the restricted modules over the twisted Heisenberg-Virasor algebra.

H.T. is partially supported by the Fundamental Research Funds for the Central
Universities (135120008). Y.Y. is partially supported by the National Natural
Science Foundation of China (11771279, 12071136, 11671138). K.Z. is partially
supported by the National Natural Science Foundation of China (11871190) and
NSERC (311907-2020).

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