††thanks: These authors equally contributed††thanks: These authors equally
contributed

# Bulk-Measurement-Induced Boundary Phase Transition in Toric Code and Gauge-
Higgs Model

Yoshihito Kuno  Graduate School of Engineering Science, Akita University,
Akita 010-8502, Japan  Takahiro Orito  Insitute for Solid State Physics, The
University of Tokyo, Kashiwa, Chiba, 277-8581, Japan  Ikuo Ichinose
Department of Applied Physics, Nagoya Institute of Technology, Nagoya,
466-8555, Japan

###### Abstract

Study of boundary phase transition in toric code under cylinder geometry via
bulk projective measurements is reported. As the frequency of the local
measurements for bulk qubits is increased, spin-glass type long-range order on
the boundaries emerges indicating spontaneous symmetry breaking (SSB) of
$Z_{2}$ symmetry. From the lattice-gauge-theory viewpoint, this SSB is a
signal of a transition to Higgs phase with symmetry protected topological
order. We numerically elucidate properties of the phase transition in detail,
especially its criticality and give a physical picture using a non-local gauge
invariant symmetry operator. Phase transition in the bulk is also studied and
its relationship to the boundary transition is discussed.

Introduction.— Measurements in quantum many-body system lead to nontrivial
dynamics and produces exotic phases. Measurements of entangled resource states
induce measurement-based quantum computation Raussendorf2001 ; Briegel2009
and produce long-range entanglement states or topological order
Tantivasadakarn2022 . With suitable choice of the measurement types,
measurement-only circuits generate unconventional phases of matter such as
symmetry protected topological (SPT) phases Lavasani2021 ; Klocke2022 ; KI2023
, topological orders Lavasani2021_2 ; Negari2023 , and non-trivial thermal and
critical phases Ippoliti2021 ; Sriram2023 ; KOI2023 ; Lavasani2023 ; Zhu2023 .
As a further example, hybrid random-unitary circuits exhibit measurement-
induced entanglement phase transitions Li2018 ; Skinner2019 ; Li2019 ;
Vasseur2019 ; Chan2019 ; Szyniszewski2019 ; Choi2020 ; Bao2020 ; Gullans2020 ;
Jian2020 ; Zabalo2020 ; Sang2021 ; Sang2021_v2 ; Nahum2021 ; Sharma2022 ;
Fisher2022_rev ; Block2022 ; Liu ; Richter2023 ; Sierant2023 ; Kumer2023 .

Recently, toric code Kitaev2003  was experimentally realized Google_quantum_AI
. Target geometry of the system is torus, providing two-qubit quantum memory
Kitaev2003 ; Dennis2001 ; Wang2003 . Recent studies, however, showed that
interplay of an open geometry, such as the cylinder with rough boundary (Fig.
1), and measurements can give rich physical phenomena Verresen2022 ;
Wildeboer2022 ; Negari2023 . This observation was obtained from the viewpoint
of lattice gauge theory Kogut1979 . In the quantum information aspect, edge
modes on the boundary work as qubit Verresen2022 ; Wildeboer2022 ; KI2023_2 ,
which is called subsystem code Poulin2005 ; Bacon2006 . From the gauge theory
side, it was recently suggested that the edge modes create some kind of long-
range order and it distinguishes the Higgs and confinement phases, although it
was believed for a long time that the two phases are adiabatically connected
without any transitions Fradkin1979 . This long-range order order indicates
that the Higgs phase is a SPT phase protected by a higher-form symmetry
Verresen2022 . The above developments attract lots of attention these days.

The above discussion was given for the Hamiltonian systems. In this Letter, we
shall study closed-related systems with the cylinder geometry by using
measurement-only scheme. In particular, we investigate the emergence of edge
modes and transition from the deconfined (toric code) to Higgs phases as the
measurement corresponding to ‘Higgs coupling’ is getting frequent. Our
protocol is as follows: We prepare a toric code state as the initial state,
and observe what happens as performing local projective measurements in the
bulk. The reason for performing measurements only in the bulk is that
interplay of a non-local gauge-invariant operator (NLGIO) symmetry and Higgs
condensation in the bulk induces a long-range order with SSB on boundary,
signature of the SPT. In fact by a numerical method, we verify an emergence of
the boundary long-range order as a clear measurement-induced phase transition
on the boundary. Furthermore, we find some noticeable feature of the
transition, and discuss that it is an evidence of the interplay of the NLGIO
and Higgs condensation. Finally, we investigate the phase transition in the
bulk and discuss its relationship to the boundary transition.

Figure 1: Schematic figure of cylinder system with rough boundaries. This
figure displays only the upper rough boundary. The star operator (the red
shaded object) and plaquette operator (right-blue shaded object) are
considered. The red dashed regime is an upper rough boundary regime.

Measured toric code system.— We consider a code system composed of
$L_{x}\times L_{y}$ plaquettes ($q$-lattice) and $L_{x}\times(L_{y}-1)$
vortices ($v$-lattice). Physical qubits reside on links of the $v$-lattice,
and cylinder geometry and rough boundaries in the $y$ direction are employed.
The schematic image of the system is shown in Fig. 1. In our protocol, we
consider a stabilizer state specified by a set of stabilizer generators
Nielsen_Chuang  focusing on pure states, $S_{\rm
int}=\\{\tilde{A}_{v}|v\in\mbox{all}\ v\\}+\\{\tilde{B}_{q}|q\in\mbox{all
plaquette but}\ q_{0}\\}+\\{S^{u}_{x}\\}$, where $q_{0}$ is one of single
$q$-lattice sites to avoid the redundancy to make all elements of the
stabilizer generator linearly independent. In the above, $\tilde{A}_{v}$ and
$\tilde{B}_{q}$ are the star and plaquette operators of the toric code,
defined by $\tilde{A}_{v}=\prod_{\ell_{v}\in v}\sigma^{x}_{\ell_{v}}$ and
$\tilde{B}_{q}=\prod_{\ell_{q}\in q}\sigma^{z}_{\ell_{q}}$, where $\ell_{v}\in
v$ stands for links emanating from vertex $v$, and $\ell_{q}\in q$ for links
composing plaquette $q$ and $\displaystyle{S^{u}_{x}=\prod_{\ell\in{\rm
URB}}\sigma^{x}_{\ell}}$, (URB stands for the upper rough boundary), which is
a non-local stabilizer related to the topological-symmetry generators
Wildeboer2022 . (See supplemental material (SM) supp .) This stabilizer state
corresponds to one of the toric code ground states, the Hamiltonian of which
is shown shortly.

We use this stabilizer state as an initial state and perform local
$\sigma^{z}$-projective measurement for each qubit with probability $p$ except
for the dangling links on the upper and lower rough boundaries. We call
measurement pattern sample, and we average physical quantities over samples.
Practically, under the process of projective measurement in the stabilizer
formalism Gottesman1997 ; Nielsen_Chuang , $\sigma^{z}$-projective measurement
at link $\ell$ removes one of the star operators $\tilde{A}_{v}$ residing on
the boundary vertices of $\ell$ ($v_{1}$ and $v_{2}$), and then
$\sigma^{z}_{\ell}$ becomes a stabilizer as well as the product of the star
operators $(\tilde{A}_{v_{1}}\tilde{A}_{v_{2}})$. That is, the initial toric
code state tends to lose $\tilde{A}_{v}$’s depending on the probability $p$,
and the number of $\sigma^{z}$-stabilizer generator increases instead. It
should be also remarked that $(\tilde{A}_{v_{1}}\tilde{A}_{v_{2}})$ loses
$\sigma^{x}_{\ell}$ on the $\ell$ meeting $v_{1}$ and $v_{2}$, and has six
$\sigma^{x}$’s. This plays an important role for discussion on the bulk
transition.

Numerically, the measurement process is performed by the efficient numerical
simulation by the stabilizer formalism Gottesman1997 ; Aaronson2004 .
Throughout this study, we ignore the sign and imaginary factors of
stabilizers, which give no effects on observables that we focus on. Note that
the order of the local measurement is interchangeable since all measurement
operators $\\{\sigma^{z}_{\ell}\\}$ are commutative with each other.

View point from Hamiltonian formalism.— Our measurement process is to be
compared with the toric code with magnetic fields on cylinder, Hamiltonian of
which is

$\displaystyle H_{\rm TC}$ $\displaystyle=$
$\displaystyle\sum_{v}h_{x}\tilde{A}_{v}+\sum_{\ell\notin\mbox{rough}}J_{z}\sigma^{z}_{\ell}+\sum_{q}h_{z}\tilde{B}_{q}+\sum_{\ell}J_{x}\sigma^{x}_{\ell}.$

This toric code is a gauge fixing version of a lattice gauge-Higgs model as
explained in the SM supp , and it was recently investigated from novel point
of view Verresen2022 . The gauge-Higgs model on infinite system was analyzed
in Fradkin and Shanker Fradkin1979 , where the phase diagram with Higgs,
confinement and deconfined phases was discovered. Remarkable fact observed
there is that the Higgs and confinement phases are connected without any
thermodynamic singular behaviors. After that, numerical studies have been
performed Trebst2007 ; Tupitsyn2010 ; Wu2012 . For the spatial dimension
$D\geq 2$, the Higgs phase exists for $|J_{z}|>|J_{x}|,|h_{x}|,|h_{z}|$, as
the $J_{z}$-term is nothing but the hopping of the charged particle. For
$J_{x}=J_{z}=0$ and $h_{x},h_{z}<0$, the model is integrable and its ground
state is a doubly degenerate (See SM supp ), an exact ground-state multiplet.

As an important symmetry aspect, we can consider NLGIO symmetries for $H_{\rm
TC}$ with $J_{x}=0$. Two types of the symmetry are: (I)
$G_{lo,1}\equiv\prod_{\ell\in\Gamma_{0}}\sigma^{z}_{\ell}$ with an arbitrary
close loop $\Gamma_{0}$, (II)
$G_{lo,2}=\sigma^{z}_{\ell_{r1}}\biggl{[}\prod_{\ell\in\Gamma_{b}}\sigma^{z}_{\ell}\biggr{]}\sigma^{z}_{\ell_{r2}}$
where $\ell_{r1}$ and $\ell_{r2}$ are two arbitrary dangling links on the
upper rough boundary and the string in the bulk $\Gamma_{b}$ connecting the
$\ell_{r1}$ and $\ell_{r2}$ links MOFS . Both these operators satisfy $[H_{\rm
TC},G_{lo,1(2)}]=0$ for $J_{x}=0$. In particular, the second-type NLGIO
$G_{lo,2}$ and the charge play an important role in the subsequent study.
Another conserved charge is $\prod_{\ell\in{\rm rough}}\sigma^{x}_{\ell}$,
which is nothing but the global charge parity of the gauge-Higgs model, and is
an element of stabilizers of the initial state (See SMsupp .).

Initial state employed in the present numerical study is one of the exact
ground states stabilized by $S_{\rm int}$. The study in Ref. Verresen2022
discussed that by employing cylinder geometry, the Higgs phase can be
distinguished from the confinement phase, and the Higgs phase is a SPT state,
which is robust due to the magnetic one-form symmetry. Key point of this study
is to observe edge physics on the rough boundaries. For the large-$J_{z}$
limit, $\sigma^{z}_{\ell}$’s in the bulk are fixed as $\sigma^{z}_{\ell}=-1$,
and then the model $H_{\rm TC}$ reduces to an effective rough boundary
Hamiltonian similar to the transverse field Ising model (See SM supp ). Thus,
it was expected that spontaneous symmetry breaking (SSB) of $Z_{2}$ full spin-
flip symmetry ($S^{u}_{x}$) with a magnetic long-range order takes place as
the strength of the $\sigma^{z}$-term increases in $H_{\rm TC}$.

In the rest of this Letter, we shall investigate the above qualitative
expectation in our measurement protocol by the efficient stabilizer numerical
methods.

Numerical stabilizer simulation.— We study numerically the edge states on the
rough boundary emerging with long-range order. In this study, we fix $L_{y}=6$
and vary $L_{x}$.

We introduce the Edward-Anderson SG order parameter Sang2021  defined by
$\chi_{SG}=\frac{1}{L_{x}}\sum_{\ell_{x},\ell^{\prime}_{x}}C_{SG}(\ell_{x},\ell^{\prime}_{x})$
with
$C_{SG}(\ell_{x},\ell^{\prime}_{x})=\langle\psi|\sigma^{z}_{\ell_{x}}\sigma^{z}_{\ell^{\prime}_{x}}|\psi\rangle^{2}-\langle\psi|\sigma^{z}_{\ell_{x}}|\psi\rangle^{2}\langle\psi|\sigma^{z}_{\ell^{\prime}_{x}}|\psi\rangle^{2},$
where $|\psi\rangle$ is a stabilizer state and $\ell_{x}$’s are links on the
boundary. As a target observable, we observe the variance of $\chi_{SG}$
divided by $L_{x}$, $F_{v}(p,L_{x})\equiv\mbox{var}(\chi_{SG})/L_{x}$. This
quantity observes sample-to-sample fluctuations and is useful to search a
phase transition on the boundary.

Figure 2: (a) System-size dependence of $F_{v}$. (b) Scaling function. (c)
Correlation for $C_{SG}(\ell_{x},\ell^{\prime}_{x})$. Here, we chose
$\ell_{x}=0$, $L_{x}=72$ and $L_{y}=6$. We used following fitting functions:
$f_{1}(\ell_{x}^{\prime})=a\exp(-\ell_{x}^{\prime}/\xi)+b$ and
$f_{2}(\ell_{x}^{\prime})=a\ell_{x}^{\prime\eta}+b$, where $a$, $b$, $\xi$,
and $\eta$ are fitting parameters. Scatter plots represent numerical results
specified by probability $p$. We used $(3-4)\times 10^{3}$ samples for
(a)-(c).

In Ref. supp , we display the calculations of $\chi_{SG}/L_{x}$ as a function
of $p$ for various $L_{x}$, which exhibit smooth curves crossing with each
other at $p\sim 0.75$. Figure 2(a) shows calculations of $F_{v}(p,L_{x})$. We
find that the peak grows with increasing $L_{x}$ and the peak shifts toward
larger $p$ with increasing $L_{x}$. This is a typical continuous phase
transition behavior. This result indicates the existence of the boundary phase
transition. As $F_{v}(p,L_{x})$ shows, at $p\sim 0.75$, the fluctuations of
the SG order becomes simultaneously zero for all system sizes, and solid SG
order appears there. This phenomenon can be understood by the LGIO symmetry
$G_{lo,2}$ of $H_{\rm TC}$ and condensation of Higgs. As we show lately, the
SSB of the LGIO symmetry $G_{lo,2}$ is restored in this regime. The LGIO
symmetry $G_{lo,2}$ dictates for any states under consideration
$G_{lo,2}|\psi\rangle=\sigma^{z}_{\ell_{r1}}\biggl{[}\prod_{\ell\in\Gamma_{b}}\sigma^{z}_{\ell}\biggr{]}\sigma^{z}_{\ell_{r2}}|\psi\rangle\propto|\psi\rangle$.
If the condensation of Higgs boson takes place for length scale
$|\Gamma_{b}|$, $\langle\prod_{\Gamma_{b}}\sigma^{z}_{\ell}\rangle\neq 0$, we
have $\langle\sigma^{z}_{\ell_{r1}}\sigma^{z}_{\ell_{r2}}\rangle\neq 0$.
Therefore, the above numerical result indicates that perfect percolating Higgs
condensation takes place at $p\sim 0.75$.

In an intuitive picture, as $G_{lo,2}$ is an element of the stabilizer group,
the finite Higgs condensation
$\langle\prod_{\Gamma_{b}}\sigma^{z}_{\ell}\rangle\neq 0$ generates an
effective stabilizer such as $\sigma^{z}_{\ell_{r1}}\sigma^{z}_{\ell_{r2}}$.
In this sense, effective interactions on the boundary are induced by the bulk
Higgs condensation. We expect that as the above string $\Gamma_{b}$ in the
bulk is arbitrary, the critical value $p$ can decrease for larger $L_{y}$ as
the number of strings $\Gamma_{b}$’s participating the effective boundary
interactions increases. Similar mechanism works for enhancing the Higgs
condensation for larger $L_{y}$.

We identify the transition point and its criticality by applying the scaling
analysis to $F_{v}(p,L_{x})$ by means of pyfssa numerical package Pyfssa1 ;
Pyfssa2 . Here, the scaling ansatz is set as
$F_{v}(p,L_{x})=L_{x}^{\frac{\zeta}{\nu}}\Psi((p-p_{c})L_{x}^{\frac{1}{\nu}})$,
where $\Psi$ is a scaling function, $\zeta$ and $\nu$ are critical exponents
and $p_{c}$ is a critical transition probability in the thermodynamics limit.
We observe the clear data collapse as shown in Fig. 2(b). Here, the transition
probability is estimated by $p_{c}=0.757\pm 0.035$ and the exponents,
$\nu=2.54\pm 0.52$ and $\zeta=2.64\pm 0.34$. The above scaling analysis
indicates almost perfect SG order on the rough boundary for $p\geq 0.757$ for
all system sizes, including infinite $L_{x}$ peak_x=0 .

Figure 3: Partition for $S_{AF}$ [(a)] and $S_{AH}$ [(b)]. Sample-to-sample
variance for $S_{AF}$[(c)] and $S_{AH}$ [(d)]. (e) System-size dependence of
$S_{AH}$, where we averaged over all measurement samples. We used $3\times
10^{3}$ samples for (c), (d) and (e).

For further study, we observe the correlation function
$C_{SG}(\ell_{x},\ell^{\prime}_{x})$ for the $L_{x}=72$ and $L_{y}=6$ system
in the vicinity of the peak of $F_{v}(p,L_{x})$, which we denote
$p=p^{\prime}_{c}$. The results are displayed in Fig. 2 (c). For
$p>p^{\prime}_{c}$, a clear long-range SG order exists, as expected from the
above observation. On the critical point $p=p^{\prime}_{c}$, the correlator is
well-fitted by the power-law decay, estimated as
$\propto|\ell_{x}-\ell_{x}^{\prime}|^{\eta}$ with $\eta\sim 0.25$. Finally for
$p<p^{\prime}_{c}$, the result is well-fitted by the exponential decay,
estimated as $\propto\exp[-|\ell_{x}-\ell_{x}^{\prime}|/\xi]$ with $\xi\sim
3.79$, indicating a disordered phase. The analysis verifies the SG boundary
phase that is induced by the bulk measurement enhancing Higgs condensation. In
passing, we examine another fitting for $C_{SG}(\ell_{x},\ell^{\prime}_{x})$
at the criticality such as $e^{-\alpha|\ell_{x}-\ell_{x^{\prime}}|^{\beta}}$
Fisher1995 ; Fisher1999 . However, we find that the power-law fitting is
better than that.

We further investigate entanglement properties for our system. We focus on
entanglement entropy (EE) for a subsystem $A$ (the complement is subsystem
$B$) is considered. (We use logarithm with base $2$ for EE.) Here, we consider
two different partitions shown in Fig. 3(a) and 3(b). First partition is that
the subsystem $A$ is the upper rough boundary and its EE is denoted by
$S_{AF}$. In the second one, the subsystem $A$ is the half of the upper rough
boundary and its EE is denoted by $S_{AH}$. The EE can be calculated from the
number of linearly-independent stabilizers within a target subsystem $A$ and
the number of qubit of the subsystem $A$ Fattal2004 ; Nahum2017 .

The sample-to-sample variance of $S_{AF}$ divided by $L_{x}$,
$\mbox{var}[S_{AF}]/L_{x}$, is plotted in Fig. 3(c), and we find the peak and
no system-size dependence indicating a crossover. The EE $S_{AF}$ evaluates
the degree of separation between the bulk and the upper rough boundary. From
the data, the rough boundary is separated from the bulk for $p\gtrsim 0.45$,
smaller than the rough boundary transition point $p_{c}$. This behavior gives
an intuitive scenario that after the entanglement separation between the rough
boundary and bulk, the SG long-range order on the rough boundary starts to
develop.

We next observe the sample-to-sample variance of $S_{AH}$ divided by $L_{x}$,
$\mbox{var}[S_{AH}]/L_{x}$ plotted in Fig. 3(d) and find almost the same
behavior to $\mbox{var}[S_{AF}]/L_{x}$. From $S_{AH}$ plotted in Fig. 3(e), we
find that for small $p$, $S_{AH}\propto L_{x}$, and interestingly enough for
large $p$, $S_{AH}$ approaches unity, $S_{AH}=1$. It indicates two-fold
degenerate states on the upper rough boundary. We impose the parity operator
$S^{u}_{X}$ spanning over the upper rough boundary as a stabilizer Fattal2004
, and we perform measurements only bulk degrees of freedom, and therefore
$S^{u}_{X}=1$ in the whole process. The entanglement separation between the
rough boundary and bulk for large $p$ indicates that the rough boundary states
are a cat state preserving $S^{u}_{X}=1$, i.e., a superposition of the SSB
states. This phenomenon is similar to the two-fold degenerate $Z_{2}$ SSB
state in the well-studied transverse field Ising chain. The SSB indicated by
$S_{AH}\to 1$ is a signal of emergence of the Higgs=SPT phase.

Observation of bulk phase transition.— Final issue is to search a bulk
transition and clarify relation between the bulk and boundary transitions. To
this end, to investigate the size of $\tilde{A}_{v}$ clusters
$(\tilde{A}_{v_{1}}\tilde{A}_{v_{2}}\cdots)$ in the set of stabilizer
generator is quite useful. The Gauss law is destroyed at sites adjacent to
$\sigma^{z}$-measurements, whereas the stabilizer of each cluster comprises
$\sigma^{x}_{\ell}$’s on its boundary, and therefore every $\tilde{A}_{v}$
cluster measures its total inside charge. In the critical regime of the
transition from the deconfined to Higgs phases, we expect the scale
invariance, and emergence of various sizes of clusters. Simple consideration
reveals (frequency of clusters) $\propto$ (size of cluster)-1. Probability
distribution of size of clusters can be measured by the number of
$\sigma^{x}_{\ell}$ in the corresponding stabilizer generator.

The numerical results are displayed in Fig. 4. For small $p=0.30$, the size of
clusters is rather small, and for large $p=0.9$, all clusters have almost the
same magnitude. For that regime, we expect that large rectangular-shaped
clusters tend to emerge. In between, various size of clusters emerge, and the
$p=0.60$ system indicates scale-invariant. Therefore, we expect that a bulk
transition from deconfined (toric code) to Higgs phases takes place at
$p=p_{b}\simeq 0.60.$ Topological order of the initial state gets lost at
$p=p_{b}$, as the efficiency of the operator $\prod_{\rm
periodic}\sigma^{x}_{\ell}$ is destroyed by ‘voids’ inside $A_{v}$ clusters,
where ‘periodic’ denotes a closed loop on the dual lattice in the periodic
direction ($x$-direction) of the cylinder supp .

This result implies that the bulk transition observed in the present protocol
takes places earlier than the rough boundary phase transition indicated by the
peak of var$(\chi_{SG})/L_{x}$. We also observed the variance of the squared
expectation value of the local $\tilde{A}_{v}$, the results of which supports
the existence of bulk transition supp .

Figure 4: Histogram of the number of $\sigma_{x}$ in each generator of
stabilizers for $0.3\leq p\leq 0.9$. We omit the data point for zero number of
$\sigma^{x}$ and $S^{u}_{x}$.

Summarizing the results, we expect that the bulk phase transition takes places
first at $p_{b}$ and then the rough-boundary phase transition at $p_{c}$,
which is naturally regarded as the SPT transition. This observation indicates
that the regime $p_{b}<p<p_{c}$ is neither topological nor SPT. As a possible
physical picture, with increasing $p$, incoherent proliferation of e-anyon
(charge particle in the gauge-Higgs model) occurs first inducing breakdown of
the long-range entanglement at $p=p_{b}$, and then, further increase in $p$
results in the condensation of e-anyon with the boundary long-range order as
the SPT order. Similar physical picture has been shown in recent studies
Fan2023 ; Wang2023 . We obviously cannot deny the possibility that the above
two transitions take place simultaneously.

Conclusion.— This work clarified the bulk-measurement-induced boundary phase
transitions in the generalized toric code system on cylinder geometry with the
rough boundaries. Our numerical results elucidated the emergence of the SG
long-range order on the rough boundaries The criticality was estimated in
detail. The SSB with long-range order and the degeneracy on the rough
boundaries in our toric code system under projective measurement obviously
indicate that the Higgs=SPT phase is produced by the local measurement on the
bulk starting with the initial toric code (deconfined) state.

There are various interesting future directions. One of them is to study the
similar system that is closely related to a subsystem code Poulin2005 ;
Bacon2006 ; Wildeboer2022 .

Acknowledgements.— This work is supported by JSPS KAKENHI Grant Number
JP23K13026(Y.K.) and JP23KJ0360(T.O.).

## Supplemental material

###  .1 I. Toric code, lattice-gauge-Higgs model and its groundstate

We shall explain the relation of the toric code and gauge-Higgs model in two
spatial dimensions, especially under cylinder geometry. Accessible reviews of
the lattice gauge theory (LGT) are Refs. Kogut1979 ; IchinoseMatsui2014 .

The ancestor model of $H_{TC}$ in the main text is a (2+1)-D $Z_{2}$ extended
gauge-Higgs model, Hamiltonian of which is given by

$\displaystyle H_{\rm GHM}$ $\displaystyle=$
$\displaystyle\sum_{v}h_{x}X_{v}+\sum_{(q,q^{\prime})}J_{x}X_{q}\sigma^{x}_{q,q^{\prime}}X_{q^{\prime}}$
(1) $\displaystyle+$
$\displaystyle\sum_{q}h_{z}Z_{q}+\sum_{(v,v^{\prime})}J_{z}Z_{v}\sigma^{z}_{v,v^{\prime}}Z_{v^{\prime}},$

where $X$ and $Z$ are Pauli operators defined on $q$\- and $v$-lattice sites.
The first and second terms are a $Z_{2}$ matter chemical potential on vertices
and its gauge coupling term, the third and fourth terms are $Z_{2}$ matter
chemical potential on plaquette sites and its gauge coupling term. The
schematic image of the system and each terms are shown in Fig. 5.

Figure 5: Schematic figure of an extended lattice gauge-Higgs model with rough
boundaries. The lattice in the $x$-direction is periodic and in $y$-direction
is open. The red dashed lines indicate the regime upper and lower boundaries

Here, the two gauge-invariant conditions, the Gauss laws, for the physical
subspace, are given by

$\displaystyle
A_{v}|\psi\rangle=|\psi\rangle,\>\>B_{q}|\psi\rangle=|\psi\rangle.$ (2)

where

$\displaystyle A_{v}=X_{v}\prod_{\ell_{v}\in v}\sigma^{x}_{\ell_{v}}\equiv
X_{v}\tilde{A}_{v},$ $\displaystyle B_{q}=Z_{q}\prod_{\ell_{q}\in
q}\sigma^{z}_{\ell_{q}}\equiv Z_{q}\tilde{B}_{q},$ (3)

and $\ell_{v}\in v$ stands for links emanating from vertex (site) $v$, and
$\ell_{q}\in q$ for links composing plaquette (box) $q$. The $Z_{2}$-electric
matter is defined on each vertex $v$, $(X_{v},Z_{v})$, and its magnetic dual,
$(X_{q},Z_{q})$, on each dual vertex $q$ (i.e., plaquette of the original
lattice). On the other hand, the $Z_{2}$ gauge field is defined on links and
denoted by $(\sigma^{x}_{\ell},\sigma^{z}_{\ell})$,
$\sigma^{z}_{v,v^{\prime}}$ denote a gauge variable on link connecting
neighboring vertices $v$ and $v^{\prime}$, and $\sigma^{x}_{q,q^{\prime}}$
denote a gauge variable on link connecting neighboring dual vertices $q$ and
$q^{\prime}$.

There are two differences between the system given by $H_{\rm GHM}$ [Eq. (1)]
and the ordinary $Z_{2}$ gauge-Higgs LGT Fradkin1979 :

  1. (I)

Dual matter field couples with the gauge field and the coupling term,
$X_{q}\sigma^{x}_{q,q^{\prime}}X_{q^{\prime}}$ is added to the Hamiltonian
besides the ordinary $Z_{2}$ electric matter-gauge coupling. This $Z_{2}$
degrees of freedom residing on each plaquette $q$, $(X_{p},Z_{p})$,
corresponds to ‘particle’ carrying magnetic flux (magnetic charge), and its
hopping induces fluctuation of the gauge field and, therefore, confinement of
electric charges.

  2. (II)

The model has an an additional local gauge symmetry by the presence of the
magnetic-charge degrees of freedom,

$X_{q}\rightarrow X_{q}V_{q},\;\;\sigma^{x}_{q,q^{\prime}}\rightarrow
V_{q}\sigma^{x}_{q,q^{\prime}}V_{q^{\prime}},\;\;V_{q},V_{q^{\prime}}\in
Z_{2},$ (4)

and we impose additional Gauss-law constraint on the physical state, Eqs. (2)
and (3). One may wonder that the system (1) reduces to the ordinary gauge-
Higgs model by ‘integrating out’ the magnetic charge degrees of freedom via,
e.g., employing unitary gauge of the second local gauge symmetry in Eq. (4).
In fact as we show shortly, disentangling of electric and magnetic particles
generates the ordinary Hamiltonian of the gauge-Higgs model in unitary gauge.

Here, we comment that if we introduce rough and smooth boundaries on the above
Hamiltonian, it was recently proposed for describing subsystem quantum code
producing fault-tolerant qubit Wildeboer2022 , although the explicit form of
$H_{\rm GHM}$ in Eq. (1) was not shown. There, each term of the Hamiltonian
(1) is categorized as “gauge” operator in the subsystem-code literature
Poulin2005 . In this work, we focus only a cylinder geometry with rough
boundary as shown in Fig. 5. The model $H_{GHM}$ with the above open boundary
conditions is known to have two important global topological symmetries
Wildeboer2022 , generators of which are given by,

$\displaystyle P$ $\displaystyle=$
$\displaystyle\prod_{v}X_{v},\>\>S_{Z}=\prod_{q}Z_{q},$ (5)

$P$ is the parity of the total electric charge, corresponding to the global
spin flip on each vertex, and similarly, $S_{Z}$ is the parity of the total
magnetic flux per plaquette.

Furthermore, as for Ref. Verresen2022  there are also two symmetries,

$\displaystyle W_{\gamma}$ $\displaystyle=$
$\displaystyle\prod_{\ell\in\gamma}\sigma^{z}_{\ell},\>\>H_{\gamma}=\prod_{\ell\in\gamma}\sigma^{x}_{\ell}.$
(6)

$W_{\gamma}$ and $H_{\gamma}$ are called one-form symmetries, which have been
extensively studied recently Gaiotto2015 ; McGreevy2022 . Although the $J^{x}$
and $J^{z}$ terms in the Hamiltonian $H_{\rm GHM}$ [Eq. (1)] explicitly break
the one-form symmetries, it was shown that the higher-form symmetry is
generally robust and give non-trivial effect on dynamics of the system
Verresen2022 . In the $(2+1)$-D system, $H_{\gamma}$ can be regarded as ’t
Hooft loop (string) dual to Wilson loop (string) $W_{\gamma}$. In particular,
the path $\gamma$ in the one-form symmetry $W_{\gamma}$ ($H_{\gamma}$) is
arbitrary. It is easily verified that by using the Gauss laws in Eq. (3), $P$
is expressed as a ’t Hooft ‘loop’ residing on the upper and lower rough
boundaries. From the view points of these symmetries, the Higgs phase was
discussed in Verresen2022 , the Higgs phase is SPT phase from the perspective
of these symmetries. In this study, however, we focus on related but slightly
different symmetry operators LGIO’s as introduced in the main text.

Figure 6: Schematic image of phase diagram of the Hamiltonian $H_{TC}$ on the
cylinder geometry. We assume $h^{x}$ and $h^{z}$ are some constant. The dashed
arrow represents our target measurement protocol related to the parameter
sweep in the Hamiltonian formalism.

Now we show that the model $H_{\rm GHM}$ is related to a generalized toric
code. We consider the following unitary transformations Wildeboer2022 ;
Verresen2022 :

$\displaystyle U_{v}$ $\displaystyle=$ $\displaystyle
H\biggl{(}\prod_{v}\prod_{\ell\in v}(CZ)_{v,\ell}\biggr{)}H,$ (7)
$\displaystyle U_{q}$ $\displaystyle=$ $\displaystyle
H\biggl{(}\prod_{q}\prod_{\ell\in q}(CZ)_{q,\ell}\biggr{)}H,$ (8)

where $H$ is the Hadamard transformation on each link and $(CZ)_{i,j}$ is a
controlled $Z$-gate for the site $i$ and link $j$. Applying the above
transformation to $H_{\rm GHM}$, we obtain the following effective
disentangled model

$\displaystyle U_{v}U_{q}H_{\rm GHM}(U_{v}U_{q})^{\dagger}\equiv H_{\rm TC},$
$\displaystyle H_{\rm TC}$ $\displaystyle=$
$\displaystyle\sum_{v}h_{x}\tilde{A}_{v}+\sum_{q}h_{z}\tilde{B}_{q}+\sum_{\ell\notin\mbox{rough}}J_{z}\sigma^{z}_{\ell}+\sum_{\ell}J_{x}\sigma^{x}_{\ell}.$

The above unitary disentanglement corresponds to gauge fixing with unitary
gauge. Equation (LABEL:HTC2) is the Hamiltonian shown in the main text.

Phase diagram of the Hamiltonian (LABEL:HTC2) has been investigated in detail
Trebst2007 ; Tupitsyn2010 ; Wu2012 ; Verresen2022 . There are three phases,
deconfined (toric code), Higgs, and confined phases, but with periodic
boundary conditions, the Higgs and confinement phases are connected
Fradkin1979 . The schematic image is shown in Fig. 6. Here, if the cylinder
geometry is considered, the Higgs and confinement phases can be distinguished
as noted in the main text. As for Verresen2022 , in the deep Higgs phase
($J_{z}\to\infty$), $\langle\sigma^{z}_{\ell}\rangle=-1$ in the bulk, then the
Hamiltonian can be reduced to a 1D transverse field Ising model on the rough
boundary,

$\displaystyle
H_{rb}=\sum_{\langle\ell_{x},\ell^{\prime}_{x}\rangle}h_{z}\sigma^{z}_{\ell_{x}}\sigma^{z}_{\ell^{\prime}_{x}}+\sum_{\ell_{x}}J_{x}\sigma^{x}_{\ell_{x}},$
(10)

where $\ell_{x}$ is a dangling link on the (upper or lower) rough boundary,
$\langle\ell_{x},\ell^{\prime}_{x}\rangle$ represents nearest-neighbor pairs
of the dangling links on the rough boundary. This Hamiltonian $H_{rb}$
exhibits the SSB for $|h_{z}|>|J_{x}|$. Thus, the Higgs phase can be
distinguished from the other two phases by observing the boundary SSB.

We comment on the ground-state property of the model $H_{\rm TC}$ on the
cylinder. For $J_{x}=J_{z}=0$ (toric code phase), the model $H_{\rm TC}$ is
exactly solvable but its state degeneracy is different from the system with
the conventional torus geometry. By using the stabilizer
languageNielsen_Chuang , we count the number of the independent stabilizer.
Here, as to the plaquette operator $\tilde{B}_{q}$, one constraint exists
$\prod_{q\in all}B_{q}=1$, thus there is $L_{x}L_{y}-1$ independent
stabilizers of the form of $\tilde{B}_{q}$, denoted by a set of generator of
stabilizer, $\\{\tilde{B}_{q}|q\in\mbox{all plaquette but}\ q_{0}\\}$, where
$q_{0}$ is one of single $q$-lattice sites to avoid the redundancy. On the
other hand, as to the star operator $\tilde{A}_{v}$, there is $L_{x}(L_{y}-1)$
independent stabilizer generators since $\prod_{v\in all}\tilde{A}_{v}\neq 1$.
From these facts, the total $2L_{x}L_{y}-L_{x}-1$ linear independent
stabilizers exist and from the total number of link qubit, $L_{x}(2L_{y}-1)$,
the degeneracy is $2^{(L_{x}(2L_{y}-1))-(2L_{x}L_{y}-L_{x}-1)}=2$. That is,
two-fold degeneracy appears for the entire energy levels.

In the exact toric code phase for $J_{x}=J_{z}=0$, two operators $\prod_{\rm
periodic}\sigma^{x}_{\ell}$ and $\prod_{\rm cross}\sigma^{z}_{\ell}$ commute
with $H_{\rm TC}$ and anti-commute with each other, where ‘periodic’ denotes a
closed loop on the dual lattice in the periodic direction ($x$-direction) of
the cylinder and ‘cross’ a string crossing the cylinder from the upper to
lower rough boundaries. The two-fold degeneracy can be understand by the
existence of them, and therefore it is expected that this degeneracy is robust
even for local perturbations with finite $J_{x}$ and $J_{z}$ as long as the
system is in the toric-code phase.

Furthermore, we focus on the two-fold degenerate ground states of $H_{\rm TC}$
with $J_{x}=J_{z}=0$ in the toric code phase denoted by $|\psi_{G1}\rangle$
and $|\psi_{G2}\rangle$. These states can be distinguished by a string
operator $S^{u}_{x}$ as
$S^{u}_{x}|\psi_{G1(2)}\rangle=+1(-1)|\psi_{G1(2)}\rangle$. The operator
$S^{u}_{x}$ is linearly independent of all $\tilde{A}_{v}$’s and
$\tilde{B}_{q}$’s and also commute to them, thus it can be a linearly
independent additional stabilizer. In our protocol with projective
measurements, we pick up the state $|\psi_{G1}\rangle$ as an initial state
corresponding to the exact toric code phase.

The above discussion is in the Hamiltonian formalism. Change of measurement
probability in our protocol with $\sigma^{z}$-projective measurement is to be
related to the red straight parameter line in the phase diagram of $H_{TC}$
shown in Fig. 6. In practice, we first prepare the deep toric code state
$|\psi_{G1}\rangle$ and then apply $\sigma^{z}$-projective measurements with
the probability $p$ for each qubits on links. This can correspond to varying
the parameter $J_{z}$ in the Hamiltonian picture of $H_{TC}$ with $J_{x}=0$.
Thus, the case of large probability $p$ is related to the case of large
$J_{z}$ in the Hamiltonian $H_{TC}$. Thus, appearing physical phenomena in our
protocol with projective measurements can give useful insight into the
physical phenomena observed in the Hamiltonian $H_{TC}$.

###  .2 II. Additional numerical results

We show additional numerical results for the measured toric code system with
cylinder geometry by using the efficient numerical stabilizer algorithm.

Figure 7: Behavior of the average value of $\chi_{SG}/L_{x}$. The right-blue
shaded line represents the crossing point of the different system size data.
We used $3\times 10^{3}$ samples. We fix $L_{y}=6$.

###  .3 Behavior of $\chi_{SG}$

We give additional numerical results for Edward-Anderson SG order. Figure 5 is
the result of the value of $\chi_{SG}/L_{x}$ averaged over measurement samples
with various system sizes $L_{x}$ for $L_{y}=6$.

Data for the different $L_{x}$ obviously cross with each other at single point
$p\sim p_{c}$, where $pc$ is the estimated critical value by the scaling
analysis of the variance of $\chi_{SG}$. The result implies that for larger
$L_{x}$, the value of $\chi_{SG}/L_{x}$ increases more rapidly with increasing
$p$. This behavior supports the existence of the rough-boundary phase
transition.

Figure 8:  $L_{y}$ dependence of spin glass order $\chi_{SG}/L_{x}$: (a)
$L_{y}=2$, (a) $L_{y}=3$, and (c) $L_{y}=4$. Each inset panel represents
$F_{v}$. We used $4\times 10^{3}$ samples.

###  .4 $L_{y}$-dependence of rough-boundary phase transition

We observe effects of the system size in the $y$-direction, $L_{y}$, to the
rough-boundary phase transition. The $L_{y}$-dependence of $\chi_{SG}$ are
displayed in Figs. 8(a)-8(c) for $L_{y}=2,3$ and $4$. For all values of
$L_{y}$, the value of $\chi_{SG}/L_{x}$ increases more rapidly as a function
of $p$ for larger $L_{x}$. Furthermore, for $L_{y}=2$ [Fig. 8(a)], we observe
that the data of the various $L_{x}$’s do not cross with each other except
$p=1$. This indicates non-existence of boundary phase transition for a finite
$p$, while for larger $L_{y}$’s, the crossing point shifts to the lower value
of $p$ as shown in Figs. 8(b) and 8(c). Also, the insets of each panels in
Figs. 8(a)-8(c) show the behavior of $F_{v}$. These results indicate the
existence of the continuous phase transition on the rough boundary. However
for the $L_{y}=2$ case [Fig. 8(a)], the peak of $F_{v}$ approaches $p=1$,
indicating that the transition point approaches $p=1$ for the thermodynamics
limit.

These numerical results indicate that the SG-ordered phase itself disappears
as decreasing $L_{y}$, corresponding to (1+1)-D system. The reason is that in
the Hamiltonian formalism, (1+1)-D gauge-Higgs model with periodic boundary
conditions has the unique ground state in the ‘Higgs regime’ Kogut1979 ;
Verresen2022 , and the model is nothing but a specific limit of the cluster
spin model with periodic boundary conditions (i.e., no boundaries in the
$x$-direction.) Our numerical result is consistent with the consideration by
the Hamiltonian formalism.

###  .5 Entanglement entropy $S_{AF}$

Figure 9: System size dependence of $S_{AF}$, where we averaged over all
measurement samples. We used $3\times 10^{3}$ samples. We fix $L_{y}=6$.

We give further additional data, the average of entanglement entropy $S_{AF}$
(defined in the main text) over the measurement samples. The result is plotted
in Fig. 9. For small $p$, $S_{AF}\propto L_{x}$ same to the case of $S_{AH}$.
For large $p$, $S_{AF}$ approaches zero, indicating the complete entangle
separation between the bulk and the rough boundary. As increasing $p$, this
entanglement separation appears late from the formation of the bulk Higgs
phase estimated by $\mbox{var}[\bar{A}]$ in the main text.

###  .6 Variance of squared $\tilde{A}_{v}$

We further present numerical study about the expectation value of squared star
operator in our protocol,
${\bar{A}}=\frac{1}{N_{v}}\sum_{v}\langle\psi|\tilde{A}_{v}|\psi\rangle^{2}$,
where $N_{v}=L_{x}(L_{y}-1)$. In our protocol, the local
$\sigma^{z}$-measurement removes the initial $\tilde{A}_{v}$’s, and thus, its
number monotonically decreases for larger $p$. In the toric code, this means
the lack of the Gauss law and discontinuity of electric flux. Then, we
calculate the sample-to-sample variance of it, $\mbox{var}[{\bar{A}}]$. Its
system-size and $p$-dependence are plotted in Fig. 10.

Figure 10: The sample-to-sample variance of ${\bar{A}}$. The inset displays
the $L\cdot{\bar{A}}$. We used $3\times 10^{3}$ samples. We fix $L_{y}=6$.

The moderate peaks emerge and their height is smaller for larger $L_{x}$. The
reason for it is simple: Since the variance is calculated from the value of
$\langle\psi|\tilde{A}_{v}|\psi\rangle^{2}$ averaged for all vertices $v$ in
each sample, the variance of the mean is inversely proportional to $L_{x}$
although these values are much small. In fact as shown in the inset of Fig.
10, $\mbox{var}[{\bar{A}}]\times L_{x}$ exhibits no system-size dependence.
This behavior can support the presence of bulk phase transition as explained
by observing the scale-free distribution of the size of the product of
$\tilde{A}_{v}$ shown in Fig. 4 in the main text. The value of $p$ on the
moderate peak of $\mbox{var}[{\bar{A}}]\times L_{x}$ is estimated as $p\sim
0.14$. This value is small compared to $p_{b}$ and $p_{c}$. This result
implies that after weak fluctuation of ${\bar{A}}$, the scale free phenomena
for the size of stabilizer constituted by the product of $\tilde{A}_{v}$
(e-anyon cluster proliferation) emerges.

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# On invariants and equivalence of differential operators under Lie
pseudogroups actions

Valentin Lychagin & Valeriy Yumaguzhin

###### Abstract.

In this paper, we study invariants of linear differential operators with
respect to algebraic Lie pseudogroups. Then we use these invariants and the
principle of n-invariants to get normal forms (or models) of the differential
operators and solve the equivalence problem for actions of algebraic Lie
pseudogroups. As a running example of application of the methods, we use the
pseudogroup of local symplectomorphisms.

##  1\. Introduction

The counterpoint and polyphony play the crucial role, not only in music and
art, but also in science and mathematics.

So, for example, if we take the theory of invariants, then at the very
beginning to study invariants of binary forms Sylvester J.J. proposed the
construction of invariants by transvectants ([23],[18]) and Gordan P.([3]) not
only proved that the algebra of polynomial invariants of binary forms is
finitely generated, but also shown that invariants might be found by
transvectants, that are concrete differential operators.

It explains Gordon’s denial of Hilbert’s proof of the finiteness of the
invariant algebras.

In the previous publications ([10],[9],[11]), we discussed applications of
differential invariants to finding of the algebraic ones. It is related to
differences between Hilbert-Rosenlicht ([21]) and Lie-Tresse theorems ([7]),
describing the structures of algebras of algebraic and differential
invariants. This interplay between the algebraic and differential methods
seems extremely fruitful, especially in application to differential operators.

It is also worth to note that the number of basic invariants, needed to
describe regular orbits, essentially different for algebraic and differential
invariants. Thus, the Rosenlicht theorem requires the number of algebraic
invariants equals to the codimension of the regular orbit, that exponentially
grows with the degree of algebraic forms under consideration. On the other
hand, the Lie-Tresse theorem, or the n-invariant principle ([1]), requires
essentially fewer invariants. Thus, in description of linear scalar
differential operators, we need only n, equals to the dimension of the base,
invariants.

In this paper, we study orbits of scalar linear (and some class of non-linear)
differential operators under action of a Lie pseudogroups, the case of the
complete pseudogroup of all local diffeomorphisms was studied in ([14]).

We reverse, in this case, the interplay between algebraic and differential
invariants and show that to find the necessary number of differential
invariants we may use algebraic invariants of symbols of operators. Because
the number of these invariants restricted to the dimension of the base, we use
here transvectants.

The paper is organized as follows. At first, we remind the basics on Lie
pseudogroups of transformations and their invariants. Then we discuss the
principle of n-invariants and models of linear differential operators built on
invariants, being in general position. This allows us to get a solution of the
equivalence problem for differential operators with respect to a given Lie
pseudogroup of transformations. We illustrate this approach on the Lie
pseudogroup of symplectomorphisms. To this end, we introduce the symplectic
transfectants. They allow us to construct symplectic invariants of
differential operators, by using the symbols of operators only.

However, there is a class of operators where symplectic invariants of symbols
are never in general position. These operators are characterized by a
requirement that symplectic orbits of their symbols are regular and constant.
We call such operators as constant type operators ([12]).

With such types of operators, having order $k>2$ , we associate a unique
affine connection that is symplectic, i.e. preserves the symplectic structure
form, as well as the symbol. These connections we call Wagner connections
([24]), who discovered such connections associated with cubic symmetric
tensors on 2-dimensional manifolds. These connections have zero curvature, but
non-trivial torsion. We use these connections to split (or quantize) the
differential operators into the sum of symmetric tensors, and get additional
invariants by pure algebraic methods. For differential operators of the second
order, we use the Levi-Civita connection to split the differential operator
and find additional invariants.

##  2\. Lie pseudogroups and Lie equations

By a pseudogroup $\mathcal{P}$, acting on a connected manifold $M$, we mean
([22]) a collection of local diffeomorphisms, i.e. diffeomorphisms of open
sets in $M$, closed under:

  1. (1)

restrictions on open domains: $\phi\in\mathcal{P}$, $U\subset M$\- open, then
$\phi|_{U}\in\mathcal{P}$; if $\phi|_{U}\in\mathcal{P}$ for any open domain
$U\subset{\rm dom}(\phi)$, then $\phi\in\mathcal{P}$,

  2. (2)

composition: $\phi,\psi\in\mathcal{P}$, then $\phi\circ\psi\in\mathcal{P}$, if
${\rm dom}\left(\phi\right)\subset{\rm im}\left(\psi\right)$,

  3. (3)

inverse: $\phi\in\mathcal{P}\Rightarrow\phi^{-1}\in\mathcal{P}$, and ${\rm
id}_{M}\in\mathcal{P}$.

The action is said to be transitive, if for any $a,b\in M$ , there exists
$\phi\in\mathcal{P},$ such that $\phi\left(a\right)=b,$ and the action is said
to be primitive, if there are no $\mathcal{P}$ \- invariant foliations on $M.$

Let $\delta_{k}\colon D_{k}\left(M\right)\rightarrow M$ be the bundles of
$k$-jets of local diffeomorphisms. Fibres $\delta_{k}^{-1}\left(a\right),$
where $a\in M,$ consist of $k$-jets $[\phi]_{a}^{k}$ of local diffeomorphisms
$\phi$ at the point.

We have the tower of bundles

$\rightarrow\\!D_{k}(M)\overset{\delta_{k,k-1}}{\rightarrow}D_{k-1}(M)\rightarrow\cdots\rightarrow\\!D_{2}(M)\overset{\delta_{2,1}}{\rightarrow}D_{1}(M)\overset{\delta_{1,0}}{\rightarrow}M\times
M\overset{\delta}{\rightarrow}M,$

where mappings
$D_{k}\left(M\right)\overset{\delta_{k,k-1}}{\rightarrow}D_{k-1}\left(M\right)$
are the reductions of $k$-jets $[\phi]_{a}^{k}$ to the $\left(k-1\right)$-jets
$[\phi]_{a}^{k-1}.$

Given a pseudogroup $\mathcal{P},$ we define $G_{k}\subset
D_{k}\left(M\right)$ to be the following subset,

$G_{k}=\left\\{[\phi]_{a}^{k},\phi\in\mathcal{P},a\in{\rm
dom}\left(\phi\right)\right\\}.$

We say that the pseudogroup $\mathcal{P}$ is regular if all $G_{k}$ are smooth
submanifolds in $D_{k}\left(M\right)$ and $\delta_{k}:G_{k}\rightarrow M$ are
smooth subbundles of bundles $\delta_{k}:D_{k}\left(M\right)\rightarrow M.$

Remark, that the composition in the pseudogroup induces partially defined
composition and defines a Lie algebroid structure in $G_{k}.$ Taking fibres
$G_{k}$ at points $a\times a$ of the diagonal $\Delta\subset M\times M,$ we
get subgroups $G_{k}\left(a\right)$ of the differential groups
$D_{k}\left(a\right),$ formed by $k$-jets $[\phi]_{a}^{k}$ of diffeomorphisms
$\phi\in\mathcal{P},$ such that $\phi\left(a\right)=a.$

We say that a regular pseudogroup $\mathcal{P}$ is a Lie pseudogroup if the
tower

$\rightarrow
G_{k}\overset{\delta_{k,k-1}}{\rightarrow}G_{k-1}\overset{\delta_{k-1,k-2}}{\rightarrow}G_{k-2}\rightarrow\cdots\rightarrow
G_{1}\overset{\delta_{1,0}}{\rightarrow}M\times
M\overset{\delta}{\rightarrow}M,$ (1)

is the tower of smooth bundles, and it defines formally integrable equation
$G$ (called Lie equation, see [8] for more details) in the sense that the
first prolongations $G_{k}^{\left(1\right)}\subset G_{k+1}$, for all $k\geq
1.$

It also follows from the Cartan-Kuranishi prolongation theorem that in this
case there is a number $k_{0},$ called the order of the pseudogroup
$\mathcal{P},$ such that $i$-th prolongations $G_{k_{0}}^{\left(i\right)}\ \
$of $G_{k_{0}}$ coincide with $G_{k_{0}+i},$ for all $i\geq 0.$

In other words, elements of $\mathcal{P}$ are solutions of differential
equation $G_{k_{0}},$ having order $k_{0}.$

Moreover, the Cartan–Kähler theorem states that in the analytical case, i.e.
in the case when all manifolds and mappings under consideration are analytic,
the elements of pseudogroup $\mathcal{P}$ are precisely all local solutions of
equation $G_{k_{0}}.$

This restriction on $\mathcal{P}$ we’ll use only once in the paper (and we’ll
indicate it) so in the rest of the paper the pseudogroup will be a pseudogroup
of smooth local diffeomorphisms.

It is also worth to note that fibres of the bundles $D_{k}\left(M\right)$ have
the intrinsic structure of algebraic manifolds and the prolongations of local
diffeomorphisms of the manifold $M$ to these bundles are algebraic too.

In particular, the differential groups $D_{k}\left(a\right)$ are algebraic as
well as their actions.

We say, (see [7], for more details), that the Lie pseudogroup $\mathcal{P}$ is
algebraic if $G_{k}\subset D_{k}\left(M\right)$ inherit this algebraicity,
i.e. fibres $G_{k}$ are algebraic submanifolds in the fibres
$D_{k}\left(M\right)$ and the composition law is also algebraic.

In particular, groups $G_{k}\left(a\right)$ are algebraic as well as their
actions.

Remark also, that this notion of algebraicity could also be applied to jet
bundles $\pi_{k}:J^{k}\left(\pi\right)\rightarrow M$ of sections of any smooth
bundle $\pi:E\left(\pi\right)\rightarrow M.$ Namely, prolongations of point
transformations (i.e. local diffeomorphisms of $E\left(\pi\right)$ ) into
fibres of projections $\pi_{k,0}:J^{k}\left(\pi\right)\rightarrow
E\left(\pi\right)$ are rational in the standard jet coordinates, and therefore
the algebraic structure of these fibres induced by the choice of the standard
jet coordinates.

We also will call formally integrable differential equations
$\mathcal{E\subset}J^{k}\left(\pi\right)$ algebraic if the bundle
$\pi:E\left(\pi\right)\rightarrow M$ is natural and algebraic, i.e. fibres of
$\pi$ have the structure of irreducible algebraic manifolds that is invariant
under the action of the pseudogroup of local diffeomorphisms of $M$ and fibres
of the projection $\pi_{k,0}:\mathcal{E\rightarrow}E\left(\pi\right)$ are
irreducible algebraic manifolds too.

In such a setting, we define ([7]) rational differential
$\mathcal{P}-$invariants of order $\left(k+l\right)$ as
$\mathcal{P}-$invariant rational functions on the $l$-th prolongation
$\mathcal{E}^{\left(l\right)}\mathcal{\subset}J^{k+l}\left(\pi\right).$

Because we assumed that the $\mathcal{P}$ \- action is transitive on manifold
$M$, these invariants are completely defined by their values on the fibre
$\mathcal{E}_{b}^{\left(l\right)}\mathcal{\subset}J_{b}^{k+l}\left(\pi\right)$
over a fixed base point $b\in M.$

Thus, rational differential $\mathcal{P}-$invariants of order
$\left(k+l\right)$ form a field $\mathcal{F}_{l}$ that, due to the Rosenlicht
theorem ([21]), separates regular $G_{k+l}\left(b\right)$ \- orbits in
$\mathcal{E}_{b}^{\left(l\right)}$ and the transcendence degree of
$\mathcal{F}_{l}$ equals to the codimension of the regular
$G_{k+l}\left(b\right)-$ orbits.

Moreover, the Lie -Tresse theorem ([7]) states that the field of all rational
differential $\mathcal{P}-$invariants $\mathcal{F}_{\ast}$ is generated by a
finite number of $\mathcal{P}-$invariants and $\mathcal{P}-$invariant
derivations (usually, they are Tresse derivations, associated with a set of
rational differential $\mathcal{P}$-invariants, being in general position).

##  3\. Linear differentiial operators and their $\mathcal{P}$-invariants

In the paper we use the following notations, the
$C^{\infty}\left(M\right)$-modules of smooth sections of vector bundles
$\pi:E\left(\pi\right)\rightarrow M$ will be denoted by
$C^{\infty}\left(\pi\right).$

By $\pi_{k}:J^{k}\left(\pi\right)\rightarrow M$ we denote the bundles of
$k$-jets of sections of the bundle $\pi$, and by
$\pi_{k,l}:J^{k}\left(\pi\right)\rightarrow J^{l}\left(\pi\right),k>l,$ the
natural projections.

Let $\psi_{k}\colon\mathit{Diff}_{k}(M)\rightarrow M$ be the bundle of linear
scalar differential operators of order $\leq k.$

Then $C^{\infty}\left(\psi_{k}\right)={\rm Diff}_{k}\left(M\right)$ is the
module of linear scalar differential operators on $M,$ having order $\leq k.$

The following exact sequences of the modules will play the crucial role

$0\rightarrow{\rm Diff}_{k-1}\left(M\right)\rightarrow{\rm
Diff}_{k}\left(M\right)\overset{{\rm
smbl}}{\rightarrow}\Sigma_{k}\left(M\right)\rightarrow 0.$ (2)

Here $\Sigma_{k}\left(M\right)$ is the module of symmetric vector fields of
degree $k$, $\Sigma_{k}(M)=C^{\infty}(\tau_{k})$, where
$\tau_{k}:S^{k}\left(TM\right)\rightarrow M$ is the $k$-th symmetric power of
the tangent bundle and the ${\rm smbl}$-map sends operators $A\in{\rm
Diff}_{k}\left(M\right)$ to their symbols
$\sigma_{A}\in\Sigma_{k}\left(M\right)$.

Remark, that any Lie pseudogroup $\mathcal{P}$ of local diffeomorphisms of $M$
acts also on all these bundles.

Rational functions on manifolds $J^{l}\left(\psi_{k}\right)$ or
$J^{l}\left(\tau_{k}\right),$ that are invariant with respect to the prolonged
$\mathcal{P}$-actions, will be called rational $\mathcal{P}$-invariants of
differential operators or $k$-symmetric vector fields.

Because of transitivity $\mathcal{P}$-action on $M$ these invariants are
defined by their values on the fibres $J_{b}^{l}\left(\psi_{k}\right)$ or
$J_{b}^{l}\left(\tau_{k}\right)$ at a fixed basic point $b\in M.$

We denote by $\mathcal{F}_{k,l},$ or simpler $\mathcal{F}_{l}$ ,if the order
of operators under consideration is fixed, the field of rational $\mathcal{P}$
-invariants of order $\leq l,$ and by $\mathcal{F}_{k,l}^{\sigma}\ $ we denote
the field of rational $\mathcal{P}$-invariants of order $\leq l$ of the
symmetric vector fields.

Exact sequence (2) shows that $\mathcal{F}_{k,l}^{\sigma}$ are subfields of
$\mathcal{F}_{k,l}$ and this gives us an option to get
$\mathcal{P}$-invariants of differential operators in pure algebraic way.

In addition, we have the natural universal differential operator ([14])

$\displaystyle\square$ $\displaystyle:$ $\displaystyle
C^{\infty}\left(J^{\infty}\left(\psi_{k}\right)\right)\rightarrow
C^{\infty}\left(J^{\infty}\left(\psi_{k}\right)\right),$
$\displaystyle\square$ $\displaystyle:$ $\displaystyle
C^{\infty}\left(J^{l}\left(\psi_{k}\right)\right)\rightarrow
C^{\infty}\left(J^{l+k}\left(\psi_{k}\right)\right),l\geq 0,$

that allow us the extend the set of available $\mathcal{P}$-invariants.

This operator is natural (and thus it is $\mathcal{P}$-invariant), i.e.
commutes with prolongations of diffeomorphisms of $M,$ and, therefore, defines
maps

$\square:\mathcal{F}_{k,l}\rightarrow\mathcal{F}_{k,k+l},$

for $l\geq 0.$

Remark (see [14]), that in standard jet coordinates
$\left(x_{1},..,x_{n},u_{\alpha}\right)$, where $\left(x_{1},..,x_{n}\right)$
are local coordinates on $M,$ $\alpha$ are multi indices of lengths
$\left|\alpha\right|\leq k,$ and $u_{\alpha}\left(A\right)=A_{\alpha},$ if
$A=\sum\limits_{\alpha}A_{\alpha}\partial^{\alpha}$ in the coordinates
$\left(x_{1},..,x_{n}\right),$ we have

$\square=\sum\limits_{\left|\alpha\right|\leq
k}u_{\alpha}\frac{d^{\left|\alpha\right|}}{dx^{\alpha}},$

where $\displaystyle{\frac{d}{dx_{i}}}$ are the total derivatives.

The n-invariants principle that we have used in ([14]) for the pseudogroup of
all local diffeomorphisms of $M,$ could be applied practically word by word
for actions of arbitrary Lie pseudogroups.

Namely, let $I_{1},\ldots,I_{n}\in
C^{\infty}\left(J^{l}\left(\psi_{k}\right)\right)$, $n=\dim M$, be
$\mathcal{P}$-invariants that are in general position in an open set
$\mathcal{O\subset}$ $J^{l}\left(\psi_{k}\right)$, i.e.

$\widehat{d}I_{1}\wedge\cdots\wedge\widehat{d}I_{n}\neq 0$

over $\mathcal{O}$, where $\widehat{d}$ is the total differential.

Then,

$I_{\alpha}=\frac{1}{\alpha!}\square\left(I^{\alpha}\right),$

where $I^{\alpha}=I_{1}^{\alpha_{1}}\cdots I_{n}^{\alpha_{n}},$
$\left|\alpha\right|\leq k,$ are also $\mathcal{P}$-invariants.

Moreover, their values at the operator $A$ (i.e. their restrictions on the
$l$-jets of section $S_{A}$ that corresponts to operator $A$) coincide with
coefficients $A_{\alpha}$ of the operator $A$ in local coordinates
$x_{1}=I_{1}\left(A\right),...,x_{n}=I_{n}\left(A\right).$

Therefore, we get the following description of the $\mathcal{P}$-invariants.

######  Theorem 1.

Let $I_{1},\ldots,I_{n}\in C^{\infty}\left(J^{l}\left(\psi_{k}\right)\right)$,
$n=\dim M$, be rational $\mathcal{P}$-invariants of scalar linear differential
operators on $M$, that are in general position in an open set
$\mathcal{O\subset}$ $J^{l}\left(\psi_{k}\right)$.

Then all rational $\mathcal{P}$-invariants of scalar linear differential
operators over the open set $\mathcal{O}$ are just rational functions of
invariants $(I_{1},\ldots,I_{n}$, $I_{\alpha},|\alpha|\leq k)$ and their
Tresse derivatives $\displaystyle{\frac{d^{\beta}I_{\alpha}}{dI^{\beta}}}$.

##  4\.  $\mathcal{P}$-models and $\mathcal{P}$-equivalence of linear
differential operators

The above theorem allow us to construct $\mathcal{P}$-models (or
$\mathcal{P}$-normal forms) of linear differential operators.

Namely, similar to ([14]), we consider space
$\Phi_{k}=\mathbb{R}^{n}\times\mathbb{R}^{\binom{n+k}{k}}$ with coordinates
$\left(y_{1},..,y_{n},Y_{\alpha},\left|\alpha\right|\leq k\right).$

Let $I_{1},\ldots,I_{n}\in C^{\infty}\left(J^{l}\left(\psi_{k}\right)\right)$
be rational $\mathcal{P}$-invariants in general position for an open set
$\mathcal{O\subset}$ $J^{l}\left(\psi_{k}\right)$. Then any linear
differential operator $A\in{\rm Diff}_{k}\left(M\right)$ in a domain
$\mathcal{O}^{\prime}\subset M,$ where
$S_{A}\left(\mathcal{O}^{\prime}\right)\subset$ $\mathcal{O}$, defines a map

$\displaystyle\phi_{B}$ $\displaystyle:$
$\displaystyle\mathcal{O}^{\prime}\rightarrow\Phi_{k},$
$\displaystyle\phi_{B}$ $\displaystyle:$ $\displaystyle
x\in\mathcal{O}^{\prime}\mathcal{\rightarrow}\left(y_{1}=I_{1}\left(A\right),...,y_{n}=I_{n}\left(A\right),Y_{\alpha}=I_{\alpha}\left(A\right)\right).$

We call a pair $\left(A,\mathcal{O}^{\prime}\right)$ adjusted if functions
$\left(I_{1}\left(A\right),\ldots,I_{n}\left(A\right)\right)$ are coordinates
in the domain $\mathcal{O}^{\prime}$, submanifold
$\Sigma_{A}=\phi_{A}\left(\mathcal{O}^{\prime}\right)\subset\Phi_{k}$ we call
a $\mathcal{P}$-model of the operator in the domain $\mathcal{O}^{\prime}$.

Let $B\in{\rm Diff}_{k}\left(M\right)$ be another differential operator such
that the pair $\left(B,\mathcal{O}^{\prime\prime}\right)$ is adjusted for the
same $\mathcal{P}$-invariants $I_{1},\ldots,I_{n}$ and models operators $A$
and $B$ coincide,

$\phi_{A}\left(\mathcal{O}^{\prime}\right)=\phi_{B}\left(\mathcal{O}^{\prime\prime}\right).$
(3)

Let $\psi_{AB}:\mathcal{O}^{\prime}\rightarrow\mathcal{O}^{\prime\prime}$ be a
diffeomorphism, such that
$\psi_{AB}^{\ast}\left(I_{i}\left(B\right)\right)=I_{i}\left(A\right).$

######  Theorem 2 (n-invariants principle).

Let $\left(A,\mathcal{O}^{\prime}\right)$ and
$\left(B,\mathcal{O}^{\prime\prime}\right)$ be adjusted pairs for the same set
of $\mathcal{P}$-invariants $I_{1},\ldots,I_{n}$.

Then operators $A,B\in{\rm Diff}_{k}\left(M\right)$ are
$\mathcal{P}$-equivalent in the open sets $\mathcal{O}^{\prime}$ and
$\mathcal{O}^{\prime\prime}$ if and only if their $\mathcal{P}$-models
coincide (3) and $\psi_{AB}\in\mathcal{P}$.

######  Remark 3.

In the case, when the Lie equation completely defines the pseudogroup
$\mathcal{P}$, the last requirement of the theorem could be made to be more
constructive. Namely, diffeomorphism $\psi_{AB}$ should satisfy the Lie
equation.

##  5\. $\mathcal{P}$\- invariants of symbols

At first, we remark, that symbols of linear differential operators from ${\rm
Diff}_{k}\left(M\right),$ at a point $a\in M,$ are symmetric tensors
$\sigma\in S^{k}T_{a}\left(M\right)$.

To describe $\mathcal{P}$\- invariants of symbols, we’ll fix a base point
$b\in M$ and denote by $V$ the cotangent space $T_{b}^{\ast}\left(M\right).$

Then the symbols of differential operators are elements of the symmetric power
$S^{k}V^{\ast},$ and, hence, could be considered as homogeneous polynomial
functions of degree $k$ on vector space $V.$

Thus, the action of the pseudogroup $\mathcal{P}$ on symbols is reduced to the
action of the algebraic Lie group $G_{1}\left(b\right)\subset{\rm
GL}\left(V^{\ast}\right)$ on the space of these polynomials.

There are various algorithmic methods to find such invariants, see, for
examples ([4],[18],[20]).

We proposed in ([9],[11]) methods of finding algebraic invariants by using
differential invariants. Thus, this paper, in some sense, conclude the
interplay between differential and algebraic invariants.

The most important are actions of primitive pseudogroups, where the Lie
algebras $\mathfrak{g}_{1}$ ( also called linear isotropy algebras) of the
correspondent Lie groups $G_{1}\left(b\right)$ are given in the Cartan’s
classification list (see,[22],[5]).

W’ll consider here the most common cases of these Lie algebras:
$\mathfrak{g}_{1}=\mathfrak{sl}\left(V^{\ast}\right)$
$\mathfrak{sl}\left(V^{\ast}\right)$ for the pseudogroup of volume preserving
diffeomorphisms, and $\mathfrak{g}_{1}=\mathfrak{sl}\left(V^{\ast}\right)$ for
the pseudogroup $\mathfrak{S}$ of symplectomorphisms.

The cases $\mathfrak{g}_{1}=\mathfrak{sl}\left(V^{\ast}\right)$ and
$\mathfrak{g}_{1}=\mathfrak{gl}\left(V^{\ast}\right)$ were considered in
([14]), for this reason, we consider here the cases
$\mathfrak{g}_{1}=\mathfrak{sp}\left(V^{\ast}\right)$ (and also
$\mathfrak{g}_{1}=\mathfrak{so}\left(V^{\ast}\right))$ to illustrate the
method of transvectants.

Remark, that the codimensions of regular $\mathfrak{g}_{1}-$ orbits in
$S^{k}T\left(M\right)$ significantly exceed $\dim M,$ when $k>2,$ and,
therefore, we may use invariants of the symbols in realization of the
$n$-invariant principle.

This observation explains the use of transvectants because we are interested
in practical methods of finding $n$ invariants, but not in the problem of
finding all possible invariants.

The case of the second order operators could be elaborated by using of the
Levi-Civita connections and related to them quantizations, as it was done in
([14]).

###  5.1. Symplectic transvectants

Let $\left(M,\omega\right)$ be a symplectic manifold, where
$\omega\in\Omega^{2}\left(M\right)$ is the structure form, and $\mathcal{P}$ =
$\mathfrak{S}$ the pseudogroup of local symplectomorphisms.

Then the Lie equation, defining this pseudogroup, is the following

$\mathfrak{S}=\left\\{\phi\in{\rm
Diffeo}(M),\phi_{\ast}(\omega)=\omega\right\\},$

and the linear isotropy algebra $\mathfrak{g}_{1}={\rm
sp}\left(V^{\ast}\right).$

Let $\left\\{e_{1},..,e_{n},f_{1},...,f_{n}\right\\}\subset
V=T_{b}^{\ast},\dim M=2n,$ be a canonical basis for the structure form
$\omega_{b}\in\Lambda^{2}\left(V\right),$ i.e.

$\omega_{b}=\sum\limits_{i=1}^{n}e_{i}\wedge f_{i}.$

Then,as we have seen, symbols of the linear differential operators on $M$ at
the point $b\in M,$ are symmetric tensors $\sigma\in S^{k}V^{\ast},$ that we
consider as homogeneous polynomials on $V.$

Denote by $\mathcal{S}=\oplus_{k\geq 0}S^{k}V^{\ast}$ the polynomial algebra
on $V$, and the structure form $\omega_{b}$ we present as bi-differential
operator

$\widehat{\omega}:S\mathcal{\otimes S\rightarrow S\otimes S},$

that acts as follows

$\widehat{\omega}\left(P\otimes
Q\right)=\frac{1}{2}\sum\limits_{i=1}^{n}\left(e_{i}\left(P\right)\otimes
f_{i}\left(Q\right)-f_{i}\left(P\right)\otimes e_{i}\left(Q\right)\right),$

where we denoted by $e_{i}\left(P\right),f_{i}\left(P\right)$ the directional
derivatives of polynomial $P$ along the basis vectors $e_{i}$ and $f_{i}.$

Then by symplectic transvectant of order $r$ we mean the following bi-
differential operator

$P\otimes Q\in S\otimes\mathcal{S}\rightarrow[P,Q_{r}]_{r}\in\mathcal{S},$

where

$[P,Q]_{r}=\mu\big{(}\widehat{\omega}^{r}(P\otimes Q)\big{)},$

and $\mu:\mathcal{S\otimes S\rightarrow S}$ is the multiplication in the
algebra $\mathcal{S}.$

Remark that operators $\widehat{\omega}$ and $\mu$, as well as the symplectic
transfectants, are ${\rm sp}-$ invariants.

In the canonical coordinates $\left(x_{1},...,x_{n},y_{1},...,y_{n}\right)$ on
$V$ the symplectic transfectants have the following form:

$[P,Q]_{r}\\!=\\!2^{-r}\sum\limits_{l=0}^{r}\sum\limits_{l_{1}+\ldots+l_{n}=l}\sum\limits_{m_{1}+\ldots+n_{n}=r-l}\left(-1\right)^{r-l}\binom{r}{l}\binom{l}{l_{1}...l_{n}}\\\
\times\binom{r-l}{m_{1}\cdots m_{n}}\frac{\partial^{r}P}{\partial
x^{l}\partial y^{m}}\frac{\partial^{r}Q}{\partial x^{m}\partial y^{l}},$

where

$\frac{\partial^{r}P}{\partial x^{l}\partial
y^{m}}=\frac{\partial^{r}P}{\partial x_{1}^{l_{1}}\cdots\partial
x_{n}^{l_{n}}\partial y_{1}^{m_{1}}\cdots\partial y_{n}^{m_{n}}}.$

Remark, that

$\deg\left([P,Q]_{r}\right)=\deg\left(P\right)+\deg\left(Q\right)-2r,$

$[P,Q]_{1}$ coincides with the Poisson bracket, and

$4[P,Q]_{2}=\sum\limits_{i=1}^{n}\left(\frac{\partial^{2}P}{\partial
x_{i}^{2}}\frac{\partial^{2}Q}{\partial
y_{i}^{2}}+\frac{\partial^{2}Q}{\partial
x_{i}^{2}}\frac{\partial^{2}P}{\partial y_{i}^{2}}\right)\\\
+2\sum\limits_{i\neq j}\left(\frac{\partial^{2}P}{\partial x_{i}\partial
x_{j}}\frac{\partial^{2}Q}{\partial y_{i}\partial
y_{j}}+\frac{\partial^{2}Q}{\partial x_{i}\partial
x_{j}}\frac{\partial^{2}P}{\partial y_{i}\partial
y_{j}}\right)-2\sum\limits_{i,j}\frac{\partial^{2}P}{\partial x_{i}\partial
y_{j}}\frac{\partial^{2}Q}{\partial y_{i}\partial x_{j}}.$

The following statement follows directly from the definition of the
transfectant.

######  Proposition 4.

The transvectants mappings $P\times Q\rightarrow$ $[P,Q]_{k}$ are bilinear
symmetric mappings if the order $k$ is even, and skew symmetric if the order
$k$ is odd.They are trivial if $k>\min\left(\deg P,\deg Q\right)$.

Thus symplectic invariants of symbols, having order $k,$ are ${\rm
sp}\left(V\right)$ \- invariant polynomial functions on the space of symbols
$S^{k}V^{\ast}$.

Namely, they do produce zero order symplectic differential invariants of
linear differential operators.

To find these ${\rm sp}$-invaiants we take a symbol $P\in
S^{p}V^{\ast},$having order $p,$ and remark that the transfectants generate
the linear operators

$Q\in S^{q}V^{\ast}\rightarrow[P,Q]_{k}\in S^{p+q-2k}V^{\ast},$

where $k\leq\min\left(p,q\right).$

Assume now, that the order $p$ is even. Then, we get operators

$A_{P,q}:S^{q}V^{\ast}\rightarrow S^{q}V^{\ast},\ \
A_{P,q}:Q\rightarrow[P,Q]_{\frac{p}{2}},$

for all $q\geq\displaystyle\frac{p}{2}$.

For general order $p,$ we substitute tensor $P\in S^{p}V^{\ast}$ by the
transfectants $P_{2l}=[P,P]_{2l},$ where $2l<p,$ and get operators

$A_{P,l}:S^{q}V^{\ast}\rightarrow S^{q}V^{\ast},\ \
A_{P,l}:Q\rightarrow[P_{2l},Q]_{p-2l},$

where $q\geq p-2l.$

######  Theorem 5.

Functions $I_{l,k}:S^{p}V^{\ast}\rightarrow\mathbb{R}$,

$I_{l,k}\left(P\right)={\rm Tr}\left(A_{P,l}^{k}\right),$

where $p-q\leq 2l\leq p,\ 1\leq k\leq\binom{2n+q-1}{q},$ for general degree
$p,$and

$J_{k,q}\left(P\right)={\rm Tr}A_{P,q}^{k},$

where $2q\geq p,\ 1\leq k\leq\displaystyle\binom{2n+q-1}{q}$, for even degree
$p,$ are $\mathfrak{sp}$ \- invariant polynomials on $S^{p}V^{\ast}$ of degree
2k.

Bilinear forms

$P\otimes Q\rightarrow[P,Q]_{p}\in\mathbb{R}$

are nondegenerate ${\rm sp}-$ invariant skew symmetric 2-forms on
$S^{p}V^{\ast},$ if $p$ is odd, and symmetric if $p$\- even.

######  Remark 6.

This theorem shows that the maximal degree of these invariant polynomials
equals to

$k_{\max}=\binom{3n-1}{n},$

and the minimal is $k_{\min}=2.$

###  5.2. Regular ${\rm sp}$-orbits

The Lie algebra ${\rm sp}\left(V\right)$ we identify with space
$S^{2}\left(V^{\ast}\right)$ equipped with the Poisson bracket $P,Q\in
S^{2}\left(V^{\ast}\right)\rightarrow\left(P,Q\right)_{1}\in
S^{2}\left(V^{\ast}\right)$ given by the first order transvectant .

Let $\left(x_{1},...,x_{n},y_{1},...,y_{n}\right)$ be the canonical
coordinates in the symplectic space $V,$ then we choose the basic in
$S^{2}\left(V^{\ast}\right)$ as union of the following disjoint sets
$B^{+}\cup B^{-}\cup C,$ where $B^{+}=\left\\{y_{i}y_{j},i,j=1,...,n,i\leq
j\right\\},B^{-}=\left\\{x_{i}x_{j},i,j=1,...,n,i\leq
j\right\\},C=\left\\{x_{i}y_{j},i,j=1,...,n\right\\}.$

Any quadric $Q\in S^{2}\left(V^{\ast}\right)$ generates the Hamiltonian
derivation $X_{Q}:S^{k}\left(V^{\ast}\right)\rightarrow
S^{k}\left(V^{\ast}\right),$ where $X_{Q}\left(P\right)=[P,Q]_{1},$ or in
canonical coordinates

$X_{Q}=\sum\limits_{i=1}^{n}\left(\frac{\partial Q}{\partial
y_{i}}\frac{\partial}{\partial x_{i}}-\frac{\partial Q}{\partial
x_{i}}\frac{\partial}{\partial y_{i}}\right).$

Then vector fields

$\displaystyle b_{ij}^{+}$ $\displaystyle=$ $\displaystyle
X_{y_{i}y_{j}}=y_{i}\frac{\partial}{\partial
x_{j}}+y_{j}\frac{\partial}{\partial x_{i}},$ $\displaystyle b_{ij}^{-}$
$\displaystyle=$ $\displaystyle X_{x_{i}x_{j}}=-x_{i}\frac{\partial}{\partial
y_{j}}-x_{j}\frac{\partial}{\partial y_{i}},$ $\displaystyle c_{ij}$
$\displaystyle=$ $\displaystyle X_{x_{i}y_{j}}=x_{i}\frac{\partial}{\partial
x_{j}}-y_{j}\frac{\partial}{\partial y_{i}},$

give us a basis into Lie algebra ${\rm sp}\left(V\right).$

To estimate dimensions of ${\rm sp}$-orbits in to
$S^{k}\left(V^{\ast}\right),$ we have to estimate dimensions of subspaces
$\left\\{B^{+}\left(P\right),B^{-}\left(P\right),C\left(P\right)\right\\}$
into $S^{k}\left(V^{\ast}\right),$ generated a polynomial $P\in
S^{k}\left(V^{\ast}\right).$

To this end, we denote by $S_{i}\left(x\right)$ and $S_{j}\left(y\right)$ the
spaces of homogeneous polynomials in $x$ and $y,$ having degrees $i$ and $j$
respectively.

Then we have the following direct decomposition

$S^{k}\left(V^{\ast}\right)=\oplus_{\alpha+\beta=k}S_{\alpha,\beta},$

where

$S_{\alpha,\beta}=S_{\alpha}\left(x\right)\otimes S_{\beta}\left(y\right),$

and

$\displaystyle b_{ij}^{+}$ $\displaystyle:$ $\displaystyle
S_{\alpha\beta}\rightarrow S_{\alpha-1,\beta+1},\ $ $\displaystyle b_{ij}^{-}$
$\displaystyle:$ $\displaystyle S_{\alpha\beta}\rightarrow
S_{\alpha+1,\beta-1},\ \ $ $\displaystyle c_{ij}$ $\displaystyle:$
$\displaystyle S_{\alpha\beta}\rightarrow S_{\alpha,\beta}.$

Take now a polynomial $P\in S^{k}\left(V^{\ast}\right)$ of the form:
$P=P_{0}+P_{1},$ where $P_{0}\in S_{k}\left(x\right),P_{1}\in
S_{k}\left(y\right).$

Then,

$\displaystyle B^{+}\left(P\right)$ $\displaystyle=$ $\displaystyle
B^{+}\left(P_{0}\right)\subset S_{k-1},_{1},\ $ $\displaystyle
B^{-}\left(P\right)$ $\displaystyle=$ $\displaystyle
B^{-}\left(P_{1}\right)\subset S_{1,k-1},$ $\displaystyle C\left(P_{0}\right)$
$\displaystyle\subset$ $\displaystyle S_{k,0},\ C\left(P_{1}\right)\subset
S_{0,k}.$

Thus, $B^{+}\left(P_{0}\right)$ and $B^{-}\left(P_{1}\right),$ belong to
different vector spaces and are linear independent, when $k\geq 3,$ and
therefore

$\dim\left(B^{+}\left(P_{0}\right)\oplus
B^{-}\left(P_{1}\right)\right)=n\left(n+1\right),$

for general polynomials $P_{0},P_{1}.$

Moreover, dimensions $C\left(P_{0}\right)\subset S_{k,0}$ and
$C\left(P_{1}\right)\subset S_{0,k}$ equals of dimensions of
$\mathfrak{gl}\left(n\right)$-orbits of $P_{0}\in S_{k}\left(x\right)\ $and
$P_{1}\in S_{k}\left(y\right)$ and, therefore, equals $n^{2},$ when $k\geq 3.$

To see this, it is enough to take polynomials
$P_{0}\left(x\right),P_{1}\left(y\right)$ such that the polynomials
$\det\left\|x_{i}\displaystyle\frac{\partial P_{0}}{\partial x_{j}}\right\|$
and $\det\left\|y_{i}\displaystyle\frac{\partial P_{1}}{\partial
y_{j}}\right\|$ do not equal to zero.

Summarizing, we get the following

######  Proposition 7.

Regular $\mathfrak{sp}\left(V\right)-$ orbits in $S^{k}\left(V^{\ast}\right)$
has dimension $\dim\mathfrak{sp}\left(V\right)=\dim
S^{2}\left(V^{\ast}\right),$ when $k\geq 3.$

######  Remark 8.

Codimension of regular $\mathfrak{sp}\left(V\right)-$ orbits in
$S^{2}\left(V^{\ast}\right)$ equals to the dimension of the Cartan algebra,
i.e. $n=\displaystyle\frac{\dim V}{2}$. The $\mathfrak{sp}\left(V\right)-$
invariants are traces of the even powers of the operators $X_{P}.$ Stabilizers
of regular $\mathfrak{sp}\left(V\right)$ \- orbits in
$S^{k}\left(V^{\ast}\right),k\geq 3,$ are discrete algebraic groups, and,
therefore, they are finite.

######  Example 9.

Let $\dim V=4$ and $k=3.$ Then regular $\mathfrak{sp}\left(V\right)-$ orbits
in $S^{3}\left(V^{\ast}\right)$ has codimension $10.$ Let $P\in
S^{3}\left(V^{\ast}\right),$ then $P_{2}=[P,P]_{2}\in
S^{2}\left(V^{\ast}\right)$ and $I_{4}(P)={\rm
Tr}\left(X_{P_{2}}^{2}\right),I_{8}\left(P\right)={\rm
Tr}\left(X_{P_{2}}^{4}\right),$ are $\mathfrak{sp}\left(V\right)-$ invariants.
Moreover, we have operators $A_{P}:S^{q}\left(V^{\ast}\right)\rightarrow
S^{q}\left(V^{\ast}\right),$ where $A_{P}\left(Q\right)=[P_{2},Q]_{1},$ and
traces of their powers ${\rm Tr}\left(A_{P}^{2l}\right)$ give us
$\mathfrak{sp}\left(V\right)-$ invariants. In particular, for the case $k=3,$
we get 10 $\mathfrak{sp}\left(V\right)-$ invariants$:{\rm
Tr}\left(A_{P}^{2l}\right),$ $l=1,..,10.$

###  5.3. Metric transvectants and metric invariants

The above approach could be applied word by word to description of
$\mathfrak{so}-$ invariant polynomials on an Eucledean vector space
$\left(V,g\right),$ where $g\in S^{2}\left(V\right)$ is a metric on the dual
space.

Let $\left\\{e_{1},..,e_{n}\right\\}\subset V,\dim V=n,$ be an orthonormal
basis for the structure form $g\in S^{2}\left(V\right),$ i.e.

$g=\sum\limits_{i=1}^{n}e_{i}\otimes e_{i}.$

Then the bi-differential operator $\widehat{g}:\mathcal{S\otimes S\rightarrow
S\otimes S}$ act as follows

$\widehat{g}\left(P\otimes
Q\right)=\sum\limits_{i=1}^{n}\left(e_{i}\left(P\right)\otimes
e_{i}\left(Q\right)\right),$

and the metric transvectants are

$\left(P,Q\right)_{m}=\mu\left(\widehat{g}^{m}\left(P\otimes Q\right)\right),$

and the have the following expression in the orthonormal coordinates:

$\left(P,Q\right)_{m}=\sum\limits_{m_{1}+\cdots+m_{n}=m}\binom{m}{m_{1}...m_{n}}\frac{\partial^{m}P}{\partial
x_{1}^{m_{1}}\cdots\partial x_{n}^{m_{m}}}\frac{\partial^{m}Q}{\partial
x_{1}^{m_{1}}\cdots\partial x_{n}^{m_{m}}}.$

Thus,

$\left(P,Q\right)_{1}=\sum\limits_{i=1}^{n}\frac{\partial P}{\partial
x_{i}}\frac{\partial Q}{\partial x_{i}},$

and

$\left(P,Q\right)_{2}=\sum\limits_{i=1}^{n}\frac{\partial^{2}P}{\partial
x_{i}^{2}}\frac{\partial^{2}Q}{\partial x_{i}^{2}}+2\sum\limits_{i\neq
j}\frac{\partial^{2}P}{\partial x_{i}\partial
x_{j}}\frac{\partial^{2}Q}{\partial x_{i}\partial x_{j}}.$

Remark, that, as above, the metric transvectants generate the linear operators

$\displaystyle Q$ $\displaystyle\in$ $\displaystyle
S^{q}V^{\ast}\rightarrow(P,Q)_{m}\in S^{p+q-2m}V^{\ast},$ $\displaystyle m$
$\displaystyle\leq$ $\displaystyle\min\left(p,q\right),$

and we get linear operators

$\displaystyle B_{P,q}$ $\displaystyle:$ $\displaystyle
S^{q}V^{\ast}\rightarrow S^{q}V^{\ast}$ $\displaystyle B_{P,q}$
$\displaystyle:$ $\displaystyle Q\rightarrow(P,Q)_{m},$ $\displaystyle p$
$\displaystyle=$ $\displaystyle 2m,$

in the case when $p=\deg\left(P\right)$ is even and $p=2m,$ $2q\geq p.$

For general degree $p,$ we substitute tensor $P\in S^{p}V^{\ast}$ by the
transvectants $P_{l}=(P,P)_{l},$ where $l<p,$ and get operators

$\displaystyle B_{P,l}$ $\displaystyle:$ $\displaystyle
S^{q}V^{\ast}\rightarrow S^{q}V^{\ast},$ $\displaystyle B_{P,l}$
$\displaystyle:$ $\displaystyle Q\rightarrow(P_{l},Q)_{p-l},$

where $q\geq p-l.$

In the case, $l=p$ we get also ${\rm so}\left(g\right)$ \- invariant binary
forms
$K_{p}(P,Q)=(P,Q)_{p}$ on $S^{p}V^{\ast}$.

######  Theorem 10.

Functions $M_{l,k}:S^{p}V^{\ast}\rightarrow\mathbb{R}$,

$\displaystyle M_{l,k}\left(P\right)$ $\displaystyle=$ $\displaystyle{\rm
Tr}\left(B_{P,l}^{k}\right),$ $\displaystyle p-q$ $\displaystyle\leq$
$\displaystyle l\leq p,\ 1\leq k\leq\binom{n+q-1}{q},$

for general degree $p,$ quadratic polynomials $K_{p},$ and functions
$N_{k,q}:S^{p}V^{\ast}\rightarrow\mathbb{R}$

$\displaystyle N_{k,q}\left(P\right)$ $\displaystyle=$ $\displaystyle{\rm
Tr}(B_{P,q}^{k}),$ $\displaystyle 2q$ $\displaystyle\geq$ $\displaystyle p,\
1\leq k\leq\binom{n+q-1}{q},$

for even degree $p,$ are ${\rm so}\left(V\right)-$ invariant polynomials on
$S^{p}V^{\ast}$ of degree 2k.

Invariant nondegenerate symmetric forms
$K_{p}\left(P,Q\right)=\left(P,Q\right)_{p}$ on $S^{p}V^{\ast}$ realize
representations ${\rm so}\left(V\right)\rightarrow{\rm
so}\left(S^{p}V^{\ast}\right).$

##  6\. Constant type differential operators

###  6.1. Connections and quantizations

At first, we shortly remind the quantization procedure, that we have used in
([12]).

Let $\nabla$ be a connection in the cotangent bundle
$\tau^{\ast}:T^{\ast}M\rightarrow M$ and let $\nabla^{\otimes k}$ be extension
of this connection on the bundles of symmetric tensor products
$\tau_{k}^{\ast}:S^{k}T^{\ast}M\rightarrow M.$

Denote by
$d_{\nabla}:\Sigma^{k}\left(M\right)\rightarrow\Sigma^{k}\left(M\right)\otimes\Sigma^{1}\left(M\right)$
the covariant differential of the last connection.

Taking symmetrization of these covariant differentials, we get derivations
$d_{\nabla}^{s}$ into the symmetric algebra
$\Sigma^{\cdot}\left(M\right)=\oplus_{k\geq 0}\Sigma^{k}\left(M\right),$ where

$d_{\nabla}^{s}:\Sigma^{k}\left(M\right)\overset{d_{\nabla}}{\rightarrow}\Sigma^{k}\left(M\right)\otimes\Sigma^{1}\left(M\right)\overset{\text{Sym}}{\rightarrow}\Sigma^{k+1}\left(M\right),$

for $k\geq 1,$ and
$d_{\nabla}^{s}=d:C^{\infty}\left(M\right)\rightarrow\Sigma^{1}\left(M\right),$
for $k=0.$

Then, the $k$-th power of $d_{\nabla}^{s}$ defines a $k$-th order operator
$\left(d_{\nabla}^{s}\right)^{k}:C^{\infty}\left(M\right)\rightarrow\Sigma^{k}\left(M\right).$

It is easy to check, that the symbol of this operator at a differential form
$\theta\in\Sigma^{1}\left(M\right)$ equals to the $k$-th power
$\theta^{k}\in\Sigma^{k}\left(M\right).$

Let now $H\in\Sigma_{k}\left(M\right)$ be a symmetric contravariant tensor.

Denote by $Q\left(H\right)\in{\rm Diff}_{k}\left(M\right)$ the following
$k$-th order differential operator

$Q\left(H\right)\left(f\right)=\frac{1}{k!}\left\langle
H,\left(d_{\nabla}^{s}\right)^{k}\left(f\right)\right\rangle,$

where
$\left\langle\cdot,\cdot\right\rangle:\Sigma_{k}\left(M\right)\otimes\Sigma^{k}\left(M\right)\rightarrow
C^{\infty}\left(M\right)$ is the standard pairing between contra and covariant
tensors.

It is easy to check, that the symbol $Q\left(H\right)$ equals $H,$ and,
therefore, the correspondence $Q:H\in\Sigma_{k}\left(M\right)\rightarrow
Q\left(H\right)\in{\rm Diff}_{k}\left(M\right),$ that we call quantization,
splits exact sequence (2).

On the other hand, let $A\in{\rm Diff}_{k}\left(M\right)$ be a differential
operator and let $\sigma_{A}\in\Sigma_{k}\left(M\right)$ be it symbol.

Let $A_{1}=A-Q\left(\sigma_{A}\right)\in{\rm Diff}_{k-1}\left(M\right)$ and
let $\sigma_{1}\in\Sigma_{k-1}\left(M\right)$ be its symbol. Then,
$A_{2}=A_{1}-Q\left(\sigma_{1}\right)\in{\rm Diff}_{k-2}\left(M\right),$ and,
continue this way we get tensors $\sigma_{i}\in\Sigma_{i}\left(M\right),$
$i=0,1,2...,k-1,$ that are calling subsymbols, and representation of the
initial operator $A$ in the form

$A=Q\left(\sigma_{A}\right)+\Sigma_{i=0}^{k-1}Q\left(\sigma_{i}\right),$ (4)

where

$\sigma\left(A\right)=[\sigma_{A},\sigma_{k-1},...,\sigma_{1},\sigma_{0}]\in\Sigma_{k}\left(M\right)\oplus\Sigma_{k-1}\left(M\right)\oplus\cdots\\\
\oplus\Sigma_{1}\left(M\right)\oplus\Sigma_{0}\left(M\right)$

is called total symbol of the operator.

###  6.2. Constant type operators and associated connections

Let now $\Gamma$ be a transitive Lie pseudogroup on manifold $M$ and let
$A\in{\rm Diff}_{k}\left(M\right)$ be a differential operator on $M.$

Denote by $O_{a}\subset S^{k}T_{a}M$ the $G_{1}\left(a\right)$-orbit of the
symbol $\sigma_{A,a}\in S^{k}T_{a}M,$ where $G_{1}\left(a\right)\subset{\rm
End}\left(T_{a}\right)$ , is the linear isotropy group  (1).

Denote by $[\psi]_{a,b},$ $a,b\in M,$ the $1$-jet of a diffeomorphism
$\psi\in\Gamma,$ such that $\psi\left(a\right)=b.$

Remark also, that $[\psi]_{a,b}\circ[\widetilde{\psi}]_{b,a}\in
G_{1}\left(b\right),$ for any $\widetilde{\psi}\in\Gamma,$ such that
$\widetilde{\psi}\left(b\right)=a.$

We say that an operator $A\in{\rm Diff}_{k}\left(M\right)$ has the same type
at points $a,b\in M,$ if for diffeomorphisms $\psi\in\Gamma,$
$\psi\left(a\right)=b,$ we have

$[\psi]_{a,b}\left(\sigma_{A,a}\right)\in O_{b}.$ (5)

We also say that an operator $A\in{\rm Diff}_{k}\left(M\right)$ has a constant
$\Gamma$-type, if

$[\psi]_{a,b}\left(O_{a}\right)=O_{b},$

for any points $a,b\in M$ , and all diffeomorphisms $\psi\in\Gamma,$ such that
$\psi\left(a\right)=b.$

An affine connection $\nabla$ on a $\Gamma$-manifold $M$ is said to be a
$\Gamma$-connection if the $\nabla-$parallel transports along paths,
connecting points $a,b\in M,$ are elements of $G_{1}\left(a,b\right).$

######  Theorem 11.

Let $\Gamma$ be a transitive algebraic Lie pseudogroup on manifold $M$ and let
$A\in{\rm Diff}_{k}\left(M\right)$ be a differential operator on $M$ of
constant $\Gamma$-type, such that the stabilizers of the linear isotropy
groups of the symbol $\sigma_{A}$ are finite (or equally, $\dim O_{a}=\dim
G_{1}(a)$ for all $a\in M$).

Then there exists and unique an affine $\Gamma$-connection on $M,$ that
preserves the symbol $\sigma_{A}\in\Sigma_{k}\left(M\right)$ of the operator.

###### Proof.

Let $U\ni a$ be a neighborhood, that we assume to be reasonably small,
$G_{1}\left(U,U\right)=\delta_{1,0}^{-1}\left(U\times U\right)\subset
G_{1},$and let $f:U\times U\rightarrow$ $G_{1}\left(U,U\right)$ be a section
of the bundle $\delta_{1,0}$ over $U\times U.$ Then elements
$f\left(b,a\right)\in G_{1}(b,a)$ satisfy condition (5), and
$f\left(a,a\right)\in G_{1}\left(a\right).$ Therefore, elements
$\widetilde{f}\left(b,a\right)=f(a,a)^{-1}\cdot f\left(b,a\right)$ satisfy
condition (5) and, in addition, $\widetilde{f}\left(a,a\right)$ is the unit
element of the group $G_{1}\left(a\right).$
Let $St_{a}\subset G_{1}\left(a\right)$ be the stabilizer of the symbol
$\sigma_{A,a}$ and $\pi:G_{1}\left(a\right)\rightarrow
O_{a}=G_{1}\left(a\right)/St_{a}$ be the natural covering. Take such a
neighborhood $V_{a}\subset O_{a}$, that $\pi^{-1}\left(V_{a}\right)\cap
St_{a}$ contains only the unit element. Assume now, that neighborhood $U$ is
so small, that $\widetilde{f}\left(b,a\right)\left(\sigma_{A,b}\right)\in
V_{a},$ for all $b\in U.$
Then, by the construction, we have such a unique map $\lambda:U\rightarrow
G_{1}\left(a\right),$ where $\lambda\left(a\right)$ is the unit of
$G_{1}\left(a\right)$, that
$\widetilde{f}\left(b,a\right)\left(\sigma_{A,b}\right)=\lambda\left(a\right)\sigma_{A,a}$.
Thus, the family of isomorphisms
$T_{b,a}=\lambda\left(a\right)^{-1}\widetilde{f}\left(a,b\right)\in
G_{1}(b,a)$ is uniquely determined, preserves the symbols:
$T_{b,a}\left(\sigma_{A,b}\right)=\sigma_{A,a},$ and, therefore, determines
the required affine connection (we call it Wagner $\Gamma$-connection,(cf.
[24],[13])). ∎

Let $A\in{\rm Diff}_{k}\left(M\right)$ be, as in the above theorem, a
differential operator on $M$ of constant $\Gamma$-type and let $\nabla$ be the
Wagner connection, associated with this operator.

Let $Q:\Sigma_{i}\left(M\right)\rightarrow{\rm Diff}_{i}\left(M\right)$ be the
quantization, associated with the Wagner connection.

Denote by

$\sigma\left(A\right)=[\sigma_{A},\sigma_{k-1},...,\sigma_{1},\sigma_{0}]\in\Sigma_{k}\left(M\right)\oplus\Sigma_{k-1}\left(M\right)\oplus\cdots\\\
\oplus\Sigma_{1}\left(M\right)\oplus\Sigma_{0}\left(M\right)$

the total symbol of the operator.

Remark, that $\Gamma$-equivalent operators have $\Gamma$-equivalent total
symbols.

######  Theorem 12.

Let $A_{1},A_{2}\in{\rm Diff}_{k}\left(M\right)$ be, as in the above theorem,
differential operators on $M$ of constant $\Gamma$-type and let
$\nabla_{1},\nabla_{2}$ be the Wagner connections, associated with these
operators. Then operators $A_{1},A_{2}$ are $\Gamma$-equivalent if and only if
their total symbols $\sigma_{A_{1}},\sigma_{A_{2}}\in\Sigma_{k}\left(M\right)$
are $\Gamma$-equivalent.

###### Proof.

It is enough to note that $\Gamma$-equivalence of total symbols implies the
$\Gamma$-equivalence of the principal symbols and, therefore,
$\Gamma$-equivalence of the Wagner connections and spltting (4). ∎

######  Remark 13.

The pseudogroup of local symplectomorphisms, $\mathfrak{S}$, satisfies the
requirements of the above theorems, and, therefore, any regular linear
differential operator $A\in{\rm Diff}_{k}\left(M\right)$ of the constant
$\mathfrak{S}$-type, having order $k\geq 3,$ defines a symplectic connection
on the symplectic manifold that preserves the symbol
$\sigma_{A}\in\Sigma_{k}\left(M\right).$

###  6.3. The second order operators

In the case of the second order linear differential operators, we have in
hands two practical tools. At first, to get invariants in the case of regular
symbols, one can use the Levi-Civita connection, that is naturally associated
with the operators, and the correspondent quantization.

On the other hand, at least for some pseudogroups, the symbols themselves have
algebraic invariants with respect to the linear isotropy group, as well as
differential invariants ([16]).

Altogether, this allows us to get the necessary number of applicable for any
pseudogroup differential invariants and apply the $n$-invariant principle.

###  6.4. Weakly nonlinear operators

In papers ([14],[17]) we have studied some class of nonlinear operators, that
we call weakly nonlinear.

In local coordinates $\left(x_{1},..,x_{n}\right)$ these operators have the
following form: $A_{w}\left(f\right)=\sum\limits_{\left|\alpha\right|\leq
k}a_{\alpha}\left(x,f(x)\right)\partial^{\alpha},$ where coefficients
$a_{\alpha}\left(x,u\right),$ as functions on the space of zero order jets
$\mathbf{J}^{0}\left(M\right)$ belong (at any point $x\in M$ ) to a fixed
finite algebraic extension of the field $\mathbf{Q}\left(u\right)$ of rational
in $u$ functions.

We have shown how to get natural differential invariants for such classes of
operators from invariants of linear differential operators on
$\mathbf{J}^{0}\left(M\right)$ of the form
$A=\sum\limits_{\left|\alpha\right|\leq
k}a_{\alpha}\left(x,u\right)\partial^{\alpha}.$

In the case of pseudogroup $\Gamma,$ different from the pseudogroup of all
local diffeomorphisms of $M$ , we have an additional option to use algebraic
invariants of the symbols $\sigma_{A}$ with respect to the linear isotropy
group and then apply the $n$-invariant principle.

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