LUXE Collaboration

# Towards the use of X-ray Free-Electron Laser electron beams to study Quantum
Chromo-Dynamics

Eugene Bulyak<EMAIL_ADDRESS><EMAIL_ADDRESS>National Science
Center ‘Kharkiv Institute of Physics and Technology’, 1 Academichna str,
Kharkiv, Ukraine
V.N. Karazin National University, 4 Svodody sq., Kharkiv, Ukraine  Svitozar
Serkez<EMAIL_ADDRESS>European XFEL, Holzkoppel 4, 22869 Schenefeld,
Germany  Gianluca Aldo Geloni  European XFEL, Holzkoppel 4, 22869 Schenefeld,
Germany

###### Abstract

X-ray free-electron lasers (XFELs) utilize high-density and high-energy
electron bunches which are well-suited to produce Compton back-scattering
radiation. Here we study back-scattered radiation pulses produced by the
interaction of XFEL electron beams and an optical laser. We discuss cost-
effective setups to study such processes, taking advantage of the existing
conventional as well as proposed XFEL infrastructure. We estimate parameters
of possible experiments and compare them with other projects under
construction.

##  I Introduction

The purpose of X-Ray Free-Electron Lasers (XFEL) is to generate extremely
brilliant, ultra-short and almost spatially coherent pulses of X-rays at
wavelengths down to the order of 1 Å, in order to exploit them for scientific
experiments in a variety of disciplines spanning physics, chemistry, materials
science and biology, [1].

However, the electron beams of XFELs possess unique characteristics in energy
and phase-space density, which may also be applied for experiments in other
areas of physics. In particular, if these beams collide with intense pulses of
quasi-monochromatic polarized photons, scattered high-energy polarized gamma-
ray photons are produced, which are extremely useful for paricle physics
experiments and, in particular, Quantum Chromo-Dynamics (QCD).

Multi-GeV gamma-ray photons are useful tools in the investigation of the
structures and properties of hadrons via their interaction with a target
nucleon or nucleus. In particular, they are suitable for hadron studies in the
strange sector. The search for QCD exotics uses data from a wide range of
experiments and production mechanisms. A few of such setups are currently
under construction, see [3, 4]. These setups do not cover the gamma energy
range of 2.89–8.4 GeV.

Here we propose to employ the electron beams from the European XFEL
accelerator to produce Compton back-scattered radiation pulses of polarized
gamma-ray photons with the maximum energy of up to 4 GeV for QCD experiments.

After this introduction, the second section of this paper briefly surveys
properties of Compton back-scattering radiation with emphasis on gamma-rays
production. The third section presents a schematic setup of a Compton source
based on the European XFEL electron beam to provide capabilities for QCD
experiments with photons in the multi–GeV energy range. Finally, the paper is
concluded with a summary.

##  II Compton back-scattering process at high electron energy

Compton scattering, at variance with the classical Thomson process with low
energy incident photons in the electron rest frame, exhibits modification of
the spectra and angular distribution of the scattered photons due to recoil of
the electrons.

A scheme of Compton back-scattering process is presented in Fig. 1.

electronincident photon$\psi$scattered photon$\phi$ Figure 1: Scheme of
scattering the laser pulse.

Usually, Compton backscattered photons are referred to as quasi-monochromatic.
In fact, due to the strong correlation between the scattering angle $\psi$ and
the photon energy, proper collimation enables one to select a well-defined
energy range.

###  II.1 Kinematics

The relation between the energy of a gamma-ray photon and the angle of
emission determined from kinematics – the energy-momentum conservation law –
reads (see, e.g., [5, 6]):

$\displaystyle\omega$
$\displaystyle\approx\frac{2\gamma^{2}_{*}(1+\cos\phi)\omega_{\text{las}}}{1+\gamma^{2}_{*}\psi^{2}}\;;$
(1)
$\displaystyle\gamma_{*}:=\frac{\gamma}{\sqrt{1+2(1+\cos\phi)\gamma\omega_{\text{las}}}}\;,$
(2)

where $\omega$ is the energy of the scattered quanta, $\omega_{\text{las}}$ is
the energy of laser photons, $\gamma$ is the energy of electrons, $\phi$ and
$\psi\ll 1$ are the crossing angle and the scattering angle of photons, see
Fig. 1.

Here we make use the natural system of units for particle and atomic physics,
$\hbar=c=m_{e}=1$. Therefore $\gamma:=E_{e}/m_{e}c^{2}$ is the Lorentz-factor
of electrons, $\omega:=\epsilon/m_{e}c^{2}$ equivalent Lorentz-factor of the
scattered photons of energy $\epsilon$,
$\omega_{\text{las}}:=\hbar\omega_{\text{las}}/m_{e}c^{2}$ the equivalent
Lorentz factor for the laser photons.

The energy of the recoiled electron becomes

$\gamma^{\prime}=\gamma-\omega=\gamma\,\left(\frac{1+\gamma^{2}\psi^{2}}{1+2\gamma\omega_{\text{las}}(1+\cos\phi)}\right)\;,$

and the energy of laser photon in the electron rest frame, see [7, 8] is

$\eta=(1+\cos\phi)\gamma\omega_{\text{las}}\approx
2\gamma\omega_{\text{las}}\;.$ (3)

Under the assumption of small angles, $\phi\ll 1,\psi\ll 1$, Eq. (1) yields a
simple expression for the maximum energy of the scattered quanta:

$\omega^{\mathrm{max}}=\gamma\frac{4\gamma\omega_{\text{las}}}{1+4\gamma\omega_{\text{las}}}\;,$
(4)

which reveals that the recoil effect is governed by the term
$4\gamma\omega_{\text{las}}=2\eta$. For relatively weak electron recoil,
$4\gamma\omega_{\text{las}}\ll 1$, the maximum energy of the scattered photons
scales linearly with the laser photon energy and quadratically with the
electron energy,

$\omega\approx 4\gamma^{2}\omega_{\text{las}}\,,$

as illustrated in Fig. 2 that is computed for the energy range of the European
XFEL accelerator and laser photons with an energy of 1 eV.

Figure 2: Maximum energy of back-scattered gammas vs. energy of electrons.

Accordingly, the _minimal energy_ of recoiled electron is

$\gamma^{\prime}_{\text{min}}=\frac{\gamma}{1+4\gamma\omega_{\text{las}}}\;.$

Another parameter deduced from kinematics is the angle (measured from the
incident electron trajectory) at which scattered photons possess the half-
maximum energy, $\omega_{1/2}=\omega_{\text{max}}/2$:

$\psi_{1/2}=\frac{\sqrt{1+2\gamma\omega_{\text{las}}(1+\cos\phi)}}{\gamma}=\frac{1}{\gamma_{*}}\;.$
(5)

In the low-energy Thomson’s limit, $\gamma_{*}\approx\gamma$, and the energy
of gamma-rays drops by half at the emission angle of $1/\gamma$. In the
electron rest frame it corresponds to the scenario when a photon scatters
perpendicularly to the incident direction.

###  II.2 Spectrum and cross-section

A high-energy approximation for the Klein-Nishina formula was derived by
Arutyunian and Tumanian [9] (see also [10]). One has the following dependence
of the cross section on the frequency of the scattered photons:

$\displaystyle\frac{\,\mathrm{d}\sigma_{\text{Com}}}{\,\mathrm{d}\omega}$
$\displaystyle=\frac{3\sigma_{\text{Th}}}{16\gamma^{2}\omega_{\text{las}}}\left[\frac{\omega^{2}}{4\gamma^{2}\omega_{\text{las}}^{2}(\gamma-\omega)^{2}}-\frac{\omega}{\gamma\omega_{\text{las}}(\gamma-\omega)}+\frac{\gamma-\omega}{\gamma}+\frac{\gamma}{\gamma-\omega}\right]\mathrm{\Theta}\left(\frac{4\omega_{\text{las}}\gamma^{2}}{4\omega_{\text{las}}\gamma+1}-\omega\right)\;,$
(6) while the integral is $\displaystyle\sigma_{\text{Com}}$
$\displaystyle=\frac{3\sigma_{\text{Th}}}{32\gamma^{3}\omega_{\text{las}}^{3}}\left\\{\frac{4\gamma\omega_{\text{las}}(2\gamma\omega_{\text{las}}(\gamma\omega_{\text{las}}+4)(2\gamma
w+1)+1)}{(4\gamma\omega_{\text{las}}+1)^{2}}+\left[2\gamma\omega_{\text{las}}\gamma\omega_{\text{las}}-1)-1\right]\log(4\gamma\omega_{\text{las}}+1)\right\\}\;,$
(7)

where $\sigma_{\text{Th}}=8\pi r_{0}^{2}/3$ is the Thomson cross section
($r_{0}$ the classical electron radius), $\mathrm{\Theta}(\cdot)$ is the
Heaviside’s Theta function (unit step).

The Compton radiation spectrum given by Eq. (6) depends on the energy of the
incident laser photon measured in the electron reference frame,
$2\gamma\omega_{\text{las}}$. The total cross section, Eq. (7), decreases with
the energy of the incident laser photon.

Figure 3: Rescaled spectra of Compton radiation $S$ defined in Eq. (8) for an
electron energy $E_{e}=16.5\,\text{GeV}$ and an incident laser photon energy
of 1 eV (blue curve), compared with the Thomson spectrum (orange curve).
$\zeta=\omega/\omega_{\text{max}}$. The integral Compton cross section is
about 0.81 times the integral Thomson cross section).

###  II.3 Polarization of gammas

As is seen from Fig. 3, for low energies of laser photons
$\epsilon_{\text{las}}\lesssim 10\,\text{eV}$ the spectrum of Compton photons
resembles the Thomson spectrum, which has a simple form (see Fig. 4 and [11]):

$S(\zeta)=\frac{3}{2}\left[1-2\zeta(1-\zeta)\right];\quad 0\leq\zeta\leq 1\;,$
(8)

where $\zeta:=\omega/\omega^{\text{max}}$ is the energy of the emitted gamma-
ray photon normalized to its maximum. The low-energy end of the spectrum,
being of little significance for gamma sources, scales as
$\omega^{\text{min}}\sim\omega_{\text{las}}/2\gamma^{2}$ and is approximated
with zero.

$0$$0.25$$0.5$$0.75$$0$$0.25$$0.5$$0.75$$1$$1.25$$1.5$$\zeta=\omega/\omega^{\mathrm{max}}$magnitude
Figure 4: Thomson photon spectrum (black) and its constituents: ‘positively
polarized’ (red) and ‘negatively polarized’ (blue) gamma-ray photons.

The spectrum described by Eq. (8) may be decomposed into a sum of two sub-
spectra with ‘orthogonal’ polarization components, see [12]. For a circular
laser polarization, they are $S_{\text{neg}}$ – the negative polarization
constituent with the same polarization as the incident radiation, and
$S_{\text{pos}}$ – the positive constituent with the opposite polarization,
see Fig. 4:

$S_{\text{pos}}=3\zeta^{2}/2\;;\quad S_{\text{neg}}=3(1-\zeta)^{2}/2\;.$

The degree of polarization $\rho$ is defined as

$\rho(\zeta):=\frac{S_{\text{pos}}-S_{\text{neg}}}{S_{\text{pos}}+S_{\text{neg}}}=\frac{3\zeta}{2-\zeta+2\zeta^{2}}\;.$

It should be emphasized that the degree of polarization has probabilistic
sense: if, for instance, $\rho=0.7$, one obtains 85% of gamma-ray photons
‘positively polarized’ and 15% ‘negatively polarized’.

The dependence of the degree of polarization $\rho$ and of the normalized
gamma-ray photon energy $\zeta$ as a function of the emission angle
(multiplied by $\gamma_{*}$) is presented in Fig. 5.

Figure 5: Relative energy of gamma ray photons (blue curve) and degree of
polarization (orange) vs. emission angle.

As it can be seen from the figure, at the angle $\gamma_{*}\psi=1$, the photon
energy decreases by half, and the degree of polarization is zero, i.e.
positively- and negatively-polarized photons are equal in numbers. Upon
further increase of the observation angle, the photon energy continues to drop
while the number of negatively polarized photons prevails over the positive
ones.

Compton radiation is quasi-monochromatic: if the laser beam is scattered by a
zero-emittance electron beam and observed through an angular filter (a pinhole
or a mask selecting a narrow “ring” around the radiation axis that matches the
electron trajectory) , every photon scattered by an electron with a given
energy at a definite angle to its trajectory has a fixed energy and degree of
polarization.

In contrast to a single electron or a monoenergetic filament electron beam,
realistic electron beams with finite energy spread and emittance, scatter-off
radiation with larger spectral bandwidth; at XFELs the contribution of
emittance is dominant [12].In the case of multi-GeV electrons the energy half-
maximum angle (zero degree of polarization) $\psi_{1/2}$ is rather small. For
example, for $E_{e}=16.5\,\text{GeV}$ and $\epsilon=1\,\text{eV}$ the photon
energy is diminished by half at $\psi_{1/2}\approx 35\,\mu\text{rad}$.

The correlation of the photon degree of polarization with its energy is
independent of electron’s trajectory direction. A Coincidence scheme for
defining the gamma-ray photon energy and polarization through measurement of
the electron energy that emitted it is therefore justified.

###  II.4 Yield of photons

The yield of gamma-ray photons per single crossing of the incident photon
pulse with the electron bunch, both assumed having a 3D Gaussian shape [11],
reads

$Y=\frac{N_{\text{pp}}N_{e}\,\sigma_{\text{C}}}{2\pi\sqrt{\left({\sigma^{\prime}}_{z}^{2}+\sigma_{z}^{2}\right)\left[\sigma_{x}^{2}+{\sigma^{\prime}}_{x}^{2}+\left(\sigma_{y}^{2}+{\sigma^{\prime}}_{y}^{2}\right)\tan^{2}\frac{\phi}{2}\right]}}\;,$
(9)

where $N_{\text{pp}},N_{e}$ are populations of the incident radiation pulse
and of the bunch, respectively, $\sigma_{x,y,z},\sigma^{\prime}_{x,y,z}$ are
their horizontal, longitudinal, and vertical rms dimensions and
$\sigma_{\text{C}}$ is the total Compton cross-section in Eq. (7).

yzx$\phi$y’ Figure 6: Scheme of bunch–pulse crossing.

##  III Compton sources at the European XFEL

Owing to the high density and high energy electron beams needed to drive X-ray
FELs, we propose to adapt them as Compton sources of gamma-ray photons in the
GeV energy range, e.g., for experiments in Quantum Chromo-Dynamics.

Figure 7 presents a scheme of such experimental setup. It is important that
the electron spectrum upon interaction with the radiation pulse consists of
two distinct parts: the recoiled and the undisturbed fractions. The recoiled
fraction is small, less than $10^{-4}$ of the total.

magnet spectrometer electronsto dumpphotonstargetcoinc.radiatortagger Figure
7: Scheme of experimental setup.

‘Photon tagging’ assigns the energy loss of _individual electron_ to the
energy of the emitted photon in coincidence, as depicted in Fig. 8.

$\epsilon_{\gamma}$non-scattered electrons$E_{\text{el}}$coincidence Figure 8:
Relation of the electron’s (red) to gamma’s (blue) spectra.

###  III.1 Radiation source: infrared laser

Below we discuss whether a ten $\mu$A - order average current, deliverable by
the accelerator of the European XFEL, is able to produce sufficient flux of
scattered GeV photons.

The number of back-scattered photons per crossing is proportional to the
populations and a geometrical factor. The latter is inversely proportional to
the dimensions of bunch and photon pulse. The bunch population is by three
orders of magnitude smaller than that in LEPS (storage ring SPring-8, [3]),
but it is compensated by a much larger population of the laser photons stored
in resonator. In addition, the small transverse size of low-emittance electron
bunches at the European XFEL enhances the yield as well.

The key element of the laser system proposed for the European XFEL is a
resonator. The resonator is capable of storing up to 10–20 thousand pulses of
100 Watt infrared laser. The stored energy may reach levels as high as 30 mJ
per pulse, [13].

The main parameters of the proposed gamma source together with those under
construction, LEPS2 [3] and GlueX [4], are listed in Table 1. The range of
gamma-photon energy for the European XFEL-based source is taken such that the
polarization at the bottom edge equals to 1/2:

$\rho(\zeta_{1/2})=\frac{1}{2}\Rightarrow\zeta_{1/2}=\frac{3-\sqrt{3}}{2}\approx
0.63\;.$

The polarization is linear for GlueX, while for other projects it is
determined by the polarization of laser. Corresponding parameters in each
column are marked with the same color.

Table 1: Main parameters of gamma sources.  experiment | LEPS2 | GlueX | XFEL+laser
---|---|---|---
parameter | (SPring8) | (CEBAF) | (European XFEL)
$E_{e}$, GeV | 8 | 12 | 4.0
|  |  | 8.0
|  |  | 11.5
|  |  | 14.0
|  |  | 16.5
|  |  | 17.5
$I_{\text{beam}}$, A | 0.1 | $1.5\times 10^{-7}$ | $2.7\times 10^{-5}$
radiator | lasers | diamond | las+res
|  | $\Delta=50\,\mu\text{m}$ |
$\lambda_{\text{las}}$, nm | 355 | $\sim 2\times 0.15$ | 1064
| 266 |  |
$\epsilon_{\gamma}$, GeV | 1.3–2.39 | 8.4–9 | 0.15–0.23
| 1.6–2.89 |  | 0.55–0.87
|  |  | 1.09–1.72
|  |  | 1.57–2.47
|  |  | 2.11–3.33
|  |  | 2.62–4.14
$\rho$, % | $\lesssim 98$ | (35–40) | 50–100
$W_{\text{laser}}$, W | 8 |  | $\lesssim 10^{6}$
| 0.8 |  |
flux, phot/s | $10^{6}$ | $10^{7}$ |  $\lesssim 10^{9}$ 111assuming $10^{3}$ events per second
| $10^{5}$ |  |

###  III.2 Capability of XFEL accelerator to produce GeV photons

Comparing the European XFEL accelerator to the storage ring SPring-8, [14],
where the GeV-photons Compton source LEPS2 has been built, one can see a 4
orders of magnitude larger current of the electrons circulating in the ring,
$\sim 300\,\text{mA}$, as compared with $\sim 10\,\mu\text{A}$ in the linac.
On the other hand, due to a factor two higher maximum electron energy of the
linac (17.5 GeV), the European XFEL could produce four times higher photon
energy with the same laser wavelength. Moreover, the quality of the linac beam
– in terms of transverse emittances and the bunch length – allows for
substantial reduction of the geometric factor [the denominator in (9)].

Finally, for the realistic case of a moderate but still higher energy than
attained in [14], the laser resonator of the kind developed for Compton x-ray
sources – see, e.g., [15, 16, 17] – and the sources of polarized positrons for
ILC and CLIC projects of lepton colliders [18, 19, 20] may be employed. The
optical resonator is capable of increasing the power of a YAG laser (1 $\mu$m
wavelength) by $(2\dots 4)\times 10^{3}$ times, see e.g. papers on Compton
X-ray compact sources [21, 22].

Making use of the parameters of the European XFEL accelerator [23], we
estimated the performance of a Compton source of polarized GeV photons. We
took the parameters of a laser system that was considered for the projects of
lepton colliders ILC and CLIC. Currently, a similar system is under
construction at the IJC laboratory (Orsay, France) for the ThomX Compton
source.

The dependency of the maximum energy of photons on the electron energy is
presented in Fig.9.

$5$$7.5$$10$$12.5$$15$$17.5$$20$$1$$2$$3$$4$$E_{e}$,
GeV$E^{\mathrm{max}}_{\gamma}$, GeV Figure 9: The gamma-photon maximal energy
as a function of the electron energy for $E_{\text{laser}}=1\,\text{eV}$.

A 16.5 GeV electron beam of the European XFEL, assuming a laser resonator
storing 1 mJ laser pulse at 1.164 eV photon energy (providing 8 degrees
crossing angle) produces a Compton spectrum with the maximum energy $\sim 4$
GeV with an efficiency of $10^{-4}$ photons per electron-crossing. The data
from estimations and simulations are listed in Table 2.

Simulations were done with a Monte–Carlo code initially written for study of
beam dynamics in Compton storage rings. Spectra of the photon flux are
presented in Fig. 10.

Table 2: Simulated parameters of Compton source parameter | value | units
---|---|---
electron energy | 16.5 | GeV
bunch charge | 1 | nC
bunch length | 24 |  $\mu$m
bunch tr. $\sigma_{x}/\sigma_{z}$ at IP | 20/20 |  $\mu$m
energy of photon | 1.164 | eV
pulse energy | 1 | mJ
pulse trans. $\sigma^{\prime}_{x}/\sigma^{\prime}_{z}$ at IP | 20/20 |  $\mu$m
pulse long. $\sigma^{\prime}_{y}$ at IP | 50 |  $\mu$m
crossing angle | 8 | deg
max. gammas energy | 3.88 | GeV
half max angle $\psi_{1/2}$ | 35 |  $\mu$rad
yield per crossing | $3.5\times 10^{4}$ | gammas
$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$0$$20$$40$$60$$80$$100$$\zeta=\omega/\omega^{\mathrm{max}}$photons/bin
Figure 10: Simulated photon spectrum (black) and its constituents: ‘positively
polarized gammas’ (red) and ‘negatively polarized’ (blue). 16.5 GeV electrons
crossed by 1.164 eV laser at 8 deg.

Our simulations show that the spectrum is rather uniform, and the fraction of
negatively polarized gammas negligible for $\zeta\gtrsim 0.8$.

##  IV Summary and Discussion

Installation of a laser resonator and an electron tagger system at the
European XFEL makes it possible to carry out QCD experiments. The energy range
of Compton backscattered gamma-ray photons, their flux and polarization are in
range with those at experimental setups under construction and fill the
‘energy gap’ between the most modern experimental setups .

Realization of the proposal would be a “stepping stone” to LUXE project [7]
dedicated to the study of interaction between relativistic electrons with
powerful laser pulses.

###### Acknowledgements.

Authors would like to thank Serguei Molodtsov for his interest and support.
Computations were done with help of Wolfram Mathematica Cloud package [24],
provided to one of coauthors (EB).

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# 3D Boson representation of affine Yangian of ${\mathfrak{gl}}(1)$ and 3D
cut-and-join operators

Wang Na†***Corresponding author<EMAIL_ADDRESS>, Zhang Can† Wu Ke‡
†School of Mathematics and Statistics, Henan University, Kaifeng, 475001,
China
‡School of Mathematical Sciences, Capital Normal University, Beijing 100048,
China

###### Abstract

We have constructed 3D Bosons. In this paper, we show the 3D Bosonic Fock
space, which is isomorphic to the vector space of 3D Young diagrams as graded
vector spaces. We use 3D Bosons to represent the generators of the affine
Yangian of ${\mathfrak{gl}}(1)$ and define the 3D cut-and-join operators. Then
we discuss the 3D Boson representation of $W$-operators in matrix models.

Keywords:  3D Young diagrams, $W_{1+\infty}$ algebra, Affine Yangian, Schur
functions, Jack polynomials.

##  1 Introduction

Schur functions and Jack polynomials defined on 2D Young diagrams are
attractive research objects. Schur functions were used to determine
irreducible characters of highest weight representations of the classical
groups[1, 2, 3]. Recently they appear in mathematical physics, especially in
integrable models. In [4], the group in the Kyoto school uses Schur functions
in a remarkable way to understand the KP and KdV hierarchies. In [5, 6],
Tsilevich and Sułkowski, respectively, give the realization of the phase model
in the algebra of Schur functions and build the relations between the
$q$-boson model and Hall-Littlewood functions. In [7], we build the relations
between the statistical models, such as phase model, and KP hierarchy by using
2D Young diagrams and Schur functions. In [8], the authors show that the
states in the $\beta$-deformed Hurwitz-Kontsevich matrix model can be
represented as the Jack polynomials.

3-Jack polynomials defined on 3D Young diagrams (plane partitions) are a
generalization of Schur functions and Jack polynomials. We have constructed
3-Jack polynomials in [9]. 3D Young diagrams arose naturally in crystal
melting model[10, 11]. 3D Young diagrams also have many applications in many
fields of mathematics and physics, such as statistical models, number theory,
representations of some algebras (Ding-Iohara-Miki algebras, affine Yangian,
etc). 3-Jack polynomials behave the same with 3D Young diagrams when we
consider the MacMahon representation of the affine Yangian of
${\mathfrak{gl}}(1)$.

We have constructed the 3D Bosons in [12]. In this paper, we firstly construct
the 3D Bosonic Fock space. As the 2D Bosonic Fock space is isomorphic to the
space of 2D Young diagrams or the space of Schur functions defined on 2D Young
diagrams, the 3D Bosonic Fock space is isomorphic to the space of 3D Young
diagrams or the space of 3-Jack polynomials defined on 3D Young diagrams.

The MacMahon representation space of affine Yangian of ${\mathfrak{gl}}(1)$ is
also the space of 3D Young diagrams[13]. The main result of this paper is that
we use 3D Bosons to realize the generators of the affine Yangian of
${\mathfrak{gl}}(1)$. In [12], we have obtained the representation of
$W_{1+\infty}$ algebra, which have actions on 3D Young diagrams, by 3D Bosons,
then we can see the relations between the operators in affine Yangian of
${\mathfrak{gl}}(1)$ and $W_{1+\infty}$ algebra since they both can be
represented by 3D Bosons.

Use the operators $\psi_{j}$ in affine Yangian of ${\mathfrak{gl}}(1)$, we
define 3D cut-and-join operators. 2D cut-and-join operators are operators
commutative with each other, which have prominent applications in matrix
models[8], the simplest one is

$\frac{1}{2}\sum_{n,m=1}^{\infty}\left((n+m)p_{n}p_{m}\frac{\partial}{\partial
p_{n+m}}+nmp_{n+m}\frac{\partial^{2}}{\partial p_{n}\partial p_{m}}\right).$

Schur functions are the common eigenstates of 2D cut-and-join operators. In
this paper, we use 3D Bosons to represent 3D cut-and-join operators. The
eigenstates are 3-Jack polynomials, we show that by some examples. The
$W$-operators of the 3D generalizations of some matrix models can also be
represented by 3D Bosons.

The paper is organized as follows. In section 2, we recall the $W_{1+\infty}$
algebra and the affine Yangian of ${\mathfrak{gl}}(1)$. In section 3, we
recall 3D Bosons and give the 3D Bosonic Fock space. In section 4, we use 3D
Bosons to represent the generators $\psi_{3},\ e_{0},\ f_{0}$ of affine
Yangian of ${\mathfrak{gl}}(1)$. In section 5, we construct 3D cut-and-join
operators and use 3D Bosons to represent the operators in some matrix models.

##  2 $W_{1+\infty}$ algebra and affine Yangian of ${\mathfrak{gl}}(1)$

In this section, we recall the $W_{1+\infty}$ algebra and the affine Yangian
of ${\mathfrak{gl}}(1)$, which all have the representations on 3D Young
diagrams.

###  2.1 Affine Yangian of ${\mathfrak{gl}}(1)$

Let $h_{1},h_{2}$ and $h_{3}$ be three complex numbers satisfying
$h_{1}+h_{2}+h_{3}=0$. Define

$\displaystyle\sigma_{2}$ $\displaystyle=$ $\displaystyle
h_{1}h_{2}+h_{1}h_{3}+h_{2}h_{3},$ $\displaystyle\sigma_{3}$ $\displaystyle=$
$\displaystyle h_{1}h_{2}h_{3}.$

We associate $h_{1},\ h_{2},\ h_{3}$ to $y,\ x,\ z$-axis respectively.

The affine Yangian $\mathcal{Y}$ of ${\mathfrak{gl}}(1)$ is an associative
algebra with generators $e_{j},f_{j}$ and $\psi_{j}$, $j=0,1,\ldots$ and the
following relations[13, 14]

$\displaystyle\left[\psi_{j},\psi_{k}\right]=0,$ (1)
$\displaystyle\left[e_{j+3},e_{k}\right]-3\left[e_{j+2},e_{k+1}\right]+3\left[e_{j+1},e_{k+2}\right]-\left[e_{j},e_{k+3}\right]$
$\displaystyle\quad+\sigma_{2}\left[e_{j+1},e_{k}\right]-\sigma_{2}\left[e_{j},e_{k+1}\right]-\sigma_{3}\left\\{e_{j},e_{k}\right\\}=0,$
(2)
$\displaystyle\left[f_{j+3},f_{k}\right]-3\left[f_{j+2},f_{k+1}\right]+3\left[f_{j+1},f_{k+2}\right]-\left[f_{j},f_{k+3}\right]$
$\displaystyle\quad+\sigma_{2}\left[f_{j+1},f_{k}\right]-\sigma_{2}\left[f_{j},f_{k+1}\right]+\sigma_{3}\left\\{f_{j},f_{k}\right\\}=0,$
(3) $\displaystyle\left[e_{j},f_{k}\right]=\psi_{j+k},$ (4)
$\displaystyle\left[\psi_{j+3},e_{k}\right]-3\left[\psi_{j+2},e_{k+1}\right]+3\left[\psi_{j+1},e_{k+2}\right]-\left[\psi_{j},e_{k+3}\right]$
$\displaystyle\quad+\sigma_{2}\left[\psi_{j+1},e_{k}\right]-\sigma_{2}\left[\psi_{j},e_{k+1}\right]-\sigma_{3}\left\\{\psi_{j},e_{k}\right\\}=0,$
(5)
$\displaystyle\left[\psi_{j+3},f_{k}\right]-3\left[\psi_{j+2},f_{k+1}\right]+3\left[\psi_{j+1},f_{k+2}\right]-\left[\psi_{j},f_{k+3}\right]$
$\displaystyle\quad+\sigma_{2}\left[\psi_{j+1},f_{k}\right]-\sigma_{2}\left[\psi_{j},f_{k+1}\right]+\sigma_{3}\left\\{\psi_{j},f_{k}\right\\}=0,$
(6)

together with boundary conditions

$\displaystyle\left[\psi_{0},e_{j}\right]=0,\left[\psi_{1},e_{j}\right]=0,\left[\psi_{2},e_{j}\right]=2e_{j},$
(7)
$\displaystyle\left[\psi_{0},f_{j}\right]=0,\left[\psi_{1},f_{j}\right]=0,\left[\psi_{2},f_{j}\right]=-2f_{j},$
(8)

and a generalization of Serre relations

$\displaystyle\mathrm{Sym}_{(j_{1},j_{2},j_{3})}\left[e_{j_{1}},\left[e_{j_{2}},e_{j_{3}+1}\right]\right]=0,$
(9)
$\displaystyle\mathrm{Sym}_{(j_{1},j_{2},j_{3})}\left[f_{j_{1}},\left[f_{j_{2}},f_{j_{3}+1}\right]\right]=0,$
(10)

where $\mathrm{Sym}$ is the complete symmetrization over all indicated indices
which include $6$ terms.

The affine Yangian $\mathcal{Y}$ has a representation on 3D Young diagrams. As
in our paper [15], we use the following notations. For a 3D Young diagram
$\pi$, the notation $\Box\in\pi^{+}$ means that this box is not in $\pi$ and
can be added to $\pi$. Here “can be added” means that when this box is added,
it is still a 3D Young diagram. The notation $\Box\in\pi^{-}$ means that this
box is in $\pi$ and can be removed from $\pi$. Here “can be removed” means
that when this box is removed, it is still a 3D Young diagram. For a box
$\Box$, we let

$h_{\Box}=h_{1}y_{\Box}+h_{2}x_{\Box}+h_{3}z_{\Box},$ (11)

where $(x_{\Box},y_{\Box},z_{\Box})$ is the coordinate of box $\Box$ in
coordinate system $O-xyz$. Here we use the order $y_{\Box},x_{\Box},z_{\Box}$
to match that in paper [13].

Following [13, 14], we introduce the generating functions:

$\displaystyle e(u)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{e_{j}}{u^{j+1}},$ $\displaystyle f(u)$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{\infty}\frac{f_{j}}{u^{j+1}},$ (12)
$\displaystyle\psi(u)$ $\displaystyle=$ $\displaystyle
1+\sigma_{3}\sum_{j=0}^{\infty}\frac{\psi_{j}}{u^{j+1}},$

where $u$ is a parameter. Introduce

$\psi_{0}(u)=\frac{u+\sigma_{3}\psi_{0}}{u}$ (13)

and

$\displaystyle\varphi(u)=\frac{(u+h_{1})(u+h_{2})(u+h_{3})}{(u-h_{1})(u-h_{2})(u-h_{3})}.$
(14)

For a 3D Young diagram $\pi$, define $\psi_{\pi}(u)$ by

$\displaystyle\psi_{\pi}(u)=\psi_{0}(u)\prod_{\Box\in\pi}\varphi(u-h_{\Box}).$
(15)

In the following, we recall the representation of the affine Yangian on 3D
Young diagrams as in paper [13] by making a slight change. The representation
of affine Yangian on 3D Young diagrams is given by

$\displaystyle\psi(u)|\pi\rangle$ $\displaystyle=$
$\displaystyle\psi_{\pi}(u)|\pi\rangle,$ (16) $\displaystyle e(u)|\pi\rangle$
$\displaystyle=$
$\displaystyle\sum_{\Box\in\pi^{+}}\frac{E(\pi\rightarrow\pi+\Box)}{u-h_{\Box}}|\pi+\Box\rangle,$
(17) $\displaystyle f(u)|\pi\rangle$ $\displaystyle=$
$\displaystyle\sum_{\Box\in\pi^{-}}\frac{F(\pi\rightarrow\pi-\Box)}{u-h_{\Box}}|\pi-\Box\rangle$
(18)

where $|\pi\rangle$ means the state characterized by the 3D Young diagram
$\pi$ and the coefficients

$E(\pi\rightarrow\pi+\square)=-F(\pi+\square\rightarrow\pi)=\sqrt{\frac{1}{\sigma_{3}}\operatorname{res}_{u\rightarrow
h_{\square}}\psi_{\pi}(u)}.$ (19)

Specially,

$\displaystyle\psi_{1}|\pi\rangle=0,\ \psi_{2}|\pi\rangle=2|\pi||\pi\rangle,\
\psi_{3}|\pi\rangle=\sum_{\Box\in\pi}(6h_{\Box}+2\psi_{0}\sigma_{3})|\pi\rangle,$
$\displaystyle\psi_{4}|\pi\rangle=\sum_{\Box\in\pi}(12h_{\Box}^{2}-2\sigma_{2}+6h_{\Box}\psi_{0}\sigma_{3})|\pi\rangle,$
(20)

where $|\pi|$ is the box number of $\pi$.

In the following of this paper, we treat
$E(\pi\rightarrow\pi+\square)|\pi+\square\rangle$ as one element and still
denote it by $|\pi+\square\rangle$, then 3D Young diagrams depend on the box
growth process. 3-Jack polynomials $J_{\pi}$ behave the same as $\pi$ exactly.

###  2.2 $W_{1+\infty}$ algebra

For any 3D Young diagram, we cut it into slices by the plane $z=j$, every
slice is a 2D Young diagram. Then 3D Young diagrams can be treated as a series
of 2D Young diagrams. When we consider the 3D Young diagrams which have at
most $N$ layers in $z$-axis direction, we suppose
$\psi_{0}=-\frac{N}{h_{1}h_{2}}$. Let $a_{j,n}$ be the 2D Bosons associated to
the 2D Young diagrams on the slice $z=j$ of 3D Young diagrams with the
relation

$[a_{j,n},a_{k,m}]=-\frac{1}{h_{1}h_{2}}\delta_{j,k}n\delta_{n+m,0}.$ (21)

The operators $V_{j,n}$ of the $W_{1+\infty}$ algebra can be represented by
$a_{j,n},\ j=1,2,\cdots,N$, which corresponds to that a 3D Young diagram can
be represented by a series of 2D Young diagrams. This means that the operators
$V_{j,n}$ have the representation on the space of 3D Young diagrams.

The fields

$J_{j}(z)=\sum_{n\in\mathbb{Z}}a_{j,n}z^{-n-1}\ \ \text{and}\ \
V_{j}(z)=\sum_{n\in\mathbb{Z}}V_{j,n}z^{-j-n}.$ (22)

Let[12]

$\displaystyle V_{1}(z)$ $\displaystyle=$ $\displaystyle
J_{1}(z)+J_{2}(z)+\cdots+J_{N}(z),$ (23) $\displaystyle V_{2}(z)$
$\displaystyle=$
$\displaystyle-\frac{h_{1}h_{2}}{2}\sum_{j=1}^{N}:J_{j}(z)J_{j}(z):-\frac{h_{1}h_{2}\alpha_{0}}{2}\sum_{j=1}^{N}(N+1-2j)J_{j}^{\prime}(z),$
(24) $\displaystyle V_{3}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{3}h_{1}^{2}h_{2}^{2}\sum_{j=1}^{N}:J_{1}(z)^{3}:-\frac{1}{2}\alpha_{0}h_{1}^{2}h_{2}^{2}\sum_{j<k}J_{j}J_{k}^{\prime}(z)+\frac{1}{2}\alpha_{0}h_{1}^{2}h_{2}^{2}\sum_{j<k}J_{j}^{\prime}J_{k}(z)$
(25)
$\displaystyle+\frac{1}{2}\alpha_{0}h_{1}^{2}h_{2}^{2}\sum_{j=1}^{N}(N+1-2j)J_{j}^{\prime}J_{j}(z)$
$\displaystyle-\alpha_{0}^{2}h_{1}^{2}h_{2}^{2}\sum_{j=1}^{N}\left(\frac{(j-1)(N-j)}{2}-\frac{(N-1)(N-j)}{12}\right)J_{j}^{\prime\prime}(z),$

and[16]

$\displaystyle V_{4}(z)$ $\displaystyle=$ $\displaystyle-
h_{1}^{3}h_{2}^{3}\left(\frac{1}{4}\sum_{j=1}^{N}J_{j}J_{j}J_{j}J_{j}(z)-\frac{\alpha_{0}}{2}\sum_{j<k}J_{j}J_{j}J_{k}^{\prime}(z)-{\alpha_{0}}\sum_{j<k<l}(l-3)J_{j}J_{k}^{\prime}J_{l}(z)\right.$
$\displaystyle+\frac{\alpha_{0}}{2}\sum_{j<k}J_{j}J_{k}^{\prime}J_{k}(z)+\frac{\alpha_{0}}{2}\sum_{j<k}J_{j}^{\prime}J_{j}J_{k}(z)+\frac{\alpha_{0}}{2}\sum_{j<k}J_{j}^{\prime}J_{k}J_{k}(z)$
$\displaystyle-{\alpha_{0}}\sum_{j<k<l}(l-3)J_{j}^{\prime}J_{k}J_{l}(z)+\frac{\alpha_{0}}{2}\sum_{j=1}^{N}(N+1-2j)J_{j}^{\prime}J_{j}J_{j}(z)$
$\displaystyle-\frac{\alpha_{0}^{2}}{4}\sum_{j<k}(N+1-2k)J_{j}J_{k}^{\prime\prime}(z)-\frac{\alpha_{0}^{2}}{4}\sum_{j<k}(N+1-2j)J_{j}^{\prime\prime}J_{k}(z)$
$\displaystyle\left.-\alpha_{0}^{2}\sum_{j<k}(k-j)J_{j}^{\prime}J_{k}^{\prime}(z)+\frac{3}{20h_{1}h_{2}}\sum_{j=1}^{N}J_{j}^{\prime}J_{j}^{\prime}(z)-\frac{1}{10h_{1}h_{2}}\sum_{j=1}^{N}J_{j}^{\prime\prime}J_{j}(z)+\cdots\right).$

Specially, when $N=1,h_{1}=1,h_{2}=-1$, 3D Young diagrams become 2D Young
diagrams and 3-Jack polynomials become Schur functions. We denote $V_{j}(z)$
in this special case by $V_{j}^{2DS}(z)$. Then

$\displaystyle{V}_{1}^{2DS}(z)$ $\displaystyle=$ $\displaystyle J_{1}(z),$
(26) $\displaystyle{V}_{2}^{2DS}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}:J_{1}(z)^{2}:,$ (27) $\displaystyle{V}_{3}^{2DS}(z)$
$\displaystyle=$ $\displaystyle\frac{1}{3}:J_{1}(z)^{3}:,$ (28)
$\displaystyle{V}_{4}^{2DS}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{4}:J_{1}(z)^{4}:-\frac{3}{20}:J_{1}^{\prime}(z)^{2}:+\frac{1}{10}:J_{1}^{\prime\prime}(z)J_{1}(z):$
(29)

with the OPE

$J_{1}(z)J_{(}w)\sim\frac{1}{(z-w)^{2}}.$

This special case $V_{j}^{2DS}(z)$ of the $W_{1+\infty}$ algebra have
representation on the space of 2D Young diagrams or the Schur functions
defined on 2D Young diagrams. The general case $V_{j}(z)$ of the
$W_{1+\infty}$ algebra have representation on the space of 3D Young diagrams
or the 3-Jack polynomials defined on 3D Young diagrams.

The OPEs of $V_{j}(z)V_{k}(w)$ can be found in [12, 16]. From the OPEs, the
relations $[V_{j,m},V_{k,n}]$ can be obtained. We list the first few of them:

$\displaystyle[V_{1,m},V_{1,n}]=$
$\displaystyle-\frac{N}{h_{1}h_{2}}m\delta_{m+n,0}=\psi_{0}m\delta_{m+n,0},$
$\displaystyle[V_{1,m},V_{2,n}]=$ $\displaystyle mV_{1,m+n},$
$\displaystyle[V_{2,m},V_{2,n}]=$
$\displaystyle(m-n)V_{2,m+n}-(\psi_{0}\sigma_{2}+\psi_{0}^{3}\sigma_{3}^{2})\frac{m^{3}-m}{12}\delta_{m+n,0},$
$\displaystyle[V_{1,m},V_{3,n}]=$ $\displaystyle 2mV_{2,m+n},$
$\displaystyle[V_{2,m},V_{3,n}]=$
$\displaystyle-\frac{1}{6}(\sigma_{2}+\sigma_{3}^{2}\psi_{0}^{2})(m^{3}-m)V_{1,m+n}+(2m-n)V_{3,m+n}.$

##  3 3D Bosons and 3D Bosonic Fock space

We have shown the method to construct the 3D Bosons in [12]. In this section,
we recall 3D Bosons and give the 3D Bosonic Fock space.

We denote 3D Bosons by $b_{n,j}$. When 3D Young diagrams have at most $N$
layers in the $z$-axis direction, $j$ in $b_{n,j}$ equals $1,2,\cdots,N$,
which means that $b_{n,j}=0$ when $j>N$. Specially, when $N=1$, $b_{n,j}$
becomes $b_{n,1}$, which is the normal 2D Bosons. This special case
corresponds to that 2D Young diagrams can be treated as the 3D Young diagrams
which have one layer in $z$-axis direction. 3D Bosons $b_{n,j}$ can be
represented by 2D Bosons $a_{j,n}$:

$\displaystyle b_{n,1}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{N}a_{j,n},$
$\displaystyle b_{n,2}$ $\displaystyle=$ $\displaystyle-
h_{1}h_{2}(1-\frac{1}{N})\sum_{j=1}^{N}\sum_{k+l=n}:a_{j,k}a_{j,l}:+\frac{2h_{1}h_{2}}{N}\sum_{j<k}\sum_{m+l=n}:a_{j,m}a_{k,l}:$
$\displaystyle+h_{3}\sum_{j=1}^{N}(N+1-2j)(-n-1)a_{j,n},$ $\displaystyle
b_{n,3}$ $\displaystyle=$ $\displaystyle
6h_{1}^{2}h_{2}^{2}\left(-\sum_{j<k<l}\sum_{m+p+q=n}:a_{j,m}a_{k,p}a_{l,q}:-\alpha_{0}\sum_{j<k}\sum_{m+q=n}(j-1)(-m-1):a_{j,m}a_{k,q}:\right.$
$\displaystyle-\alpha_{0}\sum_{j<k}\sum_{m+q=n}(k-2)(-q-1):a_{j,m}a_{k,q}:-\frac{\alpha_{0}^{2}}{2}\sum_{j=1}^{N}(j-1)(j-2)(-n-1)(-n-2)a_{j,n}$
$\displaystyle+\frac{N-2}{N}\sum_{j=1}^{N}\sum_{k<l}\sum_{m+p+q=n}:a_{j,m}a_{k,p}a_{l,q}:+\frac{(N-2)\alpha_{0}}{N}\sum_{j,k=1}^{N}\sum_{m+q=n}(k-1)(-q-1):a_{j,m}a_{k,q}:$
$\displaystyle-\frac{(N-1)(N-2)}{3N^{2}}\sum_{j,k,l=1}^{N}\sum_{m+p+q=n}:a_{j,m}a_{k,p}a_{l,q}:+\frac{(N-2)\alpha_{0}^{2}}{2}\sum_{j=1}^{N}(j-1)(-n-1)(-n-2)a_{j,n}$
$\displaystyle+\frac{(N-2)\alpha_{0}}{2}\sum_{j<k}\sum_{m+q=n}(-m-q-2):a_{j,m}a_{k,q}:-\frac{(N-1)(N-2)\alpha_{0}^{2}}{12}\sum_{j=1}^{N}(-n-1)(-n-2)a_{j,n}$
$\displaystyle\left.-\frac{(N-1)(N-2)\alpha_{0}}{2N}\sum_{j,k=1}^{N}\sum_{m+q=n}(-m-1):a_{j,m}a_{k,q}:\right).$

The relations between 3D Bosons can be obtained from the OPEs
$B_{j}(z)B_{k}(w)$[12], we list some of them:

$\displaystyle[b_{m,1},b_{n,1}]=$ $\displaystyle\ \psi_{0}m\delta_{m+n,0},$
(30) $\displaystyle[b_{m,1},b_{n,j\geq 2}]=$ $\displaystyle\ 0,$ (31)
$\displaystyle[b_{m,2},b_{n,2}]=$ $\displaystyle\
2(m-n)b_{m+n,2}-2(1+\sigma_{2}\psi_{0}+\psi_{0}^{3}\sigma_{3}^{2})\frac{m^{3}-1}{6}\delta_{m+n,0},$
(32) $\displaystyle[b_{m,2},b_{n,3}]=$ $\displaystyle\ 2(2m-n)b_{m+n,3}.$ (33)

Denote the vacuum state associated to 2D Bosons $a_{j,n}$ by $|0\rangle_{j}$,
and Define

$|0\rangle=\sum_{j=1}^{N}|0\rangle_{j}.$

We know that $b_{n,j}|0\rangle=0$ for any $n>0$, and $b_{n,j}|0\rangle=0$ for
any $n\leq 0,j>-n$. Denote the algebra generated by 3D Bosons $b_{n,j}$ by
$\mathfrak{B}$. Define the 3D Bosonic Fock space by

$\mathfrak{B}\cdot|0\rangle:=\\{a|0\rangle\ |\ a\in\mathfrak{B}\\}.$ (34)

Then 3D Bosonic Fock space has a basis

$\\{b_{-n_{1},j_{1}}b_{-n_{2},j_{2}}\cdots b_{-n_{r},j_{r}}|0\rangle\ |\
0<n_{1}\leq n_{2}\leq\cdots\leq n_{r},j_{i}\leq n_{i}\\}.$ (35)

Note that

$b_{-n,1}b_{-m,j\geq 2}|0\rangle=b_{-m,j\geq 2}b_{-n,1}|0\rangle,$

which is the same with that for 2D, but $b_{-n,i\geq 2}b_{-m,j\geq
2}|0\rangle$ and $b_{-m,j\geq 2}b_{-n,i\geq 2}|0\rangle$ may do not equal to
each other. For example,

$b_{-2,2}b_{-3,2}|0\rangle=b_{-3,2}b_{-2,2}|0\rangle+2b_{-5,2}|0\rangle.$

Define the degree of $b_{-n,j}$ by $n$, and the degree of $|0\rangle$ by zero.
Then 3D Bosonic Fock space is a graded space. The degree zero part is
$\mathbb{C}|0\rangle$. The degree one part is $\mathbb{C}b_{-1,1}|0\rangle$. A
basis of degree $2$ part is
$\\{b_{-1,1}^{2}|0\rangle,b_{-2,1}|0\rangle,b_{-2,2}|0\rangle\\}$. The 3D
Bosonic Fock space is isomorphic to the space of 3D Young diagrams, it is also
isomorphic to the space of 3-Jack polynomials. Let

$P_{n,k}=b_{-n,k}|0\rangle$ (36)

as in [12], and we denote

$b_{-n_{1},j_{1}}b_{-n_{2},j_{2}}\cdots b_{-n_{r},j_{r}}|0\rangle$

by $P_{n_{1},j_{1}}P_{n_{2},j_{2}}\cdots P_{n_{r},j_{r}}$, which explains the
variables in 3-Jack polynomials. Note that for any 3-Jack polynomial
$J_{\pi}$, only $P_{n,1}J_{\pi}$ equals the normal multiplication of $P_{n,1}$
and $J_{\pi}$, $P_{n,j\geq 2}J_{\pi}$ equals the action $b_{-n,j\geq 2}\cdot
J_{\pi}$. We give an example. Let

$\displaystyle a_{j,-n}|0\rangle_{j}=p_{j,n}|0\rangle_{j},\ \
a_{j,n}|0\rangle_{j}=-\frac{1}{h_{1}h_{2}}\frac{\partial}{\partial
p_{j,n}}|0\rangle_{j},\ \ n>0,$

where $p_{j,n}$ is the normal power sum on the slice $z=j$ of 3D Young
diagrams. We know that[17]

$\displaystyle P_{2,1}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{N}p_{j,2},$
$\displaystyle P_{2,2}$ $\displaystyle=$ $\displaystyle-
h_{1}h_{2}\sum_{j=1}^{N}p_{j,1}^{2}+\frac{h_{1}h_{2}}{N}(\sum_{j=1}^{N}p_{j,1})^{2}-\sum_{j=1}^{N}(N-2j+1)h_{3}p_{j,2}.$

It can be calculated that $P_{2,1}P_{2,2}$ equals the normal multiplication

$\sum_{j=1}^{N}p_{j,2}\left(-h_{1}h_{2}\sum_{j=1}^{N}p_{j,1}^{2}+\frac{h_{1}h_{2}}{N}(\sum_{j=1}^{N}p_{j,1})^{2}-\sum_{j=1}^{N}(N-2j+1)h_{3}p_{j,2}\right),$

but $P_{2,2}P_{2,2}$ does not equal

$\left(-h_{1}h_{2}\sum_{j=1}^{N}p_{j,1}^{2}+\frac{h_{1}h_{2}}{N}(\sum_{j=1}^{N}p_{j,1})^{2}-\sum_{j=1}^{N}(N-2j+1)h_{3}p_{j,2}\right)^{2}.$

##  4 3D Boson representation of affine Yangian of ${\mathfrak{gl}}(1)$

In this section, we use 3D Bosons to represent the affine Yangian of
${\mathfrak{gl}}(1)$. From the relations in the affine Yangian of
${\mathfrak{gl}}(1)$, we only need to represent the operators $\psi_{3},\
e_{0},\ f_{0}$.

######  Theorem 4.1.

The affine Yangian of ${\mathfrak{gl}}(1)$ can be represented by the 3D Bosons
$b_{n,j}$ in the following way:

$\displaystyle e_{0}$ $\displaystyle=$ $\displaystyle b_{-1,1},$ (37)
$\displaystyle f_{0}$ $\displaystyle=$ $\displaystyle-b_{1,1},$ (38)
$\displaystyle\psi_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}b_{0,3}+\frac{3}{\psi_{0}}\sum_{n>0}(b_{-n,1}b_{n,2}+b_{-n,2}b_{n,1})$
(39)
$\displaystyle+\frac{3}{\psi_{0}^{2}}\sum_{n,m>0}(b_{-n,1}b_{-m,1}b_{n+m,1}+b_{-n-m,1}b_{n,1}b_{m,1})$
$\displaystyle+3\sigma_{3}\sum_{n>0}nb_{-n,1}b_{n,1}-\sigma_{3}\psi_{0}\frac{\psi_{2}}{2},$

with

$\psi_{2}=b_{0,2}+\frac{2}{\psi_{0}}\sum_{n>0}b_{-n,1}b_{n,1}.$ (40)

From [18], we know that the operators of the affine Yangian of
${\mathfrak{gl}}(1)$ can be represented by a series of 2D Bosons $a_{j,n}$:

$\displaystyle\psi_{2}$ $\displaystyle=$
$\displaystyle-2h_{1}h_{2}\sum_{j=1}^{N}\sum_{n>0}a_{j,-n}a_{j,n},$ (41)
$\displaystyle\psi_{3}$ $\displaystyle=$ $\displaystyle
3h_{1}^{2}h_{2}^{2}\sum_{j=1}^{N}\sum_{n,m>0}(a_{j,-n-m}a_{j,n}a_{j,m}+a_{j,-n}a_{j,-m}a_{j,n+m})$
(42)
$\displaystyle+6\sigma_{3}\sum_{j<k}\sum_{n>0}na_{j,-n}a_{k,n}+(-4N+6j-3)\sigma_{3}\sum_{j=1}^{N}\sum_{n>0}a_{j,-n}a_{j,n}$
$\displaystyle+3\sigma_{3}\sum_{j=1}^{N}\sum_{n>0}na_{j,-n}a_{j,n},$

and

$\displaystyle e_{0}=\sum_{j=1}^{N}a_{j,-1},\ f_{0}=-\sum_{j=1}^{N}a_{j,1}.$
(43)

It is clear that the relations (37) and (38) hold. The relations (39) and (40)
can be proved by direct calculation, since $b_{n,j}$ can be represented by
this series of 2D Bosons $a_{j,n}$. In the following, we obtain (39) and (40)
by a simpler way. We need two lemmas.

######  Lemma 4.2.

The representation of $V_{3,0}$ by a series of 2D Bosons $a_{j,n}$ is

$\displaystyle V_{3,0}$ $\displaystyle=$ $\displaystyle
h_{1}^{2}h_{2}^{2}\sum_{j=1}^{N}\sum_{n,m>0}(a_{j,-n-m}a_{j,n}a_{j,m}+a_{j,-n}a_{j,-m}a_{j,n+m})$
(44)
$\displaystyle-\sigma_{3}\sum_{j<k}\sum_{n>0}\left(na_{k,-n}a_{j,n}-na_{j,-n}a_{k,n}\right)$
$\displaystyle-\sigma_{3}\sum_{j=1}^{N}\sum_{n>0}(N-2j+1)a_{j,-n}a_{j,n}.$

This result can be obtained from (25).

######  Lemma 4.3.

The representation of $V_{3,0}$ by 3D Bosons $b_{n,j}$ is

$\displaystyle V_{3,0}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}b_{0,3}+\frac{1}{\psi_{0}}\sum_{n>0}(b_{-n,1}b_{n,2}+b_{-n,2}b_{n,1})$
(45)
$\displaystyle+\frac{1}{\psi_{0}^{2}}\sum_{n,m>0}(b_{-n,1}b_{-m,1}b_{n+m,1}+b_{-n-m,1}b_{n,1}b_{m,1}).$

This result is obtained from

$V_{3}(z)=-\frac{1}{6}B_{3}(z)+\frac{1}{\psi_{0}}B_{1}B_{2}(z)+\frac{1}{3\psi_{0}^{2}}B_{1}B_{1}B_{1}(z).$
(46)

Note that this relation is slightly different from that in [12] since
$V_{3}(z)$ here equals $V_{3}(z)$ in [12] multiplied by $-1$.

The proof of (39).

$\displaystyle\psi_{3}-3V_{3,0}$ $\displaystyle=$ $\displaystyle
3\sigma_{3}\sum_{j<k}\sum_{n>0}\left(na_{k,-n}a_{j,n}+na_{j,-n}a_{k,n}\right)$
$\displaystyle+3\sigma_{3}\sum_{j=1}^{N}\sum_{n>0}na_{j,-n}a_{j,n}-N\sigma_{3}\sum_{j=1}^{N}\sum_{n>0}a_{j,-n}a_{j,n}$
$\displaystyle=$ $\displaystyle
3\sigma_{3}\sum_{n>0}b_{-n,1}b_{n,1}-\sigma_{3}\psi_{0}\frac{\psi_{2}}{2}.$

The proof of (40). It holds since

$\psi_{2}=2V_{2,0},$ (47)

and

$V_{2}(z)=\frac{1}{2}B_{2}(z)+\frac{1}{2\psi_{0}}B_{1}B_{1}(z).$ (48)

This relation is obtained in [12].

We give some examples about the eigenstates of $\psi_{2}$ and $\psi_{3}$ by
(40) and (39). The eigenstates of $\psi_{2}$: For 3D Young diagram of one box,

$\psi_{2}b_{-1,1}|0\rangle=\frac{2}{\psi_{0}}b_{-1,1}b_{1,1}b_{-1,1}|0\rangle=2b_{-1,1}|0\rangle.$

For 3D Young diagram of two boxes,

$\displaystyle\psi_{2}b_{-1,1}^{2}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\psi_{0}}b_{-1,1}b_{1,1}b_{-1,1}^{2}|0\rangle=4b_{-1,1}^{2}|0\rangle,$
$\displaystyle\psi_{2}b_{-2,1}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\psi_{0}}b_{-2,1}b_{2,1}b_{-2,1}|0\rangle=4b_{-2,1}|0\rangle,$
$\displaystyle\psi_{2}b_{-2,2}|0\rangle$ $\displaystyle=$ $\displaystyle
b_{0,2}b_{-2,2}|0\rangle=4b_{-2,2}|0\rangle.$

For 3D Young diagram of three boxes,

$\displaystyle\psi_{2}b_{-1,1}^{3}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\psi_{0}}b_{-1,1}b_{1,1}b_{-1,1}^{3}|0\rangle=6b_{-1,1}^{3}|0\rangle,$
$\displaystyle\psi_{2}b_{-1,1}b_{-2,1}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\psi_{0}}b_{-1,1}b_{1,1}b_{-1,1}b_{-2,1}|0\rangle+\frac{2}{\psi_{0}}b_{-2,1}b_{2,1}b_{-1,1}b_{-2,1}|0\rangle$
$\displaystyle=$ $\displaystyle 6b_{-1,1}b_{-2,1}|0\rangle,$
$\displaystyle\psi_{2}b_{-1,1}b_{-2,2}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\psi_{0}}b_{-1,1}b_{1,1}b_{-1,1}b_{-2,2}|0\rangle+b_{0,2}b_{-1,1}b_{-2,2}|0\rangle$
$\displaystyle=$ $\displaystyle 6b_{-1,1}b_{-2,2}|0\rangle,$
$\displaystyle\psi_{2}b_{-3,1}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\psi_{0}}b_{-3,1}b_{3,1}b_{-3,1}|0\rangle=6b_{-3,1}|0\rangle,$
$\displaystyle\psi_{2}b_{-3,2}|0\rangle$ $\displaystyle=$ $\displaystyle
b_{0,2}b_{-3,2}|0\rangle=6b_{-3,2}|0\rangle,$
$\displaystyle\psi_{2}b_{-3,3}|0\rangle$ $\displaystyle=$ $\displaystyle
b_{0,2}b_{-3,3}|0\rangle=6b_{-3,3}|0\rangle.$

The eigenstates of $\psi_{3}$: For 3D Young diagram of one box,

$\psi_{3}b_{-1,1}|0\rangle=3\sigma_{3}b_{-1,1}b_{1,1}b_{-1,1}|0\rangle-\sigma_{3}\psi_{0}\frac{\psi_{2}}{2}b_{-1,1}|0\rangle=2\sigma_{3}\psi_{0}b_{-1,1}|0\rangle.$

For 3D Young diagram of two boxes,

$\displaystyle\psi_{3}b_{-1,1}^{2}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{3}{\psi_{0}^{2}}b_{-2,1}b_{1,1}b_{1,1}b_{-1,1}^{2}|0\rangle+3\sigma_{3}b_{-1,1}b_{1,1}b_{-1,1}^{2}|0\rangle-\sigma_{3}\psi_{0}\frac{\psi_{2}}{2}b_{-1,1}^{2}|0\rangle$
$\displaystyle=$ $\displaystyle 6b_{-2,1}|0\rangle+3\sigma_{3}\cdot
2\psi_{0}b_{-1,1}^{2}|0\rangle-2\sigma_{3}\psi_{0}b_{-1,1}^{2}|0\rangle$
$\displaystyle=$ $\displaystyle
6b_{-2,1}|0\rangle+4\sigma_{3}\psi_{0}b_{-1,1}^{2}|0\rangle,$
$\displaystyle\psi_{3}b_{-2,1}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{3}{\psi_{0}}b_{-2,2}b_{2,1}b_{-2,1}|0\rangle+\frac{3}{\psi_{0}^{2}}b_{-1,1}b_{-1,1}b_{2,1}b_{-2,1}|0\rangle$
$\displaystyle+3\sigma_{3}2b_{-2,1}b_{2,1}b_{-2,1}|0\rangle-\sigma_{3}\psi_{0}\frac{\psi_{2}}{2}b_{-2,1}|0\rangle$
$\displaystyle=$ $\displaystyle
6b_{-2,2}|0\rangle+\frac{6}{\psi_{0}}b_{-1,1}^{2}|0\rangle+12\sigma_{3}\psi_{0}b_{-2,1}|0\rangle-2\sigma_{3}\psi_{0}b_{-2,1}|0\rangle$
$\displaystyle=$ $\displaystyle
6b_{-2,2}|0\rangle+\frac{6}{\psi_{0}}b_{-1,1}^{2}|0\rangle+10\sigma_{3}\psi_{0}b_{-2,1}|0\rangle,$
$\displaystyle\psi_{3}b_{-2,2}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{3}{\psi_{0}}b_{-2,1}b_{2,2}b_{-2,2}|0\rangle-\sigma_{3}\psi_{0}\frac{\psi_{2}}{2}b_{-2,2}|0\rangle$
$\displaystyle=$
$\displaystyle-\frac{6}{\psi_{0}}(1+\sigma_{2}\psi_{0}+\sigma_{3}^{2}\psi_{0}^{3})b_{-2,1}|0\rangle-2\sigma_{3}\psi_{0}b_{-2,2}|0\rangle,$

then we have

$\displaystyle\psi_{3}\left((1+h_{2}h_{3})\frac{1}{\psi_{0}}b_{-1,1}^{2}|0\rangle+(1+h_{2}h_{3}\psi_{0})h_{1}b_{-2,1}|0\rangle+b_{-2,2}|0\rangle\right)$
$\displaystyle=$
$\displaystyle(6h_{1}+4\sigma_{3}\psi_{0})\left((1+h_{2}h_{3}\psi_{0})\frac{1}{\psi_{0}}b_{-1,1}^{2}|0\rangle+(1+h_{2}h_{3}\psi_{0})h_{1}b_{-2,1}|0\rangle+b_{-2,2}|0\rangle\right),$

which means that

$\displaystyle\tilde{J}_{\leavevmode\hbox to19.61pt{\vbox
to10.36pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{7.11319pt}\pgfsys@lineto{4.97931pt}{9.95863pt}\pgfsys@stroke\pgfsys@invoke{
} {{{}{}}}{}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{7.11319pt}\pgfsys@lineto{12.0925pt}{9.95863pt}\pgfsys@stroke\pgfsys@invoke{
} {{{}{}}}{}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{7.11319pt}\pgfsys@lineto{19.20569pt}{9.95863pt}\pgfsys@stroke\pgfsys@invoke{
} {{{}{}}}{}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{19.20569pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{4.97931pt}{9.95863pt}\pgfsys@lineto{19.20569pt}{9.95863pt}\pgfsys@stroke\pgfsys@invoke{
} {{{}{}}}{}{{}}{}
{}{}{}\pgfsys@moveto{19.20569pt}{9.95863pt}\pgfsys@lineto{19.20569pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{1}{(h_{1}-h_{2})(h_{1}-h_{3})}\left((1+h_{2}h_{3}\psi_{0})\frac{1}{\psi_{0}}P_{1}^{2}+(1+h_{2}h_{3}\psi_{0})h_{1}P_{2,1}+P_{2,2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(h_{1}-h_{2})(h_{1}-h_{3})}\left((1+h_{2}h_{3}\psi_{0})\frac{1}{\psi_{0}}b_{-1,1}^{2}|0\rangle+(1+h_{2}h_{3}\psi_{0})h_{1}b_{-2,1}|0\rangle+b_{-2,2}|0\rangle\right)$

is the eigenstate of $\psi_{3}$ with eigenvalue
$(6h_{1}+4\sigma_{3}\psi_{0})$. Similarly, $\tilde{J}_{\leavevmode\hbox
to17.47pt{\vbox to13.2pt{\pgfpicture\makeatletter\hbox{\hskip
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{}{}{}\pgfsys@moveto{0.0pt}{7.11319pt}\pgfsys@lineto{9.95863pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{4.97931pt}{9.95863pt}\pgfsys@lineto{12.0925pt}{9.95863pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{12.0925pt}{2.84544pt}\pgfsys@lineto{12.0925pt}{9.95863pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{9.95863pt}{12.80365pt}\pgfsys@lineto{17.07182pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{17.07181pt}{5.69089pt}\pgfsys@lineto{17.07181pt}{12.80408pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ and
$\tilde{J}_{\leavevmode\hbox to12.49pt{\vbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{14.22638pt}\pgfsys@lineto{7.11319pt}{14.22638pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{7.11319pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{7.11319pt}\pgfsys@lineto{7.11319pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@lineto{4.97931pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} {{{}{}}}{}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{14.22638pt}\pgfsys@lineto{12.0925pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} {{{}{}}}{}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{7.11319pt}\pgfsys@lineto{12.0925pt}{9.95863pt}\pgfsys@stroke\pgfsys@invoke{
} {{{}{}}}{}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{12.0925pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{12.0925pt}{17.78297pt}\pgfsys@lineto{12.0925pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} {{{}{}}}{}{{}}{}
{}{}{}\pgfsys@moveto{4.97931pt}{17.07182pt}\pgfsys@lineto{12.0925pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ are the eigenstates
with eigenvalues $(6h_{2}+4\sigma_{3}\psi_{0})$ and
$(6h_{3}+4\sigma_{3}\psi_{0})$ respectively.

##  5 3D cut-and-join operators and some matrix models

In this section, we construct 3D cut-and-join operators and use 3D Bosons to
represent the operators in some matrix models.

###  5.1 2D/3D cut-and-join operators

Since Schur functions defined on 2D Young diagrams are the eigenstates of the
2D cut-and-join operators, in this subsection, we consider the special case
$N=1,h_{1}=-1,h_{2}=-1$ of $W_{1+\infty}$ algebra and affine Yangian of
$\mathfrak{gl}(1)$, where $N$ is the at most layers of 3D Young diagrams in
$z$-axis direction. Under this special case, 3D Young diagrams become 2D Young
diagrams, and 3-Jack polynomials become Schur functions.

We denote $a_{1,n}$ by $a_{n}$, clearly

$[a_{m},a_{n}]=m\delta_{m+n,0}$

specially, which have an representation on Schur functions by

$a_{-n}=p_{n},\ \ \ a_{n}=n\frac{\partial}{\partial p_{n}}$

for $n>0$. The first two examples of the cut-and-join operators in terms of
the power sums $p_{n}$ are[19]

$\displaystyle W_{1}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}np_{n}\frac{\partial}{\partial p_{n}},$
$\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{n,m=1}^{\infty}\left((n+m)p_{n}p_{m}\frac{\partial}{\partial
p_{n+m}}+nmp_{n+m}\frac{\partial^{2}}{\partial p_{n}\partial p_{m}}\right).$

We use the notation $\psi_{j}^{2DS}$ to denote $\psi_{j}$ in the special case
$N=1,h_{1}=-1,h_{2}=-1$, and the same for other operators in the affine
Yangian of $\mathfrak{gl}(1)$ and the operators in the $W_{1+\infty}$ algebra.
Clearly,

$\displaystyle\frac{1}{2}\psi_{2}^{2DS}=V_{2,0}^{2DS}=W_{1},$ (49)
$\displaystyle\frac{1}{3}\psi_{3}^{2DS}=V_{3,0}^{2DS}=W_{2}.$ (50)

Then, we can treat $W_{j=1,2}$ is the reduction of $V_{j+1,0}$ or
$\frac{1}{j+1}\psi_{j+1}$ from 3D to 2D. We will define the 3D cut-and-join
operators from $\psi_{j}$ for two reasons. The first one is that it can be
checked that 3-Jack polynomials are not the eigenstates of $V_{3,0}$, while
they are the eigenstates of all $\psi_{j}$. For the second reason, we consider
the special case $N=1$ and $h_{1},h_{2}$ arbitrary. We denote $\psi_{j}$ in
this special case by $\psi_{j}^{2D}$ and similarly for other operators. Then

$\displaystyle V_{3,0}^{2D}$ $\displaystyle=$ $\displaystyle
h_{1}^{2}h_{2}^{2}\sum_{n,m>0}(a_{-n-m}a_{n}a_{m}+a_{-n}a_{-m}a_{n+m})$
$\displaystyle\psi_{3}^{2D}$ $\displaystyle=$ $\displaystyle
3h_{1}^{2}h_{2}^{2}\sum_{n,m>0}(a_{-n-m}a_{n}a_{m}+a_{-n}a_{-m}a_{n+m})$ (51)
$\displaystyle+\sigma_{3}\sum_{n>0}(3n-1)a_{-n}a_{n}$

with $[a_{n},a_{m}]=-\frac{1}{h_{1}h_{2}}n\delta_{n+m,0}$. The eigenstates of
$\psi_{3}^{2D}$ are Jack polynomials, and in the deformed Hurwitz-Kontsevich
model, the $W$-operator

$\displaystyle\frac{1}{2}\sum_{k,l=1}^{\infty}\left(klp_{k+l}\frac{\partial}{\partial
p_{k}}\frac{\partial}{\partial
p_{l}}-h_{1}h_{2}(k+l)p_{k}p_{l}\frac{\partial}{\partial p_{k+l}}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{k=1}^{\infty}((h_{1}+h_{2})(k-1)+2\psi_{0}\sqrt{\beta}N)kp_{k}\frac{\partial}{\partial
p_{k}}$

equals

$\frac{1}{6}\psi_{3}^{2D}+\frac{1}{2}(\psi_{0}\sqrt{\beta}N-\frac{1}{3}\psi_{0}\sigma_{3})\psi_{2}^{2D}$

with $a_{-n}=p_{n},\ a_{n}=-\frac{1}{h_{1}h_{2}}\frac{\partial}{\partial
p_{n}}$ for $n>0$.

For this two reasons, we define 3D cut-and-join operators as follows.

######  Definition 5.1.

The 3D cut-and-join operators $W_{n}^{3D}$ are defined by

$W_{n}^{3D}=\frac{1}{n+1}\psi_{n+1}.$ (52)

For example,

$\displaystyle W_{1}^{3D}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\psi_{2}=\frac{1}{2}b_{0,2}+\frac{1}{\psi_{0}}\sum_{n>0}b_{-n,1}b_{n,1},$
$\displaystyle W_{2}^{3D}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\psi_{3}=-\frac{1}{6}b_{0,3}+\frac{1}{\psi_{0}}\sum_{n>0}(b_{-n,1}b_{n,2}+b_{-n,2}b_{n,1})$
$\displaystyle+\frac{1}{\psi_{0}^{2}}\sum_{n,m>0}(b_{-n,1}b_{-m,1}b_{n+m,1}+b_{-n-m,1}b_{n,1}b_{m,1})$
$\displaystyle+\sigma_{3}\sum_{n>0}nb_{-n,1}b_{n,1}-\frac{1}{6}\sigma_{3}\psi_{0}b_{0,2}-\frac{1}{3}\sigma_{3}\sum_{n>0}b_{-n,1}b_{n,1}.$

From the properties of $\psi_{j}$, we know that every two 3D cut-and-join
operators are commutative, and the 3-Jack polynomials are their eigenstates.

###  5.2 Some matrix models

In this subsection, we use 3D Bosons to represent the $W$-operators in matrix
models.

We recall the partition function hierarchy first. The Hurwitz-Kontsevich model
[20]

$\displaystyle Z_{0}\\{p\\}=\int_{\tilde{N}\times\tilde{N}}\sqrt{{\rm
det}\left(\frac{{\rm sinh}(\frac{\phi\otimes
I-I\otimes\phi}{2})}{\frac{\phi\otimes I-I\otimes\phi}{2}}\right)}d\phi
e^{-\frac{1}{2t}{\rm Tr}\phi^{2}-\frac{\tilde{N}}{2}{\rm
Tr}\phi-\frac{1}{6}t\tilde{N}^{3}+\frac{1}{24}t\tilde{N}+{\rm
Tr}(e^{\phi}\psi)},$ (53)

where $\psi$ is an $\tilde{N}\times\tilde{N}$ matrix and the time variables
$p_{k}={\rm Tr}\psi^{k}$. Here $\tilde{N}$ has no relations with $N$ before.
It is generated by the exponent of the Hurwitz operator $\hat{W}_{0}$ acting
on the function $e^{p_{1}/e^{t\tilde{N}}}$[8],

$\displaystyle Z_{0}\\{p\\}=e^{t\hat{W}_{0}}\cdot
e^{p_{1}/e^{t\tilde{N}}}=\sum_{\lambda}e^{tc_{\lambda}}S_{\lambda}\\{p_{k}=e^{-t\tilde{N}}\delta_{k,1}\\}S_{\lambda}\\{p\\},$
(54)

where $t$ is a deformation parameter and $S_{\lambda}$ are Schur functions,
the Hurwitz operator $\hat{W}_{0}$ is given by

$\displaystyle\hat{W}_{0}=\frac{1}{2}\sum_{k,l=1}^{\infty}\big{(}(k+l)p_{k}p_{l}\frac{\partial}{\partial
p_{k+l}}+klp_{k+l}\frac{\partial}{\partial p_{k}}\frac{\partial}{\partial
p_{l}}\big{)}+\tilde{N}\sum_{k=1}^{\infty}kp_{k}\frac{\partial}{\partial
p_{k}},$ (55)

and $c_{\lambda}=\sum_{(i,j)\in\lambda}(\tilde{N}-i+j)$.

Introduce

$E_{1}=[\hat{W}_{0},p_{1}],\ \ \ \hat{W}_{-1}=[\hat{W}_{0},E_{1}],$ (56)

the operators

$\hat{W}_{-n}=\frac{1}{(n-1)!}\underbrace{[\hat{W}_{-1},[\hat{W}_{-1},\cdots[\hat{W}_{-1}}_{n-1},E_{1}]\ldots]],\
\ n\geq 2$

give the partition function hierarchies which include the Gaussian hermitian
one-matrix model, $\tilde{N}\times\tilde{N}$ complex matrix model[8].

In the following, we use 3D Bosons to represent the 3D generations of these
$W$-operators. The 3D Hurwitz-Kontsevich model [17]

${Z}_{0}^{3D}\\{p\\}=e^{t\hat{{W}}_{0}^{3D}}\cdot
e^{\frac{P_{1,1}}{\psi_{0}e^{tM}}},$ (57)

where

$\displaystyle\hat{{W}}_{0}^{3D}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{N}\sum_{k,l=1}^{\infty}\big{(}klp_{i,k+l}\frac{\partial}{\partial
p_{i,k}}\frac{\partial}{\partial
p_{i,l}}-h_{1}h_{2}(k+l)p_{i,k}p_{i,l}\frac{\partial}{\partial
p_{i,k+l}}\big{)}$ (58)
$\displaystyle+(h_{1}+h_{2})\sum_{i_{1}<i_{2}}\sum_{k>0}k^{2}p_{i_{1},k}\frac{\partial}{\partial
p_{i_{2},k}}$
$\displaystyle+\frac{1}{2}\sum_{j=1}^{N}\sum_{k=1}^{\infty}\big{(}(h_{1}+h_{2})(k-2N+2j-1)+2\psi_{0}\sqrt{\beta}\tilde{N}\big{)}kp_{j,k}\frac{\partial}{\partial
p_{j,k}}.$

This operator $\hat{{W}}_{0}^{3D}$ can be represented by 3D Bosons

$\displaystyle\hat{{W}}_{0}^{3D}$ $\displaystyle=$
$\displaystyle-\frac{1}{12}b_{0,3}+\frac{1}{2\psi_{0}}\sum_{n>0}(b_{-n,1}b_{n,2}+b_{-n,2}b_{n,1})$
(59)
$\displaystyle+\frac{1}{2\psi_{0}^{2}}\sum_{n,m>0}(b_{-n,1}b_{-m,1}b_{n+m,1}+b_{-n-m,1}b_{n,1}b_{m,1})+\frac{1}{2}\sigma_{3}\sum_{n>0}nb_{-n,1}b_{n,1}$
$\displaystyle+\frac{1}{2}\left(\psi_{0}\sqrt{\beta}\tilde{N}-\frac{1}{2}\psi_{0}\sigma_{3}\right)\left(b_{0,2}+\frac{2}{\psi_{0}}\sum_{n>0}b_{-n,1}b_{n,1}\right).$

The 3D Boson representation of other operators can be obtained from this
equation, for example,

$\displaystyle E_{1}^{3D}$ $\displaystyle=$
$\displaystyle[W_{0}^{3D},b_{-1,1}]$ (60) $\displaystyle=$
$\displaystyle\frac{1}{2}b_{-1,2}+\frac{1}{\psi_{0}}\sum_{n>0}b_{-n-1}b_{n,1}+\psi_{0}\sqrt{{\beta}}\tilde{N}b_{-1,1}.$

## Data availability statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Declaration of interest statement

The authors declare that we have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

This research is supported by the National Natural Science Foundation of China
under Grant No. 12101184 and No. 11871350, and supported by the Key Scientific
Research Project in Colleges and Universities of Henan Province No. 22B110003.

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