A Calogero model for the non-Abelian
quantum Hall effect

Jean-Emile Bourgine†, Yutaka Matsuo∗

†School of Mathematics and Statistics
University of Melbourne
Parkville, Victoria 3010, Australia
<EMAIL_ADDRESS>
∗Department of Physics & Trans-scale Quantum Science Institute
& Mathematics and Informatics Center, The University of Tokyo
Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
<EMAIL_ADDRESS>
A model of the non-Abelian fractional quantum Hall effect is obtained from the
diagonalization of the matrix model proposed by Dorey, Tong, and Turner in
[1]. The Hamiltonian is reminiscent of a spin Calogero-Moser model but
involves higher-order symmetric representations of the non-Abelian symmetry.
We derive the energy spectrum and show that the Hamiltonian has a triangular
action on a certain class of wave functions with a free fermion expression. We
deduce the expression of the ground states eigenfunctions and show that they
solve a Knizhnik-Zamolodchikov equation. Finally, we discuss the emergence of
Kac-Moody symmetries in the large $N$ limit using the level-rank duality and
confirm the results obtained previously in [1].

##  1 Introduction

Quantum systems with topologically protected states called non-Abelian anyons
are the best candidates for constructing fault-tolerant quantum computing
devices [2]. States of this type are encountered in models of the non-Abelian
fractional quantum Hall effect (FQHE) [3]. The FQHE refers to the observation
of plateaux in the Hall conductivity of a system of electrons in a strong
magnetic field associated with fractional values of the filling factor. The
non-Abelian model considers the possibility of assigning spin-like degrees of
freedom to electrons to encode various physical properties like the band
structure, the presence of higher Landau levels, or a physical spin. This
model reveals excited states with non-trivial statistics that exhibit the
characteristics of topologically protected states [3]. While non-Abelian
anyons have not been observed yet in condensed matter systems, they have been
obtained very recently in quantum processors [4][5].

Three-dimensional Chern-Simons theory offers competing descriptions of the
FQHE at large distances. For instance, Susskind proposed in [6] a description
of the abelian FQHE using a non-commutative $U(1)$ Chern-Simons theory at
level $k$, and recovered in this way the Laughlin wave function corresponding
to the filling factor $\nu=1/(k+1)$. Subsequently, Polychronakos introduced a
$U(N)$ matrix model as a regularization of Susskind’s model to describe the
microscopic dynamics of a droplet of $N$ electrons [7]. In [8], building on
recent results on the moduli space of vortices [9], Tong re-interpreted
Polychronakos’s matrix model as a description of vortices in an Abelian
commutative Chern-Simons theory. Following this interpretation, Dorey Tong and
Turner (DTT) introduced an extension of Polychronakos’s matrix model with an
additional $U(p)$ symmetry to render the dynamics of vortices in the non-
Abelian Chern-Simons theory.

One of the interests of the matrix model is to relate the Chern-Simons theory
descriptions to the conformal field theory (CFT) approach to FQHE wave
functions initiated by Moore and Read [10]. Indeed, Polychronakos’s $U(N)$
matrix model reduces after diagonalization to a Calogero Hamiltonian, which
describes the dynamics of the vortices. The ground state wave function of this
model is known to coincide with the Laughlin wave function, which is
identified with a conformal block of the rational torus CFT in [10].
Similarly, DTT recovered from the non-Abelian model the Blok-Wen wave
functions identified with the correlators of Wess-Zumino-Witten (WZW) models
with $SU(p)_{k}\times U(1)_{k^{\prime}}$ Kac-Moody symmetry [3]111For a more
recent study on the relation between FQHE and Kac-Moody symmetry, see, for
example, [11][12].. In fact, the Kac-Moody currents of the CFT can be directly
constructed from the matrix model, as shown in [13, 14].

The method followed by DTT in [1] for the diagonalization and quantization of
the matrix model is based on a matrix version of holomorphic coordinates and
differs from Polychronakos’ approach. In this paper, we revisit the DTT matrix
model following Polychronakos’s original approach and show that the dynamics
of the vortices are governed by a spin Calogero Hamiltonian involving higher-
order symmetric representations. To be specific, the order of the $U(p)$ spin
representation is identified with the Chern-Simons level $k$. A generating
family of wave functions called wedge states [15][16] is introduced, and we
prove that the action of the Hamiltonian on these states is triangular. We
deduce from this result the spectrum of our model and the expression of the
ground state wave functions, recovering the Blok-Wen wave functions in the
absence of degeneracies. More generally, all ground state eigenfunctions are
shown to obey the Knizhnik-Zamolodchikov (KZ) equation characteristic of WZW
conformal blocks.

At level $k>1$, the wedge states are no longer linearly independent due to
Plücker relations. We solve this problem using a fermionic representation of
the wave functions, which highlights the underlying algebraic structure. In
the large $N$ limit, the wedge states become the exact eigenstates of the
Hamiltonian. The free fermions are identified with the fields of a 2D CFT with
the conformal embedding
$\widehat{\mathfrak{gl}}(pk)_{1}\supset\widehat{\mathfrak{sl}}(p)_{k}\otimes\widehat{\mathfrak{sl}}(k)_{p}\otimes\widehat{\mathfrak{u}}(1)$.
This type of CFT has been studied previously in the context of level-rank
duality [17]. The Plücker relations project out the dual Kac-Moody
$\widehat{\mathfrak{sl}}(k)_{p}$ algebra, and one recovers the
$\widehat{\mathfrak{sl}}(p)_{k}\otimes\widehat{\mathfrak{u}}(1)$ symmetry of
the system in agreement with [1]. The primary fields of the Kac-Moody algebra,
which appears in the KZ equation, are identified with vertex operators. They
are used to translate the wedge states into the states of the fermionic Fock
space. In this way, the spin Calogero model is directly related to the 2D
conformal field theory.

One of our main motivations is the well-known integrability properties of
Calogero-type models, which offers the prospect of applying the algebraic
methods developed in this context, like the Algebraic Bethe Ansatz [18] or the
Lax formalism [19], to the problem of the FQHE. While we do not prove these
properties in this paper, we strongly suspect that our Hamiltonian possesses a
Yangian symmetry realized in a similar way as in the lower order version
studied in [20] (see also [19] for an approach using the Lax pair). This
expectation is consistent with the well-known fact that Yangian invariance
generates Kac-Moody symmetries in the large $N$ limit [21, 22, 23]. We hope to
be able to come back to these important questions in a future publication.

This paper is organized as follows. Section two contains the derivation of the
Hamiltonian by diagonalization of the DTT matrix model. The analysis of its
spectrum and eigenfunctions is presented in section three, and the free
fermion formalism and the large $N$ limit are explained in section four.
Technical details have been gathered in the appendix. Appendix A contains the
proof of our main results, namely the triangular action of the Hamiltonian and
the KZ equation for the ground states eigenfunction. Appendix B presents the
eigenfunctions of the Hamiltonian for the simplest case of $N=2$ particles,
and Appendix C is a brief reminder of the construction of the Gelfand-Zetlin
basis, which applies to the $k=1$ case. Finally, Appendix D presents an
explicit check of the conformal embedding by comparing the explicit expression
of characters in the case $p=k=2$.

##  2 Derivation of the Hamiltonian

In this section, we first present our quantum model and then its derivation
from the diagonalization of the DTT matrix model.

###  2.1 Definition of the model

The system consists of $N$ particles with coordinates $x_{a}$ carrying a
$U(p)$ spin structure defined using the $Np$ oscillators
$[\varphi_{i,a},\varphi_{j,b}^{\dagger}]=\delta_{i,j}\delta_{a,b}$. In this
paper, we take the convention that indices $a,b,c,d\in\\{1,\cdots N\\}$ label
the particles, and $i,j\in\\{1,\cdots,p\\}$ their spin states. Each particle
carries a symmetric representation $[k]$ of $U(p)$ obtained from the action of
$k$ creation operators
$\varphi_{i_{1},a}^{\dagger}\cdots\varphi_{i_{k},a}^{\dagger}\ket{\emptyset}$
on the vacuum $\ket{\emptyset}$ satisfying

$\varphi_{i,a}\ket{\emptyset}=0,\quad\forall i\in\\{1,\cdots p\\},\quad\forall
a\in\\{1,\cdots,N\\}.$ (2.1)

In addition to the integers $N,p,k$, the model depends on the parameter $B$
that confines the particles in a harmonic potential. The Hamiltonian of the
system reads222We note some similarity with the Calogero model obtained in
[24] by dimensional reduction of 2d Super Yang-Mills. However, here the
interaction term is bosonic, while it is fermionic in [24].

${\mathcal{H}}=\sum_{a=1}^{N}\left(-\dfrac{\partial^{2}}{\partial
x_{a}^{2}}+B^{2}x_{a}^{2}\right)+2\sum_{\genfrac{}{}{0.0pt}{}{a,b=1}{a<b}}^{N}\dfrac{J_{a,b}J_{b,a}}{(x_{a}-x_{b})^{2}},\quad
J_{a,b}=\sum_{i=1}^{p}\varphi_{i,a}^{\dagger}\varphi_{i,b}.$ (2.2)

This model is expected to describe the non-Abelian quantum Hall states with
filling factor $\nu=p/(k+p)$. We note that the generators $J_{a,b}$ satisfy
the relations of the $\mathfrak{gl}(N)$ Lie algebra
$[J_{a,b},J_{c,d}]=\delta_{a,d}J_{c,b}-\delta_{b,c}J_{a,d}$, and the
Hamiltonian commutes with the diagonal operators $J_{a,a}$ for $a=1,\cdots,N$.
These operators are diagonal on spin states
$\varphi_{i_{1},a}^{\dagger}\cdots\varphi_{i_{k},a}^{\dagger}\ket{\emptyset}$
in the representation $[k]$, with the eigenvalue $k$. The $U(N)$ constraints
of our model impose to fix this integer $k$ to be the same for each particle
$a$. The Hamiltonian 2.2 also exhibits a global $U(p)$ invariance which is
generated by the operators

$K_{i,j}=\sum_{a=1}^{N}\varphi_{i,a}^{\dagger}\varphi_{j,a},\quad[K_{i,j},K_{k,l}]=\delta_{j,k}K_{i,l}-\delta_{i,l}K_{k,j}.$
(2.3)

#### Abelian model

Setting $p=1$, we recover the Abelian model introduced by Polychronakos in [7]
to describe the Laughlin states at filling factor $\nu=1/(k+1)$. In this case,
there is no spin structure as every particle carries the fixed spin state
$(\varphi_{a}^{\dagger})^{k}\ket{\emptyset}$. The action of the interaction
term in the Hamiltonian 2.2 simplifies since

$J_{a,b}J_{b,a}\prod_{c=1}^{N}(\varphi_{c}^{\dagger})^{k}\ket{\emptyset}=k(k+1)\prod_{c=1}^{N}(\varphi_{c}^{\dagger})^{k}\ket{\emptyset},$
(2.4)

and we recover the Calogero-Moser Hamiltonian with a Gaussian potential,

${\mathcal{H}}^{(p=1)}=\sum_{a=1}^{N}\left(-\dfrac{\partial^{2}}{\partial
x_{a}^{2}}+B^{2}x_{a}^{2}\right)+2k(k+1)\sum_{\genfrac{}{}{0.0pt}{}{a,b=1}{a<b}}^{N}\dfrac{1}{(x_{a}-x_{b})^{2}}.$
(2.5)

This Hamiltonian is known to be integrable [25].

#### Spin Calogero-Moser

Considering instead the case $k=1$ for arbitrary $p$, the particles carry the
fundamental representation of $U(p)$. This is usually implemented by assigning
the spin state $\ket{i_{a}}=\varphi_{i_{a},a}^{\dagger}\ket{\emptyset}$ to the
particle $a$, with the label $i_{a}\in\\{1,\cdots,p\\}$. In this case, the
interaction term also simplifies as the product $J_{a,b}J_{b,a}$ acting on
such spin states can be rewritten using the permutation $P_{a,b}$ exchanging
the spin labels $i_{a}$ and $i_{b}$,

$J_{a,b}J_{b,a}\prod_{c=1}^{N}\varphi_{i_{c},a}^{\dagger}\ket{\emptyset}=(1+P_{a,b})\prod_{c=1}^{N}\varphi_{i_{c},a}^{\dagger}\ket{\emptyset}.$
(2.6)

Then, the Hamiltonian 2.2 reduces to the spin Calogero-Moser Hamiltonian at
coupling $\lambda=1$ with an extra Gaussian potential,

${\mathcal{H}}^{(k=1)}=\sum_{a}\left(-\dfrac{\partial^{2}}{\partial
x_{a}^{2}}+B^{2}x_{a}^{2}\right)+2\sum_{\genfrac{}{}{0.0pt}{}{a,b=1}{a<b}}^{N}\dfrac{1+P_{a,b}}{(x_{a}-x_{b})^{2}}.$
(2.7)

In the absence of the Gaussian term, this model is also a well-known
integrable system which exhibits the Yangian symmetry $Y(\mathfrak{sl}_{p})$
[26, 19, 27].

#### Remark

Another remarkable value, albeit somewhat trivial, is obtained at $k=0$. In
this case, particles carry no spin degree of freedom, and the interaction term
of the Hamiltonian 2.2 vanishes. We are left with $N$ decoupled quantum
harmonic oscillators. If we look for an antisymmetric wave function so that
the true matrix model wave function is symmetric, the ground state of our
Hamiltonian is333Without the anti-symmetrization requirement, we would simply
have  $\Psi_{0}(\boldsymbol{x})=e^{-\frac{B}{2}\sum_{a}x_{a}^{2}},\quad
E_{0}=NB.$ (2.8)

$\Psi_{0}(\boldsymbol{x})=\Delta(\boldsymbol{x})e^{-\frac{B}{2}\sum_{a}x_{a}^{2}},\quad
E_{0}=N^{2}B.$ (2.9)

This is the Slater determinant of the wave functions of $N$ independent
quantum harmonic oscillators. It coincides with the Laughlin wave function of
filling factor $\nu=1$.

###  2.2 Diagonalization of the DTT matrix model

The Hamiltonian 2.2 has been obtained from the diagonalization of the matrix
quantum mechanics proposed in [1] to describe the dynamics of vortices in a
$U(p)$ Chern-Simons theory. It is a non-Abelian generalization of the quantum
Hall droplet matrix model proposed by Polychronakos in [7], which was shown in
[8, 28] to also describe the vortices of the $U(1)$ Chern-Simons theory. The
non-Abelian matrix model is formulated in terms of dynamical fields consisting
in a $N\times N$ complex matrix $Z(t)$ and $p$ $N$-dimensional vectors
$\varphi_{i}(t)$ ($i=1\cdots p$). The matrix $Z(t)$ transforms in the adjoint
representation $Z(t)\to U(t)Z(t)U(t)^{\dagger}$ under $U(N)$ symmetry, and the
vectors $\varphi_{i}(t)$ in the fundamental representation $\varphi_{i}(t)\to
U(t)\varphi_{i}(t)$. The model also contains a non-dynamical gauge field
$\alpha$ which imposes the (classical) constraint

$\frac{B}{2}[Z,Z^{\dagger}]+\sum_{i=1}^{p}\varphi_{i}\varphi_{i}^{\dagger}=(k+p)\mathbbm{1}_{N}.$
(2.10)

The action of this model reads [1]

$\mathcal{S}=\int
dt\left[\dfrac{1}{2}iB\operatorname{\text{tr}}(Z^{\dagger}{\mathcal{D}}_{t}Z)+i\sum_{i=1}^{p}\varphi_{i}^{\dagger}{\mathcal{D}}_{t}\varphi_{i}-(k+p)\operatorname{\text{tr}}\alpha-\omega\operatorname{\text{tr}}(Z^{\dagger}Z)\right],$
(2.11)

where ${\mathcal{D}}_{t}$ is the covariant derivative

${\mathcal{D}}_{t}Z=\partial_{t}Z-i[\alpha,Z],\quad{\mathcal{D}}_{t}\varphi_{i}=\partial_{t}\varphi_{i}-i\alpha\varphi_{i}.$
(2.12)

The model depends on the parameters $B$ (background magnetic field), $k$
(Chern-Simons level), and $\omega$ (strength of the harmonic trap).

The complex matrix $Z$ can be decomposed into a sum of Hermitian matrices

$Z=X_{1}+iX_{2},\quad Z^{\dagger}=X_{1}-iX_{2},$ (2.13)

and isolating the dependence in the gauge field, the previous action written
in a more explicit form,

$\displaystyle\begin{split}&\mathcal{S}=\int
dt\left(-\dfrac{B}{2}\operatorname{\text{tr}}\left(X_{1}\dot{X}_{2}-X_{2}\dot{X}_{1}\right)+i\sum_{i=1}^{p}\varphi_{i}^{\dagger}\dot{\varphi}_{i}-\omega\operatorname{\text{tr}}(X_{1}^{2}+X_{2}^{2})+\operatorname{\text{tr}}\left[\alpha
G(X_{1},X_{2},\varphi_{i},\varphi_{i}^{\dagger})\right]\right),\\\
&\text{with}\quad
G(X_{1},X_{2},\varphi_{i},\varphi_{i}^{\dagger})=-iB[X_{1},X_{2}]+\sum_{i=1}^{p}\varphi_{i}\varphi_{i}^{\dagger}-(k+p)\mathbbm{1}_{N},\end{split}$
(2.14)

and we used the dot notation to indicate the time derivative. When $p=1$, this
is precisely the form of the action studied by Polychronakos in [7]. In
contrast with the work of DTT, we will work in the framework where
$(X_{1},X_{2})$ are the canonical conjugate fields, instead of
$(Z,Z^{\dagger})$. One of our motivations is to investigate the connection
with quantum integrable systems. Indeed, the dynamics of the eigenvalues of
the matrix $X_{1}$ is given by the Calogero-Moser Hamiltonian 2.5 when $p=1$
[7]. As we will show shortly, in the non-Abelian model, this dynamics is
defined by the Hamiltonian 2.2, which we suspect is also integrable. A second,
more physical, motivation is the interpretation of these eigenvalues as the
effective electron coordinates in the quantum droplet model, and so the
position of Chern-Simons vortices. In this way, we expect the wave functions
derived in the next section to give a more physical description of the
corresponding model.

#### Diagonalization

The hermitian matrix $X_{1}$ is diagonalized by decomposing
$X_{1}(t)=\Omega(t)^{\dagger}x(t)\Omega(t)$ with $(x)_{a,b}=x_{a}\delta_{a,b}$
a diagonal matrix and $\Omega\in\text{SU}(N)$ a unitary matrix containing the
$N^{2}-N$ angular degrees of freedom. This decomposition is introduced using
the global $U(N)$ symmetry, and we denote with a prime the transformed
quantities

$X_{2}^{\prime}=\Omega
X_{2}\Omega^{\dagger},\quad\varphi_{i}^{\prime}=\Omega\varphi_{i},\quad\varphi_{i}^{\prime\dagger}=\varphi_{i}^{\dagger}\Omega^{\dagger},\quad\alpha^{\prime}=\Omega\alpha\Omega^{\dagger},\quad
G^{\prime}=\Omega G\Omega^{\dagger}.$ (2.15)

After transformation, the dynamical terms in the Lagrangian read

$\displaystyle\begin{split}&\operatorname{\text{tr}}X_{1}\dot{X_{2}}=\operatorname{\text{tr}}x\dot{X}_{2}^{\prime}+\operatorname{\text{tr}}L[x,X^{\prime}_{2}],\quad\operatorname{\text{tr}}\dot{X_{1}}X_{2}=\operatorname{\text{tr}}\dot{x}X^{\prime}_{2}-\operatorname{\text{tr}}L[x,X^{\prime}_{2}],\quad\varphi_{i}^{\dagger}\dot{\varphi}_{i}=\varphi_{i}^{\prime\dagger}\dot{\varphi}^{\prime}_{i}-\varphi_{i}^{\prime\dagger}L\varphi^{\prime}_{i},\end{split}$
(2.16)

where we introduced the anti-hermitian matrix
$L=\dot{\Omega}\Omega^{\dagger}=-\Omega\dot{\Omega}^{\dagger}$. Replacing
these terms in the action, we find after integration by part

$\displaystyle\begin{split}&\mathcal{S}=\int
dt\left(B\operatorname{\text{tr}}\dot{x}X^{\prime}_{2}+i\sum_{i=1}^{p}\varphi_{i}^{\prime\dagger}\dot{\varphi^{\prime}}_{i}-\omega\operatorname{\text{tr}}(x^{2}+(X_{2}^{\prime})^{2})-B\operatorname{\text{tr}}L[x,X^{\prime}_{2}]-i\sum_{i=1}^{p}\varphi_{i}^{\prime\dagger}L{\varphi^{\prime}}_{i}+\operatorname{\text{tr}}\alpha^{\prime}G^{\prime}\right),\\\
&\text{with}\quad
G^{\prime}=-iB[x,X_{2}^{\prime}]+\sum_{i=1}^{p}\varphi_{i}^{\prime}\varphi_{i}^{\prime\dagger}-(k+p)\mathbbm{1}_{N}.\end{split}$
(2.17)

From now on, we drop the prime on the variables to lighten the notation.

In the action 2.17, the dynamical fields are the eigenvalues $x_{a}(t)$, the
components $\varphi_{i,a}(t)$ of the vector fields, and the angular degrees of
freedom contained in $\Omega(t)$. Following the standard method reviewed in
[29], the anti-hermitian matrix $L$ can be decomposed over the generators of
$su(N)$,

$L=\sum_{a=1}^{N-1}\dot{\omega}_{a}H_{a}+\sum_{\genfrac{}{}{0.0pt}{}{a,b=1}{a<b}}^{N}(\dot{\omega}_{a,b}T_{a,b}+\dot{\theta}_{a,b}\tilde{T}_{a,b}).$
(2.18)

We denoted the Cartan generators $H_{a}=i(e_{a,a}-e_{a+1,a+1})$, and the anti-
hermitian matrices $T_{a,b}=i(e_{a,b}+e_{b,a})/\sqrt{2}$ and
$\tilde{T}_{a,b}=(e_{a,b}-e_{b,a})/\sqrt{2}$ (here $e_{a,b}$ denotes the
elements of the canonical basis of GL$(N)$). Using this decomposition, we can
compute the momentum associated to each degree of freedom,

$\displaystyle\begin{split}&p_{a}=\dfrac{\delta{\mathcal{L}}}{\delta\dot{x}_{a}}=BX_{2\
a,a},\quad\pi_{i,a}=\dfrac{\delta{\mathcal{L}}}{\delta\dot{\varphi}_{i,a}}=i\varphi_{i,a}^{\dagger},\quad\Pi_{a}=\dfrac{\delta{\mathcal{L}}}{\delta\dot{\omega}_{a}}=J_{a,a}-J_{a+1,a+1},\\\
&\Pi_{a,b}=\dfrac{\delta{\mathcal{L}}}{\delta\dot{\omega}_{a,b}}=\dfrac{1}{\sqrt{2}}\left(J_{a,b}+J_{b,a}-iB(x_{a}-x_{b})(X_{2\
a,b}-X_{2\ b,a})\right),\\\
&\tilde{\Pi}_{a,b}=\dfrac{\delta{\mathcal{L}}}{\delta\dot{\theta}_{a,b}}=-\dfrac{i}{\sqrt{2}}\left(J_{a,b}-J_{b,a}+iB(x_{a}-x_{b})(X_{2\
a,b}+X_{2\ b,a})\right),\end{split}$ (2.19)

where $J_{a,b}$ is defined in 2.2. We deduce the Hamiltonian, which reduces to
a potential term since the Lagrangian depends linearly on time derivatives.
Setting $\alpha=0$, we find

${\mathcal{H}}=\omega\operatorname{\text{tr}}(x^{2}+(X_{2})^{2})=\omega\sum_{a=1}^{N}\left(x_{a}^{2}+\dfrac{p_{a}^{2}}{B^{2}}\right)+2\dfrac{\omega}{B^{2}}\sum_{a<b}\dfrac{(J_{a,b}-\Pi_{a,b}^{+})(J_{b,a}-\Pi_{a,b}^{-})}{(x_{a}-x_{b})^{2}},$
(2.20)

with $\Pi^{\pm}_{a,b}=(\Pi_{a,b}\pm i\tilde{\Pi}_{a,b})/\sqrt{2}$.

#### Quantization

We now impose the canonical quantization conditions for the angular variables
and the vector fields,

$[\omega_{a},\Pi_{b}]=i\delta_{a,b},\quad[\omega_{a,b},\Pi_{c,d}]=i\delta_{a,c}\delta_{b,d},\quad[\varphi_{i,a},\pi_{j,b}]=i\delta_{i,j}\delta_{a,b}.$
(2.21)

In particular, $\varphi_{i,a}$ and $\varphi_{i,a}^{\dagger}$ are conjugate
variables, $[\varphi_{i,a},\varphi_{j,b}^{\dagger}]=\delta_{i,j}\delta_{a,b}$.
However, we note that due to the Vandermonde determinant in the measure coming
from the diagonalization $dX_{1}=d\Omega dx\Delta(\boldsymbol{x})$, the
momentum $p_{a}$ acts on wave functions as (see, e.g., [29])

$p_{a}=-i\Delta(\boldsymbol{x})^{-1}\partial_{a}\Delta(\boldsymbol{x}),\quad\Delta(\boldsymbol{x})=\prod_{\genfrac{}{}{0.0pt}{}{a,b=1}{a<b}}^{N}(x_{a}-x_{b}).$
(2.22)

After quantization, the constraints $G_{a,b}=0$ must be imposed on physical
wave functions. In the canonical variables, these constraints take the form

$G_{a,b}=\left\\{\begin{array}[]{ll}J_{a,a}-k&(a=b),\\\
\Pi_{a,b}^{-}&(a<b),\\\ \Pi_{b,a}^{+}&(a>b).\end{array}\right.$ (2.23)

Thus, physical states must obey

$\Pi_{a}\ket{\text{phys}}=\Pi_{a,b}\ket{\text{phys}}=\tilde{\Pi}_{a,b}\ket{\text{phys}}=0,\quad
J_{a,a}\ket{\text{phys}}=k\ket{\text{phys}}.$ (2.24)

These relations imply that they have no dependence in the angular variables
$\omega_{a}$, $\omega_{a,b}$, and $\theta_{a,b}$. Since $J_{a,a}$ counts a
number of modes, the last condition imposes the quantization $k\in\mathbb{Z}$
of the Chern-Simons level. From the relation
$J_{a,b}J_{b,a}+J_{a,a}=J_{b,a}J_{a,b}+J_{b,b}$, we also deduce that
$J_{a,b}J_{b,a}\equiv J_{b,a}J_{a,b}$ on physical states. Thus, the action of
the Hamiltonian simplifies on the physical states444We note that the canonical
quantization condition implies that the conjugate variables $\varphi_{i,a}$
and $\varphi_{i,a}^{\dagger}$ both commute with the momenta $\Pi_{a,b}$ and
$\tilde{\Pi}_{a,b}$, and so $J_{a,b}$ commutes with $\Pi_{a,b}^{\pm}$ in 2.20.

${\mathcal{H}}=\dfrac{\omega}{B^{2}}\left(\sum_{a=1}^{N}\left(-\Delta(\boldsymbol{x})^{-1}\partial_{a}^{2}\Delta(\boldsymbol{x})+B^{2}x_{a}^{2}\right)+2\sum_{a<b}\dfrac{J_{a,b}J_{b,a}}{(x_{a}-x_{b})^{2}}\right),$
(2.25)

and it indeed reduces to the Hamiltonian given in equ. 2.2 upon rescaling the
energies by a factor $\omega^{-1}B^{2}$, and a conjugation with the
Vandermonde
${\mathcal{H}}\to\Delta(\boldsymbol{x}){\mathcal{H}}\Delta(\boldsymbol{x})^{-1}$.
The latter simply amounts to multiplying the wave functions by the
antisymmetric factor $\Delta(\boldsymbol{x})$.

##  3 Spectrum and eigenfunctions

###  3.1 Abelian model

Before studying the general case, we recall the results obtained for the
Abelian model ($p=1$). As previously observed, the Hamiltonian 2.5 describes
the Calogero-Moser system with coupling $k\in\mathbb{Z}$ and a Gaussian
potential. For this system, the ground state is well-known,555We discard the
wave function singular at coincident points which leads to negative energies
for $k>0$,
$\Psi_{0}(\boldsymbol{x})=\prod_{a<b}(x_{a}-x_{b})^{-k}e^{-\frac{B}{2}\sum_{a}x_{a}^{2}},\quad
E_{0}=NB-kN(N-1)B.$ (3.1)

$\Psi_{0}(\boldsymbol{x})=\prod_{a<b}(x_{a}-x_{b})^{k+1}e^{-\frac{B}{2}\sum_{a}x_{a}^{2}},\quad
E_{0}=N^{2}B+kN(N-1)B.$ (3.2)

Upon analytic continuation, $\Psi_{0}(x)$ coincides with the Laughlin wave
function with filling factor $\nu=1/(k+1)$.

This model has been studied by Polychronakos in the context of the abelian
FQHE in [7], and we recall here some of his results. States of the model are
labeled by $N$ positive integers $n_{a}$ satisfying the constraint
$n_{a+1}-n_{a}\geq k+1$, and the corresponding energy is666To recover our
convention, we simply need to replace $\omega=2B$ in the formulas (45) and
(47) of [7]. Comparing instead with the ground state energy found by DTT in
[1], we need to discard an overall shift of $N^{2}B$. We will find the same
overall shift of the energies in the non-Abelian model as well.

$E_{\boldsymbol{n}}=B\sum_{a=1}^{N}(2n_{a}+1).$ (3.3)

For the ground state, the labels take the values $n_{a}=(a-1)(k+1)$ which
gives precisely the energy $E_{0}$ written in 3.2. To count the excited
states, we can introduce the labels $r_{a}=n_{a+1}-n_{a}-(k+1)$ with
$a=1,\cdots N-1$, together with $r_{0}=n_{1}$, for which the previous
condition becomes $r_{a}\geq 0$. In these variables, the energy reads

$E_{\boldsymbol{n}}=E_{0}+2B\sum_{a=1}^{N}ar_{N-a}.$ (3.4)

We can interpret the variables $r_{N-a}$ as counting the multiplicities of the
columns of height $a$ of a Young diagram with a total of $\sum_{a}ar_{N-a}$
boxes, each column being restricted to contain at most $N$ boxes. In the large
$N$ limit, this restriction drops, and the corresponding character is simply
the $u(1)$ WZW character

$\chi(q)=\sum_{n=0}^{\infty}p(n)q^{E_{0}+2Bn}=\dfrac{q^{E_{0}}}{\prod_{j=1}^{\infty}(1-q^{2Bj})},$
(3.5)

where $p(n)$ is the number of partitions of $n$.

###  3.2 Non-Abelian model

In the following, for the sake of simplicity, we introduce a polynomial
representation of the Fock space associated to the variables $\varphi_{i,a}$,

$\varphi^{\dagger}_{i_{1},a_{1}}\cdots\varphi^{\dagger}_{i_{n},a_{n}}|0\rangle\quad\rightarrow\quad
y_{i_{1},a_{1}}\cdots y_{i_{n},a_{n}}.$ (3.6)

In this representation, the operators $\varphi_{i,a}$ act as the derivatives
$\partial_{i,a}=\partial/\partial y_{i,a}$. The $\mathfrak{gl}(N)$ generators
are represented by

$J_{a,b}=\sum_{i=1}^{p}y_{i,a}\partial_{i,b}\,.$ (3.7)

The condition (2.24) implies that physical states are described by a
homogeneous polynomial of degree $k$ in $y_{i,a}$ for each particle
$a=1,\cdots,N$.

In this section, we derive the eigenvalues and study the eigenfunctions of the
Hamiltonian (2.2). To simplify the expression of wave functions, it is
convenient to modify the Hamiltonian and absorb the Gaussian part,

$\tilde{{\mathcal{H}}}=e^{\frac{B}{2}\sum_{a}x_{a}^{2}}{\mathcal{H}}e^{-\frac{B}{2}\sum_{a}x_{a}^{2}}=\sum_{a}\left(-\frac{\partial^{2}}{\partial
x_{a}^{2}}+2Bx_{a}\frac{\partial}{\partial x_{a}}\right)+NB+\sum_{a\neq
b}\frac{J_{a,b}J_{b,a}}{(x_{a}-x_{b})^{2}}.$ (3.8)

We further use the rescaling of coordinates $x_{a}\to B^{-1/2}x_{a}$ and
energies $\tilde{{\mathcal{H}}}\to B^{-1}\tilde{{\mathcal{H}}}$ to set $B=1$.

The diagonalization of the Hamiltonian 3.8 is treated explicitly in appendix B
in the case of two particles (i.e. $N=2$) with $k,p$ generic. The general
treatment for an arbitrary number of particles given below follows from a
careful analysis of specific cases at lower $N$ which are not presented in
this paper to avoid redundancy.

#### Wedge states

In order to write down the wave functions of the Hamiltonian 3.8, we introduce
the wedge product notation for the following $N\times N$ determinants,

$[y_{i_{1}}x^{n_{1}}\wedge\cdots\wedge
y_{i_{N}}x^{n_{N}}]=\det\left(\begin{array}[]{ccc}y_{i_{1},1}(x_{1})^{n_{1}}&\cdots&y_{i_{N},1}(x_{1})^{n_{N}}\\\
\vdots&&\vdots\\\
y_{i_{1},N}(x_{N})^{n_{1}}&\cdots&y_{i_{N},N}(x_{N})^{n_{N}}\end{array}\right).$
(3.9)

It satisfies the linearity and the anti-symmetry properties,

$\displaystyle[\cdots\wedge(\lambda_{1}u_{1}+\lambda_{2}u_{2})\wedge\cdots]=\lambda_{1}[\cdots\wedge
u_{1}\wedge\cdots]+\lambda_{2}[\cdots\wedge
u_{2}\wedge\cdots],\quad\lambda_{1},\lambda_{2}\in\mathbb{C}$ (3.10)
$\displaystyle[\cdots\wedge\underset{\hat{a}}{u_{1}}\wedge\cdots\wedge\underset{\hat{b}}{u_{2}}\wedge\cdots]=-[\cdots\wedge\underset{\hat{a}}{u_{2}}\wedge\cdots\wedge\underset{\hat{b}}{u_{1}}\wedge\cdots]\,.$
(3.11)

The second property implies that
$[\cdots\wedge\underset{\hat{a}}{u}\wedge\cdots\wedge\underset{\hat{b}}{u}\wedge\cdots]=0$.

The elementary blocks for constructing wave functions are the elements of the
following family of determinants indexed by $N$-tuple integers
$\boldsymbol{r}=(r_{1},\cdots,r_{N})$,

$\mathcal{Y}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=\det_{a,b}(y_{i_{b},a}x_{a}^{n_{b}})=[y_{i_{1}}x^{n_{1}}\wedge\cdots\wedge
y_{i_{N}}x^{n_{N}}].$ (3.12)

We call these determinants wedge states. The components $r_{a}$ of the index
${\boldsymbol{r}}$ are related to the parameters $(n_{a},i_{a})$ of the
particle $a$ by Euclidean division $r_{a}=pn_{a}+i_{a}-1$, $1\leq i_{a}\leq
p$. This notation comes from the fact that monomials $y_{i,a}(x_{a})^{n}$ can
be represented as a power of another variable $z_{a}^{r}$, where $n$ and $i-1$
are respectively the quotient and remainder of the Euclidean division of
$r=pn+i-1$ by $p$. Such a notation is used, e.g., in [30]. In these variables,
the determinants $\mathcal{Y}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})$
take the simpler forms

$\mathcal{Y}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=\det_{a,b}z_{a}^{r_{b}}=[z^{r_{1}}\wedge\cdots\wedge
z^{r_{N}}].$ (3.13)

By definition, the determinants vanish if $r_{a}=r_{b}$ for $a\neq b$, and up
to permutation we can assume $0\leq r_{1}<r_{2}<\cdots<r_{N}$. Finally, we
mention the Plücker relations for bilinear combinations of the wedge products
states777See, e.g., [31], p211. In the context of the soliton equation, it
appeared as the Hirota bilinear equation for the tau function [15].,

$\sum_{a=0}^{N}(-1)^{a}[z^{r_{1}}\wedge\cdots\wedge z^{r_{N-1}}\wedge
z^{s_{a}}][z^{s_{0}}\wedge\cdots\wedge\cancel{z^{s_{a}}}\wedge\cdots\wedge
z^{s_{N}}]=0,$ (3.14)

for arbitrary pair $\boldsymbol{r}=(r_{1},\cdots,r_{N-1})$ and
$\boldsymbol{s}=(s_{0},\cdots,s_{N})$. It produces nontrivial quadratic
relations for arbitrary choice of $\boldsymbol{r}$ and $\boldsymbol{s}$ if
$\boldsymbol{r}\not\subset\boldsymbol{s}$.

#### Eigenfunctions as a product of wedge states

To label the eigenstates, we introduce a set of $k$ sequences of $N$ non-
negative integers,

$\displaystyle\mathfrak{r}$
$\displaystyle=\left\\{\boldsymbol{r}^{(1)},\cdots,\boldsymbol{r}^{(k)}\right\\}$
(3.15) $\displaystyle\boldsymbol{r}^{(\alpha)}$
$\displaystyle=\\{r^{(\alpha)}_{1},\cdots,r^{(\alpha)}_{N}\\},\quad 0\leq
r^{(\alpha)}_{1}<r^{(\alpha)}_{2}<\cdots<r^{(\alpha)}_{N}\,,\quad\alpha=1,\cdots,k\,.$
(3.16)

We suppose that the re-ordering of $\boldsymbol{r}^{(\alpha)}$ in
$\mathfrak{r}$ does not give a new element. Let $\mathfrak{R}$ be the set of
all possible $\mathfrak{r}$. For each element $\mathfrak{r}\in\mathfrak{R}$,
we define a function of the variables $x_{a},y_{i,a}$

$\displaystyle\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle=\Delta(\boldsymbol{x})\prod_{\alpha=1}^{k}\mathcal{Y}_{\boldsymbol{r}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y})\,.$
(3.17)

It can also be seen as an element in $\mathbb{C}[z_{1},\cdots,z_{N}]$ using
the previous translation rule. We define the level of
$\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$ as the eigenvalue of the
operator $D=\sum_{a=1}^{N}x_{a}\partial_{a}$,

$D\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})=|\mathfrak{r}|\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y}),\quad\text{with}\quad|\mathfrak{r}|=\sum_{\alpha=1}^{k}|{\boldsymbol{r}}^{(\alpha)}|,\quad|{\boldsymbol{r}}^{(\alpha)}|=\sum_{a=1}^{N}\lfloor{r^{(\alpha)}_{a}}/{p}\rfloor=\sum_{a=1}^{N}n^{(\alpha)}_{a},$
(3.18)

where $\lfloor\bullet\rfloor$ is the floor function. The last equality is
obtained from the correspondence
$r_{a}^{(\alpha)}=pn_{a}^{(\alpha)}+i_{a}^{(\alpha)}-1$.

The functions $\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$ also
satisfy the second condition of (2.24), namely
$J_{a,a}\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})=k\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$,
since each factor
$\mathcal{Y}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})$ obeys the
property

$J_{a,a}{\mathcal{Y}}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=\sum_{i=1}^{p}y_{i,a}\partial_{i,a}\mathcal{Y}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=\mathcal{Y}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})$
(3.19)

for any $a$. In this sense, it gives an element of the physical space. We
define the Hilbert space $\mathfrak{H}$ as the set of functions spanned by
$\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$. It is important to note
that the functions $\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$ for
$\mathfrak{r}\in\mathfrak{R}$ are not linearly independent if $k\geq 2$, since
the Plücker relations (3.14) produce nontrivial relations among the product of
determinants.

The main result of this section is the general theorem below. [l]General form
of the eigenstates The Hamiltonian action (3.8) is triangular on the set of
functions $\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$ defined in
(3.17),

$\tilde{{\mathcal{H}}}\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})=E(\mathfrak{r})\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})+\sum_{|\mathfrak{s}|=|\mathfrak{r}|-2}C(\mathfrak{r},\mathfrak{s})\psi_{\mathfrak{s}}(\boldsymbol{x},\boldsymbol{y}),\quad
E(\mathfrak{r})=2|\mathfrak{r}|+N^{2}\,,$ (3.20)

where $C(\mathfrak{r},\mathfrak{s})$ is a numerical coefficient. As an
immediate consequence of this formula, one obtains the eigenfunctions of the
Hamiltonian through an appropriate recombination of the form,

$\Psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})=\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})+\sum_{|\mathfrak{s}|\leq|\mathfrak{r}|-2}D(\mathfrak{r},\mathfrak{s})\psi_{\mathfrak{s}}(\boldsymbol{x},\boldsymbol{y})\,.$
(3.21)

These eigenfunctions have eigenvalue $E(\mathfrak{r})$.

The theorem (3.20) is one of the main results presented in this paper. Its
proof is technical and relatively lengthy, it is given in appendix (A.1). The
second statement is a simple corollary obtained by a recursion procedure.
Indeed, since the coefficients $C(\mathfrak{r},\mathfrak{s})$ are finite (as
shown in the proof) and $E(\mathfrak{r})$ is a strictly positive integer,
taking the combination

$\Psi_{\mathfrak{r}}^{(1)}(\boldsymbol{x},\boldsymbol{y})=\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})-\sum_{|\mathfrak{s}|=|\mathfrak{r}|-2}\frac{C(\mathfrak{r},\mathfrak{s})}{E(\mathfrak{s})}\psi_{\mathfrak{s}}(\boldsymbol{x},\boldsymbol{y}),$
(3.22)

we find that $\tilde{{\mathcal{H}}}$ has a triangular action on
$\Psi_{\mathfrak{r}}^{(1)}(\boldsymbol{x},\boldsymbol{y})$, and the r.h.s. now
contains a linear combination of $\Psi_{\mathfrak{s}}^{(1)}$ with
$|\mathfrak{s}|=|\mathfrak{r}|-4$. We can continue this recursive procedure
until we arrive at the ground states, which are eigenstates since
$|\mathfrak{r}|$ is minimal. While the functions
$\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$ do not form a basis due
to their redundancy, this fact is irrelevant here since we do not use
independence. This set of functions provides a large class of solutions, and
it is natural to conjecture that general eigenfunctions can be expressed as a
linear combination of such functions. The equivalence relations among
eigenstates, such as those following from the Plücker relations (3.14), will
be discussed in section 5 below.

#### Ground state wave function and degeneracies

The ground states correspond to the labels $\mathfrak{r}$ for which
$|\mathfrak{r}|$ is minimal. As a consequence of the triangular form 3.20 of
the Hamiltonian action, the corresponding wave functions
$\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$ are eigenfunctions of the
system. We first examine the case $k=1$, and observe that the ground state is
unique when $p$ divides $N$. It is obtained for
${\boldsymbol{r}}_{0}=(0,1,2,\cdots,N-1)$, and the corresponding wedge state
is

${\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})=[y_{1}\wedge\cdots\wedge
y_{p}\wedge y_{1}x\wedge\cdots\wedge y_{p}x\wedge\cdots\wedge
y_{1}x^{m-1}\wedge\cdots\wedge y_{p}x^{m-1}],\qquad(N=mp).$ (3.23)

In the general case $N=mp+q$ with $0\leq q\leq p-1$, the ground state is
$\left(\genfrac{}{}{0.0pt}{}{p}{q}\right)$-fold degenerate which corresponds
to a choice of spin for the $q$ extra particles,

${\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})=[y_{1}\wedge\cdots\wedge
y_{p}\wedge y_{1}x\wedge\cdots\wedge y_{p}x\wedge\cdots\wedge
y_{1}x^{m-1}\wedge\cdots\wedge y_{p}x^{m-1}\wedge
y_{i_{1}}x^{m}\wedge\cdots\wedge y_{i_{q}}x^{m}],\qquad(N=mp+q),$ (3.24)

with $1\leq i_{1}<i_{2}<\cdots<i_{q}\leq p$. We denote $\mathfrak{R}_{0}$ the
set of all possible ground states label, i.e. the set of labels
${\boldsymbol{r}}\in\mathfrak{R}$ that minimize $E({\boldsymbol{r}})$. The
ground state energy for $N=mp+q$ is

$E({\boldsymbol{r}}_{0})=pm(m-1)+2qm+N^{2}.$ (3.25)

When $k>1$, ground states wave functions are obtained by taking the product of
$k$ determinants 3.24,

$\psi_{\mathfrak{r}_{0}}(\boldsymbol{x},\boldsymbol{y})=\Delta(\boldsymbol{x})\prod_{\alpha=1}^{k}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y}),\quad
E(\mathfrak{r}_{0})=k\left(pm(m-1)+2qm\right)+N^{2}$ (3.26)

with
$\mathfrak{r}_{0}=({\boldsymbol{r}}_{0}^{(1)},\cdots,{\boldsymbol{r}}_{0}^{(k)})$
and ${\boldsymbol{r}}_{0}^{(\alpha)}\in\mathfrak{R}_{0}$. Note that the
expression of the ground state energy $E(\mathfrak{r}_{0})$ reproduces the
formula 3.2 obtained for the abelian model. It also agrees with the results
obtained in [1]. The spin indices $i_{l}^{(\alpha)}$, $\alpha=1\cdots k$ and
$l=1\cdots q$ labeling these states transform in the $k$-th fold
symmetrization of the $q$th antisymmetric representation of $SU(p)$ [1], and
the dimension of $\mathfrak{R}_{0}$ is given by the hook length formula

$\prod_{\alpha=1}^{k}\prod_{i=1}^{q}\frac{p+\alpha-i}{q+k+1-\alpha-i}$ (3.27)

In fact, $\mathfrak{R}_{0}$ has more degrees of freedom than above because of
the redundancies mentioned above.

Finally, we would like to make a comment on the excited states. The theorem
(3.20) shows that excited states are linear combinations of products of the
wedge states. They can be classified according to $U(p)$ (or
$\mathfrak{gl}(p)$) irreducible representations. In the $k=1$ case, it can be
done using the Gelfand-Zetlin basis, this is explained in appendix C. For
general $k$, we have to introduce the corresponding loop (or Kac-Moody)
algebra, and it will presented in the next section.

###  3.3 KZ equation for the ground state wave function

The wave functions of the matrix model differs from those of the Hamiltonian
by a Vandermonde factor,888We continue to omit the exponential factor
$e^{-\frac{B}{2}\sum_{a}x_{a}^{2}}$ here.

$\Phi_{\mathfrak{r}_{0}}(\boldsymbol{x},\boldsymbol{y})=\Delta(\boldsymbol{x})^{-1}\psi_{\mathfrak{r}_{0}}(\boldsymbol{x},\boldsymbol{y})=\prod_{\alpha=1}^{k}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y}),$
(3.28)

In appendix A, we prove that these wave functions satisfy the following
Knizhnik-Zamolodchikov (KZ) equation

$(p+k)\partial_{a}\Phi_{\mathfrak{r}_{0}}(\boldsymbol{x},\boldsymbol{y})=\sum_{b\neq
a}\frac{J_{a,b}J_{b,a}}{x_{a}-x_{b}}\Phi_{\mathfrak{r}_{0}}(\boldsymbol{x},\boldsymbol{y}).$
(3.29)

This is a generalization of a similar result obtained in the case $N=mp$ in
[1]. Indeed, in this case the ground state is non-degenerate, and the
corresponding wave function
$\Phi_{\mathfrak{r}_{0}}=({\mathcal{Y}}_{{\boldsymbol{r}}_{0}})^{k}$ can be
identified with the DTT wave function obtained as the conformal block
function, $\langle V(x_{1})\cdots V(x_{N})\rangle$, where $V(x)$ is the
primary field of the $k$-th order symmetric representation $[k]$, of a Wess-
Zumino-Witten model
$\widehat{\mathfrak{su}}(p)_{k}\otimes\widehat{\mathfrak{u}}(1)$. To identify
the two expressions, we can re-label the particles and write the wedge state
in the form

${\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})=\varepsilon[y_{1}\wedge
y_{1}x\wedge\cdots\wedge y_{1}x^{m-1}\wedge y_{2}\wedge
y_{2}x\wedge\cdots\wedge y_{2}x^{m-1}\wedge\cdots\cdots\wedge y_{p}\wedge
y_{p}x\wedge\cdots\wedge y_{p}x^{m-1}],$ (3.30)

where $\varepsilon$ is a sign coming from the permutation of the columns in
the determinant. Expanding the determinant gives the antisymmetrization over
all particles

${\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})=\varepsilon\sum_{\sigma\in
S_{N}}(-)^{\sigma}\prod_{a=1}^{m}\left[y_{1,\sigma(a)}(x_{\sigma(a)})^{a-1}\times
y_{2,\sigma(a+m)}(x_{\sigma(a+m)})^{a-1}\times\cdots\times
y_{p,\sigma(a+m(p-1))}(x_{\sigma(a+m(p-1))})^{a-1}\right]$ (3.31)

this operation is denoted ${\mathcal{A}}$ in [1]. It is then possible to anti-
symmetrize independently the particles in each set of $m$ particles with the
same spin. It produces the Vandermonde determinants found in the equation
(5.5) of [1],

$\displaystyle\begin{split}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})&=\dfrac{\varepsilon}{(m!)^{p}}\sum_{\sigma\in
S_{N}}(-)^{\sigma}\Bigg{[}\prod_{a=1}^{m}y_{1,\sigma(a)}\prod_{\genfrac{}{}{0.0pt}{}{a,b=1}{a<b}}^{m}(x_{\sigma(a)}-x_{\sigma(b)})\times\prod_{a=m+1}^{2m}y_{2,\sigma(a)}\prod_{\genfrac{}{}{0.0pt}{}{a,b=m+1}{a<b}}^{2m}(x_{\sigma(a)}-x_{\sigma(b)})\times\\\
&\quad\quad\times\cdots\times\prod_{a=m(p-1)+1}^{mp}y_{p,\sigma(a)}\prod_{\genfrac{}{}{0.0pt}{}{a,b=(p-1)m+1}{a<b}}^{pm}(x_{\sigma(a)}-x_{\sigma(b)})\Bigg{]}\\\
&=\dfrac{\varepsilon}{(m!)^{p}}{\mathcal{A}}\Bigg{[}\prod_{a=1}^{m}y_{1,a}\prod_{\genfrac{}{}{0.0pt}{}{a,b=1}{a<b}}^{m}(x_{a}-x_{b})\times\prod_{a=m+1}^{2m}y_{2,a}\prod_{\genfrac{}{}{0.0pt}{}{a,b=m+1}{a<b}}^{2m}(x_{a}-x_{b})\times\cdots\times\prod_{a=m(p-1)+1}^{mp}y_{p,a}\prod_{\genfrac{}{}{0.0pt}{}{a,b=(p-1)m+1}{a<b}}^{pm}(x_{a}-x_{b})\Bigg{]}.\end{split}$
(3.32)

In the next section, we show that the $U(p)$ Kac-Moody symmetry appears in the
large $N$-limit. We relate the KZ equation (3.29) to the correlation function
of the vertex operators, which intertwine the wedge states with the states of
the free fermion Fock space. The results of this subsection show that the KZ
equation holds even at finite $N$.

We note that the Calogero model ground states give a limited class of the
chiral correlators of the WZW model. For the study of more general correlators
in the context of the FQHE, see for example, [32][33].

###  3.4 Generalized Statistics

Let us focus for a moment on the vacuum wave functions in the case $k=1$ for
which the matrix model wave functions coincide with wedge states, i.e.
$\Phi_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})={\mathcal{Y}}_{\boldsymbol{r_{0}}}(\boldsymbol{x},\boldsymbol{y})$.
The latter can be written in the explicit form

$\mathcal{Y}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})=\left|\begin{array}[]{ccccccc}y_{1,1}&\cdots&y_{p,1}&\cdots&y_{i_{1},1}x_{1}^{m}&\cdots&y_{i_{q},1}x_{1}^{m}\\\
y_{1,2}&\cdots&y_{p,2}&\cdots&y_{i_{1},2}x_{2}^{m}&\cdots&y_{i_{q},2}x_{2}^{m}\\\
\vdots&&&\ddots&&&\vdots\\\
y_{1,N}&\cdots&y_{p,N}&\cdots&y_{i_{1},N}x_{N}^{m}&\cdots&y_{i_{q},N}x_{N}^{m}\end{array}\right|\,.$
(3.33)

An important feature of the wave functions
${\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})$ is that
they do not vanish when two coordinates coincide $x_{a}\to x_{b}$ because of
the presence of the extra spin variables $y_{i,a}$. Instead, when $p+1$
particles approach each other at the same time, at least two particles will
have the same spin. As a result, the two corresponding rows of the
determinants will be identical, and the determinant will vanish. More
precisely, when $N=mp+q$, all the wave functions
${\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})$ vanish
when $p+1$ particles coordinate coincide. When only $r<p+1$ particles
coincide, some of the wave functions might still vanish if $q>0$, which
partially lifts the degeneracy of the vacuum. In this way, the wedge states
describe particles obeying a generalized exclusion principle.

#### Fractional quasi-holes

Using a similar argument, it is possible to introduce wave functions
describing fractional quasi-holes,

$\mathcal{Y}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y}|\boldsymbol{\zeta})=\left|\begin{array}[]{ccccccc}y_{1,1}&\cdots&y_{p,1}&\cdots&y_{i_{1},1}x_{1}^{m+1}&\cdots&y_{i_{q},1}x_{1}^{m+1}\\\
\vdots&&&\ddots&&&\vdots\\\
y_{1,N}&\cdots&y_{p,N}&\cdots&y_{i_{1},N}x_{N}^{m+1}&\cdots&y_{i_{q},N}x_{N}^{m+1}\\\
y_{1,N+1}&\cdots&y_{p,N+1}&\cdots&y_{i_{1},N+1}\zeta_{1}^{m+1}&\cdots&y_{i_{q},N+1}\zeta_{1}^{m+1}\\\
\vdots&&&\ddots&&&\vdots\\\
y_{1,N+p}&\cdots&y_{p,N+p}&\cdots&y_{i_{1},N+p}\zeta_{p}^{m+1}&\cdots&y_{i_{q},N+p}\zeta_{p}^{m}\end{array}\right|$
(3.34)

When we set $\zeta_{1}=\cdots=\zeta_{p}=\zeta$, the wave functions
$\mathcal{Y}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y}|\boldsymbol{\zeta})$
contain a factor $\prod_{a=1}^{N}(\zeta-x_{a})$ due to the generalized
exclusion principle mentioned above, and it describes a hole of coordinate
$\zeta$. Thus, the general formula
$\mathcal{Y}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y}|\boldsymbol{\zeta})$
describes the splitting of this hole into $p$ quasi-holes, each of which has a
fractional charge $-e/p$.

#### Moore-Read state

Some of the characteristic states of the non-Abelian FQHE can also be
recovered from our general formula for the ground states wave function. For
instance, taking $p=2$, $N=2m$, and $k=2$, we find a singlet ground state with
parameters $\boldsymbol{r}_{0}=\\{0,1,\cdots,2m-1\\}$. The corresponding
matrix model wave function is

$\Phi(\boldsymbol{x},\boldsymbol{y})=(\mathcal{Y}_{\boldsymbol{r}_{0}}(\boldsymbol{x},\boldsymbol{y}))^{2}\,.$
(3.35)

It was noted in [1] that this expression is equivalent to the following
Pfaffian,999To recover our expression, we need to substitute $z_{a}\to x_{a}$
and $\zeta_{a}\to y_{1,a}/y_{2,a}$ in the equation (3.7) of [1].

$\Phi(\boldsymbol{x},\boldsymbol{y})\propto\mbox{Pf}\left(\frac{(y_{1a}y_{2b}-y_{2a}y_{1b})^{2}}{x_{a}-x_{b}}\right)\prod_{a<b}(x_{a}-x_{b})\,,$
(3.36)

The coordinate dependence of this wave function coincides with the Moore-Read
state.

It seems possible to generalize the Pfaffian formula for general $p=k$ and
$N=pm$ with $\boldsymbol{r}_{0}=\\{0,1,\cdots,pm-1\\}$ in the following way

$\Phi(\boldsymbol{x},\boldsymbol{y})=(\mathcal{Y}_{\boldsymbol{r}_{0}}(\boldsymbol{x},\boldsymbol{y}))^{p}\propto\Delta(\boldsymbol{x})\sum_{A^{1},\cdots,A^{m}}\epsilon(A^{1},\cdots,A^{m})\prod_{\alpha=1}^{m}\frac{(\det_{i,j}y_{i,a^{\alpha}_{j}})^{p}}{\prod_{1\leq
i<j\leq p}(x_{a^{\alpha}_{i}}-x_{a^{\beta}_{j}})}$ (3.37)

The summation is defined over all possible decomposition of
$\left\\{1,2,\cdots,pm\right\\}$ into $L$ sets $A^{\alpha}$ of $p$ mutually
distinct indices. Namely,
$A^{\alpha}=\left\\{a^{\alpha}_{1},\cdots,a^{\alpha}_{p}\right\\}$
($a^{\alpha}_{i}>a^{\alpha}_{j}$ for $i>j$), and
$a^{\alpha}_{i}\in\left\\{1,2,\cdots,pm\right\\}$, $A^{\alpha}\cap
A^{\beta}=\emptyset$ and
$\bigcup_{\alpha=1}^{p}A^{\alpha}=\left\\{1,2,\cdots,pm\right\\}$. The sign
$\epsilon(A^{1},\cdots,A^{m})$ depends on the sets $A^{\alpha}$. This
conjecture is a natural generalization of the Moore-Read wave function (3.36).
We have checked that the power counting for both $x$ and $y$ variables match
separately, and observed the correct behavior for the generalized statistics.
Indeed, the wave function vanishes only when $p+1$ coordinates $x_{a}$
approach each other, and such behavior is broken when the $y$ vectors become
linearly dependent.

##  4 Free fermion representation and Kac-Moody algebra

In the previous section, we have shown that the eigenstates of the Hamiltonian
can be expressed as linear combinations of products of $k$ wedge states.
However, the Plücker identities imply that this description is redundant for
$k>1$. In this section, we rewrite the eigenstates using a free fermion
formalism [15]. This has two main advantages. First, it allows us to re-
interpret the Plücker relations at finite $N$ as constraints from a loop-
algebra symmetry. Second, it brings to light the appearance of the Kac-Moody
symmetry $\widehat{\mathfrak{su}}(p)_{k}$ in the large $N$-limit.

###  4.1 Plücker identity as $\widehat{\mathfrak{su}}(k)_{+}$-constraints at
finite $N$

For a finite number $N$ of particles, we introduce a set of (non-relativistic)
free fermion oscillators, $\psi^{i,\alpha}_{n}$, $\bar{\psi}^{i,\alpha}_{n}$,
indexed by the integers $n\geq 0$, $i=1,\cdots,p$, $\alpha=1,\cdots,k$, and
satisfying

$\left\\{\psi^{i,\alpha}_{n},\bar{\psi}^{j,\beta}_{m}\right\\}=\delta_{i,j}\delta_{\alpha,\beta}\delta_{n,m},\quad\left\\{\psi^{i,\alpha}_{n},\psi^{j,\beta}_{m}\right\\}=\left\\{\bar{\psi}^{i,\alpha}_{n},\bar{\psi}^{j,\beta}_{m}\right\\}=0\,.$
(4.1)

We also define the bra and ket vacua as,

$\psi^{i,\alpha}_{n}|0\rangle=0,\quad\langle
0|\bar{\psi}^{i,\alpha}_{n}=0\,,\quad\mbox{ for }n\geq 0\,.$ (4.2)

Generic states of the free fermion Fock space are obtained by the action of
the modes $\bar{\psi}_{n}^{i,\alpha}$ on the vacuum $\ket{0}$. This Fock space
has a natural family of grading operators
$\hat{N}_{\alpha}=\sum_{n=0}^{\infty}\sum_{i=1}^{p}\bar{\psi}^{i,\alpha}_{n}\psi^{i,\alpha}_{n}$
corresponding to the fermionic numbers counting the number of modes
$\bar{\psi}_{n}^{i,\alpha}$ for each $\alpha=1\cdots k$. As a result, the
total Fock space ${\mathcal{F}}^{(k)}$ decomposes as a direct sum of subspaces
$\mathcal{F}^{(k)}_{\boldsymbol{N}}$ labeled by the vector
$\boldsymbol{N}=(N_{1},\cdots,N_{k})$ encoding the eigenvalues
$N_{\alpha}\in\mathbb{Z}^{\geq 0}$ of the fermionic number operators
$\hat{N}_{\alpha}$. These subspaces admit the following basis

$\left\\{\prod_{\alpha=1}^{k}\left(\bar{\psi}^{i_{1}^{(\alpha)},\alpha}_{n_{1}^{(\alpha)}}\cdots\bar{\psi}^{i_{N_{\alpha}}^{(\alpha)},\alpha}_{n_{N_{\alpha}}^{(\alpha)}}\right)|0\rangle,\quad
n_{a_{\alpha}}^{(\alpha)}\in\mathbb{Z}^{\geq 0},\
i_{a_{\alpha}}^{(\alpha)}\in[\\![1,p]\\!],\quad a_{\alpha}=1\cdots
N_{\alpha}\right\\}$ (4.3)

In the following, we restrict ourselves to the case
$\boldsymbol{N}_{0}=(N,\cdots,N)$ where $N$ is the number of particles, and
denote the corresponding subspace ${\mathcal{F}}_{\boldsymbol{N}_{0}}^{(k)}$.

Let $\mathcal{V}^{(k)}_{N}$ denote the space linearly generated by the wave
functions of the matrix model (i.e., omitting the Vandermonde factor in
$\psi_{\mathfrak{r}}(\boldsymbol{x},\boldsymbol{y})$). As shown in the
previous section, this space is spanned by products of $k$ wedge states (which
are not all linearly independent),

$\mathcal{V}^{(k)}_{N}=\operatorname*{\text{Span
}}_{({\boldsymbol{r}}^{(1)},\cdots,{\boldsymbol{r}}^{(k)})\in\mathfrak{R}}\left\\{\prod_{\alpha=1}^{k}\mathcal{Y}_{{\boldsymbol{r}}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y})\right\\}.$
(4.4)

We define a map from the fermionic Fock space
$\mathcal{F}^{(k)}_{\boldsymbol{N}_{0}}$ to the wave functions
$\mathcal{V}^{(k)}_{N}$ realized as a linear map101010The second identity
follows from the anti-commutation relation
$\left\\{\psi^{\alpha}(x_{a},y_{a}),\bar{\psi}^{i,\beta}_{n}\right\\}=\delta_{\alpha\beta}x_{a}^{n}y_{i,a}$.

$\displaystyle\begin{split}&\prod_{\alpha=1}^{k}\left(\bar{\psi}^{i_{1}^{(\alpha)},\alpha}_{n_{1}^{(\alpha)}}\cdots\bar{\psi}^{i_{N}^{(\alpha)},\alpha}_{n_{N}^{(\alpha)}}\right)|0\rangle\to\langle\boldsymbol{x},\boldsymbol{y}|\prod_{\alpha=1}^{k}\left(\bar{\psi}^{i_{1}^{(\alpha)},\alpha}_{n_{1}^{(\alpha)}}\cdots\bar{\psi}^{i_{N}^{(\alpha)},\alpha}_{n_{N}^{(\alpha)}}\right)|0\rangle=\varepsilon\prod_{\alpha=1}^{k}[x^{n^{(\alpha)}_{1}}y_{i^{(\alpha)}_{1}}\wedge\cdots\wedge
x^{n^{(\alpha)}_{N}}y_{i^{(\alpha)}_{N}}]\,,\end{split}$ (4.5)

using the bra state $\langle\boldsymbol{x},\boldsymbol{y}|$ defined
as111111Note that the free fermion operator $\psi^{\alpha}(x,y)$ can also be
written as
$\psi^{\alpha}(z)=\sum_{n=0}^{\infty}\sum_{i=1}^{p}\psi^{i,\alpha}_{n}z^{pn+i}$
using the translation rule given in section 3.

$\displaystyle\langle\boldsymbol{x},\boldsymbol{y}|$ $\displaystyle=\langle
0|\Psi(x_{N},y_{N})\cdots\Psi(x_{1},y_{1})$ (4.6)
$\displaystyle\Psi(x_{a},y_{a})$
$\displaystyle=\prod_{\alpha=1}^{k}\psi^{\alpha}(x_{a},y_{a}),\quad\psi^{\alpha}(x_{a},y_{a})=\sum_{n=0}^{\infty}\sum_{i=1}^{p}\psi^{i,\alpha}_{n}x_{a}^{n}y_{i,a}\,.$
(4.7)

Here $\varepsilon=\pm$ is a sign that will be irrelevant in our discussion. It
is important to note that the fermionic states 4.2 are linearly independent
and define a basis of the space $\mathcal{F}^{(k)}_{\boldsymbol{N}_{0}}$.
However, we have seen that this is not the case for the products of wedge
states obtained from the inner product with
$\langle\boldsymbol{x},\boldsymbol{y}|$. We will now show that the linear
dependence is a consequence of the invariance of the bra state under a loop
algebra that we denote $\widehat{\mathfrak{su}}(k)_{+}$.

####  $\widehat{\mathfrak{su}}(k)_{+}$-symmetry

Let $L_{n}^{\alpha\beta}$ with $n\geq 0$ and $\alpha,\beta\in[\\![1,k]\\!]$
denote the positive generators of the loop algebra of $\mathfrak{u}(k)$ in the
fermionic representation,

$L^{\alpha\beta}_{n}=\sum_{\ell=0}^{\infty}\sum_{i=1}^{p}\bar{\psi}^{i,\alpha}_{n+\ell}\psi^{i,\beta}_{\ell}\,,\qquad(n\geq
0).$ (4.8)

We note that the fermionic number operators defined earlier correspond to
$\hat{N}_{\alpha}=L_{0}^{\alpha\alpha}$. These generators satisfy the
commutation relations

$\left[L^{\alpha\beta}_{n},L^{\gamma\delta}_{m}\right]=\delta_{\beta,\gamma}L^{\alpha\delta}_{n+m}-\delta_{\alpha,\delta}L^{\gamma\beta}_{n+m}\,,$
(4.9)

and we denote $\widehat{\mathfrak{u}}(k)_{+}$ the algebra spanned by these
operators. These operator act on the free fermion Fock spaces
${\mathcal{F}}^{(k)}$, and obey the following commutation relations with the
fermionic fields,

$[L^{\alpha\beta}_{n},\psi^{\gamma}(x_{a},y_{a})]=-\delta_{\alpha,\gamma}x_{a}^{n}\psi^{\beta}(x_{a},y_{a}),\quad[L^{\alpha\beta}_{n},\Psi(x_{a},y_{a})]=-\delta_{\alpha,\beta}x_{a}^{n}\Psi(x_{a},y_{a})\,,$
(4.10)

where the second identity is a consequence of the fermionic exclusion
principle $(\psi^{\alpha}(x,y))^{2}=0$. We deduce the action of
$\widehat{\mathfrak{u}}(k)_{+}$ generators on the bra states
$\bra{\boldsymbol{x},\boldsymbol{y}}$ introduced earlier,

$\langle\boldsymbol{x},\boldsymbol{y}|L^{\alpha\beta}_{n}=\delta_{\alpha,\beta}p_{n}(\boldsymbol{x})\langle\boldsymbol{x},\boldsymbol{y}|\,,$
(4.11)

where $p_{n}(\boldsymbol{x})=\sum_{a=1}^{N}x_{a}^{n}$ is the elementary power
sum symmetric polynomials. It implies that the bra state is annihilated by the
action of the $\widehat{\mathfrak{su}}(k)_{+}$ generators defined as,

$\langle\boldsymbol{x},\boldsymbol{y}|\bar{L}^{\alpha\beta}_{n}=0\,,\qquad\bar{L}^{\alpha\beta}_{n}=L^{\alpha\beta}_{n}-\frac{\delta_{\alpha,\beta}}{k}\sum_{\gamma}L^{\gamma\gamma}_{n}.$
(4.12)

It means that the map from the fermionic Fock space
${\mathcal{F}}^{(k)}_{\boldsymbol{N}_{0}}$ to the space of wedge states
products ${\mathcal{V}}_{N}^{(k)}$ is invariant under the
$\widehat{\mathfrak{su}}(k)_{+}$ symmetry. This invariance explains the
redundancy of our description of $\mathcal{V}^{(k)}_{N}$. In the following, we
refer to the subspace of $\mathcal{F}^{(k)}_{\boldsymbol{N}}$ obtained from
the action of the generators $\bar{L}^{\alpha\beta}_{n}$ on the fermionic
vacuum as the $\widehat{\mathfrak{su}}(k)_{+}$-orbit. As we have shown, any
state from this orbit has a vanishing inner product with
$\bra{\boldsymbol{x},\boldsymbol{y}}$, and thus belongs to the kernel of the
map ${\mathcal{F}}^{(k)}_{\boldsymbol{N}_{0}}\to{\mathcal{V}}_{N}^{(k)}$.

The Plücker relations (3.14) are a consequence of the
$\widehat{\mathfrak{su}}(k)_{+}$-symmetry. To show this, we restrict ourselves
to $k=2$ and consider the action of the generator
$\bar{L}^{21}_{n}=L_{n}^{21}$ on the Fock space
${\mathcal{F}}_{\boldsymbol{N}}^{(k)}$ with $\boldsymbol{N}=(N+1,N-1)$. The
action on the basis vectors reads

$\displaystyle
L^{21}_{n}\bar{\psi}^{i_{1},1}_{n_{1}}\cdots\bar{\psi}^{i_{N+1},1}_{n_{N+1}}\bar{\psi}^{j_{1},2}_{m_{1}}\cdots\bar{\psi}^{j_{N-1},2}_{m_{N-1}}|0\rangle$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}=(-1)^{N}\sum_{a=1}^{N+1}(-1)^{a-1}\bar{\psi}^{i_{1},1}_{n_{1}}\cdots\,\cancel{\bar{\psi}^{i_{a},1}_{n_{a}}}\,\cdots\bar{\psi}^{i_{N+1},1}_{n_{N+1}}\bar{\psi}^{i_{a},2}_{n_{a}+n}\bar{\psi}^{j_{1},2}_{m_{1}}\cdots\bar{\psi}^{j_{N-1},2}_{m_{N-1}}|0\rangle$

After taking the inner product with $\langle\boldsymbol{x},\boldsymbol{y}|$
and using (4.12), we find the identity

$\sum_{a=1}^{N+1}(-1)^{a}[y_{i_{1}}x^{n_{1}}\wedge\cdots\wedge\,\cancel{y_{i_{a}}x^{n_{a}}}\wedge\cdots\wedge
y_{i_{N+1}}x^{n_{N+1}}]\cdot[y_{i_{a}}x^{n_{a}+n}\wedge
y_{j_{1}}x^{m_{1}}\wedge\cdots\wedge y_{j_{N-1}}x^{m_{N-1}}]=0.$ (4.13)

This is indeed a generalization of the Plücker relations (3.14) which are
recovered by taking $n=0$. The same relations are obtained if consider instead
the action of $\bar{L}^{12}_{n}$ on ${\mathcal{F}}_{\boldsymbol{N}}^{(k)}$
with $\boldsymbol{N}=(N-1,N+1)$.

It is also instructive to examine the action of the diagonal element
$\bar{L}_{n}^{11}=\frac{1}{2}(L_{n}^{11}-L_{n}^{22})=-\bar{L}_{n}^{22}$ on the
basis vectors of the Fock space ${\mathcal{F}}_{\boldsymbol{N}_{0}}^{(k)}$,

$\displaystyle(L^{11}_{n}-L^{22}_{n})\bar{\psi}^{i_{1},1}_{n_{1}}\cdots\bar{\psi}^{i_{N},1}_{n_{N}}\bar{\psi}^{j_{1},2}_{m_{1}}\cdots\bar{\psi}^{j_{N},2}_{m_{N}}|0\rangle$
$\displaystyle~{}~{}~{}~{}~{}~{}=\sum_{a=1}^{N}\left(\bar{\psi}^{i_{1},1}_{n_{1}}\cdots\,\bar{\psi}^{i_{a},1}_{n_{a}+n}\,\cdots\bar{\psi}^{i_{N},1}_{n_{N}}\bar{\psi}^{j_{1},2}_{m_{1}}\cdots\bar{\psi}^{j_{N},2}_{m_{N}}-\bar{\psi}^{i_{1},1}_{n_{1}}\cdots\bar{\psi}^{i_{N},1}_{n_{N}}\bar{\psi}^{j_{1},2}_{m_{1}}\cdots\bar{\psi}^{j_{a},2}_{m_{a}+n}\,\cdots\bar{\psi}^{j_{N},2}_{m_{N}}\right)|0\rangle\,.$

After taking the inner product with the bra state
$\bra{\boldsymbol{x},\boldsymbol{y}}$, we find an identity which is trivially
satisfied,

$\displaystyle\sum_{a=1}^{N}\left([y_{i_{1}}x^{n_{1}}\wedge\cdots\wedge
y_{i_{a}}x^{n_{a}+n}\wedge\cdots\wedge
y_{i_{N}}x^{n_{N}}]\cdot[y_{j_{1}}x^{m_{1}}\wedge\cdots\wedge
y_{j_{N}}x^{m_{N}}]\right.$
$\displaystyle~{}~{}~{}~{}~{}-\left.[y_{i_{1}}x^{n_{1}}\wedge\cdots\wedge
y_{i_{N}}x^{n_{N}}]\cdot[y_{j_{1}}x^{m_{1}}\wedge\cdots\wedge
y_{j_{a}}x^{m_{a}+n}\wedge\cdots\wedge y_{j_{N}}x^{m_{N}}]\right)$
$\displaystyle=(p_{n}(\boldsymbol{x})[y_{i_{1}}x^{n_{1}}\wedge\cdots\wedge
y_{i_{N}}x^{n_{N}}])[y_{j_{1}}x^{m_{1}}\wedge\cdots\wedge y_{j_{N}}x^{m_{N}}]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-[y_{i_{1}}x^{n_{1}}\wedge\cdots\wedge
y_{i_{N}}x^{n_{N}}](p_{n}(\boldsymbol{x})[y_{j_{1}}x^{m_{1}}\wedge\cdots\wedge
y_{j_{N}}x^{m_{N}}])$ $\displaystyle=0\,.$

This shows that the invariance of the bra state
$\bra{\boldsymbol{x},\boldsymbol{y}}$ under
$\widehat{\mathfrak{su}}(k)_{+}$-symmetry for $k=2$ is equivalent to the
generalized Plücker identities (4.13) obeyed by products of wedge states.

#### Ground states

The ground states wave functions (3.26) for $N=pm+q$ (with $0\leq q<p$) are
associated to the following states of
$\mathcal{F}^{(k)}_{\boldsymbol{N}_{0}}$,

$\prod_{\alpha=1}^{k}\prod_{r=1}^{q}\bar{\psi}^{i_{r}^{(\alpha)},\alpha}_{m}\prod_{\alpha=1}^{k}\prod_{i=1}^{p}\prod_{l=1}^{m-1}\bar{\psi}^{i,\alpha}_{l}|0\rangle$
(4.14)

When $q=0$, the first factor is absent, and the vacuum is unique. When $q>0$,
the vacua are labeled by the indices
$(i_{1}^{(\alpha)},\cdots,i_{q}^{(\alpha)})$ with $\alpha=1\cdots k$ taking
value in $[\\![1,p]\\!]$. The vacua form a representation $[k^{q}]$ of
$\mathfrak{su}(p)$ [1]. They belong to a dual $[q^{k}]$ representation of
$\mathfrak{su}(k)$, which is singlet, which does not belong to the
$\widehat{\mathfrak{su}}(k)_{+}$-orbit.121212We note that the orbit cannot
contain a singlet state since $L^{\alpha\beta}_{n}|\chi\rangle\neq 0$ implies
that neither $|\chi\rangle$ nor $L^{\alpha\beta}_{n}|\chi\rangle$ belong to
the singlet.

#### Dual loop symmetry and excited states

It is possible to define an action of the larger algebra
$\widehat{\mathfrak{u}}(pk)_{+}$ on the fermionic Fock space
${\mathcal{F}}^{(k)}$. This algebra is defined as the positive part of the
loop algebra of $\mathfrak{u}(pk)$, it is generated by the following operators
in the fermionic representation

$M^{i+p(\alpha-1),j+p(\beta-1)}_{n}=\sum_{\ell=0}^{\infty}\bar{\psi}^{i,\alpha}_{n+\ell}\psi^{j,\beta}_{\ell}\,,\qquad(n\geq
0)$ (4.15)

where the indices $i+p(\alpha-1)$ and $j+p(\beta-1)$ run from $1$ to $pk$.

The algebra $\widehat{\mathfrak{u}}(pk)_{+}$ contains a subalgebra
$\widehat{\mathfrak{u}}(p)_{+}$ generated by

$K^{ij}_{n}=\sum_{\ell=0}^{\infty}\sum_{\alpha=1}^{k}\bar{\psi}^{i,\alpha}_{n+\ell}\psi^{j,\alpha}_{n}\,,\qquad(n\geq
0),$ (4.16)

which is dual to the algebra $\widehat{\mathfrak{u}}(p)_{+}$ in the sense that

$\left[L^{\alpha\beta}_{n},K^{ij}_{m}\right]=0\,.$ (4.17)

More precisely, we have the decomposition [17]

$\widehat{\mathfrak{u}}(pk)_{+}\supset\widehat{\mathfrak{su}}(p)_{+}\oplus\widehat{\mathfrak{su}}(k)_{+}\oplus\widehat{\mathfrak{u}}(1)_{+}$
(4.18)

where the $\widehat{\mathfrak{u}}(1)_{+}$ subalgebra is generated by

$J_{n}=\sum_{\ell=0}^{\infty}\sum_{\alpha=1}^{k}\sum_{i=1}^{p}\bar{\psi}^{i,\alpha}_{n+\ell}\psi^{i,\alpha}_{n}=\sum_{\alpha=1}^{k}L_{n}^{\alpha\alpha}=\sum_{i=1}^{p}K_{n}^{ii}.$
(4.19)

It is obvious that the states generated by the action of
$\widehat{\mathfrak{su}}(p)_{+}\oplus\widehat{\mathfrak{u}}(1)_{+}$ on the
vacuum do not belong to the $\widehat{\mathfrak{su}}(k)_{+}$-orbit. In the
large $N$ limit, it is possible to show that these states exhaust the possible
eigenstates of the Hamiltonian 3.8, and we expect that this statement holds
also at finite $N$.

The adjoint action of the $\widehat{\mathfrak{u}}(p)_{+}$ generators on the
fermionic field $\Psi(x_{a},y_{a})$ reads

$\left[K^{ij}_{n},\Psi(x_{a},y_{a})\right]=-x_{a}^{n}y_{i,a}\partial_{j,a}\Psi(x_{a},y_{a}),$
(4.20)

which shows that the fields $\Psi(x,y)$ is the finite $N$ equivalent of the
primary fields for the Kac-Moody algebra $\widehat{\mathfrak{u}}(p)$. The KZ
equation for the ground state wave functions obtained in (3.29) is expected to
follow from this primary field condition. Indeed, at large $N$ the derivation
of the KZ equation from the primary field condition is a standard procedure
[34]. On the other hand, the derivation of the KZ equation at finite $N$ given
in the appendix A.2 does not involve the
$\widehat{\mathfrak{u}}(p)_{+}$-invariance, and it would be interesting to
provide an alternative derivation starting from the condition (4.20).

In the Abelian case ($p=1$), the algebraic structure simplifies and the Fock
space ${\mathcal{F}}^{(k)}$ is generated by the action of the
$\widehat{\mathfrak{u}}(1)_{+}$ generators on the vacuum. Thus, in the large
$N$ limit, the spectrum of the Hamiltonian (3.8) produces the
$\widehat{\mathfrak{u}}(1)$ character 3.5, in agreement with the results
obtained by Polychronakos in [7].

###  4.2 Kac-Moody algebra at large $N$ and level-rank duality

In this subsection, we study the large $N$ limit of the fermionic Fock space
${\mathcal{F}}^{(k)}$ and observe the emergence of the Kac-Moody symmetry
$\widehat{\mathfrak{su}}(p)_{k}$. The emergence of this symmetry is one of the
main results obtained by DTT in [1, 13], it was later proved rigorously in
[14]. Our spin Calogero model provides an alternative point on view which we
explain here.

In order to take the large $N$ (or, equivalently, large $m$) limit of the
wedge states, we focus first on the vacuum wave functions (3.28, 3.24)

$\displaystyle\Phi_{\mathfrak{r}_{0}}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle=\prod_{\alpha=1}^{k}\mathcal{Y}_{\boldsymbol{r_{0}^{(\alpha)}}}(\boldsymbol{x},\boldsymbol{y})$
(4.21)
$\displaystyle\mathcal{Y}_{\boldsymbol{r}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle=\left(\prod_{a=1}^{N}x_{a}^{m-1}\right)[y_{1}x^{-m+1}\wedge\cdots\wedge
y_{p}x^{-m+1}\wedge\cdots\wedge y_{1}\wedge\cdots\wedge y_{p}\wedge
xy_{i_{1}^{(\alpha)}}\wedge\cdots\wedge xy_{i_{q}^{(\alpha)}}]\,.$ (4.22)

with the decomposition $N=pm+q$, $0\leq q<p$. The factor
$\prod_{a=1}^{N}x_{a}^{m-1}$ can be absorbed by a constant shift of the
Hamiltonian, it can be ignored for the purpose of taking the large $N$ limit.
Permuting the factors in the wedge product, we obtain an expression of the
wedge states appropriate for the limit $N\to\infty$, it produces

$\mathcal{Y}_{\boldsymbol{r}_{0}^{(\alpha)}}^{\infty}=[(y_{i_{1}^{(\alpha)}}x)\wedge\cdots\wedge(y_{i_{q}^{(\alpha)}}x)\wedge
y_{1}\wedge\cdots\wedge y_{p}\wedge y_{1}x^{-1}\wedge\cdots\wedge
y_{p}x^{-1}\wedge\cdots]\,.$ (4.23)

When $q=0$, the factors $y_{i_{1}^{(\alpha)}}x$ are not present, and we
recover the usual free fermion vacuum written as a semi-infinite wedge product
[16]. We will explain below how to introduce the extra factors when $q\neq 0$.
General wedge states are obtained by replacing a finite number of factors
$y_{i}x^{-n}$ by $y_{i}x^{-n+r}$ for some $r\in\mathbb{Z}^{>0}$. The limiting
procedure is the same for these states. We denote the space of the
eigenfunctions $\Phi_{\mathfrak{r}}^{\infty}(\boldsymbol{x},\boldsymbol{y})$
built upon semi-infinite wedge products as $\mathcal{V}_{\infty}^{(k)}$. We
note that, in the large $N$ limit, the wedge states become true eigenstates of
the Hamiltonian (3.8) since the energies $E(\boldsymbol{\mathfrak{r}})$ grow
as $O(N^{2})$, and so the subleading terms in (3.22) vanish. We note that the
coefficients $C(\boldsymbol{\mathfrak{r}},\boldsymbol{\mathfrak{s}})$ are
finite, as seen from the explicit computation in Appendix A, and the number of
the sets $\boldsymbol{\mathfrak{s}}$ is also finite.

To describe the Fock space at large $N$, we introduce the free fermionic modes
$\psi^{i,\alpha}_{r}$, $\bar{\psi}^{i,\alpha}_{r}$, which still carry the
color index $i=1,\cdots,p$ and Chern-Simons level index $\alpha=1,\cdots,k$,
but now have the mode index $r\in\mathbb{Z}+\frac{1}{2}$. These operators
satisfy the anticommutation relations

$\left\\{\psi^{i,\alpha}_{r},\bar{\psi}^{j,\beta}_{s}\right\\}=\delta_{i,j}\delta_{\alpha,\beta}\delta_{r+s,0},\quad\left\\{\psi^{i,\alpha}_{r},\psi^{j,\beta}_{s}\right\\}=\left\\{\bar{\psi}^{i,\alpha}_{r},\bar{\psi}^{j,\beta}_{s}\right\\}=0\,.$
(4.24)

The Fock space is built on a vacuum $\ket{\emptyset}$ which can be seen
formally as the limit of the ground state (4.14) for $q=0$.The action of the
fermionic modes on this vacuum and the corresponding dual state reads

$\psi^{i,\alpha}_{r}|\emptyset\rangle=\bar{\psi}^{i,\alpha}_{r}|\emptyset\rangle=0,\quad\langle\emptyset|\psi^{i,\alpha}_{-r}=\langle\emptyset|\bar{\psi}^{i,\alpha}_{-r}=0\,,\qquad(r>0)$
(4.25)

We denote ${\mathcal{F}}_{\infty}^{(p,k)}$ the Fock space obtained by the
action of negative modes $\psi^{i,\alpha}_{-r}$, $\bar{\psi}^{i,\alpha}_{-r}$
on this vacuum.

The map ${\mathcal{F}}_{\boldsymbol{N}_{0}}^{(k)}\to{\mathcal{V}}_{N}^{(k)}$
defined in the previous subsection can be extended formally to the limit of
infinite number of particles, i.e. $N=mp+q$ with $m\to\infty$,
$q\in[\\![0,p-1]\\!]$ fixed. To do so, we extend the definition of the bra
state $\bra{\boldsymbol{x},\boldsymbol{y}}$ as follows,

$\displaystyle\langle\boldsymbol{x},\boldsymbol{y}|$
$\displaystyle=\lim_{m\to\infty}\langle\emptyset|:\left(\prod_{\alpha=1}^{k}\prod_{i=1}^{p}\prod_{r=1/2}^{m-1/2}\bar{\psi}^{i,\alpha}_{r}\right)\Psi(x_{mp+q},y_{mp+q})\cdots\Psi(x_{1},y_{1}):$
(4.26) $\displaystyle\Psi(x_{a},y_{a})$
$\displaystyle=\prod_{\alpha=1}^{k}\psi^{\alpha}(x_{a},y_{a}),\quad\psi^{\alpha}(x_{a},y_{a})=\sum_{r\in\mathbb{Z}+1/2}\sum_{i=1}^{p}\psi^{i,\alpha}_{r}x_{a}^{r+1/2}y_{i,a}$
(4.27)

where $:\cdots:$ is the fermionic normal ordering. We use the anti-commutation
rule,

$\left\\{\psi^{\alpha}(x,y),\bar{\psi}^{i,\beta}_{r}\right\\}=x^{-r+1/2}y_{i}\,,$
(4.28)

to show that it defines by projection a map
${\mathcal{F}}_{\infty}^{(p,k)}\to{\mathcal{V}}_{\infty}^{(k)}$ which sends
the vacuum state $\ket{\emptyset}$ to the ground state wave function (4.21)
for $q=0$. The ground states wave functions for $q\neq 0$ can be obtained from
the fermionic states

$\prod_{\alpha=1}^{k}\prod_{r=1}^{q}\bar{\psi}^{i_{r}^{(\alpha)},\alpha}_{-1/2}|\emptyset\rangle\,.$
(4.29)

####  $\widehat{\mathfrak{u}}(pk)$-symmetry and level-rank duality

The fermionic Fock space ${\mathcal{F}}_{\infty}^{(p,k)}$ admits the action of
the $\widehat{\mathfrak{u}}(pk)_{1}$ Kac-Moody algebra generated by the
operators

$J^{i\alpha;j\beta}_{n}=\sum_{s\in\mathbb{Z}+1/2}:\bar{\psi}^{i\alpha}_{s}\psi^{j,\beta}_{n-s}:\,,$
(4.30)

The indices $i=1\cdots p$, and $\alpha=1\cdots k$ can be combined into a
single index $i+p(\alpha-1)$ running from one to $pk$. This algebra contains
two commuting Kac-Moody subalgebras $\widehat{\mathfrak{u}}(p)_{k}$ and
$\widehat{\mathfrak{u}}(k)_{p}$ respectively generated by

$K^{ij}_{n}=\sum_{s\in\mathbb{Z}+1/2}\sum_{\alpha=1}^{k}:\bar{\psi}^{i\alpha}_{s}\psi^{j,\alpha}_{n-s}:\,,\quad
L^{\alpha\beta}_{n}=\sum_{s\in\mathbb{Z}+1/2}\sum_{i=1}^{p}:\bar{\psi}^{i\alpha}_{s}\psi^{i,\beta}_{n-s}:.$
(4.31)

It also contains a $\widehat{\mathfrak{u}}(1)_{pk}$ subalgebra generated by
the traces $\sum_{i}K_{n}^{ii}=\sum_{\alpha}L_{n}^{\alpha\alpha}$. For each of
these algebras, we can introduce the traceless generators, as in (4.12), to
define the corresponding $\widehat{\mathfrak{su}}(pk)_{1}$,
$\widehat{\mathfrak{su}}(k)_{p}$, $\widehat{\mathfrak{su}}(p)_{k}$ subalgebra.
The central charge of the Kac-Moody $\widehat{\mathfrak{su}}(p)_{k}$ algebra
is

$c_{p,k}=\frac{k(p^{2}-1)}{k+p},$ (4.32)

it obeys the identity $c_{pk,1}=c_{k,p}+c_{p,k}$. It implies the following
decomposition as a conformal embedding [35, 36, 37, 38],

$\widehat{\mathfrak{su}}(pk)_{1}\supset\widehat{\mathfrak{su}}(p)_{k}\oplus\widehat{\mathfrak{su}}(k)_{p}.$
(4.33)

In particular, the stress-energy tensors obtained by the Sugawara construction
coincide,

$T_{\widehat{\mathfrak{su}}(pk)_{1}}=T_{\widehat{\mathfrak{su}}(p)_{k}}+T_{\widehat{\mathfrak{su}}(k)_{p}}\,.$
(4.34)

In [17], Nakanishi and Tsuchiya used this decomposition to prove the level-
rank duality between $\widehat{\mathfrak{su}}(k)_{p}$ and
$\widehat{\mathfrak{su}}(p)_{k}$ Kac-Moody algebras. The Fock space
${\mathcal{F}}_{\infty}^{(k)}$ describes the states of $pk$ free fermions in
the NS sector. It is graded by the fermion number operator
$\hat{N_{F}}=\sum_{r\in\mathbb{Z}}\sum_{i=1}^{p}\sum_{\alpha=1}^{k}:\bar{\psi}^{i,\alpha}_{r}\psi^{i,\alpha}_{-r}:$
which coincides with the zero mode of the $\widehat{\mathfrak{u}}(1)_{pk}$
current. Accordingly, the Fock space is decomposed into

${\mathcal{F}}_{\infty}^{(p,k)}=\bigoplus_{\sigma\in\mathbb{Z}}{\mathcal{F}}^{(p,k)}_{\infty,\sigma}.$
(4.35)

We note that the state of our model with $N=mp+q$ belong to the subspace
${\mathcal{F}}^{(p,k)}_{\infty,\sigma}$ with $\sigma=qk$. For each subspace,
we have the decomposition

$\mathcal{F}^{p,k}_{\sigma}\cong\bigoplus_{\boldsymbol{\lambda}}\mathcal{W}^{\widehat{\mathfrak{su}}(p)_{k}}_{\boldsymbol{\lambda}}\otimes\mathcal{W}^{\widehat{\mathfrak{su}}(k)_{p}}_{\boldsymbol{\lambda}^{\prime}}\otimes\mathcal{W}^{\widehat{\mathfrak{u}}(1)}_{\sigma}\,.$
(4.36)

where the index $\boldsymbol{\lambda}$ labels the irreducible representation
of $\widehat{\mathfrak{su}}(p)_{k}$ This label correspond to the sequence of
integers $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{p}$ with the
conditions, (i) $\lambda_{1}-\lambda_{p}\leq k$, (ii) $0\leq\lambda_{p}<k$,
(iii) $\sum_{i=1}^{p}\lambda_{i}\equiv\sigma$ mod. $pk$. The sequences
$\boldsymbol{\lambda}$ describe the dominant integral weights
$(k-\lambda_{1}+\lambda_{p})\Lambda_{0}+\sum_{j=1}^{p-1}(\lambda_{j}-\lambda_{j-1})\Lambda_{j}$,
where $\Lambda_{j}$ is the (affine) fundamental weight of
$\widehat{su}(p)_{k}$. We denoted the corresponding representation space as
$\mathcal{W}^{\widehat{\mathfrak{su}}(p)_{k}}_{\boldsymbol{\lambda}}$.
Finally, $\boldsymbol{\lambda}^{\prime}$ is obtained by transposition of the
partition defined by $\boldsymbol{\lambda}$. We refer to section 2 of [17] for
more details, or the original paper [39] (see also [40], [41] for the similar
results).

The decomposition (4.36) implies an identity for the characters of the
corresponding representations. As an illustration, this identity is checked
explicitly in appendix D in the case $p=k=2$.

In the large $N$ limit, we expect that the
$\widehat{\mathfrak{su}}(k)_{+}$-invariance (4.12) of the bra state
$\bra{\boldsymbol{x},\boldsymbol{y}}$ is replaced by the condition

$\langle\boldsymbol{x},\boldsymbol{y}|\bar{L}^{\alpha\beta}_{n}=0,\quad n\leq
0\,,$ (4.37)

where $\bar{L}^{\alpha\beta}_{n}$ is the traceless generator of
$\widehat{\mathfrak{su}}(k)_{p}$ obtained from $L^{\alpha\beta}_{n}$. If so,
the decomposition (4.33) implies that the linearly-independent eigenfunctions
are spanned by
$\widehat{\mathfrak{su}}(p)_{k}\oplus\widehat{\mathfrak{u}}(1)_{pk}$. The
extra $\widehat{\mathfrak{u}}(1)_{pk}$ factor can be introduced by an extra
free boson. In this way, we recover the Kac-Moody symmetry of the model
observed in [1].

##  5 Discussion

In this paper, we proposed a microscopic description of the non-Abelian FQHE
using a spin Calogero Hamiltonian involving $k$th order symmetric
representations of the non-Abelian $U(p)$ symmetry. This description follows
from the diagonalization of the DTT matrix model [1] describing the vortices
of a 2+1 dimensional Chern-Simons theory with $U(p)$ gauge group. We have
introduced a family of wave functions constructed as determinants involving
both coordinates and spin dependence called wedge states. We have proved that
the Hamiltonian has a triangular action on these wave functions and deduced
its spectrum and the expression of ground-state wave functions. The wedge
states are not linearly independent when the Chern-Simons level $k$ is greater
than one, and it is necessary to quotient by the loop algebra invariance
$\widehat{\mathfrak{su}}(k)_{+}$ to find a proper eigenbasis. The action of
this symmetry algebra was obtained by re-expressing the wedge states’ wave
functions in a free fermion formalism. In this way, it was shown that the
$\widehat{\mathfrak{su}}(k)_{+}$-symmetry produces bilinear relations among
determinants, which generalize the celebrated Plücker relations.

In the large $N$ limit, where $N$ is the number of vortices, the
$\widehat{\mathfrak{su}}(k)_{+}$-invariance is expected to extend to an
invariance under the Kac-Moody $\widehat{\mathfrak{su}}(k)_{p}$ algebra dual
to the $\widehat{\mathfrak{su}}(p)_{k}$-symmetry of the model under level-rank
duality. As a result, the spectrum of the Hamiltonian reproduces the WZW
$\widehat{\mathfrak{su}}(p)_{k}$ characters, in agreement with the
observations made in [1]. In fact, the presence of this Kac-Moody symmetry is
already observed at finite $N$. Indeed, it was shown that ground state wave
functions obey a Knizhnik-Zamolodchikov equation, generalizing a similar
result obtained in [1] to the case of degenerate ground states. This proof is
independent of our observation of the
$\widehat{\mathfrak{su}}(k)_{+}$-symmetry, and it would be instructive to
relate this KZ-equation to the finite $N$
$\widehat{\mathfrak{su}}(k)_{+}$-primary field condition given in (4.20).

One of the main open questions is the integrability of our model. The standard
spin Calogero model in which particles carry a fundamental representation of
the symmetry group $U(p)$ was shown to be integrable using the Lax formalism
in [19]. In addition, this model exhibits Yangian symmetry [20], and it is
natural to expect that our model is also Yangian-invariant, albeit involving
higher-order representations. In fact, we expect our Hamiltonian to belong to
the class of degenerate integrable systems recently introduced by Reshetikhin
as a generalization of the idea of quantum integrability (see [42, 43] and the
references therein).131313The spin Calogero-Moser model is indeed an example
of this class of models. This notion refers to integrable systems in which the
invariant phase space has fewer dimensions than the usual ones. Our
construction of the wedge states by projection on the bra state described in
section 4.1 suggests some similarity with this notion. We hope to be able to
come back to this important question in a future publication.

Several studies have indicated a possible role of the $W_{1+\infty}$-algebra
and its deformations (e.g., affine Yangian) in the QHE (see, for example, [44,
45]).141414We refer to the review [46] for more details on these algebraic
structures. In this context, several deformations of the model can be
introduced, like the trigonometric or elliptic deformations, relativistic
deformations, or $\beta$-deformations of the coupling. The relativistic model
is expected to be diagonalized using the $q$-deformed wedge states introduced
in [47]. Moreover, it would be interesting to investigate the interplay
between these algebraic structures and the braiding of non-Abelian anyons.

Finally, from a more physical perspective, it was noted in [1] that the model
reproduces the Blok-Wen wave functions with filling factor $\nu=p/(k+pn)$ for
$n=1$. The KZ equation and Hamiltonian eigevalue equation can accomodate for
any $n\geq 0$ simply by conjugation with powers of the Vandermonde
determinant, but the matrix model interpretation is unclear. It seems
important to better understand the physical implication of this manipulation.

## Acknowledgments

JEB would like to thank Dominic Williamson for bringing the article [1] to his
attention and for many interesting discussions. We wish to thank Didina Serban
and Jules Lamers for a critical discussion that led to the understanding of
connections with the level-rank duality and with degenerate integrable
systems. We also thank Thomas Quella for discussions on the quantum Hall
effect, Kantaro Ohmori for explaining the recent realization of non-Abelian
anyons on quantum computers, and Y. Fukusumi for the explanation of the recent
study of the relation between the Kac-Moody symmetry and the FQHE. Finally, we
are thankful to Nick Dorey for carefully reading the manuscript and for many
interesting discussions on the relationship between their work and ours. YM is
partly supported by KAKENHI Grant-in-Aid No.18K03610, 23K03380 and 21H05190.
The authors would like to thank the hospitality of the MATRIX institute where
part of this work was carried out.

##  A Proof of identities

###  A.1 Proof of the Hamiltonian action (3.20)

We rewrite the Hamiltonian as,

$\displaystyle{\mathcal{H}}_{\Delta}$
$\displaystyle=\Delta(\boldsymbol{x})^{-1}\tilde{{\mathcal{H}}}\Delta(\boldsymbol{x})=N^{2}+\sum_{a}(-\partial_{a}^{2}+2x_{a}\partial_{a})+\sum_{a\neq
b}h_{a,b}$ (A.1) $\displaystyle\text{with}\quad h_{a,b}$
$\displaystyle=-\frac{1}{x_{a}-x_{b}}(\partial_{a}-\partial_{b})+\frac{J_{a,b}J_{b,a}}{(x_{a}-x_{b})^{2}}$
(A.2)

We have to prove that $\mathcal{H}_{\Delta}$ acting on the product of
determinants $\prod_{\alpha=1}^{k}\mathcal{Y}_{\boldsymbol{r}^{(\alpha)}}$
produces the diagonal term and the subdominant terms in (3.20). Since the
Hamiltonian is of second order in the derivatives, we can decompose

$\displaystyle\mathcal{H}_{\Delta}\prod_{\alpha=1}^{k}\mathcal{Y}_{{\boldsymbol{r}}^{(\alpha)}}$
$\displaystyle=N^{2}\prod_{\alpha=1}^{k}\mathcal{Y}_{{\boldsymbol{r}}^{(\alpha)}}$
$\displaystyle~{}~{}~{}~{}~{}+\sum_{\beta}\left(\prod_{\alpha(\neq\beta)}\mathcal{Y_{\boldsymbol{r^{(\alpha)}}}}\right)\left(\sum_{a}(-\partial_{a}^{2}+2x_{a}\partial_{a})+\sum_{a\neq
b}h_{a,b}\right)\mathcal{Y_{{\boldsymbol{r}}^{(\beta)}}}(x,y)$ (A.3)
$\displaystyle~{}~{}~{}~{}~{}+2\sum_{\beta<\gamma}\left(\prod_{\alpha(\neq\beta,\gamma)}\mathcal{Y}_{{\boldsymbol{r}}^{(\alpha)}}\right)\left(-\sum_{a}\partial_{a}\mathcal{Y}_{{\boldsymbol{r}}^{(\beta)}}\partial_{a}\mathcal{Y}_{{\boldsymbol{r}}^{(\gamma)}}+\sum_{a\neq
b}\frac{1}{(x_{a}-x_{b})^{2}}J_{a,b}\mathcal{Y}_{{\boldsymbol{r}}^{(\beta)}}J_{b,a}\mathcal{Y}_{{\boldsymbol{r}}^{(\gamma)}}\right).$
(A.4)

The first line is just a multiplication by a constant. The second line gives
the action of the derivative on a single determinant $\mathcal{Y}$. The third
line gives the second-order derivative acting on distinct determinants.

#### Computation of the action on a single determinant (A.3)

We omit the index $\beta$ in (A.3) and assume $n_{1}\leq n_{2}\leq\cdots\leq
n_{N}$ in ${\boldsymbol{r}}^{(\beta)}=\boldsymbol{r}=\\{r_{1},\cdots,r_{N}\\}$
with the translation rule $r_{a}=pn_{a}+i_{a}-1$ for $a,b=1,\cdots N$. Using
the properties

$\displaystyle\begin{split}\sum_{a}x_{a}\partial_{a}{\mathcal{Y}}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=|{\boldsymbol{r}}|{\mathcal{Y}}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y}),\quad\sum_{a}\partial_{a}^{2}{\mathcal{Y}}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=\sum_{a}n_{a}(n_{a}-1)[x^{n_{1}}y_{i_{1}}\wedge\cdots\wedge\underset{\hat{a}}{x^{n_{a}-2}y_{i_{a}}}\wedge\cdots\wedge
x^{n_{N}}y_{i_{N}}]\end{split}$ (A.5)

the terms in (A.3) can be written in the form,

$\displaystyle\left(\sum_{a}(-\partial_{a}^{2}+2x_{a}\partial_{a})+\sum_{a\neq
b}h_{a,b}\right)\mathcal{Y_{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle=2|\boldsymbol{r}|\mathcal{Y}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})$
(A.6)
$\displaystyle-\sum_{a}n_{a}(n_{a}-1)[x^{n_{1}}y_{i_{1}}\wedge\cdots\wedge\underset{\hat{a}}{x^{n_{a}-2}y_{i_{a}}}\wedge\cdots\wedge
x^{n_{N}}y_{i_{N}}]$ (A.7)
$\displaystyle+2\sum_{a<b}h_{a,b}\mathcal{Y}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})\,.$
(A.8)

Eq.(A.6) defines the diagonal term and the eigenvalue associated with
$\mathcal{Y}_{{\boldsymbol{r}}}$ since $|{\boldsymbol{r}}|=\sum_{a}n_{a}$, and
$E({\boldsymbol{r}})=N^{2}+2\sum_{a}n_{a}$. The second line (A.7) is a sum of
wedge states $\mathcal{Y}_{\mathbf{s}}$ with
$|\boldsymbol{{s}}|=|\boldsymbol{r}|-2$, which can be included in the
subdominant terms in (3.20). It remains to evaluate the third line (A.8), and
we will show that it produces a sum of wedge states of the form
$\mathcal{Y}_{\mathbf{s}}(\boldsymbol{x},\boldsymbol{y})$ with
$|\mathbf{s}|=|{\boldsymbol{r}}|-2$. To do so we use an expansion of the
determinant with respect to rows $a$ then $b$,

$\mathcal{Y}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=\sum_{c<d}(-)^{a+b+c+d}\mathcal{Y}_{\boldsymbol{r},[ab|cd]}(\boldsymbol{x},\boldsymbol{y})\left|\begin{array}[]{cc}x_{a}^{n_{c}}y_{i_{c},a}&x_{a}^{n_{d}}y_{i_{d},a}\\\
x_{b}^{n_{c}}y_{i_{c},b}&x_{b}^{n_{d}}y_{i_{d},b}\end{array}\right|,$ (A.9)

where we took $a<b$ and
$\mathcal{Y}_{\boldsymbol{r},[ab|cd]}(\boldsymbol{x},\boldsymbol{y})$ denotes
the second minor of the $N\times N$ determinant
${\mathcal{Y}}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})$ with respect
to rows $a,b$ and columns $c,d$. Since the minors
$\mathcal{Y}_{\boldsymbol{r},[ab|cd]}(\boldsymbol{x},\boldsymbol{y})$ are
independent of the variables $x_{a}$, $x_{b}$ and $y_{i,a}$, $y_{i,b}$, the
operator $h_{a,b}$ acts only on the two by two determinants, and gives

$\displaystyle
h_{a,b}\left|\begin{array}[]{cc}x_{a}^{n_{c}}y_{i_{c},a}&x_{a}^{n_{d}}y_{i_{d},a}\\\
x_{b}^{n_{c}}y_{i_{c},b}&x_{b}^{n_{d}}y_{i_{d},b}\end{array}\right|$ (A.12)
$\displaystyle=-\frac{1}{x_{a}-x_{b}}\left(\left|\begin{array}[]{cc}n_{c}x_{a}^{n_{c}-1}y_{i_{c},a}&n_{d}x_{a}^{n_{d}-1}y_{i_{d},a}\\\
x_{b}^{n_{c}}y_{i_{c},b}&x_{b}^{n_{d}}y_{i_{d},b}\end{array}\right|-\left|\begin{array}[]{cc}x_{a}^{n_{c}}y_{i_{c},a}&x_{a}^{n_{d}}y_{i_{a},d}\\\
n_{c}x_{b}^{n_{c}-1}y_{i_{c},b}&n_{d}x_{b}^{n_{d}-1}y_{i_{d},b}\end{array}\right|\right)$
(A.17)
$\displaystyle~{}~{}~{}~{}+\frac{1}{(x_{a}-x_{b})^{2}}\left(\left|\begin{array}[]{cc}x_{a}^{n_{c}}y_{i_{c},a}&x_{a}^{n_{d}}y_{i_{d},a}\\\
x_{b}^{n_{c}}y_{i_{c},b}&x_{b}^{n_{d}}y_{i_{d},b}\end{array}\right|-\left|\begin{array}[]{cc}x_{b}^{n_{c}}y_{i_{c},a}&x_{b}^{n_{d}}y_{i_{d},a}\\\
x_{a}^{n_{c}}y_{i_{c},b}&x_{a}^{n_{d}}y_{i_{d},b}\end{array}\right|\right)\,.$
(A.22)

where we traded the spin permutation for a coordinate permutation in the
second line.

Assuming $n_{c}<n_{d}$, we can use the identity,

$\displaystyle-\frac{1}{x_{a}-x_{b}}(\frac{n_{c}}{x_{a}}-\frac{n_{d}}{x_{b}})+\frac{1}{(x_{a}-x_{b})^{2}}\left(1-(x_{a}/x_{b})^{n_{d}-n_{c}}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}=\frac{n_{c}}{x_{a}x_{b}}-\sum_{l=0}^{n_{d}-n_{c}-2}(l+1)x_{a}^{n_{d}-n_{c}-2-l}x_{b}^{n_{c}-n_{d}+l}$
(A.23)

to rewrite (A.22) as the sum of determinants,

$\displaystyle
n_{c}\left|\begin{array}[]{cc}x_{a}^{n_{c}-1}y_{i_{c},a}&x_{a}^{n_{d}-1}y_{i_{d},a}\\\
x_{b}^{n_{c}-1}y_{i_{c},b}&x_{b}^{n_{d}-1}y_{i_{d},b}\end{array}\right|-\sum_{l=0}^{n_{d}-n_{c}-2}(l+1)\left|\begin{array}[]{cc}x_{a}^{n_{d}-2-l}y_{i_{c},a}&x_{a}^{n_{c}+l}y_{i_{d},a}\\\
x_{b}^{n_{d}-2-l}y_{i_{c},b}&x_{b}^{n_{c}+l}y_{i_{d},b}\end{array}\right|.$
(A.28)

Re-combining the minors and taking the summation over the indices $a,b$, we
arrive at the following sum of wedge states,

$\displaystyle n_{c}[\cdots\wedge x^{n_{c}-1}y_{i_{c}}\wedge\cdots\wedge
x^{n_{d}-1}y_{i_{d}}\wedge\cdots]-\sum_{l=0}^{n_{d}-n_{c}-2}(l+1)[\cdots\wedge
x^{n_{d}-2-l}y_{i_{c}}\wedge\cdots\wedge x^{n_{c}+l}y_{i_{d}}\wedge\cdots]\,.$
(A.29)

All the wedge states in the r.h.s. are indeed of the form
$\mathcal{Y}_{\mathbf{s}}(\boldsymbol{x},\boldsymbol{y})$ with
$|\mathbf{s}|=|{\boldsymbol{r}}|-2$. The argument in the case $n_{c}>n_{d}$
follows the same lines. Finally, the expression vanishes when $n_{c}=n_{d}$.

#### Computation of the cross-terms (A.4)

To prove the theorem 3.20, it remains to show that the third line A.4 also
produces a sum of product of wedge states of total degree $|\mathfrak{r}|-2$.
More precisely, introducing the shift operators
$T_{a}=e^{\partial_{n_{a}}}=e^{p\partial_{r_{a}}}$, and the permutation
$R_{a,b}^{[{\boldsymbol{r}},{\boldsymbol{r}}^{\prime}]}$ exchanging
$r_{a}\in{\boldsymbol{r}}$ and $r^{\prime}_{b}\in{\boldsymbol{r}}^{\prime}$,
we will show that

$\displaystyle\begin{split}&\sum_{a}\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})-\sum_{a\neq
b}\dfrac{1}{(x_{a}-x_{b})^{2}}(J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y}))(J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y}))\\\
=&\sum_{c,d}\sum_{\gamma=0}^{n_{c}-1}\sum_{\delta=0}^{n_{d}^{\prime}-1}T_{c}^{-\gamma+\delta-1}(T_{d}^{\prime})^{-\delta+\gamma-1}R_{c,d}^{[{\boldsymbol{r}},{\boldsymbol{r}}^{\prime}]}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y}).\end{split}$
(A.30)

The total degree of the terms entering in the r.h.s. is

$\left(|{\boldsymbol{r}}|-\lfloor r_{c}/p\rfloor+\lfloor
r^{\prime}_{d}/p\rfloor-\delta+\gamma-1\right)+\left(|{\boldsymbol{r}}^{\prime}|-\lfloor
r^{\prime}_{d}/p\rfloor+\lfloor
r_{c}/p\rfloor-\gamma+\delta-1\right)=|{\boldsymbol{r}}|+|{\boldsymbol{r}}^{\prime}|-2.$
(A.31)

To prove the identity LABEL:JYJY, we need the lemma,

$J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})=\sum_{i=1}^{p}y_{i,b}\partial_{i,a}{\mathcal{Y}}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=\sum_{c}(-)^{a+c}y_{i_{c},b}(x_{a}^{n_{c}}-x_{b}^{n_{c}}){\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y}).$
(A.32)

This lemma is obtained by decomposing the determinant by the minors
${\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y})$ with
respect to the row $a$ and column $c$,

$J_{b,a}{\mathcal{Y}}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=\sum_{i}\sum_{c}(-)^{a+c}y_{i,b}x_{a}^{n_{c}}\partial_{i,a}(y_{i_{c},a}){\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y})=\sum_{c}(-)^{a+c}y_{i_{c},b}x_{a}^{n_{c}}{\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y}).$
(A.33)

After anti-symmetrization, we have

$J_{b,a}{\mathcal{Y}}_{\boldsymbol{r}}(\boldsymbol{x},\boldsymbol{y})=\sum_{c}(-)^{a+c}y_{i_{c},b}(x_{a}^{n_{c}}-x_{b}^{n_{c}}){\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y})+\sum_{c}(-)^{a+c}y_{i_{c},b}x_{b}^{n_{c}}{\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y}).$
(A.34)

We then notice that the last term can be re-summed and gives the minor
expansion of the determinant
${\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})$ in which the
column $a$ has been replaced by a copy of the column $b$, and so it is
vanishing. The remaining expression gives the r.h.s. of the lemma A.32.

Next, we examine the following quantity, and apply twice the previous lemma to
write down

$\displaystyle\begin{split}&\sum_{a\neq
b}\dfrac{1}{(x_{a}-x_{b})^{2}}(J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y}))(J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y}))\\\
=&-\sum_{a\neq
b}\sum_{c,d}(-)^{a+b+c+d}y_{i_{c},b}y_{i_{d}^{\prime},a}\dfrac{x_{a}^{n_{c}}-x_{b}^{n_{c}}}{x_{a}-x_{b}}\dfrac{x_{a}^{n_{d}^{\prime}}-x_{b}^{n_{d}^{\prime}}}{x_{a}-x_{b}}{\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime},[b|d]}(\boldsymbol{x},\boldsymbol{y})\\\
=&-\sum_{a\neq
b}\sum_{c,d}(-)^{a+b+c+d}y_{i_{c},b}y_{i_{d}^{\prime},a}\sum_{\gamma=0}^{n_{c}-1}\sum_{\delta=0}^{n_{d}^{\prime}-1}x_{a}^{n_{d}^{\prime}+\gamma-\delta-1}x_{b}^{n_{c}-\gamma+\delta-1}{\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime},[b|d]}(\boldsymbol{x},\boldsymbol{y})\\\
=&-\sum_{a,b}\sum_{c,d}(-)^{a+b+c+d}y_{i_{c},b}y_{i_{d}^{\prime},a}\sum_{\gamma=0}^{n_{c}-1}\sum_{\delta=0}^{n_{d}^{\prime}-1}x_{a}^{n_{d}^{\prime}+\gamma-\delta-1}x_{b}^{n_{c}-\gamma+\delta-1}{\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime},[b|d]}(\boldsymbol{x},\boldsymbol{y})\\\
&+\sum_{a}\sum_{c,d}(-)^{c+d}y_{i_{c},a}y_{i_{d}^{\prime},a}n_{c}n_{d}^{\prime}x_{a}^{n_{c}+n_{d}^{\prime}-2}{\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime},[a|d]}(\boldsymbol{x},\boldsymbol{y}).\end{split}$
(A.35)

In the last two lines, we completed the sum with the term $a=b$. We recognize
in the last term the expansion in minors of the expression

$\displaystyle\begin{split}\sum_{a}\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})=\sum_{a}\left(\sum_{c}(-)^{a+c}y_{i_{c},a}n_{c}x_{a}^{n_{c}-1}{\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y})\right)\left(\sum_{d}(-)^{a+d}y_{i_{d}^{\prime},a}n_{d}^{\prime}x_{a}^{n_{d}^{\prime}-1}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime},[a|d]}(\boldsymbol{x},\boldsymbol{y})\right)\end{split}$
(A.36)

Thus, by re-arranging the terms, we arrive at

$\displaystyle\begin{split}&\sum_{a}\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})-\sum_{a\neq
b}\dfrac{1}{(x_{a}-x_{b})^{2}}(J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y}))(J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y}))\\\
=&\sum_{a,b}\sum_{c,d}\sum_{\gamma=0}^{n_{c}-1}\sum_{\delta=0}^{n_{d}^{\prime}-1}(-)^{a+b+c+d}y_{i_{c},b}y_{i_{d}^{\prime},a}x_{a}^{n_{d}^{\prime}+\gamma-\delta-1}x_{b}^{n_{c}-\gamma+\delta-1}{\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime},[b|d]}(\boldsymbol{x},\boldsymbol{y}).\end{split}$
(A.37)

We can now perform the summations of indices $a$ and $b$; they correspond to
the minor expansions of the following determinants

$\displaystyle\begin{split}&\sum_{a}(-)^{a+c}y_{i_{d}^{\prime},a}x_{a}^{n_{d}^{\prime}+\gamma-\delta-1}{\mathcal{Y}}_{{\boldsymbol{r}},[a|c]}(\boldsymbol{x},\boldsymbol{y})=[x^{n_{1}}y_{i_{1}}\wedge\cdots\wedge\underset{\hat{c}}{x^{n^{\prime}_{d}+\gamma-\delta-1}y_{i^{\prime}_{d}}}\wedge\cdots\wedge
x^{n_{N}}y_{i_{N}}],\\\
&\sum_{b}(-)^{b+d}y_{i_{c},b}x_{b}^{n_{c}-\gamma+\delta-1}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime},[b|d]}(\boldsymbol{x},\boldsymbol{y})=[x^{n^{\prime}_{1}}y_{i^{\prime}_{1}}\wedge\cdots\wedge\underset{\hat{d}}{x^{n_{c}-\gamma+\delta-1}y_{i_{c}}}\wedge\cdots\wedge
x^{n^{\prime}_{N}}y_{i^{\prime}_{N}}].\end{split}$ (A.38)

The first line corresponds to
${\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})$ with $r_{c}$
replaced by $r^{\prime}_{d}+\gamma-\delta-1$, and the second one to
${\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})$
with $r^{\prime}_{d}$ replaced by $r_{c}-\gamma+\delta-1$. These operations
have been encoded using the permutation
$R_{a,b}^{[{\boldsymbol{r}},{\boldsymbol{r}}^{\prime}]}$ and the shift
operator $T_{a}$ in LABEL:JYJY. It concludes the proof of the Hamiltonian
action 3.20.

###  A.2 Proof of the Knizhnik–Zamolodchikov equation

In this subsection, we use the following expansion property for rational
fractions of the form

$\dfrac{x_{a}^{\alpha}x_{b}^{\beta}-x_{a}^{\beta}x_{b}^{\alpha}}{x_{a}-x_{b}}=\begin{cases}-\sum_{\gamma=\alpha}^{\beta-1}x_{a}^{\alpha+\beta-1-\gamma}x_{b}^{\gamma}&\alpha<\beta,\\\
\sum_{\gamma=\beta}^{\alpha-1}x_{a}^{\alpha+\beta-1-\gamma}x_{b}^{\gamma}&\alpha>\beta,\end{cases},\quad\alpha,\beta\in\mathbb{Z}^{\geq
0}.$ (A.39)

#### Decomposing the KZ equation

Using the chain rule, we can write

$\displaystyle\begin{split}\partial_{a}\Phi_{\mathfrak{r}_{0}}(\boldsymbol{x},\boldsymbol{y})&=\sum_{\alpha}\prod_{\beta\neq\alpha}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\beta)}}(\boldsymbol{x},\boldsymbol{y})\
\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y}),\\\
\sum_{b\neq
a}\frac{J_{a,b}J_{b,a}}{x_{a}-x_{b}}\Phi_{\mathfrak{r}_{0}}(\boldsymbol{x},\boldsymbol{y})&=\sum_{\alpha}\prod_{\beta\neq\alpha}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\beta)}}(\boldsymbol{x},\boldsymbol{y})\sum_{b\neq
a}\frac{J_{a,b}J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}\\\
+&\sum_{\alpha\neq\beta}\prod_{\gamma\neq\alpha,\beta}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\gamma)}}(\boldsymbol{x},\boldsymbol{y})\sum_{b\neq
a}\frac{J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y})J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\beta)}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}.\end{split}$
(A.40)

In this subsection, we prove the two following identities

$\displaystyle\sum_{b\neq
a}\frac{J_{a,b}J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}=(p+1)\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y}),$
(A.41) $\displaystyle\sum_{b\neq
a}\dfrac{J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}={\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}}(\boldsymbol{x},\boldsymbol{y}),$
(A.42)

from which we deduce

$\displaystyle\begin{split}&\sum_{\alpha}\prod_{\beta\neq\alpha}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\beta)}}(\boldsymbol{x},\boldsymbol{y})\sum_{b\neq
a}\frac{J_{a,b}J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}=(p+1)\sum_{\alpha}\prod_{\beta\neq\alpha}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\beta)}}(\boldsymbol{x},\boldsymbol{y})\
\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y}),\\\
&\sum_{\alpha\neq\beta}\prod_{\gamma\neq\alpha,\beta}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\gamma)}}(\boldsymbol{x},\boldsymbol{y})\sum_{b\neq
a}\frac{J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y})J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\beta)}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}=(k-1)\sum_{\alpha}\prod_{\beta\neq\alpha}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\beta)}}(\boldsymbol{x},\boldsymbol{y})\
\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{(\alpha)}}(\boldsymbol{x},\boldsymbol{y}).\end{split}$
(A.43)

It shows the KZ relation 3.29 using the decomposition A.40 for left/right hand
sides.

#### Proof of (A.41):

To prove this identity, we first notice that the action of $J_{a,b}J_{b,a}$ on
a single determinant
${\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})$ can be
rewritten using the permutation operator $\tau_{a,b}$ exchanging the
coordinates $x_{a}$ and $x_{b}$,

$J_{a,b}J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})=(1-\tau_{a,b}){\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y}).$
(A.44)

To compute its action explicitly, we perform a double minor expansion assuming
$a<b$ for definiteness

${\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})=\sum_{c<d}(-)^{a+b+c+d}\
y_{i_{c},a}y_{i_{d},b}x_{a}^{n_{c}}x_{b}^{n_{d}}{\mathcal{Y}}_{{\boldsymbol{r}}[a,b|c,d]}(\boldsymbol{x},\boldsymbol{y})-\sum_{c>d}(-)^{a+b+c+d}\
y_{i_{c},a}y_{i_{d},b}x_{a}^{n_{c}}x_{b}^{n_{d}}{\mathcal{Y}}_{{\boldsymbol{r}}[a,b|c,d]}(\boldsymbol{x},\boldsymbol{y}),$
(A.45)

where
${\mathcal{Y}}_{{\boldsymbol{r}}[a,b|c,d]}(\boldsymbol{x},\boldsymbol{y})$
denotes the double minor with rows $a$, $b$ and columns $c$, $d$ removed. From
the previous remark, we have

$\sum_{b\neq
a}\frac{J_{a,b}J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}=A_{<}-A_{>},$
(A.46)

with

$\displaystyle\begin{split}&A_{<}=\sum_{b\neq a}\sum_{c<d}(-)^{a+b+c+d}\
y_{i_{c},a}y_{i_{d},b}\dfrac{x_{a}^{n_{c}}x_{b}^{n_{d}}-x_{a}^{n_{d}}x_{b}^{n_{c}}}{x_{a}-x_{b}}{\mathcal{Y}}_{{\boldsymbol{r}}[a,b|c,d]}(\boldsymbol{x},\boldsymbol{y}),\\\
&A_{>}=\sum_{b\neq a}\sum_{c>d}(-)^{a+b+c+d}\
y_{i_{c},a}y_{i_{d},b}\dfrac{x_{a}^{n_{c}}x_{b}^{n_{d}}-x_{a}^{n_{d}}x_{b}^{n_{c}}}{x_{a}-x_{b}}{\mathcal{Y}}_{{\boldsymbol{r}}[a,b|c,d]}(\boldsymbol{x},\boldsymbol{y}).\end{split}$
(A.47)

We note that $c<d$ implies $n_{c}\leq n_{d}$ if
${\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})$ is of the
vacuum form 3.24 (i.e. ${\boldsymbol{r}}\in\mathfrak{R}_{0}$), and conversely
$c>d$ implies $n_{c}\geq n_{d}$. Using the expansion formulas A.39, we
find151515Note that the terms $n_{c}=n_{d}$ are indeed vanishing on both
sides.

$\displaystyle\begin{split}&A_{<}=-\sum_{b\neq
a}\sum_{c<d}\sum_{\gamma=n_{c}}^{n_{d}-1}(-)^{a+b+c+d}\
y_{i_{c},a}y_{i_{d},b}x_{a}^{n_{c}+n_{d}-1-\gamma}x_{b}^{\gamma}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a,b|c,d]}(\boldsymbol{x},\boldsymbol{y}),\\\
&A_{>}=\sum_{b\neq a}\sum_{c>d}\sum_{\gamma=n_{d}}^{n_{c}-1}(-)^{a+b+c+d}\
y_{i_{c},a}y_{i_{d},b}x_{a}^{n_{c}+n_{d}-1-\gamma}x_{b}^{\gamma}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a,b|c,d]}(\boldsymbol{x},\boldsymbol{y}).\end{split}$
(A.48)

The next step is to perform the summation over $b$ which is interpreted as a
minor expansion of the determinant
${\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y})$
where some replacement has occured. In the case of $A_{<}$,161616We need to
sum here with the sign $b-1$ since the row $a<b$ has been removed from the
determinant
${\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|d]}(\boldsymbol{x},\boldsymbol{y})$.

$\sum_{b\neq
a}(-)^{b-1+d}y_{i_{d},b}x_{b}^{\gamma}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a,b|c,d]}(\boldsymbol{x},\boldsymbol{y}),$
(A.49)

the elements $y_{i_{d},b}x_{b}^{n_{d}}$ on column $d$ have been replaced by
$y_{i_{d},b}x_{b}^{\gamma}$ with $\gamma<n_{d}$. Since
${\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})$ is of
the form 3.24, this column is already present in
${\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y})$,
unless it is the one that has been removed previously, i.e. unless
$\gamma=n_{c}$ and $i_{d}=i_{c}$, in which case we find
${\mathcal{Y}}_{{\boldsymbol{r}}[a|d]}(\boldsymbol{x},\boldsymbol{y})$. Thus,

$A_{<}=\sum_{\genfrac{}{}{0.0pt}{}{c<d}{n_{c}<n_{d}}}(-)^{a+d}\
\delta_{i_{c},i_{d}}y_{i_{d},a}x_{a}^{n_{d}-1}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|d]}(\boldsymbol{x},\boldsymbol{y}).$
(A.50)

The summation over $c$ can be performed by a counting argument, it produces an
extra factor $n_{d}$,171717Decomposing $N=mp+q$, $d=\bar{d}p+d^{\prime}$,
$c=\bar{c}p+c^{\prime}$ with $c^{\prime},d^{\prime}\in[\\![0,p-1]\\!]$. In the
vacuum case ${\boldsymbol{r}}\in\mathfrak{R}_{0}$, we have $n_{c}=\bar{c}$,
$n_{d}=\bar{d}$, and $i_{c}=c^{\prime}+1$, $i_{d}=d^{\prime}+1$ for
$n_{c},n_{d}<m$ (but arbitrary for $n_{c}=m$ or $n_{d}=m$). The restrictions
on the summation impose $\bar{c}=0\cdots\bar{d}-1$ and
$c^{\prime}=d^{\prime}$.

$A_{<}=\sum_{d}(-)^{a+d}\
y_{i_{d},a}n_{d}x_{a}^{n_{d}-1}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|d]}(\boldsymbol{x},\boldsymbol{y})=\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y}).$
(A.51)

Next, we examine $A_{>}$, and interpret again the summation over $b$ as a
minor expansion of the determinant
${\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y})$
where the elements $y_{i_{d},b}x_{b}^{n_{d}}$ on column $d$ have been replaced
by $y_{i_{d},b}x_{b}^{\gamma}$. The only difference is that now
$\gamma\in[\\![n_{d},n_{c}-1]\\!]$ with $n_{c}\geq n_{d}$. In general, the
determinant vanishes since the same column is present twice, unless
$\gamma=n_{d}$ in which case we simply recover the original determinant,

$\sum_{b\neq
a}(-)^{b-1+d}y_{i_{d},b}x_{b}^{n_{d}}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a,b|c,d]}(\boldsymbol{x},\boldsymbol{y})={\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y}).$
(A.52)

Thus, we have

$A_{>}=-\sum_{\genfrac{}{}{0.0pt}{}{c>d}{n_{c}>n_{d}}}(-)^{a+c}\
y_{i_{c},a}x_{a}^{n_{c}-1}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y}).$
(A.53)

The sum over $d$ is performed by a similar counting argument which gives a
factor $pn_{c}$ since the condition $\delta_{i_{c},i_{d}}$ is not present in
this case.181818Using the same decomposition as in the previous argument, we
have $\bar{d}=0\cdots\bar{c}-1$ and $d^{\prime}=0\cdots p-1$ since
$\bar{d}<m$. We find

$A_{>}=-\sum_{c}(-)^{a+c}\
y_{i_{c},a}pn_{c}x_{a}^{n_{c}-1}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y})=-p\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y}).$
(A.54)

Combining the results A.51 and A.54 for $A_{>}$ and $A_{<}$ respectively, we
deduce that $A_{<}-A_{>}$ produces indeed
$(p+1)\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})$,
which shows the identity A.41.

#### Proof of the equation A.42

In this proof, we need the following lemma which can be derived using a
similar argument as the previous lemma A.32,

$J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})=\sum_{c,d}(-1)^{a+b+c+d}y_{i_{c},b}y_{i^{\prime}_{d},a}(x_{a}^{n_{c}}x_{b}^{n^{\prime}_{d}}-x_{a}^{n^{\prime}_{d}}x_{b}^{n_{c}}){\mathcal{Y}}_{{\boldsymbol{r}}[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y}),$
(A.55)

for $a\neq b$. The proof of this lemma is obtained using a minor expansion of
both determinants to compute explicitly the action of $J_{a,b}$,

$J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})=\sum_{c,d}(-1)^{a+b+c+d}y_{i_{c},b}y_{i^{\prime}_{d},a}x_{a}^{n_{c}}x_{b}^{n^{\prime}_{d}}{\mathcal{Y}}_{{\boldsymbol{r}}[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y}).$
(A.56)

This expression is then anti-symmetrized to produce A.55, noticing that

$\sum_{c,d}(-1)^{a+b+c+d}y_{i_{c},b}y_{i^{\prime}_{d},a}x_{a}^{n^{\prime}_{d}}x_{b}^{n_{c}}{\mathcal{Y}}_{{\boldsymbol{r}}[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y})=0$
(A.57)

since it corresponds to the minor expansion of determinants with two equal
rows (two copies of the row $b$ in the first one, and two copies of the row
$a$ in the second one).

Thus, using the lemma A.55, we can write

$\dfrac{J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}}(\boldsymbol{x},\boldsymbol{y})J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}=\sum_{c,d}(-1)^{a+b+c+d}y_{i_{c},b}y_{i^{\prime}_{d},a}\dfrac{x_{a}^{n_{c}}x_{b}^{n^{\prime}_{d}}-x_{a}^{n^{\prime}_{d}}x_{b}^{n_{c}}}{x_{a}-x_{b}}{\mathcal{Y}}_{{\boldsymbol{r}}[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y}).$
(A.58)

Next, we need to perform the summation over $b\neq a$. Assuming
$n_{c}<n^{\prime}_{d}$, we can expand the rational function as in A.39 and
obtain

$\displaystyle\begin{split}\sum_{b\neq
a}(-)^{b+d}y_{i_{c},b}\dfrac{x_{a}^{n_{c}}x_{b}^{n^{\prime}_{d}}-x_{a}^{n^{\prime}_{d}}x_{b}^{n_{c}}}{x_{a}-x_{b}}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y})=&-\sum_{\gamma=n_{c}}^{n^{\prime}_{d}-1}x_{a}^{n_{c}+n^{\prime}_{d}-1-\gamma}\sum_{b}(-)^{b+d}y_{i_{c},b}x_{b}^{\gamma}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y})\\\
&+(n^{\prime}_{d}-n_{c})(-)^{a+d}y_{i_{c},a}x_{a}^{n_{c}+n^{\prime}_{d}-1}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[a|d]}(\boldsymbol{x},\boldsymbol{y}),\end{split}$
(A.59)

where the second line corresponds to the extra term $b=a$ added to the
summation. The first line contains minor expansions of the determinant
${\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})$ in
which the elements $y_{i^{\prime}_{d},b}x_{b}^{n^{\prime}_{d}}$ of the row $d$
have been replaced by $y_{i_{c},b}x_{b}^{\gamma}$ with
$\gamma<n^{\prime}_{d}$. If
${\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})$ is
a vacuum wedge state of the form 3.24, i.e.
${\boldsymbol{r}}^{\prime}\in\mathfrak{R}_{0}$, this determinant is vanishing
and so

$\sum_{b\neq
a}(-)^{b+d}y_{i_{c},b}\dfrac{x_{a}^{n_{c}}x_{b}^{n^{\prime}_{d}}-x_{a}^{n^{\prime}_{d}}x_{b}^{n_{c}}}{x_{a}-x_{b}}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}_{0}[b|d]}(\boldsymbol{x},\boldsymbol{y})=(n^{\prime}_{d}-n_{c})(-)^{a+d}y_{i_{c},a}x_{a}^{n_{c}+n^{\prime}_{d}-1}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}[a|d]}(\boldsymbol{x},\boldsymbol{y}),$
(A.60)

This equation is also trivially satisfied if $n_{c}=n^{\prime}_{d}$.

When $n_{c}>n^{\prime}_{d}$, we need to use another expansion for the ratio
A.39, and we find

$\displaystyle\begin{split}\sum_{b\neq
a}(-)^{b+d}y_{i_{c},b}\dfrac{x_{a}^{n_{c}}x_{b}^{n^{\prime}_{d}}-x_{a}^{n^{\prime}_{d}}x_{b}^{n_{c}}}{x_{a}-x_{b}}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y})=&\sum_{\gamma=n^{\prime}_{d}}^{n_{c}-1}x_{a}^{n_{c}+n^{\prime}_{d}-1-\gamma}\sum_{b}(-)^{b+d}y_{i_{c},b}x_{b}^{\gamma}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y})\\\
&+(n^{\prime}_{d}-n_{c})(-)^{a+d}y_{i_{c},a}x_{a}^{n_{c}+n^{\prime}_{d}-1}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}[a|d]}(\boldsymbol{x},\boldsymbol{y}),\end{split}$
(A.61)

The first line still contains minor expansions of the determinant
${\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}}(\boldsymbol{x},\boldsymbol{y})$ in
which the elements $y_{i^{\prime}_{d},b}x_{b}^{n^{\prime}_{d}}$ of the row $d$
have been replaced by $y_{i_{c},b}x_{b}^{\gamma}$, but now
$n^{\prime}_{d}\leq\gamma\leq n_{c}-1$. If both
${\boldsymbol{r}},{\boldsymbol{r}}^{\prime}\in\mathfrak{R}_{0}$, then $0\leq
n_{c},n^{\prime}_{d}\leq m$ and so $\gamma<m$. Thus, this row will be already
present in
${\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y})$
unless it is the one that had been removed from
${\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}}(\boldsymbol{x},\boldsymbol{y})$,
i.e. unless $\gamma=n^{\prime}_{d}$ and $i_{c}=i^{\prime}_{d}$. In this case,
we recover the original determinant,

$\sum_{b\neq
a}(-)^{b+d}y_{i_{c},b}\dfrac{x_{a}^{n_{c}}x_{b}^{n^{\prime}_{d}}-x_{a}^{n^{\prime}_{d}}x_{b}^{n_{c}}}{x_{a}-x_{b}}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}[b|d]}(\boldsymbol{x},\boldsymbol{y})=\delta_{i_{c},i^{\prime}_{d}}x_{a}^{n_{c}-1}{\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}_{0}}(\boldsymbol{x},\boldsymbol{y})+(n^{\prime}_{d}-n_{c})(-)^{a+d}y_{i_{c},a}x_{a}^{n_{c}+n^{\prime}_{d}-1}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}[a|d]}(\boldsymbol{x},\boldsymbol{y}),$
(A.62)

Combining the results A.60 and A.62, we find

$\displaystyle\begin{split}\sum_{b\neq
a}\dfrac{J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}=&\sum_{\genfrac{}{}{0.0pt}{}{c,d}{n_{c}>n^{\prime}_{d}}}(-)^{a+c}\delta_{i_{c},i^{\prime}_{d}}x_{a}^{n_{c}-1}y_{i^{\prime}_{d},a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}_{0}}(\boldsymbol{x},\boldsymbol{y})\\\
&+\sum_{c,d}(-1)^{c+d}(n^{\prime}_{d}-n_{c})y_{i_{c},a}y_{i^{\prime}_{d},a}x_{a}^{n_{c}+n^{\prime}_{d}-1}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}[a|d]}(\boldsymbol{x},\boldsymbol{y}).\end{split}$
(A.63)

The summation over $d$ in the first line can be simplified by a counting
argument. Examining the values of $i^{\prime}_{d}$ and $n^{\prime}_{d}$ for
$a=1\cdots N$, we find a factor $n_{c}$,191919Note that the last values
$n^{\prime}_{d}=m$ and $i^{\prime}_{d}=s_{\alpha}$ are never reached thanks to
the condition $n^{\prime}_{d}<n_{c}\leq m$.

$d$ | $1$ | $\cdots$ | $p$ | $p+1$ | $\cdots$ | $2p$ | $2p+1$ | $\cdots$ | $3p$ | $\cdots$ | $\cdots$ | $(m-1)p+1$ | $\cdots$ | $mp$ | $mp+1$ | $\cdots$ | $N$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$i^{\prime}_{d}$ | $1$ | $\cdots$ | $p$ | $1$ | $\cdots$ | $p$ | $1$ | $\cdots$ | $p$ | $\cdots$ | $\cdots$ | $1$ | $\cdots$ | $p$ | $s_{1}$ | $\cdots$ | $s_{q}$
$n^{\prime}_{d}$ | $0$ | $\cdots$ | $0$ | $1$ | $\cdots$ | $1$ | $2$ | $\cdots$ | $2$ | $\cdots$ | $\cdots$ | $m-1$ | $\cdots$ | $m-1$ | $m$ | $\cdots$ | $m$

Thus, we find after cancellation and reforming the determinant

$\displaystyle\begin{split}\sum_{b\neq
a}\dfrac{J_{b,a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})J_{a,b}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}}(\boldsymbol{x},\boldsymbol{y})}{x_{a}-x_{b}}=&\sum_{c}(-)^{a+c}n_{c}x_{a}^{n_{c}-1}y_{i_{c},a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}^{\prime}_{0}}(\boldsymbol{x},\boldsymbol{y})\\\
&+\sum_{c,d}(-1)^{c+d}(n^{\prime}_{d}-n_{c})y_{i_{c},a}y_{i^{\prime}_{d},a}x_{a}^{n_{c}+n^{\prime}_{d}-1}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}[a|d]}(\boldsymbol{x},\boldsymbol{y})\\\
=&\sum_{c,d}(-1)^{c+d}n^{\prime}_{d}y_{i_{c},a}y_{i^{\prime}_{d},a}x_{a}^{n_{c}+n^{\prime}_{d}-1}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}[a|c]}(\boldsymbol{x},\boldsymbol{y}){\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}[a|d]}(\boldsymbol{x},\boldsymbol{y})\\\
=&{\mathcal{Y}}_{{\boldsymbol{r}}_{0}}(\boldsymbol{x},\boldsymbol{y})\partial_{a}{\mathcal{Y}}_{{\boldsymbol{r}}_{0}^{\prime}}(\boldsymbol{x},\boldsymbol{y}).\end{split}$
(A.64)

This is indeed the identity A.42, which concludes the proof of the KZ
equation.

##  B Explicit form of solutions for $N=2$

In this appendix, we examine the eigenvalue problem for the Hamiltonian 3.8 in
the case of two particles ($N=2$) for generic $p$ and $k$.

#### Spin wave function

The spin states of each particle $a=1,2$ take the form $y_{i_{1},a}\cdots
y_{i_{k},a}$. They belong to the order $k$ totally symmetric representation of
$U(p)$ associated to the Young diagram $[k]$. The tensor product of two
representations $[k]$ has the decomposition,

$[k]\otimes[k]=[2k,0]\oplus[2k-2,2]\oplus\cdots\oplus[k,k]=\oplus_{l=0}^{k}[k+l,k-l]\,.$
(B.1)

The spin component of the wave function expressed in terms of $y$-variables is
obtained from a function $Y^{HW}_{\boldsymbol{m}}(\boldsymbol{y})$ satisfying
the highest weight condition

$K_{i,j}Y^{HW}_{\boldsymbol{m}}(\boldsymbol{y})=0,\quad i<j,\quad
K_{i,i}Y^{HW}_{\boldsymbol{m}}(\boldsymbol{y})=m_{i}Y^{HW}_{\boldsymbol{m}}(\boldsymbol{y}),\quad
i=1,\cdots,p\,,$ (B.2)

for the $U(p)$ generators $K_{i,j}$ defined in 2.3. In particular, the highest
weight state for the irreducible representation $[k+l,k-l]$ has the wave
function

$Y^{HW}_{[k+l,k-l]}(\boldsymbol{y})=(y_{11}y_{22}-y_{12}y_{21})^{k-l}(y_{11})^{l}(y_{12})^{l}.$
(B.3)

This state has parity $(-1)^{k-l}$ under the exchange of the two particles
$1\leftrightarrow 2$. It satisfies the conditions (B.2) with
$m_{1}=k+l,m_{2}=k-l,m_{3}=\cdots=m_{p}=0$.

The operator $J_{12}J_{21}$ can be written using the second Casimir operator
of $U(p)$. It is constant on the states that belong to a given representation
$[k+l,k-l]$. Its eigenvalue can be obtained from the action on the highest
weight state of this representation,

$(J_{12}J_{21})Y^{HW}_{[k+l,k-l]}(\boldsymbol{y})=l(l+1)Y^{HW}_{[k+l,k-l]}(\boldsymbol{y})\,.$
(B.4)

We will show below that the representation with $l=0$ (i.e., $[k,k]$) has
minimal energy and corresponds to the true ground state. Instead, states in
the representation $[k+l,k-l]$ with $l=1,\cdots,k$ enter in the wave function
of excited states.

We note that there are different choices for the spin components of the wave
functions at fixed $l$. In some references on the spin Calogero model (see,
e.g., [48]), a totally symmetric wave function $l=k$ is used. The form of the
KZ equation for the vacuum wave functions depends explicitly on this choice of
spin component (see, for example, [19, 27]).

Finally, we comment on the dimension of the representation. The dimension of
the symmetric representation of order $k$ is,

$d_{p,k}=\frac{\Gamma(k+p)}{\Gamma(k+1)\Gamma(p)}\,.$ (B.5)

On the other hand, the dimension of the irreducible representation $[k+l,k-l]$
is given by the factors over hooks rule (see for example, p120 in [49]),

$\mathfrak{m}(p,k,l)=\frac{(k+p+l-1)!(k+p-l-2)!(2l+1)}{(k+l+1)!(k-l)!(p-1)!(p-2)!}.$
(B.6)

We note that $d_{p,k}^{2}=\sum_{l=0}^{k}\mathfrak{m}(p,k,l)$ in agreement with
the branching rule (B.1).

#### Inclusion of the coordinate dependence

The total wave functions can be factorized into coordinate and spin
dependence,
$\psi(\boldsymbol{x},\boldsymbol{y})=X_{l}(\boldsymbol{x})Y_{l}(\boldsymbol{y})$
where $Y_{l}(\boldsymbol{y})$ is assumed to belong to the representation
$[k+l,k-l]$. In this case, the Hamiltonian 3.8 acting on the coordinates
component $X_{l}(\boldsymbol{x})$ reduces to

$\displaystyle\tilde{{\mathcal{H}}}_{|\boldsymbol{x}}$
$\displaystyle=\sum_{a=1,2}(-\partial_{a}^{2}+x_{a}\partial_{a})+2+\frac{2l(l+1)}{(x_{1}-x_{2})^{2}}\,.$
(B.7)

We recover here the Calogero Hamiltonian, the ground state is found using the
ansatz $X_{l}(\boldsymbol{x})=(x_{1}-x_{2})^{n}$. The eigenvalue problem has a
solution only if $n=l+1$ or $n=-l$. Only solutions with $n$ positive, i.e.,
$n=l+1$, are nonsingular. The corresponding eigenvalue is $E_{l}=2l+4$, and
the representation $[k+l,k-l]$ with $l=0$ has indeed the minimal energy.

Combining spin and coordinate components, we deduce the ground state wave
functions for $l$ fixed,

$\displaystyle\psi(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle=(x_{1}-x_{2})^{l+1}(y_{11}y_{22}-y_{12}y_{21})^{k-l}y_{11}^{l}y_{12}^{l}$
(B.8) $\displaystyle=(x_{1}-x_{2})\left|\begin{array}[]{cc}y_{11}&y_{12}\\\
y_{21}&y_{22}\end{array}\right|^{k-l}\left|\begin{array}[]{cc}y_{11}&y_{12}\\\
y_{11}x_{1}&y_{12}x_{2}\end{array}\right|^{l}$ (B.13)

This expression agrees with the formula 3.17 for $N=2$. We note that for $k$
even (resp. odd), the system is fermionic (resp. bosonic) under the exchange
of the two particles, i.e. the simultaneous exchange of spin and coordinate
variables $(x_{1},y_{i,1})\leftrightarrow(x_{2},y_{i,2})$.

For the singular branch of solutions obtained by choosing $n=-l$, the
corresponding energy is $E_{l}=-2l+2$. The lowest energy among the
representations $[k+l,k-l]$ is found for $l=k$ which gives $E_{k}=-2k+2$, and
the wave function has a pole of order $(x_{1}-x_{2})^{-k}$ which grows with
$k$.

#### Excited states

The wave functions of excited states have a coordinate dependence of the form
$X_{l}(\boldsymbol{x})=x_{12}^{l+1}P(\boldsymbol{x})$ where
$P(\boldsymbol{x})$ is a symmetric polynomial in the variables
$(x_{1},x_{2})$. After conjugation, the action of the Hamiltonian 3.8 on
$P(\boldsymbol{x})$ takes the form

$\tilde{{\mathcal{H}}}_{|P}=x_{12}^{-l-1}\tilde{{\mathcal{H}}}_{|\boldsymbol{x}}x_{12}^{l+1}=2l+4+\sum_{a}(-\partial_{a}^{2}+x_{a}\partial_{a})-\frac{2(l+1)}{x_{12}}(\partial_{1}-\partial_{2}).$
(B.14)

The polynomial $P(\boldsymbol{x})$ can be decomposed on the basis of symmetric
monomials $m_{r,s}(\boldsymbol{x})=x_{1}^{r}x_{2}^{s}+x_{1}^{s}x_{2}^{r}$ with
$r\geq s\geq 0$. The action of the Hamiltonian $\tilde{{\mathcal{H}}}_{|P}$ on
the monomials reads

$\tilde{{\mathcal{H}}}_{|P}m_{r,s}(\boldsymbol{x})=2(r+s+l+2)m_{r,s}(\boldsymbol{x})+\cdots$
(B.15)

where $\cdots$ is a linear combination of
$m_{r^{\prime},s^{\prime}}(\boldsymbol{x})$ with
$r^{\prime}+s^{\prime}=r+s-2$. We recover here the triangular action of the
Hamiltonian written in 3.20 for general $N$. For each value
$E_{r,s}=2(l+r+s+2)$ of the energy, it is possible to construct the
corresponding eigenfunction as a linear combination

$\psi_{l,r,s}(\boldsymbol{x},\boldsymbol{y})=Y_{l}(\boldsymbol{y})\left(m_{r,s}(\boldsymbol{x})+\sum_{r^{\prime}+s^{\prime}\leq
r+s-2}c_{r^{\prime}s^{\prime}}m_{r^{\prime},s^{\prime}}(\boldsymbol{x})\right).$
(B.16)

From the formula (B.6) for the dimension of irreducible representations, we
deduce the partition function for a system of two particles,

$\mbox{Tr}(q^{\tilde{H}})=q^{4}\sum_{l=0}^{k}\mathfrak{m}(p,k,l)\frac{q^{2l}}{(1-q^{2})(1-q^{4})}\,.$
(B.17)

##  C Classification of eigenstates using Gelfand-Zetlin patterns

In this appendix, we focus on the case $k=1$ and describe the decomposition of
the wedge state according to irreducible representations of the Lie algebra
$u(p)$, or $\mathfrak{gl}_{p}$ upon complexification. This technique is well-
known in the context of the spin Calogero model [30]. We apply here similar
ideas to the wedge states defined in (3.12). We note that the Gelfand-Zetlin
framework also works for $k>1$, but it requires the decomposition of a tensor
product of $k$ irreducible representations, which is difficult in practice.

Irreducible highest weight finite-dimensional representations of
$\mathfrak{gl}_{p}$ are labeled by sequences of integers [50],

$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{p},.$ (C.1)

The basis of these irreducible representations is labeled by Gel’fand-Zetlin
(GZ) patterns, i.e., a set of integers filling the following table,

$\Lambda=\left(\begin{array}[]{ccccccccc}\lambda_{1,1}&&\lambda_{2,1}&&\cdots&&\lambda_{p-1,1}&&\lambda_{p,1}\\\
&\lambda_{1,2}&&\lambda_{2,2}&\cdots&&&\lambda_{p-1,2}&\\\
&&\ddots&&\vdots&&\udots&&\\\ &&&\lambda_{1,p-1}&&\lambda_{2,p-1}&&&\\\
&&&&\lambda_{1,p}&&&&\end{array}\right)$ (C.2)

with the identification $\lambda_{i,1}=\lambda_{i}$ and the requirement that
other integers satisfy $\lambda_{i,j}\geq\lambda_{i,j+1}\geq\lambda_{i+1,j}$.
These patterns are in one-to-one correspondence with the sequence of embedding
$\mathfrak{gl}_{p}\supset\mathfrak{gl}_{p-1}\supset\cdots\supset\mathfrak{gl}_{1}$.
Indeed, the integers $\lambda_{i,j}$ ($i=1,\cdots,p-j+1$) correspond to the
highest weights of the representation of the subalgebra
$\mathfrak{gl}_{p-i+1}$ embedded in the representation of
$\mathfrak{gl}_{p-i+2}$ with highest weights $\lambda_{i,j+1}$. The highest
weight state of $\mathfrak{gl}_{p}$ corresponds to the Gelfand -Zetlin label
$\Lambda_{0}$ defined by $\lambda_{i,j}=\lambda_{i}$ for any $i,j$.

By construction, the wedge state belongs to a tensor product of $N$
fundamental representations of $\mathfrak{gl}_{p}$. This tensor product
decomposes as a sum of irreducible representations labeled by integers
$\lambda_{i}$ with the constraints

$\sum_{i=1}^{p}\lambda_{i}=N\,,\quad\lambda_{p}\geq 0\,.$ (C.3)

Thus, the labels $\lambda_{i}$ define a partition
$\lambda=[\lambda_{1},\cdots,\lambda_{l}]$ of $N$ with $l=\ell(\lambda)\leq
p$.

Let $\boldsymbol{n}=(n_{1},\cdots,n_{h})$ be a set of mutually distinct non-
negative integers. The highest weight states of the irreducible
representations of $\mathfrak{gl}_{p}$ mentioned before correspond to wedge
states written in two equivalent forms,

$\displaystyle{\mathcal{Y}}_{[\Lambda_{0},\boldsymbol{n}]}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle:=[y_{1}x^{n_{1}}\wedge y_{1}x^{n_{2}}\wedge\cdots\wedge
y_{1}x^{n_{\lambda_{1}}}\wedge y_{2}x^{n_{1}}\wedge\cdots\wedge
y_{2}x^{n_{\lambda_{2}}}\wedge\cdots\wedge y_{l}x^{n_{1}}\wedge\cdots\wedge
y_{l}x^{n_{\lambda_{l}}}]$ (C.4)
$\displaystyle=\varepsilon[y_{1}x^{n_{1}}\wedge y_{2}x^{n_{1}}\cdots\wedge
y_{\lambda_{1}^{\prime}}x^{n_{1}}\wedge y_{1}x^{n_{2}}\wedge\cdots\wedge
y_{\lambda^{\prime}_{2}}x^{n_{2}}\wedge\cdots\wedge
y_{1}x^{n_{\lambda_{1}}}\wedge
y_{\lambda^{\prime}_{\lambda_{1}}}x^{n_{\lambda_{1}}}],$ (C.5)

where $\varepsilon$ is a sign of re-ordering the factors in the wedge product,
and $\lambda^{\prime}=[\lambda^{\prime}_{1},\cdots,\lambda^{\prime}_{l_{1}}]$
is the transposed partition. The first form (C.4) is convenient for discussing
the highest weight representation given below. The second form (C.5) is useful
to derive the energy of the wedge state. These two expressions illustrate the
$U(N)/U(p)$ duality of the spin Calogero system.

These states satisfy the highest weight condition with weight $\lambda$,

$K_{i,j}{\mathcal{Y}}_{[\Lambda_{0},\boldsymbol{n}]}(\boldsymbol{x},\boldsymbol{y})=0,\quad
i<j,\quad
K_{i,i}{\mathcal{Y}}_{[\Lambda_{0},\boldsymbol{n}]}(\boldsymbol{x},\boldsymbol{y})=\lambda_{i}{\mathcal{Y}}_{[\Lambda_{0},\boldsymbol{n}]}(\boldsymbol{x},\boldsymbol{y}),$
(C.6)

where the action of the $\mathfrak{gl}_{p}$ generators $K_{i,j}$ has been
defined in 2.3. The highest weight state with the lowest energy in each
representation $\lambda$ is obtained by taking
$\boldsymbol{n}=\boldsymbol{n}_{0}=\left\\{0,1,\cdots,\lambda_{l}-1\right\\}$.
The corresponding wedge state is an eigenstate of the Hamiltonian 3.8 with
energy $E(\lambda)=N^{2}+\sum_{i=1}^{l}\lambda_{i}(\lambda_{i}-1)$. A short
proof of these statements is given below.

###### Proof.

We note that the wedge state can be written in the form

${\mathcal{Y}}_{[\Lambda_{0},\boldsymbol{n}]}(\boldsymbol{x},\boldsymbol{y})=\sum_{\sigma\in
S_{n}}(-)^{\sigma}\prod_{i=1}^{l}\prod_{\alpha=1}^{\lambda_{i}}y_{i,\sigma(a_{i,\alpha})}x_{\sigma(a_{i,\alpha})}^{n_{\alpha}},$
(C.7)

with $a_{i,\alpha}=\alpha+\sum_{j=1}^{i-1}\lambda_{i}$ labeling the $N$ boxes
of the Young diagram $\lambda$.

(i) The first condition comes from the fact that
$K_{i,j}(y_{k,a}x_{a}^{n})=\delta_{j,k}\ y_{i,a}x_{a}^{n}$, but when $i<j$ the
same factor exists in the wedge product due to the condition
$\lambda_{i}\leq\lambda_{j}$. The expression for the action of $K_{i,i}$ comes
from the fact that there are $\lambda_{i}$ particles of spin $i$, i.e. there
are $\lambda_{i}$ factors $y_{i,a}$ in each term.

(ii) The lowest energy state is obtained by minimizing $|{\boldsymbol{r}}|$ in
each representation $\lambda$, since the Hamiltonian has a global
$\mathfrak{gl}_{p}$-invariance. For
${\mathcal{Y}}_{[\Lambda_{0},\boldsymbol{n}]}(\boldsymbol{x},\boldsymbol{y})$,
we have from (C.5),

$|{\boldsymbol{r}}|=\sum_{\alpha=1}^{\lambda_{1}}n_{\alpha}\lambda^{\prime}_{\alpha}\,.$
(C.8)

It is indeed minimized by $n_{\alpha}=\alpha-1$ which gives

$|{\boldsymbol{r}}|=\sum_{\alpha=1}^{\lambda_{1}}(\alpha-1)\lambda^{\prime}_{\alpha}=\sum_{i=1}^{l}\sum_{j=1}^{\lambda_{i}}(j-1)=\dfrac{1}{2}\sum_{i=1}^{l}\lambda_{i}(\lambda_{i}-1).$
(C.9)

∎

#### Examples

  * •

For the totally symmetric representation
$\lambda_{1}=N,\lambda_{2}=\cdots=\lambda_{p}=0$, $l=1$ and
$\boldsymbol{n}_{0}=\\{0,1,\cdots,N-1\\}$. The corresponding wedge state is
proportional to the Vandermonde determinant:

${\mathcal{Y}}_{[\Lambda_{0},\boldsymbol{n}_{0}]}(\boldsymbol{x},\boldsymbol{y})=[y_{1}\wedge
y_{1}x\wedge\cdots\wedge
y_{1}x^{N-1}]=\left(\prod_{a=1}^{N}y_{1,a}\right)\Delta(\boldsymbol{x})\,.$
(C.10)

Generic states of the representation $\lambda$ of $\mathfrak{gl}_{p}$ are
obtained by the application of $K_{i,j}$ which does not modify the
$x$-dependence. The Gelfand-Zetlin patterns of this representation have the
form,

$\Lambda=\left(\begin{array}[]{ccccccccc}N&&0&&\cdots&&0&&0\\\
&\lambda_{1,2}&&0&\cdots&&&0&\\\ &&\ddots&&\vdots&&\udots&&\\\
&&&\lambda_{1,p-1}&&0&&&\\\ &&&&\lambda_{1,p}&&&&\end{array}\right),\quad
0\leq\lambda_{1,p}\leq\lambda_{1,p-1}\leq\cdots\leq\lambda_{1,2}\leq N\,.$
(C.11)

The number of patterns is $\frac{(N+p-1)!}{(p-1)!N!}$, which coincides with
the dimension of the representation $[N]$ of $\mathfrak{gl}_{p}$.

  * •

In the totally anti-symmetric case with $N\leq p$, we have
$\lambda=[\overbrace{1,\cdots,1}^{N}]$ and $\boldsymbol{n}_{0}=\\{0\\}$. The
corresponding wedge state does not depend on the variables $x_{a}$,

${\mathcal{Y}}_{[\Lambda_{0},\boldsymbol{n}_{0}]}(\boldsymbol{x},\boldsymbol{y})=[y_{1}\wedge
y_{2}\wedge\cdots\wedge y_{N}]\,.$ (C.12)

The number of patterns for the representation
$\lambda=[\overbrace{1,\cdots,1}^{N},\overbrace{0,\cdots,0}^{N-p}]$ is
$\frac{p!}{(p-N)!N!}$, which equals to the dimension of the order $N$ totally-
antisymmetric representation of $\mathfrak{gl}_{p}$.

  * •

With the exception of the two previous cases, in general the coordinate and
spin dependence mix. For instance, taking $\lambda=[\lambda_{1},\lambda_{2}]$
with $\lambda_{1}+\lambda_{2}=N$, we have
$\boldsymbol{n}=\\{0,1,\cdots,\lambda_{1}-1\\}$ and

${\mathcal{Y}}_{[\Lambda_{0},\boldsymbol{n}_{0}]}(\boldsymbol{x},\boldsymbol{y})=\sum_{A^{(1)},A^{(2)}}\epsilon^{A^{(1)}A^{(2)}}\prod_{i=1}^{\lambda_{1}}y_{1,a_{1i}}\prod_{j=1}^{\lambda_{2}}y_{2,a_{2j}}\Delta(\boldsymbol{x}^{(1)})\Delta(\boldsymbol{x}^{(2)})\,,$
(C.13)

where $A^{(1)}=\left\\{a_{1,1},\cdots,a_{1,\lambda_{1}}\right\\}$ and
$A^{(2)}=\left\\{a_{2,1},\cdots,a_{2,\lambda_{2}}\right\\}$ are disjoint
division of set of integers $\left\\{1,\cdots,N\right\\}$, namely
$A^{(1)},A^{(2)}\subset\left\\{1,\cdots,N\right\\}$, $A^{(1)}\cup
A^{(2)}=\left\\{1,\cdots,N\right\\}$, $A^{(1)}\cap A^{(2)}=\emptyset$.
$\boldsymbol{x}^{(1)}=\left\\{x_{a_{1,1}},\cdots,x_{a_{1,\lambda_{1}}}\right\\}$
and
$\boldsymbol{x}^{(2)}=\left\\{x_{a_{2,1}},\cdots,x_{a_{2,\lambda_{2}}}\right\\}$
. The patterns have the form

$\Lambda=\left(\begin{array}[]{ccccccccc}\lambda_{1}&&\lambda_{2}&&\cdots&&0&&0\\\
&\lambda_{1,2}&&\lambda_{2,2}&\cdots&&&0&\\\ &&\ddots&&\vdots&&\udots&&\\\
&&&\lambda_{1,p-1}&&\lambda_{2,p-1}&&&\\\
&&&&\lambda_{1,p}&&&&\end{array}\right)$ (C.14)

where all elements but $\lambda_{i,1}$, $\lambda_{i,2}$ ($i=2,\cdots,p$)
vanish.

##  D Character decomposition and level-rank duality

In this appendix, we check the decomposition (4.36) of the fermionic Fock
space at the level of characters for $p=k=2$. In this case, irreducible
representations are labeled by the integer $l=0,1,2$. The relevant
decompositions for the DTT matrix model are

$\displaystyle\mathcal{F}^{2,2}_{0}$