#  $\mathbb{Z}_{k}^{(r)}$-Algebras, FQH Ground States, and Invariants of
Binary Forms

Hamed Pakatchi<EMAIL_ADDRESS>
###### Abstract

A prominent class of model FQH ground states is those realized as correlation
function of $\mathbb{Z}_{k}^{(r)}$-algebras. In this paper, we study the
interplay between these algebras and their corresponding wavefunctions. In the
hopes of realizing these wavefunctions as a unique densest zero energy state,
we propose a generalization for the projection Hamiltonians. Finally, using
techniques from invariants of binary forms, an ansatz for computation of
correlations
$\left<\psi(z_{1})\cdots\psi(z_{2k})\right>\prod_{i<j}(z_{i}-z_{j})^{2r/k}$ is
devised. We provide some evidence that, at least when $r=2$, our proposed
Hamiltonian realizes $\mathbb{Z}_{k}^{(2)}$-wavefunctions as a _unique_ ground
state.

organization=Massachusetts Institute of Technology, addressline=77
Massachusetts Avenue, city=Cambridge, postcode=02139, state=MA, country=USA

##  1 Introduction

Inspired by Laughlin’s seminal work [31], analyzing trial wavefunctions has
been one of the main tools to research fractional quantum Hall (FQH) systems.
A promising method to construct and study model FQH wavefunctions, especially
non-Abelian ones [47, 32], is to realize them as conformal blocks in
particular rational conformal field theories (RCFT). Moore-Read pioneered this
approach [32] via the construction of the Pfaffian state (aka Moore-Read
state) based on the Ising CFT. The Pfaffian state and its generalizations,
namely the $\mathbb{Z}_{k}$ Read-Rezayi (RR) states [38], now lay the
foundation of our understanding of non-Abelian states.

Concretely, in the CFT point of view toward _bosonic_ quantum Hall states, one
first recognizes the ‘electron’ as a _bosonic_ chiral vertex operator
$V_{e}(z)$ in an appropriate CFT. For our purposes, the underlying CFT is
always the direct product of a parafermionic theory and a free boson theory.
In an FQH system with filling fraction $\nu$, a sensible choice would be
$V_{e}(z)=\psi(z):\exp[i\phi(z)/\sqrt{\nu}]:$ [32]. Here, $\psi$ is the
parafermion that generates the cyclic symmetry (see below), $\phi$ is a free
chiral boson, and $:\>:$ symbolizes normal ordering. The trial wavefunction
for a quantum Hall droplet on the plane or the sphere is then the following
correlation function:

$\Psi(z_{1},\cdots,z_{N})\propto\left<\mathcal{O}_{\mathrm{bg}}(\infty)V_{e}(z_{1})\cdots
V_{e}(z_{N})\right>=\left<\psi(z_{1})\psi(z_{2})\cdots\psi(z_{N})\right>\prod_{i<j}(z_{i}-z_{j})^{1/\nu}$
(1)

The operator $\mathcal{O}_{\mathrm{bg}}$ represents the neutralizing
background charge that we put at infinity. With this convention about the
neutralizing background, the wavefunction $\Psi$ becomes a translational
invariant homogeneous symmetric polynomial. Ideally, one has access to a
(pseudopotential) Hamiltonian [23, 44], describing a particular FQH phase of
matter on a disk/sphere, and the manufactured wavefunction $\Psi$ would be its
_unique_ densest zero-energy eigenstate (i.e., the ground state). However,
aside from a handful of examples, as influential as those rare cases may be,
designing models satisfying the ‘uniqueness’ condition has proven difficult in
the literature. Therefore, part of the current paper’s mission would be to
develop modified Hamiltonians with the uniqueness issue in mind.

In practice, it is sometimes more feasible to procure a “nice” polynomial with
specific local properties and retroactively engineer a (pseudopotential)
Hamiltonian fitting those local properties. As a basic example, consider a
wavefunction (i.e., symmetric polynomial) $\Psi(z_{1},\cdots,z_{N})$ that
vanishes of order $2r$ when $k+1$ particles are fused but does not vanish when
fewer particles are fused. This local property automatically makes $\Psi$ the
zero-energy state to the projection Hamiltonian $\mathscr{H}_{k+1}^{2r}$ that
projects out any state where any cluster of $k+1$ particles has relative
angular momentum $<2r$ (see Ref. [43] for more details on projection
Hamiltonians). We want to strengthen this vanishing property. To do so, let
$\Psi_{n}$ denote the ground state wavefunction of a particular FQH phase when
prepared with $N=nk$ electrons. A more stringent variant of the vanishing
property would then be:

$\Psi_{n}(w^{\times
k},z_{k+1},\cdots,z_{nk})=\prod_{i=k+1}^{nk}(w-z_{i})^{2r}\Psi_{n-1}(z_{k+1},\cdots,z_{nk})$
(2)

Following the authors in Ref. [15], we say the wavefunction is a $(k,2r)$
_clustering polynomial_ if it is a conformally covariant polynomial satisfying
this local property. Superficially, clustering states are appealing because if
one takes a $k$-cluster of electrons to infinity, one recovers the ‘same’
wavefunction only with fewer particles. On a classification level, as pointed
out in Refs. [48, 49, 4], the CFT approach and clustering polynomials are
closely tied.

To further comment on clustering properties and introduce some of the key
players in the current paper, we need to glance at the underlying
parafermionic CFTs. The relevant parafermionic CFTs have a cyclic fusion
algebra, say $\mathbb{Z}_{k}$. They consist of $k$ primary fields
$\psi_{0}\equiv 1,\psi_{1}\equiv\psi,\psi_{2},\cdots,\psi_{k-1}$ and the
simple fusion rules $\psi_{a}\star\psi_{b}=\psi_{a+b\pmod{k}}$. The
parafermion $\psi\equiv\psi_{1}$ is the generator of the cyclic symmetry under
the fusion rules. The cyclic nature of the fusion algebra is responsible for
(generally weaker versions of) the clustering property of the corresponding
FQH wavefunction. Translating the fusion rules into an operator product
algebra (OPA), we have (Notation: $\psi_{a}^{+}\equiv\psi_{k-a}$):

$\displaystyle\psi_{a}(z)\psi_{b}(w)(z-w)^{h_{a}+h_{b}-h_{a+b}}=C_{a,b}\psi_{a+b}(w)+\cdots$
$\displaystyle(0<a+b<k)$ (3a)
$\displaystyle\psi^{+}_{a}(z)\psi^{+}_{b}(w)(z-w)^{h_{a}+h_{b}-h_{a+b}}=C_{a,b}\psi^{+}_{a+b}(w)+\cdots$
$\displaystyle(0<a+b<k)$ (3b)
$\displaystyle\psi_{a}(z)\psi_{a}^{+}(w)(z-w)^{2h_{a}}=1+{\textstyle\frac{2h_{a}}{c}}(z-w)^{2}L(w)+\cdots$
(3c)

Here, $C_{a,b}$ are the structure constants, $c$ is the central charge and $L$
is the energy-momentum tensor. Moreover, the ellipsis only involves fields
with a higher scaling dimension than the leading field by an integer. In
truth, with $\mathcal{W}$ being the underlying chiral algebra, all subsequent
fields are $\mathcal{W}$-descendants of the leading term. There exist many
associate solutions to the above OPA. In this paper, we are only interested in
solutions with scaling dimensions $h(\psi_{a})\equiv h_{a}=a(k-a)/2\nu$, where
$\nu=k/2r$ (with integer $k,r$); the symbol $\nu$ can be identified as the
filling fraction of the corresponding FQH wavefunction. Such a theory will be
called a _$\mathbb{Z}_{k}^{(r)}$ (current) algebra_. The special solution with
$r=1$ (i.e. $\mathbb{Z}_{k}^{(1)}$-algebra) is the classic construction of
Zamolodchikov-Fateev [52] and $\mathbb{Z}_{k}^{(1)}$ wavefunctions are the
Read-Rezayi states [38]. Thus, $\mathbb{Z}_{k}^{(r)}$ states are the
generalization of Read-Rezayi states.

Note that the electron vertex operator
$V_{e}(z)=\psi(z):\exp[i\sqrt{2r/k}\>\phi(z)]:$, with $\psi$ being the
generator of a $\mathbb{Z}_{k}^{(r)}$-algebra, has the statistics of a boson
$V_{e}(z)V_{e}(w)=V_{e}(w)V_{e}(z)$, and an integer scaling dimension equal to
$r$. To respect the spin-statistics theorem, we have disallowed half-integer
values for $r$. However, this choice would exclude states like Gaffnian [41]
from the models (i.e., $k=2,r=3/2$), and not all authors would opt for such an
omission. For reference, what we named a $\mathbb{Z}_{k}^{(r)}$-algebra would
be called a $\mathbb{Z}_{k}^{(2r)}$ theory in Ref. [15], and they do include
cases like Gaffnian as well. Regardless of allowing half-integers, as is
pointed out in Ref. [15], using the operator product expansions (of the form
$\psi_{a}(z)\psi(w)$) repeatedly, the wavefunction

$\Psi_{n}(z_{1},\cdots,z_{nk})\equiv\frac{1}{(\prod_{a=0}^{k-1}C_{1,a})^{n}}\left<\psi(z_{1})\cdots\psi(z_{nk})\right>\prod_{i<j}(z_{i}-z_{j})^{2r/k}$
(4)

can easily be seen as a $(k,2r)$ clustering polynomial. Here, the subscript
$n$ refers to the size of the system and reminds us that we have $N=nk$
particles. The current paper will unravel a secondary local property of these
wavefunctions called _separability_. The new property is concerned with the
explicit form of the projection of the wavefunction when a cluster of $k+1$
particles has a relative angular momentum of exactly $2r$. As discussed in the
summary section, we use separability to design modified Hamiltonians that
realize these wavefunctions as likely the unique, densest zero-energy state.

In the literature, initially, the main motivation to investigate
$\mathbb{Z}_{k}^{(r)}$-algebras was two conjectures about (negative rational)
$(k,\rho)$ admissible Jack polynomials [17], and their role as trial
wavefunctions [7, 6]. The Jacks are an interesting avenue for generalizing
Read-Rezayi states to non-Abelian states with more general filling fractions.
In their debut as trial wavefunctions [7], it was conjectured that these Jacks
are clustering polynomials. As for the second conjecture, before coming to
prominence as trial FQH states, these specialized Jack polynomials were first
studied in Ref. [17] as a means to describe the space of symmetric polynomials
that vanish as $k+1$ variables coalesce. In the same paper, and later on in
Ref. [18], it was conjectured that these Jacks are certain correlation
functions in the $\mathrm{WA}_{k-1}(k+1,k+\rho)$ minimal models. However,
$\mathrm{WA}_{k-1}(k+1,k+\rho)$ is a (non-unitary) special case of
$\mathbb{Z}_{k}^{\rho/2}$-algebras. Thus, the former conjecture is a
consequence of the latter one. In Ref. [16], by studying the degenerate
representations of $\mathrm{WA}_{k-1}(k+1,k+\rho)$, both conjectures were
proved. We should also mention that the conjecture regarding the clustering
property of Jacks was also proved independently via the representation theory
of Cherednik algebras in Ref. [51].

In general, FQH polynomials often have hidden structures, historically leading
to new physical insight into the quantum Hall effect. For example, we have
already mentioned how researchers use trial wavefunctions to reverse engineer
(pseudopotential) Hamiltonians. In principle, one may also use such
wavefunctions to reconstruct the underlying CFT (with various degrees of
success). Assuming a given wavefunction is indeed a conformal block, one can
obtain the central charge from the wavefunction alone [5, 15, 50]. In the
current paper, we obtain further CFT data from these wavefunctions. To what
degree and how these trial wavefunctions encode the current algebra inside
them remains to be seen. However, it is safe to say that: as our technical
ability to calculate these particular conformal blocks advances, our knowledge
about these FQH states and their hidden structures will also improve. At the
moment, aside from a handful of exceptions, the best tools currently available
are only applicable situationally and even then usually only help us to
compute the $2k$ point correlation function (the so-called principal
wavefunction)

$\Psi_{2}(z_{1},\cdots,z_{2k})\equiv\frac{1}{(\prod_{a=0}^{k-1}C_{1,a})^{2}}\left<\psi(z_{1})\cdots\psi(z_{2k})\right>\prod_{i<j}(z_{i}-z_{j})^{2r/k}$
(5)

Even this limited version of the wavefunction requires special machinery to
compute. To our knowledge, the only $\mathbb{Z}_{k}^{(r)}$-algebras with fully
computed wavefunctions of arbitrary size are the following: Read-Rezayi states
[38, 11] ($k$ arbitrary, $r=1$), Haffnian ($k=r=2$) [20], and correlation
functions of $\mathcal{N}=1$ superconformal field theory ($k=2,r=3$) [40]. As
for the computations of $\Psi_{2}$ alone, in Ref. [15], the principal
wavefunction of $\mathbb{Z}_{2}^{(5)}$, $\mathbb{Z}_{k}^{(2)}$ with $k\leq 5$
are calculated in terms of Jack polynomials. The current paper provides
additional machinery to compute principal wavefunctions by utilizing the so-
called _theory of invariants of binary forms_. In particular, we compute the
principal $\mathbb{Z}_{k}^{(2)}$ wavefunction for all values of $k$ (section
8.3) and the principal $\mathbb{Z}_{2}^{(r)}$ wavefunctions for all values of
$r$ (F).

So far, we have not mentioned the topological nature of the quantum Hall
phases of matter. A signature of quantum Hall systems is an energy gap between
the ground state and the bulk excitations. Generally speaking, to ensure such
a gap, the CFT describing the FQH system is expected to be _unitary_ and
_rational_ [36]. As for concrete evidence behind the unitarity expectation,
while outside the scope of models considered here, the Haldane-Rezayi (HR)
[22] and Gaffnian [42] are the two extensively studied examples: The CFT
behind the Gaffnian is the $\mathcal{M}(5,3)$ minimal model, which has central
charge $c=-\frac{3}{5}$, hence is non-unitary. As for the HR state, Ref. [21]
shows that it, too, is related to a non-unitary CFT. It has been argued that
the HR state is the wavefunction at a phase transition [37], and thus is not
the ground state of a gapped system (also see Ref. [24]). In the case of the
Gaffnian, the gaplessness was checked numerically (e.g., [29]) and confirmed
in Ref. [14] using Matrix Product State (MPS) formalism. In addition to non-
unitarity concerns, there is also the question of _rationality_ of the
obtained theories. As a simple non-example, consider the
$\mathbb{Z}_{2}^{(2)}$-algebra. The $\mathbb{Z}_{2}^{(2)}$ wavefunction,
dubbed the Haffnian, has been studied in detail by Green [20] (also see Ref.
[24]). In this case, $\psi(z)=i\partial\phi(z)$, with $\phi(z)$ a chiral free
boson. It is well-known that for each $\alpha\in\mathbb{R}$ the vertex
operator $V_{\alpha}(z)=:\exp[i\alpha\phi(z)]:$ is a $\psi$-primary field.
Hence, the spectrum is infinite and this CFT describes a gapless phase. As for
the general current algebras, given “large enough” $k,r$, the central charge
in a generic $\mathbb{Z}_{k}^{(r)}$-algebra is a free parameter and can assume
irrational values. Given the brief discussion about the gap and rationality,
we end this section by defining the scope and limitations of the current
paper. It is well beyond the reach of the current paper to find what values of
central charge (and possibly other free CFT data) would make the CFTs
rational. In addition, the CFTs studied here include many non-unitary
theories, requiring caution in the context of the FQH systems. Moreover, we
lack the tools to adequately analyze the existence of a gap (or lack thereof)
for the models discussed here.

##  2 Motivation and Summary of Results

The main ambition of this paper is to design a model Hamiltonian to realize
the $\mathbb{Z}_{k}^{(r)}$ wavefunction (abbreviated WF) as _the unique_
ground state. More descriptively, we are searching for a multi-particle
pseudopotential Hamiltonian [44] for which a given $\mathbb{Z}_{k}^{(r)}$-WF
is the _unique_ densest zero-energy state. This demand is necessary since
quantum Hall Hamiltonians on a disk/sphere must have a unique ground state.
Our approach can be summarized as follows:

  1. 1.

We _claim_ that the entirety of a $\mathbb{Z}_{k}^{(r)}$-algebra is encoded
inside a _single_ (explicit) polynomial $\chi$.

  2. 2.

We construct a pseudopotential Hamiltonian $H_{\chi}$ (in a natural way)
parametrized by this proxy $\chi$.

The idea is that if both the current algebra and the model Hamiltonian
describe the same quantum Hall phase of matter, then there should be a one-to-
one correspondence between them. Accordingly, with $\chi$ as an intermediate,
the respective model Hamiltonian would reflect the entirety of the current
algebra in this methodology.

The two key players in the current paper are the _minimal polynomial_ $\chi$
(the presumed proxy for a $\mathbb{Z}_{k}^{(r)}$-algebra) and the _principal_
$\mathbb{Z}_{k}^{(r)}$-wavefunction. The definitions of these polynomials are
as follows (Notation: $w^{\times a}$ means $w$ repeated $a$ times):

$\displaystyle\Psi_{2}(z_{1},\cdots,z_{2k})$
$\displaystyle=\frac{1}{g_{k}^{2}}\left<\psi(z_{1})\cdots\psi(z_{2k})\right>\prod_{i<j}(z_{i}-z_{j})^{\frac{1}{\nu}}$
(6) $\displaystyle\chi(z_{1},\cdots,z_{k+1})$
$\displaystyle=\lim_{w\to\infty}\frac{\Psi_{2}(w^{\times
k-1},z_{1},\cdots,z_{k+1})}{w^{2r(k-1)}}=\frac{1}{g_{k}}\left<\psi^{+}(\infty)\psi(z_{1})\cdots\psi(z_{k+1})\right>\prod_{i<j}(z_{i}-z_{j})^{\frac{1}{\nu}}$
(7)

where, $g_{0}=1$, and for $1\leq a\leq k$ we have
$g_{a}=\prod_{j=0}^{a-1}C_{1,j}$. The minimal polynomial $\chi$ is a
homogeneous, translational invariant, symmetric polynomial in $k+1$ variables
and degree $2r$. Such a polynomial is called a $(k+1,2r)$ _semi-invariant_ and
their linear space is denoted by $\mathcal{T}_{k+1}^{2r}$. Additionally, due
to the particular normalization we choose for the wavefunctions, the minimal
polynomials _always_ have a normalization $\chi(1,0^{\times k})=1$.

To present the Hamiltonian $H_{\chi}$, let us review a few concepts. We use
the notation $\mathscr{H}_{k+1}^{2r-1}$ for the projection operator that
projects out any state where any cluster of $k+1$ particles has relative
angular momentum $<2r$. Traditionally, this is the most prominent model
Hamiltonian used in the literature. For $k=1$ (and $r$ arbitrary), this leads
to Haldane’s (gapped) Hamiltonian [23] for Laughlin $2r$-state. For $r=1$ (and
$k$ arbitrary), one recovers the Read-Rezayi (gapped) Hamiltonian [38].
Unfortunately, the fact that $\dim\mathcal{T}_{k+1}^{2r}>1$ for almost all
$(k,r)$ pairs is a big obstacle for $\mathscr{H}_{k+1}^{2r-1}$ admitting a
unique densest zero-energy state. We would like to modify
$\mathscr{H}_{k+1}^{2r-1}$ in a natural way so that
$\dim\mathcal{T}_{k+1}^{2r}>1$ is no longer relevant to uniqueness of ground
states. The projection Hamiltonian $\mathscr{H}_{k+1}^{2r-1}$ is blind to the
fine details of the projection to the sector where $(k+1)$-cluster have
_exactly_ relative angular momentum $2r$. We assert that those fine details
matter and look for a modification that includes them. The guiding observation
is the following local property: denoting by
$\mathcal{P}_{k+1}^{2r}(z_{1},\cdots,z_{k+1})$ the projection to the sector
where variables $z_{1},\cdots,z_{k+1}$ have relative angular momentum equal to
$2r$, we show in this paper that $\mathbb{Z}_{k}^{(r)}$-WFs satisfy (Notation:
$\widehat{z}=(z_{1}+\cdots+z_{k+1})/(k+1)$ is the center-of-mass)

$\mathcal{P}_{k+1}^{2r}(z_{1},\cdots,z_{k+1})\Psi_{n}=\chi(z_{1},\cdots,z_{k+1})\prod_{i>k+1}\left(\widehat{z}-z_{i}\right)^{2r}\Psi_{n-1}\left(\widehat{z},z_{k+2},\cdots,z_{nk}\right)$
(8)

The function $\chi$ is the minimal polynomial and is independent of the size
$n$. This local property is called _separability_. Utilizing separability, we
propose the following pseudopotential Hamiltonian

$H_{\chi}=\mathscr{H}_{k+1}^{2r-1}+V\sum_{i_{1}<i_{2}\cdots<i_{k+1}}\mathcal{P}_{\chi^{\perp}}\mathcal{P}_{k+1}^{2r}(z_{i_{1}},\cdots,z_{i_{k+1}}),\qquad(V>0)$
(9)

Treating $\chi$ as a vector in $\mathcal{T}_{k+1}^{2r}$, the operator
$\mathcal{P}_{\chi^{\perp}}$ projects to the hyperplane in
$\mathcal{T}_{k+1}^{2r}$ with normal vector $\chi$. Due to construction, among
$\mathbb{Z}_{k}^{(r)}$-WFs, only those with minimal polynomial $\chi$ are the
densest zero-energy states of $H_{\chi}$. While this modification resolves the
$\dim\mathcal{T}_{k+1}^{2r}>1$ issue, we still need to study if $H_{\chi}$ has
a unique ground state. We want to clearly state that we have no proof that
$H_{\chi}$ possesses a unique densest zero-energy state. Instead, we shall
provide indirect evidence supporting the idea that $H_{\chi}$ Hamiltonians
possess a unique ground state.

We base our argument regarding the uniqueness of the ground state on a triplet
of postulates. The following diagram summarizes these postulates:

$\chi\xrightarrow{\textsc{Fix}}\Psi_{2}\xrightarrow{\textsc{Fix}}\mathbb{Z}_{k}^{(r)}\text{algebra}\xrightarrow{\textsc{Fix}}\Psi_{n}$

We read this diagram as follows: (1) knowing $\chi$ (together with the fact
that $\Psi_{1}=1$ and its consequences) is enough to completely determine
$\Psi_{2}$; (2) Determining $\Psi_{2}$ is enough to fix all of the free
parameters of the $\mathbb{Z}_{k}^{(r)}$-algebra that gives birth to
$\Psi_{2}$; (3) No two _distinct_ $\mathbb{Z}_{k}^{(r)}$-algebras can give
birth to _identical_ wavefunctions. Much of this paper is devoted to the
justification (at least for $r=1,2$) of
$\Psi_{2}\to\mathbb{Z}_{k}^{(r)}\text{algebra}$ and $\chi\to\Psi_{2}$. As
$\mathbb{Z}_{k}^{(r)}\text{algebra}\to\Psi_{n}$ (for all $n$) is the least
controversial yet hardest to check, we have postponed its study until future
work.

Let us illustrate this entire program with a well-known, familiar, and
relatively simple example: correlation functions in $\mathcal{N}=1$
superconformal field theory (SCFT). Recall that the following operator product
expansions (OPEs) describe the $\mathcal{N}=1$ SCFT, aka $\mathcal{N}=1$ super
Virasoro algebra:

$\displaystyle L(z)L(w)$
$\displaystyle=\frac{\frac{1}{2}c}{(z-w)^{4}}+\frac{2L(w)}{(z-w)^{2}}+\frac{\partial
L(w)}{z-w}+\mathrm{regular}$ $\displaystyle L(z)\psi(w)$
$\displaystyle=\frac{\frac{3}{2}\psi(w)}{(z-w)^{2}}+\frac{\partial\psi(w)}{z-w}+\mathrm{regular}$
$\displaystyle\psi(z)\psi(w)$
$\displaystyle=\frac{1}{(z-w)^{3}}+\frac{3}{c}\frac{L(w)}{z-w}+\mathrm{regular}$

Here, $c$ is the central charge and can take any real value; the first OPE
declares $L$ as the energy-momentum tensor, the second states that $\psi$ has
scaling dimension $h=3/2$, and the last describes how $\psi$ fuses with
itself. A quick check shows that this theory matches the definition of
$\mathbb{Z}_{2}^{(3)}$-algebras exactly. While we do not often have the luxury
of being able to compute wavefunctions of arbitrary size, fortunately, Simon
[40] has calculated the $\mathbb{Z}_{2}^{(3)}$-WFs $\Psi_{A}$ for an arbitrary
number of particles $N=2n$ (Notation: $z_{ij}=z_{i}-z_{j}$):

$\displaystyle f_{A}(z_{1},z_{2},w_{1},w_{2})$ $\displaystyle\equiv
A(z_{1}-w_{1})^{3}(z_{1}-w_{2})^{3}(z_{2}-w_{1})^{3}(z_{2}-w_{2})^{3}+(z_{1}-w_{1})^{4}(z_{1}-w_{2})^{2}(z_{2}-w_{1})^{2}(z_{2}-w_{2})^{4}$
$\displaystyle\Psi_{A}(z_{1},\cdots,z_{2n})$
$\displaystyle\propto\mathscr{S}\prod_{1\leq r<s\leq
n}^{n}f_{A}(z_{2r-1},z_{2r},z_{2s-1},z_{2s})$

where $\mathscr{S}$ stands for symmetrization and $A=c/3-1$ is a free
parameter. Before proceeding with the analysis, we should mention that, given
the confines of this example, our methodology partially aligns with the
arguments presented in Ref. [28], section II.A. To expand on the content of
Ref. [28], let us compute the minimal polynomial:

$\chi_{A}(z_{1},z_{2},z_{3})=\frac{1}{2(A+1)}\mathscr{S}[A(z_{1}-z_{3})^{3}(z_{2}-z_{3})^{3}+(z_{1}-z_{3})^{4}(z_{2}-z_{3})^{2}]$

Note that $\chi_{1}=\mathscr{S}[(z_{1}-z_{3})^{3}(z_{2}-z_{3})^{3}]$ and
$\chi_{2}=\mathscr{S}[(z_{1}-z_{3})^{4}(z_{2}-z_{3})^{2}]$ constitute a basis
for $\mathcal{T}_{3}^{6}$. Therefore, the knowledge of the value $A$ is
entirely the same as knowing the minimal polynomial. In other words, by fixing
$\chi_{A}$, we determine $A$, which fixes the CFT and all of the $\Psi_{n}$
wavefunctions. The Hamiltonian $H_{\chi_{A}}$, which Ref. [28] also hints at,
is then really a one-parameter family of Hamiltonians (depending on $A$),
which picks $\Psi_{A}$ as the densest zero-energy state.

If $\chi\to\Psi_{2}\to\mathbb{Z}_{k}^{(r)}\text{algebra}\to\Psi_{n}$ is indeed
true, then $\chi$ acts as a proxy for the $\mathbb{Z}_{k}^{(r)}$-algebra in
the Hamiltonian $H_{\chi}$. In other words, no two current algebras will have
the same minimal polynomial. Thus, _among wavefunctions descending from a
$\mathbb{Z}_{k}^{(r)}$-algebra_, the model Hamiltonian $H_{\chi}$ has exactly
one zero-energy state. Consequently, unless $H_{\chi}$ has a densest zero-
energy state that is not a $\mathbb{Z}_{k}^{(r)}$-WF for certain choice of
free parameters, the Hamiltonian $H_{\chi}$ has a unique ground state. Whether
such a non-CFT ground state exists is beyond the scope of this paper.
Regarding the gap, we make another _guess_ : “If a
$\mathbb{Z}_{k}^{(r)}$-algebra is unitary and has a finite spectrum, and
$\chi$ is a proxy for such a current algebra, then $H_{\chi}$ is gapped.”

###  2.1 $\Psi_{1}=1$ and Evidence for
$(\Psi_{1},)\Psi_{2}\to\mathbb{Z}_{k}^{(r)}\text{algebra}$

Upon fixing a $\mathbb{Z}_{k}^{(r)}$-algebra, one can construct its
wavefunctions in $nk$ variables ($n\geq 1$ arbitrary integer). We denote this
wavefunction by $\Psi_{n}$ and often work with the sequence
$\boldsymbol{\Psi}=(\Psi_{1},\Psi_{2},\cdots,\Psi_{n},\cdots)$. Quite
generally, if no two distinct $\mathbb{Z}_{k}^{(r)}$-algebras can lead to an
identical wavefunction sequence, the first “few” elements in this sequence
should contain all of the classifying information about the current algebra.
To expand on this comment, let $\alpha$ be some free parameter of the CFT. If
changing $\alpha$ does not alter any of the $\Psi_{n}$, we violate the
correspondence between current algebras and wavefunctions. Thus, there exist a
smallest $n_{\alpha}$ such that $\Psi_{n_{\alpha}}$ is a continuous non-
constant function of $\alpha$. Now since the number of free parameters is
finite, there exists some $l$ such that determining
$(\Psi_{1},\Psi_{2},\cdots,\Psi_{l})$ would completely fix all of the
parameters of the $\mathbb{Z}_{k}^{(r)}$-algebra. Our postulate is now stating
that the first _two_ , i.e. $\Psi_{1},\Psi_{2}$, already fix the current
algebra.

As we will prove in subsection 6.2, $\Psi_{1}$, i.e. the first wavefunction,
is _trivial_ :

$1=\frac{1}{g_{k}}\left<\psi(z_{1})\cdots\psi(z_{k})\right>\prod_{i<j}(z_{i}-z_{j})^{\frac{1}{\nu}}=\Psi_{1}(z_{1},\cdots,z_{k})$
(10)

Despite $\Psi_{1}$ being trivial as a polynomial, as a correlation function,
the above statement is highly non-trivial. As we will show, the relation
$\Psi_{1}=1$ leads to a constraint on the structure constants:

$C_{a,b}=\frac{g_{a+b}}{g_{a}g_{b}}=\prod_{j=0}^{a-1}\frac{C_{1,j+b}}{C_{1,j}}$
(11)

However, beyond this constraint, the remaining CFT Data is carried by the so-
called _principal wavefunction_ $\Psi_{2}$ (or its specialization $\chi$). The
justification of $\chi\to\mathbb{Z}_{k}^{(r)}\text{algebra}$ (this is slightly
more convenient than $\Psi_{2}\to\mathbb{Z}_{k}^{(r)}\text{algebra}$) is thus
reduced to studying how the CFT data – e.g. structure constants and central
charge, etc. – can be explicitly read off from the semi-invariant $\chi$. We
prove that since $\chi\in\mathcal{T}_{k+1}^{2r}$ (i.e. due to the structure of
$\chi$, not its specifics), together with $\chi(1^{\times a+1},0^{\times
k-a})=C_{1,a}^{2}$ and $C_{a,b}=g_{a+b}/g_{a}g_{b}$, the structure constants
$C_{a,b}$ with $a+b\leq k$ are as follows ($\Gamma$ is the gamma function)

$C_{a,b}^{2}=\frac{(a+b)!(k-a)!(k-b)!}{k!a!b!(k-a-b)!}\prod_{p=1}^{r-1}\frac{\Gamma(t_{p}+a+b)\Gamma(t_{p}+k-a)\Gamma(t_{p}+k-b)\Gamma(t_{p})}{\Gamma(t_{p}+k-a-b)\Gamma(t_{p}+a)\Gamma(t_{p}+b)\Gamma(t_{p}+k)}$
(12)

Here, $t_{1},t_{2},\cdots,t_{r-1}$ are _free_ parameters of the CFT (when
$k+1>2r$; when $k+1\leq 2r$, some of $t_{i}$ are redundant). Complementing
this observation with charge conjugation symmetry $C_{k-a,k-b}=C_{a,b}$, all
structure constants are acquired. Moving on to the central charge, using the
existing techniques in the literature [5, 15, 50], we can read off $c$ from a
different specialization of $\chi$:

$\xi(x):=\chi(1,x,0^{\times
k-1})=1-\frac{x}{\nu}+\left(\frac{\frac{1}{\nu}(\frac{1}{\nu}-1)}{2}+\frac{2h^{2}}{c}\right)x^{2}+O(x^{3})$
(13)

i.e. $c$ can be found from the coefficient of $x^{2}$ in the polynomial
$\xi(x)$.

As a quick sanity check, consider the $\mathbb{Z}_{k}^{(1)}$-algebras. In this
case, $\dim\mathcal{T}_{k+1}^{2}=1$, and the constraint $\chi(1,0^{\times
k})=1$ completely fixes the minimal polynomial:

$\chi(z_{1},\cdots,z_{k+1})=\frac{1}{k}\sum_{i<j}(z_{i}-z_{j})^{2}$

From this, one can find $c=2(k-1)/(k+2)$; also $C_{1,a}^{2}=\chi(1^{\times
a+1},0^{\times k-a})=(a+1)(k-a)/k$. Complementing this with
$C_{a,b}=g_{a+b}/g_{a}g_{b}$, the $r=1$ case of Eq. (12) is obtained.
Unsurprisingly, these results match what Zamolodchikov and Fateev have
reported upon discovering these parafermionic theories [52].

###  2.2 Evidence for $(\Psi_{1},)\chi\to\Psi_{2}$ for $r=1,2$

In this introductory subsection, we will limit ourselves to two concrete
examples: $\mathbb{Z}_{2}^{(2)}$ and $\mathbb{Z}_{3}^{(2)}$ algebras. The
current algebra $\mathbb{Z}_{2}^{(2)}$ describes a free boson (i.e.
$\psi(z)=i\partial\phi(z)$), has $c=1$ and zero degrees of freedom. In
contrast, $\mathbb{Z}_{3}^{(2)}$-algebras have one free parameter $t$. The
central charge $c$ and the structure constant $C_{1,1}$ of the theory are
given by

$c=2\left(1-\frac{3}{(2t+3)(2t+1)}\right),\qquad
C^{2}_{1,1}=\frac{4}{3}\frac{(t+1)^{2}}{t(t+2)}$

The $\mathbb{Z}_{3}^{(2)}$-algebra, and its representations, are studied in
depth in Ref. [53]. Moreover, the principal wavefunction $\Psi_{2}$ are
computed in Ref. [15] using Jack polynomials. Using these two examples, we
will demonstrate an alternative approach to computations of this type. The
principal $\mathbb{Z}_{3}^{(2)}$-WF, in particular, can capture some of the
nuances one often encounters carrying out these computations. At the same
time, it is still simple enough to discuss in an introductory and minimalistic
manner. Via these examples, we illustrate how to check $\chi\to\Psi_{2}$ by
explicitly computing both $\chi$ and $\Psi_{2}$, _but without directly using
the OPEs_.

Finding the minimal polynomial is relatively simple. When $2r=4$, we have
$\dim\mathcal{T}_{2+1}^{4}=1$ and $\dim\mathcal{T}_{3+1}^{4}=2$. We are
looking for $\chi\in\mathcal{T}_{k+1}^{4}$ satisfying $\chi(1,0^{\times
k})=1$. For $k=2$, there is only one choice:

$\chi_{2}(z_{1},z_{2},z_{3})=\frac{1}{2}\sum_{i<j}(z_{i}-z_{j})^{4}$

In contrast, when $k=3$, we have a continuous family (parametrized by $\beta$)
of semi-invariants $\chi_{3}$:

$\chi_{3}(z_{1},z_{2},z_{3},z_{4})=\frac{1}{3}\sum_{i<j}(z_{i}-z_{j})^{4}+\frac{2\beta}{3}\big{[}(z_{1}-z_{2})^{2}(z_{3}-z_{4})^{2}+(z_{1}-z_{3})^{2}(z_{2}-z_{4})^{2}+(z_{1}-z_{4})^{2}(z_{2}-z_{3})^{2}\big{]}$

Using the fact that $\chi(1,1,0,0)=C_{1,1}^{2}$, we then find that
$\beta=1/t(t+2)$. However, note that $\beta$ works just as well as $t$ for
parameterizing the CFT. By calculating $\chi_{2}(1,x,0)$ and
$\chi_{3}(1,x,0,0)$ one can rediscover the central charges that we listed
above. To summarize, we wrote down the most general semi-invariant
$\chi\in\mathcal{T}_{k+1}^{2r}$, applied the normalization condition
$1=\chi(1,0^{\times k})$, and then identified the remaining free parameters
with the CFT data through certain specializations.

The computation of $\Psi_{2}$ follows a similar methodology, albeit the
procedure is much more involved. We begin by creating an ansatz for the
principal $\mathbb{Z}_{k}^{(r)}$-WF $\Psi_{2}$. The starting point is the fact
that $\Psi_{2}$ is conformally covariant: Let $f(z)=(az+b)/(cz+d)$ be an
arbitrary Möbius map with $ad-bc\neq 0$. Then

$\Psi_{2}(f(z_{1}),\cdots,f(z_{2k}))=\prod_{i=1}^{2k}(df/dz|_{z_{i}})^{2r/2}\Psi_{2}(z_{1},\cdots,z_{2k})$

One says that $\Psi_{2}$ is a uniform state on a sphere, with $2k$ particles
and $2r$ flux quanta (aka $(2k,2r)$ uniform state). Now, there is an
isomorphism between uniform states on the sphere and so-called _binary
invariants_ (see [34] for an introduction to binary invariants). We discuss
this isomorphism in section 7. It is worth mentioning that the binary
invariant that corresponds to a uniform state (i.e., a wavefunction) is
essentially its second quantized formulation in terms of lowest Landau level
orbits. The critical point is that uniform states over a sphere with $2r$ flux
quanta ($r=1,2,3,4$) are already classified. These mathematical endeavors
began in the late 19th century and spanned several generations. While these
classification efforts are made in the literature on binary invariants,
hopefully, this paper can bridge the language barrier for the quantum Hall
research community. Reporting the end-result, the most general $(2k,4)$
uniform state, with $k=2,3$ are

$\Psi^{[k=2]}_{2}(z_{1},\cdots,z_{4})=\frac{1}{2!^{2}2!^{2}}\mathscr{S}\left[\prod_{p=0,1}(z_{2p+1}-z_{2p+2})^{4}\right]$

and

$\displaystyle\Psi^{[k=3]}_{2}(z_{1},\cdots,z_{6})$
$\displaystyle=\frac{1}{2!^{3}3!^{2}}\mathscr{S}\left[\prod_{p=0,1,2}(z_{2p+1}-z_{2p+2})^{4}\right]$
$\displaystyle+\frac{\alpha}{3!3!^{2}}\mathscr{S}\left[\prod_{p=0,1}(z_{3p+1}-z_{3p+2})^{2}(z_{3p+1}-z_{3p+3})^{2}(z_{3p+2}-z_{3p+3})^{2}\right]$

Here, $\alpha$ is a free parameter we need to fix. It turns out that
$\alpha=\beta=1/t(t+2)$. While this clean $\alpha=\beta$ relation is an
artifact of the simplicity of $3\leq k<6$ (the general $k$ is more involved),
the overall theme remains the same: For $k>2$, the
$\mathbb{Z}_{k}^{(2)}$-algebras and subspace of $\mathcal{T}_{k+1}^{4}$ with
$\chi(1,0^{\times k})=1$ are both one-dimensional. We can choose either $t$ or
$\beta$ as the parameter. The most general $(2k,4)$ uniform state is
explicitly known up to coefficients like $\alpha$. Thus, one must find the
$\alpha$ coefficients via various specializations and relate them to the CFT
data. We omit the details at this introductory stage. The sections 7 and 8
will greatly expand on the above paragraph and generalize it appropriately.

In subsection 8.3, we calculate the principal $\mathbb{Z}_{k}^{(2)}$
wavefunctions for an arbitrary value of $k$ and provide a relatively clean
presentation for it. Similar to the abovementioned examples, the calculation
relies on an ansatz for principal wavefunction that uses binary invariants. In
addition, we have extensively used the graph-theoretic point of view toward
FQH states that we developed elsewhere [35]. In subsections 8.4 and 8.5, we
have carried out the procedure for $r=3,4$ as well. However, we are not as
successful in those cases as in $r=2$. Some of the coefficients in the ansatz
remain undetermined in those cases. Determining these unknown coefficients
will require further technical innovations and possibly a better understanding
of $\mathbb{Z}_{k}^{(3)},\mathbb{Z}_{k}^{(4)}$-algebras. We also compute the
principal $\mathbb{Z}_{2}^{(r)}$ wavefunctions (arbitrary $r$) in F, although
the ansatz and method used is different from that in the main body.

##  3 Uniform States on a Sphere

We begin by reviewing the notion of uniform states (over a sphere) and their
relevance to a quantum Hall system. Consider a gas of $N$ spin-polarized
bosonic/fermionic electrons over a sphere with radius $R=1/2$. Due to a
magnetic monopole at the center, $N_{\phi}$ flux quanta is passing through the
sphere. This setup is sometimes called the Haldane sphere [23]. We are
interested in describing the states in the lowest Landau level (LLL). Using
stereographic projection (see Fig. 1), the location of a particle on the
sphere is parametrized by a complex coordinate $z$. We take the inner product
of the Hilbert space to be

$\left<\phi_{2}\vphantom{\phi_{1}}\right|\left.\phi_{1}\vphantom{\phi_{2}}\right>=\int\frac{dxdy}{(1+z\bar{z})^{2+N_{\phi}}}\phi_{1}(z)\overline{\phi_{2}(z)}$
(14)

The LLL is $(N_{\phi}+1)$-fold degenerate with an orthonormal basis
$\left|m\right>$, $0\leq m\leq N_{\phi}$, where
$\left<z\vphantom{m}\right|\left.m\vphantom{z}\right>\propto z^{m}$.

Figure 1: Stereographic Projection: Let $S,N$ be the south and north pole.
Make the complex plane tangential to $S$, oriented so that
$\hat{x}\times\hat{y}$ points inward. The intersection of the ray $NP$ with
the complex plane is the _complex coordinate_ $z_{P}$ (near zero) of the point
$P$.

Extending the inner product to the $N$-body Hilbert space, wavefunctions in
the LLL are polynomials $P(z_{1},\cdots,z_{N})$. In this article, we focus on
bosonic systems (Any fermionic wavefunction is the product of a bosonic
wavefunction and a Laughlin-Jastrow factor $\prod_{i<j}(z_{i}-z_{j})$). A
bosonic wavefunction $P(z_{1},\cdots,z_{N})$ in the LLL is a symmetric
polynomial where the degree of no variable $z_{i}$ can exceed $N_{\phi}$. We
assume that the quantum Hall wavefunctions are all confined to the LLL.

One of the characteristics of a fractional quantum Hall ground state (i.e., in
the absence of local bulk excitations) is that the one-particle density is
uniform over the sphere. The uniformity is equivalent to the wavefunction $P$
being $SO(3)$ invariant; i.e., $L^{2}P=0$ in the induced representation of
$SO(3)$ in the presence of $N_{\phi}$ flux quanta. More concretely,

$\displaystyle L_{-}P$ $\displaystyle:=\sum_{i}\partial_{i}P=0$ (15a)
$\displaystyle L_{z}P$
$\displaystyle:=\sum_{i}\left(-\frac{N_{\phi}}{2}+z_{i}\partial_{i}\right)P=0$
(15b) $\displaystyle L_{+}P$
$\displaystyle:=\left(\sum_{i}N_{\phi}z_{i}-z_{i}^{2}\partial_{i}\right)P=0$
(15c)

These equations are a variant of the Ward identities in conformal field theory
and can be reformulated in terms conformal transformations: For any arbitrary
Möbius map $f(x):=(ax+b)/(cx+d)$ (with $ad-bc\neq 0$) the wavefunction $P$
transform via

$P(f(z_{1}),\cdots,f(z_{N}))=\prod_{i=1}^{N}\left(\frac{df}{dx}\right)_{x=z_{i}}^{\frac{1}{2}N_{\phi}}P(z_{1},\cdots,z_{N})$
(16)

We say $P$ is an _$(N,N_{\phi})$ uniform state_. Furthermore, at the onset of
the quantum Hall effect, $N,N_{\phi}$ are related through the constraint
$N_{\phi}=N\nu^{-1}-\mathcal{S}$; here $\nu$ is the filling fraction (a
rational number) and $\mathcal{S}$ is the shift (an integer).

##  4 $\mathbb{Z}_{k}$ Parafermionic Conformal Field Theories

This paper is concerned with model FQH ground states that descend from
$\mathbb{Z}^{(r)}_{k}$ parafermionic current algebras. We will define the
relevant conformal field theories (CFT) in this section.

###  4.1 Zamolodchikov-Fateev Parafermions

A $\mathbb{Z}_{k}$ parafermionic CFT, with some chiral algebra $\mathcal{W}$
(see B for a review), consists of $k$ holomorphic $\mathcal{W}$-primary
fields: $1,\psi(z),\psi_{2}(z),\cdots,\psi_{k-1}(z)$. These fields make a
representation of the cyclic group $\mathbb{Z}_{k}$ via the fusion rules
$\psi_{a}\star\psi_{b}=\psi_{a+b}$ (addition done in $\mathbb{Z}_{k}$,
$\psi_{0}\equiv 1$, and $\psi_{1}\equiv\psi$). The transformation
$\psi_{a}\mapsto\psi_{a}^{+}\equiv\psi_{k-a}$ is the _charge conjugation_. We
denote the scaling dimension of $\psi_{a}$ by $h_{a}$. The relative scaling
dimension is given by

$\gamma(a,b)=h_{a}+h_{b}-h_{a+b},\qquad\gamma\equiv\gamma(1,1)$ (17)

The field space of the chiral algebra $\mathcal{W}$ can be graded as
$\mathcal{W}=\bigoplus_{\Delta\in\mathbb{N}}\mathcal{W}_{\Delta}$, where
$\mathcal{W}_{\Delta}$ is the subspace of chiral field with spin/scaling
dimension $\Delta$. We assume that $\dim\mathcal{W}_{1}=0$ and
$\mathcal{W}_{2}=\mathrm{span}(L)$, where $L$ is the energy-momentum tensor.
The operator product expansions (OPEs) between the parafermions $\psi_{a}$’s
are given by

$\displaystyle\psi_{a}(z)\psi_{b}(w)$
$\displaystyle=\frac{C_{a,b}}{(z-w)^{\gamma(a,b)}}\psi_{a+b}(w)+\cdots,\qquad(a+b\neq
0)$ (18a) $\displaystyle\psi_{a}(z)\psi_{a}^{+}(w)$
$\displaystyle=\frac{1}{(z-w)^{2h_{a}}}\left(1+\frac{2h_{a}}{c}L(w)+\cdots\right)$
(18b)

where $C_{a,b}$’s are the structure constants, and $c$ is the central charge.
The fields in the ellipsis have scaling dimensions that differ from the
leading field by a non-zero integer. We require the CFT to be symmetric under
charge conjugation; i.e., if we replace all of $\psi_{a}$’s with their charge
conjugation $\psi_{a}^{+}$, then no correlation function changes. An
associative solution to this operator product algebra (OPA) is called a
_$\mathbb{Z}_{k}$ parafermionic algebra_. The fields $\psi_{a}$’s are the
parafermions introduced by Zamolodchikov-Fateev in Ref. [52]. For these
parafermions the _(semi-)locality condition_ [33, 3] is given by

$\psi_{a}(z)\psi_{b}(w)(z-w)^{\gamma(a,b)}=\mu_{a,b}\psi_{b}(w)\psi_{a}(z)(w-z)^{\gamma(a,b)}$
(19)

where $0\neq\mu_{a,b}\in\mathbb{C}$ is called the _commutation factor_. For
the reader’s convenience, we have rederived [48] the most general solution for
scaling dimensions consistent with associativity in A.

###  4.2 $\mathbb{Z}_{k}^{(r)}$-Algebras &
$\mathbb{Z}_{k}^{(r)}$-Wavefunctions

In this paper, we will utilize and study only a sub-class of $\mathbb{Z}_{k}$
parafermionic algebras. The associative solutions of interest are those with
the following scaling dimensions:

$h_{a}=\frac{a(k-a)}{2}\gamma,\qquad\gamma=\frac{2r}{k},\qquad(r\in\mathbb{Z}_{+})$
(20)

We call any such associative solution a _$\mathbb{Z}_{k}^{(r)}$ (current)
algebra_. For these current algebras, the commutation factors are trivial: For
any pair $a,b$, we have

$\psi_{a}(z)\psi_{b}(w)(z-w)^{ab\gamma}=\psi_{b}(w)\psi_{a}(z)(w-z)^{ab\gamma}$
(21)

Alternatively, since $D(a,b)=ab\gamma-\gamma(a,b)=S_{a+b}-S_{a}-S_{b}$ is an
even integer for $\mathbb{Z}_{k}^{(r)}$-algebras (See A and Eq. (29)), we may
also write

$\psi_{a}(z)\psi_{b}(w)(z-w)^{\gamma(a,b)}=\psi_{b}(w)\psi_{a}(z)(w-z)^{\gamma(a,b)}$
(22)

Moving on to the correlation functions, corresponding to a
$\mathbb{Z}_{k}^{(r)}$-algebra we construct an infinite sequence of
wavefunctions $\boldsymbol{\Psi}=(\Psi_{1},\Psi_{2},\cdots,\Psi_{n},\cdots)$
as follows:

$\Psi_{n}(z_{1},\cdots,z_{nk})=\frac{1}{\prod_{j=0}^{k-1}C_{1,j}^{n}}\left<\psi(z_{1})\psi(z_{2})\cdots\psi(z_{nk})\right>\prod_{i<j}(z_{i}-z_{j})^{\gamma}$
(23)

Due to the semi-locality condition, the functions $\Psi_{n}$ are symmetric
polynomials. The holomorphic Ward identities further show that $\Psi_{n}$ is
an $(N,N_{\phi})$ uniform states with $N=nk$ and $N_{\phi}=(n-1)r$ (i.e.
uniform state over sphere with $N$ particles and $N_{\phi}$ flux quanta). The
filling fraction $\nu$ and the shift $\mathcal{S}$ are respectively
$\nu=\gamma^{-1}=k/2r$ and $\mathcal{S}=2r$. Finally, note that [15]

$\displaystyle\Psi_{n+1}(w^{\times k},z_{1},\cdots,z_{nk})$
$\displaystyle\equiv\lim_{w_{1}\to w}\lim_{w_{2}\to w}\cdots\lim_{w_{k}\to
w}\Psi^{(k,r)}_{n+1}(w_{1},w_{2},\cdots,w_{k},z_{1},\cdots,z_{nk})$ (24)
$\displaystyle=\prod_{i=1}^{nk}(z_{i}-w)^{2r}\Psi^{(k,r)}_{n}(z_{1},\cdots,z_{nk})$

One says the sequence $\boldsymbol{\Psi}=(\Psi_{n})_{n\geq 1}$ is _$(k,2r)$
-clustering_ (or $k$-clustering of degree $2r$).

Throughout the paper, the wavefunction $\Psi_{n}$ defined in Eq. (23) is
called a _$\mathbb{Z}_{k}^{(r)}$ -wavefunction (WF)_ of size $n$. The
$\mathbb{Z}_{k}^{(r)}$-WFs of size $n=1,2$ are called the _trivial_ and
_principal_ $\mathbb{Z}_{k}^{(r)}$-wavefunctions respectively. Moreover, a
reference to symbol $\Psi$ (without any $n$ index) will automatically mean
$n=2$; i.e. $\Psi$ is the principal $\mathbb{Z}_{k}^{(r)}$-wavefunction ($2k$
variables).

##  5 $\mathbb{Z}_{k}^{(r)}$-Wavefunctions as Model FQH Ground States

The pseudopotential formalism is easier to work with in the plane/disk
geometry. Thus, let us begin with some comments on FQH states in plane
geometry. The states with $N$ ‘bosonic electrons’ in the LLL are described by

$\Phi(z_{1},\cdots,z_{N})=P(z_{1},\cdots,z_{N})\exp\left[-\sum_{i}|z_{i}|^{2}/4\ell_{B}^{2}\right]$
(25)

where $P$ is a symmetric polynomial and $B(2\pi\ell_{B}^{2})=h/e=$ magnetic
flux quantum. Similar to the sphere, we can absorb the geometric measure (the
Gaussian) into the definition of the inner product and consider $P$ itself as
the wavefunction. If $P$ is an $(N,N_{\phi})$ uniform state over a sphere, and
if we think of the norm of $\Phi$ (in plane geometry) as a droplet, the
wetting area of $|\Phi|^{2}$ would be roughly $N_{\phi}(2\pi\ell_{B}^{2})$. In
other words, the lower $N_{\phi}$ is, the higher the density.

In plane geometry, projected to the lowest Landau level, the effective
Hamiltonians of fractional quantum Hall systems are pure multi-electron
interaction. Using $b$ as the number of electrons involved in the interaction,
in plane geometry, the effective interaction $V(z_{1},\cdots,z_{b})$ is both
translational and rotational invariant. To get from these interactions to
pseudopotentials, let us introduce some terminology. Some of the core players
in this paper are the so-called _semi-invariants_. A $(b,m)$ semi-invariant is
a polynomial with $b$ variables, that is symmetric, translational invariant,
and homogeneous of degree $m$. We denote the linear space of $(b,m)$ semi-
invariants by $\mathcal{T}_{b}^{m}$. If $\left|m,q\right>$ is an orthonormal
basis for $\mathcal{T}_{b}^{m}$ ($1\leq q\leq\dim\mathcal{T}_{b}^{m}$), the
$b$-body interaction Hamiltonian is generally of the form [43]:

$H=\sum_{i_{1}<\cdots<i_{b}}\>\sum_{m\geq
0}\>\sum_{q,q^{\prime}=1}^{\dim\mathcal{T}_{b}^{m}}\left|m,q;z_{i_{1}},\cdots,z_{i_{b}}\right>V^{[m]}_{q,q^{\prime}}\left<m,q^{\prime};z_{i_{1}},\cdots,z_{i_{b}}\right|$
(26)

The matrices $V^{[m]}$ are called the _pseudopotentials_ and should be
positive semi-definite (so that the ground state has zero energy). The
_interaction number_ $b$, together with the pseudopotentials $V^{[m]}$ are the
data of the Hamiltonian. A side-effect of the plane geometry is that it is
infinite, and thus $N_{\phi}$ (the max degree of a variable) for a zero-energy
state can be arbitrarily large. To circumvent this issue and also simulate the
more realistic geometry that is of a disk, it is customary to call the
_densest_ (smallest $N_{\phi}$) zero-energy state of $H$ as _the_ ground
state. We want to emphasize that an actual FQH Hamiltonian has to have a
_unique_ ground state in disk/sphere topology.

At this stage, the best option to study FQH states from the Hamiltonian point
of view is to design _models_. Thus, one must make a particular choice for the
interaction number $b$ and the pseudopotentials $V^{[m]}$. A realistic quantum
Hall Hamiltonian is gapped (incompressibility), has a unique ground state, and
ground state would be uniform if put on a sphere. Thus, if a model
pseudopotential is reasonable in the above sense, its densest zero-energy
state is called a _model FQH ground state_. The simplest of model Hamiltonians
is the _Haldane Hamiltonian_ [23]:

$\mathscr{H}_{2}^{2r-1}=\sum_{m=0}^{2r-1}V_{m}\sum_{i<j}\mathcal{P}_{2}^{m}(z_{i},z_{j}),\qquad(V_{m}>0)$
(27)

Here and onwards, for a general $b$,
$\mathcal{P}_{b}^{m}(z_{I})\equiv\sum_{q}\left|m,q;z_{I}\right>\left<m,q;z_{I}\right|$
is the projection operator to the sector where a $b$-cluster of electrons with
coordinates $z_{I}\equiv z_{i_{1}},\cdots,z_{i_{b}}$ has a relative angular
momentum $m$. The model ground state of $\mathscr{H}_{2}^{2r-1}$ is the
$2r$-Laughlin state. The natural generalization [41, 44] of Haldane’s
construction would be

$\mathscr{H}_{k+1}^{r-1}=\sum_{m=0}^{r-1}V_{m}\sum_{i_{1}<\cdots<i_{k+1}}\mathcal{P}_{k+1}^{m}(z_{I}),\qquad(V_{m}>0)$
(28)

We call these the _projection Hamiltonians_. For specific values of $k,r$, the
model Hamiltonian $\mathscr{H}_{k+1}^{r-1}$ works exceptionally well and has a
unique ground state. Other than Laughlin, three other examples are: Read-
Rezayi ($\nu=k/r=k/2$) [38], Gaffnian ($\nu=k/r=2/3$) [41], and Haffnian
($\nu=k/r=2/4$) [20, 43]. However, for the majority of $(k,r)$ values, we have
$\dim\mathcal{T}_{k+1}^{r-1}>1$ and thus, these Hamiltonians often do not have
a unique densest zero-energy state. This section aims to construct a family of
model Hamiltonians that have a better chance of realizing
$\mathbb{Z}_{k}^{(r)}$-wavefunctions as a _unique_ ground state (the gap
should be studied separately). The construction is based on a novel local
property called _separability_.

###  5.1 Separability

Let us first establish the required terminology. Throughout the discussion,
the quantities $k,r>0$ are two fixed integers. The notation $a=pk+q$ is
reserved, where $p\geq 0$ and $0<q\leq k$. The functions $\mathcal{N}(n)=kn$
and $\mathcal{N}_{\phi}(n)=2r(n-1)$ signify the number of particles and number
of flux quanta at “size $n$”. The multi-index $I$ stands for the set
$\\{i_{1},\cdots,i_{a}\\}$ (assuming $i_{1}<\cdots<i_{a}$). The complementary
multi-index $\overline{I}$ is such that $I\cap\overline{I}=\emptyset$ and
$I\cup\overline{I}=\\{1,2,\cdots,\mathcal{N}(n)\\}$. Given a multi-index
$J={j_{1},\cdots,j_{l}}$, we define $z_{J}={z_{j_{1}},\cdots,z_{j_{l}}}$ and
$\widehat{z}_{J}=(z_{j_{1}}+\cdots+z_{j_{l}})/l$ (i.e. the center-of-mass).
The pattern of zeros $S=(S_{1},S_{2},\cdots,S_{a},\cdots)$ [48] is understood
as the infinite integral sequence:

$S_{a}=2r\left[\frac{p(p-1)}{2}k+pq\right]$ (29)

Finally, $\boldsymbol{\Psi}$ stands for a sequence of polynomials
$\boldsymbol{\Psi}=(\Psi_{n})_{n\geq 1}$. In the current discussion, The
polynomial do not have to be $\mathbb{Z}_{k}^{(r)}$-wavefunctions. However,
the sequence $\boldsymbol{\Psi}$ needs to satisfy the following criteria:

  1. 1.

Uniformity: $\Psi_{n}$ is an $(\mathcal{N}(n),\mathcal{N}_{\phi}(n))$ uniform
state.

  2. 2.

$(k,2r)$ Clustering: We have $\Psi_{1}=1$ and $\Psi_{n+1}(w^{\times
k},z_{1},\cdots,z_{nk})=\prod_{i=1}^{nk}(w-z_{i})^{2r}\Psi_{n}(z_{1},\cdots,z_{nk})$.

  3. 3.

Squeezablity: For all $a$ and $m<S_{a}$ we have
$\mathcal{P}_{a}^{m}\Psi_{n}=0$. Furthermore,
$\mathcal{P}_{a}^{S_{a}}\Psi_{n}\neq 0$.

Ignoring the gap issue, we may think of $\Psi_{n}$ as the model ground state
of a hypothetical model Hamiltonian in $\mathcal{N}(n)$ particles. The third
condition states that the minimal relative angular momentum that any cluster
of $a$ electrons can carry in $\boldsymbol{\Psi}$ equals $S_{a}$.

On top of the previous properties, we add the following: We say
$\boldsymbol{\Psi}$ is _$a$ -separable_ if there exists a polynomial
$\chi^{(a)}\in\mathcal{T}_{a}^{S_{a}}$, independent of $n$ (for all $n>p$),
such that

$\mathcal{P}_{a}^{S_{a}}(z_{I})\Psi_{n}=\chi^{(a)}(z_{1},\cdots,z_{a})\prod_{i\in\overline{I}}(z_{i}-\widehat{z}_{I})^{2pr}\Psi_{n-p}(\widehat{z}_{I}^{\times
q},z_{\overline{I}})$ (30)

If $\chi^{(a)}$ exists, we call it the _$a$ -factor_ of $\Psi_{n}$. The
$(k+1)$-factor is called the _minimal polynomial_ of $\boldsymbol{\Psi}$ and
is simply denoted as $\chi$. We say $\boldsymbol{\Psi}$ is _fully separable_
if it is $a$-separable for all $a$. As we will show in §5.2, all
$\mathbb{Z}_{k}^{(r)}$-wavefunctions are fully separable.

The $a$-factor is an element of $\mathcal{T}_{a}^{S_{a}}$; i.e. a semi-
invariant. However, not every semi-invariant can be a factor. Note that if
$\boldsymbol{\Psi}$ is $a$-separable, then we can find its factor $\chi^{(a)}$
from $\Psi_{p+1}$ via

$\chi^{(a)}(z_{1},\cdots,z_{a})=\lim_{w\to\infty}w^{-2\mathcal{J}_{a}}\Psi_{p+1}(w^{\times
k-q},z_{1},\cdots,z_{a})$ (31)

where $2\mathcal{J}_{a}=a\mathcal{N}_{\phi}(p+1)-2S_{a}$. Hence, due to the
clustering property, the $a$-factors need also to satisfy

$\chi^{(a)}(w_{1}^{\times k},w_{2}^{\times k},\cdots,w_{p}^{\times
k},0^{\times q})=\prod_{1\leq i<j\leq
p}(w_{i}-w_{j})^{2kr}\prod_{i=1}^{p}w_{i}^{2qr}$ (32)

We denote by $\mathcal{T}^{\star}_{a}$ the subspace of
$\mathcal{T}_{a}^{S_{a}}$ satisfying (32). Now, corresponding to any
$\omega\in\mathcal{T}_{a}^{\star}$ we define a model Hamiltonian

$H_{\omega}=\mathscr{H}_{a}^{S_{a}-1}+V^{\prime}\sum_{I}\left(1-\frac{\left|\omega;z_{I}\right>\left<\omega;z_{I}\right|}{\left<\omega\vphantom{\omega}\right|\left.\omega\vphantom{\omega}\right>}\right)\mathcal{P}_{a}^{S_{a}},\qquad(V^{\prime}>0)$
(33)

If $\boldsymbol{\Psi}$ is a uniform $(k,2r)$-clustering squeezable
$a$-separable sequence with $a$-factor $\omega$, then it is a densest zero-
energy state of the Hamiltonian $H_{\omega}$ [35]. Furthermore, if
$\boldsymbol{\Psi}$ is not $a$-separable, or if it is $a$-separable but with
an $a$-factor different than $\omega$, then it would not be an eigenstate of
$H_{\omega}$. As it is clear from the definition, the Hamiltonian $H_{\omega}$
is a modification of the projection Hamiltonian $\mathscr{H}_{a}^{S_{a}-1}$.
The two cases where $a=k+1$ and $a=2k$ are of particular interest to us.
Suppose $\boldsymbol{\Psi}$ is both $(k+1)$ and $2k$-separable, with minimal
polynomial $\chi$. The $2k$-factor of $\boldsymbol{\Psi}$ is nothing but its
principal wavefunction $\Psi_{2}$. Referring to (31), it is clear that $\chi$
is fixed by $\Psi_{2}$. In other words, while $\boldsymbol{\Psi}$ is a ground
state for both $H_{\chi}$ and $H_{\Psi_{2}}$, naively, it is reasonable to
think $H_{\Psi_{2}}$ is more restrictive than $H_{\chi}$.

The above observation leads to the following question: Assuming we are only
interested in fully separable $(k,2r)$-clustering squeezable FQH ground
states, should we abandon the long-standing belief that FQH Hamiltonians of
$k$-clustering states are $(k+1)$-body interactions? After all, the $2k$-body
Hamiltonian $H_{\Psi_{2}}$ looks more promising to get a _unique_ ground
state. Alternatively, perhaps, despite the naive expectations, in actuality,
the two Hamiltonians are equivalent! We cannot give definitive answers to
these questions in this paper. However, by restricting our attention to
$\mathbb{Z}_{k}^{(r)}$-WFs (i.e., the prime example of fully separable
states), we would investigate whether or not the minimal polynomial $\chi$ can
fix the principal wavefunction $\Psi_{2}$. An affirmative answer is evidence
of the equivalence of $H_{\chi}$ and $H_{\Psi_{2}}$. This is the subject of
sections 7 and 8. The evidence gets stronger if $\chi$ fixes the entire
$\mathbb{Z}_{k}^{(r)}$-algebra. The latter requires obtaining the structure
constants, central charge, and chiral weights as data encoded inside the
minimal polynomial. We pursue this study in section 6.

###  5.2 Multi-Parafermion OPEs & Full Separability of
$\mathbb{Z}_{k}^{(r)}$-Wavefunctions

To complete the discussion, in this section, we show that
$\mathbb{Z}_{k}^{(r)}$-WFs are indeed fully separable. Define $g_{0}=1$ and
$g_{a}=\prod_{j=0}^{a-1}C_{1,j}$ for all $1\leq a\leq k$. The key quantity
that we need to compute is the $a$-parafermion OPE/field
$\Phi_{a}(z_{1},\cdots,z_{a})$ defined as

$\Phi_{a}(z_{1},\cdots,z_{a}):=\frac{1}{g_{k}^{p}g_{q}}\psi(z_{1})\psi(z_{2})\cdots\psi(z_{a})\prod_{1\leq
i<j\leq a}(z_{i}-z_{j})^{\gamma}$ (34)

Ordinarily, one computes such an OPE by repeatedly utilizing the two-point
OPEs $\psi_{a}(z)\psi_{b}(w)$ (after the associativity of a solution is
confirmed). This method is very inefficient and computationally very
expensive. Here, we will provide an alternative. Due to the simple fusion
rules $\psi_{a}\star\psi_{b}=\psi_{a+b}$, it is immediate that $\Phi_{a}$ lies
in the $\mathcal{W}$-family of $\psi_{a}$. In fact, to the lowest order, we
expect

$\Phi_{a}(z_{1},\cdots,z_{a})=\chi^{(a)}(z_{1},\cdots,z_{a})\psi_{a}\left(\widehat{z}_{a}\right)+\cdots$
(35)

where $\widehat{z}_{a}=(z_{1}+\cdots+z_{a})/a$ is the center of mass and
$\chi^{(a)}(z_{1},\cdots,z_{a})$ is (at this stage) just some function. We
claim that:

######  Theorem 1.

The function $\chi^{(a)}(z_{1},\cdots,z_{a})$ belongs to
$\mathcal{T}_{a}^{S_{a}}$.

Before proving the theorem, note that in Eq. (35), the fields appearing in
ellipsis are $\mathcal{W}$-descendants of $\psi_{a}$. If $\phi$ is one such
descendant with scaling dimension $h_{a}+n$, then it will be accompanied by a
translation invariant homogeneous symmetric polynomial of degree
$\deg\chi^{(a)}+n$ in the OPE. We will only need the leading term of this OPE
in the main body of the paper. We present an ansatz for higher order terms in
C.

######  Proof of Theorem 1 1.

Note that

$\lim_{w\to\infty}w^{-2pr(k-q)}\Psi_{p+1}(w^{\times
k-q},z_{1},\cdots,z_{a})=\left<\psi_{q}^{+}(\infty)\Phi_{a}(z_{1},\cdots,z_{a})\right>=\chi^{(a)}(z_{1},\cdots,z_{a})$
(36)

where we have used the $a$-point OPE ansatz. [Throughout, given a field $\phi$
of scaling dimension $\Delta$, the notation $\phi(\infty)$ will mean
$\lim_{w\to\infty}w^{2\Delta}\phi(w)$]. The left-most quantity in the equality
is a homogeneous translational invariant symmetric polynomial in
$z_{1},\cdots,z_{a}$. We refer the reader to A for a proof of
$\deg\chi^{(a)}=S_{a}$ .

Multi-parafermion OPEs allow us to compute the wavefunction’s leading
“relative behavior” $\Psi_{n}$. Note that (with
$\mathbf{w}=w_{1},\cdots,w_{a}$, $\mathbf{z}=z_{1},\cdots,z_{nk-a}$ and
$\widehat{w}=(w_{1}+\cdots+w_{a})/a$):

$\displaystyle\Psi_{n}(\mathbf{w},\mathbf{z})$
$\displaystyle=\frac{1}{g^{n-(p+1)}_{k}g_{k-q}}\left<\Phi_{a}(\mathbf{w})\psi(z_{1})\cdots\psi(z_{nk-a})\right>\prod_{i=1}^{a}\prod_{j=1}^{nk-a}(w_{i}-z_{j})^{\gamma}\prod_{i<j}(z_{i}-z_{j})^{\gamma}$
$\displaystyle=\frac{\chi^{(a)}(\mathbf{w})}{g_{n-(p+1)}^{k}g_{k-q}}\left<\psi_{q}(\widehat{w})\psi(z_{1})\cdots\psi(z_{nk-a})\right>\prod_{j=1}^{nk-a}(\widehat{w}-z_{j})^{a\gamma}\prod_{i<j}(z_{i}-z_{j})^{\gamma}+\cdots$
$\displaystyle=\chi^{(a)}(\mathbf{w})\prod_{i=1}^{nk-a}(z_{i}-\widehat{w})^{2pr}\Psi_{n-p}(\widehat{w}^{\times
q},\mathbf{z})+\cdots$

Here, the ellipsis consists of terms with relative angular momentum higher
than $S_{a}$ (in $\mathbf{w}$-variables). In other words, using the notation
of §5.1, $\mathcal{P}_{a}^{m}(z_{I})\Psi_{n}=0$ for all $m<S_{a}$, and

$\mathcal{P}_{a}^{S_{a}}(z_{I})\Psi_{n}=\chi^{(a)}(z_{I})\prod_{i\in\overline{I}}(z_{i}-\widehat{z}_{I})^{2pr}\Psi_{n-p}(\widehat{z}_{I}^{\times
q},z_{\bar{I}})$ (37)

This concludes the proof that $\mathbb{Z}_{k}^{(r)}$-wavefunctions are fully
separable and squeezable.

##  6 Polynomialities

As discussed in the motivation section, our main hypothesis is that
$\chi\xrightarrow{\mathrm{fix}}\Psi_{2}\xrightarrow{\mathrm{fix}}\mathbb{Z}_{k}^{(r)}\text{algebra}\xrightarrow{\mathrm{fix}}\boldsymbol{\Psi}$.
The validity of the hypothesis, in turn, makes the $H_{\chi}$ Hamiltonians
viable and natural as models. In this section, we will explore if and how the
principal (or its specialization $\chi$) can fix various CFT data; e.g.
structure constants $C_{a,b}$, central charge $c$, chiral weights $h_{a}^{W}$,
etc.

###  6.1 A General Observation

As usual, let us begin by introducing a bit of terminology. Fixing some $0\leq
b<k$, suppose $\phi(z)$ be a quasi-primary $\mathcal{W}$-descendant of
$\psi_{b}^{+}(z)$. We use the notation $\beta(\phi)=b$ and write the scaling
dimension of $\phi$ in the form $h_{\phi}=h_{b}+l(\phi)$. We call $l(\phi)$
the _level_ of $\phi$. In the study of $\mathbb{Z}_{k}^{(r)}$-wavefunctions,
especially the principal ones, correlation functions of the following form are
a common sight:

$T_{\phi}(z_{1},\cdots,z_{k+\beta(\phi)})=\frac{1}{g_{k}g_{\beta(\phi)}}\left<\phi(\infty)\psi(z_{1})\cdots\psi(z_{k+\beta(\phi)})\right>\prod_{i<j}(z_{i}-z_{j})^{\gamma}$
(38)

We call $T_{\phi}$ the principal semi-invariant corresponding to $\phi$.
Various data of the $\mathbb{Z}_{k}^{(r)}$-algebra is encoded inside of semi-
invariants of this form. We can extract these data by performing
specializations. For example, a beneficial specialization would be:

$\tau_{\phi}(a)=T_{\phi}(1^{\times a+\beta(\phi)},0^{\times
k-a})=\frac{g_{a+\beta(\phi)}g_{k-a}}{g_{k}g_{\beta(\phi)}}\left<\phi(\infty)\psi_{a+\beta(\phi)}(1)\psi^{+}_{a}(0)\right>$
(39)

Our first task in this section is to prove that $\tau_{\phi}(a)$ is a
polynomial in the (integer) cluster parameter $a$. We use the umbrella term
_Polynomiality_ when a certain $\mathbb{Z}_{k}^{(r)}$ quantity parametrized by
$a$ depends on it polynomially. The polynomialities coming from the special
cases $\phi=\psi_{b}^{+}$ (for $b\neq 0$) and $\phi=W\in\mathcal{W}$ are
studied in some detail in this section.

######  Theorem 2.

$\tau_{\phi}(a)$ is a polynomial of degree $2r\beta(\phi)+l(\phi)$ with the
following properties:

  1. 1.

It enjoys a symmetry
$\tau_{\phi}(k-a-\beta(\phi))=(-1)^{l(\phi)}\tau_{\phi}(a)$.

  2. 2.

It is divisible by $(a+\beta(\phi))(k-a)$.

  3. 3.

$\tau_{\phi}(0)=\delta_{l(\phi),0}$.

###### Proof.

For simplicity, let $b=\beta(\phi)$ and $l=l(\phi)$. Beginning with the last
condition, note that $\tau_{\phi}(0)=\left<\phi(\infty)\psi_{b}(1)\right>$
which is only non-zero if $\phi=\psi_{b}^{+}$, i.e. $l=0$. Regarding the
degree, the proof that $\deg T_{\phi}=2br+l$ can be found in C. We prove the
rest by constructing a convenient basis for $\mathcal{T}^{2br+l}_{k+b}$.
Define

$P_{n}(z_{1},\cdots,z_{k+b})=\sum_{i=1}^{k+1}\left(z_{i}-\widehat{z}\right)^{n}\in\mathcal{T}_{k+b}^{n}$
(40)

where $\widehat{z}=(z_{1},\cdots,z_{k+b})/(k+b)$ is the center-of-mass. Note
that $P_{1}=0$. Let $\pi^{*}(2br+l,k+b)$ be the set of partitions
$\lambda=(\lambda_{1},\lambda_{2},\cdots)$ that satisfy
$|\lambda|=\lambda_{1}+\cdots+\lambda_{\ell}=2br+s$ and $2\leq\lambda_{i}\leq
k+b$ for all $i$. For each $\lambda\in\pi^{*}(2br+l,k+b)$, define
$P_{\lambda}=\prod_{j}P_{\lambda_{j}}$. Then $P_{\lambda}$ is a basis for
$\mathcal{T}_{k+b}^{2br+l}$. Now note that

$\tau_{n}(a)\equiv P_{n}(1^{\times a+b},0^{\times
k-a})=\frac{(a+b)(k-a)^{n}+(-1)^{n}(k-a)(a+b)^{n}}{(k-b)^{n}}$

which is a polynomial in $a$ of degree $n$ (the coefficient of $a^{n+1}$ is
zero). Since $n>1$, $(a+b)(k-a)$ divides $\tau_{n}(a)$. Moreover,
$\tau_{n}(k-a-b)=(-1)^{n}\tau(a)$. The assertion now follows from the fact
that $T_{\phi}$ is a linear combination of $P_{\lambda}$’s, and
$P_{\lambda}$’s are themselves a product of $P_{n}$’s. ∎

Note that the polynomiality of $\tau_{\phi}(a)$ is purely a consequence of
$T_{\phi}$ being a semi-invariant.

###  6.2 Structure Constants

As a first example of polynomialities, we will shortly consider
$\phi=\psi^{+}$ (i.e. $\beta(\phi)=1$ and $l(\phi)=0$). We will use this
polynomiality to determine the structure constants $C_{a,b}$. It would be
helpful if we first proved a few general identities concerning $C_{a,b}$’s.
Firstly, from charge conjugation symmetry it is immediate that
$C_{a,b}=C_{k-a,k-b}$. The identity $C_{a,b}=C_{b,a}$ is a straightforward
consequence of the semi-locality relation (22). The next identity is obtained
by studying the following 3-point correlation function:

$C_{a+1,k-a}=\left<\psi^{+}(\infty)\psi_{a+1}(1)\psi^{+}_{a}(0)\right>$

However, utilizing the semi-locality relation (22), we may alternatively write

$\displaystyle C_{a+1,k-a}$
$\displaystyle=\left<\psi^{+}(u)\psi_{a+1}(v)\psi_{a}^{+}(w)\right>(u-v)^{\gamma(k-1,a+1)}(u-w)^{\gamma(1,a)}(v-w)^{\gamma(a+1,k-1)}$
$\displaystyle=\left<\psi^{+}(u)\psi_{a}^{+}(w)\psi_{a+1}(v)\right>(u-v)^{\gamma(k-1,a+1)}(u-w)^{\gamma(1,a)}(w-v)^{\gamma(a+1,k-1)}$
$\displaystyle=C_{k-1,k-a}$

In short, $C_{a+1,k-a}=C_{1,a}$. Putting these identities to use, we now show
that $\tau_{\psi^{+}}(a)=C_{1,a}^{2}$. Note that $T_{\psi^{+}}$, the principal
semi-invariant of $\psi^{+}$, is the minimal polynomial $\chi$. Therefore, we
have (Recall that $g_{0}=1$ and $g_{a}=\prod_{j=0}^{a-1}C_{1,j}$)

$\tau_{\psi^{+}}(a)=\chi(1^{\times a+1},0^{\times
k-a})=\frac{g_{a+1}g_{k-a}}{g_{k}}\left<\psi^{+}(\infty)\psi_{a+1}(1)\psi^{+}_{a}(0)\right>=\frac{g_{a}g_{k-a}}{g_{k}}C_{1,a}C_{a+1,k-a}=\frac{g_{a}g_{k-a}}{g_{k}}C_{1,a}^{2}$
(41)

We will shortly prove that $g_{a}g_{k-a}=g_{k}$. Therefore, as a corollary of
Theorem 2, $C_{1,a}^{2}$ is a polynomial in $a$ of degree $2r$ which is
divisible by $(a+1)(k-a)$, has the symmetry $C_{1,k-1-a}^{2}=C_{1,a}^{2}$, and
$C_{1,0}^{2}=1$. Consequently, $C_{1,a}^{2}$ has roots at $-1,k$, and at some
points $-t_{p},t_{p}+k-1$ for $1\leq p\leq r-1$. In other words,

$C_{1,a}^{2}=\frac{(a+1)(k-a)}{k}\prod_{p=1}^{r-1}\frac{(t_{p}+a)(t_{p}+k-1-a)}{t_{p}(t_{p}+k-1)}$
(42)

The parameters $t_{1},t_{2},\cdots,t_{r-1}$ are part of the classification
data of the $\mathbb{Z}_{k}^{(r)}$-algebra. When $k+1\geq 2r$, these
parameters are generically distinct. When $k+1<2r$ some of these roots will
coincide.

To find the rest of the structure constants and obtain even more identities
between the structure constant, we will utilize the trivial wavefunction:

$\Psi_{1}(z_{1},\cdots,z_{k})=\frac{1}{g_{k}}\left<\psi(z_{1})\cdots\psi(z_{k})\right>\prod_{i<j}(z_{i}-z_{j})^{\gamma}$
(43)

Using the OPEs, by performing a sequence of limits, it is straightforward to
show that

$\Psi_{1}(x^{\times a},y^{\times
k-a})=\frac{g_{a}g_{k-a}}{g_{k}}\left<\psi_{a}(x)\psi_{a}^{+}(y)\right>(x-y)^{a(k-a)\gamma}=\frac{g_{a}g_{k-a}}{g_{k}}$
(44)

and

$\displaystyle\Psi_{1}(x^{\times k-a-b},y^{\times a},z^{\times b})$
$\displaystyle=\frac{g_{k-a-b}g_{a}g_{b}}{g_{k}}\left<\psi^{+}_{a+b}(x)\psi_{a}(y)\psi_{b}(z)\right>(x-y)^{(k-a-b)a\gamma}(x-z)^{(k-a-b)b\gamma}(y-z)^{ab\gamma}$
(45)
$\displaystyle=\frac{g_{k-a-b}g_{a+b}}{g_{k}}\frac{g_{a}g_{b}}{g_{a+b}}C_{a,b}$

At the same time, the trivial wavefunction $\Psi_{1}$ is a $(k,0)$ uniform
state; hence it is a constant! Using the $a=0$ special case of Eq. (44) we
find that $\Psi_{1}=1$. Consequently, the identity (44) turns into
$g_{a}g_{k-a}=g_{k}$ for all $a$, and the identity (45) morphs into

$C_{a,b}=\frac{g_{a+b}}{g_{a}g_{b}}=\prod_{j=1}^{b-1}\frac{C_{1,a+j}}{C_{1,j}}$
(46)

Despite $\Psi_{1}$ being trivial as a polynomial, it results in a highly non-
trivial constraint between the structure constants. Combining Eqs. (42) and
the constraint (46), we find (Notation: $\Gamma$ is gamma function)

$C_{a,b}^{2}=\frac{(a+b)!(k-a)!(k-b)!}{k!a!b!(k-a-b)!}\prod_{p=1}^{r-1}\frac{\Gamma(t_{p}+a+b)\Gamma(t_{m}+k-a)\Gamma(t_{m}+k-b)\Gamma(t_{m})}{\Gamma(t_{m}+k-a-b)\Gamma(t_{m}+a)\Gamma(t_{m}+b)\Gamma(t_{m}+k)}$
(47)

Note that this is a straightforward generalization of the structure constants
of $\mathbb{Z}_{k}^{(1)}$ and $\mathbb{Z}_{k}^{(2)}$ algebras first reported
in Ref. [52]. A more symmetric formula can be provided for
$C^{2}_{a,b,c}=\left<\psi_{a}(\infty)\psi_{b}(1)\psi_{c}(0)\right>$ with
$a+b+c=k$:

$C_{a,b,c}^{2}=\frac{(a+b)!(a+c)!(b+c)!}{a!b!c!(a+b+c)!}\prod_{p=1}^{r-1}\frac{(t_{p})_{a+b}(t_{p})_{a+c}(t_{p})_{b+c}}{(t_{p})_{a}(t_{p})_{b}(t_{p})_{c}(t_{p})_{a+b+c}}$
(48)

where $(x)_{n}=x(x+1)\cdots(x+n-1)$ is the Pochhammer symbol. In what follows,
we will also use the notation $(x)_{n}^{-}=x(x-1)\cdots(x-n+1)$ for the
falling factorial.

###  6.3 Polynomiality of Chiral Weights

Let $W$ be a quasi-primary field of scaling dimension $\Delta$ in the chiral
algebra $\mathcal{W}$ (See B for a review). The fact that $\psi_{a}$ is a
$\mathcal{W}$-primary field is reflected in the OPE of $W(z)\psi_{a}(w)$ as:

$W(z)\psi_{a}(w)=\frac{h_{a}^{W}\psi_{a}(w)}{(z-w)^{\Delta}}+\frac{(W_{-1}\psi_{a})(w)}{(z-w)^{\Delta-1}}+\cdots$
(49)

The quantity $h_{a}^{W}$ is called the _$W$ -weight_ of $\psi_{a}$. In the
polynomiality formalism, if we choose $\phi=W$ (with $\beta(\phi)=0$ and
$l(\phi)=\Delta$), the above OPE can be used to determine the specialization
$\tau_{W}(a)$:

$\tau_{W}(a)=T_{W}(1^{\times a},0^{\times
k-a})=\left<W(\infty)\psi_{a}(1)\psi^{+}_{a}(0)\right>=h_{a}^{W}$ (50)

This shows that the $W$-weight $h_{a}^{W}$ is a polynomial of degree $\Delta$
in $a$ with the symmetry $h_{k-a}^{W}=(-1)^{\Delta}h_{a}^{W}$ and
$h_{0}^{W}=0$. For example, when $\Delta=2$ (which means $W=L$) we rediscover
$h_{a}=a(k-a)\gamma/2$. For a spin-$3$ field $W_{3}$ (which is necessarily a
Virasoro primary), we have

$w_{a}\equiv h_{a}^{W_{3}}=\frac{a(k-a)(k-2a)}{3}\eta$ (51)

where $\eta$ plays a similar role for $W_{3}$ that $\gamma$ plays for $L$.

######  Remark 1.

It can be shown that the field corresponding to
$\left|\partial;W,a\right>\propto(2W_{-1}L_{0}-\Delta
L_{-1}W_{0})\left|a\right>$ decouples from the $\mathbb{Z}_{k}^{(r)}$-algebra
(see D). In other words, for $a+b<k$

$\psi_{a}(z)\psi_{b}(w)=C_{a,b}\left(\psi_{a+b}(w)+\frac{a}{a+b}(z-w)\>\partial\psi_{a+b}(w)\right)+O(z-w)^{2}$
(52)

Put differently, the level one descendant $(W_{-1}\psi_{a})$ is essentially
$(\Delta h_{a}^{W}/2h_{a})\partial\psi_{a}$.

###  6.4 Conformal Blocks $\xi_{a}(x)$, Central Charge and Constraints on
Chiral Weights

The specializations $\tau_{\phi}(a)$ are not the only important quantities one
can compute from the semi-invariants $T_{\phi}$. As an example, consider the
following specialization of $\chi$:

$\displaystyle\xi_{a}(x)\equiv$ $\displaystyle\chi(1,x^{\times a},0^{\times
k-a})=\lim_{w\to\infty}w^{-2r(k-1)}\Psi_{2}(w^{\times k-1},1,x^{\times
a},0^{\times k-a})$ (53) $\displaystyle=$
$\displaystyle\left<\psi^{+}(\infty)\psi(1)\psi_{a}(x)\psi_{a}^{+}(0)\right>(1-x)^{a\gamma}x^{2h_{a}}$

This relation recognizes $\xi_{a}(x)$, a specialization of the minimal
polynomial, as a conformal block. Though this identity is not easy to work
with, it nonetheless contains information about the CFT/wavefunction.

We start by writing down the OPE of $\psi_{a}\psi_{a}^{+}$. Let $\\{W^{i}\\}$
be a basis for the field space $\mathcal{W}$, and $\Delta_{i}$ the scaling
dimension of $W^{i}$. The following formula is a special case of the formalism
introduced in Ref. [8] (also see [9]):

$\psi_{a}(x)\psi_{a}^{+}(y)(x-y)^{2h_{a}}=\sum_{i}\widehat{h}^{i}_{a}\sum_{n\geq
0}\frac{(x-y)^{\Delta_{i}+n}}{n!}\frac{(\Delta_{i})_{n}}{(2\Delta_{i})_{n}}\partial^{n}W^{i}(w)$
(54)

Here $\widehat{h}_{a}^{i}$ is defined via the relation
$h_{a}^{i}=\sum_{j}d_{ij}\widehat{h}^{j}_{a}\delta_{\Delta_{i},\Delta_{j}}$
(with $\langle
W^{i}(\infty)W^{j}(0)\rangle=d_{ij}\delta_{\Delta_{i},\Delta_{j}}$). Using
this OPE, the block $\xi_{a}(x)$ can be computed:

$\xi_{a}(x)=\left<\psi^{+}(\infty)\psi(1)\psi_{a}(x)\psi_{a}^{+}(0)\right>(1-x)^{a\gamma}x^{2h_{a}}=(1-x)^{a\gamma}\left(1+\sum_{s\geq
2}\Xi_{a}^{\Delta}(-x)^{\Delta}{}_{2}F_{1}(\Delta,\Delta;2\Delta;x)\right)$
(55)

where
$\Xi^{\Delta}_{a}:=\sum_{i}\widehat{h}_{a}^{i}h^{i}\delta_{\Delta_{i},\Delta}$,
and ${}_{2}F_{1}(a,b;c;x)$ is the Gauss hypergeometric function. The
polynomiality of chiral weights results in polynomiality of
$\Xi^{\Delta}_{a}$. To the first few orders, we can expand

$\xi_{a}(x)=1-a\gamma
x+\left\\{\frac{(a\gamma)_{2}^{-}}{2}+\Xi_{a}^{2}\right\\}x^{2}-\left\\{\frac{(a\gamma)_{3}^{-}}{6}+(a\gamma-1)\Xi_{a}^{2}+\Xi_{a}^{3}\right\\}x^{3}+\cdots$
(56)

Assuming we are given the minimal polynomial $\chi$ beforehand, we can compute
the coefficient of $x^{2},x^{3},x^{4},\cdots$ in $\xi_{a}(x)$ from $\chi$.
Knowing these coefficients allows us to inductively fix $\Xi_{a}^{\Delta}$ for
all $\Delta$. Given our current knowledge about the internal structure of
$\mathbb{Z}_{k}^{(r)}$, it is unclear if the minimal polynomial is capable of
fixing all of the chiral weights. Nonetheless, through the above discussion,
$\chi$ enforces constraints on the chiral weights of each scaling dimension
$\Delta$ via $\Xi_{a}^{\Delta}$.

The above considerations can provide us with the central charge. Since
$d_{LL}=c/2$, we find that $\widehat{h}_{a}^{L}=2h_{a}/c$ and
$\Xi_{a}^{2}=2hh_{a}/c$. This, in turn, yields:

$\xi_{1}(x)=\chi(1,x,0^{\times k-a})=1-\gamma
x+\left(\frac{\gamma(\gamma-1)}{2}+\frac{2h^{2}}{c}\right)x^{2}+O(x^{3})$

Therefore, one can read off the central charge $c$ from the coefficient of
$x^{2}$ in the specialization $\chi(1,x,0^{\times k-a})$.

###  6.5 Central Charge & First Few Chiral Weights of
$\mathbb{Z}_{k}^{(1)}$-Algebra

To conclude the polynomiality section, we will rediscover the findings of
Zamolodchikov-Fateev in Ref. [52] regarding the
$\mathbb{Z}_{k}^{(1)}$-algebras. We have already found the structure
constants:

$C_{a,b}^{2}=\frac{(a+b)!(k-a)!(k-b)!}{k!a!b!(k-a-b)!}$ (57)

We would use the fact that the chiral algebra $\mathcal{W}$ is generated by
$L,W_{3},\cdots,W_{k}$ for $\mathbb{Z}_{k}^{(1)}$-algebra. This is not
required to find the central charge but allows us to find the $W_{3}$ and
$W_{4}$-weights. For simplicity, we use the notation $W,U$ for $W_{3},W_{4}$,
and their chiral weights will be denoted by $w_{a},u_{a}$ respectively. We use
the normalization convention $d_{WW}=c/3,d_{UU}=c/4$. Other than $1,L,W,U$, up
to level 4, we also have a quasi-primary chiral field
$\Lambda=(LL)-\frac{3}{10}\partial^{2}L$. The weight of $\Lambda$ is denoted
by $\lambda_{a}$. It is straightforward to find

$\lambda_{a}=h_{a}\left(h_{a}+\frac{1}{5}\right),\qquad
d_{\Lambda\Lambda}=\frac{(5c+22)c}{10}$ (58)

To compute the central charge and the weights, we first need to find the
minimal polynomial $\chi$. For $\mathbb{Z}_{k}^{(1)}$-algebras
$\chi\in\mathcal{T}_{k+1}^{2}$ and there is exactly one such polynomial
satisfying $\chi(1,0^{\times k})=1$; namely

$\chi(z_{1},\cdots,z_{k+1})=\frac{1}{k}\sum_{1\leq i<j\leq
k+1}(z_{i}-z_{j})^{2}$ (59)

Computing the specialization $\xi_{a}(x)$, we find

$\xi_{a}(x)=\chi(1,x^{\times a},0^{\times
k-a})=1-\frac{2a}{k}x+\frac{a(k-a+1)}{k}x^{2}$ (60)

On the other hand, by interpreting $\xi_{a}(x)$ as a conformal block, we can
write

$\displaystyle\xi_{a}(x)=(1-x)^{a\gamma}$
$\displaystyle\left\\{1+\frac{hh_{a}x^{2}}{d_{LL}}{}_{2}F_{1}(2,2,4,x)-\frac{ww_{a}x^{3}}{d_{WW}}{}_{2}F_{1}(3,3,6,x)+\left[\frac{\lambda\lambda_{a}}{d_{\Lambda\Lambda}}+\frac{uu_{a}}{d_{UU}}\right]x^{4}{}_{2}F_{1}(4,4,8,x)+\cdots\right\\}$
(61)

Expanding in powers of $x$ and comparing with Eq. (60) we find

$\displaystyle c$ $\displaystyle=\frac{2(k-1)}{k+2}$ (62a) $\displaystyle
w_{a}$ $\displaystyle=\frac{a(k-a)(k-2a)}{(k-1)(k-2)}w_{1}$ (62b)
$\displaystyle w_{1}$
$\displaystyle=\frac{(k-1)}{3}\sqrt{\frac{2(k-2)(3k+4)}{k^{3}(k+2)}}$ (62c)
$\displaystyle u_{a}$
$\displaystyle=\frac{a(k-a)}{(k-1)(k-3)}\left[(a-2)(k-a-2)+\frac{(k-16)}{2(k-2)}(a-1)(k-a-1)\right]u_{1}$
(62d) $\displaystyle u_{1}$
$\displaystyle=(k-1)\sqrt{\frac{(k-2)(k-3)(2k+1)(3k+4)}{2k^{4}(k+2)(16k+17)}}$
(62e)

In principle, if some information is known about the chiral algebra of
$\mathbb{Z}_{k}^{(r)}$ algebras, a similar procedure could determine some of
the chiral weights in the general case. However, we will refrain from doing so
for $r>1$. This is because we have incomplete information about the
$\mathcal{W}$-algebra in these cases. Moreover, the formulas for these weights
are significantly more complicated for larger $r$.

##  7 Anstaz for Principal $\mathbb{Z}_{k}^{(r)}$-Wavefunctions

To determine the principal $\mathbb{Z}_{k}^{(r)}$-wavefunction $\Psi$, the
idea is first to build a basis for the space of $(2k,2r)$ uniform states;
which we denote by $\mathcal{U}_{2k}^{2r}$. If $\mathfrak{f}_{m}$ is such a
basis, then we can write:

$\Psi_{\mathrm{ansatz}}(z_{1},\cdots,z_{2k})=\frac{1}{k!^{2}}\sum_{m}\alpha_{m}\mathfrak{f}_{m}(z_{1},\cdots,z_{2k})$

We call $\alpha_{m}$ the _ansatz constants_ or $\alpha$-constants of $\Psi$
(with respect to the basis $\mathfrak{f}_{m}$). Our design for the basis is an
application of classical invariant theory, in particular, invariants of binary
forms. We will spend some time in this section, briefly but adequately,
introducing the necessary mathematics. The discussion on the basis alone has
no intersection with conformal field theory. The CFT essence of the
$\mathbb{Z}_{k}^{(r)}$-WFs leaves its mark on the ansatz constants, not the
basis. In other words, the general program for finding principal wavefunctions
is as follows:

  1. 1.

Fix some $r$ and construct a set of “generators” for
$\mathcal{U}^{2r}=\bigoplus_{k\geq 0}\mathcal{U}_{2k}^{2r}$.
Despite the word “generator” needing clarification, this method yields an
explicit basis $\mathfrak{f}_{m}$ for $\mathcal{U}_{2k}^{2r}$.

  2. 2.

Writing $\Psi_{\mathrm{ansatz}}=k!^{-2}\sum_{m}\alpha_{m}\mathfrak{f}_{m}$,
fix $\alpha_{m}$ so that $\Psi_{\mathrm{ansatz}}$ matches the principal
$\mathbb{Z}_{k}^{(r)}$-WF $\Psi$.
This is done by performing various specializations of $\Psi_{\mathrm{ansatz}}$
and relating the $\alpha$-constants to CFT data.

In this paper, we will present the basis for $r=1,2,3,4$. The process of
determination of the ansatz constants is fully carried out for $r=1,2$ (i.e.,
we explicitly find the principal wavefunction). For $r=3,4$, we cannot
determine all of the $\alpha$-constants. It is unclear if this is because
$\chi\xrightarrow{\mathrm{fix}}\Psi$ fails in this case, or if there is extra
structure (currently unknown to us) in $\mathbb{Z}_{k}^{(r)}$-algebras ($r>2$)
that allows us to fix the rest of the $\alpha$-constants. The development of
tools for finding the $\alpha$-constants, together with the explicit
computation of principal wavefunctions, is the subject of section 8.

###  7.1 Invariants of Binary Forms

A _binary $n$-form_ (or a binary $n$-ic) is a degree $n$ homogeneous
polynomial in two formal variables $\mathbf{X}=(X,Y)^{t}$ (hence the name
_binary_). By convention, we write this polynomial in the following fashion:

$Q_{n}(\mathbf{X}\mid\mathbf{a})=\sum_{j=0}^{n}\binom{n}{j}a_{j}X^{j}Y^{n-j}$
(63)

We call $\mathbf{a}=(a_{0},a_{1},\cdots,a_{n})^{t}$ the _coefficient vector_
of the binary $n$-ic. The group $\mathrm{GL}_{2}(\mathbb{C})$ naturally acts
on $\mathbf{X}$ via matrix multiplication (i.e. for
$M\in\mathrm{GL}_{2}(\mathbb{C})$, we have
$\mathbf{X}\mapsto\mathbf{X}^{\prime}=M\mathbf{X}$). Using the binary $n$-ic,
we can find a linear representation
$R:\mathrm{GL}_{2}(\mathbb{C})\to\mathrm{GL}_{n+1}(\mathbb{C})$ which acts on
the coefficients. This is done via the following implicit equality

$\forall M\in\mathrm{GL}_{2}(\mathbb{C}):\qquad Q_{n}(M\mathbf{X}\mid
R_{M}\mathbf{a})=Q_{n}(\mathbf{X}\mid\mathbf{a})$ (64)

The representation $R$ is called the _induced representation_ for the
coefficients. A polynomial in the coefficients
$I(a_{0},a_{1},\cdots,a_{n})\in\mathbb{C}[a_{0},\cdots,a_{n}]$ is called an
_invariant of the binary $n$-ic_ if for all $M\in\mathbf{GL}_{2}$ we have

$I(\mathbf{a})=(\det
M)^{\mathrm{wt}(I)}I(R_{M}\mathbf{a}),\qquad\mathrm{wt}(I)\in\mathbb{N}$ (65)

The quantity $\mathrm{wt}(I)$ is called the _weight_ of $I$. Working out the
special case $M=\lambda\,\mathrm{Id}_{2}$ ($\mathrm{Id}_{s}$ the $s\times s$
identity matrix, $0\neq\lambda\in\mathbb{C}$), we have
$R_{M}=\lambda^{-n}\,\mathrm{Id}_{n+1}$ and therefore $I$ is necessarily
homogeneous. If $m$ is the degree of $I$, then $2\mathrm{wt}(I)=nm$. The space
of invariants of binary $n$-ics with degree $m$ is denoted by
$\mathcal{B}_{n}^{m}$. We have $\mathcal{B}_{n}^{0}=\mathbb{C}$ and
$\mathcal{B}_{n}^{m_{1}}\mathcal{B}_{n}^{m_{2}}\subset\mathcal{B}_{n}^{m_{1}+m_{2}}$.
Consequently, the direct sum $\mathcal{B}_{n}=\bigoplus_{m\geq
0}\mathcal{B}_{n}^{m}$ is a _graded $\mathbb{C}$-algebra_ called the _algebra
of invariants of the binary $n$-ic_.

##### Example: Binary Quadratics

Utilizing the definitions, the binary 2-form/quadratic is
$Q_{2}(X,Y)=a_{0}X^{2}+2a_{1}XY+a_{2}Y^{2}$. In this case, the induced
representation of $M\in\mathrm{GL}_{2}$ can be found to be

$M=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}\mapsto
R_{M}=\frac{1}{(\alpha\delta-\beta\gamma)^{2}}\begin{pmatrix}\alpha^{2}&-2\alpha\beta&\beta^{2}\\\
-\alpha\gamma&\alpha\delta+\beta\gamma&-\beta\delta\\\
\gamma^{2}&-2\gamma\delta&\delta^{2}\end{pmatrix}$ (66)

The _discriminant_ $\Delta=a_{0}a_{2}-a_{1}^{2}$ is an invariant of the binary
quadratic ($\deg\Delta=\mathrm{wt}(\Delta)=2$) and has the smallest possible
degree. It can be shown (see the remark in F) that
$\mathcal{B}_{2}=\mathbb{C}[\Delta]$ (as a graded algebra); in other words,
the invariants of the binary quadratic are $1,\Delta,\Delta^{2},\cdots$. The
invariant $\Delta^{k}$ will be used to construct the principal
$(k,1)$-wavefunction (aka $\mathbb{Z}_{k}$ Read-Rezayi state for $N=2k$).

##### Example: Binary Cubics

The binary cubic form is $Q_{3}=a_{0}Y^{3}+3a_{1}XY^{2}+3a_{2}XY+a_{3}X^{3}$.
Similar to binary quadratics, the algebra $\mathcal{B}_{3}$ is generated by
the cubic discriminant $\Delta_{3}$ (see F); i.e. up to scaling, the only
invariants of binary cubics are $1,\Delta_{3},\Delta_{3}^{2},\cdots$. The
cubic discriminant is of degree $4$ (weight $6$) and is given by

$\Delta_{3}=a_{0}^{2}a_{3}^{2}-6a_{0}a_{1}a_{2}a_{3}+4a_{0}a_{2}^{3}-3a_{1}^{2}a_{2}^{2}+4a_{1}^{3}a_{3}$
(67)

Compactly, we have $\mathcal{B}_{3}=\mathbb{C}[\Delta_{3}]$.

###  7.2 First vs Second Quantization: The Isomorphism
$\mathcal{B}_{n}^{m}\simeq\mathcal{U}_{m}^{n}$

To construct an ansatz for principal $\mathbb{Z}_{k}^{(r)}$-wavefunctions, we
will first construct an isomorphism between the spaces $\mathcal{B}_{n}^{m}$
and $\mathcal{U}_{m}^{n}$. As we will describe in this subsection, this
isomorphism is a restriction of the isomorphism between first quantization
(uniform states) to second quantization (binary invariants). Let us begin by
exploring this quantum language in the context of states lying in the lowest
Landau level.

####  7.2.1 First and Second Quantization

The lowest Landau level consists of $N_{\phi}+1$ ‘orbitals’ $\left|m\right>$
with $0\leq m\leq N_{\phi}$. The $m$th orbital carries a $L_{z}$-angular
momentum $m$. Let $\nu_{m}$ be the number of bosons occupying the $m$th
orbital and $N=\nu_{0}+\nu_{1}+\cdots+\nu_{N_{\phi}}$ the total number of
particles. We often abbreviate all of the information in a single array
$\nu=(\nu_{0},\nu_{1},\cdots,\nu_{\phi})$, referred to as the bosonic
occupation. In the second quantized language, the state corresponding to $\nu$
is represented as

$\left|\nu\right>=\>a_{0}^{\nu_{0}}a_{1}^{\nu_{1}}\cdots
a_{N_{\phi}}^{\nu_{N_{\phi}}}\left|0\right>\equiv\mathbf{a}^{\nu}\left|0\right>$
(68)

Here, $a_{m}$ is interpreted as the operator creating a boson at $m$th
orbital. The first quantized formulation can now be obtained as follows. Let
$N_{m}=\nu_{0}+\nu_{1}+\cdots+\nu_{m-1}$ be the number of particle with
angular momentum $<m$ (with $N_{0}=0$). We define the corresponding _symmetric
monomial_ as (caution: our convention is a bit different than the standard
one)

$m_{\nu}(z_{1},z_{2},\cdots,z_{N})=\mathscr{S}\left[\prod_{m=0}^{N_{\phi}}\prod_{j=1}^{\nu_{m}}z_{N_{m}+j}^{m}\right]$
(69)

where $\mathscr{S}$ is the symmetrization operator. Then, one finds that

$\left<z_{1},z_{2},\cdots,z_{N}\vphantom{\nu}\right|\left.\nu\vphantom{z_{1},z_{2},\cdots,z_{N}}\right>=\mathscr{C}(N,N_{\phi})\>N!\>m_{\nu}(z_{1},\cdots,z_{N})$
(70)

where $\mathscr{C}(N,N_{\phi})$ is a universal normalization (not dependent on
$\nu$) which is irrelevant for our considerations.

The inverse process, going from first quantization to second quantization, is
called the _umbral evaluation_ in the literature on classical invariant theory
(going back $\sim$ 150 years ago, way before the discovery of quantum
mechanics, let alone quantum Hall effect). The main idea, which is purely
algebraic, is as follows. Let
$\mathscr{A}=\mathbb{C}[a_{0},a_{1},a_{2},\cdots]$ be the space of polynomials
in infinite variables. The umbral evaluation is the map
$\mathscr{U}:\mathbb{C}[z_{1},\cdots,z_{N}]\to\mathscr{A}$ linearly extending
the following:

$\mathscr{U}[z_{1}^{p_{1}}z_{2}^{p_{2}}\cdots
z_{N}^{p_{N}}]=a_{p_{1}}a_{p_{2}}\cdots a_{p_{N}}$ (71)

As a consequence, we find that
$\mathscr{U}[m_{\nu}(z_{1},\cdots,z_{N})]=N!\>\mathbf{a}^{\nu}$, where
$\nu=(\nu_{0},\cdots,\nu_{N_{\phi}})$ is any occupation (i.e.
$\sum_{i}\nu_{i}=N$) in an LLL with $N_{\phi}$ flux quanta. As we next
explain, this first/second quantization isomorphism descends to an isomorphism
$\mathcal{Q}:\mathcal{U}_{N}^{N_{\phi}}\xrightarrow{\sim}\mathcal{B}_{N_{\phi}}^{N}$.
The symbol $\mathcal{Q}$ is reminder that the isomorphism is a ‘quantization’
(from first to second). However, it is often easier to work with
$\mathcal{Q}^{-1}$ (a ‘de-quantization’ if you will).

####  7.2.2 Symmetrized/Umbralized Graph Polynomials

Let us start by introducing _(regular)-graph-monomials_. In a previous work
[35], we studied the connection between graph theory and model FQH ground
states in some depth. Here, we will review the basic definitions. For our
purposes, a (weighted directed) graph $G$ is a pair of data $(V,\mu)$. The set
$V$ is called the node-set, and we take it to be $V=\\{1,2,\cdots,m\\}$. We
often use the notation $|G|$ (instead of $|V|$) for the number of nodes in
$G$. The function $\mu:V\times V\to\mathbb{Z}_{\geq}$ is the multiplicity
function of $G$. We understand $\mu(x,y)$ as the multiplicity of an arrow
$x\to y$ in graph $G$. We further specialize to graphs where
$\mu(i,j)\mu(j,i)=0$ for any pair $i,j\in V$ (i.e. all arrows between $i,j$ go
in the same direction). A graph $G$ is _$n$ -regular_ (or regular of degree
$n$) if for all $j\in V$ we have

$n=\sum_{i\in V}\mu(i,j)+\mu(j,i)$ (72)

Given an $n$-regular graph $G$, we define its _graph-monomial_ as

$\mathscr{M}[G](z_{1},\cdots,z_{m})=\prod_{i,j=1}^{m}(z_{i}-z_{j})^{\mu(i,j)}$
(73)

Suppose $G$ is such that $I=(\mathscr{U}\circ\mathscr{M})[G]\neq 0$. Then $I$
is a polynomial of the form

$I=\sum_{\begin{subarray}{c}\nu_{0}+\nu_{1}+\cdots+\nu_{n}=m\\\
\nu_{1}+2\nu_{2}+\cdots+n\nu_{n}=nm/2\end{subarray}}N_{\vec{\nu}}[G]\prod_{j=0}^{n}a_{j}^{\nu_{j}}$

with $N_{\vec{\nu}}[G]\in\mathbb{Z}$. For future convenience, we define
$u(G)=\mathrm{gcd}(N_{\vec{\nu}}[G])$ (greatest common divisor of all
coefficients). These are combinatorical factors that reflect the symmetries of
$G$. If $(\mathscr{U}\circ\mathscr{M})[G]=0$, let $u(G)=\infty$. It is
straightforward to show that $u(G_{1}\sqcup G_{2})=u(G_{1})u(G_{2})$ with
$\sqcup$ being the disjoint union. We define

$\Upsilon[G]=\frac{1}{u(G)}(\mathscr{U}\circ\mathscr{M})[G],\qquad\Psi[G]=\frac{1}{u(G)}(\mathscr{S}\circ\mathscr{M})[G]$
(74)

called _umbralized graph polynomial_ (UMP) and _symmetrized graph monomial_
(SGP) respectively. While using the symbol $\Psi$ for SGP is an abuse of
language, we use it to remind ourselves that $\Psi[G]$ is wavefunction.
Clearly, $\mathcal{Q}^{-1}(\Upsilon[G])=\Psi[G]$ ($\mathcal{Q}$ being the
‘quantization’ isomorphism). This can be extended to an isomorphism
$\mathcal{Q}^{-1}:\mathcal{B}_{n}^{m}\xrightarrow{\sim}\mathcal{U}_{m}^{n}$ as
a corollary to the following theorem:

######  Theorem 3.

Throughout, all graphs are $n$-regular in $m$ nodes and sums are finite. The
following are true:

  1. 1.

For any invariant $I\in\mathcal{B}_{n}^{m}$, there exists graphs $G_{i}$ and
$t_{i}\in\mathbb{C}$ such that $X=\sum_{i}t_{i}\Upsilon[G_{i}]$.

  2. 2.

For any uniform state $X\in\mathcal{U}_{m}^{n}$, there exists graphs $G_{i}$
and $t_{i}\in\mathbb{C}$ such that $X=\sum_{i}t_{i}\Psi[G_{i}]$.

  3. 3.

Let $P=\sum_{i}t_{i}\mathscr{M}[G_{i}]$ for some graphs $G_{i}$ and
$t_{i}\in\mathbb{C}$. Then $\mathscr{U}[P]=0$ if and only if
$\mathscr{S}[P]=0$.

For a proof of part 1, see [30, Theorem 3.1 – Part I] or [34, Ch. 6–7]. For a
proof of part 2, see [13, §88–89] or [35]. Finally, part 3 can be found in
[30, Lemma 3.4 – Part I]. We emphasize that regular graphs (and their graph-
monomials) act as a middleman in the isomorphism
$\mathcal{Q}^{-1}:\mathcal{B}_{n}^{m}\xrightarrow{\sim}\mathcal{U}_{m}^{n}$.
The relation between the three concepts is shown in the diagram (2).

Figure 2: This diagram symbolically illustrates the relation between regular
graphs, binary invariants, and uniform states.

######  Remark 2.

There exists yet another isomorphism
$\mathscr{E}:\mathcal{U}_{m}^{n}\xrightarrow{\sim}\mathcal{B}_{m}^{n}$. In the
literature on binary invariants, the combination of the two isomorphisms
$\mathscr{E}\circ\mathcal{Q}^{-1}:\mathcal{B}_{n}^{m}\xrightarrow{\sim}\mathcal{B}_{m}^{n}$
is called _Hermite reciprocity theorem_. We do not need this isomorphism in
the current paper. Nonetheless, since this secondary mapping might also be
relevant for future research, the details are provided in F.

###  7.3 About the Structure of the $\mathcal{B}_{n}$ Algebra

The algebra of invariants of binary forms is very well-studied and quite old.
In 1868, Gordan proved $\mathcal{B}_{n}$ is finitely generated [19], though
not utilizing the language of graded algebras. About twenty years later,
Hilbert, in two revolutionary papers [25, 26], not only proved that
$\mathcal{B}_{n}$ is finitely generated as a graded algebra but also pioneered
some of the foundations of modern commutative algebra. In his honor, the
generators of $\mathcal{B}_{n}$ are called a _Hilbert basis_ (though the
Hilbert basis is a set of generators, not a basis). Additionally, in Ref.
[27], the authors proved that $\mathcal{B}_{n}$ algebras are Cohen-Macaulay.
This subsection will briefly discuss some of the consequences of these
properties that are relevant to us. To avoid breaking continuity, we do not
include the technical definitions in the main body of the paper. Instead, we
review the notions used here in E. The key takeaway is the unique
decomposition (75), in which $\mathfrak{I}^{(b)}_{\nu}$ are a basis for
$\mathcal{B}_{n}^{\deg F}$.

Throughout, the notations $D$ and $\mu$ are used for the Krull dimension and
multiplicity of $\mathcal{B}_{n}$, respectively. Let $I_{1},\cdots,I_{D}$
(with degrees $d_{1},\cdots,d_{D}$ respectively) be a homogeneous system of
parameters (hsop). The full set of generators (i.e. the Hilbert basis) is
denoted by $I_{1},\cdots,I_{D},J_{1},\cdots,J_{g}$. Since $\mathcal{B}_{n}$ is
a finitely generated Cohen-Macaulay graded $\mathbb{C}$-algebra, there exists
a set of invariants

$\mathcal{Z}=\\{1,Z_{1},Z_{2},\cdots,Z_{\mu-1}\\}$

such that $\mathcal{Z}$ is a basis for $\mathcal{B}_{n}$ as a
$\mathbb{C}[I_{1},\cdots,I_{D}]$-module. We have $J_{j}\in\mathcal{Z}$ for all
$1\leq j\leq g$. In general, any element in $\mathcal{Z}$ is a monomial
$\prod_{j=1}^{g}J_{j}^{a_{j}}$ for some integer sequence
$a=(a_{1},a_{2},\cdots,a_{g})$. We use the notation $e_{b}\equiv\deg Z_{b}$
(with $Z_{0}\equiv 1$). Finally, any invariant $F\in\mathcal{B}_{n}$ has a
_unique_ decomposition

$F=\sum_{b=0}^{\mu-1}\>\sum_{\begin{subarray}{c}\nu_{1},\cdots,\nu_{D}\geq
0\\\ \nu_{1}d_{1}+\cdots+\nu_{D}d_{D}=\deg
F-e_{b}\end{subarray}}\tilde{\alpha}^{(b)}_{\nu}\mathfrak{I}^{(b)}_{\nu},\qquad\mathfrak{I}^{(b)}_{\nu}:=Z_{b}\prod_{i=1}^{D}I_{i}^{\nu_{i}},\qquad(\tilde{\alpha}^{(b)}_{\nu}\in\mathbb{C})$
(75)

Note that once the Hilbert basis is concretely constructed (and the invariants
$Z_{s}$ are identified), the computation of $F$ reduces to finding the complex
numbers $\tilde{\alpha}^{(b)}_{\nu}$. We can use this, in conjunction with the
isomorphism
$\mathcal{Q}^{-1}:\mathcal{B}_{n}^{m}\xrightarrow{\sim}\mathcal{U}_{m}^{n}$,
to obtain an ansatz for uniforms states; in particular, an ansatz for
principal $\mathbb{Z}_{k}^{(r)}$-WFs. Some further properties about
$\mathcal{B}_{n}$ can be found in E.

Let us now describe the algebras of binary invariants relevant to us. From now
on, if there is no danger of ambiguity, the notations $I_{d}$ and $J_{d}$
represent a basic invariant of degree $d$. The information we are about to
present is the findings of various authors. However, for an alternative
reference listing all that will follow, see Ref. [12].

##### Example: Binary Quadratics and Cubics

As discussed before, $\mathcal{B}_{2}=\mathbb{C}[\Delta]$ ($\deg\Delta=2$) and
$\mathcal{B}_{3}=\mathbb{C}[\Delta_{3}]$ ($\deg\Delta_{3}=4$), with
$\Delta,\Delta_{3}$ the quadratic/cubic discriminants. The Poincaré series for
these two examples are $P_{2}(t)=(1-t^{2})^{-1}$ and
$P_{3}(t)=(1-t^{4})^{-1}$.

##### Example: Binary Quartics (see [34])

The algebra $\mathcal{B}_{4}$ has a Hilbert basis consisting of two invariants
$I_{2},I_{3}$ of degree $2,3$ respectively. There are no relations between
$I_{2},I_{3}$ (meaning $I_{2},I_{3}$ constitute an hsop). The Poincaré series
is obtained to be

$P_{4}(t)=\frac{1}{(1-t^{2})(1-t^{3})}$ (76)

##### Example: Binary sextics (see [19])

The algebra $\mathcal{B}_{6}$ has a Hilbert basis $I_{2},I_{4},I_{6},I_{10}$
(hsop), together with $J_{15}$. One computes $\mu(6)=2$, meaning, $J_{15}$
squares to a polynomial in $I_{2},I_{4},I_{6},I_{10}$. In other words,
$\mathcal{B}_{6}=(1+J_{15})\mathbb{C}[I_{2},I_{4},I_{6},I_{10}]$ and the
Poincaré series is

$P_{6}(t)=\frac{1+t^{15}}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{10})}$ (77)

##### Example: Binary octavics (see [39])

The algebra $\mathcal{B}_{8}$ has a Hilbert basis
$I_{2},I_{3},I_{4},I_{5},I_{6},I_{7}$ (hsop) and $J_{8},J_{9},J_{10}$. The
Poincaré series of the octavics is

$P_{8}(t)=\frac{1+t^{8}+t^{9}+t^{10}+t^{18}}{(1-t^{2})(1-t^{3})(1-t^{4})(1-t^{5})(1-t^{6})(1-t^{7})}$
(78)

The space $\overline{\mathcal{B}}_{8}=\mathcal{B}_{8}/(\mathrm{hsop})$ is
$5$-dimensional (i.e. $\mu(8)=5$), and has a basis
$1,J_{8},J_{9},J_{10},J_{8}J_{10}$ (that is with our choice of Hilbert basis
in §8.5). Note that $J_{8}J_{10}$ is not basic.

###  7.4 A Graphic Ansatz for Principal $\mathbb{Z}_{k}^{(r)}$-Wavefunctions

Irrespective of which $n$ is chosen, it is always possible to choose the
Hilbert basis of $\mathcal{B}_{n}$ to be graphic. Concretely, if $D$ is Krull
dimension of $\mathcal{B}_{n}$, one can find $D$ $n$-regular graphs
$G_{1},G_{2},\cdots,G_{D}$ such that
$\mathcal{I}=\\{I_{i}\equiv\Upsilon(G_{i})\mid 1\leq i\leq D\\}$ is a
homogeneous system of parameters. One can then augment the hsop with $g$
$n$-regular graphs $G^{\prime}_{1},\cdots,G^{\prime}_{g}$ so that, with
$\mathcal{J}=\\{J_{i}\equiv\Upsilon(G^{\prime}_{i})\mid 1\leq i\leq g\\}$, the
union $\mathcal{I}\cup\mathcal{J}$ constitutes a full Hilbert basis for
$\mathcal{B}_{n}$. This is called a _graphic_ (Hilbert) basis. Similarly, let
$H_{0},H_{1},\cdots,H_{\mu-1}$ be $n$-regular graphs such that

$\mathcal{Z}=\\{\Upsilon(H_{i})\mid 0\leq i\leq\mu-1\\}$

is a basis for $\mathcal{B}_{n}$ as a $C[I_{1},\cdots,I_{D}]$-module. We have
$H_{0}=\emptyset$ (no nodes in $H_{0}$). The $H_{i}$ graphs are some disjoint
unions of $G^{\prime}_{i}$ graphs. In accordance with our previous notations,
we denote the number of nodes of $G_{i},G^{\prime}_{i}$, and $H_{i}$ by
$d_{i},d^{\prime}_{i}$, and $e_{i}$ respectively.

The principal $\mathbb{Z}_{k}^{(r)}$-wavefunction $\Psi$ is a $(2k,2r)$
uniform state. Therefore, $\mathcal{Q}(\Psi)$ is an element of
$\mathcal{B}_{2r}^{2k}$. Using the graphic Hilbert basis of
$\mathcal{B}_{2r}$, we can write $\Psi$ in the form

$\Psi(z_{1},\cdots,z_{2k})=\frac{1}{k!^{2}}\sum_{b=0}^{\mu-1}\sum_{\begin{subarray}{c}\nu_{1},\cdots,\nu_{D}\geq
0\\\
\nu_{1}d_{1}+\cdots+\nu_{D}d_{D}=2k-e_{b}\end{subarray}}\alpha^{(b)}_{\nu}\mathfrak{f}_{\nu}^{(b)}(z_{1},\cdots,z_{2k}),\qquad\mathfrak{f}_{\nu}^{(b)}:=\Psi\left[H_{b}\sqcup\bigsqcup_{i=1}^{D}G_{i}^{\sqcup\nu_{i}}\right]$
(79)

In particular, $\mathfrak{f}_{\nu}^{(b)}$ is a basis for
$\mathcal{U}_{2k}^{2r}$. The above is called the ansatz for principal
$\mathbb{Z}_{k}^{(r)}$-WFs. we call the complex numbers $\alpha^{(b)}_{\nu}$
the _ansatz constant_ or $\alpha$-constants of $\Psi$. We may also find the
corresponding invariant:

$\mathcal{Q}(\Psi)=\frac{1}{k!^{2}}\sum_{b=0}^{\mu-1}\sum_{\begin{subarray}{c}\nu_{1},\cdots,\nu_{D}\geq
0\\\
\nu_{1}d_{1}+\cdots+\nu_{D}d_{D}=2k-e_{b}\end{subarray}}\alpha^{(b)}_{\nu}\Upsilon[H_{b}]\prod_{i=1}^{D}\Upsilon[G_{i}]^{\nu_{i}}$
(80)

In general, our strategy in §8 would be to first find a (convenient) graphic
Hilbert basis for $\mathcal{B}_{2r}$, and then try to find as many of the
$\alpha$-constants of the principal $\mathbb{Z}_{k}^{(r)}$-wavefunction as we
can.

##### Example: Binary Quadratics

We have seen before that the invariants of binary quadratics are generated by
the quadratic discriminant $\Delta=a_{0}a_{2}-a_{1}^{2}$. Graphically, we have

$\Delta\equiv\frac{1}{2}(\mathscr{U}\circ\mathscr{M})[G]\equiv\frac{1}{2}(\mathscr{U}\circ\mathscr{M})\left[\vbox{
\hbox{ \includegraphics[scale={.33}]{discriminant2} }
}\right]=\frac{1}{2}\mathscr{U}[(z_{1}-z_{2})^{2}]$ (81)

For this simple example, let us explicitly work out the effect of
$\mathscr{U}$-operator:

$\mathscr{U}[(z_{1}-z_{2})^{2}]=\mathscr{U}[z_{1}^{2}z_{2}^{0}+z_{1}^{0}z_{2}^{2}-2z_{1}z_{2}]=a_{2}a_{0}+a_{0}a_{2}-2a_{1}a_{1}=2(a_{0}a_{2}-a_{1}^{2})$

Therefore $u(G)=2$, and $\Delta=\Upsilon[G]$. The principal
$(k,1)$-wavefunction is then equal to:

$\Psi^{(k,1)}(z_{1},\cdots,z_{2k})=\frac{1}{k!^{2}}\Psi\left[\vbox{ \hbox{
\includegraphics[scale={.4}]{Delta3} }
}\right]=\frac{1}{2^{k}k!^{2}}\mathscr{S}\Big{[}\prod_{i=1}^{k}(z_{i}-z_{k+i})^{2}\Big{]}$
(82)

which is nothing but the principal (i.e. $N=2k$) $\mathbb{Z}_{k}$ Read-Rezayi
state [38, 11].

##### Example: Binary Cubics

To compute the principal wavefunctions of $\mathbb{Z}_{k}^{(r)}$ (with our
particular definition of it), we only need to study invariants of binary
$2r$-ics (i.e. binary _even_ -ics). Nonetheless, let us take a moment and
discuss the simplest binary _odd_ -ic; namely the binary cubic. We have
already seen that $\mathcal{B}_{3}$ is generated by the cubic discriminant
$\Delta_{3}$ given by Eq. (67). We have

$\Delta_{3}\equiv\frac{1}{2}(\mathscr{U}\circ\mathscr{M})[G]\equiv\Upsilon\left[\vbox{
\hbox{ \includegraphics[scale={.33}]{discriminant3} } }\right]$ (83)

The $(4,3)$ uniform state corresponding to $\Delta_{3}$ is the principal (i.e.
$N=4$) Gaffnian wavefunction [41]. The $(4\ell,3)$ uniform state corresponding
to $\Delta_{3}^{\ell}$ is the principal (i.e. $N=4\ell$) Jack with parameter
$\alpha=-(2\ell+1)/2$ [7].

##  8 About Principal $\mathbb{Z}_{k}^{(r)}$-Wavefunctions with $2\leq r\leq
4$

In this section, we will present a graphic basis for
$\mathcal{B}_{4},\mathcal{B}_{6}$, and $\mathcal{B}_{8}$. In addition, we will
develop some technical tools to find the ansatz constants of the principal
$\mathbb{Z}_{k}^{(r)}$-WFs, with $r=2,3,4$, based on the free parameters of
$\mathbb{Z}_{k}^{(r)}$-algebra. The first tool we will introduce is the
tricoloring homomorphism (§8.1), which is our own design. This machinery
allows us to find a subset of $\alpha$-constants systematically. The second
concept we introduce in this section is that of graph residue (§8.2). Residues
help facilitate visualizing the link between principal
$\mathbb{Z}_{k}^{(r)}$-wavefunctions and their corresponding minimal
polynomial $\chi$. Finally, we will present the detailed discussion of
principal $\mathbb{Z}_{k}^{(r)}$-wavefunctions for $r=2,3,4$ at the end.

######  Remark 3.

In the main body of the paper, we always fix $r$ first and then study the
entire $k$-family (i.e., all $k$ are allowed) simultaneously. It is also
possible to fix $k$ and consider the $r$-family. The tools introduced in this
section do not apply to the latter approach. In F, we present the special case
$k=2$, i.e. the paired states ($k=1$ are the Laughlin states).

###  8.1 Tricoloring Homomorphism $K$

Let us start by fixing some terminology. Throughout, let $X,Y,Z$ be three
formal variables. We use the notations $E_{1}=X+Y+Z$, $E_{2}=XY+XZ+YZ$, and
$E_{3}=XYZ$; the elementary symmetric polynomials in three variables. The ring
of symmetric polynomials $\Lambda_{3}=\mathbb{C}[E_{1},E_{2},E_{3}]$ is to be
understood as a graded algebra with degrees $\deg E_{i}=i$. The _even
subalgebra_ is defined as $\mathcal{B}_{n}^{E}=\bigoplus_{k\geq
0}\mathcal{B}_{n}^{2k}$. We understand $\mathcal{B}_{n}^{2k}$ as the vector
space of homogeneous elements of pseudo-degree $k$ (not $2k$). With $A,B$
being two graded algebras, we say a homomorphism $\rho:A\to B$ is _degree-
preserving_ if $\deg_{A}x=\deg_{B}\rho(x)$ for any homogeneous element $x\in
A$.

In this subsection, we will build a degree preserving homomorphism
$K:\mathcal{B}_{2r}^{E}\to\Lambda_{3}$ called the _tricoloring homomorphism_.
Tricoloring homomorphism, at its core, is the generating function for 3-point
specializations of the uniforms states. More concretely, let $P$ be an
arbitrary $(2k,2r)$ uniform state. For any triple of integer $a,b,c\geq 0$
such that $a+b+c=k$ we define the specialization $P_{a,b,c}$ as

$P_{a,b,c}:=\lim_{w\to\infty}w^{-2r(k-a)}P(w^{\times k-a},1^{\times
k-b},0^{\times k-c})$ (84)

Due to how $P$ transforms under Möbius transformations (being a uniform
state), it is straightforward to see that $P_{a,b,c}$ is symmetric with
respect to the permutations of the indices. This allows us to construct a
symmetric polynomial in $X,Y,Z$:

$K[P](X,Y,Z)=\sum_{\begin{subarray}{c}a,b,c\geq 0\\\
a+b+c=k\end{subarray}}\frac{P_{a,b,c}}{(k-a)!(k-b)!(k-c)!}X^{a}Y^{b}Z^{c}$
(85)

Clearly, $K[P]$ acts as a generating function for the specializations
$P_{a,b,c}$. Note that $K[P]$ is homogeneous of degree $a+b+c=k$. By abuse of
notation, let $K[I]\equiv K[\mathcal{Q}^{-1}(I)]$. So far, we have constructed
a collection of linear mappings $\mathcal{B}_{2r}^{2k}\to\Lambda_{3}^{k}$
given by $I\mapsto K[I]$. To show that $K$ actually extends to a homomorphism
$K:\mathcal{B}_{2r}^{E}\to\Lambda_{3}$ we need to explain the naming
“tricoloring”.

Let $G$ be a $2r$-regular graph with nodes $V={1,2,\cdots,|G|}$ (with $|G|$ a
positive integer, not necessarily even). Let us illustrate the meaning of
$K[G]\equiv K[\Psi[G]]$ in terms of graphs. A (proper) _tricoloring_ of $G$ is
a map $\eta:V\to\\{\infty,1,0\\}$ so that no two adjacent nodes have the same
color (the colors being $\infty,1,0$). In contrast to the typical definition
of graph-coloring, we assume the colors are ordered: $\infty>1>0$. Given an
arrow $e=x\to y$ in $G$, the quantity
$\mathrm{sgn}(e)=\mathrm{sgn}(\eta(x)-\eta(y))\in\\{-1,+1\\}$ is called the
sign of $e$ induced by $\eta$. The sign of the tricoloring $\eta$ is then
defined as $\mathrm{sgn}(\eta)=\prod_{i,j=1}^{|G|}\mathrm{sgn}(i\to
j)^{\mu(i,j)}$ (i.e. the multiplication of all arrow signs). If we denote by
$v_{\infty},v_{1},v_{0}$ as the number of nodes colored by $\infty,1,0$
respectively, then

$K[G]\equiv
K[\Upsilon[G]]=\frac{1}{u(G)}\sum_{\eta}\mathrm{sgn}(\eta)X^{|G|/2-v_{\infty}(\eta)}Y^{|G|/2-v_{1}(\eta)}Z^{|G|/2-v_{0}(\eta)}$
(86)

Here, the sum is over all proper tricolorings $\eta$. $K[G]$ is homogeneous of
degree $|G|/2$ as a function in $X,Y,Z$. A consequence of regularity is that
$v_{i}\leq|G|/2$ for any of $i=\infty,1,0$. Therefore, when $|G|=2k$ is even,
this is a polynomial, as expected. However, when $|G|$ is odd, it is
$K[G]/\sqrt{E_{3}}$ that is a symmetric polynomial. An important property of
$K$ is that $K[G_{1}\sqcup G_{2}]=K[G_{1}]K[G_{2}]$. In other words,

$K[\Upsilon[G_{1}]\Upsilon[G_{2}]]=K[\Upsilon[G_{1}\sqcup G_{2}]]=K[G_{1}\cup
G_{2}]=K[G_{1}]K[G_{2}]$ (87)

Combining this observation with Theorem 3 shows that
$K:\mathcal{B}_{2r}^{E}\to\Lambda_{3}$ is indeed a degree-preserving
homomorphism of graded algebras.

Though the setting of the definition of tricoloring homomorphism is general,
it is the action of $K$ on the principal $\mathbb{Z}_{k}^{(r)}$-WFs that
reveals their true strength. Going to the specializations (84), with find

$\Psi_{a,b,c}=\lim_{w\to\infty}w^{-2r(k-a)}\Psi(w^{\times k-a},1^{\times
k-b},0^{\times
k-c})=\frac{g_{k-a}g_{k-b}g_{k-c}}{g_{k}^{2}}\left<\psi_{a}^{+}(\infty)\psi^{+}_{b}(1)\psi^{+}_{c}(0)\right>=C_{a,b,c}^{2}$
(88)

The formula for $C_{a,b,c}^{2}$ in Eq.(48) now yields

$\displaystyle k!^{2}K[\Psi]$
$\displaystyle=k!^{2}\sum_{\begin{subarray}{c}a,b,c\geq 0\\\
a+b+c=k\end{subarray}}\frac{C_{a,b,c}^{2}}{(k-a)!(k-b)!(k-c)!}X^{a}Y^{b}Z^{c}$
(89) $\displaystyle=\sum_{\begin{subarray}{c}a,b,c\geq 0\\\
a+b+c=k\end{subarray}}\binom{a+b+c}{a,b,c}\prod_{p=1}^{r-1}\frac{(t_{p})_{a+b}(t_{p})_{a+c}(t_{p})_{b+c}}{(t_{p})_{a}(t_{p})_{b}(t_{p})_{c}(t_{p})_{a+b+c}}X^{a}Y^{b}Z^{c}$

On the other hand, using the graphic ansatz (80) we find ($G_{i},H_{b}$, etc.
being the same as before)

$k!^{2}K[\Psi]=\sum_{b=0}^{\mu-1}\>\sum_{\begin{subarray}{c}\nu_{1},\cdots,\nu_{D}\geq
0\\\
\nu_{1}d_{1}+\cdots+\nu_{D}d_{D}=2k-e_{b}\end{subarray}}\alpha_{\nu}^{(b)}K[H_{b}]\prod_{i=1}^{D}K[G_{i}]^{\nu_{i}}$
(90)

This leads to what we call the _tricoloring identity_ :

$\sum_{\begin{subarray}{c}a,b,c\geq 0\\\
a+b+c=k\end{subarray}}\binom{a+b+c}{a,b,c}\prod_{p=1}^{r-1}\frac{(t_{p})_{a+b}(t_{p})_{a+c}(t_{p})_{b+c}}{(t_{p})_{a}(t_{p})_{b}(t_{p})_{c}(t_{p})_{a+b+c}}X^{a}Y^{b}Z^{c}=\sum_{b=0}^{\mu-1}\>\sum_{\begin{subarray}{c}\nu_{1},\cdots,\nu_{D}\geq
0\\\
\nu_{1}d_{1}+\cdots+\nu_{D}d_{D}=2k-e_{b}\end{subarray}}\alpha_{\nu}^{(b)}K[H_{b}]\prod_{i=1}^{D}K[G_{i}]^{\nu_{i}}$
(91)

We will use this identity to determine a significant number of ansatz
constants $\alpha_{\nu}^{(b)}$, though not necessarily all, in our analysis of
principal $\mathbb{Z}_{k}^{(r)}$-wavefunctions with $r=2,3,4$.

###  8.2 Graph Residue and Residual Basis

As mentioned before, a partial reason to explicitly find the principal
$\mathbb{Z}_{k}^{(r)}$ wavefunction $\Psi$ is to investigate whether
$\chi\to\Psi_{2}$ is true. In such a study, it would be convenient to define a
basis for $\mathcal{T}_{k+1}^{2r}$ that is naturally obtained from the Hilbert
basis of $\mathcal{B}_{2r}$-algebras. We will develop this basis in this
subsection.

In general, the minimal polynomial $\chi$ is a specialization of the principal
wavefunction:

$\chi(z_{1},\cdots,z_{k+1})=\lim_{w\to\infty}w^{-2r(k-1)}\Psi(w^{\times(k-1)},z_{1},\cdots,z_{k+1})$

On the other hand, the principal wavefunction $\Psi$ is a superposition of
symmetrized graph polynomials $\Psi[\Gamma]$, with $\Gamma$ a $2r$-regular
graph in $2k$ nodes. Hence, we routinely need to calculate specializations of
the form:

$\chi_{\Gamma}(z_{1},\cdots,z_{k+1})\equiv\frac{1}{k!^{2}}\lim_{w\to\infty}w^{-2r(k-1)}\Psi[\Gamma](w^{\times
k-1},z_{1},\cdots,z_{k+1})$ (92)

We would like to reduce this algebraic calculation to a graph-theoretic one.
To do so, we need to introduce a few notions.

##### Co-Independence Number

Let $\Gamma=(V,\mu)$ be an arbitrary graph. A subset of nodes $S$ in $\Gamma$
is called an _independent set_ if no two nodes in $S$ are adjacent. The
_independent number_ $\alpha(\Gamma)$ is defined as the size of the largest
independent set. The independent sets $S$ with $|S|=\alpha(\Gamma)$ are called
_maximum_. We denote the number of maximum independent sets by $m(\Gamma)$.
Note that
$\alpha(\Gamma_{1}\sqcup\Gamma_{2})=\alpha(\Gamma_{1})+\alpha(\Gamma_{2})$ and
$m(\Gamma_{1}\sqcup\Gamma_{2})=m(\Gamma_{1})m(\Gamma_{2})$. While the
definitions so far are standard, the following definition of _co-independence
number_ $\varepsilon(\Gamma)$ is tailored for our specific purposes:

$\varepsilon(\Gamma)\equiv\frac{|\Gamma|}{2}-\alpha(\Gamma)$ (93)

Using this notion, if $\Gamma$ has $2k$ nodes, a necessary condition for
$\chi_{\Gamma}\neq 0$ is $\varepsilon(\Gamma)=0,1$. If $\Gamma$ is a regular
graph, its co-independence number is non-negative. The quantity
$\varepsilon(\Gamma)$ is an integer/half-integer iff $|\Gamma|$ is even/odd
respectively. Moreover, we have
$\varepsilon(\Gamma_{1}\sqcup\Gamma_{2})=\varepsilon(\Gamma_{1})+\varepsilon(\Gamma_{2})$.

##### Graph Residue

Suppose $\Gamma$ is a $2r$-regular graph with $\varepsilon(\Gamma)=1$, say in
$2k$ nodes. Given any maximum independent set $S$ (i.e., $|S|=k-1$), the graph
$\Gamma-S$ is obtained from $\Gamma$ by deleting the nodes in $S$ (and all
arrows with one end in $S$). We also define $n(S)$ as the number of arrows of
the form $(V-S)\to S$ (i.e., arrows ending in $S$). We say $\Gamma$ is
reductive if for all maximum independent sets $S$:

  1. 1.

$n(S)=$even.

  2. 2.

There exists a graph $\Gamma^{\star}$ with $k+1$ nodes, called the _residue_
of $\Gamma$, such that $\Gamma-S$ is isomorphic to $\Gamma^{\star}$.

If $\Gamma$ is reductive in $2k$ nodes, then the corresponding $\chi$
polynomial is relatively simple (normalization chosen for future convenience)

$\chi_{\Gamma}(z_{1},\cdots,z_{k+1})=\frac{1}{kk!}\frac{m(\Gamma)}{u(\Gamma)}(\mathscr{S}\circ\mathscr{M})[\Gamma^{\star}]$
(94)

Moreover, since $\Gamma^{\star}$ has $2r$ arrows, this polynomial is an
element of $\mathcal{T}_{k+1}^{2r}$ as expected. Note that if
$\varepsilon(\Gamma)=1$, then $\chi_{\Gamma}(1,0^{\times k})=0$.

Let us now study the concepts mentioned above in the context of the graphic
bases of $\mathcal{B}_{2r}$. We want to establish three rules of thumb for a
“good” graphic basis. We do not know if these rules can always be satisfied,
but in our explicit construction for $r=2,3,4$, it is possible to abide by
them. Suppose $\mathcal{B}_{2r}$ has $N(d)\neq 0$ basic invariants of degree
$d$. We denote the corresponding graphs by $G_{d}^{(i)}$, with $1\leq i\leq
N(d)$. We will omit the superindex $(i)$ if $N(d)=1$. Due to ansatz (80), we
have a special interest in graphs $\Gamma$ with an even number of nodes that
are some disjoint union of basic graphs. We call such a graph _constructible_.
To get to the first rule, let us first recite a fact: The algebra
$\mathcal{B}_{2r}$ always has exactly one basic invariant of degree two.
Graphically, this invariant is represented by

$G_{2}=\vbox{ \hbox{ \includegraphics{I1} } }$ (95)

and clearly has $\varepsilon(G_{2})=0$. For the first rule, we desire that
“$\varepsilon(G_{d}^{(i)})>0$ for all $d>2$”. This would make $G_{2}$ the only
basic graph with a co-independence number of zero. As the second rule, we put
in place: “any constructible graph either has $\varepsilon>1$ or is
reductive”. Among all of the constructible graphs which do not have a
component equal to $G_{2}$, only two types have co-independence number one;
namely

$\displaystyle\mathcal{G}^{(i,j)}_{d_{1},d_{2}}=G_{d_{1}}^{(i)}\sqcup
G_{d_{2}}^{(j)}$ $\displaystyle\text{if
}\varepsilon(G_{d_{1}}^{(i)})=\varepsilon(G_{d_{2}}^{(j)})=\frac{1}{2}$ (96a)
$\displaystyle\mathcal{G}^{(i)}_{d}=G_{d}^{(i)}$ $\displaystyle\text{if
}\varepsilon(G_{d}^{(i)})=1$ (96b)

In general, any constructible graph $\Gamma$, which has $2k$ nodes and
satisfies $\varepsilon(\Gamma)=1$, is of the form $\Gamma=H\sqcup
G_{2}^{\sqcup(k-l)}$, where $H$ is one of the above graphs satisfying
$|H|=2l\leq 2k$. Computing the $\chi$ polynomial corresponding to
$\Gamma^{\star}$ (recall that by hypothesis such $\Gamma$ is reductive)

$\chi_{H\sqcup
G_{2}^{\sqcup(k-l)}}(z_{1},\cdots,z_{k+1})=\frac{1}{k(k)^{-}_{l}}\frac{m(H)}{u(H)}\sum_{i_{1}<i_{2}<\cdots<i_{l+1}}(\mathscr{S}\circ\mathscr{M})[H^{\star}](z_{i_{1}},\cdots
z_{i_{l+1}})$ (97)

reduces, basically, to the computation of symmetrized graph polynomial of the
residue $H^{\star}$. From now on, if $H=\mathcal{G}^{(i,j)}_{d_{1},d_{2}}$
(resp. $\mathcal{G}^{(i)}_{d}$), we use the notation
$\chi_{d_{1},d_{2}}^{(i,j)}$ (resp. $\chi_{d}^{(i)}$) instead of
$\chi_{H\sqcup G_{2}^{\sqcup(k-l)}}$. There is one more constructible graph
$\Gamma$ in $2k$ variables with non-vanishing $\chi_{\Gamma}$; namely
$\Gamma=G_{2}^{\sqcup k}$. The $\chi_{2}\equiv\chi_{G_{2}^{\sqcup k}}$ has a
simple universal form:

$\chi_{2}(z_{1},\cdots,z_{k+1})=\frac{1}{k}\sum_{i<j}(z_{i}-z_{j})^{2r}$ (98)

This brings us to the third rule: “For any $k\geq 1$, the collection

$\mathscr{B}=\\{\chi_{2}\\}\cup\\{\chi_{d_{1},d_{2}}^{(i,j)}\neq 0\mid
d_{1},d_{2}=\text{odd},\;d_{1}+d_{2}\leq 2k\\}\cup\\{\chi_{d}^{(i,j)}\neq
0\mid d=\text{even},\;2<d\leq 2k\\}$ (99)

is not redundant (i.e., there is no polynomial appearing twice), is linearly
independent, and spans $\mathcal{T}_{k+1}^{2r}$.” If these three rules can be
satisfied, which is the case at least for $r=1,2,3,4$, then we can find a
natural basis for $\mathcal{T}_{k+1}^{(2r)}$ directly from the Hilbert basis.
We call this the _residual basis_ of $\mathcal{T}_{k+1}^{(2r)}$.

######  Remark 4.

Due to universality of $\chi_{2}$ we can compute the specialization
$\xi^{2}_{a}=\chi_{2}(1,x^{\times a},0^{\times k-a})$ in general:

$\xi^{2}_{a}(x)=\frac{(a+1)(k-a)}{k}+\frac{a(k-a)}{k}\left(\frac{(x-1)^{2r}}{k-a}+x^{2r}-1\right)=1-\frac{2ar}{k}x+\cdots$
(100)

We will use this formula on a few occasions.

######  Remark 5.

Regarding the graph drawing conventions:
Before starting our construction of Hilbert bases, we need to clarify our
drawing convention for the graphs to appear. Let $G=(V,\mu)$ be a graph.
Recall that we only care about those graphs which satisfy $\mu(i,j)\mu(j,i)=0$
for any pair $i,j\in V$. This means that all arrows shared between $i,j$ will
have the same orientation. In our convention, if $i,j$ is such that
$\mu(i,j)+\mu(j,i)=2m$, then we will draw these as $2m$ _undirected_ edges
between $i,j$. One is free to orient those edges in either of the two ways (as
long as $\mu(i,j)\mu(j,i)=0$). We can afford this ambiguity since the
invariant $\Upsilon[G]$ is insensitive to inverting the direction of an arrow
with even multiplicity.

###  8.3 Binary Quartics ($2r=4$) and the Principal $(k,2)$-Wavefunction

##### Hilbert Basis

The algebra of invariants of the binary quartics $\mathcal{B}_{4}$ has two
basic invariants $I_{2},I_{3}$ with degrees $2,3$ respectively. The graphs for
these invariants are

$G_{2}=\vbox{ \hbox{ \includegraphics{quartic_2} } },\qquad G_{3}=\vbox{
\hbox{ \includegraphics{quartic_3} } }$ (101)

Computing the image of $G_{2},G_{3}$ under the tricoloring homomorphism $K$ we
have

$K[G_{2}]=E_{1},\qquad K[G_{3}]=\sqrt{E_{3}}$ (102)

The co-independence number of these graphs are $\varepsilon(G_{2})=0$ (as
expected) and $\varepsilon(G_{3})=1/2$. We have $u(G_{2})=m(G_{2})=2$,
$u(G_{3})=6$ and $m(G_{3})=3$. Using the ansatz (80), the principal
$\mathbb{Z}_{k}^{(2)}$-wavefunction can be written as

$\Psi=\frac{1}{k!^{2}}\sum_{m=0}^{\lfloor
k/3\rfloor}\alpha_{m}\Psi\left[\vbox{ \hbox{
\includegraphics[scale={.75}]{quartic_2} } }^{\sqcup(k-3m)}\bigsqcup\vbox{
\hbox{ \includegraphics[scale={.75}]{quartic_3} } }^{\sqcup(2m)}\right]$ (103)

We now proceed to find the constants $\alpha_{m}$.

##### Tricoloring Identity

To determine the $\mathbb{C}$-numbers $\alpha_{m}$, we put the tricoloring
identity (91) to use. We find that

$k!^{2}K[\Psi]=\sum_{\begin{subarray}{c}a,b,c\geq 0\\\
a+b+c=k\end{subarray}}\binom{k}{a,b,c}\frac{(t)_{k-a}(t)_{k-b}(t)_{k-c}}{(t)_{a}(t)_{b}(t)_{c}(t)_{k}}X^{a}Y^{b}Z^{c}=\sum_{m=0}^{\lfloor
k/3\rfloor}\alpha_{m}E_{1}^{k-3m}E_{3}^{m}$ (104)

Expanding the right-hand side in powers of $X,Y,Z$, this identity reduces to
(recall that $(x)_{n}^{-}=x(x-1)\cdots(x-n+1)$ is the falling factorial, while
$(x)_{n}=x(x+1)\cdots(x+n-1)$ is the Pochhammer symbol)

$\sum_{m=0}^{b}\alpha_{m}\frac{(b)_{m}^{-}(a)_{m}^{-}(k-a-b)_{m}^{-}}{(k)_{3m}^{-}}=\frac{(t+a)_{b}(t+k-b-a)_{b}}{(t)_{b}(t+k-b)_{b}}$
(105)

At the same time, due to the Saalschütz’s theorem (see [2, §2.2]), for any
non-negative integer $b$, we have the hypergeometric identity

${}_{3}F_{2}(-b,-a,b+a-k;t,1-k-t;1)\equiv\sum_{m=0}^{b}\binom{b}{m}\frac{(a)_{m}^{-}(k-a-b)_{m}^{-}}{(t)_{m}(t+k-1)_{m}^{-}}=\frac{(t+a)_{b}(t+k-a-b)_{b}}{(t)_{b}(t+k-b)_{b}}$
(106)

where ${}_{3}F_{2}(\alpha_{1},\alpha_{2};\alpha_{3},\beta_{1},\beta_{2};x)$ is
the generalized Gauss hypergeometric function. Consequently, the ansatz
constants are found to be

$\alpha_{m}=\frac{1}{m!}\frac{(k)_{3m}^{-}}{(t)_{m}(t+k-1)_{m}^{-}}$ (107)

To summarize, the principal $(k,2)$-wavefunction is the following polynomial:

$\Psi=\frac{1}{k!}\sum_{m=0}^{\lfloor
k/3\rfloor}\frac{1}{(k-3m)!m!}\frac{1}{(t)_{m}(t+k-1)_{m}^{-}}\>\Psi\left[\vbox{
\hbox{ \includegraphics[scale={.75}]{quartic_2} } }^{\sqcup(k-3m)}\sqcup\vbox{
\hbox{ \includegraphics[scale={.75}]{quartic_3} } }^{\sqcup 2m}\right]$ (108)

Note that $\varepsilon(G_{2}^{\sqcup k-3m}\sqcup G_{3,3}^{m})=m$. So only the
summands for $m=0,1$ contribute to the minimal polynomial of the
$(k,2)$-algebra.

##### Minimal Polynomial

It is obvious from the high degree of symmetry of $G_{3}$ that it leads to
reductive graphs. The residue of $G_{3,3}=G_{3}\sqcup G_{3}$ is then obtained
to be

$G_{3,3}^{\star}=\vbox{ \hbox{ \includegraphics[scale={.5}]{r2_star} } }$
(109)

We have $\chi_{2}(1,0^{\times k})=1$ and $\chi_{3,3}(1,0^{\times k})=0$. This,
in particular, shows that $\chi_{2},\chi_{3,3}$ are linearly independent. They
also span $\mathcal{T}_{k+1}^{4}$ since $\dim\mathcal{T}_{k+1}^{4}=2$ (for
$k\geq 3$). Thus, our three rules of thumb are satisfied. The minimal
polynomial can now be written as

$\chi=\chi_{2}+\beta\chi_{3,3}$

for some coefficient $\beta$. To find $\beta$ and the central charge $c$,
consider the specialization $\xi_{a}=\chi(1,x^{\times a},0^{\times k-a})$. We
calculate

$\displaystyle\xi_{a}^{2}\equiv\chi_{2}(1,x^{\times a},0^{\times
k-a})=\frac{(a+1)(k-a)}{k}+\frac{a(k-a)}{k}\left(\frac{(x-1)^{4}}{k-a}+x^{4}-1\right)$
(110) $\displaystyle\xi_{a}^{3,3}\equiv\chi_{2}(1,x^{\times a},0^{\times
k-a})=\frac{(a+1)_{2}^{-}(k-a)_{2}^{-}}{k(k)_{3}^{-}}x^{2}+\frac{(a)_{2}^{-}(k-a)_{2}^{-}}{k(k)_{3}^{-}}x^{2}\left(\frac{2(x-1)^{2}}{k-a-1}+x^{2}-1\right)$
(111)

and $\xi_{a}(x)=\xi_{a}^{2}(x)+\beta\xi^{3,3}_{a}(x)$. Now, since
$\xi_{a}(1)=C_{1,a}^{2}$, we can find $\beta$ as a function of $t$:

$\beta=\frac{(k)_{3}^{-}}{t(t+k-1)}$ (112)

On the other hand, interpreting $\xi_{1}(x)$ as a conformal block, we have
$\xi_{1}(x)=1-\gamma x+[\gamma(\gamma-1)/2+2h^{2}/c]x^{2}+O(x^{3})$.
Therefore, as discussed in subsection 6.4, since

$\xi_{1}(x)=1-\frac{4}{k}x+\left(\frac{6}{k}+\frac{2(k-1)(k-2)}{kt(t+k-1)}\right)x^{2}-\frac{4}{k}x^{3}+x^{4}$
(113)

we can read off the central charge

$c=\frac{4(k-1)t(t+k-1)}{(k+2t-2)(k+2t)}$ (114)

This is the same central charge Zamolodchikov-Fateev report in Ref. [52]. One
can, in principle, use the formula for $\xi_{a}(x)$, in conjunction with the
conformal block interpretation of $\xi_{a}(x)$ to find certain constraints on
the chiral weights of $\mathbb{Z}_{k}^{(2)}$-algebra. Nevertheless, due to the
complexity of the formulas, we will refrain from doing so.

######  Remark 6.

We have shown that, in the case of $\mathbb{Z}_{k}^{(2)}$-algebras, both the
minimal polynomial and the principal wavefunction is parametrized by $t$. In
other words, both polynomials carry the same information. The major agents
responsible for making this equivalence happen are (1) the constraint
$C_{a,b}=g_{a+b}/g_{a}g_{b}$, (2) the polynomiality of $C_{1,a}^{2}$.
Polynomiality is a consequence of the separability of the principal
wavefunction. We can reformulate the hidden constraint in the form ($a+b\leq
k$)

$\chi^{(k+b)}(1^{\times a+b},0^{\times k-a})=\chi(1^{\times b+1},0^{\times
k-b})\prod_{s=1}^{a-1}\frac{\chi(1^{\times s+b+1},0^{\times
k-s-b})}{\chi(1^{\times s+1},0^{\times k-s})}$ (115)

These puts some non-trivial constraints on the $(k+b)$-factors $\chi^{(k+b)}$.
It is unclear whether these constraints are also a result of separability. If
they are not, then $\mathbb{Z}_{k}^{(r)}$-WFs have even more structure than
originally anticipated. This raises the question: Suppose $\chi$ is the
minimal polynomial of $\mathbb{Z}_{k}^{(r)}$-algebra for some $t$. Is it
possible to build a separable $(k,4)$-clustering uniform sequence of
wavefunctions that is different from the $\mathbb{Z}_{k}^{(r)}$ sequence but
has minimal polynomial $\chi$? If such a sequence exists, then the Hamiltonian
$H_{\chi}$ will not have a unique ground state.

###  8.4 Binary Sextics ($2r=6$) and Principal $(k,3)$-Wavefunctions

Invariant | $I_{2}$ | $I_{4}$ | $I_{6}$ | $I_{10}$
---|---|---|---|---
Graph |  |  |  |
$u$ | $2$ | $24$ | $6$ | $8$
$m$ | $2$ | $4$ | $3$ | $1$
$K$ | $E_{1}$ | $0$ | $E_{3}$ | $E_{2}E_{3}$
$\varepsilon$ | $0$ | $1$ | $1$ | $1$
Residue | N/A |  |  |
Table 1: A graphic Hilbert basis for the algebra of invariants of binary
sextics. For each graph, its $u$-value, $m$-value, co-independence number
$\varepsilon$, and its image under $K$ is listed. The residue for graphs with
$\varepsilon=1$ is included as well. The colorings in the first row represent
all possible maximum independent sets (the number of colors is the $m$-value).
Note that removing each color leads to the graph residue.

As discussed previously, invariants of binary sextics have a Hilbert basis
$I_{2},I_{4},I_{6},I_{10}$ (hsop) together with an extra invariant $J_{15}$.
The degree-15 invariant $J_{15}$ squares to a polynomial in
$I_{2},I_{4},I_{6},I_{10}$. Since we are only interested in invariants with
even degrees, we may ignore $J_{15}$. We have gathered our choice for the
basic graphs in Table (1).

##### Minimal Polynomial

Although one can check the linear independence of
$\chi_{2},\chi_{4},\chi_{6},\chi_{10}$ directly, we prove this by computing
the specializations $\xi^{d}_{a}\equiv\chi_{d}(1,x^{\times a},0^{\times
k-a})$.

$\displaystyle\xi^{2}_{a}(x)$
$\displaystyle=\frac{(a+1)(k-a)}{k}+\frac{a(k-a)}{k}\left(\frac{(x-1)^{6}}{k-a}+x^{6}-1\right)$
(116a) $\displaystyle\xi^{4}_{a}(x)$
$\displaystyle=\frac{a(k-a)}{k^{2}(k-1)}x^{2}(x-1)^{2}$ (116b)
$\displaystyle\xi_{a}^{6}(x)$
$\displaystyle=\frac{(a+1)_{2}^{-}(k-a)_{2}^{-}}{k(k)_{3}^{-}}x^{3}+\frac{(a)^{-}_{2}(k-a)^{-}_{2}}{k(k)_{3}^{-}}x^{3}\left(\frac{2(x-1)^{3}}{k-a-1}+x^{3}-1\right)$
(116c) $\displaystyle\xi_{a}^{10}(x)$
$\displaystyle=\frac{(a+1)_{3}^{-}(k-a)}{k(k)_{5}^{-}}x^{4}+\frac{(a)_{3}^{-}(k-a)_{3}^{-}}{k(k)_{5}^{-}}x^{4}\left(\frac{3(x-1)^{2}}{k-a-2}+x^{2}-1\right)$
(116d)

Let us also write

$\chi=\beta_{2}\chi_{2}+\beta_{4}\chi_{4}+\beta_{6}\chi_{6}+\beta_{10}\chi_{10}$
(117)

for some coefficients $\beta_{d}$. To check linear independence, we put
$\chi\to 0$. Now $\xi_{a}^{2},\xi_{a}^{4},\xi_{a}^{6}$ and $\xi_{a}^{10}$ are
respectively $O(1),O(x^{2}),O(x^{3})$ and $O(x^{4})$. This immediately shows
linear independence. However, the true advantage of the residual basis is that
it grants us a certain degree of interpretability for the coefficients
$\beta_{d}$. To illustrate, since $\xi_{a}^{2}$ is the only specialization
surviving when $a=0$ (in fact, $\xi_{0}^{2}=1$) the coefficient $\beta_{2}$ is
controlled by the normalization of $\chi$ (namely, $\beta_{2}=1$). Realizing
$\xi_{a}(x)=\chi(1,x^{\times a},0^{\times k-a})$ as a conformal block, one
interprets $\beta_{4},\beta_{6}$ to be controlled by the central charge and
the $W_{3}$-weight of $\psi$ respectively. Finally, note that $\xi_{a}^{10}=0$
when $a=1$. Using the polynomiality of chiral weight, we can write the
$W_{4}$-weight of $\psi_{a}$ in the form

$h^{W_{4}}_{a}=\frac{a(k-a)}{k-3}\left[-\frac{(a-2)(k-a-2)}{(k-1)}w_{1}+\frac{(a-1)(k-a-1)}{2(k-2)}w_{2}\right]$

where $w_{1},w_{2}$ are the $W_{4}$-weights of $\psi,\psi_{2}$ respectively.
The coefficient $\beta_{10}$ is controlled by $w_{2}$. However, the better way
to interpret $\beta_{6},\beta_{10}$ would be in terms of structure constant
parameters $t_{1},t_{2}$. Note that $\xi_{a}(1)=C_{1,a}^{2}$. We have
$\xi_{a}^{4}(1)=0$; i.e. $\beta_{4}$ is a priori independent of $t_{1},t_{2}$.
The coefficients $\beta_{6},\beta_{10}$ are just an alternative
parametrization to $t_{1},t_{2}$. In other words, while $\beta_{4}$ is
controlled by the central charge, the $W_{3}$ and $W_{4}$ weights are
controlled by the central charge $c$ and the structure constant parameters
$t_{1},t_{2}$. As this analysis shows, the vectors
$\chi_{2},\chi_{4},\chi_{6},\chi_{10}$ of the residual basis have (to some
degree) specific roles in the language of CFT.

##### Tricoloring Homomorphism

Using the ansatz (80), we can write the $(k,3)$-wavefunction in the form

$\Psi=\frac{1}{k!^{2}}\sum_{\nu_{2}+2\nu_{4}+3\nu_{6}+5\nu_{10}=k}\alpha(\nu_{4},\nu_{6},\nu_{10})\Psi[G_{2}^{\sqcup\nu_{2}}\sqcup
G_{4}^{\sqcup\nu_{4}}\sqcup G_{6}^{\sqcup\nu_{6}}\sqcup
G_{10}^{\sqcup\nu_{10}}]$ (118)

In terms of ansatz coefficients, the $\beta$-coefficients of the minimal
polynomial $\chi$ (117) are $\beta_{4}=\alpha(1,0,0)$,
$\beta_{6}=\alpha(0,1,0)$ and $\beta_{10}=(0,0,1)$. The normalization also
forces $\alpha(0,0,0)=1$. As summarized in Table 1, we have
$K(G_{2})=E_{1},K(G_{4})=0,K(G_{6})=E_{3}$ and $K(G_{10})=E_{2}E_{3}$. The
tricoloring identity (91) now yields

$\sum_{\begin{subarray}{c}a,b,c\geq 0\\\
a+b+c=k\end{subarray}}\binom{k}{a,b,c}\prod_{i=1,2}\frac{(t_{i})_{k-a}(t_{i})_{k-b}(t_{i})_{k-c}}{(t_{i})_{a}(t_{i})_{b}(t_{i})_{c}(t_{i})_{k}}X^{a}Y^{b}Z^{c}=\sum_{\nu_{2}+3\nu_{6}+5\nu_{10}=k}\alpha(0,\nu_{6},\nu_{10})E_{1}^{\nu_{2}}E_{2}^{\nu_{10}}E_{3}^{\nu_{6}+\nu_{10}}$
(119)

Although we do not have a closed formula, this identity can be used to compute
the constants $\alpha(0,\nu_{6},\nu_{10})$ in terms of $t_{1},t_{2}$.
Unfortunately, however, except for $\alpha(1,0,0)$, we do not know of a
general formalism that allows us to compute $\alpha(\nu_{4},\nu_{6},\nu_{10})$
with $\nu_{4}\neq 0$.

######  Remark 7.

The machinery that allows us to extend our knowledge of $\alpha(0,1,0)$ and
$\alpha(0,0,1)$ is the tricoloring identity. Tricoloring identity takes
advantage of the constraint $C_{a,b}=g_{a+b}/g_{a}g_{b}$. As discussed in
Remark 6, this is highly non-trivial condition in terms of $(k+b)$-factors. In
other words, this constraint reveals some hidden structure in the principal
$\mathbb{Z}_{k}^{(r)}$-polynomial. If we did not know about this structure, we
would not have had much success in analysing the principal
$\mathbb{Z}_{k}^{(2)}$ wavefunctions either. It is possible that
$\mathbb{Z}_{k}^{(r)}$-algebra/wavefunctions, at least when $r\geq 3$, possess
extra structure/hidden constraints that would allow us to strengthen the
method presented in this paper and, for example, let us compute all of the
constants $\alpha(\nu_{4},\nu_{6},\nu_{10})$.

######  Remark 8.

A possible approach for finding the previously unknown ansatz constants (at
least a few more of them) is through conformal blocks similar to $\xi_{a}(x)$.
Define

$\displaystyle\xi_{a,b}(x)$
$\displaystyle=\lim_{w\to\infty}w^{-2r(k-b)}\Psi(w^{\times(k-b)},1^{\times
b},x^{\times a},0^{\times k-a})$ (120)
$\displaystyle=\left<\psi_{b}^{+}(\infty)\psi_{b}(1)\psi_{a}(x)\psi_{a}^{+}(0)\right>(1-x)^{ab\gamma}x^{2h_{a}}$
$\displaystyle=(1-x)^{ab\gamma}\left[1+\sum_{s\geq
2}\left(\sum_{i}\widehat{h}_{a}^{i}h^{i}_{b}\delta_{s,\Delta_{i}}\right)\right](-x)^{s}{}_{2}F_{1}(s,s;2s;x)$

The idea is to use the chiral weights’ polynomialities to reduce this block’s
unknowns to a “handful” of constants. Unfortunately, this method comes with
severe limitations. Aside from the fact that a lot of tedious computations are
involved in this method, not knowing any details about the chiral algebras is
restrictive as well. For one, at the very least, we need to know how many
simple fields exist at each level. Equally problematic is that to compute the
weight of a qNOP of two simple fields (neither being the energy-momentum
tensor), some limited knowledge of the Lie algebra of the modes is required.

Invariant | $I_{2}$ | $I_{3}$ | $I_{4}$ | $I_{5}$
---|---|---|---|---
Graph |  |  |  |
$u$ | $2$ | $6$ | $16$ | $120$
$\varepsilon$ | $0$ | $1/2$ | $1$ | $3/2$
$K$ | $E_{1}$ | $\sqrt{E_{3}}$ | $0$ | $0$

Invariant | $I_{6}$ | $I_{7}$ | $J_{8}$
---|---|---|---
Graph |  |  |
$u$ | $8$ | $8$ | $4$
$\varepsilon$ | $1$ | $1/2$ | $1$
$K$ | $0$ | $E_{2}\sqrt{E_{3}}$ | $0$

Invariant | $J_{9}$ | $J_{10}$
---|---|---
Graph |  |
$u$ | $8$ | $32$
$\varepsilon$ | $3/2$ | $2$
$K$ | $0$ | $0$

Table 2: A graphic Hilbert basis for the algebra of invariants of binary
octavics. For each graph, its $u$-value, its co-independence number
$\varepsilon$, and its image under $K$ are also listed. One reads from this
table that $I_{5},J_{9},J_{10}$ have $\varepsilon>1$ and therefore will not
contribute to the residual basis. One also reads that the only basic
invariants with $K\neq 0$ are $I_{2},I_{3}$ and $I_{7}$.

###  8.5 Binary Octavics ($2r=8$) & Principal $(k,4)$ Wavefunctions

Our inability to compute the ansatz constants gets even worse in the case of
$\mathbb{Z}_{k}^{(4)}$-algebras. Despite this, the brief analysis of binary
octavics brings to light new oddities we should expect for $r\geq 4$.
$\mathcal{B}_{8}$ has an hsop consisting of invariants
$I_{2},I_{3},I_{4},I_{5},I_{6},I_{7}$. In addition to the hsop, the Hilbert
basis contains invariants $J_{8},J_{9},J_{10}$. The algebra $\mathcal{B}_{8}$
can be written as

$\mathcal{B}_{8}=(1+J_{8}+J_{9}+J_{10}+J_{8}J_{10})\>\mathbb{C}[I_{2},I_{3},I_{4},I_{5},I_{5},I_{6},I_{7}]$
(121)

This is the first instance that we encounter a situation where we need to keep
track of the $J$-invariants appearing on a Hilbert basis. This will be a
common sight in $r\geq 4$. The graphic representation of these invariants can
be found in Table (2). The residual basis is described in Fig. 3. A general
invariant $F$ of even degree $2k$ can now be written in the form

$\displaystyle k!^{2}F$