# Quartic Hamiltonians, and higher Hamiltonians at next-to-leading order, for
the affine $\bm{\mathfrak{sl}_{2}}$ Gaudin model

Tommaso Franzini111email<EMAIL_ADDRESS>Charles Young222email
<EMAIL_ADDRESS>Department of Physics, Astronomy and Mathematics,
University of Hertfordshire, Hatfield, AL10 9AB, United Kingdom

###### Abstract

In this work we will use a general procedure to construct higher local
Hamiltonians for the affine $\mathfrak{sl}_{2}$ Gaudin model. We focus on the
first non-trivial example, the quartic Hamiltonians. We show by direct
calculation that the quartic Hamiltonians commute amongst themselves and with
the quadratic Hamiltonians which define the model.

We go on to introduce a certain next-to-leading-order semi-classical limit of
the model. In this limit, we are able to write down the full hierarchy of
higher local Hamiltonians and prove that they commute.

###### Contents

  1. 1 Introduction
  2. 2 Vacuum Verma modules for $\widehat{\mathfrak{sl}}_{2}$
  3. 3 Construction of higher Hamiltonians
  4. 4 Quartic Hamiltonian
  5. 5 Higher local Hamiltonians to sub-leading order
  6. A Full expression for $\mathsf{A}_{1,3}(z,w)$ and $\mathsf{B}_{1,3}(z,w)$

##  1 Introduction

Gaudin models represent interesting theories that find applications in several
contexts in mathematical physics. In particular, they provide a general
framework to study rich classes of classical and quantum integrable 1+1
dimensional integrable theories [Gau76, Gau14, Vic18, Lac18]. Recently, it has
also been found that classical untwisted Gaudin models provide a dual
description of such theories to the one given by 4D holomorphic Chern-Simons
theories [Vic21, LV21].

In this paper we will focus on _quantum Gaudin models_. To define them, one
needs a collection of distinct points
$\\{z_{1},\dots,z_{N}\\}\in\mathbb{C}\mathbb{P}^{1}$ on the Riemann sphere and
a Kac-Moody algebra $\mathfrak{g}$, which can be of finite or affine type.
These models are defined by the quadratic Hamiltonians, namely

$\mathscr{H}_{i}=\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\kappa_{ab}\frac{I^{a(i)}I^{b(j)}}{z_{i}-z_{j}},\qquad
i=1,\dots,N,$ (1)

in the (suitably completed, in the affine case) tensor product
$U(\mathfrak{g})^{\otimes N}$. Here $I^{a(i)}$ are the generators of
$\mathfrak{g}$, defined at site $i$, and $\kappa$ is the invariant bilinear
form on $\mathfrak{g}$.

Finite-type quantum Gaudin models have been deeply studied. In particular, it
is known that the quadratic Hamiltonians (1) are part of a larger family of
mutually commuting operators $\mathscr{B}\in U(\mathfrak{g}^{\otimes N})$
called the Bethe or Gaudin subalgebra [Fre04, Tal04, Ryb06]. Moreover, these
higher Hamiltonians can be diagonalized with an elegant form of Bethe Ansatz,
where the eigenvector is the Schechtman-Varchenko vector and the eigenvalues
are encoded as functions on a space of
$\prescript{L}{}{{{\mathfrak{g}}}}$-opers, the Langlands dual algebra of
$\mathfrak{g}$ [FFR94, MTV06, MTV07, MV02, MV04].

On the other hand, affine-type Gaudin models are still far from being fully
understood.

One approach to affine Gaudin models was proposed in a recent paper [KL21]. In
this work the authors propose an integrable model called _generalized_ affine
$\mathfrak{sl}_{2}$ Gaudin model, which reproduces the usual
$\widehat{\mathfrak{sl}}_{2}$ model in a certain limit. Their construction is
based on a new realization of the subalgebra $U_{q}(\mathfrak{b}_{-})$ through
a new class of vertex operators, and fits affine Gaudin models into the
general procedure first given in the seminal work [BLZ97].

Another approach, first proposed in the pioneering work [FF07], is to treat
affine Gaudin models in closer analogy with their finite-type cousins. In
particular, the spectrum of higher Hamiltonians should be described by
suitable functions on a space of opers. Some further conjectures of how this
might work, at least for the local Hamiltonians, were made in [LVY18], where
it was conjectured that the eigenvalues of higher local Hamiltonians of the
affine Gaudin models, as well as the Hamiltonians themselves, are given by
hypergeometric-type integrals in the spectral plane, namely

$\widehat{Q}_{n}^{\gamma}=\int_{\gamma}\mathscr{P}(z)^{-n/2}\varsigma_{n}(z)_{[0]}\text{d}z,$
(2)

where $n$ lives is a (multi)set of indices given by the exponents of the
algebra $\mathfrak{g}$, $\mathscr{P}$ is a certain multi-valued function
defined by the data of the levels $k_{i}$ of the modules attached to the
marked points $z_{i}$, $\gamma$ is a Pochhammer contour in the spectral plane
around any two of these points (see e.g. fig. 1) and $\varsigma_{n}(z)_{[0]}$
can be thought as the Hamiltonian density.

Figure 1: An example of Pochhammer contour $\gamma$ around any two marked
points.

The key step in computing the higher Hamiltonians is to characterize these
Hamiltonian densities, which are obtained by defining a suitable state
$\varsigma_{n}(z)\in\mathcal{V}$ for each given exponent $n$. In order to do
that, it is possible to exploit the general properties consistent Hamiltonians
must obey: they have to commute with the generators
$\\{I^{\alpha}_{n}\\}_{n\in\mathbb{Z}}^{\alpha=1,\dots,\dim\mathring{\mathfrak{g}}}$
of the algebra $\mathfrak{g}$ defining the model as well as amongst themselves
(here $\mathring{\mathfrak{g}}$ denotes the underlying finite algebra). As we
will see in section 4.7, this is equivalent to the following requirements

$\begin{split}\Delta I^{\alpha}_{n\geq 0}\varsigma_{m}(z)=0&\mod\text{twisted
derivatives},\\\ \varsigma_{n}(z)_{(0)}\varsigma_{m}(z)=0&\mod\text{twisted
derivatives and translates},\end{split}$ (3)

where $\Delta I^{\alpha}_{n\geq 0}$ represents the diagonal action of the
positive modes of the generators of the algebra $\mathfrak{g}$ and the zero
mode $\varsigma_{n}(z)_{(0)}$ is intended in the vertex algebra sense (see
section 2.4). We give a precise definition of translation and twisted
derivative of a state in sections 2.4 and 4.4.

The general expectation is that there exists a state $\varsigma_{m}(z)$ for
every exponent $m$ of the affine algebra $\mathfrak{g}$, and that it takes the
following form

$\varsigma_{m}(z)=t_{i_{1},\dots,i_{m+1}}I^{i_{1}}_{-1}(z)\cdots
I^{i_{m+1}}_{-1}(z)\ket{0}+\text{quantum corrections},$ (4)

where $I(z)=\sum_{i}I^{(i)}/(z-z_{i})$ and $t$ is a certain symmetric
invariant tensor of $\mathring{\mathfrak{g}}$. This particular structure is
justified by the semi-classical counterpart of these models, which have been
thoroughly studied [Eva+99, Eva01, LMV17]; in particular, the precise choice
of symmetric invariant tensors needed to ensure that the Hamiltonians Poisson-
commute is well understood and related to Drinfel’d-Sokolov reduction [Eva01,
DS85]. In the very simplest cases, including the cubic Hamiltonian in type
$\widehat{\mathfrak{sl}}_{M\geq 3}$, there are no quantum corrections needed
[LVY20]. The first case in which quantum corrections are present appeared in
the recent paper [KLT22]: it is the example of the quartic Hamiltonian in type
$\widehat{\mathfrak{sl}}_{2}$. In this paper, we consider this example in
detail. In particular, we carry out the full computation to show that the
resulting quartic Hamiltonians commute amongst themselves (see theorem 4.5.1
and proposition 4.7.2).

This direct calculation reveals the following fact: it turns out that _any_
density $\varsigma_{3}(z)$ obeying the first of the two conditions in eq. 3
(i.e. the one which ensures the Hamiltonian commutes with the generators of
the affine algebra), automatically also obeys the second condition (which is
needed for the Hamiltonians to commute amongst themselves) at least for
$n=1,3$, as shown in corollary 4.5.1 below. We should stress that this
property is, for the moment, highly non-obvious and suggestive; it would be
good to get a more systematic understanding of why it should be true.

Already in this case of the exponent $n=3$, i.e. of quartic Hamiltonians, the
direct computations needed are very lengthy. This is especially true of the
computations needed to show the mutual commutativity of the quartic
Hamilonians. For higher exponents $n\geq 5$, direct calculations become
computationally difficult even with the aid of computer algebra, but we are
able to prove a result for all exponents by truncating to the next-to-leading
order in $\hbar$. To introduce the dependence from the formal parameter
$\hbar$, we perform a re-scaling of the generators, namely
$I\to\tilde{I}\equiv\hbar I$, $k\to\tilde{k}\equiv k$, in such a way that
every time we perform a commutation, we introduce a factor of $\hbar$. This
procedure allows us to identify different quantum corrections by their $\hbar$
dependence. In this spirit, we will then work modulo terms of order
$\hbar^{2}$. We show that, modulo such terms, the Hamiltonian density for each
exponent $2n-1$, $n\in\mathbb{Z}_{\geq 1}$, of $\widehat{\mathfrak{sl}}_{2}$,
takes the following form

$\begin{split}\tilde{\varsigma}&{}_{2n-1}(z)=t_{i_{1},\dots,i_{2n}}\tilde{I}^{i_{1}}_{-1}(z)\tilde{I}^{i_{2}}_{-1}(z)\cdots\tilde{I}^{i_{2n}}_{-1}(z)\ket{0}\\\
&\qquad+\hbar\frac{n(2n+1)(2n-2)}{(2n-1)}t_{i_{1},\dots,i_{2n-4}}f^{abc}\tilde{I}^{a}_{-2}(z)\tilde{I}^{b\prime}_{-1}(z)\tilde{I}^{(c}_{-1}(z)\tilde{I}^{i_{1}}_{-1}(z)\cdots\tilde{I}^{i_{2n-4})}_{-1}(z)\ket{0}\end{split}$

and prove that the resulting Hamiltonians commute up to and including terms of
order $\hbar^{2}$.

***

The paper is organized as follows.

In section 2 we recall the main ideas behind the theory of Kac-Moody algebras,
their local completion and the concept of vacuum Verma module. We also recall
the definition and the main features of vertex algebras.

In section 3 we define the algebra of observables of the Gaudin model and we
recall the definition of invariant tensor of an algebra $\mathfrak{g}$,
focusing in particular on the case of $\widehat{\mathfrak{sl}}_{2}$.

In section 4 we define the meromorphic states $I(z)$, which represent the
building blocks for the states we want to define. We also present their
commutation relations and we describe the gradation they give rise, which will
be of fundamental importance to prove the main results of the paper. After
that, we characterize the space of states that vanish under the diagonal
action of zero modes: this is the first step to define higher Gaudin
Hamiltonians. At this point we will present all quantum corrections to the
quartic state and we will explain why they appear in this specific example and
not in the other known ones. In the following subsection, we restrict the
number of possible states by asking they be singular up to twisted derivative
under the diagonal action of positive modes. We focus on the quadratic and
quartic states. Afterwards, we prove the main result of the paper, where we
also ask that the 0th product (in the vertex algebra sense) of these states
vanishes modulo twisted derivatives and translates. We will show that there is
one quartic state, up to re-scaling and modulo the addition of translates and
twisted derivatives, that satisfies this last requirement. At this point,
having a precise definition of the quartic state, we describe the general
construction of the higher Hamiltonians, in the spirit of [LVY18]. We will
explain in detail why we ask for these properties and why they are important
at the level of the quantum Hamiltonians.

In section 5 we try to solve the same problem from a different point of view.
Instead of focusing on one specific Hamiltonian, trying to work out its
explicit definition, we want to characterize all Hamiltonians at arbitrary
$n$, but working at sub-leading order. In order to do that we introduce the
formal parameter $\hbar$ by making a particular re-scaling of the generators
of the algebra. We will then prove similar theorems to those of the previous
sections, working modulo terms at order $\hbar^{3}$.

Acknowledgements. The authors would like to thank Sylvain Lacroix for helpful
discussions and for providing many detailed comments. The research of Charles
Young is supported by the Leverhulme Trust, Research Project Grant number
RPG-2021-092.

##  2 Vacuum Verma modules for $\widehat{\mathfrak{sl}}_{2}$

###  2.1 Loop realization of $\widehat{\mathfrak{sl}}_{2}$

We define the loop algebra
$\mathfrak{sl}_{2}[t,t^{-1}]=\mathfrak{sl}_{2}\otimes\mathbb{C}[t,t^{-1}]$ as
the algebra of Laurent polynomials in a formal variable $t$ with coefficient
in the finite-dimensional Lie algebra $\mathfrak{sl}_{2}$. The Lie brackets on
this algebra are given by

$[a\otimes f(t),b\otimes g(t)]=[a,b]_{\mathfrak{sl}_{2}}\otimes f(t)g(t),$ (5)

where $f(t)$ and $g(t)$ are arbitrary Laurent polynomials in
$\mathbb{C}[t,t^{-1}]$.

Let
$(\cdot|\cdot):\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}\rightarrow\mathbb{C}$
be the canonically normalized symmetric invariant bilinear form on
$\mathfrak{sl}_{2}$. It is given by taking the trace in the defining
representation:

$(a|b)\vcentcolon=\Tr(ab),$ (6)

It is possible to extend the loop algebra by a one-dimensional central element
$\mathbb{C}\mathsf{k}$,

$0\longrightarrow\mathbb{C}\mathsf{k}\longrightarrow\widehat{\mathfrak{sl}}_{2}\longrightarrow\mathfrak{sl}_{2}[t,t^{-1}]\longrightarrow
0.$ (7)

This extension is called the affine Lie algebra $\widehat{\mathfrak{sl}}_{2}$,
whose commutation relations are

$\displaystyle[a\otimes f(t),b\otimes g(t)]=[a,b]_{\mathfrak{sl}_{2}}\otimes
f(t)g(t)-(\text{res}_{t}f\text{d}g)(a|b)\mathsf{k},$ (8)
$\displaystyle[\mathsf{k},\cdot]=0.$ (9)

We shall use the notation

$a_{n}\vcentcolon=a\otimes t^{n},\quad\text{for $a\in\mathfrak{sl}_{2}$ and
$n\in\mathbb{Z}$}.$ (10)

The commutation relations can be equivalently written as

$[a_{m},b_{n}]=[a,b]_{n+m}-n\delta_{n+m,0}(a|b)\mathsf{k}.$ (11)

We can add to this algebra a one-dimensional derivation $\mathsf{d}$, such
that $[\mathsf{d},\mathsf{k}]=0$ and $[\mathsf{d},a\otimes f(t)]=a\otimes
t\partial_{t}f(t)$, for all $a\in\mathfrak{sl}_{2}$ and
$f(t)\in\mathbb{C}[t,t^{-1}]$. It is possible to show that this algebra is
isomorphic to the Kac-Moody algebra over $\mathbb{C}$ of type
$\mathsf{A}_{1}^{(1)}$, see e.g. [Kac90, ch. 7],

$\mathfrak{g}=\widehat{\mathfrak{sl}}_{2}\oplus\mathbb{C}\mathsf{d}.$ (12)

###  2.2 $\widehat{\mathfrak{sl}}_{2}$ as a Kac-Moody algebra

The Cartan matrix for the Kac-Moody algebra of type ${}^{1}\mathsf{A}_{1}$ is
defined as
$A=(a_{i,j})_{i,j=0}^{1}=(2\delta_{i,j}-\delta_{i+1,j}-\delta_{i-1,j})_{i,j=0}^{1}$.
The Cartan decomposition is given by
$\mathfrak{g}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{+}$. The
Chevalley-Serre generators are $\\{e_{i}\\}_{i=0}^{1}\subset\mathfrak{n}_{+}$,
$\\{f_{i}\\}_{i=0}^{1}\subset\mathfrak{n}_{-}$ while
$\\{\check{\alpha}_{i}\\}_{i=0}^{1}\subset\mathfrak{h}$ and
$\\{\alpha_{i}\\}_{i=0}^{1}\subset\mathfrak{h^{\ast}}$ are respectively a
basis for the Cartan subalgebra of simple coroots of $\mathfrak{g}$ and a
basis for the dual Cartan subalgebra of simple roots of $\mathfrak{g}$. The
latter are related via the canonical pairing between the Cartan algebra and
its dual,
$\langle\cdot,\cdot\rangle:\mathfrak{h}^{\ast}\times\mathfrak{h}\rightarrow\mathbb{C}$

$\langle\alpha_{i},\check{\alpha}_{j}\rangle=a_{i,j}.$ (13)

The fundamental commutation relations in $\mathfrak{g}$ are

$\begin{gathered}=\langle\alpha_{i},x\rangle
e_{i},\qquad[x,f_{i}]=-\langle\alpha_{i},x\rangle f_{i},\\\
[x,x^{\prime}]=0,\qquad[e_{i},f_{j}]=\check{\alpha}_{i}\delta_{ij},\end{gathered}$
(14)

where $x,x^{\prime}\in\mathfrak{h}$ and $i,j=0,1$, together with the Serre
relations

$(\text{ad}e_{i})^{1-a_{ij}}e_{j}=0,\qquad(\text{ad}f_{i})^{1-a_{ij}}f_{j}=0.$
(15)

The Kac-Moody algebra $\mathfrak{g}$ has a central element
$\mathsf{k}=\sum_{i=0}^{1}\check{\alpha}_{i}$, which spans a one-dimensional
centre. A basis for the Cartan subalgebra $\mathfrak{h}$ is given by the
coroots $\\{\check{\alpha}_{i}\\}_{i=0}^{1}$ together with the derivation
element $\mathsf{d}$, which by definition satisfies

$\langle\alpha_{i},\mathsf{d}\rangle=\delta_{i,0}.$ (16)

If we remove the 0th row and column from $A$, we obtain the Cartan matrix
corresponding to the finite dimensional Lie algebra $\mathfrak{sl}_{2}$. This
subalgebra of $\mathfrak{g}$ is generated by $e_{1}\in\mathfrak{n}_{+}$,
$f_{1}\in\mathfrak{n}_{-}$ and $\check{\alpha}_{1}\in\mathfrak{h}$.

###  2.3 Local completion and vacuum Verma modules

For any $k\in\mathbb{C}$, let us define $U_{k}(\widehat{\mathfrak{sl}}_{2})$
as the quotient of the universal enveloping algebra
$U(\widehat{\mathfrak{sl}}_{2})$ of $\widehat{\mathfrak{sl}}_{2}$ by the two-
sided ideal generated by $\mathsf{k}-k$. For each $n\in\mathbb{Z}_{\geq 0}$,
let us introduce the left ideal
$J_{n}=U_{k}(\widehat{\mathfrak{sl}}_{2})\cdot(\mathfrak{sl}_{2}\otimes
t^{n}\mathbb{C}[t])$. The inverse limit

$\widetilde{U_{k}}(\widehat{\mathfrak{sl}}_{2})=\varprojlim\faktor{U_{k}(\widehat{\mathfrak{sl}}_{2})}{J_{n}}$
(17)

is a complete topological algebra, called the local completion of
$U_{k}(\widehat{\mathfrak{sl}}_{2})$ at level $k$. With this definition, the
elements of $\widetilde{U_{k}}(\widehat{\mathfrak{sl}}_{2})$ are possibly
infinite sums of the type $\sum_{m\geq 0}a_{m}$ of elements $a_{m}\in
U_{k}(\widehat{\mathfrak{sl}}_{2})$ which do truncate to finite sums when
working modulo any $J_{n}$.

A module $\mathscr{M}$ over $\widehat{\mathfrak{sl}}_{2}$ is said to be smooth
if, for all $a\in\mathfrak{sl}_{2}$ and all $v\in\mathscr{M}$, $a_{n}v=0$ for
sufficiently large $n$. A module $\mathscr{M}$ has level $k$ if $\mathsf{k}-k$
acts as zero on $\mathscr{M}$. Any smooth module of level $k$ over
$\widehat{\mathfrak{sl}}_{2}$ is also a module over the completion
$\widetilde{U_{k}}(\widehat{\mathfrak{sl}}_{2})$.

We can identify the subalgebra of positive modes
$\mathfrak{sl}_{2}[t]\oplus\mathbb{C}\mathsf{k}\subset\widehat{\mathfrak{sl}}_{2}$
and introduce the one-dimensional representation $\mathbb{C}\ket{0}^{k}$
defined by

$(\mathsf{k}-k)\ket{0}^{k}=0,\qquad\qquad a_{n}\ket{0}^{k}=0\quad\text{for all
$n\geq 0$, $a\in\mathfrak{sl}_{2}$}.$ (18)

We define $\mathbb{V}_{0}^{k}$, the vacuum Verma module at level $k$, as the
induced smooth $\widehat{\mathfrak{sl}}_{2}$-module

$\mathbb{V}_{0}^{k}=U(\widehat{\mathfrak{sl}}_{2})\otimes_{U(\mathfrak{sl}_{2}[t]\oplus\mathbb{C}\mathsf{k})}\mathbb{C}\ket{0}^{k}$
(19)

This vector space is spanned by monomials of the form $a_{p}\cdots
b_{q}\ket{0}^{k}$, with $a,\dots,b\in\mathfrak{sl}_{2}$ and strictly negative
mode numbers $p,\dots,q\in\mathbb{Z}_{<0}$. We call these vectors states.

Let us denote by $[T,\cdot]$ the derivation on
$U_{k}(\widehat{\mathfrak{sl}}_{2})$ defined by $[T,a_{n}]=-na_{n-1}$ and
$[T,1]=0$. By setting $T(X\ket{0}^{k})=[T,X]\ket{0}^{k}$ for any $X\in
U_{k}(\widehat{\mathfrak{sl}}_{2})$, one can interpret $T$ as a translation
operator $T:\mathbb{V}_{0}^{k}\rightarrow\mathbb{V}_{0}^{k}$: the reason for
this identification will be clear in the next section.

###  2.4 Vertex algebra structure

As we now recall, the vacuum Verma module defined in the previous section has
the structure of vertex algebra. Namely, we have the state-field map
$Y(\cdot,x)$, which for every state $A\in\mathbb{V}_{0}^{k}$ associates a
formal power series in the variable $x$,

$\begin{split}Y(\cdot,x):\mathbb{V}_{0}^{k}&\longrightarrow\operatorname{Hom}(\mathbb{V}_{0}^{k},\mathbb{V}_{0}^{k}((x)))\\\
A&\longmapsto Y(A,x)=\sum_{n\in\mathbb{Z}}A_{(n)}x^{-n-1}\end{split}$ (20)

where $A_{(n)}\in\operatorname{End}(\mathbb{V})$ is the $n$th mode of $A$. By
definition, if $A=a_{-1}\ket{0}^{k}$ for some $a\in\mathfrak{sl}_{2}$, then
$A_{(n)}=a_{n}$ for all $n\in\mathbb{Z}$, i.e.

$Y(a_{-1}\ket{0}^{k},x)=\sum_{n\in\mathbb{Z}}a_{n}x^{-n-1}.$ (21)

The fields for all other states can be obtained with the following properties

$Y(TA,x)=\partial_{z}Y(A,x),\qquad\qquad Y(A_{(-1)}B,x)=:Y(A,x)Y(B,x):$ (22)

where we have introduced the normal ordered product between fields

$:Y(A,x)Y(B,x):\,=\left(\sum_{m<0}A_{(m)}z^{-m-1}\right)Y(B,x)+Y(B,x)\left(\sum_{m\geq
0}A_{(m)}x^{-m-1}\right).$ (23)

In fact, any state $C\in\mathbb{V}_{0}^{k}$ can be written as $C=a_{-n}B$, and
using eqs. 22 and 23 we can always explicitly compute $Y(C,x)$. Summarising
what we have introduced so far, we have

  * •

a space of states $\mathbb{V}_{0}^{k}$,

  * •

a vacuum vector $\ket{0}^{k}\in\mathbb{V}_{0}^{k}$,

  * •

a translation operator $T\in\operatorname{End}(\mathbb{V}_{0}^{k})$,

  * •

the state field map $Y(\cdot,x)$ as in eq. 20.

This structure, together with some additional properties (see e.g. [FB01,
§2.4.4]), defines a vertex algebra on $\mathbb{V}_{0}^{k}$.

##  3 Construction of higher Hamiltonians

###  3.1 The algebra of observables

Let us introduce a set of complex numbers $\bm{k}=\\{k_{i}\\}_{i=1}^{N}$,
where $N\in\mathbb{Z}_{>0}$ and $k_{i}\neq-2$ for all $i=1,\dots,N$. Consider
the following tensor product of vacuum Verma modules

$\mathbb{V}_{0}^{\bm{k}}=\mathbb{V}_{0}^{k_{1}}\otimes\dots\otimes\mathbb{V}_{0}^{k_{N}}.$
(24)

This space can be interpreted as a module over the direct sum of $N$ copies of
$\widehat{\mathfrak{sl}}_{2}$. Let us denote by
$A^{(i)}\in\widehat{\mathfrak{sl}}_{2}^{\oplus N}$ the copy of
$A\in\widehat{\mathfrak{sl}}_{2}$ in the $i$th direct summand.

Let us denote by $\mathbb{C}\ket{0}^{\bm{k}}$ the one-dimensional vacuum
representation of the “positive modes” Lie subalgebra
$(\mathfrak{sl}_{2}[t]\oplus\mathbb{C}\mathsf{k})^{\oplus
N}\subset\widehat{\mathfrak{sl}}_{2}^{\oplus N}$, defined by
$(\mathsf{k}^{(i)}-k_{i})\ket{0}^{\bm{k}}=0$ and
$a_{n}^{(i)}\ket{0}^{\bm{k}}=0$ for all $n\geq 0$, $a\in\mathfrak{sl}_{2}$ and
$i=1,\dots,N$. Therefore $\mathbb{V}_{0}^{\bm{k}}$ is the induced
$\widehat{\mathfrak{sl}}_{2}^{\oplus N}$-module, namely

$\mathbb{V}_{0}^{\bm{k}}=U(\widehat{\mathfrak{sl}}_{2}^{\oplus
N})\otimes_{U(\mathfrak{sl}_{2}[t]\oplus\mathbb{C}\mathsf{k})^{\oplus
N}}\mathbb{C}\ket{0}^{\bm{k}}.$ (25)

Repeating similar arguments to those of the previous sections, we can define
$U_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$ as the quotient of
$U(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$ by the two-sided ideal generated
by $\mathsf{k}^{(i)}-k_{i}$ for all $i=1,\dots,N$. We have the isomorphism

$U_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})\cong
U_{k_{1}}(\widehat{\mathfrak{sl}}_{2})\otimes
U_{k_{2}}(\widehat{\mathfrak{sl}}_{2})\otimes\dots\otimes
U_{k_{N}}(\widehat{\mathfrak{sl}}_{2}).$ (26)

Thanks to this fact, $A^{(i)}\in\widehat{\mathfrak{sl}}_{2}^{\oplus N}\subset
U_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$ can be presented as

$A^{(i)}=\mathds{1}\otimes\dots\otimes\mathds{1}\otimes
A\otimes\mathds{1}\otimes\dots\otimes\mathds{1},$ (27)

where $A$ is acting as the $i$th tensor factor. Again, we can introduce the
left ideals $J_{n}^{N}\in U_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$
generated by $a_{r}^{(i)}$ for all $r\geq n$, $a\in\mathfrak{sl}_{2}$ and
$i=1,\dots,N$. Let $\widetilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus
N})=\varprojlim U_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})/J_{n}^{N}$
be the inverse limit. This space is a complete topological algebra and

$\widetilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus
N})\cong\widetilde{U}_{k_{1}}(\widehat{\mathfrak{sl}}_{2})\widehat{\otimes}\cdots\widehat{\otimes}\widetilde{U}_{k_{N}}(\widehat{\mathfrak{sl}}_{2}),$
(28)

where $\hat{\otimes}$ denotes the completed tensor product. This space
$\widetilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$ is called the
algebra of observables of the Gaudin model.

The tensor product $\mathbb{V}_{0}^{\bm{k}}$ is again a vertex algebra. The
state-field map
$Y(\cdot,x):\mathbb{V}_{0}^{\bm{k}}\rightarrow\operatorname{Hom}(\mathbb{V}_{0}^{\bm{k}},\mathbb{V}_{0}^{\bm{k}}((x)))$
is just as in section 2.4 but decorated with the extra index (i).

Let us introduce the map
$\Delta:\widehat{\mathfrak{sl}}_{2}\hookrightarrow\widehat{\mathfrak{sl}}_{2}^{\oplus
N}$, which is the diagonal embedding of $\widehat{\mathfrak{sl}}_{2}$ into
$\widehat{\mathfrak{sl}}_{2}^{\oplus N}$, defined as

$\Delta x=\sum_{i=1}^{N}x^{(i)},\qquad\text{for all
}x\in\widehat{\mathfrak{sl}}_{2}.$ (29)

It extends to an embedding
$\Delta:U(\widehat{\mathfrak{sl}}_{2})\hookrightarrow
U(\widehat{\mathfrak{sl}}_{2}^{\oplus N})\cong
U(\widehat{\mathfrak{sl}}_{2})^{\otimes N}$. It is easy to check that

$[\Delta X_{m},\Delta
Y_{n}]=\Delta[X,Y]_{n+m}-n\delta_{n+m,0}(X|Y)\sum_{i=1}^{N}\mathsf{k}^{(i)},$
(30)

where $(\cdot|\cdot)$ is the usual Killing form as in eq. 6.

Therefore $\Delta$ descends to an embedding of the quotients
$\Delta:U_{|\bm{k}|}(\widehat{\mathfrak{sl}}_{2})\hookrightarrow
U_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$, where
$|\bm{k}|=\sum_{i=1}^{N}k_{i}$, and hence of their completions

$\Delta:\widetilde{U}_{|\bm{k}|}(\widehat{\mathfrak{sl}}_{2})\hookrightarrow\widetilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus
N}).$ (31)

###  3.2 Invariant tensors on $\mathfrak{sl}_{2}$

Let $\\{I^{a}\\}_{a=1}^{3}$ be a basis of $\mathfrak{sl}_{2}$, and let
$\\{I_{a}\\}_{a=1}^{3}$ be its dual basis with respect to the non-degenerate
Killing form of eq. 6. Let $f^{ab}{}_{c}$ denote the structure constants, so
that

$[I^{a},I^{b}]=f^{ab}{}_{c}I^{c}.$ (32)

Here and in what follows we employ the summation convention on Lie algebra
indices. Thanks to the non-degeneracy of the bilinear form, we may suppose our
basis is chosen in such a way that

$(I^{a}|I^{b})=\delta^{ab}.$ (33)

By doing this, we no longer have to distinguish between upper and lower
indices. The structure constants are then

$f^{ab}{}_{c}=f^{abc}=i\sqrt{2}\epsilon^{abc},$ (34)

where $\epsilon^{abc}$ is the usual Levi-Civita symbol.

(Concretely, in the defining representation we have

$I^{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\qquad
I^{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}\qquad
I^{3}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (35)

Using (6), it is easy to check that eq. 33 holds.)

Recall that for any finite-dimensional Lie algebra $\mathring{\mathfrak{g}}$,
a tensor
$t:\mathring{\mathfrak{g}}\times\dots\times\mathring{\mathfrak{g}}\rightarrow\mathbb{C}$
is _invariant_ if

$t([a,x],y,\dots,z)+t(x,[a,y],\dots,z)+\dots+t(x,y,\dots,[a,z])=0,\qquad\text{for
all }x\in\mathring{\mathfrak{g}},$ (36)

or equivalently, if its components $t^{a_{1}\dots
a_{n}}:=t(I^{a_{1}},\dots,I^{a_{n}})$ satisfy

$f^{ca_{1}}{}_{b}t^{ba_{2}\dots a_{n}}+f^{ca_{2}}{}_{b}t^{a_{1}b\dots
a_{n}}+\dots+f^{ca_{n}}{}_{b}t^{a_{1}a_{2}\dots b}=0,$ (37)

where the indices take values from $1$ to $\dim\mathring{\mathfrak{g}}$. In
our case of $\mathfrak{sl}_{2}$, the ring of invariant tensors is generated by
$\delta^{ab}$ and $f^{abc}$. We shall need the following syzygy relations
between them:

$\begin{gathered}f^{abc}f^{cde}=2(\delta^{ae}\delta^{bd}-\delta^{ad}\delta^{be}),\qquad\qquad
f^{abc}f^{abd}=-4\delta^{cd},\\\
f^{abc}\delta^{de}-f^{bcd}\delta^{ae}+f^{cda}\delta^{be}-f^{dab}\delta^{ce}=0.\end{gathered}$
(38)

Note in particular the last of these, which will play a crucial role in the
explicit calculations of the next sections. It can also be generalized to
higher rank tensors (see e.g. [Itō93, §369 F]).

##  4 Quartic Hamiltonian

###  4.1 Meromorphic states

Let us introduce a set $\\{z_{1},\dots,z_{N}\\}$ of $N\in\mathbb{Z}_{>0}$
points $z_{i}\in\mathbb{C}$ in the complex plane, chosen to be pairwise
distinct, $z_{i}\neq z_{j}$ whenever $i\neq j$. For any element
$A\in\widehat{\mathfrak{sl}}_{2}$ we introduce the
$\widehat{\mathfrak{sl}}_{2}^{\oplus N}$-valued meromorphic functions

$A(z):=\sum_{i=1}^{N}\frac{A^{(i)}}{z-z_{i}}.$ (39)

We are allowed to take derivatives of such functions, which will be denoted by
$A^{\prime}(z)$ or, in general, for each $p\geq 0$,

$A^{[p]}(z):=\left(\frac{d}{dz}\right)^{p}A(z)=\sum_{i=1}^{N}(-1)^{p}p!\frac{A^{(i)}}{(z-z_{i})^{p+1}}.$
(40)

Considering two of these functions with different spectral parameters, we get
the following commutation relations

$\begin{split}[A^{[p]}(z),B^{[q]}(w)]=&(-1)^{p+1}(p+q)!\frac{[A,B](z)-[A,B](w)}{(z-w)^{p+q+1}}\\\
&+\sum_{k=1}^{p}(-1)^{p+1-k}\binom{p}{k}(p+q-k)!\frac{[A,B]^{[k]}(z)}{(z-w)^{p+q+1-k}}\\\
&-\sum_{k=1}^{q}(-1)^{p+1}\binom{q}{k}(p+q-k)!\frac{[A,B]^{[k]}(w)}{(z-w)^{p+q+1-k}}.\end{split}$
(41)

By taking the limit $w\to z$, we get the commutation relations for the same
spectral parameter, namely

$[A^{[p]}(z),B^{[q]}(z)]=-[A,B]^{[p+q+1]}(z).$ (42)

We see that these $A^{[p]}(z)$, for $A\in\widehat{\mathfrak{sl}}_{2}$ and
$p\geq 0$, span a Lie algebra of $\widehat{\mathfrak{sl}}_{2}^{\oplus
N}$-valued meromorphic functions of $z$ with poles at the marked points.

It is helpful to be able to treat this as an abstract Lie algebra. Thus, let
$\mathfrak{L}$ denote the Lie algebra over $\mathbb{C}$ with basis consisting
of $I_{n}^{a[p]}(z)$ and $\mathsf{k}^{[p]}(z)$, for $n\in\mathbb{Z}$,
$p\in\mathbb{Z}_{\geq 0}$ and $a\in\\{1,2,3\\}$ with the non-vanishing Lie
brackets given by

$[I^{a[p]}_{m}(z),I^{b[q]}_{n}(z)]=-\frac{p!q!}{(p+q+1)!}(f^{ab}_{c}I^{c[p+q+1]}_{m+n}(z)-n\delta^{ab}\delta_{m+n,0}\mathsf{k}^{[p+q+1]}(z)).$
(43)

Let $\mathfrak{L}_{+}$ denote subalgebra generated by $I_{n}^{a[p]}(z)$ for
$n\geq 0$, $p\in\mathbb{Z}_{\geq 0}$ and $a\in\\{1,2,3\\}$, and let

$\mathcal{V}:=U(\mathfrak{L})\otimes_{U(\mathfrak{L}_{+})}\mathbb{C}\ket{0}$
(44)

denote the module over $\mathfrak{L}$ induced from the trivial one-dimensional
module $\mathbb{C}\ket{0}$ over $\mathfrak{L}_{+}$.

We call the $\mathcal{V}$ the _space of meromorphic states_. It is again a
vertex algebra, with the state-field map as in section 2.4 but decorated with
extra indices. For each $z\in\mathbb{C}\setminus\\{z_{1},\dots,z_{N}\\}$, one
has the homomorphism of Lie algebras
$\mathfrak{L}\to\widehat{\mathfrak{sl}}_{2}^{\oplus N}$ given by evaluating at
$z$. It gives rise to a map $\mathcal{V}\to\mathbb{V}_{0}^{\bm{k}}$ of vertex
algebras.

There is a bi-gradation of $\mathfrak{L}$ in which $X_{-n}^{[p]}(z)$ (for any
$X\in\mathfrak{sl}_{2}$) has weight $(n,p+1)$ and $\mathsf{k}^{[p]}(z)$ has
weight $(0,p+1)$. This yields a bi-gradation of $\mathcal{V}$

$\mathcal{V}=\bigoplus_{n\geq 0,p\geq 0}\mathcal{V}_{n,p}.$ (45)

For each $n$, let $\mathcal{V}_{n}:=\mathcal{V}_{n,n}$ denote the subspace of
grade $(n,n)$. We call elements of $\mathcal{V}_{n}$ _homogeneous meromorphic
states of degree $n$_.

###  4.2 Diagonal action of the zero modes of $\widehat{\mathfrak{sl}}_{2}$

There is an evident diagonal action of the Lie algebra
$\widehat{\mathfrak{sl}}_{2}$ on the $\mathfrak{L}$-module $\mathcal{V}$,
defined in the same way as the action on $\mathbb{V}_{0}^{\bm{k}}$ in eq. 29.
In particular, for any $X\in\mathfrak{sl}_{2}$, the zero modes stabilize each
subspace $\mathcal{V}_{n,p}$, namely

$\Delta X_{0}:\mathcal{V}_{n,p}\rightarrow\mathcal{V}_{n,p}.$ (46)

An important fact is that every state in $\mathcal{V}_{n}$ properly contracted
with an $\mathfrak{sl}_{2}$-invariant tensor vanishes under the diagonal
action of the zero modes. This follows directly from the defining property of
invariant tensors in eq. 37. Let denote with
$\mathcal{V}_{n}^{\mathfrak{sl}_{2}}$ the invariant subspace. We can
characterize this space for small $n$:

  * •

for $n=0$,
$\mathcal{V}_{0}^{\mathfrak{sl}_{2}}=\mathcal{V}_{0}=\mathbb{C}\ket{0}$.

  * •

for $n=1$, $\mathcal{V}_{1}^{\mathfrak{sl}_{2}}=\\{0\\}$. Indeed, elements of
$\mathcal{V}_{1}$ are of the form $t_{a}I^{a}_{-1}(z)\ket{0}$. Such an element
is in $\mathcal{V}^{\mathfrak{sl}_{2}}_{1}$ if and only if $t_{a}$ are the
components of an $\mathfrak{sl}_{2}$-invariant tensor of rank 1. But there are
no nonzero such tensors.

  * •

for $n=2$, $\mathcal{V}_{2}^{\mathfrak{sl}_{2}}$ has dimension 1 and it is
spanned by the state

$\varsigma_{1}(z)=\delta_{ab}I^{a}_{-1}(z)I^{b}_{-1}(z)\ket{0}.$ (47)

  * •

for $n=3$, $\mathcal{V}^{\mathfrak{sl}_{2}}_{3}$ has dimension 2 and it is
spanned by the states

$\begin{split}f^{abc}I^{a}_{-1}(z)I^{b}_{-1}(z)I^{c}_{-1}(z)\ket{0}=&f^{abc}\frac{1}{2}(I^{a}_{-1}(z)I^{b}_{-1}(z)-I^{b}_{-1}(z)I^{a}_{-1}(z))I^{c}_{-1}(z)\ket{0}\\\
=&f^{abc}f^{abd}I^{d\prime}_{-2}(z)I^{c}_{-1}(z)\ket{0}\\\
=&-4I^{c\prime}_{-2}(z)I^{c}_{-1}(z)\ket{0},\end{split}$ (48)

and

$I^{c}_{-2}(z)I^{c}_{-1}(z)\mathsf{k}(z)\ket{0}.$ (49)

  * •

for $n=4$, $\mathcal{V}_{4}^{\mathfrak{sl}_{2}}$ has dimension 14. Below, we
will make use of the following explicit choice of basis:

$\begin{gathered}\mathsf{v}_{1}:=\delta^{(ab}\delta^{cd)}I^{a}_{-1}(z)I^{b}_{-1}(z)I^{c}_{-1}(z)I^{d}_{-1}(z),\qquad\mathsf{v}_{2}:=f^{abc}I^{a}_{-2}(z)I^{b\prime}_{-1}(z)I^{c}_{-1}(z),\\\
\mathsf{v}_{3}:=I^{a\prime\prime}_{-3}(z)I^{a}_{-1}(z),\qquad\mathsf{v}_{4}:=I^{a}_{-3}(z)I^{a\prime\prime}_{-1}(z),\qquad\mathsf{v}_{5}:=I^{a\prime\prime}_{-2}(z)I^{a}_{-2}(z),\\\
\mathsf{v}_{6}:=I^{a\prime}_{-3}(z)I^{a\prime}_{-1}(z),\qquad\mathsf{v}_{7}:=I^{a\prime}_{-2}(z)I^{a\prime}_{-2}(z),\qquad\mathsf{v}_{8}:=I^{a}_{-3}(z)I^{a}_{-1}(z)\mathsf{k}^{\prime}(z),\\\
\mathsf{v}_{9}:=I^{a}_{-2}(z)I^{a}_{-2}(z)\mathsf{k}^{\prime}(z),\qquad\mathsf{v}_{10}:=I^{a\prime}_{-3}(z)I^{a}_{-1}(z)\mathsf{k}(z),\\\
\mathsf{v}_{11}:=I^{a}_{-3}(z)I^{a\prime}_{-1}(z)\mathsf{k}(z),\qquad\mathsf{v}_{12}:=I^{a\prime}_{-2}(z)I^{a}_{-2}(z)\mathsf{k}(z),\\\
\mathsf{v}_{13}:=I^{a}_{-3}(z)I^{a}_{-1}(z)\mathsf{k}(z)^{2},\qquad\mathsf{v}_{14}:=I^{a}_{-2}(z)I^{a}_{-2}(z)\mathsf{k}(z)^{2}.\end{gathered}$
(50)

Note that to write these terms we have to choose an ordering prescription. In
this work we sort level first in ascending order from left to right and after
that, for a given level, we sort derivatives in descending order from left to
right. For example
$f^{abc}I^{a}_{-2}I^{b\prime\prime}_{-2}I^{c\prime}_{-3}=f^{abc}I^{c\prime}_{-3}I^{b\prime\prime}_{-2}I^{a}_{-2}+\text{terms
obtained from commutations}$.

###  4.3 Top terms

We can see from the above construction that in the case $n=2$ and $n=4$, there
is a particular state, that we will call _top term_ , which is the state in
$\mathcal{V}_{n}^{\mathfrak{sl}_{2}}$ that contains exactly $n$ generators:

$\delta_{ab}I^{a}_{-1}(z)I^{b}_{-1}(z)\ket{0},\qquad\delta_{(ab}\delta_{cd)}I^{a}_{-1}(z)I^{b}_{-1}(z)I^{c}_{-1}(z)I^{d}_{-1}(z)\ket{0}.$
(51)

We do not have such state for $n=3$, because we can always use the commutation
relations to reduce the number of generators, as shown in eq. 48. This pattern
continues, as we now describe.

Notice that the universal enveloping algebra $U(\mathfrak{L})$ has an
increasing filtration

$F_{0}U(\mathfrak{L})\subseteq F_{1}U(\mathfrak{L})\subseteq\dots\subseteq
U(\mathfrak{L}),$ (52)

in which the generators $I^{a[p]}_{n}(z)$ count as $+1$ and the generators
$\mathsf{k}^{[p]}(z)$ count as $0$, cf. the commutation relations of
$\mathfrak{L}$ in eq. 43. For example $I^{a}_{-1}(z)I^{a\prime}_{-2}(z)\in
F_{2}$, and $I^{a}_{-1}(z)I^{a\prime}_{-2}(z)\mathsf{k}(z)\in F_{2}$ as well.
It gives rise to a corresponding filtration, $F_{0}\mathcal{V}\subseteq
F_{1}\mathcal{V}\subseteq\dots\subseteq\mathcal{V}$, on the space
$\mathcal{V}$ of meromorphic states.

Observe that if $\mathsf{v}\in\mathcal{V}_{N}$ then $\mathsf{v}\in
F_{N}\mathcal{V}_{N}$. We see that

$\displaystyle\mathsf{v}$ $\displaystyle\equiv t_{i_{1}\dots
i_{N}}I^{i_{1}}_{-1}(z)\dots I^{i_{N}}_{-1}(z)\ket{0}\mod
F_{N-1}\mathcal{V}_{N}$ $\displaystyle\equiv t_{(i_{1}\dots
i_{N})}I^{i_{1}}_{-1}(z)\dots I^{i_{N}}_{-1}(z)\ket{0}\mod
F_{N-1}\mathcal{V}_{N},$ (53)

for some $\mathfrak{sl}_{2}$ tensor $t_{i_{1},\dots,i_{N}}$, where the
brackets around the indices denote the operation of symmetrization,

$t_{(i_{1},\dots,i_{n})}=\frac{1}{n!}\sum_{\sigma\in\mathcal{S}_{n}}t_{\sigma(i_{1})\dots\sigma(i_{n})}$
(54)

(and we may symmetrize without loss of generality because the non-symmetric
pieces fall into $F_{N-1}$, as for example in eq. 48). Let us call
$t_{(i_{1}\dots i_{N})}I^{i_{1}}_{-1}(z)\dots I^{i_{N}}_{-1}(z)\ket{0}$ the
_top term_ of the state $\mathsf{v}\in\mathcal{V}_{N}$.

If this state $\mathsf{v}\in\mathcal{V}$ is $\mathfrak{sl}_{2}$-invariant,
$\mathsf{v}\in\mathcal{V}^{\mathfrak{sl}_{2}}$, then $t_{(i_{1}\dots i_{N})}$
is a symmetric invariant tensor. Nonzero such tensors exist only in even
degrees, and up to rescaling they are, explicitly,

$\begin{gathered}t_{i_{1},i_{2}}=\delta_{i_{1}i_{2}}\\\
t_{i_{1},i_{2},i_{3},i_{4}}=\delta_{(i_{1}i_{2}}\delta_{i_{3}i_{4})}\\\
t_{i_{1},i_{2},i_{3},i_{4},i_{5},i_{6}}=\delta_{(i_{1}i_{2}}\delta_{i_{3}i_{4}}\delta_{i_{5}i_{6})}\\\
\dots{}\end{gathered}$

In what follows, our interest is in meromorphic states
$\mathsf{v}\in\mathcal{V}^{\mathfrak{sl}_{2}}$ that have nonzero top term (in
other words states whose principal symbol has maximal degree) and that are
$\mathfrak{sl}_{2}$-invariant.

###  4.4 Singular vectors up to twisted derivative

Let us define the twisted derivative operator of degree $j\in\mathbb{Z}$ with
respect to the spectral parameter $z$,

$D_{z}^{(j)}=\left(\partial_{z}-\frac{j}{2}\mathsf{k}(z)\right).$ (55)

Note that this operator sends $\mathcal{V}_{n,p}\to\mathcal{V}_{n,p+1}$ in the
bigradation we introduced above.

We will say that a vector $\mathsf{v}\in\mathcal{V}^{\mathfrak{sl}_{2}}_{n}$
is _singular up to twisted derivatives_ if for all $x\in\mathfrak{sl}_{2}$ we
have

$\Delta
x_{m}\mathsf{v}=0\qquad\textup{mod}\>D^{(n-1)}_{z}\mathcal{V}_{n-m,n-1}.$ (56)

for _all_ non-negative modes $x_{m}$, $m\geq 0$. This defines a subspace

$\mathcal{V}_{n}^{\text{sing}}\subset\mathcal{V}_{n}^{\mathfrak{sl}_{2}}$ (57)

of vectors singular up to twisted derivatives.

######  Proposition 4.4.1.

The space of singular vectors $\mathcal{V}^{\text{sing}}_{2}$ is spanned by
the quadratic state $\varsigma_{1}(z)$ defined in (47).

###### Proof.

We need to show that

$\Delta I^{r}_{k}\varsigma_{1}(z)=0\quad\mod
D^{(1)}_{z}\mathsf{G}^{r}_{k}(z),$ (58)

for some meromorphic states $\mathsf{G}_{k}^{r}\in\mathcal{V}_{2-k,1}$, for
all $k\geq 0$ and $r=1,2,3$. For $k=0$ there is nothing to check since $\Delta
I^{r}_{0}\varsigma_{1}(z)=0$ identically, by the definition of
$\mathcal{V}^{\mathfrak{sl}_{2}}_{2}$. It is enough to check the action of the
first modes $I^{r}_{1}$, since any higher modes can be expressed in terms of
their brackets, i.e. $I_{2}^{r}=-\frac{1}{4}f^{rbc}[I^{b}_{1},I^{c}_{1}]$ etc.
From direct calculations we get that

$\Delta I^{r}_{1}\varsigma_{1}(z)=D^{(1)}_{z}\mathsf{G}^{r}_{1}(z),$ (59)

where

$\mathsf{G}^{r}_{1}(z)=-4I^{r}_{-1}(z)\ket{0}.$ (60)

∎

More non-trivially, for $n=4$ we have the following result.

######  Proposition 4.4.2.

The space of singular vectors $\mathcal{V}^{\text{sing}}_{4}$ is of dimension
7. A choice of basis is given by the state

$\begin{split}\varsigma_{3}(z)=\Big{[}&\delta_{(ab}\delta_{cd)}I^{a}_{-1}(z)I^{b}_{-1}(z)I^{c}_{-1}(z)I^{d}_{-1}(z)+\frac{20}{3}f_{abc}I^{a}_{-2}(z)I^{b\prime}_{-1}(z)I^{c}_{-1}(z)\\\
&+\frac{40}{9}I^{a}_{-3}(z)I^{a\prime\prime}_{-1}(z)-\frac{20}{3}I^{a\prime\prime}_{-2}(z)I^{a}_{-2}(z)+\frac{40}{9}I^{a\prime}_{-3}(z)I^{a\prime}_{-1}(z)\\\
&\phantom{+}\qquad-\frac{10}{3}I^{a\prime}_{-2}(z)I^{a\prime}_{-2}(z)-\frac{20}{3}I^{a}_{-3}(z)I^{a}_{-1}(z)\mathsf{k}^{\prime}(z)\Big{]}\ket{0},\end{split}$
(61)

together with the double translate state

$T^{2}\Big{(}I^{a\prime\prime}_{-1}(z)I^{a}_{-1}(z)\ket{0}-I^{a\prime}_{-1}(z)I^{a\prime}_{-1}(z)\ket{0}-\frac{3}{4}I^{a}_{-1}(z)I^{a}_{-1}(z)\mathsf{k}^{\prime}(z)\ket{0}\Big{)}$
(62)

and the following twisted derivative states

$\displaystyle
D^{(3)}_{z}\Big{(}I^{a}_{-3}(z)I^{a\prime}_{-1}(z)\ket{0}\Big{)},\quad
D^{(3)}_{z}\Big{(}I^{a\prime}_{-3}(z)I^{a}_{-1}(z)\ket{0}\Big{)},\quad
D^{(3)}_{z}\Big{(}I^{a\prime}_{-2}(z)I^{a}_{-2}(z)\ket{0}\Big{)},$ (63)
$\displaystyle
D^{(3)}_{z}\Big{(}I^{a}_{-3}(z)I^{a}_{-1}(z)\mathsf{k}(z)\ket{0}\Big{)},\quad
D^{(3)}_{z}\Big{(}I^{a}_{-2}(z)I^{a}_{-2}(z)\mathsf{k}(z)\ket{0}\Big{)}.$

###### Proof.

Let $s(z)\in\mathcal{V}_{4}^{\mathfrak{sl}_{2}}$. We may write it in our basis
(50),

$s(z)=\sum_{i=1}^{14}\xi_{i}\mathsf{v}_{i}(z)$ (64)

for some coefficients $\xi_{i}\in\mathbb{C}$ with $i=1,\dots,14$, and ask what
conditions the requirement of being singular up to twisted derivatives, (56),
places on these coefficients. It is enough to demand that

$\Delta I^{r}_{k}s(z)=0\mod D^{(3)}_{z}\mathsf{G}^{r}_{k}(z)$ (65)

for some meromorphic states $\mathsf{G}^{r}_{k}(z)\in\mathcal{V}_{4-k,3}$, for
all $k\geq 0$ and $r=1,2,3$. For zero modes there is nothing to check since
$\Delta I^{a}_{0}s(z)=0$ exactly, by definition of
$\mathcal{V}_{4}^{\mathfrak{sl}_{2}}$. It is then enough to check the action
of first modes, $I^{r}_{1}$, since any higher modes can be expressed in terms
of their brackets, $I^{r}_{2}=-\frac{1}{4}f^{rbc}[I^{b}_{1},I^{c}_{1}]$ etc.
So we are to check under what conditions

$\Delta I^{r}_{1}s(z)=D^{(3)}_{z}\mathsf{G}^{r}_{1}(z)$ (66)

for some $\mathsf{G}^{a}_{1}(z)\in\mathcal{V}_{3,3}$. By direct calculation,
one finds that solutions exist precisely if the coefficient $\xi_{i}$ obey the
relations

$\begin{gathered}\xi_{2}=\frac{20}{3}\xi_{1},\qquad\xi_{3}=\frac{20}{3}\xi_{1}-\xi_{4}+2\xi_{5}+\xi_{6}-2\xi_{7},\qquad\xi_{9}=-\frac{5}{4}\xi_{1}-\frac{3}{8}\xi_{5}+\frac{3}{8}\xi_{7}-\frac{2}{3}\xi_{14},\qquad\\\
\xi_{10}=\frac{5}{3}\xi_{1}+\frac{3}{2}\xi_{4}-\frac{3}{2}\xi_{5}-\frac{3}{2}\xi_{6}+\frac{3}{2}\xi_{7}+\xi_{8},\qquad\xi_{11}=\frac{55}{3}\xi_{1}-\frac{3}{2}\xi_{4}+\frac{3}{2}\xi_{5}-\frac{3}{2}\xi_{7}+\xi_{8},\qquad\\\
\xi_{12}=-\frac{15}{2}\xi_{1}-\frac{3}{4}\xi_{5}-\frac{3}{4}\xi_{7}-\frac{4}{3}\xi_{14},\qquad\xi_{13}=-\frac{55}{4}\xi_{1}-\frac{9}{8}\xi_{5}+\frac{9}{8}\xi_{7}-\frac{3}{2}\xi_{8}.\end{gathered}$
(67)

When they do obey these relations, the required functions
$\mathsf{G}^{r}_{1}(z)$ are given by

$\begin{split}\mathsf{G}^{r}_{1}(z)=\Big{[}&\rho_{1}I^{(a}_{-1}(z)I^{r}_{-1}(z)I^{a)}_{-1}(z)\\\
&+f^{rab}\left(\rho_{2}I^{a}_{-2}(z)I^{b\prime}_{-1}(z)+\rho_{3}I^{a\prime}_{-2}(z)I^{b}_{-1}(z)+\rho_{4}I^{a}_{-2}(z)I^{b}_{-1}(z)\mathsf{k}(z)\right)\\\
&+\rho_{5}I^{r}_{-3}(z)\mathsf{k}(z)^{2}+\rho_{6}I^{r}_{-3}(z)\mathsf{k}^{\prime}(z)+\rho_{7}I^{r\prime}_{-3}(z)\mathsf{k}(z)+\rho_{8}I^{r\prime\prime}_{-3}(z)\Big{]}\ket{0},\end{split}$
(68)

where

$\begin{gathered}\rho_{1}=-\frac{8}{3}\xi_{1},\qquad\rho_{2}=\frac{20}{3}\xi_{1}-\xi_{4}+\xi_{5},\qquad\rho_{3}=\xi_{4}-\xi_{5}-\xi_{6}+2\xi_{7},\\\
\rho_{4}=-\frac{55}{6}\xi_{1}-\frac{3}{4}\xi_{5}+\frac{3}{4}\xi_{7}-\xi_{8}-\frac{4}{3}\xi_{14}\qquad\rho_{5}=\frac{55}{6}\xi_{1}+\frac{3}{4}\xi_{5}-\frac{3}{4}\xi_{7}+\xi_{8},\qquad\rho_{6}=\xi_{4},\\\
\rho_{7}=5\xi_{1}-\xi_{4}+\frac{3}{2}\xi_{5}+\xi_{6}-\frac{3}{2}\xi_{7}+\frac{8}{3}\xi_{14},\qquad\rho_{8}=-\frac{100}{9}\xi_{1}-\frac{4}{3}\xi_{5}-\frac{2}{3}\xi_{7}.\end{gathered}$

The basis reported in the proposition can be obtained by the one defined by
the restrictions (67) by a change of basis. ∎

The proposition above is in agreement with the calculation of the quartic
Hamiltonian density $S_{4}(z)$ (the analogue of our $\varsigma_{3}(z)$),
recently presented in [KLT22]. In the present conventions, the latter is given
by

$\begin{split}S_{4}(z)=\Big{[}&\delta^{(ab}\delta^{cd)}I^{a}_{-1}(z)I^{b}_{-1}(z)I^{c}_{-1}(z)I^{d}_{-1}(z)+\frac{20}{3}f^{abc}I^{a}_{-2}(z)I^{b\prime}_{-1}(z)I^{c}_{-1}(z)\\\
&-\frac{40}{9}I^{a\prime\prime}_{-3}(z)I^{a}_{-1}(z)-\frac{140}{9}I^{a\prime\prime}_{-2}(z)I^{a}_{-2}(z)+\frac{40}{3}I^{a\prime}_{-3}(z)I^{a\prime}_{-1}(z)\\\
&\phantom{+}\qquad-\frac{10}{3}I^{a\prime}_{-2}(z)I^{a\prime}_{-2}(z)+5I^{a}_{-2}(z)I^{a}_{-2}(z)\mathsf{k}(z)^{2}\Big{]}\ket{0}\end{split}$
(69)

and it does333To match conventions, note that for us
$\delta_{(ab}\delta_{cd)}=\frac{1}{3}\left(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}\right)$
(70) and in [KLT22] the tensor called $\tau_{3}^{abcd}$ is given by
$\tau_{3}^{abcd}=\frac{1}{16}\left(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}\right).$
(71) We thank Sylvain Lacroix for clarifying discussions on this point. indeed
lie in the space $\mathcal{V}^{\text{sing}}_{4}$.

###  4.5 Hamiltonian densities

Now, to state the main result of the paper, we need two reintroduce rational
functions of two different spectral parameters, $z$ and $w$, cf. eq. 41 and
eq. 42.

Recall the Lie algebra $\mathfrak{L}\equiv\mathfrak{L}_{(z)}$ over
$\mathbb{C}$ from section 4.1. Let $\mathfrak{L}_{(z,w)}$ be the Lie algebra
with generators $I_{n}^{a[p]}(z)$, $I_{n}^{a[p]}(w)$, $\mathsf{k}^{[p]}(z)$
and $\mathsf{k}^{[p]}(w)$ for $a=1,2,3$, $n\in\mathbb{Z}$ and
$p\in\mathbb{Z}_{\geq 0}$ and commutation relations

$\begin{split}[I_{m}^{a[p]}(z),I_{n}^{b[q]}(w)]=&(-1)^{p+1}(p+q)!\frac{f^{ab}_{c}(I^{c}_{m+n}(z)-I^{c}_{m+n}(w))-n\delta_{m+n,0}\delta^{ab}(\mathsf{k}(z)-\mathsf{k}(w))}{(z-w)^{p+q+1}}\\\
&+\sum_{j=1}^{p}(-1)^{p+1-j}\binom{p}{j}(p+q-j)!\frac{f^{ab}_{c}I^{c[j]}_{m+n}(z)-n\delta_{n+m,0}\delta^{ab}\mathsf{k}^{[j]}(z)}{(z-w)^{p+q+1-k}}\\\
&-\sum_{k=1}^{q}(-1)^{p+1}\binom{q}{k}(p+q-k)!\frac{f^{ab}_{c}I^{c[j]}_{m+n}(w)-n\delta_{n+m,0}\delta^{ab}\mathsf{k}^{[j]}(w)}{(z-w)^{p+q+1-k}},\end{split}$
(72)

together with eq. 43 for the generators with parameter $z$ and the analogue
with parameter $w$. This Lie algebra $\mathfrak{L}_{(z,w)}$ and its modules
are defined over the ground ring $\mathbb{C}[(z-w)^{-1}]$ of polynomials in
$(z-w)^{-1}$. We have the vertex algebra $\mathcal{V}_{(z,w)}$ defined
analogously to eq. 44 and the two obvious embedding maps of vertex algebras
$\mathcal{V}\hookrightarrow\mathcal{V}_{(z,w)}$, which we write as
$\mathsf{v}\mapsto\mathsf{v}(z)$ and $\mathsf{v}\mapsto\mathsf{v}(w)$.

Moreover, there is a natural notion of “expanding around $z=w$”. Namely, there
is a homomorphism $\mathfrak{L}_{(z,w)}\to\mathfrak{L}_{(z)}((w-z))$ of Lie
algebras over $\mathbb{C}[(z-w)^{-1}]$ defined by

$I^{a[p]}_{m}(w)=I^{a[p]}_{m}(z)+I^{a[p+1]}_{m}(z)(w-z)+\frac{1}{2}I^{a[p+2]}_{m}(z)(w-z)^{2}+\dots$
(73)

which is motivated by considering the Taylor expansion $\iota_{w-z}A(w)$ of
the function $A(w)$ from eq. 39. This gives rise to a map
$\mathcal{V}_{(z,w)}\to\mathcal{V}_{(z)}((w-z))$. We say a state
$\mathsf{v}\in\mathcal{V}_{(z,w)}$ is _regular at $z=w$ modulo translates_ if
there exists $Z\in\mathcal{V}_{(z,w)}$ such that the image of $\mathsf{v}-TZ$
under this map has no singularities in $(z-w)$.

Recall from eqs. 47 and 61 the definitions of the quadratic state
$\varsigma_{1}\in\mathcal{V}_{2}^{\textup{sing}}$ and of the vector
$\varsigma_{3}\in\mathcal{V}_{4}^{\textup{sing}}$, respectively.

######  Theorem 4.5.1.

The elements $\varsigma_{1}\in\mathcal{V}_{2}^{\textup{sing}}$ and
$\varsigma_{3}\in\mathcal{V}_{4}^{\textup{sing}}$ obey the relations

$\displaystyle\varsigma_{1}(z)_{(0)}\varsigma_{1}(w)$
$\displaystyle=(D_{z}^{(1)}-D_{w}^{(1)})\mathsf{A}_{1,1}(z,w)+T\mathsf{B}_{1,1}(z,w),$
(74a) $\displaystyle\varsigma_{1}(z)_{(0)}\varsigma_{3}(w)$
$\displaystyle=(3D_{z}^{(1)}-D_{w}^{(3)})\mathsf{A}_{1,3}(z,w)+T\mathsf{B}_{1,3}(z,w),$
(74b) $\displaystyle\varsigma_{3}(z)_{(0)}\varsigma_{1}(w)$
$\displaystyle=(D_{z}^{(3)}-3D_{w}^{(1)})\mathsf{A}_{3,1}(z,w)+T\mathsf{B}_{3,1}(z,w),$
(74c) $\displaystyle\varsigma_{3}(z)_{(0)}\varsigma_{3}(w)$
$\displaystyle=(3D_{z}^{(1)}-3D_{w}^{(3)})\mathsf{A}_{3,3}(z,w)+T\mathsf{B}_{3,3}(z,w),$
(74d)

where $\mathsf{A}_{i,j}(z,w)$ and $\mathsf{B}_{i,j}(z,w)$ are elements of
$\mathcal{V}_{(z,w)}$. Moreover, $\mathsf{A}_{i,j}(z,w)$, $i,j\in\\{1,3\\}$,
are regular at $z=w$ modulo translates.

###### Proof.

The two statements of the theorem follow from direct calculations. In
particular, when $m=n=1$, we get

$\mathsf{A}_{1,1}(z,w)=\frac{8}{z-w}I^{a}_{-2}(z)I^{a}_{-1}(w)\ket{0},\qquad\mathsf{B}_{1,1}(z,w)=\frac{8}{(z-w)^{2}}I^{a}_{-1}(z)I^{a}_{-1}(w)\ket{0}.$
(75)

We have computed $\mathsf{A}_{1,3}(z,w)$, $\mathsf{B}_{1,3}(z,w)$,
$\mathsf{A}_{3,3}(z,w)$ and $\mathsf{B}_{3,3}(z,w)$ explicitly, with the aid
of the computer algebra system FORM [Ver13, Kui+13]. The expressions for
$\mathsf{A}_{1,3}(z,w)$ and $\mathsf{B}_{1,3}(z,w)$, are given in appendix A.
The expressions for $\mathsf{A}_{3,3}(z,w)$ and $\mathsf{B}_{3,3}(z,w)$ are
extremely lengthy (more that 500 terms in total), and we do not reproduce them
here.

Once the expression of $\varsigma_{1}(z)_{(0)}\varsigma_{3}(w)$ is known, i.e.
the functions $\mathsf{A}_{13}(z,w)$ and $\mathsf{B}_{13}(z,w)$ are found, it
can be shown that the theorem is automatically satisfied for the product
$\varsigma_{3}(z)_{(0)}\varsigma_{1}(w)$. This comes from the property of the
$n$th product between two states $a,b$ of a vertex algebra, namely

$a_{(n)}b=-\sum_{k=0}^{\infty}\frac{1}{k!}(-1)^{k+n}T^{k}(b_{(n+k)}a).$ (76)

Therefore, by swapping two states in a 0th product, we obtain
$\mathsf{A}_{31}(z,w)=-\mathsf{A}_{13}(w,z)$ and a series of terms which are
nothing but translates and therefore can be absorbed in the definition of
$\mathsf{B}_{31}(z,w)=-\mathsf{B}_{13}(w,z)+\sum_{k=0}^{\infty}(-1)^{k}T^{k}(\varsigma_{1}(w)_{(k+1)}\varsigma_{3}(z))$.

To prove the second part of the theorem one can expand according to eq. 73 and
the result follows from direct calculation. ∎

Having established this statement for the particular choice of quartic density
$\varsigma_{3}$, we automatically get the following property for _any_ element
of $\mathcal{V}^{\text{sing}}_{4}$. It is a slightly weaker property, because
the condition on the twisted derivative terms on the right hand side is less
rigid. As we shall see in section 4.7 below, it is sufficient for defining
consistent Hamiltonians.

######  Corollary 4.5.1.

For _any_ element $\mathsf{v}_{3}\in\mathcal{V}^{\text{sing}}_{4}$, one has

$\displaystyle\varsigma_{1}(z)_{(0)}\mathsf{v}_{3}(w)$
$\displaystyle=D_{z}^{(1)}\mathsf{A}^{\@slowromancap
i@}_{1,3}(z,w)+D_{w}^{(3)}\mathsf{A}^{\@slowromancap
ii@}_{1,3}(z,w)+T\mathsf{B}_{1,3}(z,w),$ (77)
$\displaystyle\mathsf{v}_{3}(z)_{(0)}\varsigma_{1}(w)$
$\displaystyle=D_{z}^{(3)}\mathsf{A}^{\@slowromancap
i@}_{3,1}(z,w)+D_{w}^{(1)}\mathsf{A}^{\@slowromancap
ii@}_{3,1}(z,w)+T\mathsf{B}_{3,1}(z,w),$ (78)
$\displaystyle\mathsf{v}_{3}(z)_{(0)}\mathsf{v}_{3}(w)$
$\displaystyle=D_{z}^{(3)}\mathsf{A}^{\@slowromancap
i@}_{3,3}(z,w)+D_{w}^{(3)}\mathsf{A}^{\@slowromancap
ii@}_{3,3}(z,w)+T\mathsf{B}_{3,3}(z,w).$ (79)

where $\mathsf{A}^{\@slowromancap i@,\@slowromancap ii@}_{i,j}(z,w)$ and
$\mathsf{B}_{i,j}(z,w)$ are elements of $\mathcal{V}_{(z,w)}$. Moreover,
$\mathsf{A}^{\@slowromancap i@,\@slowromancap ii@}_{ij}(z,w)$,
$i,j\in\\{1,3\\}$, are regular at $z=w$ modulo translates.

###### Proof.

We already know from theorem 4.5.1 that there exists an element,
$\varsigma_{3}(z)$, satisfying these relations. But we saw in proposition
4.4.2 that every element $\mathsf{v}_{3}(z)$ of
$\mathcal{V}^{\text{sing}}_{4}$ is proportional to $\varsigma_{3}(z)$ up to
the addition of certain translates and twisted derivatives.

It follows from the property (76) that if we add to $\varsigma_{3}(z)$ any
translate then the statement of theorem 4.5.1 still holds, the only difference
being a re-definition of the states $\mathsf{B}(z,w)$. And it is evident that,
if we add to $\varsigma_{3}(z)$ any linear combination of the twisted
derivatives in eq. 63 then the resulting vector $\mathsf{v}_{3}(z)$ still
obeys the weaker relations given above. (One might worry about introducing
singularities at $z=w$, but note that for _any_ meromorphic states $a(z)$ and
$b(z)$, the product $a(z)_{(0)}b(w)$ is regular at $z=w$, as is manifest if we
expand $b(w)$ about $w=z$ in the spectral plane before taking the vertex-
algebra product:
$a(z)_{(0)}b(w)=a(z)_{(0)}\left(b(z)+(w-z)b^{\prime}(z)+\dots\right)=a(z)_{(0)}b(z)+(w-z)a(z)_{(0)}b^{\prime}(z)+\dots$.)
∎

###  4.6 Gaudin Hamiltonian

Let us define the following state at non-critical level, i.e. $k_{i}\neq-2$,

$s_{1}(z)=\frac{1}{2}\Big{(}\varsigma_{1}(z)+4D^{(1)}_{z}\omega(z)\Big{)}\in\mathbb{V}_{0}^{\bm{k}},$
(80)

where $\varsigma_{1}(z)$ is now the image in $\mathbb{V}_{0}^{\bm{k}}$ of the
density defined in (47) and where

$\omega(z):=\sum_{i=1}^{N}\frac{1}{z-z_{i}}\left(\frac{1}{2(k_{i}+2)}I^{a(i)}_{-1}I^{a(i)}_{-1}\ket{0}^{\bm{k}}\right),$
(81)

the term in the brackets being the Segal-Sugawara vector at site $i$.

It is possible to show (see [LVY20]), that the operator $(s_{1}(z))_{(0)}$ is
the image in $\tilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$ of

$\sum_{i=1}^{N}\frac{\mathscr{C}^{(i)}}{(z-z_{i})^{2}}+\sum_{i=1}^{N}\frac{\mathscr{H}_{i}}{z-z_{i}}\in\tilde{U}(\mathfrak{g}^{\oplus
N}),$ (82)

where

$\mathscr{C}^{(i)}:=(\mathsf{k}^{(i)}+2)\mathsf{d}^{(i)}+\frac{1}{2}I^{a(i)}_{0}I^{a(i)}_{0}+\sum_{n>0}I^{a(i)}_{-n}I^{a(i)}_{-n}$
(83)

is the $i$th copy of the quadratic Casimir operator of $\mathfrak{g}$ in
$\tilde{U}(\mathfrak{g}^{\oplus N})$ and $\mathscr{H}_{i}$ are the
Hamiltonians in (1).

######  Theorem 4.6.1.

Given the images in $\mathbb{V}_{0}^{\bm{k}}$ of the densities
$\varsigma_{i}$, $i\in\\{1,3\\}$, we have

$s_{1}(z)_{(0)}\varsigma_{i}(w)=-\frac{1}{2}D^{(1)}_{w}\mathsf{A}_{1,i}(z,w)+T\Big{(}\frac{1}{2}\mathsf{B}_{1,i}(z,w)+2D^{(1)}_{z}\frac{\varsigma_{i}(w)}{z-w}\Big{)},$
(84)

with $\mathsf{A}_{1,i}(z,w)$ and $\mathsf{B}_{1,i}(z,w)$ being the images in
$\mathbb{V}_{0}^{\bm{k}}$ of the meromorphic states in theorem 4.5.1.

###### Proof.

The result follows from direct calculations, using the definitions of
$\mathsf{A}_{1,1}$, $\mathsf{B}_{1,1}$, $\mathsf{A}_{1,3}$, $\mathsf{B}_{1,3}$
in (75) and appendix A, respectively. ∎

As we will see in the next section, this requirement is sufficient to ensure
the commutativity of local Hamiltonians, arising from the densities
$\varsigma_{1}(z)$ and $\varsigma_{3}(z)$, with the usual quadratic Gaudin
Hamiltonians which define the model.

###  4.7 Commuting Hamiltonians

In this section, we will simply recall the ideas presented in [LVY18].
Consider two states $X,Y\in\mathbb{V}_{0}^{\bm{k}}$ and their formal zero
modes
$X_{(0)},Y_{(0)}\in\widetilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus
N})$. It is possible to show the following vertex algebra identity (see e.g.
[FB01])

$[X_{(0)},Y_{(0)}]=(X_{(0)}Y)_{(0)}.$ (85)

This means that if one is able to find a family of operators whose 0th product
vanishes (or that can be expressed as a translation, since $(TZ)_{(0)}=0$ by
definition), then their formal zero modes form a commutative subalgebra of the
algebra of observables $\widetilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}^{\oplus
N}_{2})$.

The meromorphic function which is obtained by acting with $\mathsf{k}(z)$ on
the module $\mathbb{V}_{0}^{\bm{k}}$, _i.e._ setting the central elements to
numbers,

$k(z)=\sum_{i=1}^{N}\frac{k_{i}}{z-z_{i}},$ (86)

has a special role and it is called the _twist function_ of the model. Let us
define also

$\mathscr{P}(z)\vcentcolon=\prod_{j=1}^{N}(z-z_{j})^{k_{j}}.$ (87)

The function $\mathscr{P}^{1/2}$ is multi-valued. It becomes single-valued on
a certain multi-sheeted cover of $\mathbb{C}\setminus\\{z_{1},\dots,z_{N}\\}$.
Let $\gamma$ be any closed contour in this cover. For example, $\gamma$ could
be the lift to this cover of a Pochhammer contour in
$\mathbb{C}\setminus\\{z_{1},\dots,z_{N}\\}$ around any two of the marked
points. Then $\mathscr{P}^{n/2}$, for any integer $n$, is single-valued along
$\gamma$, and one can introduce the integral

$\int_{\gamma}\mathscr{P}(z)^{-n/2}f(z)\text{d}z$ (88)

which has the fundamental property that, for any meromorphic function $f(z)$
which is non-singular along $\gamma$

$\int_{\gamma}\mathscr{P}(z)^{-n/2}\left(\frac{d}{dz}-\frac{n}{2}k(z)\right)f(z)\text{d}z=\int_{\gamma}\frac{d}{dz}(\mathscr{P}^{-n/2}f(z))\text{d}z=0,$
(89)

cf. eq. 55. (See e.g. [LVY18] for the details.)

Let us now define the following object in
$\widetilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$,

$Q_{n}^{\gamma}=\int_{\gamma}\mathscr{P}(z)^{-n/2}\varsigma_{n}(z)_{(0)}\text{d}z,$
(90)

for $n=1,3$, where $\varsigma_{n}(z)$ are now the images in
$\mathbb{V}_{0}^{\bm{k}}$ of the densities we have defined in the previous
section.

######  Proposition 4.7.1.

The operators
$Q_{n}^{\gamma}\in\widetilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus
N})$ commute amongst themselves, with the generators of
$\widehat{\mathfrak{sl}}_{2}$, and with the quadratic Hamiltonians
$\mathscr{H}_{i}$.

###### Proof.

We can use eq. 85 to compute

$\begin{split}[Q_{m}^{\gamma},Q_{n}^{\eta}]=&\int_{\gamma}\int_{\eta}\mathscr{P}(z)^{-m/2}\mathscr{P}(w)^{-n/2}[\varsigma_{m}(z)_{(0)},\varsigma_{n}(w)_{(0)}]\text{d}z\text{d}w\\\
=&\int_{\gamma}\int_{\eta}\mathscr{P}(z)^{-m/2}\mathscr{P}(w)^{-n/2}(\varsigma_{m}(z)_{(0)}\varsigma_{n}(w))_{(0)}\text{d}z\text{d}w,\end{split}$
(91)

but we know from theorem 4.5.1 that the 0th product between those states can
be expressed as a sum of twisted derivatives and translations. It is now
straightforward to check that the result of the commutator is zero: on one
side because $(TX)_{(0)}=0$ for every state $X\in\mathbb{V}_{0}^{\bm{k}}$, on
the other because of the property (89).

To prove the second statement, we recall that
$I^{a}_{n}=(I^{a}_{-1}\ket{0}^{\bm{k}})_{(n)}$ and the general property that
for any two states $A,B$ of a vertex algebra we have

$[A_{(m)},B_{(n)}]=\sum_{k\geq 0}\binom{m}{k}(A_{(k)}B)_{(m+n-k)},\qquad
m,n\in\mathbb{Z}$ (92)

where

$\binom{m}{k}=\frac{m(m-1)\dots(m-k+1)}{k!},\quad
k\in\mathbb{Z}_{>0};\qquad\binom{m}{0}=1.$ (93)

Note that this represents a generalization of eq. 85, see e.g. [VY17, FB01].
At this point we can consider the generators $\\{I_{n}^{a}\\}_{a=1}^{3}$ with
$n\in\mathbb{Z}$ and we get

$\begin{split}[I^{a}_{n},Q_{m}^{\gamma}]=&\int_{\gamma}\mathscr{P}(z)^{-m/2}[I^{a}_{n},\varsigma_{m}(z)_{(0)}]\text{d}z\\\
=&\int_{\gamma}\sum_{k\geq
0}\binom{n}{k}\mathscr{P}(z)^{-m/2}(I^{a}_{k}\varsigma_{m}(z))_{(n-k)}\text{d}z.\end{split}$
(94)

It is now straightforward to check that the result is zero, by using property
(56), described in the relevant cases in propositions 4.4.1 and 4.4.2, and the
property (89). ∎

####  4.7.1 Fourier modes

Even though the operators $Q^{\gamma}_{m}$ we have just defined have all the
right characteristics to be well-defined Hamiltonians as pointed out in
proposition 4.7.1, there is one last subtlety about these objects, related to
the fact that we want their action on highest weight modules to be
diagonalisable. In fact, considering
$X^{(i)}_{n}\in\widetilde{U}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$ such
that $\deg{X^{(i)}_{n}}=n$ and setting $\deg(\ket{0}^{\bm{k}})=0$, induces a
$\mathbb{Z}_{\leq 0}$-gradation on the product of vacuum Verma modules, called
the homogeneous gradation. Therefore if $X\in\mathbb{V}_{0}^{\bm{k}}$ with
degree $\deg(X)=k$, the degree of its modes is $\deg(X_{(m)})=1+k+m$. The
objects we have constructed $\varsigma_{n}(z)$, by definition, have
$\deg(\varsigma_{n}(z))=-n-1$, therefore $\deg(\varsigma_{n}(z)_{(0)})=-n$.

This shows that in the homogeneous gradation these operators have degree $\neq
0$: this means that if we consider a module over
$U(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$ which has a trivial subspace of
grade $n$ for large $n$, then the operator
$\int_{\gamma_{n}}\mathscr{P}(z)^{-n/2}\varsigma_{n}(z)_{(0)}dz$ has a non-
zero eigenvalue.

A way to overcome this issue is to consider the notion of Fourier mode
$X_{[n]}\in\widetilde{U}(\widehat{\mathfrak{sl}}_{2}^{\oplus N})$ of the state
$X\in\mathbb{V}_{0}^{\bm{k}}$: they have the property that we are looking for,
namely $\deg(X_{[n]})=n$. Additionally, they satisfy a similar relation to
(85),

$[X_{[0]},Y_{[0]}]=(X_{(0)}Y)_{[0]},$ (95)

with $(TX)_{[0]}=0$. One has $(x^{(i)}_{-1}\ket{0}^{\bm{k}})_{[n]}=x_{n}$ for
$x\in\mathfrak{sl}_{2}$ and it is possible to show that the following
recursive formula holds:

$(A_{(-n)}B)_{[m]}=((A\otimes
f(t))B)_{[m]}+\sum_{k>0}c_{k}A_{[-k]}B_{[k+m]}+\sum_{k\leq
0}c_{k}B_{[k+m]}A_{[-k]},$ (96)

where $f(t)$ is the Taylor series in $t:=u-v$ given by

$f=\frac{1}{(n-1)!}(-\partial_{u})^{n-1}\Big{(}\frac{1}{u-v}-\iota_{u-v}\frac{e^{v}}{e^{u}-e^{v}}\Big{)}.$
(97)

and where the coefficients $c_{k}$ are defined by the requirement that
$\sum_{k>0}c_{k}(\frac{z}{w})^{k}$ and $-\sum_{k\leq 0}c_{k}(\frac{z}{w})^{k}$
are the expansions, for $|z|<|w|$ and $|w|<|z|$ respectively, of the function

$\frac{1}{(n-1)!}(-w\partial_{w})^{n-1}\frac{z}{w-z}.$ (98)

The first relevant examples are

$\displaystyle(A_{(-1)}B)_{[m]}=\begin{aligned}
\frac{1}{2}(A_{(0)}B)_{[m]}-\frac{1}{12}(A_{(1)}&B)_{[m]}+\dots\\\
&+\sum_{k>0}A_{[-k]}B_{[k+m]}+\sum_{k\leq 0}B_{[k+m]}A_{[-k]}\end{aligned}$
(99a) $\displaystyle(A_{(-2)}B)_{[m]}=\begin{aligned}
\frac{1}{12}(A_{(0)}B)_{[m]}-\frac{1}{240}&(A_{(2)}B)_{[m]}+\dots\\\
+&\sum_{k>0}kA_{[-k]}B_{[k+m]}+\sum_{k\leq
0}(-k)B_{[k+m]}A_{[-k]}\end{aligned}$ (99b)

where $A,B\in\mathbb{V}_{0}^{\bm{k}}$. These formulae are the Fourier-analog
of the normal ordered product formula eq. 23, and they allow one to compute by
recursion the Fourier modes of a general state $X\in\mathbb{V}_{0}^{\bm{k}}$.

Property (95) means that if the vertex algebra 0th product of $X$ and $Y$
vanishes their Fourier zero-modes generate a commutative subalgebra, in
homogeneous degree zero, of $\widetilde{U}(\widehat{\mathfrak{sl}}_{2}^{\oplus
N})$. We let

$\widehat{Q}_{n}^{\gamma}=\int_{\gamma}\mathscr{P}(z)^{-n/2}\varsigma_{n}(z)_{[0]}\text{d}z,$
(100)

for $n=1,3$. By the same logic as for proposition 4.7.1, we have the
following.

######  Proposition 4.7.2.

The operators
$\widehat{Q}_{n}^{\gamma}\in\widetilde{U}_{\bm{k}}(\widehat{\mathfrak{sl}}_{2}^{\oplus
N})$ have homogeneous degree 0 and they commute amongst themselves, with the
generators of $\widehat{\mathfrak{sl}}_{2}$, and with the quadratic
Hamiltonians $\mathscr{H}_{i}$.

##  5 Higher local Hamiltonians to sub-leading order

In the previous section, we have shown that it is possible to define quartic
local Hamiltonians which commute among themselves and with the quadratic ones,
together with the generators of $\mathfrak{sl}_{2}$. Following the same steps,
one could in principle try to construct the Hamiltonians for every exponent of
$\widehat{\mathfrak{sl}}_{2}$. However, the direct calculation (already
lengthy in the case of $\varsigma_{3}(z)_{(0)}\varsigma_{3}(w)$, as we noted
above) becomes computationally very demanding. What we shall do instead is
work to next-to-leading order in a certain semiclassical limit, which will at
least give a strong consistency check on the existence of the expected
Hamiltonian densities.

Thus, let us introduce a formal parameter $\hbar$ and work over
$\mathbb{C}[[\hbar]]$. So in particular, all vector spaces above are now to be
regarded as modules over $\mathbb{C}[[\hbar]]$. We consider the following
rescaled generators:

$\displaystyle I^{a}\longrightarrow\tilde{I}^{a}$ $\displaystyle:=\hbar
I^{a},$ $\displaystyle\mathsf{k}\longrightarrow\widetilde{\mathsf{k}}$
$\displaystyle:=\mathsf{k}.$ (101)

With this re-scaling the commutation relations become

$[\tilde{I}^{a[p]}_{m}(z),\tilde{I}^{b[q]}_{n}(z)]=-\frac{p!q!}{(p+q+1)!}(\hbar
f^{ab}_{c}\tilde{I}^{c[p+q+1]}_{m+n}(z)-\hbar^{2}n\delta^{ab}\delta_{m+n,0}\widetilde{\mathsf{k}}^{[p+q+1]}(z)).$
(102)

At this point we can identify the various quantum corrections by their $\hbar$
dependence and work grade by grade. We shall work at next-to-leading order,
i.e. the next order beyond the usual semi-classical calculation of Poisson
brackets. Thus, we consider the densities of Hamiltonians up to and including
the leading quantum corrections at order $\hbar$, and we compute commutators
up to and including terms of order $\hbar^{2}$.

######  Remark.

It is worth remarking that the classical limit, section 5, that we take is not
quite the standard one which recovers the usual classical Gaudin model (cf.
[KLT22] for a very complete discussion of that limit), because for us the
central charges remain $O(1)$ in the limit. From our present perspective this
is simply for computational convenience – this limit produces the simplest
possible non-trivial check, and had we rescaled the central charges there
would be more potential quantum correction terms already at next-to-leading
order. But it might be interesting to consider this classical limit in its own
right.

Having introduced the formal parameter $\hbar$, there is a gradation on the
enveloping algebras in which $\tilde{I}^{a}$ and $\hbar$ have grade one and
$\widetilde{\mathsf{k}}$ has grade zero. Recall that $\mathcal{V}_{n}$ denotes
the space of homogeneous meromorphic states of degree $n$, eq. 45. Let now
$\widetilde{\mathcal{V}}_{n}\subset\mathcal{V}_{n}$ denote the subspace
consisting of states that are also of grade $n$ in this new gradation (_i.e._
which are sums of terms having exactly $n$ factors of $\tilde{I}^{a}$ or
$\hbar$).

######  Proposition 5.0.1.

Modulo terms of order $\hbar^{2}$ there is, up to rescaling, exactly one state
$\tilde{\varsigma}_{2n-1}\in\widetilde{\mathcal{V}}_{2n}^{\mathfrak{sl}_{2}}$,
$n\in\mathbb{Z}_{\geq 1}$, such that, for all $x\in\mathfrak{sl}_{2}$ and
$m\in\mathbb{Z}_{\geq 0}$,

$\Delta
x_{m}\tilde{\varsigma}_{2n-1}=0\qquad\textup{mod}\>\hbar^{2}D^{(2n-1)}_{z}\mathcal{V}_{2n-m,2n-1},\quad\textup{mod}\>\hbar^{3}\mathcal{V}_{2n-m,2n}.$
(103)

Explicitly, modulo terms in $\hbar^{2}\mathcal{V}_{2n}$,

$\begin{split}\tilde{\varsigma}&{}_{2n-1}(z)=t_{i_{1},\dots,i_{2n}}\tilde{I}^{i_{1}}_{-1}(z)\tilde{I}^{i_{2}}_{-1}(z)\cdots\tilde{I}^{i_{2n}}_{-1}(z)\ket{0}\\\
&\qquad+\hbar\frac{n(2n+1)(2n-2)}{(2n-1)}t_{i_{1},\dots,i_{2n-4}}f^{abc}\tilde{I}^{a}_{-2}(z)\tilde{I}^{b\prime}_{-1}(z)\tilde{I}^{(c}_{-1}(z)\tilde{I}^{i_{1}}_{-1}(z)\cdots\tilde{I}^{i_{2n-4})}_{-1}(z)\ket{0}.\end{split}$
(104)

###### Proof.

Given the basis $\\{I^{r}\\}_{r=1}^{3}$ for $\mathfrak{sl}_{2}$, we need to
show that there exist a function $\mathsf{G}_{m}(z)$ such that

$\Delta
I^{r}_{m}\tilde{\varsigma}_{2n-1}(z)=D^{(2n-1)}_{z}\mathsf{G}^{r}_{m}(z)+\mathcal{O}(\hbar^{3}).$
(105)

For $m=0$, this is always true thanks to the invariance of the tensor (37).
For the same reasons explained in the previous section, the only relevant
check that one needs to make is the one for $m=1$. From direct calculation, we
get

$\mathsf{G}^{r}_{1}(z)=-\frac{4n}{(2n-1)}\hbar^{2}t_{i_{1},\dots,i_{2n-2}}\tilde{I}^{(r}_{-1}(z)\tilde{I}^{i_{1}}_{-1}(z)\cdots\tilde{I}^{i_{2n-2})}_{-1}(z)\ket{0}.$
(106)

Note that this result is in accordance with the exact ones obtained for the
quadratic ($n=1$) and the leading order of quartic ($n=2$) states, cf. eqs. 60
and 68. ∎

(Observe that this is consistent with proposition 4.4.2 because the vectors in
eqs. 62 and 63 all come with factors of $\hbar^{2}$ in the limit.)

We can now state the following theorem

######  Theorem 5.0.1.

Let $\tilde{\varsigma}_{n}(z)$ be as in eq. 104 above, for all odd
$m,n\in\mathbb{Z}_{\geq 1}$. We have

$\begin{split}\tilde{\varsigma}_{m}(z)_{(0)}\tilde{\varsigma}_{n}(w)=(nD^{(m)}_{z}-mD^{(n)}_{w})\mathsf{A}_{m,n}(z,w)+T\mathsf{B}_{m,n}(z,w)+\mathcal{O}(\hbar^{3}),\end{split}$
(107)

where
$\mathsf{A}_{m,n}(z,w),\mathsf{B}_{m,n}(z,w)\in\widetilde{\mathcal{V}}_{(z,w)}$
are given by

$\begin{split}\mathsf{A}_{m,n}(z,w)=&\zeta_{m,n}\frac{\hbar^{2}}{z-w}t_{i_{1},\dots,i_{m-1}}t_{j_{1},\dots,j_{n-1}}\\\
&\times
T\Big{(}\tilde{I}_{-1}^{(a}(z)\tilde{I}_{-1}^{i_{1}}(z)\dots\tilde{I}_{-1}^{i_{m-1})}(z)\Big{)}\tilde{I}_{-1}^{(a}(w)\tilde{I}_{-1}^{j_{1}}(w)\dots\tilde{I}_{-1}^{j_{n-1})}(w)\ket{0}+\mathcal{O}(\hbar^{3}),\end{split}$
(108)
$\begin{split}\mathsf{B}_{m,n}(z,w)=&\zeta_{m,n}\frac{\hbar^{2}}{(z-w)^{2}}t_{i_{1},\dots,i_{m-1}}t_{j_{1},\dots,j_{n-1}}\\\
&\times\tilde{I}_{-1}^{(a}(z)\tilde{I}_{-1}^{i_{1}}(z)\dots\tilde{I}_{-1}^{i_{m-1})}(z)\tilde{I}_{-1}^{(a}(w)\tilde{I}_{-1}^{j_{1}}(w)\dots\tilde{I}_{-1}^{j_{n-1})}(w)\ket{0}+\mathcal{O}(\hbar^{3}),\end{split}$
(109)

where

$\zeta_{m,n}=\frac{2(m+1)(n+1)}{mn}.$ (110)

Moreover, $\mathsf{A}_{m,n}(z,w)$ is a regular function for $z=w$ modulo
translates and modulo terms proportional to $\hbar^{3}$.

###### Proof.

The 0th mode of $\varsigma_{m}(z)\in\widetilde{\mathcal{V}}_{m+1}$ can be
inferred from a purely combinatorial reasoning. Let us start with the top term
$\tilde{\varsigma}_{m}^{\textup{TT}}(z)$ of (104). We know that computing the
0th mode, the number of generators in any term we get does not change, but the
result will be a state of total depth $m$ and therefore we know there must be
at least one generator with a positive mode. We can also use the fact that we
are working at leading order in $\hbar$, therefore we could get at least one
$\tilde{I}_{0}$, one $\tilde{I}_{1}$ or a term with two $\tilde{I}_{0}$, every
other term will be $\mathcal{O}(\hbar^{3})$. The only thing to fix is the
combinatorial factor describing the number of possible ways to write such
terms. The result is

$\begin{split}\tilde{\varsigma}_{m}^{\textup{TT}}(z)_{(0)}=t_{i_{1},\dots,i_{m+1}}\Big{[}&\frac{(m+1)!}{(m-1)!}\tilde{I}^{i_{1}}_{-2}(z)\tilde{I}^{i_{2}}_{-1}(z)\dots\tilde{I}^{i_{m}}_{-1}(z)\tilde{I}^{i_{m+1}}_{1}(z)\\\
&+\frac{(m+1)!}{(m)!}\tilde{I}^{i_{1}}_{-1}(z)\dots\tilde{I}^{i_{m}}_{-1}(z)\tilde{I}^{i_{m+1}}_{0}(z)\\\
&+\frac{(m+1)!(m-1)}{2(m-1)!}\tilde{I}^{i_{1}}_{-2}(z)\tilde{I}^{i_{2}}_{-1}(z)\dots\tilde{I}^{i_{m-1}}_{-1}(z)\tilde{I}^{i_{m}}_{0}(z)\tilde{I}^{i_{m+1}}_{0}(z)\Big{]}+\mathcal{O}(\hbar^{3}).\end{split}$

With similar arguments we can compute the 0th mode of the correction term
$\tilde{\varsigma}_{m}^{\textup{C}}(z)$ of (104), the result reads

$\begin{split}\tilde{\varsigma}_{m}^{\textup{C}}(z)_{(0)}=&-\hbar\xi(m-2)!t_{i_{1},\dots,i_{m-3}}f^{abc}\tilde{I}_{-2}^{a}(z)\tilde{I}_{-1}^{(b}(z)\tilde{I}_{-1}^{i_{1}}(z)\dots\tilde{I}_{-1}^{i_{m-3})}(z)\tilde{I}^{c}_{0}(z)\\\
&+\hbar\xi
2(m-2)(m-3)!t_{i_{1},\dots,i_{m-3}}f^{abc}\tilde{I}_{-2}^{a}(z)\tilde{I}_{-1}^{b\prime}(z)\tilde{I}_{-1}^{(i_{1}}(z)\dots\tilde{I}_{-1}^{i_{m-3})}(z)\tilde{I}_{0}^{c}(z)\\\
&+\hbar\xi(m-2)(m-3)(m-3)!t_{i_{1},\dots,i_{m-5}}f^{abc}\tilde{I}_{-2}^{a}(z)\tilde{I}_{-1}^{b\prime}(z)\\\
&\hskip
128.0374pt\times\tilde{I}_{-1}^{(c}(z)\tilde{I}_{-1}^{d}(z)\tilde{I}^{i_{1}}_{-1}(z)\dots\tilde{I}_{-1}^{i_{m-5})}\tilde{I}_{0}^{d}(z)\\\
&-\hbar\xi(m-2)(m-3)(m-3)!t_{i_{1},\dots,i_{m-5}}f^{abc}\tilde{I}_{-2}^{d}(z)\tilde{I}_{-1}^{a\prime}(z)\\\
&\hskip
128.0374pt\times\tilde{I}_{-1}^{(b}(z)\tilde{I}_{-1}^{d}(z)\tilde{I}^{i_{1}}_{-1}(z)\dots\tilde{I}_{-1}^{i_{m-5})}(z)\tilde{I}_{0}^{c}(z)+\mathcal{O}(\hbar^{3})\\\
\end{split}$

where $\xi=\frac{(m+2)(m+1)(m-1)}{2m}$. At this point, acting with what we
have obtained on $\varsigma_{n}(w)$ and using repeatedly the commutation
relations (41), we obtain eq. 107. ∎

##  Appendix A Full expression for $\mathsf{A}_{1,3}(z,w)$ and
$\mathsf{B}_{1,3}(z,w)$

The explicit expressions for $\mathsf{A}_{1,3}(z,w)$ and
$\mathsf{B}_{1,3}(z,w)$ in $\mathcal{V}_{(z,w)}$ obtained by direct
calculations are

$\begin{split}\mathsf{A}_{1,3}(z,w)&=\Big{[}\frac{8}{3}\frac{1}{z-w}I^{a}_{-2}(z)I^{(a}_{-1}(w)I^{b}_{-1}(w)I^{b)}_{-1}(w)\\\
&+f^{abc}\Big{(}\frac{80}{9}\frac{1}{z-w}I^{a}_{-2}(z)I^{b}_{-2}(w)I^{c\prime}_{-1}(w)-\frac{80}{9}\frac{1}{(z-w)^{2}}I^{a}_{-2}(z)I^{b}_{-2}(w)I^{c}_{-1}(w)\\\
&\hskip
56.9055pt+\frac{160}{9}\frac{1}{z-w}I^{a}_{-3}(z)I^{b\prime}_{-1}(w)I^{c}_{-1}(w)\Big{)}\\\
&-\frac{80}{9}\frac{1}{z-w}I^{a}_{-2}(z)I^{a}_{-3}(w)\mathsf{k}^{\prime}(w)-\frac{80}{3}\frac{1}{z-w}I^{a}_{-4}(z)I^{a}_{-1}(w)\mathsf{k}^{\prime}(w)\\\
&+\frac{320}{27}\frac{1}{(z-w)^{3}}I^{a}_{-2}(z)I^{a}_{-3}(w)+\frac{160}{9}\frac{1}{z-w}I^{a}_{-4}(z)I^{a\prime\prime}_{-1}(w)\\\
&+\frac{320}{9}\frac{1}{(z-w)^{3}}I^{a}_{-4}(z)I^{a}_{-1}(w)+\frac{160}{27}\frac{1}{(z-w)^{2}}I^{a}_{-2}(z)I^{a\prime}_{-3}(w)\\\
&-\frac{160}{9}\frac{1}{z-w}I^{a}_{-3}(z)I^{a\prime\prime}_{-2}(w)-\frac{160}{9}\frac{1}{(z-w)^{2}}I^{a}_{-3}(z)I^{a\prime}_{-2}(w)\\\
&-\frac{160}{9}\frac{1}{(z-w)^{2}}I^{a}_{-4}(z)I^{a\prime}_{-1}(w)\Big{]}\ket{0}.\end{split}$

$\begin{split}\mathsf{B}_{1,3}(z,w)=&\Big{[}\frac{8}{3}\frac{1}{(z-w)^{2}}I^{a}_{-1}(z)I^{(a}_{-1}(w)I^{b}_{-1}(w)I^{b)}_{-1}(w)\\\
&+f^{abc}\Big{(}-\frac{160}{9}\frac{1}{(z-w)^{3}}I^{a}_{-1}(z)I^{b}_{-2}(w)I^{c}_{-1}(w)-\frac{80}{9}\frac{1}{(z-w)^{2}}I^{a}_{-1}(z)I^{b\prime}_{-1}(w)I^{c}_{-2}(w)\\\
&\hskip
56.9055pt-\frac{160}{9}\frac{1}{(z-w)^{2}}I^{a}_{-2}(z)I^{b\prime}_{-1}(w)I^{c}_{-1}(w)\Big{)}\\\
&-\frac{160}{9}\frac{1}{(z-w)^{2}}I^{a}_{-1}(z)I^{a}_{-3}(w)\mathsf{k}^{\prime}(w)+\frac{1120}{27}\frac{1}{(z-w)^{2}}I^{a}_{-1}(z)I^{a\prime\prime}_{-3}(w)\\\
&+\frac{640}{27}\frac{1}{(z-w)^{3}}I^{a}_{-1}(z)I^{a\prime}_{-3}(w)-\frac{320}{9}\frac{1}{(z-w)^{2}}I^{a}_{-2}(z)I^{a\prime}_{-2}(w)\\\
&+\frac{320}{9}\frac{1}{(z-w)^{4}}I^{a}_{-1}(z)I^{a}_{-3}(w)-\frac{640}{9}\frac{1}{(z-w)^{3}}I^{a}_{-3}(z)I^{a\prime}_{-1}(w)\\\
&+\frac{320}{3}\frac{1}{(z-w)^{4}}I^{a}_{-3}(z)I^{a}_{-1}(w)\Big{]}\ket{0}.\end{split}$

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