# Are prime numbers and quadratic residues random?

Michael Blank  Institute for Information Transmission Problems RAS (Kharkevich
Institute);National Research University “Higher School of Economics”; e-mail:
<EMAIL_ADDRESS>
(March 7, 2024)

###### Abstract

Appeals to randomness of various number-theoretic constructions are regularly
found in scientific publications. It is enough to mention such famous names as
V.I. Arnold, M. Kac, Ya.G. Sinai, and T. Tao. Unfortunately, all this comes
down to various, although often quite non-trivial, heuristics. I will describe
a novel analytical approach to address this question. As an application, the
expected positive answer to the question about the randomness of quadratic
residues and the unexpected negative answer in the case of prime numbers will
be given. Technically, the proposed approach is based on a fundamentally new
construction of the entropy of the trajectory of a dynamical system, which is
in some way intermediate between the classical Kolmogorov-Sinai metric entropy
and topological entropy.

2020 Mathematics Subject Classification. Primary: 37A44; Secondary: 37A35,
11N05, 11K65.
Key words: entropy, ergodic theory, randomness, complexity, prime number,
quadratic residue.

##  1 Introduction

Due to the obvious observable complexity of various number-theoretic
constructions and the variety of patterns of numbers that arise, both
specialists in this theory and other mathematicians interested in similar
problems build purely random models to describe them. Let us note the Cramer
model (see, for example, [13]), the substitution of random numbers in a series
that determines the zeta function (see, for example, [12]), the analysis of
hidden periodicities of emerging sequences and geometric properties of the
Poisson process (see [23]), or construction of a natural invariant measure
concentrated on the set of square-free numbers (see [8]). These and other
approaches will be discussed in Sections 5,6.

The range of conclusions is also interesting: from Arnolds denial of the
randomness of quadratic residues to Cramer’s model, which asserts the
randomness of prime numbers. It seems that the point here is to highlight some
purely specific properties to the detriment of others. In particular, in the
case of the Cramer’s model, everything is based on the selection of the
desired probability distribution, without taking into account the fact that
this characteristic alone, although undoubtedly important, does not completely
determine the random process.

The purpose of this article is to offer a quantitative answer to the question
of the complexity/randomness of a single purely deterministic series of points
of the types discussed above. Of course, such approaches are well known, but
they all have serious drawbacks, since they are either non-constructive or
completely ad hoc. Perhaps the most beautiful among them was proposed by A.N.
Kolmogorov. His main idea was to treat the sequence under study as the
trajectory of some dynamical system and reduce the question of the complexity
of the sequence to the analysis of the “simplest” dynamical system that
generates it.

We will follow the same idea, but remembering that different trajectories of
the same dynamical system can exhibit qualitatively different behavior, we
will come up with a new concept of “local” dynamical entropy $h_{\rm loc}$
(see Section 3), independent of the choice of the invariant measure (unlike
the Kolmogorov-Sinai metric entropy) and making it possible to study even
dynamical systems that do not have such a measure (see, for example, [2], for
the discussion of such systems).

In what follows, we will refine this concept to analyze the
complexity/randomness of individual binary series representing the number
theoretic constructions mentioned above.

In order to make a distinction between static and dynamic entropy-like
characteristics we use the uppercase letter $H$ in the static setting and the
lowercase letter $h$ in the dynamic one. By $\lim^{\pm}$ we mean upper and
lower limits correspondingly and drop the signs if they coincide.

The entropy-like characteristic of randomness $h(\vec{x})$ takes values in
${\mathbb{R}}_{+}$. The value $h(\vec{x})=\infty$ is interpreted as a pure
randomness, while the opposite value $h(\vec{x})=0$ corresponds to a non-
random sequence $\vec{x}$.

The paper is organized as follows. In Section 2 we recall the classical
definitions related to the concept of Shannon entropy of a discrete
distribution and demonstrate that from the point of view of small
perturbations this functional exhibits a number of unexpected features. In
Section 3 we deal with various dynamical versions of the concept of entropy,
starting with the famous Kolmogorov-Sinai metric entropy of a dynamical
system. To take care about complexity properties of individual trajectories,
we introduce here a pair of completely new notions of local and information
entropies of a trajectory. By specifying these new entropies for the case of a
discrete phase space in Section 4 we introduce new measures of
complexity/randomness of sequences of points from finite alphabet and compare
them to some known approachers. The remaining part of the paper is devoted to
the application of these constructions to prove in Section 5 the nonrandom
nature of the set of prime numbers and in Section 6 the randomness of
quadratic residues.

The author is grateful to L.Bassalygo, A.Kalmynin, S.Pirogov, A.Shen,
M.Tsfasman, and A.Vershik for useful discussions of the issues raised in the
article.

##  2 Information (Shannon) entropy

Let $(\Omega,\Sigma)$ be a measurable space with the Borel $\sigma$-algebra
$\Sigma$ of measurable subsets.

Throughout this section we assume that the space $\Omega$ is discrete, i.e.,
$\Omega:=\\{\omega_{1},\omega_{1},\dots\\}$, and $\Sigma:=2^{\Omega}$. By
$\vec{p}:=(p_{1},p_{2},\dots)$ we denote a distribution (non-necessarily
probabilistic) on $\Omega$, i.e. $p_{i}\geq 0~{}\forall i$, but
$||\vec{p}||:=\sum_{i}p_{i}$ may differ from $1$. Denote $H(C):=-C\log C$ for
any constant $C\geq 0$, $H(0)=0$.

###### Definition 2.1

The (Shannon) entropy of $\vec{p}$ is defined as
$H(\vec{p}):=-\sum_{i}p_{i}\log p_{i}$.

###### Lemma 2.1

For distributions $\vec{p},\vec{q}$ on $\Omega$ with $||\vec{q}||\ll 1$

$H(\vec{p}+\vec{q})=H(\vec{p})+|\log(\inf_{i}p_{i})+1|\cdot||\vec{q}||+o(||\vec{q}||\cdot|\log(\inf_{i}p_{i})|)$
(1)
$\displaystyle|H(\vec{p}+\vec{q})-H(\vec{p})|\leq\min\\{H(\vec{p})+H(||\vec{p}||)+H(||\vec{q}||)+||\vec{q}||\log(\\#\Omega),$
(2) $\displaystyle~{}2|\log(\inf_{i}p_{i})+1|\cdot||\vec{q}||\\}$

###### Corollary 2.2

The functional $H:\ell_{1}(\Omega)\to{\mathbb{R}}_{+}\cup\\{\infty\\}$ is
continuous on the entire space $\ell_{1}(\Omega)$ if and only if
$\\#\Omega<\infty$.

This simple result is important from both theoretical and practical points of
view; in particular, it demonstrates that Khinchins classical axiomatic
entropy construction [Kh] cannot be extended from finite space to an infinite
one. In practice, this makes it possible control the accuracy of entropy
calculations when we know the distribution only approximately, which will be
very handy in Sections 5,6.

Surprisingly, Lemma 2.1 seems new. At least, I was not able to find results of
this sort in numerous publications devoted to the concept of entropy.

Proof of Lemma 2.1. The key point here is the observation that for
$\varphi(t):=H(t),~{}t\in{\mathbb{R}}_{+}$ and $0\leq\varepsilon\ll 1$ one has

$\varphi(t+\varepsilon)-\varphi(t)=-\varepsilon(\log t+1)+o(\varepsilon|\log
t|)$ (3)

and that in the case of finite $\Omega$ the uniform distribution maximizes the
entropy. $\sqcup\\!\\!\\!\\!\sqcap$

In the sequel we will need the following important result of perturbation
type.

###### Lemma 2.3

Let $\vec{p},\vec{p}^{(N)},~{}N\in{\mathbb{Z}}_{+}$ be probability
distributions such that $\vec{p}^{(N)}_{i}=0~{}\forall i>N$, and let

$\varepsilon_{N}:=\sum_{i}|p^{(N)}_{i}-p_{i}|\leq CN^{-\alpha}\quad\forall
N\in{\mathbb{Z}}_{+}$

for some $C,\alpha\in{\mathbb{R}}_{+}$. Then
$H(\vec{p}^{(N)})\stackrel{{\scriptstyle
N\to\infty}}{{\longrightarrow}}H(\vec{p})$. If the assumptions above are
violated, namely either $\vec{p}^{(N)}$ are not supported by segments $[1,N]$,
or $\limsup_{N\to\infty}N^{\alpha}\varepsilon_{N}=\infty~{}\forall\alpha>0$,
then there may be no convergence of entropies.

Proof. Fix some $t>0$ (to be specified later) and denote by
$I^{t}:=(i^{t}_{1},i^{t}_{2},\dots)$ those indices, for which
$p_{i^{t}_{1}}\geq t$. We estimate separately the contribution to the
difference of entropies coming from the indices $j\in I^{t}$ and from the
others, namely, in the former case we use the inequality (3), while in the
latter case we make an estimate from above by zeroing the corresponding
$p_{j}$. Setting $N_{t}:=\\#I^{t}$, we get

$\displaystyle|H(\vec{p}^{(N)})-H(\vec{p})|\\!\\!\\!$
$\displaystyle\leq-\varepsilon_{N}(\log t+1)+o(\varepsilon_{N}|\log
t|)-\sum_{j\not\in
I^{t}}\frac{\varepsilon_{N}}{N-N_{t}}\log(\frac{\varepsilon_{N}}{N-N_{t}})$
$\displaystyle\leq 2\varepsilon_{N}|\log
t|-\varepsilon_{N}\log\varepsilon_{N}+\varepsilon_{N}\log(N-N_{t})$
$\displaystyle\leq 2CN^{-\alpha}|\log
t|-CN^{-\alpha}\log(CN^{-\alpha})+CN^{-\alpha}\log N$ $\displaystyle\leq
N^{-\alpha}\left(2C|\log t|+C\alpha\log(C^{-\alpha}N)+C\log
N\right)\stackrel{{\scriptstyle N\to\infty}}{{\longrightarrow}}0.$

Note that the convergence does not depend on the choice of $t$.

In the case of slow convergence of distributions one easily constructs
situations when the upper and lower bounds for the difference of entropies
differ significantly, which leads to the desired counterexample.
$\sqcup\\!\\!\\!\\!\sqcap$

##  3 Dynamical entropy in ergodic theory

Let us give a brief account on classical approaches to the construction of
entropy-like characteristics of a discrete time dynamical system, defined by a
measurable map $f$ from a measurable space $(X,\Sigma,\mu)$ into itself. We
start with the Kolmogorov-Sinai construction (see, for example, [5] for
details).

###### Definition 3.1

Given a pair of finite measurable partitions $\Delta,\Delta^{\prime}$ of
$(X,{\cal B},\mu)$ by their common refinement one means
$\Delta\bigvee\Delta^{\prime}:=\\{\Delta_{i}\cap\Delta^{\prime}_{j}:~{}\mu(\Delta_{i}\cap\Delta^{\prime}_{j})>0\\}$.

Let $\mu\in{\cal M}_{f}$ (the set of all $f$-invariant measures). Making the
refinement of $\\{f^{-1}\Delta_{i}\\}$ we again get a finite measurable
partition which we denote by $f^{-1}\Delta$. The $n$-th refinement
$\Delta^{n}$ of the partition $\Delta$ can be defined inductively

$\Delta^{n}:=\Delta^{n-1}\bigvee f^{-1}\Delta^{n-1},~{}~{}\Delta^{0}:=\Delta.$

###### Definition 3.2

The conditional Kolmogorov-Sinai entropy of a partition is defined as

$h_{\mu}(f|\Delta):=\liminf\limits_{n\to\infty}\frac{1}{n}H_{\mu}(\Delta^{n})=\lim\limits_{n\to\infty}\frac{1}{n}H_{\mu}(\Delta^{n}),$

where $H_{\mu}(\Delta):=-\sum_{i}\mu(\Delta_{i})\ln\mu(\Delta_{i})$ is the
entropy of the discrete distribution $\\{\mu(\Delta_{i})\\}$.

###### Definition 3.3

The Kolmogorov-Sinai metric entropy of the dynamical system $(f,X,\Sigma,\mu)$
is $h_{\mu}(f):=\sup_{\Delta}h_{\mu}(f|\Delta)$.

Alternative approaches are known for continuous maps $f\in C^{0}(X,X)$, where
$(X,\rho)$ is a compact metric space.

###### Definition 3.4

The $n$-th Bowen metric $\rho_{n}$ on $X$ is defined as
$\rho_{n}(u,v):=\max\left\\{\rho\left(f^{k}(u),f^{k}(v)\right):\;k=0,\ldots,n-1\right\\}$.

Let $B_{\varepsilon}^{n}(x)$ be the open ball of radius $\varepsilon$ in the
metric $\rho_{n}$ around $x$.

###### Definition 3.5

The Brin-Katok measure-theoretical entropy of a measure $\mu\in{\cal
M}_{f}(X)$ at a point $u\in X$ is $h_{\mu}(f,u):=-\lim\limits_{\varepsilon\to
0}\lim\limits_{n\to\infty}\frac{1}{n}\log\mu(B_{\varepsilon}^{n}(u))$.

Roughly speaking $h_{\mu}(f,u)$ measures the exponential rate of decay of the
measure of points that stay $\varepsilon$-close to the point $u$ under forward
iterates of the map $f$.

###### Theorem 3.1

[3] $h_{\mu}(f,u)$ is well defined for an ergodic measure $\mu$ and does not
depend on $u$ for $\mu$-a.e $u\in X$.

A topological version (independent on the choice of the measure $\mu$) is
available in the case of a continuous map $f$. In this setting
$\Delta:=\\{\Delta_{i}\\}_{1}^{r}$ is a covering of $X$ by open sets. Define a
transition matrix $M:=\\{m_{ij}\\}$, where $m_{ij}=1$ if $\Delta_{i}\cap
f^{-1}\Delta_{j}\neq\emptyset$ and $=0$ otherwise.

Then on the Cantor set $X_{M}$ (the space of sequences with the alphabet
${\cal A}:=\\{1,2,\dots,r\\}$ with the transition matrix $M$) the left shift
map $\sigma$ defines a symbolic dynamical system.

Denoting by $A_{\Delta}^{n}$ the set of all admissible words of length $n$
(i.e. different pieces of length $n$ of all trajectories of $(\sigma,X_{M})$)
and by $\\#A_{\Delta}^{n}$ – the number of such, we set

$h_{{\rm
top}}(f|\Delta):=\liminf_{n\to\infty}\frac{1}{n}\log(\\#A_{\Delta}^{n})=\lim_{n\to\infty}\frac{1}{n}\log(\\#A_{\Delta}^{n}).$

###### Definition 3.6

The topological entropy is defined as $h_{{\rm top}}(f):=\sup_{\Delta}h_{{\rm
top}}(f|\Delta)$.

Note that the construction of Kolmogorov-Sinai metric entropy (as well as of
Brin-Katok entropy) are based on the choice of $f$-invariant measure $\mu$
(and depends on it), and the construction of topological entropy makes sense
only for continuous maps. On the other hand, a general measurable dynamical
system needs not to have even a single invariant measure (not speaking about
the assumption on continuity). Discussion of dynamical systems having no
invariant measures can be found, for example, in [2].

To overcome these difficulties we propose yet another entropy-like
constructions.

Let $\Delta:=\\{\Delta_{i}\\}$ be a partition (covering) of $X$ by measurable
sets. We refer to the indices of $\Delta_{i}$ as an alphabet ${\cal A}$, which
needs not to be finite. We say that on a starting segment of length $N$ of a
given trajectory $\vec{x}:=(x_{1},x_{2},\dots)$ of our system there is a word
$\vec{w}:=(w_{1},\dots,w_{n})$ composed of the letters $w_{i}\in{\cal A}$, if
there is $i$ such that

$x_{i+j}\in\Delta_{w_{j}}~{}~{}\forall j<n+1,i+j\leq N.$

Denote by $L(\vec{x},\vec{w},N)$ the number of occurrences of a word $\vec{w}$
in the starting piece of length $N$ of the trajectory $\vec{x}$, and let
$\vec{p}(\vec{x},n,N)=(p_{1},p_{2},\dots)$ be a distribution (frequency) of
all such words of length $n$.

###### Definition 3.7

By a conditional local entropy of the trajectory $\vec{x}$ we mean

$h_{{\rm
loc}}^{\pm}(\vec{x}|\Delta):=\lim_{n\to\infty}^{\pm}\lim_{N\to\infty}^{\pm}H(\vec{p}(\vec{x},n,N)).$

Here $\pm$ refers to the upper and lower limits, and

$H(\vec{p}(\vec{x},n,N)):=\sum_{i=1}p_{i}\log p_{i}$

is the entropy of the distribution $\vec{p}(\vec{x},n,N)$.

It is clear that this construction is something intermediate between metric
and topological entropy, but works for any measurable map and provides
information about “complexity” of individual trajectories. The “locality” of
$h_{{\rm loc}}$ is trajectory-wise, unlike the Brin-Katok entropy which is
point-wise.

In the case of a continuous map one a topological type approach to the same
problem is known (see [11, 10]), which can be formulated with very slight
modifications for the measurable case under discussion in the present paper.

Denote by $L(\vec{x},n,N)$ the number of different words of length $n$ in the
starting piece of length $N$ of the trajectory $\vec{x}$.

###### Definition 3.8

By conditional information entropy of the trajectory $\vec{x}$ we mean

$h_{{\rm
info}}^{\pm}(\vec{x}|\Delta):=\lim_{n\to\infty}^{\pm}\lim_{N\to\infty}^{\pm}\frac{1}{n}\log
L(\vec{x},n,N).$

Finally, we define the unconditional versions of entropies under consideration
as follows:

$h_{{\rm loc}}^{\pm}(\vec{x}):=\sup_{\Delta}h_{{\rm
loc}}^{\pm}(\vec{x}|\Delta),\quad h_{{\rm
info}}^{\pm}(\vec{x}):=\sup_{\Delta}h_{{\rm info}}^{\pm}(\vec{x}|\Delta).$ (4)

###### Definition 3.9

We say that a sequence of points $\vec{x}:=(x_{1},x_{2},\dots),~{}~{}x_{i}\in
X$ is typical with respect to a probabilistic measure $\mu$, if

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}1_{A}(x_{i})=\mu(A)\quad\forall
A\in\Sigma.$

In other words, the sequence $\vec{x}$ is distributed according to the measure
$\mu_{\vec{x}}=\mu$.

###### Lemma 3.1

For a periodic sequence $\vec{x}$ the measure $\mu_{\vec{x}}$ is well defined,
$h_{{\rm info}}(\vec{x})=0$, but $h_{{\rm loc}}(\vec{x})$ may be positive.

Proof. The claim about the measure $\mu_{\vec{x}}$ is obvious and we discuss
only statements about the entropies. Let $\vec{x}$ be $\ell$-periodic. Then
$L(\vec{x},n,N)\leq\ell~{}~{}\forall n,N$, which implies the claim. Consider
now the 2-periodic sequence $\vec{x}:=(010101\dots)$. Then for each given $n$
there are exactly 2 different sub-words of length $n$, each with the frequency
$1/2+o(1/N)$. The local entropy of this sequence is equal to $\log 2>0$.
$\sqcup\\!\\!\\!\\!\sqcap$

It is known that under reasonably mild assumptions on the dynamical system
$(f,X,\Sigma)$ $h_{{\rm top}}(f)=\sup_{\mu}h_{\mu}(f)$, where the supremum is
taken over all ergodic $f$-invariant measures. Therefore one might expect
something similar about the connection between $h_{{\rm info}}(\vec{x})$ and
$h_{{\rm loc}}(\vec{x})$. The claims of Lemma 3.1 demonstrate that this is
absolutely not the case. The reason is the locality of the construction in
distinction to the global definition of the metric and topological entropies.

If the measure $\mu_{\vec{x}}$ is well defined, one can explore the
connections between our newly defined entropy-like characteristics and more
classical approaches. However, this is beyond the scope of the present paper
and will be studied in a separate article. Note also that our local entropy
can be easily modified to work with bi-infinite trajectories of measurable
semi-groups, which will also be studied elsewhere.

In the next section we apply above construction to study “complexity” of
sequences of numbers, considered as trajectories of unknown dynamical systems,
and later to some number-theoretic constructions.

##  4 Complexity

One of the main concepts of complexity theory, introduced by A.N. Kolmogorov,
was to reduce the question of the complexity of a given sequence of points $x$
to analysis of the complexity of a dynamical system admitting it (i.e. this
sequence is the trajectory of such a system). Naturally, there are many
dynamical systems admitting a given sequence, and Kolmogorov proposed to take
into account the “simplest” among them. To implement this concept, he uses a
universal Turing machine to describe any dynamical system living in a finite
phase space, and the complexity of such machine is described by the minimum
length of the program generating it [9, 10, 11]. The beauty of this approach
comes at the price of being completely non-constructive. In practice, this
type of complexity can only be calculated for some toy examples.

Apart from the non-constructiveness of this approach there are two other
important issues. First, for a given sequence there might be no dynamical
systems, admitting it as a trajectory. Second, different trajectories of the
same dynamical system may demonstrate very different qualitative properties.

An alternative (let’s call it algorithmic) concept (see [10, 29, 28, KS, 11])
is based on the idea to treat a given sequence as an unordered set of points
and boils down to various dimension-like characteristics of this set. A
serious disadvantage here is that important information about the order of
items is lost.

In what follows we try to make use of Kolmogorov’s idea of the “dynamical
origin” of the sequence under study.

Let ${\cal A}:=\\{a_{1},a_{2},\dots\\}$ be at most a countable collection of
“letters” (to which we refer as an alphabet), equipped with the complete
$\sigma$-algebra $\Sigma:=2^{\cal A}$, and let
$\vec{x}:=\\{a_{k_{i}}\\}_{i\in{\mathbb{Z}}_{+}}$ be a sequence composed of
the letters from this alphabet.

For a map $f:{\cal A}\to{\cal A}$, admitting the sequence $\vec{x}$ as a
trajectory, we may apply the notions of the local and information entropies,
defined by the relations (4). There are two important observations here.
First, these notions do not depend on the choice of the map $f$ admitting
$\vec{x}$. Second, in the present setting the supremum over all measurable
partitions/covers can be easily calculated due to the non-negativity of
conditional entropies and the existence of the most fine partition, which
coincides with the partition into points. Thus we get

$h_{{\rm
loc}}^{\pm}(\vec{x}):=\lim_{n\to\infty}^{\pm}\lim_{N\to\infty}^{\pm}H(\vec{p}(\vec{x},n,N)).$
(5) $h_{{\rm
info}}^{\pm}(\vec{x}):=\lim_{n\to\infty}^{\pm}\lim_{N\to\infty}^{\pm}\frac{1}{n}\log
L(\vec{x},n,N).$ (6)

In the sequel we will need the following important perturbation type result.

###### Lemma 4.1

Let $\vec{p},\vec{p}^{(N)},~{}N\in{\mathbb{Z}}_{+}$ be probability
distributions, corresponding to the sequences
$\vec{x},\vec{x}^{(N)},~{}N\in{\mathbb{Z}}_{+}$ and let
$\varepsilon_{N}:=\sum_{i}|p^{(N)}_{i}-p_{i}|\leq CN^{-\alpha}$ for some
$C,\alpha\in{\mathbb{R}}$. Then $h_{{\rm
loc}}(\vec{x}^{(N)})\stackrel{{\scriptstyle
N\to\infty}}{{\longrightarrow}}h_{{\rm loc}}(\vec{x})$.

Proof. This is a direct corollary of Lemma 2.3. $\sqcup\\!\\!\\!\\!\sqcap$

It is worth noting that a functional qualitatively similar to (5) is known in
the literature on information theory under the name “finite-state dimension”
(see [28, 29] and further references therein). It was introduced in 2004 as a
finite-state version of classical Hausdorff dimension, and it measures the
lower asymptotic density of information in an infinite sequence over a finite
alphabet. The connection to any version of dynamical entropy has never been
discussed in this context.

##  5 Spatial distribution of prime numbers

In publications on number theory (see, for example, [12, 13, 14, 15, 16, 17])
we often read that random models provide heuristic support for various
conjectures and that prime numbers are believed to behave pseudo-randomly in
many ways and do not follow any simple pattern. An important example of a
purported statement about the pseudo-randomness of primes (known as the Cramer
model) is the Hardy-Littlewood conjecture for k-tuples, namely that the number
of occurrences of different patterns in primes can be approximated by treating
them as a sequence of random numbers generated by independently counting each
$k\in{\mathbb{Z}}_{+}$ as “prime” with probability $1/\log k$. For a detailed
discussion of this and a number of other examples of this kind, see [16].

Consider the sequence of prime numbers $\pi:=(1,2,3,5,7,11,13,\dots)$ and
match it to the binary sequence $\vec{b}$, such that $b_{\pi_{i}}=1~{}\forall
i\in{\mathbb{Z}}_{+}$ and $b_{j}=0~{}\forall j\not\in\pi$.

###### Theorem 5.1

$h_{{\rm loc}}(\vec{b})=0$.

To prove this result we discuss connections of some known statistics of primes
to similar statistics of finite binary words.

###### Conjecture 5.1

[19] Let $\vec{a}:=(a_{1},\dots,a_{k}$ be distinct integers which do not cover
all residue classes to any prime modulus. Then the number of integers $0<m\leq
N$ for which $m-a_{1},\dots,m-a_{k}$ are all primes satisfies the asymptotic
formula

$L_{k}(N,\vec{a})\approx C(k)N/(\log N)^{-k}.$ (7)

This conjecture is a close relative to the Hardy-Littlewood conjectures, but
at present only partial results in this direction have been rigorously proven.

###### Definition 5.1

A pair of consecutive prime numbers, separated by a single composite number
are called prime twins.

Denote by $L_{1}(N)$ and $L_{2}(N)$ the number of primes and prime twins in
$1,2,\dots,N$ correspondingly.

###### Theorem 5.2

[19] $L_{1}(N)<C\frac{N}{\log N},~{}~{}L_{2}(N)<C\frac{N}{\log^{2}N}$ for some
$C<\infty$ and all $N\gg 1$.

Let us compare these statistics with similar information about general binary
sequences.

###### Lemma 5.2

Let $Q(n,\vec{w})$ be the number of all binary words of length $n$, containing
the given sub-word $\vec{w}$. Then
(a) $Q(n+1,11)=Q(n,11)+Q(n-1,11)$,
(b) $Q(n+1,101)=Q(n,101)+Q(n-1,101)+Q(n-2,101)$.

Proof. We proceed by induction on $n$. If the new letter on the $(n+1)$-th
position is $0$ no new sub-word $11$ may show up, while if the new letter is
$1$ the new (in comparison to $n$) words of length $n+1$, containing $11$ show
up iff the $n$-th letter is $1$. This proves statement (a). Statement (b) is
proved in a similar way. $\sqcup\\!\\!\\!\\!\sqcap$

###### Corollary 5.3

  * (a)

$Q(n,11)$ are Fibonacci numbers, satisfying the limit relations
$z_{n}/z_{n+1}\to C$, $z_{n+1}/(z_{n+1}+z_{n})\to C$, where $C^{2}+C=1$,
$1/C=(1+\sqrt{5})/2<2$, $z_{n}\approx C^{-n}$.

  * (b)

$Q(n,101)$ are tribonacci numbers, satisfying the limit relations
$z_{n}/z_{n+1}\to C$, $z_{n+1}/(z_{n+1}+z_{n}+z_{n-1})\to C$, where
$C^{3}+C^{2}+C=1$, $1/C\approx 1.839286755<2$ and $z_{n}\approx C^{-n}$. The
tribonacci constant
$\frac{1}{3}\left(1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}\right)\approx
1.83929.$ The other two roots are complex conjugates.

For a given infinite binary sequence $\vec{b}$, denote by $M_{N}$ the number
of ones among the first $N$ letters of $\vec{b}$. By a zero word we will mean
a word, consisting of zeros only.

###### Lemma 5.4

Let $Q(n,N)$ be the fraction of zero sub-words of length $n<N$ in the first
$N$ letters of the sequence $\vec{b}$. Assume that $\frac{M_{N}}{N}\leq
CN^{-\alpha},~{}\alpha>0$. Then $Q(n,N)\geq 1-2CN^{-\alpha}$.

Proof. The binary sequence of length $N$, having the smallest number of zero
sub-words of length $n$ can be realized as follows: $0\dots 01~{}0\dots
01~{}\dots~{}0\dots 01~{}0\dots 0$. Each of $M_{N}$ blocks $0\dots 01$
consists of $(n-1)$ zeros and $1$ in the end. Thus the number of zero sub-
words of length $n$ is equal to $N-M_{N}n-n$, while the total number of zero
sub-words of length $n$ is $N-n$. Therefore the frequency

$Q(n,N)=\frac{N-M_{N}n-n}{N-n}=1-\frac{M_{N}n}{N-n}=1-\frac{M_{N}}{N}\frac{n}{1-n/N}.$

This gives the result. $\sqcup\\!\\!\\!\\!\sqcap$

Recalling that the number of primes below $N$ satisfies the inequality
$L_{1}(N)<C\frac{N}{\log(N)}$ (see Theorem 5.2), the application of Lemma 5.4
to the binary sequence $\vec{b}$ implies the claim of Theorem 5.1.

###### Remark 5.5

The argument above can be falsely interpreted as a claim that zero density of
ones in the binary sequence $\vec{b}$ implies that $h_{{\rm loc}}(\vec{b})=0$.
In fact, there are binary sequences $\vec{b}$ with zero density of ones, but
having $h_{{\rm loc}}(\vec{b})>0$, as well as binary sequence $\vec{b}$ with
the density of ones arbitrary close to 1, but having $h_{{\rm
loc}}(\vec{b})=0$.

Proof. To construct the first counter-example, observe that the non-
contradicting to the zero density faster growth of order $N^{\alpha}$ with
$0<\alpha<1$ of the number of ones below $N$ in $\vec{b}$ breaks down the
proof of Lemma 2.3.

The second counter-example can be realized as the sequence consisting of ones
only. $\sqcup\\!\\!\\!\\!\sqcap$

###### Remark 5.6

The validity of Conjecture 5.1 with the uniform bound $C(k)\leq
C<\infty~{}~{}\forall k$ implies that $0<h_{{\rm info}}(\vec{b})<\log 2$.

##  6 Spatial distribution of quadratic residues

Patterns formed by quadratic residues and non-residues modulo a prime have
been studied since the 19th century [21] and still they continue to attract
attention of contemporary mathematicians [23, 27, 22, 20] from various points
of view. For a detailed historical overview of the concept of quadratic
residues, we refer the reader to the monograph [24], and to the modern
analysis from an algebraic-geometric point of view - to [20].

Probably V.I. Arnold [23] was the first to discuss these matters from the
point of view of randomness, albeit at a heuristic level. Arnold’s negative
answer to this question stands in stark contrast to S. Wright’s [24] positive
answer, who used a completely different heuristic approach based on the
Central Limit Theorem. S. Wright argues also that the positive answer follows
from earlier results due to P. Kurlberg and Z. Rudnick [25, 26] about the
distribution of spacings between quadratic residues.

###### Definition 6.1

An integer $q$ is called a quadratic residue modulo $\pi$ if it is congruent
to a perfect square modulo $\pi$; i.e., if there exists an integer $\ell$ such
that: $\ell^{2}\equiv q~{}({\rm mod}~{}\pi)$. Otherwise, $q$ is called a
quadratic non-residue modulo $\pi$.

For $\pi=19$, the 9 quadratic residues are $(1,4,5,6,7,9,11,16,17)$, while
another 9 numbers $(2,3,8,10,16,12,13,14,15,18)$ are quadratic non-residues. A
number of other examples with their analysis can be found in [24].

For an odd prime $\pi$ consider the finite sequence $1,2,\dots,\pi-1$.
Replacing each number in this sequence with $1$ if it is a quadratic residue
modulo $\pi$, and $0$ otherwise, we get the binary word
$\vec{b}^{(\pi)}:=(b_{1},\dots,b_{\pi-1})$.

In this section we will be interested in the “randomness” of the sequence of
these binary words growing as $\pi\to\infty$ . Note that longer words here do
not include shorter ones, and that none of the complexity-type concepts
discussed in the literature capture situations of this kind. Therefore we need
a new definition in terms of a scheme of series.

###### Definition 6.2

Let $\vec{B}:=(\vec{b}^{m})_{m\in{\mathbb{Z}}_{+}}$ be a sequence of binary
words with $|\vec{b}^{m}|\stackrel{{\scriptstyle
m\to\infty}}{{\longrightarrow}}\infty$. By the local entropy of $\vec{B}$ we
mean $h_{{\rm loc}}(\vec{B}):=\lim_{m\to\infty}h_{{\rm loc}}(\vec{b}^{m})$,
and by the information entropy $h_{{\rm
info}}(\vec{B}):=\lim_{m\to\infty}h_{{\rm info}}(\vec{b}^{m})$.

###### Theorem 6.1

Let $\vec{B}$ be a sequence of binary words represented by quadratic residues,
namely $\vec{b}^{m}:=\vec{b}^{(\pi_{m})}$, where $\pi_{m}$ is the $m$-th prime
number. Then $h_{{\rm loc}}(\vec{B})=\infty,~{}h_{{\rm info}}(\vec{B})>0$.

To prove this result we need some known information about the statistics of
quadratic residues.

Let $L(\vec{w},\vec{b})$ be the number of occurrences of the binary word
$\vec{w}$ in a finite binary sequence $\vec{b}$. The following result on the
asymptotic equidistribution of these quantities for quadratic residues was
proved in [27] (see also a discussion in [20]).

###### Theorem 6.2

[27] For each binary word $\vec{w}$ of length $n:=|\vec{w}|\leq\pi$ we have

$|L(\vec{w},\vec{b}^{(\pi)})-\pi
2^{-|\vec{w}|}|<(|\vec{w}|-1)\sqrt{\pi}+|\vec{w}|/2.$

Proof of Theorem 6.1. By Theorem 6.2 the distribution of sub-words of the same
length in $\vec{b}^{m}$ is asymptotically uniform with accuracy or order
$1/\sqrt{\pi_{m}}$. Setting $N:=\pi_{m},n:=|\vec{w}|$, we see that the point-
wise modulus of difference between the theoretical uniform distribution
$\vec{p}^{u}$ of words of length $n$ and the observed distribution
$\vec{p}^{o}$ cannot exceed $CN^{-1/2}$. Thus it seems that we are in a
position to apply the result of Lemma 2.3.Unfortunately this is not the case.
Indeed, the $\ell_{1}$-norm of the difference of distributions is of order
$NN^{-1/2}=N^{1/2}\stackrel{{\scriptstyle
N\to\infty}}{{\longrightarrow}}\infty$.

Therefore, it is necessary to adjust the approach. Denote by
$\\{\varepsilon_{i}:=p_{i}^{u}-p_{i}^{o}\\}_{i}$ the point-wise differences of
distributions. Then

$|\varepsilon_{i}|\leq CN^{-1/2},\quad\sum_{i}\varepsilon_{i}=0.$

The equality above follows from the fact the both distributions are
probabilistic.

Using the same arguments as in the proof of Lemma 2.1, we get

$|H(\vec{p}^{u})-H(\vec{p}^{o})|=\sum_{i}\varepsilon_{i}|\log
N|+o(1/N)=o(1/N)\stackrel{{\scriptstyle N\to\infty}}{{\longrightarrow}}0.$

This proves that

$h_{{\rm loc}}(\vec{B})=\lim_{N\to\infty}H(\vec{p}^{u})=\infty.$

The positivity of $h_{{\rm info}}(\vec{B})$ follows from the observation that
by Theorem 6.2 all finite binary patterns have positive frequencies.
$\sqcup\\!\\!\\!\\!\sqcap$

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Perfect Integrability and Gaudin Models

Perfect Integrability and Gaudin Models††This paper is a contribution to the
Special Issue on Representation Theory and Integrable Systems in honor of
Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th
birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html

Kang LU

K. Lu

Department of Mathematics, University of Denver, 2390 S. York St., Denver, CO
80210, USA<EMAIL_ADDRESS>https://kanglu.me

Received August 26, 2020, in final form December 02, 2020; Published online
December 10, 2020

We suggest the notion of perfect integrability for quantum spin chains and
conjecture that quantum spin chains are perfectly integrable. We show the
perfect integrability for Gaudin models associated to simple Lie algebras of
all finite types, with periodic and regular quasi-periodic boundary
conditions.

Gaudin model; Bethe ansatz; Frobenius algebra

82B23; 17B80

##  1 Introduction

Quantum spin chains are important models in integrable system. These models
have numerous deep connections with other areas of mathematics and physics. In
this article, we would like to suggest the notion of perfect integrability for
quantum spin chains.

Let us recall Gaudin models and XXX spin chains. Let $\mathfrak{g}$ be a
simple (or reductive) Lie (super)algebra and $G$ the corresponding Lie group.
Let $\mathscr{A}_{\mathfrak{g}}$ be an affinization of $\mathfrak{g}$ where
the universal enveloping algebra $\mathrm{U}(\mathfrak{g})$ of $\mathfrak{g}$
can be identified as a Hopf subalgebra of $\mathscr{A}_{\mathfrak{g}}$. Here
$\mathscr{A}_{\mathfrak{g}}$ is either the universal enveloping algebra of the
current algebra $\mathrm{U}(\mathfrak{g}[t])$ which describes the symmetry for
Gaudin models, or Yangian $\mathrm{Y}(\mathfrak{g})$ associated to
$\mathfrak{g}$ for XXX spin chains. In both cases the algebra
$\mathscr{A}_{\mathfrak{g}}$ has a remarkable commutative subalgebra called
the _Bethe algebra_. We denote the Bethe algebra by
$\mathscr{B}_{\mathfrak{g}}$. The Bethe algebra $\mathscr{B}_{\mathfrak{g}}$
commutes with $\mathrm{U}(\mathfrak{g})$. Take any finite-dimensional
irreducible representation $M$ of $\mathscr{A}_{\mathfrak{g}}$, then
$\mathscr{B}_{\mathfrak{g}}$ acts naturally on the space of singular vectors
$M^{\mathrm{sing}}$. Let $\mathscr{B}_{\mathfrak{g}}(M^{\mathrm{sing}})$ be
the image of $\mathscr{B}_{\mathfrak{g}}$ in
$\mathrm{End}(M^{\mathrm{sing}})$. The problem is to study the spectrum of
$\mathscr{B}_{\mathfrak{g}}(M^{\mathrm{sing}})$ acting on
$M^{\mathrm{sing}}$.111The reason these models are called spin chains is that
$M$ is usually a tensor product of evaluation modules where each factor
corresponds to a particle of some spin.

With the agreement with the philosophy of geometric Langlands correspondence,
it is important to understand and describe the finite-dimensional algebra
$\mathscr{B}_{\mathfrak{g}}(M^{\mathrm{sing}})$ and the corresponding scheme
$\mathsf{Spec}(\mathscr{B}_{\mathfrak{g}}(M^{\mathrm{sing}}))$. Or more
generally, find a geometric object parameterizing the eigenspaces of
$\mathscr{B}_{\mathfrak{g}}$ when $M$ runs over all finite-dimensional
irreducible representations (up to isomorphism). In Gaudin models, the
underlying geometric objects are described by the sets of monodromy-free
${}^{L}\mathfrak{g}$-opers with regular singularities of prescribed residues
at evaluation points, see [4, 21], where ${}^{L}\mathfrak{g}$ is the Langlands
dual of $\mathfrak{g}$. Moreover, when $\mathfrak{g}=\mathfrak{gl}_{N}$, the
algebra $\mathscr{B}_{\mathfrak{g}}(M^{\mathrm{sing}})$ is interpreted as the
space of functions on the intersection of suitable Schubert cycles in a
Grassmannian variety, see [16]. This interpretation gives a relation between
representation theory and Schubert calculus useful in both directions which
has important applications in real algebraic geometry, see [13, 16].

Any finite-dimensional unital commutative algebra $\mathcal{B}$ is a module
over itself induced by left multiplication. We call this module the _regular
representation of_ $\mathcal{B}$. The dual space $\mathcal{B}^{*}$ is
naturally a $\mathcal{B}$-module which is called the _coregular
representation_. A Frobenius algebra is a finite-dimensional unital
commutative algebra whose regular and coregular representations are
isomorphic, see Section 2.6.

Let $V$ be a finite-dimensional $\mathcal{B}$-module. Let $\mathcal{B}(V)$ be
the image of $\mathcal{B}$ in $\mathrm{End}(V)$. We say that the
$\mathcal{B}$-module $V$ is _perfectly integrable_ if $\mathcal{B}$ acts on
$V$ cyclically and the algebra $\mathcal{B}(V)$ is a Frobenius algebra. Note
that in this case, the $\mathcal{B}(V)$-module $V$ is isomorphic to the
regular and coregular representations of $\mathcal{B}(V)$.

Based on the extensive study of quantum spin chains, see the evidences from
[2, 4, 10, 15, 16, 17, 21], the following conjecture is expected to hold.

######  Conjecture 1.1.

The $\mathscr{B}_{\mathfrak{g}}$-module $M^{\mathrm{sing}}$ is perfectly
integrable.

In fact there is a family of commutative Bethe algebras
$\mathscr{B}_{\mathfrak{g}}^{\mu}$ depending on an element
$\mu\in\mathfrak{g}^{*}$ (resp. $\mu\in G$). From here to Conjecture 1.2, we
use the parenthesis to indicate the modifications for XXX spin chains. If
$\mu\in\mathfrak{g}^{*}$ (resp. $\mu\in G$) is a regular semi-simple element,
we say that the corresponding spin chain has regular quasi-periodic boundary
condition. Moreover, if $\mu=0$ (resp. $\mu=\mathrm{Id}$), then the algebra
$\mathscr{B}_{\mathfrak{g}}^{0}$ (resp.
$\mathscr{B}_{\mathfrak{g}}^{\mathrm{Id}}$) coincides with the algebra
$\mathscr{B}_{\mathfrak{g}}$ considered above. If $\mu=0$ (resp.
$\mu=\mathrm{Id}$), we say that the corresponding spin chain has periodic
boundary condition.

For regular quasi-periodic spin chains the Bethe algebra does not commute with
$\mathrm{U}(\mathfrak{g})$ and one replaces $M^{\mathrm{sing}}$ with $M$.
Denote by $\mathscr{B}_{\mathfrak{g}}^{\mu}(M)$ the image of
$\mathscr{B}_{\mathfrak{g}}^{\mu}$ in $\mathrm{End}(M)$. Let $\mathfrak{h}$ be
the Cartan subalgebra of $\mathfrak{g}$ and $H$ the Cartan subgroup of $G$.

######  Conjecture 1.2.

If $\mu\in\mathfrak{h}^{*}$ $($resp. $\mu\in H)$ is regular, then the
$\mathscr{B}_{\mathfrak{g}}^{\mu}$-module $M$ is perfectly integrable.

For more general $\mu\in\mathfrak{g}^{*}$, one has to replace
$M^{\mathrm{sing}}$ or $M$ with an appropriate subspace of $M$ depending on
$\mu$, see Conjecture 2.9 in Section 2.8.

When Conjectures 1.1, 1.2, and 2.9 hold, we say that the corresponding quantum
spin chains are _perfectly integrable_.

The perfect integrability was shown for

  * •

Gaudin models of $\mathfrak{gl}_{N}$ in [15, 16] with periodic and regular
quasi-periodic boundary conditions;

  * •

XXX (resp. XXZ) spin chains of $\mathfrak{gl}_{N}$ associated to irreducible
tensor products of vector representations in [17] (resp. [18]) with periodic
and regular quasi-periodic boundary conditions;

  * •

XXX spin chains of $\mathfrak{gl}_{1|1}$ associated to cyclic tensor products
of polynomial representations in [10] with periodic and regular quasi-periodic
boundary conditions;

  * •

XXX spin chains of $\mathfrak{gl}_{m|n}$ associated to irreducible tensor
products of vector representations in [2] with periodic boundary condition.

Our main result confirms Conjectures 1.1 and 1.2 for Gaudin models of all
finite types, see Theorem 2.8. We deduce Theorem 2.8 from [4, Corollary 5],
[21, Theorem 3.2], and [6, Theorem 8.1.5].

Our suggestion to call the situations in Conjectures 1.1 and 1.2 “perfect
integrability” is motivated by Lemma 1.3 below.

Let $\mathcal{B}$ be a finite-dimensional unital commutative algebra. Let $V$
be a finite-dimensional $\mathcal{B}$-module and
$\mathcal{E}\colon\mathcal{B}\to\mathbb{C}$ a character, then the
$\mathcal{B}$-eigenspace and generalized $\mathcal{B}$-eigenspace associated
to $\mathcal{E}$ in $V$ are defined by

$\bigcap_{a\in\mathcal{B}}\ker(a|_{V}-\mathcal{E}(a))\qquad\text{and}\qquad\bigcap_{a\in\mathcal{B}}\big{(}\bigcup_{m=1}^{\infty}\ker(a|_{V}-\mathcal{E}(a))^{m}\big{)},$

respectively. Let $\mathcal{B}(V)$ be the image of $\mathcal{B}$ in
$\mathrm{End}(V)$.

######  Lemma 1.3.

If the $\mathcal{B}$-module $V$ is perfectly integrable, then every
$\mathcal{B}$-eigenspace in $V$ has dimension one, and there exists a
bijection between $\mathcal{B}$-eigenspaces in $V$ and
$\mathsf{Specm}(\mathcal{B}(V))$ – the subset of closed points in
$\mathsf{Spec}(\mathcal{B}(V))$. Moreover, each generalized
$\mathcal{B}$-eigenspace is a cyclic $\mathcal{B}$-module, and the algebra
$\mathcal{B}(V)$ is a maximal commutative subalgebra in $\mathrm{End}(V)$ of
dimension $\dim V$.

This lemma easily follows from general well-known facts about regular and
coregular representations of a finite-dimensional unital commutative algebra,
see, e.g., [16, Section 3.3]. We provide a proof of Lemma 1.3 in Section 2.6.

Note that we expect that the dimensions of eigenspaces are one from the
general philosophy of Bethe ansatz conjecture. The integrability in any sense
always asserts that the algebra of Hamiltonians is maximal commutative. And we
also expect that the Bethe algebra has geometric nature based on the geometric
Langlands correspondence [6].

In the case of Gaudin models, it is proved in [21, Theorem 3.2] (resp. [4,
Corollary 5]) that $\mathscr{B}_{\mathfrak{g}}$ (resp.
$\mathscr{B}^{\mu}_{\mathfrak{g}}$ with regular $\mu$) acts cyclically on
$M^{{\mathrm{sing}}}$ (resp. $M$). For generic values of evaluation parameters
(in the periodic case or in the case of generic regular
$\mu\in\mathfrak{h}^{*}$) the action of Bethe algebra is diagonalizable and we
immediately obtain that eigenspaces have dimension one. However, we cannot
make such a conclusion for arbitrary parameters. Indeed, if a linear operator
acts cyclically on a vector space then all its eigenspaces have dimension one.
But the same result fails if we replace a single operator by a set of
commuting linear operators, as the following simple example shows.

######  Example.

Let
$\mathscr{A}=\mathbb{C}[x_{1},x_{2}]/\big{\langle}x_{1}^{2},x_{2}^{2},x_{1}x_{2}\big{\rangle}$.
Consider the regular representation $\mathscr{A}$. Then the eigenspace
corresponding to the trivial character is spanned by $x_{1}$ and $x_{2}$ which
is two-dimensional.

We supplement the results of [4] and [21] with the nondegenerate symmetric
bilinear form on $M$ which makes $\mathscr{B}_{\mathfrak{g}}^{\mu}(M)$
Frobenius which allows us to use Lemma 1.3. The bilinear form comes from the
tensor product of Shapovalov forms on $M$, we show that all elements of Bethe
algebra $\mathscr{B}_{\mathfrak{g}}^{\mu}(M)$ with $\mu\in\mathfrak{h}^{*}$
are symmetric with respect to this form, see Lemma 2.6.

In the rest of the paper, we only deal with Gaudin models. We refer the
readers to [7] for details about the Bethe algebra of Yangian
$\mathrm{Y}(\mathfrak{g})$ (XXX spin chains). We expect the conjectures with
proper modifications also hold for XXZ and XYZ spin chains.

##  2 Perfect integrability of Gaudin models

###  2.1 Feigin–Frenkel center

In this section, we recall the definition of Feigin–Frenkel center and its
properties.

Let $\mathfrak{g}$ be a complex simple Lie algebra of rank $r$. Consider the
affine Kac–Moody algebra $\widehat{\mathfrak{g}}$,

$\widehat{\mathfrak{g}}=\mathfrak{g}\big{[}t,t^{-1}\big{]}\oplus\mathbb{C}K,\qquad\mathfrak{g}\big{[}t,t^{-1}\big{]}=\mathfrak{g}\otimes\mathbb{C}\big{[}t,t^{-1}\big{]},$

where $\mathbb{C}\big{[}t,t^{-1}\big{]}$ is the algebra of Laurent polynomials
in $t$. For $X\in\mathfrak{g}$ and $s\in\mathbb{Z}$, we simply write $X[s]$
for $X\otimes t^{s}$. Let $\mathfrak{g}_{-}=\mathfrak{g}\otimes
t^{-1}\mathbb{C}\big{[}t^{-1}\big{]}$ and
$\mathfrak{g}[t]=\mathfrak{g}\otimes\mathbb{C}[t]$.

Let $h^{\vee}$ be the _dual Coxeter number_ of $\mathfrak{g}$. Define the
module $V_{-h^{\vee}}(\mathfrak{g})$ as the quotient of
$\mathrm{U}(\widehat{\mathfrak{g}})$ by the left ideal generated by
$\mathfrak{g}[t]$ and $K+h^{\vee}$. We call the module
$V_{-h^{\vee}}(\mathfrak{g})$ the _Vaccum module at the critical level over_
$\widehat{\mathfrak{g}}$. The vacuum module $V_{-h^{\vee}}(\mathfrak{g})$ has
a vertex algebra structure.

Define the subspace $\mathfrak{z}(\widehat{\mathfrak{g}})$ of
$V_{-h^{\vee}}(\mathfrak{g})$ by

$\mathfrak{z}(\widehat{\mathfrak{g}})=\\{v\in
V_{-h^{\vee}}(\mathfrak{g})\,|\,\mathfrak{g}[t]v=0\\}.$

Using the PBW theorem, it is clear that $V_{-h^{\vee}}(\mathfrak{g})$ is
isomorphic to $\mathrm{U}(\mathfrak{g}_{-})$ as vector spaces. There is an
injective homomorphism from $\mathfrak{z}(\widehat{\mathfrak{g}})$ to
$\mathrm{U}(\mathfrak{g}_{-})$. Hence $\mathfrak{z}(\widehat{\mathfrak{g}})$
is identified as a commutative subalgebra of $\mathrm{U}(\mathfrak{g}_{-})$.
We call $\mathfrak{z}(\widehat{\mathfrak{g}})$ the _Feigin–Frenkel center_.

There is a distinguished element
$S_{1}\in\mathfrak{z}(\widehat{\mathfrak{g}})$ given by

$S_{1}=\sum_{a=1}^{\dim\mathfrak{g}}X_{a}[-1]^{2},$

where $\\{X_{a}\\}$ is an orthonormal basis of $\mathfrak{g}$ with respect to
the Killing form. The element $S_{1}$ is called the _Segal–Sugawara vector_.

######  Proposition 2.1 ([20]).

The subalgebra $\mathfrak{z}(\widehat{\mathfrak{g}})$ is the centralizer of
$S_{1}$ in $\mathrm{U}(\mathfrak{g}_{-})$.

Let $e_{1},\dots,e_{r}$, $h_{1},\dots,h_{r}$, $f_{1},\dots,f_{r}$ be a set of
Chevalley generators of $\mathfrak{g}$. Let
$\varpi\colon\mathfrak{g}\to\mathfrak{g}$ be the Cartan anti-involution
sending $e_{1},\dots,e_{r}$, $h_{1},\dots,h_{r}$, $f_{1},\dots,f_{r}$ to
$f_{1},\dots,f_{r}$, $h_{1},\dots,h_{r}$, $e_{1},\dots,e_{r}$, respectively.
Let $\widehat{\varpi}$ be the anti-involution on $\widehat{\mathfrak{g}}$
defined by

$\widehat{\varpi}\colon\
\widehat{\mathfrak{g}}\to\widehat{\mathfrak{g}},\qquad
X[s]\mapsto\varpi(X)[s],$

for all $X\in\mathfrak{g}$ and $s\in\mathbb{Z}$. We also call
$\widehat{\varpi}$ the Cartan anti-involution.

######  Corollary 2.2.

The Feigin–Frenkel center $\mathfrak{z}(\widehat{\mathfrak{g}})$ is invariant
under $\widehat{\varpi}$.

###### Proof.

Since by Proposition 2.1, $\mathfrak{z}(\widehat{\mathfrak{g}})$ is the
centralizer of $S_{1}$ in $\mathrm{U}(\mathfrak{g}_{-})$, the statement
follows from the fact that $\widehat{\varpi}(S_{1})=S_{1}$. ∎

###  2.2 Affine Harish-Chandra homomorphism

Let $\mathfrak{n}_{+}$ be the nilpotent Lie subalgebra generated by
$e_{1},\dots,e_{r}$. Let $\mathfrak{n}_{-}$ be the nilpotent Lie subalgebra
generated by $f_{1},\dots,f_{r}$. Let $\mathfrak{h}$ be the Cartan subalgebra
generated by $h_{1},\dots,h_{r}$. One has the triangular decomposition
$\mathfrak{g}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{+}$.

The Lie algebra $\mathfrak{g}$ is considered as a subalgebra of
$\widehat{\mathfrak{g}}$ via identifying $X\in\mathfrak{g}$ with
$X[0]\in\widehat{\mathfrak{g}}$. The Lie subalgebra $\mathfrak{h}$ acts on
$\widehat{\mathfrak{g}}$ adjointly and hence acts on
$\mathrm{U}(\mathfrak{g}_{-})$. Let
$\mathrm{U}(\mathfrak{g}_{-})^{\mathfrak{h}}$ be the centralizer of
$\mathfrak{h}$ in $\mathrm{U}(\mathfrak{g}_{-})$.

Let $J$ be the intersection of $\mathrm{U}(\mathfrak{g}_{-})^{\mathfrak{h}}$
and the left ideal of $\mathrm{U}(\mathfrak{g}_{-})$ generated by
$t^{-1}\mathfrak{n}_{-}[t^{-1}]$. Then we have the direct sum of vector
spaces,

$\displaystyle\mathrm{U}(\mathfrak{g}_{-})^{\mathfrak{h}}=\mathrm{U}\big{(}t^{-1}\mathfrak{h}\big{[}t^{-1}\big{]}\big{)}\oplus
J.$ (2.1)

Hence we have the projection

$\mathfrak{f}\colon\
\mathrm{U}(\mathfrak{g}_{-})^{\mathfrak{h}}\to\mathrm{U}\big{(}t^{-1}\mathfrak{h}\big{[}t^{-1}\big{]}\big{)}.$

It is clear that $\mathfrak{f}$ is a homomorphism of algebras. We call
$\mathfrak{f}$ the affine Harish-Chandra homomorphism. We use the same letter
$\mathfrak{f}$ for the restriction map
$\mathfrak{f}\colon\mathfrak{z}(\widehat{\mathfrak{g}})\to\mathrm{U}\big{(}t^{-1}\mathfrak{h}\big{[}t^{-1}\big{]}\big{)}$.

The following proposition is a part of [6, Theorem 8.1.5].

######  Proposition 2.3.

The homomorphism
$\mathfrak{f}\colon\mathfrak{z}(\widehat{\mathfrak{g}})\to\mathrm{U}\big{(}t^{-1}\mathfrak{h}\big{[}t^{-1}\big{]}\big{)}$
is injective.

Using Proposition 2.3, we improve Corollary 2.2 to the following proposition.

######  Proposition 2.4.

For any element $S\in\mathfrak{z}(\widehat{\mathfrak{g}})$, we have
$\widehat{\varpi}(S)=S$.

The proposition was proved in [14, Proposition 8.4] for type A and in [8,
Proposition 6.1] for types B and C.

###### Proof.

Now take $S\in\mathfrak{z}(\widehat{\mathfrak{g}})$ and write the
decomposition of $S$ as in (2.1), $S=S_{\mathfrak{h}}+S_{\mathfrak{j}}$, where
$S_{\mathfrak{h}}\in\mathrm{U}\big{(}t^{-1}\mathfrak{h}\big{[}t^{-1}\big{]}\big{)}$
and $S_{\mathfrak{j}}\in J$. Then
$\widehat{\varpi}(S)=\widehat{\varpi}(S_{\mathfrak{h}})+\widehat{\varpi}(S_{\mathfrak{j}})$.
Note that $\widehat{\varpi}$ fix elements in
$\mathrm{U}\big{(}t^{-1}\mathfrak{h}\big{[}t^{-1}\big{]}\big{)}$ and
$S_{\mathfrak{h}}\in\mathrm{U}\big{(}t^{-1}\mathfrak{h}\big{[}t^{-1}\big{]}\big{)}$
we have $\widehat{\varpi}(S_{\mathfrak{h}})=S_{\mathfrak{h}}$. Note also that
$\widehat{\varpi}$ maps
$\mathrm{U}\big{(}t^{-1}\mathfrak{n}_{+}\big{[}t^{-1}\big{]}\big{)}$ to
$\mathrm{U}\big{(}t^{-1}\mathfrak{n}_{-}\big{[}t^{-1}\big{]}\big{)}$ and
$\mathrm{U}\big{(}t^{-1}\mathfrak{n}_{-}\big{[}t^{-1}\big{]}\big{)}$ to
$\mathrm{U}\big{(}t^{-1}\mathfrak{n}_{+}\big{[}t^{-1}\big{]}\big{)}$, we have
$\widehat{\varpi}(S_{\mathfrak{j}})\in J$ since $J$ is the intersection of the
$\mathfrak{h}$-centralizer $\mathrm{U}(\mathfrak{g}_{-})^{\mathfrak{h}}$ with
the left ideal of $\mathrm{U}(\mathfrak{g}_{-})$ generated by
$t^{-1}\mathfrak{n}_{-}\big{[}t^{-1}\big{]}$ and also the intersection of
$\mathrm{U}(\mathfrak{g}_{-})^{\mathfrak{h}}$ with the right ideal of
$\mathrm{U}(\mathfrak{g}_{-})$ generated by
$t^{-1}\mathfrak{n}_{+}\big{[}t^{-1}\big{]}$. It follows that

$\mathfrak{f}(S)=S_{\mathfrak{h}}=\mathfrak{f}\circ\widehat{\varpi}(S).$

Note that by Corollary 2.2 both $S$ and $\widehat{\varpi}(S)$ are elements in
$\mathfrak{z}(\widehat{\mathfrak{g}})$. Since by Proposition 2.3 the
homomorphism
$\mathfrak{f}\colon\mathfrak{z}(\widehat{\mathfrak{g}})\to\mathrm{U}\big{(}t^{-1}\mathfrak{h}\big{[}t^{-1}\big{]}\big{)}$
is injective, we conclude that $S=\widehat{\varpi}(S)$, completing the proof.
∎

###  2.3 Gaudin models

We recall the construction of Gaudin models from, e.g., [19, 21]. The
coproduct of $\mathrm{U}(\mathfrak{g}_{-})$ is given by

$\Delta\colon\ X[s]\mapsto X[s]\otimes 1+1\otimes X[s],\qquad
X\in\mathfrak{g},\qquad s<0.$

Let $\ell$ be a positive integer. Using the iterated coproduct, one has the
homomorphism

$\mathrm{U}(\mathfrak{g}_{-})\to\mathrm{U}(\mathfrak{g}_{-})^{\otimes(\ell+1)}.$

For any $z\in\mathbb{C}^{\times}$, one gets the _evaluation map_ at $z$

$\varphi_{z}\colon\
\mathrm{U}(\mathfrak{g}_{-})\to\mathrm{U}(\mathfrak{g}),\qquad X[s]\mapsto
z^{s}X.$

For any $\mu\in\mathfrak{g}^{*}$, one obtains the character

$\psi_{\mu}\colon\ \mathrm{U}(\mathfrak{g}_{-})\to\mathbb{C},\qquad
X[s]\mapsto\delta_{s,-1}\mu(X),$

Fix a sequence of pairwise distinct nonzero complex numbers
$\bm{z}=(z_{1},\dots,z_{\ell})$. Then using these three homomorphisms, one
obtains a new homomorphism

$\varphi_{\bm{z},\mu}\colon\
\mathrm{U}(\mathfrak{g}_{-})\to\mathrm{U}(\mathfrak{g})^{\otimes\ell},\qquad\varphi_{\bm{z},\mu}(X[s])=\sum_{a=1}^{\ell}z_{a}^{s}(X)_{a}+\delta_{s,-1}\mu(X),$

where $(X)_{a}=1^{\otimes(a-1)}\otimes X\otimes 1^{\otimes(\ell-a)}$.

Set $u-\bm{z}=(u-z_{1},\dots,u-z_{\ell})$. Define the _Gaudin algebra_ as a
subalgebra generated by elements in
$\varphi_{u-\bm{z},\mu}(\mathfrak{z}(\widehat{\mathfrak{g}}))\subset\mathrm{U}(\mathfrak{g})^{\otimes\ell}$
for all $u\in\mathbb{C}\setminus\\{z_{1},\dots,z_{\ell}\\}$. The Gaudin
algebra is commutative and it is denoted by $\mathcal{A}_{\bm{z},\mu}$. When
$\mu=0$, the Gaudin algebra commutes with the diagonal action of
$\mathrm{U}(\mathfrak{g})$, see, e.g., [19, Proposition 3].

Let $\bm{\lambda}=(\lambda_{1},\dots,\lambda_{\ell})$ be a sequence of
dominant integral weights. Denote by $V_{\lambda_{i}}$ the finite-dimensional
irreducible $\mathfrak{g}$-module of highest weight $\lambda_{i}$. We set
$V_{\bm{\lambda}}=\otimes_{i=1}^{\ell}V_{\lambda_{i}}$ and

$\displaystyle(V_{\bm{\lambda}})^{\mathrm{sing}}=\\{v\in
V_{\bm{\lambda}}\,|\,\mathfrak{n}_{+}v=0\\},\qquad\mathcal{M}_{\bm{\lambda},\mu}=\begin{cases}(V_{\bm{\lambda}})^{\mathrm{sing}},&\text{if
}\mu=0,\\\ V_{\bm{\lambda}},&\text{if }\mu\in\mathfrak{h}^{*}\text{ is
regular}.\end{cases}$

Here we identify $\mathfrak{h}^{*}$ with the subspace of $\mathfrak{g}^{*}$
consisting of all elements annihilating
$\mathfrak{n}_{+}\oplus\mathfrak{n}_{-}$. By the construction of
$\mathcal{A}_{\bm{z},\mu}$, $\mathcal{M}_{\bm{\lambda},\mu}$ is an
$\mathcal{A}_{\bm{z},\mu}$-module.

Let $\mathcal{A}_{\bm{z},\mu}$ be the algebra of Hamiltonians and
$\mathcal{M}_{\bm{\lambda},\mu}$ the spin chain. We call the corresponding
integrable system the _Gaudin model_. We say that the Gaudin model has
_periodic boundary condition_ if $\mu=0$ and _regular quasi-periodic boundary
condition_ if $\mu\in\mathfrak{h}^{*}$ is regular. We would like to study the
spectrum of $\mathcal{A}_{\bm{z},\mu}$ acting on
$\mathcal{M}_{\bm{\lambda},\mu}$.

The following theorem is obtained in [4, Corollary 5] for any regular
$\mu\in\mathfrak{g}^{*}$ and in [21, Theorem 3.2] for $\mu=0$.

######  Theorem 2.5.

If $\mu\in\mathfrak{h}^{*}$ is regular or if $\mu=0$, then the space
$\mathcal{M}_{\bm{\lambda},\mu}$ is cyclic as an
$\mathcal{A}_{\bm{z},\mu}$-module.

###  2.4 Bethe algebra

Note that our definition of Gaudin models is slightly different from that in
Introduction. In this section, we define the Bethe algebra in
$\mathrm{U}(\mathfrak{g}[t])$ and clarify this point.

Following, e.g., [12, Section 5], we recall the definition of Bethe algebra
using Feigin–Frenkel center $\mathfrak{z}(\widehat{\mathfrak{g}})$. Note that
the Feigin–Frenkel center $\mathfrak{z}(\widehat{\mathfrak{g}})$ is in
$\mathrm{U}(\mathfrak{g}_{-})$ while the Bethe algebra is in
$\mathrm{U}(\mathfrak{g}[t])$.

For $\mathcal{X}\in\mathfrak{g}$ and $\mu\in\mathfrak{g}^{*}$, define the
current $\mathcal{X}^{\mu}(u)$ by

$\mathcal{X}^{\mu}(u)=\mu(\mathcal{X})+\sum_{r\geqslant
0}\mathcal{X}[r]u^{-r-1}\in\mathrm{U}(\mathfrak{g}[t])\big{[}\big{[}u^{-1}\big{]}\big{]}.$

For any element $a$ of the form

$a=\sum\mathcal{X}_{1}[-s_{1}-1]\mathcal{X}_{2}[-s_{2}-1]\cdots\mathcal{X}_{k}[-s_{k}-1]\in\mathfrak{z}(\widehat{\mathfrak{g}})$

for some $k\in\mathbb{Z}_{>0}$, $\mathcal{X}_{i}\in\mathfrak{g}$,
$s_{i}\in\mathbb{Z}_{\geqslant 0}$, define a series in $u^{-1}$ whose
coefficients are in $\mathrm{U}(\mathfrak{g}[t])$ by

$\displaystyle\sum\frac{(-1)^{k}}{s_{1}!s_{2}!\cdots
s_{k}!}\partial_{u}^{s_{1}}\mathcal{X}_{1}^{\mu}(u)\partial_{u}^{s_{2}}\mathcal{X}_{2}^{\mu}(u)\cdots\partial_{u}^{s_{k}}\mathcal{X}_{k}^{\mu}(u)\in\mathrm{U}(\mathfrak{g}[t])\big{[}\big{[}u^{-1}\big{]}\big{]}.$
(2.2)

The _Bethe algebra_ $\mathscr{B}^{\mu}_{\mathfrak{g}}$ is the subalgebra of
$\mathrm{U}(\mathfrak{g}[t])$ generated by the coefficients of all such series
of form (2.2) as $a$ runs over $\mathfrak{z}(\widehat{\mathfrak{g}})$.

The Bethe algebra $\mathscr{B}^{\mu}_{\mathfrak{g}}$ (or Feigin–Frenkel center
$\mathfrak{z}(\widehat{\mathfrak{g}})$) is considered as the universal Gaudin
algebra, see, e.g., [7]. The Gaudin algebra $\mathcal{A}_{\bm{z},\mu}$ in
$\mathrm{U}(\mathfrak{g})^{\otimes\ell}$ can also be obtained from
$\mathscr{B}^{\mu}_{\mathfrak{g}}$ by taking the $(\ell-1)$-th fold coproduct
and then applying to the $i$-th factor the evaluation map at $z_{i}$ for every
$1\leqslant i\leqslant\ell$. In particular, the image of the Gaudin algebra
$\mathcal{A}_{\bm{z},\mu}$ acting on $V_{\bm{\lambda}}$ coincides with that of
Bethe algebra $\mathscr{B}_{\mathfrak{g}}^{\mu}$ acting on tensor product of
evaluation modules $V_{\bm{\lambda}}$ with evaluation points at
$\bm{z}=(z_{1},\dots,z_{\ell})$, see [21, Propositions 2.3 and 2.5] or [3, 5,
19]. Note that in this case, all finite-dimensional irreducible
$\mathrm{U}(\mathfrak{g}[t])$-modules are tensor products of evaluation
modules with pairwise distinct evaluation parameters.

###  2.5 Shapovalov form

For a dominant integral weight $\lambda$, there is a unique nondegenerate
symmetric bilinear form $S_{\lambda}$ on $V_{\lambda}$ such that

$\mathcal{S}_{\lambda}(v_{\lambda},v_{\lambda})=1,\qquad\mathcal{S}_{\lambda}(Xv,w)=\mathcal{S}_{\lambda}(v,\varpi(X)w),$

where $v_{\lambda}$ is a highest weight vector of $V_{\lambda}$ and $v,w\in
V_{\lambda}$. We call $\mathcal{S}_{\lambda}$ the _Shapovalov form on
$V_{\lambda}$_. The Shapovalov form $\mathcal{S}_{\lambda}$ is positive
definite on the real part of $V_{\lambda}$.

The Shapovalov forms $\mathcal{S}_{\lambda_{i}}$ induce a nondegenerate
symmetric bilinear form
$\mathcal{S}_{\bm{\lambda}}=\otimes_{i=1}^{\ell}\mathcal{S}_{\lambda_{i}}$ on
$V_{\bm{\lambda}}$. The restriction of $\mathcal{S}_{\bm{\lambda}}$ on the
singular subspace $(V_{\bm{\lambda}})^{\mathrm{sing}}$ is also nondegenerate.

Suppose $\mu\in\mathfrak{h}^{*}$, then it is clear that

$\displaystyle\mathcal{S}_{\bm{\lambda}}(\varphi_{\bm{z},\mu}(X[s])v,w)=\mathcal{S}_{\bm{\lambda}}(v,\varphi_{\bm{z},\mu}(\varpi(X)[s])w)=\mathcal{S}_{\bm{\lambda}}(v,\varphi_{\bm{z},\mu}\circ\widehat{\varpi}(X[s])w),$
(2.3)

for all $v,w\in V_{\bm{\lambda}}$ and $X\in\mathfrak{g}$.

Let
$\rho_{\bm{\lambda},\bm{z},\mu}\colon\mathcal{A}_{\bm{z},\mu}\to\mathrm{End}(\mathcal{M}_{\bm{\lambda},\mu})$
be the representation of the natural action of $\mathcal{A}_{\bm{z},\mu}$ on
$\mathcal{M}_{\bm{\lambda},\mu}$. Let $\mathfrak{A}_{\bm{\lambda},\bm{z},\mu}$
be the image of $\mathcal{A}_{\bm{z},\mu}$ under
$\rho_{\bm{\lambda},\bm{z},\mu}$.

######  Lemma 2.6.

Let $a\in\mathfrak{A}_{\bm{\lambda},\bm{z},\mu}$ and
$v,w\in\mathcal{M}_{\bm{\lambda},\mu}$. If $\mu\in\mathfrak{h}^{*}$, then we
have $\mathcal{S}_{\bm{\lambda}}(av,w)=\mathcal{S}_{\bm{\lambda}}(v,aw)$.

###### Proof.

The statement follows from (2.3) and Proposition 2.4. ∎

###  2.6 Frobenius algebra

Let $A$ be a finite-dimensional commutative unital algebra. If there exists a
nondegenerate symmetric bilinear form $(\cdot,\cdot)$ on $A$ such that

$(ab,c)=(a,bc)\qquad\text{for all }a,b,c\in A,$

then it is clear that the regular and coregular representations of $A$ are
isomorphic. Thus $A$ is a Frobenius algebra.

We prepare the following lemma for the proof of the main theorem. Suppose $A$
is a unital commutative algebra acting on a finite-dimensional space $V$,
$\rho\colon A\to\mathrm{End}(V)$. Let $\mathfrak{A}$ be the image of $A$ under
$\rho$ in $\mathrm{End}(V)$. Clearly, $\mathfrak{A}$ is a finite-dimensional
unital commutative algebra.

######  Lemma 2.7.

Suppose $A$ acts on $V$ cyclically. If there is a nondegenerate symmetric
bilinear form $(\cdot|\cdot)$ on $V$ such that

$(av|w)=(v|aw),\qquad\text{for all }a\in\mathfrak{A},\quad v,w\in V,$

then the algebra $\mathfrak{A}$ is a Frobenius algebra. In particular, the
$A$-module $V$ is perfectly integrable.

###### Proof.

Let $v^{+}$ be a cyclic vector of the action of $\mathfrak{A}$ on $V$. Define
a linear map $\xi$ by

$\xi\colon\ \mathfrak{A}\to V,\qquad a\mapsto av^{+}.$

Clearly, $\xi$ is surjective.

We claim that $\xi$ is injective. Indeed, suppose that $a\in\ker\xi$, then
$a\in\mathrm{End}(V)$ and $av^{+}=0$. Hence
$aa^{\prime}v^{+}=a^{\prime}av^{+}=0$ for all $a^{\prime}\in\mathfrak{A}$,
namely $a\,\xi(\mathfrak{A})=0$. Since $\xi(\mathfrak{A})=V$, we conclude that
$aV=0$. Therefore $a=0$, which implies $\xi$ is injective and hence a
bijection. Then it is clear that $\xi$ defines an $\mathfrak{A}$-module
isomorphism between the regular representation of $\mathfrak{A}$ and the
$\mathfrak{A}$-module $V$.

Define a bilinear form $(\cdot,\cdot)$ on $\mathfrak{A}$ as follows,

$(a,b)=(av^{+}|bv^{+}),\qquad\text{for all }a,b\in\mathfrak{A}.$

Since $(\cdot|\cdot)$ is symmetric, so is $(\cdot,\cdot)$. Because
$(\cdot|\cdot)$ is nondegenerate and $\xi$ is bijective, the form
$(\cdot,\cdot)$ is nondegenerate as well. For $a,b,c\in\mathfrak{A}$, we also
have

$(ab,c)=(abv^{+}|cv^{+})=(bav^{+}|cv^{+})=(av^{+}|bcv^{+})=(a,bc).$

Hence $\mathfrak{A}$ is a Frobenius algebra. ∎

Since Lemma 1.3 is central to the results of the present paper, we also
provide a detailed proof for it.

###### Proof of Lemma 1.3.

To simplify the notation, we write $\mathfrak{B}$ for $\mathcal{B}(V)$. If the
$\mathcal{B}$-module $V$ is perfectly integrable, then the
$\mathfrak{B}$-module $V$ is isomorphic to the regular representation
$\mathfrak{B}$ and coregular representation $\mathfrak{B}^{*}$. Note that the
$\mathcal{B}$-eigenspaces are essentially the same as
$\mathfrak{B}$-eigenspaces, thus we shall use $\mathfrak{B}$-eigenspaces
instead.

Let $\psi\in\mathfrak{B}^{*}$ be a $\mathfrak{B}$-eigenvector for the
coregular representation $\mathfrak{B}^{*}$ with the eigenvalue
$\xi_{\psi}\in\mathfrak{B}^{*}$, namely $a\psi=\xi_{\psi}(a)\psi$ for any
$a\in\mathfrak{B}$. It is clear that $\xi_{\psi}$ is a character of
$\mathfrak{B}$.

On one hand, by definition of coregular representation, we have
$(a\psi)(1)=\psi(a\cdot 1)=\psi(a)$ for any $a\in\mathfrak{B}$. On the other
hand, since $\psi$ is a $\mathfrak{B}$-eigenvector, we have

$(a\psi)(1)=\big{(}\xi_{\psi}(a)\psi\big{)}(1)=\xi_{\psi}(a)\psi(1)$

for any $a\in\mathfrak{B}$. Therefore $\psi(a)=\xi_{\psi}(a)\psi(1)$ for any
$a\in\mathfrak{B}$, which means the $\mathfrak{B}$-eigenvector $\psi$ is
proportional to the corresponding eigenvalue $\xi_{\psi}$. This shows that
every $\mathfrak{B}$-eigenspace in $V$ has dimension one.

For any character $\xi\in\mathfrak{B}^{*}$ and any $a,b\in\mathfrak{B}$, we
have

$(a\xi)(b)=\xi(ab)=\xi(a)\xi(b).$

Therefore, any character $\xi\in\mathfrak{B}^{*}$ is a
$\mathfrak{B}$-eigenvector with the eigenvalue $\xi$.

Note that the characters of $\mathfrak{B}$ are parameterized by their kernels,
that is the maximal ideals of $\mathfrak{B}$. Combining the facts above, we
conclude that there is a bijection between $\mathfrak{B}$-eigenspaces and
$\mathsf{Specm}(\mathfrak{B})$, namely the maximal ideals of $\mathfrak{B}$.

We then show that each generalized $\mathfrak{B}$-eigenspace is a cyclic
$\mathfrak{B}$-module. We call a finite-dimensional commutative algebra $A$
_local_ if it has a unique maximal ideal $\mathfrak{m}$. Hence it has a unique
character $\zeta\in A^{*}$. Moreover, $\mathfrak{m}$ is nilpotent, see [1,
Proposition 8.6]. Therefore, the local algebra $A$ itself as the regular
representation is the generalized $A$-eigenspace corresponding to the
eigenvalue $\zeta$ as $a-\zeta(a)\in\ker\zeta=\mathfrak{m}$. Note that every
finite-dimensional commutative algebra is a direct sum of local algebras, see
[1, Theorem 8.7]. In addition, each local summand is a generalized
$A$-eigenspace with the corresponding eigenvalue uniquely determined by the
summand, see the first paragraph of the proof of [1, Theorem 8.7]. This part
now follows from the fact that $V$ is isomorphic to the regular representation
of $\mathfrak{B}$.

It is clearly that the algebra $\mathfrak{B}$ is maximal commutative in
$\mathrm{End}(V)$. The last statement follows from the first half of the proof
of Lemma 2.7. ∎

###  2.7 Perfect integrability of Gaudin models

The following is our main theorem which asserts Gaudin models with periodic
and regular quasi-periodic boundary conditions are perfectly integrable.

######  Theorem 2.8.

If $\mu\in\mathfrak{h}^{*}$ is regular or if $\mu=0$, then the
$\mathcal{A}_{\bm{z},\mu}$-module $\mathcal{M}_{\bm{\lambda},\mu}$ is
perfectly integrable.

###### Proof.

By Theorem 2.5, Gaudin algebra $\mathcal{A}_{\bm{z},\mu}$ acts on
$\mathcal{M}_{\bm{\lambda},\mu}$ cyclically. Recall that
$\rho_{\bm{\lambda},\bm{z},\mu}\colon\mathcal{A}_{\bm{z},\mu}\allowbreak\to\mathrm{End}(\mathcal{M}_{\bm{\lambda},\mu})$
and
$\mathfrak{A}_{\bm{\lambda},\bm{z},\mu}=\rho_{\bm{\lambda},\bm{z},\mu}(\mathcal{A}_{\bm{z},\mu})$.
It remains to show that $\mathfrak{A}_{\bm{\lambda},\bm{z},\mu}$ is Frobenius.

By Lemma 2.6, we can apply Lemma 2.7 for the case
$A=\mathcal{A}_{\bm{z},\mu}$,
$\mathfrak{A}=\mathfrak{A}_{\bm{\lambda},\bm{z},\mu}$,
$V=\mathcal{M}_{\bm{\lambda},\mu}$, and
$(\cdot|\cdot)=\mathcal{S}_{\bm{\lambda}}(\cdot,\cdot)$. Therefore we conclude
that the algebra $\mathfrak{A}_{\bm{\lambda},\bm{z},\mu}$ is a Frobenius
algebra. ∎

Theorem 2.8 gives the following important facts. By Theorem 2.8, Lemma 1.3,
and [21, Corollary 3.3], we see that the joint eigenvectors (up to
proportionality) of the Gaudin algebra in $V_{\bm{\lambda}}^{\mathrm{sing}}$
are in one-to-one correspondence with monodromy-free
${}^{L}\mathfrak{g}$-opers on the projective line with regular singularities
at the points $z_{1},\dots,z_{\ell},\infty$ and the prescribed residues at the
singular points. Here $z_{1},\dots,z_{\ell}$ are arbitrary pairwise distinct
complex numbers, cf. [21, Corollary 3.4]. Similarly, when $\mathfrak{g}$ is of
type B or C (resp. G2), one deduces from [11, Theorem 4.5] (resp. [9, Theorem
5.8]) that there exists a bijection between joint eigenvectors (up to
proportionality) of the Gaudin algebra in $V_{\bm{\lambda}}^{\mathrm{sing}}$
and self-dual (resp. self-self-dual) spaces of polynomials in a suitable
intersection of Schubert cells in Grassmannian.

###  2.8 Conjecture for general $\bm{\mu\in\mathfrak{g}^{*}}$

For an arbitrary $\mu\in\mathfrak{g}^{*}\cong\mathfrak{g}$, there exists an
element $g\in G$ such that $g\mu g^{-1}$ is in the negative Borel part
$\mathfrak{b}_{-}=\mathfrak{n}_{-}\oplus\mathfrak{h}$. Thus, without loss of
generality, we can assume that $\mu\in\mathfrak{b}_{-}$.

Let $\mathfrak{z}_{\mu}(\mathfrak{g})$ be the centralizer of $\mu$ in
$\mathfrak{g}$. It is known that $\mathcal{A}_{\bm{z},\mu}$ commutes with the
diagonal action of $\mathfrak{z}_{\mu}(\mathfrak{g})$, see [19, Proposition
4].

Let $V_{\bm{\lambda}}$ be as before. Define $\mathcal{M}_{\bm{\lambda},\mu}$
as a subspace of $V_{\bm{\lambda}}$ by

$\mathcal{M}_{\bm{\lambda},\mu}:=\\{v\in V_{\bm{\lambda}}\,|\,xv=0,\text{ for
all }x\in\mathfrak{z}_{\mu}(\mathfrak{g})\cap\mathfrak{n}_{+}\\}.$

Then $\mathcal{A}_{\bm{z},\mu}$ acts on $\mathcal{M}_{\bm{\lambda},\mu}$.

######  Conjecture 2.9.

The $\mathcal{A}_{\bm{z},\mu}$-module $\mathcal{M}_{\bm{\lambda},\mu}$ is
perfectly integrable.

### Acknowledgements

The author is grateful to E. Mukhin and V. Tarasov for interesting discussions
and helpful suggestions. The author also thanks the referees for their
comments and suggestions that substantially improved the first version of this
paper. This work was partially supported by a grant from the Simons Foundation
#353831.

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