# Bounds on Dao numbers and applications to regular local rings

Antonino Ficarra  Department of Mathematics and Computer Sciences, Physics and
Earth Sciences, University of Messina, Viale Ferdinando Stagno d’Alcontres 31,
98166 Messina, Italy<EMAIL_ADDRESS>,  Cleto B. Miranda-Neto
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João
Pessoa, PB, Brazil<EMAIL_ADDRESS>and  Douglas S. Queiroz  Departamento
de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa, PB,
Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The so-called Dao numbers are a sort of measure of the asymptotic behaviour of
full properties of certain product ideals in a Noetherian local ring $R$ with
infinite residue field and positive depth. In this paper, we answer a question
of H. Dao on how to bound such numbers. The auxiliary tools range from
Castelnuovo-Mumford regularity of appropriate graded structures to reduction
numbers of the maximal ideal. In particular, we substantially improve previous
results (and answer questions) by the authors. As an application, we provide
new characterizations of when $R$ is regular; for instance, we show that this
holds if and only if the maximal ideal of $R$ can be generated by a
$d$-sequence (in the sense of Huneke) if and only if the third Dao number of
any (minimal) reduction of the maximal ideal vanishes.

###### Key words and phrases:

Full ideal, Castelnuovo-Mumford regularity, Rees algebra, reduction number,
regular local ring

###### 2020 Mathematics Subject Classification:

Primary: 13C05, 13C13; Secondary: 13A30, 13H99

##  1\. Motivation: Dao’s problem on the fullness of certain ideals

Throughout this note, by a ring we mean a commutative, Noetherian, unital
ring. Let $R$ be either a local ring with residue field $K$ and maximal ideal
$\mathfrak{m}$, or a standard graded algebra over a field $K$ having a unique
graded maximal ideal $\mathfrak{m}$. We will assume throughout that $K$ is
infinite and $\mathrm{depth}\ R>0$ (i.e., $\mathfrak{m}$ contains a non-
zerodivisor), and in addition $I\subset R$ stands for an ideal which we assume
to be homogeneous whenever $R$ is graded.

In this paper we focus on the properties of $\mathfrak{m}$-fullness, fullness,
and weak $\mathfrak{m}$-fullness (to be recalled in the next section) of
certain ideals. More precisely, we are interested in the so-called Dao numbers
of the given ideal $I$, i.e., three non-negative integers
${\mathfrak{d}}_{i}(I)$, $i=1,2,3$, defined below, which in some sense provide
a measure for the asymptotic behaviour of the full properties of certain
product ideals involving $I$.

######  Definition 1.1.

The Dao numbers of $I$ are defined as:

$\displaystyle{\mathfrak{d}}_{1}(I)$ $\displaystyle=$ $\displaystyle{\rm
min}\\{t\geq 0\mid\mbox{$I\mathfrak{m}^{k}$ is $\mathfrak{m}$-full for all
$k\geq t$\\}};$ $\displaystyle{\mathfrak{d}}_{2}(I)$ $\displaystyle=$
$\displaystyle{\rm min}\\{t\geq 0\mid\mbox{$I\mathfrak{m}^{k}$ is full for all
$k\geq t$\\}};$ $\displaystyle{\mathfrak{d}}_{3}(I)$ $\displaystyle=$
$\displaystyle{\rm min}\\{t\geq 0\mid\mbox{$I\mathfrak{m}^{k}$ is weakly
$\mathfrak{m}$-full for all $k\geq t$\\}}.$

It is worth observing that, if $(R,\mathfrak{m},K)$ and $I$ are as above, then
as shown in [17, Proposition 2.2] the basic relations among the Dao numbers of
$I$ are

${\mathfrak{d}}_{2}(I)\leq{\mathfrak{d}}_{1}(I)={\mathfrak{d}}_{3}(I).$

Our motivation is the following problem suggested by H. Dao, which was first
addressed in [17] and then in [6].

######  Question 1.2.

([4, Question 4.5]) Can we find good lower and upper bounds for the
${\mathfrak{d}}_{i}(I)$’s?

In the case where $I$ is a reduction of $\mathfrak{m}$, a lower bound for
${\mathfrak{d}}_{3}(I)$ is given by ${\rm r}_{I}(\mathfrak{m})$, the reduction
number of $\mathfrak{m}$ with respect to $I$, which follows immediately from
[17, Theorem 3.4] (the question as to whether ${\rm
r}_{I}(\mathfrak{m})\leq{\mathfrak{d}}_{2}(I)$ remains unanswered). Otherwise,
for a general $I$ (not necessarily a reduction of $\mathfrak{m}$), a more
elaborated lower bound was established in [6, Proposition 1.8].

So, in the present paper, our main goal is to answer the upper bound part of
Dao’s question, by using two fundamental numerical invariants in commutative
algebra: the Castelnuovo-Mumford regularity (of appropriate graded structures)
and, again, the reduction number. The connection between these numbers and
Dao’s problem was first exploited in [17] and later developed even further
(concerning specifically the Castelnuovo-Mumford regularity) in [6]. The
present work establishes, in fact, generalizations and substantial
improvements of the results given in these two papers. Additionally, we shall
finally derive, as an application of some of our results, new
characterizations of regular local rings.

##  2\. Auxiliary notions and basic properties

In this section, we invoke some basic concepts and facts which we shall freely
use in this note (without explicit mention).

###  2.1. Full properties of ideals

Let $K$ be an infinite field and $(R,\mathfrak{m})$ be either a local ring
with residue field $K$ or a standard graded $K$-algebra having a unique
homogeneous maximal ideal $\mathfrak{m}$. Assume $\mathrm{depth}\ R>0$, and
let $I\subset R$ stand for an ideal (homogeneous whenever $R$ is graded).

######  Definition 2.1.

The following notions are central in this paper:

  1. (a)

$I$ is $\mathfrak{m}$-full if $I\mathfrak{m}:x=I$ for some element
$x\in\mathfrak{m}\setminus{\mathfrak{m}}^{2}$;

  2. (b)

$I$ is full if $I:x=I:\mathfrak{m}$ for some element
$x\in\mathfrak{m}\setminus{\mathfrak{m}}^{2}$;

  3. (c)

$I$ is weakly $\mathfrak{m}$-full if $I\mathfrak{m}:\mathfrak{m}=I$.

For completeness, we recall a few interesting properties. It is clear that
$\mathfrak{m}$-full ideals are weakly $\mathfrak{m}$-full. If $I$ is
$\mathfrak{m}$-primary, then $I$ is weakly $\mathfrak{m}$-full if and only if
$I$ is basically full in the sense of [8]. Moreover, $\mathfrak{m}$-full
ideals satisfy the so-called Rees property, and if $R$ is a normal domain then
any integrally closed ideal is $\mathfrak{m}$-full; see [24] (also [7]).

###  2.2. Castelnuovo-Mumford regularity

Let $S=\bigoplus_{m\geq 0}S_{m}$ be a finitely generated standard graded
algebra over a ring $S_{0}$. As usual, by standard we mean that $S$ is
generated by $S_{1}$ as an $S_{0}$-algebra. We write $S_{+}=\bigoplus_{m\geq
1}S_{m}$ for the ideal generated by all elements of $S$ of positive degree.
For a graded $S$-module $A=\bigoplus_{m\in{\mathbb{Z}}}A_{m}$ satisfying
$A_{m}=0$ for all $m\gg 0$, we let $a(A)=\textrm{max}\\{m\in{\mathbb{Z}}\ |\
A_{m}\neq 0\\}$ if $A\neq 0$, and $a(A)=-\infty$ if $A=0$.

Now, for a finitely generated graded $S$-module $N\neq 0$ and an integer
$j\geq 0$, we take $A=H_{S_{+}}^{j}(N)$ and use the notation

$a_{j}(N)\,:=\,a(H_{S_{+}}^{j}(N)),$

where $H_{S_{+}}^{j}(-)$ stands for the $j$-th local cohomology functor with
respect to the ideal $S_{+}$. It is known that $H_{S_{+}}^{j}(N)$ is a graded
module with $H_{S_{+}}^{j}(N)_{n}=0$ for all $n\gg 0$ (see, e.g., [1,
Proposition 15.1.5(ii)]). Thus, $a_{j}(N)<+\infty$.

######  Definition 2.2.

Maintain the above setting and notations. The Castelnuovo-Mumford regularity
of $N$ is defined as

$\mathrm{reg}_{S}N\,:=\,\mathrm{max}\\{a_{j}(N)+j\,\mid\,j\geq 0\\}.$

It is well-known that $\mathrm{reg}\,N$ governs the complexity of the graded
structure of $N$ and is relevant in commutative algebra and algebraic
geometry, for example in the study of degrees of syzygies over polynomial
rings (see, e.g., [1, Chapter 15]).

######  Remark 2.3.

A classical instance of interest is when $S=\mathcal{R}(J)$, the Rees algebra
of an ideal $J$ in a ring $R$ (to be recalled in the next subsection), which
is known to be a finitely generated standard graded $R$-algebra. In
particular, we can consider the case where $R$ is local and $J=\mathfrak{m}$,
the maximal ideal of $R$.

We recall below a few basic rules about this invariant. For details, we refer
to [2, p. 277, (a), (c) and (d)], [5, Corollary 20.19] and [10, Lemma 3.1].

  * (i)

As usual, given an integer $j$, we denote by $N(j)$ the module $N$ with
degrees shifted by $j$, that is, $N(j)_{i}=N_{i+j}$ for all $i$. Then,

$\mathrm{reg}_{S}N(j)=\mathrm{reg}_{S}N-j.$

  * (ii)

Let $0\rightarrow M\rightarrow N\rightarrow P\rightarrow 0$ be a short exact
sequence of finitely generated graded $S$-modules. Then:

    * $\bullet$

$\mathrm{reg}_{S}N\leq\mathrm{max}\\{\mathrm{reg}_{S}M,\mathrm{reg}_{S}P\\},$
with equality if $\mathrm{reg}_{S}P\neq\mathrm{reg}_{S}M-1$ or $M_{k}=0$ for
$k\gg 0$.

    * $\bullet$

$\mathrm{reg}_{S}M\leq\mathrm{max}\\{\mathrm{reg}_{S}N,\mathrm{reg}_{S}P+1\\},$
with equality if $\mathrm{reg}_{S}N\neq\mathrm{reg}_{S}P.$

    * $\bullet$

$\mathrm{reg}_{S}P\leq\mathrm{max}\\{\mathrm{reg}_{S}N,\mathrm{reg}_{S}M-1\\},$
with equality if $\mathrm{reg}_{S}N\neq\mathrm{reg}_{S}M.$

  * (iii)

If $N_{j}=0$ for all $j\gg 0$, then $\mathrm{reg}_{S}N=a(N)$.

###  2.3. Rees algebra and Dao module

Here, we consider some useful graded structures. Let $R$ be a ring and $J$ an
ideal of $R$.

######  Definition 2.4.

The Rees algebra of $J$ is the graded ring

$\mathcal{R}(J)=\bigoplus_{k\geq 0}J^{k}=R\oplus J\oplus J^{2}\oplus\cdots$

######  Notation 2.5.

Given an $R$-module $M$, it is customary to write
$\mathcal{R}(J,M)=\bigoplus_{k\geq 0}J^{k}M$ for the Rees module of $J$
relative to $M$. In this paper, if $I$ is another $R$-ideal, we will be
particularly interested in

$\mathcal{R}(J,I)=\bigoplus_{k\geq 0}IJ^{k}.$

Note $\mathcal{R}(J,I)=I\mathcal{R}(J)$, the extension of $I$ to the ring
$\mathcal{R}(J)$, which is therefore a finitely generated ideal of
$\mathcal{R}(J)$.

Now let $(R,\mathfrak{m})$ be as in Section 1. Its associated graded ring is
defined by $\mathrm{gr}_{\mathfrak{m}}(R)=\bigoplus_{k\geq
0}\mathfrak{m}^{k}/\mathfrak{m}^{k+1}$. In [6], the following graded structure
is introduced for a given ideal $I$ of $R$.

######  Definition 2.6.

The Dao module of $I$ is given by

$\mathfrak{D}_{\mathfrak{m}}(I)=\bigoplus_{k\geq
0}\frac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k}},$

which is a graded $\mathcal{R}(\mathfrak{m})$-module.

######  Remark 2.7.

The $k$th component of the Dao module vanishes if and only if the ideal
$I{\mathfrak{m}}^{k}$ is weakly $\mathfrak{m}$-full. Since

$\mathfrak{D}_{\mathfrak{m}}(I)_{k}=0\quad\mbox{for\, all}\quad
k\geq{\mathfrak{d}}_{3}(I),$

it follows that $\mathfrak{D}_{\mathfrak{m}}(I)$ has finite length and
therefore is a finitely generated graded $\mathcal{R}(\mathfrak{m})$-module,
satisfying

$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathfrak{D}_{\mathfrak{m}}(I)={\mathfrak{d}}_{3}(I)-1$

whenever ${\mathfrak{d}}_{3}(I)\geq 1$ (e.g., if $I$ is not weakly
$\mathfrak{m}$-full).

###  2.4. Ratliff-Rush operation

Let $I$ be an ideal of a ring $R$.

######  Definition 2.8.

The Ratliff-Rush closure $\widetilde{I}$ of the ideal $I$ is given by

$\widetilde{I}=\bigcup_{m\geq 1}\,I^{m+1}:I^{m}.$

This is an ideal of $R$ containing $I$ which in fact refines the integral
closure of $I$, so that $\widetilde{I}=I$ whenever $I$ is integrally closed.
For details, see [18].

Now suppose $I$ contains a regular element, i.e., a non-zerodivisor on $R$.
Then it is well-known that $\widetilde{I}$ is the largest ideal that shares
with $I$ the same sufficiently high powers; hence,

$\widetilde{I^{m}}=I^{m}\quad\mbox{for\, all}\quad m\gg 0.$

This enables us to consider the following helpful number (inspired by [19,
Proposition 4.2]).

######  Notation 2.9.

If $I$ contains a regular element, we set

$s(I)=\mathrm{min}\,\\{n\geq 1\,\mid\,\widetilde{I^{i}}=I^{i}\ \mathrm{for\
all}\ i\geq n\\}.$

######  Remark 2.10.

Let us invoke a couple of useful properties. First, according to [14, Lemma
2.2] we can write

$\widetilde{I^{k+1}}:I=\widetilde{I^{k}}\quad\mbox{for \,all}\quad k\geq 0.$

Moreover, if $\mathrm{gr}_{I}(R)=\bigoplus_{k\geq 0}I^{k}/I^{k+1}$ denotes the
associated graded ring of $I$, then by [19, Remark 1.6] we get that
$\widetilde{I^{k}}=I^{k}$ for all $k\geq 0$ (i.e., $s(I)=1$) if and only if

$\mathrm{depth}\,\mathrm{gr}_{I}(R)>0.$

###  2.5. Reduction number

One last auxiliary notion is in order.

######  Definition 2.11.

Let $J$ be a proper ideal of a ring $R$. An ideal $I\subset J$ is said to be a
reduction of $J$ if $IJ^{r}=J^{r+1}$ for some integer $r\geq 0$. Such a
reduction $I$ is minimal if it is minimal with respect to inclusion. If $I$ is
a reduction of $J$, we define the reduction number of $J$ with respect to $I$
as the number

${\rm r}_{I}(J)=\mathrm{min}\,\\{m\in\mathbb{N}\,\mid\,IJ^{m}=J^{m+1}\\},$

and the reduction number of $J$ as

${\rm r}(J)=\mathrm{min}\,\\{{\rm r}_{I}(J)\,\mid\,\mbox{$I$ is a minimal
reduction of $J$}\\}.$

Of special interest in this paper will be the case where $(R,\mathfrak{m})$ is
a local ring and $J=\mathfrak{m}$.

##  3\. Upper bounds on Dao numbers via Castelnuovo-Mumford regularity

Before establishing the main results of the section, we fix a piece of
notation. For a graded $R$-module $M=\bigoplus_{k\geq 0}M_{k}$ and an integer
$\ell\geq 0$, we can consider the truncation
$M_{\geq\ell}=\bigoplus_{k\geq\ell}M_{k}$. Specifically, with notation as in
(2.5) and in case $(R,\mathfrak{m},K)$ is a local ring or a standard graded
$K$-algebra, we will be interested in the truncation

$\mathcal{R}(\mathfrak{m},I)_{\geq 1}=\bigoplus_{k\geq 1}I{\mathfrak{m}}^{k}.$

Here we are interested in tackling the upper bound part of Question 1.2 in
terms of the Castelnuovo-Mumford regularity of appropriate graded structures.
The first result in this direction, in case $R$ is local and $I$ is a
reduction of $\mathfrak{m}$, was proved in [17] and can be stated as follows.

######  Theorem 3.1.

$($[17, Theorem 3.10]$)$ Let $(R,\mathfrak{m})$ be a local ring with infinite
residue field and ${\rm depth}\,R>0$, and let $I$ be a reduction of
$\mathfrak{m}$. Then,

${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m}).$

Later, in [6], the following general answer to Question 1.2 was provided.

######  Theorem 3.2.

$($[6, Theorem 1.1]$)$ Let $(R,\mathfrak{m},K)$ be either a local ring or a
standard graded $K$-algebra, with $K$ infinite and $\mathrm{depth}\,R>0$. Let
$I\subset R$ be an ideal $($homogeneous if $R$ is graded$)$. Then,

${\mathfrak{d}}_{3}(I)\leq\mathrm{max}\\{\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m},I),\
\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m},I)_{\geq
1}:_{\mathcal{R}(R)}\mathfrak{m}\\}.$

Here, we prove the following result, which establishes [6, Conjecture 0.1] in
its full generality.

######  Theorem 3.3.

Let $R$ and $I$ be as in Theorem 3.2. Then,

${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m},I).$

Proof.  We may suppose $\mathfrak{d}_{3}(I)>0$, otherwise there is nothing to
prove. Let
$\mathfrak{M}=\mathcal{R}(\mathfrak{m})_{+}=\bigoplus_{k>0}\mathfrak{m}^{k}$
be the homogeneous maximal ideal of $\mathcal{R}(\mathfrak{m})$. It was shown
in [6, proof of Theorem 1.3] that

(1) $\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))\ =\
0:_{\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)}\mathfrak{M}\ =\
\bigoplus_{k\geq
0}\frac{(I\mathfrak{m}^{k+1}:\mathfrak{m})\cap\mathfrak{m}^{k}}{I\mathfrak{m}^{k}}.$

Now, since

$\frac{(I\mathfrak{m}^{k+1}:\mathfrak{m})\cap\mathfrak{m}^{k}}{I\mathfrak{m}^{k}}\
\subset\
\frac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k}}\quad\mbox{for\,
all}\quad k\geq 0,$

we have

(2) $\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))\
\subset\ \mathfrak{D}_{\mathfrak{m}}(I)$

as a graded $\mathcal{R}(\mathfrak{m})$-submodule of
$\mathfrak{D}_{\mathfrak{m}}(I)$. Next, let
$t=\max\\{k\mid\mathfrak{D}_{\mathfrak{m}}(I)_{k}\neq 0\\}$. Then,
$t=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathfrak{D}_{\mathfrak{m}}(I)=\mathfrak{d}_{3}(I)-1$.
Because $\mathfrak{M}\cdot\mathfrak{D}_{\mathfrak{m}}(I)_{t}=0$, it follows
that

$\mathfrak{D}_{\mathfrak{m}}(I)_{t}\ \subset\
[0:_{\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)}\mathfrak{M}]_{t}\
=\ \mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{t}$

and note that the opposite inclusion always holds by equation (2). Therefore,

$\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{t}=\mathfrak{D}_{\mathfrak{m}}(I)_{t}\neq
0$

and consequently

$\displaystyle\mathfrak{d}_{3}(I)-1\ =\ t\ $ $\displaystyle\leq\
\max\\{j\mid\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{j}\neq
0\\}$ $\displaystyle\leq\
\textrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)$
$\displaystyle=\
\textrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)-1.$

The assertion follows. $\square$

Naturally, we can ask when the inclusion (2) is actually an equality.

######  Proposition 3.4.

Let $R$ and $I$ be as in Theorem 3.2. If $R$ is a standard graded $K$-algebra
or $\mathrm{depth}\,\mathrm{gr}_{\mathfrak{m}}(R)>0$, then
$\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))=\mathfrak{D}_{\mathfrak{m}}(I)$.

Proof.  In view of equation (1), it suffices to show that

$I\mathfrak{m}^{k+1}:\mathfrak{m}\ \subset\ \mathfrak{m}^{k}\quad\mbox{for\,
all}\quad k\geq 0.$

First, if $R$ is a standard graded $K$-algebra, we can represent it as a
quotient $S/J$ where $S=K[x_{1},\dots,x_{n}]$ is a standard graded polynomial
ring and $J\subset S$ is a homogeneous ideal. Now let $f\in
I\mathfrak{m}^{k+1}:\mathfrak{m}$ be an homogeneous element. Then

$(x_{1}+J)f\in I\mathfrak{m}^{k+1}\subset\mathfrak{m}^{k+1}$

and so $\deg f+1\geq k+1$. Thus $f\in\mathfrak{m}^{k}$ and consequently
$I\mathfrak{m}^{k+1}:\mathfrak{m}\subset\mathfrak{m}^{k}$ for all $k\geq 0$.
Otherwise, if $\mathrm{depth}\,\mathrm{gr}_{\mathfrak{m}}(R)>0$ then, as we
know, $\widetilde{\mathfrak{m}^{k}}=\mathfrak{m}^{k}$ for all $k\geq 0$.
Consequently,

$I\mathfrak{m}^{k+1}:\mathfrak{m}\subset\mathfrak{m}^{k+1}:\mathfrak{m}=\widetilde{\mathfrak{m}^{k+1}}:\mathfrak{m}=\widetilde{\mathfrak{m}^{k}}=\mathfrak{m}^{k}.$

$\square$

We conclude this section with some other consequences of Theorem 3.3.

######  Corollary 3.5.

Let $R$ and $I$ be as in Theorem 3.2. If $\mathfrak{d}_{3}(I)>0$, then
$\mathfrak{d}_{3}(I)$ is equal to the highest socle degree of
$\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)$ increased by one, that
is,

$\mathfrak{d}_{3}(I)\ =\
\max\\{j\mid\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{j}\neq
0\\}+1.$

Proof.  Set
$h=\max\\{j\mid\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{j}\neq
0\\}$. In the proof of Theorem 3.3 we have seen that $\mathfrak{d}_{3}(I)\leq
h+1$. On the other hand, since $\mathfrak{d}_{3}(I)>0$, we can write
$\mathfrak{d}_{3}(I)=\max\\{j\mid\mathfrak{D}_{\mathfrak{m}}(I)_{j}\neq
0\\}+1$. By equation (2) we have an inclusion
$\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))\subset\mathfrak{D}_{\mathfrak{m}}(I)$,
so that

$\mathfrak{d}_{3}(I)\ =\ \max\\{j\mid\mathfrak{D}_{\mathfrak{m}}(I)_{j}\neq
0\\}+1\ \geq\ h+1,$

thus completing the proof. $\square$

Next, we estimate the Castelnuovo-Mumford regularity of some natural blowup
algebras that are related to the Dao module of $I$.

######  Definition 3.6.

The associated graded module of $\mathfrak{m}$ relative to $I$ is the
$\mathcal{R}(\mathfrak{m})$-module

$\mathrm{gr}_{\mathfrak{m}}(I)=\mathcal{R}(\mathfrak{m},I)/\mathfrak{m}\mathcal{R}(\mathfrak{m},I)=\bigoplus_{k\geq
0}I\mathfrak{m}^{k}/I\mathfrak{m}^{k+1}.$

Note that, for each $k\geq 0$, we have the following commutative diagram with
exact rows and columns:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\mathfrak{m}^{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\mathfrak{m}^{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\mathfrak{m}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\mathfrak{m}^{k+1}:\mathfrak{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cfrac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cfrac{I\mathfrak{m}^{k}}{I\mathfrak{m}^{k+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cfrac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cfrac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

######  Notation 3.7.

For simplicity, we set

$\mathcal{P}_{\mathfrak{m}}(I)=\mathcal{R}(\mathfrak{m},I)_{\geq
1}:_{\mathcal{R}(R)}\mathfrak{m}\quad\mbox{and}\quad\mathcal{Q}_{\mathfrak{m}}(I)=(\mathcal{R}(\mathfrak{m},I)_{\geq
1}:_{\mathcal{R}(R)}\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)_{\geq 1}.$

Taking the direct sum in the above diagram, we obtain the following
commutative diagram of finitely generated graded
$\mathcal{R}(\mathfrak{m})$-modules with exact rows and columns:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{R}(\mathfrak{m},I)_{\geq
1}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{R}(\mathfrak{m},I)_{\geq
1}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{R}(\mathfrak{m},I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{P}_{\mathfrak{m}}(I)(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{D}_{\mathfrak{m}}(I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{gr}_{\mathfrak{m}}(I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Q}_{\mathfrak{m}}(I)(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{D}_{\mathfrak{m}}(I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

The lemma below is a special case of [25, Corollary 3].

######  Lemma 3.8.

There is an equality
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathrm{gr}_{\mathfrak{m}}(I)$.

Now we prove:

######  Corollary 3.9.

Let $R$ and $I$ be as in Theorem 3.2. Then,

$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I),\,\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)\
\leq\ \mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)+1.$

In particular,

$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)+1$

if either $\mathfrak{d}_{3}(I)=0$ or
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)>\mathfrak{d}_{3}(I)>0$.

Proof.  There are short exact sequences of finitely generated graded
$\mathcal{R}(\mathfrak{m})$-modules

$0\rightarrow\mathcal{R}(\mathfrak{m},I)\rightarrow\mathcal{P}_{\mathfrak{m}}(I)(1)\rightarrow\mathfrak{D}_{\mathfrak{m}}(I)\rightarrow
0,$
$0\rightarrow\mathrm{gr}_{\mathfrak{m}}(I)\rightarrow\mathcal{Q}_{\mathfrak{m}}(I)(1)\rightarrow\mathfrak{D}_{\mathfrak{m}}(I)\rightarrow
0.$

Recall that Lemma 3.8 gives
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathrm{gr}_{\mathfrak{m}}(I)$.

First, assume that $\mathfrak{d}_{3}(I)=0$. Then,
$\mathfrak{D}_{\mathfrak{m}}(I)=0$. In this case,
$\mathcal{P}_{\mathfrak{m}}(I)(1)\cong\mathcal{R}(\mathfrak{m},I)$ and
$\mathcal{Q}_{\mathfrak{m}}(I)(1)\cong\mathrm{gr}_{\mathfrak{m}}(I)$, so that

$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I)-1=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathrm{gr}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)-1.$

Now, suppose $\mathfrak{d}_{3}(I)>0$. In particular
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathfrak{D}_{\mathfrak{m}}(I)=\mathfrak{d}_{3}(I)-1$.
On the other hand, Theorem 3.3 yields

$\mathfrak{d}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathrm{gr}_{\mathfrak{m}}(I)$

and thus the above exact sequence implies

$\displaystyle\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I)(1)\
$ $\displaystyle\leq\
\max\\{\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I),\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathfrak{D}_{\mathfrak{m}}(I)\\}$
$\displaystyle=\
\max\\{\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I),\mathfrak{d}_{3}(I)-1\\}$
$\displaystyle=\
\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I).$

Similarly,
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)(1)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)$.
Hence,

$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I),\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)+1.$

Finally, if we suppose that
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)>\mathfrak{d}_{3}(I)>0$,
we derive

$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)-1\neq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathfrak{D}_{\mathfrak{m}}(I),$

and the equalities
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)+1$
follow. $\square$

Next, we strengthen [6, Proposition 1.6] by finding the relation between the
regularity of an ideal as in (2.5) and that of the ambient Rees algebra
$\mathcal{R}(J)$.

######  Lemma 3.10.

Let $(R,\mathfrak{m},K)$ be either a local ring or a standard graded
$K$-algebra. Let $I,J$ be ideals of $R$. Then,

$\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J)\leq\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I),$

with equality if $I$ is a reduction of $J$.

Proof.  Since $\mathcal{R}(J,I)$ is a homogeneous ideal of $\mathcal{R}(J)$,
the short exact sequence

$0\rightarrow\mathcal{R}(J,I)\rightarrow\mathcal{R}(J)\rightarrow\frac{\mathcal{R}(J)}{\mathcal{R}(J,I)}\rightarrow
0$

yields

$\displaystyle\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J)$ $\displaystyle\leq$
$\displaystyle\mathrm{max}\Big{\\{}\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I),\mathrm{reg}_{\mathcal{R}(J)}\frac{\mathcal{R}(J)}{\mathcal{R}(J,I)}\Big{\\}}$
$\displaystyle=$
$\displaystyle\mathrm{max}\Big{\\{}\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I),\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I)-1\Big{\\}}$
$\displaystyle=$ $\displaystyle\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I).$

For the equality part, the proof of
$\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J)\geq\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I)$
is exactly the same as the one given in [6, Proposition 1.6] for the case
$J=\mathfrak{m}$. $\square$

######  Remark 3.11.

Notice that, as a direct consequence of Theorem 3.3 and Lemma 3.10, we
rediscover Theorem 3.1.

In our view, and in connection to Lemma 3.10, it is worth asking the
following.

######  Question 3.12.

Let $(R,\mathfrak{m},K)$ be either a local ring or a standard graded
$K$-algebra. Let $I\subset J$ be ideals of $R$. When does the condition
$\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J)=\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I)$
force $I$ to be a reduction of $J$?

Inspired by Theorem 3.1 and Theorem 3.3, we might wonder whether the
comparison

${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$

holds in general. However, in the case where $I$ is not a reduction of
$\mathfrak{m}$, the relationship with the regularity of
$\mathcal{R}(\mathfrak{m})$ may become rather wild, as we can see in the
examples below.

######  Example 3.13.

Let $(R,\mathfrak{m})$ be a local ring with infinite residue field and ${\rm
depth}\,R>0$. By [20, Proposition 1.5], we have
$\mathfrak{m}^{n+1}:\mathfrak{m}=\mathfrak{m}^{n}$ for all
$n\geq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$.
Now set $I={\mathfrak{m}}^{k}$ for any given $k\geq 2$ (note $I$ cannot be a
reduction of $\mathfrak{m}$). We can write

(3)
$I\mathfrak{m}^{n}=\mathfrak{m}^{n+k}=\mathfrak{m}^{n+k+1}:\mathfrak{m}=I\mathfrak{m}^{n+1}:\mathfrak{m}$

for all
$n\geq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$ and
therefore
${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$.
Furthermore notice that, by using (3) whenever
$n\geq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})-k\geq
0$, we obtain

${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})-k<\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m}).$

Finally, if
$k\geq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$
then it is easy to see that ${\mathfrak{d}}_{3}(I)=0$. In particular, if
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})\leq 2$
then ${\mathfrak{d}}_{3}(I)=0$.

######  Example 3.14.

Let $R$ be as in [17, Example 4.3]. As observed there,
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})=8$.
Consider the ideal $I={\mathfrak{m}}^{2}$. By Example 3.13 above, we can write

${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})-2=6,$

whereas, on the other hand, a computation shows $I{\mathfrak{m}}^{5}\neq
I{\mathfrak{m}}^{6}:\mathfrak{m}$. Therefore, we must have
${\mathfrak{d}}_{3}(I)=6$.

We are also able to illustrate the opposite situation, as follows.

######  Example 3.15.

Let $I=(x^{a},y^{a})\subset R=k[\\![x,y]\\!]$, where $k$ is a field and $a\geq
2$. Clearly, $I$ is not a reduction of $\mathfrak{m}=(x,y)$. From [4, Example
4.1] we can write

${\mathfrak{d}}_{3}(I)=a-1>0=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m}),$

where the last equality holds because $R$ is a regular local ring (see Theorem
5.3 for a more general statement).

##  4\. Approach (and a conjecture) via reduction numbers

We begin recalling the following result in dimension one.

######  Proposition 4.1.

$($[17, Corollary 3.7]$)$ If $(R,\mathfrak{m})$ is a one-dimensional Cohen-
Macaulay local ring with infinite residue field, then

$\mathfrak{d}_{3}(I)={\rm r}(\mathfrak{m})$

for any minimal reduction $I$ of $\mathfrak{m}$.

It is worth pointing out that this proposition is no longer true if the
reduction $I$ is not required to be minimal, as exemplified in [17, Example
4.2].

In higher dimension, there is the conjecture below.

######  Conjecture 4.2.

$($[17, Conjecture 3.8]$)$ If $(R,\mathfrak{m})$ is a Cohen-Macaulay local
ring with infinite residue field and ${\rm dim}\,R\geq 2$, then

$\mathfrak{d}_{3}(I)={\rm r}_{I}(\mathfrak{m})$

for any minimal reduction $I$ of $\mathfrak{m}$.

Our main purpose in this section is to provide partial answers to this
conjecture. A crucial tool in this investigation is given by the following
result.

######  Theorem 4.3.

$($[17, Theorem 3.4]$)$ Let $(R,\mathfrak{m})$ be a local ring with infinite
residue field and ${\rm depth}\,R>0$, and let $I$ be a reduction of
$\mathfrak{m}$. Then

$\mathfrak{d}_{3}(I)=\mathrm{max}\\{{\rm
r}_{I}(\mathfrak{m}),s(\mathfrak{m})-1\\}.$

Now we can illustrate Conjecture 4.2 in the case ${\rm dim}\,R=2$.

######  Example 4.4.

Let $(R,\mathfrak{m})$ be the local ring of a rational triple point as in [17,
Example 4.1], where we highlighted that $s(\mathfrak{m})=1$. Then, by Theorem
4.3, we obtain $\mathfrak{d}_{3}(I)={\rm r}_{I}(\mathfrak{m})$ for any
reduction (in particular, minimal reduction) $I$ of $\mathfrak{m}$.

The above example actually shows that Conjecture 4.2 is true in the case
$s(\mathfrak{m})=1$, or equivalently, ${\rm depth}\,{\rm
gr}_{\mathfrak{m}}(R)>0$ (see also Proposition 4.9 below). Furthermore, it is
clear that $\widetilde{\mathfrak{m}}=\mathfrak{m}$, which forces
$s(\mathfrak{m})\neq 2$. Therefore, in tackling the conjecture we can suppose
$s(\mathfrak{m})\geq 3$.

######  Theorem 4.5.

Conjecture 4.2 holds true in case $s:=s(\mathfrak{m})\geq 3$ and

$I\widetilde{\mathfrak{m}^{s-2}}=I\mathfrak{m}^{s-2}.$

Proof.  By virtue of Theorem 4.3, it suffices to show that ${\rm
r}_{I}(\mathfrak{m})\geq s-1$. Suppose, by way of contradiction, that

${\rm r}_{I}(\mathfrak{m})\leq s-2.$

Since $I$ is a minimal reduction of $\mathfrak{m}$, we can apply [14,
Proposition 2.4 and Theorem 2.10] to obtain
$I\widetilde{\mathfrak{m}^{k}}=\widetilde{\mathfrak{m}^{k+1}}$ for all
$k\geq{\rm r}_{I}(\mathfrak{m})$. Consequently,

$\widetilde{\mathfrak{m}^{s-1}}=I\widetilde{\mathfrak{m}^{s-2}}=I\mathfrak{m}^{s-2}=\mathfrak{m}^{s-1},$

which contradicts the definition of $s$. $\square$

Since $\mathfrak{m}=\widetilde{\mathfrak{m}}$ is always true, the case
$s(\mathfrak{m})=3$ is a straightforward consequence of Theorem 4.5.

######  Corollary 4.6.

Conjecture 4.2 is true if $s(\mathfrak{m})=3$.

######  Example 4.7.

Let $K$ be an infinite field and

$R=\frac{K[\\![x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7}]\\!]}{(x_{1}^{2},x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{2}x_{3},x_{2}x_{4},x_{3}x_{4},x_{2}^{3}-x_{1}x_{5},x_{3}^{3}-x_{1}x_{6},x_{4}^{3}-x_{1}x_{7})},$

which is a three-dimensional Cohen-Macaulay local ring. Notice that

$x_{1}\in\widetilde{\mathfrak{m}^{2}}\setminus\mathfrak{m}^{2}$

and $\widetilde{\mathfrak{m}^{n}}=\mathfrak{m}^{n}$ for all $n\geq 3$. Thus,
$s(\mathfrak{m})=3$. By Corollary 4.6, we obtain $\mathfrak{d}_{3}(I)={\rm
r}_{I}(\mathfrak{m})$ for any minimal reduction $I$ of $\mathfrak{m}$.

When $s:=s(\mathfrak{m})\geq 4$, it may be difficult to determine whether
$I\widetilde{\mathfrak{m}^{s-2}}=I\mathfrak{m}^{s-2}$. However, by employing a
different argument, we are able to establish Conjecture 4.2 in the case
$s(\mathfrak{m})=4$.

######  Proposition 4.8.

Conjecture 4.2 is true if $s(\mathfrak{m})=4$.

Proof.  Suppose ${\rm r}_{I}(\mathfrak{m})\leq s(\mathfrak{m})-2=2$. Then, by
[3, Theorem 3.6], the ring $\mathrm{gr}_{\mathfrak{m}}(R)$ must be Cohen-
Macaulay. Hence ${\rm depth}\,\mathrm{gr}_{\mathfrak{m}}(R)={\rm
dim}\,\mathrm{gr}_{\mathfrak{m}}(R)={\rm dim}\,R>0$. But this implies
$s(\mathfrak{m})=1$, which is a contradiction. Therefore,

${\rm r}_{I}(\mathfrak{m})\geq s(\mathfrak{m})-1$

and the result follows from Theorem 4.3. $\square$

We close the section with yet another affirmative case of the conjecture.

######  Proposition 4.9.

Conjecture 4.2 is true if $R$ has minimal multiplicity.

Proof.  For such $R$, the ring ${\rm gr}_{\mathfrak{m}}(R)$ is Cohen-Macaulay
(see [22, Theorem 2]), hence its depth is equal to ${\rm dim}\,R\geq 2>0$. In
this case, as we already know, the conjecture holds true. $\square$

##  5\. Application: Regular local rings

In this section, as applications of some of our results, we provide new
characterizations of regular local rings, and describe a potential approach to
the long-standing Zariski-Lipman conjecture about derivations.

###  5.1. Characterizations of regular local rings

Our result in this part is Theorem 5.3 below. We observe that the implication
(a) $\Rightarrow$ (d) recovers [17, Corollary 3.11]. Moreover, a crucial fact
here (which, as far as we know, is new) is given by (b) $\Rightarrow$ (a),
i.e., $(R,\mathfrak{m})$ must be regular if $\mathfrak{m}$ is generated by a
$d$-sequence (see [11], [12] for the definition of this type of sequence and
its properties), which in particular solves the problem suggested in [17,
Remark 3.12]. Finally, the equivalence between assertions (a) and (e) reveals
the curious role played by ${\mathfrak{d}}_{3}(I)$ in regard to the theory of
regular local rings, which can be re-expressed by means of equivalence to the
structural assertion (f).

For the proof, the following two interesting facts will be useful.

######  Lemma 5.1.

([23, Corollary 5.2]) Let $(R,\mathfrak{m})$ be a local ring with infinite
residue field. Then, $\mathfrak{m}$ is generated by a $d$-sequence if and only
if $\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m})=0$.

######  Lemma 5.2.

([15, Theorem 1.1]) Let $(R,\mathfrak{m})$ be a local ring with ${\rm
depth}\,R>0$. If $Q$ is a parameter ideal of $R$ such that $Q^{n}$ is
$\mathfrak{m}$-full for some $n\geq 1$, then $R$ is regular.

######  Theorem 5.3.

Let $(R,\mathfrak{m},K)$ be either a local ring or a standard graded
$K$-algebra, with $K$ infinite and $\mathrm{depth}\,R>0$. The following
assertions are equivalent:

  * (a)

$R$ is regular;

  * (b)

$\mathfrak{m}$ is generated by a $d$-sequence;

  * (c)

$s(\mathfrak{m})=1$ and ${\rm r}_{I}(\mathfrak{m})=0$, for any (minimal)
reduction $I$ of $\mathfrak{m}$;

  * (d)

${\mathfrak{d}}_{1}(I)={\mathfrak{d}}_{2}(I)={\mathfrak{d}}_{3}(I)=0$, for any
(minimal) reduction $I$ of $\mathfrak{m}$;

  * (e)

${\mathfrak{d}}_{3}(I)=0$, for any (minimal) reduction $I$ of $\mathfrak{m}$;

  * (f)

$\mathcal{R}(\mathfrak{m},I)=(\mathcal{R}(\mathfrak{m},I)_{\geq
1}:_{\mathcal{R}(R)}\mathfrak{m})(1)$, for any (minimal) reduction $I$ of
$\mathfrak{m}$.

Proof.  (a) $\Rightarrow$ (b) If $R$ is regular, then $\mathfrak{m}$ is
generated by a regular sequence, which is therefore a $d$-sequence.

(b) $\Rightarrow$ (c) According to [20, Theorem 2.1(ii)], we have ${\rm
max}\\{\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m}),1\\}\geq
s(\mathfrak{m})$. Hence, using Lemma 5.1, we obtain $s(\mathfrak{m})=1$. In
order to deal with ${\rm r}_{I}(\mathfrak{m})$, consider first the case where
the reduction $I$ is minimal. Applying [20, Theorem 1.3] and [21, p. 12], we
derive

$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m})\geq{\rm
r}_{I}(\mathfrak{m}),$

which gives ${\rm r}_{I}(\mathfrak{m})=0$. Now, if $I$ is not minimal, then it
necessarily contains a minimal reduction $J$ of $\mathfrak{m}$, so that ${\rm
r}_{I}(\mathfrak{m})\leq{\rm r}_{J}(\mathfrak{m})=0$.

(c) $\Rightarrow$ (d) This follows readily from the fact (proved in [17,
Theorem 3.4]) that

${\mathfrak{d}}_{3}(I)\leq\mathrm{max}\\{{\rm
r}_{I}(\mathfrak{m}),s(\mathfrak{m})-1\\}.$

Now, by [17, Proposition 2.2], the vanishing of ${\mathfrak{d}}_{3}(I)$ forces
that of ${\mathfrak{d}}_{1}(I)$ and ${\mathfrak{d}}_{2}(I)$.

(d) $\Rightarrow$ (e) This is obvious.

(e) $\Rightarrow$ (a) In the present setting, $\mathfrak{m}$ admits a
reduction $Q$ which is a parameter ideal (see, e.g., [13, Exercise 8.11(ii)]).
As ${\mathfrak{d}}_{3}(Q)={\mathfrak{d}}_{1}(Q)$, we obtain

${\mathfrak{d}}_{1}(Q)=0$

and consequently $Q$ is $\mathfrak{m}$-full. We are now in a position to apply
Lemma 5.2 with $n=1$ to conclude that $R$ is regular.

It remains to prove that assertions (a) and (f) are equivalent. To this end,
simply note that (f) holds if and only if
$(\mathcal{R}(\mathfrak{m},I))_{k}=(\mathcal{R}(\mathfrak{m},I)_{k\geq
1}:_{\mathcal{R}(R)}\mathfrak{m})(1)_{k}$ for all $k\geq 0$ and any (minimal)
reduction $I$ of $\mathfrak{m}$, which means

$I\mathfrak{m}^{k}=I\mathfrak{m}^{k+1}:\mathfrak{m}\quad\mbox{for\, all}\quad
k\geq 0.$

This is, by definition, tantamount to ${\mathfrak{d}}_{3}(I)=0$, which as we
have shown above is equivalent to $R$ being regular. $\square$

###  5.2. Potential approach to a classical conjecture

For a field $K$ and a $K$-algebra $R$, we write as usual ${\rm Der}_{K}(R)$
for the module of $K$-derivations of $R$, i.e., the additive maps $D\colon
R\to R$ that vanish on $K$ and satisfy Leibniz rule: $D(\alpha\beta)=\alpha
D(\beta)+\beta D(\alpha)$ for all $\alpha,\beta\in R$. Now assume that $R$ is
a positive-dimensional local ring which is either

(4) ${K}[x_{1},\ldots,x_{m}]_{\mathfrak{q}}/I\ \ \ ({\mathfrak{q}}\in{\rm
Spec}\,{K}[x_{1},\ldots,x_{m}])\quad\mbox{or}\quad{K}[\\![x_{1},\ldots,x_{m}]\\!]/I,$

with $I$ a proper radical ideal and $x_{1},\ldots,x_{m}$ indeterminates over a
field ${K}$ of characteristic zero. In this setting, there is the following
long-held classical problem.

######  Conjecture 5.4.

(Zariski-Lipman) Let $R$ be as in (4). If ${\rm Der}_{K}(R)$ is free, then $R$
is regular.

This problem has an interesting long history, featuring in particular a strong
geometric counterpart, and remains open in some cases. For details and
references, see [9, Section 2] (in particular, see [9, Theorem 2.3] for a
simple proof of the Zariski-Lipman conjecture in the graded case).
Additionally, [16, Section 4] shows a relation between this conjecture and the
$\mathfrak{m}$-full property of ideals in the (open) two-dimensional local
case, which makes it somewhat natural to expect further connections.

We point out that, under the hypotheses of the conjecture, $R$ must be a
(normal) domain, and we can write

${\rm Der}_{K}(R)=\bigoplus_{1\leq j\leq t}RD_{j}\leavevmode\nobreak\
\leavevmode\nobreak\ \cong R^{t},\quad t={\rm dim}\,R,$

for some free basis $\\{D_{j}\\}$ consisting of precisely $t$ derivations.
Conversely, if ${\rm Der}_{K}(R)$ admits a free basis
$\\{D_{1},\ldots,D_{s}\\}$, then necessarily $s=t$.

######  Remark 5.5.

Let $(R,\mathfrak{m})$ be as in (4). If, as above, the $R$-module ${\rm
Der}_{K}(R)$ is free, then our guess is that, for each minimal reduction $I$
of $\mathfrak{m}$, there exist $\alpha_{j}^{(I)},\beta_{j}^{(I)}\in R$,
$j=1,\ldots,t$, such that the element given by

${\ell}^{(I)}=\sum_{j=1}^{t}\beta_{j}^{(I)}D_{j}(\alpha_{j}^{(I)})$

satisfies ${\ell}^{(I)}\in\mathfrak{m}\setminus{\mathfrak{m}}^{2}$ and
$I{\mathfrak{m}}^{k+1}\colon({\ell}^{(I)})=I{\mathfrak{m}}^{k}$ for all $k\geq
0$. This would confirm the validity of Conjecture 5.4, because such conditions
yield $I{\mathfrak{m}}^{k}$ to be $\mathfrak{m}$-full for all $k\geq 0$, which
means $\mathfrak{d}_{1}(I)=0$ and hence, as we already know,
$\mathfrak{d}_{3}(I)=0$. Now, Theorem 5.3 ensures that $R$ is regular.

Acknowledgements. The second-named author was partially supported by CNPq
(grants 406377/2021-9 and 313357/2023-4).

## References

  * [1] M. Brodmann, R. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Univ. Press, Cambridge, 1998.
  * [2] W. Bruns, A. Conca, C. Raicu, M. Varbaro, Determinants, Gröbner bases and cohomology, Springer, 2022.
  * [3] A. Corso, C. Polini, M. E. Rossi, Depth of associated graded rings via Hilbert coefficients of ideals, J. Pure Appl. Algebra 201 (2005), 126 - 141.
  * [4] H. Dao, On colon operations and special types of ideals, Palestine J. Math. 10 (2021), 383-388.
  * [5] D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Springer-Verlag, 1995.
  * [6] A. Ficarra, Dao numbers and the asymptotic behaviour of fullness, 2024, preprint arxiv.org/abs/2402.05555.
  * [7] S. Goto, F. Hayasaka, Finite homological dimension and primes associated to integrally closed ideals, Proc. Amer. Math. Soc. 130 (2002), 3159-3164.
  * [8] W. Heinzer, L. J. Ratliff, D. E. Rush, Basically full ideals in local rings, J. Algebra 250 (2002), 371-396.
  * [9] J. Herzog, The module of differentials, Lecture notes, Workshop on Commmutative Algebra and its Relation to Combinatorics and Computer Algebra, International Centre for Theoretical Physics (Trieste, Italy, 1994).
  * [10] L. T. Hoa, N. D. Tam, On some invariants of a mixed product of ideals, Arch. Math. 94 (2010), 327-337.
  * [11] C. Huneke, On the symmetric and Rees algebras of an ideal generated by a $d$-sequence, J. Algebra 62 (1980), 268-275.
  * [12] C. Huneke, The theory of $d$-sequences and powers of ideals, Adv. Math. 46 (1982), 249-279.
  * [13] C. Huneke, I. Swanson, Integral Closure of Ideals, Rings and Modules, London Math. Soc. Lecture Note Ser. 336, Cambridge Univ. Press, 2006.
  * [14] A. Mafi, Ratliff-Rush ideal and reduction numbers, Comm. Algebra 46 (2018), 1272–1276.
  * [15] N. Matsuoka, On $\mathfrak{m}$-full powers of parameter ideals, Tokyo J. Math. 29 (2006), 405-411.
  * [16] C. B. Miranda-Neto, Free logarithmic derivation modules over factorial domains, Math. Res. Lett. 24 (2017), 153-172.
  * [17] C. B. Miranda-Neto, D. S. Queiroz, Dao’s question on the asymptotic behaviour of fullness, 2023, preprint arxiv.org/abs/2308.03997.
  * [18] L. J. Ratliff, D. E. Rush, Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978), 929-934.
  * [19] M. E. Rossi, I. Swanson, Notes on the behaviour of the Ratliff-Rush filtration, Contemp. Math. 331 (2003), 313-328.
  * [20] M. E. Rossi, D. T. Trung, N. V. Trung, Castelnuovo–Mumford regularity and Ratliff–Rush closure, J. Algebra 504 (2018), 568–-586.
  * [21] M. E. Rossi, G. Valla, Hilbert functions of filtered modules, UMI Lect. Notes 9, Springer, 2010.
  * [22] J. D. Sally, On the associated graded ring of a local Cohen-Macaulay ring, J. Math. Kyoto Univ. 17 (1977), 19–21.
  * [23] N. V. Trung, The Castelnuovo regularity of the Rees algebra and the associated graded ring, Trans. Amer. Math. Soc. 350 (1998), 2813–2832.
  * [24] J. Watanabe, $\mathfrak{m}$-full ideals, Nagoya Math. J. 106 (1987), 101-111.
  * [25] N. Zamani, Regularity of the Rees and Associated Graded Modules, Eur. J. Pure Appl. Math. 7 (2014), 429-436.


# Conformal invariance and multifractality at Anderson transitions in
arbitrary dimensions

Jaychandran Padayasi  Ilya A. Gruzberg<EMAIL_ADDRESS>The Ohio State
University, Department of Physics, Columbus OH, 43210, USA

###### Abstract

Electronic wave functions at Anderson transitions exhibit multifractal scaling
characterized by a continuum of generalized multifractal exponents
$\Delta_{\gamma}$ with vector indices $\gamma=(q_{1},\ldots,q_{n})$. In a
field theory description of the transitions, there are corresponding
multifractal operators $\mathcal{O}_{\gamma}$ with scaling dimensions
$\Delta_{\gamma}$. Assuming conformal invariance and using the conformal
bootstrap framework, we derive a constraint that implies that the generalized
multifractal spectrum $\Delta_{\gamma}$ must be quadratic in all $q_{i}$ in
any dimension $d>2$. As several numerical studies have shown deviations from
parabolicity, we argue that conformal invariance is likely absent at Anderson
transitions in dimensions $d>2$.

††preprint: APS/123-QED

Introduction. Continuous Anderson transitions (ATs) between metals and
insulators, as well as transitions between topologically different localized
phases are a major focal point in the study of disordered systems [1]. Despite
major breakthroughs in the scaling theory [2], symmetry classification [3],
critical properties at ATs have eluded a rigorous theoretical treatment
because of the strongly-coupled nature of the critical points.

A remarkable property of ATs is the multifractality of critical wave
functions, or the local density of states $\nu(r)$: their moments scale (in
any spatial dimension $d$) with the system size $L$ as
$\overline{\nu^{q}(r)}\sim L^{-\Delta_{q}}$, where the exponents $\Delta_{q}$,
called the multifractal (MF) spectrum, depend on the continuous (complex)
parameter $q$ in a non-linear way. (The bar denotes disorder averaging.) Also,
there are more general combinations $P_{\gamma}$ of critical wave functions
[4, 5, 6, 7] labeled by vectors $\gamma=(q_{1},\ldots,q_{n})$ of complex
numbers $q_{i}$, with scaling dimensions $\Delta_{\gamma}$ 111Notice that what
we denote as $\Delta_{q}$ and $\Delta_{\gamma}$ here is usually denoted by
$x_{q}$ and $x_{\gamma}$ in other papers on Anderson multifractality..

Field theories of Anderson localization are non-linear sigma models whose
target spaces are cosets $\mathcal{G}/\mathcal{K}$ of certain Lie supergroups
[9, 10, 11, 1, 12]. In these models, the generalized MF observables
$P_{\gamma}$ are represented by gradientless composite operators
$\mathcal{O}_{\gamma}$. A key fact is that $\mathcal{O}_{\gamma}$ can be
constructed as highest-weight vectors under the action of the Lie superalgebra
of $\mathcal{G}$ with weights $\gamma$. Then the $\mathcal{G}$-symmetry of the
target space (not broken at the critical point) strongly restricts the fusion
of MF operators and implies certain symmetries of the generalized
($\Delta_{\gamma}$) and simple ($\Delta_{q}$) MF spectra [13, 4].
Specifically, the MF operators $\mathcal{O}_{\gamma}$ satisfy 1)  _Abelian
fusion_ (addition of highest weights)

$\displaystyle\mathcal{O}_{\gamma_{1}}\times\mathcal{O}_{\gamma_{2}}\sim\mathcal{O}_{\gamma_{1}+\gamma_{2}}+\ldots,$
(1)

where the ellipses denote derivatives of
$\mathcal{O}_{\gamma_{1}+\gamma_{2}}$; 2)  _Weyl symmetry_ of the MF spectra
$\Delta_{\gamma}=\Delta_{w\gamma}$, $w\in W$, where $W$ is the Weyl group that
acts in the space of weights $\gamma$ and is generated by the sign inversions
of ${\tilde{q}}_{j}\equiv q_{j}+c_{j}/2$ and interchanges of ${\tilde{q}}_{i}$
and ${\tilde{q}}_{j}$:

$\displaystyle q_{i}$ $\displaystyle\rightarrow-c_{i}-q_{i},$ $\displaystyle
q_{i}\rightarrow q_{j}+(c_{j}-c_{i})/2.$ (2)

The parameters $c_{i}$ are the coefficients of the half-sum of the positive
roots $\rho_{b}=\sum_{j=1}^{n}c_{j}e_{j}$ in a standard basis $e_{j}$. They
are known for all families of symmetric superspaces [14, 13, 4, 5, 6, 7, 15].
A consequence of Eq. (2) is the existence of the operator
$\mathcal{O}_{-\rho_{b}}=\mathcal{O}_{(-c_{1},-c_{2},\ldots,-c_{n})}$ with
vanishing scaling dimension $\Delta_{-\rho_{b}}=0$. The simple MF spectrum
$\Delta_{q}$ corresponds to $\gamma=(q,0,0,\ldots,0)$. In this case one
denotes $c_{1}=-q_{*}$, and the Weyl symmetry reduces to
$\Delta_{q}=\Delta_{q_{*}-q}$. The Weyl symmetry is fully supported by
numerical and analytical results for various symmetry classes and dimensions
$d\geq 2$ [16, 17, 14, 18, 19, 20, 21, 22, 23, 24, 25, 5, 6, 7, 15].

As for other critical points, the scale invariance at ATs may be enhanced to
conformal invariance (though this is not guaranteed [26, 27]), in which case
critical properties are described by a conformal field theory (CFT). In this
letter we derive a general constraint on the MF spectra in an _arbitrary_
dimension $d$ and for _any_ symmetry class, that results from the properties
(1)–(2) and the assumption of presence of conformal symmetry at ATs. Our main
result is that in this situation the MF spectra have exactly parabolic form:

$\displaystyle\Delta_{\gamma}$ $\displaystyle=-b\sum_{i}q_{i}(q_{i}+c_{i}),$
$\displaystyle\Delta_{q}$ $\displaystyle=bq(q_{*}-q),$ (3)

where the overall prefactor $b$ cannot be determined from symmetry
considerations alone.

Our result is very general, and should apply to other situations where
multifractality and conformal invariance may both be present, such as critical
points in disordered systems [28, 29], turbulence [30, 31, 32], and other
systems.

Multifractality and CFT in $d=2$. Two-dimensional (2D) CFTs possess the
infinite-dimensional Virasoro symmetry. In this setting, the ellipses in Eq.
(1) represent Virasoro descendants, and lead to a single Virasoro block in a
4-point function of MF operators, and a Vafa-Lewellen [33, 34] constraint on
the MF spectra. The unique solution of this constraint subject to the
symmetries (2) is the parabolic spectrum (3) [35, 36, 5]. The MF operators
appear as vertex operators in a 2D Coulomb gas theory (Gaussian free field
with a background charge).

Both the simple and the generalized parabolicity (3) were tested analytically
and numerically, and were found to be violated at 2D ATs in various symmetry
classes [21, 22, 37, 5, 6, 7, 15]. This has led to the understanding that
conformal invariance might be lost at these critical points. Similarly,
numerical studies of multifractality in $d=3,4,5$ have found strong deviations
from parabolicity [25, 38, 39, 40] but there has not been any prediction in
$d>2$ from a CFT perspective. Our paper provides such a prediction.

The conformal bootstrap program [41] has brought the study of higher-
dimensional CFTs into the limelight with extensive work on both analytical and
numerical fronts. The bootstrap philosophy attempts to solve the _crossing
symmetry_ conditions coming from associativity of the operator product
expansion (OPE) of the CFT operators, with minimal inputs from global symmetry
and expected fusion rules for the operators. Crossing symmetry relates three
possible ways (or channels) of reducing a 4-point function
$\langle\prod_{i=1}^{4}O_{i}(x_{i})\rangle$ to 2-point functions by replacing
pairs of operators with their OPEs (Fig. 1). The $s$-channel fusion ($1\to 2$,
$3\to 4$) and the $t$-channel fusion ($1\to 4$, $2\to 3$) result in expansions
of the 4-point function that have overlapping regions of convergence and give
the crossing equation

$\displaystyle\sum_{O_{s}}\lambda_{12}^{O_{s}}\lambda_{34}^{O_{s}}W_{O_{s}}=\sum_{O_{t}}\lambda_{14}^{O_{t}}\lambda_{23}^{O_{t}}W_{O_{t}}.$
(4)

$\displaystyle\sum_{O_{s}}$ $O_{1}$$O_{2}$$O_{4}$$O_{3}$$O_{s}$ =
$\displaystyle\sum_{O_{t}}$ $O_{1}$$O_{2}$$O_{4}$$O_{3}$$O_{t}$

Figure 1: Schematic representation of the $s$-$t$ crossing equation.

Here the factors $W_{O}$ are fully determined by conformal symmetry, while the
CFT data $\\{\Delta_{i},\lambda_{ijk}\\}$ consisting of the spectrum of
scaling dimensions and the OPE coefficients are to be found. Solutions of the
crossing equations determine consistent CFTs fully defined by the CFT data,
circumventing the need for a Lagrangian description.

The $s$ and $t$ channels are obtained from each other by interchanges of
indices of the operators (and their points of insertion): $s\leftrightarrow
t\equiv 1\leftrightarrow 3$. Accordingly, starting with a function
$f^{(s)}\equiv f(x_{1},x_{2};x_{3},x_{4})$ of four ordered arguments we
obtain, by permuting $1\leftrightarrow 3$, another function $f^{(t)}\equiv
f(x_{3},x_{2};x_{1},x_{4})$. Using this notation, we can write the 4-point
function as a product of a conformally-covariant kinematic factor
$\mathbb{K}_{4}^{(c)}$ and a $G$-function (different in each channel)

$\displaystyle\Big{\langle}\prod_{i=1}^{4}O_{i}(x_{i})\Big{\rangle}=\mathbb{K}_{4}^{(s)}G^{(s)}=\mathbb{K}_{4}^{(t)}G^{(t)}.$
(5)

The $G$-functions depend on the cross ratios

$\displaystyle u$
$\displaystyle=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}},$
$\displaystyle v$
$\displaystyle=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}},$ where
$\displaystyle x_{ij}$ $\displaystyle=x_{i}-x_{j},$ (6)

and operator labels. The cross ratios get transformed upon crossing so that

$\displaystyle G^{(s)}$ $\displaystyle=G_{12,34}(u,v),$ $\displaystyle
G^{(t)}$ $\displaystyle=G_{32,14}(v,u).$ (7)

In terms of the $G$-functions the $s$-$t$ crossing equation is

$\displaystyle G^{(s)}u^{-\frac{\Delta_{1}+\Delta_{2}}{2}}$
$\displaystyle=G^{(t)}v^{-\frac{\Delta_{2}+\Delta_{3}}{2}}.$ (8)

Much of the bootstrap formalism is geared towards solving Eq. (8) self-
consistently for unitary CFTs. Since any putative CFT for MF correlators
contains infinitely many relevant operators and, thus, is non-unitary, we
resort to novel, unorthodox methods that focus on the $G$-function and various
physical inputs (similar in spirit to the “inverse bootstrap” method in [42]).
We start by studying the Coulomb gas theories in the language of modern
conformal bootstrap and use them as signposts to generalize the notion of
Abelian fusion to higher dimensions. Then we show that the generalized Abelian
fusion and crossing symmetry together yield a constraint on the spectrum of
scaling dimensions in any $d$ that is analogous to the Vafa-Lewellen
constraints [33, 34] known in CFT in $d=2$. Finally, additional physical
assumptions specific to MF observables allow us to solve the constraint,
leading to a quadratic dependence of the MF spectrum $\Delta_{\gamma}$ on
$q_{i}$ in any dimension $d$.

Coulomb gas theories with global conformal blocks. In $d=2$, the Coulomb gas
theories arise out of breaking the U(1) symmetry of the free boson $\phi$ by
including a background charge $Q$ in the action [43, 44, 45]. We will be
interested in the correlators of the vertex operators $V_{\alpha}\sim
e^{d\alpha\phi}$. A Coulomb gas CFT can be defined [44, 45] in any dimension
$d\in\mathbb{N}$ by considering an action with a possibly non-local kinetic
term $\propto\phi(-\square)^{\frac{d}{2}}\phi$. Such CFTs also arise as limits
of generalized free fields, where the scaling dimension of the free boson is
tuned to $\Delta_{\phi}=0$. In this limit $\langle\phi\phi\rangle$ is
logarithmic in any dimension which allows us to study vertex operators.
Following the conventions in [44, 45], the spectrum and the multi-point
functions in such higher-dimensional Coulomb gas theories are

$\displaystyle\Delta_{\alpha}$ $\displaystyle=d\alpha(Q-\alpha),$
$\displaystyle\Big{\langle}\prod V_{\alpha_{i}}(x_{i})\Big{\rangle}$
$\displaystyle=\prod_{i<j}|x_{ij}|^{-2d\alpha_{i}\alpha_{j}},$ (9)

where the correlators satisfy the charge neutrality $\sum{\alpha_{i}}=Q$. Our
first goal is to derive the OPE of vertex operators in terms of primaries of
the global conformal group by studying the conformal block expansion.

Consider a four-point function of vertex operators which can be written in the
form (5) with the $G$-function

$\displaystyle G^{(s)}_{\text{CG}}$
$\displaystyle=u^{\frac{d}{2}(\alpha_{3}+\alpha_{4})(\alpha_{1}+\alpha_{2})}v^{-d\alpha_{2}\alpha_{3}}$
$\displaystyle=u^{\frac{1}{2}\Delta_{\alpha_{1}+\alpha_{2}}}v^{\frac{1}{2}(\Delta_{\alpha_{2}+\alpha_{3}}-\Delta_{\alpha_{2}}-\Delta_{\alpha_{3}})}.$
(10)

This function is explicitly crossing symmetric [satisfies Eq. (8)] and has a
convergent conformal block expansion [41] in the $s$-channel in any
dimensionality $d$,

$\displaystyle
G^{(s)}_{\text{CG}}=\sum_{O}\lambda_{12}^{O}\lambda_{34}^{O}\,g_{\Delta_{O},l_{O}}(u,v).$
(11)

The conformal blocks $g_{\Delta_{O},l_{O}}$ are known analytically in even
dimensions. (Their dependence on the dimensions $\Delta_{1}$–$\Delta_{4}$ of
the external operators will not be important for us.) The blocks are often
written in conformal frame coordinates $(z,\bar{z})$ related to the cross
ratios $(u,v)$ by

$\displaystyle u$ $\displaystyle=z\bar{z},$ $\displaystyle v$
$\displaystyle=(1-z)(1-\bar{z}).$ (12)

In the $s$-channel limit, $x_{12}\approx x_{34}\ll x_{13}\approx x_{24}\approx
x_{23}\approx x_{14}$, and thus $u\rightarrow 0$, $v\rightarrow 1$, see Eq
(6). This corresponds to $z,\bar{z}\rightarrow 0$, and the $G$-function
(Conformal invariance and multifractality at Anderson transitions in arbitrary
dimensions) has the form

$\displaystyle G^{(s)}$
$\displaystyle=(z\bar{z})^{\frac{\Delta^{(s)}}{2}}f(z,\bar{z}),$ (13)

where $f(z,\bar{z})$ is a Taylor series symmetric in $(z,\bar{z})$, and
$\Delta^{(s)}=\Delta_{\alpha_{1}+\alpha_{2}}$. In the Supplementary material
[46] we use the leading behavior of the conformal blocks in the $s$-channel
[47] to show that any $G$-function of the form (13) admits the conformal block
expansion

$\displaystyle G^{(s)}=\sum_{n,l\geq
0}\mu^{(n,l)}g_{\Delta^{(s)}+2n+l,l}(z,\bar{z})$ (14)

in arbitrary dimensions $d\geq 2$. Conversely, any $G$-function that can be
expanded as in Eq. (14) can also be written in the form of Eq. (13).

Let us write global primaries in any dimension as $[\Delta,l]$ specifying
their scaling dimension $\Delta$ and spin $l$. Then we say that the expansion
(14) contains just one twist family [42] consisting of the leading primary
$[\Delta^{(s)},0]$ and subleading operators $[\Delta^{(s)}+2n+l,l]$ which are
constructed from its derivatives. The superscript of the product of the OPE
coefficients $\mu^{(n,l)}\equiv\lambda_{1,2}^{(n,l)}\lambda_{3,4}^{(n,l)}$
identifies the operator $[\Delta^{(s)}+2n+l,l]$ in the twist family.

Expanding the Coulomb gas $G$-function (Conformal invariance and
multifractality at Anderson transitions in arbitrary dimensions) in global
conformal blocks gives the OPE of $V_{\alpha_{1}}\times V_{\alpha_{2}}$ as
[46]

$\displaystyle[\Delta_{\alpha_{1}},0]\times[\Delta_{\alpha_{2}},0]\sim\sum_{n,l\geq
0}\lambda^{(n,l)}_{1,2}[\Delta_{\alpha_{1}+\alpha_{2}}+2n+l,l],$ (15)

where $n,l$ are non-negative integers, and the $(n,l)=(0,1)$ term is absent in
the OPE. For two identical operators ($\alpha_{1}=\alpha_{2}$), their OPE is
completely specified by the conformal block expansion since we can extract the
squared OPE coefficients (see [46] for explicit expressions in the $d=2$ and
$d=4$ cases).

Generalized Abelian fusion. In the strict sense, Abelian fusion (1) cannot
hold in CFTs in $d>2$, since an OPE written with finitely many global
conformal primaries cannot satisfy crossing [48, 49, 50, 51]. Thus, we need to
generalize the notion of Abelian fusion to $d>2$. Global conformal block
expansions of Coulomb gas correlators exhibit certain features that we adopt
as the _definition_ of Abelian fusion in $d>2$:

  * •

All primary multifractal operators of the CFT can be grouped into twist
families.

  * •

The OPE of any two leading multifractal primaries contains only one twist
family

$\displaystyle[\Delta_{1},0]\times[\Delta_{2},0]\sim\sum_{n,l\geq
0}\lambda_{1,2}^{(n,l)}[\Delta+2n+l,l].$ (16)

This definition is more general than Eq. (15), where the three operators
involved, $V_{\alpha_{1}}$, $V_{\alpha_{2}}$, an $V_{\alpha_{1}+\alpha_{2}}$,
are labeled by continuous parameters that add upon fusion.

The generalized Abelian fusion (16) and the related conformal block expansion
(14) constrain a general scalar 4-point $G$-function to have the form (13).
Since $z\bar{z}=u$ and $z+\bar{z}=u+1-v$ form a basis in the ring of symmetric
functions of $z$ and $\bar{z}$, Eq. (13) can also be written as

$G^{(s)}=u^{\frac{\Delta^{(s)}}{2}}\sum_{n\geq 0}f^{(s)}_{n}(v)u^{n}.$ (17)

All powers of $u$ in this expansion differ from the leading power
$\Delta^{(s)}/2$ by non-negative integers $n$, consistent with the even
integer gaps between the spin-zero conformal blocks in Eq. (14). The functions
$f_{n}(v)$ are arbitrary so far, and quantities with the superscript $(s)$
depend on the external dimensions in a channel-covariant manner. At this
point, one can make a further simplification by assuming that the functions
$f_{r}(v)$ can be represented as (possibly infinite) sums of power laws in
$v$, i.e.

$G^{(s)}=u^{\frac{\Delta^{(s)}}{2}}\sum_{n,m\geq
0}C_{nm}v^{\sigma^{(s)}_{m}}u^{n}$ (18)

where $\sigma_{m}$ are unrelated real numbers and the coefficients $C_{nm}$ do
not depend on the cross-ratios $(u,v)$ or the external dimensions. The Coulomb
gas theories are of this form with a single term, see Eq. (Conformal
invariance and multifractality at Anderson transitions in arbitrary
dimensions). The generalized free field correlators [50] with
$\Delta_{\phi}\geq 0$ are similarly composed of sums of power-laws in $u$ and
$v$ although they do not satisfy Abelian fusion (integer gaps in powers of
$u$). The above ansatz made an appearance for the case of a correlator of
identical operators in Ref. [42] which discussed the idea of building
crossing-symmetric $G$-functions. The issue of its generality was raised and
left unanswered there. Similarly, the authors of Ref. [51] use a version where
the coefficients $C_{nm}$ are functions of $\log u,\log v$ (the logarithms
come from anomalous dimensions of the subleading operators in the twist
family). As our definition of Abelian fusion, Eq. (16), exactly fixes the
dimensions of all subleading operators, the logarithms are unnecessary in our
treatment.

Constraints on the $G$-function from crossing. Now we explore constraints on
the Abelian $G$-function (18) imposed by crossing symmetry. Substituting this
function into the crossing equation (8), we obtain

$\displaystyle
u^{\frac{\Delta^{(s)}}{2}-\frac{\Delta_{1}+\Delta_{2}}{2}}\sum_{n,m\geq
0}C_{nm}v^{\sigma^{(s)}_{m}}u^{n}$
$\displaystyle=v^{\frac{\Delta^{(t)}}{2}-\frac{\Delta_{2}+\Delta_{3}}{2}}\sum_{n,m\geq
0}C_{nm}u^{\sigma^{(t)}_{m}}v^{n}.$ (19)

This equation enforces a structure on the $G$-function that is understandable
in terms of crossing symmetric building blocks [42], such that any truncation
up to $(N,M)$ of the double sum in Eq. (18) is also crossing symmetric. In
this case all terms in Eq. (18) fall into two groups: single terms that are
crossing-symmetric by themselves, and pairs of terms that get exchanged under
crossing. The result is the $G$-function

$\displaystyle G^{(s)}(u,v)$
$\displaystyle=u^{\frac{\Delta^{(s)}}{2}}v^{\frac{\Delta^{(t)}}{2}-\frac{\Delta_{2}+\Delta_{3}}{2}}\bigg{(}\sum_{k\geq
0}S_{k}(uv)^{k}$ $\displaystyle\quad+\sum_{j\geq 1,k\geq
0}D_{jk}(uv)^{k}(u^{j}+v^{j})\bigg{)}.$ (20)

where we use $S_{k}$ and $D_{jk}$ to represent the coefficients of the
crossing-symmetric terms and pairs respectively. Thus, we adopt Eq. (Conformal
invariance and multifractality at Anderson transitions in arbitrary
dimensions) as the generic form of the Abelian $G$-function (18) that also
satisfies $s$-$t$ crossing.

Excluding the spin-1 operator. Focusing on the last part of the puzzle, we
expand the $G$-function (Conformal invariance and multifractality at Anderson
transitions in arbitrary dimensions) in conformal blocks in arbitrary
dimensions to first few orders in $z$ and $\bar{z}$. By construction, the
first block that appears in the expansion is $[\Delta^{(s)},0]$. The product
of the OPE coefficients of the leading block is read off as $\mu^{(0,0)}\equiv
S_{0}$. The coefficient $\mu^{(0,1)}$ of the spin-1 block $[\Delta^{(s)}+1,1]$
can be obtained by matching the coefficients of the series for the order
$\sim(z\bar{z})^{\Delta^{(s)}/2}(z+\bar{z})$ as

$\displaystyle\frac{\mu^{(0,1)}}{S_{0}}$
$\displaystyle=\frac{\Delta_{2}+\Delta_{3}-\Delta^{(t)}}{2}-\frac{(\Delta^{(s)}-\Delta_{12})(\Delta^{(s)}+\Delta_{34})}{4\Delta^{(s)}}$
$\displaystyle+\frac{\Delta_{2}+\Delta_{3}-\Delta^{(t)}}{2}\sum_{j\geq
1}\frac{D_{j0}}{S_{0}}-\sum_{j\geq 1}\frac{jD_{j0}}{S_{0}}.$ (21)

Both sums over $j$ must converge for the OPE coefficient to be well-defined,
and thus we introduce

$\displaystyle\sum_{j\geq 1}\frac{D_{j0}}{S_{0}}$
$\displaystyle\equiv\mathcal{P},$ $\displaystyle\sum_{j\geq
1}\frac{jD_{j0}}{S_{0}}$ $\displaystyle\equiv\mathcal{Q}.$ (22)

In the context of MF correlators, we identify
$\Delta^{(s)}\equiv\Delta_{\gamma_{1}+\gamma_{2}}$,
$\Delta^{(t)}\equiv\Delta_{\gamma_{3}+\gamma_{2}}$. The neutrality condition
$\sum_{i}\gamma_{i}=-\rho_{b}$ fixes
$\Delta_{4}=\Delta_{-\rho_{b}-\gamma_{1}-\gamma_{2}-\gamma_{3}}=\Delta_{\gamma_{1}+\gamma_{2}+\gamma_{3}}$.

This spin-1 operator cannot appear in any OPE of two identical scalar
operators on general grounds. Indeed, $O(x_{1})O(x_{2})$ is even with respect
to the interchange $x_{1}\leftrightarrow x_{2}$ but a spin-1 operator must
appear in the OPE as $\sim(x_{1}-x_{2})\cdot\partial
O_{\Delta^{(s)}}((x_{1}+x_{2})/2)$ which is odd. Exploiting this fact, we set
$O_{2}\equiv O_{1}$ in which case $\mu^{(0,1)}=0$, and Eq. (Conformal
invariance and multifractality at Anderson transitions in arbitrary
dimensions) becomes a constraint on $\Delta$’s:

$\displaystyle\Delta^{(s)}+\Delta_{3}-\Delta_{4}+4\mathcal{Q}=2(\Delta_{1}+\Delta_{3}-\Delta^{(t)})(1+\mathcal{P}).$
(23)

Furthermore, the continuity of MF spectra allows us to choose
$\gamma_{1}=\gamma_{2}=\epsilon e_{i}$, where $e_{i}=(0,\ldots,1,\ldots,0)$
(unit in the $i$-th place), with $\epsilon\ll 1$, and $\gamma_{3}=\gamma$ in
Eq. (23), which gives

$\displaystyle\Delta_{2\epsilon
e_{i}}+\Delta_{\gamma}-\Delta_{\gamma+2\epsilon e_{i}}+4\mathcal{Q}$
$\displaystyle\quad=2(\Delta_{\epsilon
e_{i}}+\Delta_{\gamma}-\Delta_{\gamma+\epsilon e_{i}})(1+\mathcal{P}).$ (24)

Now we expand in orders of $\epsilon$. At order $\epsilon^{0}$ we get
$4\mathcal{Q}=(1+2\mathcal{P})\Delta_{0}=0$, because $\Delta_{0}=0$ by
definition. At order $\epsilon$ we get
$\mathcal{P}\partial_{q_{i}}(\Delta_{0}-\Delta_{\gamma})=0$ which implies
either $\mathcal{P}=0$ or
$\partial_{q_{i}}\Delta_{0}=\partial_{q_{i}}\Delta_{\gamma}$. The second
condition [together with Eq. (2)] yields the trivial spectrum
$\Delta_{\gamma}=\text{const}=0$. Hence, we consider the case $\mathcal{P}=0$.
Then at order $\epsilon^{2}$ we get
$\partial_{q_{i}}^{2}(\Delta_{0}-\Delta_{\gamma})=0$, which implies that
$\Delta_{\gamma}$ is a quadratic polynomial in all $q_{i}$. Any such
polynomial that vanishes at $\gamma=0$ and satisfies the symmetry properties
(2) is exactly of the form given in Eq. (3) [5]. Thus, we arrive at our main
result:

> _The only MF spectrum $\Delta_{\gamma}$ which satisfies generalized Abelian
> fusion and crossing symmetry has the form given in Eq. (3)._

Going back to Eq. (23), we substitute $\mathcal{P}=\mathcal{Q}=0$ and the
quadratic solution for $\Delta_{\gamma}$ to find that the constraint

$2\Delta_{\gamma_{1}}+\Delta_{\gamma_{3}}-2\Delta_{\gamma_{1}+\gamma_{3}}-\Delta_{2\gamma_{1}}+\Delta_{2\gamma_{1}+\gamma_{3}}=0$
(25)

correctly picks out Abelian CFTs in $d\geq 2$, and thus is the appropriate
generalization of the 2D Vafa-Lewellen constraint with a single exchanged
Virasoro primary.

Summary and Outlook. Using conformal invariance, we have shown that any
Abelian CFT in $d>2$ must be intimately related to the Coulomb gas theory, and
have a quadratic spectrum. To summarize, the set of assumptions used to prove
the exact parabolicity of the MF spectrum in $d\geq 2$ is the following:

  1. 1.

The MF operators $\mathcal{O}_{\gamma}$ are global primaries in some CFT, so
that their correlators solve the crossing equations.

  2. 2.

The fusion of two MF operators is of the general form (16) (generalized
Abelian fusion).

  3. 3.

The $G$-functions of the charge-neutral 4-point correlators of MF operators
admit an expansion of the form
$G^{(s)}=\sum_{n,m}C_{nm}u^{\Delta_{\gamma_{1}+\gamma_{2}}/2+n}v^{\sigma_{m}}$.

The second and third assumptions are our strongest, and are fundamentally
related to each other. As in the case of weakly perturbed CFTs [51], it
remains to be seen if the generalized Abelian CFT defined here could be
perturbed so that the derivative operators gain anomalous dimensions.

Finally, let us discuss the implication of the first assumption, that of
conformal invariance. As we summarised earlier, the perturbative analytical
results in $d=2+\epsilon$ and numerical simulations in $d=3,4,5$ have shown
that the MF spectra for generic Anderson transitions are in fact, not
parabolic. In the light of our result, it follows that conformal symmetry is
likely lost at Anderson transitions. The only alternative is that the
symmetries of the sigma models that were used to deduce Abelian fusion and
Weyl symmetry are lost at the critical point. We believe this to be a very
unlikely scenario that would contradict the vast body of literature on
Anderson transitions. Thus, we identify Anderson transitions as examples of
systems where scale invariance does not imply conformal invariance.

Going forward, we believe that it is very important to understand the
relationship between scale and conformal invariance, especially for non-
unitary theories. Anderson transitions may provide a fertile ground for such
understanding. Moreover, it would be interesting to translate our analysis to
other physical systems exhibiting multifractal scaling. Finally, it is
exciting to consider possible generalizations of rational CFTs to $d>2$. This
line of inquiry may provide important answers in the much bigger project of
tracking the evolution of CFTs with $d$.

Acknowledgements. We thank Dalimil Mazáč, Marco Meineri, Alexander Mirlin,
Yaron Oz, Lorenzo Quintavalle, Sylvain Ribault, and Bernardo Zan for useful
discussions. This research was supported by Grant No. 2020193 from the United
States-Israel Binational Science Foundation (BSF).

## References

  * Evers and Mirlin [2008] F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008), arXiv:0707.4378 [cond-mat.mes-hall] .
  * Abrahams  _et al._ [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979).
  * Altland and Zirnbauer [1997] A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997), arXiv:cond-mat/9602137 .
  * Gruzberg  _et al._ [2013] I. A. Gruzberg, A. D. Mirlin, and M. R. Zirnbauer, Classification and symmetry properties of scaling dimensions at Anderson transitions, Phys. Rev. B 87, 125144 (2013).
  * Karcher  _et al._ [2021] J. F. Karcher, N. Charles, I. A. Gruzberg, and A. D. Mirlin, Generalized multifractality at spin quantum Hall transition, Annals Phys. 435, 168584 (2021), arXiv:2107.06414 [cond-mat.dis-nn] .
  * Karcher  _et al._ [2022a] J. F. Karcher, I. A. Gruzberg, and A. D. Mirlin, Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables, Phys. Rev. B 105, 184205 (2022a), arXiv:2203.12617 [cond-mat.dis-nn] .
  * Karcher  _et al._ [2022b] J. F. Karcher, I. A. Gruzberg, and A. D. Mirlin, Generalized multifractality at metal-insulator transitions and in metallic phases of two-dimensional disordered systems, Phys. Rev. B 106, 104202 (2022b), arXiv:2206.12226 [cond-mat.dis-nn] .
  * Note [1] Notice that what we denote as $\Delta_{q}$ and $\Delta_{\gamma}$ here is usually denoted by $x_{q}$ and $x_{\gamma}$ in other papers on Anderson multifractality.
  * Efetov [1983] K. Efetov, Supersymmetry and theory of disordered metals, Advances in Physics 32, 53 (1983).
  * Efetov [1997] K. Efetov,  _Supersymmetry in Disorder and Chaos_ (Cambridge University Press, Cambridge, England, 1997).
  * Mirlin [2000] A. D. Mirlin, Statistics of energy levels and eigenfunctions in disordered systems, Physics Reports 326, 259 (2000).
  * Wegner [2016] F. Wegner,  _Supermathematics and its Applications in Statistical Physics. Grassmann Variables and the Method of Supersymmetry_ (Springer, Heidelberg, Germany, 2016).
  * Gruzberg  _et al._ [2011] I. A. Gruzberg, A. W. W. Ludwig, A. D. Mirlin, and M. R. Zirnbauer, Symmetries of multifractal spectra and field theories of Anderson localization, Phys. Rev. Lett. 107, 086403 (2011).
  * Mirlin  _et al._ [2006] A. D. Mirlin, Y. V. Fyodorov, A. Mildenberger, and F. Evers, Exact Relations between Multifractal Exponents at the Anderson Transition, Phys. Rev. Lett. 97, 046803 (2006), arXiv:cond-mat/0603378 [cond-mat.mes-hall] .
  * Karcher  _et al._ [2023] J. F. Karcher, I. A. Gruzberg, and A. D. Mirlin, Generalized multifractality in two-dimensional disordered systems of chiral symmetry classes, Phys. Rev. B 107, 104202 (2023).
  * Evers  _et al._ [2003] F. Evers, A. Mildenberger, and A. D. Mirlin, Multifractality at the spin quantum Hall transition, Phys. Rev. B 67, 041303 (2003), arXiv:cond-mat/0203134 [cond-mat.mes-hall] .
  * Mirlin  _et al._ [2003] A. D. Mirlin, F. Evers, and A. Mildenberger, Wavefunction statistics and multifractality at the spin quantum Hall transition, Journal of Physics A Mathematical General 36, 3255 (2003), arXiv:cond-mat/0208451 [cond-mat.mes-hall] .
  * Mildenberger  _et al._ [2007] A. Mildenberger, A. R. Subramaniam, R. Narayanan, F. Evers, I. A. Gruzberg, and A. D. Mirlin, Boundary multifractality in critical one-dimensional systems with long-range hopping, Phys. Rev. B 75, 094204 (2007), arXiv:cond-mat/0611713 [cond-mat.mes-hall] .
  * Mildenberger and Evers [2007] A. Mildenberger and F. Evers, Wave function statistics at the symplectic two-dimensional Anderson transition: Bulk properties, Phys. Rev. B 75, 041303 (2007), arXiv:cond-mat/0608560 [cond-mat.mes-hall] .
  * Obuse  _et al._ [2007] H. Obuse, A. R. Subramaniam, A. Furusaki, I. A. Gruzberg, and A. W. W. Ludwig, Multifractality and Conformal Invariance at 2D Metal-Insulator Transition in the Spin-Orbit Symmetry Class, Phys. Rev. Lett. 98, 156802 (2007), arXiv:cond-mat/0609161 [cond-mat.dis-nn] .
  * Obuse  _et al._ [2008] H. Obuse, A. R. Subramaniam, A. Furusaki, I. A. Gruzberg, and A. W. W. Ludwig, Boundary multifractality at the integer quantum Hall plateau transition: implications for the critical theory, Phys. Rev. Lett. 101, 116802 (2008), arXiv:0804.2409 [cond-mat.mes-hall] .
  * Evers  _et al._ [2008] F. Evers, A. Mildenberger, and A. D. Mirlin, Multifractality at the Quantum Hall Transition: Beyond the Parabolic Paradigm, Phys. Rev. Lett. 101, 116803 (2008), arXiv:0804.2334 [cond-mat.mes-hall] .
  * Vasquez  _et al._ [2008] L. J. Vasquez, A. Rodriguez, and R. A. Römer, Multifractal analysis of the metal-insulator transition in the three-dimensional Anderson model. I. Symmetry relation under typical averaging, Phys. Rev. B 78, 195106 (2008), arXiv:0807.2217 [cond-mat.dis-nn] .
  * Rodriguez  _et al._ [2008] A. Rodriguez, L. J. Vasquez, and R. A. Römer, Multifractal analysis of the metal-insulator transition in the three-dimensional Anderson model. II. Symmetry relation under ensemble averaging, Phys. Rev. B 78, 195107 (2008), arXiv:0807.2209 [cond-mat.dis-nn] .
  * Rodriguez  _et al._ [2011] A. Rodriguez, L. J. Vasquez, K. Slevin, and R. A. Römer, Multifractal finite-size scaling and universality at the Anderson transition, Phys. Rev. B 84, 134209 (2011), arXiv:1107.5736 [cond-mat.dis-nn] .
  * Riva and Cardy [2005] V. Riva and J. Cardy, Scale and conformal invariance in field theory: a physical counterexample [rapid communication], Physics Letters B 622, 339 (2005), arXiv:hep-th/0504197 [hep-th] .
  * Nakayama [2015] Y. Nakayama, Scale invariance vs conformal invariance, Phys. Reports 569, 1 (2015).
  * Ludwig [1990] A. W. W. Ludwig, Infinite hierarchies of exponents in a diluted ferromagnet and their interpretation, Nuclear Physics B 330, 639 (1990).
  * Duplantier and Ludwig [1991] B. Duplantier and A. W. W. Ludwig, Multifractals, operator-product expansion, and field theory, Phys. Rev. Lett. 66, 247 (1991).
  * Bernard  _et al._ [2006] D. Bernard, G. Boffetta, A. Celani, and G. Falkovich, Conformal invariance in two-dimensional turbulence, Nature Physics 2, 124 (2006), arXiv:nlin/0602017 [nlin.CD] .
  * Bernard  _et al._ [2007] D. Bernard, G. Boffetta, A. Celani, and G. Falkovich, Inverse turbulent cascades and conformally invariant curves, Phys. Rev. Lett. 98, 024501 (2007), arXiv:nlin/0609069 [nlin.CD] .
  * Falkovich [2007] G. Falkovich, Conformal invariance in hydrodynamic turbulence, Russian Mathematical Surveys 62, 497 (2007).
  * Vafa [1988] C. Vafa, Toward Classification of Conformal Theories, Phys. Lett. B 206, 421 (1988).
  * Lewellen [1989] D. C. Lewellen, Constraints for Conformal Field Theories on the Plane: Reviving the Conformal Bootstrap, Nucl. Phys. B 320, 345 (1989).
  * Bondesan  _et al._ [2017] R. Bondesan, D. Wieczorek, and M. R. Zirnbauer, Gaussian free fields at the integer quantum Hall plateau transition, Nucl. Phys. B 918, 52 (2017), arXiv:1612.04109 [cond-mat.dis-nn] .
  * Zirnbauer [2019] M. R. Zirnbauer, The integer quantum Hall plateau transition is a current algebra after all, Nuclear Physics B 941, 458 (2019), arXiv:1805.12555 [math-ph] .
  * Puschmann  _et al._ [2021] M. Puschmann, D. Hernangómez-Pérez, B. Lang, S. Bera, and F. Evers, Quartic multifractality and finite-size corrections at the spin quantum Hall transition, Phys. Rev. B 103, 235167 (2021), arXiv:2104.12079 [cond-mat.dis-nn] .
  * Ujfalusi and Varga [2015] L. Ujfalusi and I. Varga, Finite-size scaling and multifractality at the Anderson transition for the three Wigner-Dyson symmetry classes in three dimensions, Phys. Rev. B 91, 184206 (2015), arXiv:1501.02147 [cond-mat.dis-nn] .
  * Lindinger and Rodríguez [2017] J. Lindinger and A. Rodríguez, Multifractal finite-size scaling at the Anderson transition in the unitary symmetry class, Phys. Rev. B 96, 134202 (2017), arXiv:1707.05701 [cond-mat.dis-nn] .
  * Tarquini  _et al._ [2017] E. Tarquini, G. Biroli, and M. Tarzia, Critical properties of the Anderson localization transition and the high-dimensional limit, Phys. Rev. B 95, 094204 (2017), arXiv:1612.04753 [cond-mat.dis-nn] .
  * Poland  _et al._ [2019] D. Poland, S. Rychkov, and A. Vichi, The conformal bootstrap: theory, numerical techniques, and applications, Rev. Mod. Phys. 91, 015002 (2019), arXiv:1805.04405 [hep-th] .
  * Li [2018] W. Li, Inverse Bootstrapping Conformal Field Theories, JHEP 01, 077, arXiv:1706.04054 [hep-th] .
  * Di Francesco  _et al._ [1997] P. Di Francesco, P. Mathieu, and D. Senechal,  _Conformal Field Theory_, Graduate Texts in Contemporary Physics (Springer-Verlag, New York, 1997).
  * Levy and Oz [2018] T. Levy and Y. Oz, Liouville Conformal Field Theories in Higher Dimensions, JHEP 06, 119, arXiv:1804.02283 [hep-th] .
  * Kislev  _et al._ [2022] A. C. Kislev, T. Levy, and Y. Oz, Odd dimensional nonlocal Liouville conformal field theories, Journal of High Energy Physics 2022, 150 (2022), arXiv:2206.10884 [hep-th] .
  * [46] J. Padayasi and I. A. Gruzberg, Supplementary material.
  * Dolan and Osborn [2001] F. A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599, 459 (2001), arXiv:hep-th/0011040 .
  * Rattazzi  _et al._ [2008] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12, 031, arXiv:0807.0004 [hep-th] .
  * Simmons-Duffin [2017] D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03, 086, arXiv:1612.08471 [hep-th] .
  * Fitzpatrick  _et al._ [2013] A. L. Fitzpatrick, J. Kaplan, D. Poland, and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12, 004, arXiv:1212.3616 [hep-th] .
  * Alday and Zhiboedov [2016] L. F. Alday and A. Zhiboedov, Conformal Bootstrap With Slightly Broken Higher Spin Symmetry, JHEP 06, 091, arXiv:1506.04659 [hep-th] .
  * Hogervorst and Rychkov [2013] M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87, 106004 (2013), arXiv:1303.1111 [hep-th] .
  * Padayasi [2023] J. Padayasi, Coulomb gas expansion notebooks (2023).

## Supplementary material

###  .1 Subleading behavior of global conformal blocks

Start with the Casimir equation solved by the spin-0 conformal blocks

$\mathcal{D}g_{\Delta,0}^{\Delta_{12},\Delta_{34}}(z,\bar{z})=C_{\Delta,0}g_{\Delta,0}^{\Delta_{12},\Delta_{34}}(z,\bar{z})$
(26)

where $\mathcal{D}$ is the Casimir operator of $SO(d+1,1)$:

$\mathcal{D}=\mathcal{D}_{z}+\mathcal{D}_{\bar{z}}+2(d-2)\frac{z\bar{z}}{z-\bar{z}}\left[(1-z)\partial_{z}-(1-\bar{z})\partial_{\bar{z}}\right],$
(27)
$\mathcal{D}_{z}=2z^{2}(1-z)\partial_{z}^{2}-(2+\Delta_{34}-\Delta_{12})z^{2}\partial_{z}+\frac{\Delta_{12}\Delta_{34}}{2}z$
(28)

and $C_{\Delta,0}\equiv\Delta(\Delta-d)$ is the eigenvalue of the Casimir
operator. Based on the leading form of $g_{\Delta,0}$ and the symmetricity
with respect to $z$ and $\bar{z}$, we use the following power series as an
ansatz for the solution:

$g_{\Delta,0}(z,\bar{z})=(z\bar{z})^{\alpha}\sum_{r,s=0}^{\infty}\frac{\kappa_{rs}}{1+\delta_{rs}}(z^{r}\bar{z}^{s}+z^{s}\bar{z}^{r}).$
(29)

For the subleading behavior of the block, we need to compute the coefficient
$\kappa_{10}$. Substituting the ansatz into the Casimir equation and
extracting the terms at leading order gives

$C_{\Delta,0}=\Delta(\Delta-d)=4\alpha(\alpha-1)-2(d-2)\alpha$ (30)

which is solved by $\alpha=\Delta/2$ as expected. $\kappa_{00}$ is a
normalization that we can set to unity. Now considering the subleading order
terms, we have

$\displaystyle
2\kappa_{10}\bigg{(}(\alpha+1)_{2}-(\alpha)_{2}-(d-2)\alpha\bigg{)}-2(\alpha)_{2}-\alpha(2+\Delta_{34}-\Delta_{12})+\frac{\Delta_{12}\Delta_{34}}{2}=\kappa_{10}\Delta(\Delta-d).$
(31)

$\kappa_{10}$ is read off from the equation as

$\kappa_{10}=\frac{(\Delta-\Delta_{12})(\Delta+\Delta_{34})}{4\Delta}$ (32)

with no dependence on dimension $d$.

###  .2 Global conformal block decomposition of Coulomb gas-like correlator

Consider a $G$-function which admits a series expansion at
$z,\bar{z}\rightarrow 0$ of the form

$G(z,\bar{z})=(z\bar{z})^{p}\sum_{i,j=0}^{\infty}\mathscr{C}_{ij}z^{i}\bar{z}^{j}$
(33)

with $\mathscr{C}_{ij}=\mathscr{C}_{ji}$. The Coulomb gas correlator
(Conformal invariance and multifractality at Anderson transitions in arbitrary
dimensions) is a particular example, with

$2p=\Delta_{\alpha_{1}+\alpha_{2}}\hskip 36.135pt\text{and}\hskip
36.135pt\mathscr{C}_{ij}=\binom{i+d\alpha_{2}\alpha_{3}-1}{i}\binom{j+d\alpha_{2}\alpha_{3}-1}{j}.$
(34)

Let us show that the given $G(z,\bar{z})$ can be expanded in terms of global
conformal blocks in any dimension $d$ with a single double trace family

$[2p+2n+l,l].$

In other words, we show that

$G(z,\bar{z})=\sum_{n,l\geq 0}\mu^{(n,l)}g_{2p+2n+l,l}(z,\bar{z})$ (35)

where $\mu^{(n,l)}$ is a product of two OPE coefficients. The proof will be
based upon the idea that the expansion Eq. 35 can be uniquely fixed order-by-
order in powers of $z$ and $\bar{z}$.

As noted in the main text, The leading order behavior of the global conformal
blocks in any dimensions is

$g_{\Delta,l}\sim\mathcal{N}_{d,l}(z\bar{z})^{\Delta/2}\text{Geg}_{l}^{d/2-1}\left(\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}\right)$
(36)

where we work with the normalization

$\mathcal{N}_{d,l}=\frac{l!}{\left(d/2-1\right)_{l}}.$ (37)

Focusing on the double-trace blocks $g_{2p+2n+l,l}$, we know further that the
subleading terms should be expandable as a power series

$g_{\Delta+2n+l,l}(z,\bar{z})=(z\bar{z})^{\Delta/2}\left(\mathcal{N}_{d,l}(z\bar{z})^{l/2+n}\text{Geg}_{l}^{d/2-1}\left(\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}\right)+\sum_{r+s>(n+l/2)}\frac{\kappa^{(n,l)}_{rs}}{1+\delta_{rs}}z^{r}\bar{z}^{s}\right)$
(38)

where $r,s$ are integers and $\kappa_{rs}=\kappa_{sr}$ [52].

The Gegenbauer polynomial for $l=0$ is just 1, so for the leading degree, the
block $g_{2p,0}\sim(z\bar{z})^{p}$ has the correct power-law to match
$G(z,\bar{z})=\mathscr{C}_{00}(z\bar{z})^{p}+\ldots$.

Next, consider degree 1 terms in the series,
$(z\bar{z})^{p}(\mathscr{C}_{10}(z+\bar{z})+\ldots)$. On the conformal blocks
side, the subleading contribution from the leading block at this degree can be
balanced by adding a new term $[2p+1,1]$ with the correct power-law behavior,

$g_{2p+1,1}\sim(z\bar{z})^{p+\frac{1}{2}}\left(\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}\right).$
(39)

The OPE coefficient is read off by equating the two series upto this order,

$\mu^{(1,1)}=\mathscr{C}_{10}-\mu^{(0,0)}\frac{\partial}{\partial
z}\left.\frac{g_{2p,0}(z,\bar{z})}{(z\bar{z})^{p}}\right|_{z=\bar{z}=0}=\mathscr{C}_{10}-\mathscr{C}_{00}\frac{(2p-\Delta_{12})(2p+\Delta_{34})}{8p}$
(40)

where we have used the subleading coefficient of the spin-zero block derived
in .1. Now, consider an arbitrary term in the $G$-function expansion
$\sim\mathscr{C}_{ij}z^{i}\bar{z}^{j}$. There is always a conformal block in
the double-trace family (in this case $g_{2p+i+j,i-j}$) with the correct
leading behavior to fix the series at this order. Looking closely at Eq. 36,
we have (with $j=n$ and $i-j=l$)

$\displaystyle g_{2p+2n+l,l}$
$\displaystyle\sim(z\bar{z})^{p+n+l/2}\frac{l!}{(d/2-1)_{l}}\text{Geg}_{l}^{d/2-1}\left(\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}\right)$
(41)
$\displaystyle=(z\bar{z})^{p+n+l/2}\frac{l!}{(d/2-1)_{l}}\sum_{k=0}^{\lfloor\frac{l}{2}\rfloor}\frac{\left(d/2-1\right)_{l-k}}{k!(l-2k)!}(-1)^{k}\left(\frac{z+\bar{z}}{\sqrt{z\bar{z}}}\right)^{l-2k}$
(42)
$\displaystyle=(z\bar{z})^{p}\sum_{k=0}^{\lfloor\frac{l}{2}\rfloor}\frac{(-1)^{k}(l-2k)_{2k}}{k!(d/2+l-k-1)_{k}}(z\bar{z})^{k+n}(z+\bar{z})^{l-2k}.$
(43)

Setting $k=0$ in Eq. (43) yields a term proportional to
$(z\bar{z})^{p}(z^{i}\bar{z}^{j})$ with coefficient 1 as desired. There are
additional terms even at the leading order in the conformal block, but all of
them are of the same degree in $z,\bar{z}$: $i+j$. Therefore, for each degree,
the terms in the series of the same degree must be fixed in descending order
in $l$, so that the $l=0$ or $l=1$ term is fixed last. For example, in the
quadratic degree, there are two terms in the G-function series,
$\mathscr{C}_{20}(z\bar{z})^{p}(z^{2}+\bar{z}^{2})$ and
$\mathscr{C}_{11}(z\bar{z})^{p}(z\bar{z})$. Of the two corresponding blocks
$g_{2p+2,2}$ and $g_{2p+2,0}$,
$g_{2p+2,2}\sim(z\bar{z})^{p}(z^{2}+\bar{z}^{2}+z\bar{z})$ contributes to the
$z\bar{z}$ term but the block $g_{2p+2,0}\sim(z\bar{z})^{p+1}$ does not feed
back into the former.

In summary, the recursive process can be codified into a formula for
calculating $\mu^{(n,l)}$ given $C_{ij}$ as

$\mu^{(j,i-j)}=\mathscr{C}_{ij}-\frac{1}{i!j!}\frac{\partial^{i}}{\partial
z^{i}}\frac{\partial^{i}}{\partial\bar{z}^{j}}\left(\frac{1}{(z\bar{z})^{p}}\left[\sum_{2\alpha<(i+j)}\sum_{\beta=0}^{i+j-1-2\alpha}\mu^{(\alpha,\beta)}g_{2p+2\alpha+\beta,\beta}+\sum_{\alpha=0}^{j-1}\mu^{(\alpha,i+j-2\alpha)}g_{(2p+i+j,i+j-2\alpha)}\right]\right)_{z=\bar{z}=0}.$
(44)

Using the form of double-trace blocks in Eq. 38, this becomes

$\mu^{(j,i-j)}=\mathscr{C}_{ij}-\left[\sum_{2\alpha<(i+j)}\sum_{\beta=0}^{i+j-1-2\alpha}\mu^{(\alpha,\beta)}\kappa^{(\alpha,\beta)}_{ij}+\sum_{\alpha=0}^{j-1}\mu^{(\alpha,i+j-2\alpha)}\mathscr{K}_{\alpha}\right].$
(45)

where the numbers $\mathscr{K}_{\alpha}$ are given by

$\mathscr{K}_{\alpha}=\sum_{k=j-\alpha}^{\lfloor\frac{i+j}{2}-\alpha\rfloor}\binom{i+j-2\alpha-2k}{i-\alpha-k}\frac{(-1)^{k}(i+j-2\alpha-2k)_{2k}}{k!(d/2+i+j-2\alpha-k-1)_{k}}.$
(46)

Of course, to calculate $\mu^{(j,i-j)}$ more explicitly one requires the
coefficients $\kappa_{ij}$ in the definition of the conformal blocks, which is
not available in generic dimensions. But we still have the result

> Any G-function of the form $G(z,\bar{z})=(z\bar{z})^{p}f(z,\bar{z})$ where
> $f(z,\bar{z})$ has a convergent Taylor series expansion at $z=\bar{z}=0$ can
> be expanded in conformal blocks from a single twist family $[2p+2n+l,l]$.

###  .3 Coulomb gas OPE coefficients

Following the procedure outlined in .2, we performed the global conformal
block expansion of the Coulomb Gas $G$-function in $d=2$ and $d=4$. In this
section we present some interesting observations and the first few
coefficients in the OPE of identical scalars.

Consider the Coulomb Gas $G$-function reproduced here for convenience,

$G(u,v)=u^{\frac{d}{2}(\alpha_{3}+\alpha_{4})}v^{-d\alpha_{2}\alpha_{3}}.$
(47)

One sees that it can be rewritten as a series expansion of the kind discussed
in .2,

$G(z,\bar{z})=(z\bar{z})^{\frac{\Delta_{\alpha_{1}+\alpha_{2}}}{2}}\left(\sum_{i,j=0}^{\infty}\binom{i+d\alpha_{2}\alpha_{3}-1}{i}\binom{j+d\alpha_{2}\alpha_{3}-1}{j}z^{i}\bar{z}^{j}\right)$
(48)

where we have used
$\Delta_{\alpha_{1}+\alpha_{2}}=d(\alpha_{1}+\alpha_{2})(\alpha_{3}+\alpha_{4})$
and charge neutrality. At leading order in any $d$, the coefficient
$\mathscr{C}_{00}=1$ in the series above. Thus the leading block in the
expansion is $[\Delta_{\alpha_{1}+\alpha_{2}},0]$ with coefficient
$\mu^{(0,0)}=1$. At the next degree, the only block to be considered is
$[\Delta_{\alpha_{1}+\alpha_{2}}+1,1]$, with the product OPE coefficient given
by Eq. 40, which turns out to be

$\mu^{(0,1)}=\binom{d\alpha_{2}\alpha_{3}}{1}-\frac{(\Delta_{\alpha_{1}+\alpha_{2}}-\Delta_{12})(\Delta_{\alpha_{1}+\alpha_{2}}+\Delta_{34})}{4\Delta_{\alpha_{1}+\alpha_{2}}}.$
(49)

As the discussion pans out in the key argument of the main text, we notice
upon simplification that generically, for all the Coulomb Gas theories,

$\mu^{(0,1)}=0.$ (50)

Let us stress that for $d=2$, it is expected that this OPE coefficient
vanishes in the global block expansion. It is a manifestation of the result
that in a Verma module, there are no quasiprimaries at level 1. But as we have
shown, this fact explicitly generalizes to higher dimensions for Coulomb Gas
theories.

Computing the expansion to higher degrees in $d=2$, we observe that the twist-
two $(n=1)$ family of operators $[\Delta_{\alpha_{1}+\alpha_{2}}+l+2,l]$ is
missing from the OPE (their OPE coefficients vanish for any $G$-function
independent of $\alpha_{i}$). Additionally, as we expect, in the OPE of
identical operators, the coefficients of odd-spin operators vanish. For
example,

$\mu^{(0,3)}=\frac{2\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}(\alpha_{1}-\alpha_{2})(\alpha_{3}-\alpha_{4})}{3(1+(\alpha_{1}+\alpha_{2})(\alpha_{3}+\alpha_{4}))(2+(\alpha_{1}+\alpha_{2})(\alpha_{3}+\alpha_{4}))}$
(51)

and we see that $\mu^{(0,3)}=0$ if $\alpha_{1}=\alpha_{2}$ or
$\alpha_{3}=\alpha_{4}$. The non-zero coefficients in the OPE of two identical
scalars are tabulated in Table 1. (For the case of non-identical scalars, see
[53]).

$(n,l)$ | $(\lambda^{(n,l)})^{2}$
---|---
$(0,0)$ | 1
$(0,2)$ | $\dfrac{2\alpha^{4}}{8\alpha^{2}+1}$
$(0,4)$ | $\dfrac{\alpha^{4}(2\alpha^{2}+1)^{2}}{2(64\alpha^{4}+64\alpha^{2}+15)}$
$(2,0)$ | $\dfrac{4\alpha^{8}}{(8\alpha^{2}+1)^{2}}$
$(0,6)$ | $\dfrac{\alpha^{4}(2\alpha^{4}+3\alpha^{2}+1)^{2}}{3(8\alpha^{2}+5)(8\alpha^{2}+7)(8\alpha^{2}+9)}$
$(2,2)$ | $\dfrac{\alpha^{8}(2\alpha^{2}+1)^{2}}{(8\alpha^{2}+1)(8\alpha^{2}+3)(8\alpha^{2}+5)}$
$(0,8)$ | $\dfrac{\alpha^{4}(4\alpha^{6}+12\alpha^{4}+11\alpha^{2}+3)^{2}}{24(8\alpha^{2}+7)(8\alpha^{2}+9)(8\alpha^{2}+11)(8\alpha^{2}+13)}$
$(2,4)$ | $\dfrac{2\alpha^{8}(2\alpha^{4}+3\alpha^{2}+1)^{2}}{3(8\alpha^{2}+1)(8\alpha^{2}+5)(8\alpha^{2}+7)(8\alpha^{2}+9)}$
$(4,0)$ | $\dfrac{\alpha^{8}(2\alpha^{2}+1)^{4}}{4(64\alpha^{4}+64\alpha^{2}+15)^{2}}$
Table 1: All non-zero OPE coefficients up to degree 8 $(n=8)$ in $d=2$ for the
coulomb gas correlator $\langle
V_{\alpha}V_{\alpha}V_{\alpha}V_{\alpha}\rangle$ in the $s$-channel with
$\alpha\equiv Q/4$ to satsify charge neutrality.

Similarly, for $d=4$, we obtain the global conformal block expansion of the
$G$-function. We notice that the odd spin OPE coefficients still vanish, the
twist-two family is no longer missing in the OPE. We also see that not all OPE
coefficient squares are positive definite, owing to the non-unitary nature of
the theory. The coefficients are tabulated in Table 2 .

$(n,l)$ | $(\lambda^{(n,l)})^{2}$
---|---
$(0,0)$ | 1
$(0,2)$ | $\dfrac{8\alpha^{4}}{16\alpha^{2}+1}$
$(1,0)$ | $\dfrac{8\alpha^{4}}{1-16\alpha^{2}}$
$(0,4)$ | $\dfrac{2\alpha^{4}(4\alpha^{2}+1)^{2}}{(16\alpha^{2}+3)(16\alpha^{2}+5)}$
$(2,2)$ | $-\dfrac{2\alpha^{4}(4\alpha^{2}+1)^{2}}{3+64(4\alpha^{4}+\alpha^{2})}$
$(2,0)$ | $\dfrac{64\alpha^{8}}{256\alpha^{4}-1}$
$(0,6)$ | $\dfrac{4\alpha^{4}(8\alpha^{4}+6\alpha^{2}+1)^{2}}{3(16\alpha^{2}+5)(16\alpha^{2}+7)(16\alpha^{2}+9)}$
$(1,4)$ | $-\dfrac{4\alpha^{4}(8\alpha^{4}+6\alpha^{2}+1)^{2}}{3(16\alpha^{2}+3)(16\alpha^{2}+5)(16\alpha^{2}+7)}$
$(2,2)$ | $\dfrac{16\alpha^{8}(4\alpha^{2}+1)^{2}}{(16\alpha^{2}-1)(16\alpha^{2}+3)(16\alpha^{2}+5)}$
$(3,0)$ | $-\dfrac{16\alpha^{8}(4\alpha^{2}+1)^{2}}{(16\alpha^{2}+1)^{2}(16\alpha^{2}+3)}$
Table 2: All non-zero OPE coefficients up to degree 6 in $d=4$ for the coulomb
gas correlator $\langle V_{\alpha}V_{\alpha}V_{\alpha}V_{\alpha}\rangle$ in
the $s$-channel with $\alpha\equiv Q/4$ to satsify charge neutrality.