# Elementary bounded generation for ${\rm SL}_{n}$ for global function fields
and $n\geq 3$

Alexander A. Trost  Fakultät für Mathematik, Ruhr Universität Bochum, D-44780
Bochum, Germany<EMAIL_ADDRESS>
###### Abstract.

This paper shows that the group ${\rm SL}_{n}(R)$ is boundedly elementary
generated for $n\geq 3$ and $R$ the ring of algebraic integers in a global
function field. Contrary to previous statements for number fields and earlier
statements for global function fields, the bounds proven in this preprint for
elementary bounded generation are independent of the underlying global
function field and only depend on the integer $n.$ Combining our main result
with earlier results, we further establish that elementary bounded generation
always has bounds independent from the global field in question, only
depending on $n.$

##  1\. Introduction

Classically, a group $G$ is called boundedly generated if there are finitely
many cyclic groups $Z_{1},\dots,Z_{L}$ such that $G=Z_{1}\cdots Z_{L}.$ This
property was introduced, among else, as a possible unified approach to proving
the Congruence Subgroup Property of Arithmetic Groups as stated in the (now
disproven) Rapinchuk-Conjecture [6, Question A] and has applications
concerning the representation theory of the group in question. There are
papers proving this property for various classical, arithmetic groups like
${\rm SL}_{n}(\mathbb{Z})$ in a paper by Carter and Keller [5] or for more
general arithmetic (twisted) Chevalley groups by Tavgen [15]. For ${\rm
SL}_{2}(R)$ for $R$ a ring of algebraic integers with infinitely many units
there are papers by Carter, Keller and Paige written up by Morris [10] and
another one by Rapinchuk, Morgan and Sury [9]. More recently, there was also a
result for certain isotropic orthogonal groups over number fields [7]. In case
of Chevalley groups like ${\rm SL}_{n}$ or ${\rm Sp}_{2n}$, bounded generation
results are usually proven by showing first that each element of the group can
be written as some bounded product of elementary matrices (or the
corresponding generalization in the group, so called root elements) and then
analyzing the subgroups given by root elements more closely. The first
property is commonly referred to as bounded generation by root elements or
bounded elementary generation. For linear groups in positive characteristic
classical bounded generation is only possible, if the group is virtually
abelian [1, Theorem 1]. That being said, the weaker property of bounded
generation by root elements still holds in positive characteristic for
Chevalley groups defined over global function fields:

######  Theorem 1.1.

[16, Theorem 1.3]Let $K$ be a global function field that is a finite extension
of $\mathbb{F}_{q}(T)$ with ring of integers ${\mathcal{O}}_{K}$ and field of
constants $\mathbb{F}_{q}.$ Further, let $\Phi$ be an irreducible root system
of rank at least $2.$ Then the simply-connected, split Chevalley group
$G(\Phi,{\mathcal{O}}_{K})$ is boundedly generated by root elements. More
precisely, there is a constant $L(q,k)\in\mathbb{N}$ only depending on $q$ and
$k:=[K:\mathbb{F}_{q}(T)]$ such that each element of
$G(\Phi,{\mathcal{O}}_{K})$ can be written as a product of $L(q,k)\cdot{\rm
rank}(\Phi)$ many root elements.

There are also some earlier results: The earliest known to me is the result
[12, Theorem 2] by Clifford and Queen. More recently, Nica has presented the
following result:

######  Theorem 1.2.

[11, Theorem 2] Let $\mathbb{F}$ be a finite field and $n\geq 3.$ Then ${\rm
SL}_{n}(\mathbb{F}[T])$ is boundedly generated by root elements.

However, most such results concerning bounded generation by root elements
depend on the global function field in question: Cliffords requires rather
artificial conditions on the class number of the global function field and the
degrees of certain primes and Nica’s result crucially uses the fact that
$\mathbb{F}[T]$ is a principal ideal domain. The main goal of this preprint is
to generalize such results to all rings of integers in global function fields
in a way that avoids the dependencies on the function field in question:

######  Theorem 1.3.

Let $K$ be a global function field with ring of integers ${\mathcal{O}}_{K}$
and $n\geq 3$. Then ${\rm SL}_{n}({\mathcal{O}}_{K})$ is boundedly generated
by elementary matrices. More precisely, each element of ${\rm
SL}_{n}({\mathcal{O}}_{K})$ can be written as a product of at most
$1/2(3n^{2}-n)+50$ elementary matrices.

######  Remark 1.4.

For the sake of completeness, we should mention that Carter and Keller claim
in [5, Remarks] that their (at the time unpublished) manuscript [4] also
contains a result for rings of integers in global function fields. The
publication [4] does not seem to be available though and the problem of
bounded elementary generation in this case is still considered to be open in
much later publications like [11, P. 404], [14, P. 523] and [8, 9. Final
Remarks].

Similar to Nica’s Theorem 1.2, Theorem 1.3 arises from an adaption of the
method of Carter-Keller in [5] to the global function field setup. The
resulting bounds in Theorem 1.3 do not depend on the global function field in
question however. The underlying reason for this difference between global
function fields and number fields seems to be that for a number field $K$ the
Carter-Keller argument required handling the ramified primes of the field
extension $K|\mathbb{Q}$ one at a time in order to ensure that various
different cyclotomic field extensions of $K$ appearing in the proof are in a
certain sense compatible with one another by controlling their ramification
behavior. In contrast, for global function fields, one is not concerned with
cyclotomic but rather constant field extensions of the function field in
question. These constant field extensions (if chosen correctly) are almost
automatically compatible in the required sense as they are always unramified.
The structure of the paper is straightforward: In Section 2, we introduce the
necessary background on global function fields, their constant field
extensions and class field theory and in Section 3, we prove Theorem 1.3. In
the last section, we sketch how these arguments can be used to answer
questions from [9] and [8] concerning elementary bounded generation and pose a
conjecture.

## Acknowledgments

The first part of the research for this preprint was done during a research
visit at the Mathematische Forschungsinstitut Oberwolfach and the second part
during a visit at the Otto-von-Guerricke University Magdeburg. I am very
grateful for the support I received at these places.

##  2\. Basic definitions and notions

###  2.1. Global function fields and their integers

In this subsection, we introduce global function fields, their rings of
integers and their primes. For a good introduction to the topic, we refer the
reader to Rosen’s book [13]. We begin with the definition of global function
fields:

######  Definition 2.1.

A global function field over $\mathbb{F}_{q}$ is a finite field extension $K$
of the rational function field $\mathbb{F}_{q}(T)$ for $\mathbb{F}_{q}$ a
finite field with $q$ elements. Then the ring of integers ${\mathcal{O}}_{K}$
of $K$ is defined as the integral closure of $\mathbb{F}_{q}[T]$ in $K.$

######  Remark 2.2.

  1. (1)

Equivalently, one can define global function fields as the field of functions
$\mathbb{F}_{q}(C)$ of a nonsingular, geometrically integral, affine curve $C$
defined over the finite field $\mathbb{F}_{q}.$

  2. (2)

The principal example of a global function over $\mathbb{F}_{q}$ is the field
of rational functions $\mathbb{F}_{q}(T).$ The associated curve is the affine
line $\mathbb{A}^{1}$ of the field $\mathbb{F}_{q}.$

In the rest of the paper, we will always replace the field $\mathbb{F}_{q}$ by
its algebraic closure $\mathbb{F}$ in $K$ and consequently set
$q:=|\mathbb{F}|$. We leave it as an exercise to check that $\mathbb{F}$ is a
finite field extension of $\mathbb{F}_{q}$. Assuming that $\mathbb{F}$ is
algebraically closed in $K$, we call $\mathbb{F}$ the field of constants of
$K$. Later, we will also need the concept of primes in global function fields,
not to be confused with the prime ideals of the corresponding ring of
integers:

######  Definition 2.3.

Let $K$ be a global function field with field of constants $\mathbb{F}$.

  1. (1)

Then a prime of $K$ is a discrete valuation domain $R_{{\mathcal{P}}}$ with
fraction field $K$ such that $R_{{\mathcal{P}}}$ contains $\mathbb{F}$ and has
maximal ideal ${\mathcal{P}}.$

  2. (2)

We define the function

${\rm ord}_{{\mathcal{P}}}:K^{*}\to\mathbb{Z}$

as follows: If $a\in R_{{\mathcal{P}}}-\\{0\\}$ is given, then ${\rm
ord}_{{\mathcal{P}}}(a)\in\mathbb{Z}$ is defined as the maximal
$n\in\mathbb{N}_{0}$ such that $a\in{\mathcal{P}}^{n}.$ If $a\in K^{*}$ is not
an element of $R,$ then ${\rm ord}_{{\mathcal{P}}}(a)$ is defined as $-{\rm
ord}_{{\mathcal{P}}}(a^{-1}).$

######  Remark 2.4.

Especially later on, we will often refer to ‘the prime ${\mathcal{P}}$ of $K$’
rather than ‘to the prime $R_{{\mathcal{P}}}$ of $K$.’

Furthermore, for a finite extension $K\mid L$ of global function fields and
$R_{{\mathcal{P}}}$ a prime of $K,$ the intersection
$S_{p}:=R_{{\mathcal{P}}}\cap L$ is a prime of the global function field $L.$
In this case, we refer to the prime ${\mathcal{P}}$ of $K$ as covering (or
lying over) the prime $p$ of $L.$ Let $x_{p}\in S_{p}$ be the generator of the
maximal ideal $p$ in $S_{p}.$ We then define $e({\mathcal{P}},p):={\rm
ord}_{{\mathcal{P}}}(x_{p})$. The integer $e({\mathcal{P}},p)$ does not depend
on the choice of $x_{p}$ and is called the ramification index of
${\mathcal{P}}$ over $p.$ Furthermore, the integer $f({\mathcal{P}}\mid
p):=[R_{{\mathcal{P}}}/{\mathcal{P}}:S_{p}/p]$ is called the relative degree
of ${\mathcal{P}}$ over $p.$

As mentioned already, the field $\mathbb{F}(T)$ is a global function field in
its own right. It has two distinct types of primes: First, for a monic,
irreducible polynomial $f(T)\in\mathbb{F}[T]$, consider the localization
$R_{f}:=\mathbb{F}[T]_{f(T)\mathbb{F}[T]}.$ This local ring is a prime of
$\mathbb{F}(T).$ Second, consider the ring $A^{\prime}:=\mathbb{F}[T^{-1}]$
and its localization $R_{\infty}:=A^{\prime}_{T^{-1}A^{\prime}}.$ The ring
$R_{\infty}$ is also a prime of $\mathbb{F}(T)$ and is commonly called the
prime ${\mathcal{P}}_{\infty}$ at infinity of $\mathbb{F}(T)$. We leave it as
an exercise to the reader to check that these two types make up all primes of
$\mathbb{F}(T)$.

Also note, that each maximal ideal $\mathfrak{P}$ of the ring of integers
${\mathcal{O}}_{K}$ of a global function field $K$ defines a prime of $K$,
namely the localization
$R_{\mathfrak{P}}:=({\mathcal{O}}_{K})_{\mathfrak{P}}.$ We leave it as an
exercise to check that the primes of $K$ arising in this manner are precisely
the ones not covering the prime ${\mathcal{P}}_{\infty}$ of $\mathbb{F}(T)$
and that the map

${\rm MaxSpec}({\mathcal{O}}_{K})\to\\{{\mathcal{P}}\text{ prime of
}K\mid{\mathcal{P}}\text{ does not cover
}{\mathcal{P}}_{\infty}\\},\mathfrak{P}\mapsto R_{\mathfrak{P}}$

is a bijection. By way of this bijection, we will usually identify maximal
ideals of ${\mathcal{O}}_{K}$ with their associated primes.

######  Remark 2.5.

Identifying maximal ideals in ${\mathcal{O}}_{K}$ and certain primes of $K$ is
less natural than it might seem: Contrary to number fields, where there is a
canonical choice for the infinite primes and so the identification between
maximal ideals of ${\mathcal{O}}_{K}$ and the finite primes is canonical, for
global function one has a choice in concerning which set $S$ of primes are
considered to be at infinity and a different choice of $S$ yields a ring of
integers ${\mathcal{O}}_{K}^{(S)}$ different from ${\mathcal{O}}_{K}$. Our
implicit choice of $S$ amounts to those primes in $K$ covering the prime
${\mathcal{P}}_{\infty}$ in $\mathbb{F}(T).$ This choice recovers
$\mathbb{F}[T]$ as the ring of integers in $\mathbb{F}(T)$ and
${\mathcal{O}}_{K}$ as the normal closure of $\mathbb{F}[T]$ in $K$ analogous
to the situation for number fields where $\mathbb{Z}$ is the ring of integers
in $\mathbb{Q}$ and ${\mathcal{O}}_{K}$ its normal closure. It would be
possible to state the results of the preprint for ${\mathcal{O}}_{K}^{(S)}$
for other sets $S$ but as long as $S\neq\emptyset$ holds this would not change
the statement of Theorem 1.3 nor the argument while introducing more notation,
so we decided against doing so.

###  2.2. Class field theory

In this subsection, we will give a brief introduction to results and
definitions from class field theory of global function fields needed later on.
The theory is very similar to the corresponding theory of algebraic number
fields, but there are technical differences related to the absence of infinite
places (e.g. Remark 2.5). Let $K$ be a global function field and
$\mathfrak{m}$ a non-zero ideal in ${\mathcal{O}}_{K}$. Then define

$J^{\mathfrak{m}}_{K}:=\\{\mathfrak{a}\mid\mathfrak{a}\text{ non-zero
fractional ideal in }{\mathcal{O}}_{K}\text{ coprime to }\mathfrak{m}\\}.$

Recall that a fractional ideal in ${\mathcal{O}}_{K}$ is an
${\mathcal{O}}_{K}$-submodule $\mathfrak{a}$ of $K$ such that there is a
$\lambda\in{\mathcal{O}}_{K}$ with
$\lambda\mathfrak{a}\subset{\mathcal{O}}_{K}.$ Also a fractional ideal
$\mathfrak{a}$ is called coprime to $\mathfrak{m}$, if its prime factorization
shares no prime divisor with $\mathfrak{m}$ including negative powers
${\mathcal{Q}}^{-k}$ appearing as prime divisors of $\mathfrak{m}$. We
consider $J^{\mathfrak{m}}_{K}$ as a group with multiplication given by
products of fractional ideals. Further, set

$P^{\mathfrak{m}}_{K}:=\\{a{\mathcal{O}}_{K}\mid a\in
K^{*},a{\mathcal{O}}_{K}\in J^{\mathfrak{m}}_{K}\\}.$

Then the quotient ${\rm
Cl}^{\mathfrak{m}}_{K}:=J^{\mathfrak{m}}_{K}/P^{\mathfrak{m}}_{K}$ is called
the ray class group modulo $\mathfrak{m}.$ Next, consider an Galois extension
of global function fields $L|K$ and let ${\mathcal{P}}$ an unramified prime in
$K.$ Then let $\mathfrak{P}$ be a prime in $L$ lying over ${\mathcal{P}}$ and
let $S_{\mathfrak{P}}\subset L$ be the discrete valuation ring associated with
the prime $\mathfrak{P}$. Then there is an element $(\mathfrak{P},L|K)\in{\rm
Gal}(L|K)$ defined uniquely by the property

$\forall a\in S_{\mathfrak{P}}:(\mathfrak{P},L|K)(a)\equiv
a^{N({\mathcal{P}})-1}\text{ mod }\mathfrak{P}.$

For two primes $\mathfrak{P}_{1},\mathfrak{P_{2}}$ lying over ${\mathcal{P}}$,
the two elements $(\mathfrak{P}_{1},L|K),(\mathfrak{P}_{2},L|K)$ of ${\rm
Gal}(L|K)$ are conjugate to each other in ${\rm Gal}(L|K).$ Hence for an
abelian extension $L|K$, $(\mathfrak{P},L|K)$ does not depend on the choice of
$\mathfrak{P}$ and so we set $({\mathcal{P}},L|K):=(\mathfrak{P},L|K).$ For a
fractional ideal $\mathfrak{a}\subset K$ factorizing as
$\mathfrak{a}=\prod_{{\mathcal{P}}}{\mathcal{P}}^{n_{{\mathcal{P}}}}$ with
primes ${\mathcal{P}}$ unramified in an abelian extension $L|K$, one defines

$(\mathfrak{a},L|K):=\prod_{{\mathcal{P}}}({\mathcal{P}},L|K)^{n_{{\mathcal{P}}}}\in{\rm
Gal}(L|K)$

Then we have the following famous theorem due to Artin:

######  Theorem 2.6.

[2, Proposition 1.3,Theorem 1.4] Let $K$ be a global function field and
$\mathfrak{m}$ a non-zero ideal in ${\mathcal{O}}_{K}$. Then there is an
abelian field extension $K_{\mathfrak{m}}|K$ which is unramified outside of
prime divisors of $\mathfrak{m}$ such that ${\rm Cl}^{\mathfrak{m}}_{K}$ is
isomorphic to ${\rm Gal}(K_{\mathfrak{m}}|K)$ with an isomorphism given by
$(\cdot,K_{\mathfrak{m}}|K):{\rm Cl}^{\mathfrak{m}}_{K}\to{\rm
Gal}(K_{\mathfrak{m}}|K),\mathfrak{a}\mapsto(\mathfrak{a},K_{\mathfrak{m}}|K)$.

######  Remark 2.7.

Fixing an algebraic closure $\overline{K}$ of $K$, the field extension
$K_{\mathfrak{m}}|K$ is also unique, when considered in $\overline{K}.$

###  2.3. Finite fields and constant field extensions

In the proofs to follow we will need various statements concerning finite
fields and so-called constant field extensions of global function fields. In
particular, we need the following two lemmas describing the composita and
intersections of finite fields and constant field extensions. While these
lemmas are standard, we will recall their proofs for the convenience of the
reader.

######  Lemma 2.8.

Let $\mathbb{E},\mathbb{E}_{1},\mathbb{E}_{2},\mathbb{E}_{3},\mathbb{F}$ be
finite fields such that $\mathbb{F}\subset\mathbb{E}_{i}\subset\mathbb{E}$
holds for $i=1,2,3.$ Further, set $a_{i}:=[\mathbb{E}_{i}:\mathbb{F}]$ for
$i=1,2.$ Then $\mathbb{E}_{1}\cdot\mathbb{E}_{2}$ and
$\mathbb{E}_{1}\cap\mathbb{E}_{2}$ are the unique subextensions $L,K$ of
$\mathbb{E}|\mathbb{F}$ with $[L:\mathbb{F}]={\rm lcm}(a_{1},a_{2})$ and
$[K:\mathbb{F}]={\rm gcd}(a_{1},a_{2})$ respectively. In particular, if
$\mathbb{E}_{3}\cap\mathbb{E}_{i}=\mathbb{F}$ holds for $i=1,2$, then
$\mathbb{E}_{3}\cap(\mathbb{E}_{1}\cdot\mathbb{E}_{2})=\mathbb{F}$ holds as
well.

###### Proof.

The first step is to note that for an extension $\mathbb{E}|\mathbb{F}$ of
finite fields with $[\mathbb{E}:\mathbb{F}]=a$ that for each divisor $b$ of
$a$ there is a unique field $\mathbb{F}_{b}$ with
$\mathbb{F}\subset\mathbb{F}_{b}\subset\mathbb{E}$ and
$[\mathbb{F}_{b}:\mathbb{F}]=b$: Namely, set $q:=\mathbb{F}.$ Then
$\mathbb{E}$ is the splitting field of the polynomial $f_{a}(Z):=Z^{q^{a}}-Z$
defined over $\mathbb{F}.$ But $f_{b}(Z):=Z^{q^{b}}-Z$ is a divisor of
$f_{a}(Z)$ and hence the splitting field $\mathbb{F}_{b}$ of $f_{b}$ must be
contained in $\mathbb{E}.$ Further, the sum and products of roots of
$f_{b}(Z)$ are also roots of $f_{b}(Z).$ Hence $\mathbb{F}_{b}$ consists
precisely of the $q^{b}$ many distinct roots of $f_{b}(Z).$ Thus
$[\mathbb{F}_{b}:\mathbb{F}]=b$ must hold. On the other hand, if there is a
different field $\mathbb{F}_{b}^{\prime}$ with
$[\mathbb{F}_{b}^{\prime}:\mathbb{F}]=b,$ then $x^{q^{b}}-x=0$ must hold for
all $x\in\mathbb{F}_{b}^{\prime}$ and so $\mathbb{F}_{b}^{\prime}$ must also
be contained in $\mathbb{F}_{b}.$ We next note that
$[\mathbb{E}_{1}\cdot\mathbb{E}_{2}:\mathbb{F}]$ is a multiple of both
$[\mathbb{E}_{1}:\mathbb{F}]=a_{1}$ and $[\mathbb{E}_{2}:\mathbb{F}]=a_{2}$
and so of ${\rm lcm}(a_{1},a_{2}).$ Thus $\mathbb{E}_{1}\cdot\mathbb{E}_{2}$
contains $\mathbb{F}_{{\rm lcm}(a_{1},a_{2})}.$ But
$\mathbb{E}_{1}=\mathbb{F}_{a_{1}}$ and $\mathbb{E}_{2}=\mathbb{F}_{a_{2}}$
must also be contained in $\mathbb{F}_{{\rm lcm}(a_{1},a_{2})}$. Hence
$\mathbb{E}_{1}\cdot\mathbb{E}_{2}$ is also contained in $\mathbb{F}_{{\rm
lcm}(a_{1},a_{2})}.$ Thus $\mathbb{F}_{{\rm
lcm}(a_{1},a_{2})}=\mathbb{E}_{1}\cdot\mathbb{E}_{2}$ and so
$[\mathbb{E}_{1}\cdot\mathbb{E}_{2}:\mathbb{F}]={\rm lcm}(a_{1},a_{2})$ holds.
On the other hand, note that there is a unique field
$\mathbb{F}_{{\gcd}(a_{1},a_{2})}$ contained in $\mathbb{E}$ with
$[\mathbb{F}_{{\gcd}(a_{1},a_{2})}:\mathbb{F}]={\gcd}(a_{1},a_{2}).$ But
$\mathbb{F}_{{\gcd}(a_{1},a_{2})}$ must also be contained in both
$\mathbb{E}_{1}$ and $\mathbb{E}_{2}$. Thus
$\mathbb{F}_{{\gcd}(a_{1},a_{2})}\subset\mathbb{E}_{1}\cap\mathbb{E}_{2}.$ But
$[\mathbb{E}_{1}\cap\mathbb{E}_{2}:\mathbb{F}]$ must divide both $a_{1}$ and
$a_{2}$ and hence ${\gcd}(a_{1},a_{2})$. This implies
$\mathbb{F}_{{\gcd}(a_{1},a_{2})}=\mathbb{E}_{1}\cap\mathbb{E}_{2}.$ For the
last claim, note that $\mathbb{E}_{3}\cap\mathbb{E}_{i}=\mathbb{F}$ for
$i=1,2$ implies by the second claim that $a_{3}$ must be coprime to $a_{1}$
and $a_{2}.$ But then

$[\mathbb{E}_{3}\cap(\mathbb{E}_{1}\cdot\mathbb{E}_{2}):\mathbb{F}]={\rm
gcd}(a_{3},[\mathbb{E}_{1}\cdot\mathbb{E}_{2}:\mathbb{F}])={\rm
gcd}(a_{3},{\rm lcm}(a_{2},a_{3}))$

must be $1$ and so the last claim follows. ∎

######  Lemma 2.9.

Let $K$ be a global function field with $\mathbb{F}$ its field of constants.
Further, let $\mathbb{E}_{1},\mathbb{E}_{2}$ be two finite field extension of
$\mathbb{F}$ contained in some algebraic closure of $K$. Define $K_{1}$ as the
composita of $K\cdot\mathbb{E}_{1}$ and define $K_{2}$ similarly. Then
$K_{1}\cdot K_{2}=K\cdot(\mathbb{E}_{1}\cdot\mathbb{E}_{2})$ and $K_{1}\cap
K_{2}=K\cdot(\mathbb{E}_{1}\cap\mathbb{E}_{2})$ holds.

###### Proof.

The first statement is clear. For the second statement set
$a_{i}:=[\mathbb{E}_{i}:\mathbb{F}]$ for $i=1,2$. According to [13,
Proposition 8.1], we have $[K_{i}:K]=a_{i}$ for $i=1,2$ and consequently the
second statement follows from Lemma 2.8. ∎

##  3\. Bounded generation by root elements in positive characteristic

We introduce the following notations: For a rational prime $p$ and a natural
number $n,$ we set $v_{p}(n)$ as the maximal $i\in\mathbb{N}_{0}$ such that
$p^{i}$ divides $n.$ Further, for an element $b\in{\mathcal{O}}_{K}-\\{0\\},$
we define $\epsilon(b)$ as the exponent of the finite group
$({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})^{*}.$ The crucial step to derive
Theorem 1.3 are to prove global function field versions of the two technical
lemmas [5, Lemma 3,4] of Carter and Keller. We begin with [5, Lemma 3]:

######  Lemma 3.1.

Let $K$ be a global function field and with $\mathbb{F}$ its field of
constants and set $m:=|\mathbb{F}|-1.$ Then up to multiplication by three
elementary matrices in ${\rm SL}_{2}({\mathcal{O}}_{K})$ the first row of each
$A\in{\rm SL}_{2}({\mathcal{O}}_{K})$ can be assumed to have the form
$(a^{m},b)$ with $b{\mathcal{O}}_{K}$ a prime ideal.

###### Proof.

Let

$A=\begin{pmatrix}a_{1}&b_{1}\\\ c_{1}&d_{1}\end{pmatrix}$

be given. If $a_{1}=0,$ then $c_{1}$ is a unit in $R$ and hence picking any
principal, prime ideal $b{\mathcal{O}}_{K}$, the matrix
$E_{12}(c_{1}^{-1})AE_{12}(-b_{1}+b)$ has the desired form. The case $b_{1}=0$
is resolved in a similar manner. So assume wlog. that $a_{1},b_{1}\neq 0.$ Let
$S=\\{R_{\infty}^{(1)},\dots,R_{\infty}^{(L)}\\}$ be the set of primes
covering the prime ${\mathcal{P}}_{\infty}$ of $\mathbb{F}(T)$. Then according
to [3, A16], for the function

$\left(\frac{\cdot,\cdot}{R_{\infty}^{(1)}}\right)_{m}:K_{R_{\infty}^{(1)}}^{*}\times
K_{R_{\infty}^{(1)}}^{*}\to\mathbb{F}^{*}$

there are $u,w\in K_{R_{\infty}^{(1)}}^{*}$ such that

$\left(\frac{u,w}{R_{\infty}^{(1)}}\right)_{m}$

generates $\mathbb{F}^{*}.$ Using Dirichlet’s Theorem, we can choose a
principal prime ideal $a_{2}{\mathcal{O}}_{K}$ satisfying the following three
properties:

$\displaystyle a_{2}\equiv a_{1}\text{ mod }b_{1}{\mathcal{O}}_{K},\forall
i=2,\dots,L:a_{2}\equiv 1\text{ mod }R_{\infty}^{(i)}\text{ and }a_{2}\equiv
w\text{ mod }(R_{\infty}^{(1)})^{N}$

with $N$ chosen so large that $a_{2}/w$ has an m.th root in
$K_{R_{\infty}^{(1)}}.$ Then note

$\displaystyle\left(\frac{u,a_{2}}{R_{\infty}^{(1)}}\right)_{m}=\left(\frac{u,a_{2}/w}{R_{\infty}^{(1)}}\right)_{m}\cdot\left(\frac{u,w}{R_{\infty}^{(1)}}\right)_{m}=\left(\frac{u,w}{R_{\infty}^{(1)}}\right)_{m}.$

Hence there is a positive integer $k$ such that

$\displaystyle\left(\frac{b_{1},a_{2}}{R_{\infty}^{(1)}}\right)_{m}\cdot\left(\frac{u,a_{2}}{R_{\infty}^{(1)}}\right)_{m}^{k}=1.$

Set $v:=u^{k}.$ Then using Dirichlet’s Theorem again, we may choose a
principal prime ideal $b_{2}{\mathcal{O}}_{K}$ satisfying the following three
properties

$\displaystyle b_{2}\equiv b_{1}\text{ mod }a_{2}{\mathcal{O}}_{K},\forall
i=2,\dots,L:b_{2}\equiv 1\text{ mod }R_{\infty}^{(i)}\text{ and }b_{2}\equiv
v\text{ mod }(R_{\infty}^{(1)})^{N}$

Again, the integer $N$ is chosen so large that $b_{2}/v$ is an m.th root of
unity in $K_{R_{\infty}^{(1)}}.$ Then according to the m.th power reciprocity
law, one has

$1=\prod_{{\mathcal{P}}}\left(\frac{b_{2},a_{2}}{{\mathcal{P}}}\right)_{m}$

with the product taken over all primes in $K.$ But note that due to the choice
of $a_{2}{\mathcal{O}}_{K},b_{2}{\mathcal{O}}_{K}$ as prime ideals and the
fact that $a_{2}$ and $b_{2}$ are both chosen to be congruent to $1$ modulo
the primes $R_{\infty}^{(i)}$ for $i\geq 2,$ this product reduces to

$1=\left(\frac{b_{2},a_{2}}{b_{2}{\mathcal{O}}_{K}}\right)_{m}\cdot\left(\frac{b_{2},a_{2}}{a_{2}{\mathcal{O}}_{K}}\right)_{m}\cdot\left(\frac{b_{2},a_{2}}{R_{\infty}^{(1)}}\right)_{m}$

according to [3, A16]. But as $b_{2}/v$ is an m.th root in
$K_{R_{\infty}^{(1)}},$ this yields

$\left(\frac{b_{2},a_{2}}{a_{2}{\mathcal{O}}_{K}}\right)_{m}\cdot\left(\frac{b_{2},a_{2}}{R_{\infty}^{(1)}}\right)_{m}=\left(\frac{b_{2},a_{2}}{a_{2}{\mathcal{O}}_{K}}\right)_{m}\cdot\left(\frac{v,a_{2}}{R_{\infty}^{(1)}}\right)_{m}=\left(\frac{b_{2},a_{2}}{a_{2}{\mathcal{O}}_{K}}\right)_{m}\cdot\left(\frac{u^{k},a_{2}}{R_{\infty}^{(1)}}\right)_{m}=\left(\frac{b_{2},a_{2}}{a_{2}{\mathcal{O}}_{K}}\right)_{m}\cdot\left(\frac{u,a_{2}}{R_{\infty}^{(1)}}\right)_{m}^{k}=1$

This implies

$\left(\frac{a_{2}}{b_{2}{\mathcal{O}}_{K}}\right)_{m}^{-1}=\left(\frac{a_{2}}{b_{2}{\mathcal{O}}_{K}}\right)_{m}^{-{\rm
ord}_{b_{2}{\mathcal{O}}_{K}}(b_{2}{\mathcal{O}}_{K})}=\left(\frac{b_{2},a_{2}}{b_{2}{\mathcal{O}}_{K}}\right)_{m}=1$

Hence $a_{2}$ is an m.th power modulo $b_{2}{\mathcal{O}}_{K}$ and so there is
an $a\in{\mathcal{O}}_{K}$ such that $a^{m}\equiv a_{2}\text{ mod
}b_{2}{\mathcal{O}}_{K}.$ To finish note that as $a_{2}-a_{1}\in
b_{1}{\mathcal{O}}_{K},b_{2}-b_{1}\in a_{2}{\mathcal{O}}_{K}$ and
$a^{m}-a_{2}\in b_{2}{\mathcal{O}}_{K}$, there are $x,y,z\in{\mathcal{O}}_{K}$
such that $a_{2}=a_{1}+xb_{1},b_{2}=b_{1}+ya_{2}$ and $a^{m}=a_{2}+zb_{2}.$
Hence the matrix $AE_{21}(x)E_{12}(y)E_{21}(z)$ has the desired form. ∎

######  Remark 3.2.

Note, that the bound in the global function field case is three rather than
four as in the number field case [5, Lemma 3]. The reason for this is that one
of the "preprocessing" steps from the number field case served to make $b_{1}$
prime as integers to the number $m$ of roots of unity in $K$. However, this
step has no analog for global function fields.

Next, we will derive the global function field version of [5, Lemma 4]:

######  Lemma 3.3.

Let $K$ be a global function field with $\mathbb{F}$ its field of constants.
Let $b\in{\mathcal{O}}_{K}-\\{0\\}$ and a principal ideal
$\mathfrak{a}\subset{\mathcal{O}}_{K}$ be given such that $b{\mathcal{O}}_{K}$
is a prime ideal and $\mathfrak{a}+b{\mathcal{O}}_{K}={\mathcal{O}}_{K}$. Then
for every $u\in{\mathcal{O}}_{K}^{*},$ there exists an element
$c\in{\mathcal{O}}_{K}$ such that $bc\equiv u\text{ mod }\mathfrak{a}$ and
such that ${\rm gcd}(\epsilon(b),\epsilon(c))=|\mathbb{F}|-1=m.$

###### Proof.

First, we consider the finite field
$\mathbb{E}:={\mathcal{O}}_{K}/b{\mathcal{O}}_{K}.$ For each rational prime
$p\in\mathbb{N}$ choose $e_{p}\in\mathbb{N}_{0}$ maximal such that $p^{e_{p}}$
divides $m.$ Note that $({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})^{*}$ contains
$\mathbb{F}^{*}$ as a subgroup and so $p^{e_{p}}$ divides
$|({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})^{*}|=\epsilon(b).$ Thus
$v_{p}(\epsilon(b))\geq e_{p}$ holds for all rational primes $p$. Set
$S:=\\{p_{1},\dots,p_{l}\\}$ as the finite set of primes $p$ such that
$v_{p}(\epsilon(b))>e_{p}.$ If $S=\emptyset$, then $\epsilon(b)=m$ and so
picking any element $c\in{\mathcal{O}}_{K}$ with $bc\equiv u\text{ mod
}\mathfrak{a}$ we are done. So we assume in the following that
$S\neq\emptyset.$ For each $p\in S$, pick a primitive $p^{e_{p}+1}$.th root of
unity $\xi_{p}$ in $\mathbb{E}$ and consider the finite fields
$\mathbb{E}_{i}:=\mathbb{F}[\xi_{p_{i}}]$ for $i=1,\dots,l$. We will next
inductively describe a process to find subfields
$\mathbb{L}_{1}^{(t)},\dots,\mathbb{L}_{t}^{(t)}$ of
$\mathbb{E}_{1},\dots,\mathbb{E}_{t}$ for all $t\leq l$ such that
$\mathbb{L}_{i}^{(t)}\cap\mathbb{L}_{j}^{(t)}=\mathbb{F}$ or
$\mathbb{L}_{i}^{(t)}=\mathbb{L}_{j}^{(t)}\neq\mathbb{F}$ holds for all $1\leq
i,j\leq t.$ First, if $t=1$, there is nothing to do. So assume $t>1$ and
assume further that the subfields
$\mathbb{L}_{1}^{(t-1)},\dots,\mathbb{L}_{t-1}^{(t-1)}$ are already chosen as
required. Then we check whether there is an $i$ between $1$ and $t-1$ such
that $\mathbb{L}_{i}^{(t-1)}\cap\mathbb{E}_{t}\neq\mathbb{F}.$ If there is
such an $i$, then we set
$\mathbb{L}_{i}^{(t)}:=\mathbb{L}_{t}^{(t)}:=\mathbb{L}_{i}^{(t-1)}\cap\mathbb{E}_{t}$
for all such $i\leq t$ and set $\mathbb{L}_{h}^{(t)}:=\mathbb{L}_{h}^{(t-1)}$
for all other $h\leq t-1$. However, if
$\mathbb{E}_{t}\cap\mathbb{L}_{i}^{(t-1)}=\mathbb{F}$ holds for all $i\leq t$,
then we set $\mathbb{L}_{i}^{(t)}:=\mathbb{L}_{i}^{(t-1)}$ for $i\leq t-1$ and
$\mathbb{L}_{t}^{(t)}:=\mathbb{E}_{t}.$ Clearly, these choices of
$\mathbb{L}_{i}^{(t)}$ have the required properties. Next, consider the
constant field extensions $K_{1}:=\mathbb{L}_{1}^{(l)}\cdot
K,\dots,K_{l}:=\mathbb{L}_{l}^{(l)}\cdot K$ of $K$. For simplicity, we will
also assume in the following that all the fields $K_{1},\dots,K_{l}$ are
actually distinct. Also consider the compositum $L:=\prod_{i=1}^{l}K_{l}.$
Next, consider the field $B_{l}:=K_{l}\cap\prod_{i=1}^{l-1}K_{i}.$ Note that
according to Lemma 2.9, we know that

$\displaystyle
B_{l}=K_{l}\cap\prod_{i=1}^{l-1}K_{i}=(K\cdot\mathbb{L}_{l}^{(l)})\cap(\prod_{i=1}^{l-1}K\cdot\mathbb{L}_{i}^{(l)})=K\cdot(\mathbb{L}_{l}^{(l)}\cap\prod_{i=1}^{l-1}\mathbb{L}_{i}^{(l)}).$

However, by the construction of the $\mathbb{L}_{i}^{(l)}$, we know that
$\mathbb{L}_{l}^{(l)}\cap\mathbb{L}_{i}^{(l)}=\mathbb{F}$ holds for all
$i=1,\dots,l-1$ and hence
$(\mathbb{L}_{l}^{(l)}\cap\prod_{i=1}^{l-1}\mathbb{L}_{i}^{(l)})=\mathbb{F}$
according to the last statement in Lemma 2.8. Thus $B_{l}=K.$ Hence we have an
isomorphism of Galois groups:

${\rm Gal}(L\mid K)\to{\rm Gal}(K_{l}|K)\times{\rm
Gal}(\prod_{i=1}^{l-1}K_{i}|K),\sigma\mapsto(\sigma|_{K_{l}},\sigma|_{\prod_{i=1}^{l-1}K_{i}}).$

Iterating this argument, we obtain that ${\rm Gal}(L|K)=\prod_{i=1}^{l}{\rm
Gal}(K_{i}|K).$ Also note that $L$ is a constant field extension of $K$ and
hence an unramified field extension according to [13, Proposition 8.5]. Next,
consider the ray class field $K_{\mathfrak{a}}$ associated to the ideal
$\mathfrak{a}.$ By assumption $b{\mathcal{O}}_{K}$ and $\mathfrak{a}$ are
coprime and so $b{\mathcal{O}}_{K}$ is unramified in $K_{\mathfrak{a}}|K.$
Hence $b{\mathcal{O}}_{K}$ is also unramified in the extension
$L_{\mathfrak{a}}|K$ for $L_{\mathfrak{a}}:=K_{\mathfrak{a}}\cdot L.$ Thus we
may consider the element
$\sigma:=(b{\mathcal{O}}_{K},L_{\mathfrak{a}}|K)\in{\rm
Gal}(L_{\mathfrak{a}}|K).$ Next, we define the constant field extensions
$L_{p}:=K\cdot\mathbb{F}[\xi_{p}]$ of $K$ for $p\in S$ and consider the
restrictions $\sigma_{p}:=\sigma|_{L_{p}}=(b{\mathcal{O}}_{K},L_{p}|K)$ for
$p\in S.$

######  Claim 1.

One has $\sigma_{p}={\rm id}_{L_{p}}$ for $p\in S.$

Let $Q$ be a prime in ${\mathcal{O}}_{L_{p}}$ covering $b{\mathcal{O}}_{K}.$
Then
${\mathcal{O}}_{L_{p}}/Q=({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})[\overline{\xi}_{p}]$
holds for $\overline{\xi}_{p}$ the image of $\xi_{p}$ in
${\mathcal{O}}_{L_{p}}/Q$ according to [13, Proposition 8.10]. But note that
$v_{p}(\epsilon(b))>e_{p}$ and so $p^{e_{p}+1}$ divides
$\epsilon(b)=|({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})^{*}|.$ This implies in
particular that
$\overline{\xi}_{p}^{|({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})^{*}|}=1$ holds
and thus $\overline{\xi}_{p}$ is a root of the polynomial
$F(z):=z^{|({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})^{*}|}-1\in({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})[z].$
But the roots of this polynomial are precisely the elements of
$({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})^{*}$ and so $\overline{\xi}_{p}$ is an
element of $({\mathcal{O}}_{K}/b{\mathcal{O}}_{K})^{*}.$ Thus
${\mathcal{O}}_{L_{p}}/Q={\mathcal{O}}_{K}/b{\mathcal{O}}_{K}$. Hence the
relative degree $f(Q|b{\mathcal{O}}_{K})$ is equal to $1.$ But then [13,
Proposition 9.10] implies $\sigma_{p}=(b{\mathcal{O}}_{K},L_{p}|K)=1.$

Next, note that for $i=1,\dots,l,$ each group ${\rm Gal}(K_{i}|K)={\rm
Gal}(\mathbb{L}_{i}^{(l)}|\mathbb{F})$ is non-trivial by construction and so
as ${\rm Gal}(L|K)=\prod_{i=1}^{l}{\rm Gal}(K_{i}|K)$ is a quotient of ${\rm
Gal}(L_{\mathfrak{a}}|K)$, we may choose an element $\sigma_{1}\in{\rm
Gal}(L_{\mathfrak{a}}|K)$ with $\sigma_{1}|_{K_{i}}\neq{\rm id}_{K_{i}}$ for
all $i=1,\dots,l.$ Set further $\sigma_{2}:=\sigma\sigma_{1}^{-1}.$ Then
according to Tchebotarevs density theorem [13, Theorem 9.13A], we may choose
two primes ${\mathcal{P}}_{1},{\mathcal{P}}_{2}$ unramified in
$L_{\mathfrak{a}}|K$ such that
$({\mathcal{P}}_{j},L_{\mathfrak{a}}|K)=\sigma_{j}^{-1}$ holds for $j=1,2.$
Note that as the $L_{p}$ for $p\in S$ are constant field extensions of $K,$
the primes ${\mathcal{P}}_{j}$ are automatically unramified in $L_{p}|K$ for
all $p\in S.$ So we can also consider $({\mathcal{P}}_{j},L_{p}|K)\in{\rm
Gal}(L_{p}|K)$ for all $p\in S.$ But note that by the construction of the
$K_{i}$, there is an $i=1,\dots,l$ such that $K_{i}\subset L_{p}$. But
$({\mathcal{P}}_{j},L_{p}|K)$ maps to
$({\mathcal{P}}_{j},K_{i}|K)=\sigma_{j}|_{K_{i}}\neq{\rm id}_{K_{i}}$ under
the restriction homomorphism ${\rm Gal}(L_{p}|K)\to{\rm Gal}(K_{i}|K).$ Hence
$({\mathcal{P}}_{j},L_{p}|K)=\sigma_{j}|_{L_{p}}^{-1}\neq{\rm id}_{L_{p}}$
holds for $j=1,2$ and $p\in S.$ Thus [13, Proposition 9.10] implies that
neither ${\mathcal{P}}_{1}$ nor ${\mathcal{P}}_{2}$ split completely in the
extension $L_{p}|K$ for any $p\in S.$ But similar to the proof of the claim
above this implies that $v_{p}(|{\mathcal{O}}_{K}/{\mathcal{P}}_{j}|-1)=e_{p}$
for $j=1,2$ and $p\in S$, because
$v_{p}(|{\mathcal{O}}_{K}/{\mathcal{P}}_{j}|-1)>e_{p}$ would imply that
${\mathcal{P}}_{j}$ does split completely in the unramified field extension
$L_{p}|K$. Next, choose $c_{0}\in{\mathcal{O}}_{K}$ such that $bc_{0}\equiv
u\text{ mod }\mathfrak{a}.$ But this implies

$\displaystyle({\mathcal{P}}_{1}\cdot{\mathcal{P}}_{2},K_{\mathfrak{a}}|K)$
$\displaystyle=({\mathcal{P}}_{1},K_{\mathfrak{a}}|K)\cdot({\mathcal{P}}_{2},K_{\mathfrak{a}}|K)=(\sigma_{1}^{-1}|_{K_{\mathfrak{a}}})\cdot(\sigma_{2}^{-1}|_{K_{\mathfrak{a}}})=\sigma|_{K_{\mathfrak{a}}}^{-1}=(b{\mathcal{O}}_{K},K_{\mathfrak{a}}|K)^{-1}$
$\displaystyle=(c_{0}{\mathcal{O}}_{K},K_{\mathfrak{a}}|K).$

Next, Theorem 2.6 implies that there is a $\lambda\in K^{*}$ multiplicatively
congruent to $1$ modulo $\mathfrak{a}$ such that
$c_{0}{\mathcal{O}}_{K}=\lambda{\mathcal{P}}_{1}\cdot{\mathcal{P}}_{2}.$ So
setting $c:=c_{0}\lambda^{-1}$, one obtains
$c{\mathcal{O}}_{K}={\mathcal{P}}_{1}\cdot{\mathcal{P}}_{2}.$ Hence
$\epsilon(c)={\rm
lcm}(\epsilon({\mathcal{P}}_{1}),\epsilon({\mathcal{P}}_{2}))$ holds. Then let
$p$ be a rational prime. If $p\in S,$ then
$v_{p}(\epsilon(c))=\min\\{v_{p}(\epsilon({\mathcal{P}}_{1})),v_{p}(\epsilon({\mathcal{P}}_{2}))\\}=e_{p}.$
On the other hand, if $p\notin S,$ then $v_{p}(\epsilon(b))=e_{p}$. Hence
$v_{p}({\rm gcd}(\epsilon(b),\epsilon(c)))=e_{p}$ holds for all rational
primes $p$ and so ${\rm gcd}(\epsilon(b),\epsilon(c))=m.$ ∎

######  Remark 3.4.

We would like to consider the field extension $\mathbb{F}[\xi_{p}\mid p\in
S]\cdot K$ of $K$ in the above argument and show that ${\rm
Gal}(\mathbb{F}[\xi_{p}\mid p\in S]\cdot K|K)$ decomposes as the direct
product $\prod_{p\in S}{\rm Gal}(L_{p}|K).$ However, this strategy doesn’t
work as the various $L_{p}$ do in some cases intersect in non-trivial ways.
The construction of the $K_{i}$ is done to circumvent precisely this problem.

Using these two lemmas, one can now prove Theorem 1.3:

###### Proof.

Let $A\in{\rm SL}_{n}({\mathcal{O}}_{K}).$ Then using the stability argument
from the proof of [5, Main Theorem], one can transform $A$ into an element of
the form

$A_{1}=\begin{pmatrix}\begin{matrix}a_{1}&b_{1}\\\
c_{1}&d_{1}\end{matrix}&0\\\ 0&I_{n-2}\end{pmatrix}$

by multiplication with $1/2(3n^{2}-n-10)$ elementary matrices. Using Lemma
3.1, one can then transform $A_{1}$ into a matrix

$A_{2}=\begin{pmatrix}\begin{matrix}a^{m}&b\\\ c_{2}&d_{2}\end{matrix}&0\\\
0&I_{n-2}\end{pmatrix}$

by multiplication with three elementary matrices. But note that by Lemma 3.3,
we can find an element $c\in{\mathcal{O}}_{K}$ such that $bc\equiv-1\text{ mod
}a{\mathcal{O}}_{K}$ and such that ${\rm gcd}(\epsilon(b),\epsilon(c))=m.$
Consequently, there is an element

$A_{3}=\begin{pmatrix}\begin{matrix}a&b\\\ c&d_{3}\end{matrix}&0\\\
0&I_{n-2}\end{pmatrix}^{m}$

of ${\rm SL}_{n}({\mathcal{O}}_{K}).$ Then according to [5, Lemma 1], one can
transform $A_{2}$ into $A_{3}$ by multiplication with $16$ elementary matrices
and according to [5, Lemma 2], the matrix $A_{3}$ can be transformed into the
identity matrix by multiplication with $36$ elementary matrices. Thus in
total, the initial matrix $A$ is a product of at most
$1/2(3n^{2}-n-10)+3+16+36=1/2(3n^{2}-n)+50$ elementary matrices. ∎

##  4\. What remains

After some research activity in the last fifteen or so years bounded
generation by root elements for split Chevalley groups defined over rings of
integers in global fields is now essentially understood: On the qualitative
side, the Carter-Keller-Paige approach presented by Morris [10] serves as a
uniform framework to prove bounded generation results first for ${\rm SL}_{n}$
of number fields and as seen in [16] also for global function fields and the
other split Chevalley groups.

On the quantitative side, the main question previously was whether there are
only bounds for elementary bounded generation depending on the global field in
question or not. We denote the minimal number of elementary matrices needed to
write any element of ${\rm SL}_{n}(R)$ by $\nu_{n}(R)$. For $n\geq 3,$
standard stable range considerations yield

$\nu_{n}(R)\leq 1/2(3n^{2}-n)-27+\nu_{3}(R)$

for $R$ the ring of S-algebraic integers in a global field $K$ with $S$ a non-
empty set of valuations of $K$ containing all archimedean places of $K.$ We
fix $K,S$ and $R$ like this in the following. As seen by Rapinchuk, Morgan and
Sury [9] if $K$ is a number field but not imaginary quadratic or $\mathbb{Z}$,
one has $\nu_{3}(R)\leq 16$. This preprint establishes in Theorem 1.3 for a
global function field $K$ that $\nu_{3}(R)\leq 62$ holds. The remaining cases
are the imaginary quadratic integers and $\mathbb{Z}$. But the ring of
imaginary quadratic integers $R_{D}$ in the number field
$\mathbb{Q}[\sqrt{-D}]$ are the rings of all algebraic integers of a number
field of degree two over $\mathbb{Q}.$ Thus [10, Theorem 1.2] implies that
there is a natural number $L^{\prime}(2,3)$ independent of the precise $D$
such that $\nu_{3}(R_{D})\leq L^{\prime}(2,3)$. Thus setting
$L(2,3):=\max\\{L^{\prime}(2,3),62,v_{3}(\mathbb{Z})\\}-27$, we obtain the
following:

######  Theorem 4.1.

There is an $L(2,3)\in\mathbb{N}$ such that for each global field $K$, each
non-empty set $S$ of valuations of $K$ containing all its archimedean places
and $R$ the ring of all $S$-algebraic integers in $K$, the inequality
$\nu_{n}(R)\leq L(2,3)+1/2(3n^{2}-n)$ holds for all $n\geq 3$.

So bounded elementary generation has bounds completely independent of the
precise global field $K$ and the precise set $S$. This answers questions
raised in [9] and mentioned in a preprint by Kunyavskii, Plotkin and Vavilov
[8, 9. Final Remarks]. On the other hand, the precise values for $\nu_{n}(R)$
themselves remain a mystery: A simple counting argument shows that
$\nu_{n}(R)$ being quadratic in $n$ is at least the correct asymptotic, but
the bound $L(2,3)$ above is non-explicit as it arises (among else) from a
model-theoretic compactness argument and it would be desirable to replace it
with some explicit number independent of $D.$ Additionally, it is very
unpleasant that three very different line of arguments are needed to
seperately treat the three cases of global function fields, imaginary
quadratic number fields and the rest. Thus it stands to reason that there is a
better method to treat them all at once. Given Theorem 4.1, one might even
nurture the following hope:

######  Conjecture 4.2.

There is a function $f:\mathbb{N}\to\mathbb{N}$ such that for any $K,S$ and
$R$ as in Theorem 4.1, one has $\nu_{n}(R)=f(n)$ for all $n\geq 3.$

Lastly, we should mention that while the main result in this manuscript,
Theorem 1.3, is only stated for ${\rm SL}_{n}$, it is all but clear that the
result will also generalize to all the other split Chevally groups.

## References

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  * [2] Roland Auer  “Ray class fields of global function fields with many rational places”  In _Acta Arith._ 95.2, 2000, pp. 97–122  DOI: 10.4064/aa-95-2-97-122
  * [3] H. Bass, J. Milnor and J.-P. Serre  “Solution of the congruence subgroup problem for ${\rm SL}_{n}\,(n\geq 3)$ and ${\rm Sp}_{2n}\,(n\geq 2)$”  In _Inst. Hautes Études Sci. Publ. Math._ , 1967, pp. 59–137  URL: http://www.numdam.org/item?id=PMIHES_1967__33__59_0
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#  TIT/HEP-682 March 2021 Flavor symmetries of six-dimensional ${\cal
N}=(1,0)$ theories from AdS/CFT correspondence

Shota Fujiwara111E-mail<EMAIL_ADDRESS>Yosuke
Imamura222E-mail<EMAIL_ADDRESS>and Tatsuya Mori333E-mail:
<EMAIL_ADDRESS>

Department of Physics, Tokyo Institute of Technology,
Tokyo 152-8551, Japan

We calculate the superconformal indices of a class of six-dimensional ${\cal
N}=(1,0)$ superconformal field theories realized on M5-branes at
$\mathbb{C}^{2}/\mathbb{Z}_{k}$ singularity by using the method developed in
previous works of the authors and collaborators. We use the AdS/CFT
correspondence, and finite $N$ corrections are included as the contribution of
M2-branes wrapped on two-cycles in $S^{4}/\mathbb{Z}_{k}$. We confirm that the
indices are consistent with the expected flavor symmetries.

###### Contents

  1. 1 Introduction
  2. 2 6d $(1,0)$ theories
    1. 2.1 Setup
    2. 2.2 Flavor symmetries
  3. 3 Indices from M-theory in $AdS_{7}\times\bm{S}^{4}/\mathbb{Z}_{k}$
    1. 3.1 Conjectural formula
    2. 3.2 The bulk contribution
    3. 3.3 Wrapped M2-branes on $\bm{S}^{4}/\mathbb{Z}_{k}$
  4. 4 Results and consistency check
    1. 4.1 Results for $N=1$
      1. 4.1.1 $k=2$
      2. 4.1.2 $k=3$
    2. 4.2 Results for $N=2$
      1. 4.2.1 $k=2$
      2. 4.2.2 $k=3$
    3. 4.3 Results for $N=3$
      1. 4.3.1 $k=2$
      2. 4.3.2 $k=3$
  5. 5 Conclusions and discussions

##  1 Introduction

Recent progress of quantum field theory has enabled us quantitative analyses
of strongly coupled field theories. In particular, for supersymmetric theories
realized in string/M-theories we can use different powerful methods like
dualities and localization. In this paper we will study a class of such
theories: six-dimensional ${\cal N}=(1,0)$ superconformal field theories
realized on a stack of $N$ M5-branes placed at the fixed locus of the
$\mathbb{C}^{2}/\mathbb{Z}_{k}$ orbifold. A theory in this class has
holographic description: M-theory in $AdS_{7}\times S^{4}/\mathbb{Z}_{k}$. A
quiver gauge theory description is also useful to describe the tensor branch
of the theory [1, 2, 3, 4], and the root of the tensor branch, where the
superconformal symmetry restores, corresponds to the strong gauge coupling
limit.

An interesting property of such ${\cal N}=(1,0)$ theories is non-trivial
flavor symmetry depending on $k$ and $N$ [5, 6, 7]. A subtle point is that the
symmetry may be different from what is read off from the corresponding quiver
gauge theory. It was proposed that a certain discrete symmetry of the quiver
gauge theory, which is not manifest perturbatively, is gauged in the strong
coupling limit and as a result the flavor symmetry is reduced [6]. For
example, in the case of $N=k=2$, although the flavor symmetry of the quiver
gauge theory is $SO(8)$, that of the superconformal theory is $SO(7)$ [5]. See
Table 1 for the flavor symmetries for different $k$ and $N$.

Table 1: The flavor symmetries of interacting ${\cal N}=(1,0)$ theories. | $N=2$ | $N\geq 3$
---|---|---
$k=2$ | $SO(7)$ | $SU(2)_{a}\times SU(2)_{b}\times SU(2)_{F}$
$k\geq 3$ | $SU(2k)$ | $SU(k)_{a}\times SU(k)_{b}\times U(1)_{F}$

The purpose of this paper is to confirm such flavor symmetries on the AdS
side.

For generic values of $k$ and $N$ the flavor symmetry is $SU(k)_{a}\times
SU(k)_{b}\times U(1)_{F}$. In addition, we also have $SU(2)_{R}$ symmetry.
These symmetries are manifest on the AdS side; $U(1)_{F}\times SU(2)_{R}$ is
the isometry of $S^{4}/\mathbb{Z}_{k}$, and two $SU(k)$ symmetries are
associated with the two $A_{k-1}$ singularities at the fixed points. In the
case of $k=2$ and $N\geq 3$, $U(1)_{F}$ is enhanced to $SU(2)_{F}$, and this
is also understood as the isometry of $S^{4}/\mathbb{Z}_{2}$.

The enhancement for $N=2$ is more interesting. This symmetry enhancement is
not manifest on the AdS side, and it is interesting to study how this is
realized. We confirm this symmetry enhancement by calculating the
superconformal index using the AdS/CFT correspondence. This cannot be seen in
the large $N$ limit, and to confirm such symmetry enhancement we need to
include finite $N$ corrections.

To calculate the finite $N$ corrections to the superconformal index we use the
method developed in the previous work [8] of the authors and collaborators for
$(2,0)$ theories. See also [9, 10, 11, 12] for similar analysis of the 4d
superconformal index. We calculate the corrections as a contribution from
wrapped M2-branes. We focus only on the single-wrapping contribution. For a
configuration with more than one wrapped M2-branes its contribution is given
in the form of integral over gauge fugacities. We have not yet understood how
to choose integration contours in the integrals, and we leave the analysis of
such contributions for future work.

This paper is organized as follows. In the next section, we summarize basic
properties of the theory we study. In particular, we briefly explain the
flavor symmetry. In Section 3, we give a formula we use to calculate the
index. In Section 4 by using the formula we calculate the index for small $k$
and $N$. For $N=1$ and arbitrary $k$ the ${\cal N}=(1,0)$ theory is free and
we can directly calculate the index without using the holographic description,
and we can confirm the formula gives the correct index. This analysis is done
for small $k$ in Section 4.1. Results for $N=2$ and $N=3$ are shown in
subsection 4.2 and 4.3, respectively, and the consistency with the flavor
symmetries in Table 1 is confirmed. The last section is devoted to
discussions.

##  2 6d $(1,0)$ theories

###  2.1 Setup

An ${\cal N}=(1,0)$ theory we discuss is defined as the theory on $N$
M5-branes placed at $A_{k-1}$ singularity. We consider M-theory in the
background
$\mathbb{R}^{1,5}\times\mathbb{C}^{2}/\mathbb{Z}_{k}\times\mathbb{R}_{T}$. Let
$X_{\mu}$ ($\mu=0,1,\ldots,5$), $z_{i}$ ($i=1,2$), and $x_{5}$ be the
coordinates of $\mathbb{R}^{1,5}$, $\mathbb{C}^{2}$, and $\mathbb{R}_{T}$,
respectively. We also define $x_{m}$ ($m=1,2,3,4$) by

$\displaystyle z_{1}=x_{1}+ix_{2},\quad z_{2}=x_{3}+ix_{4}.$ (1)

Let $R_{ab}$ ($a,b=1,\ldots,5$) be the generators of the rotation group
$SO(5)_{R}$ in the $x_{a}$ space. We define the orbifold by $\mathbb{Z}_{k}$
generated by

$\displaystyle\exp\left(\frac{2\pi i}{k}(R_{12}-R_{34})\right).$ (2)

This acts on $(z_{1},z_{2},x_{5})$ as

$\displaystyle(z_{1},z_{2},x_{5})\rightarrow(e^{2\pi i/k}z_{1},e^{-2\pi
i/k}z_{2},x_{5}).$ (3)

We put $N$ M5-branes at $x_{1}=\cdots=x_{5}=0$. If it were not for the
orbifolding the $A_{N-1}$-type ${\cal N}=(2,0)$ theory would be realized on
the worldvolume of the M5-branes. The orbifolding breaks the
$\mathcal{N}=(2,0)$ supersymmetry down to $\mathcal{N}=(1,0)$. At the same
time, the $SO(5)_{R}$ symmetry is broken to $SU(2)_{R}\times U(1)_{F}$ for
$k\geq 3$. $U(1)_{F}$ is replaced by $SU(2)_{F}$ for $k=2$. The $SU(2)_{R}$ is
the $R$-symmetry of the 6d $(1,0)$ SCFTs, while $U(1)_{F}$ or $SU(2)_{F}$ does
not act on the ${\cal N}=(1,0)$ supercharges and is treated as a flavor
symmetry. In addition, the orbifold singularity provides $SU(k)$ flavor
symmetry. The singular locus $\mathbb{R}^{1,5}\times\mathbb{R}_{T}$ is devided
by the M5-branes at $x_{5}=0$ into two parts: the $x_{5}>0$ part and the
$x_{5}<0$ part. Correspondingly, we have two copies of $SU(k)$ symmetry which
we denote by $SU(k)_{a}$ and $SU(k)_{b}$. In summary, the bosonic global
symmetry is

$\displaystyle SO(2,6)_{\rm conf}\times SU(2)_{R}\times G_{\rm flavor},$ (4)

where $SO(2,6)_{\rm conf}\times SU(2)_{R}$ is the bosonic subgroup of the 6d
$\mathcal{N}=(1,0)$ superconformal symmetry $OSp(8|2)$ and the flavor symmetry
$G_{\rm flavor}$ is generically given by

$\displaystyle G_{\rm flavor}=U(1)_{F}\times SU(k)_{a}\times SU(k)_{b}.$ (5)

We define the superconformal index as follows. Let $H$ and $J_{ij}$
($i,j=1,\ldots,6$) be the generators of $SO(2)_{H}\times SO(6)_{\rm
spin}\subset SO(2,6)_{\rm conf}$, and take $H$, $J_{12}$, $J_{34}$, and
$J_{56}$ as Cartan generators. $H$ is the Hamiltonian and $J_{ij}$ are spins.
We also take $R_{12}$ and $R_{34}$ as $SO(5)_{R}$ Cartan generators. To define
the index we need to choose one component of the supercharge. We take the
component with the following quantum numbers:

$\displaystyle{\cal
Q}:(H,J_{12},J_{34},J_{56};R_{12},R_{34})=(+\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2};+\tfrac{1}{2},+\tfrac{1}{2}).$
(6)

Note that ${\cal Q}$ is invariant under the orbifold group $\mathbb{Z}_{k}$
generated by (2). The anticommutation relation between ${\cal Q}$ and its
hermitian conjugate ${\cal Q}^{\dagger}$ is

$\displaystyle\Delta\equiv\\{{\cal Q},{\cal
Q}^{\dagger}\\}=H-(J_{12}+J_{34}+J_{56})-2(R_{12}+R_{34}).$ (7)

Then we define the superconformal index by

$\displaystyle{\cal I}(q,y_{a},u,a_{i},b_{i})=\mathop{\rm
tr}\nolimits\left[(-1)^{F}x^{\Delta}q^{H+\frac{1}{3}(J_{12}+J_{34}+J_{56})}y_{1}^{J_{12}}y_{2}^{J_{34}}y_{3}^{J_{56}}u^{R_{12}-R_{34}}\prod_{i=1}^{k-1}a_{i}^{F_{a,i}}b_{i}^{F_{b,i}}\right].$
(8)

Due to the boson/fermion cancellation only the BPS operators with $\Delta=0$
contribute to the index, and hence the index does not depends on $x$. The
choice of ${\cal Q}$ breaks the $SO(6)_{\rm spin}$ to $U(1)_{\rm spin}\times
SU(3)_{\rm spin}$, and $y_{1}$, $y_{2}$, and $y_{3}$ are the $SU(3)_{\rm
spin}$ fugacities constrained by $y_{1}y_{2}y_{3}=1$. $F_{a,i}$ and $F_{b,i}$
are respectively Cartan generators of $SU(k)_{a}$ and $SU(k)_{b}$. $u$ is the
fugacity for $U(1)_{F}$ generated by $R_{12}-R_{34}$. Note that for $k=1$ this
index agree with the ${\cal N}=(2,0)$ superconformal index in [8].

###  2.2 Flavor symmetries

For the analysis of the operator spectrum and the flavor symmetry of the
theory it is convenient to consider the quiver gauge theories realized in the
tensor branch. By taking $U(1)_{F}$ orbits as M-theory circles we can regard
the system as a type IIA brane configuration. $N$ M5-branes become $N$
NS5-branes, and the $A_{k-1}$ singularity becomes a stack of $k$ D6-branes.

The $(2,0)$ tensor multiplet on an M5-brane separate into a $(1,0)$ tensor
multiplet and a $(1,0)$ hypermultiplet on the corresponding NS5-brane, and the
scalar component in the $(1,0)$ tensor multiplet corresponds to the location
of the NS5-branes in the $x_{5}$ direction. At a generic point in the tensor
branch all the NS5-branes are separated one by one in the $x_{5}$ direction,
and a linear quiver gauge theory is realized on the D6-branes.

The worldvolume of the stack of D6-branes is divided into $N+1$ parts by the
NS5-branes. We label the NS5-branes by $i=1,2,\ldots,N$. The D6-branes
suspended between two NS5-branes $i$ and $i+1$ give $SU(k)_{i}$ gauge group
while two semi-infinite parts of D6-branes give the flavor symmetries
$SU(k)_{a}\equiv SU(k)_{0}$ and $SU(k)_{b}\equiv SU(k)_{N}$. Let
$(h_{i},\widetilde{h}_{i})$ be the hypermultiplet arising from open strings
crossing the $i$-th NS5-brane. $h_{i}$ and $\widetilde{h}_{i}$ belong to the
bi-fundamental representations $(k,\overline{k})$ and $(\overline{k},k)$,
respectively, of $SU(k)_{i-1}\times SU(k)_{i}$. The $SU(N)$ groups and the
hypermultiplets are depicted as the linear quiver diagram in Figure 1. In
addition, we also have degrees of freedom that are implicit in the diagram; in
each gauge node there exists a tensor multiplet corresponding to the degrees
of freedom of the NS5-brane.

Figure 1: The linear quiver diagram of the gauge theory realized in the tensor
branch is shown.

Among different gauge invariant operators let us focus on two classes of
operators. The first class includes operators defined by

$\displaystyle S_{ij}=(h_{0})_{ia}(\widetilde{h}_{0})_{aj},\quad
S^{\prime}_{ij}=(\widetilde{h}_{N})_{ia}(h_{N})_{aj}.$ (9)

The other class includes

$\displaystyle
L_{ij}=(h_{0})_{ia_{1}}(h_{1})_{a_{1}a_{2}}\cdots(h_{N})_{a_{N}j},\quad
L^{\prime}_{ij}=(\widetilde{h}_{N})_{ia_{N}}\cdots(h_{1})_{a_{2}a_{1}}(h_{0})_{a_{1}j}.$
(10)

These operators play an important role when we discuss the flavor symmetry.

The operators $S_{ij}$ and $S^{\prime}_{ij}$ belong to the adjoint
representations of $SU(k)_{0}$ and $SU(k)_{N}$, respectively. They have
dimension $4$ and are the primary operators of the current multiplets of the
generic flavor symmetry

$\displaystyle G_{\rm flavor}=SU(k)_{0}\times SU(k)_{N}\times U(1)_{F}.$ (11)

(Although there are $N$ classical $U(1)$ symmetries only one of them is
anomaly free.)

The operators $L_{ij}$ and $L^{\prime}_{ij}$, which belong to the bi-
fundamental representations $(k,\overline{k})_{+1}$ and
$(\overline{k},k)_{-1}$ of $G_{\rm flavor}$ in (11), respectively, have
dimension $2N$. These operators appear in the spectrum only when $N$ is
finite, and play a role similar to baryonic operators in four-dimensional
quiver gauge theories. They are expected to correspond to wrapped M2-branes on
the gravity side. Indeed, their dimension coincide with the mass of an
M2-brane wrapped around a large $S^{2}/\mathbb{Z}_{k}$ in the unit of the
inverse AdS radius.

The flavor symmetry (11) is enhanced if one of $k$ or $N$ becomes $2$. If
$k=2$ and $N\geq 3$ $U(1)_{F}$ is enhanced to $SU(2)_{F}$. Correspondingly,
the index is written in terms of $SU(2)_{F}$ characters. This symmetry is
manifest on the gravity side as the isometry of $S^{4}/\mathbb{Z}_{2}$.

The enhancement for $N=2$ is more interesting. If $N=2$ the operators $L$ and
$L^{\prime}$ have dimension $4$ as well as $S$ and $S^{\prime}$, and they give
additional current multiplets. As the result the flavor symmetry (11) is
enhanced to $SU(2k)$ for $k\geq 3$. On the gravity side, this enhancement
should be realized when we include the contribution of wrapped M2-branes.

The $k=N=2$ case is most interesting. In this case there are eight $SU(2)$
gauge symmetry doublet in the hypermultiplets, and we can write down $28$
gauge invariant dimension $4$ operators forming the $SO(8)$ adjoint
representation. They correspond to the $SO(8)$ global symmetry of the quiver
gauge theory. However, it is known that the symmetry is reduced to $SO(7)$ in
a highly non-trivial way [5], and it would be nice if we can reproduce this
flavor symmetry on the gravity side by the index calculation.

##  3 Indices from M-theory in $AdS_{7}\times\bm{S}^{4}/\mathbb{Z}_{k}$

###  3.1 Conjectural formula

Based on the idea explained in detail in [8] we propose the formula of the
index for 6d $(1,0)$ theories

$\displaystyle\mathcal{I}^{(1,0)}_{N,k}=\mathcal{I}^{\mathrm{bulk}}\sum_{n_{1},n_{2}=0}^{\infty}\mathcal{I}_{(n_{1},n_{2})}^{\mathrm{M2}}.$
(12)

This formula gives the index as the combination of contributions from objects
in the dual geometry $AdS_{7}\times S^{4}/\mathbb{Z}_{k}$, where the internal
space $\bm{S}^{4}/\mathbb{Z}_{k}$ is defined by

$\displaystyle|z_{1}|^{2}+|z_{2}|^{2}+x_{5}^{2}=1,$ (13)

together with the identification by the $\mathbb{Z}_{k}$ action (3). ${\cal
I}^{\rm bulk}$ is the contribution of Kaluza-Klein modes in the bulk. We also
include in ${\cal I}^{\rm bulk}$ the contribution from localized modes at the
fixed points of the orbifold. ${\cal I}^{\rm M2}_{(n_{1},n_{2})}$ are
contributions of wrapped M2-branes in the internal space. $n_{1}$ and $n_{2}$
are numbers of M2-brane wrapped on the two specific two-cycles $z_{1}=0$ and
$z_{2}=0$, respectively. As we will explain in 3.3 the $q$ expansion of ${\cal
I}_{(n_{1},n_{2})}$ starts from order $q^{2(n_{1}+n_{2})N}$ terms. In the
large $N$ limit all contributions but ${\cal I}_{(0,0)}=1$ decouple and the
formula reduces to
$\mathcal{I}^{(1,0)}_{N=\infty,k}=\mathcal{I}^{\mathrm{bulk}}$. On the other
hand, if $N$ is finite, all sectors labelled by $(n_{1},n_{2})$ contribute to
the index. ${\cal I}_{(n_{1},n_{2})}$ for each $(n_{1},n_{2})$ is calculated
as the index of the theory realized on the wrapped M2-branes by the standard
localization formula. If $n_{1}+n_{2}\geq 2$ the formula includes non-trivial
gauge integrals, and unfortunately we have not yet found systematic rules for
the integration contours. For this reason we leave the analysis of
$n_{1}+n_{2}\geq 2$ for future work and in this paper we focus only on the
single-wrapping sectors $(n_{1},n_{2})=(1,0)$ and $(0,1)$. Namely, we consider
the formula

$\displaystyle\mathcal{I}^{(1,0)}_{N,k}=\mathcal{I}^{\rm grav}_{N,k}+{\cal
O}(q^{4N}),\quad$ (14)

where ${\cal I}^{\rm grav}_{N,k}$ is defined by

$\displaystyle\mathcal{I}^{\rm
grav}_{N,k}=\mathcal{I}^{\mathrm{bulk}}\left(1+\mathcal{I}_{(1,0)}^{\mathrm{M2}}+\mathcal{I}_{(0,1)}^{\mathrm{M2}}\right).$
(15)

With the conjectural formula (14), we can calculate the index for an arbitrary
$N$ and $k$ up to the expected error of order $q^{4N}$.

###  3.2 The bulk contribution

Let us first consider ${\cal I}^{\rm bulk}$, which gives the large $N$ index.
This is given by the plethystic exponential of the single-particle index,
which is the sum of two contributions: the supergravity Kaluza-Klein modes in
the internal space $\bm{S}^{4}/\mathbb{Z}_{k}$ and the vector multiplets
localized at the two fixed points of $\bm{S}^{4}/\mathbb{Z}_{k}$.

The contribution of the Kaluza-Klein modes in $AdS_{7}\times\bm{S}^{4}$
without orbifolding has already been studied in [13] and is given by
$\mathop{\rm Pexp}\nolimits i_{\rm KK}$ with the single-particle index

$\displaystyle i_{\rm KK}$
$\displaystyle=\frac{q^{2}\chi_{[1]}^{u}-q^{\frac{8}{3}}\chi_{[0,1]}^{y}+q^{\frac{16}{3}}\chi_{[1,0]}^{y}-q^{6}\chi_{[1]}^{u}}{(1-uq^{2})(1-u^{-1}q^{2})(1-y_{1}q^{\frac{4}{3}})(1-y_{2}q^{\frac{4}{3}})(1-y_{3}q^{\frac{4}{3}})},$
(16)

where $\chi_{[n]}^{u}$ are the $SU(2)$ characters defined by

$\displaystyle\chi_{[n]}^{u}=\frac{u^{n+1}-u^{-(n+1)}}{u-u^{-1}},$ (17)

and $\chi_{[m_{1},m_{2}]}^{y}$ are the $SU(3)$ characters of the
representations with Dynkin labels $[m_{1},m_{2}]$. The Kaluza-Klein modes in
the orbifold $S^{4}/\mathbb{Z}_{k}$ is obtained by picking up the
$\mathbb{Z}_{k}$ invariant modes from the modes in $S^{4}$ [14].
Correspondingly, the single-particle index for the orbifold is given by ${\cal
P}_{k}i_{\rm KK}$, where ${\cal P}_{k}$ is the projection operator associated
with the $\mathbb{Z}_{k}$ orbifold which acts on a function of the fugacity
$u$ as

$\displaystyle{\cal P}_{k}f(u)=\frac{1}{k}\sum_{l=0}^{k-1}f(e^{2\pi il/k}u).$
(18)

The other contribution we need to include in the single-particle index comes
from two $A_{k-1}$ singularities on $\mathbb{C}^{2}/\mathbb{Z}_{k}$ at
$(z_{1},z_{2},x_{5})=(0,0,\pm 1)$, where the 7d $SU(k)_{a}\times SU(k)_{b}$
vector multiplets are localized. In general, a gauge field in the bulk of AdS
corresponds to a flavor symmetry on the boundary, and the corresponding
current multiplet contributes to the index. The corresponding single-particle
index is $i_{F}(\chi_{\mathrm{adj.}}^{a}+\chi_{\mathrm{adj.}}^{b})$, where
$\chi^{a/b}_{\mathrm{adj.}}$ are characters of adjoint representations of the
global $SU(k)_{a/b}$ symmetries and $i_{F}$ is given by

$\displaystyle
i_{F}=\frac{q^{4}}{(1-q^{\frac{4}{3}}y_{1})(1-q^{\frac{4}{3}}y_{2})(1-q^{\frac{4}{3}}y_{3})}.$
(19)

Note that $i_{F}$ is independent of $u$ and we do not have to perform the
$\mathbb{Z}_{k}$ projection.

By combining two contributions, we can calculate the index for the large $N$
limit. For example, for $k=2$ we obtain

$\displaystyle\mathop{\rm Pexp}\nolimits({\cal P}_{2}i_{\rm
KK}+i_{F}(\chi^{a}_{[2]}+\chi^{b}_{[2]}))$
$\displaystyle=1-\chi_{[0,1]}^{y}q^{\frac{8}{3}}+(\chi^{u}_{[2]}+\chi^{a}_{[2]}+\chi^{b}_{[2]}-\chi_{[1,1]}^{y})q^{4}$
$\displaystyle+((2+\chi^{u}_{[2]}+\chi^{a}_{[2]}+\chi^{b}_{[2]})\chi_{[1,0]}^{y}-\chi_{[2,1]}^{y})q^{\frac{16}{3}}+{\cal
O}(q^{\frac{20}{3}}).$ (20)

This includes the contribution of the “center of mass” free tensor multiplet.
The existence of such a decoupled free sector is suggested by the coefficient
“$2$” of the term $\chi_{[1,0]}^{y}q^{\frac{16}{3}}$ in the above expansion,
which is identified as the contribution of two copies of stress-tensor
multiplets. Such a free tensor multiplet exists for all $k$ and $N$, and we
always remove its contribution in the following calculation. Namely, we define
${\cal I}^{\rm bulk}$ in (14) by

$\displaystyle{\cal I}^{\rm bulk}$ $\displaystyle=\mathop{\rm
Pexp}\nolimits({\cal P}_{k}i_{KK}+i_{F}(\chi^{a}_{\rm adj}+\chi^{b}_{\rm
adj})-i_{\rm tensor}),$ (21)

where the single-particle index $i_{\rm tensor}$ of the free tensor multiplet
is given by [13]

$\displaystyle i_{\mathrm{tensor}}$
$\displaystyle=\frac{-q^{\frac{8}{3}}\chi^{y}_{[0,1]}+q^{4}}{(1-q^{\frac{4}{3}}y_{1})(1-q^{\frac{4}{3}}y_{2})(1-q^{\frac{4}{3}}y_{3})}.$
(22)

For $k=2,3$ (21) gives

$\displaystyle{\cal I}^{\rm bulk}_{k=2}$ $\displaystyle=\mathop{\rm
Pexp}\nolimits({\cal P}_{2}i_{KK}+i_{F}(\chi^{a}_{[2]}+\chi^{b}_{[2]})-i_{\rm
tensor})$
$\displaystyle=1+(\chi^{u}_{[2]}+\chi^{a}_{[2]}+\chi^{b}_{[2]})q^{4}+(1+\chi^{u}_{[2]}+\chi^{a}_{[2]}+\chi^{b}_{[2]})\chi_{[1,0]}^{y}q^{\frac{16}{3}}+{\cal
O}(q^{\frac{20}{3}}),$ (23) $\displaystyle{\cal I}^{\rm bulk}_{k=3}$
$\displaystyle=\mathop{\rm Pexp}\nolimits({\cal
P}_{3}i_{KK}+i_{F}(\chi_{[1,1]}^{a}+\chi_{[1,1]}^{b})-i_{\rm tensor})$
$\displaystyle=1+(1+\chi_{[1,1]}^{a}+\chi_{[1,1]}^{b})q^{4}+(2+\chi_{[1,1]}^{a}+\chi_{[1,1]}^{b})\chi_{[1,0]}^{y}q^{\frac{16}{3}}+(-\chi_{[1]}^{u}+\chi_{[3]}^{u})q^{6}+{\cal
O}(q^{\frac{20}{3}}).$ (24)

These are interpreted as the indices in the large $N$ limit. The $q^{4}$ terms
in each index is the contribution of the flavor current multiplets, and we can
read off the expected flavor symmetries $SU(2)^{3}$ for $k=2$ and
$SU(3)^{2}\times U(1)$ for $k=3$. We also confirm that all other terms are
consistent with these flavor symmetries.

###  3.3 Wrapped M2-branes on $\bm{S}^{4}/\mathbb{Z}_{k}$

Next we consider the contribution of M2-branes. The worldvolume of a BPS
M2-brane is described by the intersection of $\bm{S}^{4}$ and a holomorphic
surface [15, 16]

$\displaystyle f(z_{1},z_{2})=0.$ (25)

The consistency with the $\mathbb{Z}_{k}$ orbifolding require the function $f$
to satisfy

$\displaystyle f(e^{2\pi i/k}z_{1},e^{-2\pi i/k}z_{2})=e^{2\pi
iw/k}f(z_{1},z_{2}),$ (26)

where $w\in\mathbb{Z}/k\mathbb{Z}$ is the topological wrapping number.

In [8], in which the system without $\mathbb{Z}_{k}$ orbifolding was studied,
it was proposed that we can take only M2-brane configurations given by
monomials of the form $f(z_{1},z_{2})=z_{1}^{n_{1}}z_{2}^{n_{2}}$ and were
shown that the formula passes some non-trivial checks. Let us adopt the same
assumption. The function $f(z_{1},z_{2})=z_{1}^{n_{1}}z_{2}^{n_{2}}$ gives the
system with $n_{1}$ M2-branes wrapped on $z_{1}=0$ and $n_{2}$ M2-branes
wrapped on $z_{2}=0$. The total topological wrapping number is $w=n_{1}-n_{2}$
mod $k$. ${\cal I}^{\rm M2}_{(n_{1},n_{2})}$ in (12) is the contribution from
the specific wrapping sector with $(n_{1},n_{2})$. We focus on the two sectors
$(1,0)$ and $(0,1)$. In the absence of the $\mathbb{Z}_{k}$ orbifolding, the
contribution of the $(1,0)$ sector, a single M2-brane wrapped on $z_{1}=0$, is
[8]

$\displaystyle(q^{2}u)^{N}\mathop{\rm Pexp}\nolimits
i^{\mathrm{M2}}_{z_{1}=0},$ (27)

with the single-particle index $i^{\mathrm{M2}}_{z_{1}=0}$ given by

$\displaystyle i^{\rm M2}_{z_{1}=0}$
$\displaystyle=\frac{q^{-2}u^{-1}-q^{\frac{2}{3}}u^{-1}\chi_{[0,1]}^{y}+q^{\frac{4}{3}}\chi_{[1,0]}^{y}-q^{4}}{1-q^{2}u^{-1}}.$
(28)

To obtain the index for the $\mathbb{Z}_{k}$ orbifold, we need two
modifications. First, we perform the $\mathbb{Z}_{k}$ projection on the
single-particle index. Second, we insert the character of the $SU(k)_{a}\times
SU(k)_{b}$ bi-fundamental representation because the wrapped M2-brane couples
to the localized vector multiplets at the fixed points
$(z_{1},z_{2},x_{5})=(0,0,\pm 1)$. As the result, the contribution of the
$(1,0)$ sector is given by

$\displaystyle\mathcal{I}^{\mathrm{M2}}_{(1,0)}$
$\displaystyle=(q^{2}u)^{N}\chi^{a}_{\rm
fund.}\chi^{b}_{\rm\overline{fund.}}\mathop{\rm
Pexp}\nolimits\left[\mathcal{P}_{k}i^{\mathrm{M2}}_{z_{1}=0}\right],$ (29)

where $\chi^{a/b}_{\rm fund.}$ ($\chi^{a/b}_{\rm\overline{fund.}}$) are
characters of (anti-) fundamental representations of the $SU(k)_{a/b}$
symmetries. The contribution of the other sector $(0,1)$ is given from (29) by
the replacement $u\rightarrow u^{-1}$, $a_{i}\rightarrow a_{i}^{-1}$, and
$b_{i}\rightarrow b_{i}^{-1}$. $u\rightarrow u^{-1}$ is the Weyl reflection of
$SO(5)_{R}$ exchanging $z_{1}$ and $z_{2}$. The inversion of $a_{i}$ and
$b_{i}$ are necessary because the cycle $z_{2}=0$ has the opposite topological
wrapping number to the cycle $z_{1}=0$; the former has $w=+1$ while the latter
has $w=-1$. After the replacement we obtain

$\displaystyle\mathcal{I}^{\mathrm{M2}}_{(0,1)}$
$\displaystyle=(q^{2}u^{-1})^{N}\chi^{a}_{\rm\overline{fund.}}\chi^{b}_{\rm
fund.}\mathop{\rm
Pexp}\nolimits\left[\mathcal{P}_{k}i^{\mathrm{M2}}_{z_{2}=0}\right],$ (30)

where $i^{\mathrm{M2}}_{z_{2}=0}=i^{\mathrm{M2}}_{z_{1}=0}|_{u\rightarrow
u^{-1}}$.

If $k=2$ the flavor characters appearing in (29) and (30) are the same,
$\chi^{a}_{\rm
fund.}\chi^{b}_{\rm\overline{fund.}}=\chi^{a}_{\rm\overline{fund.}}\chi^{b}_{\rm
fund.}$, and
$\mathcal{I}^{\mathrm{M2}}_{(1,0)}+\mathcal{I}^{\mathrm{M2}}_{(0,1)}$ is
invariant under the $SU(2)_{R}$ Weyl reflection $u\rightarrow u^{-1}$. This is
consistent with the symmetry enhancement $U(1)_{F}\rightarrow SU(2)_{F}$.

##  4 Results and consistency check

Now we are ready to calculate the index of the 6d $(1,0)$ theories by using
our formula (15) for different values of $k$ and $N$.

###  4.1 Results for $N=1$

The theory with $N=1$ is the free theory consisting of the “center-of-mass”
tensor multiplet and hypermultiplets belonging to the bi-fundamental
representation of $SU(k)_{a}\times SU(k)_{b}$. The index with the tensor
multiplet contribution removed is given by

$\displaystyle\mathcal{I}^{(1,0)}_{N=1}=\mathop{\rm
Pexp}\nolimits\left[i_{\mathrm{hyper}}(\chi^{a}_{\rm
fund.}\chi^{b}_{\rm\overline{fund.}}u+\chi^{a}_{\rm\overline{fund.}}\chi^{b}_{\rm
fund.}u^{-1})\right],$ (31)

where $i_{\mathrm{hyper}}$ is given by [13]

$\displaystyle i_{\mathrm{hyper}}$
$\displaystyle=\frac{q^{2}}{(1-q^{\frac{4}{3}}y_{1})(1-q^{\frac{4}{3}}y_{2})(1-q^{\frac{4}{3}}y_{3})}.$
(32)

Let us compare the index on the gravity side based on the formula (15) with
the free theory result from (31). As we do not include the multiple-wrapping
M2-branes, the errors should start at $q^{4}$ terms and we check the agreement
up to errors of this order.

####  4.1.1 $k=2$

On the gravity side (15) yields

$\displaystyle\mathcal{I}_{N=1,k=2}^{\mathrm{grav}}$
$\displaystyle=1+\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{u}_{[1]}q^{2}+\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{u}_{[1]}\chi_{[1,0]}^{y}q^{\frac{10}{3}}+(\chi^{a}_{[2]}+\chi^{b}_{[2]}+\chi^{u}_{[2]})q^{4}+\mathcal{O}(q^{\frac{14}{3}}).$
(33)

Expanding (31) with $k=2$ we obtain

$\displaystyle\mathcal{I}^{(1,0)}_{N=1,k=2}$
$\displaystyle=\mathcal{I}_{N=1,k=2}^{\mathrm{gr}}+\chi^{a}_{[2]}\chi^{b}_{[2]}\chi^{u}_{[2]}q^{4}+\mathcal{O}(q^{\frac{14}{3}}).$
(34)

We can see the agreement up to the error of ${\cal O}(q^{4})$ as expected.

####  4.1.2 $k=3$

For $k=3$ the result on the gravity side is

$\displaystyle\mathcal{I}_{N=1,k=3}^{\mathrm{grav}}$
$\displaystyle=1+\left(u\chi_{[1,0]}^{a}\chi_{[0,1]}^{b}+u^{-1}\chi_{[0,1]}^{a}\chi_{[1,0]}^{b}\right)q^{2}+\left(u\chi_{[1,0]}^{a}\chi_{[0,1]}^{b}\chi_{[1,0]}^{y}+u^{-1}\chi_{[0,1]}^{a}\chi_{[1,0]}^{b}\chi_{[1,0]}^{y}\right)q^{\frac{10}{3}}$
$\displaystyle+\left(1+\chi_{[1,1]}^{a}+\chi_{[1,1]}^{b}+u^{2}\chi_{[0,1]}^{a}\chi_{[1,0]}^{b}+u^{-2}\chi_{[1,0]}^{a}\chi_{[0,1]}^{b}\right)q^{4}+\mathcal{O}(q^{\frac{14}{3}}).$
(35)

On the CFT side we obtain

$\displaystyle\mathcal{I}^{(1,0)}_{N=1,k=3}$
$\displaystyle=\mathcal{I}_{N=1,k=3}^{\mathrm{grav}}+\Bigg{(}u^{2}\chi_{[2,0]}^{a}\chi_{[0,2]}^{b}+u^{-2}\chi_{[0,2]}^{a}\chi_{[2,0]}^{b}+\chi_{[1,1]}^{a}\chi_{[1,1]}^{b}\Bigg{)}q^{4}+\mathcal{O}(q^{\frac{14}{3}}).$
(36)

Again we can find agreement up to expected error terms of order $q^{4}$.

###  4.2 Results for $N=2$

When $N=2$, the generic flavor symmetry $SU(k)_{a}\times SU(k)_{b}\times U(1)$
is enhanced to $SU(2k)$ for $k\geq 3$ and $SO(7)$ for $k=2$ [5, 17, 6]. Then,
the indices should be written in terms of the characters of the enhanced
symmetries. Let us confirm this for $k=2$ and $k=3$. The expected errors due
to double-wrapping configurations are of order $q^{8}$, and we show the
results below the order.

####  4.2.1 $k=2$

We consider the $k=N=2$ case first. In this case the index should be written
in terms of the $SO(7)$ character $\chi^{SO(7)}_{[l_{1},l_{2};l_{3}]}$. The
last component of the Dynkin labels corresponds to the short root. The formula
(14) gives

$\displaystyle{\cal I}^{\rm grav}_{N=k=2}$
$\displaystyle=1+\chi^{SO(7)}_{[0,1;0]}q^{4}+(1+\chi^{SO(7)}_{[0,1;0]})\chi^{y}_{[1,0]}q^{\frac{16}{3}}$
$\displaystyle+\left((1+\chi^{SO(7)}_{[0,1;0]})\chi^{y}_{[2,0]}+(1-\chi^{SO(7)}_{[1,0;0]})\chi^{y}_{[0,1]}\right)q^{\frac{20}{3}}+{\cal
O}(q^{8}).$ (37)

This is correctly expanded in terms of $SO(7)$ characters. We also confirm
that it is not written in terms of characters of $SO(8)$, the symmetry of the
corresponding quiver gauge theory.

####  4.2.2 $k=3$

For $k=3$ and $N=2$ the expected flavor symmetry is $SU(6)$. The formula (14)
gives

$\displaystyle\mathcal{I}_{N=2,k=3}^{\mathrm{grav}}$
$\displaystyle=1+\chi^{SU(6)}_{[1,0,0,0,1]}q^{4}+(1+\chi^{SU(6)}_{[1,0,0,0,1]})\chi_{[1,0]}^{y}q^{\frac{16}{3}}+\chi^{SU(6)}_{[0,0,1,0,0]}q^{6}$
$\displaystyle+\left(1+\chi_{[1,0,0,0,1]}^{SU(6)}\right)\chi_{[2,0]}^{y}q^{\frac{20}{3}}+\chi^{SU(6)}_{[0,0,1,0,0]}\chi_{[1,0]}^{y}q^{\frac{22}{3}}+\mathcal{O}(q^{8}),$
(38)

and this is correctly written in terms of $SU(6)$ characters.

###  4.3 Results for $N=3$

In this subsection we show the results for $N=3$ calculated on the gravity
side. Because we do not have results we can compare, we give the results
simply as predictions. The expected errors are of order $q^{12}$, and we show
the results below the order.

####  4.3.1 $k=2$

The global symmetry for $N=3$ and $k=2$ is $G_{\rm flavor}=SU(2)_{a}\times
SU(2)_{b}\times SU(2)_{F}$. The formula (14) gives

$\displaystyle\mathcal{I}_{N=3,k=2}^{\mathrm{grav}}$
$\displaystyle=1+(\chi^{a}_{[2]}+\chi^{b}_{[2]}+\chi^{u}_{[2]})q^{4}+(\chi^{a}_{[2]}\chi^{y}_{[1,0]}+\chi^{b}_{[2]}\chi^{y}_{[1,0]}+\chi^{u}_{[2]}\chi^{y}_{[1,0]}+\chi^{y}_{[1,0]})q^{\frac{16}{3}}$
$\displaystyle+\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{u}_{[3]}q^{6}+(\chi^{y}_{[0,1]}+\chi^{a}_{[2]}\chi^{y}_{[2,0]}+\chi^{b}_{[2]}\chi^{y}_{[2,0]}+\chi^{y}_{[2,0]}+\chi^{u}_{[2]}(\chi^{y}_{[2,0]}-\chi^{y}_{[0,1]}))q^{\frac{20}{3}}$
$\displaystyle+\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{u}_{[3]}\chi^{y}_{[1,0]}q^{\frac{22}{3}}+(\chi^{a}_{[4]}+\chi^{a}_{[2]}\chi^{b}_{[2]}+\chi^{b}_{[4]}+2\chi^{u}_{[4]}+\chi^{y}_{[1,1]}+\chi^{a}_{[2]}\chi^{y}_{[3,0]}+\chi^{b}_{[2]}\chi^{y}_{[3,0]}+\chi^{y}_{[3,0]}$
$\displaystyle\hskip
142.26378pt+\chi^{u}_{[2]}(\chi^{a}_{[2]}+\chi^{b}_{[2]}-\chi^{y}_{[1,1]}+\chi^{y}_{[3,0]}-1)+2)q^{8}$
$\displaystyle+(\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{u}_{[3]}\chi^{y}_{[2,0]}-\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{u}_{[1]}\chi^{y}_{[0,1]})q^{\frac{26}{3}}$
$\displaystyle+(2\chi^{a}_{[2]}\chi^{y}_{[1,0]}+\chi^{a}_{[4]}\chi^{y}_{[1,0]}+2\chi^{a}_{[2]}\chi^{b}_{[2]}\chi^{y}_{[1,0]}+2\chi^{b}_{[2]}\chi^{y}_{[1,0]}+\chi^{b}_{[4]}\chi^{y}_{[1,0]}+2\chi^{u}_{[4]}\chi^{y}_{[1,0]}$
$\displaystyle\quad+2\chi^{y}_{[1,0]}+\chi^{y}_{[2,1]}+\chi^{a}_{[2]}\chi^{y}_{[4,0]}+\chi^{b}_{[2]}\chi^{y}_{[4,0]}+\chi^{y}_{[4,0]}$
$\displaystyle\quad+\chi^{u}_{[2]}(2\chi^{a}_{[2]}\chi^{y}_{[1,0]}+2\chi^{b}_{[2]}\chi^{y}_{[1,0]}+2\chi^{y}_{[1,0]}-\chi^{y}_{[2,1]}+\chi^{y}_{[4,0]}))q^{\frac{28}{3}}$
$\displaystyle+(\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{u}_{5}-\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{u}_{[1]}\chi^{y}_{[1,1]}+\chi^{u}_{[3]}(2\chi^{a}_{[1]}\chi^{b}_{[1]}+\chi^{a}_{[3]}\chi^{b}_{[1]}+\chi^{a}_{[1]}\chi^{y}_{[3,0]}\chi^{b}_{[1]}+\chi^{a}_{[1]}\chi^{b}_{[3]}))q^{10}$
$\displaystyle+(3\chi^{a}_{[2]}\chi^{y}_{[0,1]}+\chi^{a}_{[2]}\chi^{b}_{[2]}\chi^{y}_{[0,1]}+3\chi^{b}_{[2]}\chi^{y}_{[0,1]}-\chi^{y}_{[0,1]}+3\chi^{a}_{[2]}\chi^{y}_{[2,0]}+2\chi^{a}_{[4]}\chi^{y}_{[2,0]}$
$\displaystyle\qquad+3\chi^{a}_{[2]}\chi^{b}_{[2]}\chi^{y}_{[2,0]}+3\chi^{b}_{[2]}\chi^{y}_{[2,0]}+2\chi^{b}_{[4]}\chi^{y}_{[2,0]}+6\chi^{y}_{[2,0]}+\chi^{u}_{[4]}(3\chi^{y}_{[2,0]}-2\chi^{y}_{[0,1]})$
$\displaystyle\qquad+\chi^{y}_{[3,1]}+\chi^{a}_{[2]}\chi^{y}_{[5,0]}+\chi^{b}_{[2]}\chi^{y}_{[5,0]}+\chi^{y}_{[5,0]}+\chi^{u}_{[2]}(3\chi^{y}_{[0,1]}+3\chi^{a}_{[2]}\chi^{y}_{[2,0]}+3\chi^{b}_{[2]}\chi^{y}_{[2,0]}$
$\displaystyle\qquad+3\chi^{y}_{[2,0]}-\chi^{y}_{[3,1]}+\chi^{y}_{[5,0]}))q^{\frac{32}{3}}$
$\displaystyle+(2\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{u}_{[5]}\chi^{y}_{[1,0]}+\chi^{u}_{[1]}(2\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{y}_{[1,0]}-\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{y}_{[2,1]})+\chi^{u}_{[3]}(6\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{y}_{[1,0]}$
$\displaystyle\qquad+2\chi^{a}_{[3]}\chi^{b}_{[1]}\chi^{y}_{[1,0]}+2\chi^{a}_{[1]}\chi^{b}_{[3]}\chi^{y}_{[1,0]}+\chi^{a}_{[1]}\chi^{b}_{[1]}\chi^{y}_{[4,0]}))q^{\frac{34}{3}}+\mathcal{O}\left(q^{12}\right).$
(39)

####  4.3.2 $k=3$

The global symmetry for $N=3$ and $k=3$ is $G_{\rm flavor}=SU(3)_{a}\times
SU(3)_{b}\times U(1)_{F}$. The formula (14) gives

$\displaystyle\mathcal{I}_{N=3,k=3}^{\mathrm{grav}}$
$\displaystyle=1+(\chi^{a}_{[1,1]}+\chi^{b}_{[1,1]}+1)q^{4}+(\chi^{a}_{[1,1]}\chi^{y}_{[1,0]}+\chi^{b}_{[1,1]}\chi^{y}_{[1,0]}+2\chi^{y}_{[1,0]})q^{\frac{16}{3}}$
$\displaystyle+(u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}+u^{3}+u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}+u^{-3})q^{6}+(\chi^{a}_{[1,1]}\chi^{y}_{[2,0]}+\chi^{b}_{[1,1]}\chi^{y}_{[2,0]}+2\chi^{y}_{[2,0]})q^{\frac{20}{3}}$
$\displaystyle+(u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}\chi^{y}_{[1,0]}+u^{3}\chi^{y}_{[1,0]}+u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}\chi^{y}_{[1,0]}+u^{-3}\chi^{y}_{[1,0]})q^{\frac{22}{3}}$
$\displaystyle+(\chi^{a}_{[1,1]}\chi^{b}_{[1,1]}+\chi^{a}_{[1,1]}\chi^{y}_{[3,0]}+2\chi^{a}_{[1,1]}+\chi^{a}_{[2,2]}+\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}$
$\displaystyle\qquad+\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}+2\chi^{b}_{[1,1]}+\chi^{b}_{[2,2]}+\chi^{b}_{[1,1]}\chi^{y}_{[3,0]}+2\chi^{y}_{[3,0]}+2)q^{8}$
$\displaystyle+(-u^{3}\chi^{y}_{[0,1]}+u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}\chi^{y}_{[2,0]}+u^{3}\chi^{y}_{[2,0]}-u^{-3}\chi^{y}_{[0,1]}+u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}\chi^{y}_{[2,0]}+u^{-3}\chi^{y}_{[2,0]})q^{\frac{26}{3}}$
$\displaystyle+(\chi^{a}_{[0,3]}\chi^{y}_{[1,0]}+5\chi^{a}_{[1,1]}\chi^{y}_{[1,0]}+\chi^{a}_{[2,2]}\chi^{y}_{[1,0]}+\chi^{a}_{[3,0]}\chi^{y}_{[1,0]}+\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}\chi^{y}_{[1,0]}+\chi^{b}_{[0,3]}\chi^{y}_{[1,0]}$
$\displaystyle\qquad+\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}\chi^{y}_{[1,0]}+2\chi^{a}_{[1,1]}\chi^{b}_{[1,1]}\chi^{y}_{[1,0]}+5\chi^{b}_{[1,1]}\chi^{y}_{[1,0]}+\chi^{b}_{[2,2]}\chi^{y}_{[1,0]}+\chi^{b}_{[3,0]}\chi^{y}_{[1,0]}$
$\displaystyle\qquad+4\chi^{y}_{[1,0]}+\chi^{a}_{[1,1]}\chi^{y}_{[4,0]}+\chi^{b}_{[1,1]}\chi^{y}_{[4,0]}+2\chi^{y}_{[4,0]})q^{\frac{28}{3}}$
$\displaystyle+(u^{3}\chi^{a}_{[1,1]}+u^{3}\chi^{a}_{[0,2]}\chi^{b}_{[0,1]}+2u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}+u^{3}\chi^{a}_{[2,1]}\chi^{b}_{[0,1]}+u^{3}\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}+u^{3}\chi^{b}_{[1,1]}$
$\displaystyle\qquad+u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[1,2]}+u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[2,0]}-u^{3}\chi^{y}_{[1,1]}+u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}\chi^{y}_{[3,0]}+u^{3}\chi^{y}_{[3,0]}+u^{3}$
$\displaystyle\qquad+u^{-3}\chi^{a}_{[1,1]}+u^{-3}\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}+u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[0,2]}+2u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}+u^{-3}\chi^{a}_{[1,2]}\chi^{b}_{[1,0]}$
$\displaystyle\qquad+u^{-3}\chi^{a}_{[2,0]}\chi^{b}_{[1,0]}+u^{-3}\chi^{b}_{[1,1]}+u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[2,1]}-u^{-3}\chi^{y}_{[1,1]}+u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}\chi^{y}_{[3,0]}$
$\displaystyle\qquad+u^{-3}\chi^{y}_{[3,0]}+u^{-3})q^{10}$
$\displaystyle+(\chi^{a}_{[0,3]}\chi^{y}_{[0,1]}+3\chi^{a}_{[1,1]}\chi^{y}_{[0,1]}+\chi^{a}_{[3,0]}\chi^{y}_{[0,1]}-\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}\chi^{y}_{[0,1]}+\chi^{b}_{[0,3]}\chi^{y}_{[0,1]}-\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}\chi^{y}_{[0,1]}$
$\displaystyle\qquad+\chi^{a}_{[1,1]}\chi^{b}_{[1,1]}\chi^{y}_{[0,1]}+3\chi^{b}_{[1,1]}\chi^{y}_{[0,1]}+\chi^{b}_{[3,0]}\chi^{y}_{[0,1]}+\chi^{y}_{[0,1]}+\chi^{a}_{[0,3]}\chi^{y}_{[2,0]}+8\chi^{a}_{[1,1]}\chi^{y}_{[2,0]}$
$\displaystyle\qquad+2\chi^{a}_{[2,2]}\chi^{y}_{[2,0]}+\chi^{a}_{[3,0]}\chi^{y}_{[2,0]}+\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}\chi^{y}_{[2,0]}+\chi^{b}_{[0,3]}\chi^{y}_{[2,0]}+\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}\chi^{y}_{[2,0]}$
$\displaystyle\qquad+3\chi^{a}_{[1,1]}\chi^{b}_{[1,1]}\chi^{y}_{[2,0]}+8\chi^{b}_{[1,1]}\chi^{y}_{[2,0]}+2\chi^{b}_{[2,2]}\chi^{y}_{[2,0]}+\chi^{b}_{[3,0]}\chi^{y}_{[2,0]}+9\chi^{y}_{[2,0]}+\chi^{a}_{[1,1]}\chi^{y}_{[5,0]}$
$\displaystyle\qquad+\chi^{b}_{[1,1]}\chi^{y}_{[5,0]}+2\chi^{y}_{[5,0]})q^{\frac{32}{3}}$
$\displaystyle+(2u^{3}\chi^{a}_{[1,1]}\chi^{y}_{[1,0]}+2u^{3}\chi^{a}_{[0,2]}\chi^{b}_{[0,1]}\chi^{y}_{[1,0]}+6u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}\chi^{y}_{[1,0]}+2u^{3}\chi^{a}_{[2,1]}\chi^{b}_{[0,1]}\chi^{y}_{[1,0]}$
$\displaystyle\qquad+u^{3}\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}\chi^{y}_{[1,0]}+2u^{3}\chi^{b}_{[1,1]}\chi^{y}_{[1,0]}+2u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[1,2]}\chi^{y}_{[1,0]}+2u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[2,0]}\chi^{y}_{[1,0]}$
$\displaystyle\qquad+4u^{3}\chi^{y}_{[1,0]}-u^{3}\chi^{y}_{[2,1]}+u^{3}\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}\chi^{y}_{[4,0]}+u^{3}\chi^{y}_{[4,0]}+2u^{-3}\chi^{a}_{[1,1]}\chi^{y}_{[1,0]}$
$\displaystyle\qquad+u^{-3}\chi^{a}_{[1,0]}\chi^{b}_{[0,1]}\chi^{y}_{[1,0]}+2u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[0,2]}\chi^{y}_{[1,0]}+6u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}\chi^{y}_{[1,0]}+2u^{-3}\chi^{a}_{[1,2]}\chi^{b}_{[1,0]}\chi^{y}_{[1,0]}$
$\displaystyle\qquad+2u^{-3}\chi^{a}_{[2,0]}\chi^{b}_{[1,0]}\chi^{y}_{[1,0]}+2u^{-3}\chi^{b}_{[1,1]}\chi^{y}_{[1,0]}+2u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[2,1]}\chi^{y}_{[1,0]}+4u^{-3}\chi^{y}_{[1,0]}$
$\displaystyle\qquad-u^{-3}\chi^{y}_{[2,1]}+u^{-3}\chi^{a}_{[0,1]}\chi^{b}_{[1,0]}\chi^{y}_{[4,0]}+u^{-3}\chi^{y}_{[4,0]})q^{\frac{34}{3}}+\mathcal{O}\left(q^{12}\right).$
(40)

##  5 Conclusions and discussions

In this paper we calculated the superconformal index of ${\cal N}=(1,0)$
theories realized on $N$ M5-branes at the $\mathbb{C}^{2}/\mathbb{Z}_{k}$
singularity. We used the holographic description of the theories, M-theory on
$AdS_{7}\times S^{4}/\mathbb{Z}_{k}$. For small $k$ and $N$ we confirmed that
the indices are consistent with the expected flavor symmetries in Table 1.
Namely, each of the index is expanded in terms of characters of the flavor
symmetry. To see the symmetry enhancement at $N=2$ it is crucial to include
the finite $N$ corrections due to wrapped M2-branes. We use the formula (15)
including the single-wrapping contributions.

In fact for the $N=1$ case the 6d index was reproduced from 5d abelian quiver
theories in [18]. It is nice if one can generalize their results and compare
directly with our results for $N\geq 2$.

There are many ways of extension. We consider only the simple class of ${\cal
N}=(1,0)$ theories realized on a D6-NS5 system. It is known that this is
generalized by introducong D8-branes [2, 3, 4]. Because the corresponding
supergravity solutions are known [19, 20, 21] it would be possible to apply
our method to these theories. Inclusion of orientifold planes and M9-planes
[1, 22] may also be interesting.

In this paper we focus only on the single-wrapping contributions. This is
because we have not yet understood how to determine the integration contours
in the gauge integrals that we need to perform to calculate the contribution
of multiple branes. In the case of $(2,0)$ theory we can define Schur-like
index and it was found in [8] that by adopting an appropriate pole selection
rule we can reproduce the known Schur-like index. Although we cannot define
the Schur-like index for ${\cal N}=(1,0)$ theory, unfortunately, this strongly
suggests that we can reproduce the all order index by including multiple-brane
contributions.

## Acknowledgments

We would like to thank R. Arai and D. Yokoyama for wonderful discussions and
useful comments.

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