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Published for SISSA by Springer

Received: March 28, 2017

Revised: April 29, 2017

Accepted: June 6, 2017

Published: June 12, 2017

Refined counting of necklaces in one-loop

N = 4 SYM

Ryo Suzuki

ICTP South American Institute for Fundamental Research,

Instituto de F́ısica Teórica, UNESP — Universidade Estadual Paulista,

Rua Dr. Bento Teobaldo Ferraz 271, 01140-070, São Paulo, SP, Brazil

E-mail: rsuzuki.mp@gmail.com

Abstract: We compute the grand partition function of N = 4 SYM at one-loop in the

SU(2) sector with general chemical potentials, extending the results of Pólya’s theorem.

We make use of finite group theory, applicable to all orders of perturbative 1/Nc expansion.

We show that only the planar terms contribute to the grand partition function, which is

therefore equal to the grand partition function of an ensemble of XXX 1
2

spin chains. We

discuss how Hagedorn temperature changes on the complex plane of chemical potentials.

Keywords: AdS-CFT Correspondence, Lattice Integrable Models, Supersymmetry and

Duality

ArXiv ePrint: 1703.05798

Open Access, c© The Authors.

Article funded by SCOAP3.
doi:10.1007/JHEP06(2017)055

mailto:rsuzuki.mp@gmail.com
https://arxiv.org/abs/1703.05798
http://dx.doi.org/10.1007/JHEP06(2017)055


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Contents

1 Introduction 1

2 Tree-level counting 4

2.1 Permutation basis of gauge-invariant operators 4

2.2 Sum over partitions 5

2.3 Power enumeration theorem 7

2.4 Counting single-traces 8

2.5 Partition function at finite Nc 10

3 One-loop counting 11

3.1 Mixing matrix 11

3.2 Partition form 12

3.2.1 First term 12

3.2.2 One-loop generating function 14

3.3 Totient form 15

3.3.1 First term 15

3.3.2 One-loop generating function 15

3.4 Comparison with Bethe Ansatz 17

4 Hagedorn transition 18

4.1 Grand partition function 18

4.2 Hagedorn temperature 19

5 Conclusion and outlook 21

A Notation 22

B Details of derivation 22

B.1 Second term in partition form 23

B.2 Second term in totient form 26

1 Introduction

The N = 4 super Yang-Mills theory (SYM) has attracted a lot of attention owing to

its simple and profound structure. Besides being the primary example of the AdS/CFT

correspondence [1], this theory is believed to be integrable in the planar limit [2]. The

integrability enables us to predict various observables at any values of the ’t Hooft coupling;

see [3] for a review.

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As a parallel development, alternative methods have been developed to uncover the

non-planar structure of N = 4 SYM with the gauge group U(Nc) or SU(Nc), based on

finite group theory [4]. New bases of gauge-invariant operators have been discovered,

which diagonalize the tree-level two-point functions at finite Nc [5–8]; see [9] for a review.

With numerous approaches at hand to study individual operators, let us ask questions

complementary to the above line of development. We reconsider the statistical property of

N = 4 SYM, namely the grand partition function including perturbative 1/Nc corrections.

In [10], the tree-level partition function of N = 4 SYM on R × S3 was computed to

investigate its phase space structure. The free energy has the expansion F = N2
c F0 +

F1 + . . . , where F0 = 0 in the confined or low-temperature phase, and F0 > 0 in the

deconfined or high-temperature phase. In the confined phase, the density of states increases

exponentially as the energy increases, leading to the singularity of the partition function

at a finite temperature. There the vacuum undergoes the so-called Hagedorn transition to

the deconfined phase [11, 12]. In the dual supergravity, it is argued that a thermal scalar

in AdS5 × S5 becomes tachyonic at a finite temperature, and condensates into the AdS

blackhole [13].

Below we consider the grand partition function of N = 4 SYM in the low-temperature

phase at one-loop. It amounts to summing up the one-loop anomalous dimensions of all

gauge-invariant operators. The problem simplifies a lot by noticing that we do not need

to take an eigenbasis of the dilatation operator to compute the trace. The main problem

is how to take the trace efficiently in the general setup. The one-loop partition function

without chemical potential has been obtained by using Pólya enumeration theorem in [14].

The Hagedorn transition in the pp-wave/BMN limit was studied in [15, 16]. The grand

partition function with a one-parameter family of chemical potential was given in [17], and

the phase space near the critical chemical potentials was studied in [18].

However, at one-loop the Pólya-type formulae are known only for single-variable cases,

which makes it difficult to obtain the grand partition function with general chemical poten-

tials. In this paper, we incorporate the fully general chemical potentials in the SU(2) sector,

by using finite group theory which is valid to all orders of perturbative 1/Nc expansion.1

In the planar limit, the dilatation operator of N = 4 SYM at one-loop in the SU(2)

sector is the Hamiltonian of XXX 1
2

spin chain. The (canonical) partition function of the

spin chain can be computed by using string hypothesis in the thermodynamic limit. Our

work provides alternative derivation based on the microscopic counting of states, if we take

care of subtle differences to be discussed in section 3.4.

Main results. Let us briefly summarize the main results of this paper. We consider the

grand partition function of N = 4 SYM in the confined phase, with the gauge group U(Nc)

up to one-loop in the SU(2) sector. The theory is put on R × S3 , where R is the radial

1A similar quantity was computed in [19, 20], that is the grand partition function with general chemical

potential at one-loop in the SU(2) sector including O(N2
c ) term. This result involves a multi-dimensional

integral, and less explicit than our counting formula. Another formula was obtained in [21], namely the

trace of the one-loop dimension over the product of two fundamental representations of psu(2, 2|4). This

quantity is a building block in the Pólya-type formula, but not identical to the grand partition function.

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direction of R4 and the radius of S3 is set to unity. We write the grand partition function as

Z(β, ~ω) = tr
(
e−βD+

∑
i ωiJi

)
, D = D0 + λD2 + . . . , (1.1)

where D is the dilatation operator and Ji are the R-charges. We take the trace over all

gauge-invariant local operators in the SU(2) sector, made out of complex scalars {W,Z}.
The grand partition function (1.1) has the weak coupling expansion

Z(β, x, y) = ZMT
0 (x, y)− 2λβZMT

2 (x, y) +O(λ2), λ =
Nc g

2
YM

16π2
, (1.2)

where the operators WmZn are weighted by xmyn. We apply finite group theory to compute

(the perturbative part of) ZMT
0 and ZMT

2 . It will turn out that only the planar term

contributes at one-loop in this setup, even though our methods are valid to all orders of

perturbative 1/Nc corrections.

On top of 1/Nc corrections, there are non-perturbative corrections coming from fi-

nite Nc constraints. At finite Nc , certain combinations of operators vanishes when their

canonical dimension of an operator exceeds Nc . Let us write the exact grand partition

function as

Zexact
Nc (β, x, y) =

∑
m,n≥0

N (p)
m,n(Nc)x

myn −
∑
m,n≥0
m+n>Nc

N (np)
m,n (Nc)x

myn, (1.3)

where the second sum represents the subtraction at finite Nc . If we set x = y = e−β , then

the second sum is of order e−βNc at large β , which are non-perturbative in view of 1/Nc

expansion. Our methods capture the first sum in (1.3), which may contain the corrections

of order 1/N `
c at `-loop. To simplify the notation, we write e.g. ZMT

0 (x, y) instead of

Z
MT(p)
0 (x, y) throughout the paper.

We obtain two expressions of ZMT
2 in section 3, which we call Partition form and

Totient form. The ZMT
2 in Partition form is written as a sum over partitions of operator

length L , r = [1r1 , 2r2 , . . . , LrL ] :

ZMT
2 (x, y) = Nc

∞∑
L=0

∑
r`L

∞∏
k=1

(xk + yk)rk

{
L−

L∑
a=1

θ>(ra)−
L/2∑
a=1

a (ra + 1)θ>(r2a)

− 2
L∑
a<b

θ>(L+ 1− a− b)θ>(ra)θ>(rb)−
L/2∑
a=1

θ>(ra − 1)

}
, (1.4)

where θ>(x) = 1 for x > 1, and vanishes otherwise. The ZMT
2 in Totient form involves Eu-

ler’s totient function Tot(d), which counts the number of relatively prime positive integers

less than d :

ZMT
2 (x, y) = Nc

∞∏
h=1

1

1− xh − yh

×

[ ∞∑
k=1

2

( ∞∑
d=1

Tot(d)
xkdykd

1− xkd − ykd
−
∞∑
L=2

L−1∑
m=1

xkmyk(L−m) δ(gcd(m,L), 1)

)]
. (1.5)

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The equivalence of two results can be checked by expanding both series around the zero

temperature, corresponding to the limit of small x, y. When the gauge group is SU(Nc),

an overall factor of (1− x)(1− y) should be multiplied.

It is straightforward to compute the Hagedorn temperature by using Totient form (1.5).

The Hagedorn temperature TH(λ) in the SU(2) sector is a function of the chemical poten-

tials (ω1, ω2) = (log x+ 1
T , log y + 1

T ), which is given by

TH(λ) =



1

log(eω1 + eω2)

[
1 +

4λeω1+ω2

(eω1 + eω2)2

]
(eω1 > 0 and eω2 > 0, eω1 + eω2 ≥ 1) ,

2

log(e2ω1 + e2ω2)

[
1 +

4λe2ω1+2ω2

(e2ω1 + e2ω2)2

]
(
eω1 < 0 or eω2 < 0, e2ω1 + e2ω2 ≥ 1

)
,

(1.6)

which is valid at large Nc , due to the second sum in (1.3). Roughly said, the first line

represents the deconfinement of W and Z, whereas the second line that of W 2 and Z2.

This result shows that the complex chemical potentials change the location of the Hagedorn

transition, as will be discussed further in section 4.

2 Tree-level counting

We introduce two methods of computing the generating function of the number of gauge-

invariant operators in N = 4 SYM with U(Nc) gauge group. This generating function is

equal to the grand partition function at tree-level.

2.1 Permutation basis of gauge-invariant operators

General gauge-invariant local operators of N = 4 SYM can be specified by an element of

permutation group. We introduce the elementary fields of N = 4 SYM

WA = {∇sΦI ,∇sF,∇sF̄ ,∇sψ,∇sψ̄}, (s ≥ 0), (2.1)

and their polynomials

OA1...AL
α = tr L

[
αWA1WA2 . . .WAL

]
,

=

Nc∑
a1,a2,...,aL=1

(WA1)a1aα(1)(W
A2)a2aα(2) . . . (W

AL)aLaα(L)
, (α ∈ SL).

(2.2)

There is an equivalence relation coming from the relabeling (ai , Ai)→ (aγ(i) , Aγ(i)),

OA1...AL
α = OAγ(1),...,Aγ(L)

γαγ−1 , (∀γ ∈ SL). (2.3)

Each of the equivalence class uniquely specifies a gauge-invariant operator.

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In the SU(2) sector, we restrictWA to the elementary fields to a pair of complex scalars

{W,Z}. We use the gauge degrees of freedom (2.3) to set

Oα = tr L(αWmZn), WAi =

W (1 ≤ i ≤ m)

Z (m+ 1 ≤ i ≤ m+ n).
(2.4)

The residual gauge degrees of freedom is

Oα = Oγαγ−1 , (∀γ ∈ Sm × Sn). (2.5)

This equivalence class uniquely specifies a gauge-invariant operator in the SU(2) sector.

The number of such multi-trace operators at a fixed (m,n) is given by summing over all

solutions of (2.5),2

NMT
m,n =

1

m!n!

∑
α∈Sm+n

∑
γ∈Sm×Sn

δm+n(α−1γαγ−1), (2.6)

where

δL(σ) =

1 (σ = 1 ∈ SL)

0 (otherwise)
(2.7)

The generating function of the number of multi-trace operators is defined by

ZMT
0 (x, y) =

∞∑
m,n=0

Nm,n x
myn, (2.8)

which will be computed below.

2.2 Sum over partitions

First we evaluate NMT
m,n in (2.6). Let us write γ = γW · γZ with γW ∈ Sm and γZ ∈ Sn .

Suppose that γW , γZ have the cycle structure p ` m, q ` n, respectively.

p ` m ⇔ p = [1p1 , 2p2 , . . .mpm ],

m∑
k=1

k pk = m. (2.9)

We also define

rk = pk + qk , r ` m+ n. (2.10)

We look for the general solutions of the condition α−1γαγ−1 = 1 for a fixed γ, which

we call stabilizer Stab(γ). Let us parametrize γ by

γ =
m+n∏
k=1

rk∏
h=1

(g
(k)
h,1 . . . g

(k)
h,k), (2.11)

2This formula is called Burnside’s lemma. The number of operators is related to the large Nc limit of

the tree-level two-point functions [22].

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where (g
(k)
h,1 . . . g

(k)
h,k) is the cyclic permutation defined in appendix A. The identity (A.4) gives

α−1γα =
m+n∏
k=1

rk∏
h=1

(α(g
(k)
h,1) . . . α(g

(k)
h,k)), (2.12)

and the stabilizer condition is solved by

α(g
(k)
h,k) = g

(k)
σ(h),τh(k) , σ ∈ Srk , τh ∈ Zk , (h = 1, 2, . . . , rk). (2.13)

Thus, for each γ, α should belong to the direct product of the wreath product groups,

α ∈
m+n∏
k=1

Srk [Zk] ≡ Stab(γ),
∣∣∣Stab(γ)

∣∣∣ =
m+n∏
k=1

krk rk! . (2.14)

The symbol |G| means the order of the group G.

The number of permutations in Sm with the cycle structure p ` m is given by the

orbit-stabilizer theorem,

|Tp| =
|Sm|∏m

k=1 |Spk [Zk]|
=

m!∏
k k

pk pk!
. (2.15)

We can rewrite NMT
m,n in (2.6) as

NMT
m,n =

1

m!n!

∑
p`m
q`n

|Tp| |Tq|
∣∣∣Stab(γ)γ∈Tp×Tq

∣∣∣ =
∑
p`m
q`n

∏
k

(pk + qk)!

pk! qk!
.

(2.16)

Consider the generating function (2.8). The double sum
∑

m

∑
p`m can be transformed

to an infinite product
∏∞
k=1

∑∞
pk=0 , and thus

ZMT
0 (x, y) =

∞∏
k=1

∞∑
pk=0

∞∑
qk=0

(pk + qk)!

pk!qk!
xkpkykqk =

∞∏
k=1

∞∑
rk=0

(xk + yk)rk =

∞∏
k=1

1

1− xk − yk
.

(2.17)

The first few terms read

ZMT
0 (x, y) = 1 + (x+ y) + 2

(
x2 + xy + y2

)
+
(
3x3 + 4x2y + 4xy2 + 3y3

)
+
(
5x4 + 7x3y + 10x2y2 + 7xy3 + 5y4

)
+ . . . . (2.18)

The series gives the number of multi-trace operators in the SU(2) sector of N = 4 SYM

with U(Nc) gauge group. For SU(Nc) theories, we subtract the terms with p1 > 0 or q1 > 0

in (2.17), which gives

Z̃MT
0 (x, y) = (1− x)(1− y)

∞∏
k=1

1

1− xk − yk
. (2.19)

At finite Nc, fewer terms contribute to the generating function (2.8), which modifies the

expansion (2.18). The precise expression will be reviewed in section 2.5. Our formula (2.17)

is valid up to the order xmyn with m+ n ≤ Nc .

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2.3 Power enumeration theorem

We review another derivation of the tree-level generating function based on Pólya Enumer-

ation Theorem [10].

Consider a single-trace operator with length p. We define the domain D = {1, 2, . . . , p}
and the range R = {Z,W}. A single-trace operator is graphically equivalent to a necklace,

that is the map D → R modulo the action of the cyclic group Zp acting on D,

Single-trace operator ↔ Necklace = Map (Zp\D → R) . (2.20)

We associate the weights in R by c(x, y) = x + y, where xmyn corresponds to the opera-

tor WmZn. Then, Pólya Enumeration Theorem says that the generating function of the

number of graphs (2.20) is given by

ZST
0 (x, y) =

∑
p

ZZp

(
c(x, y), c(x2, y2), . . . , c(xp, yp)

)
, (2.21)

Here ZZp(s1, s2, . . . , sp) is the cycle index of the cyclic group,

ZZp(s1, s2, . . . , sp) =
1

p

∑
h|p

Tot(h) s
p/h
h , (2.22)

where we take a sum over h such that p/h is a positive integer, and Tot(h) is Euler’s totient

function defined by

Tot(h) =

h∑
d=1

δ(gcd(d, h), 1). (2.23)

By combining (2.21) with (2.22) and writing p = hs, we get

ZST
0 (x, y) =

∑
h

∑
s

Tot(h)
(xh + yh)s

hs
= −

∑
h

Tot(h)

h
log
(

1− xh − yh
)
. (2.24)

The first few terms read

ZST
0 (x, y) = (x+ y) +

(
x2 + xy + y2

)
+
(
x3 + x2y + xy2 + y3

)
+
(
x4 + x3y + 2x2y2 + xy3 + y4

)
+
(
x5 + x4y + 2x3y2 + 2x2y3 + xy4 + y5

)
+ . . . . (2.25)

The generating function of multi-trace operator is given by the plethystic exponential

of the single-trace generating function,

ZMT
0 (x, y) = exp

(∑
m=1

ZST
0 (xm, ym)

m

)
=
∏
d=1

1

1− xd − yd
, (2.26)

where we used
∑

j|d Tot(j) = d. This result agrees with (2.17). For SU(Nc) theories, we

subtract the p = 1 term in (2.21),

Z̃MT
0 (x, y) = exp

(
−
∞∑
n=1

xn + yn

n

) ∞∏
k=1

1

1− xk − yk
. (2.27)

in agreement with (2.19).

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2.4 Counting single-traces

For later purposes, we rederive the generating function of the number of single-trace oper-

ators by counting the solutions (2.6) under the constraint α ∈ ZL , with L = m+ n.

We will obtain

NST
m,n =


L∑
d=1

d|m, d|n

(L/d)!

(m/d)!(n/d)!

Tot(d)

L
(m 6= 0, L)

1 (m = 0, L),

(2.28)

which is derived as follows. Suppose m and n are divisible by a positive integer d,

(m,n,L) = (dm′, dn′, d`), m′ + n′ = `, (d = 1, 2, . . . , L). (2.29)

The upper bound of d is L if mn = 0, and Min(m,n) otherwise. Choose µ̃ ∈ Z`d = Zm′d ×Zn
′
d

from Sm × Sn and write α ∈ ZL as

α =
(
a1 . . . a` µ̃

κ(a1) . . . µ̃κ(a`) µ̃
2κ(a1) . . . µ̃2κ(a`) . . . µ̃

(d−1)κ(a1) . . . µ̃(d−1)κ(a`)
)
,

1 ≤ κ < d, gcd(κ, d) = 1.
(2.30)

This set of (µ̃, κ, α) is the general solution to the conditions

α = µ̃αµ̃−1 and α ∈ ZL . (2.31)

Let us parametrize µ = µ̃ as

µ̃ =



(m̃11 m̃12 . . . m̃1d)
...

(m̃m′1 m̃m′2 . . . m̃m′d)

(m̃m′+1,1 m̃m′+1,2 . . . m̃m′+1,d)
...

(m̃`1 m̃`2 . . . m̃`d)


∈ Zm

′
d × Zn

′
d . (2.32)

The number of possible µ̃ chosen from Sdm′ × Sdn′ is3
(dm′)!

dm′m′!

(dn′)!

dn′ n′!
(m′, n′ > 0)

(`− 1)! (m′n′ = 0).

(2.33)

For each µ̃ , we sum over α as parametrized in (2.30). For this purpose we identify

{a1 , . . . , a`} with some of {m̃hk} in (2.32). In order to avoid double counting, we fix

a1 = m̃11 and choose

ah = m̃σ(h)kh (σ ∈ S`−1 , 1 ≤ kh ≤ d) for each 2 ≤ h ≤ d, (2.34)

3The condition µ̃ ∈ Zm
′

d × Zn
′
d is equivalent to µ̃ ∈ T[dm

′
] × T[dn

′
], and the order of the latter group is

given by the orbit-stabilizer theorem (2.15).

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The number of choices of a2 . . . a` is4

d`−1 (`− 1)! . (2.35)

The number of possible κ is Tot(d). Thus, the number of possible α, divided by |Sm×Sn| is

1

|Sm × Sn|
∑

µ∈Sm×Sn

∑
α∈T[m+n]

δm+n

(
α−1µαµ−1

)
=

L∑
d=1

d|m, d|L

(m′ + n′)!

m′!n′!

Tot(d)

L
, (2.36)

which is (2.28). This result is formally correct when m′n′ = 0 thanks to
∑

d|L Tot(d) = L.

Therefore, the tree-level generating function is given by

ZST
0 (x, y) =

∑
L

L∑
m=0

L∑
d=1

d|m, d|L

xmyn
(L/d)!

(m/d)!((L−m)/d)!

Tot(d)

L
. (2.37)

To simplify it, we apply the formulae

∞∑
L

L∑
m=0

L−1∑
d=1

d|m, d|L

fd(m,L−m) =
∞∑
d=1

∞∑
L

L∑
m=0

fd(dm, d(L−m)),

L∑
m=0

xdmyd(L−m) L!

m!(L−m)!
= (xd + yd)L,

(2.38)

to obtain

ZST
0 (x, y) =

∞∑
d=1

∑
L

Tot(d)

d

(xd + yd)L

L
, (2.39)

which is (2.24).

Consider an example. If (m,n) = (3, 3) and d = 3, we find

S3 × S3 ⊃ Z1
3 × Z1

3 3 µ̃ =

{[
(123)

(456)

]
,

[
(132)

(456)

]
,

[
(123)

(465)

]
,

[
(132)

(465)

]}
, (2.40)

which is consistent with {3!/(31 1!)}2 = 4 in (2.33). For each µ̃ we generate

α = (a1 a2 µ̃
κ(a1) µ̃κ(a2) µ̃2κ(a1) µ̃2κ(a2)), (κ = 1, 2). (2.41)

The possible choices of (a1, a2) are (1, 4), (1, 5), (1, 6). Thus

1

|S3 × S3|
∑

µ∈Z3×Z3

∑
α∈T[6]

δ6

(
α−1µαµ−1

)
=

4× 2× 3

3!2
=

2

3
, (2.42)

which agrees with (2.36).

4In other words, we remove the redundancy coming from the overall translation of α ∈ ZL .

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2.5 Partition function at finite Nc

The exact tree-level partition function of N = 4 SYM at finite Nc can be computed

precisely in various methods. We briefly review these arguments, to understand the finite

Nc corrections in (1.3).

A straightforward method is to compute the grand partition function is to evaluate

the path integral of N = 4 SYM action [12]. Let a0 be the zero-mode of the gauge field A0

and U = exp(iβa0). The grand partition function in the complete PSU(2, 2|4) sector is

Zcomplete
Nc

(w) =

∫
dUSU(Nc) exp

(
1

n

∞∑
n=1

{
ζB(wn) + (−1)n+1ζF (wn)

}
tradj (Un)

)
(2.43)

where ζB(w), ζF (w) are functions of chemical potentials, and dUSU(Nc) is the SU(Nc) Haar

measure.5

The complete partition function (2.43) can be reduced to the one in the SU(2) sector

by setting

ζB(wn) = xn + yn, ζF (wn) = 0, (2.44)

which gives Zexact
Nc

(β, x, y) in (1.3). It turns out that the resulting expression is identical to

the Molien-Weyl formula which gives the Hilbert-Poincaré series of GL(Nc) invariants [23].

For example, the Molien-Weyl formula for the gauge group U(Nc) with q variables, corre-

sponding to the SU(q) sector, can be written as6

ZSU(q)
Nc

(x1, . . . , xq) =
1

(2πi)Nc−1

1∏q
i=1(1− xi)n

∮
U

dt1
t1

. . .

∮
U

dtNc−1

tNc−1

n−1∏
r=1

r∏
k=1

χ+
k,r(1, t)

φk,r(x, t)
,

χ±k,r(α, t) = 1− α
r∏
j=k

t±1
j , φk,r(x, t) =

q∏
`=1

χ+
k,r(x`, t)χ

−
k,r(x`, t), (2.45)

where U is the counterclockwise contour of unit radius.7 It can also be written as [5, 26]

ZSU(q)
Nc

(x1, . . . , xq) =
∑
L=0

∑
R`L

∑
Λ`L

Row(Λ)≤q

C(R,R,Λ)sΛ(x1, . . . xq), (2.46)

where C(R,R,Λ) is the Clebsch-Gordan multiplicity defined by R ⊗ R = ⊕ΛC(R,R,Λ) as

SL-modules, and sΛ(x1, . . . xq) is the Schur polynomial. We sum over the partitions Λ

having at most q rows, since Λ is related to the SU(q) global symmetry. At finite Nc we

should sum over the partitions R having at most Nc rows in (2.46).

One can evaluate the formula (2.45) or (2.46) explicitly when Nc and q are small. At

(Nc, q) = (2, 2) we obtain

ZSU(2)
Nc

(x, y) = (ZMT
0 )exact

Nc=2(x, y) =
1

(1− xy)

2∏
k=1

1

(1− xk)(1− yk)
, (2.47)

in agreement with [27] for q = 2. The formula also reproduces the q > 2 cases in [18].

5This is the result for SU(Nc) gauge group. For U(Nc), we replace tradj (Un) by tradj (Un) + 1. Recall

that the U(1) part of N = 4 SYM is free since all interactions are of commutator type.
6The explicit form of the Molien-Weyl formula depends on the choice of basis of the (adjoint) represen-

tation of U(Nc). The convention of [24] is used here for efficient evaluation.
7This formula is elaborated further as the highest weight generating function [25].

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3 One-loop counting

We compute the sum of anomalous dimensions at one-loop in the SU(2) sector in two ways,

which we call Partition form and Totient form. The corresponding generating function gives

the partition function at one-loop.

3.1 Mixing matrix

The dilatation operator of N = 4 SYM is given by [28, 29],

D =
∑
n=0

λnD2n = tr (WW̌ + ZŽ)− 2λ

Nc
: tr [W,Z][W̌ , Ž] : +O(λ2). (3.1)

Let Hm,n be the Hilbert space of all gauge-invariant operators in the SU(2) sector with the

R-charges (m,n). We define the mixing matrix as

D2Oα ≡
2

Nc
(M2)α

β Oβ . (3.2)

On the permutation basis introduced in section 2.1, the mixing matrix inside Hm,n takes

the form [30]

(M2)α
β =

L∑
i 6=j

[
δL([β−1][α]Jiα(j)K)− δL([β−1](ij)[α](ij)Jiα(j)K)

]
,

=
1

m!n!

L∑
i 6=j

∑
µ∈Sm×Sn

δL

(
µβ−1µ−1

{
α− (ij)α(ij)

}
Jiα(j)K

)
.

(3.3)

where we introduced the notation L = m+ n,

JijK =

{
(ij) (i 6= j)

Nc (i = j),
(3.4)

and denoted the equivalence class by

[α] =
1

|Sm × Sn|
∑

γ∈Sm×Sn

γαγ−1. (3.5)

We will evaluate the sum of one-loop dimensions at a fixed (m,n),

〈M2〉m,n =
∑

α,β∈Hm,n

(M2)α
β δβ

α . (3.6)

Since the gauge-invariant operator is in one-to-one correspondence with the equivalence

class (3.5), we can rewrite the sum (3.6) as

〈M2〉m,n =
1

(m!n!)2

∑
α,β∈SL

∑
γ1,γ2∈Sm×Sn

(M2)γ1αγ−1
1

γ2βγ
−1
2 δL

(
γ2β

−1γ−1
2 γ1αγ

−1
1

)
. (3.7)

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According to (3.3), the mixing matrix is invariant inside the same conjugacy class. By

writing γ−1
2 γ1 ≡ γ, we find

〈M2〉m,n =
1

m!n!

∑
α,β∈SL

∑
γ∈Sm×Sn

(M2)α
β δL

(
β−1γαγ−1

)
,

=
1

m!n!

∑
α∈SL

∑
γ∈Sm×Sn

(M2)α
γαγ−1

,

=
∑
α∈SL

(M2)α
α ,

=
1

m!n!

L∑
i 6=j

∑
α∈SL

∑
µ∈Sm×Sn

δL

(
µα−1µ−1 {α− (ij)α(ij)} Jiα(j)K

)
.

(3.8)

Let us inspect the argument of the δ-function. Recall that any permutation can be

decomposed into the product of transpositions, like (1234) = (34)(23)(12). The number

of transpositions defines the parity of a permutation, which is conserved at any orders of

perturbative 1/Nc expansion.8 In particular, odd powers of transpositions cannot become

the identity, and only the planar term i = α(j) contributes in (3.8). Thus,

〈M2〉m,n =
Nc

m!n!

L∑
i 6=j

∑
α∈SL

∑
µ∈Sm×Sn

δL(iα(j))

×
{
δL(µα−1µ−1α)− δL

(
µα−1µ−1(ij)α(ij)

)}
, (3.9)

where L = m+n. The generating function of the sum of one-loop dimensions is defined by

ZMT
2 (x, y) ≡

∞∑
m,n=0

〈M2〉m,n xmyn. (3.10)

3.2 Partition form

We evaluate the sum of dimensions (3.9) by generalizing the methods used in section 2.2.

3.2.1 First term

Consider the first term of (3.9),

〈M2〉(1st)
m,n =

Nc

m!n!

L∑
i 6=j

∑
α∈SL

∑
µ∈Sm×Sn

δL(iα(j)) δL(µα−1µ−1α). (3.11)

We denote the cycle type of µ by p ` m, q ` n and define rk = pk + qk . We parametrize µ

by µ =
∏m+n
k=1

∏rk
h=1

(
m

(k)
h,1m

(k)
h,2 . . .m

(k)
h,k

)
as in (2.11). The condition µα−1µ−1α = 1 imposes

that α should belong to the stabilizer of µ.

Suppose that i and j are part of the cycle of µ of length-a and length-b, respectively.

There are L(L− 1) choices of {i, j}, which can be written as

L(L− 1) =
L∑

a,b=1

ab rarb −
L∑
a=1

ara =
∑
a 6=b

ab rarb +
∑
a

ara(ara − 1). (3.12)

8Conversely said, finite Nc constraints mix permutations with different parity.

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From (2.13) we see that α ∈ Stab(µ) permutes {1, 2, . . . , L} only among those having the

same cycle length in µ. Thus, the condition i = α(j) results in a = b, so we neglect the

terms a 6= b in (3.12).

Define the number of solutions of the two δ-functions in (3.11) for a given (i, j, µ) by

Nsol(a, µ) =
∑
α∈SL

δ(iα(j)) δL(µ−1α−1µα)
∣∣∣
i,j ∈ length-a cycle

. (3.13)

This can be rewritten as

Nsol(a, µ) =
∑

α∈Stab(µ)

δ(iα(j))
∣∣∣
i,j ∈ length-a cycle

, Stab(µ) =

L∏
k=1

Srk [Zk] . (3.14)

If we introduce α = α0 (ij), then

Nsol(a, µ) =
∑

α0∈Stab(µ)

δ(iα0(i))
∣∣∣
i∈ length-a cycle

. (3.15)

Since the group Stab(µ) acts transitively on Sra [Za] ⊂ Sara , the isotropy group satisfies

the property

(Sra [Za])(i) ≡
{
g(i) = i | g ∈ Sra [Za]

}
,

∣∣∣(Sra [Za])(i)
∣∣∣ =

1

ara

∣∣∣Sra [Za]
∣∣∣. (3.16)

Thus, the number of solutions in (3.11) for a given µ ∈ Tp × Tq is

Nsol(p, q) ≡
L∑
i 6=j

∑
α∈SL

δ(iα(j)) δL(µ−1α−1µα)

=

L∑
a=1

ara (ara − 1)Nsol(a, µ)

=

L∑
a=1

θ>(ra) (ara − 1)
∣∣∣Stab(µ)

∣∣∣
µ∈Tp×Tq

,

(3.17)

where

θ>(x) =

{
1 (x > 0)

0 (x ≤ 0).
(3.18)

Proceeding as in (2.17), we obtain the generating function for the first term.

Z
(1st)
2 (x, y) = Nc

∑
m,n

∑
p`m,q`n

|Tp||Tq|
xmyn

m!n!
Nsol(p, q)

= Nc

∑
m,n

∑
p`m,q`n

∏
k=1

(xk)pk(yk)qk
(pk + qk)!

pk! qk!

∑
a

θ>(ra) (ara − 1)

= Nc

∞∑
L=0

∑
r`L

∏
k=1

(xk + yk)rk

(
L−

L∑
a=1

θ>(ra)

)
. (3.19)

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j 1 2 3 4 5 6 7 8 9 10 11

α(j)

1144 348 996

2144 448 1096

548 1196

648

748

848

Table 1. Table of α(j) for 1 ≤ j ≤ 11 and α ∈ S2[Z1] · S3[Z2] · S1[Z3]. The symbol ab means that

the number a appears b times.

Consider an example. Let µ be (1)(2)(3, 4)(5, 6)(7, 8)(9, 10, 11), having r = [12, 23, 31] `
11. Then Stab(µ) = S2[Z1]·S3[Z2]·S1[Z3], which has the order 2×48×3 = 244. Choose i 6= j

from {1, 2, . . . , 11}. The list {α(j) |α ∈ Stab(µ)} for all j is summarized in table 1. The list

shows that the number of solutions to i = α(j) is precisely given by the formula (3.16), like

(i, j) = (1, 2), #sol = 144 =
∣∣∣1 · S3[Z2] · S1[Z3]

∣∣∣ (3.20)

(i, j) = (3, 4), (3, 5), . . . , (7, 8), #sol = 48 =
∣∣∣S2[Z1] · S2[Z2] · S1[Z3]

∣∣∣ (3.21)

(i, j) = (9, 10), (10, 11), (9, 11), #sol = 96 =
∣∣∣S2[Z1] · S3[Z2] · 1

∣∣∣. (3.22)

3.2.2 One-loop generating function

We will analyze the second term of the mixing matrix in appendix B.1. Here we summarize

the results by combining (3.19) and (B.17).

The generating function of the sum of one-loop dimensions over all multi-trace opera-

tors in the SU(2) sector is given by

ZMT
2 (x, y) = Nc

∞∑
L=0

∑
r`L

∞∏
k=1

(xk + yk)rk

{
L−

L∑
a=1

θ>(ra)−Θ(r)

}
,

= Nc

∞∑
L=0

∑
r`L

∞∏
k=1

(xk + yk)rk

{
L−

L∑
a=1

θ>(ra)−
L/2∑
a=1

a (ra + 1)θ>(r2a)

− 2
L∑
a<b

θ>(L+ 1− a− b)θ>(ra)θ>(rb)−
L/2∑
a=1

θ>(ra − 1)

}
.

(3.23)

The summand can be negative for some r ` L, though the sum becomes non-negative if

we sum over all r ` L.9 The first few terms read

ZMT
2 (x, y)

Nc
= 6x2y2 +

(
10x3y2 + 10x2y3

)
+
(
26x4y2 + 36x3y3 + 26x2y4

)
+
(
44x5y2 + 84x4y3 + 84x3y4 + 44x2y5

)
9Negative terms are needed to kill the coefficients of BPS terms.

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+
(
84x6y2 + 176x5y3 + 254x4y4 + 176x3y5 + 84x2y6

)
+
(
134x7y2 + 348x6y3 + 548x5y4 + 548x4y5 + 348x3y6 + 134x2y7

)
+ . . .

(3.24)

The term 6x2y2 is responsible for the one-loop dimensions of the SU(2) Konishi descendant,

which is 3Nc g
2
YM/(4π

2).

3.3 Totient form

We compute the generating function of the sum of one-loop dimensions in another way.

First, we compute the one-loop generating function for single-trace operators by imposing

α ∈ ZL , as done in section 2.4. Then, we conjecture the generating function for multi-

traces, by writing the plethystic exponential of the single-trace results.

3.3.1 First term

Let d ≥ 1 be a divisor of m and n as in (2.29). Specify the cycle type of µ to p = [dm
′
] and

q = [dn
′
] and α to [L] simultaneously. Consider the first term of the one-loop mixing matrix:

〈M2〉(1st)
m,n =

Nc

m!n!

L∑
i 6=j

∑
α∈T[L]

L∑
d=1

d|m, d|L

∑
µ∈T

[dm
′
]
×T

[dn
′
]

δ(iα(j)) δL(α−1µαµ−1). (3.25)

The number of solutions to the stabilizer condition α = µαµ−1 is given by (2.36). For each

(j, µ = µ̃) and α given by (2.30), there is only one i ∈ {1, 2, . . . , L} satisfying i = α(j).

Hence the sum over i, j gives a factor of L, leading to

〈M2〉(1st)
m,n = Nc

L∑
d=1

d|m, d|L

(m′ + n′)!

m′!n′!
Tot(d). (3.26)

Note that d = L is possible only when mn = 0.

3.3.2 One-loop generating function

The second term of the one-loop mixing will be computed in appendix B.2. By combining

the results (3.26) and (B.45), we obtain the one-loop generating function for single-trace

operators as10

ZST
2 (x, y) = Nc

∞∑
L=2

L∑
m=0

xmyn

(
L∑
d=1

d|m, d|L

(L/d)!

(m/d)! (n/d)!
Tot(d)

− (1− δ(mn, 0)) 2 δ(gcd(m,n), 1)− δ(mn, 0) Tot(L)

−
L−1∑
d=1

d|m, d|L

Tot(d)

{
(L/d− 2)!

(m/d− 2)!(n/d)!
+

(L/d− 2)!

(m/d)!(n/d− 2)!

})
, (3.27)

10The sum over L begins with L = 2, because L = 1 are BPS operators.

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with n = L −m. On the first line, d = L is possible only if mn = 0. This contribution is

cancelled exactly by the last term on the second line. It follows that

ZST
2 (x, y)

Nc
= − 2

∞∑
L=2

L−1∑
m=1

xmyn δ(gcd(m,n), 1) (3.28)

+
∞∑
L=2

L∑
m=0

xmyn
L−1∑
d=1

d|m, d|L

(L/d)!

(m/d)! (n/d)!
Tot(d)

×
{

1− m(m/d− 1)

L(L/d− 1)
− n(n/d− 1)

L(L/d− 1)

}
.

We apply the formula (2.38) and simplify the second line as

ZST
2 (x, y)

Nc
= − 2

∞∑
L=2

L−1∑
m=1

xmyL−m δ(gcd(m,L), 1) (3.29)

+
∞∑
d=1

Tot(d)
∞∑
L=2

L∑
m=0

xdmyd(L−m) L!

m! (L−m)!

×
{

1− m(m− 1)

L(L− 1)
− (L−m)(L−m− 1)

L(L− 1)

}
,

= − 2

∞∑
L=2

L−1∑
m=1

xmyL−m δ(gcd(m,L), 1) + 2

∞∑
d=1

Tot(d)

∞∑
L=2

xdyd(xd + yd)L−2,

= − 2

∞∑
L=2

L−1∑
m=1

xmyL−m δ(gcd(m,L), 1) + 2

∞∑
d=1

Tot(d)
xdyd

1− xd − yd
. (3.30)

The first few terms are

ZST
2 (x, y)

Nc
= 6x2y2 + 4x2y2(x+ y) + 2x2y2

(
5x2 + 8xy + 5y2

)
+ 2x2y2

(
4x3 + 9x2y + 9xy2 + 4y3

)
+ 2x2y2

(
7x4 + 14x3y + 24x2y2 + 14xy3 + 7y4

)
+ . . . . (3.31)

In the degeneration limit x = y = z, the generation function (3.30) becomes

ZST
2 (z)

Nc
= −2

∞∑
L=2

zL Tot(L) +

∞∑
d=1

Tot(d)
2z2d

1− 2zd
,

= 2

{
z −

∞∑
L=1

Tot(L) zL
1− 3zL

1− 2zL

}
,

(3.32)

in perfect agreement with [14].

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We conjecture that the one-loop generating function for multi-traces is given by the

plethystic exponential of the singe-trace results (3.30):

ZMT
2 (x, y) = ZMT

0 (x, y)
∞∑
k=1

ZST
2 (xk, yk), (3.33)

= 2Nc

∞∏
h=1

1

1− xh − yh
∞∑
k=1

( ∞∑
d=1

Tot(d)
xkdykd

1− xkd − ykd

−
∞∑
L=2

L−1∑
m=1

xkmyk(L−m) δ(gcd(m,L), 1)

)
.

One can check that its expansion in small x, y agrees with (3.24). The first line is general-

ization of the single-variable case discussed in [14].

3.4 Comparison with Bethe Ansatz

We compare our results with the prediction of Bethe Ansatz Equations (BAEs) for XXX 1
2

spin chain. The single-trace operators of N = 4 SYM in the SU(2) sector correspond to

the level-matched and physical solutions of BAE. The Bethe roots of the level-matched

solutions satisfy

Q(i/2) = Q(−i/2), Q(v) =
∏
j

(v − uj), (3.34)

and in the physical solutions the second Q-function Q̃(v) must be a polynomial in v, as

clarified in [31, 32].

A solution of BAEs is called regular if no Bethe roots are located at infinity. Regular

BAE solutions correspond to the SU(2) highest weight states. If we denote a state with

WmZn by |m,n〉, the highest weight states satisfy

S− |m,n〉HWS = 0, S+ |m,n〉 = |m+ 1, n− 1〉 , (3.35)

where {S±,S3} are the SU(2) generators. We also need m ≥ n to count the BAE solutions

correctly.

We should include exceptional solutions whose energy superficially diverges due to the

Bethe roots at v = ±i/2. In such cases, we should regularize the BAEs by introducing

twists and by carefully taking the zero-twist limit. The results are shown in table 2.

A bit of arithmetic is needed to compare the two sets of numbers in table 2. First,

consider the second row. At (m,n) = (3, 2) we have 10 = 4 + 6, where 6 comes from the

multi-trace (m,n) = (2, 2) + (1, 0). At (m,n) = (4, 2), we have

26 = 10 + (4 + 6 + 6),

Multi-traces (4, 2) = {(3, 2) + (1, 0), (2, 2) + (2, 0), (2, 2) + (1, 0) + (1, 0)}
(3.36)

Next, consider the third row. At (m,n) = (3, 3) we have

36 = 6 + (10) + (2× 6 + 4 + 4)

Descendants (4, 2)→ (3, 3)

Multi-traces (3, 3) = {(2, 2) + (1, 1) . . . , (3, 2) + (0, 1), (2, 3) + (1, 0)} ,
(3.37)

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n

m
1 2 3 4 5

1 0 0 0 0 0

2 6 10 26 44

3 36 84 176

n

m
1 2 3 4 5

1 0 0 0 0 0

2 6 4 10 8

3 6 10 14

Table 2. Average SU(2) one-loop dimensions for WmZn with m ≥ n, in the unit of Nc g
2
YM/(8π

2).

Left table shows the sum over all U(Nc) multi-trace operators, and Right table shows the sum over

all single-trace SU(2) highest weight states.

where . . . means other possible partitions, namely (1, 1) or (1, 0)+(0, 1). At (m,n) = (4, 3)

we have

84 = 10 + (8 + 10) + (10 + 6 + 2× 4 + 2× 4 + 4× 6)

Descendants {(5, 2)→ (4, 3), (4, 2)→ (3, 3) + (1, 0)}
Multi-traces

{
(4, 2) + (0, 1), (3, 3) + (1, 0), (3, 2) + (1, 1) . . . ,

(2, 3) + (1, 1) . . . , (2, 2) + (2, 1) . . .
}
.

(3.38)

The (canonical) partition function of XXX 1
2

spin chain of length L has been computed

in [33]. He also showed that the partition function gives the character of su(2) affine Kac-

Moody algebra at level one in the large L limit, as conjectured in [34]. Their analysis

slightly differs from ours in three points: we compute 〈M2〉 rather than 〈eβM2〉, sum over

the level-matched states, and consider the grand partition function by summing over L.11

4 Hagedorn transition

We compute the grand partition function of N = 4 SYM in the SU(2) sector at one-loop at

large Nc based on the above results. Then we determine the Hagedorn temperature of the

N = 4 SYM in the SU(2) sector, namely the smallest temperature T ≥ 0 at which the grand

partition function diverges, as in (1.6). We will see that the Hagedorn temperature has

numerous branches depending on the value of the chemical potential on the complex plane.

4.1 Grand partition function

Consider the grand partition function of N = 4 SYM in (1.1),

Z(β, ~ω) = tr
(
e−βD+

∑
i ωiJi

)
, D = D0 + λD2 + . . . . (4.1)

The trace is taken over the Hilbert space of all gauge-invariant operators in the SU(2)

sector. The partition function has the weak coupling expansion

Z(β, ~ω) = tr
(
e−βD0+

∑
i ωiJi − λβD2 e

−βD0+
∑
i ωiJi + . . .

)
= ZMT

0 (x, y)− 2λ

Nc
β ZMT

2 (x, y) + . . . ,
(4.2)

11The level-matching condition may be included by introducing another chemical potential coupled to

the total momentum.

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where we used (3.2). We assign the R-charge Ji to complex scalars as

Z : Ji = δi1 , W : Ji = δi2 , NZ =
D0 + J1 − J2

2
, NW =

D0 − J1 + J2

2
. (4.3)

The partition function (1.1) depends on (β, ω1, ω2), whereas the generating functions

in (4.2) depends on (x, y). The two sets of variables are related by12

x = e−β+ω1 , y = e−β+ω2 , w = e−β , β =
1

T
. (4.4)

In particular, the low-temperature expansion corresponds to the expansion in small x, y.

The computation below is valid at large Nc , due to the non-perturbative corrections

in (1.3).

4.2 Hagedorn temperature

We introduce

x ≡ e−β x̃, y ≡ e−β ỹ, (4.5)

and vary T at a fixed (x̃, ỹ). The tree-level generating function (2.17) has simple poles at

T∗ =
k

log (x̃k + ỹk)
, (k = 1, 2, . . . ), (4.6)

and the one-loop generating function (3.33) has double poles at the same location. By using

(x̃+ ỹ)− (x̃k + ỹk)1/k =
(x̃+ ỹ)k − (x̃k + ỹk)∑k−1

j=0(x̃+ ỹ)k−1−j(x̃k + ỹk)j/k
≥ 0,

for (x̃, ỹ) ∈ R+ = {x̃ ≥ 0 and ỹ ≥ 0, x̃+ ỹ ≥ 1} ,

(4.7)

The term k = 1 gives the smallest value of T∗ inside the region R+ . The condition x̃+ỹ ≥ 1

in R+ guarantees T∗ ≥ 0.

We assume that the Hagedorn temperature and the partition function are expanded as

TH(λ) = T∗
[
1 + t1 λ+O(λ2)

]
,

Z(β, ~ω) =
c

T − TH(λ)
=

c

T − T∗

[
1 +

λT∗t1
T − T∗

+O(λ2)

]
.

(4.8)

Let us expand the partition function (4.2) around the pole (4.6) with k = 1 and compare

the result with the above expansion. We find that

TH(λ) =
1

log(x̃+ ỹ)

[
1 +

4λx̃ỹ

(x̃+ ỹ)2

]
, (x̃, ỹ) ∈ R+ . (4.9)

Consider the region outside R+. When we cross the line x̃ + ỹ = 1, then T∗ becomes

negative for all k. As we approach x̃ → 0 keeping x̃ + ỹ ≥ 1, all simple poles in (4.6)

12The tree-level grand partition (1.1) is invariant under the simultaneous shift of (β, ω1, ω2) by ν. This

redundancy is broken at one-loop.

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Figure 1. Plots of Re Ω at λ = 0.05. The left figure shows the plot with (T = 5/3, x̃, ỹ), the middle

figure with (T, x̃ = ỹ), and the right figure with (T, x̃ = −ỹ).

accumulate at T∗ = 1/ log(ỹ). Let us take either x̃ and ỹ negative, where the chemical

potentials (4.4) are shifted by πi. When |x̃|, |ỹ| are large enough, the pole (4.6) with k = 2

becomes the closest to the origin among those giving T∗ > 0. Thus, in the region

R− =
{
x̃ ≤ 0 or ỹ ≤ 0, x̃2 + ỹ2 ≥ 1

}
, (4.10)

the one-loop Hagedorn temperature is given by

TH(λ) =
2

log(x̃2 + ỹ2)

[
1 +

4λx̃2ỹ2

(x̃2 + ỹ2)2

]
, (x̃, ỹ) ∈ R− . (4.11)

More generally, if we put (x̃, ỹ) ∈ C2 on

Arg x̃ =
2π

p1
, Arg ỹ =

2π

p2
, p = lcm(p1, p2), (p1, p2 ∈ Z≥1 , p� N2

c ), (4.12)

the Hagedorn temperature is given by the pole (4.6) at k = p. When p = O(N2
c ), the

Hagedorn transition may take place around 1/ log |x̃+ ỹ|, because the free energy becomes

O(N2
c ) without hitting the pole.

In figure 1, the plots of the grand partition function Ω ≡ −T logZ are shown as

a function of (T, x̃, ỹ). The left figure at a fixed T shows that the singularity of Ω is

associated with the boundary of R± . By comparing the middle figure (T, x̃ = ỹ) and the

right figure (T, x̃ = −ỹ), we find that the former is not invariant under the flip x̃ ↔ −x̃,

whereas the latter is invariant. This pattern is consistent with R± .

For comparison with the literature, we vary T at a fixed (ω̃1 = ω1/β , ξ = y/x). It

follows that

TH(λ) =


1− ω̃1

log(1 + ξ)

[
1 +

4λ ξ

(1 + ξ)2 (1− ω̃1)
+ . . .

]
(x̃, ỹ) ∈ R+ ,

2 (1− ω̃1)

log(1 + ξ2)

[
1 +

4λ ξ2

(1 + ξ2)2 (1− ω̃1)
+ . . .

]
(x̃, ỹ) ∈ R− .

(4.13)

The first line agrees with [14] when ξ = 1, ω̃1 = 0, and with [17] when ξ = 1.13

13Note that (1.2) of [14] is the Hagedorn temperature of the entire N = 4 SYM. The Hagedorn temper-

ature in the SU(2) sector can be found e.g. in [35].

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Let us make a few remarks on the Hagedorn transition.

First, the physical partition function should not diverge. This means that the system

turns into the deconfined phase around the Hagedorn temperature, when the free energy

becomes O(N2
c ). In order to inspect the details of the phase transition, we need to eval-

uate (2.43) in the large Nc limit as in [10]. In the SU(2) sector, it is expected that the

system is described by N2
c + 1 harmonic oscillators around the Hagedorn temperature [18].

Second, the parameter region x̃ < 0 or ỹ < 0 can be interpreted as the insertion of the

number operator (−1)NZ or (−1)NW to the grand partition function (1.1), which makes

Z or W effectively a fermion. The pole at k = 1 disappears when the scalar becomes

fermionic. The pole at k = 2 still contributes to the divergence because Z2 or W 2 are

bosonic. Similarly, when the transition takes place at k = p as in (4.12), Zp,W p are

effectively bosonic. This pattern indicates that only effective bosons form a condensate

inside which U(Nc) degrees of freedom are liberated from the confinement.

Third, the grand partition function at finite Nc is a smooth function of the temperature,

and no transition should happen [11]. This can be checked by evaluating ZSU(2)
Nc

(x, y)

in (2.45). For example, ZSU(2)
Nc=2 (x, y) with x = y = e−β in (2.47) is regular for any β > 0.

More generally, it is conjectured that the denominator of ZSU(2)
Nc

(x, y) at any Nc < ∞ is

always a product of the factors (1 − xayb) for some integers a, b ≥ 0 [24]. Hence, the

Hagedorn temperature is infinite at finite Nc .

5 Conclusion and outlook

In this paper, we computed the grand partition function of N = 4 SYM in the SU(2) sector

at one-loop by making use of finite group theory. Only the planar terms contribute in this

setup, though our result is valid to all orders of perturbative 1/Nc expansion. We derived

two expressions for the one-loop generating function, called Partition form and Totient

form. Based on Totient form we computed the Hagedorn temperature which depends

on general values of the chemical potentials. We argued how the Hagedorn temperature

changes when the chemical potentials are complex.

As future directions of research, one can consider the grand partition functions in more

general situations, such as finite Nc corrections at one-loop, larger sectors of N = 4 SYM,

or higher order in λ in the SU(2) sector. It is also interesting to study superconformal field

theories other than N = 4 SYM, such as β-deformed and γ-deformed theories [36], ABJM

model [37], theories with 16 supercharges [38], and quiver gauge theories [39]. Our counting

methods should be applicable to integrable models like q-deformed Hubbard model [40],

which is a generalization of XXZ spin chain.

Another topic is to develop group-theoretical techniques to study multi-point func-

tions. It is well known that the OPE limit of four-point functions in N = 4 SYM yields

the sum of the anomalous dimensions of intermediate operators, weighted by the square

of OPE coefficients. Such objects have been studied by conformal bootstrap [41] and inte-

grability methods [42–44]. The effects of 1/Nc corrections in such a limit is worth further

investigation.

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a b c da b c a b c

da bca bcab c

Figure 2. Permutations represented as diagrams. Left figure shows (123) ∈ S3 , Middle figure

shows (12)(23) = (132), and Right figure shows (123)(34)(132) = (24).

Acknowledgments

This work is supported by FAPESP grants 2011/11973-4, 2015/04030-7 and 2016/01343-

7, 2016/25619-1. RS thanks Sanjaye Ramgoolam for discussions and comments on

the manuscript.

A Notation

A permutation cycle is denoted by (i1i2 . . . i`). We define the action of permutation by

keeping track of the position of a list. Algebraically, it means

σ : {v(1), . . . , v(n)} 7→ {v′(1), . . . , v′(n)}, v′(n) = v(σ(n)). (A.1)

For example, we have

σ = (123) : v = {a, b, c} 7→ v′ = {b, c, a}, (A.2)

as shown in figure 2. As corollaries, we find

(12) · (23) = (132), (132) · {1} = (23) ·
(

(12) · {1}
)

= (23) · {2} = {3},

(123) · (34) · (132) = (24),
(A.3)

and in general

α · β (n) = β(α(n)), α−1(ij)α = (α(i)α(j)). (A.4)

In Mathematica, Permute[] gives v′(n) = v(σ(n)). For example,

Permute[{a,b,c},Cycles[{{1,2,3}}]] = {c,a,b}

PermutationReplace[{1,2,4},Cycles[{{1,2,3}}]] = {2,3,4} (A.5)

PermutationProduct[Cycles[{{3,4}}],Cycles[{{1,2,3}}]] = Cycles[{{1,2,3,4}}]

B Details of derivation

The second term of (3.9) can be rewritten as

〈M2〉(2nd)
m,n =

Nc

m!n!

∑
α

L∑
i 6=j=1

∑
µ∈Sm×Sn

δ(iα(j)) δL

(
µ0α

−1µ−1
0 α

)
, µ0 = (ij)µ. (B.1)

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This quantity will be computed below. The result is called Partition form when we sum

over α ∈ SL, and Totient form when α ∈ ZL .

B.1 Second term in partition form

We evaluate 〈M2〉(2nd)
m,n in the following steps:

1) Choose µ ∈ Tp × Tq ⊂ Sm × Sn
2) Generate µ0 = (ij)µ by summing over (i, j)

3) Solve the two δ-function constraints simultaneously

1) We denote the cycle type of µ by r ` L. We have rk = pk + qk for 1 ≤ k ≤ L, and∑
µ∈Sm×Sn

f(µ) =
∑
p`m

∑
q`n
|Tp||Tq|f([µ]). (B.2)

2) The cycle type of µ0 depends on how i and j appear inside µ. Let us parametrize

the permutation cycles of µ which contain i , j by (x1 . . . xa). We write (y1 . . . yb) if

i, j belong to different cycles, including the case a = b. Then

(ij)µ ∼

(ij) (x1 . . . xl−1 i xl+1 . . . xa−1j) = (x1 . . . xl−1 i)(xl+1 . . . xa−1 j)

(ij) (x1 . . . xa−1i)(y1 . . . yb−1j) = (x1 . . . xa−1 i y1 . . . yb−1 j)
(B.3)

Thus, the transposition (ij) relates the cycle type of µ and µ0 as

{rl , ra−l , ra} → {rl + 1, ra−l + 1, ra − 1}, if j = µa−l(i) (1 ≤ l ≤ a− 1),

(B.4)

{ra , rb , ra+b} → {ra − 1, rb − 1, ra+b + 1}, if j 6= µm(i) (∀m ∈ Z), (B.5)

{ra , r2a} → {ra − 2, r2a + 1}, if j 6= µm(i) (∀m ∈ Z). (B.6)

Let us count the number of (i, j) corresponding to each line of (B.3). As for the

first line, i.e. splitting, we choose a cycle of length a and split it into l + (a − l), for

1 ≤ l ≤ a − 1. There are ara ways to choose i, and the choice of j is unique for

a given (i, l). As for the second line, i.e. joining, we choose two different cycles of

length a and b. If a 6= b, there are ab rarb ways to identify i , j . And if a = b, there

are a2 ra(ra − 1) ways. In total, we have14

L∑
a=1

a−1∑
l=1

a ra +

L∑
a 6=b=1

ab rarb +

L∑
a=1

a2 ra(ra − 1),

=
L∑
a=1

a(a− 1) ra +
L∑

a,b=1

ab rarb −
L∑
a=1

a2 ra

= L(L− 1).

(B.7)

14We used
∑
a ra = L and rc = 0 for c > L.

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Thus, we replace the sum over (i, j) in (B.1) by the sums over a, b, l shown above.

Schematically, the sum of dimensions is given by

〈M2〉(2nd)
m,n = [ ra → rl + ra−l ]split + [ ra + rb → ra+b ]join + [ 2 ra → r2a ]join . (B.8)

3) We label all possible choices of (i, j) by ζ = 1, 2, . . . , L(L− 1), and denote the cycle

type of µ
(ζ)
0 by r̄(ζ). For each r̄(ζ) we choose α as

α ∈ Stab(µ
(ζ)
0 ) =

L∏
k=1

S
r̄
(ζ)
k

[Zk], (B.9)

to solve the δ-function constraint (B.1). The ζ belongs to either of the two groups

in (B.3).

Next, we solve the planarity condition i = α(j) for the three cases (B.4)–(B.6). Recall

that i and j belong to the cycle of the same length in µ0 to solve the conditions

i = α(j) and α ∈ Stab(µ0), as discussed in section 3.2.1.

As for the splitting case (B.4), only the process r2l → rl + rl can solve the condition

i = α(j). The stabilizer of µ0 is

Stab(µ0)split = Srl+2[Zl] · Sr2l−1[Z2l] ·
∏
k 6=l,2l

Srk [Zk]. (B.10)

There are 2l r2l choices of i, j, including the interchange i ↔ j.15 In order to solve

i = α(j) when i, j appear in the cycles of length l, we write

α = (ij)α0 , α0(i) = i. (B.11)

Here α0 freezes the cycle of i but not of j, which restricts Stab(µ0)split down to

α0 ∈ Stab′(µ0)split = Srl+1[Zl] · Sr2l−1[Z2l] ·
∏
k 6=l,2l

Srk [Zk]. (B.12)

The number of solutions in the splitting process is given by

N ′sol(µ)split =

L/2∑
l=1

θ>(r2l) 2l r2l

∣∣∣Stab′(µ0)split

∣∣∣ =

L/2∑
l=1

θ>(r2l) l (rl + 1)
∣∣∣Stab(µ)

∣∣∣.
(B.13)

As for the joining cases (B.5) and (B.6), any a, b can solve the condition i = α(j).

The stabilizer of µ0 is

Stab(µ0)join =


Sra−1[Za] · Srb−1[Zb] · Sra+b+1[Za+b]

∏
k 6=a,b,a+b

Srk [Zk] (a 6= b)

Sra−2[Za] · Sr2a+1[Z2a] ·
∏

k 6=a,2a
Srk [Zk] (a = b).

(B.14)

15The interchange i↔ j does not change the transposition (ij), but we need to sum over all i 6= j in (B.1).

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For a 6= b there are ab rarb choices of i, j, and for a = b there are a2 ra(ra−1) choices,

including the interchange i ↔ j. In order to solve i = α(j) when i, j appear in the

cycle of length a+ b, we restrict Stab(µ0)join down to

α0 ∈ Stab′(µ0)join =


Sra−1[Za] · Srb−1[Zb] ·

∏
k 6=a,b

Srk [Zk] (a 6= b)

Sra−2[Za] ·
∏
k 6=a

Srk [Zk] (a = b).

(B.15)

The number of solutions in the joining process is given by

N ′sol(µ)join =
∑
a 6=b

a+b≤L

θ>(ra)θ>(rb) ab rarb

∣∣∣Stab′(µ0)a 6=bjoin

∣∣∣
+

L/2∑
a=1

θ>(ra − 1) a2ra(ra − 1)
∣∣∣Stab′(µ0)a=b

join

∣∣∣
=

{ ∑
a 6=b

a+b≤L

θ>(ra)θ>(rb) +

L/2∑
a=1

θ>(ra − 1)

}∣∣∣Stab(µ)
∣∣∣. (B.16)

4) In total, we have

N ′sol ≡ N ′sol(µ)split +N ′sol(µ)join = Θ(r)
∣∣∣∏
k

Srk [Zk]
∣∣∣,

Θ(r) ≡
L/2∑
a=1

a (ra + 1)θ>(r2a) +

L∑
a<b

2 θ>(L+ 1− a− b)θ>(ra)θ>(rb)

+

L/2∑
a=1

θ>(ra − 1),

where we removed the constraint a + b ≤ L by inserting θ>(L + 1 − a − b). Follow-

ing (2.15)–(2.17), we obtain the generating function for the second term as

Z
SU(2),(2nd)
2 (x, y) = Nc

∑
m,n

∑
p`m,q`n

|Tp||Tq|
xmyn

m!n!
Nsol ,

= Nc

∞∑
L=0

∑
r`L

∏
k=1

(xk + yk)rk Θ(r).

(B.17)

Consider an example. Let µ be (1)(23)(456), having r = [11, 21, 31] ` 6. Then Stab(µ) =

Z1 · Z2 · Z3 , which has the order 1× 2× 3 = 6. The possible splitting process is

µ0 = (23)µ = (32)µ = (1)(2)(3)(456), Stab(µ0) = S3[Z1] · Z3 . (B.18)

We have (i, j) = (2, 3) or (3, 2). The list of {α(j) |α ∈ Stab(µ0)} is {16, 26, 36} for j = 2, 3,

in the notation of table 1. Thus, the number of solutions is

N ′sol(µ)split = 6× 2 = 2
∣∣∣Stab(µ)

∣∣∣, (B.19)

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which agrees with (B.13). The possible joining processes are

µ0 =


(123)(456), . . . (ij) = (12), (13)

(23)(1456), . . . (ij) = (14), (15), (16)

(1)(23456), . . . (ij) = (24), (25), (26), (34), (35), (36),

(B.20)

and their stabilizers are

Stab(µ0) =


S2[Z3] r̄ = [32]

Z2 · Z4 r̄ = [21, 41]

Z1 · Z5 r̄ = [11, 51].

(B.21)

The lists of α(j) is

j = 2, α(j) =
{

13, 23, 33, 43, 53, 63
}
, r̄ = [32] (B.22)

j = 4, α(j) =
{

12, 42, 52, 62
}
, r̄ = [21, 41] (B.23)

j = 4, α(j) =
{

21, 31, 41, 51, 61
}
, r̄ = [11, 51]. (B.24)

Thus, the number of solutions is

N ′sol(µ)join = 3× 4 + 2× 6 + 1× 12 = 36 = 6
∣∣∣Stab(µ)

∣∣∣, (B.25)

which agrees with (B.16).

B.2 Second term in totient form

We evaluate 〈M2〉(2nd)
m,n in the following steps:

1) Choose µ ∈ Tp × Tq ⊂ Sm × Sn
2) Generate µ = (ij)µ0 from µ0 in Z`d
3) Generate µ0 = (ij)µ by reverting the last step

4) Solve the two δ-function constraints simultaneously

1) We denote the cycle type of µ ∈ Sm × Sn by p ` m and q ` n. Let m,n be divisible

by d ≥ 1 as in (2.29).

2) Given µ0 ∈ Z`d parametrized as in (2.32), we generate µ = (ij)µ0 . We classify two

cases depending on whether i, j belong to the same or different cycles of µ0 ,

Same : (i, j) = (m̃kh, m̃kd), (1 ≤ k ≤ `, 2 ≤ h ≤ d)

Different : (i, j) = (m̃kd, m̃k′d), (1 ≤ k 6= k′ ≤ `)
(B.26)

which correspond to

(ij)(m̃k1 . . . m̃kd)→ (m̃k1 . . . m̃k,h−1 i)(m̃k,h+1 . . . m̃k,d−1 j), (B.27)

(ij)(m̃k1 . . . m̃kd)(m̃k′1 . . . m̃k′d)→ (m̃k1 . . . m̃kd i m̃k′1 . . . m̃k′d−1 j). (B.28)

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In terms of cycle types of µ0 and µ, these processes can be written as

Same: [d`] → [h, d− h, d`−1]

Different: [d`] → [d`−2, 2d]
(B.29)

We assume ` = 1 in the Same case and ` ≥ 2 in the Different case, because we will

see later that other cases do not contribute. These conditions are equivalent to d = L

and d < L, respectively.

3) We revert the above argument, and generate µ0 ∈ Z`d from µ ∈ Sm × Sn . For the

Same case, we choose

µ ∈

{
Zm × Zn , (mn 6= 0)

Zh × ZL−h (mn = 0),
µ0 ∈ ZL . (B.30)

First, consider the case mn 6= 0. There are (m− 1)!(n− 1)! ways to choose Zm × Zn
from Sm × Sn . Then we choose (i, j) from (Zm ,Zn) or (Zn ,Zm). Using the formula

(m,L)(1, · · · ,m)(m+ 1, · · · , L) = (1, · · · ,m,m+ 1, · · · , L) ∈ ZL , (B.31)

we generate (ij)µ = µ0 ∈ ZL . There are 2mn ways to choose (i, j), giving us the

multiplicity

(m− 1)!(n− 1)! · 2mn = 2m!n!. (B.32)

For any choices we obtain

j =

{
µn0 (i) (if m+ 1 ≤ j ≤ L)

µm0 (i) (if 1 ≤ j ≤ n) .
(B.33)

Next, consider the case mn = 0. The number of choices of µ ∈ Zh×ZL−h from SL is

L!

h (L− h)

(
1 ≤ h < L

2

)
,

L!

2(L/2)2

(
h =

L

2

)
, (B.34)

and there are 2h(L− h) ways to choose (i, j) for any h, giving the multiplicity16

L!

2h (L− h)
· 2h(L− h) = L!, (h = 1, 2, . . . L− 1) . (B.35)

For any choices, we have j = µh0(i) or µL−h0 (i).

For the Different case, we set

µ ∈
(
Zm

′−2
d × Z2d

)
× Zn

′
d or Zm

′
d ×

(
Zn
′−2
d × Z2d

)
, µ0 ∈ Zm

′+n′

d = Z`d . (B.36)

The case with mn = 0 is allowed. The number of choices of
(
Zm

′−2
d × Z2d

)
× Zn′d

from Sm × Sn is

|Sm × Sn|
|Stab(Zm′−2

d × Z2d)× Stab(Zn′d )|
=

m!n!

dm′+n′−2 (2d) (m′ − 2)! (n′)!
. (B.37)

16We added an extra factor of 1/2 by extending the summation range of h.

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The other case m′ ↔ n′ can be treated similarly. Then we choose (i, j) from Z2d and

use the formula

(mdm2d)(m1m2 . . .m2d) = (m1m2 . . .md)(md+1md+2 . . .m2d) ∈ Z2
d , (B.38)

to generate µ0 ∈ Z`d . There are 2d ways to choose such (i, j) from Z2d , which gives

the multiplicity

m!n!

dm′+n′−2 (2d) (m′ − 2)! (n′)!
(2d) =

m!n!

d`−2 (m′ − 2)! (n′)!
(B.39)

4) Given (i, j, µ0), we construct α via (2.30). This equation is repeated below:

α =
(
a1 . . . a` µ̃

κ(a1) . . . µ̃κ(a`) µ̃
2κ(a1) . . . µ̃2κ(a`) . . . µ̃

(d−1)κ(a1) . . . µ̃(d−1)κ(a`)
)
,

1 ≤ κ < d, gcd(κ, d) = 1.

(B.40)

Now we solve i = α(j), keeping in mind that (a1, a2, . . . a`) in (B.40) belong to the

different cycles of µ̃ ≡ µ0 .

For the Same case, namely when i, j belong to the same cycle of µ0 , the equation

i = α(j) has a solution only if

` = 1, d = L, i = µ̃κ(j) ⇔ α = µ̃κ. (B.41)

From (B.33) we find κ = m or n for mn 6= 0, and κ = h, L− h for mn = 0.

For the Different case, the general solution is

j = µ̃ωκ(aξ), i = µ̃ωκ(aξ+1), or j = µ̃ωκ(a`), i = µ̃(ω+1)κ(a1),

` ≥ 2, d ≤ L/2, (1 ≤ ξ ≤ `− 1, 0 ≤ ω ≤ d− 1).
(B.42)

We use the overall translation of α ∈ ZL to fix j = a1 and i = a2 , for both solutions

in (B.42). Then α is given by

α = (j i a3 . . . a` µ̃(j)µ̃κ(i) . . . µ̃κ(a`) . . . µ̃
(d−1)κ(j) . . . µ̃(d−1)κ(a`)) (B.43)

The number of choices of a3 . . . a` or a2 . . . a`−1 is

d`−2 (`− 2)! (B.44)

5) We summarize the above calculation. The second term of the sum of dimensions

consists of two parts:

〈M2〉(2nd)
m,n =

Nc

m!n!

(
NSame
m,n +NDifferent

m,n

)
. (B.45)

The number of solutions in the splitting case is

NSame
m,n =

2m!n! δ(gcd(m,n), 1), (mn 6= 0)

L! Tot(L) (mn = 0).
(B.46)

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where we used (2.23) in the last line. The number of solutions in the joining case is17

NDifferent
m,n =

L−1∑
d=1

d|m, d|n

m!n!

d`−2 (m′ − 2)! (n′)!
d`−2 (`− 2)! Tot(d) +

(
m′ ↔ n′

)
,

=

L−1∑
d=1

d|m, d|n

m!n! Tot(d)

{
(L/d− 2)!

(m/d− 2)!(n/d)!
+

(L/d− 2)!

(m/d)!(n/d− 2)!

}
.

(B.47)

Consider some examples. Suppose (m,n) = (7, 2) and consider the Same case,

µ ∈ Z7 × Z2 , µ0 ∈ Z9 . (B.48)

There are 6! × 1! = 720 ways to choose such µ from S7 × S2 . Then we choose (i, j) from

two cycles, like

(ij)µ = (79)(1234567)(89) = (123456789) = µ0 ∈ Z9 . (B.49)

There are 2×7×2 = 28 ways to choose (i, j), and all of them satisfy j = µ2
0(i) or j = µ7

0(i).

The solution of µ0α
−1µ−1

0 α = 1 is

α = (aµκ0(a)µ2κ
0 (a) . . . µ8κ

0 (a)), gcd(κ, 9) = 1. (B.50)

The condition i = α(j) requires κ = 2, 7. Therefore, the number of Same solutions is

720× 28 = 20160 = 7!× 2!× 2, (B.51)

in agreement with (B.46).

If µ ∈ S6 × S3 , then there is no contribution from the Same case due to gcd(3, 6) 6= 1.

In the Different case we can generate µ0 ∈ Z3
3 . There are 5! × 2! = 240 ways to choose

µ ∈ Z6 × Z3 from S6 × S3. We choose (i, j) from Z6 , like

(ij)µ = (36)(123456)(789) = (123)(456)(789) = µ0 ∈ Z3
3 . (B.52)

There are 6 choices of (i, j). The solution of µ0α
−1µ−1

0 α = 1 can be written as

α = (a1a2a3 µ
κ
0(a1)µκ0(a2)µκ0(a3) . . . µ2κ

0 (a3)), κ ∈ {1, 2}. (B.53)

We look for the solutions of i = α(j). By overall translation we put

(i, j) = (a2, a1). (B.54)

If (i, j) = (3, 6), there are 3 ways to choose a3 from (789). The multiplicity for other (i, j)

is identical. Therefore, the number of Different solutions is

240× 6× 3× Tot(3) = 6!× 3!× 2, (B.55)

which agrees with (B.47).

17If m′ ≤ 1 or n′ ≤ 1, one of the two terms in (B.47) vanishes due to (−1)! =∞.

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Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

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	Introduction
	Tree-level counting
	Permutation basis of gauge-invariant operators
	Sum over partitions
	Power enumeration theorem
	Counting single-traces
	Partition function at finite N(c)

	One-loop counting
	Mixing matrix
	Partition form
	First term
	One-loop generating function

	Totient form
	First term
	One-loop generating function

	Comparison with Bethe Ansatz

	Hagedorn transition
	Grand partition function
	Hagedorn temperature

	Conclusion and outlook
	Notation
	Details of derivation
	Second term in partition form
	Second term in totient form