Soryushiron Kenkyu

素粒子論研究 ･ 電子版 Vol. 34 (2021) No. 3

Doctoral Thesis

Finite N Corrections
to the Superconformal Index
in AdS/CFT Correspondence

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

February, 2021

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Abstract
In this thesis, we propose a new calculation method of the superconformal index by using
the Anti-de Sitter (AdS)/Conformal Field Theory (CFT) correspondence.
AdS/CFT is a correspondence between a four-dimensional conformal field theory (CFT)and type IIB string theory on AdS5 × M5 (AdS), where M5 is a five-dimensional compact
manifold. This is the strong/weak duality, and one can use AdS/CFT to study the strongly
coupled field theories from the theory on the AdS side. There are many AdS/CFTs
depending on the compact manifold M5 . The simplest example is M5 = S 5 , and thecorresponding CFT is N = 4 U (N ) supersymmetric Yang-Mills (SYM) theory. There arealso AdS/CFTs in which the Lagrangian on the CFT side is not known. The interesting
example is AdS/CFT between a Zk S-fold theory and type IIB string theory on AdS5 ×
S 5 /Zk , where the S-fold theory is an N = 3 superconformal field theory (SCFT) andalways strongly coupled. In this case, AdS/CFT is a significant tool to study the S-foldtheories. To understand strongly coupled field theories, the study of theories for which
the Lagrangian is not known is quite important.
It is possible to understand AdS/CFT from D3-branes in type IIB string theory, where
the D3-brane is a 3+1-dimensional object. A rank N CFT is realized on the worldvolume of N D3-branes, while D3-branes produce the AdS geometry. This is the origin of
AdS/CFT. In the large N limit, the AdS side is well described by classical gravity, andthere are many applications to the analysis of strongly coupled theories, even for the theories for which the Lagrangian is not known. On the other hand, if N is finite, it is believedthat the theory on the AdS side is described by quantum gravity. Since no one knows
quantum gravity, people think that it is diﬃcult to calculate the physical quantities on
the CFT side from the AdS side when N is finite.
However, it may be possible to calculate the superconformal index by using AdS/CFT
due to the simplification by the supersymmetry even though N is finite. The superconformal index has the information of the BPS operator spectrum of a given SCFT and
is independent of the coupling constant. Thus, if we calculate the superconformal index
for an SCFT, one can immediately obtain the non-trivial information of the SCFT eventhough the Lagrangian is not known. In fact, the calculation method of the superconformal index on the AdS side in the large N limit is already known, and the application to
S-fold theories has also been performed by using AdS/CFT. Furthermore, the candidates
that contribute to the superconformal index on the AdS side in the finite N region are
known: the D3-branes wrapped on three-cycles in the compact manifold M5 .

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Abstract

Inspired by these previous researches, we develop the calculation method of the superconformal index on the AdS side when N is finite. To achieve this purpose, we adopt thefollowing strategy. First, we use the well-known AdS/CFT: the correspondence between
N = 4 U (N ) SYM and type IIB string theory on AdS5 × S 5 . In this case, the superconformal index is calculable using the Lagrangian on the CFT side. The results thenprovide us a hint to search for the calculation method of the superconformal index on the
AdS side in the finite N region. Concretely, we analyze the single wrapped D3-branes andcompare the results with the CFT results. Then, we propose the calculation method of
the superconformal index on the AdS side in the finite N region. We see that our results
agree with the CFT results up to contributions from the multiple wrapped D3-branes.
Second, we apply our method to S-fold theories. Then, we can predict the superconformal index of the finite rank S-fold theories from the AdS side. Since it is expected thatthe rank one and two S-fold theories are equivalent to N = 4 SYM, we can carry out
the consistency check. Besides this check, recently, the superconformal index of the rank
three Z3 S-fold theory was calculated using the renormalization group flow, and we canconfirm the correctness of our method for this case. All the results given in this thesis are
consistent with the known facts.

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Acknowledgment
First of all, the author expresses his gratitude to his supervisor Yosuke Imamura for hispatient and kind guidance throughout his five-year career in the particle theory groupin Tokyo Institute of Technology. The author also gives his thanks to the faculty of theparticle theory laboratory in Tokyo Institute of Technology, Katsushi Ito and Yoshiyuki
Watabiki. The author appreciates his collaborators for discussions and their guidance,
Shota Fujiwara, Yosuke Imamura, Tatsuya Mori, and Daisuke Yokoyama. He also thanks
to the following people for stimulating discussions, Testuji Kimura, Yoshishige Kobayashi,
Shoichi Kanno, Masaya Yata, Yugo Abe, Naoto Kan, Yusuke Kanayama, Hirotaka Kato,
Hongfei Shu, Takafumi Okubo, Saki Koizumi, Yusuke Nagatsuka, Takashi Hagiwara, Kohei Fujikura, Kensuke Akita, Stephan Hübsch, Kohei Kuroda, Takayasu Kondo, Shota
Fujiwara, Tatsuya Mori, Yota Endo, Keita Imaizumi, Rei Masuda, Keisuke Sawaizumi,
Kodai Kitahara, Shingo Horikawa, Shuichi Murayama, Sumiya Umezu, Mingshuo Zhu,
Ibuki Hashimoto, Jingjing Yang and all the members of the fundamental physics group.
The author thanks all the universities and institutes that have provided the nice research
environments, especially thanks to the Center for Field Theory and Particle Physics inFudan University, the particle theory group in Rikkyo University, Komaba particle theory group in University of Tokyo, Riken iTHEMS, OCU research group for mathematicalphysics in Osaka City University, and the particle theory group in Saitama University
for inviting the author as a speaker of a seminar and stimulating discussions with themembers of those universities and institutes.
The author appreciates the financial support from the Sasakawa Scientific Research
Grant of The Japan Science Society.
Finally, the author thanks his parents and sister for their warm support in all aspects.

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Contents
Chapter 1
1.1
1.2
1.3
1.4

1.5

1.6
Chapter 2
2.1

2.2

2.3

2.4

Chapter 3

Introduction
Partition function . . . . . . . . . . . . . . . . . . . . . . . . . . .
Witten index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Superconformal field theory . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Conformal symmetry . . . . . . . . . . . . . . . . . . . . .
1.4.2 Superconformal symmetry . . . . . . . . . . . . . . . . . . .
1.4.3 Deformation of CFTs . . . . . . . . . . . . . . . . . . . . .
1.4.4 Theories for which the Lagrangian is unknown . . . . . . .
Superconformal index . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Chiral ring . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Superconformal index for Lagrangian theories . . . . . . . .
BPS partition function . . . . . . . . . . . . . . . . . . . . . . . . .

9
13
16
18
20
20
23
26
28
29
30
32
34

AdS/CFT CorrespondenceFour-dimensional N = 4 supersymmetric Yang-Mills theory . . . .
2.1.1 N = 1, 2 decomposition . . . . . . . . . . . . . . . . . . . .
2.1.2 Coulomb branch and Higgs branch . . . . . . . . . . . . . .
2.1.3 BPS partition function . . . . . . . . . . . . . . . . . . . .
2.1.4 Superconfromal index . . . . . . . . . . . . . . . . . . . . .
Type IIB string theory . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Closed strings . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Open strings . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Type IIB supergravity . . . . . . . . . . . . . . . . . . . . .
The correspondence . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 D3-brane picture . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Black 3-brane solution . . . . . . . . . . . . . . . . . . . . .
Agreement of the index in the large N limit . . . . . . . . . . . . .
2.4.1 BPS partition function . . . . . . . . . . . . . . . . . . . .
2.4.2 Superconformal index . . . . . . . . . . . . . . . . . . . . .

37
37
41
42
44
49
54
55
56
57
61
62
64
65
66
66
68

Finite N Corrections to the Superconformal Index

73

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Contents
3.1
3.2

BPS partition function . . . . . . . . . . . . . . . . . . . . . . . . .
Superconformal index . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Supersymmetry on wrapped D3-branes . . . . . . . . . . .
3.2.2 Superconformal index of wrapped D3-brane . . . . . . . . .
3.2.3 Comparison with the CFT side . . . . . . . . . . . . . . . .

75
83
83
85
88

Chapter 4
4.1

Four-Dimensional S-fold Theories
N = 4 orientifold theory . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 AdS/CFT correspondence in the orientifold background . .
4.1.2 BPS partition function . . . . . . . . . . . . . . . . . . . .
4.1.3 Superconformal index . . . . . . . . . . . . . . . . . . . . .
S-fold theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 AdS/CFT correspondence in the S-fold background . . . .
4.2.2 Properties of S-fold theories . . . . . . . . . . . . . . . . . .
BPS partition function of S-fold theories . . . . . . . . . . . . . . .
Large N limit on the AdS side . . . . . . . . . . . . . . . . . . . .
4.4.1 BPS partition function . . . . . . . . . . . . . . . . . . . .
4.4.2 Superconformal index . . . . . . . . . . . . . . . . . . . . .
Finite N corrections . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 BPS partition function . . . . . . . . . . . . . . . . . . . .
4.5.2 Superconformal index . . . . . . . . . . . . . . . . . . . . .
Consistency check . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Supersymmetry enhancement . . . . . . . . . . . . . . . . .
4.6.2 Comparison with Zafrir’s result . . . . . . . . . . . . . . . .

91
91
92
94
97
99
102
103
106
108
109
109
113
113
117
120
120
121

Conclusions and Discussions

123

4.2

4.3
4.4

4.5

4.6

Chapter 5

Appendix A Conventions and Notations
127
A.1
Spinor convention for four-dimensional theories . . . . . . . . . . . 127
A.2
Notations of group theory . . . . . . . . . . . . . . . . . . . . . . . 129
A.2.1 Dynkin label . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.2.2 Character formula . . . . . . . . . . . . . . . . . . . . . . . 130
Bibliography

131

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Chapter 1

Introduction
Quantum field theories (QFT) has been extensively studied in elementary particle physics.
In particular, the weak coupling region of QFT is well understood by the perturbation
method. On the other hand, it is diﬃcult to analyze the strong coupling region of QFT
because the perturbation method is not applicable. Nevertheless, strongly coupled field
theories often appear in particle physics. For example, quantum chromodynamics (QCD)
is an asymptotically free theory and becomes the strongly coupled theory in the low
energy region. One of the quite important problems in QCD is to understand color
confinement. So far, no one understands color confinement in an analytical way, eventhough the lattice QCD suggests that color confinement should occur. To solve the color
confinement problem, the understanding of the strongly coupled field theory is essential.
Although the strong coupling region of the realistic QFT is very diﬃcult to analyze,
one can learn many things about the strongly coupled physics from the supersymmetricfield theories. The supersymmetry is the symmetry between bosonic fields and fermionicfields. Ths supersymmetry algebra is generated by N Lorentz spinor generators. The
restriction for the theory is stronger as N increases. It is known that N is smaller thanor equal to four for four-dimensional field theory without gravity.
N = 1 supersymmetric theories are important for phenomenology. Since the restrictionof N = 1 supersymmetry for the theory is relatively weak, one can construct various
phenomenological models with N = 1 supersymmetry. However, N = 1 supersymmetry
is not enough to solve the theory. On the other hand, N = 4 supersymmetry gives
the quite severe constraint to the theory, and it is believed that N = 4 supersymmetric
Yang-Mills (SYM) theory [1, 2] is the only theory having N = 4 supersymmetry in fourdimension. Therefore, an N = 4 theory is uniquely specified by the gauge group only.
Although this theory cannot be the realistic theory, it is still interesting as a toy model
to learn the strongly coupled physics because the quantum corrections to the theory aresuppressed due to the boson/fermion cancellation. Furthermore, since the β-function ofthis theory is zero, N = 4 SYM also has the conformal symmetry. In total, N = 4 SYM
has the N = 4 superconformal symmetry.
For N = 2 supersymmetric theories, the restriction is not so strong. However, it isenough to calculate the low energy eﬀective action, which is obtained by integrating out

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Chapter 1 Introduction

the massive modes in the theory. In fact, Seiberg and Witten determined the low energy
eﬀective action of N = 2 SU (2) gauge theory [3, 4] in 1994. They used a certain Riemannsurface to obtain the low energy eﬀective action. Such a Riemann surface is called the
Seiberg-Witten (SW) curve. The understanding of the non-perturbative eﬀect in the
supersymmetric field theory has been greatly progressed by their success. For instance,
one can explain the electric charge confinement via the monopole condensation by breaking
N = 2 to N = 1.
In addition to the understanding of the non-perturbative eﬀects of QFT, the diversity
of QFT is recognized more than before by Seiberg and Witten’s analysis. The significant
fact is that there are many conformal field theories (CFT) that the construction of the
Lagrangian is uncertain. Since it looks like that these theories have no tunable parameters, there is no general procedure that gives the Lagrangian (or, there are few tunable
parameters that can help us find the Lagrangian). The famous examples of such theories
are Argyres-Douglas (AD) theories [5, 6]. AD theories arise at the superconformal fixed
point in which the mutually non-local BPS particles become massless [5–9]. Note that, inprinciple, the existence of the ultra-violet (UV) Lagrangian that flows to a target theory
in the renormalization group (RG) flow is enough to analyze the theory. In fact, the UV
Lagrangian that flows to a certain AD theory was discovered by Maruyoshi and Song [10],and some physical quantities were discussed. Although it is quite diﬃcult to analyze theories for which neither the Lagrangian nor UV Lagrangian are known, one may be able
to analyze them by using dualities.
A duality is a correspondence between outwardly diﬀerent but physically equivalent
theories. In the duality, it often occurs that a strongly coupled theory corresponds toa weakly coupled theory. In such a situation, the duality enables us to analyze thestrongly coupled theory via the dual weakly coupled theory. For example, in Seiberg andWitten’s analysis, the electric-magnetic duality, an example of strong/weak duality, plays
an important role.
Another significant example of strong/weak duality is the AdS/CFT correspondence
proposed by Maldacena [11] in 1997. Roughly speaking, the AdS/CFT correspondence isa correspondence between a d-dimensional CFT and a d + 1-dimensional gravity theory
on Anti-de Sitter (AdS) spacetime. This correspondence can be understood throughthe string theory. The string theory is a theory of strings defined on ten-dimensional
spacetime, and there are p + 1-dimensional objects called Dp-branes as well as strings.
The p + 1-dimensional field theory is realized on the worldvolume of Dp-branes [12],whereas the AdS spacetime is realized near the D-branes [13]. Although this observation
is far from the proof of AdS/CFT, it is enough to convince us that AdS/CFT holds.
It is known that the strong coupling region on the CFT side corresponds to the classicalgravity on the AdS side. Thus, AdS/CFT provides us a powerful tool to investigate thestrong coupling region of QFT. One can construct AdS/CFT that the theory on the CFT
side is a QCD-like theory by arranging the D-branes in the string theory. The SakaiSugimoto model [14, 15] is known as an example of such the case. One can calculate theSoryushiron Kenkyu

11
mass of mesons in the Sakai-Sugimoto model, and it is a good example of applications of
AdS/CFT to QCD. It is also known that if the rank of CFT (number of D-branes in thestring theory setup) is finite, the corresponding gravity theory is quantum gravity. Then
it is possible to investigate quantum gravity by using AdS/CFT. However, this fact is an
obstacle to study the strongly coupled CFT with the finite rank when we use AdS/CFT.
Physical quantities protected by the supersymmetry may overcome this obstacle dueto the simplification by the supersymmetry. Candidates of such quantities are the superconformal index and the BPS partition function [16]. The superconformal index is akind of partition functions defined in a superconformal field theory (SCFT). It has information regarding the operator spectrum of the theory. Also, the superconformal index
does not depend on the coupling constant and is calculable by using the Lagrangian on
the CFT side. Once we calculate the superconformal index, we can obtain the non-trivial
information of the theory. Thus the calculation of the superconformal index for SCFTs,
for which the Lagrangian is not known, is the first goal of the analysis of such SCFTs.
Similarly, the BPS partition function also has information about the operator spectrumof the theory. Moreover, the structure of the BPS partition function is much simpler than
the superconformal index, and it is useful to speculate the structure of the superconformal
index.
In this thesis, we would like to calculate the superconformal index of the finite rank
SCFTs, for which the Lagrangian is not known, by using AdS/CFT in the finite N region,
where N is the rank of a given SCFT. Then, we need to develop the calculation methodof the superconformal index on the AdS side when N is finite. This research also helps toexpand the scope of applications of AdS/CFT.
For this purpose, it is useful to consider the simplest AdS/CFT: the correspondence
between N = 4 U (N ) SYM and type IIB string theory on AdS5 × S 5 . In this case,the Lagrangian on the CFT side is known, and the superconformal index is calculable.
The concrete expression of the superconformal index provides us a hint to search for the
calculation method of the superconformal index on the AdS side. First, AdS/CFT in
the N → ∞ limit (large N limit) is well studied because the corresponding theory on
the CFT side is classical gravity. Actually, the superconformal index on the AdS side
was calculated as contributions of Kaluza-Klein (KK) modes on S 5 , and the agreementwith that on the CFT side was confirmed [16]. Second, when N is finite, there are newcontributions coming from the D3-branes wrapping on three-cycles in S 5 as well as KK
modes. The existence of supersymmetric configurations of D3-branes expanded in S 5
was first shown in [17]. The calculation method of the superconformal index for suchwrapped D3-branes has not been known so far, and we propose the formula to calculate
the contributions of wrapped D3-branes on the AdS side. Based on the formula, we also
confirm the agreement of the superconformal index with that on the CFT side up to acertain order of the parameters in the index. A similar analysis of the BPS partition
function is also given.
Once we develop the calculation method of the superconformal index on the AdS side

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Chapter 1 Introductionin the finite N region, we can apply our method to SCFTs for which the Lagrangian is notknown. We focus on the S-fold theories as examples of such theories. Namely, we calculate
the superconformal index of the S-fold theories as well as the BPS partition function. An
S-fold theory is an N = 3 SCFT and constructed by Garcia-Etxebarria and Regalado [18]in 2015. It is well known that any N = 3 theories connected with the free theory bytuning the parameters have N = 4 supersymmetry automatically. So the existence of
genuine N = 3 theories that do not have N = 4 supersymmetry were uncertain until
the discovery of S-fold theories, where a genuine N = 3 theory is an N = 3 theory thatnever has the N = 4 supersymmetry. Since S-fold theories are constructed by using D3branes in type IIB string theory, we can consider the AdS/CFT correspondence for S-foldtheories [19, 20]. Thus, it is possible to investigate S-fold theories by using AdS/CFT. We
apply our formula mentioned in the last paragraph to find the superconformal index of
S-fold theories. Because it is conjectured that some of the S-fold theories are equivalent
to N = 4 SYM with a certain gauge group (Aharony-Tachikawa conjecture) [20], we
can compare our result with the index of corresponding N = 4 SYM. Moreover, the UV
Lagrangian of some of the S-fold theories were given by Zafrir [21], and the index was
calculated by using the UV Lagrangian. We can also compare our result with Zafrir’s
result.
The organization of this thesis is as follows.
In the rest of this chapter, we review some basic concepts to define the superconformal
index and the BPS partition function. In particular, we review the Witten index, theconformal and superconformal symmetry. Then we define the superconformal index and
the BPS partition function. We also discuss SCFTs for which the Lagrangian is notknown.
In Chapter 2, we review the AdS/CFT correspondence between N = 4 U (N ) SYM andtype IIB string theory on AdS5 × S 5 . We start from the review of N = 4 SYM and givethe superconformal index and the BPS partition function of N = 4 U (N ) SYM. After
that, we see the elementary facts of type IIB string theory. Then we consider AdS/CFT
correspondence. The agreement of the index and the BPS partition function in the large
N limit is also reviewed.
In Chapter 3, we calculate the finite N corrections to the BPS partition function andthe superconformal index following the author’s and his collaborators’ paper [22]. Afterthe calculation, we compare our results with those on the CFT side.
In Chapter 4, we review the S-fold theories following [18–20]. First, we consider theorientifold and calculate the BPS partition function of N = 4 SO(N ) SYM. Then we define
the S-fold and S-fold theories. We calculate the finite N corrections to the BPS partition
function and the superconformal index following the author’s and his collaborators’ papers[22, 23]. The results of orientifold are compared with those on the CFT side. For Sfold theories, we also carry out the consistency check regarding the Aharony-Tachikawa
conjecture [20] and the Zafrir’s analysis [21].
Chapter 5 is devoted to conclusions and discussions.

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1.1 Partition function

13

We give the conventions for four-dimensional theories and notations of group theory inAppendix A.
In this thesis, we denote Napier’s number, imaginary unit, and derivative in Roman
type. We use the notation “Tr” only for traces of gauge group indices. Otherwise, we use“tr” for traces.

1.1

Partition function

A partition function is very important because we can read oﬀ the non-trivial informationof a quantum theory. For example, the thermal partition function gives us the energy
spectrum of a theory. The thermal partition function of a theory with the Hamiltonian
H is given byZ(β) = trH (e−βH ) = trH (q H ) = Z(q),q = e−β ,(1.1)

where the trace is taken over the Hilbert space of the theory, and β = 1/kB T is theinverse temperature. Actually, the coeﬃcients and exponents of the Taylor expansion ofZ(q) with respect to q have information of the energy spectrum of the theory.
There is another expression of the partition function: the path integral form. Let us
consider a quantum mechanical system of a particle. To see the path integral form, let us
rewrite the partition function (1.1) by position eigenstates |x⟩:∫Z(β) =

dx⟨x|e−βH |x⟩.

(1.2)

This is reminiscent of the transition amplitude. If we identify the Boltzmann factor
with the time evolution operator, this is the transition amplitude with the time evolutiont = −iβ. Thus, the partition function can be written as the path integral with the periodic
boundary condition:∫Z(β) =[ ∫Dx exp i]−iβdtL(x, ẋ) ,x(0) = x(−iβ),

(1.3)

0

where L(x, ẋ) is the Lagrangian obtained by the Legendré transformation of the Hamiltonian.
In a quantum field theory, the fundamental degrees of freedom are not positions ofparticles but fields. Then the partition function of a quantum field theory is given by∫Z=

DΦeiS[Φ] .

(1.4)

The time direction is compactified to S 1 .
In the analysis of the quantum field theory, we often want to know the correlation

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Chapter 1 Introduction

function because it encodes the scattering amplitudes through the Lehmann-SymanzikZimmermann reduction formula. As it is well known we add a source term J(x) coupled
to the field Φ in the action, and the correlation function ⟨Φ(x1 )Φ(x2 ) · · · ⟩ is1
⟨Φ(x1 )Φ(x2 ) · · · ⟩ =Z[J = 0](−iδδJ(x1 ))(

δ−i
δJ(x2 ))
· · · Z[J]

.

(1.5)

J=0

Then, we can read oﬀ what particles are present and how they interact with each other.
This is one of the main goals of particle physics.
The benefits of the partition function are not limited to those mentioned above. For example, if the number of particles N is conserved, we can obtain the information regarding
the number of particles as well as energy spectrum from the grand partition function:
Ξ(β, µ) = tr(e−βH e−µN ) = tr(q H uN ) = Z(q, u),q = e−β ,u = e−µ ,(1.6)

where the trace is taken over all possible N values as well as Hilbert space with theconstant N and µ is the chemical potential. We refer to u as the fugacity. Thus the
grand partition function is a generalization of the partition function by adding a number
operator N , which commutes with the Hamiltonian.
Similarly, in general, we can add more information of the spectrum to the partition
function by using the symmetries of the theory. If there are symmetries commuting
with the Hamiltonian, we can generalize the partition function as follows because thesymmetries do not change the structure of the eigenstates of the Hamiltonian:
1 F2
Z(q, u1 , u2 , · · · ) = trV (q H uF
1 u2 · · · ),(1.7)

where F1 , F2 , · · · commute each other and u1 , u2 , · · · are also called the fugacities as ananalogy of the grand partition function. The trace is taken over the Hilbert space of thetheory. The commutativity means that we can choose Fj as the Cartan generators of thesymmetries.
Actually, we can regard the partition function as a character in the context of the grouptheory. The character of a representation R of a group G is defined as the trace over the
representation space VR :
χR (g) = trVR R(g) =∑[R(g)]ii ,

(1.8)

i

where R(g) is a representation matrix of an element g ∈ G. The partition function has
the same structure. The Hilbert space is VR . Hence the partition function is represented
1 F2
as a character of the state space with g = q H uF
1 u2 · · · .
Let us see the generalized partition function in the concrete example: 3d harmonic

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1.1 Partition function

15

oscillators. The Hamiltonian is given byH=

3∑a†i ai ,

(1.9)

i=1

where we neglect the zero-point energy for simplicity. The ordinary thermal partition
function is now
Z(q) =

1= 1 + 3q + 6q 2 + 10q 3 + 15q 4 + · · · .
(1 − q)3(1.10)

Then we can read oﬀ the energy spectrum of this theory from the coeﬃcients of fugacity
q as promised. As you know, there is an SU (3) global symmetry of this system. Since
the rank of SU (3) is two, we can define two fugacities coupled to the Cartan generatorsof SU (3). The definition of the generalized partition function is
1 R2Z(q, u1 , u2 ) = tr(q H uR
1 u2 ),(1.11)

where R1 and R2 are the SU (3) Cartan generators. For the fundamental representation
they are R1 = diag(1, −1, 0) and R2 = diag(0, 1, −1). When the energy eigenvalue isn without the zero-point energy, it turns out that degenerating states belong to (n, 0)
representation of SU (3). The notation (n, 0) means the Dynkin label of SU (3), and its
definition is given in Appendix A.2. Hence the partition function isZ(q, u1 , u2 ) =

∞
∑
n=0

q n χ(n,0) (u1 , u2 ) =1,(1 − qu1 )(1 − q uu21 )(1 − q u12 )(1.12)

where χ(n,0) is defined by (A.34). Expanding the partition function with respect to q, we
can find the energy spectrum with SU (3) charge information. Therefore, the generalized
partition function tells us information about the theory in more detail.
The partition function (1.11) is expressed as the character as we introduced before. The
Hamiltonian (1.9) is time independent, and it behaves as U (1) symmetry of the theory.
So we can define the U (3) symmetry by combining this U (1) symmetry with the SU (3)symmetry. The number of the fugacities is the rank of U (3), namely three. It is convenient
to introduce linearly independent Cartan generators RX , RY , and RZ of U (3) instead ofH, R1 , and R2 . The relation among them is H = RX + RY + RZ , R1 = RX − RY , andR2 = RY − RZ . Let x, y, and z be the corresponding fugacities to RX , RY , and RZ .
Then the new fugacities are related with the old ones as x = qu1 , y = qu2 /u1 , z = q/u2 .
The definition (1.11) of the partition function is replaced byZ(x, y, z) = trV (xRX y RY z RZ ).

(1.13)

This is just the character of U (3) (see (A.27)). Note that we have the (n, 0) representation

Soryushiron Kenkyu

16

Chapter 1 Introduction

for n-th excited states, and we have to sum contributions from all possible representations.
Therefore we haveZ(x, y, z) =

∞
∑
n=0

χ(n,0) (x, y, z) =1,
(1 − x)(1 − y)(1 − z)

(1.14)

where χ(n,0) is defined by (A.32). This agrees with (1.12) with the replacement x =qu1 , y = qu2 /u1 , z = q/u2 .
Because the 3d harmonic oscillator is solvable, there are no obstacles to calculate thepartition function. In the case of an interacting quantum field theory, if the theory is
weakly coupled, we can use the perturbative expansion to obtain the partition function.
It is very diﬃcult to obtain the partition function in the strong coupling region due to thenon-perturbative eﬀects. However, we are often able to calculate the partition function
exactly for supersymmetric theories. Then, we understand quantum field theories moredeeply.

1.2

Witten index

In the presence of supersymmetry, we can define an important physical quantity that does
not depend on the coupling constant. It is the Witten index [24]. The independence ofthe coupling constant enables us to calculate the Witten index exactly even for the strong
coupling region, and we can learn something about the non-perturbative eﬀects of thetheory. Hence it is worthwhile to consider the Witten index.
In the supersymmetric theory, the supersymmetry algebra is defined asb † , Q}b = ∆,
b{Q(1.15)b is nilpotent, and the form of ∆b depends on the detail of the theory. In the superwhere Q
b is nothing but the Hamiltonian. In other theories such
symmetric quantum mechanics, ∆b is a certain combination of the symmetry generators.
as superconformal field theories, ∆b commutes with Q
b and Qb † , an action of Q
b and Qb † on an eigenstate of ∆b doesBecause ∆not change its eigenvalue. Let us focus on an eigenstate |ψ⟩ with an eigenvalue ∆. Then
we can show that ∆ is positive-definite:2bbb † |ψ⟩|2 ≥ 0.
∆ = ⟨ψ|∆|ψ⟩= |Q|ψ⟩|+ |Q(1.16)In what follows, we omit the hat symbol of operators for simplicity unless there is noconfusion.
Now the Witten index of a supersymmetric theory with the supersymmetry algebra(1.15) is defined by
IWitten (β) = trH [(−1)F e−β∆ ],(1.17)

Soryushiron Kenkyu

1.2 Witten index

17

∆

∆

∆=0

∆=0
coupling g1 , IWitten = 1

coupling g2 , IWitten = 1

Fig. 1.1 The structure of states in a supersymmetric theory with supersymmetry
algebra (1.15). Here a white node stands for a bosonic state, while a black nodestands for a fermionic state. The coupling constant is g1 on the left side and g2 on
the right side. The structure of states also changes if we change the coupling constant.
However, the Witten index does not change because the excitation is always a pairof bosonic and fermionic states with the same eigenvalue.

where F is the fermion number operator and (−1)F assigns the minus sign to fermionicstates. Thus the Witten index can also be written as
IWitten (β) =∑bosonse−β∆n −∑

e−β∆n ,

(1.18)

fermions

where n is a label of each state.
As we mentioned at the beginning of this section, the Witten index does not depend on
the coupling constant. To see this, let us consider representations of the supersymmetry
algebra√(1.15). For ∆√ > 0 states, we† can define a creation and annihilation operator††
a = Q/ ∆, a = Q / ∆ satisfying {a , a} = 1. This is just the two-dimensional Cliﬀord
algebra, and we can define the Cliﬀord vacuum as usual: a|Ω⟩ = 0. We assume that
the Cliﬀord vacuum is a bosonic state. Due to the nilpotency of the supercharges, wehave a two dimensional representation: (|Ω⟩, a† |Ω⟩). This indicates that all states with
∆ > 0 form a pair between a bosonic state and a fermionic state. Because supercharges
commute with ∆, this pair of the states has the same eigenvalue for ∆. This meansthat contributions from ∆ > 0 are always canceled. Therefore, the Witten index is only
received from contributions of ∆ = 0 states. If we change the coupling constant, somestates may be excited. However, this excitation is always the pair of bosonic and fermionicstates (see Fig. 1.1). Hence the excitation does not aﬀect the Witten index, and thatis why the Witten index is independent of the coupling constant. Strictly speaking, this
discussion is applicable to any continuous parameter of the theory. In particular, theWitten index is an RG flow invariant.
The fact that the Witten index has contributions from ∆ = 0 states only shows that

Soryushiron Kenkyu

18

Chapter 1 Introduction

the Witten index is independent of β. Actually, we can also show it more directly.
†dIWitten (β) = trH [(−1)F {Q† , Q}e−β{Q ,Q} ] = 0,dβ(1.19)

where we used the cyclic symmetry of the trace and (−1)F Q = −Q(−1)F .
According to the above discussion, we also express the Witten index as
IWitten (β) = #(bosonic states with ∆ = 0) − #(fermionic states with ∆ = 0).

(1.20)

This expression may be more practical for the calculation.
We have seen that the Wittne index receives the contribution from ∆ = 0 states. Let|ψ⟩ be state with ∆ = 0. In fact, this state satisfies the following condition:
Q|ψ⟩ = Q† |ψ⟩ = 0.

(1.21)

We call this condition the Bogomol’nyi-Prasad-Sommerfield (BPS) condition and the state|ψ⟩ satisfying (1.21) is called the BPS state. This condition is a necessary and suﬃcientcondition to be ∆ = 0. In this sense, the Witten index has the information of the BPS
state spectrum of the theory.
As in the case of the thermal partition function, it is possible to generalize the Witten
index to include the conserved charges associated with the global symmetries of the theory.
However, we have an additional constraint: the generators of global symmetries should
commute with the supercharges; otherwise, the discussion of independence of β is invalid.
Then the generalized Witten index is defined by
1 F2I(q, uj ) = trV [(−1)F q ∆ uF
1 u2 · · · ],(1.22)

where q = e−β , and we introduced the fugacities related to the symmetries. As we noted,
we have constraints [Q, Fj ] = 0. Again, this index tells us about the more detailed
information of the BPS spectrum of the theory, and the index can be calculable even forthe strong coupling region.

1.3

Localization

In this section, we explain how to calculate the Witten index in the Lagrangian theory.
As we saw in the previous section, the partition function can be expressed as the path
integral formalism in the quantum field theory. This is also the case for the Witten index.
Due to the trace over the states, a spacetime manifold in which the path integral is carriedout has a form X × S 1 , where the period of S 1 is the inverse temperature β. To ensure
the convergence of the path integral, the space manifold X must be compact.
In the path integral formalism, boundary conditions for fields are very important. In
the thermal partition function, we impose the anti-periodic boundary condition of S 1

Soryushiron Kenkyu

1.3 Localization

19

for fermionic fields. Conversely, we have the fermion number operator (−1)F in thedefinition of the Witten index, and we impose the periodic boundary condition even for
fermionic fields as well as bosonic fields. Furthermore, the generalized Witten index has
additional operators related to global symmetries. Then we have to include the eﬀects
of these additional operators on the boundary conditions. Let us consider a general
trace tr[(−1)F e−β∆ O]. In the original Witten index O = 1 while in the generalized one
1 F2
O = uF1 u2 · · · . Then the boundary condition becomese−β∆ OΦ(τ )O−1 eβ∆ = eβ∂τ OΦ(τ )O−1 e−β∂τ = OΦ(τ + β)O−1 = Φ(τ ),(1.23)

where Φ is a field, and τ is a coordinate in S 1 . With this boundary condition, the Witten
index in the path integral formalism is∫
IWitten =X ×S 1DΦe−S[Φ] ,(1.24)

where we performed the Wick rotation appropriately. Although it is diﬃcult to carry out
this path integral in general, the localization method [25] enables us to calculate the path
integral exactly in the presence of the fermionic symmetry (e.g., supersymmetry).
In this thesis, we would like to apply the localization method to the Witten index.
However, the localization method itself can be applied for various partition functions in acompact manifold M with a fermionic symmetry. So let us review the general argumentof the localization method in this subsection.
Let S be the action with fermionic symmetry: δF S = 0, where δF is nilpotent or its
square becomes a certain bosonic symmetry δB with δB S = 0. Here we would like toevaluate the following path integral:∫
Z=
M

DΦe−S[Φ] .

(1.25)

Instead of this, we consider the following deformed path integral:∫Z(t) =M

DΦe−S[Φ]−tδF V [Φ] ,(1.26)

where V is a Grassmann odd functional satisfying δB V = δF2 V = 0. We can reproduce Z
by Z(t = 0). In fact, it is easy to show that Z(t) does not depend on t:
dZ(t)
=−
dt
=−

∫
∫M
M

DΦδF V e−S−tδF V()
DΦδF V e−S−tδF V = −δF(∫MDΦδF V e−S−tδF V)= 0,(1.27)

where we assumed the fermionic symmetry δF is not anomalous. Thus, we can evaluate
Z by Z(t) with any t. The simplest value of t to find Z is the t → ∞ limit. In this limit,

Soryushiron Kenkyu

20

Chapter 1 Introduction

the path integral is dominated by saddle points of the deformation term δF V with theassumption δF V ≧ 0. Then, only configurations satisfying δF V [Φ0 ] = 0 contribute to the1b In theintegral. Let us expand S[Φ] and δF V [Φ] around Φ0 with the fluctuation t− 2 Φ.
t → ∞ limit, residual path integral becomes
Z=

∑∫
Φ0

M

b

b −S[Φ0 ]−ΦδF V
DΦe

(2)

b
[Φ0 ]Φ

,(1.28)

where δF V (2) [Φ0 ] is the second-order derivative of the functional δF V [Φ]. It is happily
the Gauss integral. We can perform the integral, and the result is the determinant of
the linear operator δF V (2) . However, we have to pay attention to the existence of zero
eigenvalues. If the operator δF V (2) has zero eigenvalues, we have to eliminate these zero
eigenvalues and the integral remains for the corresponding component of the vector Φ0 .
Let a be such a component. Then, the final result is
Z=

∑∫
Φ0

da

DetF (δF V (2) [Φ0 ]),Det′B (δF V (2) [Φ0 ])(1.29)

where the prime in Det indicates the elimination of the zero eigenvalues. The important
point of this result is that there are only finite number integrals. We call this determinant
the one-loop determinant because it corresponds to the one-loop diagram in the absenceof the non-perturbative eﬀects.

1.4Superconformal field theory

A theory with the superconformal symmetry is called a superconformal field theory(SCFT). The superconformal symmetry is essential for the definition of the superconformal index. First, we review the conformal symmetry, and second, we generalize it tosuperconformal symmetry to define the superconformal index in the latter subsection.

1.4.1

Conformal symmetry

The 4d conformal transformation in R1,3 is generated by translations Pµ , Lorentz transformations Mµν , dilatation D, and special conformal transformations Kµ . They satisfy

Soryushiron Kenkyu

1.4 Superconformal field theory21

the following commutation relations:[D, Pµ ] = iPµ ,(1.30a)[D, Kµ ] = −iKµ ,(1.30b)[Pµ , Kµ ] = −2iMµν + 2iηµν D,(1.30c)[Mµν , Mρσ ] = iηµρ Mνσ − iηµσ Mνρ − iηνρ Mµσ + iηνσ Mµρ ,(1.30d)[Mµν , Pρ ] = iηµρ Pν − iηνρ Pµ ,(1.30e)[Mµν , Kρ ] = iηµρ Kν − iηνρ Kµ ,(1.30f)where all the generators are Hermite. If we define generators MAB (A, B = 0, 1, · · · , 5)
as



Mµν
− 21 (Pν − Kν ) − 12 (Pν + Kν )
,
MAB =  21 (Pµ − Kµ )
0
D
1
−D
0
2 (Pµ + Kµ )(1.31)

then the commutation relations (1.30) are unified into a simple commutation relation:[MAB , MCD ] = iηAC MBD − iηAD MBC − iηBC MAD + iηBD MAC(1.32)

with ηAB = diag(−1, 1, 1, 1, 1, −1). This means that the conformal algebra is isomorphic
to SO(2, 4).
In CFTs, it is useful to consider the radial quantization instead of the vertical quantization. In the radial quantization, we define the theory on R × S 3 instead of R1,3 , and we
regard R as the time direction. The theory on R × S 3 is related with the original theory
on R1,3 by the Weyl transformation of the metric:′gµν (x) → gµν(x) = e2w(x) gµν (x),(1.33)

where w(x) is a certain real function of x. Note that in this manipulation, we needappropriate Wick rotations. Since the Weyl transformation preserves the structure of theconformal symmetry, the information of the original theory on R1,3 is mapped into thetheory on R × S 3 . Now we can avoid the infra-red (IR) divergence due to the infinite
volume of each time slice because the time slice is now S 3 . This is the benefit of the radialquantization.
Now operators inserted at the origin in R1,3 are mapped into states in the Hilbert spaceof the theory in R × S 3 . Namely, we have a one-to-one correspondence between operators
at the origin O(0) in R1,3 and states|O⟩ ≡ O(0)|0⟩(1.34)

in R × S 3 , where |0⟩ is the conformal invariant vacuum of a CFT. This is the so-called

Soryushiron Kenkyu

22

Chapter 1 Introduction

operator-state correspondence. Thus we can find the operator spectrum of the theory in
R1,3 in terms of states in R × S 3 . In what follows, the term “an operator” indicates “anoperator inserted at the origin”; otherwise, we will explain the meaning of “an operator.”
It is useful to classify operators O(0) in terms of representations of conformal algebra.
First of all, an operator with the scaling dimension E is transformed under the dilatationas[D, O(0)] = iEO(0).

(1.35)

Then, from the conformal algebra (1.30), we find that Pµ and Kµ have a role of raisingand lowering the scaling dimension respectively:[D, [Pµ , O(0)]] = i(E + 1)[Pµ , O(0)],(1.36)[D, [Kµ , O(0)]] = i(E − 1)[Kµ , O(0)].

(1.37)

These are followed by the Jacobi identity. In the unitary theory, there is a lower boundto the scaling dimension. Thus the action of Kµ terminates at a certain value of thescaling dimension. An operator with this scaling dimension is called the primary operator.
Namely, the primary operator satisfies[Kµ , O(0)] = 0.

(1.38)As in the highest weight construction of the group theory, we can use the primary operator
as a starting point to construct the conformal representation. Note that in the state
picture, we call the corresponding state the primary state.
Before discussing conformal representation, let us give the conformal transformation
laws for operators inserted at a generic point x. This operator O(x) is related to theoperator inserted at the origin byO(x) = eiPµ x O(0)e−iPµ x .
µµ(1.39)

Then the conformal transformation laws of an operator O(x) are given by [26][Pµ , O(x)] = −i∂µ O(x),(1.40a)[Mµν , O(x)] = [i(xµ ∂ν − xν ∂µ ) + Σµν ]O(x),(1.40b)[Kµ , O(x)] = [i(x2 ∂µ − 2xµ xν ∂ν − 2Exµ ) − 2xν Σµν ]O(x),(1.40c)[D, O(x)] = i(xµ ∂µ + E)O(x),(1.40d)where Σµν is a spin matrix corresponding to the Lorentz transformation.

Soryushiron Kenkyu

1.4 Superconformal field theory

23

Representations of conformal algebra
A conformal representation is uniquely specified by the Cartan charges of a primary
operator. We have already seen that the conformal algebra is isomorphic to so(2, 4). Its
subalgebra u(1) × so(4) corresponds to the dilatation and the Lorentz transformation in
R1,3 respectively. Let jL and jR be the left- and right-handed spin. The dilatation charge
is E. Then this primary operator is labeled by jL , jR , and E. We denote the label as[jL , jR ]E .

(1.41)

We can construct a tower of operators (or states) by acting Pµ . The operators (or states)
generated by acting Pµ on the primary operator (state) are called descendants. The
labeling (1.41) can also be used for descendants.
There is one thing we have to pay attention to when the construction of representation.
In fact, the zero norm states may appear in the tower of the representation. Corresponding
operators are not physical. In order to make the theory unitary, we have to remove suchstates from the tower. Then such the tower is called a short representation.
As an example of a short representation, let us consider free massless primary scalar
field ϕ : [0, 0]1 . We can construct descendants by acting Pαα̇ ≡ 12 (σ µ )αα̇ Pµ : [ 12 , 12 ]1 . The
result isPαα̇
ϕ : [0, 0]1 −−→ ∂αα̇ ϕ : [ 12 , 21 ]2

P

P

αα̇
αα̇
−−
→ ∂{α{α̇ ∂β}β̇} ϕ : [1, 1]3 ⊕ ϵαβ ϵα̇β̇ ∂αα̇ ∂β β̇ ϕ : [0, 0]3 . −−
→ ··· ,

(1.42)

where ∂αα̇ = 12 (σ µ )αα̇ ∂µ . Other irreducible combinations are vanished because ∂αα̇ ’s arecommutative each other. Now the representation [0, 0]3 is in fact absence physically since
it corresponds to the equation of motion of ϕ:*1
ϵαβ ϵα̇β̇ ∂αα̇ ∂β β̇ ϕ ∝ ∂ 2 ϕ = 0.

(1.43)In the state picture, the corresponding state is the zero norm state. It is possible to show
this statement directly by calculating its norm. Hence we have to subtract this operator(or state) from the tower of descendants, and the representation becomes short.

1.4.2

Superconformal symmetry

In the presence of the supersymmetry and the conformal symmetry, the symmetry isenhanced to the superconformal symmetry. What supersymmetries survive on a generalmanifold is obtained by the Killing spinor equations. In the flat space, we have the*1 The presence of a free massless scalar field means that we assume the presence of its LagrangianL = ∂µ ϕ∂ µ ϕ and as a result we have the equation of motion for ϕ.

Soryushiron Kenkyu

24

Chapter 1 Introduction

following Killing spinor equations:∂µ ϵI = −iσµ κI ,∂µ ϵI = −iσ µ κI ,

(1.44)

I
where ϵαI (I = 1, · · · , N ) is a parameter of the supersymmetry transformation and κ isan arbitrary spinor. The general solutions are given by

ϵI = ηI − ixµ σµ ζ I ,κI = ζ I ,(1.45)

where ηI and ζ I are constant spinors. The first term corresponds to a supercharge QIα ,I
and the second term corresponds to a supercharge S α̇ . We also have their Hermite conα̇
jugations: QI and SIα . Namely, the Hermiticity of the supercharges isα̇

I(QIα )† = QI ,(SIα )† = S α̇ .

(1.46)

They generate the following anti-commutation relations:α̇{QI , QJβ } = δIJ P α̇ β ,(1.47a)J{SIα , S β̇ } = −δIJ K α β̇ ,i α J
δβ δI D + iδIJ M α β + δβα RJ I ,
2
i
α̇
J
α̇
{QI , S β̇ } = δβ̇α̇ δIJ D + iδIJ M β̇ − δβ̇α̇ RJ I ,2{SIα , QJβ } =(1.47b)(1.47c)(1.47d)where RI J are generators of R-symmetry U (N )R (N ≤ 3) and SU (4)R (N = 4) which
are the rotation of the supercharges. For the convenience of a supersymmetric theory, we
introduced the spinor indices for P, K, and M generators by
1 µ(σ )αβ̇ Pµ ,21Kαβ̇ = −Kβ̇α = (σ µ )αβ̇ Kµ ,21βMα β = (σ µν )α Mµν ,41α̇M β̇ = (σ µν )α̇ β̇ Mµν .
4Pαβ̇ = −Pβ̇α =(1.48)(1.49)(1.50)(1.51)The transformation law of the fundamental and ant-fundamental representation of the
R-symmetry is
1I
ϕJ − δJI ϕK ,[RI J , ϕK ] = δK
4

1
[RI J , ϕK ] = −δJK ϕI + δJI ϕK .
4

(1.52)

Soryushiron Kenkyu

1.4 Superconformal field theory

25

Then the generators of R-symmetry satisfy
I K[RI J , RK L ] = δL
R J − δJK RI L .

(1.53)

The commutation relation (1.52) indicates that for the N = 4 case, all the commutation
relations of RK K vanish, so that the R-symmetry group is not U (4)R but SU (4)R .
Other commutators with supercharges are the following:
i I
Q ,
2 α
i
[D, SIα ] = − SIα ,
2

i α̇
Q ,
2 I
i I
I
[D, S α̇ ] = − S α̇ ,2α̇[D, QIα ] =[D, QI ] =[K α β̇ , QIγ ] = −δγα S β̇ ,I

[K α β̇ , QI ] = −δβ̇γ̇ SIα ,γ̇α̇[P α̇ β , S γ̇ ] = −δγα QIβ .

I[P α̇ β , SIγ ] = −δβγ QI ,(1.54)(1.55)(1.56)(1.57)As in the conformal symmetry, these commutation relations are unified with (1.30) intoa simple relation. To see this, let us define the generators T A B (A = (α, α̇, I)) as


N αβ


T A B =  P α̇ β
QIβ

K α β̇
α̇

N β̇
Sβ̇I

α

SJ



α̇ 
−QJ  ,
RI J

(1.58)wherei1
N α β = − δβα D + iM α β − δβα RK K ,24i α̇
1 α̇ Kα̇α̇N β̇ = δβ̇ D − iM β̇ − δβ̇ R K .
2
4(1.59)(1.60)

Then the superconformal algebra is unified into the following simple relation:
A C
C A
T A B T C D − ωT C D T A B = ωδD
T B − δBT D,

ω = (−1)(A+B)(C+D) .

(1.61)(1.62)We defined (−1)A as +1 or −1 depending on whether A is bosonic or fermionic (spinor)
index. Note that the trace of T A B is zero:α̇

N α α + N α̇ + RK K = 0.

(1.63)

Therefore the 4d superconformal algebra is isomorphic to SU (2, 2|N ) and for N = 4
isomorphic to P SU (2, 2|4) which is defined by removing the U (1)R factor.

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26

Chapter 1 Introduction

Representations of the superconformal algebra
As in the case of the conformal representation, it is possible to define the superconformal
primary operator satisfying
α[Sα̇I , O(0)] = [S I , O(0)] = 0.

(1.64)

Note that a conformal primary operator is not always a superconformal primary. However, its converse is true. Since the generator K α α̇ is roughly SS, a superconformal
primary operator is always a conformal primary. And operators constructed by acting
Q on a superconformal primary operator are all conformal primary operators because K
and S commute each other.
α̇Then we can use supercharges QIα and QI as raising operators. From the commutation
relations (1.54) and (1.55), we find that the raising operators raise the scaling dimension
Iby half and the lowering operators SαI and S α̇ lower the scaling dimension by half.
By using the raising operators Q and Q, we also construct the tower of operators.
Since we have already discussed conformal representations constructed by acting Pα̇α ,we here restrict our target to the conformal primary operators. Then we can regard thesupercharges Q and Q as nilpotent raising operators. Thus the tower must terminate if we
use up all the raising operators. Then we call this representation long. Otherwise, there
are certain non-physical operators (zero norm states), and the representation becomesshort. We can consider conformal descendants for each conformal primary constructed bya superconformal primary, and finally, we obtain a superconformal representation.
The superconformal primary operators (states) and Q-excited (Q-excited) operators(states) are labeled by not only the Lorentz spins jL , jR and the scaling dimension E but
also the R-charges, where R-charges are the Cartan charges of the R-symmetry. So we
use the notation(R-charges)[jL , jR ]E(1.65)

for the components of the superconformal representations.
We give the general structure of the superconformal representation in Fig. 1.2. For
simplicity of the notation, we used the state picture.

1.4.3

Deformation of CFTs

In relativistic theories, it is believed that CFTs are realized at a fixed point of a renormalization group (RG) flow in which the beta-function is zero. Here we would like toconsider deformations of a CFT. A deformation of a CFT is defined as an infinitesimal
shift of this CFT from a fixed point in the space of RG flows. Roughly speaking, there
are three classes of deformations:• Adding local operators to the Lagrangian.

Soryushiron Kenkyu

1.4 Superconformal field theory|p⟩Pα̇α y

QI

α
−−−−
→

27

α̇

QIα |p⟩Pα̇α yQI or QI−−α−−−−→· · · −−−−→Pα̇α ypossible)|p⟩(All
combinations of Q and QPα̇α yα̇

QI

QI or Q

possibleαIPα̇α |p⟩ −−−−
→ Pα̇α QIβ |p⟩ −−α−−−−→· · · −−−−→ Pα̇α (All)|p⟩
combinations of Q and Q








Pα̇α yPα̇α yPα̇α yPα̇α y

..
.

..
.

..
.

..
.

Fig. 1.2 An example of a general superconformal representation. |p⟩ stands for thesuperconformal primary state. In terms of operators QIα |O⟩ is replaced by [QIα , O].
The horizontal sequences correspond to the tower of Q-excited and Q-excited states,
and the vertical sequences correspond to the conformal descendants. The states
located in the first row are all conformal primary.

• Gauging of a global symmetry.
• Moving onto a moduli space of vacua.
In this thesis, we focus on the first one: adding a local operator to the Lagrangian.
The adding of a local operator to the Lagrangian is the most common way to deform a
CFT. This deforms the Lagrangian L of a CFT as follows:δL = gO.

(1.66)

Here g is an infinitesimal coupling constant, and O is a local operator existing in theoriginal CFT. For the original (undeformed) CFT g = 0. This deformation can also be
defined even though we do not know the Lagrangian of the original CFT, by using theconformal perturbation theory [27].
Now we would like to restrict the class of an operator O we treat in this thesis. First,
we require that a deformation preserves the Lorentz symmetry, namely, an operator O
must be a scalar. Second, we require that an operator O is a conformal primary of aconformal representation. Since O exists in the original CFT, it is clear that O belongs toa certain conformal representation. However, if O were a conformal descendant, it would
not change the bulk physics because a descendant operator can be written as derivativesof a certain operator. Thus O should be a conformal primary.
Let ∆O be the conformal dimension of a deformation operator O. Then we can classify
the deformation operator in three types by the value of ∆O .
• Relevant operators (∆O < 4): In this case, the corresponding coupling constantis called a relevant coupling constant. Then a CFT at g = 0 is an ultraviolet(UV) fixed point. An RG flow is initiated by turning on the coupling constant,and the coupling constant in the infrared (IR) region grows. Thus the conformal

Soryushiron Kenkyu

28

Chapter 1 Introduction
perturbation is eventually violated.
• Irrelevant operators (∆O > 4): The corresponding coupling constant is called anirrelevant coupling constant. Now a CFT at g = 0 is realized at an IR fixed point towhich the irrelevant coupling gets smaller. The deformed CFT might be an eﬀective
theory of a certain UV theory.
• Marginal operators (∆O = 4): The corresponding coupling constant is called amarginal coupling constant and dimensionless. The marginal deformation conserves the conformal symmetry at the leading order of the conformal perturbation.
Then the deformation leads to a nearby fixed point for a small g enough. Including the eﬀect of higher-order corrections, the marginal deformations are divided
into marginally relevant, marginally irrelevant, and exactly marginal. An exactly
marginal deformation preserves the conformal symmetry exactly.

Let us consider an exactly marginal deformation in detail. An exactly marginal deformation enables us to deform a CFT continuously, and parameters (coupling constants) ofcontinuous deformations form a space of deformations for a CFT. Such a space is calleda conformal manifold. Some of interacting CFTs can be constructed by continuous deformations from a free theory. Conversely, if there is no marginal deformation in a given
CFT, this CFT does not connect with any free theories. Furthermore, even though there
are marginal deformations, CFTs are not necessarily connected to a free theory.

1.4.4

Theories for which the Lagrangian is unknown

For an interacting CFT connecting to a certain free theory, we can in principle give theLagrangian by exactly marginal deformations. However, for a CFT which is not connected
to a free theory, there is no general prescription to give the Lagrangian.
There may be the UV Lagrangian that flows to a target CFT. Although we do notknow the Lagrangian of a target CFT, we can find the information about the CFT fromthe UV Lagrangian. In fact, although the Lagrangian is not known for AD theories,
UV Lagrangians were found, and several physical quantities were discussed using the UV
Lagrangian [10]. However, unfortunately, there is also no general prescription to find theUV Lagrangian.
In summary, it is quite diﬃcult to discuss CFTs that are not connected to a free theory.
The only way to discuss such a CFT is to use a duality.
In this thesis, we mainly focus on the N = 4 SYM and the S-fold theories. The
former has the Lagrangian. On the other hand, the latter theories are CFTs that are
not connected to a free theory. Then, it seems that there is no way to calculate physicalquantities of S-fold theories*2 . However, fortunately, there is AdS/CFT for S-fold theories,
and we may calculate the physical quantities of the S-fold theories through type IIB string*2 Recently, UV theories of a part of S-fold theories were found [21], so they enable us to calculate

physical quantities of such a part of S-fold theories. However, our calculation method we will givein the latter part of this thesis can be applied for general S-fold theories and is still useful.

Soryushiron Kenkyu

1.5 Superconformal index

29

theory on the AdS side. This analysis will be discussed in Chapter 4.

1.5

Superconformal index

So far, we have considered the general properties of CFTs and SCFTs. We also haveseen that there is no prescription to give the Lagrangian and UV Lagrangian for CFTs
that are not connected to a free theory. In this case, the only way to investigate suchtheories is to use a duality, as we saw in the last subsection. In fact, AdS/CFT enables us
to analyze the S-fold theories. However, AdS/CFT is a strong/weak duality, and this isan obstruction to calculate various physical quantities. So we focus on the Witten index
because it does not depends on the coupling constant. If a theory is an SCFT, the Witten
index is specifically referred to as the superconformal index [16].
The superconformal index itself is defined in the state picture, namely defined on R×S 3 ,which ensures the convergence of the path integral. To define the superconformal index,
we need the superconformal algebra on R×S 3 instead of that on R1,3 . The superconformal
algebra on R × S 3 is related to that on R1,3 by the analog of the Wick rotation. Although
we do not give the derivation, the transformation of the generators of the superconformal
algebra from R1,3 to R × S 3 is given byαα

P α̇ α → −P α̇ α ,S I → −S I ,M α β → −iJ α β ,M β̇ → iJ β̇ ,

α̇

α̇

α̇

QI → −QI ,α̇

D → iH,(1.67)and the Hermiticity of generators is changed. Concretely, the Hermiticity is given by
H † = H,(P α̇ β )† = K β α̇ ,(J α β )† = J β α ,(J β̇ )† = J α̇ ,(QIα )† = SIα ,(QI )† = S α̇ .

α̇

β̇

α̇

I

(1.68)The important commutation relation to define the superconformal index is ∆ = {Q† , Q},α̇=1̇where Q is one of the supercharges. We choose Q as QI=1 . Let ∆ be the commutation1̇

relation between Q1 and its Hermite conjugate we have1̇1

∆ = 2{Q1 , S 1̇ } = H − 2jR − R1 1 ,(1.69)

where the factor 2 in front of the anti-commutation relation is just the convention to make1̇our notation simpler. jR is the right-handed spin operator J 1̇ and H is the Hamiltonianin the radial quantization.
Roughly speaking, the superconformal index is the Witten index in regard to the “supersymmetry (1.69)”. As we saw in Sec. 1.2, we can add more fugacities corresponding
to global symmetries commuting with Q and Q† . For superconformal theories, a part ofSoryushiron Kenkyu

30

Chapter 1 Introduction

the conformal symmetry and other global symmetries commute with Q and Q† . Notethat the R-symmetry does not commute with Q and Q† and only its subgroup commute
with them. We define flavor symmetries as global symmetries other than the conformal
symmetry, which commute with Q and Q† including the subgroup of the R-symmetry.
Then the superconformal index is defined as[
ISCI = tr (−1)F x∆ q H+jR y 2jL

∏

]
i
uF
,
i

(1.70)iwhere Fi is the Cartan generators of the flavor symmetries (or their linear combination)commuting with each other as well as Q and Q† . The contributions come from the ∆ = 0
states (BPS states) as we saw in Sec. 1.2. Namely, the index does not depend on x.
As we mentioned before, the index is defined on R × S 3 . Since we have the operatorstate correspondence, it is possible to translate the BPS state spectrum in R × S 3 into
the BPS operator spectrum in R1,3 , where the BPS operator OBPS is defined as operator
satisfying
1̇1[Q1 , OBPS ] = [S 1̇ , OBPS ] = 0.

(1.71)

Then the index also encodes the information of the BPS operator spectrum of the theory
on R1,3 .
Thus, since the index encodes the BPS spectrum and is independent of the coupling
constant, it may be possible to find the BPS spectrum of the theory using AdS/CFTeven for theories for which the Lagrangian is unknown. One of the goals of quantum field
theories is to investigate its low energy eﬀective theory (CFT) and to determine whattypes of operators and interactions exist. The fact that the combination of the index and
AdS/CFT may actually achieve part of that goal is surprising and shows the importanceof the index.

1.5.1

Chiral ring

Before explaining how to calculate the superconformal index, we need to comment on theproperties of the BPS operators. States on R × S 3 correspond to operators inserted into
the origin on R1,3 . In general, the composite operators are ill-defined at the same pointin the spacetime. To make composite operators well-defined, we have to consider certain
regularizations or operator product expansions (OPEs). Otherwise, the operator-statecorrespondence does not work for composite operators. However, the BPS operators are
well-defined even for the composite operators inserted into the same point.
This statement can be understood in terms of the correlation function among BPS

Soryushiron Kenkyu

1.5 Superconformal index

31

operators. Actually, we can show∂⟨O(x1 )O2 (x2 ) · · · On (xn )⟩ = 0,∂xµ1(1.72)

where Oi (xi ) is a BPS operator inserted at a point xi . This equation states that thecorrelation function among BPS operators does not depend on the coordinates. Thus it
is possible to define the correlation function even for x1 = x2 = · · · = xn . This is thestatement we mentioned above.
Let us prove the statement (1.72). Then, it is useful to consider the chiral ring [28].
The chiral ring is defined in an N = 1 supersymmetric theory. If an operator O satisfies[Qα̇ , O] = 0(1.73)

for a certain Qα̇ , we say that the operator O belongs to the chiral ring. Namely, thechiral ring is a set of operators satisfying (1.73). We call operators satisfying (1.73) chiral
operators.
The chiral operators are independent of the spacetime coordinates, up to Qα̇ commutators. From the commutation relation (1.40a) and (1.47a) we havei(σ µ )αα̇ ∂µ O(x) = −(σ µ )αα̇ [Pµ , O(x)] = 2{Qα̇ , [Qα , O(x)]}.

(1.74)

This implies that a correlation function among the chiral operators is independent of thespacetime coordinates:i µ∂(σ )αα̇ µ ⟨O1 (x1 )O2 (x2 ) · · · On (xn )⟩2∂x1= ⟨{Qα̇ , [Qα , O1 (x)1 ]}O2 (x2 ) · · · On (xn )⟩=−n∑⟨[Qα , O1 (x1 )]O2 (x2 ) · · · [Qα̇ , Ok (xk )] · · · On (xn )⟩ = 0.

(1.75)

k=2

Therefore we finished the proof. Then we can take points of inserted operators freely.
This enables us to do the cluster decomposition as follows:⟨O1 (x1 )O2 (x2 ) · · · On (xn )⟩ = ⟨O1 ⟩⟨O2 ⟩ · · · ⟨On ⟩.

(1.76)

Since the condition of the BPS operators (1.71) is just the chiral ring (1.73) for α̇ = 1,
the composite BPS operators make sense even for the same point. This is one of thereasons why we focus on the BPS operators.

Soryushiron Kenkyu

32

1.5.2

Chapter 1 Introduction

Superconformal index for Lagrangian theories

As the end of this section, we review how to calculate the superconformal index in an
SCFT with its Lagrangian. In Sec. 1.3, we saw that the Witten index can be calculatedby the localization technique. Likewise, the superconformal index can also be in principle
calculated by the localization technique using its Lagrangian. However, we have a more
physical way called “free field counting” to obtain the same result with the localization.
In this subsection, we review this way instead of the localization.
Since the superconformal index does not depend on the coupling constant g, we canevaluate the index at g = 0. Then the problem to calculate the index is what kinds of
BPS operators exist in the theory. This can be read oﬀ from the Lagrangian. Then theprocedure to find the index is the same as the harmonic oscillators.
In the case of the single harmonic oscillator, we have a state tower|0⟩, a† |0⟩, (a† )2 |0⟩, · · · , (a† )k |0⟩, · · · ,(1.77)

where |0⟩ a vacuum state. Then we call a† |0⟩ a single-particle state and a state whose
form is (a† )k |0⟩ is called a multi-particle state for k > 1. Similarly, when we have a BPS
operator O(x) which is not a composite operator in a CFT on R1,3 , we have a state tower
on R × S 3 as
|0⟩CFT , O(0)|0⟩CFT , (O(0))2 |0⟩CFT , · · · , (O(0))k |0⟩CFT , · · · ,(1.78)

where |0⟩CFT is a conformal invariant vacuum state, and we call O(0)|0⟩CFT and(O(0))k |0⟩CFT a single-particle BPS state and a multi-particle BPS state, respectively.
Correspondingly, we call a BPS operator O(x) a single-particle BPS operator, and O(x)kis called a multi-particle BPS operator. If there are several kinds of BPS operators, a
BPS operator that is not made of any other operators is called a single-particle BPS
operator. Conversely, a composite BPS operator is called a multi-particle BPS operator.
The first step of the free field counting is to consider the superconformal index of singleparticle states. To do it, we need to list all the single-particle states. Let mn (q, y, ui ) be amonomial of a contribution from a certain single-particle state to the index, where n is alabel of a single-particle state. Then we can obtain the single-particle index by summingup all monomials corresponding to single-particle states in the list:
isp (q, y, ui ) =∑cn mn (q, y, ui ),(1.79)nwhere cn is an integer corresponding to the number of a monomial mn (q, y, ui ).
The next step is to consider the multi-particle states. We have seen that the state
structure (1.78) is the same as that of the single harmonic oscillator (1.77) for each singleparticle BPS state. Namely, each single-particle BPS state corresponding to a monomial

Soryushiron Kenkyu

1.5 Superconformal index

33

mn (q, y, ui ) forms a single harmonic oscillator. Then a partition function of the single
harmonic oscillator represented by a monomial mn (q, y, ui ) without the zero point energy
is given by
mn (q, y, ui ) →1.
1 − mn (q, y, ui )(1.80)

This manipulation is called plethystic exponential. We denote this manipulation as
Pexp(mn (q, y, ui )) =1.
1 − mn (q, y, ui )(1.81)

When the theory has some harmonic oscillators, the total partition function is given by
the product of each partition function:
Pexp

(
∑
n

)
cn mn (q, y, ui )

=

∏

1
.
cn
(1
−
m
(q,
y,
u
))
n
i
n

(1.82)It seems that this manipulation is valid only for bosonic operators. However, in fact,
this is valid even for fermionic operators due to the extra minus sign. In the ordinary
partition function in which the temporal boundary condition of a fermionic operator isanti-periodic, the fermionic contribution is given by 1 + mn , where mn is a monomial
corresponding to a fermionic contribution. On the other hand, in the index, the temporalboundary condition is periodic as in the case of bosons, and we have an extra minus sign infront of the monomial: −mn . The index is then given by 1 − mn , which is the same result
as applying the plethystic exponential to −mn . Therefore, the plethystic exponential(1.82) is applicable to both bosonic and fermionic single-particle BPS operators.
One more manipulation is needed to find the index for gauge theories. In general,
the BPS operators are charged under the gauge interaction, and the index would dependon the gauge fugacity za (a = 1, · · · , rankG), where G is the gauge group. However,
only the gauge invariant BPS operators contribute in gauge theories because the index
is a kind of partition function. Thus the index is gauge invariant and independent of
the gauge fugacities. Also, when we discuss the AdS/CFT correspondence, there is nocorresponding symmetry to the gauge symmetry on the AdS side. Then the physicalquantities on the CFT side, which we should compare with corresponding quantities on
AdS sides, should be gauge invariant. Conversely, this is also why we focus on the gauge
invariant BPS operators and the superconformal index.
In order to extract the gauge invariant components, we have to perform the gauge
integral with the Haar measure defined by
rankG
∏ I
(
)
dza
1
Pexp χGdµ ≡adj (za ) − rankG ,|WG | a=1 |za |=1 2πiza
G∫(1.83)

Soryushiron Kenkyu

34

Chapter 1 Introduction

where |WG | is the order of the Weyl group WG and χGadj (za ) is the character of the adjoint
representation of the gauge group G. Actually, the Haar measure is obtained by the zeromode integral in the localization. Finally, the index that counts the gauge invariant BPS
operators for Lagrangian theories is given by∫
ISCI (q, y, ui ) =dµPexp (isp (q, y, ui , za )) ,(1.84)

G

where isp (q, y, ui , za ) is the single-particle index including the gauge fugacities. In thisway, we can find the superconformal index of an SCFT with its Lagrangian.
We have an additional comment for the plethystic exponential (1.82). There is anotherexpression of the plethysitc exponential. To obtain this expression, we rewrite the righthand side of (1.82) by using f = exp(log f ):[Pexp(isp (q, y, ui )) = exp −∑]cn log(1 − mn (q, y, ui ))

n

= exp

[
]
∞
∑∑
cn mn (q, y, ui )k
n k=1

= exp

[∞
∑1
k=1

k

k

]

isp (q k , y k , uki ) ,(1.85)∑

where isp (q, y, ui ) = n cn mn (q, y, ui ) with the monomial mn (q, y, ui ). We use these two
formulas (1.82) and (1.85) depending on the situation.
We have a comment for the free field counting. In fact, the free field counting is notalways valid for the calculation of the superconformal index. In general, BPS operators
can acquire the anomalous dimension. In this case, the superconformal index calculatedby the localization is not in agreement with that calculated by the free field counting. For
the N = 4 SYM, BPS operators do not acquire the anomalous dimension, and we can use
the free field counting.

1.6

BPS partition function

There is another important supersymmetric partition function named BPS partition function. It includes only contributions from the BPS primary scalar operators. Thus the discussion of the Witten index is not applicable to the BPS partition function and in general
the BPS partition function depends on the coupling constant. As a result, the form of the
BPS partition function of the interacting theory is diﬀerent from that of the free theory*3 .
However, the structure of the BPS partition function is quite simple, and we can calculate*3 The BPS spectrum of primary scalar operators is invariant for the non-zero coupling constant [16,28].

Then, all we have to do is to take into account the spectrum at g = 0 and the spectrum for g ̸= 0,
where g is a coupling constant.

Soryushiron Kenkyu

1.6 BPS partition function

35

the BPS partition function even for interacting theories. Furthermore, the BPS partition
function gives us a significant hint to search the structure of the superconformal index.
Thus, study of the BPS partition function is important.
The definition of the BPS partition function is given by[
Z(ui ) = trBPS scalars

∏

]
i
uF
,
i

(1.86)iwhere Fi are charges of the global symmetries. Like the superconformal index, we canread oﬀ the BPS primary scalar operator spectrum from this.
For free theories, the BPS partition function can be evaluated by the free field counting.
On the other hand, there is no general method to calculate the BPS partition function
for interacting theories. In particular, the localization technique is not applicable to the
BPS partition function.
In this thesis, we calculate the BPS partition function for the N = 4 SYM and S-foldtheories. The BPS partition function of the N = 4 SYM has been well studied because
the BPS partition function can be calculated as a certain invariant polynomial of the
Weyl group of a gauge group. On the other hand, the S-fold theories are not a gauge
theory, and no one had calculated the BPS partition function for S-fold theories except
for the author’s and his collaborators’ research [23]. In Chapter 4, we will explain the
BPS partition function of the S-fold theories.

Soryushiron Kenkyu

Soryushiron Kenkyu

37

Chapter 2

AdS/CFT Correspondence
In this chapter, we review the AdS/CFT correspondence between the four-dimensional(4d) N = 4 U (N ) supersymmetric Yang-Mills (SYM) theory and type IIB superstring
theory on AdS5 × S 5 . This chapter aims to introduce the essential and basic conceptsof the AdS/CFT correspondence. We also review the BPS partition function and thesuperconformal index for N = 4 U (N ) SYM. Since the large N limit of the AdS/CFT
correspondence for N = 4 SYM have been well studied, the agreement of the BPS partition function and the superconformal index were already confirmed*1 . We also see thisagreement.
First, we review N = 4 SYM and its properties. Then we calculate the BPS partition
function and the superconformal index for N = 4 U (N ) SYM. Second, we review thestring theory and its properties necessary to see the AdS/CFT correspondence. Third, we
consider the AdS/CFT correspondence by using the D3-brane, which is a 3+1-dimensional
object existing in type IIB string theory. Finally, we confirm the agreement of the BPS
partition function and the superconformal index in AdS/CFT.

2.1

Four-dimensional N = 4 supersymmetric
Yang-Mills theory

Here we would like to discuss N = 4 SYM. This theory has N = 4 supersymmetry andthe maximally supersymmetric theory without gravity. It is believed that N = 4 SYM isthe only theory having N = 4 supersymmetry. In addition to the supersymmetry, N = 4
SYM also has the conformal symmetry, and hence the theory has N = 4 superconformal
symmetry. Namely, N = 4 SYM is an SCFT.
N = 4 SYM has a vector multiplet composed of a gauge field Aµ , gauginos λI and*1 Since the structure of the BPS partition function is easy,

the agreement of the BPS partition
function in the finite N region was also confirmed [29]. However, the investigation of the BPS
partition function gives us a great insight to find the finite N corrections to the superconformal
index on the AdS side and still important. The analysis in regard to the finite N corrections will
be given in Chapter 3.

Soryushiron Kenkyu

38

Chapter 2 AdS/CFT Correspondence

4

Q
Q3
Q2
Q1
Q1
Q2
Q3
Q4

RX

RY

1
2
1
2
− 12
− 12
1
2
1
2
− 12
− 12

1
2
− 12
1
2
− 12
1
2
− 12
1
2
− 12

RZ
− 12

SO(6)R

1
2
1
2
− 12
1
2
− 12
− 12
1
2

4

4

Table 2.1 The SO(6)R Cartan charges of supercharges. QI ’s belong to 4 ∈ SO(6)R ,while QI ’s belong to 4 ∈ SO(6)R .

I

λ , and adjoint scalar fields ϕIJ = −ϕJI . I, J = 1, 2, 3, 4 are the spinor indices of theSO(6)R ≃ SU (4)R symmetry, which is the rotation symmetry of four supercharges QI .
The four-dimensional spinor conventions is given in Appendix A.1. We define the Cartan generators of SO(6)R as RX , RY , and RZ . These generators are related to the
R-symmetry generators RI J defined in subsection 1.4.2 as
1 1(R 1 + R2 2 − R3 3 − R4 4 ),2
1
RY = (R1 1 − R2 2 + R3 3 − R4 4 ),2
1
RZ = (R1 1 − R2 2 − R3 3 + R4 4 ).
2

(2.1a)

RX =(2.1b)(2.1c)The upper and lower indices of QI and QI correspond to 4 and 4, respectively. The
SO(6)R Cartan charges of supercharges are listed in Table 2.1. We also list field contentsof N = 4 SYM in Table 2.2. Note that the scalar field ϕIJ satisfies the reality condition
ϕ

IJ

=

1 IJKL
ϵ
ϕKL .
2

(2.2)

Thus ϕIJ has the three independent complex scalar fields X, Y, and Z. We define thesethree complex scalar fields as
X = ϕ12 ,Y =ϕ13 ,Z = ϕ14 .

(2.3)

For the gauge field Aµ , its field strength Fµν can be decomposed into the self dual partα̇Fα β and anti-self dual part F β̇ asFα β =

1 µν β(σ )α Fµν ,
4

F

α̇

β̇ =

1 µν α̇(σ ) β̇ Fµν .
4

(2.4)

Soryushiron Kenkyu

2.1 Four-dimensional N = 4 supersymmetric Yang-Mills theory
Fields
ϕIJ
λI
I
λ
Fµν

39

SO(6)R
6
4
4
1

Table 2.2 Fields contents of the N = 4 SYM with their representation of SO(6)R .

The Lagrangian of the N = 4 SYM on R1,3 is given by
=4
LN
SYM =

[
1
1
1
I
IJ
Tr − Fµν F µν + iλ σ µ Dµ λI + Dµ ϕ Dµ ϕIJ
2
g
2
4
]
1 IJ KL
1
1 I J
θ
IJ
Tr(Fµν Feµν ),+ λI [λJ , ϕ ] + λ [λ , ϕIJ ] + [ϕ , ϕ ][ϕIJ , ϕKL ] +
2
2
32
16π 2
(2.5)

where Feµν is defined by1Feµν = ϵµνρσ Fρσ .
2

(2.6)

The Lagrangian (2.5) is invariant under the N = 4 supersymmetric transformation givenbyIδQ Aµ (x) = i(η I σ µ λI + ηI σµ λ ),(2.7a)1i
JK
δQ λI (x) = − Fµν σ µν ηI + Dµ ϕIJ σµ η J − [ϕIJ , ϕ ]ηK ,2
2(2.7b)L

δQ ϕIJ (x) = 2i(ηI λJ − ηJ λI − ϵIJKL η K λ ).

(2.7c)

Here the action δQ on a field Φ(x) is defined asδQ Φ(x) ≡ [ηI QI + η I QI , Φ(x)].

(2.8)

Furthermore, the Lagrangian (2.5) is also invariant under the super-conformal transformationIδS Aµ (x) = −xν λI σµ σ ν ζ I − xν λ σ µ σν ζ I ,δS λI (x) =(2.9a)i1
JKFµν xλ σ µν σ λ ζ I − iDµ ϕIJ xν σ µ σ ν ζ J + 2iϕIJ ζ J + [ϕIJ (x), ϕ ]xµ σµ ζ K ,2
2(2.9b)LδS ϕIJ (x) = −2xµ (ζ I σ µ λJ − ζ J σ µ λI − ϵIJKL ζ K σµ λ ).

(2.9c)

Soryushiron Kenkyu

40

Chapter 2 AdS/CFT Correspondence

Here we also define theIδS Φ(x) ≡ [ζ I SI + ζ I S , Φ(x)].

(2.10)

We can show that the transformations δQ and δS satisfy the N = 4 superconformal
algebra. Let us focus on δQ as an example. Then we can calculate a commutation′
relation [δQ , δQ
]Φ(x) as follows:′[δQ , δQ
]Φ(x) = [[ηI QI + η I QI , ηI′ QI + η ′I QI ], Φ(x)],(2.11)′
where δQ
Φ(x) is defined by (2.8) with the replacement η → η ′ and η → η ′ . When wechoose Φ(x) as the adjoint scalar field ϕIJ , we find′′[δQ , δQ
]ϕIJ = 2i∂µ ϕIJ ηK σ µ η ′K − 2i∂µ ϕIJ ηK
σµ ηK .

(2.12)

This means thatα̇α̇[{QI , QJβ }, ϕKL ] = [−2δIJ (σ µ )β Pµ , ϕKL ].

(2.13)

Thus we derived the same commutation relation as (1.47a). In this way, we can obtainall the anti-commutation relations (1.47) among fermionic generators. We can also showthat the Lagrangian (2.5) is invariant under the conformal transformation using (1.40).
Therefore, the Lagrangian (2.5) indeed has N = 4 superconformal symmetry on R1,3 .
The beta function of N = 4 SYM is zero. Since the theory includes four Weyl spinor
fields λI and three complex scalar fields defined by (2.3), the one-loop beta function
becomes
(
)
11 1
2
g3
β(g) ∝ −− · 3 − · 4 = 0,(2.14)
16π 2 33
3
and this is consistent with the superconformal symmetry.
It is often useful to define the complex coupling constant τ as
τ=

θ
4πi
+ 2.
2π
g

(2.15)Let us consider the marginal deformation of N = 4 SYM. To see it, we rescale thedefinition of the fields as
Aµ → gAµ ,ϕIJ → gϕIJ ,λI → gλI .

(2.16)

Soryushiron Kenkyu

2.1 Four-dimensional N = 4 supersymmetric Yang-Mills theory
Fields(X, λ2 )(Y, λ3 )(Z, λ4 )(Aµ , λ1 )N = 4 (Q1,2,3,4 )N = 2 (Q1,4 )hypermultiplet

vector multiplet
vector multiplet

41

N = 1 (Q1 )chiral multiplet
chiral multiplet
chiral multiplet
vector multiplet

Table 2.3 The N = 2 and N = 1 decomposition of the N = 4 vector multiplet.
The N = 2 supersymmetry is generated by Q1,4 and its conjugate and the N = 1
supersyymetry is generated by Q1 and its conjugate.

Then the Lagrangian (2.5) is changed as
[
1
1
I
IJ
L = Tr − Fµν F µν + iλ σ µ Dµ λI + Dµ ϕ Dµ ϕIJ
24]g 2 IJ KL
g
g I J
IJ
+ λI [λJ , ϕ ] + λ [λ , ϕIJ ] + [ϕ , ϕ ][ϕIJ , ϕKL ] ,2
2
32(2.17)

where we neglect the topological term Fµν Feµν . Then we can split the Lagrangian into twoparts: the free part and the interacting part. The conformal dimension of the interacting
part is four, and g is dimensionless. Therefore, all interacting terms are regarded as themarginal deformations, and N = 4 SYM has the marginal deformations and connects toa free theory. Note that N = 4 U (1) Maxwell theory is always a free theory because allinteracting terms vanish.

2.1.1

N = 1, 2 decomposition

It is helpful to decompose the N = 4 vector multiplet into the N = 1, 2 multiplets. Herewe choose Q1,4 and Q1,4 as supercharges of the N = 2 subalgebra. For N = 1 subalgebra
we use Q1 and Q1 . Then the N = 4 vector multiplet splits into the N = 2 vector multiplet
and the N = 2 hypermultiplet as follows:
N =4
=2
N =2
(X, Y, Z, λI , Fαβ )vector→ (Z, λ1,4 , Fαβ )Nvector ⊕ (X, Y, λ2,3 )hyper ,(2.18)

where we only consider the self dual part of the gauge field and its superpartner. Similarly,
the N = 4 vector multiplet splits into the N = 1 vector multiplet and three N = 1 chiralmultiplets as
N =1
N =1
=4
N =1
N =1
(X, Y, Z, λI , Fαβ )Nvector → (λ1 , Fαβ )vector ⊕ (X, λ2 )chiral ⊕ (Y, λ3 )chiral ⊕ (Z, λ4 )chiral .
(2.19)

We summarize these decomposition in Table 2.3.
For calculating the BPS partition function, it is convenient to classify the BPS operatorsof N = 4 SYM in terms of the N = 1, 2 language. First, we define the chiral primary

Soryushiron Kenkyu

42

Chapter 2 AdS/CFT Correspondence
Operators
TrX
TrY
TrZ

Supercharges
Q3,4 , Q1,2
Q2,4 , Q1,3
Q2,3 , Q1,4

Table 2.4 Examples of chiral primary operators and supercharges annihilating them.

operators as superconformal primary operators annihilated by a certain combination ofsupercharges QI , QI . Examples of the chiral primary operators are the adjoint scalar
operators TrX, TrY, and TrZ. The trace is taken in the adjoint representation of thegauge group and necessary for making operators gauge invariant. The super-conformaltransformation of the adjoint scalar fields (2.9c) vanishes at the origin. Thus the adjoint
scalar operators (inserted at the origin) are indeed the superconformal primary operators.
Also, the supersymmetry transformation rules up to numerical coeﬃcients, which can beread oﬀ from (2.7c), areI[QIα , ϕJK ] ∝ δJI λKα − δK
λJα ,

α̇

[QI , ϕJK ] ∝ ϵIJKL λ

Lα̇.

(2.20)

Then we can see the adjoint scalar operators TrX, TrY, TrZ defined by (2.3) are annihilated
by some of the supercharges. We list which supercharges annihilate the adjoint scalar
operators in Table 2.4. That is why the adjoint scalar operators TrX, TrY, TrZ are thechiral primary operators. Note that Q1 annihilates all the adjoint scalar operators TrX,
TrY, TrZ.
kk-BPS operators”. The 16-BPS operators are
Next, we would like to define the term “ 16α̇

the BPS operators annihilated by k supercharges of {QIα , QI |I = 1, 2, 3, 4, α = 1, 2, α̇ =1̇, 2̇}. The number of total supercharges are sixteen, so k can take value from 1 to 16.
Let us consider the several examples. First, operators consisting of Z are annihilated by
α̇
1
Q2,3
α and Q1,4 , so they are 2 -BPS operators. Second, operators consisting of X and Y areα̇annihilated by Q4α and Q1 , so they are 14 -BPS operators. Finally, operators consisting ofα̇

X, Y, and Z are annihilated by Q1 , so they are 18 -BPS operators.

2.1.2

Coulomb branch and Higgs branchHere we would like to discuss the Coulomb branch and the Higgs branch. They are themoduli space of a given theory, which is defined as the set of vacua. Let us consider agauge theory with the gauge group G and a potential term V (ϕ), where ϕ is a scalar fieldof this theory. We assume that the minimum value of V (ϕ) is zero. Then the moduli spaceis given by a set of vacua, which is specified by the vacuum expectation value (VEV) of
ϕ satisfying V (ϕ) = 0. Furthermore, we need to remove the gauge redundancy from a set

Soryushiron Kenkyu

2.1 Four-dimensional N = 4 supersymmetric Yang-Mills theory

43of vacua. Then the moduli space M is defined by
M ≡ {ϕ|V (ϕ) = 0}/G.

(2.21)Let us consider an N = 2 gauge theory. Then, there is the N = 2 vector multiplet
composed of (ϕ, λ, ψ, Aµ ), where ϕ is an adjoint scalar field, λ and ψ are Weyl spinor
fields, and Aµ is the gauge field. In general, the adjoint scalar field ϕ has a potential term
V (ϕ) ∝ Tr[ϕ, ϕ]2(2.22)The minimum condition V (ϕ) = 0 gives the VEV of ϕ as
⟨ϕ⟩ =

rankG
∑

ca H a ,(2.23)

a=1

where ca is a complex parameter and H a is the Cartan generator of the gauge group G.
The residual gauge redundancy is the Weyl transformation of G. Then the moduli space isgiven by CrankG /W, where W is the Weyl group of G. This moduli space is parameterized
by the gauge invariant Casimir operators and called the Coulomb branch. Also, we call
Casimir operators parameterizing the Coulomb branch the Coulomb branch operators.
A similar discussion can be made for the matter sector. Here we consider an N = 2gauge theory with a hypermultiplet composed of (ψ qe, q, qe, ψq ), where ψ qe, ψq are Weyl
spinor fields and q, qe are complex scalar fields. The hypermultiplet belongs to a certain
representation of G. If there is a potential term for a field in the hypermultiplet, the VEVof this field defines vacua, which forms the moduli space. This moduli space is called
the Higgs branch. The Casimir operators parameterizing the Higgs branch are called
the Higgs branch operators. We can also define the mixed branch by using both vector
multiplet and hypermultiplet.
Now let us apply above discussion to N = 4 SYM with the gauge group G. From theN = 2 perspective, the theory has the vector multiplet (Z, λ1,4 , Fαβ ) and the hypermultiplet (X, Y, λ2,3 ) as we have seen in (2.18). The potential term of adjoint scalar fields
X, Y, Z is given by
V (X, Y, Z) =

1
1
IJ
KL
[ϕ
,
ϕ
][ϕ
,
ϕ
]
=
−
KL
IJ
32g 2
2g 2

∑

[Φ, Φ′ ]2 .

(2.24)

Φ,Φ′ =X,Y,Z,X,Y,Z

Since Z sits on the vector multiplet, the minimum condition
1
V (X, Y, Z) ⊃ V (Z) = − 2 [Z, Z]2 = 0g(2.25)

gives the Coulomb branch of this theory. Similarly, since X and Y sit on the hypermultiplet,Soryushiron Kenkyu

44

Chapter 2 AdS/CFT Correspondence

the minimum condition
V (X, Y, Z) ⊃ V (X, Y) = −

1
2g 2

∑

[Φ, Φ′ ]2 = 0(2.26)

Φ,Φ′ =X,Y,X,Y

gives the Higgs branch. The minimum condition
V (X, Y, Z) = 0(2.27)

gives the mixed branch. The moduli spaces for the Coulomb branch, the Higgs branch,
the mixed branch are CrankG /WG , C2rankG /WG , C3rankG /WG , respectively.

2.1.3

BPS partition function

In this subsection, we would like to discuss the BPS partition function of the N = 4 U (N )SYM. According to the discussion in Sec. 1.6, we can define the BPS partition function
by using the global symmetries of N = 4 SYM, namely SO(6)R . Thus the BPS partition
function is defined byZ(x, y, z) = trBPS (xRX y RY z RZ ),(2.28)

where the trace is taken over the gauge invariant BPS operators consisting of the adjoint
scalar fields X, Y, and Z. The BPS condition for an operator O is here given by[Q1 , O] = 0.

(2.29)

An operator O satisfying this condition is a 18 -BPS operator. In this sense, the BPS
partition function (2.28) is called the 18 -BPS partition function. Then the BPS partition
function have the information of the BPS operator spectrum for operators consisting of
X, Y, Z. Note that we do not consider the operators including X, Y, and Z because thesefields do not satisfy the BPS condition (2.29).
We can impose another BPS condition for the trace. If we impose[Q2 , O] = [Q3 , O] = [Q1 , O] = [Q4 , O] = 0(2.30)as BPS conditions, the operators satisfying these conditions are 21 -BPS operators andcomposed of Z. Because the R-charges of Z are (RX , RY , RZ ) = (0, 0, 1), the corresponding BPS partition function is derived by setting x = y = 0 in Z(x, y, z). We call this BPS
partition function the 21 -BPS partition function. Also, since Z sits in the N = 2 vector
multiplet, this BPS partition function is often called the Coulomb branch Hilbert series.
We can also impose the following BPS conditions:[Q4 , O] = [Q4 , O] = 0.

(2.31)

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2.1 Four-dimensional N = 4 supersymmetric Yang-Mills theory

45

Then the operators satisfying these conditions are 14 -BPS operators and composed of X
and Y. Because the R-charges of X and Y are (RX , RY , RZ ) = (1, 0, 0), (0, 1, 0), thecorresponding BPS partition function is derived by setting z = 0 in Z(x, y, z). We callthis BPS partition function the 41 -BPS partition function. Also, since X and Y belong to
the N = 2 hypermultiplet, this BPS partition function is often called the Higgs branch
Hilbert series.
In this section, we are interested in the N = 4 U (N ) SYM. In this case, the gauge
invariant BPS operators are simply trace operators consisting of X, Y, Z. Although thedeterminant operators are also gauge invariant operators, we do not consider them because
the determinant operators can be written in terms of the trace operators. The reason is
as follows. For the determinant operators in the N = 4 U (N ) SYM, we use two epsilon
tensors like ϵij··· ϵkl··· Φik Φjl · · · . Then the epsilon tensors can be written in terms of the
Kronecker deltas. Thus the determinant operators can be written by a certain combinationof the trace operators.
We also comment on the Pfaﬃan operators. It is roughly adjoint fields contracted by
one epsilon tensor. The precise definition is given in (4.6). In the U (N ) case, this cannot
be a gauge invariant operator since it is impossible to contract all the indices of adjoint
fields by one epsilon tensor. Thus, the Pfaﬃan operators are absent in the U (N ) gauge
theory. However, the gauge invariant Pfaﬃan operators exist in the SO(2N ) gauge theory,and they play a significant role in Chapter 4.
Free theory
Here we discuss the BPS partition function of the N = 4 U (N ) SYM at g = 0. In thiscase, we can use the free field counting to find the BPS partition function. Because allgauge invariant BPS operators are obtained as a combination of X, Y, Z, the BPS partition
function for g = 0 can be derived from the following formula:∫
U (N )ZU (N ) (x, y, z) =U (N )∫

dµPexp[(x + y + z)χadj ],(2.32a)

(
)
dzi ∏
zj
1−
,
zk
|zi |=1 2πizi(2.32b)1 ∏dµ =N ! i=1
U (N )N

U (N )

χadj

=

N
∑
zi

z
i,j=1 j

∫

j̸=k

,

(2.32c)where the Haar measure (2.32b) is obtained by the formula (1.83). Therefore, the formula(2.32) contains terms for operators corresponding to all combinations of gauge invariant

Soryushiron Kenkyu

46

Chapter 2 AdS/CFT Correspondence

BPS operators. We show the BPS partition functions for small ranks below:
ZUfree(1) (x, y, z) =ZUfree(2) (x, y, z) =1,
(1 − x)(1 − y)(1 − z)

(2.33)
1 + xyz(1 − x)(1 − y)(1 − z)(1 − x2 )(1 − y 2 )(1 − z 2 )(1 − xy)(1 − yz)(1 − zx).

(2.34)Let us consider the physical meaning of the BPS partition function given above. For
simplicity, we set x = y = z = q. Expanding ZUfree(2) (q, q, q) with respect to q, we find
2
3
ZUfree(2) (q, q, q) = 1 + 3q + 12q + 29q + · · · .

(2.35)

The coeﬃcients describe the number of BPS operators having the same conformal dimension in each term because the BPS condition (2.29) means RX + RY + RZ = E where E
is the conformal dimension. Let us focus on the term 12q 2 as an example. Actually, we
can find the following BPS operators with E = 2:
Tr(X2 ),

Tr(Y2 ),

Tr(Z2 ),Tr(XY),

Tr(YZ),

Tr(ZX),(TrX)2 ,(TrY)2 ,(TrZ)2 ,(TrX)(TrY),(TrY)(TrZ),(TrZ)(TrX).

(2.36)The number of these BPS operators is 12. Thus the BPS partition function indeed has
the information of the BPS operator spectrum as coeﬃcients of the fugacities.
For N = 4 SU (N ) SYM, the BPS partition function can be obtained by dividing the
U (N ) BPS partition function (2.32) by the U (1) BPS partition function (2.33):
ZSU (N ) =ZU (N ).
ZU (1)(2.37)For example, the SU (2) BPS partition function is given by
free
ZSU
(2) =

1 + xyz.
(1 − x2 )(1 − y 2 )(1 − z 2 )(1 − xy)(1 − yz)(1 − zx)(2.38)

Although the 21 -BPS partition function can be obtained by ZUfree(N ) (0, 0, z), we have11another derivation for the 2 -BPS partition function. Since the 2 -BPS operators consist of
Z only, we can find all the 21 -BPS operators directly. These are {TrZ|n = 1, · · · , N } andtheir composite operators. Namely, we can regard the single trace operators {TrZn } as
the single-particle operators of the 21 -BPS operators. Thus the 21 -BPS partition function
becomes
(N
)
N
∑
∏
1
n
ZUfree
(0,
0,
z)
=
Pexp
.
(2.39)
z
=
(N )
n
1
−
z
n=1
n=1

Soryushiron Kenkyu

2.1 Four-dimensional N = 4 supersymmetric Yang-Mills theory

47

Actually, we can show that∫
U (N )U (N )
dµPexp(zχadj ) =

N
∏

1
.
1 − znn=1(2.40)

Interacting theory
Let us consider the BPS partition function for the interacting theory, g ̸= 0. In general,
the form of the BPS partition function in g ̸= 0 is diﬀerent from that at g = 0 because someof the BPS operators acquire the quantum correction, and they are no longer the BPS
operator in g ̸= 0. However, the BPS partition functions in the weak coupling region and
the BPS partition functions in the strong coupling region agree with each other [16, 28].
For U (1) Maxwell theory, all fields in the Lagrangian (2.17) are commutative with eachother, so there is no interaction terms and the U (1) BPS partition function in g ̸= 0 is
the same as that at g = 0.
First, the 21 -BPS partition function is the same as that in the free theory (2.39), because
the ingredient to make the 12 -BPS operators is the only Z, and we need not take theinteraction into account.
It is not so easy to find the 18 -BPS partition function for the interacting theory because
there are non-trivial relations called syzygies [30] among the single trace operators. For
example, in the U (2) SYM, we find the following relation
Tr({X, Y}Z) = (TrX)(TrYZ) + (TrY)(TrZX) + (TrZ)(TrXY) − (TrX)(TrY)(TrZ). (2.41)
and the operator Tr({X, Y}Z) does not contribute to the U (2) BPS partition function.
A more systematic way to count the independent BPS operators satisfying (2.29) is
to use the chiral ring generated by Q1 . As we have seen in subsection 1.5.1, the chiralring is defined in an N = 1 theory. Now we regard N = 4 SYM as an N = 1 theory,
where supercharges are Q1 and Q1 . Then the interaction terms regarding the adjoint
scalars are treated as the potential terms (2.24), and they should be reproduced by thesuperpotential from the N = 1 perspective. The superpotential W (Φi ) that reproduce
the potential term should satisfy an equation
δW∝ g 2 ϵijk Tr[Φj , Φk ],δΦi(2.42)

where Φi=x,y,z are N = 1 chiral multiplets including the scalar field X, Y, Z, respectively.
Furthermore, it follows thatδW
∝ [Φ∗i ]FδΦi(2.43)

from an equation of motion for a general homogeneous superpotential consisting of a chiralmultiplet Φi , where [ ]F means picking up the F -term of the anti-chiral multiplet Φ∗i .

Soryushiron Kenkyu

48

Chapter 2 AdS/CFT Correspondence

Thanks to the F -term condition, we regard the commutation relation [Φi , Φj ] as zero, andfor the adjoint scalar fields, we have
[X, Y] = [Y, Z] = [Z, X] = 0.

(2.44)

Therefore, we regard these scalar fields as Cartan valued adjoint matrices of the gauge
group. This condition is actually the same as the minimum condition of the potential (2.24). Thus the BPS operators consisting of X, Y, Z parametrize the moduli space
C3N /WU (N ) , where WU (N ) = SN is the Weyl group of U (N ).
Now we can diagonalize X, Y, Z simultaneously and let xi , yi , zi be the diagonal components. The Weyl transformation acts on xi , yi , zi as xσ(i) , yσ(i) , zσ(i) , where σ ∈ SN .
Then the 18 -BPS operators are polynomials of these N variables invariant under the Weyltransformation. We can use the following polynomials as bases of the invariant polynomial.
N
∑ ∏

mx

my

mz

i
i
i
xσ(i)
yσ(i)
zσ(i)
,

(2.45)σ∈SN i=1

where the set of three integers {mxi , myi , mzi } labels the basis. We can regard (2.45) as the∑wave function of the three-dimensional harmonic oscillator. The symmetrization σ∈SN
is now interpreted as the Bose statistics. Therefore, the wave function (2.45) is regarded
as that of the system with N bosonic particles in the harmonic potential.
The 18 -BPS grand partition function is the same as the grand partition function of thissystem and given by1
8 -BPSΞU (∗) (x, y, z; t) =

∞
∑

1

-BPS

ZU8 (N ) (x, y, z)tN

N =0
∞
∏

1
= Pexp=1 − txp y q z r
p,q,r=0(t(1 − x)(1 − y)(1 − z)).

(2.46)

The 12 -BPS and 14 -BPS grand partition functions can be obtained by setting x = y = 0
and z = 0 respectively. In particular, we can show that1
2 -BPSΞU (∗)∞∏∞∑
1(z; t) ==1 − tz r
r=0

N =0

(N
∏

1
1 − znn=0)tN(2.47)

by using the q-binomial theorem, and the coeﬃcient of tN indeed agrees with (2.39).
It is easier to obtain the BPS partition functions for small ranks by expanding the grandSoryushiron Kenkyu

2.1 Four-dimensional N = 4 supersymmetric Yang-Mills theory

49

partition function (2.46) with respect to t. The results for N = 1, 2 are
1,(1 − x)(1 − y)(1 − z)
1
1 + xy + yz + zx
-BPS
ZU8 (2) =.
(1 − x)(1 − y)(1 − z)(1 − x2 )(1 − y 2 )(1 − z)2
1-BPS

ZU8 (1)=(2.48)(2.49)In these examples, we see that the U (1) BPS partition function indeed agrees with thatof the free theory (2.33), as we mentioned at the beginning of this sub-subsection. On
the other hand, the U (2) partition function is completely diﬀerent from the free U (2)partition function (2.34).
To see the diﬀerence between (2.34) and (2.49) clearer, let us expand the U (2) BPS
partition function (2.49) with respect to x = y = z ≡ q. The result is
1-BPS

ZU8 (2) (q, q, q) = 1 + 3q + 12q 2 + 28q 3 + · · · .

(2.50)We see that the term 28q 3 deviates from the term in (2.35) by q 3 . The origin of thisdeviation comes from the fact that a degree three BPS operator Tr([X, Y]Z) is no longer
contributes to the BPS partition function because of the commutativity (2.44).

2.1.4

Superconfromal index

Another example of the physical quantities we would like to consider is the superconformal
index. As in the case of the BPS partition function, the superconformal index has theinformation of the BPS spectrum of the theory. Unlike the BPS partition function, thesuperconformal index includes contributions from scalar, fermion, gauge field, and theirdescendant operators, as well as chiral primary scalar operators. Thus the superconformal
index has richer information than the BPS partition function.
As we have seen in Sec. 1.5, the superconformal index is defined as the Witten index
regarding the N = 4 superconformal algebra on R × S 3 . The algebra among fermionic
generators are given byα̇{QI , QJα } = δIJ P α̇ β ,(2.51a)J{SIα , S β̇ } = δIJ K α β̇ ,

(

)

1 α J
1
δβ δI H + δIJ J α β + δβα RJ I − δIJ RK K ,2
4()
1 α̇ J
1 J K
α̇
J
J α̇
α̇
J
{QI , S β̇ } = δβ̇ δI H − δI J β̇ − δβ̇ R I − δI R K .
2
4{SIα , QJβ } =(2.51b)(2.51c)(2.51d)Soryushiron Kenkyu

50

Chapter 2 AdS/CFT Correspondence

Now the superconformal index I of the N = 4 SYM on R × S 3 is defined as
I = tr[(−1)F x∆ q H+jR y 2jL uRX −RY v RY −RZ ],(2.52)

where ∆ is given by11̇

∆ = 2{S 1̇ , Q1 } = H − 2jR − (RX + RY + RZ ).

(2.53)The trace is taken over the Hilbert space of the theory on R × S 3 . Here H is the Hamil1̇

tonian in the radial quantization, and we defined jL = −J 2 2 and jR = J 1̇ . The SO(6)Rgenerators RX , RY , and RZ are defined in (2.1). The BPS condition of an operator O
for the superconformal index (2.52) is1̇

1[Q1 , O] = [S 1̇ , O] = 0.

(2.54)

This condition is equivalent to the condition ∆ = 0, and the index (2.52) does not dependon x.
It is convenient to expand the index (2.52) with respect to the fugacity q. Then, we
can classify the BPS operators in order from the operator having the smallest conformal
dimension. In this case, the index respects the left-handed spin symmetry SU (2)jL and1̇

part of SO(6)R . Since we choose a supercharge Q1 to define the index (2.52), SO(6)R is
broken to SU (3)R × U (1). This SU (3)R is generated by RX − RY and RY − RZ , while1̇

this U (1) is generated by RX + RY + RZ . The supercharge Q1 has a non-trivial charge
under this U (1) (see Table 2.1), so the index only respects this SU (3)R symmetry. As
a result, the index expanded by q can be written in terms of the SU (3)R character andthe SU (2)jL character. The SU (3) character is defined in (A.34), and we define SU (2)jLcharacter as
χJn = χn (y),(2.55)

where χn (a) is defined in (A.33).
It seems that the BPS condition (2.54) is diﬀerent from what we used for the BPS
partition function (2.29). In the case of the BPS partition function, since we only consider
1
the chiral primary operators, the condition [S 1̇ , O] = 0 is automatically satisfied. Thus,
the chiral primary operators that we have considered in the previous subsection satisfy
the BPS condition (2.54), and they also contribute to the index (2.52).
Let us calculate the index (2.52) for the N = 4 U (N ) SYM following the procedure
given in subsection 1.5.2. To do it, our task is to find the single-particle index of thistheory, and we need the list of all fields and equations of motion with charges of U (1)H ×
SU (2)jL × SU (2)jR × SO(6)R . We show this list in Table 2.5 and Table 2.6, and we use(R ,R ,R )the notation [jL , jR ]E X Y Z , where E is the conformal dimension (eigenvalue of H).

Soryushiron Kenkyu

2.1 Four-dimensional N = 4 supersymmetric Yang-Mills theory
Fields
X
Y
Z
X
Y
Z
λ11
λ12
λ21
λ22
λ31
λ32
λ41
λ42
41̇
λ4
42̇

λ

31̇

λ

32̇

λ

21̇

λ

22̇

λ

11̇

λ

12̇

λ

F11
F12
F22
F
F

1̇1̇
1̇2̇
2̇2̇

F
∂11̇
∂12̇
∂21̇
∂22̇

(R ,RY ,RZ )[jL , jR ]E X∆

(1,0,0)[0, 0]1(0,1,0)[0, 0]1(0,0,1)[0, 0]1(−1,0,0)[0, 0]1(0,−1,0)[0, 0]1(0,0,−1)[0, 0]1(1,1,1)[ 12 , 0] 32 2 22(1,1,1)1[− 2 , 0] 32 2 2
2( 1 ,− 1 ,− 1 )[ 12 , 0] 32 2 2
2( 1 ,− 1 ,− 1 )1[− 2 , 0] 32 2 2
2(− 1 , 1 ,− 1 )[ 12 , 0] 3 2 2 2
2(− 1 , 1 ,− 1 )1[− 2 , 0] 3 2 2 2
2(− 1 ,− 1 , 1 )[ 12 , 0] 3 2 2 2
2(− 1 , 1 ,− 1 )1[− 2 , 0] 3 2 2 2
2( 1 , 1 ,− 1 )[0, 12 ] 32 2 2
2
1 1
1
1 ( 2 , 2 ,− 2 )[0, − 2 ] 3
2
1
1 1
1 ( 2 ,− 2 , 2 )[0, 2 ] 32
1
1 1
1 ( 2 ,− 2 , 2 )[0, − 2 ] 3
2
1 1 1
1 (− 2 , 2 , 2 )[0, 2 ] 32
1 1 1
1 (− 2 , 2 , 2 )[0, − 2 ] 32111
1 (− 2 ,− 2 ,− 2 )[0, 2 ] 32(− 1 ,− 1 ,− 1 )[0, − 12 ] 3 2 2 2
2(0,0,0)[1, 0]2(0,0,0)[0, 0]2(0,0,0)[−1, 0]2(0,0,0)[0, 1]2(0,0,0)[0, 0]2(0,0,0)[0, −1]2(0,0,0)[ 12 , 12 ]1(0,0,0)[ 12 , − 12 ]1(0,0,0)[− 12 , 12 ]1(0,0,0)[− 12 , − 12 ]1

0
0
0
2
2
2

∂2X = 0
∂2Y = 0∂2Z = 0
∂2X = 0
∂2Y = 0∂2Z = 00(∂λ)11̇ = 00(∂λ)12̇ = 02(∂λ)21̇ = 02(∂λ)22̇ = 02(∂λ)31̇ = 02(∂λ)32̇ = 02(∂λ)41̇ = 02(∂λ)42̇ = 00(∂λ)41 = 0

2(∂λ)42 = 00(∂λ)31 = 02(∂λ)32 = 00(∂λ)21 = 02(∂λ)22 = 02(∂λ)11 = 04(∂λ)12 = 0

2
2
2
0
2
4
0
2
0
2

Table 2.5 Charges for fields in N = 4
SYM.

EOM

(∂F )1̇1 = 0(∂F )1̇2 = 0(∂F )2̇4 = 0(∂F )2̇2 = 0(∂F )1̇1 = 0(∂F )1̇2 = 0(∂F )2̇1 = 0(∂F )2̇2 = 0ϵ∂[γα̇ ∂β̇]Fγϵ ≡ 0[α β]∂γ̇ ∂ϵ̇ F

γ̇ ϵ̇

≡0

51

[jL , jR ]E X

(R ,RY ,RZ )∆(1,0,0)[0, 0, 0]3(0,1,0)[0, 0, 0]3(0,0,1)[0, 0, 0]3(−1,0,0)[0, 0, 0]3(0,−1,0)[0, 0, 0]3(0,0,−1)[0, 0, 0]3(1,1,1)[0, 12 ] 52 2 22(1,1,1)[0, − 12 ] 52 2 2
2( 1 ,− 1 ,− 1 )[0, 12 ] 52 2 2
2111
1 ( 2 ,− 2 ,− 2 )[0, − 2 ] 5
2(− 1 , 1 ,− 1 )[0, 12 ] 5 2 2 2
21 1
11 (− 2 , 2 ,− 2 )[0, − 2 ] 5
2(− 1 ,− 1 , 1 )[0, 12 ] 5 2 2 2
21
1 1
1 (− 2 ,− 2 , 2 )[0, − 2 ] 5
2( 1 , 1 ,− 1 )[ 12 , 0] 52 2 2
2( 1 , 1 ,− 1 )1[− 2 , 0] 52 2 2
2( 1 ,− 1 , 1 )[ 12 , 0] 52 2 2
2( 1 ,− 1 , 1 )1[− 2 , 0] 52 2 22(− 12 , 21 , 21 )1
[ 2 , 0] 5
2(− 1 , 1 , 1 )[− 12 , 0] 5 2 2 22(− 12 ,− 21 ,− 12 )1
[ 2 , 0] 5
2(− 1 ,− 1 ,− 1 )[− 12 , 0] 5 2 2 22
1 1 (0,0,0)[ 2 , 2 ]3(0,0,0)[ 12 , − 21 ]3(0,0,0)[− 12 , 12 ]3(0,0,0)[− 12 , − 12 ]3(0,0,0)[ 12 , 12 ]3(0,0,0)[ 12 , − 21 ]3(0,0,0)[− 12 , 12 ]3(0,0,0)[− 12 , − 12 ]3(0,0,0)[0, 0]4(0,0,0)[0, 0]4

2
2
2
4
4
4
0
2
2
4
2
4
2
4
2
2
2
2
2
2
4
4
2
4
2
4
2
4
2
4
4
4

Table 2.6 Charges for equations of motion(EOM) in N = 4 SYM.

Soryushiron Kenkyu

52

Chapter 2 AdS/CFT Correspondence

Now we pick up the BPS fields with ∆ = 0 to calculate the single-particle index. Other
contributions are automatically canceled due to the boson-fermion cancellation. We should
also include the descendants generated by ∂11̇ and ∂21̇ because these derivatives satisfy
the BPS condition. Let m(q, y, u, v) be a monomial corresponding to the contribution ofa certain BPS primary operator OBPS . Its descendants are obtained by multiplying thederivatives repeatedly. For example, if a descendant has k ∂11̇ ’s and l ∂21̇ ’s, this descendant
field is (∂11̇ )k (∂21̇ )l OBPS . Since the monomials corresponding to the derivatives ∂11̇ and3
3∂21̇ are q 2 y and q 2 y −1 respectively, the monomial corresponding to this descendant field
3
3
is given by (q 2 y)k (q 2 y −1 )l m(q, y, u, v). Therefore, all the contributions from the primary
field and descendant fields becomes∞∑(q 2 y)k (q 2 y −1 )l m(q, y, u, v) =3m(q, y, u, v)

3

k,l=0

3

3

(1 − q 2 y)(1 − q 2 y −1 ).

(2.56)

Gathering all the BPS contributions, including their descendants, gives the following
formula for the single-particle index:
3

isp =

χ(1,0) q − χJ1 q 2 − χ(0,1) q 2 + 2q 3
3

3

(1 − q 2 y)(1 − q 2 y −1 ),(2.57)

where χ(m,n) and χJn are the SU (3)R character and SU (2)jL character defined by (A.34)
and (2.55), respectively.
According to the formula (1.84), the superconformal index of the N = 4 U (N ) SYM is
now given by∫
U (N )IU (N ) =U (N )

dµPexp(isp χadj ),(2.58)U (N )

where isp is defined in (2.57). The Haar measure is given by (2.32b) and χadj is given by(2.32c). Note that this formula is valid even for the interacting theory because the index
does not depend on the coupling constant. The indices for small ranks up to some ordersof q are given by3

IU (1) = 1 + χ(1,0) q − χJ1 q 2 + (χ(2,0) − χ(0,1) )q 2 + (χ(3,0) − χ(1,1) + 1 − χJ2 )q 3
7

+ χJ1 χ(0,1) q 2 + (χ(4,0) − χ(2,1) + χ(1,0) − χJ2 χ(1,0) )q 4
11

+ (χJ2 χ(0,1) + χ(2,0) − χ(3,1) + χ(5,0) )q 5 + (−χJ1 χ(1,0) − χJ3 χ(1,0) )q 2
13

+ (χJ2 χ(1,1) + χ(3,0) − χ(4,1) + χ(6,0) )q 6 + O(q 2 ),(2.59)

Soryushiron Kenkyu

2.1 Four-dimensional N = 4 supersymmetric Yang-Mills theory
3

53

5

IU (2) = 1 + χ(1,0) q − χJ1 q 2 + (−χ(0,1) + 2χ(2,0) )q 2 − χJ1 χ(1,0) q 2
7+ (2 − χJ2 − χ(1,1) + 2χ(3,0) )q 3 + χJ1 (χ(0,1) − χ(2,0) )q 2
9+ (−χJ2 χ(1,0) + χ(1,0) − 2χ(2,1) + 3χ(4,0) )q 4 + χJ1 (1 + 2χ(1,1) − χ(3,0) )q 2
+ (−2χJ2 χ(2,0) − 2χ(0,1) + χ(2,0) − 2χ(3,1) + 3χ(5,0) )q 5
11

+ χJ1 (χ(1,0) + 2χ(2,1) − χ(4,0) )q 2

+ (χJ2 (χ(1,1) − χ(3,0) ) − χ(1,1) + 2χ(3,0) − 3χ(4,1) + 4χ(6,0) )q 6
13

+ (−χJ3 χ(2,1) + χJ1 (−3χ(0,1) − χ(1,2) + 2χ(3,1) − χ(5,0) ))q 2+ (χJ2 (3χ(0,2) + 2χ(1,0) + 2χ(2,1) − 2χ(4,0) ) + 4χ(1,0) + 2χ(4,0)− 3χ(5,1) + 4χ(7,0) )q 7
+ (χJ5 + χJ3 (−1 − 2χ(1,1) − χ(3,0) ) + χJ1 (−1 − 5χ(1,1)
15− χ(2,2) − 2χ(3,0) + 2χ(4,1) − χ(6,0) ))q 2+ (χJ4 (χ(0,1) − χ(2,0) ) + χJ2 (χ(1,2) + 4χ(2,0) + 3χ(3,1) − 2χ(5,0) )17+ 3χ(0,1) + χ(1,2) + χ(2,0) + 3χ(5,0) − 4χ(6,1) + 5χ(8,0) )q 8 + O(q 2 ), (2.60)

Soryushiron Kenkyu

54

Chapter 2 AdS/CFT Correspondence
3

5

IU (3) = 1 + χ(1,0) q − χJ1 q 2 + (−χ(0,1) + 2χ(2,0) )q 2 − χJ1 χ(1,0) q 2
7+ (2 − χJ2 − χ(1,1) + 3χ(3,0) )q 3 + χJ1 (χ(0,1) − 2χ(2,0) )q 2
+ (−χJ2 χ(1,0) + χ(0,2) + 2χ(1,0) − 2χ(2,1) + 4χ(4,0) )q 4
9

+ χJ1 (1 + χ(1,1) − 2χ(3,0) )q 2
+ (χJ2 (χ(0,1) − 2χ(2,0) ) − 2χ(0,1) + 2χ(2,0) − 2χ(3,1) + 5χ(5,0) )q 5
11

+ χJ1 (χ(1,0) + 3χ(2,1) − 3χ(4,0) )q 2

+ (−χ(0,3) − 4χ(1,1) + χ(2,2) + 3χ(3,0) − 3χJ2 χ(3,0) − 4χ(4,1) + 7χ(6,0) )q 6
13

+ χJ1 (χ(0,1) + χ(1,2) + 3χ(2,0) + 3χ(3,1) − 3χ(5,0) )q 2

+ (χJ2 (−χ(0,2) − χ(1,0) + χ(2,1) − 4χ(4,0) ) − χ(0,2) + χ(1,0) − 5χ(2,1)+ 3χ(4,0) − 4χ(5,1) + 8χ(7,0) )q 7
15

+ (χJ5 − χJ3 χ(3,0) + χJ1 (1 − χ(1,1) + 3χ(3,0) + 5χ(4,1) − 4χ(6,0) ))q 2+ (χJ2 (χ(0,1) + 2χ(1,2) + 2χ(2,0) + χ(3,1) − 5χ(5,0) ) + 3χ(0,1) + χ(0,4)+ χ(1,2) + χ(2,0) − χ(2,3) − 5χ(3,1) + χ(4,2) + 4χ(5,0) − 6χ(6,1) + 10χ(8,0) )q 8
+ (χJ1 (−4χ(0,2) − 5χ(1,0) − χ(1,3) − 6χ(2,1) + 3χ(4,0) + 5χ(5,1) − 4χ(7,0) )17+ χJ3 (−3χ(0,2) − χ(1,0) − χ(2,1) − χ(4,0) ) + χJ5 χ(1,0) )q 2

+ (3χ(0,3) + 8χ(1,1) + 2χ(2,2) + 6χ(3,0) − 5χ(4,1) + 5χ(6,0) − 7χ(7,1) + 12χ(9,0)+ χJ2 (6 + χ(0,3) + 8χ(1,1) + 3χ(2,2) + 4χ(3,0) + 3χ(4,1) − 6χ(6,0) )+ χJ4 (χ(1,1) − χ(3,0) − 1))q 9+ (χJ1 (−8χ(0,1) − 8χ(1,2) − 8χ(2,0) − χ(2,3) − 10χ(3,1) − χ(4,2) + χ(5,0) + 7χ(6,1) − 5χ(8,0) )19+ χJ3 (−4χ(0,1) − 2χ(1,2) − 2χ(2,0) − 2χ(3,1) − 2χ(5,0) ) + χJ5 (−χ(0,1) + 2χ(2,0) ))q 2+ (−4χ(0,2) − χ(0,5) + 3χ(1,0) + 2χ(1,3) + 12χ(2,1) + χ(2,4) + 3χ(3,2) + 6χ(4,0) − χ(4,3)− 5χ(5,1) + χ(6,2) + 6χ(7,0) − 9χ(8,1) + 14χ(10,0)+ χJ2 (2χ(0,2) + 4χ(1,0) + 7χ(2,1) + 5χ(3,2) + 7χ+ 4χ(5,1) − 7χ(7,0) )21+ χJ4 (−χ(0,2) + χ(1,0) − χ(4,0) ))q 10 + O(q 2 ).

(2.61)In the next chapter, we will reproduce these results up to certain orders of the fugacity q
from the AdS side. Before going to this reproduction, we will review the essential conceptsof type IIB string theory in the next section.

2.2

Type IIB string theory

The superstring theory is a theory of strings that propagate in ten-dimensional (10d)spacetime. It is well known that there are five diﬀerent anomaly free superstring theories:
type I, type IIA, type IIB, Heterotic SO(32), and Heterotic E8 ×E8 . These string theories

Soryushiron Kenkyu

2.2 Type IIB string theory55

are related to each other by the string duality. In this section, we review type IIB theories,
which has N = (2, 0) supersymmetry, because our interest in this thesis is the AdS/CFT
correspondence regarding type IIB string theory.
There are two kinds of strings: closed strings and open strings. In the 10d spacetime, astring sweeps a two-dimensional (2d) surface Σ called a worldsheet. Let σ a (a = 0, 1) be
the coordinates on the worldsheet, where σ 0 and σ 1 stand for the temporal and spatial
coordinates, respectively. We normalize σ 1 so that σ 1 ∈ [0, π]. Then the motion ofstrings is described in the 10d coordinates X M (σ) (M = 0, 1, · · · , 9), a map from the
2d worldsheet into the 10d spacetime called the target space. There are also fermionic
partners of X M , denoted by ψ M (σ), which is a two-component Dirac spinor in 2d.
The bosonic part of the string action is the Polyakov action given by
1
Sstr = −
4πℓ2s

∫

√
d2 σ −gg ab (σ)∂a X M ∂b XM ,(2.62)

Σ

where g ab (σ) is a metric on the worldsheet. ℓs is called the string length, a uniqueparameter of the string theory.
The general solution of the equation of motion obtained from (2.62) is a superpositionof left- and right-moving waves
M
X M (σ) = XLM (σ 1 + σ 0 ) + XR
(σ 1 − σ 0 ).

(2.63)

The corresponding fermionic partners to the left- and right-moving waves are the lower
and upper component of ψ M (σ). Then we denote ψ M (σ) as)( M 1
ψR (σ − σ 0 ).
ψ (σ) =
M
(σ 1 + σ 0 )ψLM(2.64)

Although we omitted the fermionic part of the action in (2.62), we can find the coordinateM
dependence of ψL,R
by using the equation of motion of the fermionic fields. We refer to
M
M(XL/R , ψL/R ) as the left/right moving sector.

2.2.1

Closed strings

Let us first consider the closed strings. Since the discussion of the right moving sector isparallel to that of the left moving sector, we focus on the left moving sector.
Here we discuss the boundary condition of fields. For the bosonic field XLM , the periodicity of the closed string means XLM (σ 1 + σ 0 + π) = XLM (σ 1 + σ 0 ). On the other hand,Mthere are two choices of the boundary condition for the fermionic field ψL: the periodic

Soryushiron Kenkyu

56

Chapter 2 AdS/CFT Correspondence
Sectors
NS-NS
NS-R
R-NS
R-R

Fieldsϕ, BM N , gM N
λM
L , χRλM
L , χR
C, CM N , CM N RS

Table 2.7 The massless spectrum of the type IIB theory. The NS-NS and R-R sectors
are made of bosonic fields, whereas NS-R and R-NS sectors are made of fermionicfields.

and anti-periodic boundary conditions
M
M
Ramond (R) : ψL
(σ 1 + σ 0 + π) = ψL
(σ 1 + σ 0 ),M
MNeveu-Schwarz (NS) : ψL
(σ 1 + σ 0 + π) = −ψL(σ 1 + σ 0 ).

(2.65a)(2.65b)Fermionic fields satisfying (2.65a) are called Ramond (R) fermions, and fermionic fields
satisfying (2.65b) are called Neveu-Schwarz (NS) fermions.
Now we can do the oscillator expansion for the bosonic field XLM and the fermionicM. If we impose the canonical commutation relations for the coeﬃcients of the
field ψL
oscillator expansion, we can perform the quantization of the theory. Then, we obtain thefinite number of massless states and the infinite number of massive states. However, we
are interested in the low energy behavior of the string theory, so we focus on the masslessstates here. Combining the left moving sector and the right moving sector, we have four
sectors depending on the choice of the boundary condition of the fermionic fields: NS-NS,
NS-R, R-NS, and R-R sector. The NS-NS and R-R sectors have the 10d bosonic fields,
and NS-R and R-NS sectors have 10d fermionic fields. The massless fields of type IIB
string theory appearing in these sectors are summarized in Table 2.7.
We can find many massless fields in Table 2.7. In the NS-NS sector, we have a scalar
field ϕ called the dilaton, 2-form field BM N called “B-field”, and symmetric tensor fieldgM N called graviton. In the NS-R and R-NS sectors, we have two left-handed gravitinos
λM
L and two right-handed dilatinos χR . The existence of two gravitinos belonging to thesame chirality shows that the theory indeed has the N = (2, 0) supersymmetry. In the
R-R sector, we have p-form fields with p = 0, 2, 4, and they are called R-R p-form fields.

2.2.2

Open strings

So far, we have reviewed the massless spectrum of closed strings. There are also open
strings as well as closed strings. For open strings, there are two kinds of boundary conditions: Dirichlet and Neumann boundary conditions. For bosonic fields, these boundary

Soryushiron Kenkyu

2.2 Type IIB string theory

57

Fig. 2.1 An open string connecting a D-brane with the other D-brane. Its worldsheet
is a cylinder.

conditions are expressed as
X M |σ1 =0,π = const. : Dirichlet boundary conditions,∂X M= 0 : Neumann boundary conditions.
∂σ 1 σ1 =0,π(2.66)(2.67)

We can choose diﬀerent boundary conditions for each M . When the Dirichlet boundary
condition is imposed on the 9 − p bosonic fields, endpoints of strings can move in ap + 1-dimensional hypersurface called a Dp-brane. This hypersurface is specified by theequations (2.66).
The spectrum of open strings is also obtained by the canonical quantization. We canalso consider NS and R fermions. The quantization gives us the finite number of masslessstates and the infinite number of massive states. Again, we are interested in the masslessstates, and it is convenient to regard the massless fields as fields living on Dp-branes.
Namely, we can realize a p + 1-dimensional quantum field theory on Dp-branes in the
low-energy limit. In this sense, D-branes are useful to investigate quantum field theories.
So, in what follows, we will see the properties of Dp-branes.

2.2.3

D-branes

In the last subsection, we defined a Dp-brane as a p + 1-dimensional hypersurface where
open strings can end. However, D-branes have various properties as physical objects that
go beyond just the hypersurface.
Coupling to R-R fields
In fact, D-branes are electrically coupled to R-R fields [31]. This fact can be understood
through the open-closed duality in the string theory. Suppose a situation where two flat
D-branes are placed with a certain distance, as in Fig. 2.1. Then an open string can
be stretched between two D-branes. The shape of the worldsheet of this open string isSoryushiron Kenkyu

58

Chapter 2 AdS/CFT Correspondence

Fig. 2.2 A closed string propagating between two D-branes. Its worldsheet is also
a cylinder. We can regard this situation as the radiation and absorption of a closed
string by D-branes.

a cylinder connecting a D-brane with the other one. However, this situation can also be
regarded as a propagation of a closed string between two D-branes, as in Fig. 2.2. In thissense, we can say that D-branes are the source of closed strings. Then we have an electric
coupling term of a single Dp-brane to the R-R p + 1-form field Cp+1 as well as the kinetic
term of the R-R field as follows:∫∫1
Fp+2 ∧ ∗Fp+2 −SRR = − 2
Cp+1 ,(2.68)
2ge
WV
where the integral of the coupling term is taken over the p+1-dimensional worldvolume of a
Dp-brane. Note that p takes the values −1, 1, 3 for type IIB string theory*2 . Fp+2 = dCp+1
is the field strength of the R-R field. The Hodge dual ∗ of a k-form field Ak is defined by
)
1
M1
Mk
AM1 ···Mk dX∧ · · · ∧ dX
∗Ak = ∗k!
1=
ϵM ···Mk N1 ···N10−k AM1 ···Mk dX N1 ∧ · · · ∧ dX N10−k .
k!(10 − k)! 1((2.69)

ge is the unit charge of a single Dp-brane defined by
7−pge = (2πℓs ) 2 .

(2.70)

Then the definition of the electric charge of a single Dp-branes should be given by1
Qele =
geI∗Fp+2 = ge .

(2.71)

S 8−p*2 Here we do not discuss type IIA string theory in detail, p can take the values 0, 2 in type IIA string

theory.

Soryushiron Kenkyu

2.2 Type IIB string theory

59
D-branes
D(−1)
D1
D3
D5
D7

electric
C0
C2
C4
e6
C
e8
C

magnetic
e8
C
e6
C
C4
C2
C0

Table 2.8 A table for R-R p+1-forms and electrically coupled Dp-branes. Dp-branes
can also magnetically couple to R-R 7 − p-forms.

Therefore, the electric charge of Dp-branes is the unit charge ge times the number of
Dp-branes.
It is possible to consider the magnetic version of above discussion. The magnetic fluxeF8−p is defined by the Hodge dual of the electric flux Fp+2 as
1
1 e
F8−p =∗ Fp+2 ,gm
ge(2.72)

where gm is the unit magnetic charge of a single Dp-brane. The unit electric charge geand the unit magnetic charge gm satisfy the Dirac quantization condition
ge gm = 2π.

(2.73)

Then the definition of the magnetic charge of a single Dp-brane should be given by1
Qmag =
gm

I∗Fe8−p = gm .

(2.74)

S p+2

Using (2.72) and the Dirac quantization condition (2.73), we can show thatI
Fp+2 ∈ 2πZ.

(2.75)

Finally, we can define Dp-branes with p = 5, 7 as objects magnetically coupled to R-R
fields in type IIB string theory. Thus we find Dp-branes with odd p in type IIB string
ep
theory. They are summarized in Table 2.8. In this Table we defined dual R-R fields C
ep .
satisfying Fep+1 = dC(p, q)-string
In addition to the D1-brane, we can consider the electrically charged object for the B-field.
This object is called the fundamental string (F1-string). In this context, we may call theD1-brane the D1-string. Furthermore, we can also consider the one-dimensional object,a bound state of p F1-strings and q D1-strings. This object is called (p, q)-string, where

Soryushiron Kenkyu

60

Chapter 2 AdS/CFT Correspondence(p, q) charges are given by
∫
p=

∫
∗H3 ,

q=

∗F3 .

(2.76)

Here we defined the fluxes H3 = dB2 and F3 = dC2 .
Dirac-Born-Infeld action
We have seen that D-branes couple to the R-R fields. Besides this, we can also consider
the motion of D-branes. In the case of a relativistic particle, its motion is described by
its worldline. Likewise, the motion of D-branes is described by its worldvolume. Thus theaction of D-branes is schematically given by∫SDp ∝ −dp ξ√
− det Gab ,(2.77)

where ξa (a = 0, · · · , p − 1) are coordinates on Dp-branes and Gab is the induced metricdefined by
Gab =∂X M ∂X N
gM N .
∂ξ a ∂ξ b(2.78)

Here X M (ξ) (M = 0, · · · , 9) are 10d coordinates having a role of a map from p + 1dimensional worldvolume to 10d target space and gM N is the 10d metric.
The existence of the 10d metric gM N in the induced metric shows that D-branes couple
to the graviton in the NS-NS sector. Then it is easy to expect that D-branes couple to not
only the graviton but also all fields in the NS-NS sector, namely the dilaton and B-field.
Actually, this is the case.
Besides the NS-NS sector in the bulk, an open string living on a D-brane has a vector
field. This vector field is a U (1) gauge field on the D-brane.
Including all fields coupled to the D-brane, we obtain the correct action for the Dp-brane
called the Dirac-Born-Infeld (DBI) action as follows:
DBI
SDp
=−

1
(2π)p ℓp+1
s∫

dp+1 ξe−ϕ√− det(Gab + Bab + 2πℓ2s Fab ).

(2.79)

Because of the supersymmetry, there should also be coupling to the fermionic terms that
we omitted here.
The vacuum expectation value of the dilaton ϕ gives the string coupling constant oftype II string theories as gs = e⟨ϕ⟩ . Although it is believed that gs should be determinedby physics, we have no way to do it up to now because we may need the non-perturbative

Soryushiron Kenkyu

2.2 Type IIB string theory

61

formulation of the string theory. Then the tension of a D-brane is given by
Tp =

1
.
(2π)p ℓp+1
gss

(2.80)

In summary, the action of Dp-branes is made with the DBI action and the coupling to
the R-R fields. In fact, the coupling to the R-R field is generalized to the Chern-Simons(CS) type action
]
∫ [
∑
2
CS
SDp
=
e2πℓs F2 +B2 ∧Ckk,(2.81)

p+1

where [ ]p+1 indicates that we pick up only p + 1-form. The R-R coupling term wehave already discussed is included in the first term of the expansion of the exponential.
Therefore, the action of a single Dp-brane is
DBICS
SDp = SDp
+ SDp
.

2.2.4

(2.82)

Type IIB supergravity

Finally, let us review type IIB supergravity. This is the low energy eﬀective theory oftype IIB string theory and only includes the massless fields in the string theory. The
massless fields of type IIB string theory are summarized in Table 2.7. The action of type
IIB supergravity is given by
SUGRA
SIIB
=

(
)
∫
√
1
1 M N ∂M τ ∂N τ
10d X −G R − G2κ2
2(Imτ )2∫∫
1
1− 2|τ |2 H3 ∧ ∗H3 − 2F3 ∧ ∗F3
4κ
4κ
∫
∫
1
1
Reτ F3 ∧ ∗H3 − 2Reτ H3 ∧ ∗F3
− 2
4κ
4κ
∫
∫
1
1
′
′
− 2
F5 ∧ ∗F5 − 2
C4 ∧ H3 ∧ F3 + (fermion terms),
4κ
4κ

(2.83)

where κ is the 10d Newton constant defined by
κ2 =(2π)7 ℓ8s.
2(2.84)

τ = C0 + ie−ϕ is the axio-dilaton field. Fluxes H3 , F3 , and F5′ are defined as
H3 = dB2 ,F3 = dC2 ,11
F5′ = dC4 + B2 ∧ dC2 − C2 ∧ dB2 .
22(2.85)

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62

Chapter 2 AdS/CFT Correspondence

We rescaled the metric gM N as
GM N = e−ϕ/2 gM N .

(2.86)

The action (2.83) is invariant under the following SL(2, R) transformationaτ + bτ → τ′ =,cτ + d
( )
( ′)
( )
C2
C2
C2
→
=M
,
′
B2
B2
B2

(2.87a)(2.87b)C4 → C4′ = C4 ,(2.87c)GM N → G′M N = GM N ,(2.87d)()
a b
M=∈ SL(2, R).
c d(2.88)

whereThus type IIB supergravity has the SL(2, R) symmetry.
Because the 2-form fields form an SL(2, R) doublet, the (p, q) charges also form anSL(2, R) doublet. Then, due to the flux quantization (2.75), the SL(2, R) symmetry is
broken to SL(2, Z).
This symmetry maps type IIB string theory with the string coupling constant gs = e⟨ϕ⟩onto that with 1/gs . This can be seen by considering a matrix
(
M=

)
0 −1
1 0

(2.89)

with C0 = 0. Then the transformation (2.87a) means
gs →

1
.
gs

(2.90)

Therefore, type IIB supergravity has a strong/weak self duality called the S-duality. The
S-duality holds even for type IIB string theory.

2.3

The correspondence

We have finished preparing to discuss the AdS/CFT correspondence between 4d N = 4
SYM and type IIB string theory on AdS5 × S 5 . Here, let us confirm the precise statementof AdS/CFT. The AdS/CFT correspondence states that

Soryushiron Kenkyu

2.3 The correspondence63

The AdS/CFT correspondence
4d N = 4 U (N ) supersymmetric Yang-Mills theory with coupling constant gYM
is equivalent totype IIB superstring theory with string length ℓs and string coupling constant gs on
AdS5 × S 5 whose radius is L.
Note that the AdS/CFT correspondence is still a conjecture.
There are two free parameters on both sides. On the CFT side, we have the rank ofgauge group N and the coupling constant gYM . On the AdS side, we have the string
length ℓs and the AdS radius L, which is the typical scale of the AdS system. They are
related by the following two independent equations:
L4
,
ℓ4s
L4
4πN = 4 ,
ℓP

2
2gYM
N=

(2.91a)(2.91b)where ℓP is the Planck length defined by ℓ4P = ℓ4s gs . Using this definition of the Planck2= 2πgs from (2.91). In this duality, all the physics on the CFTlength, it follows that gYM
side is mapped to the physics on the AdS side, and vise versa.
Let us see the AdS/CFT correspondence in various parameter region using (2.91).
2• gYM
N ≫ 1 and N ≫ 1, or correspondingly, L ≫ ℓs and L ≫ ℓP
In this case, on the CFT side, the system is described by the strongly coupled
N = 4 U (N ) SYM with the N → ∞ limit (large N limit)*3 . On the AdS side, thesystem is described by the classical gravity, and we can neglect string excitation
states whose mass is proportional to 1/ℓs . Namely, the system on the AdS side iswell described by the type IIB supergravity. Thus, the AdS/CFT correspondence inthis parameter region gives us a strong tool to investigate strongly coupled quantum
field theories from the classical supergravity. This is one of the motivations for theinvestigation of the AdS/CFT correspondence.
2
N ≲ 1 and N ∼ 1, or correspondingly, L ≲ ℓs and L ∼ ℓP
• gYM
In this case, on the CFT side, the system is described by the weakly coupled N = 4
U (N ) SYM, where we can use the perturbation technique. On the AdS side, thesystem is described by the quantum gravity, and the string excitation states haveto be taken into account. This situation might enable us to study the quantum
gravity through AdS/CFT. This is another motivation for the investigation of the
AdS/CFT correspondence.

*3 We often regard N ≫ 1 as the large N limit, namely N → ∞ limit.

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64

Chapter 2 AdS/CFT Correspondence
CFT sideparameter region
description
2
gYM
N ≲1
weak coupling
2
gYM
N ≫1
strong couplingN ∼1
finite NN ≫1large NAdS sideparameter region
description
L ≲ ℓsall states necessary
L ≫ ℓsonly massless statesL ∼ ℓP
quantum gravity
L ≫ ℓP
classical gravity

Table 2.9 The possible situations depending on the parameter region in AdS/CFT.
We might be able to make four situations combining the upper parameter relation(2.91a) and lower parameter relation (2.91b).

We summarize the possible situations depending on the parameters in Table 2.9.
The AdS/CFT correspondence arises from two faces of D3-branes in type IIB string
theory: the open string perspective and closed string perspective. In what follows, we
review how to obtain the correspondence from D3-branes.

2.3.1

D3-brane picture

In this subsection, we discuss the open string perspective and focus on fields on D3branes. As a setup, we consider a stack of N coincident D3-branes in flat 10d Minkowski
spacetime. These D3-branes are flat, and embedding of D3-branes into the 10d target
space is given by
X µ = xµ (µ = 0, 1, 2, 3),

X i+3 = 0 (i = 1, · · · , 6),(2.92)

where xµ is the worldvolume coordinates on N D3-branes.
For a single D3-brane (N = 1 in the above setup), we mentioned just above (2.79)
that a 4d U (1) gauge theory arises on the D3-brane. Actually, in the low energy region
Eob ≪ ℓ−1
we can obtain the action of the 4d Maxwell theory by expanding the DBI
saction (2.79) with respect to ℓ2s , where Eob is the energy scale of the observation. This
fact can be explicitly seen by expanding the DBI action (2.79) with respect to ℓ2s underthe backgrounde⟨ϕ⟩ = gs ,BM N = 0,

gM N = ηM N = diag(−1, 1, · · · , 1).

(2.93)In addition to this background of the NS-NS fields, we assume the following backgroundfor the R-R fields
Ck = 0 for k ̸= 4.

(2.94)In this setup, coordinates X i+3 (i = 1, · · · , 6) look like scalar fields living on the D3brane from the viewpoint of the worldvolume theory. We define these scalar fields as

Soryushiron Kenkyu

2.3 The correspondence65

ϕi = X i+3 /2πℓ2s . Then, expanding the DBI action (2.79), we find the following action:
DBI
SD3
≃−

1
(2π)3 ℓ4s gs

∫

1
d x+
2πgs
4

∫

(

1
1
d x − Fµν F µν − ∂µ ϕi ∂ µ ϕi424)
+ O(ℓ2s ), (2.95)

The first term is merely a constant proportional to the worldvolume, and does not aﬀect
the physics, so we neglect it. Then, this is equivalent to the bosonic part of the N = 4
2
U (1) Maxwell theory if we identify 2πgs = gYM. More precisely, the axio-dilaton field τ
is identified as the complex coupling constant τ define in (2.15).
As we mentioned in the first paragraph in Sec 2.2, the supersymmetry of type IIB
theory is N = (2, 0). Now, half of the supersymmetry is broken due to the existence of
the D3-brane. Namely, the D3-brane is a half BPS object, and we have the 4d N = 4
supersymmetry on this D3-brane. Since the DBI action (2.79) and its low energy expansion(2.95) include only the bosonic terms, we have to add the fermionic terms to (2.95) sothat the action is invariant under N = 4. Note that since we have eight degrees of
freedom for bosons, the fermions also have the same degrees of freedom. Performing thesupersymmetric completion for (2.95), we find the following action:
S=

1
2
gYM

∫

(

)
1
1
µν
µ
I
i µ i
d x − Fµν F + iλI σ ∂µ λ − ∂µ ϕ ∂ ϕ ,4
2
4(2.96)

where λI (I = 1, 2, 3, 4) is the Weyl spinor belonging to the spinor representation ofSO(6)R with the positive chirality. This is the complete action of N = 4 U (1) Maxwell
theory.
In the case of N coincident D3-branes, open strings have the Chan-Paton (CP) factors[32]. The CP factor runs from 1 to N , corresponding to the number of D3-branes, andis assigned to each endpoint of strings. We can regard the CP factor as the index of
the gauge group U (N ), and an open string that ends on D3-branes is transformed as theadjoint representation of U (N ). Therefore, there is the U (N ) gauge field on N coincident
D3-branes, and the N = 4 U (N ) SYM is realized.

2.3.2

Black 3-brane solution

Here we discuss the closed string perspective. For simplicity, we do not take into accountthe string massive states. Besides this, we take the large N limit to avoid the diﬃculty ofthe quantum gravity. In this case, N D3-branes are viewed as a massive charged object
under the R-R 4-form filed C4 . Then the supergravity description is valid.
Now we are considering the type IIB supergravity. From the action (2.83), we obtainthe 10d Einstein equation. Corresponding to (2.93) and (2.94), we assume the ansatz
satisfying the following properties for the solution of the Einstein equation:
1. The solution is time independent.
2. R-R fields Ck = 0 for k ̸= 4.

Soryushiron Kenkyu

66

Chapter 2 AdS/CFT Correspondence
3. B-field BM N = 0.
4. The solution preserves the SO(1, 3) × SO(6) isometry as well as sixteen supercharges.
5. The metric gM N , R-R 4-form field C4 , and the dilaton ϕ has non-trivial solutions.
6. At infinity, gM N = ηM N and eϕ = gs .

The solution is then given by
r2
L2
µ
ν
η
dX
dX
+
δij dX i dX j
µν
L2
r2
L2
= 2 (dz 2 + ηµν dX µ dX ν ) + L2 dΩ25 ,z

ds23-brane =(2.97)

with the 5-form flux
F5 = 16πℓ4s N ϵ5 ,(2.98)

where µ, ν = 0, · · · , 3 and i, j = 4, · · · , 9. The radial coordinate r is defined by r2 =∑i 2
5i (X ) . ϵ5 is the volume form of S defined by r = L. In this solution we took the nearhorizon limit r ≪ L. The metric (2.97) is nothing but the metric of AdS5 × S 5 . Then
this system describes closed strings propagating in AdS5 × S 5 spacetime, namely, type
IIB string theory on AdS5 × S 5 .
So far, we have seen two perspectives in N coincident D3-branes, and we obtained the
4d N = 4 U (N ) SYM from the open string perspective and type IIB string theory on
AdS5 × S 5 from the closed string perspective. In this way, we can reach the idea of the
AdS/CFT correspondence.

2.4

Agreement of the index in the large N limit

We have seen the idea of the AdS/CFT correspondence in the last section. In this section,
we confirm the agreement of the BPS partition function and the superconformal index
on both sides in the large N limit. Although the BPS partition function depends on the
coupling constant, this dependence is quite simple. It is known that the BPS partition
function of the N = 4 U (N ) SYM in the weak coupling region is the same as that on
the strong coupling region [16, 28]. Thus, this coupling dependence does not matter. On
the contrary, the investigation of the BPS partition function gives us significant hints tostudy the superconformal index, so it is worthwhile to review the BPS partition function
as well as the superconformal index.

2.4.1

BPS partition function

First we consider the 18 -BPS partition function defined in (2.28). Since this quantity iseasier to calculate than the superconformal index, it is a good practice before calculating

Soryushiron Kenkyu

2.4 Agreement of the index in the large N limit67

the superconformal index. We calculate the BPS partition function on both sides.
CFT side
Before calculating on the CFT side, we have to pay attention to the fact that the BPS
partition function depends on the coupling constant. On the AdS side, it is diﬃcult
to consider the zero coupling limit because of the string massive states. Thus we only
consider the non-zero coupling case. Then the BPS partition function on the CFT side isgiven by the series coeﬃcients of the grand partition function (2.46).
Now it is easy to evaluate the BPS partition function in the large N limit from the
grand partition function (2.46). It can be read oﬀ from the grand partition function byusing a formulaa∞ = lim (1 − t)f (t)t→1(2.99)∑∞
for a series expansion f (t) = n=0 an tn with |t| < 1. Then the BPS partition function of
N = 4 U (N ) SYM in the large N limit is1
8 -BPS

1
8 -BPSZU (∞) = lim (1 − t)ΞU (∗) (x, y, z; t) = Pexp
t→1

(

)
1
−1 .
(1 − x)(1 − y)(1 − z)

(2.100)AdS side
Let us consider the AdS side in which the system is described by type IIB supergravity. We want to know the corresponding object on the AdS side to the chiral primary
operators, contributing to the BPS partition function. Actually, Witten noted that thecorresponding objects are Kaluza-Klein modes on S 5 [33], which are scalar particles withthe angular momentum on S 5 . Their spectrum has already been analyzed by [34]. Thus,
we here confirm the agreement of the spectrum on both sides in terms of the BPS partition
function.
There are the same global symmetry SO(2, 4) × SO(6) on the AdS side as that on the
CFT side. This symmetry is the isometry of AdS5 × S 5 spacetime. On the CFT side,
SO(2, 4) is the conformal symmetry, and SO(6) is the R-symmetry of N = 4 supersymmetry. Then the KK modes are also classified by this symmetry. Here we use the notation(R ,R ,R )[jL , jR ]E X Y Z , which we introduced below (2.55), to classify the KK modes under the
global symmetry U (1)H × SU (2)jL × SU (2)jR × SO(6)R ⊂ SO(2, 4) × SO(6). In terms of(n,0,0)representation.
this notation, the KK modes belong to [0, 0]nIn the case of the superconformal index (2.52), we saw that the index respects theSU (3)R ⊂ SO(6)R symmetry because the particular choice of the supercharge breaks thesymmetry as SO(6)R → SU (3)R × U (1). This SU (3)R is generated by RX − RY and
RY − RZ . The index does not respects this U (1) symmetry because the supercharges arecharged under this U (1). However, in the case of the BPS partition function, since we
do not consider the superpartner of the KK modes (or primary scalars on the CFT side),
SU (3)R × U (1) = U (3)R symmetry generated by RX , RY , and RZ is preserved. Namely,

Soryushiron Kenkyu

68

Chapter 2 AdS/CFT Correspondence

the BPS partition function can be written in terms of the U (3)R character. The U (3)character for the (m, n) representation is given by
U (3)χ(m,n)R = χ(m,n) (x, y, z),(2.101)

where χ(m,n) (a, b, c) is defined in (A.32). Then the KK modes with E = n ≥ 1 belong to(n, 0) representation under U (3)R , and the single-particle BPS partition function is justthe summation of the character of the (n, 0) representation over n ≥ 1:
iKK
BPS =

∞
∑

U (3)

χ(n,0)R =

n=1

1
− 1.
(1 − x)(1 − y)(1 − z)

(2.102)The BPS partition function is given by its plethystic exponential and therefore coincides
with that on the CFT side, (2.100).
The next problem is how to include the finite N corrections into this expression. We
will discuss the finite N corrections to the BPS partition function in the next chapter.

2.4.2

Superconformal index

Second, we consider the superconformal index in the large N limit. Unlike the BPS
partition function, we do not care about the coupling constant because the index does notdepend on the coupling constant.
CFT side
Now we consider the large N limit of the localization formula (2.58). This can be obtainedby the saddle point method. In order to use it, we need to perform the variable change
zi → eiθi and pick up the leading term in the large N limit from the expression. The
expression of the index after the variable change becomes




dθi
1 ∑
1
n
n

exp −
(1 − isp (x )) cos[n(θi − θj )] − N isp (x )  ,IU (N ) =

N ! −π i=1 2π
n
n=1
∫ π ∏
N



∞
∑

i̸=j

(2.103)where an argument x in isp (x) stands for (q, y, u, v). If we take the large N limit, terms∑with i,j is of order N 2 , and it is a leading order contribution to the index. On theother hand, the term N isp is of order N and can be neglected. Likewise, since the Weyl
factor 1/N ! is about exp(−N log N + N ) in the large N limit, we also neglect it. Afterthe computation, we have to normalize the index so that I(x = 0) = 1.

Soryushiron Kenkyu

2.4 Agreement of the index in the large N limit

69

Then the index in the large N limit is given byN ≫1

IU (N ) ≈




∞
∑
∑
dθi
1
exp −(1 − isp (xn )) cos[n(θi − θj )] .
2πn−π i=1
n=1

∫ π ∏
N

(2.104)i̸=jNow, we introduce the eigenvalue density ρ(θ) given by1 ∑δ(θ − θi ).
N i=1N

ρ(θ) =
Then, the summation(2.105)∑i is replaced by the integration with the eigenvalue density:

∫

∑

→N

dθρ(θ).

(2.106)i

Under this manipulation, the integration over θi is interpreted as a summation over aconfiguration of the eigenvalue density ρ(θ). Therefore
∏

dθi →

∏

dρ(x)dρ∗ (x) ≡ DρDρ∗ .

(2.107)xiMore precisely, we should take into account the Jacobian related with θi → ρ(θ). However,the Jacobian is irrelevant in the large N limit [35], and we neglect it.
We now have the following integral:N ≫1∫DρDρ∗ e−Seff ,
∫
N2
Seﬀ =
dθdθ′ ρ(θ)ρ∗ (θ′ )V (θ − θ′ ),2π∞∑
1
V (θ) = 2π
(1 − isp (xn )) cos(nθ).
nn=1

IU (N ) ≈(2.108a)(2.108b)(2.108c)

If we carry out the Fourier expansion for ρ(θ) as
ρ(θ) =∞∑

ρn e−inθ ,(2.109)n=0

the eﬀective action becomes
Seﬀ = N2∞∑
|ρn |2n=1n(1 − isp (xn )).

(2.110)This eﬀective action Seﬀ has a critical point at ρn≥1 = 0, ρ0 = 1. Thus we can evaluate

Soryushiron Kenkyu

70

Chapter 2 AdS/CFT Correspondence(R ,RY ,RZ )[jL , jR ]E X

Condition

Contributionn≥1

q n χ(n,0)

n≥1−q n+ 2 χJ1 χ(n−1,0)n≥1−q n+1 χ(n−1,1)

n≥1
n≥2
n≥2

q n+2 χ(n−1,0)
q n+1 χ(n−2,0)3
q n+ 2 χJ1 χ(n−2,1)n≥2−q n+ 2 χJ1 χ(n−2,0)n≥3−q n+2 χ(n−3,1)n≥3

q n+3 χ(n−3,0)(n,0,0)[0, 0]n(n− 1 , 1 , 1 )[ 21 , 0]n+ 12 2 2
21
1 1
1 (n− 2 , 2 ,− 2 )[0, 2 ]n+ 12(n−1,0,0)[0, 1]n+1(n−1,1,1)[0, 0]n+1(n−1,1,0)[ 12 , 12 ]n+1
(n− 3 , 1 , 1 )[ 12 , 1]n+ 32 2 2
2(n− 3 , 3 , 1 )[0, 12 ]n+ 32 2 22(n−2,1,1)[0, 1]n+2

1

5

Table 2.10 The KK modes on S 5 with ∆ = 0. This list contains only the highest
weight Cartan charges. Note that we have to sum over the full representations as thecharacter for SU (3)R and SU (2)jL since the index respects these symmetries. The
SU (3)R symmetry is generated by RX − RY and RY − RZ . The characters for thesesymmetries are defined in (A.34) and (2.55).

the large N index by the Gaussian integral around this critical point. Note that theclassical contribution vanishes. After the Gaussian integral and normalizing the index sothat I(x = 0) = 1, we finally have
IU (∞) =∞∏1

1 − isp (xn )n=1.

(2.111)

Next, we will see this expression exactly coincides with the index calculated from the
Kaluza-Klein modes on S 5 on the AdS side.
AdS side
The large N limit of the BPS partition function on the AdS side was obtained by thesummation of the KK modes on S 5 . The superconformal index is a generalization of the
BPS partition function by including fermionic fields, gauge fields, and their descendants.
This generalization is a superconformal completion in the context of the algebra. Thus we
can find the superconformal index for the KK modes by considering the superconformal(n,0,0)(n,0,0). After the superconformal completion of [0, 0]n
we havecompletion of [0, 0]nrepresentations listed in Table 2.10 with its contribution to the superconformal index.
These representations are also found by the S 5 compactification of type IIB string theory.
See the reference [34].

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2.4 Agreement of the index in the large N limit

71

Let us denote all contributions listed in Table 2.10 by sn :1

sn = χ(n,0) q n − χJ1 χ(n−1,0) q n+ 2 − (χ(n−1,0) − χ(n−2,0) )q n+1
3

5+ χJ1 χ(n−2,1) q n+ 2 + (χ(n−1,0) − χ(n−3,1) )q n+2 − χJ1 χ(n−2,0) q n+ 2 + χ(n−3,0) q n+3(2.112)
with n ≥ 1. Although we cannot take a negative value for (m, n) in χ(m,n) essentially,
we can do it formally by using the Weyl character formula, and this generates the correct
spectrum for the KK modes. Physically, terms including χ(m,n) with negative m or ncorrespond to the equations of motions of the KK particles. Then the single-particle
index is given by
iKK
sp =

1

∞
∑

sn
3
3
(1 − q 2 y)(1 − q 2 y −1 ) n=1
1
1
1
1
1
+
=
−
−
− 1,
3
v +
1
1 − uq 1 − u q 1 − v q 1 − q 2 y 1 − q 32 y −1

(2.113)The multi-particle index is just the plethystic exponential of the single-particle index(2.113):KK
IAdS= Pexp(iKK
sp ) =∞∏(1 − q 2 n y n )(1 − q 2 n y −n ),n(1 − un q n )(1 − uv n q n )(1 − v −n q n )
n=1
3

3

which exactly coincides with the large N index (2.111) on the CFT side.

(2.114)

Soryushiron Kenkyu

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73

Chapter 3

Finite N Corrections to theSuperconformal Index
In this chapter, we discuss the finite N corrections to the BPS partition function andthe superconformal index following the author’s and his collaborators’ paper [22]. At the
beginning, let us review the fact that the KK index (2.114) partially reproduces the finite
N index on the CFT side as well. To see this, we expand (2.114) with respect to q as
3

5

KK
IAdS
= 1 + χ(1,0) q − χJ1 q 2 + (−χ(0,1) + 2χ(2,0) )q 2 − χJ1 χ(1,0) q 2
7+ (2 − χJ2 − χ(1,1) + 3χ(3,0) )q 3 + (χJ1 χ(0,1) − 2χJ1 χ(2,0) )q 2
9+ (χ(0,2) + 2χ(1,0) − χJ2 χ(1,0) − 2χ(2,1) + 5χ(4,0) )q 4 + O(q 2 ).

(3.1)

Comparing it with the finite N index on the CFT side (2.59), (2.60) and (2.61), we findthe following fact:
3

KK
N +1
IUCFT
+ O(q N + 2 ).
(N ) = IAdS − χ(N +1,0) q(3.2)

Our question is what is the corresponding object of this extra term −χ(N +1,0) q N +1 + · · ·on the AdS side.
KK
On the CFT side, the meaning of this extra term is clear. Since we proved IUCFT(∞) = IAdS ,above relation is written into
3

CFT
N +1
IUCFT
+ O(q N + 2 ).
(N ) = IU (∞) − χ(N +1,0) q(3.3)

Then, the extra term means that there are additional constraints for BPS operators. For
simplicity, let us consider the N = 2 case. We have following BPS trace operatorsTrΦ,

Tr(Φ2 ),

Tr(Φ3 ),

··· ,

(3.4)

where Φ is one of the adjoint scalar fields X, Y, Z. In the large N limit, all these operators
are independent. However, in the N = 2 case, some operators in (3.4) are not independent

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Chapter 3 Finite N Corrections to the Superconformal Index

operators. Since the gauge group is U (2), Ta = (1, σi ) and hence we have a following
relation*1 :Tr(Φ3 ) =3
1
Tr(Φ2 )TrΦ − (TrΦ)3 .
2
2

(3.5)

Thus, the operator Tr(Φ3 ) is not the independent operator, and we must not count thiscontribution in the calculation of the index. This indicates that Tr(Φn ) with n > N are
not independent operators, namely
Eop ≤ N,

(3.6)

where Eop is the conformal dimension of the independent trace operators. Then we haveto subtract the contributions of operators, which can be written by other independent
operators, from the large N index to obtain the finite N index. More precisely, if we
consider all the combinations of the scalar fields X, Y, Z to construct Eop = 3 operators,
we find the following ten scalar operators of the form Tr(Φ3 ):Tr(X3 ),

Tr(X2 Y),

Tr(XY2 ),Tr(Y3 ),

Tr(X2 Z),

Tr(XZ2 ),Tr(Y2 Z),

Tr(YZ2 ),Tr(XYZ),

Tr(Z3 ),

(3.7)

where we ignored the order of the operators. These operators form the (3, 0) representationof SU (3)R ⊂ SO(6)R . Summing the contributions of these operators, we obtain χ(3,0) q 3 .
Hence, the extra term in (3.5) can be understood as the subtraction of the not-independent
operators from the large N limit.
Again, what is the corresponding object of the extra terms on the AdS side? To solve
this problem is the main theme of this thesis. Let us recall the parameter relation (2.91b).
Since the D3-brane tension T3 defined in (2.80) is proportional to ℓ−4
P , the relation (2.91b)
can be rewritten as
N = 2π 2 T3 L4 .

(3.8)

If N ≫ 1, TD3 ≫ L−4 , that means a D3-brane on the AdS side is too heavy to participate
in the physics. On the other hand, if N is of order unity (finite N ), TD3 ∼ L−4 and we
may need to include the eﬀect of D3-branes on the AdS side.
Even if D3-branes appear on the AdS side in the finite N region, how do we realize
the condition (3.6)? The answer is to wrap D3-brane on a three-cycle in S 5 [17]. Sucha D3-brane is called a giant graviton or simply a wrapped D3-brane. In this thesis, we
adopt the name “wrapped D3-brane.” Because the worldvolume of a wrapped D3-brane isbounded by the maximal circle of S 5 , namely S 3 , we can reproduce the condition (3.6) on
the AdS side. Note that the energy of a wrapped D3-brane is proportional to its volume.

*1 This relation is the simple version of (2.41).

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3.1 BPS partition function

75

Now the condition is regarded as
3volume of a wrapped D3-brane ≤ Vol(SMax) = 2π 2 L3 .

(3.9)

On the AdS side, the conformal dimension is translated as the energy of the corresponding
object times the AdS radius L. Therefore, we obtain the condition
ED3≤ 2π 2 L3T3 L⇐⇒

ED3 ≤ N,

(3.10)

where ED3 is the energy of a wrapped D3-brane scaled by the AdS radius L. Therefore we
expect that the wrapped D3-branes may give the finite N corrections to the index. The
preliminary investigation is given by [19]. In what follows, we compute the finite N BPS
partition function and superconformal index on the AdS side by using wrapped D3-brane
as well as KK modes following [22]. After the calculation, we compare our results withthose on the CFT side.

3.1

BPS partition function

In this section, we reproduce the BPS partition function for the interacting N = 4 U (N )SYM on the AdS side, because the calculation of the BPS partition function gives us
many instructive experiences for the index in spite of their ease. The reproduction of the
BPS partition function shows non-trivial evidence for the AdS/CFT correspondence. Our
strategy in the finite N region is to consider the wrapped D3-branes discussed above. Thissection is also meant to be a preparation for the analysis of the superconformal index on
the AdS side that will be discussed in the next section.
The first problem is what shape of wrapped D3-branes contribute to the BPS partition
function. This problem is elegantly solved by Mikhailov in his beautiful paper [36]. Heshowed that the any 18 -BPS wrapped D3-branes are given by an intersection between aholomorphic surface in C3 and S 5 ⊂ C3 . Let be X, Y and Z be complex coordinates on
C3 . Then S 5 is expressed by|X|2 + |Y |2 + |Z|2 = L2 = 1,

(3.11)

where L is the radius equal to the AdS radius*2 . Any holomorphic surfaces are given bya holomorphic function f (X, Y, Z) on C3 . Mikhailov claims that the worldvolume of any
wrapped D3-branes is given by a holomorphic surface
f (X, Y, Z) = 0*2 We take the L = 1 unit here.

(3.12)

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Chapter 3 Finite N Corrections to the Superconformal Index

imposed by (3.11). The Taylor expansion of the holomorphic function isf (X, Y, Z) =

∞
∑

p

q

r

cpqr X Y Z =∞∑∑cpqr X p Y q Z r .

(3.13)

n=0 p+q+r=np,q,r=0

The value of coeﬃcients cpqr determines the configuration of wrapped D3-branes. Then wetreat the coeﬃcients cpqr as dynamical variables. To obtain the BPS partition function of
wrapped D3-branes, we need the quantization of wrapped D3-branes. This can be carriedout by the geometric quantization [29, 37].
Since the overall factor is irrelevant for f and the configuration, we can regard coeﬃcients as projective coordinates of CP ∞ . There are two complicated issues. One is that
there may be diﬀerent holomorphic functions that give the same D3-brane configurations.
The other is that there may be holomorphic surfaces that do not intersect with S 5 . Even
if we take into account the issues, the result is the same as what we obtained by neglecting
the issues [29]. As a result, we can treat the coeﬃcients as independent variables.
Thanks to the result of the geometric quantization, we can treat each term for fixed nin (3.13) independently. Let us focus on the degree n holomorphic surface given by∑cpqr X p Y q Z r = 0.

(3.14)

p+q+r=n

We define n-ple wrapped D3-branes as D3-branes represented by (3.14). Let ZnD3 becontributions to the BPS partition function from n-ple wrapped D3-branes. Then the
BPS partition function on the AdS side can be written in the form of
(
AdS
KK
ZN
= ZBPS

1+

∞
∑

)
ZnD3

.

(3.15)n=1
KK
ZBPS
is the BPS partition function in the large N limit on the AdS side
KK
ZBPS
= Pexp(iKK
BPS ),(3.16)

where iKKBPS is given by (2.102).
In our CFT analysis, we have calculated the BPS partition function in the exact form
like (2.48) and (2.49). However, here we would like to calculate the finite N corrections
to the index in each power of n. Then it is convenient to define the q fugacity associatedwith the energy E of wrapped D3-branes in the unit of AdS radius L. (E is the conformal
dimension of the corresponding operators on the CFT side.) Then the definition of the
BPS partition function (2.28) is modified asZ(x, y, z; q) = tr(q H xRX y RY z RZ ),(3.17)

Soryushiron Kenkyu

3.1 BPS partition function

77

Note that the fugacity q is redundant because of the BPS condition
E − (RX + RY + RZ ) = 0.

(3.18)

Although It is then not inherently necessary to define the BPS partition function, we cansee the finite N corrections due to the wrapped D3-branes by expanding the BPS partition
function with respect to q.
Let us consider the n = 1 case as a first non-trivial finite N correction to the BPS
partition function. In this case, the most general holomorphic surface is
aX + bY + cZ = 0.

(3.19)The phase space is thus CP 2 whose homogeneous coordinates are (a, b, c). Our strategy
that we take here is to treat Z = 0 as a ground state and others as its excitations. In
order to do it, we consider the D3-brane action (2.82) for the wrapped D3-brane locatedZ = 0. The DBI action describes a 4d field theory on the worldvolume, and the CS action
comes from the couping to the background R-R 4-form field C4 .
First, let us consider the DBI action. It is easier to work in the polar coordinates thanthe Cartesian coordinates in C3 . The metric of AdS5 × S 5 with the polar coordinates is
ds210d = − cosh2 rdt2 + dr2 + sinh2 rdΩ23+ dθ2 + sin2 θdϕ2 + cos2 θ[dψ12 + sin2 ψ1 (dψ22 + sin2 ψ2 dψ32 )],(3.20)

where (t, r, Ω3 ) forms AdS5 while (θ, ϕ, ψ1 , ψ2 , ψ3 ) forms S 5 . Z = 0 means θ = 0 andr = 0 in the polar coordinates. Because our purpose is to consider the fluctuations on thewrapped D3-brane, we calculate the induced metric of the D3-brane and expand r and θ
around r = 0, θ = 0. The DBI action with the induced metric up to quadratic terms of rand θ is then∫√
D3
SDBI = −T3
d4 x −g0
R×S 3



∑
1
× 1 − (|Φ̇W |2 − |∇ΦW |2 ) + 3|ΦZ |2 − |ΦX |2 − |ΦY |2  , (3.21)
2
W =X,Y,Z

where xa = (t, ψ1 , ψ2 , ψ3 ) is the worldvolume coordinates of the wrapped D3-brane and√−g0 = sin2 ψ1 sin ψ2 is its volume element. The D3-brane tension T3 is given by
T3 =

1
N
=
.
4
3
(2π) ℓP
2π 2 L4(3.22)

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78

Chapter 3 Finite N Corrections to the Superconformal Index

We introduced three complex scalar fields as|ΦX |2 + |ΦY |2 = r2 .

ΦZ = θeiϕ ,(3.23)

These fluctuation modes must carry the non-trivial spin charges jL and jR because scalar
fields ΦX and ΦY represent fluctuations of AdS5 directions. The BPS partition function
does not have contributions with the non-trivial spin charge, so we neglect them in thecalculation of the BPS partition function.
Next, we consider the CS action. Because the CS action describes the coupling to theR-R 4-form field, the action is given by*3
∫
D3
SCS
=

R×S 3

C4

(3.24)

To find the explicit form of the R-R 4-form field C4 , we can use the flux quantization
condition∫
dC4 = 2πN
(3.25)
S5

followed by (2.75). Using the polar coordinates, we find
C4 =

N
4 √
ϕ̇
cos
θ −g0 dt ∧ dψ1 ∧ dψ2 ∧ dψ3 .
2π 2(3.26)

Expanding cos4 θ up to quadratic terms, the CS action becomes∫√N
d4 x −g0 ϕ̇(1 − 2θ2 )
2
2π
∫
∫
N
iN
4 √
4 √
d
x
−g
ϕ̇
+
d
x −g0 (Φ∗Z Φ̇Z − ΦZ Φ̇∗Z )
=
0
2
2
2π
2π

D3
SCS
=

(3.27)

In summary, the D3-brane action is given by the classical action and fluctuations:
D3
D3
S D3 = Scl+ Sfluc,(3.28a)
∫
N
D3
4 √
Scl
=
(3.28b)dx −g0 (ϕ̇ − 1),
2π 2
[
∫
N
1
D3
4 √
Sfluc =
d x −g0 (|Φ̇X |2 − |∇ΦX |2 + |Φ̇Y |2 − |∇ΦY |2 + |Φ̇Z |2 − |∇ΦZ |2 )
2
2π
2
]
3
1
2
2
2
∗
∗
+ |ΦZ | − (|ΦX | + |ΦY | ) + i(ΦZ Φ̇Z − ΦZ Φ̇Z ) .
(3.28c)22

Let us consider the classical contribution. The energy of the D3-brane corresponds to*3 Other R-R fields are set to be zero.

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3.1 BPS partition function

79

the potential term in the classical Lagrangian. Namely,
N
E=
2π 2

∫

√
d3 x −g0 = N,

(3.29)

S3

in the unit of the AdS radius. The wrapped D3-brane feels the R-R flux, so that thewrapped D3-brane has a non-trivial angular momentum. In this configuration, this is just
RZ corresponding to the R-charge on the CFT side. This is calculated as
∂Lcl
N
RZ =
=
2π 2
∂ ϕ̇

∫

√
d3 x −g0 = N.

(3.30)

S3

Therefore, the Z = 0 configuration satisfies the BPS condition E = RZ . This result is
consistent with Mikhailov’s claim. This information tells us that the classical contribution
to the BPS partition function is (qz)N .
Let us move on to the calculation of the fluctuation modes on the wrapped D3-branes.
As we mentioned below (3.23), the fluctuation modes related with ΦX and ΦY is irrelevant
for the BPS partition function, so we only take into account terms including ΦZ . The
equation of motion for ΦZ is−Φ̈Z + ∇2 ΦZ + 3ΦZ + 4iΦ̇Z = 0,

(3.31)

where ∇2 is the Laplacian on S 3 . If we assume the solution takes the form e−iEt Yl using
the spherical harmonics Yl on S 3 we have the following second-order equations for E:(E − l + 1)(E + l + 3) = 0,(3.32)

where we used the eigenvalue of the Laplacian given by −l(l + 2). The solution corresponding to the BPS fluctuation modes is
E = l − 1,

(3.33)

because (l − 1) − (−l − 3) = 2l + 2 > 0. From the BPS condition (3.18) we find RX + RY +RZ = l − 1. The fluctuation modes around Z = 0 must also be given by the holomorphic
function, and this holomorphic function has the formZ∝∞∑

X pY q .

(3.34)

p,q=0

According to [36], the time dependence of a holomorphic surface is(X(t), Y (t), Z(t)) = (eit X(0), eit Y (0), eit Z(0)).

(3.35)

If we assume that the initial configuration for each mode is given by Z(0) = X(0)p Y (0)q ,Soryushiron Kenkyu

80

Chapter 3 Finite N Corrections to the Superconformal Index

the Z(t) at time t becomesZ(t) = e−i(p+q−1)t X(t)p Y (t)q .

(3.36)

Now we recognize that the E = −1 at l = 0 comes from Z itself while l = p + q modes(l)(p,q,0)come from X p Y q *4 . We denote this scalar mode as ϕ : [0, 0]l−1 . The single-particle
BPS partition function coming from these scalar modes on the wrapped D3-brane locatedZ = 0 is then
iBPS
=
sp

∞
∑

q p+q−1 z −1 xp y q =p,q=0(qz)−1.
(1 − qx)(1 − qy)(3.37)

Hence, the single-particle BPS partition function on the wrapped D3-brane is obtained as
(
D3
ZZ=0
= (qz)N Pexp(qz)−1(1 − qx)(1 − qy)).

(3.38)The Taylor expansion with respect to q is
iBPS
=
sp

(x y)
1
+
+ O(q).
+qzzz(3.39)

Thus we have a negative energy mode and two zero energy modes. We call a negative
energy mode a tachyonic mode and call a zero energy mode a zero mode simply. The
existence of a tachyonic mode implies that there are lower energy modes than the Z = 0
configuration. Actually, there are smaller worldvolume configurations than Z = 0 since
it is a great circle of S 5 . Mathematically, the plethystic exponential of a tachyonic term
does not work as far as z < 1, q < 1. So we temporarily impose z > 1, q > 1 and treat theplethystic exponential of a tachyonic term as follows:
(
Pexp

1
qz

)
=

qz
1
=−
= −qzPexp(qz).
−1
1 − (qz)
1 − qz(3.40)In this treatment, the tachyonic term hence serves to raise the overall power of q by one.
Now that the first term of the finite N corrections is not (qz)N but −(qz)N +1 . This canbe understood as follows. On the CFT side, there exist finite number independent chiralprimary operators in the finite N region, while there exist infinite number operators in thelarge N limit. For U (2) theory as an example, we have only two independent operators
TrZ and TrZ2 for the 21 -BPS sectors. Then the diﬀerence between the BPS partition
function for the U (2) theory and the U (∞) theory starts from (qz)2+1 and the minus sign
can be interpreted as a subtraction of extra operators from the large N theory to become
the U (2) theory. On the other hand, we have no physical interpretation on the AdS side.

*4 Note that the scalar fields ΦW =X,Y,Z are proportional to the coordinates X, Y, Z.

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3.1 BPS partition function

81

Up to now, we are only able to handle it mathematically.
The zero modes are also important. They never describe higher energy configurations
because of no q-dependence. Instead of this, they correspond to the degenerating groundstates other than Z = 0. Actually, the plethystic exponential of the zero modes contains
all degenerating states:−(qz)N +1 Pexp

(x
z

+

y)
(qz)N +1=−z
(1 − xz )(1 − yz )
(
)(
)
x x2
y y2
N +1 N +1= −qz
1 + + 2 + ···1 + + 2 + ··· .
zzzz(3.41)

Indeed, this formula includes not only all degenerating ground states but also extra states
as an infinite series. This fact reflects that the configuration space is CP 2 . We have toconsider other patches of CP 2 in addition to Z = 0. Actually, the patches including Z = 0
is spanned by coordinates (a/c, b/c) but never includes X = 0 and Y = 0 since we haveto set to be c = 0 to realize X = 0 and Y = 0.
The discussion for other patches is essentially the same as that of Z = 0. To obtain thefluctuation modes coming from X = 0 and Y = 0, only we have to do is to exchange thefugacities (x, y, z) symmetrically. Actually, this manipulation corresponds to the Weylreflection of the U (3)R symmetry. We call this manipulation the Weyl completion.
After the Weyl completion, the BPS partition functions corresponding to Y = 0 and
Z = 0 are given by()(qx)−1,(1 − qy)(1 − qz)()(qy)−1
D3
N
ZY =0 = (qy) Pexp
(1 − qz)(1 − qx)

D3
ZX=0= (qx)N Pexp

(3.42)(3.43)

To cancel the extra terms in (3.41) using the zero modes in the single-particle partition
function for X = 0 and Y = 0, all we have to do is to sum (up three patches. )To see this2
procedure does work, let us focus on the term −(qz)N +1 1 + xz + xz2 + · · · in (3.41).
N +1

The extra terms in this expression are actually canceled by the term − (qx)1− zin X = 0:x(qz)N +1−
= −q N +1 (z N +1 + z N x + · · · + zxN + xN +1 + xN +2 z −1 + · · · ),x
1− z(3.44)

q N +1 xN +1 xz(qx)N +1−=
= q N +1 (xN +2 z −1 + · · · ),
z
x
1− x
1− z

(3.45)

where we performed the Taylor expansion of the denominator in a region xz < 1. Notethat this cancellation can always occur since the summation of two expressions is the Weyl

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82

Chapter 3 Finite N Corrections to the Superconformal Index

character formula of U (2) ⊂ U (3)R :−(qz)N +1(qx)N +1−
= −q N +1 (z N +1 + z N x + · · · + zxN + xN +1 ).
1 − xz
1 − xz(3.46)How to realize the cancellation between three patches, including Y = 0 ? In fact, thesummation over three patches also becomes the Weyl character formula of U (3)R :−(qx)N +1(qy)N +1(qz)N +1
U (3)R−−= −q N +1 χ(N +1,0)(x, y, z). (3.47)(1 − xy )(1 − xz ) (1 − yz )(1 − xy ) (1 − xz )(1 − yz )Here we include all zero modes to respect the U (3)R symmetry between x, y, z.
Now the formula of the finite N corrections to the BPS partition function is the summation of each wrapped D3-brane BPS partition function over the three patches:
D3
D3
Z1D3 = ZX=0
+ ZYD3=0 + ZZ=0.

(3.48)We expect that this must give the finite N corrections to the BPS partition function
coming from the single wrapped D3-brane configurations. Likewise, in the evaluation of
the classical energy in the single wrapped D3-brane case, the classical energy of the double
wrapping configuration is 2N in the unit of the AdS radius, the contribution to the BPS
partition function starts from O(q 2N ). Note that this evaluation is just the estimation.
There may be a contribution from tachyonic terms as in the single wrapping case, andthe first contribution of the double wrapping configurations may be a higher power thanq 2N .
Now the finite N BPS partition function up to the single wrapping on the AdS side isgiven byAdS
KK
ZN
= ZBPS(1 + Z1D3 ),(3.49)

where we do not include the higher wrapping than the single wrapping. We expect thatthe formula (3.49) is valid up to O(q 2N +δ2 ), where δ2 is an eﬀect of tachyonic terms on
the double wrapping.
Let us confirm the validity of this formula for small ranks, N = 1, 2. For simplicity, we
set x = y = z = 1 in the definition (3.17), or x = y = z = q in (2.28). On the CFT side,
expanding (2.48) and (2.49) up to some orders, we find
2
3
4
5
6
7
8
ZUCFT
(1) = 1 + 3q + 6q + 10q + 15q + 21q + 28q + 36q + O(q ),(3.50)

2
3
4
5
6
ZUCFT
(2) = 1 + 3q + 12q + 28q + 66q + 126q + 236q

+ 396q 7 + 651q 8 + 1001q 9 + O(q 10 ).

(3.51)

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3.2 Superconformal index

83

On the other hand, our formula (3.49) gives
AdS
7
8
ZN
=1 = (· · · identical terms with (3.50) · · · ) − 216q + O(q ),(3.52)AdS
910ZN=2 = (· · · identical terms with (3.51) · · · ) + 539q + O(q ).

(3.53)

From this comparison, we find δ2 = 5. A detail analysis suggests that δn may be given
by 12 n(n + 3) for the 81 -BPS sector. Thus, our formula indeed gives the correct finite N
corrections up to the expected order. Although we could go on to the multiple wrapping,next, we will apply the analysis for the single wrapping to the superconformal index.
Before the ending of this section, there are some comments for other BPS sectors. In
fact, the analysis for 21 -BPS and 14 -BPS sectors immediately obtained by setting x = y = 0
and z = 0, respectively. The eﬀect of tachyonic terms may become δn = 12 n(n + 1) and11
1
2 n(n + 2) for the 2 -BPS and 4 -BPS sectors. It is worthwhile to mention that the analysis
for the 12 -BPS sector can be done in all orders of the fugacity, including the all wrapped
D3-branes. Hence we can exactly show that δn = 12 n(n + 1) for the 12 -BPS sector.

3.2

Superconformal index

This section aims to reproduce the superconformal index of the N = 4 U (N ) SYM byconsidering wrapped D3-branes on the AdS side, as we did in the last section for the BPS
partition function. We will calculate the finite N corrections to the superconformal index
by using the same technique in the previous section.
In the last chapter, we first calculated the BPS partition function on the AdS side.
Then only KK modes contributed to the BPS partition function. On the other hand,
there were contributions from not only the KK modes but also their superpartners in thecase of the superconformal index. Thus, in order to include all the contributions to thesuperconformal, what we have to do is to perform the supersymmetric completion of the
primary scalar modes.

3.2.1

Supersymmetry on wrapped D3-branes

Let us consider the Z = 0 configuration of the wrapped D3-brane as in the BPS partition
function. According to the Mikhailov discussion [36], this configuration is 12 -BPS. We givethe conserved supercharges for Z = 0 as well as X = 0 and Y = 0 in Table 3.1.
The N = 4 supersymmetry is partially broken on the wrapped D3-brane Z = 0. In
particular, there is no P and K symmetries. The SO(6) R-symmetry is also broken toSO(4)XY × SO(2)Z = SU (2)R × SU (2)R × U (1)Z because of a choice of the Z = 0
configuration. The Cartan generators areR=

1(RX − RY ),2

R=

1(RX + RY ),2

RZ ,(3.54)

respectively. The supercharge quartet of SO(6)R is then broken to two doublets of SU (2)RSoryushiron Kenkyu

84

Chapter 3 Finite N Corrections to the Superconformal Index
Configurations
X=0
Y =0
Z=0

Supercharges
Q3,4 , Q1,2
Q2,4 , Q1,3
Q2,3 , Q1,4

Table 3.1 The preserved supercharges on X = 0, Y = 0, and Z = 0 configurationsof the wrapped D3-brane, respectively. Due to the loss of part of supercharges in
the N = 4 superalgebra, we have no longer P and K symmetries on the wrapped
D3-brane.

QI and QI
Q4
Q3
Q2
Q1
Q1
Q2
Q3
Q4

RX

RY

1
2
1
2
− 12
− 12
1
2
1
2
− 12
− 12

1
2
− 12
1
2
− 12
1
2
− 12
1
2
− 12

RZ
− 12

1
2
1
2
− 12
1
2
− 12
− 12
1
2

R
0

R

1
2
− 12

0
0
− 21

0
0
1
2
− 21

0

1
2

1
2

0
0
− 12

Table 3.2 The R-charges of supercharges. From this, we can find that (Q2 , Q3 )form a doublet in SU (2)R with RZ = 12 and (Q1 , Q4 ) form a doublet in SU (2)R with
RZ = 21 on Z = 0 configuration. The plus sign of the RZ charge is consistent because
Z = 0 is not an anti-holomorphic surface but a holomorphic surface.

and SU (2)R as
QIα : 4 ∈ SO(6)R → (2, 1) 12 ⊕ (1, 2)− 12 ∈ SU (2)R × SU (2)R × U (1)Z .

(3.55)

We summarize R-charges of supercharges in Table 3.2.
Let A and Ȧ be the indicies of SU (2)R and SU (2)R , respectively. We define
A=1
I=3
QA
, QA=2) = (QI=2
α = (Qααα , Qα ),
α̇

α̇

α̇

α̇

(3.56a)α̇QȦ = (QȦ=1̇ , QȦ=2̇ ) = (QI=1 , QI=4 ).

(3.56b)Then the unbroken algebra among fermionic generators becomes
α
α B
B α
α B
2{SA
, QB
β } = δβ δA (H − RZ ) + 2δA J β + 2δβ R A ,
α̇

Ḃ

α̇

Ḃ

Ḃ
Ḃ
2{QȦ , S β̇ } = δβ̇α̇ δȦ
J β̇ − 2δβ̇α̇ R Ȧ .
(H − RZ ) − 2δȦ

(3.57a)(3.57b)This can be derived from N = 4 supersymmetry algebra (2.51) by restricting the supercharges to those on Z = 0 configuration.

Soryushiron Kenkyu

3.2 Superconformal index

85

In the algebra (3.57), we can obtain the bounds
α=22{SA=2, QB=2β=2 } = E − 2jL − (2R + RZ ) ≥ 0,α̇=1̇(3.58)Ḃ=1̇

2{QȦ=1̇ , S β̇=1̇ } = E − 2jR − (2R + RZ ) ≥ 0,

(3.59)The lower bound (3.59) is the same as our BPS bound (2.53), if we rewrite the R- andR-charges by (3.54). So we construct a representation starting from the primary scalar(l)mode. Since we have already seen that the modes ϕ indeed reproduce the BPS partition
function in the finite N region, it turns out that these modes correspond to the primary(l)scalar operators on the CFT side. On the AdS side, ϕ should also be the superconformal
primary of the superconformal algebra. Thus we can construct a representation of the(l)algebra (3.57) starting from ϕ .

Superconformal index of wrapped D3-brane

3.2.2

Since we have obtained the symmetry algebra (3.57), we can construct a representationof modes on the wrapped D3-brane to find the superconformal index. Since ϕsuperconformal primary, it satisfies
(l)

Ȧ

(l)

α
[SA
, ϕ ] = [S α̇ , ϕ ] = 0.

(l)is the(3.60)α̇The superpartners are obtained by acting supercharges QA
α and QȦ to this scalar mode.
(R,R,RZ )Let us adopt the notation [jL , jR ]E
ϕ
ϕ

(l)

(l)

A
( l , l ,−1) Qα

2 2: [0, 0]l−1( l−1 , l ,− 21 )−−→ [ 12 , 0]l−21 22α̇( l , l ,−1) QȦ

2 2: [0, 0]l−1

to represent these modes. Then we obtain( l+1 , l ,− 12 )⊕ [ 21 , 0]l−21 2,(3.61).

(3.62)

2( l , l−1 ,− 21 )−−→ [0, 12 ]l−2 1 2
2( l , l+1 ,− 21 )⊕ [0, 12 ]l−2 1 2
2The second representation in the right hand side actually corresponds a null state. Thus
we have only the first representation. We denote these fermionic modes as ψ and χ.
Namely, we findϕ(l) QA
α

−−→ ψ

(l)

( l−1 , l ,− 1 ): [ 12 , 0]l−21 2 2 ,
2

α̇

ϕ

(l) QȦ( l , l−1 ,− 12 )−−→ χ(l) : [0, 12 ]l−2 1 2.

(3.63)

2

By repeating this process, we have a multiplet of the massless modes on the wrapped
D3-brane Z = 0. We summarize all the modes in Table 3.3 and the structure of themultiplet in Fig. 3.1.
We have finished the classification of the massless modes on the wrapped D3-brane.
Next, let us compute the single-particle index of the worldvolume theory on the wrapped

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Chapter 3 Finite N Corrections to the Superconformal Index
Modes

E

[jL ]

[jR ]

[R]

[R]

RZ

(l)

l−1
l − 12000[ 2l ][ 2l ]

−1
− 21

F
ψ (l)
w(l)

l
l − 12l0(l)

l + 12ll + 12l+1[ 2l ]
l−1
[ 2 ]
[ l−2
2 ]
[ 2l ]
[ l−1
2 ]
l−2
[ 2 ]
[ 2l ]
[ l−1
2 ]
l−2
[ 2 ]

ϕ
χ(l)
(l)

ψ
F (l)
χ(l)
ϕ(l)

[ 12 ][ 2l ]

00[ 21 ][ 12 ][ 21 ][ 12 ]

00
0[ 12 ]

[ l−1
2 ]
l−1
[ 2 ]
[ l−1
2 ]
l−2
[ 2 ]
[ l−2
2 ]
l−2
[ 2 ]

0
0
0

0
− 21
0
1
2

0
1
2

1

Table 3.3 The massless modes arising in the worldvolume theory on the wrapped
D3-brane Z = 0. [j] stands for a spin j representation of a corresponding SU (2).
Some of these modes saturate the BPS bound (3.59).

ϕ
↗

↖

ψ
↗

χ
↖

F

↗

↖

w
↖

↗

F
↖

↗

χ

ψ
↗

↖
ϕ

Fig. 3.1 The multiplet structure of the massless modes on the wrapped D3-brane
Z = 0. The right diagonal up arrow stands for the action of QA
α , and the left diagonalα̇up arrow stands for the action of QȦ .

D3-brane. Here it is convenient to rewrite (2.52) in terms of R and R as
I = tr[(−1)F x∆ q E+jR y 2jL u2R v R−R−RZ ].

(3.64)

Then all we have to do is to sum up the contributions of the BPS modes of modes shownin Table 3.3. From this table, we find the BPS modes shown in Table 3.4.
The single-particle index obtained from Table 3.4 is(1 − vq )(1 − yq 2 )(1 − y −1 q 2 )
3

iZ=0
=1−
sp

(1 − uq)(1 − uv q)

3.

(3.65)

This is just the index of a U (1) theory realized on the Z = 0 wrapped D3-brane. Thus

Soryushiron Kenkyu

3.2 Superconformal index(R,R,RZ )[jL , jR ]E([ 2l ], 2l ,−1)[0, 0]l−1([ l ], l−1 ,− 1 )[0, 21 ]l−21 2 22([ l−1 ], l ,− 1 )[[ 12 ], 0]l− 12 2 22l−2 l−1
1 1 ([ 2 ], 2 ,0)[[ 2 ], 2 ]l([ l−2 ], l ,0)[0, 0]l 2 2([ l−2 ], l−1 , 1 )[0, 12 ]l+ 12 2 2
287
Condition

Range of R

Contributionl≥0− 2l → 2lq l−1 u2R v 2 −R+1l≥1− 2l → 2l−q l u2R v 2 −R+ 2

l≥1

l−1
− l−1
2 → 2

−q l− 2 χJ1 u2R v 2 −R+ 2

l≥1

l−1
− l−1
2 → 2

q l+ 2 χJ1 u2R v 2 −R

l≥2

l−2
− l−2
2 → 2

q l u2R v 2 −R

l≥2

l−2
− l−2
2 → 2

−q l+1 u2R v 2 −R− 2

l

l−1

1

1

l

1

l−1

1

l

l−1

1

Table 3.4 The BPS massless modes arising on the wrapped D3-brane. Summing up
all contributions shown here, we find the single-particle index for the fluctuations of
the wrapped D3-brane Z = 0.

the multi-particle index is the plethystic exponential of (3.65):
D3
IZ=0= (v −1 q)N Pexp(iZ=0
sp ),(3.66)

where the factor (v −1 q)N is the classical contribution obtained by (3.29) and (3.30). Again,
we have a tachyonic mode and zero modes in the single-particle index as in the case of
the BPS partition function:isp =1
v −1 q

(
+

u−1 v
+
v −1
v −1
u

)
1

+ O(q 2 ).

(3.67)( l , l ,−1)2 2Actually, these modes come from the modes l = 0, 1 in [0, 0]l−1
shown in the first lineof Table 3.4. Since these modes contributed to the BPS partition function, the situationis the exactly same as the BPS partition function (3.39). Hence we take the analytic
continuation for the tachynic mode,

(
Pexp

1
v −1 q)=1
v −1 q=−= −v −1 qPexp(v −1 q)
1 − (v −1 q)−11 − v −1 q

(3.68)

as (3.40), and perform the Weyl completion
D3
D3
I D3 = IX=0
+ IYD3=0 + IZ=0.

(3.69)

Note that this summation becomes the Weyl character formula of the SU (3)R character.
We expect that (3.69) gives the first non-trivial finite N corrections to the index on
the AdS side. Since we only consider the single wrapped D3-branes, there should be theerror coming from the double wrapping contributions. The energy of D3-branes of double

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Chapter 3 Finite N Corrections to the Superconformal Index

wrapping is about 2N . Then the index on the AdS side isAdSIN= I KK (1 + I D3 ) + O(q 2N ).

(3.70)

We conjecture that this index is equivalent to that on the CFT side, namely
KK
IUCFT(1 + I D3 ) + O(q 2N ),(N ) = I(3.71)

where IUCFT(N ) is given by (2.58).

3.2.3

Comparison with the CFT side

Let us consider the comparison with the CFT side to check the correctness of our formula(3.70) for N = 1, 2, 3. The results on the CFT side was already given in (2.59), (2.60) and(2.61). Comparing these results with our formula (3.70), we find
AdS
IN
=1 = (· · · identical terms with (2.59) · · · )13+ 2(1 + χ(2,2) + χ(6,0) )q 6 + O(q 2 ),(3.72)AdS
IN=2 = (· · · identical terms with (2.60) · · · )17+ 2(χ(0,4) + χ(2,0) + χ(4,2) + χ(8,0) )q 8 + O(q 2 ),(3.73)AdS
IN=3 = (· · · identical terms with (2.61) · · · )21+ 2(χ(0,2) + χ(2,4) + χ(4,0) + χ(6,2) + χ(10,0) )q 10 + O(q 2 ).

(3.74)

There results are consistent with our conjecture (3.71). Furthermore, the results suggestthat the relation of the index between the CFT side and the AdS side is
KK
IUCFT(1 + I D3 ) + O(q 2N +4 )(N ) = I(3.75)

rather than (3.71). The shift q 2N → q 2N +4 of the error may be interpreted as the eﬀectof the tachyonic modes in the double wrapping contributions. Let us consider the specificconfiguration given by Z 2 = 0 as an example of a double wrapping configuration. Roughly
speaking, the finite N corrections coming from this configuration is the U (2) index withthe single-particle index (3.65). Thus we have
IZD32 =0 = (v −1 q)2N
I
∼

I

dz
−1
Pexp[iZ=0+ 2 + z)]
sp (z
2πiz
|z|=1

(v −1 q)2N
dz
· · · ∼ (v −1 q)2N +4 + · · · ,1
2 (1 − 1 z)(1 − 1 z −1 )
2πiz
(1
−
)
|z|=1
v −1 qv −1 qv −1 q(3.76)and this is consistent with the shift q 2N → q 2N +4 .

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3.2 Superconformal index

89

In summary, we confirmed that our formula (3.69) and (3.70) does work at least for
N = 1, 2, 3, up to the double wrapping configurations. We expect that the formula works
even for arbitrary N . In the next chapter, we apply the formula to the S-fold theories tofind the BPS partition function and the superconformal index.

Soryushiron Kenkyu

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91

Chapter 4

Four-Dimensional S-fold
Theories
It is well known that any N = 3 theories that have the weak coupling limit automaticallyhave the N = 4 supersymmetry. Then, an N = 3 theory that never has the N =4 supersymmetry is called a genuine N = 3 theory. In the paper [38], Aharony andEvtikhiev discussed some properties of genuine N = 3 theories under the assumptionthat genuine N = 3 theories exist. After their work, Garcia-Etxebarria and Regalado
discovered explicit examples of genuine N = 3 theories [18] as a generalization of theorientifold theory. We call these theories S-fold theories following [20].
In this chapter, we investigate the S-fold theories by using the AdS/CFT correspondence. In particular, we compute the BPS partition function and the superconformal
index following the author’s and his collaborators’ papers [22, 23]. Since the S-fold theories have no weak coupling limit, namely no marginal deformation, we cannot use the freefield counting. Alternatively, we use the method developed in the last chapter to calculate
the finite N corrections to the index. After the calculation, we do the consistency check.
It was conjectured in [20] that some of the S-fold theories are equivalent to N = 4 SYM
with a certain gauge group. Then we can compare our result with that of the corresponding N = 4 SYM. Also, recently Zafrir found the UV Lagrangian for some of the S-foldtheories and calculated the index [21]. So we compare our results with Zafrir’s result.

4.1

N = 4 orientifold theory

Before discussing the S-fold theories, we would like to review the orientifold and the
AdS/CFT correspondence regarding the orientifold.
An orientifold action in type II string theory is defined by I · Ω · J, where I is the sign
flip of coordinates of R9−p , Ω is the orientation reversal of the worldsheet of strings, and J
is the fermion number operator (−1)FL of left moving spacetime fermions for p = 2, 3, 6, 7,
otherwise the identity operator. Then an orientifold p-plane (Op-plane) is defined as afixed plane under the orientifold action. In other words, an Op-plane is a p+1-dimensional

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Chapter 4 Four-Dimensional S-fold Theories

plane of type II string theory on R1,p × R9−p /(I · Ω · J). We consider the p = 3 case here.
Let us see the orientifold action for p = 3 in more detail. We consider an O3-plane isembedded in R1,3 × C3 , where C3 is orthogonal to the O3-plane. Unlike the orbifold, theorientifold acts on fields as well as coordinates. First, the orientifold acts on the transverse
coordinates X, Y, Z ∈ C3 as(X, Y, Z) → (−X, −Y, −Z).

(4.1)

Since SO(6)R is the rotation symmetry of the transverse direction of D3-branes, thistransformation can be realized by −1 ∈ SO(6)R .
Second, the orientifold acts on the massless fields in type IIB string theory as*1
(

C2
B2

)

(
→

)
−C2
,
−B2

(4.2)

and other bosonic fields are invariant. Therefore, the orientifold action can be realized bytaking M = −1 in the SL(2, R) transformation (2.87). Note that if we take account of
the flux quantization (2.75), SL(2, R) is broken to SL(2, Z).

4.1.1

AdS/CFT correspondence in the orientifold background

The AdS/CFT correspondence in the presence of the O3-plane is first discussed by Witten[39]. To discuss it, we place N D3-branes parallel to O3-plane. The realized 4d theory on
D3-branes should be invariant under the orientifold action. Since the orientifold action has
the orientation reversal, we expect that the realized theory is N = 4 with the SO or Sp
type gauge group. If we take the near horizon limit, the geometry becomes AdS5 × S 5 /Z2
due to the orientifold action (4.1).
Let H3 and F3 be the field strength of B2 and C2 , respectively. In the presence of theO3-plane, H3 and F3 are not globally defined 3-form fields but twisted 3-form fields. Thenthese twisted 3-form fields are classified by the twisted sheaf cohomologye ⊕ Z)
e = Z2 ⊕ Z2 ,H 3 (S 5 /Z2 , Z(4.3)

e is a twisted sheaf of integers. This is called the discrete torsion of 3-form fields
where Z
H3 and F3 . The realizable theories on N D3-branes are now classified by Z2 ⊕ Z2 .
Let us call the two types of discrete torsion θNS and θRR . The four possible models
correspond to values(θNS , θRR ) = (0, 0), (1, 0), (0, 1), (1, 1).
It is clear that θNS and θRR form a doublet under SL(2, Z).
*1 The massless fields of type IIB string theory are listed in Table 2.7.

(4.4)
Then a model with

Soryushiron Kenkyu

4.1 N = 4 orientifold theory

93(θNS , θRR )(0, 0)(1, 0)(0, 1)(1, 1)

Gauge group
SO(2N )
Sp(N )
SO(2N + 1)
Sp(N )

Table 4.1 A table of the classification of the discrete torsion (θNS , θRR ) with corresponding gauge groups.

(θNS , θRR ) = (0, 0) corresponds to the N = 4 SO(2N ) SYM because this theory is self
dual under SL(2, Z) (known as the Montonen-Olive duality [40]). Other models can also
be classified in this way. We summarized the correspondence between (θNS , θRR ) and thegauge group in Table 4.1.
In summary, we have found the following AdS/CFTs [39]:4d N = 4 SO, Sp gauge theory⇐⇒

Type IIB string theory on AdS5 × S 5 /Z2 .
(4.5)

Pfaﬃan operators and wrapped D3-branes
In Chapter 3, we analyzed wrapped D3-branes on three-cycle in S 5 , and they indeed gave
the finite N corrections to the BPS partition function and the superconformal index. In
the orientifold theory, these wrapped D3-branes also have an essential role in finding thefinite N corrections.
Actually, Witten pointed out that a Pfaﬃan operator on the CFT side corresponds toa wrapped D3-brane on a non-trivial three-cycle in S 5 /Z2 [39]. The Pfaﬃan operator isdefined as N adjoint scalars contracted by an epsilon tensor. In the N = 4 SO(2N ) SYM,
this is given byPf(Φ) =

1
ϵi ···i (Φ)i1 i2 · · · (Φ)i2N −1 i2N ,N ! 1 2N

(4.6)

where Φ is one of the adjoint scalars X, Y, Z. It is possible to generalize this definition,
including diﬀerent types of fields, for example, ϵXY · · · Z. Then we refer to this type ofoperator as the Pfaﬃan type operator. We can also consider the composite operators of
Pfaﬃans and trace operators. From the definition (4.6), it turns out that the Pfaﬃan
operators exist only for N = 4 SO(2N ) SYM.
Note that the square of the Pfaﬃan operator is the determinant operator*2 : Pf(Φ) ∝det Φ. This is reminiscent of Z2 structure. In fact, the wrapped D3-branes on three-cycle*2 As we pointed out in subsection 2.1.3, the determinant operators can be written by trace operators.

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Chapter 4 Four-Dimensional S-fold Theories

in S 5 /Z2 are classified by the third homology group
H3 (S 3 /Z2 , Z) = Z2 .

(4.7)

Its charge is called the winding number (mod 2) of the wrapped D3-brane.
As a final comment in this subsection, the wrapped D3-branes on trivial cycles arealways present because they correspond to the trace operators on the CFT side [19]. For
example, the double wrapping is allowed to exist because they have a trivial charge of
H3 (S 5 /Z2 , Z) = Z2 . This implies that the first non-trivial finite N corrections come fromthe double wrapping for N = 4 SO(2N +1) and Sp(N ) SYMs. This is beyond our purposein this thesis, and we leave it for future work.

4.1.2

BPS partition function

Now, we give a short explanation about the BPS partition function of the interacting
N = 4 SO(N ) and Sp(N ) SYMs. The calculation on the AdS side will be given later.
1
2 -BPS partition function
First let us consider the 12 -BPS partition function.

The single-particle 21 -BPS partition
function is given by the Coulomb branch operators of the theory, namely the Casimir
operators. The Casimir operators of SO(2N + 1) and Sp(N ) group have a conformal
dimension 2, 4, · · · , 2N . They correspond to the 21 -BPS operators {TrZ2n |n = 1, · · · , N }.
Thus the 12 -BPS partition function is
1
2 -BPS

1
2 -BPSZSO(2N +1) (z) = ZSp(N ) (z) = Pexp

(N
∑

)
x

2n

=

n=1

N
∏

1
.
2n
1
−
x
n=1

(4.8)

The equality of SO(2N + 1) and Sp(N ) is consistent with the SL(2, Z) duality. This
result is the same as the U (N ) 12 -BPS partition function with the replacement z → z 2 .
Therefore, the grand partition function is also obtained by the same replacement for (2.47)
as follows:
1
2 -BPS

1
2 -BPS2

ΞSO(odd) (z; t) = ΞU (∗) (z ; t) =

(
)
1
Z2
≡
Pexp
I
(z)t
,
m=0
2n
1
−
tz
n=0
∞
∏

(4.9)

Z2
where the single-particle grand partition function Im=0
is given by
Z2
Im=0
(z) =

1
1
=
2
1−z
2

(

1
1
+
1−z
1+z

)
.

(4.10)On the other hand, N = 4 SO(2N ) SYM has not only trace operators but also a Pfaﬃan
operator Pf(Z), the conformal dimension of them are 2, 4, · · · , 2N − 2 and N . Note thatthe trace operator TrZ2N is constructed by Pf(Z)2 , it is not the independent operator,

Soryushiron Kenkyu

4.1 N = 4 orientifold theory

95

and we do not consider it. Then the 12 -BPS partition function is obtained by removing afactor (1 − x2N )−1 and multiplying a factor (1 − xN )−1 to (4.8). Thus we have
1

-BPS

2
ZSO(2N) (z) =

11
1 − x2N 12 -BPS
N
2 -BPS
Z
(z)
=
(1
+
x
)Z
SO(2N
+1)
SO(2N
+1) (z).
N
1−x(4.11)-BPS

2
The latter part, xN ZSO(2N+1) (z), is the contributions from the Pfaﬃan type operators.
1

-BPS

2
xN is the contribution from Pf(Z) and ZSO(2N+1) (z) receives the contributions from othertrace operators. Now we can immediately find the grand partition function from (4.9):

1
2 -BPSΞSO(even) (z; t) =∞∑1
2 -BPSNZSO(2N ) (z)t

N =0

=

∞
∑

1

-BPS

N
2
(1 + xN )ZSO(2N+1) (z)tN =0

1
2 -BPS

1
2 -BPS

Z2
Z2= ΞSO(odd) (z; t) + ΞSO(odd) (z; zt) ≡ Pexp(Im=0(z)t) + Pexp(Im=1(z)t),(4.12)

where
Z2
Im=1
(z) =

1
z
=
1 − z2
2

(

1
1
−
1−z
1+z

)
.

(4.13)

In summary, the 12 -BPS partition function can be unified into the following form
1

-BPS

2
ΞSO(∗)(z; t) =∑Z2(z)t),
Pexp(Im(4.14)

pm=0

where m = 0, 1. p = 0 corresponds to the trivial discrete torsion, and p = 1 corresponds
to the non-trivial torsion.
1
8 -BPS partition function
Next, let us consider the 18 -BPS partition function. In the interacting theory, we can findthe 18 -BPS partition function by considering the chiral ring. Then, all the adjoint scalars

are commutable, and they are valued in the Cartan subgroup of the gauge group. Thus
the 18 -BPS partition function should be the invariant series under the Weyl group.
First, let us see the SO(2N + 1) case. Then we choose the Cartan basis as
X = diag(ix1 σ2 , · · · , ixN σ2 , 0),

(
)
0 −i
σ2 =
,
i 0

(4.15)

where the last component 0 is a 1 × 1 block matrix and others are 2 × 2 block matrices. Y
and Z have the same structure. The Weyl transformation contains a sign flip xk → −xk(k = 1, · · · , N ) as well as the permutation of xk ’s. This sign flip is realized by SO(3) ⊂Soryushiron Kenkyu

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Chapter 4 Four-Dimensional S-fold Theories

SO(2N + 1), whose action is

−xk

xk





SO(3)
−
−−−→ xk

−xk

0


.

(4.16)

0

To make the BPS partition function invariant under this sign flip and the permutation,
we modify the single grand particle index of U (N ) theory I(x, y, z) as follows:I(x, y, z) →1
Z2
(I(x, y, z) + I(−x, −y, −z)) ≡ Im=0(x, y, z).
2(4.17)

Then the grand partition function is18ΞSO(odd)(x, y, z; t) = Pexp(I0Z2 (x, y, z)t).

-BPS(4.18)

When x = y = 0, it reproduces the 12 -BPS grand partition function of N = 4 SO(2N + 1)
SYM. The 81 -BPS partition function of Sp(N ) is the same as (4.18).
Next let us see the SO(2N ) case. In this case, the Cartan valued adjoint scalar fields
are given by removing the last zero component from (4.15). Thus, there is no longer
an SO(3) transformation like (4.16). However, we can perform an SO(4) transformation
whose action is(xk , xl ) → (−xk , −xl )(4.19)

for arbitrary xk and xl with k ̸= l. Since this transformation flips the sign of two variables
simultaneously, the single-particle grand partition function does not have to be invariant
Z2
under (4.19). Therefore, in addition to Im=0
, we can consider
Z2
Im=1
=

1
(I(x, y, z) − I(−x, −y, −z)).
2(4.20)

So we find the 18 -BPS partition function for N = 4 SO(2N ) SYM:
1

8
ΞSO(even)(x, y, z; t) = Pexp(I0Z2 (x, y, z)t) + Pexp(I1Z2 (x, y, z)t).

-BPS(4.21)

Again, setting x = y = 0, it reproduces the 21 -BPS grand partition function (4.12).
In summary, similarly to the 12 -BPS grand partition function (4.14), we can write the1
8 -BPS partition function for N = 4 SO(N ) and Sp(N ) SYMs as
1

-BPS

8
(x, y, z; t) =ΞSO(∗)∑

Z2
Pexp(Im(x, y, z)t).

(4.22)

pm=0

The large N limit of the BPS partition function can be obtained by the formula (2.99).

Soryushiron Kenkyu

4.1 N = 4 orientifold theory

97

The result is
1

8
ZSO(∞)(x, y, z) = Pexp(I0Z2 (x, y, z) − 1).

-BPS(4.23)

Z2
It seems that the Im=1
is absent in the large N limit. The reason is as follows. In the large
N limit, the number of indices in the epsilon tensor is infinite. So we cannot define thePfaﬃan operators. In other words, the conformal dimension of the Pfaﬃan operator is oforder N . Then they decouple from the BPS partition function in N → ∞. That is why
Z2
the corresponding term Im=1
does not appear. This is consistent because the wrapped
D3-branes are not the physical objects in the large N limit because their mass is of orderN.

Consistency check
Before going on, let us do a consistency check of our result (4.22). This can be done byusing the group theory isomorphisms. We know the following isomorphisms:
SO(2) ≃ U (1),SO(3) ≃ SU (2), SO(4) ≃ SU (2) × SU (2), SO(6) ≃ SU (4). (4.24)

Thus the BPS partition functions should be the same as those of theories with these gauge
groups. The BPS partition functions of these gauge theories are obtained by (4.22), andwe can check the following facts:
1

-BPS

1

-BPS

8
ZSO(2)= ZU8 (1) ,

4.1.3

1

1

-BPS

-BPS

8
8
ZSO(3)
= ZSU
(2) ,

1

1

-BPS

-BPS

2
8
8
ZSO(4)= (ZSU(2) ) ,

1

1

-BPS

-BPS

8
8
ZSO(6)= ZSU(4) .
(4.25)

Superconformal index

Let us consider the superconformal index for N = 4 SO(N ) SYM as well. The calculationof the index on the CFT side can be performed by the free field counting. The diﬀerent
point of the index from the N = 4 U (N ) SYM is only the gauge group. Thus, all we haveto do is to replace the character of the adjoint representation and the Haar measure by
those of SO(N ).
The characters of the adjoint representation of SO(2N ) and SO(2N + 1) are given by
SO(2N )
χadj(za ) = N +

∑ za
a̸=b

SO(2N +1)
χadj(za ) = N +

zb

∑ za
a̸=b

zb

+

∑(
a>b

+

∑(
a>b

1
za z b +za zb
1
za z b +
za zb

)
,
)
+

(4.26)
N (
∑
a=1

1
za +
za

)
,

(4.27)

where za (a = 1, · · · , N ) are the gauge fugacities. The Haar measure can be found fromthe formula (1.83).

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Chapter 4 Four-Dimensional S-fold Theories

For convenience, we give examples of the index up to rank three.
3

7

ISO(2) = 1 + χ(1,0) q − χJ1 q 2 + (χ(2,0) − χ(0,1) )q 2 + (χ(3,0) − χ(1,1) + 1 − χJ2 )q 3 + O(q 2 ),5

7

ISO(4) = 1 + 2χ(2,0) q 2 − 2χJ1 χ(1,0) q 2 + (2 − 2χ(1,1) )q 3 + 2χJ1 (χ(0,1) + χ(2,0) )q 2
9

+ (−2χJ2 χ(1,0) + χ(0,2) + χ(2,1) + 3χ(4,0) )q 4 − 4χJ1 (χ(1,1) + χ(3,0) )q 2
11

+ (χJ2 (3χ(0,1) + 3χ(2,0) ) + χ(0,1) − 2χ(1,2) + χ(2,0) − 4χ(3,1) )q 5 + O(q 2 ),5

7

ISO(6) = 1 + χ(2,0) q 2 − χJ1 χ(1,0) q 2 + (1 − χ(1,1) + χ(3,0) )q 3 + χJ1 χ(0,1) q 2
9+ (χ(0,2) + χ(1,0) − χJ2 χ(1,0) − χ(2,1) + 2χ(4,0) )q 4 + χJ1 (−χ(1,1) − χ(3,0) )q 2
+ (−χ(0,1) + 2χJ2 χ(0,1) + 2χ(2,0) − χ(3,1) + χ(5,0) )q 5
11

+ (χJ1 (χ(0,2) + χ(1,0) + χ(2,1) ) − χJ3 χ(1,0) )q 2

+ (1 − 2χ(1,1) + χ(3,0) − χ(4,1) + 3χ(6,0) + χJ2 (−1 − 2χ(1,1) − χ(3,0) ))q 6
13+ (χJ1 (−χ(0,1) + χ(2,0) − 2χ(5,0) ) + χJ3 (2χ(0,1) + χ(2,0) ))q 2+ (−χJ4 χ(1,0) + χJ2 (2χ(0,2) + 2χ(1,0) + χ(2,1) )15+ χ(1,0) + χ(4,0) − 2χ(5,1) + 2χ(7,0) )q 7 + O(q 2 ),(4.28)

Soryushiron Kenkyu

4.2 S-fold theories

99

5

7

ISO(3) = 1 + χ(2,0) q 2 − χJ1 χ(1,0) q 2 + (1 − χ(1,1) )q 3 + χJ1 (χ(0,1) + χ(2,0) )q 2
9+ (−χJ2 χ(1,0) + χ(4,0) )q 4 + O(q 2 ),5

7

ISO(5) = 1 + χ(2,0) q 2 − χJ1 χ(1,0) q 2 + (−χ(1,1) + 1)q 3 + χJ1 (χ(2,0) + χ(0,1) )q 2
9+ (2χ(4,0) + χ(0,2) − χJ2 χ(1,0) )q 4 − χJ1 (−2χ(1,1) − 2χ(3,0) )q 2
+ (2χJ2 χ(0,1) − χ(0,1) − χ(1,2) + χ(2,0) + χ(5,0) )q 5
11

+ (−χJ3 χ(1,0) + χJ1 (2χ(0,2) + χ(1,0) + 4χ(2,1) + 2χ(4,0) ))q 2+ (χJ2 (−1 − 4χ(1,1) − 3χ(3,0) ) − 1 + χ(0,3) − 3χ(1,1) + χ(2,2) + χ(4,1)
13+ 2χ(6,0) )q 6 + O(q 2 ),5

7

ISO(7) = 1 + χ(2,0) q 2 − χJ1 χ(1,0) q 2 + (1 − χ(1,1) )q 3 + χJ1 (χ(0,1) + χ(2,0) )q 2
9+ (−χJ2 χ(1,0) + χ(0,2) + 2χ(4,0) )q 4 + χJ1 (−2χ(1,1) − 2χ(3,0) )q 2
+ (χJ2 (2χ(0,1) + χ(2,0) ) − χ(1,2) + 2χ(2,0) − 2χ(3,1) )q 5
11

+ (−χJ3 χ(1,0) + χJ1 (2χ(0,2) + χ(1,0) + 4χ2,1) + 2χ(4,0) ))q 2+ (χJ2 (−1 − 4χ(1,1) − 3χ(3,0) ) + χ(0,3) − 3χ(1,1) + 2χ(2,2) + χ(4,1) + 3χ(6,0) )q 6
13

+ (χJ3 (2χ(0,1) + 2χ(2,0) ) + χJ1 (−χ(0,1) − 5χ(1,2) − χ(2,0) − 6χ(3,1) − 4χ(5,0) ))q 2+ (χJ4 (−χ(1,0) ) + χJ2 (5χ(0,2) + 4χ(1,0) + 8χ(2,1) + 4χ(4,0) ) + 3χ(0,2) + 4χ(1,0)
15− 2χ(1,3) + 4χ(2,1) − 3χ(3,2) + 3χ(4,0) − 4χ(5,1) )q 7 + O(q 2 ).

(4.29)

Finally, we calculate the index in the large N limit. Like the BPS partition function,
the large N index is the same for SO(2N ) and SO(2N + 1) because we cannot distinguishthese two theories in N → ∞. The procedure of the calculation process is almost thesame as that of the U (N ) theory: the saddle point method. This analysis is given in [19],and the result is
CFT
ISO(∞)= Pexp(iSO(∞)),
sp

iSO(∞)=
sp(1 − isp (q, y, u, v))2
1 KK
1
isp +
−
,
2
4(1 − isp (q 2 , y 2 , u2 , v 2 )) 4

(4.30)

where iKK
sp is defined in (2.113), and isp is defined in (2.57).

4.2

S-fold theories

In this section, we would like to define the S-fold theories and investigate the propertiesof the S-fold theories.
An S-fold action is defined as the Zk generalization of the orientifold action [18]. First,

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Chapter 4 Four-Dimensional S-fold Theories

the S-fold action acts on the coordinates as((X, Y, Z) → (ωk−1 X, ωk Y, ωk Z),

ωk = exp

2πi
k

)
.

(4.31)

This is actually the Zk generalization of (4.1). This transformation is realized by
(
exp

2πi
S
k

)
∈ SO(6)R ,S = −RX + RY + RZ ,(4.32)

where RX , RY , and RZ are the SO(6)R Catran generators defined by (2.1).
To define the S-fold action for massless fields in type IIB string theory, it is convenient
to regard the type IIB supergravity with the axio-dilaton field τ as the SL(2, R)/U (1)non-linear sigma model [41,42]. In this formalism, we introduce an SL(2, R)-valued scalar
field
)
( x
v2 v2y∈ SL(2, R).
(4.33)V =v1x v1y
It is also useful to define complex scalar fields va (a = 1, 2) as
va = vax + ivay

(4.34)The scalar field V is transformed under the global SL(2, R) symmetry and the local
U (1) ≃ SO(2) symmetry as
V → gV,
V →Vh

−1

,

g ∈ SL(2, R),(4.35)h ∈ SO(2).

(4.36)

Although there are unphysical degrees of freedom in the scalar field V in this formalism,
we can remove them by the gauge fixing of the local U (1) symmetry. The axio-dilaton
field is given by the gauge invariant combination
τ=

v2
.
v1

(4.37)Let Vb be the expectation value of V . It breaks SL(2, R) × U (1) into U (1)Y . Thissymmetry is determined by the equation
Vb = g Vb h−1 .

(4.38)

This global U (1)Y symmetry plays a significant role in the S-fold action. The fermionicfields in type IIB string theory are transformed non-trivially under U (1)Y , while thebosonic fields are invariant. We summarize the charge of each field under U (1)Y in Table
4.2.

Soryushiron Kenkyu

4.2 S-fold theories

101
fieldsva=1,2
λM
L
χR

Y
−2
−1
−3

Table 4.2 U (1)Y charges of each massless field of type IIB string theory. We only
show the fields with non-trivial U (1)Y charges.

k
2
3
4
6

τ
any
πi
e3
i
2πi
e 3

Table 4.3 The value of τ for each S-fold theory. In k = 2, τ can take any value, andthis is consistent to the orientifold theories.

The non-trivial U (1)Y charge of the gravitino λM
L implies that the supercharges also
carry the non-trivial U (1)Y charge. Let Y be the U (1)Y generator. The commutation
relation between Y and QI , which generates the N = 4 supersymmetry on the CFT side,
is given by[Y, QI ] = −QI .

(4.39)

If we take into account the flux quantization for the 3-form fluxes (2.75), U (1)Y is broken
to its discrete subgroup Zk . Let q a , a = 1, 2 be the magnetic charges of the (p, q)-string
associated with the 3-form fields:
∫
∫
1
1
2
1
q =
H3 ,
q =
F3 ,
(4.40)
2π
2π
where the H3 and F3 are the 3-form fluxes associated with the R-R 2-form fields C2 andB2 . Then the quantized charge q C = vba q a form the charge lattice with modulus τ = vb2 /bv1 .
Now the charge lattice must be invariant under Zk . k = 2 is possible for the generic valueof τ while k = 3, 4, 6 are possible only for(k−2
τ = exp iπk).

(4.41)We classify the value of τ in Table 4.3 and show an example of the invariant charge latticefor k = 3 in Fig. 4.1.

Soryushiron Kenkyu

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Chapter 4 Four-Dimensional S-fold Theories
Im
C

v2
Re

v1

Fig. 4.1 An example of the invariant string charge lattice defined by q C = vba q a forπi
k = 3. To make the lattice invariant under Z3 , the modulus τ = vb2 /b
v1 must be e 3
fixed.

Now we define the Zk S-fold action R for k = 3, 4, 6 as [18][2πiR = exp
k

(
)]
Y
S−
,
2

(4.42)

where S is defined in (4.32).

4.2.1

AdS/CFT correspondence in the S-fold backgroundLet us consider a stack of N parallel D3-branes in the S-fold background defined by
R1,3 × C3 /R, where R is the S-fold action (4.42). We define a 4d rank N Zk S-fold theory
as the worldvolume theory on the D3-branes [18]. If we take the near horizon limit, thegeometry becomes AdS5 × S 5 /Zk due to the S-fold action (4.31).
As in the orientifold case, there are several variants of S-fold theories due to the discrete
torsion. In the S-fold background, the discrete torsion associated with the 3-form fluxes
is given by [19, 20]

 Z3
3
5
^
H (S /Zk , Z⊕ Z) =
Z 2
Z1

for k = 3,
for k = 4,
for k = 6.

(4.43)For k = 3, the torsion group is Z3 , so we have three variants about the Z3 S-fold. Theories corresponding to non-trivial elements of Z3 are related by the S-duality, as in theorientifold case. For k = 4, there are two variants. However, for k = 6, there is only one
variant. We summarize this knowledge in Table 4.4.
Then an S-fold theory is specified by the value of k, N , and discrete torsion. We denoteSoryushiron Kenkyu

4.2 S-fold theories

103
k
2
3
4
6

discrete torsion
Z2 ⊕ Z2
Z3
Z2
Z1# of variants
43
2
1

Table 4.4 A table for the discrete torsion for S-fold theories. As in the orientifold
case, the two variants with the non-trivial torsion in the Z3 S-fold are related by the
S-duality.

the theory by S(N, k, p), where p = 0, 1. p = 0 corresponds to the trivial torsion, andp = 1 corresponds to a non-trivial torsion. The AdS/CFT correspondence regarding S-foldtheories is [19, 20]
4d Zk S-fold theory⇐⇒

Type IIB string theory on AdS5 × S 5 /Zk .

(4.44)

for k = 3, 4, 6.
Like the orientifold case, there are also Pfaﬃan-like operators in the S-fold theories withthe trivial torsion, called the generalized Pfaﬃan operators [20]. Corresponding objects
are the wrapped D3-branes on a non-trivial three-cycle in S 5 /Zk , and they are classified
by the third homology group
H3 (S 5 /Zk , Z) = Zk .

(4.45)

Corresponding charge to Zk is called the winding number of wrapped D3-branes. These
wrapped D3-branes give the finite N corrections to the index [19].

4.2.2Properties of S-fold theories

In this subsection, we review some remarkable properties of S-fold theories summarized
as follows:• S-fold theories have N = 3 supersymmetry [18].
• S-fold theories have no marginal deformation [18, 38]*3 . Then, the Lagrangian of
S-fold theories is not known.
• It was conjectured that the rank one and two S-fold theories are equivalent to N = 4
SYMs with rank one and two gauge groups [20] (Aharony-Tachikawa conjecture).
Let us see these properties in more detail below.

marginal deformations that do not respect N = 3 supersymmetry are notprohibited. In fact, the marginal deformation that breaks N = 3 to N = 1 was discussed in [21].

*3 Precisely speaking,Soryushiron Kenkyu

104

Chapter 4 Four-Dimensional S-fold Theories

Q1
Q2
Q3
Q4

RX

RY

RZ

S

1
2
1
2
− 12
− 12

1
2
− 12
1
2
− 12

1
2
− 12
− 12
1
2

1
2
− 32
1
2
1
2

Y
1
1
1
1

S − Y /20−200Table 4.5 S and Y charges of supercharges of an N = 4 supersymmetry, where
S = −RX + RY + RZ . If we impose the Zk identification generated by (4.42) for
supercharges, Q2 and Q2 are projected out, and an N = 3 supersymmetry is realized.
The removed charge is just a convention, and if we use another convention for S,
corresponding another supercharge would be removed.

N = 3 supersymmetry
The S-fold theories are obtained by imposing the invariance under the S-fold action (4.42)
on the worldvolume theory on N D3-branes, namely N = 4 SYM. The behavior of theN = 4 supercharges QI are shown in Table 4.5. From this table, if we impose the
S-fold invariance on supercharges, we see that Q2 is projected out. Then the N = 3
supersymmetry is realized in S-fold theories.
No marginal deformation
It is known that genuine N = 3 theories have no marginal deformation [38]. Thus the
S-fold theories also have no marginal deformation, and it was shown in [18]. Then, theS-fold theories are not connected to a free theory, and the Lagrangian is not known. This
fact is consistent with the fixing of the complex coupling constant τ defined in (2.15).
Note that marginal deformations that do not respect N = 3 supersymmetry are allowed.
Aharony-Tachikawa conjecture
It is known that if a genuine N = 3 theory has a Coulomb branch operator with dimension one or two, the N = 3 supersymmetry is non-trivially enhanced to the N = 4
supersymmetry [38]. This is also the case for some of the S-fold theories. To see this, let
us investigate the Coulomb branch operators of S-fold theories following [20].
As we saw in subsection 2.1.2, the Coulomb branch operators parametrize the Coulomb
branch of a theory. In N = 4 U (N ) SYM, the Coulomb branch is C/WU (N ) , where
WU (N ) = SN is the Weyl group of U (N ). From the viewpoint of the N D3-branes
system, the Weyl transformation is regarded as the exchanging of N D3-branes locatedat zi ∈ C ⊂ C3 (i = 1, · · · , N ). In the S-fold case, C3 is replaced by C3 /Zk , and the Weyl

Soryushiron Kenkyu

4.2 S-fold theories

105

transformation is given by [20](zi , zj ) ↔ (zj , zi ),(4.46a)(zi , zj ) → (ωk zi , ωk−1 zj ),(4.46b)zi → ωkr zi ,(4.46c)where r is a divisor of k. r = k corresponds to the trivial discrete torsion and r ̸= 0
corresponds to a non-trivial torsion. Now the first transformation is SN generated by the∑symmetric polynomials σ∈SN zσ(i) . However, in order to make polynomials invariant
under (4.46b) and (4.46c) as well, we need zi → zik . We also find that (z1 z2 · · · zN )l is
invariant, where l = k/r. Thus the dimension of Coulomb branch operators is [20]k, 2k, 3k, · · · , (N − 1)k; N l.

(4.47)Let us consider the non-trivial supersymmetry enhancement of S-fold theories. Fromthe list of the dimension of the Coulomb branch operators (4.47), we find that the supersymmetry enhancement always occurs for k = 2, and this is consistent with the orientifold.
We also find that for N = 1, 2 with k = 3, 4, 6 and l = 1, or equivalently r = k there
exists the Coulomb branch operators with dimension two. All these S-fold theories havethe trivial discrete torsion. Thus, we expect that (N, k, l) = (1, k, 1), (2, k, 1) cases areequivalent to certain N = 4 theories*4 . In terms of the notation S(N, k, p) introduced insubsection 4.2.1, (1, k, 1), (2, k, 1) cases are S(1, k, 0) and S(2, k, 0). Since it is believedthat any N = 4 theory is an N = 4 SYM, these S-fold theories may be equivalent tocertain N = 4 SYM.
For N = 1, namely the rank one case, the only possible gauge group is U (1). So it isexpected that a rank one S-fold theories with trivial torsion are equivalent to the N = 4
Maxwell theory. In order to determine the gauge group of rank two case, let us focus on
the dimension of all independent single-particle Coulomb branch operators: 2, 3 for k = 3,
ECBO =
2, 4 for k = 4,
2, 6 for k = 6.

(4.48)

These spectra agree with those of SU (3), SO(5), and G2 , respectively. Therefore, it isexpected that rank two S-fold theories with the trivial torsion are equivalent to N = 4
SYMs with the gauge group SU (3), SO(5), and G2 (shown in Table 4.6). This was conjectured by Aharony and Tachikawa in [20], so we call this conjecture Aharony-Tachikawa(AT) conjecture. We will confirm AT conjecture by using the BPS partition function andthe superconformal index.

*4 We can always construct the Coulomb branch operators with dimension two from those with di-mension one.

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Chapter 4 Four-Dimensional S-fold Theories
theoryS(1, k, 0)S(2, 3, 0)
S(2, 4, 0)S(2, 6, 0)

GU (1)SU (3)
SO(5)
G2

ECBO
1
2,3
2,4
2,6

Table 4.6 A list of the gauge group and corresponding S-fold theories. For rank one,all S-fold theories with trivial torsion are enhanced to the N = 4 Maxwell theory.
For the rank two case, we can obtain the corresponding gauge group by comparing
the spectrum of the Coulomb branch operators.

4.3

BPS partition function of S-fold theories

In this section, we find the BPS partition function of S-fold theories by generalizing theresults of N = 4 SO(N ) SYM. This analysis is not on the AdS side but the CFT side. Thisis possible even though the Lagrangian is unknown because the BPS partition function
is constructed as the invariant polynomial under the Weyl transformation. However, thisis not the case for the superconformal index. In general, we have no idea to find thesuperconformal index on the CFT side. We have to use the AdS/CFT correspondence toobtain the index of S-fold theories.
Since the BPS partition function counts the primary scalars of the theory, it is enoughto consider the Zk orbifold (4.31). In N = 3 we no longer use the term 12 , 14 , 18 -BPS
because the number of supercharges were changed. Instead of these terms, we simply callthe BPS partition function with three variables (x, y, z) as the BPS partition function,and the BPS partition function with the specialization of z = 0 and x = y = 0 called the
Higgs branch Hilbert series and the Coulomb branch Hilbert series.
Let us generalize the Weyl transformation (4.46) for three coordinates (xi , yi , zi ) ∈3
C /Zk . Each of variables specifies the location of N D3-branes in the S-fold backgrounds.
The generalization is simply given by{(xi , yi , zi ), (xj , yj , zj )} ↔ {(xj , yj , zj ), (xi , yi , zi )},(4.49a){(xi , yi , zi ), (xj , yj , zj )} → {(ωk−1 xi , ωk yi , ωk zi ), (ωk xj , ωk−1 yj , ωk−1 zj )},(4.49b)(xi , yi , zi ) → (ωk−p xi , ωkp yi , ωkp zi ),(4.49c)where p is a divisor of k again.
Let us remind that the oscillator formalism we have used in subsection 2.1.3, in which
we interpret (xi , yi , zi ) as coordinates of each oscillator. The first transformation is simply
SN , so this indicates that we can treat each oscillator as a boson. The second condition

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4.3 BPS partition function of S-fold theories

107

requires that all single wave functions should satisfy a common boundary condition
Ψ(ωk−1 x, ωk y, ωk z) = ωkm Ψ(x, y, z),(4.50)

where m ∈ {1, 2, · · · , k − 1}. It specifies to which element of Zk corresponds. Thus theinvariant single-particle BPS partition function can be calculated as∑ZkIm(x, y, z) =xnx y ny z nz

−nx +ny +nz =m mod k

ωk−ml
1∑
,
=
k
(1 − ωk−l x)(1 − ωkl y)(1 − ωkl z)
k−1(4.51)

l=0

Zk
where nx , ny , nz ≥ 0. Note that Im(x, y, z) satisfies
Zk
Zk
Im(ωk−1 x, ωk y, ωk z) = ωkm Im(x, y, z).

(4.52)

To obtain the grand partition function, we finally have to take the third condition(4.49c) into account. Actually, this condition is related to the discrete torsion. If p = kcorresponding to the trivial torsion, the condition (4.49c) does not make sense. It requirespm = 0 mod k. Thus we haveΞS(∗,k,p) (x, y, z; t) =∑Zk
Pexp(Imt),(4.53)

pm=0

where we adopt p = 0 instead of p = k for the trivial torsion. S(N, k, p) is a label of a rank
N Zk S-fold theory with p, which corresponds to the discrete torsion. This expression
indeed reproduces the orientifold case (4.22). We can interpret the non-zero m sectors as
the contributions from the generalized Pfaﬃans.
Let us find the BPS partition function of S-fold theories in the large N limit. By using
the formula (2.99), we immediately obtain
ZS(∞,k,p) (x, y, z) = Pexp(I0Zk (x, y, z) − 1).

(4.54)Let us find the BPS partition function of S-fold theories in the large N limit. By using theformula (2.99), we immediately obtain (4.54). The non-zero m sectors decouple because
the generalized Pfaﬃans have a conformal dimension larger than or equal to N . On the
AdS side, this means that the corresponding wrapped D3-branes becomes too heavy toparticipate in the physics.
Supersymmetry enhancement
Let us check the supersymmetry enhancement in terms of the BPS partition function. To
see this, we pick up N = 1, 2 BPS partition functions from the grand partition function.

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Chapter 4 Four-Dimensional S-fold Theories

For N = 1, we have
ZS(1,k,0) (x, y, z) =

k−1
∑

Zk
Im
= I(x, y, z) = ZU (1) (x, y, z).

(4.55)

m=0

Thus this is consistent with the supersymmetry enhancement of rank one S-fold theories.
For N = 2, we must obtain the BPS partition functions of N = 4 SYM withSU (3), SO(5), G2 gauge groups. The results obtained from the grand partition function(4.53) are
ZS(2,3,0) =16(1 − x)2 (1 − y)2 (1 − z)2+

ZS(2,4,0) =ZS(2,6,0) =

1
1

+

1
2(1 − x2 )(1 − y 2 )(1 − z 2 )1

12(1 − x)2 (1 − y)2 (1 − z)2++

12(1 − x2 )(1 − y 2 )(1 − z 2 )3(1 + x + x2 )(1 + y + y 2 )(1 + z + z 2 )8(1 − x)2 (1 − y)2 (1 − z)2++1(4.56)
18(1 + x)2 (1 + y)2 (1 + z)2++2(1 − x2 )(1 − y 2 )(1 − z 2 ),14(1 + x2 )(1 + y 2 )(1 + z 2 ),(4.57)

1
12(1 + x)2 (1 + y)1 (1 + z)2+16(1 − x + x2 )(1 − y + y 2 )(1 − z + z 2 )16(1 + x + x2 )(1 + y + y 2 )(1 + z + z 2 ),(4.58)and they are indeed equal to ZSU (3) , ZSO(5) , ZG2 , respectively. The ZG2 can be calculatedby the Mölien series. Therefore, we checked the supersymmetry enhancement in terms of
the BPS partition function.

4.4

Large N limit on the AdS side

Before going to the calculation of the finite N corrections to the BPS partition function
and the superconformal index of S-fold theories, we need to find the expression of the large
N limit as a zeroth correction. In the case of the BPS partition function, we can comparewith the results calculated in the last section. On the other hand, the superconformal
index is non-calculable on the CFT side, the result on the AdS side is the unique expression
we can find. The analysis of the index on the large N limit was already done by Imamura
and Yokoyama [19], then we review it.

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4.4 Large N limit on the AdS side

4.4.1

109

BPS partition function

Now let us consider the BPS partition function of S-fold theories in the large N limit on
the AdS side. As in the case of U (N ), the contributions in the large N limit come fromthe KK modes on S 5 /Zk in Zk S-folds. Since the BPS partition functions count only the
primary scalars and they form an (n, 0, 0) representation of SO(6)R which is real, it isenough to consider the eﬀect of Zk orbifold (4.31).
To realize the Zk orbifold for the BPS partition function, we impose the Zk invarianceof the single-particle BPS partition function of KK modes on S 5 : I(x, y, z) − 1. For x, y, z
fugacities, the Zk orbifold is defined as(x, y, z) → (ωk−1 x, ωk y, ωk z).

(4.59)The single-particle BPS partition function of KK modes should be invariant under Zk
orbifold. Thus we have
1 ∑I(ωk−1 x, ωk y, ωk z) − 1 = I0Zk (x, y, z) − 1,
Pk I(x, y, z) − 1 =
k m=0
k−1

(4.60)

where the orbifold projection Pk is defined by
1 ∑f (ωk−1 x, ωk y, ωk z).
Pk f (x, y, z) =
k m=0
k−1

(4.61)

Its plethystic exponential agrees with (4.54). Therefore we proved that the BPS partition
function of S-fold theory in the large N limit is the same on both sides.

4.4.2

Superconformal index

Now let us consider the superconformal index of S-fold theories in the large N limit on the
AdS side. For calculations of the index, we adopt the same technique as the BPS partition
function calculated in the last subsection. Namely, we perform the S-fold projection forthe single-particle index of KK modes on S 5 given in (2.113).
However, there is a diﬀerent point from the last subsection: we count not only primaryscalars but also other fields constructed by the supersymmetric completion in the index.
The problem is that other fields are in general charged under Zk ⊂ U (1)Y because supercharges have a non-trivial charge under U (1)Y . Thus we have to use R given in (4.42)
when the projection. To do it, we add a new fugacity η regarding R to the single-particle
index and define the “refined” single-particle index
eisp (η) = trsp [(−1)F x∆ q E+jR y 2jL uRX −RY v RY −RZ η S−Y /2 ],(4.62)

Soryushiron Kenkyu

110

Chapter 4 Four-Dimensional S-fold Theories

where S = −RX + RY + RZ . Before the S-folding, the supersymmetry is N = 4, so we
may expect that the refined index would respect the SU (3)R ⊂ SU (4)R symmetry, whichis generated by RX − RY and RY − RZ . However, S does not commute with this SU (3)R ,and the refined index is not represented by SU (3)R characters. To avoid this troublesome
calculation, we split S into SU (3)R Cartan parts and an SU (3)R invariant part as follows:421
S = −RX + RY + RZ = − (RX − RY ) − (RY − RZ ) + (RX + RY + RZ ),3
3
3(4.63)

where RX + RY + RZ is invariant under SU (3)R . Then we can write the refined index(4.62) in the form respecting SU (3)R :[)RX −RY ()RY −RZ ]
(
4
2
F
∆
E+j
2j
−
−
R
eisp (η) = trsp (−1) x q
y L uη 3
vη 3
ηT ,
T ≡

1
Y
(RX + RY + RZ ) + .
3
2(4.64a)(4.64b)So we use the variables uη − 3 and vη − 3 instead of u and v for SU (3)R characters.
Now we can calculate the refined single-particle index of KK modes from the Table 2.10
and the T charges. We list T charges in Table 4.7 as well as Y charges, corresponding to
KK modes listed in Table 2.10. The result is
4

eiKK
sp (η) =

2

1
3

3

η
2
2 −1 )(1 − uη q)(1 − vηu q)(1 − v q)(1 − q y)(1 − q y[()()
u vη η
v
u
η
−1 32
×
+
+
q − η(y + y )q − (1 + η)
+
q2
+
η
u
v
η vη u
(
)
7
u η2
−1
2 3
−1
+ (η + 1 + 2η + η )q + (y + y ) v + +
q2
v
u
(
)
]
2
2
vη
η
4
−1 29
6
− u+
+
q − (1 + η)(y + y )q + ηq .
(4.65)
uv

Then the single-particle index of KK modes in a Zk S-fold theory labeled by S(∞, k, p) is
obtained as
eKK
iKK
S(∞,k,p) = Pk isp ,(4.66)

where Pk is a projection operator defined as
1∑
f (η = ωkl ).
Pk f (η) =
k
k−1

l=0

(4.67)

Soryushiron Kenkyu

4.4 Large N limit on the AdS side

111
(R ,RY ,RZ )[jL , jR ]E X
(n,0,0)[0, 0]n(n− 21 , 12 , 21 )n+ 12
(n− 21 , 12 ,− 21 )
n+ 12

Y
0

[ 12 , 0]−1[0, 12 ]1(n−1,0,0)[0, 1]n+1(n−1,1,1)[0, 0]n+1(n−1,1,0)[ 12 , 12 ]n+1

2
−2
0

[ 12 , 1]n+ 32 2 2

1[0, 12 ]−1(n− 3 , 1 , 1 )2(n− 23 , 32 , 21 )n+ 32

(n−2,1,1)[0, 1]n+2

0

T
n
3
n+2
3
n−2
3
n−4
3
n+4
3
n
3
n−2
3
n+2
3
n
3

Table 4.7 A table for T charges and Y charges for KK modes listed in Table 2.10.
4
2
T commutes with SU (3)R symmetry, and we use the variables uη − 3 and vη − 3 forSU (3)R characters.

The multi-particle index is now
KK
IS(∞,k,p)= Pexp(iKK
S(∞,k,p) ).

(4.68)

Note that the diﬀerence of the discrete torsion is absent in the large N limit, so the result
is the same for all possible p.
We give some comments for each k = 2, 3, 4, 6 following.
k = 2 orientifold
If we take k = 2, this is the orientifold case. Then we can compare the KK index with
the index on the CFT side in the large N limit. Actually, we can show that
AdS
CFT
IS(∞,k=2,p)= ISO(∞),(4.69)

where the right hand side is given by (4.30).
k = 3, 4, 6 S-foldIn this case, the index has a diﬀerent behavior from the index of N = 4 theories. In
N = 4 theories, there is the SU (4)R symmetry, and the index respects SU (3)R due tothe choice of the supercharge which is used to define the index. However, now, there is
only an N = 3 supersymmetry. Then the index respects U (2)R ⊂ U (3)R and the index
is represented by the U (2)R character χn (u−1 v, v −1 ) defined by (A.31). Indeed, the KK

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112

Chapter 4 Four-Dimensional S-fold Theories
13

indices for k = 3, 4, 6 up to q 2 become
57KKIS(∞,3,p)= 1 + uχ1 q 2 − uχJ1 q 2 + (−1 + u3 − uχ2 + χ3 )q 3 + χJ1 (2uχ1 − χ2 )q 2
1
+ (−uχJ2 − u + 2χ1 − u3 χ1 − χ2 + 2u2 χ2 )q 4
u91
+ χJ1 (−2 + u3 + χ1 − 2u2 χ1 − uχ2 + χ3 )q 2
u1
+ (χJ2 (2uχ1 − χ2 ) − + 3u2 − 2uχ1 + 2u4 χ1 − 2u2 χ3 + 2uχ4 )q 5u111
+ (−uχJ3 + χJ1 (3u − 2u4 + χ1 − u3 χ1 − χ2 + 5u2 χ2 − 3uχ3 ))q 2
u1
1+ (χJ2 (−2 + u3 + χ1 − 4u2 χ1 + χ3 ) + 1 − 3u3 + 2u6 − χ1 − 4u2 χ1
uu1+ 2 χ2 + 7uχ2 − 3u4 χ2 − 4χ3 + 4u3 χ3 − 2uχ5 + 2χ6 )q 6
u13
2
+ O(q ),
(4.70)

5

7

KK
IS(∞,4,p)= 1 + uχ1 q 2 − uχJ1 q 2 + (−1 − uχ2 )q 3 + 2uχJ1 χ1 q 2+ (−u + u4 − uχJ2 + χ1 + 2u2 χ2 + χ4 )q 4
9

+ χJ1 (−2 − 2u2 χ1 − uχ2 − χ3 )q 2
+ (2uχJ2 χ1 + 2u2 − 2uχ1 − u4 χ1 + χ2 −1
χ3 − 2u2 χ3 )q 5
u11
1
χ2 + 5u2 χ2 + χ4 ))q 2
u1+ (χJ2 (−2 − 4u2 χ1 − uχ2 − χ3 ) + u3 − χ1 − 4u2 χ1 + 2u5 χ1
u13
2
6+ 4uχ2 + 3u χ3 + 2uχ5 )q + O(q 2 ),+ (−uχJ3 + χJ1 (3u + u4 + χ1 +5

(4.71)

7

KK
IS(∞,6,p)= 1 + uχ1 q 2 − uχJ1 q 2 + (−1 − uχ2 )q 3 + 2uχJ1 χ1 q 2
9+ (−u − χJ2 + χ1 + 2u2 χ2 )q 4 + χJ1 (−2 − 2u2 χ1 − uχ2 )q 2
+ (2u2 − 2uχ1 + 2uχJ2 χ1 − 2u2 χ3 )q 5
11

+ (−uχJ3 + χJ1 (3u + χ1 + 5u2 χ2 ))q 2+ (χJ2 (−2 − 4u2 χ1 − uχ2 ) + u6 − 4u2 χ1 + 4uχ2 + 3u2 χ3 + χ6 )q 6
13

+ O(q 2 ).

(4.72)

The fact that the index is represented by the U (2)R character indicates that the corresponding theory has N = 3 supersymmetry. Namely, we can read oﬀ the supersymmetry

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4.5 Finite N corrections

113

of the theory from the characters of the R-symmetry.

4.5

Finite N corrections

Let us go to the topic of the finite N corrections to the BPS partition function andsuperconformal index of S-fold theories. First, we consider the finite N corrections to the
BPS partition function. After that, we move on to the discussion of the superconformal
index.

4.5.1

BPS partition function

We consider the finite N corrections to the BPS partition functions of S-fold theories
by considering the single wrapped D3-branes on S 3 /Zk ⊂ S 5 /Zk . Fortunately, we know
the BPS partition functions on the CFT side exactly, and we can compare the finite N
corrections with the CFT results. Note that wrapped D3-branes are only present when
the discrete torsion is trivial: p = 0, so we only consider the trivial torsion case here. The
discussions are based on the author’s and his collaborators’ paper [22].
In the last chapter, we have already done the analysis of finite N corrections for the
N = 4 U (N ) SYM, in which a generic configuration of the single wrapped D3-branes isgiven byf (X, Y, Z) = aX + bY + cZ = 0.

(4.73)

Then the complex coeﬃcients (a, b, c) form CP 2 as a phase space of single-wrapped D3branes. Actually, the following analysis is similar to N = 4 U (N ) SYM case.
Now, we have an additional constraint for the holomorphic surface f (X, Y, Z) = 0 in the
S-fold background. Since the overall factor does not aﬀect the shape of the holomorphic
surface, a Zk S-fold identification gives a constraint
f (ωk−1 X, ωk Y, ωk Z) = ωkm f (X, Y, Z),(4.74)

where an integer m is the winding number of wrapped D3-branes. This is classified by
H3 (S 5 /Zk , Z) = Zk , so m is an integer modulo k.
In fact, the constraint (4.74) restricts the possible form of the holomorphic function,that is, the possible shape of wrapped D3-branes, depending on the value of k. In what
follows, we consider k = 2 orientifold and k = 3, 4, 6 S-folds separately.
k = 2 orientifold
In this case, the most general form of the single-wrapping holomorphic surface satisfying
the constraint (4.74) is
aX + bY + cZ = 0.

(4.75)

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114

Chapter 4 Four-Dimensional S-fold Theories

This is actually the same as the AdS5 × S 5 case. Thus the discussion is almost the same.
We can expect that the single-particle BPS partition function has at least two zero modes
because the phase space is CP 2 . And the total BPS partition function may be obtainedby the summation over the X = 0, Y = 0, and Z = 0 configurations.
Let us check these expectations. The single-particle BPS partition function of the Z = 0
configuration can be derived by the projection (4.61) with k = 2 for (3.37):
x+y
Z2
iBPS,
= P2 iBPS
=
sp
Z=0
2
z(1 − q x2 )(1 − q 2 y 2 )(x y)++ O(q 2 ).
=zz

(4.76)

We found two zero modes. This is consistent with the fact that the phase space is CP 2 .
We have already analyzed this situation in the last chapter, and we know that in thiscase, the summation over the X = 0, Y = 0, and Z = 0 configurations give the total BPS
partition function. The BPS partition function of these configurations are related eachother by the Weyl completion (x, y, z) → (y, z, x) and (x, y, z) → (z, x, y).
A diﬀerent point from the AdS5 × S 5 case is the absence of a tachyonic mode. Thiscan be understood as the eﬀect of the orientifold. In the S 5 case, the configuration X = c
can exist, and this is the lower energy configuration than X = 0, where c is a nonzero constant. However, in the orientifold case, such configurations are forbidden by theorientifold projection. That is why a tachyonic mode is absent.
Now the finite N corrections to the BPS partition function of the single wrapping isgiven by
Z2
Z2
Z2
ZZD3
= (qx)N Pexp(iBPS,) + (qz)N Pexp(iBPS,) + (qz)N Pexp(iBPS,
),
X=0
Y =0
Z=0
2

(4.77)and the BPS partition function on the AdS side is
ZZAdS= ZZKK(1 + ZZD3),2
2
2(4.78)

where ZZKK= Pexp(I0Z2 − 1). ZZKKZZD3must correspond to the Pfaﬃan sector of the2
2
2
BPS partition function of N = 4 SO(2N ) SYM. We checked the following relation up to
N = 3:CFT
ZS(N,2,1)= ZZAdS+ O(q 2N +2 ).
2(4.79)

k = 3, 4, 6 S-folds
In fact, the constraint (4.74) restricts the possible form of the holomorphic function, thatis, the possible shape of wrapped D3-branes. Then the most general single wrapped

Soryushiron Kenkyu

4.5 Finite N corrections

115

D3-branes are represented by holomorphic surfaces
X = 0,(4.80)

bY + cZ = 0,(4.81)

and we cannot consider aX + bY + cZ = 0 because it does not satisfy (4.74). The first
configurations have the winding number m = −1, and the second one has m = 1.
First, let us see the configuration given by (4.80). The first task is to find the singleparticle BPS partition function on the wrapped D3-brane X = 0. Actually, this is very
easy to do. All we have to do is to perform the projection (4.61) to the single-particle
BPS partition function of X = 0:(qx)−1.
(1 − qy)(1 − qz)(4.82)

Thus we have
BPS, Zk
iX=0
= Pk

(

(qx)−1(1 − qy)(1 − qz))(qωk−1 x)−11 ∑=.
k m=0 (1 − qωk y)(1 − qωk z)
k−1(4.83)

Hence the total contribution is obtained by its plethystic exponential and the classicalcontribution (qx)N :
Zk
Zk
ZX=0
= (qx)N Pexp(iBPS,).
X=0(4.84)Let us check whether the tachyonic mode and zero modes exist or not for k = 3 as an
example. The expansion of (4.83) with k = 3 is
Z3
iBPS,
=
X=0

y 2 + yz + z 2(y 2 + yz + z 2 )q=
q + O(q 2 ).
3
3
3
3
x(1 − q y )(1 − q z )x(4.85)

Thus there are only zero modes. This is, in fact, the natural result because the presenceof configurations with energy smaller than X = 0 is forbidden by the constraint (4.74).
Second, we consider the configuration (4.81). Let us focus on the Y = 0 by taking(b, c) = (1, 0). Then the single-particle BPS partition function can be derived by the Weyl
completion (x, y, z) → (y, z, x) before the projection (4.61). The result is
Zk
iYBPS,= Pk=0((qy)−1(1 − qz)(1 − qx))(qωk y)−11 ∑=.
k m=0 (1 − qωk z)(1 − qωk−1 x)
k−1(4.86)Let us check the presence of tachyonic and zero modes in the single-particle BPS partition

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116

Chapter 4 Four-Dimensional S-fold Theories

function. The expansion of the k = 3 case as an example is
Z3
iBPS,
=
Y =0

z + x2 q + xz 2 q 2zx2= + q + O(q 2 ).
333
3y(1 − q z )(1 − q x )yy(4.87)

Now we found a zero mode. When we perform the plethystic exponential, this term withthe classical contribution gives a factor
( )
( N
)
(qy)N
z
N
(qy) Pexp=y + y N −1 z + · · · + yz N −1 + z N + y −1 z N +1 + · · · .
z =q
y
1− y
(4.88)N

This is evaluated in the region | yz | < 1. The infinite series includes unphysical terms.
However, they are canceled by the Z = 0 contribution in the same fugacity region | yz | < 1:(qz)N Pexp(y)z

= −q N (y −1 z N +1 + · · · ).

(4.89)

Then the summation over these terms gives()
q N y N + y N −1 z + · · · + yz N −1 + z N .

(4.90)

These terms are summarized into the U (2) character χN (y, z). This is an analogy of thefact that the summation over the X = 0, Y = 0, and Z = 0 configurations gave the U (3)character in the orientifold and S 5 cases. Here the summation over the contributionsof Y = 0 and Z = 0 configurations gives all the single wrapping contributions with thewinding number m = 1.
We have seen that the k = 3 case as an example. We can also do the same analysis fork = 4, 6 cases. Finally, we find the same expression of the final result as the orientifold
and S 5 case:
Zk
Zk
Zk
ZZD3
= (qx)N Pexp(iBPS,) + (qz)N Pexp(iBPS,) + (qz)N Pexp(iBPS,
),
X=0
Y =0
Z=0
k

(4.91)

where the first term gives rise to m = −1 sector, and other terms give m = 1 sector. We
can also check the following result up to N = 3.
CFT
ZS(N,3,1)= ZZAdS+ O(q 2N +1 ),k(4.92)CFT
ZS(N,4,1)= ZZAdS+ O(q 2N ),k(4.93)CFT= ZZAdS
ZS(N,6,1)+ O(q 2N ).
k(4.94)

Soryushiron Kenkyu

4.5 Finite N corrections
Modes
ϕ

(l)

χ(l)
ψ (l)
w(l)
F (l)
χ(l)

117
(R,R,RZ )[jL , jR ]E([ 2l ], 2l ,−1)[0, 0]l−1([ l ], l−1 ,− 1 )[0, 12 ]l−21 2 22([ l−1 ], l ,− 1 )[[ 12 ], 0]l− 12 2 22l−2 l−1
1 1 ([ 2 ], 2 ,0)[[ 2 ], 2 ]l([ l−2 ], l ,0)[0, 0]l 2 2([ l−2 ], l−1 , 1 )[0, 12 ]l+ 12 2 22YContribution0

q l−1 (uη −2 )2R v 2 −R+1 η −1

1

−q l (uη −2 )2R v 2 −R+ 2 η −1

−1

−q l− 2 χJ1 (uη −2 )2R v 2 −R+ 20

q l+ 2 χJ1 (uη −2 )2R v 2 −R

−2q l (uη −2 )2R v 2 −R η−1−q l+1 (uη −2 )2R v 2 −R− 2 η

l

l−1

1

1

l

1

l−1

1

l

l−1

1

Table 4.8 The BPS massless modes arising on the wrapped D3-brane. Summing allcontributions shown here, we find the refined single-particle index for the fluctuationsof the wrapped D3-brane Z = 0.

4.5.2

Superconformal index

Here we consider the finite N corrections to the superconformal index of S-fold theories.
Unlike the BPS partition function, there is no general formula to find the index of S-foldtheories due to the lack of the Lagrangian, so our analysis is the first predictions of theindex.
The strategy is almost the same as that of the BPS partition function. The only problem
we have to solve is to obtain the refined single-particle index of wrapped D3-branes for
X = 0, Y = 0, and Z = 0 configurations. Then all we have to do is to find the U (1)Ycharges for the fluctuation modes on wrapped D3-branes, which are listed in Table 3.3.
To find the U (1)Y charges for massless modes on wrapped D3-branes, let us focus on
the Z = 0 configuration. We know all the massless modes shown in Table 3.3 and thestructure of the supersymmetry multiplet shown in Fig. 3.1. So it is enough to find thell
U (1)Y charge of ϕ . Because ϕ has already appeared in the BPS partition function, its
transformation low is the Zk orbifold projection. It states that there is no U (1)Y chargelfor ϕ . From this fact, we can find the U (1)Y charge for all massless fields on Z = 0, as
shown in Table 4.8.
The refined single-particle index of the Z = 0 configuration, obtained from the information of Table 3.3 and Table 4.8, is then13v −1
q − (y + y −1 )q 2 − ( ηu2 + uv )q + (y + y −1 )q 2 + v(η −1 + η)q 2 − ηq 3
ηZ=0
eisp (η) =.
(1 − uη q)(1 − vηu q)(4.95)

Soryushiron Kenkyu

118

Chapter 4 Four-Dimensional S-fold Theories

The single-particle index of the Z = 0 configuration in a Zk S-fold background iskiZZ=0= PkeiZ=0
sp (η).

(4.96)For X = 0 and Y = 0 configurations, the refined single-particle index cannot be obtainedby the ordinary Weyl completion for the refined single-particle index due to the existenceof the η fugacity. This is because the corresponding generator to the η fugacity is −RX +RY +RZ −Y /2, and it does not commute with SU (3)R generators RX −RY and RY −RZ .
Instead of the SU (3)R character, we define the “refined SU (3)R character” by including
the η fugacity asχ′(m,n) (u, v, η) = tr[uRX −RY v RY −RZ η −RX +RY +RZ −Y /2 ].

(4.97)

Then the refined character corresponding to the representation to which (Q2 , Q3 , Q4 )belong isχ′(1,0) (u, v, η) =

u
v
1
+ + .
2
η
u v

(4.98)

This implies that our new Weyl completion should be defined as follows:
)
(
)
v 1 u
u v 1
, ,
→
, ,
,
Z=0→X=0:
η2 u vu v η2
)
(
)
(
1 u v
u v 1
Z=0→Y =0:
, ,
→
, ,
.
η2 u vv η2 u((4.99)(4.100)

By using this rule, we can obtain the refined single-particle index of X = 0 and Y = 0
configurations:
η −1

u
eiX=0
sp (η) =eiYsp=0 (η) =q− (y + y −1 )q 2 − ( uv + v1 )q + (y + y −1 )q 2 + u−1 (η + η 3 )q 2 − ηq 3,
η(1 − vη
u q)(1 − v q)
(4.101)
1

3

1
3
u −1
− (y + y −1 )q 2 − ( v1 + ηu2 )q + (y + y −1 )q 2 + uv (η −1 + η)q 2 − ηq 3vη q.
(1 − ηv q)(1 − uη q)(4.102)In what follows, we see the orientifold case and k = 3, 4, 6 S-fold cases separately.
k = 2 orientifold
Actually, the flow of the discussion is the same as the BPS partition function. The mostgeneral configuration of the single wrapping is aX + bY + cZ = 0, and the summation over
X, Y, Z = 0 gives the SU (3)R character corresponding to the fact that the phase space isSoryushiron Kenkyu

4.5 Finite N corrections

119

CP 2 . Then the index of the wrapped D3-branes is given by
IZD3= (qu)N Pexp(P2eiX=0
sp (η)) +2( )N( v )N1
Y =0eq
Pexp(P2 isp (η)) + q
Pexp(P2eiZ=0
sp (η)).
uv(4.103)

Then the superconformal index on the orientifold background is
AdS
IS(N,2,0)= IZKK(1 + IZD3),2
2(4.104)

where IZKK
IZD3should correspond to the Pfaﬃan sector on the CFT side. We checked the22following relations up to N = 3:
CFT
AdS
2N +1
ISO(2N).
) = IS(N,2,0) + O(q(4.105)k = 3, 4, 6 S-folds
Again, the procedure to find the finite N corrections to the index is the same as the BPS
partition function. In this case there is two winding sectors m = −1 and m = 1. The firstone and second one correspond to X = 0 and bY + cZ = 0 configuration, respectively.
Since the phase space of m = 1 sector is CP 1 , the single-particle index for Y = 0 and
Z = 0 has the corresponding zero mode, and summation over two configurations of m = 1
gives the U (2)R character χn (u−1 v, v −1 ) defined in (A.31). This fact reflects an N = 3
supersymmetry of S-fold theories for k = 3, 4, 6.
Our formula of the finite N corrections for a Zk S-fold background is now given by
AdS
IS(N,k,0)= IZKK(1 + IZD3
),
k
k

IZD3
= (qu)N Pexp(PkeiX=0
sp (η)) +k(( v )N1
Pexp(PkeiYsp=0 (η)) + q
q
u
v

)N

(4.106)
Pexp(PkeiZ=0
sp (η)).
(4.107)This formula gives the prediction of the index for S-fold theories up to an order fromwhich the double wrapping starts.
Let us make sure that the index is indeed written by the U (2)R character as evidenceof an N = 3 supersymmetry. We take up S(3, 3, 0) as an example here. Then our formulaSoryushiron Kenkyu

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Chapter 4 Four-Dimensional S-fold Theories

gives
5

AdS
IS(3,3,0)= 1 + uχ1 q 2 − uχJ1 q 2 + (−1 + 2u3 − uχ2 + 2χ3 )q 3

+ χJ1 (2uχ1 − 2χ2 )q 2 + (−u − uχJ2 + 3χ1 − 2u3 χ1 − 2u−1 χ2 + 4u2 χ2 )q 4
91
+ χJ1 (−2 + 2u3 + 2 χ1 − 4u2 χ1 − uχ2 + 2χ3 )q 2
uJ
+ (χ2 (2uχ1 − 2χ2 ) + 6u2 − 4uχ1 + 4u4 χ1 − 4u2 χ3 + 4uχ4 )q 57

+ (5uχJ1 − 4u4 χJ1 − uχJ3 − χJ1 χ1 − 2u3 χJ1 χ1 − 2u−1 χJ1 χ2
11+ 9u2 χJ1 χ2 − 6uχJ1 χ3 )q 2

+ (2 − 6u3 + 4u6 + 2u3 χJ2 − 4u−1 χ1 − 5u2 χ1 + 2u−1 χJ2 χ1− 6u2 χJ2 χ1 + 3u−2 χ2 + 14uχ2 − 6u4 χ2 + uχ2J χ2 − 8χ3+ 8u3 χ3 + 2χJ2 χ3 + u−1 χ4 − 4uχ5 + 4χ6 )q 6
+ (6u−1 χJ1 + 3u2 χJ1 + u2 χJ3 − 3u−2 χJ1 χ1 − 17uχJ1 χ1+ 12u4 χJ1 χ1 + 2uχJ3 χ1 + 16χJ1 χ2 − 12u3 χJ1 χ2 − 2χJ3 χ2− 4u−1 χJ1 χ3 − 6u2 χJ1 χ3 + 12uχJ1 χ4 − 5χJ1 χ5 )q 2 + O(q 7 ).
13

(4.108)This index is never unified into the SU (3)R character. Thus a theory S(3, 3, 0) indeed has
an N = 3 supersymmetry.

4.6

Consistency check

Let us check the consistency with some known results. We can do two checks. The first one
is to check the supersymmetry enhancement in terms of the index. If the enhancement
occurs, the index should be unified into the SU (3)R characters rather than the U (2)Rcharacter. The second check is to compare our result of S(3, 3, 0) with Zafrir’s result
calculated recently. He proposed an N = 1 model is dual to S(3, 3, 0), and calculated theindex of this model. So his calculation should agree with our prediction given in (4.108).

4.6.1

Supersymmetry enhancement

First, let us consider the supersymmetry enhancement of rank one and two S-fold theories.
It is expected from the general discussion of an N = 3 superconformal algebra, andAharony and Tachikawa conjectured that the rank one and two S-fold theories with k =3, 4, 6 are equivalent to the N = 4 SYM with the gauge group U (1), SU (3), SO(5) andG2 .
Here we focus on the S(2, 6, 0) as an example. It is expected that this theory is equivalent

Soryushiron Kenkyu

4.6 Consistency check121

to the N = 4 G2 SYM. Our formula (4.107) gives the following amazing result:
AdS
IS(2,6,0)= 1 + (u2 + uχ1 + χ2 ) q 2 − χJ1 (u + χ1 ) q 2 − (u−1 χ1 + u2 χ1 + uχ2 ) q 3|{z}| {z }|{z}5

χ(2,0)

χ(1,1) −1χ(1,0)+ χJ1 (u−1 + u2 + 2uχ1 + χ2 ) q
|
{z
}

7
2

χ(0,1) +χ(2,0)+ (−uχJ2 + χ1 + u3 χ1 − χJ2 χ1 + 2u2 χ2 + uχ3 )q 4
− χJ1 (2 + u3 + u−1 χ1 + 3u2 χ1 + 3uχ2 )q 2
9

+ (−u−1 + u−1 χJ2 + u2 χJ2 + 3uχJ2 χ1 − 3χ2 − 2u3 χ2 + χJ2 χ2 − 2u2 χ3 − uχ4 )q 5
11

+ ((3u + 7χ1 + 4u3 χ1 + 5u2 χ2 + 4uχ3 )χJ1 − (u + χ1 )χJ3 )q 2

+ (−4 − 2u3 + u6 − 5χJ2 − 2u3 χJ2 + (3u−1 + u5 − u−1 χJ2 − 5u2 χJ2 )χ1
13+ (−u + 2u4 − 6uχJ2 )χ2 + (2 + 4u3 )χ3 + 2u2 χ4 + uχ5 + χ6 )q 6 + O(q 2 ).
(4.109)The first few terms are indeed summarized into the SU (3)R characters. We expect that the
double wrapping starts from q 4 because the double wrapping has an order of 2N (N = 2)
energy. If we take into account the multiple wrapping, the index would be written interms of SU (3)R characters more. Note that the summarizing way is quite non-trivial.
In the level of the phase space of holomorphic surfaces, N = 4 supersymmetry could notappear. However, the enhancement is indeed present. We can also show that
AdS
CFT
IS(2,6,0)= IG+ O(q 4 ).
2

(4.110)Thus, we confirmed the Aharony-Tachikawa conjecture in the terms of the superconformal
index.
We also confirmed the following relations:
3
AdS
IS(1,3,0)= IUCFT(1) + O(q ),(4.111)AdS
2IS(1,4,0)= IUCFT(1) + O(q ),(4.112)AdS
2IS(1,6,0)= IUCFT(1) + O(q ),(4.113)AdS
CFT
5IS(2,3,0)= ISU(3) + O(q ),(4.114)AdS
CFT
IS(2,4,0)= ISO(5)+ O(q 4 ).

(4.115)All results are consistent with the emergence of the double wrapping.

4.6.2

Comparison with Zafrir’s result

Very recently, Zafrir discovered certain Lagrangian theories that flow to N = 3 SCFTs
with a free chiral field [21]. Since the superconformal index is an RG flow invariant, itSoryushiron Kenkyu

122

Chapter 4 Four-Dimensional S-fold Theories

is possible to calculate the index of the system, including an N = 3 SCFT and a freefield. Then we need to remove the latter contribution. This can be done by using theprocedure given in [43]. Note that the procedure given by Zafrir is not applicable to all
N = 3 theories. He showed the cases of S(3, 1, 1) and S(3, 3, 0). The latter case is therank 3 Z3 S-fold theory with the trivial torsion, so we compare our result (4.108) with theresult given in [21]. The Zafrir’s result of the S(3, 3, 0) theory is given by [21]
1

2

2

4

1

IZafrir =1 + 2(pq) 3 − (pq) 3 (p + q) + 6pq − 2(pq) 3 (p + q) + 6(pq) 3 − (pq) 3 (p2 + q 2 )
5

2

4

1

+ (pq)(p + q) + 8(pq) 3 − 2(pq) 3 (p2 + q 2 ) − 9(pq) 3 (p + q) − (pq) 3 (p3 + q 3 )52+ 29(pq)2 + 5(pq)(p2 + q 2 ) − 6(pq) 3 (p + q) − (pq) 3 (p3 + q 3 ) + · · · .

(4.116)This is written in the “standard notation” [44,45]. In order to compare it with (4.108), we
need the following replacement from the standard notation (p, q) to our notation (q, y, u, v)
with u = v = 1:3

p → q 2 y,

q → q 2 y −1 .
3

(4.117)

Then we find the agreement
IS(3,3,0) |u=v=1 = IZafrir + O(q 6 ),(4.118)where IS(3,3,0) is given in (4.108). The error is consistent with the double wrapping contributions. This is the first non-trivial confirmation of our predictions for S-fold theories
and ensures the correctness of our calculations of the finite N corrections to the index for
S=fold theories.

Soryushiron Kenkyu

123

Chapter 5Conclusions and Discussions
In this chapter, we give the conclusions of this thesis and the discussions.
In this thesis, we developed the calculation method of the superconformal index on the
AdS side in the finite N region of the AdS/CFT correspondence. Then we applied our
method to the S-fold theories whose Lagrangian is not known. For this purpose, we started
from the review of basic concepts of the partition function, Witten index, superconformal
symmetry, superconformal index, and the BPS partition function in Chapter 1.
In Chapter 2, we reviewed N = 4 SYM and type IIB string theory to discuss the
AdS/CFT correspondence between these theories. We also gave the concrete expressionsof the superconformal index and the BPS partition function of N = 4 U (N ) SYM. Thenwe gave the precise statement of the AdS/CFT correspondence. We also performed thecalculation of the index and the BPS partition function on the AdS side in the large N
limit and confirmed the agreement with those on the CFT side.
In Chapter 3, we studied the finite N corrections to the BPS partition function andthe superconformal index. It is expected that D3-branes wrapped on three-cycles in S 5give the finite N corrections [17, 19, 29]. Then we analyzed the single wrapped D3-braneby considering massless fields living on the wrapped D3-brane. First, we considered thefinite N corrections to the BPS partition function to study the structure of the finite N
corrections on the AdS side. We calculated the single-particle BPS partition function of
three single wrapped D3-branes whose configurations are X = 0, Y = 0, and Z = 0. We
found that there are a tachyonic term and two zero modes. The tachyonic term is thenegative power of the fugacity q, and we needed the special treatment as
Pexp(q −1 ) =q
1=−= −qPexp(q).
1 − q −1
1−q

(5.1)

The zero modes are related to the U (3)R symmetry and have the unphysical infinite
contributions. These contributions are canceled each other by summing the contributions
from X = 0, Y = 0, and Z = 0. The significant observation is that the summationof these three contributions is just the Weyl character formula. Then we proposed theSoryushiron Kenkyu

124

Chapter 5 Conclusions and Discussions

formula which gives the finite N corrections to the BPS partition function as
AdS
AdS
D3
D3
D3
2N +5
Zfinite).
N = Zlarge N (1 + ZX=0 + ZY =0 + ZZ=0 ) + O(q(5.2)

We confirmed that results obtained by this formula agree with the results on the CFT side
up to q 2N +5 . It is expected that the error comes from the double wrapping contributions.
After the computation of the BPS partition function, we considered the superconformal
index. The same technique as the case of the BPS partition function can be applied, andwe proposed a similar formula to (5.2):
AdS
AdS
D3
D3
D3
2N +4
Ifinite).
N = Ilarge N (1 + IX=0 + IY =0 + IZ=0 ) + O(q(5.3)

We also confirmed that the results given by this formula on the AdS side agree with theresults on the CFT side up to the expected order of the fugacity q.
In Chapter 4, we investigated the orientifold theories and the S-fold theories. First, we
reviewed the orientifold in type IIB string theory, and after that, we defined the S-foldaction and the S-fold theories. We constructed the BPS partition functions for S-foldtheories by using the oscillator formalism that we developed in Chapter 2. Moreover, weapplied our formula (5.3) with the minor modification due to the S-fold invariance. In [20],it is expected that the rank one S-fold theories are equivalent to N = 4 U (1) Maxwell
theory and rank two S-fold theories for k = 3, 4, 6 are equivalent to N = 4 SYM withSU (3), SO(5), and G2 gauge groups, respectively (Aharony-Tachikawa (AT) conjecture).
We confirmed the agreement of the index for these S-fold theories with the index for theseN = 4 theories. The UV Lagrangian for S(3, 3, 0) S-fold theories was found, and theindex of this theory was calculated by Zafrir [21]. So we finally confirmed the agreementof our prediction with the result given in [21].
There are many unsolved problems. First, we have no clear physical explanation of thetreatment of the tachyonic mode (5.1). To ensure the convergence of the left-hand side of(5.1), we need the condition q > 1, while we need q < 1 for the right-hand side of (5.1).
Up to now, all we can do is to regard this treatment as the analytic continuation. We
hope that there is a certain physical meaning of this treatment.
Second, the multiple wrapping contributions are not included in the formula (5.3). For
AdS/CFT regarding N = 4 U (N ) SYM, we expect that contributions of the n-ple wrapped
D3-branes may be obtained by considering the U (n) gauge theory on the wrapped D3branes. However, it is necessary to perform the gauge integral like (3.76), and the pole
selection rule of this integral is unclear.
If we consider the Schur limit [46] of the superconformal index defined by
ISchur (q, u) =lim
1y→q − 2 ,v→1

I(q, y, u, v),(5.4)

the structure of the gauge integral becomes much simpler. The index (5.4) is called the
Schur index. Then the analysis of the multiple wrapped D3-branes is simplified, and weSoryushiron Kenkyu

125
performed the analysis in the author’s and his collaborators’ paper [47]. Our analysis
shows that there are “the intersection string contributions” as well as the multiple wrapping contributions. Note that in the Schur limit, the contributions from the configurations
including Z vanish. Let us consider the configuration X n−k Y k = 0. Then we have theU (n − k) × U (k) gauge theory. The intersection string between X n−k = 0 and Y k = 0
corresponds to the bi-fundamental field of the U (n − k) × U (k) theory. Roughly speaking,the contributions of the n-ple wrapped D3-branes is thenD3
In,N
= q nNn ∫∑
k=0

U (n−k)×U (k)U (n−k)dµPexp(iX=0
sp χadjU (k)+ iYsp=0 χadj + iintsp χbi-fund ),(5.5)

where iint
sp is the single-particle index of the contributions from the intersection string. We
confirmed that the results up to n = 4 completely agree with the CFT results [47]. We
expect that the same structure holds even for the superconformal index.
There are also various extensions of our analysis [22] that are not contained in thisthesis. These extensions were given by the author’s and his collaborators’ papers [48–50].
The significant benefit of our method to compute the finite N corrections is that it
is applicable for any AdS/CFT in principle. The simplest extension is the quiver gauge
theories realized on D3-branes put on Abelian orbifolds. AdS/CFT for orbifold S 5 /Γwas suggested in [51, 52]. Thus we can apply our formula to this AdS/CFT to calculate
the superconformal index. Since the quiver gauge theories have the explicit Lagrangian,the index is calculable. That is, our task is the confirmation of the agreement of theindex on both sides. In particular, the D3-branes wrapped on the non-trivial threecycles in S 5 /Γ correspond to the baryonic operators defined as operators including thedeterminant operator. This situation is quite similar to the S-fold theories. Then, thewrapped D3-brane contributions may be obtained by an appropriate projection of I D3
associated with the orbifold. In fact, our analysis with the orbifold projection does work,and the agreement up to the double wrapping contributions was confirmed in [48].
We also generalized the analysis of the orbifold quiver gauge theories to the toric quivergauge theories [49]. The toric quiver gauge theories are realized on D3-branes put on atoric Calabi-Yau (CY) three-fold. The realized gauge theory depends on toric CY threefolds, and there is a systematic prescription [53, 54] to determine the gauge theory fromthe toric data of the CY three-fold. The corresponding AdS dual is given by type IIB
string theory on AdS5 × SE5 , where SE5 is the five-dimensional Sasaki-Einstein manifold.
The simple example is the Klebanov-Witten theory [55] and its AdS dual is type IIB string
theory on AdS5 × T 1,1 . The index is also calculable for toric quiver gauge theories, so
the task is also the confirmation of the agreement of the index on both sides. Note thatN = 4 U (N ) SYM and the orbifold quiver gauge theories are simple examples of toricquiver gauge theories. Thus we can guess the relationship between our method presented
in this thesis and the toric data. In fact, we proposed the generalization of our method tothe toric CY cases and showed that all the results are consistent with the CFT results [49].
Our method is also applicable in other dimensions. In [50], we analyzed theSoryushiron Kenkyu

126

Chapter 5 Conclusions and Discussions

three-dimensional Aharony-Bergman-Jaﬀeris-Maldacena (ABJM) theory [56] and thesix-dimensional (6d) N = (2, 0) theories. The ABJM theory is realized on M2-branes inM-theory, while 6d N = (2, 0) theories are realized on M5-branes. The corresponding
AdS duals are M-theory on AdS4 × S 7 and AdS7 × S 4 . We calculated the finite N
corrections to the index as contributions from M5-branes in S 7 . Because the Lagrangianof the ABJM theory is known, and we confirmed the agreement of the index on bothsides. On the other hand, the Lagrangian of 6d N = (2, 0) theories are unknown, so we
predicted the index from the contributions of M2-branes in S 4 . For the rank one theory(realized on one M5-brane), it is known that the theory is free, and we can calculate theindex. In this case, we confirmed the agreement of the index on both sides.
All of these analyses show that our method is very powerful in calculating the index forvarious AdS/CFTs.
There are various future directions. Up to now, our results of S-fold theories are still
the prediction except for few examples. Thus the check of our results is important.
A way to do it is to use the N = 2 SCFT/chiral algebra correspondence [57]. This
correspondence comes from the fact that the four-dimensional N = 2 superconformal
algebra contains the two-dimensional chiral algebra. Concretely, the Schur index of an
N = 2 SCFT is equivalent to the vacuum character of the corresponding two-dimensional
chiral algebra [57]. This means that if we identify the chiral algebra corresponding to
S-fold theories, we may be able to calculate the Schur index of S-fold theories from thevacuum character. Our prediction for the Schur index is also given in [22], so we cancompare these results. Note that the chiral algebra corresponding to rank one S-foldtheories were already given in [58]. However, the rank one S-fold theory with the trivial
torsion is the free theory, and we need the chiral algebras corresponding to higher rank
S-fold theories.
Another future direction is to apply our method to AdS/CFT regarding AD theories.
Some of the AdS dual to AD theories are given in [59, 60]. Then we may be able tocalculate the finite N corrections to the index as well as the contributions of KK modes.
Another interesting example is AdS/CFT regarding TN theories [61]. The superconformal
index of TN theories has not been given yet, and we may be able to calculate the index
for these theories by using their AdS dual given in [62].

Soryushiron Kenkyu

127

Appendix A

Conventions and Notations
In this appendix we summarize the spinor conventions for four-dimensional theories andthe notations for group theory.

A.1

Spinor convention for four-dimensional theories

In this section we would like to explain our spinor notation. Our spinor convention isalmost the same as that used in [63].
In four-dimensional (4d) theories, the four-vector xµ (µ = 0, 1, 2, 3) is the position
vector. The flat Minkowski metric is denoted by ηµν . We use the following convention:ηµν = diag(−1, 1, 1, 1).

(A.1)

The 4d gamma matrices γ µ satisfy the Cliﬀord algebra{γ µ , γ ν } = −2η µν .

(A.2)

Let ψα be a left-handed two component Weyl spinor and let χα̇ be a right-handed
two-component Weyl spinor. The Hermite conjugate of Weyl spinors are(ψα )† = ψ α̇ ,(χα̇ )† = χα .

(A.3)

The contractions of spinors are given byψξ = ψ α ξα ,α̇χζ = χα̇ ζ .

(A.4)

The anti-symmetric epsilon tensor raises and lowers the spinor indices asψ α = ϵαβ ψβ ,

ψα = ϵαβ ψ β ,(A.5)

χα̇ = ϵα̇β̇ χβ̇ ,χα̇ = ϵα̇β̇ χβ̇ ,(A.6)

Soryushiron Kenkyu

128

Appendix A Conventions and Notations

where the anti-symmetric epsilon tensor satisfies
ϵ12 = ϵ21 = −1,(A.7)

ϵ12 = ϵ21 = 1(A.8)

ϵαβ ϵβγ = ϵγβ ϵβα = δαγ ,(A.9)ϵα̇β̇ ϵβ̇ γ̇ = ϵγ̇ β̇ ϵβ̇ α̇ = δα̇γ̇ .

(A.10)

The four-Pauli matrices σ µ and σ µ are defined by(σ µ )α̇α = (1, −σ i ),(σ µ )αα̇ = (1, σ i ),(A.11)

where
(
0
σ =
1
1

)
1
,
0

(
2

σ =

)
0 −i
,
i 0

(
3

σ =

)
1 0
.
0 −1

(A.12)

The four-Pauli matrices satisfy(σ µ )αα̇ (σ µ )β̇β = −2δαβ δα̇β̇ .

(A.13)The gamma matrices γ µ are related with the four-Pauli matrices as
(
µ

γ =

0
σµ

)
σµ
.
0

(A.14)

It is convenient to define the anti-symmetric gamma and four-Pauli matrices as
( µν
)
1 µ ν
σ
0
γ = [γ , γ ] =
,
0
σ µν
2
1
β
β
(σ µν )α = (σ µ σ ν − σ ν σ µ )α ,21
(σ µν )α̇ β̇ = (σ µ σ ν − σ ν σ µ )α̇ β̇ .
2µν

(A.15)(A.16)(A.17)

The four-component Dirac spinor ΨD is made of a left-handed Weyl spinor ψα and aright-handed Weyl spinor χα̇ as
( )
ψα
ΨD =
.
χα̇

(A.18)The Dirac conjugate ΨD is defined asΨD = Ψ†D γ 0 .

(A.19)

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129

The Majorana condition is defined by
(
−ϵαβ
C=
0

ΨD = ΨTD C,

0−ϵα̇β̇

)= −iγ 2 γ 0 .

(A.20)The Majorana spinor satisfying the Majorana condition is
(
ΨM =

ψα

ψ

α̇

)
.

(A.21)The Dirac and Majorana kinetic terms are given by
LD = iΨD γ µ ∂µ ΨD ∼ iψσ µ ∂µ ψ + iχσ µ ∂µ χ,(A.22)iΨM γ µ ∂µ ΨM ∼ iψσ µ ∂µ ψ,2(A.23)

LM =where the symbol “∼” means that we neglect total derivatives.

A.2

Notations of group theory

Here we would like to define and fix the notations for group theory. We only consider Liegroups.

A.2.1

Dynkin label

Let us consider a Lie group G with simple roots αj (j = 1, · · · , rankG). Let µj be thefundamental weight satisfying
2αj · µk
= δjk .
(αj )2(A.24)

Then the highest weight of an arbitrary irreducible representation of a Lie group G canalways be written as
µ=

rankG
∑

ℓj µj ,

(A.25)

j=1

where ℓj is a non-negative integer called the Dynkin label. Thus we can label an irreducible
representation by the Dynkin label. We often use the notation(ℓ1 , · · · , ℓrankG )to express an irreducible representation with the highest weight µ =(A.26)∑j j
j ℓ µ of G.

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A.2.2

Appendix A Conventions and Notations

Character formula

A character χR (u) of a representation R of a Lie group G is defined byχR (u) = trR 

uj j  ,

rankG
∏

H

(A.27)

j=1

where Hj is the Cartan generator of G. There is a formula to calculate the character fora given representation R with the highest weight µ. It is given by∑σ(µ+ρ)σ∈W sign(σ)uχR (u) = ∏
,
α
−α
2
2 )
α∈R+ (u − u(A.28)

where R+ is a set of positive roots and W is a set of Weyl reflections. ρ is defined by
ρ=

1 ∑
α.
2

(A.29)α∈R+

We used the vector notation
uµ =

rankG
∏

µ

uj j .

(A.30)

j=1

The formula (A.28) is called the Weyl character formula.
For convenience, we show the Weyl character formula for U (2) and U (3). The U (2)character χn (a, b) is given by
χn (a, b) =an+1 − bn+1.
a−b(A.31)The U (3) character χ(m,n) (a, b, c) is given byam+1
abc
χ(m,n) (a, b, c) = −bm+1(a − b)(b − c)(c − a) m+1c

1 a−n−1
1 b−n−1 .
1 c−n−1(A.32)

Then the SU (2) character χn (u) and the SU (3) character χ(m,n) (u, v) are given by
χn (u) = χn (u, u−1 ),χ(m,n) (u, v) = χ(m,n) (u, u−1 v, v −1 ).

(A.33)(A.34)

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