Some aspects of abelian and nonabelian T-duality and the gauge/gravity correspondence

178 Pages·2016·1.95 MB·English

(cid:0)(cid:44)(cid:44)
(cid:104)(cid:40)(cid:104)(cid:40)(cid:88)(cid:80)(cid:80)(cid:88)(cid:104)(cid:40)(cid:16)(cid:72)(cid:72)(cid:80)(cid:88)(cid:104)(cid:40)(cid:16)(cid:8)(cid:44)(cid:17)(cid:8)(cid:101)(cid:64)(cid:108)(cid:81)(cid:10)(cid:37)(cid:0)(cid:10)(cid:10)(cid:37)(cid:8)(cid:44)(cid:17)(cid:108)(cid:81)(cid:0)(cid:37)(cid:81)(cid:17) IFT InstSiªtuotPeafourloTShteaotreeUticnaivlePrhsiytsyics
(cid:101)(cid:101)(cid:64)(cid:64)(cid:108)
Ph.D. Thesis IFT(cid:21)D.001/16
Some aspects of abelian and nonabelian T-duality
and the gauge/gravity correspondence
Thiago Rocha Araujo
Ph.D. Advisor:
Horatiu Stefan Nastase
February 2016
to Jeane & David
i
(cid:16)The fundament upon which all our knowledge and learning rests is the
inexplicable.(cid:17)
Arthur Schopenhauer
ii
Acknowledgements
I would like to thank my advisor Horatiu Nastase; his support was fundamental in my PhD
studiesandmostofthetopicsofthisthesisIlearnedfromhim. Inmy(cid:28)rstyearasaPhDstudent,I
usedtothinkthatinordertomakesomeoriginalcontributiontothis(cid:28)eldIshouldknoweverything
about string theory, but Horatiu told me that it is always wiser to pick a small problem and my
knowledge on the subject will grow around the problem. It was a great advice and now I know
that in every aspect of life, we can apply a version of this advice. . . be calm.
The IFT is an amazing place to make theoretical physics, and I could learn a fraction of it
from great physicists, in particular, professor Nathan Berkovits, JosØ Fracisco Gomes, Abraham
Zimerman and Ricardo D’Elia Matheus, I would like to thank all of them. Furthermore, I thank
all the people of IFT that helped me all these years, Rosane, Luzinete, Neila, Rodrigo and Helen,
just to name some of them. Also, I would like to thank the CNPq for funding support and the
Brazilian taxpayers.
Ihavetothankallmyfriends. InparticulartoprofessorStoianZlatev,whohelpedmetostudy
high energy physics when I was an undergraduate student, and to Prieslei Goulart, Guilherme
Martins, Reginaldo, Cristian, Paulo, Wesley and Jonas. We had good moments talking about
physics, heavy metal, religion, politics and everything else, serious and silliness, there were no
prejudice in our conversations.
I would like to thank all my family. My cousin Fabio and my sisters Vanessa and Julia for
the friendship. My grandma and my aunts Maria, Lucia and Luciene were very important in my
education. Also,Iwanttothankmyparents-in-lawTiagoandIzabelandmybrother-in-lawAlison.
Thank you all.
Iamgratefultomyparents,JeaneandDavid,forlovingme. WhenIwasakidmyparentsused
to explain me several things about the nature: that it is impossible to reach the light speed, how
gravity works, the mysterious numbers smaller than zero and so forth. We watched great science
(cid:28)ction movies together: The (cid:29)ight of the navigator and Contact, made me wanted to understand
why the time changes when we travel near the speed of light. These small things developed my
taste for physics. I dedicate this thesis for you two.
Finally, my beloved wife Aline made my life easier. Thank you for the love, the patience and
friendship. You are the most important part of my life and you made me a better man. I really
love you. . . and seven years later I can still quote Nietzsche: (cid:16)What is done out of love always
takes place beyond good and evil.(cid:17)
iii
Abstract
In this thesis we study properties of type II supergravity solutions generated by
abelian and nonabelian T-duality. Also we determine, through the gauge/gravity
conjecture, some aspects of the (cid:28)eld theory dual to the supergravity solutions ob-
tained by T-dualization. We consider three distinct types of backgrounds solutions,
namely, backgrounds that are dual to con(cid:28)ning (cid:28)eld theories, backgrounds dual to
conformal (cid:28)eld theories and those dual to nonrelativistic (cid:28)eld theories. We conclude
this thesis with an analysis of Wilson loops on backgrounds with nonrelativistic
symmetries.
Keywords: AdS/CFT; gauge/gravity; supergravity; string theory.
Areas: High Energy Physics - Theory.
iv
Resumo
Nessa tese estudamos propriedades de solu(cid:231)ıes de supergravidade tipo II obtidas
atravØs da dualidade T abeliana e nªo abeliana. TambØm determinamos, atravØs
da conjectura gauge/gravidade, aspectos da teoria de campos dual a essas solu(cid:231)ıes
obtidas por dualidade T. Consideramos trŒs tipos distintos de solu(cid:231)ıes: duais a
teorias de campos que con(cid:28)nam, duais a teoria de campos conformes e duais a
teoria de campos nªo-relativistas. Concluimos essa tese com uma anÆlise dos la(cid:231)os
de Wilson em solu(cid:231)ıes com simetria nªo relativista.
Palavras Chaves: AdS/CFT; gauge/gravidade; supergravidade; teoria de cordas.
`reas do conhecimento: F(cid:237)sica de Altas Energias - Teoria.
v
Disclaimer
Chapters 3,4,5 and 6 of this thesis are based, respectively, on the following
research papers
1. Araujo, Thiago R. and Nastase, Horatiu Comments on the T-dual of the
gravity dual of D5-branes on S3, JHEP, 04, 2015, 081,
doi=10.1007/JHEP04(2015)081, [hep-th/1409.7333];
2. Araujo, Thiago R. and Nastase, Horatiu N = 1 SUSY backgrounds with
an AdS factor from non-Abelian T duality, Physical Review, D91, 2015, 12,
doi=10.1103/PhysRevD.91.126015, [hep-th/1503.00553];
3. Araujo, Thiago R. and Nastase, Horatiu, Non-Abelian T-duality for non-
relativistic holographic duals, JHEP, 1511, 2015, 203,
doi=10.1007/JHEP11(2015)203, [hep-th/1508.06568];
4. Araujo, Thiago R., Revisiting Wilson loops for non-relativistic backgrounds,
Physical Review, D92, 2015, 12,
doi=10.1103/PhysRevD.92.126007, [hep-th/1509.02011].
vi
Contents
1 Introduction 1
2 Dualities 6
2.1 Dualities in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Dualities in QFT and String Theory . . . . . . . . . . . . . . . . . . 7
2.2.1 Quantum (cid:28)eld theory dualities . . . . . . . . . . . . . . . . . . 9
2.2.2 String theory dualities . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 String-QFT duality . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Nonabelian T-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Nonabelian T-duality action on RR (cid:28)elds . . . . . . . . . . . . 28
3 D5-branes on S3 30
3.1 Wrapped (cid:28)vebranes on a three-cycle . . . . . . . . . . . . . . . . . . . 32
3.2 Deformed Maldacena-Nastase solution . . . . . . . . . . . . . . . . . 34
3.3 D4-brane solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 NS-NS sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 R-R sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.3 Brane charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Field theory aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.1 Wilson loops . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.2 Gauge coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.3 Nonlocality and entanglement entropy . . . . . . . . . . . . . 51
3.4.4 Domain walls . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 N = 1 AdS Backgrounds 59
4.1 Warped AdS solution . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5
vii
4.1.1 Nonabelian T-dual model . . . . . . . . . . . . . . . . . . . . 62
4.2 Flowing from AdS to AdS . . . . . . . . . . . . . . . . . . . . . . . 68
5 3
4.2.1 AdS solution and its nonabelian T-dual . . . . . . . . . . . . 69
3
4.2.2 Domain Wall and its nonabelian T-dual . . . . . . . . . . . . 73
4.3 Dual conformal (cid:28)eld theories, central charges and RG (cid:29)ow . . . . . . 75
4.3.1 Page charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.2 Central charges . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Nonabelian T-duality for nonrelativistic holographic duals 82
5.1 Nonabelian T duality revisited . . . . . . . . . . . . . . . . . . . . . . 84
5.2 Galilean Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.1 Galilean type-IIA solution and its NATD . . . . . . . . . . . . 88
5.2.2 Galilean solution in massive type-IIA and its NATD . . . . . . 95
5.3 Lifshitz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 Homogeneous Space T(1,1) . . . . . . . . . . . . . . . . . . . . 99
5.3.2 Sasaki-Einstein Space . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Holographic Dual Field Theory . . . . . . . . . . . . . . . . . . . . . 105
5.4.1 Quantized Charges for Galilean Solutions . . . . . . . . . . . . 105
5.4.2 Quantized Charges for Lifshitz Solutions . . . . . . . . . . . . 109
5.4.3 Wilson Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6 More on Wilson loops for nonrelativistic backgrounds 119
6.1 Short review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1.1 Quark-antiquark system . . . . . . . . . . . . . . . . . . . . . 120
6.1.2 Drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Schr(cid:246)dinger backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.1 Constant compact direction . . . . . . . . . . . . . . . . . . . 122
6.2.2 Nonconstant compact direction . . . . . . . . . . . . . . . . . 125
6.3 Lifshitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7 Conclusions 133
A Type II superstring 136
A.1 Highlights on the RNS and GS formalisms . . . . . . . . . . . . . . . 136
A.1.1 RNS formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 136
viii
A.1.2 GS formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.2 Triality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A.3 Type II supermultiplet . . . . . . . . . . . . . . . . . . . . . . . . . . 142
A.4 Type II supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.4.1 D= 11 and the type IIA supergravity . . . . . . . . . . . . . . 144
A.4.2 Type IIB Supergravity . . . . . . . . . . . . . . . . . . . . . . 146
A.4.3 Einstein and String Frame . . . . . . . . . . . . . . . . . . . . 147
ix