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156 Pages·2016·1.37 MB·English

3d-3d Correspondence for Seifert Manifolds
Thesisby
Du Pei
InPartialFulfillmentoftheRequirements
fortheDegreeofDoctorofPhilosophy
CALIFORNIAINSTITUTEOFTECHNOLOGY
Pasadena,California
2016
DefendedMay26th
ii
©2016
DuPei
ORCID:https://orcid.org/0000-0001-5587-6905
Allrightsreservedexceptwhereotherwisenoted.
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Tomyfamily.
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ACKNOWLEDGEMENTS
IhavespentfiveamazinglyenrichingyearsatCaltechand Ithankall ofmyfriends
whohaveaddedtomyexperiencealongthiswonderfuljourney.
First, I consider myself extremely fortunate to have Professor Sergei Gukov as
myadvisor. Goingfarbeyondhisdutyinacademicmentorship,Sergeihasprovided
mewithimmeasurablehelp,guidanceandencouragementandmadeeveryeffortto
carvemeintoagoodresearcher. Ineveryway,heepitomizesthegreatestvirtuesof
anadvisorandafriend.
I am grateful to Professor Anton Kapustin, Yi Ni, Hirosi Ooguri, John Schwarz
and Cumrun Vafa, not just for serving on my thesis/candidacy committee but also
fortheenlighteningacademicexchangeswehadduringthewholefiveyears. Iwish
to thank my collaborators including Jørgen Ellegaard Andersen, Martin Fluder,
Daniel Jafferis, Monica Kang, Ingmar Saberi, Wenbin Yan and Ke Ye. It is so fun
working with you guys and I have learned so much from you! And I also want
to thank Ning Bao, Francesco Benini, Sungbong Chun, Hee-Joong Chung, Tudor
Dimofte, Lei Fu, Abhijit Gadde, Temple He, Enrico Herrmann, Sam van Leuven,
Qiongling Li, Wei Li, Vyacheslav Lysov, Brendan McLellan, Daodi Lu, Tristan
Mckinney,SatoshiNawata,HyungrokKim,MuratKoloğlu,PetrKravchuk,Tadashi
Okazaki, Chris Ormerod, Tony Pantev, Chan Youn Park, Daniel S. Park, Abhishek
Pathak, Ana Peón-Nieto, Jason Pollack, Pavel Putrov, Brent Pym, Grant Remmen,
Tom Rudelius, Laura Schaposnik, Shu-Heng Shao, Caili Shen, Chia-Hsien Shen,
Jaewon Song, Bogdan Stoica, Kung-Yi Su, Piotr Sulkowski, Yifan Wang, Richard
Wentworth, Dan Xie, Masahito Yamazaki, Huang Yang, Xinyu Zhang and Peng
Zhaoforinterestingdiscussionsandconversations,andCarolSilbersteinforhelping
withallkindsoflogisticissues.
Thisdissertationcontainsthefruitsofnotjustmyfiveyearsasagraduatestudent
at Caltech but also the twenty-seven years since I was born, during the entirety of
which I was consistently provided an extraordinary education about math, physics,
nature,languagesandlife. Therearenowordsthatcouldeverdescribemygratitude
towards those whose effort and dedication made all of this possible. Among them,
my middle school teacher Jun Shao and my high school teacher Jiangtao Gao had
the greatest impact on my decision to become a physicist, and I wish to give them
specialthanksforsettingmeonthisbeautifuljourney.
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Lastly, I would like to thank my family. My dad Bailin Pei has the biggest
intellectual influence on me, and it is he who taught me how to think critically,
showed me the excitement of discovery and, step by step, led me onto the road of
pursuingknowledge. MymomFeiChenalwaysgivesfullrespectandunconditional
support for my decisions. Her care, love and trust have sustained me through every
difficulty and hardship. Her optimism, kindness and profound enthusiasm for life
have helped shape me into the person I am. Beside my parents, I owe an equally
immense debt to my wife Honggu Fan. She was on the course of becoming an
exceptionalscholarbut,withoutanyhesitation,chosetosacrificeherowncareerto
ensurethatIcouldstayfullycommittedtothepursuitofmine. Eversinceourbaby
was born, she has spend every second awake doing what is the best for me and our
child. Icanhardlyimagineanyonewhocouldbetterexemplifythebestattributesof
awifeortheseofamother. Also,Iwouldliketothankmyparents-in-law,HuaFan
andShifangChen,andmycousins,aunts,unclesandgrandparentsfortheenjoyable
time we have shared. And I want to thank Rae Pei, my baby daughter, for being
amazinglyeasygoingandagreeablewhenIwasworkingonresearchandwritingthe
thesis.
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ABSTRACT
In this dissertation, we investigate the 3d-3d correspondence for Seifert manifolds.
This correspondence, originating from string theory and M-theory, relates the dy-
namics of three-dimensional quantum field theories with the geometry of three-
manifolds.
WefirststartinChapter2withthesimplestcasesanddemonstratetheextremely
rich interplay between geometry and physics even when the manifold is just a
direct product M = (cid:6) (cid:2) S1. In this particular case, by examining the problem
3
from various vantage points, we generalize the celebrated relations between 1) the
Verlinde algebra, 2) quantum cohomology of the Grassmannian, 3) Chern-Simons
theory on (cid:6)(cid:2) S1 and 4) the index theory of the moduli space of flat connections to
a completely new set of relations between 1) the “equivariant Verlinde algebra” for
a complex group, 2) the equivariant quantum K-theory of the vortex moduli space,
3) complex Chern-Simons theory on (cid:6) (cid:2) S1 and 4) the equivariant index theory of
themodulispaceofHiggsbundles.
InChapter3wemoveonestepupincomplexitybylookingatthenextsimplest
example of M = L(p;1). We test the 3d-3d correspondence for theories that are
3
labeledbylensspaces,reachingafullagreementbetweentheindexofthe3dN = 2
“lensspacetheory”T[L(p;1)]andthepartitionfunctionofcomplexChern-Simons
theoryon L(p;1).
The two different types of manifolds studied in the previous two chapters also
have interesting interactions. We show in Chapter 4 the equivalence between two
seemingly distinct 2d TQFTs: one comes from the “Coulomb branch index” of the
class S theory on L(k;1) (cid:2) S1, the other is the “equivariant Verlinde formula” on
(cid:6)(cid:2)S1. Wecheckthisrelationexplicitlyfor SU(2) anddemonstratethatthe SU(N)
equivariant Verlinde algebra can be derived using field theory via (generalized)
Argyres-Seibergdualities.
In the last chapter, we directly jump to the most general situation, giving a
proposalforthe3d-3dcorrespondenceofanarbitrarySeifertmanifold. Weremark
onthehugeclassofnoveldualitiesrelatingdifferentdescriptionsofT[M ]withthe
3
same M andsuggestwaysthatourproposalcouldbetested.
3
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PUBLISHED CONTENT AND CONTRIBUTIONS
This thesis is based on preprints [1], [2] and [3], which are adapted for Chapters 2,
3 and 4 respectively, as well as ongoing work such as [4] and [5]. Each of them is
theoutcomeofactiveandfruitfulinteractionsbetweenmeandmycollaborators.
[1] Sergei Gukov and Du Pei. “Equivariant Verlinde formula from fivebranes
andvortices”(2015).arXiv:1501.01310[hep-th].url:http://arxiv.
org/abs/1501.01310.
[2] Du Pei and Ke Ye. “A 3d-3d appetizer” (2015). arXiv: 1503.04809
[hep-th].url:http://arxiv.org/abs/1503.04809.
[3] SergeiGukov,DuPei,WenbinYan,andKeYe.“EquivariantVerlindealge-
brafromsuperconformalindexandArgyres-Seibergduality”(2016).arXiv:
1605.06528[hep-th].url:http://arxiv.org/abs/1605.06528/.
[4] Daniel Jafferis, Sergei Gukov, Monica Kang, and Du Pei. “Chern-Simons
theoryatfractionallevel”(workinprogress).
[5] JørgenEllegaardAndersenandDuPei.“VerlindeformulaforHiggsbundle
modulispaces”(workinprogress).
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TABLE OF CONTENTS
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
PublishedContentandContributions . . . . . . . . . . . . . . . . . . . . . . vii
TableofContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
ListofIllustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
ChapterI:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
ChapterII: M = (cid:6)(cid:2) S1 andtheequivariantVerlindeformula . . . . . . . . . 4
3
2.1 FromtheVerlindeformulatoitsequivariantversion . . . . . . . . . 4
2.2 FivebranesonRiemannsurfacesand3-manifolds . . . . . . . . . . . 9
2.3 Branesandvortices . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 EquivariantintegrationoverHitchinmodulispace . . . . . . . . . . 24
2.5 (cid:12)-deformedcomplexChern-Simons . . . . . . . . . . . . . . . . . . 29
2.6 Anewfamilyof2dTQFTs . . . . . . . . . . . . . . . . . . . . . . . 37
2.7 t-deformationandcategorificationoftheVerlindealgebra . . . . . . 49
ChapterIII: M = L(p;1) anda“3d-3dappetizer” . . . . . . . . . . . . . . . 63
3
3.1 Testingthe3d-3dcorrespondence . . . . . . . . . . . . . . . . . . . 63
3.2 Chern-Simonstheoryon S3 andfreechiralmultiplets . . . . . . . . 65
3.3 3d-3dcorrespondenceforlensspaces . . . . . . . . . . . . . . . . . 70
ChapterIV:Whenalensspacetalksto (cid:6)(cid:2) S1 . . . . . . . . . . . . . . . . 84
4.1 EquivalencebetweentwoTQFTs . . . . . . . . . . . . . . . . . . . 84
4.2 EquivariantVerlindealgebraandCoulombbranchindex . . . . . . . 87
4.3 Acheckoftheproposal . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4 SU(3) equivariantVerlindealgebrafromtheArgyres-Seibergduality 107
ChapterV:Generalizationanddiscussion . . . . . . . . . . . . . . . . . . . 126
5.1 Goingonestepfurther . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2 Thetheory L(p;q) anditsdualities . . . . . . . . . . . . . . . . . . 127
5.3 Couplinglensspacetheoriestothestar-shapedquiver . . . . . . . . 130
AppendixA: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.1 ComplexChern-Simonstheoryonlensspaces . . . . . . . . . . . . 133
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LIST OF ILLUSTRATIONS
Number Page
2.1 Agenus-2Riemannsurfacedecomposedintotwopairsofpants. . . . 8
2.2 CP1 asacirclefibration. . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Thelensspace L(k;1) asatorusfibration. . . . . . . . . . . . . . . . 17
2.4 TheNS5-D3-(1,k)branesystemintypeIIBstringtheory. . . . . . . 18
2.5 The(1,k)-braneinfigure2.4asaboundstateofanNS5-braneand k
D5-branes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 The NS5-D2-NS5-D4 brane system in Type IIA string theory ob-
tainedbydimensionallyreducingthesysteminfigure2.4. . . . . . . 21
2.7 The“fusiontetrehedron”. . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 IllustrationofArgyres-Seibergduality. . . . . . . . . . . . . . . . . 109
4.2 IllustrationofgeometricrealizationofArgyres-SeibergdualityforT
3
theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 TheWeylalcoveforthechoiceofholonomyvariablesatlevel k = 3. . 111
4.4 TheillustrationofthenilpotentconeinMH((cid:6)0;3;SU(3)). . . . . . . 121
D
4.5 TheDynkindiagramfor E . . . . . . . . . . . . . . . . . . . . . . . 122
6
4.6 IllustrationofgeneralizedArgyres-SeibergdualityfortheT theories. 123
N
4.7 IllustrationofthegeometricrealizationofgeneralizedArgyres-Seiberg
dualityforT theories. . . . . . . . . . . . . . . . . . . . . . . . . . 123
N
5.1 The3dmirrordescriptionofT[(cid:6)g;n (cid:2) S1]. . . . . . . . . . . . . . . . 127
5.2 The“lensspacetheory”T[L(p;q)]. . . . . . . . . . . . . . . . . . . 128
5.3 The“Seiferttheory”. . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4 ThetheorylabeledbythePoincaréfakesphere. . . . . . . . . . . . . 132
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LIST OF TABLES
Number Page
2.1 Thespectrumof5d N = 2super-Yang-Millstheoryon S2 (cid:2)(cid:6)(cid:2) S1. . 13
2.2 Buildingblocksofa2dTQFT. . . . . . . . . . . . . . . . . . . . . . 53
[ ]
3.1 Thesuperconformalindexofthe“lensspacetheory”T L(p;1);U(N) . 78
[ ]
3.2 The S3 partitionfunctionofT L(p;1);U(N) . . . . . . . . . . . . . 81
b [ ]
3.3 The S3 partitionfunctionofT L(p;1);U(N) (continued). . . . . . 82
b [ ]
3.4 ThecomparisonbetweentheS3partitionfunctionofT L(p;1);U(2)
b
andthe“naive”partitionfunctionoftheGL(2;C)Chern-Simonstheory. 83
4.1 Comparing Z and Z for SU(2). . . . . . . . . . . . . . . . . . . 107
EV CB