BPS Approaches to Anyons, Quantum Hall States and Quantum Gravity

315 Pages·2017·2.01 MB·English

BPS Approaches to Anyons, Quantum
Hall States and Quantum Gravity
Carl Peter Turner
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Trinity College
June 2017
Thisdissertationissubmittedforthedegreeof
DoctorofPhilosophy

BPS Approaches to Anyons, Quantum Hall States and Quantum Gravity
CarlPeterTurner
We study three types of theories, using supersymmetry and ideas from string theory
astoolstogainunderstandingofsystemsofmoregeneralinterest.
Firstly, we introduce non-relativistic Chern-Simons-matter field theories in three di-
mensionsandstudytheiranyonicspectruminaconformalphase. Thesetheorieshave
supersymmetric completions, which in the non-relativistic case suffices to protect cer-
tain would-be BPS quantities from corrections. This allows us to compute one-loop
exact anomalous dimensions of various bound states of non-Abelian anyons, analyse
someinterestingunitarityboundviolations,andtestsomerecentlyproposedbosoniza-
tiondualities.
Secondly, we turn on a chemical potential and break conformal invariance, putting
thetheoryintotheregimeoftheFractionalQuantumHallEffect(FQHE).Thisisillus-
trated in detail: the theory supports would-be BPS vortices which model the electrons
of the FQHE, and they form bag-like states with the appropriate filling fractions, Hall
conductivities, and anyonic excitations. This formalism makes possible some novel
explicit computations: an analytic calculation of the anyonic phases experienced by
Abelianquasiholes;analyticrelationshipstotheboundaryWess-Zumino-Wittenmodel;
andderivationsofawideclassofQHEwavefunctionsfromabulkfieldtheory. Wealso
furthertestthethree-dimensionalbosonizationdualitiesinthisnewsetting. Alongthe
way,weaccumulatenewdescriptionsoftheQHE.
Finally,weturnawayfromflatspaceandinvestigateaproblemin(3+1)-dimensional
quantumgravity. Wefindthatevenasaneffectivetheory,thetheoryhasenoughstruc-
turetosuggesttheinclusionofcertaingravitationalinstantonsinthepathintegral. An
explicit computation in a minimally supersymmetric case illustrates the principles at
work, and highlights the role of a hitherto unidentified scale in quantum gravity. It
also is an interesting result in itself: a non-perturbative quantum instability of a flat
supersymmetricKaluza-Kleincompactification.

Thisthesisisdedicatedtothememoryofmygrandmother,VeraTurner
Thankyou,Grandma,forthemanyyearsofwarmthandjoyyoubroughtusall

Declaration
This dissertation is the result of my own work and includes nothing which is the out-
come of work done in collaboration except as declared in the Preface and specified in
thetext.
ItisnotsubstantiallythesameasanythatIhavesubmitted,or,isbeingconcurrently
submittedforadegreeordiplomaorotherqualificationattheUniversityofCambridge
oranyotherUniversityorsimilarinstitutionexceptasdeclaredinthePrefaceandspec-
ified in the text. I further state that no substantial part of my dissertation has already
been submitted, or, is being concurrently submitted for any such degree, diploma or
other qualification at the University of Cambridge or any other University of similar
institutionexceptasspecifiedinthetext.
This document is mostly adapted from two papers on non-relativistic anyons with
conformal symmetry [1, 2]; four papers [3, 4, 5, 6] on quantum Hall physics; and one
paper[7]onquantumgravity. Thesesevenpapersarebroadlyreflectedinthestructure
ofthisdocument,withsuperconformalanyonsformingPartII,thequantumHallwork
comprisingPartsIII,IV,andVandthequantumgravityworkmakingupPartVI.There
issomeunpublishedmaterialonthebosonizationofanyonsinPartII.Theonlysection
ofworkforwhichIwasnotprimarilyresponsibleistheworkrecapitulatedinChapter
18,forwhichNickDoreywaschieflyresponsible.
CarlTurner
June5,2017

Acknowledgements
DoingaPhDcanbechallenging,butthelastfewyearshavebeenaveryhappytimefor
me,andforthatIowethankstomanypeople.
Firstly, I owe a huge amount to my supervisor, the excellent David Tong, who must
not go unacknowledged. He has been an unending source of resourcefulness and en-
thusiasminequalmeasures,andthatmorethananythingelsehasmadedoingaphysics
PhD a real pleasure. My other collaborators – Nick Dorey, Nima Doroud and Ðorđe
Radičević – also deserve significant thanks for their valuable input, as do many excel-
lent physicists who have passed on some of their wisdom, including Ofer Aharony,
Siavash Golkar, Guy Gur-Ari, Amihay Hanany, Sean Hartnoll, Stephen Hawking, An-
dreasKarch,SangminLee,SungjayLee,NickManton,MalcolmPerry,ChrisPope,Har-
veyReall,NathanSeiberg,AndrewSingleton,andKennyWong.
Specialthanksgotothosewhowerealsokindenoughtohostvisitstoseveralamaz-
ing places: Kimyeong Lee at KIAS; Shiraz Minwalla at TIFR; and Shamit Kachru at
Stanford. Of course, the people here in the CMS, especially my fellow students, have
made being in DAMTP a real pleasure; they get a definite 5/5. I particularly want to
thank James, Jack and Eduardo for making the office a welcoming and cheerful place,
andprovidingmoralsupportinthefaceofadversity,bureaucracyandpoorlydesigned
websites.
Imustalsoexpressmygratitudetomyfamilyforalltheirsupport;Iwishmygrand-
mother, to whose memory this thesis is dedicated, were still here to read this. Most
especially I have a great debt to my parents, whose kindness and generosity got me
here;theirimplicitbeliefhashelpedmethroughsometoughtimes. Thankyou.
Finally, I must mention the fine people who I have been lucky enough to surround
myself with in Cambridge. My old friends from my time as an undergraduate – es-
pecially Alex, Amit, Fiona, George, Guy, Hannah, James, Jana, Jonathan, Josh, Rach,
Rob, Steph, Tasha and Ted – take a huge amount of credit for making Cambridge such
a happy place for me. In more recent years, the Cambridge University Ceilidh Band and
itssatelliteoutfitshavebeenaconstantsourceofamazingfun,phenomenalmusicand
incrediblefriends,allofwhomhaveaplaceinmyheart. ItalsohelpedmefindAlice,to
whomIamgratefulforlaughter,comfort,music,food,andagreatmanyotherthings.
Thankyouall.