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A hitchhiker's guide to quantum field theoretic aspects of $\mathcal{N}=4$ SYM theory and its deformations PDF

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Preview A hitchhiker's guide to quantum field theoretic aspects of $\mathcal{N}=4$ SYM theory and its deformations

A hitchhiker’s guide to quantum field theoretic aspects of N = 4 SYM theory and its deformations based on my 7 1 D i s s e r t a t i o n 0 2 n a J 9 eingereicht an der 1 ] Mathematisch-Naturwissenschaftlichen Fakult¨at h t der Humboldt-Universit¨at zu Berlin - p e von h [ Jan Fokken 2 v 5 [email protected] 8 7 0 Institut fu¨r Mathematik und Institut fu¨r Physik, Humboldt-Universita¨t zu Berlin, 0 IRIS-Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany . 1 0 7 1 : v i X r a Fu¨r sie, die mich fing, den Sirenen entriss, auf mich wartete und ging, und doch niemals verließ. Zusammenfassung In den vergangenen Jahrzehnten gab es enormen Fortschritt im Verst¨andnis der Struktur der N = 4 SYM Theorie in vier Raumzeitdimensionen, welcher viele Werkzeuge fu¨r die ef- fiziente Berechnung von Observablen hervorgebracht hat. Mit Hilfe von Integrabilit¨atsmeth- odenwurdeesprinzipiellm¨oglichdieanomalenDimensionenzusammengesetzterOperatoren im’tHooftLimesexaktzuberechnen.InspiriertdurchdieseFortschrittegehenwirderFrage nach,welcheVoraussetzungenerfu¨lltseinmu¨ssen,damitObservableneinerTheoriemitHilfe dieserneuenWerkzeugeberechnetwerdenk¨onnen.InsbesondereuntersuchenwiramBeispiel der einparametrischen β- und dreiparametrischen γ -deformierten Abk¨ommlinge der N = 4 i SYM Theorie, ob die anomalen Dimensionen zusammengesetzter Operatoren auch in diesen weniger symmetrischen Theorien durch Integrabilit¨atsmethoden erhalten werden k¨onnen. Fu¨r die deformierten Theorien stellt sich heraus, dass nicht alle ihre Wechselwirkungen alsAbk¨ommlingederundeformiertenWechselwirkungenverstandenwerdenk¨onnen.Umper- sistente Divergenzen in perturbativen Entwicklungen zu vermeiden, fu¨hren wir zus¨atzliche sogenannte Mehrspurwechselwirkungen ein. Fu¨r die γ -Deformation zeigen wir durch feyn- i mandiagrammatische Berechnung der relevanten Einschleifenkorrekturen im ’t Hooft Limes, dass diese nichtvererbten Wechselwirkungen laufende Kopplungskonstanten besitzen, welche die konforme Invarianz der quantisierten Theorie brechen. Daru¨ber hinaus untersuchen wir den Einfluss der nichtvererbten Wechselwirkungen auf die anomalen Dimensionen zusam- mengesetzter Operatoren am Beispiel der Operatoren tr(cid:0)φL), indem wir ihre anomalen Di- i mensionen bis zur fu¨hrenden Wickel ( Wrapping“) Schleifenordnung K = L berechnen. Fu¨r ” L ≥ 3 lassen sich so die Ergebnisse von integrabilit¨atsbasierten Methoden reproduzieren. Fu¨r L = 2 finden wir jedoch die endliche und renormierungsschemenabh¨angige anomale Di- mension im Kontrast zum divergenten integrabilit¨atsbasierten Ergebnis. Basierend auf den feldtheoretischen Daten aus der β- und der γ -Deformation schlagen wir einen Test vor, i welcher kl¨aren soll ob Supersymmetrie und/oder exakte konforme Invarianz notwendige Be- dingungen fu¨r die in der N = 4 SYM Theorie gefundene Quantenintegrabilit¨at sind. Auch fu¨r die β-Deformation analysieren wir das Auftreten von nichtvererbten Mehr- spurbeitr¨agen. Aus der vollst¨andigen Wechselwirkungsstruktur leiten wir einen Algorithmus ab,dererlaubtdenEinflussvonMehrspurkopplungenaufdieanomalenDimensionenzusam- mengesetzterOperatorenaufEinschleifenebeneinder’tHooftKopplungkonsistentimSpin- kettenbild abzubilden. Hiermit konstruieren wir den vollst¨andigen Dilatationsoperator der konformen β-Deformation im ’t Hooft Limes auf Einschleifenebene. AbschließendnutzenwirunsereErgebnisse,umdenp´olyatheoretischenAnsatzzurBerech- nung der thermalen Einschleifen-Zustandssumme auf dem kompakten Raum S3 ×R in den deformierten Theorien nutzbar zu machen. Unsere Ergebnisse zeigen, dass die Deconfine- ” ment“-Phasenu¨bergangstemperatur der deformierten Theorien auf Einschleifenniveau mit jener der undeformierten N = 4 SYM Theorie u¨bereinstimmt und wir vermuten, dass dieser Befund sogar im nichtperturbativen Bereich Bestand hat. Zus¨atzlich zu den Forschungsergebnissen enth¨alt diese Arbeit die vollst¨andige Wirkung inklusivederSymmetriegeneratorenderN = 4SYMTheorie,derβ-undderγ -Deformation. i Wir wiederholen allgemeine Techniken zur Renormierung zusammengesetzter Operatoren und elementarer Felder, gehen auf das weit verbreitete dimensionale Reduktionsschema ein und wie relevante UV Divergenzen in Niederschleifenintegralen effizient bestimmt werden k¨onnen.IndiesemZusammenhangleitenwirdieFeynmanRegelnalleruntersuchtenTheorien herundstellendasWerkzeugFokkenFeynPackagevor,welchesdieseRegelninMathematica implementiert. Alle Rechnungen in dieser Dissertation wurden mit FokkenFeynPackage durchgefu¨hrt,sodassdieseArbeiteinemunabh¨angigenTestallerFeynman-diagrammatischen Rechnungen in den Publikationen [11–44] darstellt. v Abstract OverthelastdecadestremendousprogresswasmadeinunderstandingthestructureofN = 4 SYM theory in four-dimensional spacetime and many tools for the efficient calculation of observablesinthistheoryweredeveloped.Theanomalousdimensionsofcompositeoperators in the ’t Hooft limit became in principle accessible by means of integrability-based methods. Inspiredbythesefindings,weinvestigatewhichprerequisitesmustbefulfilledforobservables of a theory to be calculable by the means of these new tools. In particular, we focus on the one-parameter β- and the three-parameter γ -deformed descendents of N = 4 SYM theory i toanalysewhethertheanomalousdimensionsofcompositeoperatorsintheselesssymmetric theories can also be obtained by the means of integrability. In the deformed theories it turns out that not all interactions originate from the interac- tions in the undeformed theory. Additionally, we have to include so-called multi-trace inter- actions to prevent persistent divergences in perturbative expansions. For the γ -deformation, i we show by an explicit feynman-diagrammatic one-loop calculation that these non-inherited interactions have running coupling constants which spoil the conformal invariance of the quantised theory, even in the ’t Hooft limit. Furthermore, we investigate the impact of these non-inherited interactions on the anomalous dimensions of composite operators, by perturbatively calculating the K = L loop leading order wrapping corrections to the op- erators tr(cid:0)φL). We reproduce the findings from integrability for L ≥ 3 and find the finite i renormalisation-scheme-dependent anomalous dimension of the L = 2 states in contrast to the integrability-based methods which yield a divergent result. Based on the field-theoretic datafromtheβ-andγ -deformation,weproposeatesttodeterminewhethersupersymmetry i and/or exact conformal invariance are necessary prerequisites of the quantum integrability found for N = 4 SYM theory. For the β-deformation, we also analyse the occurrence of non-inherited multi-trace con- tributions.Fromthefullinteractionstructure,wederiveanalgorithmwhichallowstoconsis- tentlyincludemulti-tracecouplingsthataffectanomalousdimensionsofcompositeoperators atone-looporderinthe’tHooftcouplinginthespin-chainpicture.Thisleadstothecomplete one-loop dilatation operator of the conformal β-deformation in the ’t Hooft limit. Finally, we employ our findings to generalise the P´olya-theoretic approach to the ther- mal one-loop partition function of N = 4 SYM theory on S3 ×R to be also applicable in the deformed theories. We find that the deconfinement phase-transition temperature in the deformed theories is the same as in the undeformed N = 4 SYM theory at one-loop level and we conjecture that it remains the same even non-perturbatively. In the context of this thesis, we employ various field-theoretic aspects of N = 4 SYM theory and its deformations. Therefore, we provide the action and symmetry generators of N = 4 SYM theory, the β-, and the γ -deformation. Furthermore, we review the general i techniques for the renormalisation of elementary fields and composite operators in a unified setting and discuss the relation to the dilatation operator. We include a detailed description of the widely used dimensional reduction scheme and discuss how the UV divergence of log- arithmically divergent integrals may be extracted with relatively little effort. In this context, we derive the Feynman rules for N = 4 SYM theory, the β- and the γ -deformation and i present the tool FokkenFeynPackage which implements these rules into Mathematica. All calculationsinthisthesisarecarriedoutusingthistoolandhenceitprovidesanindependent test of all Feynman-diagrammatic calculations in [11–44]. vi List of own publications This thesis is based on the following publications: [11] J. Fokken, C. Sieg and M. Wilhelm, Non-conformality of γ -deformed N = 4 SYM i theory, JJ..PPhhyyss.. AA4477 ((22001144)) 445555440011, [aarrXXiivv::11330088..44442200 [[hheepp--tthh]]]. [22] J.Fokken,C.SiegandM.Wilhelm,The complete one-loop dilatation operator of planar realβ-deformedN = 4SYMtheory,JJHHEEPP 11440077 ((22001144)) 115500,[ aarrXXiivv::11331122..22995599 [[hheepp--tthh]]]. [33] J. Fokken, C. Sieg and M. Wilhelm, A piece of cake: the ground-state energies in γ -deformed N = 4 SYM theory at leading wrapping order, JJHHEEPP 11440099 ((22001144)) 7788, i [ aarrXXiivv::11440055..66771122 [[hheepp--tthh]]]. [44] J. Fokken and M. Wilhelm, One-Loop Partition Functions in Deformed N = 4 SYM Theory, JJHHEEPP 0033 ((22001155)) 001188, [ aarrXXiivv::11441111..77669955 [[hheepp--tthh]]]. • In addition, this thesis contains the presentation and manual of the Mathematica tool FokkenFeynPackage. I developed the FokkenFeynPackage to provide an efficient Mathematica implementation of the Feynman rules of N = 4 SYM theory and its de- formations that were presented in this thesis. This tool will be made publicly available together with this thesis. vii Contents Zusammenfassung vv Abstract vvii List of own publications vviiii 1 Introduction and Overview 1133 1.1 Introduction 1133 1.2 Overview 1188 2 The classical theories 2211 2.1 N = 1 SYM theory in ten dimensions 2222 2.1.1 The action 2222 2.1.2 Dimensional reduction to four dimensions 2244 2.2 N = 4 SYM theory in four dimensions 2277 2.3 Symmetries of N = 4 SYM theory 2299 2.3.1 Conformal symmetry algebra so(4,2) 3300 2.3.2 Internal symmetry algebra R 3311 2.3.3 SUSY transformations involving Q 3322 2.3.4 Special SUSY transformations involving S 3333 2.3.5 Commutation relations of the symmetry algebra of psu(2,2|4) 3344 2.3.6 The spinor or oscillator representation 3355 2.4 Deformations of N = 4 SYM theory 3366 2.4.1 The deformations 3377 2.4.2 The deformed single-trace action 3388 2.4.3 Multi-trace parts of the action 3399 2.4.4 Symmetries of the deformed models 4422 2.5 Composite operators 4422 2.5.1 Normal ordering 4433 2.5.2 Building blocks 4444 2.5.3 Symmetry transformations of composite operators 4455 3 Renormalisation and the quantised theories 4477 3.1 The path integral approach 4488 3.2 Massless ϕ3-theory 4499 3.2.1 The bare theory 4499 3.2.2 The renormalised theory 5511 viii 3.2.3 Renormalisation group equation 5544 3.2.4 Composite operator insertions 5555 3.3 N = 4 SYM theory and its deformations 6600 3.3.1 The renormalised theories 6600 3.3.2 Composite operator insertions 6622 3.3.3 Calculating Green’s functions in N = 4 SYM theory 6633 3.4 The ’t Hooft limit 6688 3.4.1 Finite-size effects 7700 3.5 The (asymptotic) planar one-loop dilatation generator 7722 4 Applications 7755 4.1 Non-conformality of the γ -deformation 7766 i 4.1.1 One-loop renormalisation of Qii 7777 Fii 4.1.2 Immediate implications for the AdS/CFT correspondence 8811 4.2 Ground-state energies at leading wrapping order in the γ -deformation 8822 i 4.2.1 Identifying deformation-dependent diagrams 8844 4.2.2 Finite-size corrections to the ground state 8866 4.3 The complete one-loop dilatation operator of the planar β-deformation 9922 4.3.1 (Q1,Q2)-neutral states 9933 4.3.2 (Q1,Q2)-charged states 9955 4.3.3 The SU(N) dilatation operator 9966 4.3.4 The U(N) dilatation operator 9999 4.3.5 Immediate implications for the AdS/CFT correspondence 110000 4.4 The thermal one-loop partition functions of the deformed theories 110000 4.4.1 Partition functions via P´olya theory 110033 4.4.2 Ingredients of the P´olya-theoretic approach 110066 4.4.3 Partition function and Hagedorn temperature 111100 4.4.4 Immediate implications for the AdS/CFT correspondence 111111 5 Summary, conclusion, and outlook 111133 5.1 Summary and conclusion 111133 5.2 Outlook 111155 Acknowledgements 111177 Appendix 111199 A Conventions and list of used symbols and abbreviations 111199 B Clifford algebras in various dimensions 112233 B.1 General properties 112244 B.2 Construction of a d-dimensional Clifford algebra 112244 B.3 Four-dimensional Minkowski Clifford algebra 112266 B.4 Six-dimensional Euclidean Clifford algebra 112277 B.5 Ten-dimensional Minkowski Clifford algebra 112288 C Spinors in various dimensions 112299 C.1 General properties 112299 ix C.2 Spinors in four-dimensional Minkowski space 113311 C.3 Spinors in six-dimensional Euclidean space 113322 C.4 Spinors in ten-dimensional Minkowski space 113333 D Kaluza-Klein compactification 113344 E The conformal algebra 113377 E.1 Conformal transformations of coordinates 113377 E.2 Conformal transformations of fields 113388 F Comparison of the field and the oscillator representation 114400 G Derivation of Feynman rules 114422 G.1 The action and general setting 114422 G.2 Propagators and the free theory 114444 G.3 Interactions and the full theory 114477 G.4 Feynman rules 115500 G.5 Feynman rules for real scalars 115522 H Equations of motion and the Bianchi identity 115522 I The FokkenFeynPackage 115533 J Dimensional renormalisation schemes 116655 J.1 Dimensional regularisation 116666 J.2 Dimensional reduction 116688 J.3 The subtraction procedure and renormalisation schemes 116699 K Evaluating Feynman integrals 117700 K.1 Wick rotation and Euclidean space integrals 117700 K.2 Ultraviolet and infrared divergences in scalar integrals 117711 K.3 Tensor integrals 117766 K.4 Products of σ-matrices 117788 L Fourier transformation of the free two-point function 117799 M The harmonic action 118800 N Scalar one-loop self-energy 118811 O Coupling tensor identities for the γ -deformation 118822 i P Calculation of (cid:104)PDL≥3(w,y)(cid:105) 118833 2 (1) Q Calculation of Z (x) 118877 f.s.c. R Summation identities 118877 References 220011 x

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