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Integrability in Weakly Coupled Super Yang-Mills Theory: Form Factors, On-Shell Methods and Q-Operators

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Integrability in Weakly Coupled Super Yang-Mills Theory: Form Factors, On-Shell Methods and Q-Operators Integrability in weakly coupled super Yang-Mills theory: form factors, on-shell methods and Q-operators Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Physik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät der Humboldt-Universität zu Berlin von David Meidinger, M.Sc. Präsidentin der Humboldt-Universität zu Berlin Prof. Dr.-Ing. Dr. Sabine Kunst Dekan der Mathematisch-Naturwissenschaftlichen Fakultät Prof. Dr. Elmar Kulke Gutachter Prof. Dr. Matthias Staudacher Prof. Dr. Jan Plefka Prof. Dr. Dmytro Volin Tag der mündlichen Prüfung: 26.01.2018 Hiermit erkläre ich, die Dissertation selbstständig und nur unter Verwendung der angegebenen Hilfen und Hilfsmittel angefertigt zu haben. Ich habe mich anderwärts nicht um einen Doktorgrad beworben und besitze einen entsprechenden Doktorgrad nicht. Ich erkläre die Kenntnisnahme der dem Verfahren zugunde liegenden Promotionsordnung der Mathematisch-Naturwissenschaftlichen Fakultät I der Humboldt-Universität zu Berlin vom 27. Juni 2012. ZUSAMMENFASSUNG Diese Arbeit untersucht die superkonforme N = 4 Yang-Mills-Theorie (N = 4 SYM) bei schwacher Kopplung, mit dem Ziel eines tieferen Verständnisses von Größen der Theorie als Zustände des integrablen Modells, von dem vermutet wird, dass es der planaren Theorie zu Grunde zu liegt. Die Grundlage hierfür bilden moderne On-Shell-Methoden, insbesondere für Formfaktoren und nicht-planare On-Shell-Diagramme. Die Untersuchung von Streuam- plituden hat Verknüpfungen zwischen On-Shell-Methoden und der Yangschen Invarianz der Theorie aufgezeigt; die am besten vom Blickwinkel der Integrabilität aus verstande- nen Observablen sind hingegen die Zweipunktfunktionen zusammengesetzter Operatoren. Formfaktoren interpolieren zwischen diesen Größen und sind deshalb ein vielversprechen- der Untersuchungsgegenstand für eine einheitliche Beschreibung der integrablen Struktur der Theorie. Zu diesem Zweck leiten wir On-Shell-Diagramme für Formfaktoren des chiralen Energie- Impuls-Tensor-Multipletts aus der Britto-Cachazo-Feng-Witten-Rekursion her, und untersu- chen deren Eigenschaften. Sie erlauben die Herleitung eines Graßmannschen Integrals für diese Formfaktoren, d.h. einer Darstellung als Konturintegral in Graßmann-Mannigfaltig- keiten. Für Komponenten mit nächst-zu-maximaler Helizität (NMHV) bestimmen wir die entsprechende Integrationskontur. Dies erlaubt es das Integral mit einer Twistor-String- inspirierten Formulierung in Beziehung zu setzen. Mit Hilfe dieser On-Shell-Methoden zeigen wir dass sowohl Formfaktoren des chiralen Energie-Impuls-Tensor-Multipletts, als auch On-Shell-Funktionen mit Einfügungen beliebiger Operatoren Eigenzustände integ- rabler Transfermatrizen sind. Die entsprechenden Eigenwertgleichungen verallgemeinern die Yangsche Invarianz der On-Shell-Funktionen von Amplituden. Wir zeigen weiterhin dass ein Teil der Yangschen Symmetrien trotz der Operatoreinfügung erhalten bleibt. Da die On-Shell-Diagramme von Formfaktoren in einem gewissen Sinne nichtplanar sind, er- weitern wir unsere Untersuchung auf allgemeine nichtplanare On-Shell-Funktionen. Wir zeigen dass diese Funktionen, obwohl sie erst in höheren Ordnungen der 1=N-Entwick- lung auftreten, einen Teil der Yangschen Symmetrien erhalten. Weitere, von Transfermatri- zen generierte Symmetrien können ebenfalls hergeleitet werden, und zeigen insbesondere dass Diagramme auf Zylindern als Intertwiner fungieren. Alle untersuchten Größen lassen sich als Graßmannsche Integrale darstellen, deren Integranden, wie wir zeigen, selbst als Zustände integabler Spinketten zu verstehen sind. On-Shell-Diagramme basieren auf Größen der führenden Ordnung der Störungsrech- nung. Als einen Schritt hin zur Berechnung der Eigenzustände des integrablen Modells zu höheren Schleifenordnungen untersuchen wir zusammengesetzte Operatoren. Hier erlaubt die Quanten-Spektral-Kurve (Quantum Spectral Curve, QSC) die nichtperturbative Berech- nung ihres Spektrums, liefert jedoch keine Information zur Form der Zustände selbst. Die QSC kann als algebraisches Q-System verstanden werden, welches in Operatorform durch Baxter Q-Operatoren formulierbar sein sollte. Diese Formulierung enthielte die vollständi- ge Information über die Eigenzustände. Um auf eine solche Operatorform der QSC hinzuar- beiten untersuchen wir die Q-Operatoren nichtkompakter Superspinketten, insbesondere jene welche N = 4 SYM zur führenden Schleifenordnung beschreibt. Wir entwickeln ein effiziente Methode zur Berechnung der Matrixelemente der Q-Operatoren. Dies erlaubt es das gesamte Q-System durch endliche Matrizen für jeden Anregungssektor zu realisieren, und liefert die Grundlage für perturbative Rechnungungen mit der QSC in Operatorform. SUMMARY This thesis investigates weakly coupled N = 4 superconformal Yang-Mills theory (N = 4 SYM), aiming at a better understanding of on-shell and off-shell quantities as states of the integrable model that is conjectured to underlie the planar limit of the theory. The basis for this investigation are modern on-shell methods, in particular for form factors and non- planar on-shell functions. The study of scattering amplitudes has revealed links between on-shell methods and the Yangian symmetry of the model; the most well-understood ob- jects from the viewpoint of integrability however are the two-point correlators of composite operators. Form factor are quantities interpolating between off-shell correlation functions and on-shell scattering amplitudes, and are therefore a promising subject of study for a unified description of the integrable structure of the theory. We use the Britto-Cachazo-Feng-Witten recursion relations to develop on-shell diagrams for form factors of the chiral stress-tensor multiplet, and investigate their properties. The diagrams then allow to derive a Graßmannian integral for these form factors, i.e. a repre- sentation as a contour integral inside a Graßmannian. We show how to obtain the contour of this integral for next-to-maximally-helicity-violating (NMHV) form factors, and use this knowledge to relate the integral to a twistor string inspired formulation. Based on these on-shell methods, we show that both form factors of the chiral stress-tensor multiplet as well as on-shell functions with insertions of arbitrary operators are eigenstates of integrable transfer matrices. The corresponding eigenvalue equations can be seen as symmetries gen- eralizing the Yangian invariance of amplitude on-shell functions. In addition, a part of these Yangian symmetries remain unbroken by the operator insertion. Motivated by the fact that on-shell diagrams of form factors can be understood as being nonplanar, we con- sider general nonplanar on-shell functions. Despite appearing in subleading contributions of the 1=N-expansion, we prove that these quantities exhibit a partial Yangian invariance, inherited from the integrability of the planar theory. We also derive symmetries generated by transfer matrices, and show that on-shell diagrams on cylinders can be understood as in- tertwiners. All these quantities can be written as Graßmannian integrals and interestingly, their integrands can be regarded as states of integrable spin chains themselves. On-shell diagrams involve quantities at lowest order in perturbation theory. To make progress towards the calculation of the higher loop eigenstates of the integrable model, we consider single trace operators, for which the Quantum Spectral Curve (QSC) exists as a beautiful and concise way of determining the spectrum of these states non-perturbatively. This formulation however carries no information about the form of the states. The QSC is an algebraic Q-system, for which an operatorial form in terms of Baxter Q-operators should exist, containing the complete information on the eigenstates in addition to the eigenval- ues. To initiate the developments of an operatorial formulation of the QSC, we investigate the Q-operators of non-compact super spin chains, in particular those of the integable spin chain describing N =4 SYM at the one-loop level, and devise efficient methods to evaluate their matrix elements. This allows to obtain the entire Q-system in terms of finite dimen- sional matrices for each magnon sector. These can be used as input data for perturbative calculations using the QSC in operatorial form. CONTENTS 1 Introduction 3 Superconformal Yang-Mills theory and the AdS/CFT correspondence 3 Integrability in N =4 SYM 5 Scattering amplitudes: on-shell methods and integrability 8 Overview 10 I On-shell methods for form factors 2 On-shell methods for scattering amplitudes 17 Kinematics & on-shell superelds 17 Scattering amplitudes 21 On-shell techniques 24 On-shell diagrams 28 The Graßmannian integral 31 3 On-shell diagrams for form factors 35 Form factors 36 The chiral stress-tensor multiplet 37 From BCFW to on-shell diagrams 38 Inverse soft limits 39 Cyclicity and equivalence moves 40 A relation to amplitude diagrams 40 Multiple top-cell diagrams 42 Permutations 44 Leading singularities and on-shell functions 45 4 Form factors as Graßmannian integrals 47 Graßmannian geometry and the o-shell momentum 48 Gluing the operator into on-shell diagrams 49 Gluing the top-form 51 Twistor and momentum twistor space 55 More examples 60 5 The BCFW contour for NMHV 65 The NMHV top-forms 66 Extracting the geometry from diagrams 67 The contour in closed form 71 6 The Graßmannian and the connected prescription 75 Brief review of the connected prescription for form factors 76 Graßmannian integrals from the connected
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