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Matrix Field Theory Mathematik Matrix Field Theory Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik und Informatik der Mathematisch-Naturwissenschaftlichen Fakult¨at der Westf¨alischen Wilhelms-Universit¨atM¨unster arXiv:2005.07525v1 [math-ph] 15 May 2020 vorgelegt von Alexander Hock aus Dschangi-Dscher/Kirgisistan -2020- Dekan: Prof. Dr. Xiaoyi Jiang Erster Gutachter: Prof. Dr. Raimar Wulkenhaar Zweiter Gutachter: Prof. Dr. Gernot M¨unster Tag der m¨undlichen Pr¨ufung: Tag der Promotion: Abstract This thesis studies matrix field theories, which are a special type of matrix models. First, the different types of applications are pointed out, from (noncommutative) quantum field theory over 2-dimensional quantum gravity up to algebraic geometry with explicit com- putation of intersection numbers on the moduli space of complex curves. The Kontsevich model, which has proved the Witten conjecture, is the simplest exam- ple of a matrix field theory. Generalisations of this model will be studied, where different potentials and the spectral dimension are introduced. Because they are naturally embed- ded into a Riemann surface, the correlation functions are graded by the genus and the number of boundary components. The renormalisation procedure of quantum field theory leads to finite UV-limit. We provide a method to determine closed Schwinger-Dyson equations with the usage of Ward-Takahashi identities in the continuum limit. The cubic (Kontsevich model) and the quartic (Grosse-Wulkenhaar model) potentials are studied separately. The cubic model is solved completely for any spectral dimension < 8, i.e. all cor- relation functions are derived explicitly. Inspired by topological recursion, we propose creation and annihilation operators by differential and residue operators. The exact re- sults are confirmed by perturbative computations with Feynman graphs renormalised by Zimmermann's forest formula. The number and the amplitudes of the graphs grow facto- rially, which is known as renormalon problem. However, these series are convergent since the exact results are provided. A further differential operator is derived to determine all free energies. Additionally, by the theorem of Kontsevich, the intersection numbers of the moduli space of complex curves Mg;b are found. For the quartic model, the 2-point function is derived for any spectral dimension < 6 explicitly. The first step is to derive an angle function which is, after analytic continuation, interpreted as an effective measure. On the 4-dimensional noncommutative Moyal space, the effective measure is given by a hypergeometric function. Its asymptotic behaviour arcsin(λπ) 1 changes the spectral dimension effectively to 4 − 2 π for jλj < π . This dimension drop prevents the quantum field theoretical 4-dimensional Φ4-model on the Moyal space from the triviality problem. After combinatorial analysis, an explicit (not recursive) formula for any planar N-point function is provided. The evident difference between the cubic and the quartic model is of algebraic- geometric nature. Computing correlation functions via topological recursion needs the spectral curve as initial data. This algebraic curve has for the cubic model only one branch point which coincides with the pole of the stable correlation functions. However, the quartic model has a spectral curve which admits infinitely many branch points in the continuum limit. Zusammenfassung Diese Dissertation besch¨aftigt sich mit Matrix-Feldtheorien, einer speziellen Form der Ma- trixmodelle. Zun¨achst werden unterschiedliche Anwendungsm¨oglichkeiten hervorgehoben, die von (nichtkommutativer) Quantenfeldtheorie ¨uber 2-dimensionale Quantengravitation bis hin zur algebraischen Geometrie mit expliziter Berechnung von Schnittzahlen auf dem Modulraum komplexer Kurven reichen. Das Kontsevich-Modell ist ein paragdigmatisches Beispiel einer Matrix Feldtheorie, mit der Wittens Vermutung bewiesen wurde. Es werden Verallgemeinerungen dieses Modells betrachtet, die durch die Wahl eines anderen Potentials und durch Einf¨uhrung der spektralen Dimension. Die Korrelationsfunktionen werden durch das Geschlecht und die Zahl der Randkomponenten unterschieden, da diese eine nat¨urliche Darstellung auf Riemannschen Fl¨achen besitzen. Um dem UV Limes Bedeutung zu verleihen, werden Renormierungsmethoden aus der Quantenfeldtheorie verwendet. Wir zeigen, wie geschlossene Schwinger-Dyson-Gleichungen f¨urdie Korrelationsfunk- tionen mit Hilfe von Ward-Takahashi-Identit¨atenim Kontiuumslimes bestimmt werden k¨onnen. Die spezielle Wahl eines kubischen (Kontsevich-Modell) und eines quartischen (Grosse-Wulkenhaar-Modell) Potentials wird separat betrachtet und untersucht. Das kubische Modell wird vollst¨andigf¨ureine spektrale Dimension < 8 gel¨ost,d.h. es werden alle Korrelationsfunktionen explizit berechnet. Die Erzeuger- und Vernichterope- ratoren werden als Differential- und Residuumsoperator angegeben, wobei die Konstruk- tion durch topologische Rekursion inspiriert wurde. Die Resultate werden durch st¨orungs- theoretische Rechnungen best¨atigt,in denen in Feynman-Graphen entwickelt wird, die durch Zimmermanns Waldformel renormiert werden. Die Anzahl der Graphen und die Amplitude der Graphen steigen mit O(n!), welches als Renormalon-Problem bezeich- net wird; dennoch konvergieren diese Reihen, da wir sie konkret angeben. Es wird ein Differentialoperator angegeben, der die freien Energien berechnet und somit nach dem Theorem von Kontsevich die Schnittzahlen auf dem Modulraum der komplexen Kurven Mg;b erzeugt. F¨urdas quartische Modell geben wir die 2-Punkt-Funktion f¨urdie spektrale Dimen- sion < 6 explizit an. Hierzu wird zun¨achst eine Winkelfunktion berechnet, die nach kom- plexer Fortsetzung als effektives Maß interpretiert werden kann. Auf dem 4-dimensionalen nichtkommutativen Moyalraum ist das effektive Maß durch eine hypergeometrische Funk- arcsin(λπ) 1 tion gegeben, die die spektrale Dimension effektiv zu 4 − 2 π f¨ur jλj < π ¨andert. Durch diese effektive Anderung¨ wird das Trivialit¨atsproblemdes quantenfeldtheoretischen Φ4-Modells auf dem 4-dimensionalen Moyalraum verhindert. Schließlich zeigen wir nach kombinatorischer Analyse, wie jede planare N-Punkt-Funktion im quartischen Model ex- plizit (nicht rekursiv) berechnet werden kann. Die starken Unterschiede zwischen dem kubischen und quartischen Modell sind algebraisch-geometrischer Natur. Die Bestimmung der Korrelationsfunktionen durch topologische Rekursion bedient sich einer algebraischen Kurve, die im kubischen Modell nur einen Verzweigungspunkt hat, der mit der Polstelle der stabilen Korrelationsfunk- tionen ¨ubereinstimmt. Das quartische Modell hingegen hat im Kontinuumslimes un- endlich viele Verzweigungspunkte, die zu unendlich vielen Zweigen f¨uhren. Danksagung Zu Beginn m¨ochte ich mich bei meinem Betreuer Raimar Wulkenhaar bedanken, der sich jedes Mal die Zeit f¨urDiskussion mit mir genommen hat. Seine vorausgehenden Ar- beiten und seine Unterst¨utzung waren maßgeblich an der Anfertigung dieser Dissertation beteiligt. Desweiteren will ich mich bei Harald Grosse, Akifumi Sako, J¨orgSch¨urmannund Roland Speicher bedanken, die mir durch die unterschiedlichsten Unterhaltungen ver- schiedene neue Blickwinkel auf meine Arbeit er¨offnethaben. Ihre Hinweise haben daf¨ur gesorgt, dass meine Arbeit an der Dissertation kontinuierlich voranging. Die angenehme Arbeitsatomsph¨are mit Alexander Stottmeister, Carlos I. P´erez- S´anchez, Jins de Jong, Johannes Branahl, Johannes Th¨urigenund Romain Pascalie war durch intensive Diskussionen erf¨ullt. Finanziell wurde ich w¨ahrend meiner Promotion unterst¨utztdurch das Mathematische Institut M¨unsterund die DFG mit dem Cluster of Excellence "Mathematics M¨unster", dem RTG 2149 und dem SFB 878. Ein großer Dank geht auch an meine Eltern und meine Freunde, die mich schon immer auf meinem Weg in der Physik und der Mathematik unterst¨utzthaben. Doch am meisten bedanke ich mich bei meinen beiden S¨ohnen,Philipp und Maxim, sowie meiner geliebten Frau Caroline. to my wife Caroline Contents 1 Introduction 1 1.1 Quantum Field Theory . .1 1.2 Quantum Field Theory on Noncommutative Geometry . .2 1.3 2D Quantum Gravity . .3 1.4 Algebraic Geometry . .4 1.5 Outline of the Thesis . .5 2 Matrix Field Theory 7 2.1 Perturbation Theory . .9 2.1.1 Ribbon Graphs . 11 2.1.2 Perturbative Expansion . 12 2.1.3 Feynman Rules . 14 2.2 Schwinger-Dyson Equation and Ward-Takahashi Identity . 16 2.3 Large N ;V -Limit . 21 2.4 Renormalisation . 23 2.4.1 Zimmermann's Forest Formula . 24 2.5 QFT on the Moyal Space and Matrix Field Theory . 26 3 Cubic Interaction 31 3.1 Schwinger-Dyson Equations . 31 3.2 Solution of the Schwinger-Dyson Equations . 37 3.2.1 Large N ;V -Limit . 37 3.2.2 The Planar 1-Point Function . 38 3.2.3 Analytic Continuation . 44 3.2.4 Solution for χ = 2 − 2g − b ≥ −1 ................... 46 3.2.5 Solution for b > 1 via Boundary Creation Operator . 47 3.2.6 Solution for b = 1 and g > 0...................... 50 3.2.7 Link to Topological Recursion . 53 3.3 Free Energy . 54 3.3.1 A Laplacian to Compute Intersection Numbers . 56 3.3.2 Deformed Virasoro Algebra . 60 3.4 Over-Renormalisation . 63 3.5 Summary . 64 4 Quartic Interaction 66 4.1 Schwinger-Dyson Equations . 67 4.2 Solution of the Planar 2-Point Function . 76 4.2.1 Solution of the Angle Function . 78 4.2.2 Solution of the 2-Point Function . 83 4.2.3 Solution for Finite Matrices (D = 0) . 87 4.2.4 Solution on the D = 4 Moyal Space . 93 4.3 Higher Order Correlation Functions . 98 4.4 Explicit Form of the Planar
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