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Planar Homotopy Algebras and Open-String Field Theory Planar Homotopy Algebras and Open-String Field Theory Christoph Mario Chiaffrino M¨unchen2020 Planar Homotopy Algebras and Open-String Field Theory Christoph Mario Chiaffrino Dissertation an der Fakult¨atf¨urPhysik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Christoph Mario Chiaffrino aus Oderding M¨unchen, den 14.09.2020 Erstgutachter: Prof. Dr. Ivo Sachs Zweitgutachter: Dr. Olaf Hohm Tag der m¨undlichen Pr¨ufung:26.11.2020 Zusammenfassung Diese Dissertation besch¨atigtsich mit der Frage nach der Existenz einer offenen Stringfeldthe- orie, welche auf der Quantenebene ohne eine Kopplung an den geschlossenen String konsis- tent ist. Dies soll durch eine Einschr¨ankungauf planare Feynman Graphen geschehen. Ziel ist eine Formulierung einer solchen Theorie in der mathematischen Sprache der Homotopieal- gebren zu finden. Anschließend untersuchen wir ob sich eine solche Formulierung auch auf allgemeine Eichfeldtheorien, insbesondere im Limes großer Eichgruppen ¨ubertragen l¨asst. Zum Schluss gehen wir auf Probleme einer solchen Formulierung ein, und wie sich diese durch eine Erweiterung weg von Beschr¨ankungauf rein planare Graphen beheben lassen. Diese Arbeit soll gleichzeitig als ausgedehnte Einf¨uhrungin die Theorie des Batalin- Vilkovisky Formalismus dienen. Wir betrachten verschiedene mathematische Aspekte dieses Formalismus und dessen Bezug zu Homotopiealgebren. v Abstract This thesis is concerned about the existence of a open-string field theory that is consistent at the quantum level without coupling to the closed string. We want to achieve this via a restriction to planar Feynman graphs. Our aim is to formulate this theory in the mathemat- ical language of homotopy algebras. We further ask whether such a formulation is applies also to general gauge theories, in particular in the limit of large gauge groups. Finally, we will discuss the problems of such a formulation, as well as how these can be solved by lifting the restriction to planar diagrams only. This work should also serve as an extensive introduction to the Batalin-Vilkovisky for- malism. We look at different mathematical aspects of this formalism and its relation to homotopy algebras. vii Danksagung An vorderster Stelle m¨ochte ich meinem Betreuer Ivo Sachs f¨urall die Zeit und M¨uhen danken, die er w¨ahrendder Zeit, die ich in seiner Forschungsgruppe verbracht habe, aufge- bracht hat. Ich kann mir kaum einen besseren Doktorvater vorstellen. Ebenso danke ich Sebastian Konopka, Ted Erler und Brano Jurˇco,deren fachliche Expertisen mir sehr geholfen haben. Des Weiteren erinnere ich mich gerne zur¨uck an meine gute Gesellschaft im B¨uro A328 und bei der Mittagspause, namentlich Marc Schneider, Leila Mirzagholi, Cecilia Gi- avoni, Maximilian K¨ogler,Ludwig Eglseer, Charlie Mattschas, Maximilian Urban und Katrin Hammer. Ein großes Dankesch¨ongeht auch an Herta f¨urihre tolle Arbeit. Ich danke meiner Mutter Rosa, sowie meinen Schwestern Daniela, Christina und Martina f¨urall die Liebe und Unterst¨utzung. Ich habe die beste Familie der Welt! Mein Dank gilt auch Willi und Darco f¨urihre Hilfe. ix Contents 1. Introduction1 1.1. Gauge Theories and the Batalin-Vilkovisky Formalism . .1 1.2. BV Formalism in String Field Theory . .3 1.2.1. Open-Closed String Field Theory . .4 1.2.2. A Simple String Field Theory? . .4 1.3. Homotopy Algebras in Field Theories . .5 1.4. Quantum A1-Algebras and Planar Field Theories . .5 1.5. Overview . .6 2. Homotopy Algebras and the Batalin-Vilkovisky Formalism9 2.1. Homotopy Algebras . 11 2.2. Diagrammatic Representation of Products . 14 2.3. Bar Construction of Homotopy Associative and Homotopy Lie Algebras . 15 2.3.1. Tensor (Co-)Algebra . 15 2.3.2. Symmetric Tensor (Co-)Algebra . 17 2.3.3. A Definition of Homotopy Associative and Homotopy Lie Algebras . 18 2.3.4. The Homological Perturbation Lemma and the Homotopy Transfer Theorem . 21 2.3.5. The Minimal Model . 24 2.3.6. A1-Products as Polytopes . 26 2.3.7. Cyclic Homotopy Algebras . 27 2.4. The Batalin-Vilkovisky Formalism of Classical Gauge Theory . 29 2.4.1. Quasi-Isomorphisms in Field Theory . 29 2.4.2. Homotopy Intersections . 30 2.4.3. Homotopy Quotients . 33 2.4.4. The Batalin-Vilkovisky Formalism . 34 2.4.5. Solving the Classical Master Equation . 39 2.4.6. Symplectic Geometry in BV . 44 2.4.7. More Field Theory Examples of BV Theories . 45 2.5. From Batalin-Vilkovisky to Homotopy Algebras . 46 2.5.1. Example: Deformations in φ3-theory . 48 2.5.2. General Deformations . 50 2.5.3. Chern-Simons and Yang-Mills Theory as L1-Algebras . 52 2.5.4. From L1-Algebras to A1-Algebras in Field Theory . 55 2.5.5. Chern-Simons and Yang-Mills Theory as A1-Algebras . 56 3. Quantum Homotopy Algebras and the Batalin-Vilkovisky Formalism 59 3.1. Path Integral without Gauge Symmetries - The Twisted de Rham Complex . 59 3.2. Gauge Symmetries . 63 3.3. General Quantum BV Formalism . 65 xi Contents 3.4. Integration and Gauge Fixing . 66 3.4.1. Perturbative Evaluation of Integrals . 68 3.4.2. Example: Gauge Fixing Electromagnetism without Trivial Pairs . 71 3.5. An Anomaly Computation . 72 3.5.1. Canceling the Anomaly by Introducing New Particles . 75 3.6. From Quantum BV to Quantum Homotopy Algebras . 76 3.6.1. Higher Order Coderivations over Commutative Coalgebras . 76 3.6.2. Quantum Homotopy Lie Algebras . 78 3.6.3. Quantum Homotopy Associative Algebras . 80 4. Geometric Vertices and String Field Theory 83 4.1. Unoriented Riemann Surfaces with Boundary . 83 4.2. Local Parametrizations . 84 4.3. Geometric Vertices and Sewing Operations . 85 4.4. A (Very Brief) Introduction to String Field Theory . 88 4.5. Stubs in String Field Theory and Regularization . 89 4.6. The Green-Schwarz Anomaly in String Field Theory . 93 4.6.1. Cancellation at the One-Loop Level . 93 4.6.2. The Closed String Reappearing at Two Loops . 99 4.7. Does the Loop A1-algebra know about the L1-algebra? . 100 5. A planar quantum A1-algebra 103 5.1. (Co-)derivations in Higher Orders for Noncommutative Algebras . 103 5.2. Some Examples of Derivations in Higher Orders . 109 5.3. The Anomalous Behavior of the Braces with Respect to Invertible Elements . 110 5.4. Definition of Planar Quantum Homotopy Algebras . 111 5.5. The Large N Limit . 112 5.6. Gauge Invariance of Planar Amplitudes . 116 5.7. A Larger Subsector of Loop Homotopy Associative Algebras . 121 6. Conclusion and Outlook 123 A. Graded Vector Spaces and the D´ecalage Isomorphism 125 B. Chain complexes over Vector Spaces 129 References . 131 xii 1. Introduction 1.1. Gauge Theories and the Batalin-Vilkovisky Formalism During the development of physics in the last century, quantum field theories crystallized as the most effective tool to describe and predict physical processes. In high energy physics, the standard model puts quarks and leptons as the building blocks of matter in our world. It also describes their interaction in terms of the strong and the electroweak forces. This standard model of particles and forces is formulated as a quantum field theory (QFT). But even away from this fundamental level QFT is dominant as a language. When we zoom out of the scale of quarks and gluons, we enter the realm of hadrons. In this picture, the atomic nucleus is made up of baryons (protons and neutrons, but also more exotic objects are possible), which are held together by mesons (most importantly pions). This model is also described by a quantum field theory. At even larger scales, quantum field theories appear in solid state physics. Finally, at astronomical and cosmological scales, we use general relativity, which can be treated as a classical field theory. Since quantum field theory is ubiquitous in physics, a proper understanding of it is of course necessary. A major complication arises when the theory is a gauge theory. This means that the theory is invariant under certain variations in the field variables. These variations are called gauge symmetries of the theory. The physical quantities in a gauge theory are independet of whether we use one set of fields or a gauge equivalent one. \Unfortunately", all the experimentally tested fundamental theories are gauge theories. Needless to say, it is expected that a more fundamental theory is a gauge theory as well, which may possess even more complicated gauge structure. A potential candidate is string (field) theory, which has a gauge symmetry that is infinitely more complex (in at least two ways) than that of the standard model. A very simple example of a gauge theory is quantum electrodynamics, the theory of elec- trons (or any electrically charged particle) interacting through electromagnetic forces. One manifestation of this simplicity lies in the fact that photons, the mediators of the electro- magnetic force, are electrically neutral (not charged) and therefore do not influence each other directly. They can only do so using electrons and positrons as mediators. Because of that reason the techniques we explore in this work are not needed in quantum electrody- namics. Nevertheless, they can of course also be applied in this case. Things can change when one considers theories with more than one force carrying particle (gauge bosons). Let us add weak interactions to the game. We get three additional gauge bosons, abbreviated by the letters W +;W −;Z. In addition to being electrically charged, they also possess a \charge" called the weak isospin. All particles with weak isospin (like quarks and the gauge bosons themselves) interact via the three gauge particles. In this way the gauge particles can scatter even without creating fermions (matter particles) or photons. The fundamental interaction among gauge bosons leads to inconsistencies when we treat weak interactions the same way as electromagnetic interactions.
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