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Research Collection Research Collection Doctoral Thesis Non-Abelian Orbifold Theory and Twisted Modules for Vertex Operator Algebras Author(s): Gemünden, Thomas Publication Date: 2020 Permanent Link: https://doi.org/10.3929/ethz-b-000446462 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library diss. eth no. 26903 NON-ABELIANORBIFOLDTHEORYANDTWISTEDMODULES FORVERTEXOPERATORALGEBRAS A thesis submitted to attain the degree of doctor of sciences of eth zurich¨ (Dr. sc. ETH Z¨urich) presented by thomas gemunden¨ Master of Mathematics, University of Cambridge Master of Arts, University of Cambridge born on 05.02.1994 citizen of Germany accepted on the recommendation of Prof. Dr. G. Felder, Referent Prof. Dr. C. A. Keller, Korreferent Prof. Dr. N. Scheithauer, Korreferent 2020 Thomas Gem¨unden, Non-Abelian Orbifold Theory and Twisted Modules for Vertex Operator Algebras Abstract The topic of this thesis is the theory of holomorphic extensions of orbifolds of holomorphic vertex operator algebras. We develop an orbifold theory for non-abelian groups of the form Zq o Zp and give an explicit formula for the characters of the holomorphic extensions. Additionally, we show how groups of the form Zq o Zp can be realised as automorphism groups of lattice vertex operator algebras. Furthermore, we construct the action of automorphisms on twisted modules for lattice vertex operator algebras and give explicit expressions for the corresponding graded trace functions. Using these results, we conduct a survey of orbifolds by cyclic groups and groups of the form Zq o Zp of lattice vertex operator algebras associated to extremal lattices in dimensions 48 and 72. This is the first systematic survey of holomorphic vertex operator algebras at higher central charge, as well as the first systematic survey of non-abelian orbifold theories. We construct around 200 new holomorphic vertex operator algebras at central charge c = 48 and c = 72. These include those with the lowest number of low-weight states currently known among holomorphic vertex operator algebras of the respective central charges. These allow us to prove that there exist holomorphic vertex operator algebras that cannot be constructed as abelian orbifolds of lattice vertex operator algebras. Finally, we investigate the notion of `large-central charge limit' of a family of vertex operator algebras. We define a vertex algebra that can be considered such a large-central charge limit for a family of permutation orbifolds and give a sufficient condition for its existence. 3 Zusammenfassung Das Thema dieser Abhandlung ist die Theorie der holomorphen Erweiterungen der Orbifaltigkeiten von holomorphen Vertexoperatoralgebren. Wir entwickeln eine Theorie der Orbifaltigkeiten f¨urnicht-abelsche Gruppen der Form Zq o Zp und bestimmen einen expliziten Ausdruck f¨urdie Charaktere der holomorphen Erweiterungen. Zus¨atzlich, zeigen wir, wie Gruppen der Form Zq o Zp als Automorphismusgruppen von Gittervertexoperatoralgebren realisiert werden k¨onnen. Weiterhin konstruieren wir die Wirkung von Automorphismen auf die verdrehten Moduln von Gitter- vertexoperatoralgebren und bestimmen die dazugeh¨origengradierten Spurfuntionen. Mithilfe dieser Resultate unternehmen wir eine Untersuchung der Orbifaltigkeiten von zyklischen Gruppen und Gruppen der Form Zq o Zp f¨urGittervertexoperatoralgebren von extremalen Gittern in den Dimensionen 48 und 72. Dies ist die erste systematische Untersuchung von holomorphen Vertexoper- atoralgebren gr¨ossererzentraler Ladung, sowie die erste systematische Untersuchung von nicht-abelschen Orbifaltigkeiten. Wir konstruieren etwa 200 neue holomorphe Vertexoperatoralgebren der zentralen Ladungen 48 und 72. Darunter sind die holomorphen Vertexoperatoralgebren mit der jeweils kleinsten bekannten Anzahl von leichten Zust¨andenunter holomorphen Vertexoperatoralgebren ihrer zentralen Ladung. Diese erlauben uns zu beweisen, dass holomorphe Vertexoperatoralgebren existieren, die nicht als abelsche Orbifaltigkeiten von Gittervertexoperatoralgebren konstruiert werden k¨onnen. Zuletzt untersuchen wir das Konzept des `Grenzwertes zur grossen zentralen Ladung` einer Familie von Vertexoperatoralgebren. Wir definieren eine Vertexalgebra, die als Grenzwert zur grossen zentralen Ladung einer Familie von Permutationsorbifaltigkeiten angesehen werden kann und geben eine hinreichende Bedingung f¨urihre Existenz. 4 Acknowledgements I would like to express my gratitude to Prof. Dr. Christoph Keller for his excellent supervision and his patience and encouragement throughout a remarkably eventful doctorate. I would like to thank Prof. Dr. Giovanni Felder for valuable discussions and advise, helpful comments on this text and finally for taking me in as his student. I am grateful to Prof. Dr. Nils Scheithauer for invaluable advise and helpful comments on this text. I would like to thank Prof. Dr. Gerald H¨ohnfor his kind hospitality during my stay in Manhattan. Finally, I would like to thank my colleague Simon Brun, for his fantastic work as a group organiser. My work is supported by the Swiss National Science Foundation Project Grant 175494. 5 Contents Introduction 8 1 Preliminaries and Definitions 13 1.1 Formal Calculus . 13 1.2 Vertex Operator Algebras . 13 1.3 Automorphisms of Vertex Operator Algebras . 14 1.4 Modules and Twisted Modules . 14 1.5 Regularity and Rationality . 16 1.6 Holomorphic Vertex Operator Algebras . 17 1.7 Homomorphisms, Schur's Lemma and Action of Automorphisms on Twisted Modules . 17 1.8 Characters, Graded Trace Functions and Modular Invariance . 18 1.9 Intertwining operators, Fusion and Vertex Tensor Categories . 19 2 Orbifold Theory 22 2.1 Holomorphic Extensions of Vertex Operator Algebras . 22 2.2 Cyclic Orbifolds . 25 2.3 Non-Abelian Orbifolds . 27 2.3.1 Groups with Periodic Cohomology . 28 2.3.2 Semidirect Products of Cyclic Groups Zq oφ Zp .................... 29 2.3.3 Characters of Holomorphic Extensions for Zq oφ Zp ................. 30 3 Lattice Vertex Operator Algebras 33 3.1 Lattices . 33 3.1.1 Integral Lattices . 33 3.1.2 Lattice theta-functions and Modularity . 34 3.1.3 Lattice Automorphisms . 35 3.1.4 Smith Normal Form and Lattice Quotients . 36 3.2 Construction of Lattice Vertex Operator Algebras . 37 3.3 Automorphisms of Lattice Vertex Operator Algebras . 39 3.4 Graded Trace Functions for Lattice Vertex Operator Algebras . 40 3.5 Finite Automorphism Groups for Lattice Vertex Operator Algebras . 41 3.5.1 Lifting Semidirect Products of Cyclic Groups Zq oφ Zp ................ 43 4 Twisted Modules for Lattice Vertex Operator Algebras 47 4.1 Construction of Twisted Modules . 47 4.2 The Vacuum Representation . 49 4.3 A Basis for VL(^g)......................................... 52 4.4 Action of Automorphisms on Twisted Modules . 53 4.5 Graded Trace Functions for Twisted Modules . 53 4.6 Examples: Orbifolds of VE8 ................................... 55 4.6.1 An S3-Orbifold . 55 4.6.2 A Z2 × Z2-Orbifold . 55 4.6.3 A Z3 × Z3-Orbifold . 56 5 Cyclic Orbifolds of Lattice Vertex Operator Algebras at c = 48 and c = 72 57 5.1 Computational Aspects | Characters of Holomorphic Extensions for Cyclic Groups . 57 5.1.1 Eta-Quotients and Modular Transformations of Lattice Theta-Functions . 57 5.1.2 Computing Characters of Cyclic Orbifolds . 60 5.2 The Orbifold Algorithm . 61 5.3 Cyclic Orbifolds of Extremal Lattices in d =48........................ 62 5.3.1 Cyclic Orbifolds for the Lattice P48m .......................... 63 5.3.2 Cyclic Orbifolds for the Lattice P48n .......................... 63 6 5.3.3 Cyclic Orbifolds for the Lattice P48p .......................... 65 5.3.4 Cyclic Orbifolds for the Lattice P48q .......................... 65 5.4 Cyclic Orbifolds for the Lattice Γ72 ............................... 66 6 Non-Abelian Orbifolds of Lattice Vertex Operator Algebras at c = 48 and c = 72 68 6.1 Zq oφ Zp-Orbifolds for d =48.................................. 68 6.1.1 Zq oφ Zp-Orbifolds for the lattice P48p ......................... 68 6.1.2 Zq oφ Zp-Orbifolds for the lattice P48q ......................... 68 6.1.3 Zq oφ Zp-Orbifolds for the lattice P48m ......................... 69 6.1.4 Zq oφ Zp-Orbifolds for the lattice P48n ......................... 69 6.2 Zq oφ Zp-Orbifolds for the Lattice Γ72 ............................. 70 6.3 Bounds for Abelian Orbifolds . 70 7 The Large-N-Limit for Permutation Orbifolds 72 7.1 Oligomorphic Families of Permutation Groups . 72 7.1.1 Unitary Vertex Operator Algebras . 72 7.1.2 Oligomorphic Orbifolds . 73 7.1.3 Space of states . 74 7.1.4 Structure Constants . 75 7.1.5 An Example: SN ..................................... 77 7.2 The Limit of GL(N; q)-Orbifolds . 78 Bibliography 81 7 Preface The topic of this text are certain algebraic structures, called vertex operator algebras. They can be used to construct full realisations of conformal field theories, special models in particle physics that play a central role in string theory. In informal terms, our goal will be to construct new vertex operator algebras from old ones and their symmetries and hence construct new conformal field theories. The theory of vertex operator algebras is a relatively new field of study with deep roots in different areas of mathematics and physics that appear entirely unrelated at first sight. First introduced in the 1980s by Richard Borcherds [Bor86] vertex operator algebras played a crucial role in his proof of the \monstrous moonshine"
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