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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

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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 fürnicht-abelsche Gruppen der Form Zq o Zp und bestimmen einen expliziten Ausdruck fürdie Charaktere der holomorphen Erweiterungen. Zusätzlich, zeigen wir, wie Gruppen der Form Zq o Zp als Automorphismusgruppen von Gittervertexoperatoralgebren realisiert werden können. Weiterhin konstruieren wir die Wirkung von Automorphismen auf die verdrehten Moduln von Gitter- vertexoperatoralgebren und bestimmen die dazugehörigengradierten Spurfuntionen. Mithilfe dieser Resultate unternehmen wir eine Untersuchung der Orbifaltigkeiten von zyklischen Gruppen und Gruppen der Form Zq o Zp fürGittervertexoperatoralgebren von extremalen Gittern in den Dimensionen 48 und 72. Dies ist die erste systematische Untersuchung von holomorphen Vertexoper- atoralgebren grössererzentraler 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 Zuständenunter holomorphen Vertexoperatoralgebren ihrer zentralen Ladung. Diese erlauben uns zu beweisen, dass holomorphe Vertexoperatoralgebren existieren, die nicht als abelsche Orbifaltigkeiten von Gittervertexoperatoralgebren konstruiert werden können. 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 fürihre 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(ˆg)...... 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” conjectures — a remarkable set of conjectures by Conway and Norton [CN79] revolving around an unexpected connection between the Monster finite simple group M and Klein’s j-function that was first established with the construction of the famous moonshine module V \ by Frenkel, Lepowsky and Meurman [FLM88]. This lead to various important developments regarding connections between algebraic structures and the theory of modular forms as well as in physics.

Introduction

Let V be a vertex operator algebra. A (2-dimensional) chiral conformal field theory can, roughly speaking, be considered a suitable category of V -modules together with corresponding intertwining operators satisfying certain modular invariance and convergence properties. The problem of constructing a conformal field theory from a given vertex operator algebra is hard because the representation theory of vertex operator algebras is in general poorly understood. A central problem in the field of conformal field theory is the problem of construction. One approach to construction that has been very successful in the past is orbifold theory. Let V be a vertex operator algebra and G an automorphism group of V . Then we denote the subalgebra of G-invariant elements of V by V G. The fundamental objective of orbifold theory is to construct and classify the conformal field theories associated to the vertex operator algebra V G. In the most general case, this is very much an open problem. The first challenge is to construct and study modules of V G. Since it has proved difficult to infer properties of V G-modules from those of V directly, a more fruitful approach is to study so-called g-twisted V -modules for g ∈ G that, in a sense, are considered a generalisation of V G-modules. The most successful construction method for g-twisted modules currently available associates a g-twisted V -module to a module for a certain associative algebra Ag(V ), called twisted Zhu’s algebra [DLM98, HY19]. Problems with this approach are that, in this framework, it is hard to study properties of general g-twisted modules and that Ag(V ) and its modules are, in general, hard to construct and classify themselves. Promising advances were recently made by Huang [Hua19a], who constructed a universal lower-bounded g-twisted [g] V -module McB that can be considered the vertex operator algebraic analogue to a Verma module for a finite-dimensional Lie algebra. These objects have the potential to facilitate the construction of twisted modules as well as systematic research into their properties. On the problem of constructing and studying twisted intertwining operators, much less progress has been made. Twisted intertwining operators among twisted modules associated to non-commuting automorphisms were, in fact, only introduced recently [Hua18]. The central questions in orbifold conformal field theory are the convergence of products, iterates and q-traces, the operator product expansion (also commonly referred to as associativity) and the modular invariance of twisted intertwining operators, especially in the nonabelian case. All these properties were proved by Huang in the case of untwisted intertwining operators for strongly rational vertex operator algebras [Hua05a, Hua05b]. A particularly interesting class of vertex operator algebras are the so-called holomorphic vertex operator algebras, containing among others the moonshine module V \. The defining property of a holomorphic vertex operator algebra is that it is its own unique irreducible module. The problem of constructing a conformal field theory from a holomorphic vertex operator algebra V then becomes trivial, as V is a conformal field theory by itself. The most prominent method to construct such vertex operator algebras is the lattice construction, where a holomorphic vertex operator algebra is associated to an even, self-dual lattice. In the very restricted setting where V is a holomorphic vertex operator algebra and G a finite group of automorphisms, much more is known about orbifold theory. Specifically, significant progress has recently been made on the problem of classifying extensions of the fixed-point vertex operator algebras V G by a suitable category of modules to a new holomorphic vertex operator algebra. While there has been

8 significant attention in recent years, a full theory of holomorphic extensions is only available for individual families of finite groups. A recent breakthrough in this area was the theory of cyclic orbifolds by van Ekeren, Möllerand Scheithauer [vEMS17]. They were able to give a necessary and sufficient condition for the existence of holomorphic extensions for the case where G is cyclic and showed that the characters of the extensions can be calculated by making use of the modular transformation properties of graded trace functions on V without having to construct the irreducible V G-modules explicitly. In the past years, this program of constructing holomorphic vertex operator algebras using orbifold theory has been extremely successful. A recent breakthrough was the almost complete classification and construction of all holomorphic vertex operator algebras at central charge c = 24, a research endeavour of over two decades [FLM88, DGM90b, Don93,DGM96,Lam11, LS12,Lam11,Miy13,vEMS17,Mö16,SS16,LS16a, LS16b, LL16]. The remaining piece in this classification and one of the big open problems in the theory of vertex operator algebras is a proof of the uniqueness of the moonshine module V \. Recent attempts to find a proof were unsuccessful, and no promising pathways towards finding a proof in the near future have been proposed. The theory of cyclic orbifolds was, furthermore, used in Scott Carnahan’s recent proof of the Generalized Moonshine Conjecture [Car12]. It was also shown [MS19] that at c = 24 all 70 possible holomorphic vertex operator algebras with non-trivial weight-1 space can be constructed as extensions of cyclic orbifolds of lattice vertex operator algebras. Terry Gannon and David Evans recently made significant progress on the problem of orbifolding by general finite groups in [EG18]. Building on the fact that if V is holomorphic and G finite, the module G category V − mod is equivalent to the module category of a twisted Drinfeld double Dω(G) − mod, with ω ∈ Z3(G, C∗), they were able to fully classify all possible holomorphic extensions. The cohomological twist ω acts, in a certain sense, as an obstruction to orbifolding, so that it is central for us to understand if ω is in the trivial class for a given V G. In general, this is an open problem. In particular, ω can generally not be inferred from modular data alone [MS17]. To construct orbifold vertex operator algebras, we first need to construct the irreducible V G-modules. When G is finite, they can be obtained as submodules of g-twisted V -modules. Specifically, the centraliser G CG(g) acts projectively on the irreducible g-twisted module V (g) and the irreducible V -modules arise as the multiplicities of the corresponding irreducible projective representations of CG(g); a property referred to as the Schur-Weyl-Type Duality [DLM00, DRX17]. The critical step in finding these submodules is to understand how elements of CG(g) act on the irreducible g-twisted module. Twisted modules for lattice vertex operator algebras were first explicitly described by Lepowsky [Lep85] and later again by Bakalov and Kac [BK04] so that in this case substantial calculations are possible. Another topic of current interest is the structure of holomorphic vertex operator algebras at a central charge larger than 24, where very little is known. Attempts to find a full classification at any higher central charge are widely considered hopeless because the corresponding result for even self-dual lattices already appears to be forever out of reach: The number of even self-dual lattices in higher dimensions is known to be inconceivably large, and no useful method to construct them all seems to be anywhere in sight [CS99]. However, a question that is still worth asking is whether anything can be said about which holomorphic vertex operator algebras can be constructed from a given one using the orbifold construction at higher central charge. It is, for example, known that, in principle, any even, self-dual lattice can be constructed from any other one of the same dimension using Kneser’s neighbour method [Kne57] in a finite number of steps. Using the theory of cyclic orbifolds, it is easy to show that this result can, in fact, be extended to lattice vertex operator algebras by orbifolding by inner automorphisms. Whether it might be possible to construct any holomorphic vertex operator algebra as an orbifold of a lattice vertex operator algebra, is a completely open problem. A further important application of vertex operator algebras and conformal field theory is the AdS/CFT correspondence that conjecturally maps theories of quantum gravity to certain families of vertex operator algebras that admit a large-central charge limit [HR15, BKM15, BKM16, KM19], but as of yet no mathematically precise definition of such a limit has been given. Families of particular interest are those with a small number of light states that are expected to correspond to theories of pure gravity [Wit07]. However, very few suitable vertex operator algebras are currently known, and vertex operator algebra with a minimal number of weight-2 states have been conjectured not to exist by physicists. The existence of conformal field theories with a smaller number of low-weight states has recently also been studied by Gerald Höhn[Hö19]. As orbifold theory provides good control of the low-weight subspaces, it is well-suited as a tool to construct such vertex operator algebras.

9 Structure and Results

This text is structured as follows: In the first chapter, we give a review of basic definitions and results on vertex operators algebras, their representation theory, and their automorphisms. The second chapter first reviews the theory of holomorphic extensions of orbifolds of holomorphic vertex operator algebras and the theory of cyclic orbifolds. We show that the finite groups with periodic cohomology are exactly the groups for which the character of the holomorphic extension can be computed without the construction of twisted modules by using the modular invariance of graded trace functions. Then we discuss the orbifold theory of non-abelian groups with periodic cohomology of the form Zq o Zp. In particular, we show that:

3 ∗ Theorem. Let Zq o Zp be a finite group with periodic cohomology. Then H (Zq o Zp, C ) is cyclic, and 3 ∗ [ω] ∈ H (G, C ) is trivial if and only if the type 0 condition is satisfied for the cyclic subgroups Zp and Zq. Furthermore, we give an explicit expression for the character of the holomorphic extension in terms of characters of orbifolds by cyclic subgroups. In the third chapter, we discuss the construction of lattice vertex operator algebras, their automorphism groups, and graded trace functions. Then we study the properties of finite automorphism groups of lattice vertex operator algebras. In particular, we construct groups of the form Zq oφ Zp as automorphism groups of lattice vertex operator algebras. In the fourth chapter, we review the construction of twisted modules for lattice vertex operator algebras. We explicitly construct the lowest weight-space of an irreducible twisted module as an irreducible projective representation of a certain finite abelian group. Building on this, we construct the action of automorphisms on twisted modules for lattice vertex operator algebras and give explicit expressions for the graded trace functions on twisted modules. Finally, we construct irreducible modules for the fixed-point subalgebras for three examples of non-cyclic finite groups acting on the E8 lattice vertex operator algebra and verify that their characters exhibit the expected modular transformation properties. In the fifth chapter, we conduct a survey of cyclic orbifolds 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 in the literature, and we construct around 150 new holomorphic vertex operator algebras at central charge c = 48 and c = 72, including ones with a small number of low-weight states. In particular, we prove the following theorem: Theorem. There exists a strongly rational, holomorphic vertex operator algebra with central charge c = 48, such that dim V(1) = 0 , dim V(2) = 48 , and a strongly rational, holomorphic vertex operator algebra with central charge c = 72, such that

dim V(1) = 0 , dim V(2) = 36 dim V(3) = 408 . In the sixth chapter, we systematically construct orbifolds for groups with periodic cohomology of the form Zq o Zp of lattice vertex operator algebras associated to the same lattices. This is the ﬁrst systematic survey of holomorphic extensions for non-abelian orbifolds in the literature, and we construct around 50 new holomorphic vertex operator algebras. In particular, we prove that Theorem. A strongly rational, holomorphic vertex operator algebra with central charge c = 48, such that

dim V(1) = 0 , dim V(2) = 48 can be constructed as an orbifold of any lattice vertex operator algebra associated to one of the three extremal lattices in d = 48 with automorphisms of large order [Neb13], P48n,P48p and P48q. Furthermore, we prove that Theorem. There exists a strongly rational, holomorphic vertex operator algebra with central charge c = 72 and dim V(1) = 0 , dim V(2) = 12 , dim V(3) = 200 . By the obtaining a lower bound for the dimension of the weight-2-space for abelian orbifolds of lattice vertex operator algebras we can additionally show that:

10 Corollary. There exists a strongly rational, holomorphic vertex operator algebra V at central charge −3 −1 d = 72 with character χV (τ) = q + 12q + 200 + O(q), that cannot be constructed as a cyclic orbifold of a lattice vertex operator algebra by a lifted lattice automorphism.

N The AdS/CFT correspondence maps a given theory of quantum gravity to a family {V }N∈N of vertex operator algebras, whose central charges are parametrized by N. Physicists are most interested in the ‘large central charge limit’, that is the limit N → ∞ of this family. These notions of families of vertex operator algebras and their limits have not been defined mathematically, and much less investigated systematically. In the seventh chapter, we investigate the notion of the large-central charge limit for families of permutation orbifolds. We define a vertex algebra that can be considered such a large-central charge limit and give a sufficient condition for its existence depending only on the permutation group involved. Furthermore, we investigate a family of permutation orbifolds by general linear groups GL(N, q). In particular, we show that for such orbifolds the conformal weights of all twisted modules diverge in the large-central charge limit. This thesis is based on the three papers [GK19c,GK19b] and [GK19a].

Outlook

Due to recent advances, many interesting and important open problems in the ﬁeld of orbifold conformal ﬁeld theory are now accessible [Hua20]. Some problems that will be addressed in future work are the following:

• Grading restricted quotients of universal twisted modules: In the past year, Huang has made enormous advances in the theory of twisted modules and solved some long-standing open problems [Hua19a, Hua19b, Hua19c]. Particularly with his construction of the universal twisted modules, many further open problems and conjectures on twisted modules can be solved. A very fundamental open problem that needs to be addressed is the following: As the orbifold-theoretic analogue to Verma modules, Huang’s universal twisted modules have, by construction, infinite-dimensional homogeneous subspaces. An open problem is to construct quotients of such universal twisted modules whose homogeneous subspaces are finite-dimensional and to find out under which conditions such quotients exist. This problem is an orbifold-theoretic analogue of the well-known result in the theory of finite-dimensional Lie algebras that Verma modules generated by highest weights have finite-dimensional irreducible quotients if the highest weights are dominant and integral. Apart from the fundamental importance of this question, improved construction methods for twisted modules would enable us to study orbifold theories associated to further classes of vertex operator algebras. Currently, substantial calculations in orbifold theory are limited largely to lattice vertex operator algebras. The first step towards this objective is to establish the connection between twisted Zhu’s algebras and universal twisted modules. • Convergence and modular invariance of twisted intertwining operators for lattice orbifolds: The central questions in orbifold conformal field theory are the convergence of products, iterates and q-traces, the operator product expansion (also commonly referred to as associativity) and the modular invariance of twisted intertwining operators, especially in the non-abelian case. In particular, while current construction methods for orbifold conformal field theories are, for the most part, restricted to their existence and character, solving this question would enable the construction and study of full orbifold conformal field theories. We want to approach this question by generalising Huang’s methods for (untwisted) intertwining operators [Hua05a, Hua05b] for some explicit examples of non-abelian orbifold vertex operator algebras constructed in this thesis. • The structure of lattice orbifolds: Greg Moore asked a very fundamental question: Can every holomorphic vertex operator algebra be constructed as an orbifold theory of a lattice vertex operator algebra? This is an extremely difficult problem and, as of yet, there is very little intuition for how it might best be approached. In fact, there is no evidence that the answer to this question should be ”yes”. Even for holomorphic vertex operator algebras of central charge c = 24, where, at first sight, it appears to be the case, that all theories can be constructed as orbifold theories of lattice

11 vertex operator algebras, in fact, a proof of the uniqueness of the moonshine module is still missing. On the other hand, if it were possible to prove that the answer to this problem is yes, then it would open up a potential pathway to prove the uniqueness of the moonshine module. We can reformulate this problem as follows: What types of holomorphic vertex operator algebras are orbifold theories constructed from a lattice vertex operator algebra? To approach this question, we will construct and classify further orbifold theories from lattice vertex operator algebras in order to make deductions about their general structure. Making use of our experience with orbifold theories constructed from lattice vertex operator algebras gained in previous work and the new construction method for twisted modules, we will be able to construct interesting examples that will give us some fundamental insights into this difficult open problem. • More holomorphic extensions: One of the large obstacles to the construction of further holomorphic extensions is to understand if the cohomological twist associated to a holomorphic, strongly rational vertex operator algebra and a finite automorphism group vanishes. As of yet, this problem has only been solved for a few very simple families of finite groups. In future work, we will aim to find conditions for the vanishing of the cohomological twist for further groups. A good first target will be the generalised quaternion groups and, by extension, the finite groups with periodic cohomology. • Conformal field theory with large mass-gap: A related problem is that of conformal packing [Hö19].We would like to construct vertex operator algebras, whose minimal conformal weight, that is the smallest positive conformal weight of a Virasoro highest-weight vector, is as large as possible. Examples of holomorphic vertex operator algebras with minimal conformal weight 2 are known, but none have been constructed for higher minimal conformal weights, and physicist have conjectured that they may not exist. A further research goal would be to construct a vertex operator algebra with large minimal conformal weight using techniques developed in this thesis and possibly disprove this conjecture. In a large survey of lattices and their automorphism groups, we could only find two lattice automorphism groups whose orbifolds could realise such a holomorphic vertex operator algebra. In future work, we will construct and classify the lifts of these groups and the corresponding orbifolds.

12 1 Preliminaries and Deﬁnitions

1.1 Formal Calculus

Let W be a C-vector space. We will denote the various spaces of formal series in the formal variable x as:

−1 P k • W x, x = { k∈Z akx : ak ∈ W }, the space of (doubly-infinite) formal Laurent series J K P k • W x = { k∈Z akx : ak ∈ W }, the space of formal power series J K ≥0 −1 P k • W [x, x ] = { k∈Z akx : ak ∈ W, all but finitely many ak = 0}, the space of formal Laurent polynomials • W [x] = {P a xk : a ∈ W, all but finitely many a = 0}, the space of formal polynomials k∈Z≥0 k k k

P k • W ((x)) = { k∈Z akx : ak ∈ W, ak = 0 for suﬃciently negative k}, the space of truncated formal Laurent series

P k • W {x} = { k∈C akx : ak ∈ W }, the space formal series with complex exponents For an introduction to the theory of formal power series, we refer the reader to the standard texts on vertex operator algebras such as [LL04] and [FHL93].

1.2 Vertex Operator Algebras

The following definition of a vertex operator algebra was first given in [FLM88] and is based on Borcherd’s definition of vertex algebras [Bor86]. In particular, the central defining property is the Jacobi identity. An equivalent definition in terms of the weak commutativity of vertex operators is given in [Kac98]. Definition 1.2.1. A vertex operator algebra of central charge c consists of a graded vector space (space of states) M V = V(n) with dimV(n) < ∞ for all n ∈ Z≥0 (1.2.1) n∈Z together with a linear map (vertex operator map)

−1 X −n−1 Y (·, x): V → (EndV ) x, x , v 7→ Y (v, x) = vnx (1.2.2) J K n∈Z

and two distinguished elements |0i ∈ V(0) (vacuum vector) and ω ∈ V(2) (conformal vector) such that the following axioms are satisﬁed for all u, v ∈ V :

• (truncation) umv = 0 for m 0,

• (vacuum) Y (|0i, x) = 1V , • (creation) Y (v, x)|0i = v + O(x), • (Jacobi identity) x − x x − x x−1δ 1 2 Y (u, x )Y (v, x ) − x−1δ 2 1 Y (v, x )Y (u, x ) 0 x 1 2 0 −x 2 1 0 0 (1.2.3) −1 x1 − x0 = x2 δ Y (Y (u, x0)v, x2), x2

13 Let X W −n−2 Y (ω, x) = Ln x . (1.2.4) n∈Z W Then the modes Ln satisfy the following relations: • (Virasoro) The relations of the Virasoro algebra at central charge c: c [L ,L ] = (m − n)L + δ (m3 − m)1 , (1.2.5) m n m+n n+m,0 12 V

• (translation) Y (L−1v, x) = ∂xY (v, x),

• (grading) L0|V(n) = n1V(n) and V(n) = 0 for n suﬃciently negativ. We can also rewrite the Jacobi identity as a condition for the modes. Lemma 1.2.1 ( [Kac98]). Let V be a vertex operator algebra. Then for any a, b, c ∈ V the modes satisfy the Borcherds identity

∞ ∞ ∞ X m X n X n ((a b) c = (−1)j a b c − (−1)j+n b a c. (1.2.6) j n+j m+k−j j m+n−j k+j j n+k−j m+j j=0 j=0 j=0

Deﬁnition 1.2.2. We say a vertex operator algebra V is of CFT-Type if

V(n) = 0, for all n < 0. (1.2.7)

1.3 Automorphisms of Vertex Operator Algebras

An automorphism of a vertex operator algebra is a linear bijection that preserves the vertex operators, vacuum vector and conformal vector. They will play a central role in this text.

Deﬁnition 1.3.1. Let V be a vertex operator algebra and g a bijective, linear map V → V . Then g is called a V -automorphism if for any u, v ∈ V

gY (u, x)v = Y (gu, x)gv. (1.3.1)

The group of V -automorphisms will be denoted Aut(V ).

Furthermore, we have the following notion of inner automorphisms of vertex operator algebras:

Deﬁnition 1.3.2. Let V be a vertex operator algebra and a ∈ V(1) a homogenous element of weight 1. Then the map σa = exp(2πia0) is an automorphism of V . Such automorphisms are called inner automorphisms of V . The inner automorphism group will be denoted by Inn(V ).

The group-theoretic structure of automorphism groups of vertex operator algebras has been investigated in [DG02]. For ﬁnitely-generated vertex operator algebras (cf. [FHL93]), the automorphism group is isomorphic to an algebraic group, and for strongly rational vertex operator algebras, it has been conjectured that the connected subgroup is given by Inn(V ).

1.4 Modules and Twisted Modules

Here we will review the basic deﬁnitions of modules and twisted modules for vertex operator algebras. For a very readable introduction to the theory of twisted modules refer to [GG09].

Deﬁnition 1.4.1. [DLM00][g-twisted V -module] Let V be a vertex operator algebra of central charge c ∈ C and let g be an automorphism of V of order n. Then a g-twisted V -module W is determined by the following data:

14 • (space of states) a C-graded vector space

W = ⊕λ∈CWλ

with weight wt(w) = λ for w ∈ Wλ extended linearly, such that the homogeneous subspaces are ﬁnite dimensional dim(Wλ) < ∞ and the grading is bounded from below

Wλ+r/n = {0} for r suﬃciently small and ﬁxed λ ∈ C .

• (vertex operators) a linear map

1/n −1/n YW (·, x): V → EndC(W )[[x , x ]] X W −k−1 (1.4.1) a → YW (a, x) = ak x k∈(1/n) Z

W where for each w ∈ W, ak w = 0 for suﬃciently large k, subject to the following axioms:

• (twist compatibility) For each a ∈ V r

X W −k−1 YW (a, x) = ak x (1.4.2) k=−r/n+Z

• (left vacuum axiom) YW (1, x) = idW .

• (translation axiom) For any a ∈ V, [T,YW (a, x)] = ∂xYW (a, x). • (twisted Jacobi identity) For a ∈ V r and b ∈ V , x − x x − x x−1δ 1 2 Y (a, x )Y (b, x ) − x−1δ 2 1 Y (b, x )Y (a, x ) = 0 x W 1 W 2 0 x W 2 W 1 0 0 (1.4.3) −1x1 − x0 r/n x1 − x0 x2 δ YW (Y (a, x0)b, x2) x2 x2

• (Virasoro relations) Let X W −n−2 YW (ω, x) = Ln x . (1.4.4) n∈Z W Then the modes Ln satisfy the relations of the Virasoro algebra at central charge c: m3 − m [LW ,LW ] = (m − n)LW + δ id c (1.4.5) m n m+n 12 m+n W W for m, n ∈ Z and L−1 = T and L0 w = wt(w)w for homogenous w ∈ W . Note that a 1-twisted V -module is commonly referred to as a V -module or, to emphasise the distinction, an untwisted V -module. In order to distinguish them from the other, closely related types of modules defined below, g-twisted V -modules are sometimes referred to as ordinary g-twisted V -modules. Moreover, the following alternate notions of (twisted) modules for vertex operator algebras were defined in [DLM97] and [DLM98]. Definition 1.4.2 (Weak g-twisted module). A weak g-twisted V -module is a not necessarily graded vector space M, together with a vertex operator map

1/n −1/n YW (·, x): V → EndC(W )[[x , x ]] X W −k−1 (1.4.6) a → YW (a, x) = ak x k∈(1/n) Z

such that the twisted Jacobi identity and the Virasoro relations are satisﬁed.

15 Deﬁnition 1.4.3 (Admissible g-twisted module). An admissible g-twisted V -module is a weak g-twisted L V -module W that admits a grading W = 1 W (k) such that for all homogenous v ∈ V k∈ n Z vW (m)W (k) ⊆ W (k + wt(v) − m − 1). (1.4.7)

We have the usual notions of reducibility of twisted V -modules and irreducible twisted V -modules. Note that the compatibility of the vertex operators with the grading implies the following: Deﬁnition 1.4.4. If W is an irreducible twisted V -module then the grading will be of the form M W = Wρ+k/n, (1.4.8) k∈N where we call ρ the conformal weight of W . A vertex operator algebra V has itself the structure of a V -module. As such it is often called the adjoint module. Furthermore, there is a natural notion of contragredient (read dual) twisted modules that was introduced in [FHL93, Xu00, Xu01].

0 Deﬁnition 1.4.5 (Contragredient twisted module). Let (W, YW ) be a g-twisted V -module. Let W be 0 0 graded dual of W . Then W together with the vertex operator (YW ) deﬁned by

0 0 0 xL(1) −2 L(0) −1 h(YW ) (v, x)w , wi = hw ,YW (e (−x ) v, x )wi (1.4.9) is a g−1-twisted V -module called the contragredient module.

1.5 Regularity and Rationality

The representation theory of vertex operator algebras is, in general, not well understood. Extensive general results are only available under some rather restrictive regularity conditions that we will discuss in this section. The following definition of rationality is due to [DLM97]. An alternate, but equivalent definition was given in [Zhu96]. Definition 1.5.1 (Rationality). A vertex operator algebra V is called rational if every admissible V -module is isomorphic to a direct sum of irreducible admissible V -modules. Note that by [DLM98], if V is a rational vertex operator algebra, then up to isomorphism there are only finitely many irreducible V-modules and also only finitely many irreducible g-twisted V -modules for any automorphism g. The following condition has been defined by Zhu [Zhu96]:

Definition 1.5.2 (C2-Cofiniteness). Let V be a vertex operator algebra. Then is said to be C2-cofinite if the space C2 := spanC(a−2b | a, b ∈ V ) (1.5.1) has finite codimension in V .

By the results of [Miy04] and [DLM98], C2-cofiniteness is equivalent to the semi-simplicity and finite- dimensionality of Zhu’s algebra A(V ), an associative algebra, whose irreducible modules are in 1 to 1 correspondence with irreducible V -modules. It is conjectured that rationality implies C2-cofiniteness. By [DLM00], a consequence of rationality and C2-cofiniteness of a vertex operator algebra is the rationality of its central charge and the conformal weights of its irreducible modules. Note that for vertex operator algebras the notions of simplicity and irreduciblity coincide. Definition 1.5.3. A vertex operator algebra V is called simple if it is irreducible as a V -module. Definition 1.5.4. A vertex operator algebra V is called self-contragredient if it is isomorphic to V 0 as a V -module. By [Li94], self-contragredience implies, in particular, that V can be equipped with an invariant, symmetric, bilinear form, the vertex operator algebraic analogue, to the Killing form in the theory of finite-dimensional Lie algebras.

16 Deﬁnition 1.5.5 (Regularity). A vertex operator algebra V is called regular if every weak V -module is a direct sum of irreducible V -modules.

Note that by [ABD04], if V is of CFT-Type, regularity is equivalent to rationality and C2-cofiniteness. By [DY12], regularity is equivalent to the semi-simplicity of the weak module category. In this text, we will mostly be interested in vertex operator algebras that satisfy all of these conditions. We define: Definition 1.5.6. A vertex operator algebra is strongly rational if it is simple, self-contragredient, rational, C2-cofinite and of CFT-Type. A central open conjecture in the orbifold theory of strongly rational vertex operator algebras is the following: Conjecture 1.5.1. If V is strongly rational and G ∈ Aut(V ) a finite group, then the fixed-point vertex operator algebra V G is also strongly rational. In this text, we will assume that any vertex operator algebra is strongly rational unless indicated otherwise. Note that this condition is rather restrictive. Not in particular that there are many important instances of vertex operator algebras that are not rational or C2-cofinite, such as affine vertex operator algebras at admissible level [AM09] and the bosonic ghost vertex algebra [RW15a].

1.6 Holomorphic Vertex Operator Algebras

So-called holomorphic vertex operator algebras have the deﬁning property that their adjoint module is their unique irreducible module. Deﬁnition 1.6.1 (Holomorphicity). A rational vertex operator algebra V is called holomorphic if the adjoint module V is the only irreducible V -module up to isomorphism.

That, in particular, makes them the simplest conformal field theories with only one module and one intertwining operator. As such, they have been of interest to both the mathematics [DM94, Höh96]and physics [God89,Sch93a,Sch93b,Mon94] communities for a long time. The most famous example is, of course, the moonshine module V \. So far, work has mostly focussed on holomorphic vertex operator algebras at central charge c = 24. Let us finally discuss some consequences of holomorphicity that were shown by Dong, Li and Mason [DLM00].

Lemma 1.6.1. Let V be a C2-coﬁnite, holomorphic vertex operator algebra. Then 1. its central charge c is a multiple of 8. 2. there exists a unique irreducible g-twisted V -module for every automorphism g of ﬁnite order.

1.7 Homomorphisms, Schur’s Lemma and Action of Automorphisms on Twisted Modules

In this section, we will deﬁne homomorphisms for twisted modules and present the appropriate manifesta- tion of Schur’s lemma. As a consequence, we will see that for two commuting automorphisms g and h, h induces a homomorphism of g-twisted modules (and vice versa).

Deﬁnition 1.7.1. [LL04][V -homomorphism] Let V be a vertex operator algebra and W1 and W2 (g-twisted) V -modules. Then a V -homomorphism from W1 to W2 is a linear map ψ such that

ψ(YW1 (v, x)w) = YW2 (v, x)ψ(w), (1.7.1)

for all v ∈ V and w ∈ W1. The space of V -homomorphisms from W1 to W2 will be denoted by HomV (W1,W2). If W1 = W2 = W we call ψ a V -endomorphism and the space of such maps will be denoted EndV (W ).

17 Theorem 1.7.1. [LL04][Schur’s Lemma] Let V be a vertex operator algebra and W1 and W2 irreducible (g-twisted) V -modules. Then

( ∼ ∼ C, if W1 = W2 HomV (W1,W2) = (1.7.2) 0, if W1 W2 so that in particular for any irreducible V -module W ∼ EndV (W ) = C . (1.7.3)

Now we can create new twisted modules by inserting an automorphism into the corresponding vertex operators.

Lemma 1.7.2. Let V be a vertex operator algebra, g and h automorphisms of V and (W, YW (·, x)) a −1 g-twisted V -module. Then (W, YW (h·, x)) is a h gh-twisted V -module. Hence we ﬁnd that if g and h commute, h induces a homomorphism of g-twisted modules. We are particularly interested in the case where V is holomorphic. Let V be a holomorphic vertex operator algebra, g and h automorphisms of V such that gh = hg and (W, YW (·, x)) an irreducible g-twisted V -module. Then by lemma 1.6.1 (W, YW (·, x)) and (W, YW (h·, x)) are isomorphic irreducible g-twisted V -modules.

Deﬁnition 1.7.2. Hence there exists a linear map φ(h): W → W such that

−1 φg(h)YW (v, x)φg(h) = YW (hv, x). (1.7.4)

Theorem 1.7.3. φg : CAut(V )(g) → Hom(W, W ) is a projective representation of CAut(V )(g) on W .

0 Proof. First we show that the map φg(h) is unique up to multiplication by a scalar. Let φg(h) be another −1 0 map satisfying Equation 1.7.4. Then φg (h)φg(h) ∈ EndV (W ) is a homomorphism of twisted modules as

−1 0 0 −1 φg (h)φg(h)YW (v, x)φg(h) φg(h) = YW (v, x) (1.7.5)

−1 0 and hence by Schur’s Lemma we ﬁnd that φg (h)φg(h) ∈ C. Furthermore for two automorphisms h, k ∈ CAut(V )(g) we ﬁnd that

−1 0 −1 −1 φg(hk) φg(h)φg(k)YW (v, x)φg(k) φg(h) φg(hk) = YW (v, x) (1.7.6)

−1 and hence that φg(hk) φg(h)φg(k) ∈ EndV (W ). Then there exists a cocycle cg(h, k) such that

φg(h)φg(k) = cg(h, k)φg(hk). (1.7.7)

Hence φg is indeed a projective representation.

1.8 Characters, Graded Trace Functions and Modular Invariance

In this section, we will discuss the characters and graded trace functions associated to (twisted) modules of vertex operator algebras. They are relatively simple invariants of a vertex operator algebra that encode important information about its representation theory that have been studied since the early days of conformal field theory [MS89] and orbifold theory [DHVW85,DHVW86,DGM90a]. The proof of the modular invariance of characters by Zhu [Zhu96] was a milestone in our field. Dong, Li and Mason [DLM00] later proved the modular invariance of graded trace functions. Famously, the Verlinde Formula that is going to be discussed in section 1.9 relates the modular transformation properties of characters for the modules of a vertex operator algebra to the tensor category structure of its module category. In sections 2.2 and 2.3.2, we will use the graded trace functions to make deductions about the module category of the fixed-point subalgebra V G for certain families of finite groups G. Using the action of commuting automorphisms on twisted modules we can define the following graded trace functions:

18 Definition 1.8.1 (Graded trace function). Let V be a C2-cofinite, holomorphic vertex operator algebra, g and h be commuting automorphisms and W the unique irreducible g-twisted module. Then graded trace functions are defined as LM −c/24 T (g, h, τ) = tr|W φg(h)q 0 , (1.8.1) where q = e2πiτ , τ ∈ H. Note that graded trace functions can also be defined for vertex operator algebras that are not holomorphic, when the g-twisted V -module W is h-stable, but in this text we are not interested in such cases. Special cases of these trace functions are the characters of twisted modules. Definition 1.8.2. Let W be a twisted V -module. Then its character is given by

LM −c/24 chW (τ) = tr|W q 0 (1.8.2) Proving the convergence and modular invariance of characters and graded trace function was a central problem in our ﬁeld. In the case of characters, this was achieved by Zhu in his landmark paper [Zhu96]. Miyamoto [Miy04] later strengthened the result. Theorem 1.8.1. Let V be a strongly rational vertex operator algebra and W a V -module. Then the LM −c/24 character chW (τ) = tr|W q 0 converges to a holomorphic function on the upper half-plane.

Theorem 1.8.2. Let V be a strongly rational vertex operator algebra and IrrV the set of irreducible V -modules. Then there exists a representation ρ of SL(2, Z) on the vector space generated by the characters of irreducible V -modules, such that X chW (M · τ) = ρW,U (M)chU (τ). (1.8.3)

U∈IrrV These results were generalised to graded trace function by Dong, Li and Mason [DLM00]. Note that we only state the case of holomorphic vertex operator algebras. Theorem 1.8.3. The graded trace functions T (g, h, τ) converge to meromorphic functions on the complex upper half-plane. Theorem 1.8.4 (Modular Invariance of Trace Functions). Let V be a holomorphic, strongly rational vertex operator algebra. Then the graded trace functions transform under the action of a modular α β transformation M = ∈ SL (Z) as γ δ 2 T (g, h, M · τ) = σ(g, h, M)T ((g, h) · M, τ), (1.8.4) where (g, h) · M = (gαhγ , gβhδ) and the constants σ(g, h, M) depend only on g, h and M. Note that this implies, in particular, that the graded trace functions span a representation of the modular group SL2(Z).

1.9 Intertwining operators, Fusion and Vertex Tensor Categories

In this section, we will introduce the concepts of intertwining operators, the fusion product and the tensor categorical structure of the module categories of vertex operator algebras.

Deﬁnition 1.9.1. Let (W1,Y1), (W2,Y2) and (W3,Y3) be V -modules. Then an intertwining operator of type W3 is a linear map W1W2 Y W3 (., z): W → Hom(W ,W ){z} (1.9.1) W1W2 1 2 3

satisfying the Jacobi identity for any a ∈ V and w ∈ W1 x − x x−1δ 1 0 Y W3 (Y (a, x )w, x ) 2 W1W2 W1 0 2 x2 x − x x − x = x−1δ 1 2 Y (a, x )Y W3 (w, x ) − x−1δ 2 1 Y W3 (w, x )Y (a, x ). (1.9.2) 0 W3 1 W1W2 2 0 W1W2 2 W2 1 x0 x0

19 and the translation axiom for any w ∈ W1 [L ,Y W3 (w, x)] = ∂ Y W3 (w, x) (1.9.3) −1 W1W2 x W1W2 Analogous twisted intertwining operators that play a central role in general orbifold theory have been introduced by Xu [Xu95] and Huang [Hua18]. Related to intertwining operators is the notion of tensor products of V -modules. This was first studied by Moore and Seiberg [MS88, MS89] in the early days of conformal field theory. A natural, heuristic definition of the tensor product was given by Gaberdiel [Gab94a, Gab94b] and developed further by Nahm [Nah94] and Gaberdiel and Kausch [GK96] into an algorithmic theory. However, their approach is beset by convergence issues. A rigorous construction of the fusion product was first given by Huang and Lepowsky [HL94,HL95a, HL95b, HL95c, Hua95] and extended further to logarithmic modules by Huang, Lepowsky and Zhang [HLZ10a,HLZ10b,HLZ10c,HLZ10d,HLZ10e,HLZ10f,HLZ11a,HLZ11b]. The tensor product for V -modules is defined as follows. Note that for a V -module W , we will denote its algebraic completion by W , that is W = W 0∗, the dual of the graded dual.

Deﬁnition 1.9.2. Let (W1,Y1), (W2,Y2), (W3,Y3) and (W4,Y4) be V -modules. A P (z)-tensor product is a linear map P (z) : W1 ⊗ W2 → W3 satisfying

−1 x1 − z x0 δ Y3(v, x1)(w1 P (z) w2) x0 −1 x1 − x0 −1 z − x1 = z δ (Y1(v, x0)w1) P (z) w2 + x0 δ w1 P (z) (Y2(v, x1)w2) (1.9.4) z −x0 0 subject to the universal property that for any P (z)-tensor product P (z) : W1 ⊗ W2 → W4 there exists a map η : W3 → W4 extending a homomorphism of modules such that 0 P (z) = η ◦ P (z). (1.9.5)

The module W3 is then denoted W1 P (z) W2.

Showing that such a P (z)-tensor product exists and constructing the module W1 P (z) W2 is a highly non-trivial task. Will not discuss this point in any detail, but just give a very brief breakdown of the ∗ construction following the excellent review by Kanade and Ridout [KR19]: Let W1 P (z) W2 ⊂ (W1 ⊗W2) ∗ denote the largest linear subspace of (W1 ⊗ W2) that is a V -module (in the language of [HL94] this is the subspace satisfying the compatibility condition). Then under suitable conditions W1 P (z) W2 can be identified with the graded dual of W1 P (z) W2. In particular, Huang and Lepowsky prove the following: Theorem 1.9.1. Let V be a strongly rational vertex operator algebra. Then the P (z)-tensor product P (z) exists. Note that this result has been extended by Huang, Lepowsky and Zhang to logarithmic modules. Now let us briefly discuss some tensor categorical properties of the module categories of vertex operator algebras. For an introduction to the theory of tensor categories refer to [BK01]. In the following, we will denote = P (1). With some technical difficulties we can show that:

Theorem 1.9.2. Let M, W, U ∈ V − mod. Then the tensor product satisfies ∼ ∼ • (left and right unit) V U = U V = U, ∼ • (associativity) M (W U) = (M W ) U and ∼ • (braiding) W U = U W . Furthermore, the left unit, right unit, associativity and braiding isomorphisms satisfy the relevant axoims (cf. [BK01] Theorem 1.2.5.) to endow V − mod with the structure of a braided tensor category. The tensor categorical properties of the twisted module category have similarly been studied by Kirillov [Kir02,Kir01,Kir04] and Müger[Mü04,Mü05].In particular, for strongly rational vertex operator algebra, the twisted module category can be endowed with the structure of a G-crossed tensor category in the sense of Turaev [Tur00]. If V is strongly rational, then the module category is semi-simple and hence the tensor product can be written as a direct sum of irreducible modules. This gives rise to the notion of fusion rules, the vertex operator algebraic analog to Clebsch-Gordon coefficients in the theory of finite-dimensional Lie algebras.

20 Deﬁnition 1.9.3. Let V be strongly rational and let W1 and W2 be irreducible V -modules. Then we can decompose the tensor product as M W W ∼ N W3 W . (1.9.6) 1 P (z) 2 = W1W2 3 W3∈Irr(V )

The coefficients N W3 are called fusion rules. W1W2 It was conjectured by Verlinde [Ver88] that there is a connection between the fusion category structure defined in 1.9.6 and Zhu’s modular invariance of characters defined in 1.8.3: 0 −1 Theorem 1.9.3 (Verlinde conjecture). Let V be strongly rational and let S = ∈ SL(2, Z) act 1 0 on the characters of irreducible V -modules as X chW (S · τ) = SWU chU (τ). (1.9.7) U∈Irr(V )

Then the fusion rules for the braided tensor category V − mod are given by

0 X SW1W4 SW2W4 SW W4 N W3 = 3 . (1.9.8) W1W2 SW4V W4∈Irr(V )

In particular, this implies that the S-matrix diagonalises the fusion rules. This relation was first studied by Moore and Seiberg [MS88] then proved by Huang [Hua08b]. Finally, the Verlinde formula was instrumental in Huang’s landmark proof [Hua08a] of the rigidity, balancing and non-degeneracy (cf. [BK01] Definition 3.1.1.) of the module category for strongly rational vertex operator algebras. We find that: Theorem 1.9.4. Let V be strongly rational. Then the module category V − mod can be endowed with the structure of a modular tensor category.

Note that for vertex operator algebras that are not rational or C2-coﬁnite, it is expected that suitable module categories (but not necessarily the full module category) can also be endowed with a rigid structure. In particular, this was recently shown [AW20] for the bosonic ghost vertex operator algebra.

21 2 Orbifold Theory

Let V be a vertex operator algebra and G an automorphism group of V . Then we denote the subalgebra of G-invariant elements of V by V G. The fundamental objective of orbifold theory is to study the category V G − mod of V G-modules and describe it in terms of the representation theory of the initial vertex operator algebra V and the group G. In the most general case, very little is known, and it has proved difficult to infer properties of V G-modules from those of V directly. A more fruitful approach is to study so-called g-twisted V -modules for g ∈ G. A central conjecture of orbifold theory is that the representation theory of the fixed-point vertex operator algebra V G is in a suitable sense equivalent to the study of the twisted representation theory of V . The most successful construction method for g-twisted modules currently available associates a g-twisted V -module to a module for a certain associative algebra Ag(V ), called twisted Zhu’s algebra [DLM98,HY19]. Problems with this approach are that, in this framework, it is hard to study properties of general g-twisted modules and that Ag(V ) and its modules are, in general, hard to construct and classify themselves. Promising advances were recently made by Huang [Hua19a], who constructed a universal lower-bounded [g] g-twisted V -module McB that can be considered the vertex operator algebraic analogue to a Verma module in the theory of finite-dimensional Lie algebras. A general orbifold theory should study these modules, as well as the corresponding twisted intertwining operators, but, as of yet, only very little is known. In this text, we will, however, focus on much easier problems involving only finite automorphism groups acting on strongly rational vertex operator algebras. In this case, much more is known, and the central remaining open problem is to prove the following conjecture: Conjecture 2.0.1. Let V be a strongly rational vertex operator algebra and G a finite group of automorphisms acting on V . Then the fixed-point vertex operator algebra V G is also strongly rational. It is easy to show that if V is strongly rational, V G must be self-contragredient and of CFT-Type [McR19]. Furthermore, the simplicity of V G has been established in [DM99] and by Theorem 2.1.1 V G is rational if it is C2-cofinite. Hence the missing piece in the proof of this important conjecture is the question of G C2-cofiniteness of V . Importantly, Carnahan and Miyamoto [Miy15,CM16] proved that the conjecture is true for solvable groups G. Theorem 2.0.1 ( [CM16]). Let V be a strongly rational vertex operator algebra and G a finite solvable group of automorphisms acting on V . Then the fixed-point vertex operator algebra V G is also strongly rational.

2.1 Holomorphic Extensions of Vertex Operator Algebras

G Assuming C2-cofiniteness of the fixed-point vertex operator algebra V , its representation theory can be fully described from the twisted representation theory of V and G. In particular, if V is holomorphic, we find that: Theorem 2.1.1 (Schur-Weyl-Type Duality [DLM00, DRX17]). Let V (g) be the unique irreducible g- twisted V -module and let CG(g) projectively act on V (g) with 2-cocycle cg. Then V (g) decomposes into G modules for Ccg [Cg] ⊗ V as M V (g) = V[g,χ] ⊗ W[g,χ], (2.1.1)

χ∈Irrcg (CG(g)) where Irrcg denotes the set of irreducible projective characters corresponding to the 2-cocycle cg. Then G the irreducible modules of V are exactly given by W[g,χ] for some conjugacy class g and some projective irreducible character χ of CG(g) with 2-cocycle cg.

22 In summary, the irreducible modules for V G are labeled by [g, χ] and can be obtained from the twisted modules by acting with the projector

1 X π[g,χ] = χ(h)φg(h) (2.1.2) |CG(g)| h∈Cg

In particular note that, these operators are independent of the choice of actions on the twisted modules. The problem we will be most interested in here is the following: Problem. Let V be a holomorphic vertex operator algebra and G a ﬁnite group of automorphisms. Classify and construct the holomorphic extensions of the ﬁxed-point vertex operator algebra V G.

G G The fundamental idea is to choose a suitable category C = {V ,M1,...,Mk} of V -modules and endow their direct sum C G M V = V ⊕ Mi (2.1.3) i with a vertex operator algebra structure by assembling the corresponding intertwining operators to new vertex operators. A purely vertex operator algebraic approach to this problem is extremely challenging, but Huang, Kirillov and Lepowsky [HKL15] showed that this problem is, in fact, equivalent to corresponding notion in the module category of V G: Theorem 2.1.2. Let V be a strongly rational vertex operator with module category V − mod. Then any extension Ve ⊇ V of CFT-Type corresponds to a haploid V − mod-algebra Ve with trivial twist. See also [CKM17] for the recent generalisation to vertex operator superalgebras. Particularly important modular categories arising in this context are the twisted group double Dω(G) associated to a group G and a 3-cocycle ω ∈ H3(G, C∗) and its module category Dω(G) − mod that were initally described in [DW90] and [DPR91]. We will be primarily interested in modular tensor categories of the latter type. The simple objects in ω D (G) − mod can be parametrised by pairs [g, χ], where g ∈ G is an element and χ ∈ Irrcg (CG(g)) is an irreducible projective character for the centraliser of g corresponding to the 2-cocycle

h1h2 h1 ∗ cg(h1, h2) = ω(g, h1, h2)ω(h1, h2, g )ω(h1, g , h2) . (2.1.4)

The reader should note the obvious similarity to Theorem 2.1.1. When V is holomorphic and strongly rational, and V G is again strongly rational, Kirillov showed that this analogy goes even further.

Theorem 2.1.3 ( [Kir02]). Let V be a holomorphic, strongly rational vertex operator algebra and G a ﬁnite group such that the ﬁxed-point vertex operator algebra is again strongly rational. Then there exists a 3-cocycle [ω] ∈ H3(G, C∗) that characterises the associativity isomorphisms in V G − mod such that V (g) V (h) V (k) = ω(g, h, k) V (g) V (h) V (k), (2.1.5)

G where denotes fusion in V − mod. If [ω] = 1, then the module category V G − mod is equivalent to the module category of a group double D(G) − mod. Furthermore, by analogy with the theory of conformal nets, it is believed that the following result holds in general:

Conjecture 2.1.1. The module category V G − mod is equivalent to the module category of the twisted quantum double Dω(G) − mod, where ω is the 3-cocycle deﬁned in Theorem 2.1.3. This was originally proposed in [RPD90], following up on work on the operator algebra of general orbifolds [DVVV89]. Note, however, that for reasons outlined below, we will only be interested in groups G such that ω is in the trivial class. The modular data of these categories was ﬁrst described by Coste, Gannon and Ruelle [CGR00]:

23 Theorem 2.1.4. The S- and T -matrices for the category Dω(G) − mod are given by

−1 −1 1 X cg (k , h1)cg (k , h2) S = χ (h )∗χ (h )∗ 1 1 2 2 (2.1.6) [a1,χ1],[a2,χ2] |G| 1 1 2 2 −1 −1 cg1 (g2, k1 )cg2 (g1, k2 ) gi∈cl(ai),g1g2=g2g1 −2πic/24 T[a1,χ1],[a2,χ2] = e δ[a1,χ1],[a2,χ2]χ1(a1)/χ1(1) (2.1.7)

ki ki k2 Where the ki are any solutions to gi = ai , and h1 := g2, h2 := g1 and cg is the 2-cocycle defined in Equation 2.1.4. Hence when V G − mod is equivalent to Dω(G) − mod we know its modular data and hence, by the Verlinde formula 1.9.8, its tensor category structure. Note, however, that the parametrisation of simple objects in V G − mod as defined by Theorem 2.1.1 will only agree with that in Theorem 2.1.4 for a set of specific, preferred choices of the actions φg of the centralisers on the twisted modules. In particular, we find that: Lemma 2.1.5. Let a holomorphic, strongly rational vertex operator algebra V and a finite group G be such that V G−mod is equivalent to the twisted quantum double Dω(G)−mod. Then the projective representation of CG(g) on the unique irreducible g-twisted module can be chosen such that the corresponding 2-cocycle coincides with the one given by Equation 2.1.4. ∗ Note that this leaves a remaining ambiguity by an element of Hom(CG(g), C ). Now by Theorem 2.1.2 extensions of V G correspond to haploid Dω(G) − mod-algebras. Particularly, in [EG18], Evans and Gannon proved that holomorphic extensions correspond to so so-called type 1 or extension type module categories. These have been studied in [Ost03] and classified in [EG18]. In particular, they prove that: Theorem 2.1.6 (Corollary 2 of [EG18]). Let V be a holomorphic, strongly rational vertex operator algebra and G a finite group such that the module category V G − mod is equivalent to the twisted quantum double Dω(G) − mod. Then there is a holomorphic extensions V orb(K,ψ) of V G for any choice of subgroup 2 ∗ K ⊆ G such that [ω|K ] = 1 and 2-cocycle ψ ∈ Z (K, C ) given by

orb(K,ψ) M V = W[k,βψ ·Res IndG 1] (2.1.8) k CG(k) K k

ψ g ∗ where βg (h) = ψ(g, h )ψ(h, g) and the sum runs over conjugacy class representatives of K by G. Note that if H2(G, C∗) is non-trivial, we can thus recover several possibly non-isomorphic extensions. This choice of 2-cocycle ψ when deﬁning the holomorphic extensions is usually called ‘discrete torsion’ in the physics literature. The holomorphic vertex operator algebra V orb(K,ψ) is also an extension for the ﬁxed-point vertex operator algebra V K ⊃ V G. To have a holomorphic extension of V G that is not also an extension for V K for some subgroup K ⊂ G, it is thus necessary that the cohomological twist ω is in the trivial class. Hence, as we are mostly concerned with the construction of new holomorphic vertex operator algebras, we are primarily interested in groups G such that the 3-cocycle ω is trivial. In the physics literature, groups that do satisfy this condition are referred to as ‘anomalous’. The 3-cocycle ω is also commonly referred to as H3-twist. In the case that ω is in the trivial class, we can choose the descending 2-cocycles to be trivial cg(·, ·) = 1, which means that the projective characters χ are actually linear. The extension with discrete torsion ψ = 1 then simply consists of the Cg-invariant part of the g-twisted modules Wg,

orb(G),1 M Ck V = Wk , (2.1.9) k recovering the conjectural deﬁnition of the orbifold from the physics literature. When ω = 1 the transformation matrices are given by 1 X S = χ (h )∗χ (h )∗ [a1,χ1],[a2,χ2] |G| 1 1 2 2 gi∈cl(ai),g1g2=g2g1 (2.1.10) −2πic/24 T[a1,χ1],[a2,χ2] =e δ[a1,χ1],[a2,χ2]χ1(a1)/χ1(1) Choosing the projective representations on the twisted modules such that Equation 2.1.10 holds then corresponds to imposing the following modular transformation properties of the graded trace functions:

24 G Lemma 2.1.7. Let {W[g, χ]: g ∈ G, χ ∈ Irr(CG(g))} be the irreducible V -modules satisfying Equation 2.1.10 and Theorem 2.1.1. Then the graded trace functions satisfy

T (g, h, S · τ) = T (h, g−1, τ), (2.1.11) T (g, h, T · τ) = e−2πic/24T (g, gh, τ).

Note that this fully determins all graded trace functions, whose orbits under the modular group Γ intersects the untwisted sector and hence the corresponding actions on twisted modules. The remaining ambiguity corresponds exactly to the choice of discrete torsion when defining holomorphic extensions. One of our largest remaining obstacles to constructing these holomorphic extensions is proving that the 3-cocycle ω is in the trivial class for a given finite group G acting on some holomorphic, strongly rational vertex operator algebra V . One possible approach is to construct the irreducible modules for the fixed-point vertex operator algebra V G and determine their modular data. Constraints on ω can then be obtained by comparing the result to 2.1.10. The corresponding result for cyclic groups as discovered by van Ekeren, Möllerand Scheithauer in [vEMS17] will be discussed in the next section. Note, however, that Schauenburg and Mignard [MS17] showed that, in general, it is not possible to deduce ω from the modular data alone. In particular, this is true for groups of the form Zq o Zp that will be discussed in subsection 2.3.2. The author is nonetheless relatively confident that it should be possible to distinguish whether ω is in the trivial class or not from modular data alone. Note further that for any g ∈ G the 2-cocycle cg associated with the action of automorphisms on the irreducible g-twisted module has to be in the trivial class to be consistent with [ω] = 1. Further methods to constrain ω will be discussed in subsection 2.3.1.

2.2 Cyclic Orbifolds

A significant development in the orbifold theory of vertex operator algebras was the theory of cyclic orbifolds by van Ekeren, Möllerand Scheithauer [vEMS17]. Using a vertex operator algebraic approach, they were able to give a necessary and sufficient condition for the existence of holomorphic extensions for the case where G is cyclic. In particular, they do not rely on assumptions on the tensor category structure of the module category of the fixed-point vertex operator algebra, so that they, indeed, give a full construction of holomorphic extensions for cyclic groups. Furthermore, they showed that the characters of the holomorphic extensions can be calculated by making use of the modular transformation properties of graded trace functions on V without having to construct the irreducible V G-modules explicitly. In the past years, this program of constructing holomorphic vertex operator algebras using orbifold theory has been extremely successful. In particular, the theory of cyclic orbifolds was used in Scott Carnahan’s recent proof of the Generalized Moonshine Conjecture [Car12]. It was also shown [MS19] that at c = 24 all 70 possible holomorphic vertex operator algebras with non-trivial weight-1 space can be constructed as extensions of cyclic orbifolds of lattice vertex operator algebras. In the following, we will briefly develop the theory of cyclic orbifolds from a categorical viewpoint. This is also discussed in [CGR00] and [EG18]. Let a be a generator of the cyclic group ZN with N elements. The group cohomology for cyclic groups is known and explicit forms for the generators can be given.

Theorem 2.2.1 ( [dWP95]). The third cohomology group of ZN is given by

3 ∗ ∼ H (ZN , C ) = ZN (2.2.1) and is generated by the 3-cocycle 2πi ω(aj, ak, al) = exp j[k + l − hk + li ] . (2.2.2) N 2 N where h·iN denotes reduction modulo N.

A general 3-cocycle ωr, r = 0,...N − 1, 2πir ω (aj, ak, al) = exp j[k + l − hk + li ] , (2.2.3) r N 2 N

25 descends to the 2-cocycles

k l 2πir c j (a , a ) = exp j[k + l − hk + li ] (2.2.4) a N 2 N Note further that the second cohomology group vanishes for cyclic groups:

2 ∗ Lemma 2.2.2. H (ZN , C ) = 1.

j k 2πilk/N For [a , χl] the linear characters are χˆl(a ) = e for l ∈ ZN , and the projective characters with k 2πi(rjk/N+lk)/N cocycle (2.2.4) are χl(a ) = e , so that the modular data is given by

−2πic/24 2 2 j j T[a ,χl],[a ,χl] = e exp(2πi(rj + Njl)/N ) . (2.2.5) and 1 S = χ (g )∗χ (g )∗ (2.2.6) [g1,χ1],[g2,χ2] N 1 2 2 1

From Equation 2.2.5 it follows that the conformal weight ρa of the unique irreducible a-twisted V - module V (a) lies in (1/N 2) Z and further that the associated 3-cocycle ω is in the trivial class if and only if it lies in (1/N) Z. This was also proved in [DLM00, vEMS17] without making assumptions on the tensor category structure of the module category. As far as the author knows, the structure of the module category for the cyclic orbifold first given in the literature by Coste, Gannon and Ruelle in [CGR00]. The type t ∈ ZN of the orbifold is defined as Definition 2.2.1 (Type [vEMS17]). We define the type t of an vertex operator algebra automorphism a by 2 t = N ρa mod N (2.2.7)

where ρa is the conformal weight of the a-twisted module Wa. Hence we ﬁnd that the existence of holomorphic extensions depends only on the type of the automorphism.

Corollary 2.2.3. Let V a holomorphic, strongly rational vertex operator algebra and ZN a ﬁnite, cyclic automorphism group. Then the associated 3-cocycle ω is in the trivial class if and only if r = 0.

This is called the Type-0-condition, and it is equivalent to what is called ‘level matching’ in the physics literature. Furthermore, as the second cohomology group vanishes, there is also a single extension for every cyclic group. j Now consider a cyclic group ZN of type 0 and let us choose a representation φj : ZN → End(V (a )) on the unique irreducible aj-twisted module such that Equation 2.1.11 is satisﬁed. Note that, this j representation has to be linear. Then we decompose V (g ) in φj(g) eigenspaces M V (gj) = W (j,l), (2.2.8)

l∈ZN where (j,l) j W = {w ∈ V (g ) | φj(g)v = e(l/N)v}. (2.2.9) Then by the Schur-Weyl-Type Duality the W (j,l) are exactly the N 2 irreducible V ZN -modules and the holomorphic extensions are constructed as follows:

Theorem 2.2.4. Let V be a holomorphic, strongly rational vertex operator algebra and ZN a ﬁnite cyclic automorphism group then the graded vector space

N M V orb(ZN ) = W (i,0) (2.2.10) i=1 can be endowed with the structure of a holomorphic, strongly rational vertex operator algebra.

26 As an application of orbifold theory, we are interested in the construction of holomorphic, strongly rational vertex operator algebras with a small number of low-weight states. Hence, in order to test for this property, we will need to compute the characters of our new holomorphic extensions. In the most general case, that requires the explicit construction of the twisted modules and the action of automorphisms on them. However, in the case of cyclic groups, [vEMS17] shows that this is not necessary and characters of holomorphic extensions can be computed alone from data of the initial vertex operator algebra by making use of the modular transformation properties of the graded trace functions. Recall the definition of the graded trace function T (i, j, τ) for the automorphism aj on the twisted i i module V (a ), where the representation φi : ZN → End(V (a )) is again chosen such that Equation 2.1.11 is satisfied. i j j L0−c/24 T (i, j, τ) = T (a , a , τ) = tr |V (ai) φi(a )q (2.2.11) Then it follows immediately that the characters of the irreducible V ZN -modules are given by 1 X T (i,j) (τ) = e(−jk/N)T (i, k, τ) (2.2.12) W N k∈ZN Hence we find that the character V orb(G) is given by X 1 X 1 X ch orb(G) (τ) = T (i,0) (1, τ) = T (1, i, j, τ) = T (i, j, τ) (2.2.13) V W N N i∈ZN i,j∈ZN i,j∈ZN It was shown in [vEMS17] and [Möl16]that the graded trace functions and characters are modular functions for congruence subgroups as follows: Corollary 2.2.5. For V and g as in the theorem above we find that

1. T (i, j, τ) and TW (i,j) (τ), i, j ∈ ZN , are modular functions for Γ(N),

2. T (0, j, τ), j ∈ Zn, are modular functions for Γ1(N), 3. T (0, 0, τ) is a modular functions for Γ,

4. TW (0,0) (τ) is a modular function for Γ0(N).

Here we denote SL2(Z) by Γ, and Γ(N), Γ1(N) and Γ0(N) denote the usual congruence subgroups. Note in particular that T (i, j, τ) can be obtained as a modular image of a graded trace function T (0, k, τ) evaluated on the inital vertex operator algebra V if and only if gcd(k, N) = gcd(i, j, N). It ZN follows that the characters of all irreducible V -modules and, in particular, the character chV orb(G) (τ) of the holomorphic extension can be written in terms of modular images of graded trace functions of V . For a more detailed discussion of the computational aspects of the calculation of characters of holomorphic extensions refer to Section 5.1.

2.3 Non-Abelian Orbifolds

In order to construct holomorphic extensions for more general, non-abelian groups G acting on a holomorphic, strongly rational vertex operator algebra V , we will need to address the following points: 1. We need to determine if the 3-cocycle ω associated to G and V is in the trivial class. 2. We need to construct an automorphism group of a holomorphic vertex operator algebra V that is isomorphic to G. 3. We need to construct the irreducible V G-modules and hence the action of centralisers on twisted modules. The first point will be addressed in the next section 2.3.1. We will discuss families of groups G that have properties analogous to cyclic groups in that • The H3-twist ω is in the trivial class if its restriction to every cyclic subgroup is in the trivial class, that is the Type-0-condition is satisfied for every cyclic subgroup. Note that it is clear that the Type-0-condition will be satisfied for every element if ω is in the trivial class, so that, in general, this condition is necessary, but not sufficient.

27 • Every graded trace function is a modular image of a graded trace function of the initial vertex operator algebra. This implies that the character of the holomorphic extension can be computed without the construction of the twisted modules. In particular, we will give a full classiﬁcation of groups satisfying the second property. The construction of ﬁnite automorphism groups for lattice vertex operator algebras will be discussed in section 3.5. Finally, we address the construction of twisted modules for lattice vertex operator algebras and the action of automorphisms on them in sections 4.1 and 4.4, respectively.

2.3.1 Groups with Periodic Cohomology

The problem of constraining and identifying the cohomological twist ω for orbifold vertex operator algebras has been studied extensively by Johnson-Freyd [Joh19,Joh18,JT18] using cohomological methods such as the LHS-spectral sequence, the Galois action on cohomology groups and the detection theorem. For an introduction to group cohomology refer to [Wei69] or [Car16]. In this text, we will restrict ourselves to a simple application of the last one of these methods: Lemma 2.3.1. Let G be a finite group and let S ⊆ G be a subgroup that contains a Sylow p-subgroup k ∗ for some prime p. Let H (G, C )(p) denote the p-primary part of the k-th cohomology group. Then the k ∗ k ∗ restriction map α 7→ α|S : H (G, C )(p) → H (S, C )(p) is an injection onto a direct summand. Using this result, we can immediately extend Corollary 2.2.3 on the vanishing of the cohomological twist for cyclic groups to a larger family of groups: Corollary 2.3.2. Let V a holomorphic, strongly rational vertex operator algebra and G a finite automorphism group such that for every prime p, the Sylow p-subgroup is cyclic. Then the associated 3-cocycle ω is in the trivial class if and only if the Type-0-condition is satisfied for every cyclic subgroup of G In fact, these groups fall into a larger family of groups that we will be interested in in the following, namely the groups with periodic cohomology. Definition 2.3.1 ( [Joh03,Wal13]). A finite group G with periodic cohomology satisfies the following equivalent conditions

k ∗ ∼ k+d ∗ • There exists a d ∈ Z>0 such that H (G, C ) = H (G, C ) for all k. • For every subgroup H ⊆ G, H2(H, C∗) is trivial. • Every abelian subgroup of G is cyclic. • For every odd prime p, the Sylow p-subgroup is cyclic and the Sylow 2-subgroup is either cyclic or a generalised quaternion group.

The finite groups with periodic cohomology have been fully classified [Wol11]. Note in particular that the family of groups described in Corollary 2.3.2 contains groups that are not cyclic. Note that the Type-0-condition is not sufficient for the vanishing of the cohomological obstruction for groups with periodic cohomology that do not satisfy the conditions of Corollary 2.3.2, namely groups whose Sylow 2-subgroup is a generalised quaternion group. In [FV87] the quaternion group was identified as the smallest group such that the Type-0-condition does not imply that ω is in the trivial class. In fact, it can easily be seen that this is true for any generalised quaternion group: Let Q2N be the generalised N 3 ∗ ∼ quaternion group of order 2 . Then the cohomology group is given by H (Q2N , C ) = Z2N , but the order N−1 largest cyclic subgroup of Q2N is 2 . It follows from Theorem 2.2.1 that the restriction of an element 3 ∗ N N−1 of H (Q2N , C ) of order 2 to any cyclic group is of order at most 2 . Hence the restriction of the 3 ∗ element of order 2 in H (Q2N , C ) to any cyclic subgroup is trivial. We will now show that orbifolds by finite groups with periodic cohomology share a useful property with those by cyclic groups. In the previous section, we saw that for a cyclic group any graded trace function T (g, h, τ) could be written as the modular image of a graded trace function of the form T (1, k, τ). It is easy to show that for any two commuting automorphisms g and h this possible if and only if g and h generate a cyclic group. It follows that

28 Theorem 2.3.3. Let V be a holomorphic, strongly rational vertex operator algebra, G be a ﬁnite group of automorphisms and T (., ., τ) the corresponding graded trace functions. Then there exist k ∈ G and M ∈ Γ for any commuting pair of elements g, h ∈ G such that T (g, h, τ) = T (1, k, M · τ) if and only if G has periodic cohomology.

Note that any group G with periodic cohomology has a trivial second cohomology group H2(G, C∗). This agrees with the fact that their orbifolds cannot have discrete torsion since their characters are completely ﬁxed by the untwisted modules. The converse, however, is not true: The group (Z7 × Z7) o Z3, where Z3 2 ∗ acts on both Z7-factors by squaring, has trivial H (G, C ), but does not have periodic cohomology.

2.3.2 Semidirect Products of Cyclic Groups Zq oφ Zp

In this section, we will construct the semidirect products of cyclic groups Zq oφ Zp and discuss their group cohomological properties. In particular, we will show that these groups give rise to a non-abelian family of groups with periodic cohomology and that their 3-cocycles are cohomologically trivially if and only if their restrictions to cyclic subgroups are. These properties and their structural simplicity make these groups an attractive target for a ﬁrst systematic survey of holomorphic extensions of non-abelian orbifolds. In subsection 3.5.1, we construct automorphism groups of lattice vertex operator algebras isomorphic to Zq oφ Zp. In chapter 6, we present of orbifolds of lattice vertex operator algebras corresponding to extremal lattice in dimensions 48 and 72 by groups of the form Zq oφ Zp. Deﬁnition 2.3.2. Let φ ∈ Z satisfy φp ≡ 1 (mod q). (2.3.1)

and let (φ − 1) be coprime to q. Then the following relations deﬁne a group Zq oφ Zp: aq = 1 (2.3.2) Ap = 1 (2.3.3) AaA−1 = aφ (2.3.4)

If φ = 1, then, of course, this simply gives the direct product. This family of groups contains many groups with periodic cohomology:

Theorem 2.3.4. The groups Zq oφ Zp with p and q coprime have periodic cohomology. Proof. If p and q are coprime, it is clear that any Sylow subgroup is a subgroup of one of the cyclic factors and hence cyclic itself. Hence the groups have periodic cohomology.

Now we want to describe the 3-cohomology classes for Zq oφ Zp where q and p are coprime. We will further give explicit generators for the cohomology groups so that the modular S- and T -matrices can be computed.

Lemma 2.3.5. [Wal61] The third cohomology group for the group Zq oφ Zp such that q and p are coprime is given by 3 ∗ ∼ H (Zq oφ Zp, C ) = Zp × Zr, (2.3.5) where r = (φ2 − 1, q). Furthermore, we ﬁnd explicit generators for the cohomology group.

3 ∗ Proposition 2.3.6. H (Zq oφ Zp, C ) is generated by the 3-cocycles 2πi ω ((aj,AJ )(ak,AK )(al,AL)) = exp J[K + L − hK + Li ] (2.3.6) p p2 p and 2πi ω ((aj,AJ )(ak,AK )(al,AL)) = exp φK+Lj[φLhki + hli − hφLk + li ] (2.3.7) r r2 r r r

29 Proof. ωp is a 3-cocycle and together both cocycles have the correct order to generate the group. It remains to be shown that ωr is indeed a 3-cocycle. We calculate

j J k K l L m M δωr((a ,A )(a ,A )(a ,A )(a ,A )) 2πi = exp φK+Lj[φLhki + hli − hφLk + li ] + φK+L+M j[φM hφLk + li + hmi − hφM (φLk + l) + mi ] r2 r r r r r r L+M M M L+M K M M + φ k[φ hlir + hlir − hφ l + mir] − φ (φ j + k)[φ hlir + hmir − hφ l + mir] K+L+M L+M M L+M M − φ j[φ hkir + hφ l + mir − hφ k + φ l + mir]

2πi = exp (1 − φ2M )φK+Lj(φLhki + hli − hφLk + li ) r2 r r = 1, (2.3.8)

2M L L where we make use of the facts that r divides both 1 − φ and φ hkir + hlir − hφ k + lir. For the groups listed in section 6, we ﬁnd there is a simple criterion for determining whether the cohomological twist is trivial. Corollary 2.3.7. H3(G, C∗) is cyclic, and ω ∈ H3(G, C∗) is trivial if and only if the type 0 condition is satisﬁed for the cyclic subgroups Zp and Zr.

Proof. The restrictions of ωp and ωr to the cyclic subgroups Zp and Zr ⊂ Zq respectively are exactly the generators of the corresponding third cohomology groups. Hence their contribution to the cohomological twist can be determined from the respective types as in Equation 2.2.4. There is one other case which we will use later on: namely, if p = q, and φ = 1. In that case we recover the Abelian group Zp × Zp, which does not have periodic cohomology. The generators of its cohomology group are given by the following:

Theorem 2.3.8 ( [dWP95]). For abelian groups of the form Zp × Zp the third cohomology group is given by 3 ∗ ∼ 3 H (Zp × Zp, C ) = Zp (2.3.9) and is generated by 2πi ω(1)((aj,AJ )(ak,AK )(al,AL)) = exp j[k + l − hk + li ] , (2.3.10) p2 p 2πi ω(2)((aj,AJ )(ak,AK )(al,AL)) = exp J[K + L − hK + Li ]. (2.3.11) p2 p and 2πi ω(12)((aj,AJ )(ak,AK )(al,AL)) = exp j[K + L − hK + Li ], (2.3.12) p2 p Note that this implies immediately that:

3 Corollary 2.3.9. For abelian groups of the form Zp × Zp the H -twist ω vanishes if and only if the cyclic groups hai, hAi and haAi are all of type 0.

2.3.3 Characters of Holomorphic Extensions for Zq oφ Zp

In this section, we will classify the conjugacy classes of Zq oφ Zp and derive an explicit expression for the character of the holomorphic extension in terms of characters of orbifolds by cyclic subgroups. Note that for groups with periodic cohomology, we expect that the character can always be expressed in terms of characters for cyclic orbifolds. For simplicity, we will assume that q is prime. In particular, we will find that there are three types of conjugacy classes: classes conjugate to non-central elements of the cyclic subgroup Zp, classes conjugate to non-central elements of the cyclic subgroup Zq and, finally, elements of the central subgroup that we will denote by Z p . Hence we find that if q is prime the character for r orbifolds by Z Z can be written in terms of orbifold characters for Z , the central subgroup Z p and q oφ p p r the maximal subgroup Z pq . r To do this, we introduce a slightly different notation emphasising the central subgroup Z p : Let p be r coprime to q, 1 6= φ ∈ Z as above, and r the smallest divisor of p such that φr ≡ 1 (mod q).

30 Lemma 2.3.10. The following relations deﬁne the same group Zq oφ Zp as above: aq = 1 (2.3.13) Ar = B (2.3.14) p B r = 1 (2.3.15) AaA−1 = aφ (2.3.16) BaB−1 = a (2.3.17)

Then, in particular, φr ≡ 1 (mod q). In what follows we take q to be prime. This will cover most of the cases we are interested in, and it turns out that the expression for the character we derive still holds in the cases when q is not prime. u v w A general element g ∈ Zq oφ Zp can be expressed as g = a A B with u = 0, . . . , q − 1, v = 0, . . . , r − 1 p and w = 0,..., − 1. The center of Z Z is given by Z p = hBi. r q oφ p r

Theorem 2.3.11. The conjugacy classes and centralisers of Zq oφ Zp are given by i 1. A class [A ] of length q for every i ∈ Zp with r - i with centraliser CAi = Zp.

Z∗ 2. A class [aiBj] of length r for every pair (ai,Bj) of a coset representative ai ∈ q and an (possibly hφimult j i j identity) element B ∈ Z p with centraliser C (a B ) = Z pq r Zq oφ Zp r i i 3. A central class [B ] of length 1 for every element B ∈ Z p r Proof. For a general element conjugation is given by

y v axAyauAvBwA−ya−x = auφ −x(φ −1)AvBw. (2.3.18)

First take v 6= 0. Then φv − 1 is invertible modulo q and the element auAvBw is conjugate to AvBw for i any u. The centralisers of these representatives are exactly the group Zp = hAi so that the length of [A ] is q. Next take v = 0, u 6= 0. Then y axAyauBwA−ya−x = auφ Bw. (2.3.19)

∗ q−1 p u w Zq shows that there are r × r conjugacy classes representatives a B , u ∈ hφi . The centralisers of those conjugacy classes are generated by a and B and are therefore the cyclic group Z pq = haBi. p w r Finally, u = v = 0 gives the r central classes B whose centralizer is of course the entire group Zq oφ Zp. We are now ready to give an explicit expression for the character of the holomorphic extension:

Theorem 2.3.12. Assume that ω = 1 for the orbifold Zq oφ Zp. Then the character of the orbifold V orb(Zq oφ Zp) is given by

orb(Z Z ) orb(Z ) 1 orb(Z p ) orb(Z pq ) χ q oφ p (τ) = χ p (τ) − (χ r (τ) − χ r (τ)). (2.3.20) r

G Proof. Let Wz (τ) be the G-invariant of the irreducible z-twisted module. By [EG18] the orbifold character is given by

orb(Zq oφ Zp) X Cg χ (τ) = Wg (τ) (2.3.21) g∈cl(Zq oφ Zp)) Z pq X Zq oφ Zp X Zp X r = WBj (τ) + WAj (τ) + WaiBj (τ) (2.3.22) j j Z∗ B ∈Z p A ,r j i j i q j r - a B :a ∈ ,B ∈Z p hφi r Z pq X Zq Zp X Z 1 X = W oφ (τ) + W p (τ) + W r (τ) (2.3.23) Bj Aj r aiBj j j i j i ∗ j B ∈Z p A ,r j a B :a ∈Zq ,B ∈Z p r - r

∗ ∗ where in the last line we used the fact that | Zq /hφi| = | Zq |/r. To rewrite the character invariant under the entire group Zq oφ Zp, we use the fact that it has q maximal subgroups of order p that are all

31 pq conjugate to Z = hAi, and one maximal subgroup Z pq = haBi of order . Z Z is the union of all p r r q oφ p these maximal subgroups, and their pairwise intersections all equal to the center hBi. Using the fact that the twining characters T (g, h; τ) only depend on the conjugacy class of h, we can thus decompose the ﬁrst character as

pq p p−1 r −1 r −1 Zq Zp 1 X X X W oφ (τ) = q T (Bj,Ad, τ) + T (Bj, (aB)b, τ) − q T (Bj,Bc, τ) (2.3.24) Bj pq d=0 b=0 c=0 1 Z pq Z pq pq Z p = pqW p (τ) + W r (τ) − W r (τ) (2.3.25) pq Bj r Bj r Bj Z 1 Z p Z pq = W p (τ) − W r (τ) − W r (τ). (2.3.26) Bj r Bj Bj Hence

X Z 1 Z p Z pq χorb(Zq oφ Zp)(τ) = W p (τ) − W r (τ) − W r (τ) (2.3.27) Bj r Bj Bj Bj ∈Z p r X Z 1 X Z pq + W p (τ) + W r (τ) (2.3.28) Aj r aiBj j i j i ∗ j A ,r j a B :a ∈Zq ,B ∈Z p - r X Z 1 X Z pq 1 X Z p = W p (τ) + W r (τ) − W r (τ) (2.3.29) Aj r aiBj r Bj j i j j A ∈Zp a B ∈Z pq B ∈Z p r r

orb(Z ) 1 orb(Z p ) orb(Z pq ) = χ p (τ) − (χ r (τ) − χ r (τ)). (2.3.30) r

There are some cases in section 6 for which q is not prime. They turn out to be of the form Zq o−1 Z4, and it turns out that (2.3.20) is still correct for them.

32 3 Lattice Vertex Operator Algebras

In this chapter, we will discuss the construction of lattice vertex operator algebras, as well as their automorphism groups and graded trace functions. In subsection 3.5.1, we will show in particular, that groups of the form Zq oφ Zp that where introduced in subsection 2.3.2 can be realised as automorphism groups of lattice vertex operator algebras.

3.1 Lattices

We begin with a brief review of the most important result concerning lattices, their automorphisms and lattice theta-functions. For introductory references to the theory of lattices and their automorphism groups refer to the classic [CSB+13] or [Gri11]. An online database of lattices by Gabriele Nebe and Neil Sloan is available at [Neb17].

3.1.1 Integral Lattices d Deﬁnition 3.1.1. A lattice of rank d is the Z-span of an Q-basis {b1, . . . , bd} of Q with a symmetric, positive deﬁnite, nondegenerate bilinear form h·|·i. That is

d X L = hb1, . . . , bdiZ = { vibi|vi ∈ Z}. (3.1.1) i=1 Note that the assumption that the bilinear form be positive-definite and nondegenerate is not necessary, but in this text we will only be concerned with lattices that do fullfil it. For the purpose of constructing lattice vertex operator algebra we are primarily interested in so-called integral and even lattices. Definition 3.1.2. A lattice L is called integral if hv|wi ∈ Z for all v, w ∈ L. Definition 3.1.3. A lattice L is called even if hv|vi ∈ 2 Z for all v ∈ L. Note that as hv + w|v + wi = hv|vi + 2hv|wi + hw|wi, even lattices are integral.

Definition 3.1.4. The gram matrix G of a lattice L with basis {b1, . . . , bd} is the symmetric d×d-matrix with entries Gij = hbi|bji. (3.1.2) Evidently, a lattice is integral if the entries of its gram matrix are integers and even if the diagonal entries are even. Definition 3.1.5. The determinant det(L) of a lattice L is defined to be the determinant of its gram matrix. Furthermore we define: Definition 3.1.6. A lattice L is called unimodular if det(L) = 1. Furthermore we associate to any lattice its dual lattice Definition 3.1.7. Let L be a lattice, then its dual lattice L0 is defined by L0 = {x ∈ Qd |hx|vi ∈ Z for all v ∈ L}. Definition 3.1.8. Let L be an even lattice, then the level of L is the smallest number N such that NL0 is again even. Note that if L is integral, then L ⊆ L0 and L/L0 is a finite abelian group of order det(L), often called the dual quotient. It is clear an integral lattice is unimodular if and only if L = L0. Hence integral, unimodular lattices are often called self-dual. Finally, we define Definition 3.1.9. The minimum of a lattice L is given by min(L) = min{hv|vi|v ∈ L, v 6= 0}.

33 3.1.2 Lattice theta-functions and Modularity Furthermore, even lattices give rise to special modular forms, their associated lattice theta-functions. Their existence establishes a relation between the theory of lattices and number theory and allows to prove important properties of even, self-dual lattices. They will play a central role as parts of the characters of lattice vertex operator algebras. Deﬁnition 3.1.10. The lattice theta-function of a lattice L is given by

X 1 hv|vi 2πiτ θL(τ) = q 2 , where q = e . (3.1.3) v∈L

In particular, the following modularity result holds: Lemma 3.1.1 (Hecke, Sch¨onberg [Zag08]). If L is an lattice even of rank d, level N and determinant ∆. Then its theta-function θL(τ) is a holomorphic modular form for the congruence subgroup Γ0(N) of d (−1)d∆ . weight 2 and character χ(p) = p for the congruence subgroup Γ0(N), where . denotes the Legendre symbol.

The modularity has the following profound consequences for even, unimodular lattices: Lemma 3.1.2 ( [CS99]). Let L be an even, unimodular lattice of rank d. Then

d • The lattice theta-function θL(τ) is a holomorphic modular form of weight 2 for the full modular group Γ.

• The rank d of L is a multiple of 8.

d • The minimum of L is bounded from above by min(L) ≤ 2 + 2b 24 c. We will see that lattice vertex operator algebras associated to lattices with a large minimum are a good starting point for our goal of constructing vertex operator algebras with a low number of low-weight states. Hence we are particularly interested in lattices saturating the upper bound for the minimum given in lemma 3.1.2. Definition 3.1.11. An even, unimodular lattice is called extremal if its minimum saturates the upper bound d min(L) = 2 + 2b c. (3.1.4) 24 Clearly, every even lattice of rank smaller than 24 is also extremal. By Niemeier’s classification of even, unimodular lattices of rank 24 [Nie68], the only extremal lattice at this rank is the famous Leech lattice [Lee67]. In what follows, we will be primarily interested in extremal lattices at rank 48 and 72, that have been constructed by Gabriele Nebe [Neb98,Neb14,Neb12]. Finally, define the following modified theta-functions

Deﬁnition 3.1.12. Let L be a lattice of rank d and λ, β ∈ Qd.

X hα+λ,α+λi θL+λ(τ) = q 2 (3.1.5) α∈L

and β X hα,αi 2 2πihα,βi θL(τ) = q e . (3.1.6) α∈L The following result on their modular transformation properties will prove very useful in the computation of characters of holomorphic extensions for cyclic groups.

Theorem 3.1.3 (Inversion Formula, [Iwa97]).

β −1 − 1 Rank(L) θ = (det(L)) 2 (−iz) θ 0 (τ) L τ L +β

34 3.1.3 Lattice Automorphisms

In this subsection we will discuss automorphisms of lattices and review some of their properties, as well as some associated lattices they give rise to. In section 3.3, we will show that an automorphism of a lattice can be lifted to an automorphism of the associated lattice vertex operator algebra. This is very attractive for the purpose of orbifold theory, as the structure of the lifted groups can be well controlled and a wide variety of groups can be realised thos way. Hence it is these groups of lifted automorphisms that we will use to construct new holomorphic, strongly vertex operator algebras from orbifold theory. Note that for consistency with most computer algebra systems we will adopt the convention that lattice automorphisms act to the left. Deﬁnition 3.1.13. Let L be a lattice. Then an automorphism of L is a bijective, Z-linear map g : L → L, such that hvg|wgi = hv|wi for all v, w ∈ L. The automorphism group of L will be denoted Aut(L). To any automorphism, we associate the corresponding invariant and coinvariant sublattice.

Deﬁnition 3.1.14. Let L be a lattice and g ∈ Aut(L). Then the invariant sublattice Lg ⊆ L is given by

Lg = {v : v ∈ L|vg = v}. (3.1.7)

⊥ Deﬁnition 3.1.15. Let L be a lattice and g ∈ Aut(L). Then the coinvariant sublattice Lg ⊆ L is given by ⊥ Lg = {v : v ∈ L|hv|wi = 0, ∀w ∈ Lg}. (3.1.8) For later convenience we deﬁne notion of the cycle type associated to a lattice automorphism.

Deﬁnition 3.1.16. A cycle type C of order n is a set of pairs {(t, bt)}, such that t | n, bt ∈ Z and gcd({t}) = n. As a shorthand we will write Y C := tbt . t|n

Let g be an automorphism of an integral lattice of order n. Then the characteristic polynomial of g has integer coeﬃcients and its roots are n-th roots of unity. Such a polynomial is a product of cyclotomic polynomials.

Deﬁnition 3.1.17. The n-th cyclotomic polynomial Φn is given by

Y 2iπ k Φn(x) = x − e n (3.1.9) 1≤k≤n gcd(k,n)=1

Now let g have characteristic polynomial χg, such that

Y nt χg(q) = Φt(q) (3.1.10) t|n where Φt is the t-th cyclotomic polynomial and nt ∈ N. Using the M¨obiusinversion formula Y d µ t Φt(q) = (q − 1) d , d|t where µ is the M¨obiusfunction, we can express the characteristic polynomial of g as

Y t bt χg(q) = (q − 1) (3.1.11) t|n with bt ∈ Z. From this we deﬁne the cycle type of g:

Q t bt Definition 3.1.18. Let g be a lattice automorphism with characteristic polynomial χg(q) = t|n(q −1) . Q bt Then we define the cycle type of g to be Cg = t|n t . Finally, define the projectors onto the g-invariant subspace and its orthogonal complement:

35 Deﬁnition 3.1.19. The projectors onto the g-invariant subspace and its orthogonal complement are given by n−1 1 X π = gi (3.1.12) g n i=0 and ⊥ πg = 1 − πg, (3.1.13) respectively.

3.1.4 Smith Normal Form and Lattice Quotients

In the following, we will study some applications of the Smith normal form to lattices with a special focus on operators of the form 1 − g. This will allow us to ﬁnd useful results on the decomposition of lattice into sublattices. Deﬁnition 3.1.20 (Smith normal form). Let L be an even, unimodular lattice of rank n and A an n × n-matrix with entries in the integers (or more generally, a principal ideal domain). Then there exist integral, (over the integers) invertible matrices P and Q such that the product

S = PAQ (3.1.14)

is a diagonal matrix of the form

S = diag(s1, s2, ..., sk, 0,..., 0), (3.1.15) Qn such that si ∈ Z≥0 and si|si+1 for all i and i=1 si = ±det(A). The diagonal entries si are unique up to sign and are called the elementary divisors of A. S is called the Smith normal form of A. To put it another way, there exist bases {α˜1,..., α˜k, γ1, . . . , γn−k} and {α1, . . . , αk, δ1, . . . , δn−k} of L (given by the row vectors of P and Q−1 respectively) such that A acts as

α˜iA = siαi, i = 1, . . . , k (3.1.16) and γjA = 0, j = 1, . . . , n − k. (3.1.17)

Since the lattice LA is spanned by the basis {siαi}, it follows immediately: Corollary 3.1.4. Let L be a lattice of rank n and let A act on elements of L in the coordinate basis. Then the group structure of the quotient L/LA is given by

n ∼ M L/LA = Z /si Z, (3.1.18) i where si = 0 for i > k. Lemma 3.1.5. Let g be an automorphism of L of order n, A = 1 − g and the matrices P , Q and S as in Theorem 3.1.4. Then the ﬁxed-point lattice Lg is given by

Lg = spanZ(γ1, . . . , γn−k) (3.1.19)

⊥ and Lg , its orthogonal complement in L, by

⊥ Lg = spanZ(α1, . . . , αk). (3.1.20)

Proof. Clearly, spanZ(γ1, . . . , γn−k) ⊆ Lg. So to prove the ﬁrst statement, we need to show that there is no vector v ∈ spanZ(α˜1,..., α˜k) such that v(1−g) = 0. This follows directly from the linear independence of the {αi}. ⊥ Clearly, spanZ(α1, . . . , αk) ⊆ Lg . So to prove the second statement, we need to show that there is ⊥ ⊥ no vector w ∈ spanZ(δ1, . . . , δn−k) such that w ∈ Lg . But because Rank(Lg ) = Rank(L(1 − g)) for any such vector there has to exist an integer nw such that nww ∈ L(1 − g). This contradicts the linear independence of the elements of BQ. Hence the second statement follows.

36 ⊥ Corollary 3.1.6. Lg /L(1 − g) is generated by the elements {[α1],..., [αk]} and the element [αi] has ⊥ order si. In particular, Lg /L(1 − g) is the torsion subgroup of L/L(1 − g).

⊥ It follows immediately, that Lg and Lg are so-called primitive sublattices of L. That is Lemma 3.1.7. There exist sublattices Λ and Γ in L, such that

L = Lg + Γ (3.1.21) and ⊥ L = Λ + Lg . (3.1.22)

3.2 Construction of Lattice Vertex Operator Algebras

Let L be an even lattice. Then we can associate to it a lattice vertex operator algebra VL. Lattice vertex operator algebras are one of the best studied families of vertex operator algebras and both their module categories and automorphism groups are well understood, making them an attractive target for orbifold theory. In particular, lattice vertex operator algebra are strongly rational and, if the associated lattice is even and self-dual, they are holomorphic. In fact, all holomorphic vertex operator algebras at central charge c ≤ 16 are lattice vertex operator algebras associated to even, self-dual lattice. The construction of lattice vertex operator algebras is described in all of the standard texts [LL04, FLM88, FBZ04] and we will give a brief review in the following: Definition 3.2.1. The Heisenberg current algebra is the affine Lie algebra associated with the complexified lattice h = L ⊗Z C is given by ˆ −1 h = h ⊗C C[t, t ] ⊕ C k, with the Lie bracket defined by the linear continuation of

[x(n), y(m)] = hx, yinδn+mk and [u, k] = 0 for x, y ∈ h,n, m ∈ Z and u ∈ hˆ where we use the shorthand x(n) := x ⊗ tn. We decompose into subalgebras hˆ = hˆ− ⊕ hˆ+ ⊕ hˆ ⊕ C k, (3.2.1) where hˆ ⊕ C k is the subalgebra of weight 0 and hˆ+ and hˆ− are the graded subalgebras of positive and negative weights, respectively.

Deﬁnition 3.2.2. The twisted group algebra C[L] corresponding to the lattice L is spanned by the C-basis {eα}α∈L and the multiplication is deﬁned by

eαeβ = (α, β)eα+β, (3.2.2) where : L × L → {±1} is a 2-cocycle satisfying

(α, α) = (−1)hα,αi/2 and (α, β)/(β, α) = (−1)hα,βi (3.2.3) for α, β ∈ L. We define the weight by wt(eα) = hα|αi/2. Note that an infinite number of 2-cocycles satisfy Equation 3.2.3, but they all yield isomorphic lattice vertex operator algebras so that the specific choice for is irrelevant from a structural standpoint. However, when carring out computations involving lattice vertex operator algebras working with a particular solution is convenient. Hence note that: Corollary 3.2.1. The bimultiplicative function : L × L → {±1} defined by  (−1)hαi|αj i if i > j  hα |α i/2 (αi, αj) = (−1) i i if i = j (3.2.4) 1 if i < j, where {α1, . . . , αd} is a basis for L, satisfies Equation 3.2.3.

37 These two ingredients can now be assembled to form a lattice vertex operator algebra VL.

Deﬁnition 3.2.3. The Lattice Vertex Operator Algebra VL corresponding to the lattice L is spanned by elements of the form

hk(−nk) . . . h1(−n1)eα with nk, . . . , n1 ≥ 0. The weight of this element is given by 1 n + ··· + n + hα|αi ∈ Z . 1 k 2 ≥0

Hence as a vector space VL is isomorphic to

∼ ˆ− VL = S(h ) ⊗ C[L], (3.2.5) where S(hˆ−) is the symmetric algebra of hˆ−. Note that the symmetric algebra S(hˆ−) can be endowed with the structure of a vertex operator algebra. As such, it is called the Heisenberg vertex operator ˆ− algebra. The twisted multiplication of C[L] encodes the tensor category structure of S(h )-modules and ˆ− VL is an extension of S(h ). The vacuum vector is given by the element

|0iL = e0 (3.2.6)

and the conformal vector by c 1 X ω = α (−1)α (−1)e . (3.2.7) L 2 i i 0 i=1 Finally, we can deﬁne the vertex operators for the lattice vertex operator algebra Deﬁnition 3.2.4. The vertex operators are generated by

X −n−1 YL(α(−1)e0, z) = α(z) = α(n)z , (3.2.8) n∈Z where we let the Heisenberg current algebra act on VL as

k · eα = eα (3.2.9)

h(n)eα = 0 (3.2.10)

h(0)eα = hh|αieα (3.2.11)

for all h ∈ h and n ∈ Z≥0, and

− + α YL(eα, z) = E (−α, z)E (−α, z)eαz , (3.2.12) where X −α(n) E±(−α, z) = exp z−n (3.2.13) n n∈± Z>0 α and the operator z acts on VL as

[h(m), zα] = 0 (3.2.14) α hα|βi z eβ = z eβ. (3.2.15)

Together this implies:

∼ ˆ− Theorem 3.2.2. The graded vector space VL = S(h ) ⊗ C[L] endowed with the vertex operators YL, vaccum vector |0iL and conformal vector ωL has the structure of a vertex operator algebra. Finally, we will discuss the various regularity properties of lattice vertex operator algebra. It is evident from the construction, that lattice vertex operator algebras are of CFT-Type. Furthermore, Dong [Don93] fully classiﬁed the modules for lattice vertex operator algebras. In particular, he showed that

38 Lemma 3.2.3. The irreducible modules of the vertex operator algebra VL are in 1 − 1-correspondence with the elements of the dual quotient L0/L. For a detailed discussion of the (twisted) modules of lattice vertex operator algebras also refer to chapter 4. Simplicity, self-contragredience and rationality are also proved in [Don93]. The C2-coﬁniteness of lattice vertex operator algebras is shown in [DLM00]. Altogether, we ﬁnd that:

Theorem 3.2.4. Lattice vertex operator algebras are simple, self-contragredient, rational, C2-coﬁnite and of CFT-Type, and hence strongly rational. For even, self-dual lattices we ﬁnd in particular:

Lemma 3.2.5. A lattice vertex operator VL is holomorphic if and only if L is even and self-dual.

3.3 Automorphisms of Lattice Vertex Operator Algebras

Automorphisms of lattices can be lifted to automorphisms of the associated lattice vertex operator algebras. This, in particular, allows us to construct a wide variety of ﬁnite groups as automorphism groups of vertex operator algebras. The theory of automorphisms of lattice vertex operator algebras is well understood and is described in many standard texts on vertex operator algebras such as [FLM84] and [Bor92]. Here will give a brief review. A lattice automorphism g ∈ Aut(L) can be lifted to a VL-automorphismg ˆ by setting

ghˆ k(−nk) . . . h1(−n1)1 = (hkg)(−nk) ... (h1g)(−n1)e0 (3.3.1)

on the Heisenberg current algebra. It can be readily checked that this implies that gˆ is an automorphism of the Heisenberg vertex operator algebra. Applying the equivariance condition in Deﬁnition 1.3.1 to the vertex operator Y (eα, z), we ﬁnd that a lifted lattice automorphism should act as an automorphism of the twisted group algebra C[L] such that

gˆ(eαeβ) =g ˆ(eα)ˆg(eβ). (3.3.2)

Theng ˆ satisﬁes gˆ(eα) = ug(α)eαg, (3.3.3) ∗ where ug : L → C is a function satisfying (α, β) u (α)u (β) = g g . (3.3.4) (αg, βg) ug(α + β) Hence we arrive at

Theorem 3.3.1. Let L be an even lattice and g an automorphism. Then a linear map gˆ : VL → VL satisfying Equations 3.3.1 and 3.3.2 is an automorphism of VL.

(α,β) 2 ∗ ∗ Note that (αg,βg) ∈ B (L, C ) is a 2-coboundary so that a function ug : L → C satisfying equation (3.3.4) always exists. One solution can be constructed as follows:

Lemma 3.3.2. Let {α1, . . . , αd} be a basis for L and let g : L × L → {±1} be the bimultiplicative function deﬁned by ( (αi,αj ) if i > j (αig,αj g) g(αi, αj) = (3.3.5) 1 otherwise .

Then ug(α) = g(α, α) satisﬁes Equation 3.3.4.

This solution ug is of course not unique: ∗ Lemma 3.3.3. Let ug satisfy Equation (3.3.4). Then ξgug again satisﬁes (3.3.4) for any ξg ∈ Hom(L, C ). ug ∗ Now let µg also satisfy Equation (3.3.4). Then ∈ Hom(L, C ). µg Further, we ﬁnd immediately that lifts of lattice automorphisms behave as expected under multiplication: Corollary 3.3.4. Let gˆ and hˆ be lifts of g and h, respectively. Then gˆhˆ is a lift of gh.

39 Considering lifts of the identity element in Aut(L) we find the following result: ∗ Corollary 3.3.5. The group Hom(L, C ) has a natural embedding into Aut(VL) as lifts of the identity element in Aut(L). Furthermore, the elements of Hom(L, C∗) are inner, corresponding to elements of weight 1 in the ∗ Heisenberg subalgebra of VL, that is exp(h(0)) ∈ Hom(L, C ). ∗ The restriction of ug to the fixed-point sublattice Lg is a homomorphism Lg → C . It is easy to show that the conjugacy class of a liftg ˆ in Aut(VL) is, in fact, determined by this restriction. A commonly studied class of automorphisms are so-called standard lifts were this homomorphism is the trivial one: Definition 3.3.1 (Standard Lift). A liftg ˆ such that

ug(α) = 1 for all α ∈ Lg, is called a standard lift. In particular, standard lifts exists for any lattice automorphism Lemma 3.3.6. A standard lift can be constructed from any arbitrary one by multiplying with an inner automorphism. ∗ Proof. The restriction ug|Lg to the ﬁxed-point lattice is a homomorphism. Hence for any ξg ∈ Hom(L, C ) such that ξ | = u |−1 the lift ξ (·)u (·) is a standard lift. g Lg g Lg g g This implies that any lift can be written as the product of an inner automorphism and a standard lift. Corollary 3.3.7. All lifts gˆ of g act as

2πihβ|αi gˆ(eα) = e ug(α)eαg, (3.3.6)

for some vector β ∈ h, such that the restriction of ug to the ﬁxed point lattice is trivial.

Note that the automorphism eh(0) is conjugate to e(hπg )(0). Hence we may choose β ∈ hg without loss of generality. Standard lifts have the useful properties that their twisted modules and graded trace functions have a particularly simple form and that their order is at most twice that of the original lattice automorphism. Lemma 3.3.8 (Order of lifted automorphisms). [M¨ol16,Bor92] Let gˆ be a standard lift of g of order m. If m is odd, then gˆ has order m. If m is even, then gˆ has order m if hαgm/2|αi ∈ 2 Z for all α ∈ L and order 2m otherwise.

In particular, note that if the ﬁxed-point lattice Lg is trivial for some g ∈ Aut(L), any lift gˆ ∈ Aut(VL) is a standard lift of the same order as g and conjugate to any other lift of g.

3.4 Graded Trace Functions for Lattice Vertex Operator Algebras

We can now make use of these results to give an explicit form for the graded trace function of a lifted lattice automorphism gˆ in terms of (modiﬁed) lattice theta-functions and eta-quotients. In particular, their modular transformation properties are well understood. Making of use of Theorem 2.1.7, this will prove extremely valuable in the computation of characters for holomorphic extensions, as working with the twisted modules directly is complicated and computationally expensive.

Theorem 3.4.1. Let gˆ be an automorphism of VL obtained as a lift of a lattice automorphism g ∈ Aut(L) Q bt ∗ of cycle type t|n t deﬁned by a function ug : L → C . Then the graded character for gˆ on VL is given by ϑ (τ) L0−c/24 Lg ,ug trVL gqˆ = , (3.4.1) ηg(τ) where ϑLg ,ug (τ) is the generalised theta-function of the ﬁxed-point sublattice Lg given by

X hα,αi/2 ϑLg ,ug (τ) = ug(α)q ,

α∈Lg

40 and the eta-quotient ηg(τ) is given by

Y bt ηg(τ) = η(tτ) . (3.4.2) t|n

This follows from a straightforward computation. Note that the graded character, as expected, depends

only on the restriction of ug to the ﬁxed-point lattice. As the restriction ug|Lg is a homomorphism, the

generalised theta-function ϑLg ,ug (τ) is furthermore a modiﬁed theta-function in the sense of Equation 3.1.6. For a standard lift this clearly reduces to the familiar result

ϑ (τ) L0−c/24 Lg trVL gqˆ = , ηg(τ)

where ϑLg (τ) is the ordinary theta-function of the ﬁxed-point sublattice Lg. In the notation of graded trace functions, we ﬁnd the corresponding result for the automorphismg ˆk:

ϑL k ,u k (τ) k k L0−c/24 g g T (1, gˆ , τ) = trVL gˆ q = , (3.4.3) ηgk (τ)

Qk−1 i with ugk (α) = i=0 ug(αg ). This explicit form allows us, in particular, to deduce the conformal weight of the unique irreducible gˆ-twisted VL-module

Q bt Theorem 3.4.2. Let g be a lattice automorphism of order n with cycle type t|n t and let gˆ be a lift of g as in Corollary 3.3.7. Then the the unique irreducible gˆ-twisted VL-module has conformal weight

c 1 X bt 1 ρ = − + min(L0 + β), (3.4.4) gˆ 24 24 t 2 t|n where min(L0 + β) is the squared length of a minimal element of L0 + β.

Proof. Apply the S-transformation to the twisted trace T (0, 1, τ) as deﬁned in equation (3.4.3) and use Corollary 5.1.5 and Theorem 3.1.3.

3.5 Finite Automorphism Groups for Lattice Vertex Operator Algebras

In this section we will discuss the lifting of a finite subgroup G of Aut(L) to a finite subgroup Gˆ of Aut(VL). In order to construct new orbifold vertex operator algebras, we need good control over the structure of the lifted group Gˆ. It is therefore a natural goal to find a lift Gˆ that retains the structure of G to a large extend or, ideally, is isomorphic to it. This important since automorphism groups of vertex operator algebras are not well understood in general and hence examples of non-abelian finite automorphism groups of vertex operator algebras are not available. We will consider this in general and prove that well-behaved lifts exists for a certain family of non-abelian groups.

Definition 3.5.1. The group of all lifted lattice automorphisms as above will be denoted by O(Lˆ). A complete description of the automorphism group of lattice vertex operator algebras was given in [DN99]. Let O(Lˆ) be the group of lifted lattice automorphisms as in Definition 3.5.1 and define the inner automorphism group a(0) N = he : a ∈ V(1)i. (3.5.1) Theorem 3.5.1. [DN99] Let L be a positive definite even lattice. Then

Aut(VL) = N · O(Lˆ) (3.5.2)

41 In particular, if L has no vectors of length 2 we have N ⊂ O(Lˆ) so that O(Lˆ) is the full automorphism group of the lattice vertex operator algebra. It is easy to see that O(Lˆ) is a group extension of Aut(L) of the form

1 → Hom(L, C∗) → O(Lˆ) → Aut(L) → 1, (3.5.3) where · denotes the canonical projection from O(Lˆ) onto Aut(L). In the following, ˆ· : Aut(L) → O(Lˆ) will denote a section of Aut(L) in O(Lˆ). Where there is no risk of confusion, we will, by abuse of notation, letg ˆ denote any element of O(Lˆ), or an appropriate subgroup, that projects onto g ∈ Aut(L). The 2-cocycle corresponding to the short exact sequence 3.5.3 is given by

−1 uh(·)ug(h·) s(g, h) =g ˆhˆghc = . (3.5.4) ugh(·)

It can be readily verified that s(g, h) ∈ Hom(L, C∗) and that it satisfies the appropriate cocycle condition. Now let G ⊂ Aut(L) be finite group of lattice automorphisms and let Gˆ ⊂ Aut(VL) be a group of automorphisms of the lattice vertex operator algebra such that each of its elements is a lift of some element of G. Such a finite extension Gˆ satisfies the short exact sequence

1 → H → Gˆ → G → 1, (3.5.5) where H = Gˆ ∩ Hom(L, C∗). We would, in particular, like to construct extensions Gˆ such that H has minimal order. In the case H = 1, the groups Gˆ and G are isomorphic and the isomorphism G → Gˆ is a splitting map for the short exact sequence 1 → Hom(L, C∗) → Hom(L, C∗) · G → G → 1. (3.5.6) ∗ H Note that for a finite subgroup H ∈ Hom(L, C ) the fixed-point vertex operator algebra VL is itself a H lattice vertex operator algebra associated to the lattice L = ker(H). Let Gˆ|H denote the restriction to H ˆ ∼ VL = VLH . It is clear that the restriction H is trivial, so that G|H = G, that is the extension of G splits in Aut(VLH ). In particular, we find that: Lemma 3.5.2. The following identity of fixed-point vertex operator algebras holds

Gˆ G VL = VLH . (3.5.7)

Now let G ⊂ Aut(L) be generated by elements {g1, . . . , gN } satisfying relations given by words QMi {w1, . . . , wK }, where wi = j=1 gwi,j for some numbers Mi ∈ Z≥0 and wi,j ∈ {1, . . . , n}. Then let {gˆ1,..., gˆn} be lifts to AutVL in the sense of Equation 3.3.3 and deﬁne Gˆ = hgˆ1,..., gˆni the group generated by these elements. This allows us to explicitly construct Gˆ ∩ Hom(L, C∗), the subgroup of inner automorphisms in Gˆ. Note that by Corollary 3.3.5, any word in G that evaluates to the identity gives rise to a corresponding inner automorphism in Gˆ. Particularly, we ﬁnd that

∗ Lemma 3.5.3. Every relation wi deﬁnes a homomorphism φwi ∈ Hom(L, C ) given by

M Yi φwi = gˆwi,j . (3.5.8) j=1

Proof. By Corollaries 3.3.4 and 3.3.5, φwi is a lift of the identity element in Aut(L) and therefore an element of Hom(L, C∗).

Let Hφ = hφw1 , . . . , φwK iG be the G-module generated by these homomorphisms. This is, in fact, the whole normal subgroup of inner automorphisms in Gˆ and consequently, we arrive at the following result on the structure of Gˆ: Theorem 3.5.4. The group Gˆ satisﬁes the short exact sequence

1 → Hφ → Gˆ → G → 1. (3.5.9)

That implies in particular ∗ Gˆ ∩ Hom(L, C ) = Hφ. (3.5.10)

42 ∗ Proof. Lemma 3.5.3 implies the inclusion Hφ ⊂ Gˆ ∩ Hom(L, C ). By Corollary 3.3.5 the elements of ∗ Hom(L, C ) ⊂ Aut(VL) are exactly the lifts of the identity element of G, but by Corollary 3.3.4 these are exactly the homomorphisms corresponding to the set words of in G that evaluate to the identity, that is ∗ the normal closure of the {w1, . . . , wK }. It follows that Gˆ ∩ Hom(L, C ) ⊂ Hφ.

Hence constructing a splitting map amounts to constructing lifts of the generators {gˆ1,..., gˆn}, such

that the homomorphisms φwi associated to all words vanish. Note that in practice, we will want to construct a lifting that satisfies further desirable properties, such as, importantly, the vanishing of the associated H3-twist ω as defined in Theorem 2.1.3. In order to do so, we will start from any given set of lifted generators {gˆ1,..., gˆn} and then perform a change of section ∗ gî 7→ ξgi gî for some ξgi ∈ Hom(L, C ) in the sense of Lemma 3.3.3. This corresponds to constructing a crossed homomorphism ξ : Gˆ → Hom(L, C∗), that is a map satisfying

∗ ξ(ˆgi) = ξgi (·) ∈ Hom(L, C ) (3.5.11)

and ξ(ˆghˆ) = hˆ · ξ(ˆg)ξ(ˆg), for allg, ˆ hˆ ∈ G,ˆ (3.5.12) ˆ where h · ξ(ˆg) = ξg(h·).

Under this change of section the homomorphisms φwi transform as

φwi 7→ ξ(wi)φwi . (3.5.13)

In particular, ﬁnding a splitting map amounts to constructing a crossed homomorphism ξ such that

ξ(w ) = φ−1, for all i ∈ {1,...,K}. (3.5.14) i wi

Now consider conjugation of Gˆ by an element χ of Hom(L, C∗):

χgχˆ −1 = χ(ˆg−1 · χ)−1gˆ (3.5.15)

This corresponds to a deformation by a so-called principal crossed homomorphism. It follows that Hom(L, C∗)-conjugacy classes of groups generated by lifted generators can be parametrised by first cohomology groups: Theorem 3.5.5. H1(Hom(L, C∗) · G, Hom(L, C∗)) parametrises Hom(L, C∗)-conjugacy classes of groups Gˆ generated by lifted generators. Theorem 3.5.6. H1(G, Hom(L, C∗)) parametrises Hom(L, C∗)-conjugacy classes of groups Gˆ generated by lifted generators with isomorphic normal subgroups Gˆ ∩ Hom(L, C∗). Proof. An element ξ ∈ H1(G, Hom(L, C∗)) acts on Gˆ as gˆ 7→ ξ(g)gˆ. That implies that ξ acts trivially on any 1ˆ ∈ Gˆ, such that 1ˆ = 1. By Corollary 3.3.5 and Theorem 3.5.4, the elements of H1(G, Hom(L, C∗)) are exactly those crossed homomorphisms that act trivially on Gˆ ∩ Hom(L, C∗). This implies, in particular, that there is a finite number of these Hom(L, C∗)-conjugacy classes of groups. Note that these result can be found in standard texts on group cohomology, for example [AM04], stated in various different forms.

3.5.1 Lifting Semidirect Products of Cyclic Groups Zq oφ Zp Finding a lift that splits is particularly important if G has periodic cohomology and we want Gˆ to retain it. In view of section 2.3.2, let us discuss the lifting theory of groups of the form

G = Zq oφ Zp . (3.5.16) In this section we discuss how to construct a splitting lift such that

∼ Zq\oφ Zp = Zq oφ Zp . (3.5.17)

43 This amounts to constructing liftsg ˆ and hˆ satisfying the group relations, namely

hˆq = 1 (3.5.18) gˆp = 1 (3.5.19) gˆhˆgˆ−1 = hˆφ (3.5.20)

For simplicity, we would additionally like to impose thatg ˆ and hˆ are standard lifts. We will furthermore assume that gˆ and hˆ are standards lifts without order doubling, such that the ﬁrst two group relations are automatically satisﬁed. When Zq oφ Zp has periodic cohomology, any element is conjugate to a power of one the generators so that every element is a standard lifts if the generators are. In general, this is however not the case. It is straightforward to generalize our construction for the case of non-standard lifts. To ensure (2.3.4), we use the following lemma:

ˆ −1 ˆφ Lemma 3.5.7. The relation gˆhgˆ = h is equivalent to ug and uh satisfying

φ−1 −1 −1 −1 Y i ug(αg h)ug(αg )uh(αg ) uh(αh ) = 1. (3.5.21) i=0 Proof. Sinceg ˆ acts on lattice states as

−1 gˆeα = ug(α) eαg , (3.5.22) we have

n−1 n −1 Y i −1 gˆ eα = ugn (α) eαgn = ug(αg ) eαgn (3.5.23) i=0 and −1 −1 −1 gˆ eα = ug(αg ) eαg−1 . (3.5.24) Using (3.5.23) and (3.5.24) and the fact that g−1hgh−φ = e, a straightforward computation gives

φ−1 ˆ−φ ˆ −1 −1 −1 −1 Y i h gˆhgˆ eα = ug(αg )uh(αg )ug(αg h) uh(αh )eα , (3.5.25) i=0 which establishes (3.5.21).

To obtain such ug and uh, we construct homomorphisms ξg and ξh such that the new lifts u˜g = ugξg and u˜h = uhξh do satisfy Equation 3.5.21. In order for the new u˜g and u˜h to still be standard lifts we demand that ξg(Lg) = ξh(Lh) = 1. By carefully analysis the action of the group Zq oq Zp using the Smith normal form as described in subsection 3.1.4, we ﬁnd that there is an obstruction to the existence of such ξg and ξh and we construct the homomorphisms explicitly when they exist. For all examples we consider in chapter 6, this obstruction vanishes, so that we can construct a splitting lift. For an orbifold by a group with periodic cohomology as described in Theorem 2.3.4, the character of the holomorphic extension does not depend on the precise of the lift, but is determined by gˆ and hˆ being standard lifts. Hence the mere existence of this lift is enough to compute the character. For groups that do not fall in this category, such as for instance Z3 × Z3, we will need the explicit expression for the lift, which is provided in lemma 3.5.12, since here the computation of characters does require the construction of twisted modules. ˆ Let now gˆ and h be standard lifts without order doubling and ug and uh the corresponding funtions that will not satisfy (3.5.21). We therefore want to construct homomorphisms ξg and ξh such that the new lifts u˜g = ugξg and u˜h = uhξh do satisfy Equation 3.5.21. In order for the new u˜g and u˜h to still be standard lifts we demand that ξg(Lg) = ξh(Lh) = 1. For convenience we will deﬁne the homomorphism f as the failure of gˆ and hˆ to satisfy Equation 3.5.20

φ−1 −1 −1 −1 Y i f(α) := ug(αg h)ug(αg )uh(αg ) uh(αh ). (3.5.26) i=0

44 Note that in the language of lemma 3.5.3 we have f = φghg−1h−φ . To satisfy (3.5.21), ξg and ξh must then satisfy

φ−1 −1 −1 X i ξg(αg (h − 1))ξh(α(g − h )) = f(α). (3.5.27) i=0

We now want to construct such homomorphisms ξh, ξg. To do this, we will want to use the Smith normal ∼ form on the quotient Λh = L/Lh. Let us ﬁrst establish that the action of g is well deﬁned on this quotient:

Lemma 3.5.8. gLh ⊂ Lh

Pq−1 i Proof. The projector πh = i=0 h onto the h invariant subspace commutes with g,

−1 gπhg = πh . (3.5.28)

−1 This follows from the fact that Zq is a normal subgroup of G, such that g Zq g = Zq. It follows that gLh ⊂ Lh.

Corollary 3.5.9. f(α) = 1 for all α ∈ Lh.

Proof. Follows from (3.5.26), the fact that uh is a standard lift, and Lemma 3.5.8.

We want to deﬁne ξg, ξh using the following basis of L:

Lemma 3.5.10. There exist two bases {α1, . . . , αk, γ1, . . . , γl, 1, . . . , m} and {β1, . . . , βk, δ1, . . . , δl, 1, . . . , m} of L together with positive integers si such that

φ−1 −1 X i αi(g − h ) − siβi ∈ Lh (3.5.29) i=0 φ−1 −1 X i γj(g − h ) ∈ Lh (3.5.30) i=0 φ−1 −1 X i j(g − h ) ∈ Lh . (3.5.31) i=0

Proof. Let L = Λh + Lh be a decomposition into the ﬁxed-point sublattice Lh and the primitive sublattice −1 Pφ−1 i Λh. Pick {j} to be a basis of Lh. Lemma 3.5.8 implies (3.5.31), and also establishes that g − i=0 h acts on the quotient L/Lh. We can therefore use the Smith normal form for L/Lh, meaning that there exist bases {a1, . . . , ak, c1, . . . , cl} and {b1, . . . , bk, d1, . . . , dl} of L/Lh such that

φ−1 −1 X i ai(g − h ) = sibi (3.5.32) i=0 and φ−1 −1 X i cj(g − h ) = 0, (3.5.33) i=0 −1 Pφ−1 i ∼ where {s1, . . . , sk} are the elementary divisors of g − i=0 h . Using the isomorphism Λh = L/Lh as free Z-modules, we can lift the above bases of L/Lh to bases {α1, . . . , αk, γ1, . . . , γl} and {β1, . . . , βk, δ1, . . . , δl} of Λh. (3.5.29) and (3.5.30) are then automatically satisﬁed.

−1 Pφ−1 i Note that by lemma 3.1.5 the basis {γi} spans the complement of Lh in ker((g − i=0 h )(1 − h)) −1 Pp−1 Pφ−1 and the basis {βi} spans the complement of Lh in ker(g ( i=0 Mh)(1 − h)), where Mh = g i=0 is an operator of order p on the quotient L/Lh. Let us now discuss under what conditions it might be possible to construct such a lift. We will prove the following lemma:

Lemma 3.5.11. There is no solution to (3.5.27) if f(α) 6= 1 for all α ∈ ker(g−1(h − 1)(1 − g)).

45 Proof. Note that by the group relations the equality

φ−1 X g−1(h − 1)(1 − g) = −(g−1 − hi)(1 − h) (3.5.34) i=0

−1 −1 −1 Pφ−1 i holds. Hence for all α ∈ ker(g (h−1)(1−g)), it follows that αg (h−1) ∈ Lg and α(g − i=0 h ) ∈ Lh. As ξg and ξh are both standard lifts, that is they vanish on Lg and Lh, respectively, it follows that

−1 ξg(αg (h − 1)) = 1 (3.5.35)

and φ−1 −1 X i ξh(α(g − h )) = 1. (3.5.36) i=0 for all α ∈ ker(g−1(h − 1)(1 − g)). Hence the left side of Equation 3.5.27 vanishes for all α ∈ ker(g−1(h − 1)(1 − g)) so that there can only a solution if the right side also vanishes.

Using the basis of L constructed in Lemma 3.5.10, we can now deﬁne possible solutions ξg, ξh. Note that we ﬁnd that a solution always exists if f(α) = 1 for all α ∈ ker(g−1(h − 1)(1 − g)).

Theorem 3.5.12. Let f satisfy f(γi) = 1, i = 1, . . . , l. Then a solution to Equation 3.5.27 is given by

ξg(α) = 1, for all α ∈ L (3.5.37) and 1 s ξh(βi) = f(αi) i , ξh(δj) = 1 , ξh(j) = 1 . (3.5.38)

Proof. We prove this by evaluating (3.5.27) on the basis {αi, γj, j}. Note that the ﬁrst factor in (3.5.27) vanishes because ξg = 1. We immediately have

φ−1 −1 X i f(j) = 1 = ξh(j(g − h )) . (3.5.39) i=0

By (3.5.29) we have φ−1 −1 X i f(αi) = ξh(siβi) = ξh(αi(g − h )) (3.5.40) i=0 and by (3.5.30) φ−1 −1 X i f(γi) = 1 = ξh(γi(g − h )) (3.5.41) i=0

It follows immediately that Corollary 3.5.13. There is a solution to (3.5.27) if and only if f(α) = 1 for all α ∈ ker(g−1(h−1)(1−g)). It turns out that for all the groups we consider the condition holds so that a splitting lift exists. Note also that for the abelian group Zq × Zp the condition is trivial and a splitting lift always exists.

46 4 Twisted Modules for Lattice Vertex Operator Algebras

In this chapter, we review the construction of twisted modules for lattice vertex operator algebras. In particular, we explicitly construct the lowest weight-space of an irreducible twisted module as an irreducible projective representation of a certain finite abelian group. Building on this, we construct the action of automorphisms on twisted modules for lattice vertex operator algebras and give explicit expressions for the graded trace functions on twisted modules. Finally, we construct irreducible modules for the fixed-point vertex algebra for three examples of non-cyclic finite groups and verify that their characters exhibit the expected modular transformation properties.

4.1 Construction of Twisted Modules

The gˆ-twisted VL-modules Wgˆ associated to a lattice vertex operator algebras VL and a lifted lattice automorphism gˆ were constructed in [Lep85,BK04]. Wgˆ will be assembled from lowest-weight modules for ˆ the g-twisted Heisenberg current algebra hg, just like the lattice vertex operator algebra VL was assembled from lowest-weight modules for the (untwisted) Heisenberg current algebra h. However, we will see that ˆ the twist compatibility condition 1.4.2 requires the lowest-weight spaces of the constituent hg-modules to ⊥ Lg carry projective representations for the finite abelian group L(1−g) . We will briefly review the construction of Wgˆ, using the notation of [BK04], and then give an explicit ⊥ Lg construction of the projective representation of L(1−g) on the vacuum subspace Ω0 of Wgˆ. For simplicity we will assume that gˆ is a standard lift, but the construction below can be easily generalised to non-standard lifts. Note in particular, that Li [Li96] showed that for an inner V -automorphism σh, a σhg-twisted V -module can always be constructed from a g-twisted V -module. −j Let hj be the eigenspace of g to the eigenvalue ξn and πj the projection onto hj. ˆ Definition 4.1.1. The g-twisted Heisenberg current algebra hg is given by

n−1 M j hˆ = span h(m): h ∈ h , m ∈ + Z (4.1.1) g C j n j=0

together with the bracket 0 0 [h(m), h (k)] = hπnmh|h imδm,−k. (4.1.2) To construct the twisted lattice vertex operators, we want to deﬁne the analog of the twisted group algebra 3.2.2 used in the construction of the lattice vertex operator algebra. However in order to accommodate the requirement of twist compatibility 1.4.2 the relevant structure has to be slightly larger than one might otherwise expect.

× Deﬁnition 4.1.2. The twisted lattice group Ggˆ = C × exp h0 × L, where h0 = πgh the orthogonal v(0) projection on g-invariant subspace, consists of elements of the form ce Uα, together with the relations

−1 UαUβ = (α, β)B(α, β) Uα+β, (4.1.3) where (α, β) can be chosen as in Equation 3.2.4 and

n−1 −hα|βi Y k hαgk|βi B(α, β) = n (1 − ξn) , (4.1.4) k=1

ev(0)ew(0) = e(v+w)(0), (4.1.5) v(0) −v(0) e Uαe = exp(hv|αi)Uα (4.1.6)

47 and the twist compatibility condition W Uαg = ug(α)Uα exp 2πi(bα + α (0)) , (4.1.7) where |π (α)|2 b = g ∈ C . (4.1.8) α 2 ˆ Additionally, let the twisted lattice operators act on hg by

[h(m),Uα] = δm,0hπg(h)|αiUα, for all α ∈ L, h ∈ h. (4.1.9) By [Lep85] and [BK04] we can now deﬁne twisted vertex operators using these two structures: Deﬁnition 4.1.3. The twisted vertex operators are generated by

W X W −k−1 Y (h(−1)e0, z) = h (k)z , (4.1.10) j k∈ n +Z

for h ∈ hj, and W bα M Y (eα, z) = z UαEα (z), (4.1.11) with −k −k W X z X z EW (z) = zα (0) exp αW (k) exp αW (k) (4.1.12) α −k −k 1 1 k∈ n Z<0 k∈ n Z>0 ˆ Note that by Equation 4.1.10, the gˆ-twisted module Wgˆ has the structure of an hg-module. Then as an ˆ hg-module it is induced from its lowest-weight subspace ˆ Ω = {w ∈ Wgˆ|v(m)w = 0, v(m) ∈ hg, m ≥ 0}. (4.1.13) ˆ Then by Equation 4.1.9 the operator Uβ can be considered a map from a lowest-weight hg-module ˆ of level α to a lowest-weight hg-module of level πg(β) + α. Hence the lowest-weight subspace Ω can be ˆ decomposed into homogenous lowest-weight hg-modules as M Ω = Ωα, (4.1.14)

α∈h0 such that

UβΩα = Ωπg (β)+α (4.1.15)

Now by the twist compatibility condition 4.1.7 we find that exp(2πiα(0)) acts trivially on Wgˆ for all 0 0 α ∈ Lg so that weights must lie in Lg. Using the fact that Lg = πg(L) for an even, unimodular lattice, we find that there is a single compatible grading. Note if L is not unimodular, we would obtain a family of 0 modules indexed by Lg/πg(L) and for non-standard lifts we would find a non-trivial shift in the weights. Note that forg ˆ = 1 this reduces to the familiar result for untwisted modules. ⊥ Finally, notice that twisted lattice operators associated to the lattice Lg operate on the homogenous lowest-weight spaces Ωα. Define the corresponding subgroup of Ggˆ ⊥ ⊥ Ggˆ = {cUµ : c ∈ C, µ ∈ Lg }. (4.1.16)

Note that by Equation (4.1.7), lattice operators of the form U(1−g)β act on the Ωα as scalars. Hence ⊥ Ggˆ is a central extension for the ﬁnite abelian group

⊥ N := Lg /L(1 − g) (4.1.17)

and Ωα is a projective representation for N whose isomorphism class is determined by the commutator

n−1 −1 −1 Y k −hαgk|βi C(α, β) = UαUβUα Uβ = (−ξn) . (4.1.18) k=0 In particular, Equation (4.44) in [BK04] shows that C(α, β) restricts to a nondegenerate, alternating, bimultiplicative form on N, endowing it with the structure of a symplectic module. Note that if the ⊥ commutator C is non-trivial, the Ggˆ -module Ωα and hence the lowest-weight space of the twisted module cannot be one-dimensional. Putting everything together we arrive that the following theorem:

48 Theorem 4.1.1 ( [Lep85,BK04]). Let L be an even unimodular lattice and g an automorphism and let VL be the corresponding lattice vertex operator algebra and gˆ a standard lift of g. Let Wgˆ be a lowest-weight ˆ module for the g-twisted Heisenberg current algebra hg with graded lowest weight space M Ω = Ωα. (4.1.19)

α∈πg (L)

Then Wgˆ endowed with the twisted vertex operators 4.1.10 and 4.1.11 is a gˆ-twisted VL-module. Further it is clear that:

Lemma 4.1.2. Wgˆ is irreducible if and only if Ω0 is an irreducible projective representation of N.

4.2 The Vacuum Representation

We now want to construct the vacuum representation Ω0 (or any Ωα for that matter) as an irreducible projective representation of N. We will see that Ω0 is the the unique projective representation of N with commutator C(·, ·) and that it is of dimension d(g) = p|N|. N is a finite group with the alternative, bimultiplicative, nondegenerate form C : N × N → C. As a first step, we will present some results on the structure of N. The author first came across these results in [Dav10] and has, as of yet, been unable to identify an original reference.

Lemma 4.2.1. Let {a1, . . . , ak} be a basis of N such that a1 has order n. Then there is another basis element b1 := ai, i 6= 1 such that C(a1, b1) is a primitive n-th root of unity.

Proof. Assume C(α1, αj) is at most an m-th root of unity, for some proper divisor m | n and for all j. Then C(mα1, αj) = 1 for all j and hence C is degenerate. A contradiction.

Note that this also implies that ord(b1) = ord(a1). From this we can deduce the following:

Theorem 4.2.2. N admits a basis {a1, . . . , ak, b1, . . . , bk} such that

o(ai) = o(bi) = ni, (4.2.1) where ni|ni+1 and such that C(a , b ) = δ ξli , (4.2.2) i j i,j ni where gcd(ni, li) = 1 and C(ai, aj) = C(bi, bj) = 1. (4.2.3)

Proof. By the fundamental theorem of ﬁnitely generated abelian groups there exists a basis {a1, . . . , ar} of N such ord(ai) = ni,nr = nk and ni | ni+1. Then by Lemma 4.2.1 there exists an aj =: br such that

C(ar, br) = χ. (4.2.4) where χ is a primitive n-th root of unity. Hence for any basis element ai, i 6= r, j there are positive integers ki,r and ki,j such that ki,r C(ai, ar) = χ (4.2.5) and ki,j C(ai, aj) = χ . (4.2.6) Now set a˜i = ai + ki,raj − ki,jar. (4.2.7)

Note that ki,raj and ki,jar and hence also α˜i are elements of order ni. Then {a˜1,..., a˜r−1, ar, br} is a basis of N. The theorem follows by induction.

⊥ Lemma 4.2.3. There is a basis {α1, . . . , αk, β1, . . . , βk} of L such that {[α1],..., [αk], [β1],..., [βk]} is ⊥ Lg a basis of N = L(1−g) that satisﬁes the properties from Theorem 4.2.2. Proof. The orthogonalisation in proof of Theorem 4.2.2 describes valid change of basis of L⊥.

49 To construct the irreducible projective representation (ρ, Ω0) of N we want to decompose N in the following way. Using the fact that C is a nondegenerate, bimultiplicative, alternating form by Theorem 4.2.2, we know that there exists a subgroup A ≤ N and an isomorphism γ such that ˇ γ : N → A + A [µ] 7→ (x[µ], χ[µ]) , (4.2.8) where Aˇ denotes the group of irreducible characters of A, such that the bilinear form C is given by C(γ−1(x, χ), γ−1(y, ψ)) = ψ(x)χ(y)−1. (4.2.9)

From this decomposition we can construct the sought after projective representation Ω0:

Theorem 4.2.4. Let Ω0 = C[Aˇ]. Then the unique (up to projective equivalence), irreducible representation (ρ, Ω0) of N such that the skew of its 2-cocycle is given by C is given by

ρ([µ])eψ = ψ[µ](x[µ])eχψ[µ] , (4.2.10) where (x[µ], χ[µ]) = γ([µ]). Proof. We ﬁnd that ρ satisﬁes the product relation

ρ([µ])ρ([ν])eψ = φ[ν](x[µ])ψ(x[µ]y[ν])eχ[µ]φ[ν]ψ (4.2.11)

= φ[ν](x[µ])ρ([µ + ν])eψ (4.2.12) so that ρ does indeed deﬁne a projective representation for the group N and the commutator is indeed given by C. Then the characters are given by ( P ψ(x ), if χ = 1 Tr(ρ([µ])) = ψ∈Aˇ [µ] [µ] (4.2.13) 0 otherwise. Hence the number of irreducible representations in ρ is 1 X kρk2 = Tr(ρ([µ]))Tr(ρ([µ])) (4.2.14) |A|2 [µ]∈N 1 X X = χ(x)ψ(x) (4.2.15) |A|2 x∈A χ,ψ∈Aˇ X 1 X = χ(x)χ(x) (4.2.16) |A|2 χ∈Aˇ x∈A X 1 = (4.2.17) |A| χ∈Aˇ = 1, (4.2.18) so that ρ is indeed irreducible. Finally, the uniqueness of ρ follows from the fact that dim(ρ)2 = |N|.

⊥ Lg ⊥ Finally, we want to extend this representation ρ of L(1−g) to a representation ρ˜ of Lg satisfying the twist compatibility condition (4.1.7 so that we can make the identiﬁcation ⊥ ρ˜ : Lg → End(Ω0), µ 7→ Uµ . (4.2.19) with the corresponding twisted lattice operator in the sense of 4.1.3. Due to the uniqueness of the irreducible projective representation of N the representations ρ˜ and ρ ⊥ ∗ diﬀer at most by a change of section, i.e. there is a function λ : Lg → C such that

Uµ = λ(µ)ρ([µ]). (4.2.20)

We need to be careful in our choice of λ, since the Uµ need to satisfy both (4.1.3) as well as (4.1.7). These are equivalent to λ satisfying the two conditions −1 λ(µ)λ(ν)χν (xµ) = (µ, ν)B(µ, ν) λ(µ + ν) (4.2.21) and −1 −1 λ(α(1 − g)) = ug(α) (α(1 − g), αg)B(α(1 − g), αg) exp(2πibα) (4.2.22) For convenience let us restate the twist compatibility condition in a diﬀerent format.

50 Lemma 4.2.5. The twist compatibility condition 4.1.7 is equivalent to

−1 −1 Uα(1−g) = ug(α) (α(1 − g), αg)B(α(1 − g), αg) exp(2πibα) exp(−2πiα(0)). (4.2.23)

In order to proceed to ﬁnd a λ satisfying the conditions 4.2.21 and 4.2.22 ﬁrst note the following:

Lemma 4.2.6. If the operators Uα(1−g),Uβ(1−g) satisfy the twist compatibility condition (4.1.3), then so does U(α+β)(1−g).

−1 Proof. We prove this by showing that U(α+β)(1−g) satisﬁes (4.2.23). Let ˜(., .) = (., .)B(., .) .We calculate

−1 U(α+β)(1−g) = ˜(α(1 − g), β(1 − g)) Uα(1−g)Uβ(1−g) (4.2.24) ˜(α(1 − g), αg)˜(β(1 − g), βg) = u (α)−1u (β)−1 (4.2.25) g g ˜(α(1 − g), β(1 − g))

exp(2πi(bα + bβ)) exp(−2πi(α(0) + β(0))) (4.2.26) −1 = ug(α + β) ˜(β(1 − g), αg)˜(α(1 − g), αg) (4.2.27)

˜(β(1 − g), βg)˜(α(1 − g), βg) exp(2πi(bα + bβ + hα0|β0i)) exp(−2πi(α + β)(0)) (4.2.28) −1 = ug(α + β) ˜((α + β)(1 − g), (α + β)g) exp(2πibα+β) exp(−2πi(α + β)(0)). (4.2.29)

⊥ Hence by Lemma 3.1.5 there exists a basis {µi} of L such that {siµi} is a basis of L(1 − g), where ⊥ {si} are the elementary divisors of 1 − g. Hence if we determine λ(µi) for a basis µi of Lg (and hence N)

such that all Usiµi satisfy twist compatibility, we can repeatedly apply (4.1.3) to obtain a full set of twist

compatible lattice operators. In particular, once we have ﬁxed the Uµi that a general lattice is given by

Theorem 4.2.7. Let {αi} be a basis of L and assume that we have a ﬁxed set Uαi that satisfy (4.1.3). Then the lattice operators deﬁned by

ni(ni−1) Y ninj Y n U P = (α , α )B(α , α ) U i (α , α )B(α , α ) 2 . (4.2.30) ( i niαi) i j i j αi i i i i i

Proof. Let ˆ(α, β) = (α, β)B(α, β), such that Equation 4.1.3 becomes Uα+β = ˆ(α, β)UαUβ. Then we calculate X X U P U P ˆ n α , m α = (4.2.32) ( i niαi) ( j mj αj ) i i j j i j Y n m Y n n +m m Y ni(ni−1)+mi(mi−1) Y n Y m = ˆ(α , α ) i j ˆ(α , α ) i j i j ˆ(α , α ) 2 U i U j i j i j i i αi αj i,j i

Y n n +m m +n m +n m Y ni(ni−1)+mi(mi−1) +n m Y n m = ˆ(α , α ) i j i j i j j i ˆ(α , α ) 2 i i U i i (4.2.34) i j i i αi i

nimj Y Y Y Y ˆ(αj, αi) U ni U mj = U nimi . (4.2.36) αi αj αi ˆ(α , α ) i j i i>j i j

51 ⊥ All that remains to be done is to solve 4.2.21 and 4.2.22 on a basis of Lg . An explicit expression for the λ is given by:

⊥ Theorem 4.2.8. Let {µi} be a basis of L such that siµi ∈ L(1 − g) for all i, where µ˜i(1 − g) = siµi and let λ satisfy Equations 4.2.21 and 4.2.22. Then λ(µi) is uniquely determined up to an si-th root of unity by

1 si+1 − −1 2 2πi si λ(µi) = ug(˜µi) (µi, µ˜i)B(µi, µ˜i) (µi, µi)B(µi, µi) exp( bµ˜i ) (4.2.37) si Proof. By Equation 4.2.21 we have

si(si−1) si λ(siµi) = λ(µi) ((µi, µi)B(µi, µi)) 2 , (4.2.38)

where we may choose a basis such that χµi (xµi ) and by Equation 4.2.22

λ(siµi) = λ(˜µi(1 − g)) (4.2.39) −1 −1 = ug(˜µi) (˜µi(1 − g), µ˜ig)B(˜µi(1 − g), µ˜ig) exp(2πibµ˜i ) (4.2.40) −1 −1 = ug(˜µi) (siµi, µ˜ig)B(siµi, µ˜ig) exp(2πibµ˜i ) (4.2.41) −1 −1 = ug(˜µi) (siµi, µ˜i − siµi)B(siµi, µ˜i − siµi) exp(2πibµ˜i ) (4.2.42) 2 −1 −1si si = ug(˜µi) (µi, µ˜i)B(µi, µ˜i) (µi, µi)B(µi, µi) exp(2πibµ˜i ) (4.2.43)

Hence it follows that

s si(si+1) si −1 −1 i 2 λ(µi) = ug(˜µi) (µi, µ˜i)B(µi, µ˜i) (µi, µi)B(µi, µi) exp(2πibµ˜i ), (4.2.44)

so that in particular λ(µi) is determined up to an si-th root of unity.

From this we can construct Uµi for a basis µi, and thus by (4.1.4) all the operators Uµ.

4.3 A Basis for VL(ˆg)

Finally let us give a basis for the irreducible gˆ-twisted VL-module VL(gˆ) starting from the irreducible N-module (˜ρ, Ω0). First note that by Lemma 3.1.7 there exists a lattice Λ such that

⊥ L = Λ + Lg , (4.3.1) where, in general, we cannot choose Λ to equal Lg. Let {eχ : χ ∈ Aˇ} be a basis for Ω0. Then we ﬁnd that

Theorem 4.3.1. A basis for VL(ˆg) is given by

{v1 . . . vk U e : vi ∈ hˆ , m > 0, γ ∈ Λ, χ ∈ Aˇ}. (4.3.2) −m1 −mk γ χ −mi g i The weight of such a state is given by

2 m1 + ... + mk + |(πgγ)| /2 + ρg, (4.3.3) recovering the familiar result for the character.

A general twisted lattice operator Uα then acts on (4.3.2) in the obvious way. Hence as a graded vector space VL(ˆg) is isomorphic to √ ∼ ˆ− 0 |N| VL(ˆg) = S(hg ) ⊗ C[Lg] ⊗ R . (4.3.4)

52 4.4 Action of Automorphisms on Twisted Modules

In chapter 2, we saw that the irreducible modules for the ﬁxed point vertex operator algebra V G can be constructed as V G-submodules of the irreducible g-twisted V -modules via the Schur-Weyl-Type Duality 2.1.1. In order to carry out this construction it is necessary to have an explicit description of the action of automorphisms on twisted modules. We will study this action for lattice vertex operator algebras. ˆ Let VL be a holomorphic lattice vertex operator algebra, gˆ and h commuting automorphisms and (VL(gˆ),YW (·, x)) the irreducible gˆ-twisted V -module. Then by the results of [DLM00,LL04,DRX17] there ˆ exists a linear map φgˆ(h): VL(ˆg) → VL(ˆg) such that

ˆ ˆ −1 ˆ φgˆ(h)YW (v, x)φgˆ(h) = YW (hv, x) , (4.4.1) which is unique up to multiplication by a scalar. φgˆ(·) is a projective representation of C(gˆ). It is this projective representation that we will construct here. ˆ Acting with φgˆ(h) on a vector in (4.3.2) we can use (4.4.1) to ﬁnd

ˆ ˆ −1 φgˆ(h)Uαφgˆ(h) = uh(α)Uαh (4.4.2)

and ˆ ˆ −1 φgˆ(h)v−nφgˆ(h) = (v · h)−n . (4.4.3) We therefore have ˆ ˆ ˆ Lemma 4.4.1. Let g,ˆ h ∈ Aut(VL) be two automorphisms such that gˆh = hgˆ lifted from commuting ˆ ˆ lattice automorphisms g, h ∈ Aut(L). Then VL(ˆg) carries an action φgˆ(h) of h given by

ˆ −1 ˆ φgˆ(h)v−n1 . . . v−nk Uαeχ = (vh)−n1 ... (vh)−nk uh(α) Uαhφgˆ(h)|Ω0 eχ (4.4.4) ˆ ˆ We thus need to ﬁnd the action of φgˆ(h) on Ω0. Note that φgˆ(h)|Ω0 is a d(gˆ)×d(gˆ)-matrix Ohˆ ∈ End(Ω0). ⊥ To compute Ohˆ , we note thatρ ˜(·h) is again an irreducible projective representation of Lg of dimension d(g) with the same commutator as ρ˜. Since there is only one such irreducible projective representation,

Schur’s lemma tells us that ρ˜(·h) and ρ˜(·) are related by a linear automorphism Ohˆ on Ω0, which is unique ∗ ˆ up to an overall factor in C . From this it follows immediately that φgˆ|Ω0 : h 7→ Ohˆ is indeed a projective representation of Cgˆ on Ω0. ⊥ To construct Ohˆ , we construct Uµi |Ω0 for a basis of Lg and then use O U O−1 = u (µ )U (4.4.5) hˆ µi hˆ h i µih

to obtain constraints on the matrix Ohˆ . Schur’s lemma guarantees that the solution Ohˆ is unique up to multiplication by a complex number. Repeating this for all hˆ ∈ C(gˆ) gives us the projective representation

φgˆ|Ω0 . ˆ ˆ From the matrices Oh we can read oﬀ the cocycle cgˆ(h1, h2) of this projective representation. In particular, this could now be used to place constraints on the obstructing 3-cocycle ω. We are interested only in orbifolds that allow a holomorphic extension, that is ones such that ω is trivial. In such a case, the 2-cocycle cgˆ must be in the trivial cohomology class. Hence a necessary condition for the existence of holomorphic extensions is that the φgˆ for all gˆ ∈ G diﬀer only by change of section from linear representations.

4.5 Graded Trace Functions for Twisted Modules

We now have all the ingredients to compute the graded trace function

ˆ ˆ L0−c/24 T (ˆg, h; τ) = tr|VL(ˆg)φgˆ(h)q (4.5.1) First, let us consider what conditions a state in 4.3.2 needs to satisfy in order to contribute to the trace function. ˆ ˆ Lemma 4.5.1. Let h be a VL-automorphism that commutes with gˆ and φgˆ(h) the corresponding action on VL(ˆg). Then states of the form

ˆ ˆ− {v−n1 . . . v−nk Uαeχ : n1, . . . , nk ∈ ..., α ∈ Λ, eχ ∈ C[A], v−ni ∈ hg } (4.5.2)

53 ˆ contribute to the trace of φgˆ(h) only if α ∈ Λ is such that

πg(α) = πg(hα), (4.5.3) 0 0 ⊥ that is πg(α) ∈ hg ∩ hh or equivalently (1 − h)α ∈ Lg . We denote the sublattice of Λ such that Equation 4.5.3 is satisﬁed by Λh. Proof. In order for the state to contribute to the character we require that [αh] = [α], (4.5.4) L where [α] denotes the class of α in ⊥ . In other words we have that Lg αh = α + β⊥, (4.5.5) ⊥ ⊥ for some β ∈ Lg . Let us consider the contributions of the lattice operators and of the Heisenberg modes to the graded trace function separately. Lemma 4.5.2. The lattice contribution to the graded trace function is given by

2 X |α0| ˆ 2 −1 ˆ TL(ˆg, h, τ) = q uh(α) g(α, α(h − 1))Bg(α, α(h − 1))Tr|Ω Uα(h−1)φgˆ(h) . (4.5.6)

α∈Λh

Proof. For α ∈ Λh we ﬁnd that ˆ −1 ˆ φgˆ(h)v−n1 . . . v−nk Uαeχ = (vh)−n1 ... (vh)−nk uh(α) Uαhφgˆ(h)eχ (4.5.7) −1 ˆ = (vh)−n1 ... (vh)−nk uh(α) g(α, α(h − 1))Bg(α, α(h − 1))UαUα(h−1)φgˆ(h)eχ. (4.5.8)

⊥ ˆ Note that if α(h − 1) ∈ Lg then Uα(h−1)φgˆ(h)eχ ∈ Ω0. We find that the lattice contribution to the graded trace function takes a form that resembles the modified theta-functions that occur in the untwisted case. However as far as the author knows, these new objects and particularly their automorphic properties have not been studied in the literature. i j Lemma 4.5.3. Let Ψ be the set of pairs (i, j) such that ξn and ξm are simultaneous eigenvalues of g and h. The Heisenberg contribution to the twining character is given by ∞ 1 ˆ ρg Y Y TH (ˆg, h, τ) = q j i , (4.5.9) k+ n (i,j)∈Ψ k=0 1 − ξmq where the conformal weight ρg is given by X j(n − j) ρ = dim(h ), (4.5.10) g 4n2 j j −j where hj is the eigenspace of g to the eigenvalue ξn . i j Proof. Let v ∈ h be a simultaneous eigenvector of g and h with eigenvalues ξn and ξm, respectively. Then ˆ i by Definition 4.1.1 the weights of the operators corresponding to v in hg are given by n + Z≥0. Then the contribution to the trace of h due to descendants of v is given by

i 2i i 2i ˆ ρg j n 2j n j 1+ n 2j 2+ n Tv(ˆg, h; τ) = q (1 + ξmq + ξm q + ...)(1 + ξmq + ξm q + ...) ... (4.5.11) ∞ 1 ρg Y = q j i . (4.5.12) k+ n k=0 1 − ξmq The desired result follows by multiplying over all simultaneous eigenvectors. Hence the Heisenberg contribution to the graded trace function can be written in terms of well-known generalised eta functions, whose properties have been studied, for example, in [Yan04]. Putting everything together, we ﬁnd that: ˆ Theorem 4.5.4. The graded trace function of φgˆ(h) on the twisted module VL(ˆg) is then given by ˆ ˆ ˆ T (ˆg, h; τ) = TH (ˆg, h, τ)TL(ˆg, h, τ). (4.5.13)

54 4.6 Examples: Orbifolds of VE8

As a warm-up, let us discuss orbifolds of the E8 lattice vertex operator algebra VE8. This is the smallest even unimodular lattice, which makes our computations easier. This case is strongly constrained, since the only holomorphic vertex operator algebra of central charge 8 is VE8 itself. Any orbifold thus only has two possible outcomes: either ω is not trivial, so that there is no holomorphic extension, or we recover the original unorbifolded vertex operator algebra. We will give an example for both possibilities. Investigating all 62092 conjugacy classes of subgroups of Aut(E8), we ﬁnd that under standard lifts only 14 are non-anomalous of which 10 are cyclic. The non-cyclic ones are isomorphic to Z3 × Z3, Z5 × Z5, Z3 × Z6 and Z5 × Z10. We will investigate the smallest one of these as well as anomalous groups isomorphic to S3 and Z2 × Z2.

4.6.1 An S3-Orbifold

First consider an orbifold of VE8 by a group S3 = Z3 o−1 Z2 generated by elements s and t of cycle types 4 2 2 Cs = 2 and Ct = 1 3 . S3 has three conjugacy classes, [e], [s], [t], containing elements of order 1,2, 3 ∗ and 3 respectively. From (2.3.5) we have H (S3, C ) = Z6. The cyclic subgroups generated by s and t have types rs = 1 and rt = 2, from which it follows that ω is non-trivial. We can therefore not obtain a holomorphic extension. However, we can still obtain the graded trace functions for S3

θ (τ) θ (τ) θ (τ)2 T (e, e, τ) = E8 T (e, s, τ) = D4 T (e, t, τ) = A2 (4.6.1) η(τ)8 η(2τ)4 η(τ)2η(3τ)2 θ ( τ ) θ ( τ+1 ) T (s, e, τ) = 2 D4 2 T (s, s, τ) = 2ξ D4 2 (4.6.2) τ 4 3 τ+1 4 η( 2 ) η( 2 ) θ ( τ )2 θ ( τ+1 )2 θ ( τ+2 )2 T (t, e, τ) = A2 3 T (t, t, τ) = ξ A2 3 T (t, t2, τ) = ξ2 A2 3 (4.6.3) 2 τ 2 3 2 τ+1 2 3 2 τ+2 2 η(τ) η( 3 ) η(τ + 1) η( 3 ) η(τ + 2) η( 3 ) Let us now write down the characters of the irreducible modules. There are 3 irreducible representation of S3: The trivial, signum and 2-dimensional standard representation. The other centralizers have the usual cyclic characters. Using the irreducible characters with the 2-cocycles descended from ω, we ﬁnd

T (e, e, τ) + 3T (e, s, τ) + 2T (e, t, τ) χ (τ) = (4.6.4) e,0 6 T (e, e, τ) − 3T (e, s, τ) + 2T (e, t, τ) χ (τ) = (4.6.5) e,sgn 6 T (e, e, τ) − T (e, t, τ) χ (τ) = (4.6.6) e,2 3 T (s, e, τ) − ξ T (s, s, τ) χ (τ) = 4 (4.6.7) s,0 2 T (s, e, τ) + ξ T (s, s, τ) χ (τ) = 4 (4.6.8) s,1 2 T (t, e, τ) + ξ−2T (t, t, τ) + ξ−4T (t, t2, τ) χ (τ) = 9 9 (4.6.9) t,0 3 T (t, e, τ) + ξ T (t, t, τ) + ξ2T (t, t2, τ) χ (τ) = 9 9 (4.6.10) t,1 3 T (t, e, τ) + ξ4T (t, t, τ) + ξ−1T (t, t2, τ) χ (τ) = 9 9 , (4.6.11) t,2 3 where ξn denotes a primitive n-th root of unity. Using the ordering above, the transformation matrices under T and S agree with the ones obtained in [DVVV89, CGR00] with p = 1.

4.6.2 A Z2 × Z2-Orbifold

Next we consider an orbifold of VE8 by a group Z2 × Z2 generated by elements g and h both of cyclic 4 type Cg = Ch = 2 . The types of the cyclic subgroups generated by g, h and gh are given rg = rh = 1 and rgh = 0, respectively. Thus by theorem 2.3.8 the cohomological twist ω is non-trivial. We can of course still write down the graded trace functions. There are 16 of them, which form the following orbits

55 under SL(2, Z)-transformations:

{T (e, e; τ)} , {T (g, e; τ),T (e, g; τ),T (g, g; τ)} , {T (h, e; τ),T (e, h; τ),T (h, h; τ)} , {T (gh, e; τ),T (e, gh; τ),T (gh, gh; τ)} , (4.6.12) {T (g, h; τ),T (g, gh; τ),T (h, g; τ),T (h, gh; τ),T (gh, g),T (gh, h)}.

Of those orbits, all but the last one intersect the untwisted sector and can be computed from the untwisted sectors using modular transformation properties. For the last orbit however, we need to use our full technology: that is, we needs to construct at least one irreducible twisted module, say VE8(gˆ), compute the projective representation φg(·), and compute the graded trace functions as described in sections 4 and 4.4. By matching the first few coefficients of the expressions we compute, we find experimentally that

η(τ)2 4 T (g, h; τ) = T (h, g; τ) = 2 (4.6.13) η(τ/2)η(2τ)

and η(τ)2 4 T (g, gh; τ) = T (h, gh) = e2πi/122 . (4.6.14) η(τ/2 + 1/2)η(2τ) The product gh however is not a standard lift, so that we leave the construction of the gh-twisted module and its characters for future work. We conjecture that the characters are given by

η(τ)2 4 T (gh, g; τ) = T (gh, h; τ) = e−2πi/6 . (4.6.15) η(τ/2 − 1/2)η(τ/2)

4.6.3 A Z3 × Z3-Orbifold −1 3 Now consider the Z3 × Z3 orbifold generated by two elements g and h of cycle type 1 3 . We ﬁnd that all cyclic subgroups have type 0 and hence ω = 1 . (4.6.16) 2 Again, this orbifold does not have periodic cohomology, and we have H (Z3 × Z3) = Z3 6= 0, which allows for discrete torsion. This seems to pose a bit of a puzzle: even though discrete torsion seems to allow for diﬀerent holomorphic extensions, we know that there is only one holomorphic vertex operator algebra of central charge 8. The resolution to this is that the contribution of the characters that is controlled by discrete torsion vanishes. As before, we can arrange all 81 twining characters into orbits of SL(2, Z), giving 6 orbits, one of which is simply {T (e, e; τ)}. Of the remaining ones, 4 have length 8 and intersect the untwisted sector, and corresp to the orbifolds by 4 cyclic subgroups of Z3 × Z3. The representatives for these four orbits T (e, g; τ), T (e, h; τ), T (e, gh; τ) and T (e, g2h; τ) are all equal and given by

θ (τ) T (e, g; τ) = T (e, h; τ) = T (e, gh; τ) = T (e, g2h; τ) = A2 (4.6.17) η(τ)−1η(3τ)3

The ﬁfth orbit with representative T (g, h; τ) has length 48 and does not intersect the untwisted sector. We therefore again need to use our full technology to construct a twisted module and the graded trace function explicitly. It turns out however that all graded trace functions in this orbit are constant. The reason for this is that for all pairs (g , g ) that appear in this orbit, the lattice Λ complementary to L⊥ 1 2 g1 does not have any g2-invariant vectors, so that the generalized theta-function is a constant. Moreover the generalised η functions turn out to give a constant, so that the overall character is indeed, in turn, constant. Acting with T on such a character thus simply introduces a third root of unity. When we decompose the orbit into 16 T orbits, each of them is a sum of three roots of unity, which vanishes. The upshot is thus that this orbit, whose contribution is in principle controlled by discrete torsion, never contributes to the character of the holomorphic extension. There is in fact a quicker way to see this: since this orbit does not intersect the untwisted module, and the conformal weights of the twisted are all strictly positive, no term q−1/3 can appear, but only terms with higher powers of q. There is no modular function with the right multiplier system and only higher powers, so that it follows that the sum over the orbit necessarily vanishes.

56 5 Cyclic Orbifolds of Lattice Vertex Operator Algebras at c = 48 and c = 72

In this chapter, we will present a systematic survey of cyclic orbifolds of the lattice vertex operator algebras associated the known extremal lattices in dimensions 48 and 72.

5.1 Computational Aspects — Characters of Holomorphic Extensions for Cyclic Groups

We begin with a discussion of some computational considerations.

5.1.1 Eta-Quotients and Modular Transformations of Lattice Theta-Functions

In this subsection, we will discuss the modular transformation of eta-quotients and lattice theta-functions. The computation of the characters of holomorphic extensions for cyclic orbifolds will require calculating the q-series for modular transformations of lattice theta-functions and eta-quotients. According to Lemma 3.1.1 the theta-function θL(τ) of an even lattice L of rank d, level N and determinant ∆ is a holomorphic d (−1)d∆ modular form for the congruence subgroup Γ0(N) of weight 2 and character χ(p) = p . While the modular transformation properties of theta-function are, in principle, known, they are hard to compute. Expressions that useful for computational purposes are given in [Sch09]. The computer algebra package PARI/GP [PAR19] also provides such functionality. We will use a different approach and will show in this section that all the relevant spaces of modular forms can, in fact, be spanned by eta-quotients. Definition 5.1.1. The Dedekind eta function, η(τ), is defined by the infinite product

∞ Y η(τ) := q1/24 (1 − qn). n=1 The modular transformation properties of the Dedekind eta-function are well known to be: Theorem 5.1.1 ( [Iwa97]). Under elements of SL(2, Z) η(τ) transforms as a modular form of weight 1/2 with multiplier system ϑ(γ): 1 a b η(γτ) = ϑ(γ)(cτ + d) 2 η(τ), where γ = ∈ SL(2, Z), c d where e(b/24) if c = 0  ϑ(γ) = a+d−3c 1 e 24c − 2 s(d, c) if c > 0 with s(d, c) the Dedekind sum

X ndn dn 1 s(d, c) = − − . c c c 2 0≤n

Deﬁnition 5.1.2. We deﬁne the eta-quotient ηC (τ) of cycle type C

Y bt ηC (τ) := η(tτ) t|n

57 To obtain the SL(2, Z)-transformation of such an eta-quotient we can use need the following lemma: AB Proposition 5.1.2. For every ∈ SL(2, Z) and q ∈ Z there exist an SL(2, Z)-transformation CD a b ∈ SL(2, Z) and three integers α, β, γ ∈ Z such that: c d

AB a b ατ + β η q (τ) = η . (5.1.1) CD c d γ Then 1. α = (qA, C) q 2. γ = α qA 3. a = α C 4. c = α 5. ad ≡ 1 (mod c)

ad−1 6. b = c 7. β = qBd − Db where (x, y) ≡ gcd(x, y). Note that relation 5 implies that d is the modular inverse of a modulo c which always exists as (a, c) = 1. Proof. Expanding the argument on both sides of equation 5.1.1 we obtain qAτ + qB aατ + aβ + bγ = Cτ + D cατ + cβ + dγ Equating term by term and solving the resulting relations gives the above result. Together with Theorem 5.1.1 this proposition fully determines the modular transformation properties of the eta-quotients 5.1.2. Corollary 5.1.3. Note that this result in particular implies that ατ + β Cτ + D c + d = . γ γ By the following theorem we can express a wide range of functions in terms of eta-quotient and therefore determine their modular transformation properties using Proposition 5.1.2. Theorem 5.1.4 ( [Ono04]). Let C be a cycle type satisfying the following conditions: P 1. 2k = t|n bt ≡ 0 (mod 2) P 2. t|n tbt ≡ 0 (mod 24) P n 3. t|n t bt ≡ 0 (mod 24)

Then the associated eta-quotient ηC (τ) satisﬁes aτ + b η = χ(d)(cτ + d)kη (τ), C cτ + d C a b (−1)ks for every ∈ Γ (n) with k = (1/2) P b and character χ(d) = , where s := Q tbt . c d 0 t|n t d t|n 2 c n P (d,t) bt The order of vanishing of ηC (τ) at the cusp at is given by n . d 24 t|n (d, d )dt 1 P Corollary 5.1.5. For an eta-quotient satisfying the conditions of Theorem 5.1.4 the integer 24 t|n tbt P n gives the lowest non-trivial order in its q-expansion and the integer t|n t bt gives the lowest non-trivial order in the q-expansion of its S-transformation. Table 5.1 shows bases of eta-quotients for all the spaces of modular forms relevant to our computation.

58 (N, k, χ(d)) Dim Cg (4, 12, 1) 7 [2−24448], [182−24440], [1162−24432], [1242−24424], [1322−24416], [1402−2448], [1482−24] (6, 4, 1) 5 [142−83−12624], [192−93−11619], [1142−103−10614], [1192−113−969], [1242−123−864] (6, 8, 1) 9 [182−163−24648], [1132−173−23643], [1182−183−22638], [1232−193−21633], [1282−203−20628], [1332−213−19623], [1382−223−18618], [1432−233−17613], [1482−243−1668] (6, 12, 1) 13 [1122−243−36672], [1172−253−35667], [1222−263−34662], [1272−273−33657], [1322−283−32652], [1372−293−31647], [1422−303−30642], [1472−313−29637], [1522−323−28632], [1572−333−27627], [1622−343−26622], [1672−353−25617], [1722−363−24612] (8, 6, 1) 7 [4−12824], [1−42104−14820], [1−82204−16816], [1−122304−18812], [1−162404−2088], [1−202504−2284], [1−242604−24] (9, 8, 1) 9 [1243−8], [3−8924], [115349−3], [11234], [163169−6], [133169−3], [1−33289−9], [1−63289−6], [1−123409−12] (10, 4, 1) 7 [1−102205210−4], [1−52155110−3], [21010−2], [15255−110−1] [1105−2], [1152−55−3101], [1202−105−4102] (10, 8, 1) 13 [1−202405410−8], [1−152355310−7], [1−102305210−6], [1−52255110−5], [22010−4], [152155−110−3], [1102105−210−2], [115255−310−1], [1205−4], [1252−55−5101], [1302−105−6102], [1352−155−7103], [1402−205−8104] (12, 4, 1) 9 [1−42103−124−463012−12], [3−1664012−16], [142−103−204465012−20], [182−203−244866012−24], [2−4486−4128], [2−2446−61212], [6−81216], [224−46−101220], [244−86−121224] (15, 4, 1) 8 [183−45−4158], [1−33151515−5], [1−2510], [1−13−155155], [3−21510], [15355−115−1], [1105−2], [1153−55−3151] (18, 4, 1) 13 [1−12224346−8], [1−9221336−7], [1−6218326−6], [1−3215316−5], [2126−4], [13293−16−3], [16263−26−2], [19233−36−1], [1123−4], [1152−33−561], [1182−63−662], [1212−93−763], [1242−123−864] (20, 4, 1) 12 [224−410−102020], [1−132294−105910−9202], [1−202504−205410−10204], [172−34−25−310−12010], [10−82016], [1−162404−16], [122−14−15−210−32013], [152−24−15−110−62013], [1−142374−155610−9203], [1−142334−135610−5201], [1−82204−85810−4], [1−122264−1051210−10202] (44, 2, 1) 9 [182−4], [2−448], [22−4448], [12112], [112−34411−322944−4], [1−1224−111322−2443], [132−24311−122244−1], [1−32711122−1], [1−3294−411122−3444] −23 1 1 1 1 1 1 2 −1 2 −1 (92, 1, d ) 6 [1 23 ], [2 46 ], [4 92 ], [1 2 23 46 ], [112−14−123146−1921], [2−14246−1922]

Table 5.1: Eta-quotient bases . The ﬁrst column speciﬁes the relevant space of modular forms for Γ0(N) of weight k and character χ(d). The second column shows the dimension of this space of modular forms. The third column list a set of cycle types such that the corresponding eta-quotients form a basis of the space of modular forms.

59 5.1.2 Computing Characters of Cyclic Orbifolds

Our aim is to compute characters for around 150 new cyclic orbifolds of lattice vertex operator algebras associated to extremal lattices in dimensions 48 and 72. These computations are relatively expensive, so there is a need for moderate computational optimisation that will be described in the following. In section 2.2, we showed that the character of the holomorphic extension V orb(G) is given by X 1 X 1 X ch orb(G) (τ) = T (i,0) (1, τ) = T (1, i, j, τ) = T (i, j, τ) (5.1.2) V W n n i∈Zn i,j∈Zn i,j∈Zn In order to calculate characters it will be useful to deﬁne further modular invariants by splitting the sum in (2.2.13) into orbits under the modular group Γ. Theorem 5.1.6. For t | n deﬁne X Ct(τ) = T (i, j, τ) . (5.1.3) i,j:(i,j,n)=t

Then Ct is a modular function for Γ.

Proof. For i, j ∈ Zn and M ∈ SL2(Z) we have (i, j, n) = ((i, j)M, n) so that modular transformations only permute terms in the sum in equation (5.1.3). Note that we can express the character of V orb(G) as 1 X ch orb(G) (τ) = C (τ) . V n t t|n

Recall that according to Corollary 2.2.5 for t | n, T (0, t, τ) is a modular function for Γ1(n/t). Then Ct(τ) is the sum of Γ1(n/t)-inequivalent modular images of T (0, t, τ). As should be expected, we ﬁnd that the number of terms in equation (5.1.3) is equal to the index of Γ1(n/t) in Γ: n 2 Y 1 |{(i, j) ∈ Z :(i, j, n/t) = 1}| = [Γ : Γ (n/t)] = 1 − . n/t 1 t p2 p|n/t,p prime

k Ifg ˆ acts on eα as in Equation 3.3.3 theng ˆ acts as

k gˆ (eα) = wk(α)eνkα, (5.1.4)

k−1 where wk(α) = ηg(α)ηg(gα) . . . ηg(g α). In order to calculate the Ct(τ) it will be convenient to ﬁrst introduce a further modular invariant Dt(τ) Lemma 5.1.7. For t | n deﬁne X Dt(τ) := T (0, j, τ) . (5.1.5) (j,n)=t

Then Dt is invariant under Γ0(n/t).

Proof. From Corollary 2.2.5 we know that T (0, j, τ) invariant under Γ1(n/t), with t = (n, j). T (0, t, τ) is ! ∗ ∗ mapped to T (0, j, τ), with (n, j) = t by the representative ∈ Γ1(n/t)\Γ0(n/t). Furthermore n/t j/t

[Γ0(n/t):Γ1(n/t)] = |{j ∈ Zn :(n, j) = t}| = ϕ(n/t), where ϕ is Euler’s totient function, which follows immediately from standard result on congruence subgroups as presented, for example, in [Iwa97]. Hence {T (0, j, τ):(n, j) = t, j ∈ Zn} is the set of inequivalent Γ0(n/t)-images of T (0, t, τ) and their sum is modular function for Γ0(n/t).

It follows immediately that Ct(τ) is the sum of Γ0(n/t)-inequivalent modular images of Dt(τ).

Lemma 5.1.8. The following formula holds for the modular invariant Ct X Ct(τ) = Dt(γ · τ). (5.1.6)

γ∈Γ0(n/t)/Γ1(n/t)

60 For primes we ﬁnd in particular:

i Corollary 5.1.9. For a prime p a set of representatives for Γ0(p)\Γ is given by {id} ∪ {ST : i = 0, . . . , p − 1}.

We are now ready to give a general expression for Dt as defined in Lemma 5.1.7 for lattice vertex operator algebras. Theorem 5.1.10. P n n n µ ϑKt,d (τ) d| t td td Dt(τ) = , ηνt (τ) d where Kt,d is the kernel of the restriction of wt to the fixed-point lattice Lνt . Proof. Using Equation 5.1.4 and the Möbiusinversion formula we have

X X k hα,αi/2 ηνt (τ)Dt(τ) = wt (α) q νt n α∈L (k, t )=1 n X X n Xtd = µ wkd(α) qhα,αi/2 td t νt n α∈L d| t k=1 n X n X Xtd = µ wkd(α)qhα,αi/2 td t n νt d| t α∈L k=1

n d n P td kd n Now wt is a td -th root of unity for all t and d, hence k=1 wt (α) is equal to td if α is in the kernel and vanishes otherwise. The stated result follows. Note that this implies that for cyclic orbifolds, the character of the holomorphic extension can always we written in terms of ordinary lattice theta-functions, rather than modiﬁed theta-functions in the sense of Deﬁnition 3.1.12. This is important since, due to the availability of better algorithms, the computation of ordinary theta-functions is many times faster and the modular transformation properties of ordinary theta-functions are better understoof.

Note. The lattice Kt,d will be a full rank sublattice of Lνt . This is particularly problematic when t is such that Lνt = L, as if the rank of Lνt and therefore of Kt,d is large the computational cost of calculating the theta-function to suﬃciently high order may be prohibitive. This is one limiting factor in our computations.

5.2 The Orbifold Algorithm

We are interested in lattice vertex operator algebra associated to even self-dual extremal lattices in d = 48 and d = 72. For d = 48, four such lattices are known [CS99,Neb98,Neb14], and in d = 72 one [Neb12]. Information about their automorphism groups is listed in table 5.2. As mentioned before, we are interested in extremal lattices because we want to construct vertex operator algebras with few low weight states. To do this as systematically as possible, we use the following approach. For a given lattice, we use MAGMA to ﬁrst identify all conjugacy classes of cyclic groups of Aut(L) and their generators g. For each generator we then proceed on a case by case basis.

1. In the simplest case, g and all its powers have no ﬁxed point lattices. This turns out to be a fairly common case. In that case all lifts are standard lifts, and gˆ has the same order as g. The T (0, j, τ) are simply eta-quotients, whose SL2(Z) transformation properties we know from proposition 5.1.2, so that we can obtain all T (i, j, τ) from Theorem 2.1.7. We can compute the type of all orbifolds, and only keep the ones of type 0 to construct holomorphic extensions V orb(hgˆi).

If g has a non-trivial fixed point lattice Lg, there are more options. We can still use a standard lift to obtain gˆ, but in this case it can happen by Corollary 3.3.8 that the order of gˆ is double the order of g. Again we are looking for type 0 orbifolds. If the standard lift gˆ does not have type 0, we can try to use a non-standard lift of g to obtain a vertex operator algebra automorphism which does have type 0. In the cases at hand, we could always find such a non-standard lift. We will discuss non-standard lifts below, and first discuss the case whereg ˆ and all its powers are standard lifts.

61 L Aut(L) |Aut(L)|

‡ 4 2 P48m (C5 × C5 × C3):(D8YC4) 1200 = 2 3 5 2‡ 7 2 P48n (SL2(13)YSL2(5)) · 2 524160 = 2 3 5 7 13 5 2 P48p (SL2(23) × S3) : 2 72864 = 2 3 11 23 5 P48q SL2(47) 103776 = 2 3 23 47 8 2 2 Γ72 (SL2(25) × PSL2(7)) : 2 5241600 = 2 3 5 7 13

Table 5.2: Known extremal lattices in d = 48 and 72. Taken from [Neb14]. Explicit expressions for the Gram matrix and the generators of the automorphism groups were taken from [Neb17].

2. If the order of gˆ is prime, and all powers of gˆ are standard lifts, then all the T (0, j, τ) are equal to a product of an eta-quotient and a (ordinary) lattice theta-function and by Corollary 5.1.9 all the remaining twisted characters can be obtained from the T (i, 0, τ) by applying T - transformations. In order to calculate T (i, 0, τ) we express the S-transformation of the lattice 0 theta-function θLg (τ) in terms of the theta-function of the dual lattice Lg using the inversion − 1 Rank(L ) formula 3.1.3 θ (−1/τ) = (det L ) 2 (−iτ) g θ 0 (τ). Subsequent summation over T -images Lg g Lg will only remove non-integer orders in the q-expansion of T (i, 0, τ). See also [vEMS17].

3. If the order of gˆ is not prime, but all powers of gˆ are standard lifts, then all the T (0, j, τ) are equal to a product of an eta-quotient and a (ordinary) lattice theta-function. The theorem of Hecke-Schoenberg (Lemma 3.1.1) tells us that those lattice theta-functions are modular forms of Γ0(N) for some level N of some weight k, possibly with some character χ. Alternatively, we can also apply Lemma 5.1.7 to establish this for Dt(τ). We can try to express these in terms of eta-quotients. In all cases at hand, following the approach of Rouse and Webb [RW15b] we are able to ﬁnd a basis of Mk(Γ0(n)) in terms of eta-quotients by virtue of Theorem 5.1.4 — see table 5.1 for these bases. This allows us to express Dt(τ) as sums of eta-quotients, so that we can read oﬀ the SL2(Z)-transformations.

The resulting characters of orbifold vertex operator algebras constructed in (1) through (3) are listed under ‘Standard lift without order doubling’ in the tables below. They form indeed the majority of the cases we analysed. Next let us discuss non-standard lifts. 4. In the simplest case we still take gˆ to be a standard lift, but have some gˆk that are not standard, such as in the case of order doubling described in Corollary 3.3.8. The graded trace function of gˆk thus leads to generalized theta-functions with phases. We can however use Theorem 5.1.10 to express those in terms of standard theta-functions, and use Hecke-Schoenberg again just as before. We listed these cases under ‘Standard lift with order doubling’ in the tables below. 5. Finally, we can consider cases where the standard lift for g does not give an orbifold of type 0. In this case we can consider instead more general lifts of the form of Corollary 3.3.7. The idea is to pick a vector β, which increases the order gˆ so that the resulting orbifold becomes type 0. We were able to ﬁnd at least one such β for every orbifold with a non-vanishing ﬁxed point lattice. Again we get generalized theta-functions with phases, and use Theorem 5.1.10 to rewrite them in terms of standard theta-functions. We listed these cases under ‘Non-standard lift’ in the tables below.

5.3 Cyclic Orbifolds of Extremal Lattices in d = 48

In the following we list all holomorphic extensions that we could construct. For the lattices in d = 48, this covers all cyclic orbifolds with vanishing fixed point lattice and gives one example for every cyclic orbifold with non-vanishing fixed point lattice without order doubling. We did not however systematically construct all possible lifts in those cases. For the lattice Γ72 we list orbifolds for all cyclic groups such that every element is a standard lift. In each case, the table list the order n and the cycle type Cg of the lattice automorphism g, as well as the rank of the associated fixed-point lattice Lg and the character of the holomorphic extension.

‡Y denotes central product.

62 5.3.1 Cyclic Orbifolds for the Lattice P48m

n Cg Rank(Lg) chV orb(hg˜i) (q) Standard lift without order doubling 1 148 48 q−2 + 48q−1 + 1224 + O(q) 2 1−48248 0 q−2 + 1176 + O(q) 3 1−24324 0 q−2 + 576 + O(q) 4 2−24424 0 q−2 + 576 + O(q) 4 2−24424 0 q−2 + 576 + O(q) 5 1−2510 8 q−2 + 8q−1 + 264 + O(q) 5 1−2510 8 q−2 + 8q−1 + 264 + O(q) 5 1−12512 0 q−2 + 288 + O(q) 5 1858 16 q−2 + 16q−1 + 456 + O(q) 6 1242−243−24624 0 q−2 + 1176 + O(q) 10 122−25−101010 0 q−2 + 312 + O(q) 10 122−25−101010 0 q−2 + 312 + O(q) 10 1−8285−8108 0 q−2 + 408 + O(q) 10 1122−125−121012 0 q−2 + 600 + O(q) 12 2124−126−121212 0 q−2 + 648 + O(q) 15 113−15−5155 0 q−2 + 192 + O(q) 15 163−65−6156 0 q−2 + 360 + O(q) 15 113−15−5155 0 q−2 + 192 + O(q) 15 1−4345−4154 0 q−2 + 192 + O(q) 20 264−610−6206 0 q−2 + 360 + O(q) 30 1−62636566−610−615−6306 0 q−2 + 600 + O(q) 30 142−43−4546410−415−4304 0 q−2 + 408 + O(q) 30 1−12131556−110−515−5305 0 q−2 + 312 + O(q) 30 1−12131556−110−515−5305 0 q−2 + 312 + O(q) Standard lift with order doubling 4 224 24 q−2 + 24q−1 + 1896 + O(q) 20 24104 8 q−2 + 8q−1 + 744 + O(q)

5.3.2 Cyclic Orbifolds for the Lattice P48n

n Cg Rank(Lg) chV orb(hg˜i) (q) Standard lift without order doubling 1 148 48 q−2 + 48q−1 + 1224 + O(q) 2 224 24 q−2 + 24q−1 + 648 + O(q) 2 1−48248 0 q−2 + 1176 + O(q) 2 224 24 q−2 + 24q−1 + 648 + O(q) 3 1−24324 0 q−2 + 576 + O(q) 4 2−24424 0 q−2 + 576 + O(q) 4 2−24424 0 q−2 + 576 + O(q) 4 2−24424 0 q−2 + 576 + O(q)

63 n Cg Rank(Lg) chV orb(hg˜i) (q) 4 2−24424 0 q−2 + 576 + O(q) 4 2−24424 0 q−2 + 576 + O(q) 5 1−12512 0 q−2 + 288 + O(q) 6 2−12612 0 q−2 + 288 + O(q) 6 2−12612 0 q−2 + 288 + O(q) 6 1242−243−24624 0 q−2 + 1176 + O(q) 7 1−878 0 q−2 + 192 + O(q) 10 1122−125−121012 0 q−2 + 600 + O(q) 10 2−6106 0 q−2 + 142 + O(q) 12 2124−126−121212 0 q−2 + 648 + O(q) 12 2124−126−121212 0 q−2 + 648 + O(q) 12 2124−126−121212 0 q−2 + 648 + O(q) 13 1−4134 0 q−2 + 96 + O(q) 14 2−4144 0 q−2 + 96 + O(q) 14 182−87−8148 0 q−2 + 408 + O(q) 14 2−4144 0 q−2 + 96 + O(q) 20 264−610−6206 0 q−2 + 360 + O(q) 21 143−47−4214 0 q−2 + 246 + O(q) 26 2−2262 0 q−2 + 48 + O(q) 26 142−413−4264 0 q−2 + 216 + O(q) 28 244−414−4284 0 q−2 + 264 + O(q) 28 244−414−4284 0 q−2 + 264 + O(q) 28 244−414−4284 0 q−2 + 264 + O(q) 35 125−27−2352 0 q−2 + 168 + O(q) 39 123−213−2392 0 q−2 + 168 + O(q) 42 226−214−2422 0 q−2 + 168 + O(q) 42 226−214−2422 0 q−2 + 168 + O(q) 42 1−424346−47414−421−4424 0 q−2 + 408 + O(q) 52 224−226−2522 0 q−2 + 168 + O(q) 65 115−113−1651 0 q−2 + 120 + O(q) 70 1−222527210−214−235−2702 0 q−2 + 216 + O(q) 70 2110−114−1701 0 q−2 + 120 + O(q) 78 216−126−1781 0 q−2 + 120 + O(q) 78 1−222326−213226−239−2782 0 q−2 + 216 + O(q) 84 2−2426212−214228−2422842 0 q−2 + 192 + O(q) 130 1−1215110−113126−165−11301 0 q−2 + 120 + O(q) Standard lift with order doubling 4 224 24 q−2 + 24q−1 + 1896 + O(q) 8 412 12 q−2 + 12q−1 + 936 + O(q) Non-standard lift 9 316 16 q−2 + 18q−1 + 1488 + O(q) 9 316 16 q−2 + 18q−1 + 1560 + O(q) 18 68 8 q−2 + 8q−1 + 816 + O(q) 18 68 8 q−2 + 8q−1 + 888 + O(q)

64 5.3.3 Cyclic Orbifolds for the Lattice P48p

n Cg Rank(Lg) chV orb(hg˜i) (q) Standard lift without order doubling 1 148 48 q−2 + 48q−1 + 1224 + O(q) 2 1−48248 0 q−2 + 1176 + O(q) 3 1−24324 0 q−2 + 576 + O(q) 4 2−24424 0 q−2 + 576 + O(q) 4 2−24424 0 q−2 + 576 + O(q) 4 2−24424 0 q−2 + 576 + O(q) 6 1242−243−24624 0 q−2 + 1176 + O(q) 11 14114 8 q−2 + 8q−1 + 264 + O(q) 12 2124−126−121212 0 q−2 + 648 + O(q) 12 2124−126−121212 0 q−2 + 648 + O(q) 22 1−42411−4224 0 q−2 + 216 + O(q) 23 12232 4 q−2 + 4q−1 + 168 + O(q) 33 1−23211−2332 0 q−2 + 96 + O(q) 44 2−24222−2442 0 q−2 + 96 + O(q) 46 1−22223−2462 0 q−2 + 120 + O(q) 66 122−23−26211222−233−2662 0 q−2 + 216 + O(q) 69 1−13123−1691 0 q−2 + 48 + O(q) 132 214−16−112122144−166−11321 0 q−2 + 168 + O(q) 138 112−13−16123146−169−11381 0 q−2 + 120 + O(q) Standard lift with order doubling 4 224 24 q−2 + 24q−1 + 744 + O(q) 4 224 24 q−2 + 24q−1 + 1800 + O(q) 44 22222 4 q−2 + 4q−1 + 264 + O(q) 44 22222 4 q−2 + 4q−1 + 360 + O(q) 92 21461 2 q−2 + 2q−1 + 216 + O(q) Non-standard lift 9 316 16 q−2 + 18q−1 + 1488 + O(q) 9 316 16 q−2 + 18q−1 + 1632 + O(q)

5.3.4 Cyclic Orbifolds for the Lattice P48q

n Cg Rank(Lg) chV orb(hg˜i) (q) Standard lift without order doubling 1 148 48 q−2 + 48q−1 + 1224 + O(q) 2 1−48248 0 q−2 + 1176 + O(q) 4 2−24424 0 q−2 + 576 + O(q) 23 12232 4 q−2 + 4q−1 + 168 + O(q) 46 1−22223−2462 0 q−2 + 120 + O(q) 47 11471 2 q−2 + 2q−1 + 120 + O(q)

65 n Cg Rank(Lg) chV orb(hg˜i) (q) 94 1−12147−1941 0 q−2 + 72 + O(q) Non-standard lift 9 316 16 q−2 + 18q−1 + 1560 + O(q)

5.4 Cyclic Orbifolds for the Lattice Γ72

n Cg Rank(Lg) chV orb(hg˜i) (q) Standard lift without order doubling 1 172 72 q−3 + 72q−2 + 2700q−1 + 70080 + O(q) 2 1−24248 24 q−3 + 24q−2 + 1500q−1 + 37824 + O(q) 2 1−72272 0 q−3 + 2628q−1 + 5184 + O(q) 2 124224 48 q−3 + 48q−2 + 1548q−1 + 40704 + O(q) 3 324 24 q−3 + 24q−2 + 900q−1 + 23424 + O(q) 3 324 24 q−3 + 24q−2 + 900q−1 + 23424 + O(q) 3 324 24 q−3 + 24q−2 + 900q−1 + 23424 + O(q) 4 1−24424 0 q−3 + 876q−1 + 16128 + O(q) 4 1242−24424 24 q−3 + 24q−2 + 900q−1 + 23424 + O(q) 5 112512 24 q−3 + 24q−2 + 612q−1 + 16512 + O(q) 5 1−18518 0 q−3 + 648q−1 + 13608 + O(q) 6 3−24624 0 q−3 + 876q−1 + 1728 + O(q) 6 3−24624 0 q−3 + 876q−1 + 1728 + O(q) 6 3−8616 8 q−3 + 8q−2 + 500q−1 + 12672 + O(q) 6 3868 16 q−3 + 16q−2 + 516q−1 + 13632 + O(q) 6 3−24624 0 q−3 + 876q−1 + 1728 + O(q) 7 1−12712 0 q−3 + 432q−1 + 9936 + O(q) 10 1182−185−181018 0 q−3 + 648q−1 + 5928 + O(q) 10 1−122125−121012 0 q−3 + 588q−1 + 1728 + O(q) 10 142454104 16 q−3 + 16q−2 + 356q−1 + 9792 + O(q) 10 1−62−656106 0 q−3 + 360q−1 + 7992 + O(q) 10 1−4285−4108 8 q−3 + 8q−2 + 340q−1 + 8832 + O(q) 10 162−125−61012 0 q−3 + 360q−1 + 7872 + O(q) 12 386−8128 8 q−3 + 8q−2 + 300q−1 + 7872 + O(q) 12 3−8128 0 q−3 + 292q−1 + 5376 + O(q) 13 1−6136 0 q−3 + 216q−1 + 5400 + O(q) 14 1122−127−121412 0 q−3 + 432q−1 + 3120 + O(q) 15 34154 8 q−3 + 8q−2 + 204q−1 + 5568 + O(q) 15 3−6156 0 q−3 + 216q−1 + 4536 + O(q) 20 142−4445410−4204 8 q−3 + 8q−2 + 204q−1 + 5568 + O(q) 20 164−65−6206 0 q−3 + 216q−1 + 4200 + O(q) 20 1−6264−65610−6206 0 q−3 + 216q−1 + 4536 + O(q) 20 1−4445−4204 0 q−3 + 196q−1 + 3840 + O(q) 21 3−4214 0 q−3 + 144q−1 + 3312 + O(q) 26 162−613−6266 0 q−3 + 216q−1 + 1176 + O(q)

66 n Cg Rank(Lg) chV orb(hg˜i) (q) 26 1−22−2132262 0 q−3 + 120q−1 + 3144 + O(q) 26 122−413−2264 0 q−3 + 120q−1 + 3072 + O(q) 30 3−46415−4304 0 q−3 + 196q−1 + 576 + O(q) 30 366−615−6306 0 q−3 + 216q−1 + 2040 + O(q) 35 1−25−272352 0 q−3 + 96q−1 + 2352 + O(q) 35 135−37−3353 0 q−3 + 108q−1 + 2472 + O(q) 39 3−2392 0 q−3 + 72q−1 + 1800 + O(q) 42 346−421−4424 0 q−3 + 144q−1 + 1104 + O(q) 52 1−2224−213226−2522 0 q−3 + 72q−1 + 1800 + O(q) 52 124−213−2522 0 q−3 + 72q−1 + 1560 + O(q) 70 1−323537310−314−335−3703 0 q−3 + 108q−1 + 1512 + O(q) 70 122−2527−210−214235−2702 0 q−3 + 96q−1 + 912 + O(q) 78 326−239−2782 0 q−3 + 72q−1 + 456 + O(q) 91 117−113−1911 0 q−3 + 36q−1 + 984 + O(q) 182 1−1217113114−126−191−11821 0 q−3 + 36q−1 + 408 + O(q)

67 6 Non-Abelian Orbifolds of Lattice Vertex Operator Algebras at c = 48 and c = 72

Using MAGMA [BCP97], we extracted all subgroups of Aut(L) of the form Zq oφ Zp as in deﬁnition 2.3.2 with q, p coprime. We then restrict to groups for which no lifted element gˆ had its order doubled as a standard lift. By theorem 3.5.12 in section 3.5.1, we can lift them to automorphism groups of VL which are isomorphic to Zq oφ Zp. We then only keep the ones which had [ω] = 1. For these we use Theorem 2.3.12 to compute the characters of the holomorphic extensions V orb(G). In the following tables we list the orders q and p of the cyclic subgroups, the cycle types of their generators, the parameter φ specifying the group extension and the characters of the holomorphic extensions for all such groups. Diﬀerent groups can give the same character; in these cases we did not check if the holomorphic extensions just happen to have the same character, or if they are actually isomorphic as vertex operator algebras.

6.1 Zq oφ Zp-Orbifolds for d = 48

6.1.1 Zq oφ Zp-Orbifolds for the lattice P48p

q Zq p Zp φ chV orb(Zq oφ Zp) (q) 23 12232 132 214−16−112122144166−11321 5 q−2 + 120 + O(q) 23 12232 66 122−23−26211222−233−2662 2 q−2 + 168 + O(q) 23 12232 44 2−24222−2442 5 q−2 + 48 + O(q) 23 12232 33 1−23211−2332 2 q−2 + 48 + O(q) 23 12232 22 1−42411−4224 2 q−2 + 168 + O(q) 23 12232 11 14114 2 q−2 + 4q−1168 + O(q) 23 12232 12 2124−126−121212 -1 q−2 + 120 + O(q) 23 12232 4 2−24424 -1 q−2 + 48 + O(q) 3 1−24324 4 2−24424 -1 q−2 + 576 + O(q) 11 14114 4 2−24424 -1 q−2 + 96 + O(q) 11 14114 12 2124−126−121212 -1 q−2 + 168 + O(q) 33 1−23211−2332 4 2−24424 -1 q−2 + 96 + O(q)

6.1.2 Zq oφ Zp-Orbifolds for the lattice P48q

q Zq p Zp φ chV orb(Zq oφ Zp) (q) 47 11471 46 1−22223−2462 2 q−2 + 72 + O(q) 47 11471 23 12232 2 q−2 + 2q−1 + 120 + O(q) 23 12232 4 2−24424 -1 q−2 + 48 + O(q)

68 6.1.3 Zq oφ Zp-Orbifolds for the lattice P48m

q Zq p Zp φ chV orb(Zq oφ Zp) (q) 3 1−24324 20 264−610−6206 -1 q−2 + 264 + O(q) 3 1−24324 4 2−24424 -1 q−2 + 576 + O(q) 15 1−4345−4154 4 2−24424 -1 q−2 + 192 + O(q) 5 1−12512 12 2124−126−121212 -1 q−2 + 360 + O(q) 5 1858 12 2124−126−121212 -1 q−2 + 264 + O(q) 5 1−2510 12 2124−126−121212 -1 q−2 + 216 + O(q) 5 1−2510 12 2124−126−121212 -1 q−2 + 216 + O(q) 5 1858 4 2−24424 -1 q−2 + 192 + O(q) 5 1858 4 2−24424 -1 q−2 + 192 + O(q) 5 1−12512 4 2−24424 -1 q−2 + 288 + O(q) 5 1−2510 4 2−24424 -1 q−2 + 144 + O(q) 5 1−2510 4 2−24424 -1 q−2 + 144 + O(q)

6.1.4 Zq oφ Zp-Orbifolds for the lattice P48n

q Zq p Zp φ chV orb(Zq oφ Zp) (q) 5 1−12512 52 224−226−2522 -1 q−2 + 120 + O(q) 13 1−4134 20 264−610−6206 -1 q−2 + 120 + O(q) 13 1−4134 12 2124−126−121212 -1 q−2 + 168 + O(q) 13 1−4134 12 2124−126−121212 -1 q−2 + 168 + O(q) 39 123−213−2392 4 2−24424 -1 q−2 + 96 + O(q) 3 1−24324 52 224−226−2522 -1 q−2 + 168 + O(q) 5 1−12512 28 244−414−4284 -1 q−2 + 168 + O(q) 5 1−12512 28 244−414−4284 -1 q−2 + 168 + O(q) 35 125−27−2352 4 2−24424 -1 q−2 + 96 + O(q) 7 1−878 20 264−610−6206 -1 q−2 + 168 + O(q) 7 1−878 12 2124−126−121212 -1 q−2 + 264 + O(q) 7 1−878 12 2124−126−121212 -1 q−2 + 264 + O(q) 7 1−878 12 2124−126−121212 -1 q−2 + 264 + O(q) 21 143−47−4214 4 2−24424 -1 q−2 + 192 + O(q) 21 143−47−4214 4 2−24424 -1 q−2 + 192 + O(q) 21 143−47−4214 4 2−24424 -1 q−2 + 192 + O(q) 3 1−24324 28 244−414−4284 -1 q−2 + 264 + O(q) 3 1−24324 28 244−414−4284 -1 q−2 + 264 + O(q) 3 1−24324 28 244−414−4284 -1 q−2 + 264 + O(q) 3 1−24324 26 2−2262 -1 q−2 + 84 + O(q) 7 1−878 10 2−6106 -1 q−2 + 84 + O(q) 13 1−4134 4 2244−24 -1 q−2 + 96 + O(q) 7 1−878 6 2−12612 -1 q−2 + 132 + O(q) 3 1−24324 14 2−4144 -1 q−2 + 132 + O(q) 7 1−878 4 2−24424 -1 q−2 + 192 + O(q) 7 1−878 4 2−24424 -1 q−2 + 192 + O(q)

69 q Zq p Zp φ chV orb(Zq oφ Zp) (q) 7 1−878 4 2−24424 -1 q−2 + 192 + O(q) 7 1−878 4 2−24424 -1 q−2 + 192 + O(q) 5 1−12512 4 2−24424 -1 q−2 + 288 + O(q) 5 1−12512 4 2−24424 -1 q−2 + 288 + O(q) 3 1−24324 4 2−24424 -1 q−2 + 576 + O(q)

6.2 Zq oφ Zp-Orbifolds for the Lattice Γ72

q Zq p Zp φ chV orb(Zq oφ Zp) (q) 7 1−12712 78 326−239−2782 2 q−3 + 12q−1 + 200 + O(q) 7 1−12712 30 3−46415−4304 2 q−3 + 32q−1 + 304 + O(q) 7 1−12712 30 366−615−6306 2 q−3 + 36q−1 + 568 + O(q) 7 1−12712 39 3−2392 2 q−3 + 12q−1 + 328 + O(q) 7 1−12712 15 34154 2 q−3 + 32q−1 + 848 + O(q) 7 1−12712 15 3−6156 2 q−3 + 36q−1 + 824 + O(q) 3 324 26 1−22−2132262 -1 q−3 + 48q−1 + 1344 + O(q) 3 324 26 122−413−2264 -1 q−3 + 48q−1 + 1272 + O(q) 7 1−12712 6 3−24624 2 q−3 + 144q−1 + 1040 + O(q) 7 1−12712 6 3−24624 2 q−3 + 144q−1 + 1040 + O(q) 3 324 10 142454104 -1 q−3 + 8q−2 + 152q−1 + 4320 + O(q) 3 324 10 162−125−61012 -1 q−3 + 144q−1 + 3336 + O(q) 3 324 10 1−62−656106 -1 q−3 + 144q−1 + 3456 + O(q) 3 324 10 1−4285−4108 -1 q−3 + 136q−1 + 3360 + O(q) 7 1−12712 3 324 2 q−3 + 144q−1 + 3376 + O(q) 3 324 2 1−24248 -1 q−3 + 600q−1 + 14496 + O(q) 3 324 2 124224 -1 q−3 + 24q−2 + 648q−1 + 17376 + O(q)

6.3 Bounds for Abelian Orbifolds

At d = 24 it turns out that all 71 holomorphic vertex operator algebras can be constructed as cyclic orbifolds of lattice vertex operator. It is an interesting question to ask whether that might also be the case at higher central charge. Consider a lattice vertex operator algebra at central charge d and let h be its Heisenberg subalgebra. Lemma 6.3.1. For any ﬁnite, abelian group G acting orthogonally on the Heisenberg vertex operator algebra h the dimensions of the G-invariant homogeneous subspaces are bounded from below by d dim(h ) ≥ (6.3.1) (2) 2 and dim(h(3)) ≥ d. (6.3.2) In particular, the bounds are sharp.

70 Proof. Let {vi} denote the set of simultaneous eigenvectors of the elements of G. h(2) is spanned by states of the form vi(−2)e0 and vi(−1)vj(−1)e0. Clearly, none of the former states will be invariant if G contains a fixed-point-free element. Among the latter, elements of the form d vi(−1)vi(−1)e0 are clearly G-invariant yielding at least 2 invariant elements. If G contains an element g d with no real eigenvalues and whose characteristic polynomial is squarefree then there are exactly 2 states of this form and they contain all g-invariant states at level 2. Hence the bound is sharp. h3 is spanned by states of the form vi(−3)e0, vi(−2)vj(−1)e0 and vi(−1)vj(−1)vk(−1)e0. By a similar reasoning to above the first two types of states contribute at least a total of d states. A state of the last type is invariant under an element g if the respective eigenvalues satisfy λiλjλk = 1. Now let g have even order n and let all its eigenvalues be primitive n-th roots of unity. Then the product of any 3 eigenvalues is an odd power of a primitive n-th root of unity and there no invariant states. Hence the bound dim(h(3)) ≥ d is sharp. Hence we find that for any finite abelian automorphism group of lifted lattice automorphisms the number of low-weight states is bounded from below. In particular, this provides us with an immediate answer to our question:

−3 Corollary 6.3.2. The vertex operator algebra at central charge d = 72 with character χV (τ) = q + 12q−1 + 200 + O(q) cannot be constructed as a cyclic orbifold of a lattice vertex operator algebra by a lifted lattice automorphism.

71 7 The Large-N-Limit for Permutation Orbifolds

Vertex operator algebras and their conformal ﬁeld theories play an important in the AdS/CFT correspondence [Mal98,AGM+00]. This correspondence conjecturally maps theories of quantum gravity to certain types of vertex operator algebras. Such vertex operator algebras tend to have large central charge. More precisely, the correspondence maps a given theory of quantum gravity not just to a single N vertex operator algebra V , but rather to a whole family {V }N∈N of vertex operator algebras, whose central charges are parametrized by N. Physicists are most interested in the ‘large central charge limit’, that is the limit of this family for N → ∞. These notions of families of vertex operator algebras and their limits have not been deﬁned mathematically, and much less investigated systematically. It would be intuitive to arrange the vertex operator algebras {V N } in some sequence

V 1 →? V 2 →? ... →? V ∞ (7.0.1)

such as a limit V ∞ can be defined. Note that when V N is, for example, the Heisenberg vertex operator algebra of central charge N or the N-th tensor power V ⊗N of some given vertex operator algebra V such a sequence can be defined with the maps being embeddings as vertex algebras, but there is no limit in a desirable category of vertex algebras, as the dimensions of the homogenous subspaces do not converge. A different situation presents itself when considering so-called permutation orbifolds. The starting point of a permutation orbifold is the N-th tensor power V ⊗N of some ‘seed vertex operator algebra’ V of central charge c. Then the tensor product is again a vertex operator algebra with central charge cN. The starting point of a permutation orbifold is the N-th tensor power V ⊗N of some ‘seed vertex operator algebra’ V of central charge c. Then the tensor product is again a vertex operator algebra with ⊗N central charge cN. Its automorphism group Aut(V ) contains as a subgroup the symmetric group SN , which acts by permuting the N tensor factors. Then the characters of the fixed-point subalgebras (V ⊗N )SN are known to converge, but, as of yet, the author has been unable to identify a suitable set of ‘nice’ maps to build a sequence such as (7.0.1). More generally, we can try to orbifold by any permutation group GN < SN [KS90,DMVV97,BHS98, Ban98]. A family of permutation groups {GN }N∈N thus leads to a family of vertex operator algebras. Here, we build on previous work [LM01,BKM15,HR15,BKM16] to define and investigate a certain notion of limits for permutation orbifolds of vertex operator algebras. In particular, define the notion of nested oligomorphic sequences of permutation groups [Cam09, BKM15, HR15], and show that if a certain family of group theoretic quantities converges and the seed vertex operator algebra V is unitary, the vertex operators for fixed-point subalgebras V GN converge component-wise. This allows us to define a limiting vertex algebra V¯ .

7.1 Oligomorphic Families of Permutation Groups 7.1.1 Unitary Vertex Operator Algebras For a comprehensive account of the theory of unitary vertex operator algebras, see [DL14]. Let V be a unitary vertex operator algebra of CFT type with inner product h·, ·i and anti-linear involution θ. In particular, this means that M V = C |0i ⊕ V(n) . (7.1.1) n≥1 S Let Φ := n≥0 Φn be a homogeneous orthonormal basis of V , that is for a, b ∈ Φ

ha, bi = δa,b . (7.1.2)

For simplicity let us also assume that Φ is real, that is θ(a) = a ∀a ∈ Φ. Deﬁne the structure constants

fabc := ha, bwt(c)+wt(b)−wt(a)−1ci . (7.1.3)

72 We note that f|0i|0i|0i = 1 and that fabc = 0 if exactly two of the three vectors are equal to |0i; in physics language we say that ‘1-point functions vanish’. Fixing all structure constants of the vertex operator algebra is equivalent to ﬁxing the state-ﬁeld map by

X wt(a)−wt(b)−wt(c) Y (b, z)c = z fabca (7.1.4) a∈Φ Proposition 7.1.1. Borcherds’ identity (1.2.6) is equivalent to the condition

m n n X X j2 X j3+n fedcfdab = (−1) feadfdbc − (−1) febdfdac ∀ a, b, c, e , (7.1.5) j1 j2 j3 d∈Φ d∈Φ d∈Φ where j1 = wt(b)+wt(a)−wt(d)−n−1, j2 = wt(c)+wt(b)−wt(d)−k−1, j3 = wt(c)+wt(a)−wt(d)−m−1.

Proof. First note that ji ≥ 0, implies that the sum over d ∈ Φ is ﬁnite. Now take the inner product of Borcherds’ identity (1.2.6) with the basis vector e, and insert a complete basis d. For a given basis vector d, only one term in the sum over j is non-vanishing: in the last sum for instance, the term with j = wt(c) + wt(a) − wt(d) − m − 1.

7.1.2 Oligomorphic Orbifolds

Let XN := {1, 2,... |XN |}, and GN a permutation group of XN . Let V be a unitary, strongly rational P n vertex operator algebra. We deﬁne the series a(t) = n≥0 ant with an := |Φn|, such that a(t) = c/24 chV (t)t is the shifted character of V . Note that Φ0 = {|0i} and therefore a0 = 1. We will denote by

V XN := V ⊗|XN | (7.1.6)

the tensor product vertex operator algebra, and by

V GN := (V XN )GN (7.1.7)

the ﬁxed point vertex operator algebra of the tensor product vertex operator algebra under GN , with GN acting by permuting factors. To give an expression for the character of V GN , we will use chapter 15.3 of [Cam94] here. Let gN be a N N function g : XN → Φ. We deﬁne the weight of g as X |gN | := wt(gN (x)) , (7.1.8)

x∈XN and its support as N N supp(g ) := {x ∈ XN : g (x) 6= |0i} . (7.1.9) N σ ∈ GN acts on g as (σgN )(x) := gN (xσ−1) . (7.1.10)

We deﬁne bn(GN ) to be the number of orbits of GN on functions of weight n. The (shifted) character of V GN is then given by X n ZGN (t) = bn(GN )t . (7.1.11) n≥0 This character can also written in terms of the cycle index:

Deﬁnition 7.1.1. Let GN be a permutation group on |XN | elements. For σ ∈ GN , denote the number of cycles of length k, 1 ≤ k ≤ |XN | in the cycle decomposition of σ by mk(σ). Then the cycle index of

GN is the following polynomial in the variables s1, s2, ...., s|XN |: 1 X m1(σ) m2(σ) m3(σ) mN (σ) χGN (s1, s2, s3, ...., sXN ) = s1 s2 s3 ··· s|X | (7.1.12) |GN | N σ∈GN Using Thm 15.3.2 of [Cam94] we can express the ﬁxed point character as

2 N ZGN (t) = χGN (a(t), a(t ), . . . , a(t )) . (7.1.13) Before defining oligomorphic permutation orbifolds, let us fix some notation first:

73 Deﬁnition 7.1.2. Let K ⊂ XN .

K 1. Denote by GN := {σ ∈ GN |kσ ∈ K, ∀k ∈ K} the setwise stabilizer of K. ˆK 2. Denote by GN := {σ ∈ GN |kσ = k, ∀k ∈ K} the pointwise stabilizer of K. N K ˆK N 3. Let G(K) be the permutation group deﬁned by the action of GN /GN on K. Note that G(K) is the restriction of GN to K in the natural sense. ˆK K Note that GN is a normal subgroup of GN , so that deﬁnition (3) makes sense.

Deﬁnition 7.1.3. Let the family {GN }N∈N satisfy the conditions:

1. The bn(GN ) converge for all n. 2. For every ﬁnite K, G(K)N converges to some group G(K).

N 3. G(XN−1) < GN−1 for all N.

We then say GN is nested oligomorphic. Let us give a weaker criterion for condition (1), which also motivates the name oligomorphic. Deﬁne fn(GN ) as the number of orbits of n-element subsets of XN [Cam09]. We then have

Proposition 7.1.2. Let fn(GN ) be the number of orbits of n-element subsets of XN . Suppose {GN }N∈N satisﬁes the nesting condition (3), and fn(GN ) is bounded for all n as N → ∞. Then {GN }N∈N satisﬁes (1).

Proof. Two different orbits of GN−1 are in different GN orbits due to the nesting condition. It follows N that bn(GN ) grows monotonically in N. To bound bn(GN ), note that for fixed weight n, g (x) = |0i N Pn except for at most n values of x, and that for each of those values g can take at most A(n) := j=0 aj values. With fn(GN ) the number of orbits of n-element subsets of XN , we have

n bn(GN ) ≤ A(n) fn(GN ) . (7.1.14)

It follows that if the fn(GN ) are bounded, then so are the bn(GN ), which means that they converge.

x∈XN

GN and the following vector in the ﬁxed point vertex operator algebra V(n) ,

N −1/2 X ˆ GN φgN = A(g ,N) φσgN ∈ V(n) (7.1.16) σ∈GN where A(g, N) is a normalization factor that ensures that hφgN , φgN i = 1; we will give an explicit N N expression below. Note that two φgN are automatically orthogonal if g1 and g2 are in diﬀerent orbits: the inner product between φˆN and φˆN is only non-vanishing if g = g , which means that φN and φN g1 g2 1 2 g1 g2 only have non-vanishing terms if g1 and g2 are in the same orbit. Now let GN be nested oligomorphic. We have:

N N N Proposition 7.1.3. For N large enough, we can ﬁnd representatives g of Bn with supp(g ) ⊂ XNn , such that N and gN | are independent of N. n XNn

Nn Nn Nn Proof. Pick Nn such that | Bn | has converged already. Pick representatives g of Bn , and extend N N them to g by g (j) = |0i if j∈ / XNn . These are in diﬀerent orbits due to the nesting condition. Since Nn N | Bn | has converged, we ﬁnd such a representative for all orbits in Bn .

74 Nn Nn Let us denote Bn := Bn , Xn := XNn and g := g . For N ≥ Nn, using (7.1.16) we can then deﬁne the embedding GN ιN : Bn → V(n) , g 7→ φgN , (7.1.17) where gN is from Prop 7.1.3. Now let K := supp(gN ). For the normalization factor A(gN ,N) we obtain

N X ˆ ˆ X ˆ ˆ X ˆ ˆ A(g ,N) = hφτgN , φσgN i = |GN | hφgN , φσgN i = |GN | hφgN , φσgN i τ,σ∈G σ∈G K N N σ∈GN ˆK X ˆ ˆ ˆK = |GN ||GN | hφgN , φσgN i =: A(g)|GN ||GN |, (7.1.18) σ∈G(K)N ˆ ˆ ˆ ˆ ˆ ˆ where in the ﬁrst step we use that hφτgN , φτgN i = hφgN , φgN i, in the second step that hφgN , φσgN i vanishes N N ˆ ˆ ˆ ˆ unless supp(g ) = supp(σg ) and in the last step that hφgN , φστgN i = hφgN , φσgN i if τg = g. Note that for N large enough A(g) is indeed independent of N: this follows from the fact that G(K)N converges to G(K), and all factors in Kc in the inner product give contribution 1. From the above it follows:

GN Proposition 7.1.4. For N ≥ Nn, ιN (Bn) forms an orthonormal basis for V(n) .

7.1.4 Structure Constants

Let us now work out explicit expressions for the structure constants of the ﬁxed point algebra. For gi ∈ B, deﬁne X YN C(gi) = C(g1, g2, g3) := fg1(x)g2(x)g3(x) (7.1.19) x=1

as the structure constant of the tensor vertex operator algebra. Note that since f|0i|0i|0i = 1, the choice N of N does not matter, as long as it is large enough. We have therefore suppressed the N in gi . Our goal is now to compute the structure constant of the ﬁxed point vertex operator algebra, 3 N Y −1/2 X fφN φN φN = A(gi,N) C(σigi) . (7.1.20) g1 g2 g3 i=1 σi∈GN

To do this, we establish the following expression for the structure constants. Denote the support of gi, i.e. the set for which gi(j) 6= |0i by Ki.

Theorem 7.1.5. Let GN be nested oligomorphic. We then have

N X X −1/2 fφN φN φN = M(κ, N) A(gi) C(κiσigi) , (7.1.21) g1 g2 g3 κ∈S σi∈G(Ki) where diag K1 K2 K3 S = GN \GN × GN × GN /GN × GN × GN (7.1.22) diag as a set, where GN is the diagonal subgroup of GN × GN × GN , and !1/2 |GˆK1 ||GˆK2 ||GˆK3 | M(κ, N) = N N N (7.1.23) ˆκ1K1∪κ2K2∪κ3K3 2 |GN ||GN |

Proof. First note that the sum over σi is indeed well-defined: C(κigi) is invariant under the action of diag ˆKi Ki ˆκ1K1∪κ2K2∪κ3K3 G and GN , and acting with GN simply permutes the elements of G(Ki). Similarly |GN | Ki diag is well-defined, since it is invariant under GN , which leaves Ki invariant, and GN , which simply gives an overall conjugation of the stabilizer, leaving its order invariant. To prove the theorem, first consider the quotient set

ˆK1 ˆK2 ˆK3 S1 = GN × GN × GN /(GN × GN × GN ) . (7.1.24) Note that this is not a group since the group with which we quotient is not normal. For our purposes it diag is enough that the quotient exists as a set. The diagonal subgroup GN acts on S1 as

diag ˆK1 ˆK2 ˆK3 ˆK1 ˆK2 ˆK3 τ · (σ1GN , σ2GN , σ3GN ) = (τσ1GN , τσ2GN , τσ3GN ) , (7.1.25)

75 allowing us to deﬁne the quotient set diag S2 = GN \S1 . (7.1.26)

If GN is nested oligomorphic, then S2 converges as N → ∞: that is, | S2 | converges, and for every 2 equivalence class we can ﬁnd an N-independent representative κ ∈ S2. To see this note that by the orbit- stabilizer theorem, S1 is given by the images of K1 × K2 × K3 under GN × GN × GN . Since GN is nested

oligomorphic, by the same argument as prop 7.1.3 we can ﬁnd τ such that τ(σ1K1 ∪ σ2K2 ∪ σ3K3) ⊂ XNn where XNn is independent of N for N large enough, which gives the representative κ. 2 To compute the length of an orbit [κ ] ∈ S2, we apply the orbit-stabilizer theorem to the action of diag 1 GN on S1. For a [κ ] ∈ S1, the stabilizer is given by

κ1K ∪κ1K ∪κ1K ˆ 1 1 2 2 3 3 GN , (7.1.27) so that by the orbit-stabilizer theorem the orbit has length

κ2K ∪κ2K ∪κ2K 2 ˆ 1 1 2 2 3 3 |[κ ]| = |GN |/|GN | . (7.1.28)

ˆKi diag Note that C(gi) is invariant under both GN and GN . This means we can write the 3pt function as X M¯ (κ2,N)C(κ2g) (7.1.29) 2 [κ ]∈S2 where M¯ (κ; N) is the length of the equivalence class, given by

|G ||GˆK1 ||GˆK2 ||GˆK3 | M¯ (κ; N) = N N N N . (7.1.30) ˆκ1K1∪κ2K2∪κ3K3 |GN | Again by oligomorphicity we can choose a representative of [κ2] that is N-independent. Next we use the K ˆK fact that the G(K) = GN /GN have a well-deﬁned right action on S2. We can thus decompose S2 as

diag K1 K2 K3 S := S2 /G(K1) × G(K2) × G(K3)) = GN \GN × GN × GN /(GN × GN × GN ) , (7.1.31) using the fact that as sets G N G(K) =∼ G /GK . (7.1.32) ˆK N N GN Including the normalization (7.1.18), we can write

N X X −1/2 fφN φN φN = M(κ, N) A(gi) C(κiσigi) , (7.1.33) g1 g2 g3 [κ]∈S σi∈GKi where !1/2 |GˆK1 ||GˆK2 ||GˆK3 | Y ˆKi −1/2 ¯ N N N M(κ, N) := (|GN ||GN |) M(κ, N) = (7.1.34) ˆκ1K1∪κ2K2∪κ3K3 i |GN ||GN |

Using the above results, we can now deﬁne a limiting vertex algebra in the following way:

Theorem 7.1.6. Assume V is a unitary vertex operator algebra of CFT type, GN is nested oligomorphic, and the limit N fg1g2g3 := lim fg g g . (7.1.35) N→∞ 1 2 3 exists for all g , g , g , where f N denotes the structure constant of the vertex operator algebra V GN . 1 2 3 g1g2g3 ¯ L∞ Then deﬁne V = n=0 V(n) with M V(n) := C φg (7.1.36)

g∈Bn and X |g1|−|g2|−|g3| Y (φg2 , z)φg3 = fg1g2g3 φg1 z . (7.1.37) g1 Then (V,Y ) deﬁnes a vertex algebra.

76 N Proof. For finite N we have f N = δ N N for |g | = |g |. This also holds for the limit, so that φN φN φ φg ,φg 1 2 g1 g2 |0i 1 2 indeed Y (a, z)|0i = a + O(z). Similarly, since the f N satisfy (7.1.5) for all N, and the sum is over a finite and N-independent number of terms, we can exchange sum and limit. It then follows that the limit also satisfies (7.1.5). To establish the existence of the N → ∞ limit, it is thus enough to ensure that M(κ, N) converges:

Corollary 7.1.7. If GN is nested oligomorphic and M(κ, N) converges for all κ, then the ﬁxed point vertex operator algebra V GN has a large N vertex algebra limit.

7.1.5 An Example: SN Let us briefly discuss the best known example of a permutation orbifold vertex operator algebra with ˆK a large N limit, namely GN = SN . SN is nested oligomorphic: in particular fK = 1, GN = SN−K , K GN = SN−K × SK , SN |XN−1 = SN−1. To show that the limit exist, we simply need to show that M(κ, N) converges. To evaluate M(κ, N), we need to find |κ1K1 ∪ κ2K2 ∪ κ3K3|. To do this, first note that C(κigi) vanishes if there is x ∈ XN such that κigi(x) = |0i for exactly two of the three indices i. We can thus restrict to configurations κ where for all x ∈ κ1K1 ∪ κ2K2 ∪ κ3K3, κigi(x) = |0i for either one or zero values of i. For fixed κ, define by n3(κ) := |κ1K1 ∩ κ2K2 ∩ κ3K3|, that is the number of x such that κigi(x) 6= |0i for i = 1, 2, 3. It turns out that because of this, n3(κ) completely determines |κ1K1 ∪ κ2K2 ∪ κ3K3| [BKM16]: For any subsets Ki we have

|K1 ∪ K2 ∪ K3| = |K1| + |K2| − |K1 ∩ K2| + |K3| − |(K1 ∪ K2) ∩ K3|

= |K1| + |K2| − |K1 ∩ K2| + |K3| − |K1 ∩ K3| − |K2 ∩ K3| + |K1 ∩ K2 ∩ K3|.

Due to the remark above about 1-point functions vanishing, a conﬁguration gives vanishing contribution unless the Ki satisfy

|K1| = |K1 ∩ K2| + |K1 ∩ K3| − |K1 ∩ K2 ∩ K3|

|K2| = |K2 ∩ K3| + |K1 ∩ K2| − |K1 ∩ K2 ∩ K3|

|K3| = |K3 ∩ K1| + |K2 ∩ K3| − |K1 ∩ K2 ∩ K3|.

Adding up gives 1 |K ∩ K | + |K ∩ K | + |K ∩ K | = (|K | + |K | + |K | + 3|K ∩ K ∩ K |) , (7.1.38) 1 2 2 3 3 1 2 1 2 3 1 2 3 which after substituting gives 1 |κ K ∪ κ K ∪ κ K | = (K + K + K − n (κ)) , (7.1.39) 1 1 2 2 3 3 2 1 2 3 3 so that (N − K )!(N − K )!(N − K )! 1/2 M(κ, N) = 1 2 3 . (7.1.40) 1 2 N!(N − 2 (K1 + K2 + K3 − n3(κ)))!

Using Stirling’s approximation it follows that for n3 > 0

M = O(N −n3(κ)/2) (7.1.41)

for N → ∞, and for n3 = 0 M → 1 . (7.1.42)

This shows that the family V SN has a large N limit. In fact, it even establishes a stronger result: In the large N limit, only conﬁgurations with n3(κ) = 0 contribute. This implies that SN orbifolds become free theories in the large N limit — see [BKM16].

77 7.2 The Limit of GL(N, q)-Orbifolds

Let us now discuss another class of permutation orbifolds, based on the finite groups GL(N, q). Let q N be such that Fq is a field. GL(N, q) is the general linear group of the finite vector space Fq , which has N N N | Fq | = q elements. Since any element of GL(N, q) is a bijective map, the action of GL(N, q) on Fq N N defines a permutation group acting on q elements. Now let XN = Fq (as a set). Then GL(N, q) acts FN on V q as a permutation group. We want to show that V orb(GL(N,q)) has a large N limit.

Lemma 7.2.1. Let fn be the number of orbits of n-element subsets. We then have: n2 log f (GL(N, q)) = log q + O(n log n) . (7.2.1) n 4

N Proof. Take a set K of n vectors in Fq . Pick N ≥ n. Denote by K the subspace spanned by K, and let K := dim K. We can always find a σ ∈ GL(N, q) which maps the K independent vectors in K to the unit vectors e1, e2 . . . eK . The remaining n − K linearly dependent vectors then live in the subspace spanned K K(n−K) by the ei, which contains q vectors. It follows that there are at most q different orbits. The total number of different orbits is thus bounded by

n X K(n−K) n2/4 fn ≤ q ≤ (n + 1)q . (7.2.2) K=0

On the other hand we can bound fn from below by picking K = n/2 linearly independent vectors, and choosing the remaining n/2 dependent vectors from their span, giving

n/2 q 2 f ≥ ∼ qn /4n−n . (7.2.3) n n/2

Proposition 7.2.2. The family (GL(N, q))N is nested oligomorphic. Moreover in the limit N → ∞ the bn(GN ) grow like 2 2 log bn = αn + o(n ) (7.2.4)

for some constant α. If the seed V has dim V(1) > 0, we have α = log q/4. Proof. With the same notation as above, we have

K GN ' GL(K, q) × GL(N − K, q) (7.2.5) and ˆK GN ' GL(N − K, q) (7.2.6)

such that G(K) = GL(K, q). Clearly we also have GL(N, q)|XN−1 = GL(N − 1, q), so that using proposition 7.1.2, we use lemma 7.2.1 to establish that the fn(GL(N,√ q)) are bounded. To show (7.2.4), note that in (7.1.14) we can bound A(n) < eC n, which gives an upper bound on bn of the form (7.2.4). On the other hand we can use the lower bound fn to establish the lower bound on bn. If dim V(1) > 0, then we take the function g(x) = a ∈ V(1)∀x ∈ supp(g), and else g(x) = ω ∈ V(2)∀x ∈ supp(g), with ω the conformal vector. This vector has weight n and 2n respectively, which leads to the bounds bn ≥ fn and b2n ≥ fn. We can now use nested oligomorphicity to establish that a N → ∞ limit exists: Proposition 7.2.3. The ﬁxed point vertex operator algebra for the orbifold group GL(N, q) has a large N limit. Proof. To evaluate M(κ, N), we use (7.2.6) together with the asymptotic expression

2 |GL(N, q)| = qN φ(q−1)(1 + O(q−N )) , (7.2.7) where φ(q) is Euler’s function ∞ Y φ(q) = (1 − qn). n=1

78 ˆκ1K1∪κ2K2∪κ3K3 To obtain |GN | = |GL(N − dim(K1 + K2 + K3), q)|, we repeat an argument similar to section 7.1.5. Namely we have the inequality

dim(K1 + K2 + K3) = K1 + K2 − dim(K1 ∩ K2) + K3 − dim((K1 + K2) ∩ K3)

≤ K1 + K2 + K3 − dim(K1 ∩ K2) − dim(K2 ∩ K3) − dim(K1 ∩ K3) + dim(K1 ∩ K2 ∩ K3) . (7.2.8)

Next, because again 1-point functions vanish, it follows that K1 ⊂ K2 + K3. This means that we have

K1 = dim(K1 ∩ K2) + dim(K1 ∩ K3) − dim(K1 ∩ K2 ∩ K3) , (7.2.9)

and similar for K2 and K3. Adding everything together as above gives 1 dim(K + K + K ) ≤ (K + K + K − dim(K ∩ K ∩ K )) . (7.2.10) 1 2 3 2 1 2 3 1 2 3

Deﬁning n3(κ) := dimhκ1K1 ∩ κ2K2 ∩ κ3K3i, we have 1 dimhκ K ∪ κ K ∪ κ K i ≤ (K + K + K − n (κ)) . (7.2.11) 1 1 2 2 3 3 2 1 2 3 3

For n3 > 0 this gives the asymptotic expression

M = O(q−Nn3 ) , (7.2.12)

1 and for n3 = 0 and dimhκ1K1 ∪ κ2K2 ∪ κ3K3i = 2 (K1 + K2 + K3) we have M → 1 . (7.2.13)

This again shows that three point functions converge as m → ∞, and also that V¯ is free.

By [EG18] there are holomorphic extensions V orb(GL(N,q)) for all N. We want to establish that these have the following limit: Theorem 7.2.4. Let V be a unitary holomorphic vertex operator algebra of CFT type with c = 24 N. Then the family of holomorphic orbifolds V orb(GL(N,q)) has a large N vertex algebra limit V¯ orb which is equal to the limit V¯ of the ﬁxed point vertex operator algebras V GL(N,q).

To prove this, we need to establish that any twisted modules have divergent conformal weight as N → ∞. To this end we establish the following lemmata.

Q st Lemma 7.2.5. Let g ∈ GN be a permutation of order n and cycle type Cg = t|n t acting on the holomorphic vertex operator algebra V ⊗N of central charge Nc. Then the conformal weight of the unique irreducible g-twisted module is given by

c X 1 ρ = s t − . (7.2.14) g 24 t t t|n

Proof. This follows from taking the S transform of the twining character with g inserted, and reading oﬀ the conformal weight from its expansion. Corollary 7.2.6. Let g have N − r 1-cycles. Then the conformal weight of the irreducible g-twisted module is bounded from below by rc ρ > . (7.2.15) g 32 c 1 Proof. Lemma 7.2.5 shows that for every t-cycle of g there is a contribution of 24 t − t to the conformal weight. We thus ﬁnd that the contribution per element in a t-cycle is given by

c t2 − 1 c ρ = > . (7.2.16) t 24 t2 32

79 bn

1065 ● ● 1055 ● ● ■ ● ■ 45 ■ 10 ● ■ ● ■ ● ■ 35 ■ 10 ● ■ ● ■ 25 ● ■ 10 ● ■ ■ ●■ 15 ●■ 10 ●■ ●■ ● ●■ GL(N,2) 5 ●■ 10 ●■ ●■ ■ SN

0 5 10 15 20

Figure 7.1: Growth of b for V¯ orb for S vs GL(N, 2). For concreteness we chose V = V 3 . n N E8

N Corollary 7.2.7. Let GL(N, q) act on V ⊗q by permutation. Then for any non-identity element g ∈ GL(N, q) the conformal weight of the g-twisted sector is bounded from below by

(q − 1)Nc ρ > . (7.2.17) g 32

N Proof. Let {vi} be a basis of Fq . If g ∈ GL(N, q) is not the identity there is a non-invariant basis element v1. Now for any other basis element vi either vi is non-invariant or v1 + vi is non-invariant. In particular, ∗ N if any element v is not invariant, neither is αv for any α ∈ Fq . Thus Fq contains at least (q − 1)N non-invariant elements and the desired result follows from Corollary 7.2.6.

This establishes theorem 7.2.4. Finally, let us briefly discuss the relevance of the GL(N, q) orbifold for the growth of bn. For general permutation orbifolds, the twisted modules tend to give exponential growth. More precisely, if there is a conjugacy class in GN of total non-trivial cycle length n, then usually one gets log bn ∼ 2πn [BKM15]. This is in particular what happens for the SN orbifold [Kel11]. The GL(N, q) orbifold circumvents this because all its non-trivial conjugacy classes g give modules which satisfy corollary 7.2.7. However, despite not having any contributions from the twisted modules, the bn(GL(N, q)) still grow faster than the bn(SN ), as can be seen in figure 7.1. Nonetheless, the GL(N, q) represents a new growth behavior, different from the ones found so far in the literature [KM19].

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88 Curriculum Vitae

Personal data

Name Thomas Gem¨unden Date of Birth Febuary 5, 1994 Place of Birth Bamberg, Germany Citizen of Germany

Education

2016 – 2020 ETH Zurich, Zürich, Switzerland Doctoral studies 2012 – 2016 University of Cambridge, Cambridge, United Kingdom Final degree: MMath, MA 2004 - 2012 Gymnasium Bruckmühl(grammar school) Bruckmühl,Germany Final degree: Abitur