A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture by Alan Gerardo Reyes Figueroa Instituto de Matem´atica Pura e Aplicada IMPA A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture by Alan Gerardo Reyes Figueroa Disserta¸c˜ao Presented in partial fulfillment of the requirements for the degree of Mestre em Matem´atica Instituto de Matem´atica Pura e Aplicada IMPA Maio de 2010 A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture Alan Gerardo Reyes Figueroa Instituto de Matem´atica Pura e Aplicada APPROVED: 26.05.2010 Hossein Movasati, Ph.D. Henrique Bursztyn, Ph.D. Am´ılcar Pacheco, Ph.D. Fr´ed´eric Paugam, Ph.D. Advisor: Hossein Movasati, Ph.D. v Abstract The Monster M is the largest of the sporadic simple groups. In 1979 Conway and Norton published the remarkable paper ‘Monstrous Moonshine’ [38], proposing a completely unex- pected relationship between finite simple groups and modular functions, in which related the Monster to the theory of modular forms. Conway and Norton conjectured in this paper that there is a close connection between the conjugacy classes of the Monster and the action of certain subgroups of SL2(R) on the upper half plane H. This conjecture implies that extensive information on the representations of the Monster is contained in the classical picture describing the action of SL2(R) on the upper half plane. Monstrous Moonshine is the collection of questions (and few answers) that these observations had directly inspired. In 1988, the book ‘Vertex Operator Algebras and the Monster’ [67] by Frenkel, Lepowsky and Meurman appeared. This book gave an explanation of why the Monster is related to the theory of modular forms, by showing that it acts as an automorphism group of a vertex algebra V ♮ of central charge 24, constructed in terms of the Leech lattice. Vertex operators also arise in mathematical physics contexts, such as conformal field theory and string the- ory. Although Frenkel et al. did not prove the Conway-Norton conjecture, they provided the raw material with which this conjecture could be approached. The main conjecture was eventually proved by Borcherds in 1992 in his paper ‘Monstrous moonshine and monstrous Lie superalgebras’ [12]. A key role in the proof was played by a class of infinite dimensional Lie algebras —called by Borcherds— generalized Kac-Moody algebras. Borcherds proved properties of the monster Lie algebra which turned out to be sufficient to complete the Conway-Norton conjecture. A key step in the proof was an application of the ‘no-ghost’ theorem from string theory. Borcherds also obtained many other remarkable connections between sporadic finite simple groups and modular forms in this and other papers. He was awarded a Fields medal in 1998 for his contributions. The principal interest of Moonshine is that it constitutes a new bridge between algebraic structures and modular apparatus, and a new era of collaboration between mathematics and physics. This work is a modest review of the Moonshine phenomena: the main conjectures, Borcherds’ proof, open problems and actual areas of research. vii Resumo O Monstro M ´eo maior dos grupos simples espor´adicos. Em 1979 Conway e Norton pub- licaram um remarc´avel artigo ‘Monstrous Moonshine’ [38], onde proporam uma rela¸c˜ao inesperada entre grupos finitos simples e fun¸c˜oes modulares, na qual relacionaram o Mon- stro `ateoria de formas modulares. Conway e Norton conjeturaram neste artigo que existe uma forte conex˜ao entre as classes de conjuga¸c˜ao do Monstro e a a¸c˜ao de certos subgru- pos de SL2(R) no semi-plano superior H. Tal conjetura implica que muita informa¸c˜ao das representa¸c˜oes do Monstro est´acontida na imagem que descreve a a¸c˜ao de SL2(R) no semi-plano superior. Monstrous Moonshine ´ea cole¸c˜ao de quest˜oes (e umas poucas respostas) diretamente inspiradas por essas observa¸c˜oes. Em 1988, apareceu o livro ‘Vertex Operator Algebras and the Monster’ [67] de Frenkel, Lep- owsky and Meurman. Este livro deu uma explica¸c˜ao de por que o Monstro est´arelacionado com a teoria de formas modulares, mostrando que M age como um grupo de automorfismos de uma ´algebra de vˆertices V ♮ de carga central 24, construida em termos do ret´ıculo de Leech. Os operadores vˆertice ocorrem tamb´em em contextos da f´ısica matem´atica, tais como teoria de campos conformes e teoria das cordas. Embora Frenkel et al. n˜ao provaram a conjetura de Conway-Norton, eles proporcionaram a ferramenta b´asica com a qual esta conjetura pode-se provar. A conjetura principal eventualmente foi provada por Borcherds em 1992 no seu artigo ‘Monstrous moonshine and monstrous Lie superalgebras’ [12]. Um papel chave na prova ´efeito por uma classe de ´algebras de Lie infinito dimensionales —chamadas por Borcherds— de ´algebras de Kac-Moody geralizadas. Borcherds provou propriedades da ´algebra de Lie monstro que foram suficentes para completar a conjetura Conway-Norton. Um papel fundamental na prova foi uma aplica¸c˜ao do teorema ‘no-ghost’ da teoria das cordas. Borcherds obteve tamb´em muitas outras conex˜oes entre grupos simples espor´adicos e for- mas modulares em este e outros artigos. Ele foi galardoado com a medalha Fields em 1998 pelas suas contribu¸c˜oes. O inter´es principal de Moonshine ´eque ele estabelece uma nova ponte entre as estruturas alg´ebricas e as modulares, e come¸ca uma nova era de coopera¸c˜ao entre a matem´atica e a f´ısica. Esta disserta¸c˜ao ´euma revis˜ao somera do fen´omeno Moonshine: as conjeturas principais, a prova de Borcherds, problemas abertos e ´areas atuais de pesquisa. ix Preface Monstrous Moonshine is probably one of the most esoteric achievements arising in math- ematics. The fact that the Monster has connections to other parts of mathematics shows that there is something very deep going on here. No one fully understands it, and the links to the fields of physics and geometry are tantalizing. The Moonshine connections have spawned a lot of work by a several mathematicians and mathematical physicists recently, and have opened intriguing conjectures, most of them remaining open until today. This work grew out in a attempt to explain the mysteries about the Monster and its relation with number theory, specially the j modular invariant. The idea about work with this topic was given to me by Prof. Hossein Movasati, with the purpose to have a better understand on some connections occurring between automorphic functions, geometry and physics. The original idea was to write a complete and detailed proof of the Moonshines conjectures, but by obviously reasons, a complete proof was somewhat voluminous to be given here. I have decided to restrict attention to the original Conway-Norton conjecture. Other aspects of Moonshine are mentioned only as a matter of general culture on the subject. I have written with special attention to the non initiated. In fact, I have included so many theory on the first chapters in order to give sufficient background to understand all material in the later ones, and to make easier most of the ideas of Borcherds’ proof. For the sake of background material, the proofs of theorems and propositions are omitted. The idea is to give only the proof of the Conway-Norton conjecture, making the material accessible to the level of a second year graduate student. Unfortunately, this work is by no means complete. There are several references included, with the purpose of encourage readers to specialize topics of their interest. Perhaps, the recent [72] is the most complete review of the Moon- shine phenomenon. A good resume of [72] is the paper [71]. For the non-mathematician reader, [160] is good source, basically for its divulgation style. Main differences between this work and [72, 71] is the devoted attention to the number theory of moonshine, and the detailed exposition of the proof of the Conway-Norton conjecture. In Chapter 1 I present an historical introduction, in order to facilitate a quickly access to the main conjecture. Several concepts appear, most of them explained more detailed in subsequent chapters. Chapters 2 and 3 give background material on algebra, particularly on Lie algebras. Chap- ter 2 also includes other classic material on algebra, such as representation of finite groups, modules, tensor products and some constructions. The main topic of Chapter 2 is to ex- plain affine Lie algebras (Section 2.8), which appear later. Chapter 3 introduce important concepts in Lie algebra theory, such as the Cartan subalgebras, the root systems and the xi Weyl group. Main topics of this chapter are the Theorem of Highest Weights (Section 3.7) and the Weyl Character formula (Section 3.8). Chapter 4 introduces the concept of vertex operator algebras, and relate it to the repre- sentation theory of Kac-Moody algebras, including a bit of lattice theory. At the final of Chapter 4, I give some interesting data relating Moonshine to the E8 classical Lie algebra and the Leech lattice. Chapter 5 is devoted exclusively to a partial construction of the Moonshine module V ♮ given by Frenkel-Lepowsky-Meurman. It also includes some aspects of other ‘unusual’ algebraic structures, such as the Golay code C24, the Leech lattice Λ24 and the Griess algebra B. In Chapters 6 and Chapter 7, I explain in more detail the j-function and the original Conway-Norton conjecture. This is the only chapter where the proofs are given (mainly due to the fact that my principal area is number theory, and I have decided give more attention to this). Chapter 7 includes important useful material to understand the ideas of Borcherds’ proof: congruence subgroups of SL2(R), replicable functions and the so called replication formulae.