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Zbl 1146.11026
Gannon, Terry
Moonshine beyond the monster. The bridge connecting algebra, modular forms and physics. (English)
[B] Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. xi, 477~p. \sterling~75.00; \$~130.00 (2006). ISBN 0-521-83531-3/hbk

The theme of this book is Moonshine, a mysterious connection between the largest sporadic finite simple group called the Monster and modular functions. The book contains an overview of the current status of Moonshine and a perspective on future study in mathematics and physics. The Monstrous Moonshine conjecture [{\it J. H. Conway} and {\it S. P. Norton}, Bull. Lond. Math. Soc. 11, 308--339 (1979; Zbl 0424.20010)] asserts that there exists a ${\mathbb Z}$-graded module $V$ for the Monster such that for each element$g$ of the Monster, the McKay-Thompson series $T_g$ is the Hauptmodul $J_{\Gamma_g}$ for a genus $0$ group $\Gamma_g$. Here $V = V_{-1} \oplus V_1 \oplus V_2 \oplus \cdots$ with $V_n$ being a finite dimensional module for the Monster and $T_g(\tau) = \text{ch}_{V_{-1}}(g)q^{-1} + \sum_{n \ge 1} \text{ch}_{V_n}(g)q^n$ with $q =e^{2\pi i \tau}$ is the generating function of the character $\text{ch}_{V_n}(g)$ of the action of $g$ on $V_n$. The module $V$ was constructed by [{\it I. Frenkel, J. Lepowsky} and {\it A. Meurman}, Vertex Operator Algebras and the Monster. Pure and Applied Mathematics, 134. Boston etc.: Academic Press, Inc. (1988; Zbl 0674.17001)] as the Moonshine module $V^{\natural}$, a remarkable example of vertex operator algebras (VOAs). The significance of $V^{\natural}$ is clear from the uniqueness conjecture (Conjecture 7.2.1) which asserts that $V^{\natural}$ is the unique holomorphic VOA with central charge $24$ and trivial weight one subspace. An important property of $V^{\natural}$ is that its automorphism group is exactly the Monster, which was proved by {\it R. L. Griess} [Invent. Math. 69, 1--102 (1982; Zbl 0498.20013)] and {\it J. Tits} [Invent. Math. 78, 491--499 (1984; Zbl 0548.20011)]. The Monstrous Moonshine conjecture was proved by {\it R. E. Borcherds} [Invent. Math. 109, 405--444 (1992; Zbl 0799.17014)]. In this book, Moonshine in its broader sense means ``a certain collection of related examples where algebraic structures have been associated with modular stuff ''. Here algebraic structures imply the Monster, lattices, affine algebras, etc., while modular stuff involves Hauptmoduls, theta functions, etc. VOAs connect algebraic structures and modular stuff in the algebraic meaning of Moonshine. Moonshine is also deeply related to physics, that is, conformal field theory (CFT) and string theory. VOAs should be replaced with CFT in the physical meaning. The original Monstrous Moonshine was proved by Borcherds and much progress has been achieved. For instance, {\it Y. Zhu} [J. Am. Math. Soc. 9, 237--302 (1996; Zbl 0854.17034)] developed a general theory on modularity concerning VOAs. However, the mysterious relationship between the Monster and Hauptmoduls is not understood conceptually yet. The author says `most Moonshine conjectures involving the Monster are still open.' The author seeks a new proof. That is the motivation of the book. Chapter 1, entitled ``Classical algebra'', is devoted to the basic material in algebra, geometry and analysis which will be necessary in later chapters. Included are finite groups with their representations, braid groups, lattices, manifolds, loops, elementary functional analysis, Lie groups and Lie algebras with their representations, category theory and elementary algebraic number theory. These subjects are carefully selected so that they match the main theme of the book. A number of notations and basic facts are presented at a reasonable length. Although the material treated here can be found in standard textbooks in various fields, the author intends to describe the ideas and contexts behind the definitions, theorems and their proofs. Chapter 2, entitled ``Modular stuff'', is directly related to a central theme of the book, namely, modular functions and modular forms. The subjects discussed here are intended not only for later use in the book but also for further study. The author begins with underlying geometries such as the hyperbolic plane and Riemann surfaces. The modularity of Moonshine functions was proved by Zhu's theorem through VOAs. The author expects a more elementary way to modularity. In this spirit, the author reviews classical strategies toward modularity of theta functions: Poisson summation, Dirichlet series, the heat kernel and representations of Heisenberg groups. Some subjects such as moduli spaces and Siegel modular forms are also described for future developments. Various interesting phenomena related to the number $24$ and the pattern $A$-$D$-$E$ are discussed in the final section of the chapter, which appear repeatedly throughout the book. Chapter 3, entitled ``Gold and brass: affine algebras and generalisation'', deals with important families of infinite dimensional Lie algebras, namely, the Virasoro algebra, the Heisenberg algebra, affine algebras, Kac-Moody algebras and Borcherds-Kac-Moody algebras. The structure theory and the representation theory of those Lie algebras are discussed. Among other things, the characters of their modules and the denominator identities are treated in detail. In fact, Borcherds' proof of Monstrous Moonshine is based on the denominator identity of a Borcherds-Kac-Moody algebra. A special feature is that the characters of modules for those Lie algebras enjoy modularity. Twisted representations and twisted characters of affine algebras are also discussed, which are closely related to twisted modules for VOAs and the orbifold construction. Chapter 4, entitled ``Conformal field theory: the physics of Moonshine'', treats the physical context for Moonshine, namely, CFT. The adequate physical background for Moonshine and other recently developed mathematics is supplied. The chapter consists of four sections. Sections 4.1 and 4.2 are preparatory ones. The author begins with classical physics such as classical mechanics and classical field theory, and then proceeds to quantum physics including mathematical formulation of quantum field theory. Section 4.3 presents CFT, especially rational CFT. Moreover, the orbifold construction, which is connected with finite groups, is discussed. String theory is also introduced as a motivation of those theories. Certain mathematical formulations of CFT are explored in Section 4.4. The last two sections are the heart of the book. The author says 'It was largely with the arrival of string theory that a much richer range of mathematics became relevant to physics, and it is this happy development that made this book possible.' Chapter 5, entitled ``Vertex operator algebras'', presents the notion of VOAs, which is the mathematical counterpart of the notion of chiral algebras in CFT. Section 5.1 begins with the motivation. The definition of VOAs (Definition 5.1.3) comprises a set of fairly complicated axioms. The mathematical and physical meaning of each axiom is explained in detail. Those axioms are described in terms of formal calculus. A brief introduction to formal calculus including the operator product expansion is supplied. The delta function is defined by $\delta(z) = \sum_{n \in {\mathbb Z}} z^n$ in the book. In Section 5.2, three basic examples: VOAs associated with a Heisenberg algebra, an affine algebra and a lattice are discussed. Moreover, two major constructions of a new VOA from old ones, namely, the coset construction and the orbifold construction are mentioned. In Section 5.3, the representation theory of VOAs is studied. Zhu's algebra and the notion of rationality and $C_2$-cofiniteness are introduced. A main ingredient is Zhu's theorem (Theorem 5.3.8) on modularity of characters of irreducible modules for a weakly rational and $C_2$-cofinite VOA. In the book the term ``weakly rational'' (Definition 5.3.2) is adopted for usual rational VOAs, while the term ``rational'' (Definition 6.2.3) means a stronger condition so that a rational VOA in the book corresponds to the chiral algebra of a rational CFT. Twisted modules for a VOA are also described. In Section 5.4, some connections with geometry are discussed. Chapter 6, entitled ``Modular group representations throughout the realm'', is devoted to several discrete structures appeared in CFT. Although the subjects in this chapter are not directly related to Monstrous Moonshine, they are included for future study. In Section 6.1, fusion rings (Definition 6.1.3) and Verlinde's formula (Eq. (6.1.1b)) are introduced. Then modular data (Definition 6.1.6) is discussed. Modular data consists of two unitary and symmetric matrices $S$ and $T$ satisfying certain conditions, which are relevant to the $SL_2({\mathbb Z})$ action in unitary rational CFT. The modular invariant (Definition 6.1.8) for a modular data $S$ and $T$ is discussed, which axiomatize the $1$-loop partition functions of rational CFT. The section ends with the notion of intertwining operators (Definition 6.1.9), a natural generalization of VOAs. Examples of the subjects discussed in Section 6.1 are presented in Section 6.2. For instance, modular data and fusion multiplicities of affine algebras are described in detail. Those are generalized to rational VOAs. The definition of a rational VOA (Definition 6.2.3) is different from the standard one. Other topics, such as quantum groups, modular data with respect to finite groups, knots and subfactors are also discussed. In Section 6.3, some relations between string theory and number theory are sketched. Chapter 7, entitled ``Monstrous Moonshine'', focuses on various features of the Monstrous Moonshine conjecture. The outline of the proof is presented. Some related topics are also discussed. In Section 7.1, several properties of the Monster, including the $Y_{555}$ presentation of the Bimonster, are collected and the fundamental conjecture (Conjecture 7.1.1) of Conway-Norton is explained. The replication formula which plays an important role in the proof of the conjecture, and its generalization to replicable functions (Definition 7.1.3) are discussed. The proof of the Monstrous Moonshine conjecture is divided into three steps, each of which is described in Section 7.2. The first step is the construction of the Moonshine module $V^{\natural}$ by Frenkel-Lepowsky-Meurman. The second step is the construction of the Monster Lie algebra by Borcherds, which is an example of Borcherds-Kac-Moody algebras. Using the denominator identity of the Lie algebra, Borcherds showed the replication formula for the McKay-Thompson series $T_g$ associated with any element $g$ of the Monster. Finally, in the third step, Borcherds proved that $T_g$ agrees with the corresponding Hauptmodul $J_{\Gamma_g}$. Section 7.2 ends with speculations on a second proof of the Monstrous Moonshine conjecture. Here ``a second proof'' means that it may not be based on VOAs but more directly on modularity in CFT. Section 7.3 discusses further developments and conjectures concerning Monstrous Moonshine. This book is not written in a textbook style. The ideas and the contexts beneath the theory are emphasized. Proofs are usually omitted. Instead, proper references are supplied. There are a number of exercises in each section. A list of notation, comprehensive references and index at the end of the book are quite useful. This book is suitable for graduate students and researchers in various areas of mathematics and physics such as number theory, algebra, functional analysis, conformal field theory and string theory.
[Hiromichi Yamada (Tokyo)]

MSC 2000:: *11F22 Relationship of automorphic forms to Lie algebras, etc.
20D08 Simple groups: sporadic finite groups
81T40 Two-dimensional field theories, etc.
17B69 Vertex operators
17B65 Infinite-dimensional Lie algebras
11-02 Research monographs (number theory)
20-02 Research monographs (group theory)
81-02 Research monographs (quantum theory)
17B67 Kac-Moody algebras
17B68 Virasoro and related algebras

Keywords: Moonshine; Monster; Hauptmodul; modular function

Citations: Zbl 0424.20010; Zbl 0674.17001; Zbl 0498.20013; Zbl 0548.20011; Zbl 0799.17014; Zbl 0854.17034

Cited in: Zbl 1218.20001

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