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# Moonshine beyond the monster

**2008-12-02**

The Monstrous Moonshine conjecture, proved by R. E. Borcherds [Invent. Math. 109, 405--444 (1992; Zbl 0799.17014)], became famous at least since Borcherds' Fields medal in 1998. Terry Gannon's book “Moonshine beyond the monster. The bridge connecting algebra, modular forms and physics” [Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. (2006; Zbl 1146.11026)] gives, in Hiromichi Yamada's words, “an overview of the current status of Moonshine and a perspective on future study in mathematics and physics.”

The reviewer points out that

“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.”“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.”