Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics (Cambridge Monographs on Mathematical Physics) [Hardcover]

It has been said that via articles, books, and verbal presentations one can train good mathematicians but not great ones. A good mathematician can create good mathematical expositions and formal proofs whereas a great mathematician is "insightful", i.e. has an understanding of the subject matter that allows him/her to push forward its frontiers with brilliance. Interestingly, the debate over the ability to train a great mathematician in this way has changed in recent years due to the appearance of monographs and textbooks in mathematics that go beyond mere formalism in giving insightful commentary and explanations on highly esoteric mathematical concepts. These books are still relatively rare, but this reviewer believes that they are so well written, and grant so much insight, that great mathematicians could be produced by their study.

This book is one of these, and after perusing its pages one cannot help but believe that something has been obtained that cannot be found in other books on the mathematical results that have been dubbed "Moonshine". The book is not merely an exposition, but instead expounds on what one needs to know in order to travel to the frontiers of the subject. Some readers may find some of the discussion trivial or be offended by some of the "hand waving", but the author wants to target a wide audience, with particular attention it seems paid to physicists, who can handle mathematical formalism well, but crave the "intuition" and historical motivation behind this formalism. As in other areas of mathematics, physicists who want to learn the mathematics behind Moonshine want the quickest way to the top, but they insist that the path they take must not skip any intuitive insights along the way. In addition, and as the title implies, the author wants to indulge himself in various speculations that he deems fruitful for further investigation. He wants to moonshine (aka `idle talk and speculation') beyond the Monster. Readers hungry for research problems might find these speculations very helpful.

So again, some readers may find the author spends too much time on topics of elementary nature, such as discrete groups, the notion of a character of a group, and so on, but the care he takes in discussing these concepts definitely pays off when having to delve into the more complicated ones. In this regard this reviewer is in complete agreement with the author when he writes in the beginning paragraph of Chapter 1: "What is harder to find are books that describe the ideas beneath and the context behind the various definitions, theorems, and proofs".

Here is a partial listing of some of the more interesting insights that readers can obtain from this book:

- The Jordan-Holder Theorem is viewed as a generalization of the uniqueness of prime factorizations.
- Group cohomology is the key to doing group extensions: it organizes the projective representations.
- The discussion on the character of a group representation: The key fact the author wants the reader to remember regarding the character is the notion of "twisting": the j-function is the graded dimension of the infinite-dimensional graded representation of the Monster group, but replacing the dimension of each vector space in the expansion of the j-function by the character gives the McKay-Thompson series for each element of the Monster group. This replacement of the dimension by the character (parametrized by a group element) is referred to as "twisting" in the book. In fact, the character of the identity element of a group is the dimension of the representation of the group.
- That Lie group structure theory is a generalization of linear algebra: the Bruhat decomposition of Lie groups, the Weyl group, the Borel subgroup etc.
- The Killing form can be viewed as an inner product on a (complex) Lie algebra.
- That the representation theory of simple Lie algebras can be viewed as a generalization of ordinary trigonometry.
- Dirichlet series can be associated to arithmetic objects (modular forms, algebraic varieties, etc) so that the arithmetic and relations between Dirichlet series correspond naturally to the arithmetic and relations between the arithmetic objects. The author quotes the example of the L-function of a rational elliptic curve: it tracks the number of points on the curve as the field is changed from the rational numbers to some finite field.
- The idea that central extensions are devices for converting projective representations to true representations. Central extensions play a fundamental role in the book, especially in the constructions of vertex algebras such as the Heisenberg vertex algebra, which is a central extension of the commutative Lie algebra of complex-valued functions on the punctured disc; the affine Kac-Moody algebra, which is a central extension of the Lie algebra of functions on the punctured disk with values in a simple Lie algebra; and the Virasoro algebra, which is a central extension of the Lie algebra of vector fields on the punctured disk.
- That the Virasoro algebra, one of the bread-and-butter constructions in string theory, is by its action on the moduli space of curves viewed as the "heart" of Moonshine in one part of this book. This is not too surprising given the importance of the modularity of characters and the conformal field theory interpretation of this modularity. At first the Virasoro algebra is viewed somewhat differently than the physics literature in this book, namely as a one-dimensional central extension of the Witt algebra. The author does spend a few pages connecting this viewpoint with the usual one coming from conformal field theory via the indecomposable but not irreducible Verma module. The irreducible quotients of this module are identified with the positive-energy representations. The author calculates their characters for special cases (one property being unitarity), and shows they are modular functions.
- The motivational discussion of affine algebras (or Kac-Moody algebras as they are frequently called in the physics literature) is brilliant and ranks as the best feature of the book and the best one in the literature that this reviewer is aware of. The Kac-Moody algebras are viewed as objects that generalize the notion of semi-simple Lie algebras as arising from finite crystallographic Coxeter groups to Lie algebras that arise from infinite Coxeter groups. This geometric interpretation (attributed to Robert Moody) should make the physicist reader very happy.
- The author includes a discussion of the representations of the braid group on three strands (B3). Readers may wonder why this discussion is included, as the braid group at first may seem very remote from the Monster group. But the author points out that B3 is the central extension of the special linear group of 2 x 2 matrices with integer entries. He believes that this central extension is more fundamental with respect to modularity, and insists on an alternative proof of Moonshine that involves B3. His proposal is based on the two formulations of quantum field theory, namely the "Hamiltonian" view or canonical quantization, and the "Lagrangian" one based on Feynman path integration. The author believes that the Hamiltonian formulation, at least for rational conformal field theory, explains nicely how the coefficients arise in the q-expansion. This he calls the "Virasoro side" of the modularity in Moonshine, due to its emphasis on vertex operator algebras. The path integral formulation (the "Feynman side") he argues gives a more transparent interpretation of the modularity in that the graded traces are functions over moduli spaces. In his alternate conception of the proof he wants to identify those parts of the "Feynman side" that contribute to the understanding of Moonshine. To this end he alerts the reader to the fact that B3 is the mapping class group of the chiral algebra, and he writes down the generators (the so-called Dehn twists).

Gannon's monograph is amazing! Although the main goal of the text is to discuss Monstrous Moonshine, he does so much more. In the end Gannon gives a whirl-wind tour of a large swath of modern mathematics all the time focusing on the philosophy behind the math instead of the formal details.

Even though my research area is vertex algebras and affine algebras, I learned a lot from his sections in the text which cover those areas. I definitely benefited greatly from his introduction to conformal field theory.

To sum up: This book is full of insight. I highly recommend it to anybody interested in stringy physics, lie theory, or anybody just looking to expand their horizons. Awesome book!

This is one magnificient book. If you have the knowledge of a, say, third year undergrade student in mathematics or physics (you know what a manifold or a Lie algebra is, you had some group theory and complex analysis, some Lagrangian mechanics and quantum physics), then you should be comfortable following 90 percent of the book. Sometimes, slowing down reading is required and maybe not every equation will make sense, but you will get through and always drive home the main points. Amazingly, the author nevertheless manages to guide you through a vast amount of mathematical physics where you will learn serious graduate level (and above) stuff. It is really a big whirlwind tour of the whole mathematical physics, with Monstrous Moonshine explained on the top (alone the chapters on Lie theory, modular forms and quantum field theoy are pure gold, every page packed with insight). He does all that, by emphasizing the ideas over the rigour, by being a bit more informal in his arguments than usually allowed.

Maybe some mathematicans are put off by this approach, finding it too hand-waving. But as the author states in the introduction: this is not a textbook! He goes on : "I'm trying to share with the reader my understanding (such as it is) of serval remarkable topics that fit loosely together under the motley banner of Moonshine. I hope it fills a gap in the literature, by focusing more on the ideas and less on the technical minutiae, important though they are."

And yes he fills a gap, a big gap. And I hope so much other scholars will follow his lead and will try harder translating higher level math into a less technical, more accessible form. Of course, most notably, that is what John Baez has done with his blog for over ten years now and why it is so popular. Or what in general goes on now on many blogs and internet forums. In book form, I think 'The Princeton Companion to Mathematics' is a very fine example of how far you can go in communicating advanced math to a broader educated audience.

I'm so glad Terry Gannon has now written also a book in that style. More please!