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Monstrous Moonshine

George DeRise

(written Dec 2004)

1. INTRODUCTION.   In 1978 John McKay noticed that

          196,884=196,883+1

The fact that this equality is nontrivial is called a Moonshine Conjecture. The right side is the sum of the first two dimensions of the Monster Sporadic Group M, the left side is the third coefficient in the Laurent expansion of j, the elliptic modular function. Moonshine is a slang term referring to crazy or foolish ideas, or to whiskey that is illegally distilled. At first people thought that McKay was talking nonsense in trying to connect two such disparate mathematical structures, one from algebra, the other from analysis. But it seemed unlikely that the following was fortuitous.

Coefficients of j (τ)                     dimensions of the irreducible representations of M

1                                          1

               196884                         196883+1

               21493760                     21296876+196883+1

Richard Borcherds in 1998 won the Field’s Medal for his novel proof of the Moonshine conjectures  which used many concepts from String Theory. In fact he once stated, “If the critical dimension of string theory were anything other than 26, I could not have proved the Moonshine conjectures.”  So a third seemingly unrelated concept is involved.

This survey article will give an overview of some of the ideas involved in this extremely rich branch of mathematics. In a sense it is almost anecdotal, nothing is proved and I don’t even give an adequate definition of the Monster Group, but the myriad of wonderful entanglements in this mathematical jungle will be surveyed from above as a forest and not from the ground studying the trees.

2. THE ENORMOUS THEOREM

The Classification Theorem (alias the Enormous Theorem). All the finite simple groups have been found. They can be classified into the following:

1. cyclic groups C_p , p prime  (although some purists say they are too simple to be included in the classification).
2. alternating groups A_n , n≥5 (consisting of the even permutations of the symmetric group).
3. 16 families of the Lie type (basically Lie groups defined over finite fields like Z_n sometimes twisted (whatever that means).
4. 26 Sporadic groups. (the sporadics fall outside of those infinite families).

We recall that a normal subgroup H of a group G is one satisfying gH=Hg for all g є G. If H is normal the set G/H of cosets becomes a group in its own right called the factor or quotient group of G by H. A group G  is called simple if its only normal subgroups are the trivial group and the group G itself.

In spite of the joke that group theorists are simple minded it took some ten to fifteen thousand pages of journal articles of about one hundred mathematicians more than one hundred years to “prove” the result.

            The cyclic groups are the only simple groups that are abelian. The fact that the groups A_n , n≤4 are not simple, while those for which n≥5 are simple is crucial in proving the fact that polynomial equations of degree 5 or more are, in general, not solvable in closed form using the elementary operations of addition, subtraction, multiplication, division and the extraction of radicals.

            For a simple example of a sporadic group [,] consider the two matrices,

Where the entries are taken from Z₁₁. They generate the entire group of  2³ ∙3∙5∙7∙11∙19 or 175,560matrices in the group. This was the sixth sporadic group discovered, the first after about one hundred years and is labeled J₁ after its discoverer Z. Janko. The largest sporadic group is called the Monster, M, and has

2⁴⁶∙3²⁰∙5⁹∙7⁶∙11²∙13³∙17∙19∙23∙29∙31∙41∙47∙59∙71≈8∙10⁵³ matrices in it, which is more than the number of elementary particles in the planet Jupiter.

3. What is the Monster?

To define the Monster as the largest Sporadic simple group tells us nothing about how to construct it. It can be defined [,] as the full automorphism group of a certain 196883 dimensional commutative, non-associative algebra called the Griess algebra. It can also be defined as the automorphism group of the Monster vertex algebra.   Griess’ construction wasn’t natural and covered 100 pages but in the mid 1980’s Frenkel-Lepowsky-Meurman constructed a graded infinite dimensional representation V^ц of M.     

n.b.  A Graded Lie algebra or Lie Superalgebra has its underlying vector space V graded mod 2. That is,

. The set of all B (F) vectors is a vector space itself, the only element in common is the zero. The special linear super algebra sl   (m׀n) is just the set of super linear transformations (represented by block diagonal matrices) that carry elements of B to F and F to B. This is the idea behind Supersymmetry which transforms bosons (force particles) to Fermions (matter particles) and vice-versa.

The physical interpretation of the Monster Algebra will be suggested later.

Postponing a discussion of Vertex operator algebras suffice it to say that the Monster M has a                                       196883    dimensional representation

                                                21296876       “                        “

                                                842609326      “                       “

And, of course, the trivial representation.

4. Some Complex analysis

Definition 1.  If a function f : H→C is meromorphic in the upper half plane H , it is said to be an elliptic modular function if  for every τ in H and   in SL₂ (Z) (the group of 2X2 matrices with integer elements and det=1).

Definition 2. A modular form of weight k is a holomorphic function on the upper half plane satisfying  for all   in SL₂ (Z).

It can be shown that a modular function f has a Laurent series expansion of the form:

Definition 3. The elliptic modular function j(τ) is defined as:

It should be noted that there are many alternate, but equivalent definitions of the j function. [,].  j is, in a sense, the simplest non constant modular function; any other modular function will be a rational function in j(τ).

It can be shown that j(τ) has the following Laurent series expansion:

Where q=exp(2πiτ). The constant 744 is of no significance and can be ignored because

 j-c generates the same function field as j for any constant c.

There is no time to finish this paper. Some other aspects are:

5. The primes p for which the group Γ₀ (P+) has genus 0 are the same as those giving the order of the Monster!  Ogg noticed this connection and bet a bottle of Jack Daniels to anyone who could explain this “coincidence”.
6. Borcherd’s proof involves constructing   a “Vertex Operator Algebra”. The book by Victor Kac, “Vertex Operator Algebra for Beginners”  2^nd ed. (AMS) begins with the Wightman Axioms for a Quantum Field Theory. The only thing I can glean from this is that these VOAs are related to QFT. So the “algebraic stuff”, the Monster Group etc. is related to the complex variable “stuff”, modular functions and the like via Vertex Operator Algebras”
7. STRING THEORY   Borcherd’s said, “If the critical dimension of string theory were anything other than 26, I could not have proved the Moonshine conjectures” Also he used the Brower-Goddard-Thorn “No Ghost Theorem” of String theory to understand the structure of the Monster Lie algebra. Conformal Field Theories on 2 dimensional spacetime are the building blocks of string theory. Borcherd’s VOAs are an abstraction of a Chiral Algebra in CFT. It is interesting that in the book “Conformal Field Theory” by Nutku et al one of the six essays is “Monstrous Moonshine  and the Classification of CFT”.

The bottom line? The connection between the Monster (Algebra) and the Elliptic Modular Function (Analysis)  is …

                              STRING THEORY!!

I wanted to submit this article to the MAA’s journal, but unfortunately, as usual, there is no time to do an adequate job. I can’t wait for the summer so that I fill in the gaping lacunae in my knowledge of these things.

I would like to thank the Hampton University Math Dept. in particular A. Makagon for the honor of presenting this lecture to the department.

                                                                              January 6, 2005

Mini-Course Description

I am very pleased to announce that my four hour mini-course on the interface between Mathematics and Physics has been accepted by the MAA and will be presented at the

NATIONAL

JOINT MATHEMATICS MEETINGS:YEAR 2000

MATHEMATICAL ASSOCIATION OF AMERICA (MAA)
AMERICAN MATHEMATICAL SOCIETY (AMS)
WASHINGTON D.C.

Abstract of Minicourse (slightly revised)

Modern Physics and the Mathematical World

Various popularizations of physics, read by our students are loaded with some pretty "crazy" notions, for example, black holes, quarks, antimatter, the big bang, superstrings in a 10 dimensional Universe, wormhole solutions of Einstein’s equations permitting time travel etc. Much of this material can be understood in a more formal manner, yet not using any mathematics beyond the sophomore year. This mini course, on a semi intuitive, semi technical level is a first bridge from the popularizations of modern physics to the more austere technical literature. Some problems in Quantum Mechanics are solved using MAPLE. Some demonstrations will be given.

Description of Course

Obviously only the very basic essentials of each of the disciplines can be presented within the given time constraints. I intend to focus on one or two particular examples in each discipline that will illustrate the key ideas of each topic.

1. Tantalizing stories of the great physicists will enrich the lecture.
2. Very brief excerpts of videos from PBS or the Learning Channel will enhance the presentation. (This would be subject to copyright approval).
3. Many little known connections between mathematics and physics are presented.

We begin with Classical Mechanics, starting with Newton's Law of Motion and describing its power as a unifying principle in explaining terrestrial and celestial phenomena. The same law describes the fall of an apple or the motion of a planet around the sun. We discuss why Newton's Law of Gravitation breaks down in certain regimes. An alternate approach to mechanics via an action principle is motivated by a simple example from freshman calculus. This Hamilton Principle of Least Action is a principle that is used in all branches of physics. The intimate connection between symmetry and conservation laws (Noether's Theorem) is illustrated and proved using two simple examples.

Next we turn to Electricity and Magnetism. Maxwell's wonderful equations are exhibited, and each illustrated with a simple example from the physical world. (In the Fall of 1997, I taught this as the last lecture in my vector calculus course.) Their symmetries and asymmetries are pointed out. Symmetry is a unifying idea in modern physics, and its importance is stressed in this course. Later its relation to group theory is brought out. A questions like why the charge of a proton is almost exactly equal in magnitude to that of an electron is answered later by a theory involving matrices. The existence of magnetic monopoles are related to concepts of topology.

Maxwell's Equations motivate Einstein's Special Theory of Relativity. We approach this topic mathematically; if a student comprehends translations and rotations in the first calculus course, then the same ideas hold for relativity. Since the transformations of Special Relativity define the subject it is only a matter of "upping" the dimension of space from two to four and "throwing in" some physical terms. These transformations can be thought of as imaginary rotations, (involving hyperbolic instead of trigonometric functions). What appears very mystical when explained in terms of moving spaceships in elementary physics books becomes crystal clear when presented mathematically. In special relativity the "twin paradox" is explained using Minkowski space. This is just a vector space with a "weird distance formula", and a "wrong way" triangle inequality. The mathematics "explains" why nothing can travel faster than the speed of light.

It is shown how quantum mechanics can be presented as a branch of linear algebra, (reality being described as a vector in Hilbert Space). One physicist's "two minute crash course in quantum mechanics" easily explains the paradox of "Schrödinger's Cat". A very simple solution to Schrödinger's equation can explain why there is a probability that you could walk through the wall as a wave.

On the second day we begin by reviewing some Non-Euclidean Geometry. Later we will see how some of these non intuitive notions are reflected in various theories of the universe. A bit of intuitive differential geometry is presented. We follow Riemann in his famous lecture ( a marvelous story) as he constructs his geometry, which was the foundation stone for Einstein's theory of gravity. Ultimately Einstein's philosophy was "Geometry=Physics". Einstein's field equations are exhibited and solved in a very important special case. One solution predicts the existence of black holes, another the growth, or expansion of the universe.

We then jump from the macroscopic to the microscopic and get a good basic grounding in elementary particle physics. We show how the abstract concept of a Lie Group, and its representations are physically manifested in "groups" of actual particles of matter. The so called "Standard Model" of particle physics is just a mathematical group. The group SU(5) was once thought to be the gauge group of the world.

We finish off with a discussion of some of the hypermodern concepts in physics, the grand unified theories (GUTS), supersymmetry, ( SUSY), and SUSY-GUTS; 5,10, and 26 dimensional universes, the so called extended Kaluza-Klein theories. Superstring theory (which is intimately related to topology) is a physical symmetry modeled by group theory. It could be the "theory of everything", the "TOE" that, among other things, unifies all four forces, explains the existence of space and time, weds Einstein's theory of general relativity to quantum mechanics, i.e. gives a theory of quantum gravity. At the incredibly small Planck length the whole topology of spacetime changes into a kind of "foam". The surprising relationship between the cosmological problems on the large scale and the elementary particles of the microscopic scale are pointed out.

THE LANDSCAPE AT TORONTO:    Dec 2005-Jan 2006                                                                              Recently at our Division meetings Roger and Sandy presented their exciting travel experiences and provided us with wonderful food. Since my travel experience was a bit more prosaic and the only food I would’ve provided would’ve been “Power Bars”, I decided to describe my experience in this essay.

STRINGS 2005 –TORONTO (July 10^th-July 16^th 2005)

This past summer my wife and I accompanied each other on professional conferences. After spending a few days in Boston we drove to Cape Cod where she attended a week long conference on Bi-Polar Disorders in children. The psych people sure know how to give a conference. Lectures from 9:00- 12:00 noon and then the afternoon is free. In the morning I was making headway with Zwiebach’s excellent “A First Course in String Theory” and looking through some more advanced papers on topics to be presented at the Toronto Conference that I thought I might be able to understand. Then we flew to Toronto, arriving Sunday and attending the registration at the (very prestigious) Perimeter Institute (PI) on Sunday night.

As usual the talks at the conference were way over my head, and we left Thursday morning. I wanted to stay for the lectures on Thursday and Friday and, in particular, the two popular lectures to be presented on Saturday but decided that I could accomplish so much more by studying at my level at home. Anyway the public lectures are on the net, as are most of the lectures at the conference. (Google “Strings 2005” and it’s the first hit!)

I would like to discuss

Susskind’s (Popular) lecture;

Cosmic Landscape:
String Theory and the Illusion of Intelligent Design

Susskind started the lecture and got a phone call on his cell; No, he said, he can’t delay the lecture even for such an important person… the Pope! The joke got a good laugh.

            There are about twenty physical constants in nature that, if changed, even slightly, would make our universe completely different. For example, if the electron's charge were slightly different, or if the strong nuclear force were only a tiny fraction stronger, stars would not be able to form, and so we couldn’t exist. Likewise, to allow carbon based life to arise the ratio of the number of photons to protons in the universe must lie within a very narrow range. But, the most fine tuning of all involves the cosmological constant. It is a very tiny number;

                        00000000000000000000

                        00000000000000000000

                        00000000000000000000

                        00000000000000000000

                        00000000000000000000

                        0000000000000000001

and it is denoted by Λ (cap lambda). If the 1 were put one or two places to the left galaxies could not form and so we couldn’t exist.

            Most people would agree that this can only be possible because of an Intelligent Designer, i.e. God. God is up there with the dials on Her control panel so exquisitely tuned to allow all the phenomena of this orderly Universe, including life and, in particular, a consciousness that can ponder these phenomena. A wonderfully, orderly  Universe.

            Science seeks more rational, less mythical explanations. Heretofore unknown mystical phenomena like lightning or magnetism were finally explained by means of a rational scientific theory using mathematical equations, in this case Maxwell’s equations.

            But all this fine tuning seems to imply the Anthropic Principle. The Universe must be suitable for life, otherwise we would not be here to ponder it.

            Susskind stresses the subtitle of his lecture the illusion of Intelligent Design, He and the other Landscapists at the conference explain all this fine tuning, not by fuzzy philosophy but, by physics with equations (to paraphrase him). And this requires an explanation of the (String Theory) Landscape.

            A bit of background first:

            Einstein’s Field Equation of General Relativity predicted that the Universe would contract because of gravity. He believed that the Universe should be static and so he had to add a term to his equation, the cosmological constant Λ that would be a repulsive force to counteract the force of gravity. When it was discovered that the galaxies were flying apart, i.e. the Universe was expanding, he dropped the term from his equation, calling it “the biggest blunder of my life”.

But modern physical theory still needs the cosmological constant.

            The vacuum is not really a complete void, there are quantum fluctuations giving rise to a “vacuum energy”. Since E=mc² the vacuum energy has mass and this mass will have an effect on the expansion of the Universe, tending to accelerate the expansion ( a strange mass indeed). So the vacuum energy is proportional to the cosmological constant.