What makes the Monster Lie Algebra special?

This is a post I’d been meaning to write for several years, but I was finally prompted to action after talking to some confused physicists. The Monster Lie Algebra, as a Lie algebra, has very little structure – it (or rather, its positive subalgebra) is quite close to being free on countably infinitely many generators. In addition to its Lie algebra structure, it has a faithful action of the monster simple group by Lie algebra automorphisms. However, the bare fact that the monster acts faithfully on the Lie algebra by diagram automorphisms is not very interesting: the almost-freeness means that the diagram automorphism group is more or less the direct product of a sequence of general linear groups of unbounded rank, and the monster embeds in any such group very easily.

The first interesting property of the Monster Lie Algebra has nothing to do with the monster simple group. Instead, the particular arrangement of generators illustrates a remarkable property of the modular J-function.

The more impressive property is a *particular* action of the monster that arises functorially from a string-theoretic construction of the Lie algebra. This action is useful in Borcherds’s proof of the Monstrous Moonshine conjecture, as I mentioned near the end of a previous post, and this usefulness is because the action satisfies a strong compatibility condition that relates the module structures of different root spaces.

What is a symplectic manifold, really?

I’m teaching a graduate course in symplectic geometry and GIT this semester, and am going to try to produce some posts related to lectures I’m giving there. Hopefully, this will help me think things through and put some new exposition out there on the internet.

So, obviously, the first question is “what is a symplectic manifold?” Now, wikipedia will tell you it’s a manifold equipped with a non-degenerate closed 2-form. Certainly that’s right, but it doesn’t tell a novice in symplectic geometry much. Why think about such a structure?

So let me try to put a different spin on this. This isn’t all that new of a spin (in fact, Henry Cohn wrote almost exactly the same thing here), but I don’t know of anywhere symplectic manifolds are really presented like this: I want to think of a symplectic manifold as a space where one can do a particular flavor of classical mechanics.  

Conference on Higher Gauge Theory, Quantum Gravity, and Topological Field Theory

In February there is going to be a workshop and school dedicated to exploring the interactions of Quantum Gravity, Higher Gauge Theory, and Topological Field Theory. I’m excited about the chance to share ideas and hopefully create some new mathematics.

The conference will take place in Lisbon, Portugal, and yours truly will be giving one of the mini-courses for the school (the topic is going to be the classification of extended 2D tqfts, something near and dear to my heart). Of course that makes me really excited, but I am also excited about the other topics too and I think the mix of ideas will be invigorating. For more info look below the break.

Extended TFTs

So I’ve finally managed to bang my dissertation into something more or less ready for public consumption. It is basically finished (except for some typos and spell checking).

It is available on my new website.

The title is “The Classification of Two-Dimensional Extended Topological Field Theories”.

Generalized moonshine I: Genus zero functions

This is a plug for my first arXiv preprint, 0812.3440. It didn’t really exist as an independent entity until about a month ago, when I got a little frustrated writing a larger paper and decided to package some results separately. It is the first in a series of n (where n is about five right now), attacking the generalized moonshine conjecture. Perhaps the most significant result is that nontrivial replicable functions of finite order with algebraic integer coefficients are genus zero modular functions. This answers a question that has been floating around the moonshine community for about 30 years.

Moonshine originated in the 1970s, when some mathematicians noticed apparent numerical coincidences between the theory of modular functions and the theory of finite simple groups. Most notable was McKay’s observation that 196884=196883+1, where the number on the left is the first nontrivial Fourier coefficient of the modular function j, which classifies complex elliptic curves, and the numbers on the right are the dimensions of the smallest irreducible representations of the largest sporadic finite simple group, called the monster. Modular functions and finite group theory were two areas of mathematics that were not previously thought to be deeply related, so this came as a bit of a surprise. Conway and Norton encoded the above equation together with other calculations by Thompson and themselves in the Monstrous Moonshine Conjecture, which was proved by Borcherds around 1992.

I was curious about the use of the word “moonshine” here, so I looked it up in the Oxford English Dictionary. There are essentially four definitions:

1. Light from the moon, presumably reflected from the sun (1425)
2. Appearance without substance, foolish talk (1468 – originally “moonshine in the water”)
3. A base of rosewater and sugar, or a sweet pudding (1558 cookbook!)
4. Smuggled or illegally distilled alcoholic liquor (1782)

The fourth and most recent definition seems to be the most commonly used among people I know. The second definition is what gets applied to the monster, and as far as I can tell, its use is confined to English people over 60. It seems to be most popularly known among scientists through a quote by Rutherford concerning the viability of atomic power.

I’ll give a brief explanation of monstrous moonshine, generalized moonshine, and my paper below the fold. There is a question at the bottom, so if you get tired, you should skip to that.

Woit on Geometric Representation Theory

Just wanted to point out to everyone that Peter Woit, of the blog Not Even Wrong is doing a great job blogging on the relations between representation theory of Lie groups, functions on Lie groups and differential operators. And he promises there will be physics before the end!

Gromov-Witten Invariants and Topological Field Theory

A few days ago, John Mangual requested that one of us secret blogging seminarians write a post explaining what Gromov-Witten invariants are all about.  I volunteered to do this, which puts me in an awkward position.  Gromov-Witten theory is a big subject, and there are a lot of good introductions to the subject (e.g., Behrend’s Algebraic Gromov-Witten Invariants). It’s not clear that I can do much in a single blog post.

So I’m going to limit the scope of this post somewhat.  I’d like to explain in what sense Gromov-Witten theory is a topological field theory.   Caution: This may involve some rambling.    If you’re curious, come below the fold.

Symplectic duality slides

I’ve been too lazy to write in detail about the progress in my research (well, I am writing six papers and applying to jobs, so it isn’t entirely due to laziness), but I did recently speak in the symplectic seminar at MIT, and have posted the slides on my webpage. Obviously, they’re less useful without someone to explain them, but given the current lack of an overarching paper on the subject (that’s no. 5 on the list, I promise), I thought it might be edifying. Executive summary below the cut.

Live Blogging: AJ on Gromov-Witten theory of stacks

Today AJ’s talking in the Grand Unified Seminar (representation theory, geometry, and combinatorics) on his this (joint work with his collaborator C. Teleman and his advisor E. Frenkel).  The title of the talk is “Gromov-Witten Theory for a point/C*.”  As AJ points out with delight (after the title was misintroduced without the mod C*, “It’s negative 2 dimensional!”

The outline of the talk is:

1. Gromov-Witten Invariants
2. The stack pt./C*
3. Integration on quotient stacks
4. (the unfortunately named) admissible classes
5. Bundles on nodal curves
6. Invariants are well-defined

This paper was written for our blog

I’ve recently been reading a paper which ties together a number of this blog’s themes: Canonical Quantization of Symplectic Vector Spaces over Finite Fields by Gurevich and Hadani. I’m going to try to write an introduction to this paper, in order to motivate you all to look at it. It really has something for everyone: symplectic vector spaces, analogies to physics, Fourier transforms, representation theory of finite groups, gauss sums, perverse sheaves and, yes, functions. In a later paper, together with Roger Howe, the authors use these methods to prove the law of quadratic reciprocity and to compute the sign of the Gauss sum. For the experts, Gurevich and Hadani’s result can be summarized as follows: they provide a conceptual explanation of why there is no analgoue of the metaplectic group over a finite field. Not an expert? Keep reading!