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Extended TFTs May 13, 2009

Posted by Chris Schommer-Pries in Paper Advertisement, QFT, Shamelss Self Promotion, differential geometry, low-dimensional topology, mathematical physics, tqft, websites.
10 comments

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”.

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Generalized moonshine I: Genus zero functions January 8, 2009

Posted by Scott Carnahan in Number theory, Paper Advertisement, group theory, mathematical physics, representation theory.
21 comments

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.

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Woit on Geometric Representation Theory December 22, 2008

Posted by David Speyer in D-modules, mathematical physics, representation theory.
1 comment so far

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 December 11, 2008

Posted by A.J. Tolland in mathematical physics.
9 comments

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.

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Symplectic duality slides November 24, 2008

Posted by Ben Webster in Algebraic Geometry, QFT, category O, crazy ideas, link homology, mathematical physics, talks.
5 comments

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. (more…)

Live Blogging: AJ on Gromov-Witten theory of stacks October 20, 2008

Posted by Noah Snyder in Algebraic Geometry, Algebraic Topology, liveblogging, mathematical physics, talks.
10 comments

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

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This paper was written for our blog August 21, 2008

Posted by David Speyer in Algebraic Geometry, Number theory, Paper Advertisement, mathematical physics, papers, representation theory.
8 comments

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, \theta 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!

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Double follow-up: pass the popcorn July 23, 2008

Posted by Ben Webster in crazy ideas, fields medals, mathematical physics, silliness.
23 comments

After a long and earnest (probably still on-going) discussion in comments about plagiarism, collaboration and communication far too interesting and labyrinthine for me to sum up (though who knows, maybe I can keep the cycle of followup posts going), we finally got a truly exciting and original idea:  when is somebody gonna make a movie about Ed Witten?  If string theory is confirmed at CERN, it would make a great ending (and if it isn’t, Hollywood can just change the story).

EDIT: Of course, I wrote this when my brain was too addled by thinking about string theory and the ethics of collaboration to be thinking straight.  Witten is a lovely guy, and deserves better than to have anybody make a movie about him until he decides it’s a good idea.  Besides, Thomas is right; Grothendieck is where it’s at, movie-wise, at least.

Gale and Koszul duality, together at last July 14, 2008

Posted by Ben Webster in category O, combinatorics, hyperplanes, mathematical physics, papers, the arXiv.
2 comments

So, in past posts, I’ve attempted to explain a bit about Gale duality and about Koszul duality, so now I feel like I should try to explain what they have to do with each other, since I (and some other people) just posted a preprint called “Gale duality and Koszul duality” to the arXiv.

The short version is this: we describe a way of getting a category \mathcal{C}(\mathcal{V}) (or equivalently, an algebra) from a linear program \mathcal{V} (or as we call it, a polarized hyperplane arrangement).

Before describing the construction of this category, let me tell you some of the properties that make it appealing.

Theorem. \mathcal{C}(\mathcal{V}) is Koszul (that is, it can be given a grading for which the induced grading on the Ext-algebra of the simples matches the homological grading).

In fact, this category satisfies a somewhat stronger property: it is standard Koszul (as defined by Ágoston, Dlab and Lukács.  Those of you with Springer access can get the paper here).  In short, the category has a special set of objects called “standard modules” (which you should think of as analogous to Verma modules) which make it a “highest weight category,”  such that these modules are sent by Koszul duality to a set of standards for the Koszul dual.

Of course, whenever confronted with a Koszul category, we immediately ask ourselves what its Koszul dual is.  In our case, there is a rather nice answer.

Theorem. The Koszul dual to \mathcal{C}(\mathcal{V}) is \mathcal{C}(\mathcal{V}^\vee), the category associated to the Gale dual \mathcal{V}^\vee of \mathcal{V}.

Now, part of the data of a linear program is an “objective function” (which we’ll denote by \xi) and of bounds for the contraints (which will be encoded by a vector \eta).  Stripping these way, we end up with a vector arrangement, simply a choice of a set of vectors in a vector space, which will specify the constraints.

Theorem. If two linear programs have same underlying vector arrangment, the categories \mathcal C(\mathcal V) may not be equivalent, but they will be derived equivalent, that is, their bounded derived categories will be equivalent.

Interestingly, these equivalences are far from being canonical. In the course of their construction, one actually obtains a large group of auto-equivalences acting on the derived category of \mathcal{C}(\mathcal{V}), which we conjecture to include the fundamental group of the space of generic choices of objective function.

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What’s a Stack? June 19, 2008

Posted by A.J. Tolland in Algebraic Geometry, mathematical physics.
54 comments

Algebraic stacks are essential to my research.  This is more acceptable now than it was twenty years ago, but it still presents a bit of a language barrier.  Most mathematicians, I think, don’t know what a stack is in the way that they know what a manifold or a scheme is.  So I want to use this post to explain what stacks are, with an eye towards their appearance in mathematical physics.  I won’t quite define them (see Vistoli’s notes for that), but I’ll get you a lot closer than Harris & Morrison do (see p. 139), hopefully close enough to be comfortable that you know what’s going on when someone says “stack”.

Let’s start by saying that a space is what you get when you start with a set and then add some geometry. Maybe make the set into a manifold, maybe make it into a scheme; you can choose your favorite category.   The elements of the set become the points of your space.

A stack is what you get when you start with a groupoid instead of a set, and then add geometry. A groupoid, remember, is a category whose morphisms are all isomorphisms.  This means that the points of a stack aren’t just elements of some set; they also come equipped with a bunch of relations, telling you which points are isomorphic to each other.

So why would anyone try to make a groupoid into a geometry?

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