The String Coffee Table

Well, I’m going to have dinner with Marnie and David and some other people, but I have a little time to kill, so maybe I’ll talk about Bouwknegt’s talk even though they’re also covering this one. After all, this talk was about string theory, and this is the string theory coffee table, not the “general abstract nonsense” coffee table!

The talk began with a lightning review of mirror symmetry, a subject I understand far too little off. There’s a kind of analogy that goes like this:

The A model is a topological quantum field theory that can be constructed by starting with strings propagating in a Kaehler manifold X and “twisting” theory using the complex structure. The algebra of observables in the A model is the Doulbeault cohomology of X. The category of D-branes is the bounded derived category of coherent sheaves on X.

The B model is a topological quantum field theory that can be constructed by starting with strings propagating in a Kaehler manifold X and twisting the theory using the symplectic structure. The algebra of observables in the A model is the quantum cohomology of X. The category of D-branes is the Fukaya category of X.

Mirror symmetry is supposed to relate these: the A-model on a Kaehler manifold X should be isomorphic to the B-model on some other Kaehler manifold Y…

… and vice versa, I believe!

As James Dolan pointed out, this “vice versa” would make the situation like “rot13” - that method of encoding where you go 13 places down the alphabet, so encoding something twice gets you back where you started. Is this “vice versa” really true???

Anyway, Hitchin’s “generalized geometries” are an attempt to understand this situation by finding a kind of geometrical structure on a manifold that includes both complex structures and symplectic structures as special cases, and which allows one to continuously interpolate between a complex structure and a symplectic structure.

The idea is to put complex structures not on the tangent bundle of X but on the direct sum of the tangent bundle and the cotangent bundle. There’s a way to do this starting with a complex structure on X. But, there’s also a way to do it starting with a symplectic structure on X!

One weird and interesting thing is that there’s a natural bracket operation on sections of $\mathrm{TX}\oplus T^{*}X$, called the Courant bracket. It’s skew symmetric but does not satisfy the Jacobi identity… it satisfies it up to d of some function. This reminds me of a paper by Weinstein on Courant algebroids, which made little sense to me when I first tried to read it.

Right around then Bouwknegt gave the definition of a Lie algebroid… and at this point my notes on the talk disintegrate, because I started pondering the definition and wondering if I could define a Lie 2-algebroid! Sorry. I hope Urs at least will forgive me for this, because he’s interested in this sort of thing. (Later I had a great talk with Danny Stevenson where we made a lot of progress on understanding the tangent bundle of a 2-space: it’s a category in the category of vector bundles!!! Understanding the tangent bundle of a 2-space and how its sections might form a Lie 2-algebra is probably a prerequisite for understanding Lie 2-algebroids.)

Anyway, for more on the subject of Bowknegt’s talk try these:

Hitchin’s paper

Gualteri’s thesis

John Baez