Wednesday at the Streetfest IV

July 13, 2005

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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!

John Baez

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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 $TX \oplus T^{*} X$ , 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

Posted at July 13, 2005 9:07 AM UTC

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2-(vector bundles)

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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!!!

That’s interesting.

Let’s see. Such a 2-(tangent bundle) of a 2-space would have a vector bundle of objects. This should be the ordinary tangent bundle to the space of objects of the 2-space.

Then it should have a vector bundle of morphism. This is presumeably just the tangent bundle to the space of morphisms of our 2-space.

Hm, then probably source, target and composition maps are just the vector bundle morphisms between these vector bundles that come from the differentials of the source, target and composition map of our original 2-space.

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.)

One way to understand semistrict Lie 2-algebroids is in terms of the dg-algebra duals of $L_{∞}$ -algebras. This way one can see in particular that what is called the Courant algebroid really is a 2-algebroid.

But of course it would be nice to have direct construction.

It seems easy to categorify the definition of an ordinary algebroid, once we know what a 2-(tangent bundle) and a 2-(vector bundle) is supposed to be.

So there should be a 2-space $M$ and a 2-(vector bundle) ‘ $E \to M$ ’ over it, which is apparently a category internalized in the category of vector bundles such that the base of the vector bundle of objects is the object space of $M$ and the base of the vector bundle of morphisms is the morphism space of $M$ .

Then there should be the 2-(tangent bundle) $TM$ of $M$ that you mentioned and an anchor functor $E stackto ρ TM$ , and so on.

On the other hand, what still seems to be missing is a nicer category-theoretic description of a mere algebroid. If that were available it might lead to a nicer 2-algebroid definition, too.

(BTW, it’s the A-model which gives rise to the Fukaya theory and the B-model which gives rise to the derived category of coherent sheaves.)

Posted by: Urs Schreiber on July 13, 2005 12:14 PM | Permalink | Reply to this

p-path p-algebroids

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One missing link in my thesis is this: I want to differentiate local $p$ -holonomy $p$ -functors

(1)

{hol}_{i} : 𝒫_{p} (U_{i}) \to G_{p}

that go from the $p$ -path $p$ -groupoid $𝒫_{p} (U_{i})$ to the structure $p$ -group $G_{p}$ . The result should be a $p$ -connection $p$ -functor

(2)

{con}_{i} : 𝔭_{p} (U_{i}) \to 𝔤_{p}

from the $p$ -path $p$ -algebroid $𝔤_{p} (U_{i})$ to the Lie $p$ -algebra $𝔤_{p}$ of $G_{p}$ .

Since I didn’t know (and possibly nobody knew) what the direct description of the $p$ -path $p$ -algebroid should look like, I switched to the dual description of algebroids in terms of dg-algebras, guessed the dg-algebra which should correspond to $𝔭_{p} (U_{i})$ and then showed that this (obvious) guess makes sense by demonstrating that it leads to the correct linearized cocycle relations. (That’s the content of section 13 of my thesis.)

Now that you opened my eyes, I should try to substantiate this guess by seeing if I can construct a 2-path 2-algebroid as a category internalized in the category of vector bundles.

Actually, the 1-path 1-algebroid should be nothing but the tangent bundle of base space. If true, the 2-path 2-algebroid should be nothing bu the 2-(tangent bundle) of base space, regarded as a 2-space.

Hm, maybe not. I don’t see how any 2-forms would get into the game this way. Maybe I would have to use the 2-(vector bundle) of the path space of base space.

Hm…

Posted by: Urs Schreiber on July 13, 2005 12:36 PM | Permalink | Reply to this

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July 13, 2005