Re: Hecke eigensheaves and KV 2-vectors

["Urs Schreiber" <urs.schreiber@googlemail.com>]

On 5/18/06, John Baez <baez@math.ucr.edu> wrote:

> I'd be interested to know if any of these sorts of counterexamples is
> useful in Witten's work, or he can live with 2-vector spaces.
>

I think what is needed in Witten's context is the following.

Fix some topological space X.

In the Witten/Langlands examples this is the moduli space X = Bun_G of
G-bundles over some complex curve. Hm, probably it has more structure than
just being topological. But anyway.

The relevant object of Vect-Mod (the relevant 2-vector space) is the
category of vector bundles over X.

Of course this is a slight lie. Really we want to be working with locally
free (coherent?) sheaves on X. But anyway.

So a 2-vector here is a vector bundle over X.

For Kaparanov-Voevodksy we would instead take X to be a finite set.

Next, the endomorphisms that are relevant are spans over X, i.e spaces with
two (possibly different) projections onto X. Certainly these "spaces" should
carry some appropriate extra structure.

For the application to Langlands these spans are usually called Hecke_x .

We act with these spans on our 2-vectors (vector bundles) by pulling back
along one leg and pushing forward along the other.

For the Kaparanov-Voevodksy case our spans would be vector bundles with two
projections to some finite set,. i.e matrices whose entries are vector
spaces.


> Therefore I was wondering what people knowledgeable in (higher) category
> > theory would think of this. Does my observation make sense?
>
> It makes sense to me!  Other mathematicians often respond better to
> more precisely phrased questions.  :-)



I am aware that the question is a little unspecific. The best thing about
this observation is that it fits so very nicely into the general picture
that is emerging in 2D QFT.


Actually Urs is asking about an even more general situation.
> A is a monoidal category, C is a category tensored over A,
>
T: C -> C, and we seek objects c in C equipped with isomorphisms
> Tc -> a tensor c for some fixed object a.



And, moreover, T is "A-linear". That's kind of important.


We can form the category of these, which is the pseudo-equalizer
> of T and a tensor -, and call it an "eigenspace of T".



And that's where Hecke eigensheaves, and hence the corresponding "D-branes",
should live in. I think.