Re: Hecke eigensheaves and KV 2-vectors

Re: Hecke eigensheaves and KV 2-vectors

[Urs Schreiber]

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.

Re: Hecke eigensheaves and KV 2-vectors

[John Baez]

I wrote:

> You're asking about something more general: "weak eigenobjects" of
> endofunctors T: C -> C on *monoidal* categories C.  In other words: objects
> c in C equipped with isomorphisms Tc -> a tensor c for some fixed object
> a.

Sorry!

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.

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

For Urs, A = Vect and C is a Vect-module.

Best,
jb

Re: Hecke eigensheaves and KV 2-vectors

[John Baez]

Hi -

> Now, the 2-category (bicategory) Vect-Mod contains that of
> Kapranov-Voevodsky 2-vector spaces, but is larger than that (isn't it?).

Yes.  Yetter's paper "Categorical Linear Algebra: a Setting for
Questions from Physics and Low-Dimensional Topology" characterizes
the Kapranov-Voevodsky 2-vector spaces among the Vect-modules.

(Here Vect stands for the category of finite-dimensional vector spaces
as a monoidal category with its usual tensor product.)

In Proposition 25, he says that the 2-category of 2-vector spaces
is isomorphic to the 2-category of "finitely semisimple" Vect-modules.
These are Vect-modules with a finite set of simple objects such that
every object is a finite direct sum of these.   We call this a "basis" of
simple objects.

(I would feel more comfortable if he had said these 2-categories were
2-equivalent, and this is all I really care about.)

There are a bunch of obvious Vect-modules that aren't 2-vector
spaces in Kapranov and Voevodsky's sense.

One is the category VECT of not-necessarily-finite-dimensional vector
spaces: not every object is a *finite* direct sum of copies of C.

Another is the category Vect^{infinity} for your favorite infinite
cardinal "infinity": the same problem, but for a different reason.

These counterexamples are a bit dull, in that one imagines there
could be some definition of an "infinite-dimensional" 2-vector
space that they might fit.  But, it seems that that there are two different
kinds of infinite-dimensionality at work here: VECT is "tall", while
Vect^{infinity} is "wide".  There's also VECT^{infinity}, which is
tall and wide.  So, somebody should straighten this stuff out.

Other counterexamples are neither "tall" nor "wide".  For example,
the category of finite-dimensional representations of a non-semisimple
algebra like the algebra of upper triangular matrices.  Or, the
category of representaions of a quiver.  (I think representations of
the A_n quiver are the same as representations of the algebra of
upper triangular matrices.)

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.

> 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.  :-)

> If yes, has it been observed before?

Not that I know of, but I could be missing examples from other
contexts.

Certainly category theorists think about "weak fixed points" of
endofunctors T: C -> C, that is, objects c in C equipped with
isomorphisms Tc -> c.  The category of all these weak fixed points
is something people enjoy studying: it's an example of a "weak
equalizer" and on a good day I know lots of examples.

You're asking about something more general: "weak eigenobjects" of
endofunctors T: C -> C on *monoidal* categories C.  In other words: objects
c in C equipped with isomorphisms Tc -> a tensor c for some fixed object
a.

I bet some people have studied these in some context or other.
You may have scared these people away with all your Langlands,
branes and defect lines.  Category theorists are dual to ordinary
people: they often get more confused when you surround an abstract
concept with a lot of distacting specifics.  "Could you please not
give me an example, to help me understand what you're saying?"  :-)

Best,
jb

Hecke eigensheaves and KV 2-vectors

[Urs Schreiber]

Dear category theorists,

if you don't mind, I'd like to mention a naive observation.

Attention of physicists, like myself, has recently been drawn to the
geometric Langlands conjecture - since E. Witten and A. Kapustin have
pointed out how it can be understood in terms of 2-dimensional topological
field theory.

Even after having been introduced to some basics, I hardly know anything
about geometric Langlands. But I believe I do understand some aspects of 2D
topological field theory.

In particular, I am fond of the general fact that where \C-modules (\C = the
complex numbers) appear in 1D quantum  field theory (quantum mechanics), we
see C-modules appear in 2D TFT, where now C is some abelian monoidal
category. In the most accessible cases of topological field theory we have C
= Vect.

Now, the 2-category (bicategory) Vect-Mod contains that of
Kapranov-Voevodsky 2-vector spaces, but is larger than that (isn't it?).

In general, it should make some sense to address objects in C-Mod (module
categories for C) as categorified vector spaces; and 1-morphisms in C-Mod as
categorified linear maps between these.

Interestingly, when one studies 2D quantum field theory (topological or
conformal), one finds that boundary conditions of the theory (known as
"D-branes") are described by objects of objects of C-Mod, i.e. objects of
C-module categories. In the above terminology these would be like
categorified vectors.

Moreover, there are phenomena called "defect lines" or "disorder operators"
in 2D QFT. These are known to be described by 1-morphisms in C-Mod, i.e. by
categorified linear maps.

Therefore a "defect line" may be applied to a "D-brane", much like a linear
map may be applied to a vector.

The above analogy naturally motivates to contemplate the case where the
D-brane is an eigenvector under this action, i.e. where it is sent by the
action of the defect line to itself, up to tensoring with an element in C.

This might be nothing but a play with words. But, remarkably, Witten and
Kapustin point out that the Hecke eigensheaves appearing in the context of
geometric Langlands are precisely to be identified with certain D-branes
that are categorified eigenvectors of some defect line, in the above sense.

Of course, they do not say so using category theoretic terminology. They are
addressing an audience of physicists. At one point they apologize for
mentioning the term "functor" once.

Therefore I was wondering what people knowledgeable in (higher) category
theory would think of this. Does my observation make sense? (Of course I am
glossing over a couple of technical details.) If yes, has it been observed
before? Is it useful for anything?

I'd be grateful for any kind of comments.

Best regards,
Urs Schreiber

P.S.

As before on previous occasions, I have written up some informal notes with
slightly more details on what I have in mind here:

http://golem.ph.utexas.edu/string/archives/000810.html