Hecke eigensheaves and KV 2-vectors

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

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