The n-Category Café

The basic picture is this:

The geometric Langlands duality is about a categorified Fourier transformation.

This is mentioned by Frenkel in section 4.4 of his lecture. Also see David Ben-Zvi’s comment here. The experts certainly are very well aware of this.

Still, for mere mortals such as myself, I would like to supply this statement with a dictionary that maps various concepts to each other.

Since the Fourier transformation is a linear map between vector spaces (of functionals), in order to understand what a categorified Fourier transformation might be, we need to have a vague idea of what the categorification of a linear map between linear spaces would be.

For the present purpose, the right notion is this: like a complex vector space is a module for the monoid $\mathbb{C}$, a complex 2-vector space is a module category for the monoidal category $\mathrm{Vect}_\mathbb{C}$.

The simplest example of such a module category is the category of vector spaces itself. The next simplest one is that of complex vector bundles on some space $X$. Since we want to be sophisticated, we should think of these as locally free sheaves (of sections of our vector bundles). And if you have made it that far, you should allow yourself to also consider complexes of such sheaves as certain 2-vector spaces. In particular complexes as live in derived categories of coherent sheaves on some moduli stack. There might be a technical issue or two with making usefully precise how these are “2-vector spaces”, but all I am saying here is that derived coherent sheaves are nothing but a souped-up version of vector bundles, and these we easily think of as 2-vectors.

With that picture of sheaves as 2-vectors understood, the main statement of the geometric Langlands duality is easily seen to be a candidate for a categorification of the Fourier isomorphism.

Geometric Langlands Conjecture - very vague but helpful version: There exist certain derived categories of sheaves (think: 2-vector spaces) $A$ and $B$, where $B$ is “spanned” in some sense by eigensheaves (think: eigen-2-vectors) of a certain $\mathrm{Vect}_\mathbb{C}$-linear endofunctor $H : B \to B$ (think: differentiation operator), called the Hecke operator, and where $A$ is similarly “spanned” by skyscraper sheaves (think: 2-$\delta$-distribution) such that there is an equivalence of derived categories $$c : A \stackrel{\sim}{\to} B$$ (think: Fourier transformation) which sends skycraper sheaves to Hecke eigensheaves (think: which sends $\delta$-distributions to plane waves).

The rest is details. :-)

What Witten and collaborators (Kapustin and Gukov, so far) are adding to this picture is another level of analogy. As I mention from time to time # there is a nice way to think of 2-dimensional quantum field theory as a categorification of ordinary quantum mechanics. And as quantum mechanics is about linear maps between vector (okay: Hilbert-)spaces, 2-dimensional quantum field theory is about 2-vector spaces.

It’s getting late and there would be lots of things to say here (some of which I have already said before, for instance in Fourier-Mukai, T-Duality and other linear 2-Maps). Maybe I leave it at that for the moment and just finish with presenting what was the main purpose of this entry here: the following table deserves to be written down.