In my continuing attempts to clear the backlog in my brain, I would like to tell you about the talks of Nadler and Gukov from Miami workshop which I was at a month ago. (Actually I really want to tell you about Kontsevich’s talks but I don’t think that I understand them well enough to do that.)
Ever since the work of Kapustin and Witten a couple of years ago, a TQFT interpretation of geometric Langlands has been available. However, I had never “internalized” it until these talks. It gives a nice conceptual picture which makes some constructions in geometric Langlands less mysterious and hopefully makes the whole subject a bit more accessible.
In this view of things, geometric Langlands concerns the equality of two 4D TQFTs, which will denote by A and B. A and B both depend on the choice of a semisimple algebraic group G. Or more precisely, if we want A = B, then we should have A depending on G and B depending on its Langlands dual group . They are 4D TQFTs, so they assign a number to a (closed) 4-manifold, a vector space to a 3-manifold, a category to a 2-manifold etc and related morphisms to bordisms of such objects.
I will start with a 2-manifold C. The first surprise is that A(C) and B(C) depend on more than just a topological structure for C — in particular we assume that C is actually endowed with the structure of smooth projective algebraic curve. Then we define and . Here is the moduli space of algebraic principal G bundles on C and is the moduli space of algebraic principal bundles with connection on C. To continue the explanation, D-mod means the category of modules for the sheaf of differential operators (equivalently the category of perverse sheaves) and QCoh means the category of quasi-coherent sheaves.
Let’s look at a particular example of the above. I will start with a slightly strange curve C, which in particular is not projective. Take C to be two copies of a formal disk D which are glued together on the compliment of the origin in each one. We denote this curve by E. It is a sort of algebraic geometry version of how a topologist would construct — instead of getting you get this E. Let us think about A(E) and B(E).
Every G-bundle on D is trivial. Let us pick trivializations on each copy of D in E. Comparing the trivializations gives us an element of and overall we see that .
On the other hand, let us examine bundles with connections on E. David stated this this space is the same as , though I must say that at the moment the reason is not clear to me.
Hence we see is just the usual geometric Satake correspondence which relates equivariant perverse sheaves (D-modules) on the affine Grassmannian with representation of (equivalently coherent sheaves on ).
Now, we are in the position to think about some 3-manifolds. Let C be again a smooth projective curve and fix a point . Consider the 3-manifold M which looks like but where we remove a ball around for some . Our M has three boundary components (one from each end and one from the ball), so we expect a functor — and similarly for the B version.
Now, we take the perspective that is the Satake category. This category acts on A(C) by a push-pull diagram that I won’t get in to right now.
Similarly, is the category of representations which acts on B(C) — actually I don’t quite understand this action.
Now, the statement of unramified geometric Langlands is that for any curve C, A(C) is equivalent to B(C) is a manner compatible with these actions of A(E) and B(E).
There is also a version of this for the ramified case which I had been hoping to write about as well — perhaps another time. Also Ben-Zvi and Nadler have used some intuition from this TQFT picture to give some nice applications which they hope will lead to a proof of Soergel’s conjecture for real groups.
As a final note, I should say that these TQFT are not just some mathematical fiction. According to the work of Kapustin-Witten and then later Gukov-Witten, they can actually be described in some gauge theoretic terms and the equality of A and B comes from something is physics called S-duality.