The String Coffee Table

Recall that we considered a smooth compact curve $C$ of genus $g\geq 2$ and moduli spaces of principal bundles over this curve. These have complex reductive structure group $G$ or ${}^L G$, which are Langlads duals.

The refined

Geometric Langlands Conjecture

says two things:

1) There exists an equivalence of the (bounded) derived category of coherent sheaves on the moduli stack of $G$-local systems over $C$ and the (bounded) derived category of modules for derivations on the moduli space of principal bundles with structure group the Langlands dual group ${}^L G$:

2) $c$ maps structure sheaves of points to Hecke eigensheaves.

This conjecture has been checked for special cases, but it pretty intractable for the general case.

Using some physics analogy (as that worked out by Kapustin and Witten ($\to$)) one can however see that the structures appearing in the conjecture can in some sense naturally be regarded as deformed, or, more precisely, quantized classical structures.

The idea is to pass to the “classical limit” of the geometric Langlands conjecture by de-deforming or dequantizing, in this sense, and to see if something more tractable is obtained this way. Indeed, this limiting case of geometric Langlands can be proven. Apparently this then also goes a long way towards proving the full statement.

Before being able to state the classical limit Langlands conjecture we need to identify the right classical versions of the moduli stack $\mathcal{Lo}_G$ of $G$-local structures, and the moduli space $\mathrm{Bun}_G$ of $G$-principal bundles on $C$.

(As you will have noticed, there is heavy machinerey involved in all of this, and things will not become more elementary as we proceed. I try to reproduce the statements as precisely as I can, but be please aware that inevitably some of my statements may be imprecise as given.)

Ok, so first let’s de-quantize the moduli stack $\mathcal{Lo}_G$. Recall that this consisted of pairs $(V,\nabla)$ with $V$ a principal $G$-bundle on $C$ and $\nabla$ a holomorphic, integrable connection on $V$.

We may regard such a connection as a lifting of infinitesimal symmetries of $C$ to infinitesimal symmetries of $V$ that commute with the $G$-action. This is made precise by the notion of splitting of the Atiyah sequence ($\to$).

So the connection is a splitting

of

where $A(V)$ is the (sheaf of sections of the) Atiyah bundle ($\to$).

Now, we may generalize this notion of connection as follows. A $z$-connection on $V$ for $z \in \mathbb{C}$ is a map

such that

is $z$ times the identity on the (sheaf of sections of the) tangen budle $T_C$ of $C$.

So for $z=1$ this is a 1-connection, which is a connection in the ordinary sense.

As we had a moduli stack $\mathcal{Lo}_G = \{(V,\nabla)\}$ of $G$-local systems before, we now get a more general moduli space where $\nabla$ is any $z$-connection. This is called

There is an obvious projection

A special case of interest of $z$-connections is that where $z=0$. In this case the image of $\nabla$ is in the kernel of $A(V) \to T_C$, hence $\nabla_0$ is really a map into $\mathrm{ad}(V)$ in this case.

The sub moduli space for such 0-connections is called

where $\theta \in \Gamma(C, \mathrm{ad}(V)\otimes \Omega^1_C)$ is the $\mathrm{ad}(V)$-valued 1-form connection corresponding to some 0-connection on $V$, and where we demand that the first Chern class of $V$ vanishes, $c_1(V) = 0$.

The ordinary local systems form a torsor for $\mathrm{Higgs}^0$. Notice that we have the following inclusions:

With the projection

the space $\mathrm{Higgs}^0_G$ becomes a vector bundle over $\mathrm{Bun}_G^0$ and $\mathcal{Lo}_G$ is an affine bundle over $\mathrm{Higgs}_G^0$.

The conclusion of all this is that we can hence regard $\mathcal{Lo}_G$ as a deformation of $\mathrm{Higgs}^0_G$. Moreover, it is a fact that the Higgs bundle is really just the cotangent bundle over the moduli space $\mathrm{Bun}_G$

So the classical limit of the left hand side of the geometric Langlands conjecture is obtained by replacing $\mathcal{Lo}_G$ with $\mathrm{Higgs}_G^0 = T^* \mathrm{Bun}_G$.

The right hand side is also easily dealt with. I’ll spare the details and just remark that the algebra of derivations on a space is in a very familiar way the quantization of the cotangent bundle of that space. So on the right hand side we want to replace the sheaf of modules of derivations $\mathbf{D}_{\mathrm{Bun}_{{}^L G}}-\mathrm{mod}$ with the coherent sheaf of sections of the cotangent bundle

With both classical limits taken this way, we can now state the

Classical Limit of the Geometric Langlands Conjecture

1) There is an equivalence $c$ of the derived categories of coherent sheaves of the cotangent bundle of $\mathrm{Bun}_G$ and of $\mathrm{Bun}_{{}^L G}$

2) $c$ sends structure sheaves of points to Hecke eigensheaves.

(Next there is a remark on Fourer-Moukai transforms which I cannot properly reproduce in detail. The message however is that the action of the Hecke operation (which defines Hecke eigensheaves) in the classical limit is nothing but a Fourier-Moukai transformation.)

After the original geometric Langlands conjecture has thus been made more tractable, it turns into an honest

Theorem: The geometric Langlands conjecture does hold in the classical limit.

The key to proving this is Hitchin’s abelianization.

Update, May 13: I realize that above I must have mixed up the moduli stacks and moduli spaces of Higgs bundles here and there.