The String Coffee Table

The following is a transcript of the talk, as reconstructed from my notes. Personal comments are set in italics.

The goal of the lecture is to give the statement and the proof of the geometric Langlands conjecture at the classical, non-quantum level.

There are three parts

1) Geometric Langlands Conjecture in the Classical Limit.

2) Hitchin systems.

3) Proofs.

1) The Geometric Langlands Conjecture in the Classical Limit.

Here and in the following, let $G$ be a complex reductive group.

Let $T \subset G$ be maximal torus inside $G$.

From this we obtain naturally two lattices

$\bullet$ the character lattice

$\bullet$ the co-character lattice

Two groups $G$,$G'$ are called Langlands dual if the character lattice of one is the cocharacter lattice of the other, i.e. if

and

If this is the case, we write

for the langlands dual of $G$.

This duality is in fact an involution on the category of complex reductive groups.

Examples:

1) Let $G=T$ be an affine torus itself. Then ${}^L G$ is the dual torus (by the very definition).

2) The general linear group is its own Langlands dual

3) for simple Lie algebras $g = \mathrm{Lie}(G)$ we have

for algebras of type A, D, E, F, and G

and

for algebras of type B and C.

Now, in order to state a first, slightly simplified version of the geometric Langlands conjcture, we need the following terminology.

Let $C$ be a compact smooth curve of genus $g \geq 2$.

Let $G$ be a complex reductive group, as before.

Let ${}^L G$ be its Langlands dual group.

Let $\mathrm{Bun}_G$ be the moduli space of (semistable) principal $G$-bundles on $C$.

Let $\mathrm{Loc}_G$ be the moduli space of (semistable) $G$-local systems on $C$. Such a local system is nothing but a pair $(V,\nabla)$, consisting of a principal $G$-bundle $V$ and a flat holomorphic connection $\nabla$ on $V$. This is the same as an element in

Given all that, the first version of the geometric Langlands conjecture (which turns out to be in need of refinement in order not to be trivially wrong) is this.

Claim (geometric Langlands conjecture, naïve version):

1) There exists a natural equivalence of categories between the (bounded) derived category ($\to$) of coherent sheaves on the moduli space $\mathrm{Loc}_G$, coming from the group $G$, and the (bounded) derived category of modules for the sheaf differential operators on the structure sheaf of the moduli space $\mathrm{Bun}_{{}^L G}$, coming from the Langlands dual group.

2) moreover, this equivalence sends structure sheaves of points $\mathrm{pt} \in \mathrm{Loc}_G$ to automorphic $\mathbf{D}-modules$, known as Hecke eigensheaves.

So in order to understand what this might mean, we need to know what Hecke eigensheaves are.

Hecke Eigensheaves

(for a remark on how Hecke Eigensheaves should be examples of categorified eigenvectors, see the previous entry ($\to$))

The moduli space $\mathrm{Bun}_G$ has a natural family of self-correspondences labeled by points $x \in C$.

These are denoted

and are defined as follows.

$\mathrm{Hecke}_x$ is the moduli space of triples $(V,V',\beta)$, where $V$ and $V'$ are principal $G$-bundles, and where $\beta$ is an isomorphism of these bundles over the complement of the point $x$

The projections $p$ and $q$ are defined simply by

and

We can unite all these $\mathrm{Hecke}_x$ for all $x$ into a single object

in the obvious way.

There is a fiberwise composition on $\mathrm{Hecke}_x$ given by

Next, we need to pick a dominant cocharacter of $G$. Call it $\mu$.

For every such dominant cocharacter $\mu$ we get a subspace

This subspace is that of triples $(V,V',\beta)$ which induce a certain nice isomorphism on associated locally free sheaves.

(hm, let me see if I can reproduce the definition…) Given any representation

of $G$, we get, from every triple $(V,V',\beta)$ (where, recall, $V$ and $V'$ are principal $G$-bundles) associated vector bundles

Now, using $\mu$ we can construct some sort of twisted version $E'_{(\mu,\lambda)}$ of $E'$, depending on the dominant cocharacter $\mu$ and an arbitrary dominant character $\lambda$ (hm, I realize I cannot precisely reproduce the details of this twisting at the moment, I will need to check this) and the condition on $(V,V',\beta) \in \mathrm{Hecke}_x$ to be in $\mathrm{Hecke}_x^\mu$ is that $\beta$ induces an inclusion of locally free sheaves $E'_{(\mu,\lambda)} \subset E$ for all $\lambda$.

Where we had spans

before, we now similarly get spans denoted

Again, by collecting these for all $x \in C$, we obtain

in the obvious way.

The point of all these spans here is that they can be regarded as operating on the derived category $D(\mathbf{D}_{\mathrm{Bun}_G}-\mathrm{mod})$ by first pulling sheaves on $\mathrm{Bun}_G$ back along $p^\mu$ to $\mathrm{Hecke}^\mu$ and then pushing them forward along $q^\mu$ to $\mathrm{Bun}_G \times C$

A Hecke eigensheaf is defined to be a sheaf which is something like an eigenvector under this operation ($\to$).

In formulas, we say $F \in D(\mathbf{D}_{\mathrm{Bun}_G}-\mathrm{mod})$ is a Hecke eigensheaf if with the above operation $H^\mu$ we have

(Again, I am not completely sure about my notes here. Apparently $d\mu$ denotes the dimension of the fiber of $p^\mu$.)

Now we can make point 2) of the above version of the geometric Langlands conjecture a little more precise. The conjecture is that the equivalence of categories $c$ in the first item of the conjecture is such that

is a Hecke eigensheaf for $V$ any point of $\mathrm{Bun}_G$ and $O_V$ its structure sheaf.

Now, why is this conjecture naïve? (“Naïve” is obviously relative here.) The answer is that the moduli space $\mathrm{Bun}_G$ is in general disconnected, while $\mathrm{Loc}_G$ is not. So the two categories appearing in the conjecture do not have any chance at all of being equivalent.

The first lecture ended with a sketch of how to remedy this problem.

Instead of using the moduli space $\mathrm{Loc}_G$, we should use the moduli stack $\mathcal{Lo}_G$ of $G$-local systems, or rather $\mathcal{Lo}^\mathrm{rs}_G$, that of regularly spable such systems.

It turns out that

is a gerbe, in fact a gerbe with band (“structure group”) $Z(G)$, the center of $G$. One finds that the derived category of coherent sheaves on $\mathcal{Lo}^\mathrm{rs}_G$ accordingly decomposes as

where $Z(G)^{*} = \Pi_1({}^L G)$.

(that’s the end of my notes for the moment)