The n-Category Café

In reading the following, you should be well aware that, while some detailed formulas do appear, chances are high that various imperfections and imprecise statements appear. I am not an expert on the following material. I’ll try to reproduce the transcript of the talk as well as I can, but that’s all.

As usual, some personal comments I include are set in italics.

So, these are the notes I have taken:

This is the first of two talks on aspects of the geometric Langlands duality:

1) Perspective of Integrable Models

2) Representation Theoretic Perspective and connection to CFT

Here is part 1).

The geometric Langlands duality conjecture states (see T. Pantev’s lecture I, II for details on that statement) that there is an equivalence

In words, this means, roughly, something like: given a complex curve $X$ and a suitable Lie group $G$, then gadgets on the space of all $G$-bundles on $X$ that differential operators may act on are in correspondence with vector bundles on the space of flat ${}^L G$-bundles with connection on $X$, where ${}^L G$ is the Langlands dual group to $G$.

Assuming the genus of $X$ is

one finds that the dimension of the space $\mathrm{Bun}_G(X)$ is

Now, and that’s what this talk will be about, the special $\mathcal{D}$-modules on the right side of the Langlands duality conjecture can be understood in terms of certain eigenstates of a certain integrable system, namely

The Hitchin System

Consider the cotangent space

of the space of $G$-bundles on $X$. Being a cotangent bundle, it carries a canonical 1-form. The 2-form differential of that defines a Poisson structure on the space of functions on $T^* \mathrm{Bun}_G$.

Fact i): this structure is classically integrable.

This means that there exists functions

that pairwise Poisson-commute

If we think of $\mathrm{Bun}_G$ as a configuration space of some classical physical system, then $T^* \mathrm{Bun}_G(X)$ would be the corresponding phase space and the $h_r$ would be a set of commuting “Hamiltonians” or of one Hamiltonian and a couple of mutually commuting “conserved charges”.

Fact ii): the surface of constant $\{h_r\}_r$ is almost always a torus

Next, we consider quantizing this classical setup. This proceeds essentially by geometric quantization.

(The statement was that what we do is not precisely the same as geometric quantization, but I can’t tell the difference right now.)

So, we find the canonical line bundle on $T^* \mathrm{Bun}_G(X)$ given by our symplectic form that defines the Poisson structure and we assume we can form a square root bundle of that. A polarization is chosen and this bundle descends to a bundle on $\mathrm{Bun}_G(X)$ itself.

Quantization sends the functions $\{h_r\}_r$ to operators

on the space of (suitably well-behaved) sections of that line bundle on $\mathrm{Bun}_G(X)$.

The next theorem says that we can find a quantization such that even the quantum system is integrable.

Theorem (Beilinson & Drinfeld) There exist differential operators

on sections of that line bundle such that

and such that the symbol of $H_r$ is $h_r$.

Given any such quantum integrable system we can pose the following spectral problem:

for which constants

can we find a section $\psi$ such that

for all $r$ ?

At least morally, all $\psi$ of this form are related to the Hecke eigensheaves appearing in the geometric Langlands duality.

The following theorem gives a necessary and sufficient condition for when such $\psi$ exist. The remainder of the talk will be concerned essentially with understanding what the following statement really means.

Theorem (Beilinson-Drinfeld): there exist solutions to the above spectral problem if and only if the $\mu_r$ are the $\chi_r$-character of a “globally defined” $\mathrm{Lie}({}^L G)$-oper.

Next we define what an “oper” is and what that notion of character might mean.

Let $\Omega$ be the canonical line bundle on our complex curve $X$.

Definition: An $\mathrm{sl}_n$-oper is an $n$-th order holomorphic differential operator

locally of the form

It’s called a $\mathrm{sl}(n)$-oper because by linearizing this degree $n$-differential operator we get an operator

where $A$ is the $n\times n$-matrix

which, due to the tracelessness, we can regard as taking values in $\mathrm{sl}(n)$.

(now I am being lazy and skip reproducing a little example, which, personally, I did not find all that illuminating)

Given any such “oper”, we can think of its linearized version as above, regard that as the covariant derivative of a connection and consider the monodromies of that connection. The following statement relates these monodromies to solutions of our spectral problem:

Proposition (E. Frenkel): a necessary condition for the existence of solutions $\psi$ to

for all $r$ is that the $\mathrm{sl}(2)$-oper of the form

with

for some set of integers $\{\lambda_r\}$ has trivial monodromy.

How do we find sets $\{\lambda_r\}$ such that the monodromy of the associated oper vanishes?

Answer (E. Frenkel): the $q(z)$ with this property can always be brought into the form

where

and where, finally, the $w_s$ are solutions of the famous Bethe ansatz equation

(There is no guarantee that I correctly reproduced these formulas. But I try my best.)

These Bethe Ansatz equations are close to the heart of those field theorists and string theorists working on integrability of $N=4$ super Yang-Mills and on AdS/CFT. There it is all about solving these equations.

The idea is that the ideals of sections generated by the kernel of $(H_r - \mu_r)$, which form a $\mathcal{D}$-module, i.e. a module for the algebra of differential operators on $\mathrm{Bun}_G(X)$ are closely related to the automorphic $\mathcal{D}$-modules, otherwise known as Hecke eigensheaves.

One question after the talk was: “So, what is that curious character $\rho \mapsto \chi_r(\rho)$ that appeared in the Beilinson-Drinfeld theorem?” Answer: “that’s implictly given by that proposition by E. Frenkel at the end”.