### A Little Bit of Geometric Langlands: Relation to Integrable Systems

#### Posted by Urs Schreiber

As I mentioned in Navigating Geometric Langlands by Analogies, the mathematicians and the string theorists in Hamburg have a a small series of lectures this term, where we try to explain to each other some tiny fraction of what *Langlands duality* is about.

Last time we had something on the “classical” number-theoretic aspect. I didn’t even try to report on that.

This time, Jörg Teschner spoke about *geometric* Langlands duality. After briefly mentioning what the main statement is, he concentrated on understanding one aspect of this statement using the language of *integrable systems*.

Ever since it was found that certain aspects of $N=4$ super Yang-Mills theory are governed by what are called quantum integrable systems, and due to the impact this has on understanding and testing the AdS/CFT duality, string theorists have been interested in integrable systems. That’s one reason why Jörg Teschner decided to emphasize this aspect of the talk.

Roughly, the main point he tried to make could maybe be summarized like this:

The geometric Langlands duality relates (conjecturally) two different derived categories by an equivalence.

But on top of that, the conjecture states that both these derived categories have something like a nice “basis of eigenstates of some operator” (compare my previous entry # on what such a statement would really mean) and that under the equivalence a basis vector on one side is sent to a basis vector on the other.

Now, on one of these two sides, those “basis vectors” (the *Hecke eigensheaves*) can be understood as coming from *common eigenstates of the set of commuting Hamiltonians of some integrable system*.

From this point of view of integrable systems, geometric Langlands duality seems to be a statement about when and how an integrable system admits a *separation of variables*.

If you like integrable systems, that should be interesting.

*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”.*

## Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

In the geometric quantization (or not quite) that you’re doing here, the symplectic form was exact, so the relevant line bundle was trivial… right?

I’m not sure where the square root came in (maybe because I’m used to Dolbeault rather than Spin

^{c}quantization), but anyway I hope that the square root chosen was also trivial!The equivalence

irreducible

^{L}G local systems on X ↔ 𝒟−modules on Bun_{G}(X)is best understood in the case X a punctured formal disc, more precisely Spf C[[z]][z

^{-1}], and is called the geometric Satake correspondence. Since I don’t remember having seen it discussed on this blog, I’ll mention the basics of this.In this case Bun

_{G}(X) is G((z))/G[[z]], containing (and homotopy equivalent to) the space of smooth based loops into the maximal compact subgroup of G. Bott studied this “loop Grassmannian” 50 years ago, and it’s still giving us great mathematics. The finite-dimensional Morse-Bott strata on this give a set of D-modules that I’m pretty sure are the relevant basis on the right side.The left side is just the representation category of the Langlands dual group, whose basis is the irreducible representations.

So, for example, you can get each irrep of the Langlands dual as the intersection homology of the closure of the corresponding Morse-Bott stratum.

Some limited references: Ginzburg’s mysteriously unpublished paper, Mirkovic-Vilonen, more recent developments in Anderson and Kamnitzer.