The String Coffee Table

Recall that we said that $D$-modules (modules over the algebra of differential operators on holomorphic sections of $K_{M_{\mathrm{stable}}(G,C)}^{1/2}$) on $M_{\mathrm{stable}}(G,C)$ correspond to $A$-branes on $M_{\mathrm{Hit}}(G,C)$.

Holomorphic differential operators on $M_{\mathrm{stable}}(G,C)$ can be viewed ($\to$) as a deformation of the algebra of holomorphic functions on $T^{*}{M}_{\mathrm{stable}}(G,C)\subset M_{\mathrm{Hit}}(G,C)$.

The key for establishing this relation is to realize that on $M_{\mathrm{Hit}}(G,C)$ there exists a canonical coisotropic $A$-brane.

Namely, assume that our $A$-brane is the whole of $M_{\mathrm{Hit}}(G,C)$. Given a bundle $L$, with connection $F=i{\nabla }^2$, the condition that this is an $A$-brane is

For the Hitchin moduli space there is a canonical choice, namely $F={\omega }_J$. It follows that the class of $F$ is trivial, so that we have a trivial bundle with connection.

The endomorphism algebra of the $A$-brane is the algebra of holomorphic differential operators on sections of $K_{M_{\mathrm{stable}}(G,C)}^{1/2}$.

We can think of this as the algebra of open string states, the algebra of boundary observables.

So

(personal remark: hence the category of $D$-modules is the category of modules for the algebra of open string states. Notice how this nicely harmonizes with the general interpretation of module categories as D-branes ($\to$, $\to$)).

Now, how are the Hecke operators realized in the TQFT setup?

(The following is pretty sketchy. See section 6 of the Witten-Kapustin paper for the details.)

1) The most obvious observable in a gauge theory is the Wilson loop operator.

Let $\gamma$ be some loop in target space, let $R$ be a representation of the gauge group and set

The corresponding correlalator is

However, in the present setup this observable is not BRST inavriant (except for $t=\pm i$, corresponding to the $B$-model), hence needs to be slightly modified.

The trick is to replace $A\mapsto A+w\varphi$ and to define a topological Wislon loop as

This is BRST invariant.

Next comes the obvious discussion in terms of pictures of how a loop operator in four dimensions gives rise to a point operator or a loop operator after compactification to 2D, depending on whether the loop wraps around the compactified space or not. I will not try to reproduce this. See section 6 of Witten-Kapustin.

The important point is that acting with $W_R(P)$ tensors all sheaves with a particular vector bundle $R({\xi }_p)$, where ${\xi }_p$ is a $G_{ℂ}$-bundle which is a restriction of the universal bunde on $M_{\mathrm{flat}}(G_{ℂ},C)\times C$.

Hence on skyscraper sheaves on $M_{\mathrm{flat}}(G_{ℂ},C)$ the operator acts by tensoring with a fixed vector space (the fiber of the above bundle over the point at which the skyscraper sheaf is concentrated.)

So skyscraper sheaves are eigen-objects ($\to$) of the Wilson loop operator.

It follows that the S-duals of the skyscraper sheaves should be eigen-objects of the S-duals to the Wilson operators.

In particular, it follows that fibers of the Hitchin fibration must be eigenobjects of some operation $H_R(\gamma )$, which acts on $A$-branes on $M_{\mathrm{Hit}}({}^{L}G)$.

This turn out to be ‘t Hooft operators and Hecke eigensheaves.

(personal remark: what is going on is something like a categorification of Fourier tranformations ($\to$). Skyscraper sheaves corespond to delta-functions, Hecke eigensheaves to exponential functions $\exp (\mathrm{ixn})$, Wilson operators to multiplication operators, ‘t Hoofs operators to differential operators and S-duality/Langlands duality to a categoified Fourier transformation exchanging these. Everything is performed in a generalization of Kapranov-Voevodsky 2-vector spaces ($\to$), where numbers are replaced by vector spaces, products by tensor products and sums by direct sums.

However, this is not to be confused with some remarks Kaparanov made a couple of years agon, saying that for doing Langlands on spaces $\Sigma$ of diemsnion larger than 2 will amount to replacing moduli spaces of vector bundles with moduli spaces of 2-vector bundles. This is already one categorification step further than I am referring to, here. )