## June 17, 2006

### Kapustin on SYM, Mirror Symmetry and Langlands, III

#### Posted by Urs Schreiber

The third part of the lecture.

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

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

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

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

(1)${\left({\omega }_{K}^{-1}F\right)}^{2}=-1\phantom{\rule{thinmathspace}{0ex}}.$

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 ${K}_{{M}_{\mathrm{stable}}\left(G,C\right)}^{1/2}$.

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

So

(2)$\left(\text{sheaf of open string operators}\right)\simeq D\left({M}_{\mathrm{stable}},{K}^{1/2}\right)\phantom{\rule{thinmathspace}{0ex}}.$

(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

(3)${W}_{R}\left(\gamma \right)=\mathrm{tr}\left(R\left({\mathrm{Hol}}_{\gamma }\left(A\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

The corresponding correlalator is

(4)$〈{W}_{R}\left(\gamma \right)=\int DA\mathrm{exp}\left(-S\left(A\right)\right){W}_{R}\left(A\right)〉\phantom{\rule{thinmathspace}{0ex}}.$

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

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

(5)${W}_{R}^{\mathrm{top}}\left(\gamma \right)=\mathrm{tr}\left(R\left({\mathrm{Hol}}_{\gamma }\left(A+i\varphi \right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

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 ${W}_{R}\left(P\right)$ tensors all sheaves with a particular vector bundle $R\left({\xi }_{p}\right)$, where ${\xi }_{p}$ is a ${G}_{ℂ}$-bundle which is a restriction of the universal bunde on ${M}_{\mathrm{flat}}\left({G}_{ℂ},C\right)×C$.

Hence on skyscraper sheaves on ${M}_{\mathrm{flat}}\left({G}_{ℂ},C\right)$ 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 ${H}_{R}\left(\gamma \right)$, which acts on $A$-branes on ${M}_{\mathrm{Hit}}\left({}^{L}G\right)$.

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 $\mathrm{exp}\left(\mathrm{ixn}\right)$, 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. )

Posted at June 17, 2006 10:51 AM UTC

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