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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 stable(G,C) 1 /2 ) on M stable(G,C) correspond to A-branes on M Hit(G,C).

Holomorphic differential operators on M stable(G,C) can be viewed () as a deformation of the algebra of holomorphic functions on T *M stable(G,C)M Hit(G,C).

The key for establishing this relation is to realize that on M Hit(G,C) there exists a canonical coisotropic A-brane.

Namely, assume that our A-brane is the whole of M Hit(G,C). Given a bundle L, with connection F=i 2 , the condition that this is an A-brane is

(1)(ω K 1 F) 2 =1 .

For the Hitchin moduli space there is a canonical choice, namely F=ω 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 stable(G,C) 1 /2 .

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


(2)(sheaf of open string operators)D(M stable,K 1 /2 ).

(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 (, )).

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 γ be some loop in target space, let R be a representation of the gauge group and set

(3)W R(γ)=tr(R(Hol γ(A))).

The corresponding correlalator is

(4)W R(γ)=DAexp(S(A))W R(A).

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 AA+wϕ and to define a topological Wislon loop as

(5)W R top(γ)=tr(R(Hol γ(A+iϕ))).

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(ξ p), where ξ p is a G -bundle which is a restriction of the universal bunde on M flat(G ,C)×C.

Hence on skyscraper sheaves on M flat(G ,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 () 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(γ), which acts on A-branes on M Hit(G L).

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 (). Skyscraper sheaves corespond to delta-functions, Hecke eigensheaves to exponential functions exp(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 (), 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 Σ 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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Tracked: December 12, 2006 9:09 PM

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