Anton Kapustin (Caltech) is visiting Harvard. Much like Edward Witten, he is thinking about the Langlands "program" - with a focus that is arguably more physics-oriented (i.e. S-duality-oriented) than the approach of Edward Witten.

Anton has answered many questions I had about S-duality, for example:

When you study the operators that are S-dual to given operators, why aren't you just satisfied with saying that the dual of the Wilson loop is the 't Hooft loop, among other examples?

- Wilson loop is a trace of the holonomy over a representation, and therefore the independent loops are labeled by irreducible representations of the gauge group. The dual label is less transparent for the 't Hooft loop, and similar subtleties need a more detailed treatment.

What is the S-dual theory of 3+1-dimensional maximally supersymmetric Yang-Mills theories with exceptional gauge groups?

- All exceptional groups are self-dual under S-duality, much like U(N) and SO(2N). SU(N) is dual to SU(N) / Z_N while SO(2N+1) is dual to USp(2N).

In "Baryons and AdS-CFT", Witten argued that nonperturbatively, there are two different USp(2N) gauge theories. How is the subtlety reflected in the S-dual description?

- The two USp(2N) theories are actually connected with each other. They can be described as one theory whose theta angle differs by one half of the periodicity. You can imagine that their Re(tau) differs by one in a context where the natural periodicity of Re(tau) is two instead of one.

Is there a stringy realization of all exceptional groups, and a geometric realization of their S-dualities?

- The gauge theories can be obtained as a (2,0) theory on a two-torus, and the (2,0) theories can be constructed as a decoupled limit of type IIB on an ADE singularity. This gives all simply laced groups, and the other groups may be obtained by orbifolding the Dynkin diagram - which may be achieved by having an extra circle whose holonomy is the outer automorphism. The picture has been explained by Vafa and allows one to construct G_2 as an orbifold of SO(8) by the triality symmetry, F_4 as an orbifold of E_6 by the reflection symmetry, SO(2N+1) from SO(2N+2), and USp(2N) from U(2N)

Is the S-duality group always SL(2,Z), inhereted from type IIB?

- No! For example, for SO(2N+1) and USp(2N), you generate the group by "tau goes to tau+1" and "tau goes to -1/(q tau)" where q is an integer - either 2 or 3. This generates a group whose entries are combinations of rational numbers and rational multiples of sqrt(q), and this group - that carries Hecke's name - is not a subgroup of SL(2,Z) even though it is inside SL(2,R).

How can you prove such a duality geometrically?

- You can't really see it from a geometric action on a two-torus but you may nevertheless illuminate such a group in terms of a T-duality. (Your humble correspondent did not understand the exact details how these exotic groups may occur as the T-duality groups.)

Should not there still be an SL(2,Z) inhereted from type IIB?

- In some sense, yes. There are subgroups of SL(2,Z) that act as transformations that do not change the gauge group. Note that inverting "tau" gives a different group in general. If you require the gauge group to be preserved, you obtain a smaller group - the intersection of SL(2,Z) with the Hecke group which is something like the Gamma(2) group.

Can you apply these things to less supersymmetric gauge theories?

- Yes, but the details are different. N=2 Seiberg-Witten is very different from N=1. I studied in what sense the N=1 Seiberg dualities are "electromagnetic" dualities or S-dualities because the answer is not obvious due to the fact that you must flow to the infrared before the Seiberg duality becomes fully valid. The coupling runs and you're trading two Lambda scales instead.

Is the gauge group an inherent property of the theory? Do you agree with Seiberg that it's just a redundancy that is moreover not uniquely determined?

- Partially. But in some sense, with a given choice of natural operators, the theory knows about the gauge group. For example, its Wilson lines are classified by the irreducible representations of the gauge group.

Yes, but there also exist operators that are arguably classified by the representations of other groups that can be used as gauge groups in a dual description, right?

- Yes but the explicit construction of these guys is non-trivial.

How many pages the paper of EW about these issues is going to have?

- Around 300-400.

Is it a paper then?

- No.

So what is it?

- [This answer is secret and cannot be revealed to the readers.]

## snail feedback (2) :

Dear Lubos,

thank you very much for pompous informations.

CY-fold is manifold X with a Riemannian metric, satisfying 3 conditions:

1. X is a complex manifold.

2. X is Kahler.

3. X is Ricci-flat.

Please what is relevance of

1. generalized complex manifold

2. non-Kahler manifold

3. non-Ricci-flat manifold

for (topological) string theory ?

Please are these manifolds S-dual to the honest CY-folds ?

Thank you.

Yours sincerely planckeon

Dear Lubos,

thank you very much for pompous informations.

CY-fold is manifold X with a Riemannian metric, satisfying 3 conditions:

1. X is a complex manifold.

2. X is Kahler.

3. X is Ricci-flat.

Please what is relevance of

1. generalized complex manifold

2. non-Kahler manifold

3. non-Ricci-flat manifold

for (topological) string theory ?

Please are these manifolds S (U)-dual to the honest CY-folds ?

Thank you.

Yours sincerely planckeon

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