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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 18th 2012

created stub for N=2 D=4 super Yang-Mills theory

… for the moment mainly such as to be able to make Coulomb branch and Higgs branch redirect to it, which are needed at symplectic duality

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 28th 2013

added a few more comments and pointers to the literature to N=2 D=4 super Yang-Mills theory on the construction of this theory by KK-compactification of the 6d theory on the M5-brane on a Riemann surface.

(prompted by this MO question which unfortunately was put “on hold” before I could get to it…)

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJun 28th 2013

Urs, if you know what Irina was trying to ask and can put it in standard mathematical language, I’d be happy to edit the question to get it “off hold”.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 28th 2013
• (edited Jun 28th 2013)

Hi Todd,

ah, I see, didn’t know that you have the power to do this.

This here is what Irina is trying to ask (in my paraphrase, of course :-):

There is a famous construction of $(N=2)$-supersymmetric 4-dimensional Yang-Mills field theories and their Seiberg-Witten theory from the $N=(2,0)$-superconformal 6-dimensional field theory on the worldvolume of M5-branes: by Kaluza-Klein-compactifying the latter on a Riemann surface. This construction was revived more recently in 2009 by the influential article

On page 22 of this article, around the displayed formula (2.27), the author mentions that the Kaluza-Klein compactification of the 6d theory on a Riemann surface involves a “well known twisting procedure” of the holonomy of the Riemann surface by choosing an $SO(2)$-subgroup of the $SO(5)$ group that is the “R-symmetry” group of the 6-dimensional supersymmetric field theory (the group under which its supercharges transform).

My question is: what is this “well known twisting procedure” exactly, and how does it work? Of course I know how to find $SO(2)$-subgroups of $SO(5)$, but what does such a choice amount to in the context of the construction of an $N=2$, $D=4$ SYM from the 6d-field theory on the 5-brane?

I think that’s the question. And the first approximation to an answer is probably simply this:

The twisting procedure referred to in that article was introduced and described earlier in section 3.2.1 of

(though one might say that it is not really pedagogically explained much there, either…)

These and other basics of the modern understanding of Seiberg-Witten theory from compactification of 5-branes can be found briefly surveyed in section 7 of the lecture notes

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 30th 2013
• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 30th 2013

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 19th 2018

Thought I’d separate Higgs branch from Coulomb branch. I hope I haven’t said anything wrong.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeFeb 19th 2018
• (edited Feb 19th 2018)

Thanks. I have added some missing hyperlinks. We should now complement the information in the two entries, currently either contains some points that concern both but are not stated in the other entry.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 19th 2018

There must be 100s of things to add. I’ll add one which suggest some interesting duality:

• Ben Webster, Koszul duality between Higgs and Coulomb categories $\mathcal{O}$, ( arXiv:1611.06541)
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeFeb 19th 2018

I meant just the little general information that the entries give should be the same for them. I have complemented it, now it reads for both as follows:

For N=2 D=4 super Yang-Mills theory the moduli space of vacuum expectation values (VEVs) of the theory is locally a Cartesian product between the space of moduli of the vector multiplet (the gauge field sector) and those of the hypermultiplet (the matter field sector). The former is called the Coulomb branch, and the latter the _Higgs branch. These are dual to each other to it under a version of mirror symmetry .

This is the topic of Seiberg-Witten theory.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 20th 2018

Since many of the references there mention it, I thought I’d add

Definitions of the Coulomb and Higgs branches have been extended to N=4 D=3 super Yang-Mills theory.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJul 22nd 2019
• (edited Jul 22nd 2019)

renamed entry to match more systematic naming convention

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeAug 8th 2021
• (edited Aug 8th 2021)