On the Coulomb branch of N=2 supersymmetric gauge theory

Question

The chiral ring of the Coulomb branch of a 4d N=2 supersymmetric gauge theory is given by the Casimirs of the vector multiplet scalars, and they don't have non-trivial relations; the Casimirs are always independent.

Also in Gaiotto's class of N=2 non-Lagrangian theories, the chiral ring of the Coulomb branch doesn't (seem to) have relations.

Is it a general fact? If so, how can we deduce it from the N=2 susy algebras?

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I was asked to clarify the definition of the Coulomb branch in non-Lagrangian theories; let's define them for N=2 SCFT by the fact that $SU(2)_R$ symmetry acts on the Coulomb branch operators trivially.

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Comments

Apologies for this possibly diversionary comment - can you recommend some reference starting point (hopefully pedagogical!) for this topic for beginning students? I have seen some papers and had worked on a project involving the calculation of the chiral ring but could never get my hands on a very clear exposition of the topic - it was quite hazy to pick it up from cutting edge papers!

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Which topic are you referring to? Chiral rings? This paper http://arxiv.org/abs/hep-th/0211170 contains a nice review. In general, read and understand all papers by Seiberg or by Witten. Problem solved.

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Thanks a lot for the reference. "All papers" by Seiberg and Witten is almost sounding like an hyperbole :) Can you give a more practical advice - like from which papers to start for getting a grasp of the background of what you are asking here? And what kind of pre-requisites would be required? And how much time should it take - like how fast should one be able to work through any typical paper that you have in mind? I really don't understand how to read these papers! Should I read them like I try to read the volumes by Weinberg - line by line working out every line?

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Once you finish a basic QFT textbook and a SUSY textbook, just pick whatever recent paper which motivates you most, and try to understand it. The required materials are either in the review sections or in the references in the paper. Going through it line-by-line won't work, because the author didn't intend the paper to be read that way. Rather, try to work out an example which is slightly different from what's dealt in the paper. That way, you'll learn exactly which tools are necessary, which part of the paper can be improved, and once done, it might result in your paper!

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Thanks a lot for this valuable advice! Can you say something more on how much of QFT and SUSY do you consider "basic"? That will help me judge my current level of understanding of these two topics. Its totally surprising to me when you say, "...going through it line-by-line won't work, because the author didn't intend the paper to be read that way...". So you do seem to say that there is a tremendous difference in how textbooks (like Weinberg's) have to be read and how such papers are read!

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There's a reason why textbooks and papers are called differently:) I say, you should just try reading a paper which interests you most, using various references. If you can, then you're ready; if you can't, then you're not.

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My inexperience speaks! But the metric of measurement you are suggesting is not very definitive. If I pick up that paper on chiral rings that you linked to then I am quite sure that much of it will be beyond me though I have done courses and projects in what I might think of as "basic" QFT and SUSY. The question is I want to know as to how much of knowledge and by when (years in grad school) is considered cutting edge?

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More specifically can you give some canonical references to read which explain how superpotentials are calculated (exact?) and how is it that often people seem to be almost magically be able to read off the beta-functions by looking at it.

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Well, anything which was done in the last century should be considered basic. The SUSY textbook by Terning covers basic stuffs pretty well.

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Thanks for the comments. I am not sure how to go about learning everything "done in the last century" - that sounds scary! But may be I can try reading Terning - though that is a very terse book!

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