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Context: I am a second-year (undergraduate) physics major applying for a summer research position. The investigator is working on Quiver Gauge Theories and in response to my inquiry email he told me to

"compute the HS for the Coulomb branch of 3D $\mathcal{N}=4$ $U(1)$ gauge theory with $n$ flavors."

Assuming HS is short for Hilbert Series, I think I am approaching an understanding of what the problem is asking of me, however I cannot seem to find any concrete description of what $\mathcal{N}$ refers to. Help?

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  • $\begingroup$ Second-year undergrad? Or second-year grad student? $\endgroup$
    – G. Smith
    Commented Jan 14, 2019 at 3:45
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    $\begingroup$ See en.wikipedia.org/wiki/N_%3D_4_supersymmetric_Yang–Mills_theory for an explanation of N. I would run away as fast as I could from this investigator if he or she is not going to spend time with you to get you up to speed. $\endgroup$
    – G. Smith
    Commented Jan 14, 2019 at 3:56
  • $\begingroup$ @G.Smith I'm an undergrad. The position description stated that students of all levels of knowledge are encouraged to apply, but willingness to learn is a must; therefore I suspect that this is just some test to see if I am willing to tackle a problem with which I am unfamiliar. At least, that's what I hope. $\endgroup$
    – Azimuth Zero
    Commented Jan 14, 2019 at 4:07
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    $\begingroup$ Not to rain on your parade, but 95% of undergrad research projects lead to nothing because the undergrad jumps to a super advanced topic. I’ve seen this play out a hundred times. If you don’t already know quantum field theory, it’s not going to be a productive summer, for either learning or research. If you want to learn, it would be much much more efficient to read a textbook instead. If you want to see what research is like, you could try a topic with fewer prereqs. $\endgroup$
    – knzhou
    Commented Jan 14, 2019 at 10:42
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/107327/2451 , physics.stackexchange.com/q/637204/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Aug 3 at 14:09

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In a supersymmetric gauge theory, N is the rank of the R-symmetry group. If N is 4, it means that the theory is invariant under four different but related supersymmetry transformations.

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  • $\begingroup$ I am out of my area of expertise here. If I got this wrong, I hope an expert will correct me. $\endgroup$
    – G. Smith
    Commented Jan 14, 2019 at 5:40
  • $\begingroup$ Indeed, N refers to the "flavor" of the supersymmetry generators, so, the the number of independent such. $\endgroup$
    – Cosmas Zachos
    Commented Aug 3 at 15:12

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