What is $\mathcal{N}=2$ QED?
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When people talk about N=2N=2 QED in 4d I think they normally mean a U(1)U(1)gauge theory (one N=2N=2 vector multiplet) coupled to one or more hypermultiplets (usually all with the same U(1)U(1) charge). As an example of this usage see Witten's discussion of N=4N=4QED in 3d (which can be obtained by dimensional reduction from the N=4N=44d theory) in this paper.
Similarly N=1N=1 QED consists of a (N=1N=1) vector multiplet coupled to chiral multiplets. To get from this to the N=2N=2theory you need to pick an odd number of chiral multiplets. One chiral multiplet needs to be neutral under the U(1)U(1) gauge symmetry (this will combine with the N=1N=1 to form the N=2N=2 vector multiplet), and the other ones should be paired up so the half of them have U(1)U(1)charge, eg., +1+1, and the other ones have charge −1−1. Finally you need to add a cubic superpotential with the correct coefficient. See, for example, section 12.5 in the review by Sohnius.
Similarly N=1N=1 QED consists of a (N=1N=1) vector multiplet coupled to chiral multiplets. To get from this to the N=2N=2theory you need to pick an odd number of chiral multiplets. One chiral multiplet needs to be neutral under the U(1)U(1) gauge symmetry (this will combine with the N=1N=1 to form the N=2N=2 vector multiplet), and the other ones should be paired up so the half of them have U(1)U(1)charge, eg., +1+1, and the other ones have charge −1−1. Finally you need to add a cubic superpotential with the correct coefficient. See, for example, section 12.5 in the review by Sohnius.
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