Abstract
We review some of the properties of 3d N=4 theories obtained by dimensionally reducing theories of class S. We study 3d partition functions, and certain limits thereof, for such theories, and the properties implied for these by 3d mirror symmetry.
Highlights
We review some of the properties of 3d N = 4 theories obtained by dimensionally reducing theories of class S
We study 3d partition functions, and certain limits thereof, for such theories, and the properties implied for these by 3d mirror symmetry
We discuss 3d partition functions making full use of N = 4 supersymmetry and in particular study some of the interesting limits these partition functions possess
Summary
There are several interesting limits of the 3d partition functions which one can discuss. Let us discuss some of these partition functions and limits in some simple examples, starting with N = 4 U(1) SYM with one charge 1 hypermultiplet, which we denote by SQED1 This exhibits the most basic example of N = 4 mirror symmetry, being dual to a free (twisted) hypermultiplet. In the case N = 1, this reduces to the duality above, and can be thought of as a mirror symmetry, but for general N , ordinary (as opposed to twisted) vector- and hypermultiplets appear in both the U(N ) and U(N − 1) gauge theories, so this is not a mirror symmetry We can see this explicitly by studying the index. In the Higgs limit, the index of a U(Nc) theory with Nf flavors is: IUH(Nc)Nf (n, μa; x) This can be computed by summing the finitely many poles that lie inside the unit circle.
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