# Infinitely many $$\mathcal{N}=1$$ dualities from m + 1 − m = 1

Journal of High Energy Physics, Oct 2015

We discuss two infinite classes of 4d supersymmetric theories, T N (m) and $${\mathcal{U}}_N^{(m)}$$, labelled by an arbitrary non-negative integer, m. The T N (m) theory arises from the 6d, A N − 1 type $$\mathcal{N}=\left(2,0\right)$$ theory reduced on a 3-punctured sphere, with normal bundle given by line bundles of degree (m + 1, −m); the m = 0 case is the $$\mathcal{N}=2$$ supersymmetric T N theory. The novelty is the negative-degree line bundle. The $${\mathcal{U}}_N^{(m)}$$ theories likewise arise from the 6d $$\mathcal{N}=\left(2,0\right)$$ theory on a 4-punctured sphere, and can be regarded as gluing together two (partially Higgsed) T N (m) theories. The T N (m) and $${\mathcal{U}}_N^{(m)}$$ theories can be represented, in various duality frames, as quiver gauge theories, built from T N components via gauging and nilpotent Higgsing. We analyze the RG flow of the $${\mathcal{U}}_N^{(m)}$$ theories, and find that, for all integer m > 0, they end up at the same IR SCFT as SU(N) SQCD with 2N flavors and quartic superpotential. The $${\mathcal{U}}_N^{(m)}$$ theories can thus be regarded as an infinite set of UV completions, dual to SQCD with N f = 2N c . The $${\mathcal{U}}_N^{(m)}$$ duals have different duality frame quiver representations, with 2m + 1 gauge nodes.

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Prarit Agarwal, Kenneth Intriligator, Jaewon Song. Infinitely many $$\mathcal{N}=1$$ dualities from m + 1 − m = 1, Journal of High Energy Physics, 2015, 35, DOI: 10.1007/JHEP10(2015)035