Following the conversation here, I am wondering if anyone knows of an example of dual pair with 4-dimensional N=1 SUSY which relates a non-Abelian gauge theory on one side to a theory with a Lagrangian description but no non-trivial gauge group. Cannot think of one off the top of my head, which doesn't mean it does not exist or even is well-known...
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This feels a little trivial, but I don't see why it isn't an example of what you want: Seiberg duality typically relates an $SU(N_c$) gauge theory with $N_f$ flavors to an $SU(N_f - N_c$) gauge theory. There are degenerate cases when $N_f - N_c = 1$ or $0$, which don't correspond to any dynamical gauge group in the infrared. These are usually described in terms of quantum moduli spaces (s-confining when $N_f = N_c + 1$ and chiral symmetry breaking when $N_f = N_c$), with the low energy fields given by mesons and baryons, but you can equally well describe these as the "dual" quarks and mesons of the usual Seiberg duality, in a degenerate limit without gauge fields coupling to them.
Of course, in the same sense, nonsupersymmetric QCD is dual to a theory of massless pions and no gauge symmetry.
answered Sep 24, 2011 at 4:14
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$\begingroup$ If you mean theories that are dual not just in the sense of having identical IR physics but in the stronger sense of having the same physics along an entire RG flow, then I can't think of an example where one is a gauge theory and the other isn't. $\endgroup$ Commented Sep 24, 2011 at 4:16
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$\begingroup$ Yes, I guess duality relevant to that discussion would have to be something stronger than just IR duality. Anyhow, thanks for your answer. $\endgroup$– user566Commented Sep 24, 2011 at 4:21