3d duality with adjoint matter from 4d duality

We study the Seiberg dualities with an adjoint matter for the $U(N)$ and $SU(N)$ gauge groups in three- and four-dimensions with four supercharges. The relation between three- and four-dimensional dualities is investigated. We especially derive the three-dimensional duality from four-dimensional one by the dimensional reduction including the non-perturbative effect of the $\mathbb{S}^1$-compactification. In the $U(N)$ case, we obtain the Kim-Park duality, which is known as a generalization of the Aharony duality including an adjoint matter. In the $SU(N)$ case, we obtain the duality which follows from un-gauging the $U(N)$ Kim-Park duality.

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