Three-Dimensional Supersymmetric Gauge Theories and Seiberg Duality

We study the three-dimensional supersymmetric gauge theories, their quantum dynamics and their dualities which connect very different-looking theories. These three-dimensional dualities are very similar to the four-dimensional ones. Especially we discuss how the 3d dualities appear from the four dimensional (Seiberg) ones by the dimensional reduction. Although the four-dimensional duality can generically include various matter fields like the adjoint or the tensor matters, there are a few examples in 3d. We especially elaborate on the 3d duality with an adjoint matter. In deriving the 3d duality from 4d we need to carefully treat the monopole operators which appear in the highand low-energy theories and need to carefully analyze the Coulomb branch of the moduli space of vacua, which is absent in 4d physics. This reduction from 4d is done by the introduction of the effective superpotential induced by the Kaluza-Klein (or being known as the twisted instantons) monopoles. We will give the KK-monopoles and the corresponding potentials with the various matter fields included.