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

We study the low-energy dynamics in three-dimensional N = 2 exceptional gauge theories with matters in a fundamental representation, especially focusing on confinement phases and on a quantum structure of the Coulomb branch in the moduli space of vacua. We argue that the confinement phases of these exceptional gauge theories have a single Coulomb branch. The 3d s-confinement phases for the exceptional gauge groups are associated with quantum-deformed moduli spaces of the corresponding 4d N = 1 exceptional gauge theories. * nii@itp.unibe.ch

Motivated by a recent paper by Rychkov-Tan [1], we calculate the anomalous dimensions of the composite operators at the leading order in various models including a φ 3 -theory in (6 − ) dimensions. The method presented here relies only on the classical equation of motion and the conformal symmetry. In case that only the leading expressions of the critical exponents are of interest, it is sufficient to reduce the multiplet recombination discussed in [1] to the classical equation of motion. We claim that in many cases the use of the classical equations of motion and the CFT constraint on two-and three-point functions completely determine the leading behavior of the anomalous dimensions at the Wilson-Fisher fixed point without any input of the Feynman diagrammatic calculation. The method developed here is closely related to the one presented in [1] but based on a more perturbative point of view.

We present various confinement phases in three-dimensional N = 2 Spin(N) gauge theories with vector and spinor matters. The quantum Coulomb branch in the moduli space of vacua is drastically modified when the rank of the gauge group and the matter contents are changed. In many examples, the Coulomb branch is one-or twodimensional but its interpretation varies. In some examples, the Coulomb branch becomes three-dimensional and we need to introduce a "dressed" Coulomb branch operator.

We study three-dimensional N = 2 Spin(7) gauge theories with N S spinorial matters and with N f vectorial matters. The quantum Coulomb branch on the moduli space of vacua is one-or two-dimensional depending on the matter contents. For particular values of (N f , N S ), we find s-confinement phases and derive exact superpotentials. The 3d dynamics of Spin (7) is connected to the 4d dynamics via KK-monopoles. Along the Higgs branch of the Spin(7) theories, we obtain 3d N = 2 G 2 or SU(4) theories and some of them lead to new s-confinement phases. As a check of our analysis we compute superconformal indices for these theories.