Between Symmetry and Duality in Supersymmetric Quantum Field Theories

We study two cases of interrelations between the enhancement of symmetries in the infrared (IR) and duality properties of supersymmetric quantum field theories in four dimensions. First, we discuss an $SU(2)$ $\mathcal{N}=1$ model with four flavors, singlet fields, and a superpotential. We show that this model flows to a conformal field theory with $E_6\times U(1)$ global symmetry. The enhancement of the flavor symmetry follows from Seiberg duality. The second example is concerned with an $SU(4)$ gauge theory with matter in the fundamental and antisymmetric representations. We argue that this model has enhanced $SO(12)$ symmetry in the IR, and, guided by this enhancement, we deduce a new IR duality.

Figure 1

Model with $E_6\times U(1)$ global symmetry. The cross on the edges denotes the fields $M_A$ and $M_B$. These are flipping the baryonic operators constructed from the fields associated with the corresponding edge. Flipping chiral operator $O$ means introducing chiral field ${\varphi }_O$ and coupling it as ${\varphi }_{O}O$.

Figure 2

An equivalent way to represent the theory of Fig. 1.

Figure 3

Model obtained upon deforming the quiver by admixing $M_B$ to the superpotential.

Figure 4

The model on the left corresponds to compactification of a rank 1 $E$ string on a torus with a particular flux for the global symmetry. Giving vacuum expectation values to the flip fields, one obtains the model in the middle, which is equivalent under Seiberg duality to the third model.