Gauging Discrete Symmetries of $T_N$-theories in Five Dimensions

Abstract

We study the gauging of a discrete ${\mathbb{Z}}_3$ symmetry in the five-dimensional superconformal $T_N$ theories. We argue that this leads to an infinite sequence of five-dimensional superconformal theories with either $E_6\times SU(N)$ or $SU(3)\times SU(N)$ global symmetry group. In the $M$-theory realisation of $T_N$ theories as residing at the origin in the Calabi-Yau orbifolds $\frac{{\mathbb{C}}^3}{{\mathbb{Z}}_N\times {\mathbb{Z}}_N}$ we identify the ${\mathbb{Z}}_3$ symmetry geometrically and the new theories arise from $M$-theory on the non-Abelian orbifolds $(\frac{{\mathbb{C}}^3}{{\mathbb{Z}}_N\times {\mathbb{Z}}_N})/{\mathbb{Z}}_3$. On the other hand, in the $(p,q)$ 5-brane web description in Type IIB theory, the symmetry combines the $U$-duality symmetry with a rotation in space, defining a so-called $U$-fold background, where the $E_6$ symmetry is manifest.