Finite Heisenbeg Groups and Seiberg Dualities in Quiver Gauge Theories

A large class of quiver gauge theories admits the action of finite Heisenberg groups of the form Heis(Z_q x Z_q). This Heisenberg group is generated by a manifest Z_q shift symmetry acting on the quiver along with a second Z_q rephasing (clock) generator acting on the links of the quiver. Under Seiberg duality, however, the action of the shift generator is no longer manifest, as the dualized node has a different structure from before. Nevertheless, we demonstrate that the Z_q shift generator acts naturally on the space of all Seiberg dual phases of a given quiver. We then prove that the space of Seiberg dual theories inherits the action of the original finite Heisenberg group, where now the shift generator Z_q is a map among fields belonging to different Seiberg phases. As examples, we explicitly consider the action of the Heisenberg group on Seiberg phases for C^3/Z_3, Y^{4,2} and Y^{6,3} quiver.