Finite Heisenberg 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(ZqxZq). This Heisenberg group is generated by a manifest Zq shift symmetry acting on the quiver along with a second Zq 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 Zq 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 Zq 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 C3/Z3, Y4,2 and Y6,3 quivers