Finite Heisenbeg Groups and Seiberg Dualities in Quiver Gauge Theories

TL;DR

The paper explores how finite Heisenberg groups act on quiver gauge theories and their Seiberg dual phases, revealing a consistent symmetry structure across different dual descriptions.

Contribution

It demonstrates that the finite Heisenberg group action persists across all Seiberg dual phases of a quiver gauge theory, with the shift generator mapping between different phases.

Findings

Finite Heisenberg groups act on quiver gauge theories.

The group action is preserved under Seiberg duality.

Explicit examples include C^3/Z_3, Y^{4,2}, and Y^{6,3} quivers.

Abstract

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.