Finite Heisenberg groups from non-Abelian orbifold quiver gauge theories

A large class of orbifold quiver gauge theories admits the action of finite Heisenberg groups of the form ∏iHeis(Zqi×Zqi). For an Abelian orbifold generated by Γ, the Zqi shift generator in each Heisenberg group is one cyclic factor of the Abelian group Γ. For general non-Abelian Γ, however, we find that the shift generators are the cyclic factors in the Abelianization of Γ. We explicitly show this for the case Γ=Δ(27), where we construct the finite Heisenberg group symmetries of the field theory. These symmetries are dual to brane number operators counting branes on homological torsion cycles. These brane number operators therefore do not commute. We compare our field theory results with string theory states and find perfect agreement.