Finite Heisenberg Groups from Nonabelian Orbifold Quiver Gauge Theories

TL;DR

This paper explores the structure of finite Heisenberg groups in orbifold quiver gauge theories, revealing their relation to the Abelianization of the orbifold group and confirming their physical significance through string theory correspondence.

Contribution

It explicitly constructs finite Heisenberg group symmetries for non-Abelian orbifolds, particularly elta(27), and links these symmetries to brane counting in string theory.

Findings

Finite Heisenberg groups act as symmetries in orbifold quiver gauge theories.

Shift generators correspond to cyclic factors in the Abelianization of the orbifold group.

The symmetries match string theory brane states, confirming their physical relevance.

Abstract

A large class of orbifold quiver gauge theories admits the action of finite Heisenberg groups of the form \prod_i Heis(Z_{q_i} x Z_{q_i}). For an Abelian orbifold generated by \Gamma, the Z_{q_i} shift generator in each Heisenberg group is one cyclic factor of the Abelian group \Gamma. For general non-Abelian \Gamma, however, we find that the shift generators are the cyclic factors in the Abelianization of \Gamma. We explicitly show this for the case \Gamma=\Delta(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, which therefore do not commute. We compare our field theory results with string theory states and find perfect agreement.