Central Extensions of Finite Heisenberg Groups in Cascading Quiver Gauge Theories

TL;DR

The paper explores how nonconformal generalizations of conformal quiver gauge theories lead to a central extension of the Heisenberg group, affecting operator commutation relations in the dual supergravity backgrounds.

Contribution

It introduces a new discrete symmetry group, a central extension of the Heisenberg group, acting on nonconformal quiver gauge theories and their dual supergravity descriptions.

Findings

Discrete transformations form a central extension of the Heisenberg group.

Operators counting wrapped branes do not commute due to this extension.

The group structure generalizes the conformal case where all gauge groups have equal rank.

Abstract

Many conformal quiver gauge theories admit nonconformal generalizations. These generalizations change the rank of some of the gauge groups in a consistent way, inducing a running in the gauge couplings. We find a group of discrete transformation that acts on a large class of these theories. These transformations form a central extension of the Heisenberg group, generalizing the Heisenberg group of the conformal case, when all gauge groups have the same rank. In the AdS/CFT correspondence the nonconformal quiver gauge theory is dual to supergravity backgrounds with both five-form and three-form flux. A direct implication is that operators counting wrapped branes satisfy a central extension of a finite Heisenberg group and therefore do not commute.