Sigma model Lagrangian for the Heisenberg group

Abstract

We study the Lagrangian for a sigma model based on the non-compact Heisenberg group. A unique feature of this model—unlike the case for compact Lie groups—is that the Lagrangian has to be regulated since the trace over the Heisenberg group is otherwise divergent. The resulting theory is a real Lagrangian with a quartic interaction term. In particular, in D=2 space–time dimensions, after a few non-trivial transformations, the Lagrangian is shown to be equivalent, at the classical level, to a complex cubic Lagrangian. A one-loop computation confirms that the quartic and cubic Lagrangians are equivalent at the quantum level as well.The complex Lagrangian is known to be classically equivalent to the SU(2) sigma model, with the equivalence breaking down at the quantum level. An explanation of this well-known results emerges from the properties of the Heisenberg sigma model.