Sigma Model Lagrangian for the Heisenberg Group

TL;DR

This paper investigates the sigma model based on the non-compact Heisenberg group, addressing divergence issues in the Lagrangian, and demonstrates classical and quantum equivalences to complex cubic and SU(2) sigma models.

Contribution

It introduces a regulated Lagrangian for the Heisenberg sigma model and establishes its classical and quantum equivalences to other well-known models.

Findings

The Lagrangian requires regularization due to divergence issues.

Classical equivalence between the Heisenberg and complex cubic Lagrangians.

Quantum equivalence confirmed through one-loop computations.

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 definition of 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. 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 shows that the quartic and cubic Lagrangians are equivalent at the quantum level as well. The complex Lagrangian is known to 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.