Perturbative and nonperturbative correspondences between compact and non-compact sigma-models

TL;DR

This paper compares compact and noncompact sigma-models in lattice formulations, establishing exact correspondences in their perturbative and large N expansions, highlighting differences and similarities in their behaviors.

Contribution

It introduces an exact correspondence between the expansion coefficients of compact and noncompact sigma-models for both perturbative and large N expansions on finite lattices.

Findings

Perturbative and large N expansions are asymptotic on finite lattices.

An exact coefficient correspondence is established between models.

The perturbative expansion remains valid in the infinite volume limit.

Abstract

Compact (ferro- and antiferromagnetic) sigma-models and noncompact (hyperbolic) sigma-models are compared in a lattice formulation in dimensions $d \geq 2$. While the ferro- and antiferromagnetic models are essentially equivalent, the qualitative difference to the noncompact models is highlighted. The perturbative and the large $N$ expansions are studied in both types of models and are argued to be asymptotic expansions on a finite lattice. An exact correspondence between the expansion coefficients of the compact and the noncompact models is established, for both expansions, valid to all orders on a finite lattice. The perturbative one involves flipping the sign of the coupling and remains valid in the termwise infinite volume limit. The large $N$ correspondence concerns the functional dependence on the free propagator and holds directly only in finite volume.