Theta-terms in nonlinear sigma-models

TL;DR

This paper investigates the origin and role of theta-terms in nonlinear sigma-models, linking them to nonperturbative anomalies and geometric phases across various fermionic models and dimensions.

Contribution

It reveals the nonperturbative anomaly origin of theta-terms and explores their relation to geometric phases and soliton quantum numbers in fermionic sigma-models.

Findings

Theta-terms originate from nonperturbative anomalies of current algebras.

Geometric phases in fermionic sigma-models are identified as theta-terms.

Theta-terms are related to the quantum numbers of solitons.

Abstract

We trace the origin of theta-terms in non-linear sigma-models as a nonperturbative anomaly of current algebras. The non-linear sigma-models emerge as a low energy limit of fermionic sigma-models. The latter describe Dirac fermions coupled to chiral bosonic fields. We discuss the geometric phases in three hierarchies of fermionic sigma-models in spacetime dimension (d+1) with chiral bosonic fields taking values on d-, d+1-, and d+2-dimensional spheres. The geometric phases in the first two hierarchies are theta-terms. We emphasize a relation between theta-terms and quantum numbers of solitons.