Quantum Group $\sigma$ Models

TL;DR

This paper explores quantum group sigma models, specifically $SU_q(2)$ and $U_q(2)$, using noncommutative geometry, and discusses their structure, representations, and open problems in the field.

Contribution

It introduces a framework for quantum group sigma models using noncommutative differential geometry and provides explicit examples with $U_q(2)$ matrices.

Findings

Representation of $U_q(2)$ as $2N\times 2N$ unitary matrices

Analysis of $SU_q(2)$ sigma model with real $q$

Discussion of open problems in quantum group sigma models

Abstract

Field-theoretic models for fields taking values in quantum groups are investigated. First we consider $SU_q(2)$ $\sigma$ model ($q$ real) expressed in terms of basic notions of noncommutative differential geometry. We discuss the case in which the $\sigma$ models fields are represented as products of conventional $\sigma$ fields and of the coordinate-independent algebra. An explicit example is provided by the $U_q(2)$ $\sigma$ model with $q\sp{N}=1$, in which case quantum matrices $U_q(2)$ are realised as $2N\times 2N$ unitary matrices. Open problems are pointed out.