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.