The $\sigma$-Model and Non-commutative Geometry
  In terms of non-commutative geometry, we show that the $\sigma$--model can be built up by the gauge theory on discrete group $Z_2$. We introduce a constraint in the gauge theory, which lead to the constraint imposed on linear $\sigma$ model to get nonlinear $\sigma$ model . 
Hanying Guo, Jianming Li and Ke Wu, 10 pages, Latex ASITP-93-67