Noncommutative U(1) Gauge Theory as a Non-linear Sigma Model

Noncommutative U (1) gauge theory in 4−dimensions is shown to be equivalent in some scaling limit to an ordinary non-linear sigma model in 2−dimensions. The model in this regime is solvable and the corresponding exact beta function is found. We also show that classical U (n) gauge theory on R d−2 ×R 2 θ can be approximated by a sequence of ordinary (d − 2)−dimensional Georgi-Glashow models with gauge groups U (n(L + 1)) where L + 1 is the matrix size of the regularized noncommutative plane R 2 θ .