Abstract :
Noncommutative U(1) gauge theory on the Moyal–Weyl space R2×R2θ is regularized by approximating the noncommutative spatial slice R2θ by a fuzzy sphere of matrix size L and radius R. Classically we observe that the field theory on the fuzzy space R2×S2L reduces to the field theory on the Moyal–Weyl plane R2×R2θ in the flattening continuum planar limits R,L→∞, where the ratio θ2=R2/|L|2q is kept fixed with q>3/2. The effective noncommutativity parameter is found to be given by θeff2∼2θ2(L/2)2q−1 and thus it corresponds to a strongly noncommuting space. In the quantum theory it turns out that this prescription is also equivalent to a dimensional reduction of the model where the noncommutative U(1) gauge theory in 4 dimensions is shown to be equivalent in the large L limit to an ordinary O(M) non-linear sigma model in 2 dimensions where M∼3L2. The Moyal–Weyl model defined this way is also seen to be an ordinary renormalizable theory which can be solved exactly using the method of steepest descents. More precisely, we find for a fixed renormalization scale μ and a fixed renormalized coupling constant gr2 an O(M)-symmetric mass, for the different components of the sigma field, which is non-zero for all values of gr2 and hence the O(M) symmetry is never broken in this solution. We obtain also an exact representation of the beta function of the theory which agrees with the known one-loop perturbative result.
