Se p 20 06 The One - Plaquette Model Limit of NC Gauge Theory in

It is found that noncommutative U (1) gauge field on the fuzzy sphere S 2 N is equivalent in the quantum theory to a commutative 2−dimensional U (N) gauge field on a lattice with two plaquettes in the axial gauge A 1 = 0. This quantum equivalence holds in the fuzzy sphere-weak coupling phase in the limit of infinite mass of the scalar normal component of the gauge field. The doubling of plaquettes is a natural consequence of the model and it is reminiscent of the usual doubling of points in Connes standard model. In the continuum large N limit the plaquette variable W approaches the identity 1 2N and as a consequence the model reduces to a simple matrix model which can be easily solved. We compute the one-plaquette critical point and show that it agrees with the observed value ¯ α * = 3.35. We compute the quantum effective potential and the specific heat for U (1) gauge field on the fuzzy sphere S 2 N in the 1/N expansion using this one-plaquette model. In particular the specific heat per one degree of freedom was found to be equal to 1 in the fuzzy sphere-weak coupling phase of the gauge field which agrees with the observed value 1 seen in Monte Carlo simulation. This value of 1 comes precisely because we have 2 plaquettes approximating the NC U (1) gauge field on the fuzzy sphere.