The One-Plaquette Model Limit of NC Gauge Theory in 2D

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 \bar{\alpha}_*=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.