The one-plaquette model limit of NC gauge theory in 2D

Abstract

It is found that noncommutative U ( 1 ) gauge field on the fuzzy sphere S N 2 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 2 N 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 N 2 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.