Quantum Equivalence of NC and YM Gauge Theories in 2 D and Matrix Theory

We construct noncommutative U(1) gauge theory on the fuzzy sphere S^2_N as a unitary 2N x 2N matrix model. In the quantum theory the model is equivalent to a nonabelian U(N) Yang-Mills theory on a 2 dimensional lattice with 2 plaquettes. This equivalence holds in the " fuzzy sphere" phase where we observe a 3rd order phase transition between weak-coupling and strong-coupling phases of the gauge theory. In the ``matrix'' phase we have a U(N) gauge theory on a single point.