Finite gauge theory on fuzzy CP2

We give a non-perturbative definition of U(n) gauge theory on fuzzy CP2 as a multi-matrix model. The degrees of freedom are 8 Hermitian matrices of finite size, 4 of which are tangential gauge fields and 4 are auxiliary variables. The model depends on a non-commutativity parameter 1N, and reduces to the usual U(n) Yang-Mills action on the 4-dimensional classical CP2 in the limit N→∞. We explicitly find the monopole solutions, and also certain U(2) instanton solutions for finite N. The quantization of the model is defined in terms of a path integral, which is manifestly finite. An alternative formulation with constraints is also given, and a scaling limit as Rθ4 is discussed.