Gauge Theory on Fuzzy S 2 × S 2 and Regularization on Noncommutative R 4

We define U(n) gauge theory on fuzzy S2 N × S2 N as a multi-matrix model, which reduces to ordinary Yang-Mills theory on S2×S2 in the commutative limit N → ∞. The model can be used as a regularization of gauge theory on noncommutative Rθ in a particular scaling limit, which is studied in detail. We also find topologically non-trivial U(1) solutions, which reduce to the known “fluxon” solutions in the limit of Rθ, reproducing their full moduli space. Other solutions which can be interpreted as 2-dimensional branes are also found. The quantization of the model is defined non-perturbatively in terms of a path integral which is finite. A gauge-fixed BRSTinvariant action is given as well. Fermions in the fundamental representation of the gauge group are included using a formulation based on SO(6), by defining a fuzzy Dirac operator which reduces to the standard Dirac operator on S2 × S2 in the commutative limit. The chirality operator and Weyl spinors are also introduced.