Field Theory on the q-deformed Fuzzy Sphere I

We study the q-deformed fuzzy sphere, which is related to D-branes on SU(2) WZW models, for both real q and q a root of unity. We construct for both cases a differential calculus which is compatible with the star structure, study the integral, and find a canonical frame of one-forms. We then consider actions for scalar field theory, as well as for Yang-Mills and Chern-Simons-type gauge theories. The zero curvature condition is solved.