  By considering scalar theories on the fuzzy sphere as matrix models, we
construct a renormalization scheme and calculate the one-loop effective action.
Because of UV-IR mixing, the two- and the four-point correlators at low energy
are not slowly varying functions of external momenta. Interestingly, we also
find that field theories on fuzzy RP^2 avoid UV-IR mixing and hence are much
more like conventional field theories. We calculate the one-loop beta-function
for the O(N) theory on fuzzy RP^2 at large N and show that in addition to the
trivial one, it has a nontrivial fixed point that is accessible in perturbation
theory.
