Scaling limits of the fuzzy sphere at one loop

We study the one loop dynamics of QFT on the fuzzy sphere and calculate theplanar and nonplanar contributions to the two point function at one loop. Weshow that there is no UV/IR mixing on the fuzzy sphere. The fuzzy sphere ischaracterized by two moduli: a dimensionless parameter N and a dimensionfulradius R. Different geometrical phases can obtained at different corners of themoduli space. In the limit of the commutative sphere, we find that the twopoint function is regular without UV/IR mixing; however quantization does notcommute with the commutative limit, and a finite ``noncommutative anomaly''survives in the commutative limit. In a different limit, the noncommutativeplane R^2_theta is obtained, and the UV/IR mixing reappears. This provides anexplanation of the UV/IR mixing as an infinite variant of the ``noncommutativeanomaly''.Comment: 16 pages. v2: add remarks on p.6, footnotes on p.6 and p.9 clarified in response to the referee suggestions for clearer presentation. To appear in JHE