Scaling limits of the fuzzy sphere at one loop

Abstract

We study the one loop dynamics of QFT on the fuzzy sphere and calculate the planar and nonplanar contributions to the two point function at one loop. We show that there is no UV/IR mixing on the fuzzy sphere. The fuzzy sphere is characterized by two moduli: a dimensionless parameter N and a dimensionful radius R. Different geometrical phases can obtained at different corners of the moduli space. In the limit of the commutative sphere, we find that the two point function is regular without UV/IR mixing; however quantization does not commute with the commutative limit, and a finite "noncommutative anomaly" survives in the commutative limit. In a different limit, the noncommutative plane R 2 θ is obtained, and the UV/IR mixing reappears. This provides an explanation of the UV/IR mixing as an infinite variant of the "noncommutative anomaly".

Effective Potential on Fuzzy Sphere

“…The integral of a function F ∈ S 2 N over the fuzzy sphere is given by [7][8][9][10] 12) and the inner product can be defined by…”

“…The above Feynman rule of propagator and four-legs vertices can be found in [9]. Here we need also the Feynman rule of three-legs vertices, it is described as…”

“…[9] have considered the case that the external moment L ≪ α. In this case they used the relation [14] 8) to evaluate the one-loop correction to the mass square.…”

“…The problems of the one loop renormalization of the scalar field on the fuzzy sphere have been studied in some recent papers [7][8][9][10]. The phenomena of the UL/IR mixing [11,12] on the fuzzy sphere are different from those in the noncommutative plane.…”

“…al. [9] use this to provides an explanation of the UV/IR mixing as an infinite variant of the "noncommutative anomaly".…”