Scaling Limits of the Fuzzy Sphere at one Loop

TL;DR

This paper analyzes the one-loop quantum field theory dynamics on the fuzzy sphere, demonstrating the absence of UV/IR mixing in certain limits and revealing a noncommutative anomaly in the commutative limit.

Contribution

It provides a detailed calculation of one-loop effects on the fuzzy sphere and clarifies the conditions under which UV/IR mixing occurs or is absent, highlighting the noncommutative anomaly.

Findings

No UV/IR mixing on the fuzzy sphere.

UV/IR mixing reappears in the noncommutative plane limit.

A finite noncommutative anomaly persists in the commutative limit.

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_theta 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''.