A gauge invariant UV-IR mixing and the corresponding phase transition for U(1) fields on the fuzzy sphere

From a string theory point of view the most natural gauge action on the fuzzy sphere SL2 is the Alekseev-Recknagel-Schomerus action which is a particular combination of the Yang-Mills action and the Chern-Simons-like term. The differential calculus on the fuzzy sphere is 3-dimensional and thus the field content of this model consists of a 2-dimensional gauge field together with a scalar fluctuation normal to the sphere. For U(1) gauge theory we compute the quadratic effective action and shows explicitly that the tadpole diagrams and the vacuum polarization tensor contain a gauge-invariant UV-IR mixing in the continuum limit L→∞ where L is the matrix size of the fuzzy sphere. In other words the quantum U(1) effective action does not vanish in the commutative limit and a noncommutative anomaly survives. We compute the scalar effective potential and prove the gauge-fixing-independence of the limiting model L=∞ and then show explicitly that the one-loop result predicts a first order phase transition which was observed recently in simulation. The one-loop result for the U(1) theory is exact in this limit. It is also argued that if we add a large mass term for the scalar mode the UV-IR mixing will be completely removed. It is found in this case to be confined to the scalar sector only. This is in accordance with the large L analysis of the model. Finally we show that the phase transition becomes harder to reach starting from small couplings when we increase M.