A gauge invariant UV–IR mixing and the corresponding phase transition for [formula omitted] fields on the fuzzy sphere

Abstract

From a string theory point of view the most natural gauge action on the fuzzy sphere S L 2 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.