Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds

Abstract

We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions of $$\mathbb{R}^l .$$ Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and non-compact spaces, as well as with periodic and non-periodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the non-planar parts of the Green functions is understood simply in terms of off-diagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the non-planar part of the one-loop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing. Communicated by Vincent Rivasseau