Heat-kernel approach to UV/IR mixing on isospectral deformation manifolds

摘要

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 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.