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

摘要

We work out the general features of perturbative field theory onnoncommutative manifolds defined by isospectral deformation. These (in generalcurved) `quantum spaces', generalizing Moyal planes and noncommutative tori,are constructed using Rieffel's theory of deformation quantization for actionof $\R^l$. Our framework, incorporating background field methods and tools ofQFT in curved spaces, allows to deal both with compact and non-compact spaces,as well as with periodic or not deformations, essentially in the same way. Wecompute the quantum effective action up to one loop for a scalar theory,showing the different UV/IR mixing phenomena for different kinds of isospectraldeformations. The presence and behavior of the non-planar parts of the Greenfunctions is understood simply in terms of off-diagonal heat kernelcontributions. For periodic deformations, a Diophantine condition on thenoncommutativity parameters is found to play a role in the analytical nature ofthe non-planar part of the one-loop reduced effective action. Existence offixed points for the action may give rise to a new kind of UV/IR mixing.