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

TL;DR

This paper explores perturbative quantum field theory on isospectral noncommutative manifolds, analyzing UV/IR mixing phenomena and effective actions, with implications for curved and non-compact spaces.

Contribution

It develops a unified framework for quantum field theory on isospectral deformations, extending previous models to curved, compact, and non-compact spaces, and analyzes UV/IR mixing.

Findings

Different UV/IR mixing behaviors for various deformations.

Heat kernel methods clarify non-planar Green function contributions.

Diophantine conditions influence non-planar effective action analysis.

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 for action 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 or not 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 noncommutativity 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.