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 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.

Spectral action on noncommutative torus

“…We will see how this formula can be extended to the case of a noninvertible Dirac operator and noninvertible perturbation of the form D C Q A with Q A´A C "JAJ 1 . All these results on spectral action are quite important in physics, especially in quantum field theory and particle physics, where one adds to the effective action some counterterms explicitly given by (1.6); see for instance [2], [3], [4], [11], [17], [18], [20], [22], [28], [34], [36], [37], [38].…”

Noncommutative QFT and Renormalization

“…There have already been attempts to generalise this approach to the non-commutative realm; for non-commutative φ 4 theory see [41,42]. First steps towards gauge kinetic models have been done in [43][44][45].…”

“…According to [41], the third line of the above equation corresponds to non-planar contributions. The above expressions for even dimensions other than four can easily be extracted.…”

Non-commutative Renormalization

“…They were able to identify the divergent graphs and to classify the divergences of the theories thanks to the topological data of the graphs. Then V. Gayral [90] showed that UV/IR mixing is present on all isospectral deformations (they consist in curved generalizations of the Moyal space and of the non-commutative torus). For this, he considered a scalar model (3.28) and discovered contributions to the effective action which diverge when the external momenta vanish.…”