Heat kernel and number theory on NC-torus

TL;DR

This paper fully determines the heat trace asymptotics on the noncommutative torus, revealing the deep influence of number theory, especially Diophantine conditions, on spectral properties and quantum field theory phenomena.

Contribution

It provides a comprehensive analysis of heat kernel asymptotics on NC-tori, highlighting the role of number theory and applying results to spectral action and quantum field theory.

Findings

Heat trace asymptotics depend on Diophantine conditions.

Spectral action computations are explicitly performed.

NC φ^4 theory can be renormalized at one loop despite non-local counterterms.

Abstract

The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the number-theoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit on a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing phenomenon for a scalar theory. Although we find non-local counterterms in the NC $\phi^4$ theory on $\T^4$, we show that this theory can be made renormalizable at least at one loop, and may be even beyond.