Heat Kernel and Number Theory on NC-Torus

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 in 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 $$\mathbb{T}^4$$ , we show that this theory can be made renormalizable at least at one loop, and maybe even beyond.