Heat kernel and number theory on NC-torus

Details Heat kernel and number theory on NC-torus Heat kernel and number theory on NC-torus

2006-07-12

arxiv.org/abs/hep-th/0607078v1

Commun.Math.Phys.273:415-443,2007

Physics

High Energy Physics

High Energy Physics - Theory

Scientific paper

10.1007/s00220-007-0194-6

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.

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