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 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 ${\varphi }^4$ 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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• Heat Trace Asymptotics on Noncommutative Spaces
• Heat trace and spectral action on the standard Podles sphere
• Spectral Action in Noncommutative Geometry
• An approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions
• Renormalization on noncommutative torus
• Spectral action on SU_q(2)
• Heat kernel expansion and induced action for the matrix model Dirac operator
• Nilpotent noncommutativity and renormalization
• Heat kernel of non-minimal gauge field kinetic operators on Moyal plane
• Quantum field theory on projective modules
• Spectral action in noncommutative geometry and global pseudodifferential calculus
• On space-time noncommutative U(1) model at high temperature
• Heat kernel expansion and induced action for matrix models
• Zeta-regularization and the heat-trace on some compact quantum semigroups
• Spectral action in noncommutative geometry: An example