The spectral action for Moyal planes

TL;DR

This paper extends the spectral action calculation for Moyal planes, providing an asymptotic expansion for a heat operator trace, generalizing previous results to arbitrary skew-symmetric matrices and noncommutative geometries.

Contribution

It derives the spectral action for Moyal planes with any skew-symmetric matrix, generalizing prior work on symplectic cases and expanding the understanding of noncommutative geometric models.

Findings

Asymptotic expansion for heat operator trace on Moyal planes

Spectral action computed for general skew-symmetric matrices

Generalization of Connes-Lott action to broader noncommutative spaces

Abstract

Extending a result of D.V. Vassilevich, we obtain the asymptotic expansion for the trace of a "spatially" regularized heat operator associated with a generalized Laplacian defined with integral Moyal products. The Moyal hyperplanes corresponding to any skewsymmetric matrix $\Theta$ being spectral triples, the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes is computed. This result generalizes the Connes-Lott action previously computed by Gayral for symplectic $\Theta$.