Arithmetic Structures in Spectral Models of Gravity

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The spectral action functional of noncommutative geometry provides a model of Euclidean (modified) gravity, possibly coupled to matter. The terms in the large energy asymptotic expansion of the spectral action can be computed via pseudodifferential calculus. In the case of highly symmetric spacetimes, like Robertson-Walker metrics and Bianchi IX gravitational instantons, there is a richer arithmetic structure in the spectral action, and the terms in the asymptotic expansion are expressiblein terms of periods of motives and of modular forms. This reveals a new occurrence of interesting periods and motives in high-energy physics.