Periods and motives in the spectral action of Robertson-Walker spacetimes

TL;DR

This paper demonstrates that the coefficients in the spectral action expansion for Euclidean Robertson-Walker spacetimes are periods of mixed Tate motives, linking geometric analysis with algebraic geometry.

Contribution

It reveals that the spectral action coefficients are periods of mixed Tate motives, connecting spectral geometry with motives in algebraic geometry.

Findings

Spectral action coefficients are periods of mixed Tate motives.

Involves relative motives of hyperplane and quadric complements.

Links spectral geometry with algebraic geometry motives.

Abstract

We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson-Walker spacetime are periods of mixed Tate motives, involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces and divisors given by unions of coordinate hyperplanes.