Motives and periods in Bianchi IX gravity models

TL;DR

This paper links the heat kernel expansion coefficients in Bianchi IX gravity models to algebro-geometric motives, revealing their mixed Tate structure and providing explicit computations of their classes.

Contribution

It introduces a novel connection between heat kernel coefficients in Bianchi IX models and algebraic motives, with explicit calculations of their Grothendieck classes.

Findings

Coefficients are algebro-geometric periods of motives.

Motives are shown to be mixed Tate.

Explicit computation of Grothendieck classes provided.

Abstract

We show that, when considering the anisotropic scaling factors and their derivatives as affine variables, the coefficients of the heat kernel expansion of the Dirac-Laplacian on $SU(2)$ Bianchi IX metrics are algebro-geometric periods of motives of complements in affine spaces of unions of quadrics and hyperplanes. We show that the motives are mixed Tate and we provide an explicit computation of their Grothendieck classes.