Arithmetic Structures in Spectral Models of Gravity

- Please not the changed date and time of the seminar this week -
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Abstract:

The spectral action functional is defined using a regularized trace of the Dirac operator. It provides a
model of (modified) gravity which recovers, through an asymptotic expansion, the usual Einstein-Hilbert
action with additional conformal and Gauss-Bonnet gravity terms. In cases of very regular geometries,
such as the Robertson-Walker metrics and the SU(2) Bianchi IX gravitational instantons, the full asymptotic
expansion of the spectral action can be computed recursively via pseudodifferential calculus. We show that,
in the case of Robertson-Walker spacetimes, all the coefficients of the asymptotic expansions are periods
of motives of complements of quadric hypersurfaces. In the case of the SU(2) Bianchi IX gravitational
instantons, the terms of the asymptotic expansion of the spectral action involve vector valued modular
forms. The talk is based on joint work with Farzad Fathizadeh and Wentao Fan.