On spectral action over Bieberbach manifolds

TL;DR

This paper calculates the spectral action for three-dimensional Bieberbach manifolds using two methods, revealing potential differences from the flat torus spectral action related to the eta invariant.

Contribution

It introduces two methods for computing the spectral action on Bieberbach manifolds and explores the impact of asymmetric cutoff functions on the spectral action.

Findings

Spectral action computed for Bieberbach manifolds

Potential difference from flat torus spectral action linked to eta invariant

Methods include Poisson summation and perturbative expansion

Abstract

We compute the leading terms of the spectral action for orientable three dimensional Bieberbach manifolds first, using two different methods: the Poisson summation formula and the perturbative expansion. Assuming that the cut-off function is not necessarily symmetric we find that that the scale invariant part of the perturbative expansion might differ from the spectral action of the flat three-torus by the eta invariant.