Note on a numerical equality regarding the eta invariant on Berger spheres

TL;DR

This paper reveals a numerical equality between the Dirac APS eta invariant on Berger spheres and the Dirac conformal anomaly on even-dimensional spheres, providing an explicit analytical expression and confirming a conjecture.

Contribution

It establishes a new numerical and analytical connection between eta invariants on Berger spheres and conformal anomalies, including a proof of a known conjecture.

Findings

Eta invariant on Berger spheres matches conformal anomaly on even spheres

Analytical expression involves generalized Bernoulli polynomials

Confirms a previously conjectured equality

Abstract

The Dirac APS eta invariant on a Berger sphere of dimension $2n-1$ is discovered, numerically, to coincide, up to spin factors, with the Dirac conformal anomaly on a round sphere of even dimension, $n$. The analytical expression, given in terms of a generalised Bernoulli polynomial, is shown to equal a known conjecture for the eta invariant. Weingart's generating function is also obtained with no extra work.