An analytical formula for the f-invariant of circle transfers

TL;DR

This paper presents an explicit analytical formula to compute the f-invariant of circle transfers using spectral asymmetry of twisted Dirac operators, with examples and extensions to the quaternionic case.

Contribution

It introduces a new method to calculate the f-invariant directly from the framed manifold via spectral asymmetry, including explicit examples and quaternionic cases.

Findings

Derived an explicit formula for the f-invariant of circle transfers.

Connected the f-invariant to spectral asymmetry of Dirac operators.

Extended the analysis to quaternionic cases.

Abstract

In this note, we explain how the f-invariant of a circle transfer can be computed on the framed manifold itself in terms of the spectral asymmetry of twisted Dirac operators on the base. Some explicit examples and a treatment of the quaternionic case are provided as well.