Differential K-theory and localization formula for $\eta$-invariants

TL;DR

This paper develops a localization formula in differential K-theory for $S^1$-actions and extends it to equivariant $ ext{eta}$-invariants, introducing a novel pre-$ ext{lambda}$-ring structure.

Contribution

It introduces a new localization formula in differential K-theory and constructs a pre-$ ext{lambda}$-ring structure, advancing the understanding of equivariant $ ext{eta}$-invariants.

Findings

Established a localization formula in differential K-theory for $S^1$-actions.

Extended Goette's result to compare two types of equivariant $ ext{eta}$-invariants.

Constructed a pre-$ ext{lambda}$-ring structure in differential K-theory.

Abstract

In this paper, we obtain a localization formula in differential K-theory for $S^1$-action. Then by combining an extension of Goette's result on the comparison of two types of equivariant $\eta$-invariants, we establish a version of localization formula for equivariant $\eta$-invariants. An important step of our approach is to construct a pre-$\lambda$-ring structure in differential K-theory which is interesting in its own right.