The flat Grothendieck-Riemann-Roch theorem without adiabatic techniques

TL;DR

This paper presents a simplified proof of the flat Grothendieck-Riemann-Roch theorem using local index theory and Chern-Simons forms, avoiding complex adiabatic limit calculations.

Contribution

It introduces a new proof technique that simplifies the understanding of the flat Grothendieck-Riemann-Roch theorem by eliminating adiabatic limit computations.

Findings

Proof avoids adiabatic limit of eta-invariant

Utilizes local family index theorem and Chern-Simons forms

Simplifies the conceptual framework of the theorem

Abstract

In this paper we give a simplified proof of the flat Grothendieck-Riemann-Roch theorem. The proof makes use of the local family index theorem and basic computations of the Chern-Simons form. In particular, it does not involve any adiabatic limit computation of the reduced eta-invariant.