The logarithmic residue density of a generalised Laplacian

TL;DR

This paper investigates the residue density of the logarithm of a generalized Laplacian on closed manifolds, expressing it through residues of pseudodifferential symbols, and provides proofs of special cases of the Atiyah-Singer index formula.

Contribution

It introduces a new approach to express the residue density as a sum of residues, offering a perturbative proof of specific Atiyah-Singer index cases.

Findings

Residue density defines an invariant polynomial differential form.

Provides a perturbative proof of Atiyah-Singer formula for Dirac operators.

Expresses residue density in terms of classical pseudodifferential symbol residues.

Abstract

We show that the residue density of the logarithm of a generalised Laplacian on a closed manifold defines an invariant polynomial valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulae provide a pedestrian proof of the Atiyah-Singer formula for a pure Dirac operator in dimension $4$ and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by S. Scott and D. Zagier announced in \cite{Sc2} and to appear in \cite{Sc3}. In our approach, which is of perturbative nature, we use either a Campbell-Hausdorff formula derived by Okikiolu or a non commutative Taylor type formula.