A Pseudodifferential Analytic Perspective on Getzler's Rescaling

Georges Habib^ab, Sylvie Paycha<sup>c

a</sup> Department of Mathematics, Faculty of Sciences II, Lebanese University, P.O. Box, 90656 Fanar-Matn, Lebanon
^b Université de Lorraine, CNRS, IECL, France
^c Institut für Mathematik, Universität Potsdam, Campus Golm, Haus 9, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany

Abstract: Inspired by Gilkey's invariance theory, Getzler's rescaling method and Scott's approach to the index via Wodzicki residues, we give a localisation formula for the $\mathbb{Z}_2$-graded Wodzicki residue of the logarithm of a class of differential operators acting on sections of a spinor bundle over an even-dimensional manifold. This formula is expressed in terms of another local density built from the symbol of the logarithm of a limit of rescaled differential operators acting on differential forms. When applied to complex powers of the square of a Dirac operator, it amounts to expressing the index of a Dirac operator in terms of a local density involving the logarithm of the Getzler rescaled limit of its square.

Keywords: index, Dirac operator, Wodzicki residue, spinor bundle.

MSC: 58J40, 47A53, 15A66

Received: March 8, 2023; in final form January 11, 2024; Published online January 30, 2024

Language: English

DOI: 10.3842/SIGMA.2024.010

ArXiv: 2303.04013