Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 010, 34 pages      arXiv:2303.04013      https://doi.org/10.3842/SIGMA.2024.010
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

A Pseudodifferential Analytic Perspective on Getzler's Rescaling

Georges Habib ^ab and Sylvie Paycha ^c
a) 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

Received March 08, 2023, in final form January 11, 2024; Published online January 30, 2024

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.

Key words: index; Dirac operator; Wodzicki residue; spinor bundle.

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