A Pseudodifferential Analytic Perspective on Getzler's Rescaling

Georges Habib ; Sylvie Paycha

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

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.

Détail

Keywords: index, Dirac operator, spinor bundle.
Mots-clés : Wodzicki residue

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