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Article Dans Une Revue Symmetry, Integrability and Geometry : Methods and Applications Année : 2024

A Pseudodifferential Analytic Perspective on Getzler's Rescaling

Résumé

Inspired by Gilkey's invariance theory, Getzler's rescaling method and Scott'sapproach 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.
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Dates et versions

hal-04017007 , version 1 (06-03-2023)
hal-04017007 , version 2 (31-01-2024)

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Georges Habib, Sylvie Paycha. A Pseudodifferential Analytic Perspective on Getzler's Rescaling. Symmetry, Integrability and Geometry : Methods and Applications, 2024, 20 (010), ⟨10.3842/SIGMA.2024.010⟩. ⟨hal-04017007v2⟩
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