A Pseudodifferential Analytic Perspective on Getzler's Rescaling
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 ℤ₂ -graded Wodzicki residue of the logarithm of a class of differential operators acting on sections of a spinor bundle o...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2024 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2024
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/212113 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Pseudodifferential Analytic Perspective on Getzler's Rescaling. Georges Habib and Sylvie Paycha. SIGMA 20 (2024), 010, 34 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862724411760574464 |
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| author | Habib, Georges Paycha, Sylvie |
| author_facet | Habib, Georges Paycha, Sylvie |
| citation_txt | A Pseudodifferential Analytic Perspective on Getzler's Rescaling. Georges Habib and Sylvie Paycha. SIGMA 20 (2024), 010, 34 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 ℤ₂ -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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| first_indexed | 2026-03-21T07:04:42Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212113 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T07:04:42Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Habib, Georges Paycha, Sylvie 2026-01-28T13:57:17Z 2024 A Pseudodifferential Analytic Perspective on Getzler's Rescaling. Georges Habib and Sylvie Paycha. SIGMA 20 (2024), 010, 34 pages 1815-0659 2020 Mathematics Subject Classification: 58J40; 47A53; 15A66 arXiv:2303.04013 https://nasplib.isofts.kiev.ua/handle/123456789/212113 https://doi.org/10.3842/SIGMA.2024.010 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 ℤ₂ -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. The first-named author would like to thank the Alfried Krupp Wissenschaftskolleg in Greifswald for the support. We are grateful to the Humboldt Foundation for funding a Linkage Programme between the University of Potsdam in Germany and the Lebanese University, as well as the American University of Beirut in Lebanon. We also thank the referees for their very helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Pseudodifferential Analytic Perspective on Getzler's Rescaling Article published earlier |
| spellingShingle | A Pseudodifferential Analytic Perspective on Getzler's Rescaling Habib, Georges Paycha, Sylvie |
| title | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_full | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_fullStr | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_full_unstemmed | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_short | A Pseudodifferential Analytic Perspective on Getzler's Rescaling |
| title_sort | pseudodifferential analytic perspective on getzler's rescaling |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212113 |
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