Eta forms and the odd pseudodifferential families index
Let A(t) be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration ϕ with base Y. The standard example is A+it where A is a family, in the usual sense, of first order, self-adjoint and elliptic p...
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International Press of Boston
2018
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| Online Access: | http://hdl.handle.net/1721.1/115916 https://orcid.org/0000-0002-1494-8228 |
| _version_ | 1826198913796800512 |
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| author | Melrose, Richard B Rochon, Frederic |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Melrose, Richard B Rochon, Frederic |
| author_sort | Melrose, Richard B |
| collection | MIT |
| description | Let A(t) be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration ϕ with base Y. The standard example is A+it where A is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and t∈\bbR is the `suspending' parameter. Let π\cA:\cA(ϕ)⟶Y be the infinite-dimensional bundle with fibre at y∈Y consisting of the Schwartz-smoothing perturbations, q, making Ay(t)+q(t) invertible for all t∈\bbR. The total eta form, η\cA, as described here, is an even form on \cA(ϕ) which has basic differential which is an explicit representative of the odd Chern character of the index of the family:
dη\cA=π∗\cAγA, \Ch(\ind(A))=[γA]∈H\odd(Y).(*)
The 1
-form part of this identity may be interpreted in terms of the τ invariant (exponentiated eta invariant) as the determinant of the family. The 2-form part of the eta form may be interpreted as a B-field on the K-theory gerbe for the family A with (*) giving the `curving' as the 3-form part of the Chern character of the index. We also give `universal' versions of these constructions over a classifying space for odd K-theory. For Dirac-type operators, we relate η\cA with the Bismut-Cheeger eta form. |
| first_indexed | 2024-09-23T11:11:54Z |
| format | Article |
| id | mit-1721.1/115916 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T11:11:54Z |
| publishDate | 2018 |
| publisher | International Press of Boston |
| record_format | dspace |
| spelling | mit-1721.1/1159162022-10-01T01:59:05Z Eta forms and the odd pseudodifferential families index Melrose, Richard B Rochon, Frederic Massachusetts Institute of Technology. Department of Mathematics Melrose, Richard B Rochon, Frederic Let A(t) be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration ϕ with base Y. The standard example is A+it where A is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and t∈\bbR is the `suspending' parameter. Let π\cA:\cA(ϕ)⟶Y be the infinite-dimensional bundle with fibre at y∈Y consisting of the Schwartz-smoothing perturbations, q, making Ay(t)+q(t) invertible for all t∈\bbR. The total eta form, η\cA, as described here, is an even form on \cA(ϕ) which has basic differential which is an explicit representative of the odd Chern character of the index of the family: dη\cA=π∗\cAγA, \Ch(\ind(A))=[γA]∈H\odd(Y).(*) The 1 -form part of this identity may be interpreted in terms of the τ invariant (exponentiated eta invariant) as the determinant of the family. The 2-form part of the eta form may be interpreted as a B-field on the K-theory gerbe for the family A with (*) giving the `curving' as the 3-form part of the Chern character of the index. We also give `universal' versions of these constructions over a classifying space for odd K-theory. For Dirac-type operators, we relate η\cA with the Bismut-Cheeger eta form. National Science Foundation (U.S.) (Grant DMS-0408993) 2018-05-25T19:55:56Z 2018-05-25T19:55:56Z 2010 2018-05-25T18:32:43Z Article http://purl.org/eprint/type/JournalArticle 1052-9233 2164-4713 http://hdl.handle.net/1721.1/115916 Melrose, Richard and Frédéric Rochon. “Eta Forms and the Odd Pseudodifferential Families Index.” Surveys in Differential Geometry 15, 1 (2010): 279–322 © 2010 International Press of Boston https://orcid.org/0000-0002-1494-8228 http://dx.doi.org/10.4310/SDG.2010.V15.N1.A9 Surveys in Differential Geometry Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf International Press of Boston arXiv |
| spellingShingle | Melrose, Richard B Rochon, Frederic Eta forms and the odd pseudodifferential families index |
| title | Eta forms and the odd pseudodifferential families index |
| title_full | Eta forms and the odd pseudodifferential families index |
| title_fullStr | Eta forms and the odd pseudodifferential families index |
| title_full_unstemmed | Eta forms and the odd pseudodifferential families index |
| title_short | Eta forms and the odd pseudodifferential families index |
| title_sort | eta forms and the odd pseudodifferential families index |
| url | http://hdl.handle.net/1721.1/115916 https://orcid.org/0000-0002-1494-8228 |
| work_keys_str_mv | AT melroserichardb etaformsandtheoddpseudodifferentialfamiliesindex AT rochonfrederic etaformsandtheoddpseudodifferentialfamiliesindex |