Zeta Forms and the Local Family Index Theorem
| dc.creator | Scott, Simon | |
| dc.date | 2004-06-15 | |
| dc.date | 2006-04-19 | |
| dc.date.accessioned | 2026-07-07T06:36:51Z | |
| dc.date.available | 2026-07-07T06:36:51Z | |
| dc.description | For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the classical single operator spectral zeta function and determinant. In the case where F is the curvature of a superconnection the zeta form is exact, extending to families the Atiyah-Bott-Seeley zeta function formula for the pointwise index, and equivalent to the transgression formula for the graded Chern character. The zeta-determinant form leads to the definition of the graded zeta-Chern class form. For a family of compatible Dirac operators D with index bundle Ind(D) we prove a transgression formula leading to a local density representing the Chern class c(Ind(D))in terms of the A-hat genus and twisted Chern character. Globally the meromorphically continued zeta form and zeta determinant form exist only at the level of K-theory as characteristic class maps K(B)->H*(B). | |
| dc.description | 34 pages | |
| dc.identifier | https://arxiv.org/abs/math/0406294 | |
| dc.identifier | http://arxiv.org/abs/math/0406294 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100222 | |
| dc.subject | Differential Geometry | |
| dc.subject | Analysis of PDEs | |
| dc.title | Zeta Forms and the Local Family Index Theorem | |
| dc.type | text |