Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Zeta forms and the local family index theorem

Author: Simon Scott
Journal: Trans. Amer. Math. Soc. 359 (2007), 1925-1957
MSC (2000): Primary 58J40, 58J52
Published electronically: December 19, 2006
MathSciNet review: 2276607
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Abstract: For a family $ F$ of elliptic pseudodifferential operators we show there is a natural zeta-form $ \zeta(F,S)$ and zeta-determinant form $ {det}_\zeta(F)$ in the ring of smooth differential forms on the parameterizing manifold, generalizing the classical single operator zeta-function and zeta-determinant. We show that the zeta forms extend the Atiyah-Bott-Seeley formula for the index of an elliptic operator to a family of elliptic operators, while the zeta-determinant form leads to a graded Chern class form for the index bundle. Globally, the zeta-form and zeta-determinant form exist only at the level of $ K$-theory as maps to cohomology.

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Simon Scott
Affiliation: Department of Mathematics, King’s College London, London, WC2R 2LS England

Received by editor(s): May 4, 2004
Published electronically: December 19, 2006
Article copyright: © Copyright 2006 American Mathematical Society