Zeta forms and the local family index theorem

Abstract:

For a family $F$ of elliptic pseudodifferential operators we show there is a natural zeta-form $\zeta (F,S)$ and zeta-determinant form $\operatorname {det}_\zeta (F)$ 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.