Relative zeta determinants and the geometry of the determinant line bundle

The spectral zeta -function regularized geometry of the determinant line bu ndle for a family of first-order elliptic operators over a closed manifold encodes a subtle relation between the local family's index theorem and fund amental non-local spectral invariants. A great deal of interest has been di rected towards a generalization of this theory to families of elliptic boun dary value problems. We give here precise formulas for the relative zeta me tric and curvature in terms of Fredholm determinants and traces of operator s over the boundary. This has consequences for anomalies over manifolds wit h boundary.