Relative zeta determinants and the geometry of the determinant line bundle

The spectral $\zeta$-function regularized geometry of the determinant line bundle for a family of first-order elliptic operators over a closed manifold encodes a subtle relation between the local family's index theorem and fundamental non-local spectral invariants. A great deal of interest has been directed towards a generalization of this theory to families of elliptic boundary value problems. We give here precise formulas for the relative zeta metric and curvature in terms of Fredholm determinants and traces of operators over the boundary. This has consequences for anomalies over manifolds with boundary.