  This is a note for the conference proceedings Topological and Geometrical
Problems related to Quantum Field Theory. We summarize our joint work with Dai
about eta invariants on manifolds with boundary. Then we apply these results to
prove the curvature and holonomy formulas for the natural connection on the
determinant line bundle of a family of Dirac operators. These were originally
proved by Bismut and the author--the proofs here are much simpler.
