ETA-INVARIANTS AND DETERMINANT LINES

The eta-invariant of an odd dimensional manifold with boundary is inve stigated. The natural boundary condition for this problem requires a t rivialization of the kernel of the Dirac operator on the boundary. The dependence of the eta-invariant on this trivialization is best encode d by the statement that the exponential of the eta-invariant lives in the determinant line of the boundary. Our main results are a variation al formula and a gluing law for this invariant. These results are appl ied to reprove the formula for the holonomy of the natural connection oh the determinant line bundle of a family of Dirac operators, also kn own as the ''global anomaly formula.'' The ideas developed here fit na turally with recent work in topological quantum field theory, in which gluing (which is a characteristic formal property of the path integra l and the classical action) is used to compute global invariants on cl osed manifolds from local invariants on manifolds with boundary.