Index of a family of lattice Dirac operators and its relation to the non-abelian anomaly on the lattice

In the continuum, a topological obstruction to the vanishing of the non-abelian anomaly in 2n dimensions is given by the index of a certain Dirac operator in 2n+2 dimensions, or equivalently, the index of a 2-parameter family of Dirac operators in 2n dimensions. In this paper an analogous result is derived for chiral fermions on the lattice in the Overlap formulation. This involves deriving an Index Theorem for a family of lattice Dirac operators satisfying the Ginsparg--Wilson relation. The index density is proportional to Luescher's topological field in 2n+2 dimensions.