ep - l at / 0 10 90 19 v 2 2 1 Se p 20 01 Families index theory for Overlap lattice Dirac operator

The index bundle of the Overlap lattice Dirac operator over the orbit space of lattice gauge fields is introduced and studied. Obstructions to the vanishing of gauge anomalies in the Overlap formulation of lattice chiral gauge theory have a natural description in this context. Our main result is a formula for the topological charge (Pontryargin number) of the index bundle over evendimensional spheres in the orbit space. It reduces to the topological charge of the usual (continuum) index bundle in the classical continuum limit (this is announced and sketched here; the details will be given in a forthcoming paper). Thus we see that topology of the index bundle of the Dirac operator over the orbit space A/G0 can be captured in a finite-dimensional lattice setting.