Theη-form and a generalized maslov index

Given a family {L0(b,L1(b)}b∈B of pairs of transverse Lagrangian subspaces of a hermitean symplectic vector space we define a family of Dirac operators on the unit interval and consider itsη-formη(L0,L1) εΘ*(B). To a family {L0(b,L1(b,L2(b)}b∈B of pairwise transverse Lagrangian subspaces we associate the cocycleη(L0,L1)+η(L1,L2)+η(L2,L1) which is a closed form. We identify its cohomology class with a generalization to families of the triple Maslov index.