An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary

For a family of Dirac operators acting on Hermitian Cli ord modules over the odd dimensional compact manifolds with boundary which are the bres of a bration with compact base we compute the Chern character of the index in K of the base Although we assume a product decomposition near the boundary we make no assumptions on invertibility of the bound ary family and instead obtain a family of self adjoint Fredholm operators by choice of an auxiliary family of projections respecting theZ decomposition of bundles over the boundary In case the boundary family is invertible this projection can be taken to be the Atiyah Patodi Singer projection and the resulting formula is as conjectured by Bismut and Cheeger The derivation of the index formula is e ected by the combination of the superconnec tion formalism of Quillen and Bismut the calculus of b pseudodi erential operators and suspension Introduction Let M B be a bration of Riemannian manifolds with B compact and with bres di eomorphic to a xed odd dimensional com pact manifold with boundary X Suppose that the bres carry smoothly varying spin structures and that the Riemannian metrics on the bres have smoothly varying product decompositions near the boundary Let g gz be for z B the associated family of Dirac operators and let g g z be the boundary family If g z is invertible for each z B the Atiyah Patodi Singer boundary condition makes gz into a Received July and in revised form October Research of the rst author was supported in part by the National Science Foun dation under grant DMS and that of the second author was supported in part by M U R S T Italy richard b melrose paolo piazza continuous family of self adjoint Fredholm operators and thus follow ing Atiyah and Singer in de nes an element Ind g K B Under this assumption of invertibility of the boundary family a formula for the Chern character Ch Ind g H B was conjectured by Bismut and Cheeger in In this paper using ideas similar to those used in for the even dimensional case we prove such a formula without making any assumptions on the boundary family and for the Dirac operator of general Hermitian Cli ord modules with unitary Cli ord connections To explain how we de ne a continuous family of self adjoint Fred holm operators consider for simplicity the spinor bundle but with no invertibility assumptions on the boundary family Observe rst that the restriction of the spinor bundle to the boundary of the bration is Z graded S M S S Let de ned by Cli ord multiplication in the normal direction be the parity operator on S M The boundary operator g is odd with respect to this Z grading and self adjoint