Determinants of dirac boundary value problems over odd-dimensional manifolds

We present a canonical construction of the determinant of an elliptic selfadjoint boundary value problem for the Dirac operatorD over an odd-dimensional manifold. For 1-dimensional manifolds we prove that this coincides with the -function determinant. This is based on a result that elliptic self-adjoint boundary conditions forD are parameterized by a preferred class of unitary isomorphisms between the spaces of boundary chiral spinor fields. With respect to a decompositionS¹=X⁰X¹, we explain how the determinant of a Dirac-type operator overS¹ is related to the determinants of the corresponding boundary value problems overX⁰ andX¹.