Determinants, Grassmannians and elliptic boundary problems for the Dirac operator
For a compact odd-dimensional manifold with boundary of product type near the boundary, the authors consider relations between different determinants of generalized Dirac operators. They investigate the case of a fixed Dirac operator \(A\) with varying elliptic boundary conditions parametrized by the infinite Grassmannian \({\mathcal G}r_\infty(A)\). In this context, they study the connection between the so-called \(\zeta\)-determinant and canonical determinant of the operator, which are defined as sections over the determinant line bundle over \({\mathcal G}r_\infty(A)\), i.e., associated to the family of elliptic boundary value problems. It is shown that the metrics on the determinant line bundle defined by both determinants lead to the same notion of curvature. The main result is in the case of selfadjoint boundary conditions the identification of both determinants up to a multiplicative constant. Moreover, it is shown that the phase of the canonical determinant is given as the integral of the variation of the \(\eta\)-invariant. Proofs of the results are partly sketched, for complete proofs the authors refer to a forthcoming paper.
scientific article; zbMATH DE number 1060088