The $ \zeta $-determinant and Quillen determinant for a Dirac operator on a manifold with boundary

Abstract.

Let ${\cal D}$ denote a Dirac operator on a compact odd-dimensional manifold M with boundary Y. The elliptic boundary value problem ${\cal D}_P$ is the operator ${\cal D}$ with domain determined by a boundary condition P from the smooth self-adjoint Grassmannian $Gr^*_{\infty}({\cal D})$. It has a well-defined $\zeta$-determinant (see [Wo5]). The determinant line bundle over $Gr^*_{\infty}({\cal D})$ has a natural trivialization in which the canonical Quillen determinant section becomes a function, denoted by ${\rm det}_{\cal C}\,{\cal D}_P$, equal to the Fredholm determinant of a naturally associated operator on the space of boundary sections. In this paper we show that the $\zeta$-regularized determinant ${\rm det}_{\zeta}\,{\cal D}_P$ is equal to det ${\rm det}_{\cal C}\,{\cal D}_P$ modulo a natural multiplicative constant.