The gluing formula of the zeta-determinants of Dirac Laplacians for certain boundary conditions

Abstract

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result see ( J. Geom. Phys. 57 (2007) 1951–1976) to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary conditions $\mathcal{P}_{-,\mathcal{L}_{0}}$, $\mathcal{P}_{+,\mathcal{L}_{1}}$. We next consider a double of de Rham complexes consisting of differential forms of all degrees with the absolute and relative boundary conditions. Using a similar method, we prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the absolute and relative boundary conditions.