The ratio of two zeta-determinants of Dirac Laplacians associated with unitary involutions on a compact manifold with cylindrical end

Abstract

Given two unitary involutions ${\sigma }_1$ and ${\sigma }_2$ satisfying $G{\sigma }_i=-{\sigma }_{i}G$ on $kerB$ on a compact manifold with cylindrical end, M. Lesch, K. Wojciechowski ([LW]) and W. Müller ([M]) established the formula describing the difference of two eta-invariants with the APS boundary conditions associated with ${\sigma }_1$ and ${\sigma }_2$. In this paper we establish the analogous formula for the zeta-determinants of Dirac Laplacians. For the proof of the result we use the Burghelea-Friedlander-Kappeler's gluing formula for zeta-determinants and the scattering theory developed by W. Müller in [M]. This result was also obtained independently by J. Park and K. Wojciechowski ([PW2]).

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