  For singular metrics, there is no Quillen metric formalism on cohomology
determinant. In this paper, we develop an admissible theory, with which the
arithmetic Deligne-Riemann-Roch isometry can be established for singular
metrics. As an application, we first study Weil-Petersson metrics and
Takhtajan-Zograf metrics on moduli spaces of punctured Riemann surfaces, and
then give a more geometric interpretation of our determinant metrics in terms
of Selberg zeta functions. We end this paper by proposing an arithmetic
factorization for Weil-Petersson metrics, cuspidal metrics and Selberg zeta
functions.
