The determinant of hyperbolic Maaß-Laplacian operator on arbitrary tensors and spinors can be written in terms of Selberg zeta function. Is there a corresponding formula for the determinant of the Cauchy-Riemann operators $\det\overline{\partial}_j$ which act on the space of $j$-differentials (i.e. arbitrary tensors and spinors) in terms of Selberg zeta function?

Determinants of Cauchy-Riemann operators over a Riemann surfaceand its generalizations to general families of Dirac-type operators by Bismut and Freed inThe analysis of elliptic families IandThe analysis of elliptic families II. $\endgroup$ – QGravity Aug 30 '16 at 17:271more comment