# Cauchy-Riemann Operators and Selberg Zeta Function

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?

• Why should the determinant exist? – user1688 Aug 30 '16 at 14:54
• @AntonDeitmar I think the determinant exists due to a paper by Quillen Determinants of Cauchy-Riemann operators over a Riemann surface and its generalizations to general families of Dirac-type operators by Bismut and Freed in The analysis of elliptic families I and The analysis of elliptic families II. – QGravity Aug 30 '16 at 17:27
• @AntonDeitmar Also the existence of such determinants is of vital importance for string theory. In usual, the determinant of such operators is defined in terms of determinant of corresponding Laplacians which have Selberg zeta function representations. I just want to check that if there is a direct formula that relates determinant of Cauchy-Riemann operators to the Selberg zeta function. The issue is related to holomorphic factorization of amplitudes in string theory on the moduli space of Riemann surfaces. – QGravity Aug 30 '16 at 21:21
• If I am not confused (I do not know anything about $j$-differentials), the question has been answered in some form by Ray and Singer's second paper on Annals. The setting is the classical $\bar{\partial}$-complex over the Riemann surface, and the result is also pretty involved. I have not read that paper, however. – Bombyx mori Aug 31 '16 at 0:00
• @Bombyxmori Thank you for the suggestion. Actually there they define the determinant of Cauchy-Riemann operator (more precisely Dirac operator) through holomorphic factorization and through its corresponding Laplacian. I am looking for a direct result that relates determinant of Dirac operator in terms of Selberg zeta function or any other well-defined function which is dependent on lenghts of simple closed geodesics. – QGravity Sep 9 '16 at 1:12