Holomorphic factorization of determinants of Laplacians using quasi-Fuchsian uniformization

Abstract

For a quasi-Fuchsian group $\Ga$ with ordinary set $\Omega$, and ${\Delta }_n$ the Laplacian on \n differentials on $\Ga\bkΩ$, we define a notion of a Bers dual basis ${\varphi }_1,...c,{\varphi }_{2d}$ for $\ker {\Delta }_n$. We prove that $\det {\Delta }_n/\det <{\varphi }_j,{\varphi }_k>$, is, up to an anomaly computed by Takhtajan and the second author in \cite{TT1}, the modulus squared of a holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the Selberg zeta Z(n). This generalizes the D'Hoker-Phong formula $\det {\Delta }_n=c_{g,n}Z(n)$, and is a quasi-Fuchsian counterpart of the result for Schottky groups proved by Takhtajan and the first author in \cite{MT}.

15 pages

References in corpus (1)

• Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography

Cited by in corpus (2)

• Averaging Over Narain Moduli Space
• Semiclassical 3D gravity as an average of large-c CFTs