Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray-Singer formula

Let $w$ be an Abelian differential on compact Riemann surface of genus $g\geq 1$\mathrm{.}Weobtainanexplicitholomorphicfactorizationformulafor$\zeta$-regularized determinant of the Laplacian in flat conical metrics with trivial holonomy $\mathrm{\mid }w{\mathrm{\mid }}^2$, generalizing the classical Ray-Singer result in $g=1$.