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 $g\geq 1$. We obtain an explicit holomorphic factorization formula for $\zeta$-regularized determinant of the Laplacian in flat conical metrics with trivial holonomy $|w|^2$, generalizing the classical Ray-Singer result in $g=1$. 
A. Kokotov and D. Korotkin