Variational formulas for determinant of Laplacian on higher genus polyhedral surface

Let $X$ be a Riemann surface of genus $g\ge 1$ endowed with a flat conical metric $m$ and let $\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{\Delta }$ be the $\zeta$-regularized determinant of the Friedrichs Laplacian on $(X,m)$. We derive variational formulas for $\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{\Delta }$ with respect to conical points and conical angles within a given conformal class. Integration of them leads to an explicit expression for $\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{\Delta }$ up to moduli dependent factor. The latter, in principle, can be calculated via comparison of the above result with the well-known formulas for the case of flat conical metrics with trivial holonomy.