On an infinitesimal Polyakov formula for genus zero polyhedra

Let $X$ be a genus zero compact polyhedral surface (the Riemann sphere equipped with a flat conical metric $m$). We derive the variational formulas for the determinant of the Laplacian, $\mathrm{d}\mathrm{e}\mathrm{t}{\mathrm{\Delta }}^m$, on $X$ under infinitesimal variations of the positions of the conical points and the conical angles (i. e. infinitesimal variations of $X$ in the class of polyhedra with the same number of vertices). Besides having an independent interest, this derivation may serve as a somewhat belated mathematical counterpart of the well-known heuristic calculation of $\mathrm{d}\mathrm{e}\mathrm{t}{\mathrm{\Delta }}^m$ performed by Aurell and Salomonson in the 90-s.