On a polygon version of Wiegmann-Zabrodin formula

Abstract

Let $P$ be a convex polygon in $\mathbb{C}$ and let ${\Delta }_{D,P}$ be the operator of the Dirichlet boundary value problem for the Lapalcian $\Delta =-4{\partial }_z{\partial }_{\bar{z}}$ in $P$. We derive a variational formula for the logarithm of the $\zeta$-regularized determinant of ${\Delta }_{D,P}$ for arbitrary infinitesimal deformations of the polygon $P$ in the class of polygons (with the same number of vertices). For a simply connected domain with smooth boundary such a formula was recently discovered by Wiegmann and Zabrodin as a non obvious corollary of the Alvarez variational formula, for domains with corners this approach is unavailable (at least for those deformations that do not preserve the corner angles) and we have to develop another one.