Determinants of Dirichlet and Neumann Laplacians on Constant Curvature Surfaces with Geodesic Polygonal Boundary

We express the zeta regularized determinants of the Dirichlet and Neumann Laplacians on compact Riemann surfaces with constant curvature metrics and geodesic polygonal boundary in terms of the Laplacian determinant on their closed conical doubles. The formulae apply without restrictions on the genus or number of boundary components. They involve explicit corner angle factors and a quotient of the Reidemeister and $L^2$ norms on the determinant line of cohomology. The proof combines a relative BFK gluing formula with the comparison theorem for analytic and combinatorial torsion on manifolds with boundary, due to Lück and Vishik. As an application, we give a rigorous proof of the Aurell-Salomonson formula for Euclidean polygons. We also obtain explicit determinants for spherical and hyperbolic triangles and certain other geodesic polygons from known formulae for their doubles.