2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/27426We investigate the Kosterlitz-Thouless transition for hexatic order on a free fluctuating membrane and derive both a Coulomb gas and a sine-Gordon Hamiltonian to describe it. In the former, both disclinations and Gaussian curvature contribute to the charge density. In the latter, there is a linear coupling between a scalar field and the Gaussian curvature. We derive renormalization group recursion relations that predict a transition with decreasing bending rigidity $κ$. Using the sine-Gordon model, we show that there is no KT transition on a deformable sphere.REVTEX, 11 pages with 2 postscript figures compressed using uufilesCondensed Mattertext