Kosterlitz-Thouless Transitions on Fluctuating Surfaces

We 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 $\kappa$. Using the sine-Gordon model, we show that there is no KT transition on a deformable sphere.