Topological defects on fluctuating surfaces: General properties and the kosterlitz-thouless transition

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. The Coulomb-gas Hamiltonian includes charge densities arising from disclinations and from Gaussian curvature. There is an interaction coupling the difference between these two densities, whose strenght is determined by the hexatic rigidity, and an interaction coupling Gaussian curvature densities arising form the Liouville Hamiltonian resulting from the imposition of a covariant cutoff. In the sine-Gordon Hamiltonian, there is a linear coupling between a scalar field and the Gaussian curvature. We discuss a gauge-invariant correlation function for hexatic order and the dielectric constant of the Coulomb gas. We also derive renormalization-group recursion relations that predict a transition with decreasing bending rigidity κ.