Casimir forces between beads on strings and membranes

We develop a general formalism to calculate the force between beads attached to a flat d-dimensional membrane due to the quantum fluctuations of the membrane. The interaction potential is derived as a function of d and the membrane energy density, tension, stiffness and temperature. We find that the induced interactions turn off when d exceeds a certain critical dimension. The potential is attractive in all cases where it is non-zero and at finite temperature falls off exponentially for large distances.