Moduli space coordinates and excited state g-functions

We consider the space of boundary conditions of Virasoro minimal models formed from the composition of a collection of flows generated by phi_{1,3}. These have recently been shown to fall naturally into a sequence, each term having a coordinate on it in terms of a boundary parameter, but no global parameter has been proposed. Here we investigate the idea that the overlaps of particular bulk states with the boundary states give natural coordinates on the moduli space of boundary conditions. We find formulae for these overlaps using the known thermodynamic Bethe Ansatz descriptions of the ground and first excited state on the cylinder and show that they give a global coordinate on the space of boundary conditions, showing it is smooth and compact as expected.