Bulk-induced D-brane deformations and the string coupling constant

We consider computing the on-shell disk action of open-closed string field theory as a gauge-invariant way of capturing the shift in D-brane tension that is induced by a deformation of the bulk CFT. We study the effect of bulk matter deformations (both marginal and relevant) on a wide range of boundary conditions in a number of CFTs up to subleading (two-loop) order in perturbation theory. In all analyzed examples, we find that the shift in the $g$-function of the matter boundary state is always accompanied by a boundary-independent shift in the string coupling constant, whose leading behaviour is universally proportional to the sphere two-point function of the deforming bulk operator.