A Worldsheet Derivation of the Classical Off-shell Boundary Action for the Dilaton in Half-Space

We use the method of images to present a worldsheet derivation of the sphere partition function for the dilaton in half-space to leading order in $\text{apr}$ with Neumann boundary conditions. We use Tseytlin’s sphere prescription to obtain the total (bulk and boundary) off-shell classical bosonic string action for the dilaton in half-space and show that it satisfies the requirement for a well-defined variational principle