Free-field approaches to boundary $\mathcal{W}\left[\widehat{g}\right](p,p^{\mathrm{\prime }})$ minimal models

We apply the background charged bosonic free-field approach to the rational principal quantum Drinfeld-Sokolov (QDS) $\mathcal{W}\left[\widehat{g}\right](p,p^{\mathrm{\prime }})$ minimal models with boundaries, where $g$ is a finite bosonic simple Lie algebra. Their Ishibashi states are expressed using the free bosonic Ishibashi states, by applying the Fock space resolutions. The Coulomb-gas formalism is applied to the calculations of the disk two-point correlation functions in some well-studied and lesser familiar rational QDS $\mathcal{W}\left[\widehat{g}\right](p,p^{\mathrm{\prime }})$ minimal models. The analytical expressions can be obtained by the repeated applications of the Pochhammer contour integral expression and the Taylor expansions of Lauricella’s hypergeometric functions $F_D^{(n)}$.