A free field approach to boundary $\widehat{g}_{k}$ WZW and $\widehat{g}_{k}/\widehat{u(1)}^{r}$ parafermionic theories

Abstract: We apply a free-field approach to boundary $\widehat{g}{k}$ WZW theories, and $\widehat{g}{k}/\widehat{u(1)}^{r}$ parafermionic theories. We review free-field resolution conjectures of dominant irreducible $\widehat{g}{k}$ highest-weight modules and $\widehat{g}{k}/ \widehat{u(1)}^{r}$ modules, apply them to Ishibashi states, and discuss on constraint condition on the free-field gluing automorphisms. We also review how to calculate disk one and two-point functions in $\widehat{g}{k}$ WZW and $\widehat{g}{k}/ \widehat{u(1)}^{r}$ parafermionic theories using free-field vertex operators.