Free-field approaches to boundary $\mathcal{W} \big[ \widehat{g} \big] (p,p') $ minimal models

TL;DR

The paper develops a comprehensive free-field framework for rational principal quantum Drinfeld-Sokolov $ W[ widehat{g}](p,p')$ minimal models with boundaries, realizing Ishibashi states as free-field objects and computing disk two-point functions via vertex operators and $ W$-intertwiners. Central to the method are BRST-based free-field resolutions, Euler-type integrals, and Lauricella $F_{D}^{(n)}$ expansions that solve BPZ-type equations, with explicit demonstrations across Virasoro, $ W_3$, $ W_4$, and $ W[ widehat{G}_2]$ models. The work also extends to semi-simple and coset constructions, showing factorization of correlation functions and providing coset Ishibashi-state expressions via BRST-projection and intertwiners. These results yield concrete, computable disk correlators for a broad class of diagonal and charge-conjugated $ W$ minimal models, enabling systematic analysis of boundary CFT data in these theories and facilitating connections to ADE coset realizations. Overall, the framework advances exact boundary CFT calculations in $ W$-algebras and clarifies how free-field resolutions encode physical boundary states and fusion properties.

Abstract

We apply the background charged bosonic free-field approach to the rational principal quantum Drinfeld-Sokolov (QDS) $\mathcal{W} \big[ \widehat{g} \big](p,p')$ minimal models with boundaries, where $g$ is a simple bosonic Lie algebra. We express their Ishibashi states using the free-field Ishibashi states, and calculate disk two-point correlation functions using the free-field vertex operators with intertwiner insertions. The integral results are obtained by repeatedly applying the Euler-type integral expression and the Taylor expansions of Lauricella's hypergeometric functions $F_{D}^{(n)}$. We also discuss the background free-field approach to the QDS $\mathcal{W} $ algebras related to a semi-simple Lie algebra $g$, and apply a coset free-field resolution to express diagonal ADE coset $\mathcal{W}$ minimal Ishibashi states.