On Perturbations of Unitary Minimal Models by Boundary Condition Changing Operators

TL;DR

This work analyzes boundary perturbations in unitary A-series Virasoro minimal models using $\phi_{r,r+2}$ operators on superposed boundaries, including boundary-changing cases, in the $c\to 1$ limit. It derives explicit beta-functions in a perturbative scheme with $y=\tfrac{2}{m+1}$ and maps out the fixed-point structure for two key examples: boundary-changing $\phi_{13}$ perturbations and $\phi_{35}$ perturbations on $(2,p)$. The author identifies nine fixed points in a representative multi-boundary setup and demonstrates that many fixed points organize into flows predicted by a diagrammatic ADE-based rule that corresponds to a lattice Behrend–Pearce construction. The analysis uses the boundary entropy $g$ and one-point bulk functions to constrain endpoints, yielding Cardy-endpoint decompositions and connecting the field-theory flows to lattice realizations. The results provide a perturbative framework for classifying boundary RG endpoints in minimal models and suggest a universal diagrammatic picture for boundary flows across Cardy boundaries, with potential extensions to other series and non-perturbative regimes.

Abstract

In this note we consider boundary perturbations in the A-Series unitary minimal models by phi_{r,r+2} fields on superpositions of boundaries. In particular, we consider perturbations by boundary condition changing operators. Within conformal perturbation theory we explicitly map out the space of perturbative renormalisation group flows for the example phi_{1,3} and find that this sheds light on more general phi_{r,r+2} perturbations. Finally, we find a simple diagrammatic representation for the space of flows from a single Cardy boundary condition.