Boundary reflection matrices of massive $φ_{1,3}$-perturbed unitary minimal models

TL;DR

The paper addresses the problem of determining boundary reflection matrices for the massive $A_m+(r,s)$ theories arising from integrable perturbations by bulk and boundary $φ_{1,3}$ operators. It develops a construction where reflection matrices are formed as direct sums of elementary Behrend-Pearce BYBE solutions and then matches boundary vacua to edges of the $(r,s)$ CBCs in the ABF lattice, ensuring consistency with the boundary bootstrap, the modified crossing, and non-invertible symmetries. Key contributions include explicit paired reflection matrices for general $(r,s)$, demonstrations of covariance under height-reversal and non-invertible symmetries, and detailed $m=3,4,5$ examples illustrating the boundary flows and structure. The work links bulk RSOS scattering to ABF lattice weights, clarifies how boundary RG flows under the $ ext{φ}_{1,3}$ perturbation are encoded in reflection data, and provides a framework for further TBA/TCSA analyses of these integrable boundary theories.

Abstract

We propose explicit expressions for the boundary reflection matrices of the ${\cal A}_m+(r,s)$ series of massive scattering theories, obtained by perturbing the ${\cal A}_m$ unitary minimal models with $(r,s)$ boundary conditions with both bulk and boundary $φ_{1,3}$ operators. We identify the vacua that live on the boundary with the allowed edges of the $(r,s)$ conformal boundary conditions of the $A_m$ Andrews-Baxter-Forrester model. The boundary reflection matrices are then ``direct sums'' of certain pairs of $A_{m-1}$ Behrend-Pearce solutions of the boundary Yang-Baxter equation and are consistent with the boundary bootstrap and the recently-introduced crossing, as well as the $Z_{2}$ (height-reversal), Kac table and non-invertible symmetries.

Figures (2)

• Figure 1: Schematic representation of the one-parameter family of flows of the ${\cal A}_m$ unitary minimal model as described by the boundary reflection matrix. These flows are parametrized by the boundary parameter $\xi$ and the source of the flows is associated with an initial conformal boundary condition $\hbox{CBC}_1=(r,s)$. The boundary RG flow between conformal fixed points $\hbox{CBC}_1$ and $\hbox{CBC}_2$ is on the left. On the right is a line of massive fixed points parametrized by $\xi$. The mass is introduced via the relevant perturbation (\ref{['bulkaction']}) with coupling $\hat{\lambda}$.
• Figure 2: Bulk $S$ matrix $S_{a\ b}^{d\ c}(\theta_{1}-\theta_{2})$.