Determine boundary-channel structure constants for discrete diagonal boundaries

Derive analytic expressions for the boundary-channel structure constants D_k appearing in the disc two-point functions of bulk fields in critical loop models with discrete diagonal boundary conditions characterized by σ = P_{(0,S)} and boundary spectrum ℳS_S.

Background

For discrete diagonal boundaries, the boundary spectrum is known and comprises degenerate boundary fields. However, the associated boundary structure constants D_k are not determined analytically.

The authors use numerical bootstrap to solve crossing and infer many D_k (including vanishing patterns), but a general analytic derivation is lacking. Providing formulas for D_k would complete the boundary-data needed for exact disc correlators.

References

From now on we focus on discrete boundaries, whose boundary parameters are \sigma=P_{(0,S)} with S\in\mathbb{N}*. We know the boundary spectrum ss, which is discrete, but we still do not know the structure constants D_k.

— Diagonal boundary conditions in critical loop models  (2512.10400 - Downing et al., 11 Dec 2025) in Section 3.2 Boundary channel decomposition