Prove the conjectured bulk channel decomposition of disc 2-point functions

Prove that, for critical loop models with a diagonal boundary parameter and bulk fields V_{(r_1,s_1)} and V_{(r_2,s_2)} obeying r_1 + r_2 ∈ ℕ, the disc two-point function equals the sum over P ∈ P_0 + β^{-1}ℤ of C^P_{(r_1,s_1)(r_2,s_2)} ⟨V_P⟩_σ times the corresponding t-channel Virasoro conformal block, as given in Equation (2pt-bulk).

Background

The authors propose an explicit bulk-channel expansion for disc two-point functions based on known bulk CFT input and the diagonal boundary condition solution. This expansion relies on diagonal intermediate states and OPE coefficients inferred from three-point functions.

They support the conjecture numerically in discrete-boundary cases, but a general analytic proof is not provided. Establishing this decomposition would confirm a key structural element of boundary CFT for loop models and underpin further bootstrap analyses.

References

Known results from the bulk CFT suggest that diagonal fields contribute to the bulk OPE with OPE coefficients that coincide with 3-point functions [nrj23]. The bulk OPE therefore leads to an explicit conjecture for the bulk channel decomposition of the bulk 2-point function, We implicitly assume r_1+r_2 \in \mathbb{N}, otherwise the 2-point function vanishes due to the conservation of r modulo integers in the bulk CFT.

— Diagonal boundary conditions in critical loop models  (2512.10400 - Downing et al., 11 Dec 2025) in Introduction, Bulk 2-point function: bulk channel decomposition