Boundary structure constants for the A-series Virasoro minimal models

TL;DR

This work derives a complete, field-rescaling–invariant set of boundary and bulk–boundary structure constants for A-series Virasoro minimal models on the upper half-plane by exploiting Lewellen sewing constraints and transforming conformal blocks with $F$-matrices. A recursive formula for the $F$-matrices is provided, enabling practical computation, and bulk–boundary couplings are expressed in terms of $F$-matrix elements and the modular $S$-matrix, yielding explicit formulas such as $^{(a)}B_i^{k}=e^{i\frac{\pi}{2}h_k}(\textsf{F}_{a1}^{a k a k})^{-1}\cdot \frac{S_a^{i}(k)}{S_a^{1}}$ and $C^{(a b c)k}_{ij}=\textsf{F}_{bk}\left[{a \atop i}\;{c \atop j}\right]$. The boundary structure constants reduce to the Pasquier algebra, and the identity one-point normalization is fixed by modular invariance and disc continuity, yielding $\langle 1\rangle^{(a)}=S_a^{1}$ and $g_a= S_a^{\Omega}/\sqrt{S_1^{\Omega}}$. Overall, the results provide a numerically accessible, unique (up to rescalings) BCFT data set for these models, with numerical checks supporting consistency with known bulk data.

Abstract

We consider A-series modular invariant Virasoro minimal models on the upper half plane. From Lewellen's sewing constraints a necessary form of the bulk and boundary structure constants is derived. Necessary means that any solution can be brought to the given form by rescaling of the fields. All constants are expressed essentially in terms of fusing (F-) matrix elements and the normalisations are chosen such that they are real and no square roots appear. It is not shown in this paper that the given structure constants solve the sewing constraints, however random numerical tests show no contradiction and agreement of the bulk structure constants with Dotsenko and Fateev. In order to facilitate numerical calculations a recursion relation for the F-matrices is given.