Conformal blocks and braiding matrices for RCFTs I: Virasoro minimal models
Abstract
We develop a systematic method for computing the braiding matrices of four-point functions in RCFTs. This is implemented for the case of Virasoro minimal models. The first few terms of the conformal block series are directly computed through the Shapovalov form. The Fuchsian (BPZ) ODEs for order three and higher have accessory parameters that are fixed by this direct computation. We thus bypass the derivation from the null-state condition which gets tedious for higher-level null states. The braiding F-matrices are connection matrices between the Frobenius solutions at two singular points of the ODE. The F-matrices are computed numerically at first, agreeing with the results of Dotsenko--Fateev, which uses the Coulomb-gas formalism. We find that after a change of basis, the F-matrix is rendered unitary, which determines (products of) the three-point structure constants. We conjecture that the squares of the entries of the unitary F-matrix lie in a cyclotomic extension of the rational numbers. This enables us to convert our numerical estimates for the F-matrix into exact ones. The formalism is illustrated through numerous examples in Virasoro minimal models. We also discuss how these methods can be extended to cases involving symmetries that extend the Virasoro symmetry as well as for tenable examples that arise from the holomorphic modular bootstrap program.
Related
- Virasoro Conformal Blocks and modular functor from Liouville CFT
- virasoro_blocks: Software for the 2d bootstrap
- Bulk OPE Coefficients of the E-Series Virasoro Minimal Models
- Resurgence in the Virasoro minimal string and 3d gravity
- Super Liouville theory on the torus: bootstrap and the super Virasoro minimal string
BibTeX
@misc{govindarajan2026conformal,
title = {{Conformal blocks and braiding matrices for RCFTs I: Virasoro minimal models}},
author = {Suresh Govindarajan and Aditya Jain and Akhila Sadanandan and Prasanta K. Tripathy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.19904},
eprint = {2609.19904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19904},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Conformal blocks and braiding matrices for RCFTs I: Virasoro minimal models
AU - Suresh Govindarajan
AU - Aditya Jain
AU - Akhila Sadanandan
AU - Prasanta K. Tripathy
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.19904
UR - https://arxiv.org/abs/2609.19904
AB - We develop a systematic method for computing the braiding matrices of four-point functions in RCFTs. This is implemented for the case of Virasoro minimal models. The first few terms of the conformal block series are directly computed through the Shapovalov form. The Fuchsian (BPZ) ODEs for order three and higher have accessory parameters that are fixed by this direct computation. We thus bypass the derivation from the null-state condition which gets tedious for higher-level null states. The braiding F-matrices are connection matrices between the Frobenius solutions at two singular points of the ODE. The F-matrices are computed numerically at first, agreeing with the results of Dotsenko--Fateev, which uses the Coulomb-gas formalism. We find that after a change of basis, the F-matrix is rendered unitary, which determines (products of) the three-point structure constants. We conjecture that the squares of the entries of the unitary F-matrix lie in a cyclotomic extension of the rational numbers. This enables us to convert our numerical estimates for the F-matrix into exact ones. The formalism is illustrated through numerous examples in Virasoro minimal models. We also discuss how these methods can be extended to cases involving symmetries that extend the Virasoro symmetry as well as for tenable examples that arise from the holomorphic modular bootstrap program.
ER -