Bulk OPE Coefficients of the E-Series Virasoro Minimal Models
Abstract
We present bulk-primary operator-product coefficients for every Virasoro minimal model with an E6, E7, or E8 modular invariant. Dividing by a universal product of chiral generalized-minimal-model OPE coefficients reduces the conformal bootstrap problem to a finite algebraic problem governed by root-of-unity quantum-group 6j symbols. The exceptional factors of the OPE coefficients can be chosen independent of the second Kac indices s, and their dependence on the central charge reduces to finitely many reference cases and explicit phases. A finite computer-assisted calculation yields candidate algebraic s=1 seed data and high-precision numerical evidence for existence and uniqueness modulo primary-field signs. We prove that exact existence and uniqueness for these seeds imply the corresponding result for all Kac labels within the genus-zero bulk locality and crossing equations; the finite seed step remains uncertified. The three families require $18$, $32$, and $107$ representative reduced OPE coefficients for E6, E7, and E8, respectively. We provide reconstruction conventions and coefficient tables for both unitary and nonunitary models. Our calculation assumes only Virasoro symmetry, bulk locality (permutation symmetry) and OPE associativity (crossing equations).
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BibTeX
@misc{lhoste2026bulk,
title = {{Bulk OPE Coefficients of the $E$-Series Virasoro Minimal Models}},
author = {Amaury Lhoste and Jiaxin Qiao and Masahito Yamazaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25215},
eprint = {2609.25215},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25215},
}
RIS
TY - UNPB
TI - Bulk OPE Coefficients of the E-Series Virasoro Minimal Models
AU - Amaury Lhoste
AU - Jiaxin Qiao
AU - Masahito Yamazaki
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2609.25215
UR - https://arxiv.org/abs/2609.25215
AB - We present bulk-primary operator-product coefficients for every Virasoro minimal model with an E_6, E_7, or E_8 modular invariant. Dividing by a universal product of chiral generalized-minimal-model OPE coefficients reduces the conformal bootstrap problem to a finite algebraic problem governed by root-of-unity quantum-group 6j symbols. The exceptional factors of the OPE coefficients can be chosen independent of the second Kac indices s, and their dependence on the central charge reduces to finitely many reference cases and explicit phases. A finite computer-assisted calculation yields candidate algebraic s=1 seed data and high-precision numerical evidence for existence and uniqueness modulo primary-field signs. We prove that exact existence and uniqueness for these seeds imply the corresponding result for all Kac labels within the genus-zero bulk locality and crossing equations; the finite seed step remains uncertified. The three families require $18$, $32$, and $107$ representative reduced OPE coefficients for E_6, E_7, and E_8, respectively. We provide reconstruction conventions and coefficient tables for both unitary and nonunitary models. Our calculation assumes only Virasoro symmetry, bulk locality (permutation symmetry) and OPE associativity (crossing equations).
ER -