What crossing symmetry forbids just above c=1
Abstract
What does a unitary two-dimensional conformal field theory look like just above c=1? Using the numerical conformal bootstrap, we study the four-point function of a scalar Virasoro primary ϕ such that no primary lighter than ϕ appears in its operator product expansion with itself. Slightly above c=1, crossing symmetry excludes the interval 0.17≲Δϕ≲1, although the compact boson realizes every such Δϕ at c=1. Combining this exclusion with results from the modular bootstrap, we show that the lightest primary of a compact unitary conformal field theory at c=1.0005 or $1.001$ has dimension Δgap<0.17. Further from c=1, the largest excluded central charge at fixed Δϕ has two sharp minima near Δϕ=1/2 and 4/7. At these values it lies within 2×10−7 of c=8/7, where the ℤ5 parafermion and its charge-conjugation orbifold have scalars of exactly these dimensions.
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BibTeX
@misc{nakayama2026what,
title = {{What crossing symmetry forbids just above $c=1$}},
author = {Yu Nakayama},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03217},
eprint = {2611.03217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03217},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - What crossing symmetry forbids just above c=1
AU - Yu Nakayama
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03217
UR - https://arxiv.org/abs/2611.03217
AB - What does a unitary two-dimensional conformal field theory look like just above c=1? Using the numerical conformal bootstrap, we study the four-point function of a scalar Virasoro primary ϕ such that no primary lighter than ϕ appears in its operator product expansion with itself. Slightly above c=1, crossing symmetry excludes the interval 0.17≲Δ_ϕ≲1, although the compact boson realizes every such Δ_ϕ at c=1. Combining this exclusion with results from the modular bootstrap, we show that the lightest primary of a compact unitary conformal field theory at c=1.0005 or $1.001$ has dimension Δ_gap<0.17. Further from c=1, the largest excluded central charge at fixed Δ_ϕ has two sharp minima near Δ_ϕ=1/2 and 4/7. At these values it lies within 2×10^−7 of c=8/7, where the ℤ_5 parafermion and its charge-conjugation orbifold have scalars of exactly these dimensions.
ER -