Critical Ising model as the M(4,3) minimal model of conformal field theory
Abstract
We construct a linear map from the space of spin polynomials in the critical Ising model on a lattice to the state space of the M(4,3) minimal model of conformal field theory. The kernel of the map consists precisely of null fields, i.e., spin polynomials that vanish in all correlations. For a polynomial Op which is not a null field, we define its scaling dimension ΔOp∈Z≥0∪(Z≥0+1/8), to be the lowest Loper0+Loperbar0 eigenvalue in the decomposition of its image into homogeneous components. We show that as the mesh size δ goes to zero, the rescaled field δ−ΔOpOp has a non-trivial scaling limit, when inserted in correlations with other such fields. Moreover, all these limits can be expressed in terms of correlations of primary fields of M(4,3) and the Virasoro action in the continuum. The highest weight vectors of M(4,3), the joint eigenvectors of Loper0 and Loperbar0 with the same eigenvalues $0,1/16,1/2$ for both, are the images of the constant $1$, the spin at a single lattice vertex, and the energy at a single lattice edge, respectively. For the proof, we construct a map from a larger lattice field space, accommodating also fermion and disorder insertions, to a direct sum of two fermionic Fock spaces. This direct sum contains contains the state spaces of both the M(4,3) and the free fermion CFT. The main technical ingredient of the proof is a new version of the exact fermionic current action on the space of lattice local fields due to C. Hongler, K.K. and F. Viklund, combined with some new advances in discrete complex analysis. Our results are valid on arbitrary isoradial graphs.
Related
- Region bounds, error margins and certified positivity in the numerical conformal bootstrap; Or the first mathematically rigorous bounds on the 3d Ising CFT critical exponents
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- Haagerup conformal field theory at central charge eight
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BibTeX
@misc{chelkak2026critical,
title = {{Critical Ising model as the $M(4,3)$ minimal model of conformal field theory}},
author = {Dmitry Chelkak and Konstantin Izyurov and Kalle Kytölä},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13705},
eprint = {2610.13705},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13705},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Critical Ising model as the M(4,3) minimal model of conformal field theory
AU - Dmitry Chelkak
AU - Konstantin Izyurov
AU - Kalle Kytölä
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13705
UR - https://arxiv.org/abs/2610.13705
AB - We construct a linear map from the space of spin polynomials in the critical Ising model on a lattice to the state space of the M(4,3) minimal model of conformal field theory. The kernel of the map consists precisely of null fields, i.e., spin polynomials that vanish in all correlations. For a polynomial Op which is not a null field, we define its scaling dimension Δ_Op∈Z_≥0∪(Z_≥0+1/8), to be the lowest Loper0+Loperbar0 eigenvalue in the decomposition of its image into homogeneous components. We show that as the mesh size δ goes to zero, the rescaled field δ^−Δ_OpOp has a non-trivial scaling limit, when inserted in correlations with other such fields. Moreover, all these limits can be expressed in terms of correlations of primary fields of M(4,3) and the Virasoro action in the continuum. The highest weight vectors of M(4,3), the joint eigenvectors of Loper0 and Loperbar0 with the same eigenvalues $0,1/16,1/2$ for both, are the images of the constant $1$, the spin at a single lattice vertex, and the energy at a single lattice edge, respectively. For the proof, we construct a map from a larger lattice field space, accommodating also fermion and disorder insertions, to a direct sum of two fermionic Fock spaces. This direct sum contains contains the state spaces of both the M(4,3) and the free fermion CFT. The main technical ingredient of the proof is a new version of the exact fermionic current action on the space of lattice local fields due to C. Hongler, K.K. and F. Viklund, combined with some new advances in discrete complex analysis. Our results are valid on arbitrary isoradial graphs.
ER -