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 $Δ_{\Op}\in\Z_{\geq0}\cup(\Z_{\geq0}+\frac{1}{8}),$ to be the lowest $\Loper{0}+\Loperbar{0}$ eigenvalue in the decomposition of its image into homogeneous components. We show that as the mesh size $δ$ goes to zero, the rescaled field $δ^{-Δ_{\Op}}\Op$ 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 $\Loper{0}$ and $\Loperbar{0}$ 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.