Ising model on the affine plane

We demonstrate that the Ising model on a general triangular graph with three distinct couplings K1, K2, and K3 corresponds to an affine transformed conformal field theory (CFT). Full conformal invariance of the c = 1/2 minimal CFT is restored by introducing a metric on the lattice through the map sinh(2K;) = l'; /l; which relates critical couplings to the ratio of the dual hexagonal and triangular edge lengths. Applied to a 2D toroidal lattice, this provides an exact lattice formulation in the continuum limit to the Ising CFT as a function of the modular parameter. This example can be viewed as a quantum generalization of the finite element method applied to the strong coupling CFT at a Wilson-Fisher IR fixed point and suggests a new approach to conformal field theory on curved manifolds based on a synthesis of simplicial geometry and projective geometry on the tangent planes.