2d Ising Critical Couplings from Quantum Gravity

Using an exact holographic duality formula between the inhomogeneous 2d Ising model and 3d quantum gravity, we provide a formula for "real" zeroes of the 2d Ising partition function on finite trivalent graphs in terms of the geometry of a 2d triangulation embedded in the three-dimensional Euclidean space. The complex phase of those zeroes is given by the dihedral angles of the triangulation, which reflect its extrinsic curvature within the ambient 3d space, while the modulus is given by the angles within the 2d triangles, thus encoding the intrinsic geometry of the triangulation. Our formula can not cover the whole set of Ising zeroes, but we conjecture that a suitable complexification of these "real" zeroes would provide a more thorough formula. Nevertheless, in the thermodynamic limit, in the case of flat planar 2d triangulations, our Ising zeros' formula gives the critical couplings for isoradial graphs, confirming its generality. Finally, the formula naturally extends to graphs with arbitrary valence in terms of geometry of circle patterns embedded in 3d space. This approach shows an intricate, but precise, new relation between statistical mechanics and quantum geometry.