Exactly solvable magnet of conformal spins in four dimensions

Abstract: We provide the eigenfunctions for a quantum chain of $N$ conformal spins with nearest-neighbor interaction and open boundary conditions in the irreducible representation of $SO(1,5)$ of scaling dimension $\Delta = 2 - i \lambda$ and spin numbers $\ell=\dot{\ell}=0$. The spectrum of the model is separated into $N$ equal contributions, each dependent on a quantum number $Y_a=[\nu_a,n_a]$ which labels a representation of the principal series. The eigenfunctions are orthogonal and we computed the spectral measure by means of a new star-triangle identity. Any portion of a conformal Feynmann diagram with square lattice topology can be represented in terms of separated variables, and we reproduce the all-loop "fishnet" integrals computed by B. Basso and L. Dixon via bootstrap techniques. We conjecture that the proposed eigenfunctions form a complete set and provide a tool for the direct computation of conformal data in the fishnet limit of the supersymmetric $\mathcal{N}=4\,$ Yang-Mills theory at finite order in the coupling, by means of a cutting-and-gluing procedure on the square lattice.