Exactly solvable magnet of conformal spins in four dimensions

TL;DR

This paper introduces a new exactly solvable quantum chain model of conformal spins in four dimensions, providing explicit eigenfunctions and spectral measures, enabling direct computation of conformal data in fishnet theories.

Contribution

It presents explicit eigenfunctions and spectral measures for a conformal spin chain, connecting integrable models with four-dimensional conformal field theories.

Findings

Eigenfunctions explicitly constructed for the conformal spin chain.

Spectral measure computed using a new star-triangle identity.

Eigenfunctions form a complete set for computing conformal data.

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.