Null Polygons in Conformal Gauge Theory.

We consider correlation functions of single trace operators approaching the cusps of null polygons in a double-scaling limit where so-called cusp times t_{i}^{2}=g^{2}logx_{i-1,i}^{2}logx_{i,i+1}^{2} are held fixed and the 't Hooft coupling is small. With the help of stampedes, symbols, and educated guesses, we find that any such correlator can be uniquely fixed through a set of coupled lattice PDEs of Toda type with several intriguing novel features. These results hold for most conformal gauge theories with a large number of colors, including planar N=4 SYM.

Exactly Solvable Magnet of Conformal Spins in Four Dimensions.

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 Δ=2-iλ and spin numbers ℓ=ℓ[over ˙]=0. The spectrum of the model is separated into N equal contributions, each dependent on a quantum number Y_{a}=[ν_{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 N=4 Yang-Mills theory at finite order in the coupling, by means of a cutting-and-gluing procedure on the square lattice.

Biscalar Integrable Conformal Field Theories in Any Dimension.

We propose a D-dimensional generalization of 4D biscalar conformal quantum field theory recently introduced by Gürdogan and one of the authors as a particular strong-twist limit of γ-deformed N=4 supersymmetric Yang-Mills theory. Similar to the 4D case, the planar correlators of this D-dimensional theory are conformal and dominated by "fishnet" Feynman graphs. The dynamics of these graphs is described by the integrable conformal SO(1,D+1) spin chain. In 2D, it is the analogue of Lipatov's SL(2,C) spin chain for the Regge limit of QCD but with the spins s=1/4 instead of s=0. Generalizing recent 4D results of Grabner, Gromov, Korchemsky, and one of the authors to any D, we compute exactly at any coupling a four-point correlation function dominated by the simplest fishnet graphs of cylindric topology and extract from it exact dimensions of operators with chiral charge 2 and any spin together with some of their operator product expansion structure constants.