Twisted local systems solve the (holographic) loop equation of large-N QCD_4

We construct a holographic map from the loop equation of large-N QCD in d=2 and d=4, for planar self-avoiding loops, to the critical equation of an equivalent effective action. The holographic map is based on two ingredients: an already proposed holographic form of the loop equation, such that the quantum contribution is reduced to a regularized residue; a new conformal map from the region encircled by the based loop to a cuspidal fundamental domain in the upper half-plane, such that the regularized residue vanishes at the cusp. As a check, we study the first coefficient of the beta function and that part of the second coefficient which arises from the rescaling anomaly, in passing from the Wilsonian to the canonically normalised (holographic) effective action.