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 the evaluation of 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 regu-larized residue vanishes at the cusp which is the image of the base point of the loop. The critical equation of the holographic effective action determines a unitary Abelian local system in d = 2 and a non-Abelian twisted local system in d = 4. As a check in the d = 2 theory, we study the distribution of eigenvalues of the Wilson loop implied by the critical equation. As a check in the d = 4 theory, we study the first coefficient of the beta function implied by the holographic loop equation and, as a preliminary step, that part of the second coefficient which arises from the rescaling anomaly, in passing from the Wilsonian to the canonically normalised (holographic) effective action.