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We construct a holographic map from the loop equation of large-N QCD in d=2and d=4, for planar self-avoiding loops, to the critical equation of anequivalent effective action. The holographic map is based on two ingredients:an already proposed holographic form of the loop equation, such that thequantum contribution is reduced to a regularized residue; a new conformal mapfrom the region encircled by the based loop to a cuspidal fundamental domain inthe 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 partof the second coefficient which arises from the rescaling anomaly, in passingfrom the Wilsonian to the canonically normalised (holographic) effectiveaction.Comment: 42 pages, latex; abstract shortened and a reference added as  suggested by the referee; typos in Eq.(72,76,77,82,83,84) correcte

Twisted local systems solve the (holographic) loop equation of large-NQCD4

Twisted local systems solve the (holographic) loop equation of large-NQCD₄