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We derive the anomalous conformal Ward identities for ${\cal N}=4$ SYM Wilsonloops on polygon-like contours with edges formed by circular arcs. With asuitable choice of parameterisation they are very similarly to those for localcorrelation functions. Their solutions have a conformally covariant factordepending on the distances of the corners times a conformally invariantremainder factor depending, besides on cross ratios of the corners, on the cuspangles and angles parameterising the torsion of the contours.

On anomalous conformal Ward identities for Wilson loops on polygon-like contours with circular edges