Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes

Planar gluon amplitudes in N=4N=4 SYM are remarkably similar to expectation values of Wilson loops made of light-like segments. We argue that the latter can be determined by making use of the conformal symmetry of the gauge theory, broken by cusp anomalies. We derive the corresponding anomalous conformal Ward identities valid to all loops and show that they uniquely fix the form of the finite part of a Wilson loop with n   cusps (up to an additive constant) for n=4n=4 and 5 and reduce the freedom in it to a function of conformal invariants for n⩾6n⩾6. We also present an explicit two-loop calculation for n=5n=5. The result confirms the form predicted by the Ward identities and matches the finite part of the two-loop five-gluon planar MHV amplitude, up to a constant. This constitutes another non-trivial test of the Wilson loop/gluon amplitude duality.