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

TL;DR

This paper derives conformal Ward identities for light-like Wilson loops in N=4 SYM, demonstrating their role in fixing the finite parts of Wilson loops and providing evidence for the duality with gluon amplitudes through explicit two-loop calculations.

Contribution

It derives all-loop conformal Ward identities for Wilson loops and confirms their predictions with two-loop calculations, supporting the Wilson loop/gluon amplitude duality.

Findings

Ward identities fix the finite part of Wilson loops for n=4,5

Two-loop calculations match Ward identity predictions

Results support the Wilson loop/gluon amplitude duality

Abstract

Planar gluon amplitudes in N=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=4 and n=5 and reduce the freedom in it to a function of conformal invariants for n>=6. We also present an explicit two-loop calculation for n=5. The result confirms the form predicted by the Ward identities and exactly matches the finite part of the two-loop five-gluon planar MHV amplitude. This constitutes another non-trivial test of the Wilson loop/gluon amplitude duality.