All-loop Leading Singularities of Wilson Loops

We study correlators of null, $n$-sided polygonal Wilson loops with a Lagrangian insertion in the planar limit of the ${\cal N}=4$ supersymmetric Yang-Mills theory. This finite observable is closely related to loop integrands of maximally-helicity-violating amplitudes in the same theory, and, conjecturally, to all-plus helicity amplitudes in pure Yang-Mills theory. The resulting function has been observed to have an expansion in terms of functions of uniform transcendental weight, multiplied by certain rational prefactors, called leading singularities. In this work we prove several conjectures about the leading singularities: we classify and compute them at any loop order and for any number of edges of the Wilson loop, and show that they have a hidden conformal symmetry. This is achieved by leveraging the geometric definition of the loop integrand via the Amplituhedron. The leading singularities can be seen as maximal codimension residues of the integrand, and the boundary structure of the Amplituhedron geometry restricts which iterative residues are accessible. Combining this idea with a further geometric decomposition of the Amplituhedron in terms of so-called negative geometries allows us to identify the complete set of leading singularities.