From Polygon Wilson Loops to spin chains and back

Null Polygon Wilson Loops in N=4 SYM can be computed using the Operator Product Expansion in terms of a transition amplitude on top of a color Flux tube. That picture is valid at any value of the 't Hooft coupling and is studied here in the planar limit. So far it has been efficiently used at weak coupling in cases where only a single particle is flowing. At any finite value of the coupling however, an infinite number of particles are flowing on top of the color flux tube. A major open problem in this approach was how to deal with generic multi-particle states at weak coupling. In this paper we study the propagation of any number of flux tube excitations at weak coupling. We do this by first mapping the Wilson loop expectation value into a sum of two point functions of local operators. That map allows us to translate the integrability techniques developed for the spectrum problem back to the Wilson loop. In particular, we find that the flux tube Hamiltonian can be represented as a simple kernel acting on the loop. Having an explicit representation for the flux tube Hamiltonian allows us to treat any number of particles on an equal footing. We use it to bootstrap some simple cases where two particles are flowing, dual to N²MHV amplitudes. The flux tube is integrable and therefore has other (infinite set of) conserved charges. The generating function of all of these charges is constructed from the monodromy matrix between sides of the polygon. We compute it for some simple examples at leading order in perturbation theory. At strong coupling, these monodromies were the main ingredients of the Y-system solution. To connect the weak and strong coupling computations, we study a case where an infinite number of particles are propagating already at leading order in perturbation theory. We obtain a precise match between the weak and strong coupling monodromies. That match is the Wilson loop analog of the well known Frolov-Tseytlin limit where the strong and weak coupling descriptions become identical. Hopefully, putting the weak and strong coupling descriptions on the same footing is the first step in understanding the all loop structure.