From Polygon Wilson Loops to Spin Chains and Back

TL;DR

The paper develops a multiparticle Operator Product Expansion for null polygon Wilson loops in planar N=4 SYM by mapping loop observables to two-point functions of local operators, enabling a flux-tube Hamiltonian kernel to act on arbitrary numbers of excitations. It leverages integrability via SL(2) spin chains and monodromy/R-matrix structures to generate conserved charges and define weak-coupling holonomies that analogize strong-coupling flat connections. Through explicit NMHV and N^2MHV octagon and dodecagon examples, the authors bootstrap one- and two-loop OPE discontinuities and demonstrate a precise WC-SC match in dense-particle limits, akin to a Frolov-Tseytlin limit. The work thus provides a concrete framework to relate weak- and strong-coupling descriptions and lays groundwork for all-loop, finite-coupling formulations of Wilson loop amplitudes using integrability and monodromy data.

Abstract

Null Polygon Wilson Loops (WL) 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 (FT). That picture is valid at any value of the 't Hooft coupling. So far it has been efficiently used at weak coupling (WC) 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 FT. A major open problem in this approach was how to deal with generic multi-particle states at WC. In this paper we study the propagation of any number of FT excitations at WC. We do this by first mapping the WL into a sum of two point functions of local operators. This map allows us to translate the integrability techniques developed for the spectrum problem back to the WL. E.g., the FT Hamiltonian can be represented as a simple kernel acting on the loop. Having an explicit representation for the FT 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 FT is integrable and therefore has other (infinite set of) conserved charges. The generating function of conserved charges is constructed from the monodromy (M) matrix between sides of the polygon. We compute it for some simple examples at leading order at WC. At strong coupling (SC), these Ms were the main ingredients of the Y-system solution. To connect the WC and SC computations, we study a case where an infinite number of particles are propagating already at leading order at WC. We obtain a precise match between the WC and SC Ms. That match is the WL analogue of the well known Frolov-Tseytlin limit where the WC and SC descriptions become identical. Hopefully, putting the WC and SC descriptions on the same footing is the first step in understanding the all loop structure.

Figures (14)

• Figure 1: The OPE picture. Flux tube excitations are created at the bottom and absorbed in the top. The flux is thick in spacetime but is 1+1 dimensional in AdS. Its excitations are integrable.
• Figure 2: At strong coupling the polygon Wilson loop expectation value is computed by a minimal surface area. The surface ends on the polygon at the boundary of AdS and is stretched in the AdS radial direction. The holonomy $T(u)$ of the flat connection between two edges of the polygon is drawn in red. It is a measure of all the higher conserved charges of what is flowing through it. Such holonomies are the building blocks in the Y-system strong coupling solution Alday:2010vh.
• Figure 3: A relation between a correlation function of two Wilson lines with insertions and a polygonal Wilson loop. On the left we have a correlation function of two Wilson lines with insertions ${\cal W}_\text{bot}$ and ${\cal W}_\text{top}$. We choose these to be composed of a sequence of null lines. At their tips we have scalar insertions indicated by blue dots in the figure. We take a limit where each one of the two scalars in ${\cal W}_\text{bot}$ becomes null separated from a conjugate scalar in ${\cal W}_\text{top}$. In that null limit the correlator develops a double pole due to two fast particles going along the left and right edges in the figure, indicated by the dashed lines. The residue of the double pole singularity is computed by two copies of a polygon Wilson loop. The polygon loop is composed of the two open curves in ${\cal W}_\text{bot}$ and ${\cal W}_\text{top}$, closed by the right and left edges into a closed loop.
• Figure 4: The setup of Wilson correlators considered in this paper. (b) On the right we have a square representing the OPE vacuum. In that case, the bottom and top operators, ${\cal W}_\text{top}^{(\text{square})}$ and ${\cal W}_\text{bot}^{(\text{square})}$, are composed of a Wilson line along a single null line connecting two scalars. (a) On the left we replace that single null line by a sequence of null lines connecting the two cusps in the bottom or top of the square.
• Figure 5: There is a simple class of N$^k$MHV amplitudes which we use as a laboratory for exploring the OPE physics. We denoted such components as $Z \to \bar{Z}$ components, see (\ref{['equation5']}). At tree level they are given by $k$ insertions of scalars $Z$ at the bottom cusps of the polygon and another $k$ insertions of the complex conjugate scalars $\bar{Z}$ at the top cusps. At tree level they are given by exactly$k$ free scalars flux tube excitations going from the bottom to the top part of the polygon. At one loop these particles start interacting and feeling the flux. Such components are pure examples of multiparticles in the OPE. Figure (a): NMHV octagon, (b): N$^2$MHV octagon, (c): N$^2$MHV dodecagon. The last two correspond to two particle examples. These three examples will be discussed in detail in this paper and illustrate the general method.