A Proof of the Supersymmetric Correlation Function / Wilson Loop Correspondence

TL;DR

This work proves, at the level of the integrand, a supersymmetric correlation function–Wilson loop correspondence in N=4 SYM: in the null-separation limit, the ratio of certain n-point correlators to their tree-level form equals the expectation value of a supersymmetric adjoint Wilson loop on twistor space. The authors construct a twistor-space operator representing the chiral half of the Konishi multiplet and formulate a supersymmetric twistor Wilson loop using holomorphic Wilson lines along CP^1s in the supertwistor space. The core result demonstrates that all surviving integrand contributions come from adjacent-line contractions, collapsing to the adjoint Wilson loop on a nodal curve, and in the planar limit this reproduces the square of the full planar superamplitude integrand. This establishes a robust, supersymmetric, twistor-space bridge between local operators, Wilson loops, and scattering amplitudes, with implications for regulator choices and operator flexibility.

Abstract

We prove that in the limit when its insertion points become pairwise null-separated, the ratio of certain n-point correlation functions in N=4 SYM is equal to a supersymmetric Wilson loop on twistor space, acting in the adjoint representation. In the planar limit, each of these objects reduces to the square of the complete n-particle planar superamplitude. Our proof is at the level of the integrand.

Figures (3)

• Figure 1: In a non-Abelian theory, the twistor space form of the local space-time operator ${\rm Tr}\,\Phi^2$ involves holomorphic Wilson lines (or covariant propagators) on the Riemann sphere X. (The lines are intended to indicate only the colour flow; the propagation between the points is non-local on the sphere and not given by propagation along real curves.)
• Figure 2: The $n$ generic points $(x,\theta)$ correspond to $n$ lines ($\mathbb{CP}^1$s) in $\mathbb{CP}^{3|4}$. In the limit that $(x_i-x_{i+1})\to\tilde{\lambda}_i\lambda_i$ and $\theta_i-\theta_{i+1} \to \eta_i\lambda_i$, these lines intersect in supertwistor space, forming a nodal curve $C$.
• Figure 3: The only non-vanishing contribution to the integrand ratio in the null limit comes from direct contractions between $\phi$s on adjacent Riemann spheres. The twistor propagators freeze the locations of the $\phi$s on each X$_i$. In the limit that the lines intersect, these locations are the intersection points. The remaining operator is the supersymmetric twistor Wilson loop, acting in the adjoint.