Hexagonalization of Correlation Functions

TL;DR

Hexagonalization provides a nonperturbative, integrability-based framework to compute general correlation functions in planar N=4 SYM by decomposing surfaces into hexagon form factors and gluing them with symmetry-determined weights. The authors derive weight factors incorporating cross-ratio dependence, extend the hexagon program from structure constants to higher-point functions, and test the method at one loop for four BPS operators and for a Konishi operator with three BPS partners, including ladder integrals. The results reproduce perturbative data and reveal a Mellin-like representation for mirror contributions, plus a flip-invariance property under different hexagon cuts. Together, the work suggests a powerful, partially nonperturbative link between integrability and conformal correlators, with potential extensions toward nonplanar, higher-loop, and AdS dual interpretations.

Abstract

We propose a nonperturbative framework to study general correlation functions of single-trace operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory at large $N$. The basic strategy is to decompose them into fundamental building blocks called the hexagon form factors, which were introduced earlier to study structure constants using integrability. The decomposition is akin to a triangulation of a Riemann surface, and we thus call it hexagonalization. We propose a set of rules to glue the hexagons together based on symmetry, which naturally incorporate the dependence on the conformal and the R-symmetry cross ratios. Our method is conceptually different from the conventional operator product expansion and automatically takes into account multi-trace operators exchanged in OPE channels. To illustrate the idea in simple set-ups, we compute four-point functions of BPS operators of arbitrary lengths and correlation functions of one Konishi operator and three short BPS operators, all at one loop. In all cases, the results are in perfect agreement with the perturbative data. We also suggest that our method can be a useful tool to study conformal integrals, and show it explicitly for the case of ladder integrals.

Figures (22)

• Figure 1: Hexagonalization of a four-point function: A planar four-point function can be represented as a surface with four holes. The idea of hexagonalization is to cut it into four hexagonal patches as depicted above. The contribution from each patch is given by a hexagon form factor. It is conceptually different from the usual operator product expansion. (The colors of the figure represent the two places where this work was done.)
• Figure 2: Hexagon formalism for three-point functions: The three-point function are represented as a pair of pants which is cut into two hexagons. These two hexagons are separated by bridges of lengths $\ell_{ij}\equiv (L_i+L_j-L_k)/2$ ($L_i$ is the length of $\mathcal{O}_i$). To compute the three-point function, one sums over partitions of physical magnons and the mirror states appearing on the dashed edges. Since the physical and the mirror edges intersect by $90$ degrees, the hexagon has a conical excess $\pi$ in its center.
• Figure 3: Mirror transformation. Left: The mirror transformation is an analytic continuation of the rapidity $(u\to u^{\gamma})$, which allows us to move a particle from one edge to another. Right: Alternative viewpoint on the finite size correction. Inserting a complete basis on a mirror edge is equivalent to dressing neighboring physical edges by virtual particle pairs and making them "entangled".
• Figure 4: An example of tree-level Wick contraction: A magnon $D$ can live in either one of three bridges ($\ell_{12}$, $\ell_{13}$, or $\ell_{14}$). As shown above, the bridges split the diagram into four hexagonal patches, which are depicted in different colors. Each term in the final result (\ref{['treesum2']}) can be interpreted as a contribution from one of those hexagon patches.
• Figure 5: Edge and cross ratio: There is a natural way to associate a cross ratio with each edge. For instance, the cross ratio associated with the black dotted edge is given by the expression above. $\pm$ signs signify whether the corresponding propagators appear in the numerator or in the denominator. In terms of these cross ratios, the weight factor in (\ref{['treesum2']}) reads $\mathcal{W}= 1/z$. This rule of relating an edge and a cross ratio is akin to the definition of the Fock coordinates of the Teichmuller space Fock or the Fock-Goncharov coordinates of the moduli space of flat connections FockGoncharov. Note also that the Fock-Goncharov coordinates show up in the study of four-point functions at strong coupling chi-system.