The Wilson loop — large spin OPE dictionary

Abstract

We work out the map between null polygonal hexagonal Wilson loops and spinning three point functions in large N conformal gauge theories by mapping the variables describing the two different physical quantities and by working out the precise normalization factors entering this duality. By fixing all the kinematics we open the ground for a precise study of the dynamics underlying these dualities — most notably through integrability in the case of planar maximally supersymmetric Yang-Mills theory.

Spinning hexagons

“…In [8] we considered the null-hexagon limit of six point functions of 20 operators and its respective snowflake OPE decomposition in terms of three large spin operators to derive a map between the expectation value of null hexagonal Wilson loops (WL) and the large spin limit of the three point functions considered in this paper. Combining [8] and this work, we can write a sharp formula expressing the expectation value of the null WL in any geometrical configuration to the hexagons expansion: 13 In the strict large spin limit we hope to have alternative techniques, see discussion below. 14 Some corners of the cross ratios space parametrized by the null snowflake OPE limit of the 6-pt function should be controlled by the abelian structure constants.…”

“…These operators are the same type of spinning operators we consider in the conformal bootstrap [8], being the only difference here that we do not need to consider leading twist, in fact, here they can have any twist τ , which we leave implicit. By conformal symmetry, the full three point function can be recovered from its values in the conformal frame.…”

“…By conformal symmetry, the full three point function can be recovered from its values in the conformal frame. 8 We place operators along the x 2 axis at 0, 1 and ∞. The rescaled correlator…”

“…We thus suspect this to be the case for the case of three twist two operators with very large spin and generic polarizations. Evaluating this kind of correlator should beautifully connect to the remarkable Wilson loop dualities [8,11,12]. So if our goal is to eventually prove these dualities from integrability perhaps we can ignore wrapping for now.…”

“…where the ratios of i and J i are kept fixed and determined in terms of the null hexagon cross ratios U i through (10) of [8]; the boundary conditions of the hexagon partition function H are fixed in terms of U i through (37) of [8]. Above, γ i are the anomalous dimensions of operator O i , while N λ is a spin independent normalization constant to be determined.…”