Tailoring three-point functions and integrability

Abstract

We use Integrability techniques to compute structure constants in N = 4 SYM to leading order. Three closed spin chains, which represent the single trace gauge-invariant operators in N = 4 SYM, are cut into six open chains which are then sewed back together into some nice pants, the three-point function. The algebraic and coordinate Bethe ansatz tools necessary for this task are reviewed. Finally, we discuss the classical limit of our results, anticipating some predictions for quasi-classical string correlators in terms of algebraic curves.

Computing three-point functions for short operators

“…However, there are indications that integrability will play a crucial role in determining the structure constants. In particular, there have been important results at low orders of perturbation theory that crucially rely on the integrable structure in the theory [14][15][16][17][18]. Impressively, in [15] it was shown that the weak-coupling three-point function for two semi-classical operators (with ∆ ∼ √ λ) and one short BPS operator, was consistent with the strong-coupling results derived from a semi-classical string calculation in [19,20].…”

“…Using the formulas for these correlators from appendix C.4, we get in this case a more complicated result 14) where…”

“…Using that t T,M A,M B is orthogonal to k and (Γ M C t T,M,N ) = 0, eliminating terms with ∂φ 6 usinḡ ∂φ 6ψ M e −φ 6 e ik·X = ik·∂Xψ M e −φ 6 e ik·X +∂ψ M e −φ 6 e ik·X , (B. 13) and noting that the following are spurious terms [34] 14) JHEP03(2014)096…”

Quark-anti-quark potential in N $$ \mathcal{N} $$ = 4 SYM

“…Finally, as the ladders limit allows for a simpler access to the wrapping corrections, it could also serve as a useful ground to attempt a finite-size resummation of perturbation theory for 3-point correlators [43][44][45][46][47], using Q-functions as building blocks.…”

Semiclassical correlation functions of Wilson loops and local vertex operators

“…The study and comparison of two-point functions on both sides of the AdS/CFT correspondence was crucial in order to uncover the integrable structure underlying the duality and magnify our understanding on the spectrum of anomalous dimensions. Comparison of three-point functions of non-protected local operators at weak and strong-coupling has in fact been started recently in [36][37][38][39] and has inspired exhaustive spectroscopy of three-point correlators [40], with the motivation to illuminate the possible role played by integrability in the general structure of the complete spectrum of the theory. As all order results have been exhibited for Wilson loop observables in the gauge theory [41] it could also be expected that the study of more general correlation functions with Wilson loops could be of help to clarify the operator product expansion.…”