χ-systems for correlation functions

Abstract

We consider the strong coupling limit of 4-point functions of heavy operators in N = 4 SYM dual to strings with no spin in AdS. We restrict our discussion for operators inserted on a line. The string computation factorizes into a state-dependent sphere part and a universal AdS contribution which depends only on the dimensions of the operators and the cross ratios. We use the integrability of the AdS string equations to compute the AdS part for operators of arbitrary conformal dimensions. The solution takes the form of TBA-like integral equations with the minimal AdS string-action computed by a corresponding free-energy-like functional. These TBA-like equations stem from a peculiar system of functional equations which we call a χ-system. In principle one could use the same method to solve for the AdS contribution in the N -point function. An interesting feature of the solution is that it encodes multiple string configurations corresponding to different classical saddle-points. The discrete data that parameterizes these solutions enters through the analog of the chemical-potentials in the TBA-like equations. Finally, for operators dual to strings spinning in the same equator in S 5 (i.e. BPS operators of the same type) the sphere part is simple to compute. In this case (which is generically neither extremal nor protected) we can construct the complete, strong-coupling 4-point function.

Correlation functions on the Half-BPS Wilson loop: perturbation and hexagonalization

“…Such operators are dual to the classical open strings and the correlators are described by some nontrivial classical string configuration. Determining the shape of such a configuration is generally a hard problem, but one may be able to compute its area directly without needing to know its shape by applying the methods developed in [16,[48][49][50]. It would also be interesting to look for the analogue of the "simple" four-point functions analyzed in [51,52].…”

“…The most notable success in this regard is the application of integrability to N = 4 supersymmetric Yang-Mills theory (SYM) in four dimensions [6,7]. As a result of relentless efforts in the past fifteen years, the integrability of the worldsheet was applied successfully to the computation of the spectrum [8][9][10][11], the scattering amplitudes [12], the structure constants [13], and the four- [14][15][16][17] and higher-point correlators [17,18] in the planar limit and their first nonplanar corrections [19][20][21][22].…”

“…See e.g [45]. for the analysis of the CFT data in the large charge limit 16. Note that, for three-point functions, the number of contractions between operators is determined completely by the lengths of the operators, ij = (L i + L j − L k )/2.…”

Liouville theory, AdS2 string, and three-point functions

“…Although simple as a concept, it is hardly possible to find such complicated surfaces in practice. Nevertheless, progress can be made by the ingenious use of the integrability: Surprisingly, one can directly compute these observables without knowing the explicit shapes of the surfaces [2][3][4][5] 1 .…”

“…First of all, it would be interesting to generalize our analysis to the four-point function and derive a functional relation governing the classical conformal blocks. For strings in AdS 2 , this was already achieved in [5], and it would not be so difficult to perform the same analysis in the Liouville theory since it is actually simpler 27 . It is also interesting to study correlation functions of irregular vertex operators [30].…”