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.

66 pages, 24 figures; v2: Corrections to the discussion of spikes and the constant WKB contribution. Main results are unaffected

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• Tailoring Non-Compact Spin Chains
• Fixing the Quantum Three-Point Function
• Algebraic Curves for Long Folded and Circular Winding Strings in AdS5xS5
• The Dark Knight Falters
• Octagons II: Strong Coupling
• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Three-point correlation functions from pulsating strings in AdS${}_5\times S^5$
• Three-point functions of semiclassical string states and conserved currents in AdS_4 x CP^3