Clustering and the Three-Point Function

Abstract

We develop analytical methods for computing the structure constant for three heavy operators, starting from the recently proposed hexagon approach. Such a structure constant is a semiclassical object, with the scale set by the inverse length of the operators playing the role of the Planck constant. We reformulate the hexagon expansion in terms of multiple contour integrals and recast it as a sum over clusters generated by the residues of the measure of integration. We test the method on two examples. First, we compute the asymptotic three-point function of heavy fields at any coupling and show the result in the semiclassical limit matches both the string theory computation at strong coupling and the tree-level results obtained before. Second, in the case of one non-BPS and two BPS operators at strong coupling we sum up all wrapping corrections associated with the opposite bridge to the non-trivial operator, or the "bottom" mirror channel. We also give an alternative interpretation of the results in terms of a gas of fermions and show that they can be expressed compactly as an operator-valued super-determinant.

52 pages + a few appendices; v2 typos corrected

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Cited by in corpus (27)

• Gluing Ladders into Fishnets
• Structure constants at wrapping order
• Determinant formula for the octagon form factor in $\mathcal{N}$=4 SYM
• Ten dimensional symmetry of N=4 SYM correlators
• Octagon at finite coupling
• Structure Constants in $\mathcal{N}=4$ SYM at Finite Coupling as Worldsheet $g$-Function
• Crossing bridges with strong Szego limit theorem
• From the octagon to the SFT vertex - gluing and multiple wrapping
• Asymptotic Four Point Functions
• Giant Wilson Loops and AdS${}_2$/dCFT${}_1$
• Conical Twist Fields and Null Polygonal Wilson Loops
• Structure constants of short operators in planar $\mathcal{N}=4$ SYM theory
• Functional Equations and Separation of Variables for Exact g-Function
• Wilson Loop Duality and OPE for Super Form Factors of Half-BPS Operators
• A numerical study of the F-model with domain-wall boundaries
• Yangian Symmetry in Holographic Correlators
• ABJM quantum spectral curve and Mellin transform
• Octagons II: Strong Coupling
• Three-point functions at strong coupling in the BMN limit
• Pentagon OPE resummation in N=4 SYM: hexagons with one effective particle contribution
• Spinning Hexagons
• On analytical perturbative solution of ABJM quantum spectral curve
• On the Semi-Classical Limit of Scalar Products of the XXZ Spin Chain
• Loop Equation and Exact Soft Anomalous Dimension in N=4 Super Yang-Mills
• Orthogonality of Q-Functions up to Wrapping in Planar N=4 Super Yang-Mills Theory
• Higher-Point Correlators in N=4 SYM: Ten-Dimensional Null Polygons
• Quantum Integrability and Quantum Groups: a special issue in memory of Petr P. Kulish