The Octagon as a Determinant

Abstract

The computation of a certain class of four-point functions of heavily charged BPS operators boils down to the computation of a special form factor - the octagon. In this paper, which is an extended version of the short note [1], we derive a non-perturbative formula for the square of the octagon as the determinant of a semi-infinite skew-symmetric matrix. We show that perturbatively in the weak coupling limit the octagon is given by a determinant constructed from the polylogarithms evaluating ladder Feynman graphs. We also give a simple operator representation of the octagon in terms of a vacuum expectation value of massless free bosons or fermions living in the rapidity plane.

22 pages, 1 figure, small typos corrected

References in corpus (4)

• Hexagonalization of Correlation Functions
• Determinant formula for the octagon form factor in $\mathcal{N}$=4 SYM
• Octagons I: Combinatorics and Non-Planar Resummations
• Exponential suppression with four legs and an infinity of loops

Cited by in corpus (24)

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• Ten dimensional symmetry of N=4 SYM correlators
• Exact null octagon
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• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• Structure Constants in $\mathcal{N}=4$ SYM at Finite Coupling as Worldsheet $g$-Function
• Crossing bridges with strong Szego limit theorem
• Giant Wilson Loops and AdS${}_2$/dCFT${}_1$
• Decagon at Two Loops
• An Operator Product Expansion for Form Factors III. Finite Coupling and Multi-Particle Contributions
• Functional Equations and Separation of Variables for Exact g-Function
• Effective Quantum Field Theory for the Thermodynamical Bethe Ansatz
• Hexagonalization of Fishnet integrals I: mirror excitations
• Octagon with finite bridge: free fermions and determinant identities
• Three-point functions at strong coupling in the BMN limit
• Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
• An Origin Story for Amplitudes
• Pentagon OPE resummation in N=4 SYM: hexagons with one effective particle contribution
• Stampedes II: Null Polygons in Conformal Gauge Theory
• Exploring superconformal Yang-Mills theories through matrix Bessel kernels
• Fishnet four-point integrals: integrable representations and thermodynamic limits
• Loop Equation and Exact Soft Anomalous Dimension in N=4 Super Yang-Mills
• Three-Point Functions in ABJM and Bethe Ansatz
• Higher-Point Correlators in N=4 SYM: Ten-Dimensional Null Polygons