Abstract
We continue the study of the octagon form factor which helps to evaluate a class of four-point correlation functions in mathcal{N} = 4 SYM theory. The octagon is characterised, besides the kinematical parameters, by a “bridge” of ℓ propagators connecting two nonadjacent operators. In this paper we construct an operator representation of the octagon with finite bridge as an expectation value in the Fock space of free complex fermions. The bridge ℓ appears as the level of filling of the Dirac sea. We obtain determinant identities relating octagons with different bridges, which we derive from the expression of the octagon in terms of discrete fermionic oscillators. The derivation is based on the existence of a previously conjectured similarity transformation, which we find here explicitly.
Highlights
A class of four-point functions of half-BPS operators with large R-charges and specially tuned polarisations, discovered in [9, 10], are free of divergencies and can be evaluated exactly for any value of the ’t Hooft coupling
We continue the study of the octagon form factor which helps to evaluate a class of four-point correlation functions in N = 4 SYM theory
The derivation is based on the existence of a previously conjectured similarity transformation, which we find here explicitly
Summary
The role of this subsection is to remind the notations and make the presentation selfconsistent. The parameters φ and ξ, respectively φ and θ, characterise the rotation aligning the two hexagons in the Euclidean, respectively flavour, space. The octagon represents two hexagons glued together by inserting a complete set of virtual states in the Hilbert space associated with the common mirror edge. An n-particle virtual state is characterised by the rapidities ui and the bound-state numbers ai of its particles. The contribution of such virtual state factorises into one-particle factors Waj (uj) and two-particle interactions Waj,ak (uj, uk) accounting for the hexagon weights. Χ±a (φ, φ, θ) = (−1)a sin(aφ) sin φ [2 cos φ − 2 cosh(φ ± iθ)].
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