Abstract
In this paper we consider systems of quantum particles in the 4d Euclidean space which enjoy conformal symmetry. The algebraic relations for conformal-invariant combinations of positions and momenta are used to construct a solution of the Yang-Baxter equation in the unitary irreducibile representations of the principal series ∆ = 2 + iν for any left/right spins ℓ, dot{ell} of the particles. Such relations are interpreted in the language of Feynman diagrams as integral star-triangle identites between propagators of a conformal field theory. We prove the quantum integrability of a spin chain whose k-th site hosts a particle in the representation (∆k, ℓk, dot{ell} k) of the conformal group, realizing a spinning and inhomogeneous version of the quantum magnet used to describe the spectrum of the bi-scalar Fishnet theories [1]. For the special choice of particles in the scalar (1, 0, 0) and fermionic (3/2, 1, 0) representation the transfer matrices of the model are Bethe-Salpeter kernels for the double-scaling limit of specific two-point correlators in the γ-deformed mathcal{N} = 4 and mathcal{N} = 2 supersymmetric theories.
Highlights
We construct the integrable chain starting from its transfer matrices
Such relations are interpreted in the language of Feynman diagrams as integral star-triangle identites between propagators of a conformal field theory
We prove the quantum integrability of a spin chain whose k-th site hosts a particle in the representation (∆k, k, ̇k) of the conformal group, realizing a spinning and inhomogeneous version of the quantum magnet used to describe the spectrum of the bi-scalar Fishnet theories [1]
Summary
After relabeling a → a, d → d the star integral takes the form [(x − y)y]db[y(y − z)]ca[(x − y)(y − z)]da ̇ (x − y)2(u+2)y2(−u−v)(y − z)2(v+2) In this form, illustrated, the star-triangle identity relates a conformal-invariant vertex of three spinning propagators of type G(u, x) with the product of three propagators whose spinorial indices are mutually mixed by the R-matrix. The proof follows the very same steps as for the first relation, adapted only in the variant of star-triangle identity involved, and its diagrammatic form is: The statement of the interchange relations can be read out of figure 9 and figure 10: moving the horizontal (red/green) propagator and the vertical (grey/black) propagator across the quartic scale invariant vertex, the powers (u, v, u , v ) in the vertex propagators get interchanged, and so do the spinorial structures depicted by different colors of lines.
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