Conformal quantum mechanics & the integrable spinning Fishnet

A bstract In this paper we consider systems of quantum particles in the 4 d 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, $$ \dot{\ell} $$ ℓ ̇ 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 , ℓ k , $$ \dot{\ell} $$ ℓ ̇ 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 $$ \mathcal{N} $$ N = 4 and $$ \mathcal{N} $$ N = 2 supersymmetric theories.

Abstract licence: CC BY

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