Conformal quantum mechanics & the integrable spinning Fishnet

Abstract

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, 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 , ℓ k , ℓ ̇ $$ \dot{\ell} $$ 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 N $$ \mathcal{N} $$ = 4 and N $$ \mathcal{N} $$ = 2 supersymmetric theories.

Keywords

• Conformal Field Theory
• Field Theories in Higher Dimensions
• Higher Spin Symmetry
• Lattice Integrable Models