Conformal quantum mechanics & the integrable spinning Fishnet

TL;DR

This paper constructs and proves the quantum integrability of a conformal spin chain model related to Fishnet theories, using conformal symmetry and Yang-Baxter solutions in four-dimensional quantum field theories.

Contribution

It introduces a new integrable spin chain model with conformal symmetry, connecting it to Fishnet theories and proving its quantum integrability using conformal algebraic relations.

Findings

Constructed solutions to the Yang-Baxter equation for conformal particles.

Proved quantum integrability of a conformal spin chain model.

Linked transfer matrices to Bethe-Salpeter kernels in supersymmetric theories.

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 $\Delta=2+i\nu$ for any left/right spins $\ell,\dot{\ell}$ of the particles. Such relations are interpreted in the language of Feynman diagrams as integral \emph{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 $(\Delta_k,\ell_k, \dot{ \ell}_k)$ 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. 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 $\gamma$-deformed $\mathcal{N}=4$ and $\mathcal{N}=2$ supersymmetric theories.