Conformal quantum mechanics & the integrable spinning Fishnet

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.

25 pages, 28 figures; v2: (4.27) and Fig.16 improved, typos corrected; v3,v4: as re-submitted to JHEP; Figures added, typos corrected

References in corpus (2)

• Structure Constants and Integrable Bootstrap in Planar N=4 SYM Theory
• Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables

Cited by in corpus (3)

• Separation of Variables in AdS/CFT: Functional Approach for the Fishnet CFT
• Stampedes I: Fishnet OPE and Octagon Bootstrap with Nonzero Bridges
• Correlation functions of determinant operators in conformal fishnet theory