A Novel Long Range Spin Chain and Planar N=4 Super Yang-Mills

TL;DR

This paper develops a new integrable long-range spin chain model for planar N=4 Super Yang-Mills theory, extending previous approaches to higher loops and establishing connections with string theory.

Contribution

It introduces an all-loop asymptotic Bethe ansatz and transfer matrix for the spin chain, preserving integrability and matching semi-classical string theory results.

Findings

Proposes an all-loop asymptotic Bethe ansatz.

Establishes a relation to inhomogeneous spin chains.

Finds agreement with semi-classical string theory for local charge densities.

Abstract

We probe the long-range spin chain approach to planar N=4 gauge theory at high loop order. A recently employed hyperbolic spin chain invented by Inozemtsev is suitable for the SU(2) subsector of the state space up to three loops, but ceases to exhibit the conjectured thermodynamic scaling properties at higher orders. We indicate how this may be bypassed while nevertheless preserving integrability, and suggest the corresponding all-loop asymptotic Bethe ansatz. We also propose the local part of the all-loop gauge transfer matrix, leading to conjectures for the asymptotically exact formulae for all local commuting charges. The ansatz is finally shown to be related to a standard inhomogeneous spin chain. A comparison of our ansatz to semi-classical string theory uncovers a detailed, non-perturbative agreement between the corresponding expressions for the infinite tower of local charge densities. However, the respective Bethe equations differ slightly, and we end by refining and elaborating a previously proposed possible explanation for this disagreement.