The Leigh-Strassler Deformation and the Quest for Integrability

TL;DR

This paper investigates the integrability of the scalar sector in Leigh-Strassler deformed N=4 SYM theory, identifying conditions under which the one-loop dilatation operator maps onto integrable spin chains, including a new su_q(3) subsector.

Contribution

It demonstrates the existence of an integrable su_q(3) subsector when adding an anti-holomorphic field, extending previous work on integrable sectors in deformed N=4 SYM.

Findings

Identified parameter values satisfying integrability criteria.

Found a new su_q(3) subsector with integrability.

Suggested potential for higher-loop generalizations.

Abstract

In this paper we study the one-loop dilatation operator of the full scalar field sector of Leigh-Strassler deformed N=4 SYM theory. In particular we map it onto a spin chain and find the parameter values for which the Reshetikhin integrability criteria are fulfilled. Some years ago Roiban found an integrable subsector, consisting of two holomorphic scalar fields, corresponding to the XXZ model. He was pondering about the existence of a subsector which would form generalisation of that model to an integrable su_q(3) model. Later Berenstein and Cherkis added one more holomorphic field and showed that the subsector obtained this way cannot be integrable except for the case when q=e^{i beta}, beta real. In this work we show if we add an anti-holomorphic field to the two holomorphic ones, we get indeed an integrable su_q(3) subsector. We find it plausible that a direct generalisation to a su_q(3,2) one-loop sector will exist, and possibly beyond one-loop.