Quantum Symmetries and Marginal Deformations

TL;DR

This paper explores the quantum group symmetries of N=1 marginal deformations of N=4 Super Yang-Mills, revealing a Hopf algebra invariance that could clarify differences in integrability and aid in understanding dualities.

Contribution

It uncovers a Hopf algebra symmetry in the classical Lagrangian of deformed theories, offering insights beyond traditional symmetry analysis.

Findings

Identifies a non-standard quantum deformation of SU(3) symmetry.

Provides a framework for understanding integrability differences.

Lays groundwork for future duality and finiteness studies.

Abstract

We study the symmetries of the N=1 exactly marginal deformations of N=4 Super Yang-Mills theory. For generic values of the parameters, these deformations are known to break the SU(3) part of the R-symmetry group down to a discrete subgroup. However, a closer look from the perspective of quantum groups reveals that the Lagrangian is in fact invariant under a certain Hopf algebra which is a non-standard quantum deformation of the algebra of functions on SU(3). Our discussion is motivated by the desire to better understand why these theories have significant differences from N=4 SYM regarding the planar integrability (or rather lack thereof) of the spin chains encoding their spectrum. However, our construction works at the level of the classical Lagrangian, without relying on the language of spin chains. Our approach might eventually provide a better understanding of the finiteness properties of these theories as well as help in the construction of their AdS/CFT duals.