# Marginal deformations and quasi-Hopf algebras

## TL;DR

This paper demonstrates that Leigh-Strassler marginal deformations of N=4 Super-Yang-Mills theory can be described using a quasi-Hopf algebra structure, linking their R-matrix to that of undeformed theories via a Drinfeld twist.

## Contribution

It constructs a Drinfeld twist that relates the R-matrix of deformed theories to N=4 SYM, revealing a quasi-Hopf algebraic structure underlying these marginal deformations.

## Key findings

- The R-matrix of Leigh-Strassler theories satisfies the quasi-Hopf QYBE.
- A star product relates the superpotentials of undeformed and deformed theories.
- The approach provides insights into integrability and supergravity solutions.

## Abstract

We establish the existence of a quasi-Hopf algebraic structure underlying the Leigh-Strassler N=1 superconformal marginal deformations of the N=4 Super-Yang-Mills theory. The scalar-sector R-matrix of these theories, which is related to their one-loop spin chain Hamiltonian, does not generically satisfy the Quantum Yang-Baxter Equation. By constructing a Drinfeld twist which relates this R-matrix to that of the N=4 SYM theory, but also produces a non-trivial co-associator, we show that the generic Leigh-Strassler R-matrix satisfies the quasi-Hopf version of the QYBE. We also use the twist to define a suitable star product which directly relates the N=4 SYM superpotential to that of the marginally deformed gauge theories. We expect our results to be relevant to studies of integrability (and its breaking) in these theories, as well as to provide useful input for supergravity solution-generating techniques.