Marginal deformations and quasi-Hopf algebras

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 (QYBE). 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.