The dual string σ-model of the SU _q(3) sector

In four-dimensional N = 4 super Yang-Mills (SYM) the SU(3) sub-sector spanned by purely holomorphic fields is isomorphic to the corresponding mixed one spanned by both holomorphic and anti-holomorphic fields. This is no longer the case when one considers the marginally deformed N = 4 SYM. The mixed SU(3) sector marginally deformed by a complex parameter β, i.e. SU _q(3) with q = e ^2iπβ, has been shown to be integrable at one-loop [1], while it is not the case for the corresponding purely holomorphic one. Moreover, the marginally deformed N = 4 SYM also has a gravity dual constructed by Lunin and Maldacena in [2]. However, the mixed SUq(3) sector has not been studied from the supergravity point of view. Hence in this note, for the case of purely imaginary marginal β-deformations, we compute the superstring SUq(3) σ-model in the fast spinning string limit and show that, for rational spinning strings, it reproduces the energy computed via Bethe equations.