Fast spinning strings on $ \eta $ deformed $ AdS_5 \times S^{5} $

In this paper, considering the correspondence between spin chains and string sigma models, we explore the rotating string solutions over $ \eta $ deformed $ AdS_5 \times S^{5} $ in the so called fast spinning limit. In our analysis, we focus only on the bosonic part of the full superstring action and compute the relevant limits on both $(R \times S^{3})_{\eta} $ and $(R \times S^{5})_{\eta} $ models. The resulting system reveals that in the fast spinning limit, the sigma model on $ \eta $ deformed $S^5$ could be $\textit{approximately}$ thought of as the continuum limit of anisotropic $ SU(3) $ Heisenberg spin chain model. We compute the energy for a certain class of spinning strings in deformed $S^5$ and we show that this energy can be mapped to that of a similar spinning string in the purely imaginary $\beta$ deformed background.