Fast spinning strings on $\eta$ deformed $Ad{S}_5\times S^5$

In this paper, considering the correspondence between spin chains and string sigma models, we explore the rotating string solutions over $\eta$ deformed $Ad{S}_5\times S^5$ 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 $\mathit{\text{approximately}}$ 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.