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The $\textrm{SL}(2,\mathbb{R})$ Wess-Zumino-Novikov-Witten model realisesbosonic-string theory in $\textrm{AdS}_{3}$ with pureNeveu-Schwarz-Neveu-Schwarz flux. We construct an effective action in thesemi-classical limit of the model, which corresponds to a$\textrm{SL}(2,\mathbb{R})$ spin-chain $\sigma$-model. We adopt twocomplementary points of view. Firstly, we consider the classical action. Weidentify fast and slow target-space coordinates. We impose a gauge-fixingcondition to the former. By expanding the gauge-fixed action in an effectivecoupling, we obtain the effective action for the slow coordinates. Secondly, weconsider the spin chain of the model. We postulate a set of coherent states toexpress a transition amplitude in the spin chain as a path integral. We observethat the temporal interval is discretised in terms of the step length of thespatial interval. This relationship implies that the Landau-Lifshitz limit ofthe spin chain involves both intervals. The limit yields a semi-classical pathintegral over coherent states, wherein we identify the effective action again.

The $\mathrm{SL}(2,\mathbb{R})$ Wess-Zumino-Novikov-Witten spin-chain $\sigma$-model