SL(2,R) Wess-Zumino-Novikov-Witten spin-chain σ -model

The $\mathrm{SL}(2,\mathbb{R})$ Wess-Zumino-Novikov-Witten model realizes bosonic-string theory in ${\mathrm{AdS}}_{3}$ with pure Neveu-Schwarz-Neveu-Schwarz flux. We construct an effective action the semiclassical limit of model, which corresponds to a spin-chain $\ensuremath{\sigma}$-model. adopt two complementary points view. First, we consider classical action. identify fast and slow target-space coordinates. impose gauge-fixing condition former. By expanding gauge-fixed coupling, obtain for Second, spin chain model. postulate set coherent states express transition amplitude as path integral. observe that temporal interval is discretized terms step length spatial interval. This relationship implies Landau-Lifshitz involves both intervals. yields integral over states, wherein again.