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

TL;DR

This paper derives an effective semi-classical action for the $ extrm{SL}(2, extrm{R})$ WZNW model, connecting string theory in $ extrm{AdS}_3$ with a spin-chain $\sigma$-model through classical and quantum analyses.

Contribution

It constructs a semi-classical effective action for the $ extrm{SL}(2, extrm{R})$ spin-chain $\sigma$-model from the WZNW model using classical gauge fixing and coherent state path integrals.

Findings

Derived the effective action for slow target-space coordinates.

Connected the spin chain transition amplitude to a path integral.

Identified the Landau-Lifshitz limit involving both spatial and temporal intervals.

Abstract

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