# The $SU(2)$ Wess-Zumino-Witten spin chain sigma model

## TL;DR

This paper connects classical string dynamics in a specific background to a quantum spin chain model, deriving a path integral formulation and analyzing the effects of spectral flow and gauge freedom in the semiclassical limit.

## Contribution

It demonstrates how the semiclassical $SU(2)$ spectrally flowed sectors emerge as the Landau-Lifshitz limit of the quantum spin chain, including a path integral representation and gauge considerations.

## Key findings

- Discretization of the propagator's time interval proportional to lattice spacing
- Path integral representation derived in the Landau-Lifshitz limit
- Higher order corrections are suppressed by inverse powers of the 't Hooft coupling

## Abstract

Classical strings propagating in $AdS_{3}\times S^{3}\times T^{4}$ supported with Neveu-Schwarz-Neveu-Schwarz flux are described by a Wess-Zumino-Witten model. In this note, we study the emergence of their semiclassical $SU(2)$ spectrally flowed sectors as the Landau-Lifshitz limit of the underlying quantum spin chain. We consider the propagator in the coherent state picture, and find that the time interval is discretized proportionally to the lattice spacing. In the Landau-Lifshitz limit, where both time and space become continuous, we derive a path integral representation of the propagator for each spectrally flowed sector. We prove that the arbitrariness of the global phase of coherent states is mapped to the gauge freedom of the $B$-field in the classical action. We show that higher order corrections in the Landau-Lifshitz limit are suppressed by inverse powers of the 't Hooft coupling.