Winding Strings in AdS_3

TL;DR

This work advances the understanding of string theory on $AdS_{3}$ by addressing correlation functions involving states in the first winding sector ($w=1$) of the $SL(2,\mathbb{R})$ WZW model. The authors derive Ward identities and the modified Knizhnik–Zamolodchikov and null-vector equations for amplitudes with spectral-flowed operators, and they compute concrete results: a three-point function with two $w=1$ insertions and a four-point function including one $w=1$ field, both expressed with precise spin- and cross-ratio dependence and checked against known limits. They also develop the necessary formal apparatus, including $j$-dependent coefficients from spectral-flowed five-/six-point functions and systematic integral identities, to analyze higher-point amplitudes and factorization properties. While providing substantial progress toward a consistent description of winding sectors and their factorization, the paper also highlights ongoing challenges in fully determining higher-winding amplitudes and their factorization structure, motivating future work and connections to Liouville theory via established correspondences.

Abstract

Correlation functions of one unit spectral flowed states in string theory on AdS_3 are considered. We present the modified Knizhnik-Zamolodchikov and null vector equations to be satisfied by amplitudes containing states in winding sector one and study their solution corresponding to the four point function including one w=1 field. We compute the three point function involving two one unit spectral flowed operators and find expressions for amplitudes of three w=1 states satisfying certain particular relations among the spins of the fields. Several consistency checks are performed.