---
abstract: 'We compute the contact term of the two-point function for the SL($2,\mathbb R$)-WZNW model in the winding sector. After reviewing some generalities of the model and its Euclidean counterpart, we discuss the reflection symmetry for the spectral flowed sector. We define the corresponding intertwining operator and use it to find the explicit form of the contact term of the propagator for a vertex carrying an arbitrary amount of spectral flow. Finally, we enhance the already known integral expression of a singly flowed vertex in order to reproduce such contact term directly in the space-time picture.'
author:
- 'Sergio M. Iguri[^1]'
- 'Nicolás Kovensky[^2]'
- 'Leila Maestri[^3]'
- 'Lucas Martín[^4]'
bibliography:
- 'refs.bib'
date: 'December 20, 2019'
title: 'Winding modes and the reflection symmetry in AdS$_3$'
---

Introduction
============

The study of strings propagating in AdS$_3$ has generated many interesting insights, ranging from string theory itself, integrability and AdS$_3$ quantum gravity [@Seiberg:1990eb; @Witten:1988hc; @Witten:1991yr; @Mukhi:1993zb; @Ghoshal:1995wm; @Giveon:1999px; @Giveon:1999tq; @Banados:1992wn; @Beisert:2010jr] to Liouville theory and condensed-matter physics [@Zirnbauer:1996zz; @Zirnbauer:1999ua; @Bhaseen:1999nm; @Kogan:1999hz]. When the background is purely of the NS-NS type, the worldsheet theory is given by the SL($2,\mathbb R$)-WZNW model, a theory that has been intensively studied since Maldacena’s conjecture 