Correlators in AdS_3 string theory

TL;DR

This paper analyzes two- and three-point correlators in $AdS_3$ string theory using a Coulomb gas free-field approach, first for the $SL(2,R)/U(1)$ coset with two screening operators $S_+$ and $S_-$ and then via an extension to $AdS_3$ as a product with a timelike boson to incorporate spectral-flow and winding. It shows that amplitudes computed with either screening reproduce known results and are consistent with Liouville-type dualities, and it derives new winding-violating three-point amplitudes within the spectral-flow extended framework. The AdS$_3$ formulation yields amplitudes that match prior literature for winding-conserving processes and reveals novel structures when winding is violated, highlighting the coherence of the free-field approach for non-compact target spaces. The work also discusses open problems, including four-point functions and supersymmetric extensions, with implications for the AdS$_3$/CFT$_2$ correspondence.

Abstract

The computation of two and three point functions in the Coulomb gas free field approach to string theory in the SL(2,R)/U(1) black hole background is reviewed. An interesting relation arises when comparing the results obtained using two different screening operators. The formalism is then modified to study string theory propagating in AdS$_3$ which is considered as the direct product of the SL(2)/U(1) coset times a timelike free boson. This representation allows to naturally include the spectral flow symmetry and winding number in vertex operators and correlation functions. Two and three point tachyon amplitudes are computed in this new scenario and the results coincide with previous reports in the literature. Novel expressions are found for processes violating winding number conservation.