A supersymmetric AdS$_3$ duality

TL;DR

The paper identifies a supersymmetric analogue of the AdS$_3$ duality for the $SL(2,\, ext{C})/SU(2)$ WZW model, relating it to a super-winding condensate CFT built from the bosonic dual at level $k_B=k+2$ plus three free fermions. It then shows that gauging the timelike isometry $J_3$ reduces this dual to $ abla ext{-}N=2$ Liouville theory, establishing a dimensional uplift of the SUSY FZZ duality. The argument leverages the known bosonic duality and the factorization of the SUSY $H_3^+$ WZW model, providing an explicit operator map between winding states and Liouville operators via bosonization and refermionization. The work also suggests future avenues, including localization techniques and a GLSM-based proof, with potential insights into AdS$_3$ black hole–string transitions.

Abstract

We find a dual description of the supersymmetric $SL(2, \mathbb{C})/SU(2)$ WZW model as a super-winding condensate CFT, which follows simply from the bosonic AdS$_3$ duality found in [arXiv:2104.07233]. This duality can be seen as a dimensional uplift of the mirror symmetry between the $H_3^+/U(1)$ supercoset and $\mathcal{N}=2$ super-Liouville theory