Strings on $\text{AdS}_3 \times \text{S}^3 \times \text{S}^3 \times \text{S}^1$

TL;DR

The paper establishes an exact AdS$_3$/CFT-like duality for strings on AdS$_3\times$S$^3\times$S$^3\times$S$^1$ with pure NS-NS flux and minimal flux through one S$^3$. By reformulating the worldsheet theory in a hybrid formalism based on the $\mathfrak{d}(2,1;\alpha)$ WZW-model and analyzing its representation theory, fusion, and modular properties, the authors show that the spacetime spectrum and the DDF-generated large $\mathcal{N}=4$ algebra reproduce the symmetric orbifold of the seed theory $\mathcal{S}_\kappa$ (with $\kappa=k^- -1$) in the large $N$ limit. The work also generalizes to $k^+>1$, where the dual is argued to be the symmetric orbifold of large $\mathcal{N}=4$ Liouville theory, highlighting a unified framework for AdS$_3$/CFT dualities in solvable string backgrounds. The results provide a controlled setting to probe exact holography, including modular properties, BPS spectra, and DDF operators, and suggest broader applicability to other backgrounds with less supersymmetry.

Abstract

String theory on ${\rm AdS}_3 \times {\rm S}^3 \times {\rm S}^3 \times {\rm S}^1$ with pure NS-NS flux and minimal flux through one of the two ${\rm S}^3$'s is studied from a world-sheet perspective. It is shown that the spacetime spectrum, as well as the algebra of spectrum generating operators, matches precisely that of the symmetric orbifold of ${\rm S}^3\times \mathrm{S}^1$ in the large $N$ limit. This gives strong support for the proposal that these two descriptions are exactly dual to one another.