String theory on $\boldsymbol{\text{AdS}_{\mathbf{3}}}$ and the symmetric orbifold of Liouville theory

TL;DR

This work provides a detailed, solvable realization of AdS$_3$/CFT with pure NS-NS flux by constructing a complete set of DDF operators on the worldsheet. For the bosonic theory on AdS$_3\times X$, the spacetime symmetry algebra reproduces the symmetric orbifold of Liouville theory times the internal CFT, with long-string continuum arising from the Liouville sector; in the supersymmetric case, the dual is the symmetric orbifold of (N=4 Liouville with c=6(k-1)) plus T$^4$, with k=1 eliminating the Liouville factor and reducing to Sym$^N$(T$^4$). The analysis extends to the psu$(1,1|2)_k$ WZW model and the hybrid formalism, clarifying how the spacetime N=4 algebra emerges and acts on physical states across spectral-flow sectors. Overall, the results offer a concrete, exact AdS$_3$/CFT correspondence in NS-NS backgrounds and illuminate how worldsheet representations map to the spacetime symmetric orbifold structure, including the special k=1 limit where the dual CFT is fully tractable.

Abstract

For string theory on AdS$_3$ with pure NS-NS flux a complete set of DDF operators is constructed, from which one can read off the symmetry algebra of the spacetime CFT. Together with an analysis of the spacetime spectrum, this allows us to show that the CFT dual of superstring theory on ${\rm AdS}_3 \times {\rm S}^3 \times \mathbb{T}^4$ for generic NS-NS flux is the symmetric orbifold of $({\cal N}=4$ Liouville theory$)\times \mathbb{T}^4$. For the case of minimal flux ($k=1$), the Liouville factor disappears, and we just obtain the symmetric orbifold of $\mathbb{T}^4$, thereby giving further support to a previous claim. We also show that a similar analysis can be done for bosonic string theory on ${\rm AdS}_3 \times X$.

Figures (1)

• Figure 1: The spacetime conformal weight in dependence of the internal conformal weight $h_{\rm int}+N$. For definiteness, we have chosen $k=20$. Dashed lines indicate the bottom of the continuum in the respective sector. We have plotted the discrete representation as solid lines. There are several lines which correspond to the choice of state in a particular $\mathfrak{sl}(2,\mathds{R})$ representation.