The Worldsheet Dual of the Symmetric Product CFT

Abstract

Superstring theory on ${\mathrm{AdS}}_3\times {\mathrm{S}}^3\times {\mathbb{T}}^4$ with the smallest amount of NS-NS flux (`$k=1$') is shown to be dual to the spacetime CFT given by the large $N$ limit of the free symmetric product orbifold ${\mathrm{Sym}}^N({\mathbb{T}}^4)$. To define the worldsheet theory at $k=1$, we employ the hybrid formalism in which the ${\mathrm{AdS}}_3\times {\mathrm{S}}^3$ part is described by the $\mathfrak{psu}(1,1|2{)}_1$ WZW model (which is well defined). Unlike the case for $k\ge 2$, it turns out that the string spectrum at $k=1$ does {\it not} exhibit the long string continuum, and perfectly matches with the large $N$ limit of the symmetric product. We also demonstrate that the fusion rules of the symmetric orbifold are reproduced from the worldsheet perspective. Our proposal therefore affords a tractable worldsheet description of a tensionless limit in string theory, for which the dual CFT is also explicitly known.

29+24 pages