The Worldsheet Dual of the Symmetric Product CFT

TL;DR

The paper identifies the tensionless limit of string theory on ${ m AdS}_3 \times { m S}^3 \times {\mathbb T}^4$ with NS-NS flux at ${k=1}$ as dual to the large $N$ limit of the free symmetric product ${\rm Sym}^N({\mathbb T}^4)$. Using the hybrid formalism, the worldsheet is described by the ${\frak{psu}}(1,1|2)_1$ WZW model, which lacks the long-string continuum and reproduces the symmetric-product spectrum and fusion rules via a free-field construction involving symplectic bosons and fermions. The analysis also uncovers a rich indecomposable structure (notably the ${\mathscr{T}}$ sector) and shows how spectral flow and the ${x}$-basis reconcile worldsheet and orbifold fusion rules, yielding a modular-invariant partition function and a topological-like worldsheet behavior. The results provide a tractable, explicit worldsheet dual for a tensionless AdS string and suggest broader connections to topological strings and higher-spin symmetries in AdS/CFT contexts.

Abstract

Superstring theory on ${\rm AdS}_3\times {\rm S}^3\times \mathbb{T}^4$ 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 ${\rm AdS}_3\times {\rm S}^3$ part is described by the $\mathfrak{psu}(1,1|2)_1$ WZW model (which is well defined). Unlike the case for $k\geq2$, 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.

Figures (1)

• Figure 1: The definition of the maps $s^\pm_{(i)}$.