Higher spins on AdS$_{3}$ from the worldsheet

TL;DR

The paper analyzes string theory on AdS$_3 imes{ m S}^3 imes T^4$ with pure NS-NS flux through its worldsheet WZW description to uncover higher spin structures in the tensionless limit. It shows that short (unflowed) strings do not produce massless higher spins for generic levels $k$, while the leading Regge trajectory organizes into an ${ m N}=4$ Vasiliev higher spin subsector; at the special level $k=1$ a stringy tower of massless higher spins emerges from long strings. The work also maps the spectrum into even-spin ${ m N}=4$ multiplets and analyzes how spectrally flowed sectors (long strings) contribute to the full Regge structure, including a detailed look at ${ m AdS}_3 imes{ m S}^3 imes K3$ and the role of GSO projections. The results connect worldsheet spectra with HS symmetries anticipated from holography and provide concrete dispersion relations for Regge states across sectors, offering a benchmark for future RR-flux analyses. Overall, the leading Regge sector is shown to capture a closed ${ m N}=4$ HS subsector, while massless HS either arise only at $k=1$ (long-string sector) or are absent in the NS-NS tensionless regime."

Abstract

It was recently shown that the CFT dual of string theory on ${\rm AdS}_3 \times {\rm S}^3 \times T^4$, the symmetric orbifold of $T^4$, contains a closed higher spin subsector. Via holography, this makes precise the sense in which tensionless string theory on this background contains a Vasiliev higher spin theory. In this paper we study this phenomenon directly from the worldsheet. Using the WZW description of the background with pure NS-NS flux, we identify the states that make up the leading Regge trajectory and show that they fit into the even spin ${\cal N}=4$ Vasiliev higher spin theory. We also show that these higher spin states do not become massless, except for the somewhat singular case of level $k=1$ where the theory contains a stringy tower of massless higher spin fields coming from the long string sector.

Figures (3)

• Figure 1: Lowest energy states for fixed $k=20$ as a function of the spin $s$. Crosses denote continuous representations (long strings), whereas dots correspond to discrete representations. The red dots describe the unflowed discrete states, while the different bands correspond, in turn, to the flows $\omega=1,2,\ldots,10$. The lines corresponding to the spectrally flowed representations have been artificially capped for illustration purposes; they continue all the way to $s=\pm\infty$.
• Figure 2: Schematic description of the discrete physical spectrum for fixed $n$ (and fixed $j$). The four blue dots on the $\bar{r}=0$ edge denote the top component of supermultiplets with the lowest energy for the given spin. See section \ref{['sec:LR']} below for a detailed discussion of these states.
• Figure 3: Dispersion relation of the lowest energy states for $k=200\,$. The dots correspond to the set of four states singled out in Figure \ref{['fig:diamond1']}, depicted here for different values of $n$ (i.e. different $j$), giving rise to different ranges $2n +1/2 \leq s \leq 2n+2$ of the spin. In each family of four dots, the last one describes a unique state, the corner state of Figure \ref{['fig:diamond1']}, while the first three correspond to states with higher multiplicity.