A holographic dual for string theory on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{S}^3 \times \mathrm{S}^1$

TL;DR

This work identifies a concrete CFT dual for string theory on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{S}^3 \times \mathrm{S}^1$ by proposing the symmetric orbifold $\mathrm{Sym}^{N}(\mathcal{S}_{\kappa})$ with $N=Q_1 Q_5^+$ and $\kappa = \frac{Q_5^-}{Q_5^+}-1$ (valid when $Q_5^-$ is a multiple of $Q_5^+$). Using a D-brane construction, the authors motivate the seed theory $\mathcal{S}_{\kappa}$ arising from an $\mathcal{N}=4$ WZW sector and show that the dual CFT is consistent with the known central charge and level structure, and then perform a detailed matching of the BPS spectrum between the symmetric orbifold and both the worldsheet description and supergravity. They demonstrate that the low-lying single-particle BPS states coincide across the CFT and gravity descriptions, and that the chiral ring and moduli content align with expectations from $\mathcal{N}=2$ subalgebras, with a parallel to a T$^2$-like effective description in certain limits. These results provide strong support for the proposed holographic dual and illuminate how large $\mathcal{N}=4$ structure governs BPS spectra in this AdS$_3$ background.

Abstract

The CFT dual of string theory on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{S}^3 \times \mathrm{S}^1$ is conjectured to be the symmetric orbifold of the $\mathcal{S}_κ$ theory, provided that one of the two $Q_5^\pm$ quantum numbers is a multiple of the other. We determine the BPS spectrum of the symmetric orbifold in detail, and show that it reproduces precisely the BPS spectrum that was recently calculated in supergravity. We also determine the BPS spectrum of the world-sheet theory that is formulated in terms of WZW models, and show that, apart from some gaps (which are reminiscent of those that appear in the corresponding $\mathbb{T}^4$ calculation), it also reproduces this BPS spectrum. In fact, the matching seems to work as well as for the familiar $\mathbb{T}^4$ case, and thus our results give strong support for this proposal.

Figures (2)

• Figure 1: Minimal conformal weight for a given spin
• Figure 2: Distribution of states in the intervals \ref{['intervals']} for the case of $k^+=5$, $k^-=3$. Solid markers indicate the boundary of an interval, dotted markers are multiples of $\tfrac{1}{2}k^+$, dashed markers are multiples of $\tfrac{1}{2}k^-$. One can clearly see that there is precisely one point where all markers meet and the following solid interval does not contain any other marker. Apart from this exception, every two solid markers enclose exactly one dotted or dashed marker. Hence either no marker meets or all three markers meet at one point.