Search for a holographic dual to $AdS_3\times S^3\times S^3\times S^1$

Abstract

The problem of finding a holographic dual to string theory on $AdS_3 \times \mathbf{S}^3 \times \mathbf{S}^3 \times \mathbf{S}^1$ is examined in depth. This background supports a large $\mathcal{N}=4$ superconformal symmetry. While in some respects similar to the familiar small $\mathcal{N}=4$ systems on $AdS_3\times \mathbf{S}^3\times K3$ and $AdS_3\times \mathbf{S}^3\times T^4$, there are important qualitative differences. Using an analogue of the elliptic genus for large $\mathcal{N}=4$ theories we rule out all extant proposals--in their simplest form--for a holographic duality to supergravity at generic values of the background fluxes. Modifications of these extant proposals and other possible duals are discussed.