Fortuity and Supergravity

BPS states in holographic CFTs naturally split into those describing black holes in the bulk and those that do not, with black hole states only existing above a certain energy threshold. In the context of the AdS$_3$/CFT$_2$ duality this can be seen from the agreement of the CFT and supergraviton supersymmetric indices up to a certain central charge-scaling conformal dimension. However, there also exist additional smooth horizonless bulk configurations called singletons that have not been previously accounted for when distinguishing black hole states from non-black hole states. These singletons describe bulk degrees of freedom that are non-trivial on the AdS$_3$ boundary. From a detailed analysis of BPS states in the D1-D5 system we identify singleton states in $\mathrm{Sym}^N(T^4)$ and $\mathrm{Sym}^N(K3)$ and explicitly incorporate them into a generalised supergraviton index for low levels for the case of $N=2$, leading to an enhanced matching with the CFT index in the case of $T^4$. Singleton states are monotone and, under the assumption that together with supergraviton states they span the monotone Hilbert space, the generalised supergraviton index represents the full monotone index. This allows us to define fortuitous indices for these theories and for $\mathrm{Sym}^2(K3)$ we construct the first explicit fortuitous states.