Following Black Hole States

We study $\mathcal{N}=4$ SYM at non-integer number of colours. By varying $N$ we can continuously follow states all the way from $N=\infty$ where integrability reigns to finite $N$ where quantum gravity effects dominate. As an application we consider classically $1/16$ BPS states. Quantum mechanically, these states are generically non-supersymmetric but some special states - at special values of $N$ - become super-symmetric at the quantum level as well. They are the so-called quantum black hole states studied recently using cohomology. We write down the form of the lightest BH state at $N=2$ - and follow it in $N$, both at weak coupling and - more speculatively - at strong coupling as well. At weak coupling this state has protected dimension $\Delta=19/2$ at $N=2$ and becomes a triple trace made out of Konishi and two light BPS operators at infinite $N$ with $\Delta=19/2+12\lambda+\dots$. At strong coupling we suspect it becomes a quadruple trace with dimension $\Delta \simeq 19/2+\text{integer}$.