BPS phases and fortuity in higher spin holography

We study the BPS states of U(N)ₖ× U(1)₋ₖ vector Chern-Simons theory on a sphere at weak coupling λ=N/k 1, dual to an AdS₄ higher spin gravity. Higher spin currents are well known to be anomalous at λ≠ 0. We show that these non-BPS higher spin particles form multi-particle `BPS bounds' at low energy, and interpret them as a primordial form of small black hole states. We also construct a new heavy BPS operator at $N=2$. We study the BPS phases of this system from the large N index at Planckian `temperatures'. The deconfined saddles at high temperature exist only above a threshold, similar to the BTZ black holes. The low temperature saddles are given by novel 2-cut eigenvalue distributions. Their phase transition involves subtle issues like the holomorphic anomaly and the background independence, whose studies we initiate. In particular, we obtain a lower bound on the critical temperature by studying the eigenvalue instantons.