The geometry of large charge multi-traces in $\mathcal{N}=4$ SYM

We construct a one-parameter family of half-BPS solutions of type IIB supergravity using a consistent truncation to gauged five-dimensional supergravity. For small values of the parameter, the solution reduces to the linear perturbation of AdS$_5\times S^5$ dual to the chiral primary operator in the stress-tensor multiplet, and we give evidence that the geometry is regular and asymptotes AdS in a normalisable way for arbitrarily large values of the parameter. We conjecture that the solution is the gravity dual of a ``heavy" multi-trace operator in $\mathcal{N}=4$ $SU(N)$ Super Yang-Mills made by $p$ copies of the stress-tensor chiral primary operator, with $p$ of order $N^2$ in the large $N$ limit. We perform some holographic checks supporting this duality map.