Entanglement in Typical States of Chern-Simons Theory

We compute various averages over bulk geometries of quantum states prepared by the Chern-Simons path integral, for any level $k$ and compact simple gauge group $G$. We do so by carefully summing over all topologically distinct bulk geometries which have $n$ disjoint boundary tori and a decomposition into space$\times$time of fixed spatial topology. We find that to leading order in the complexity of the state, the typical state contains many types of multiparty entanglement, proving a conjecture of Balasubramanian et al. Additionally, we compute an averaged wave function which captures the leading order statistics of boundary observables in the $n$ torus Chern-Simons Hilbert space.