Finite $N$ Hilbert Spaces of Bilocal Holography

Abstract: For vector/AdS and dS holography we establish the structure of the emergent Hilbert space. This is done through implementation of finite $N$ trace relations on the infinite collective space. For fermionic theories a finite Hilbert space is established, while for bosonic theories a space of freely acting primaries multiplied by a finite set of secondaries emerges. The Hilbert space of states obey finite $N$ cut off bounds, implying finiteness of traces and entropy.