Finite $N$ Hilbert Spaces of Bilocal Holography

TL;DR

This paper constructs finite-dimensional Hilbert spaces in bilocal holography for vector/AdS and dS models, demonstrating how finite N trace relations lead to bounded entropy and a well-defined state space.

Contribution

It introduces a method to implement finite N trace relations, establishing finite Hilbert spaces for fermionic and bosonic theories in bilocal holography.

Findings

Finite N trace relations lead to finite Hilbert spaces.

Fermionic theories have finite Hilbert spaces.

Bosonic theories produce spaces of primaries and secondaries.

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.