Towards a Finite-$N$ Hologram

TL;DR

The authors propose a framework for exact, non-perturbative solvability at finite $N$ in holographic tensor models related to SYK by leveraging a spinor (Clifford) structure in the Hilbert space. They demonstrate level-by-level diagonalization for the even-$n$ uncolored $O(n)^3$ model, recasting spectra and eigenstates as a representation-m-theoretic problem via Young tableaux and Littlewood-Richardson rules, with explicit results for Clifford levels 0, 1, and 2. The method avoids supersymmetry and aims to extend to higher levels, odd $n$, and colored tensor models, potentially enabling a coherent bulk interpretation from gauge-singlet states. Overall, the work provides a concrete, finite-$N$ approach to holographic quantum theories, offering a path toward exact, non-perturbative holography through representation theory.

Abstract

We suggest that holographic tensor models related to SYK are viable candidates for exactly (ie., non-perturbatively in $N$) solvable holographic theories. The reason is that in these theories, the Hilbert space is a spinor representation, and the Hamiltonian (at least in some classes) can be arranged to commute with the Clifford level. This makes the theory solvable level by level. We demonstrate this for the specific case of the uncolored $O(n)^3$ tensor model with arbitrary even $n$, and reduce the question of determining the spectrum and eigenstates to an algebraic equation relating Young tableaux. Solving this reduced problem is conceptually trivial and amounts to matching the representations on either side, as we demonstrate explicitly at low levels. At high levels, representations become bigger, but should still be tractable. None of our arguments require any supersymmetry.