Towards a Finite-$N$ Hologram

TL;DR

This paper proposes that certain tensor models related to the SYK model are exactly solvable at finite N by exploiting their spinor Hilbert space structure, enabling algebraic determination of spectra without supersymmetry.

Contribution

It introduces a method to solve the spectrum of specific tensor models at finite N by reducing the problem to algebraic equations involving Young tableaux.

Findings

Spectrum determined by algebraic equations on Young tableaux

Solvability extends to high levels with larger representations

No supersymmetry needed for the approach

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.