Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models

TL;DR

This work establishes a complete, representation-theoretic construction of gauge-invariant observables for bosonic and fermionic tensor models and computes their exact free-theory correlators. The gauge invariants form a closed ring with explicitly computable structure constants, enabling the reduction of arbitrary correlators to one-point functions. By organizing operators into a restricted Schur basis and analyzing their algebras, the authors solve the free theory in a form that is amenable to large-$N$ analysis. They further develop a collective field theory description, revealing an emergent dimension and local bulk dynamics that reproduce large-$N$ correlators, suggesting a holographic interpretation and opening avenues toward SYK-related holography via tensor models.

Abstract

Motivated by the close connection of tensor models to the SYK model, we use representation theory to construct the complete set of gauge invariant observables for bosonic and fermionic tensor models. Correlation functions of the gauge invariant operators in the free theory are computed exactly. The gauge invariant operators close a ring. The structure constants of the ring are described explicitly. Finally, we construct a collective field theory description of the bosonic tensor model.

Figures (1)

• Figure 1: The above figures label gauge invariant operators in the tensor model gauge theory. Black dots correspond to $\bar{\phi}^{ijk}$'s and white dots to $\phi_{ijk}$s. A line labeled by $i$ is a gauge index for $U(N_i)$. The operator on the left corresponds to $\bar{\phi}^{i_1j_1k_1}\phi_{i_2j_1k_1}\bar{\phi}^{i_2j_2k_2}\phi_{i_3j_2k_2}\bar{\phi}^{i_3j_3k_3}\phi_{i_4j_3k_3}\bar{\phi}^{i_4j_4k_4}\phi_{i_1j_4k_4}$ and the operator on the right corresponds to $\bar{\phi}^{ijk}\phi_{ijk}$.