Orthogonal Bases of Invariants in Tensor Models

TL;DR

This work develops a representation-theoretic framework to count and organize gauge-invariant tensor operators in rank-$d$ tensor models under $G_d=U(N_1)\otimes\cdots\otimes U(N_d)$. It introduces two counting schemes—finite-$N_k$ via Schur–Weyl duality and Kronecker coefficients, and large-$N$ via double cosets—to determine the number of invariants, and constructs two corresponding bases of invariant operators. The free-theory correlators are computed exactly, showing that the finite-$N$ basis diagonalizes the two-point function and that correlators are diagonal in the irrep labels with a conjectured orthogonality in copy indices; the large-$N$ basis is anticipated to yield an orthogonal structure in a companion work. Together, these results provide a concrete, tractable framework for analyzing entanglement probes and holographic connections in tensor models, with clear paths for extending to orthogonal bases and fermionic tensor systems.

Abstract

Representation theory provides a suitable framework to count and classify invariants in tensor models. We show that there are two natural ways of counting invariants, one for arbitrary rank of the gauge group and a second, which is only valid for large N. We construct bases of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite N diagonalizes the two-point function of the theory and it is analogous to the restricted Schur basis used in matrix models. We comment on future lines of investigation.