Structure of Loop Space at Finite $N$
Abstract
The space of invariants for a single matrix is generated by traces containing at most matrices per trace. We extend this analysis to multi-matrix models at finite . Using the Molien-Weyl formula, we compute partition functions for various multi-matrix models at different and interpret them through trace relations. This allows us to identify a complete set of invariants, naturally divided into two distinct classes: primary and secondary. The full invariant ring of the multi-matrix model is reconstructed via the Hironaka decomposition, where primary invariants act freely, while secondary invariants satisfy quadratic relations. Significantly, while traces with at most matrices are always present, we also find invariants involving more than matrices per trace. The primary invariants correspond to perturbative degrees of freedom, whereas the secondary invariants emerge as non-trivial background structures. The growth of secondary invariants aligns with expectations from black hole entropy, suggesting deep structural connections to gravitational systems.
Cite
@article{arxiv.2503.20097,
title = {Structure of Loop Space at Finite $N$},
author = {Robert de Mello Koch and Antal Jevicki},
journal= {arXiv preprint arXiv:2503.20097},
year = {2025}
}