From Symmetry to Structure: Gauge-Invariant Operators in Multi-Matrix Quantum Mechanics

TL;DR

This paper investigates the algebraic structure of gauge-invariant operators in multi-matrix quantum mechanics, revealing how primary and secondary invariants determine the operator space and its growth at large N.

Contribution

It introduces a method to compute the number of primary invariants via gauge fixing and compares it with Schur polynomial counting, providing insights into secondary invariants' exponential growth.

Findings

Number of primary invariants can be computed through gauge fixing.

Comparison with Schur polynomial basis supports exponential growth of secondary invariants.

Secondary invariants grow as e^{cN^2} at large N.

Abstract

Recently the algebraic structure of gauge-invariant operators in multi-matrix quantum mechanics has been clarified: this space forms a module over a freely generated ring. The ring is generated by a set of primary invariants, while the module structure is determined by a finite set of secondary invariants. In this work, we show that the number of primary invariants can be computed by performing a complete gauge fixing, which identifies the number of independent physical degrees of freedom. We then compare this result to a complementary counting based on the restricted Schur polynomial basis. This comparison allows us to argue that the number of secondary invariants must exhibit exponential growth of the form $e^{cN^2}$ at large $N$, with $c$ a constant.