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

TL;DR

This work clarifies the algebraic structure of gauge-invariant operators in multi-matrix quantum mechanics by proving that primary invariants count equals the Krull dimension, calculable via complete gauge fixing. It shows that for $d$ adjoint matrices the number of primaries is $N_P = 1 + (d-1)N^2$, and for $f$ bifundamentals $N_P = 2(f-1)N^2 + 1$ (with $f=1$ giving $N_P=N$). Beyond primaries, the paper establishes that the number of secondary invariants grows as $e^{cN^2}$ for some constant $c>0$, using restricted Schur polynomials and LR-coefficient asymptotics; this supports the interpretation of secondary invariants as encoding non-perturbative gravitational states such as black hole microstates. The results tie together geometric gauge fixing and algebraic invariant theory, providing a holographic perspective on how bulk degrees of freedom and black-hole-like states emerge from the gauge-invariant operator spectrum. The analysis opens avenues for exploring dynamical roles of these invariants in thermalization, scrambling, and quantum gravity diagnostics in multi-matrix quantum mechanics.

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.

Figures (1)

• Figure 1: A plot of the function $f(\alpha)={1\over 15}\left((15+\alpha)\log (1+{\alpha\over 15})-\alpha\log{\alpha\over 15}\right)-\alpha\log (2)$.