Bosonic Fortuity in Vector Models

TL;DR

The paper analyzes finite-$N$ gauge-invariant operators in coupled matrix–vector quantum mechanics under $U(N)$, using the Molien–Weyl formula to derive exact partition functions and reveal a two-tier invariant structure: primary invariants freely generate the space for $f\le N$, while for $f>N$ secondary invariants emerge and grow rapidly. It shows a bosonic analogue of the fortuity mechanism, with trace relations shaping the transition between secondary and primary invariants as $N$ grows. The results link invariant counting to holographic ideas in higher-spin contexts and illuminate how vector and matrix degrees of freedom interplay to form the full gauge-invariant operator space. Overall, the work provides analytic counts, asymptotics for secondary invariants, and structural insights that apply to both vector and matrix models and their holographic interpretations.

Abstract

We investigate the space of $U(N)$ gauge-invariant operators in coupled matrix-vector systems at finite $N$, extending previous work on single matrix models. By using the Molien-Weyl formula, we compute the partition function and identify the structure of primary and secondary invariants. In specific examples we verify, using the trace relations, that these invariants do indeed generate the complete space of gauge invariant operators. For vector models with $f \leq N$ species of vectors, the space is freely generated by primary invariants, while for $f > N$, secondary invariants appear, reflecting the presence of nontrivial trace relations. We derive analytic expressions for the number of secondary invariants and explore their growth. These results suggest a bosonic analogue of the fortuity mechanism. Our findings have implications for higher-spin holography and gauge-gravity duality, with applications to both vector and matrix models.