BPS spectra of $\operatorname{tr}[\Psi^p]$ matrix models for odd $p$

TL;DR

This paper computes exact BPS generating functions for fermionic matrix models with specific supercharges, revealing factorization properties, palindromic structures, and bounds on growth rates, with implications for supersymmetric quantum models.

Contribution

It provides explicit calculations and structural insights into BPS spectra of fermionic matrix models with supercharges $ ext{tr}( extPsi^p)$ for odd p, including factorization and palindromic properties.

Findings

Complete cases show spectrum factorization into a power of p, monomial, and palindromic polynomial.

Rank palindromicity is proven from exterior top-degree pairing.

Bounds on the growth rate of BPS partition functions are established, matching certain supersymmetric models.

Abstract

We compute exact finite-rank BPS generating functions for the fermionic matrix model with single-trace supercharge $Q_p=\operatorname{tr}(\Psi^p)$ at $(p,N)=(5,3),(5,4),(5,5),(7,4)$, together with partial data at $(7,5)$. In all complete computed cases, the charge-resolved spectrum exhibits an overdetermined factorization -- a power of $p$, times an onset monomial $x^{q_{\min}}$, times $(1+x)^N$, times a palindromic reduced polynomial -- despite the loss of Casimir solvability at $p\ge 5$. We prove rank palindromicity $r_R=r_{N^2-p-R}$ from the exterior top-degree pairing; at $(5,5)$, the ten low-charge ranks and the minimal divisibility condition $(1+x)\mid\mathcal{R}_{5,5}$ determine the remaining middle rank, and direct computation confirms the full generating function. For fixed $p$, the mod-$p$ Witten indices give a closed-form index floor; together with the trivial Hilbert-space upper bound, this places any accumulation point of $N^{-2}\log Z_{\mathrm{BPS}}^{(p,N)}$ in the window $[\log(2\cos\frac{\pi}{2p}),\,\log 2]$. A rank-projection tower gives rigorous lower bounds on the projection-fortuitous cohomology. In matched $\mathcal{N}=2$ SYK examples at $N_f=16$, the BPS count saturates the index floor, whereas the single-trace matrix model has nonzero index excess and broader charge support.