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

Abstract

We study the BPS cohomology and Hamiltonian of a model built from an $N\times N$ matrix $\Psi$ of complex fermions, with supercharge $Q_p={\rm Tr}(\Psi^p)$ for odd $p\ge3$. Numerical calculations give the BPS multiplicity at every fermion number for $(p,N)=(5,3),(5,4),(5,5),(7,4)$, and through fermion number six at $(7,5)$. In $Z_{BPS}^{(p,N)}(x)=\sum_Rh_Rx^R$, $x$ records fermion number and $h_R$ counts BPS states in sector $R$. In each of the four cases computed at every fermion number, factoring out the lowest power of $x$ leaves a polynomial divisible by a positive power of $p$ and by $(1+x)^N$. The quotient by these factors has nonnegative integer coefficients. Exact decoupling of ${\rm Tr}\Psi$ and the Euler characteristic prove divisibility by $(1+x)^2$. Pairing sectors of complementary fermion number gives a third factor when $N$ is odd. For odd $p\le2N-1$, the only range in which $Q_p$ can be nonzero, the same pairing proves the corresponding rank symmetry for $Q_p$. The additional factors required to reach $(1+x)^N$ remain conjectural in general. Evaluating at $x=1$ gives the total BPS count $Z_{BPS}^{(p,N)}(1)$. For each fixed odd $p$, the Witten indices imply that, for any sequence of matrix sizes $N_j\to\infty$ along which $N_j^{-2}\log Z_{BPS}^{(p,N_j)}(1)$ converges, its limit lies in $[\log(2\cos\frac{\pi}{2p}),\log 2]$. After dividing the Hamiltonian by $p^2$, normal ordering gives an alternating sum of operators $K_k$, with $K_k$ containing $k$ creation and $k$ annihilation operators. We obtain exact formulas for $K_0,K_1,K_2$ when $p=5$ and $N\ge3$. Only $K_3,K_4$ can contain spectral information beyond the number of traceless fermions and the quadratic Casimir. At $N=3$, the Hodge star maps the invariant cubic form to the quintic form up to scale, so all five terms commute. Calculations with integer matrices give nonzero commutators in the reported $N=4$ sectors.