BPS spectra of $\operatorname{tr}[Ψ^p]$ matrix models for odd $p$
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866914518120005632 |
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| author | Tierz, Miguel |
| author_facet | Tierz, Miguel |
| contents | We compute exact finite-rank BPS generating functions for the fermionic matrix model with single-trace supercharge $Q_p=\operatorname{tr}(Ψ^p)$ 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 $[\log(2\cos\fracπ{2p}),\,\log 2]$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27164 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | BPS spectra of $\operatorname{tr}[Ψ^p]$ matrix models for odd $p$ Tierz, Miguel High Energy Physics - Theory Mathematical Physics We compute exact finite-rank BPS generating functions for the fermionic matrix model with single-trace supercharge $Q_p=\operatorname{tr}(Ψ^p)$ 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 $[\log(2\cos\fracπ{2p}),\,\log 2]$. 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. |
| title | BPS spectra of $\operatorname{tr}[Ψ^p]$ matrix models for odd $p$ |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2604.27164 |