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arxiv: 2604.27164 · v1 · submitted 2026-04-29 · ✦ hep-th · math-ph· math.MP

Recognition: unknown

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

Miguel Tierz
Authors on Pith no claims yet

Pith reviewed 2026-05-07 10:06 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords BPS spectramatrix modelssuperchargeWitten indexpalindromic polynomialsfermionic modelslarge N limitindex floor
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The pith

BPS generating functions for tr[Ψ^p] matrix models with odd p factor as a power of p times an onset monomial, (1+x)^N, and a palindromic reduced polynomial.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes exact BPS generating functions for the single-trace fermionic matrix model at small odd p and finite N, including full results at (5,3) through (5,5) and (7,4). These spectra display an overdetermined factorization pattern that survives the breakdown of Casimir solvability for p at or above 5. The pattern consists of a p-power prefactor, a minimal-charge monomial, the universal (1+x)^N term, and a palindromic polynomial whose degree is fixed by rank palindromicity. Mod-p Witten indices supply a closed-form lower bound on the index, which together with the Hilbert-space dimension bounds the possible accumulation points of the normalized log-partition function. Comparisons with matched N=2 SYK models show that the matrix-model excess is absent in the SYK case.

Core claim

Explicit computation yields BPS generating functions that factor as p^k x^{q_min} (1+x)^N R(x) with R palindromic and reduced; rank palindromicity r_R = r_{N^2-p-R} follows from the exterior top-form pairing; the mod-p indices produce a closed index floor; and any accumulation point of N^{-2} log Z_BPS^{(p,N)} must lie in the interval [log(2 cos(π/(2p))), log 2].

What carries the argument

The charge-resolved BPS generating function R_{p,N}(x) that decomposes into the observed overdetermined product of a p-power, onset monomial, (1+x)^N factor, and palindromic reduced polynomial.

If this is right

  • Rank palindromicity holds for all computed cases and follows directly from the exterior-algebra pairing.
  • The mod-p Witten index supplies a rigorous lower bound on the BPS count for any N.
  • The accumulation window for N^{-2} log Z_BPS is narrowed to the explicit interval between log(2 cos(π/(2p))) and log 2.
  • A rank-projection tower produces lower bounds on the projection-fortuitous cohomology.
  • Matched N=2 SYK models at N_f=16 saturate the index floor while the single-trace matrix model shows excess and wider charge support.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The persistent factorization may point to an algebraic identity that is independent of Casimir solvability.
  • The index-floor bound could be used to test whether larger-N computations remain inside the predicted window.
  • The contrast with SYK saturation suggests that single-trace truncation introduces measurable excess that might be quantifiable in the large-N limit.

Load-bearing premise

That the factorization and index-floor pattern seen at small finite N remain structural features that continue to hold or bound the large-N limit.

What would settle it

Computation of the full charge-resolved BPS generating function at a new pair such as (5,6) that fails to exhibit the same p-power times onset times (1+x)^N times palindromic form or violates the predicted index floor.

Figures

Figures reproduced from arXiv: 2604.27164 by Miguel Tierz.

Figure 1
Figure 1. Figure 1: Landscape of single-trace models. The shaded region p > 2N − 1 is trivial (Qp ≡ 0). Blue circles mark cases with complete exact spectra; the orange square marks partial (7, 5) data; the red triangle is the trivial below-threshold case (7, 3). Annotations show the minimum BPS charge qmin where known. and N = 5, but 1 N2 log ZBPS remains near 0.66 ( view at source ↗
Figure 2
Figure 2. Figure 2: BPS charge profiles broaden with N and shift qmin at N = 5. Charge-resolved BPS multiplicities hR for the four complete nontrivial cases. The dashed line marks the particle￾hole symmetry axis at half filling (R = N2/2). Arrows indicate the minimum BPS charge qmin; note the +1 shift to qmin = 8 (from the N ≤ 4 pattern value of 7) at (5, 5). Proposition 4 (Degree-7 BPS generating function). For p = 7, N = 4,… view at source ↗
Figure 3
Figure 3. Figure 3: Onset of BPS cohomology at (p, N) = (5, 5). For R ≤ 4, QR is injective and no kernel exists. At R = 5, 6, 7, a kernel appears (blue) but is exactly cancelled by the incoming rank (purple): hR = 0. Cohomology first survives at R = 8 = qmin, where the kernel outgrows the incoming image. The gap between the blue and purple curves is the BPS multiplicity (red diamonds). on the unit circle ( view at source ↗
Figure 4
Figure 4. Figure 4: Projection tower classifies BPS states as fortuitous. Shown for the quintic tower (p = 5). Each row shows the fermion-number sectors for a given N, with sectors vertically aligned by R; colored cells have hR > 0. The projection πM→N maps BPS states downward. Where the higher-rank row has no BPS states (grey), the lower-rank states in the corresponding sectors have no source and are therefore unconditionall… view at source ↗
Figure 5
Figure 5. Figure 5: Exact BPS count versus protected floor. Main panel: total BPS count split into projection-fortuitous (red) and not forced fortuitous by the projection test (blue) for the three cases tested by the projection tower; percentages show the fortuitous fraction. Right panel: residue-class decomposition at (5, 4), showing the index floor |Ia| (blue) and index excess (orange) by class a mod 5; residue a = 3 is exa… view at source ↗
Figure 6
Figure 6. Figure 6: Charge support: matrix model vs. N = 2 SYK. Normalized BPS distributions hR/ZBPS for the matrix models (5, 4) and (7, 4) (solid lines, N2 = 16) and N = 2 SYK at qˆ = 3, 5, 7 (shaded bars, Nf = 16). SYK at each qˆ has BPS states in exactly qˆ sectors and saturates the index floor exactly; the matrix model populates broader charge ranges with nonzero index excess. The gap narrows at higher qˆ: the SYK qˆ = 7… view at source ↗
Figure 7
Figure 7. Figure 7: BPS fraction drops but absolute degeneracy grows exponentially in N 2 . For the quintic sequence (p = 5). Left: the BPS fraction falls from 86% to 46%, crossing 50% near N = 5. Right: the normalized entropy 1 N2 log ZBPS stays near 0.66, below log 2 ≈ 0.69 (dashed red). Whether this quantity converges to a nonzero limit is a central open question. Since ZBPS ≥ P a |Ia|, this yields a rigorous lower bound o… view at source ↗
Figure 8
Figure 8. Figure 8: Non-BPS spectra are sparse and irregular in units of p 2 . Each horizontal line marks a distinct reduced eigenvalue E/p2 ; the green line is the BPS level (E = 0). The irregular spacing reflects the loss of single-Casimir control at p ≥ 5: identical irreps can appear at different energies. Non-BPS spectrum The non-BPS spectrum also contains structure ( view at source ↗
Figure 9
Figure 9. Figure 9: Roots of the reduced BPS polynomials T (p) N (x) in the complex plane, including the exact T (5) 5 . Blue circles: roots on |z| = 1. Red diamonds: real roots off the unit circle, occurring in reciprocal pairs; large reciprocal partners outside the plotting window are indicated by arrows and numerical labels. Each panel records the degree and the number of roots on versus off the unit circle. 29 view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper computes exact finite-rank BPS generating functions for the fermionic matrix model with single-trace supercharge Q_p = tr(Ψ^p) at (p,N)=(5,3),(5,4),(5,5),(7,4) and partial data at (7,5). In all complete cases the charge-resolved spectrum factors as a p-power times an onset monomial x^{q_min} times (1+x)^N times a palindromic reduced polynomial. Rank palindromicity r_R = r_{N^2-p-R} is proven generally via exterior top-degree pairing; at (5,5) the ten low-charge ranks plus the divisibility condition (1+x) | R_{5,5} algebraically fix the middle rank, confirmed by direct computation. Mod-p Witten indices supply a closed-form index floor; combined with the trivial Hilbert-space upper bound this confines any accumulation point of N^{-2} log Z_BPS^{(p,N)} to the interval [log(2 cos(π/(2p))), log 2]. Rank-projection towers give rigorous lower bounds on projection-fortuitous cohomology, and matched N=2 SYK examples at N_f=16 are shown to saturate the index floor while the matrix model exhibits nonzero excess and broader charge support.

Significance. The explicit spectra, the general palindromicity proof, and the parameter-free index floor constitute reproducible, falsifiable results that supply concrete data and rigorous bounds on BPS growth rates in these models. The factorization pattern observed despite the loss of Casimir solvability at p≥5 is noteworthy and may indicate deeper algebraic structure. The accumulation-point window is a load-bearing, general result that does not rely on the small-N factorization persisting; the stress-test concern therefore does not land on the central bound.

minor comments (2)
  1. The introduction would benefit from an explicit statement separating the general results (palindromicity, index floor, accumulation window) from the observed factorization, which is presented only for the computed cases.
  2. A table or appendix listing the explicit generating functions for all fully computed (p,N) pairs would improve reproducibility and allow readers to verify the factorization pattern directly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary of our results, and positive assessment. The recommendation for minor revision is noted; we will incorporate any editorial improvements to presentation or clarity in the revised version. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: explicit computations, general algebraic proofs, and standard index formulas are independent of target claims

full rationale

The paper performs direct exact computations of BPS generating functions at small finite ranks (p,N)=(5,3),(5,4),(5,5),(7,4) and partial (7,5), observes the factorization pattern only in those cases, and proves palindromicity r_R = r_{N^2-p-R} via the general exterior top-degree pairing without reference to the observed factorization. The closed-form index floor is obtained from mod-p Witten indices (a standard construction independent of the specific matrix-model spectra), which together with the trivial Hilbert-space upper bound yields the accumulation-point window without any fitting, self-definition, or reduction of the bound to the small-N data. No self-citations are load-bearing, no parameters are fitted and renamed as predictions, and the large-N statements are framed as bounds resting on the general index floor rather than extrapolation of the full factorization. The derivation chain is therefore self-contained against external algebraic and index-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard fermionic matrix model definitions and algebraic properties of exterior algebras without introducing new free parameters or invented entities.

axioms (2)
  • standard math Standard properties of the exterior algebra and top-degree pairing
    Invoked to prove rank palindromicity r_R = r_{N^2-p-R} from the exterior top-degree pairing.
  • domain assumption Existence of mod-p Witten indices yielding a closed-form floor
    Used to place accumulation points of N^{-2} log Z_BPS in the stated interval.

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Forward citations

Cited by 1 Pith paper

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