bibkey: tierz2026bpsoddp
authors: Miguel Tierz
year: 2026
title: "BPS spectra of Tr[Ψ^p] matrix models for odd p"
doi: null
url: https://arxiv.org/abs/2604.27164v2
claim: "The BPS cohomology of the fermionic matrix model with supercharge Q_p = Tr(Ψ^p) factors as Λ(χ) ⊗ B_0 with B_0 the traceless cohomology, a graded module over the exterior algebra Λ(C^{N−1}) of the traceless diagonal creators. v1 conjectures that this module is free; v2 states the freeness B_0 ≅ Λ(C^{N−1}) ⊗ M^{(p,N)} as a sufficient explanation, not proved, of the (1 + x) factors of the computed BPS polynomials."
strata_touched:
  - D5/S3/Quantum/Fermionic/TierzTracelessFreenessRefutation
license: citation-only
triage: anchor

Tierz, BPS spectra of Tr[Ψ^p] matrix models for odd p

M. Tierz, BPS spectra of Tr[Ψ^p] matrix models for odd p, arXiv:2604.27164 [hep-th] (v1 29 April 2026; v2 25 August 2026).

Verified locator

No DOI; the arXiv record is the stable locator. Primary version read: https://arxiv.org/abs/2604.27164v2 (the latest arXiv version), with https://arxiv.org/abs/2604.27164v1 for the original conjecture. Source: https://arxiv.org/pdf/2604.27164v2 and https://arxiv.org/pdf/2604.27164v1 (text read from the PDFs):

• v2 §2, the model, the trace fermion χ and the traceless matrix A with Q_p = Tr(A^p);
• v2 §5, Eqs. (44), (45), Table 3, Corollary 3 and Conjecture 1;
• v2 §6.3 “Open questions”, the graded-module structure of B_0 and the freeness statement;
• v1 §5, the paragraph “Structural interpretation of (1 + x)^N”.

Source statements

v2 §6.3: “let B•_0 denote the cohomology of the traceless complex. Theorem 2 and the Künneth formula give an isomorphism of graded vector spaces B• ≅ Λ•(Cχ) ⊗ B•_0. Each traceless linear combination of the diagonal creation operators anticommutes with Q_p and therefore induces a map of degree one on B•_0. The traceless diagonal combinations form a vector space of dimension N−1. Their induced maps anticommute and square to zero, so they make B•_0 a graded module over Λ•(C^{N−1}). A sufficient explanation for the remaining factors of 1+x would be the existence of a graded vector space M^{(p,N)} and an isomorphism of graded modules B•_0 ≅ Λ•(C^{N−1}) ⊗ M^{(p,N)}. … Such freeness is not implied by divisibility of the Hilbert series and is not proved here.”

v1 §5, “Structural interpretation of (1 + x)^N”: “Since wedge multiplication by any creator descends to cohomology, the reduced BPS cohomology H•_red is naturally a graded module over the exterior algebra Λ•(C^{N−1}) generated by the N−1 traceless diagonal creators. The (1+x)^{N−1} factor would follow if this module were free. … (conjectural) H•_red is free over Λ•(C^{N−1}), so H•_red ≅ Λ•(C^{N−1}) ⊗ P^{(p,N)} … The first layer is proved; the second is the content of the open problem.”

v2 Eqs. (44), (45): “Z^{(5,3)}_BPS(x) = 5x²(1 + x)³(4 + 3x + 4x²), Z^{(5,4)}_BPS(x) = 125x⁴(1 + x)⁴(1 + 7x + 6x² + 7x³ + x⁴).”

v2 Conjecture 1: “For odd p ≥ 5 with p ≤ 2N−1, (1 + x)^N | Z^{(p,N)}_BPS(x) in Z[x].”