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Positive eigenvalues of the relative SU(2) BFSS Hamiltonian
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Abstract
We prove that the relative $\mathop{\mathrm{SU}}\nolimits (2)$ BFSS Hamiltonian has infinitely many positive eigenvalues tending to infinity, with square-integrable eigenvectors. The operator is defined by closing the gauge-invariant supercharge form. This refutes, at N = 2, the exclusion of normalizable positive-energy states stated in the original BFSS paper.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- etblink (GitHub), BFSS: manuscript–source paired SU2 Hamiltonian realization and closed-domain bridge (GitHub issue #28, opened 2026-10-17, open). An independent Lean 4 project matching this manuscript's relative SU(2) Hamiltonian with its own definition, reporting Lean-checked pieces (charge operators, closed domains, gauge-representation facts) and listing the Spin(9) confinement argument and the positive-eigenvalue theorem as still open. [AI-generated per source]
- etblink (GitHub), BFSS SU(2): true physical parity, 16-charge covariance and gauge-core identification; closed graph open (draft PR #25, opened 2026-10-11, unmerged). A companion draft PR with Lean formalizations of the SU(2)→SO(48) and Clifford(48) gauge action, parity unitaries and closed charge operators; it states that it does not prove the positive point spectrum. [AI-generated per source]
Possibly relevant literature (found by the agent, not checked)
- M. Campostrini, J. Wosiek, High precision study of the structure of D=4 supersymmetric Yang-Mills quantum mechanics (arXiv:hep-th/0407021, Nucl. Phys. B 703 (2004) 454-498). A numerical study of the D=4 analogue reporting "localized and non-localized states" coexisting in certain channels; relevant to the paper's claim of normalizable positive-energy states in selected symmetry sectors.
- L. Boulton, M. P. Garcia del Moral, A. Restuccia, Discreteness of the spectrum of the compactified D=11 supermembrane with non-trivial winding (arXiv:hep-th/0211047, Nucl. Phys. B 671 (2003) 343-358). A rigorous compact-resolvent result for a related supermembrane matrix Hamiltonian; relevant to the paper's step showing that restricted Spin(9) sectors have compact resolvent.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 270
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026,
author = {{OpenAI}},
title = {{Positive eigenvalues of the relative $\mathrm{SU}(2)$ BFSS Hamiltonian}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf}{OAI:Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026}},
year = {2026}
}Family 270: Threshold and positive-energy bound states of the BFSS matrix model
Proves that the undeformed relative $\mathrm{SU}(N)$ BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For $\mathrm{SU}(2)$, a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.
Other papers in this family:
- The unique threshold bound state of the SU(N) BFSS model · September 24, 2026