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The threshold bound state conjecture for SU(N) supersymmetric matrix quantum mechanics

The SU(N)\mathrm{SU}(N) BFSS model is the maximally supersymmetric matrix quantum mechanics with nine Hermitian N×NN\times N matrices and sixteen fermionic partners, obtained by reducing ten-dimensional super Yang-Mills to one dimension. De Wit, Luscher and Nicolai (1989) showed its spectrum is [0,∞)[0,\infty). Witten (1996) predicted that, after removing the center of mass, there is exactly one normalizable zero-energy state for every NN (the D0-brane bound state), a prediction central to the BFSS matrix-theory conjecture. Index computations (Yi, Sethi-Stern, Moore-Nekrasov-Shatashvili) supported it, but a rigorous proof of existence and uniqueness was open. For every N≥2N\ge2, does the relative SU(N)\mathrm{SU}(N) Hamiltonian have a kernel of dimension exactly one?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Supersymmetric matrix quantum mechanics; spectral theory
Posed by
E. Witten, Bound states of strings and p-branes, Nuclear Physics B (1996), Section 4.2; central to Banks, Fischler, Shenker and Susskind (1997)
Year posed
1996
Years open
30y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every integer N≥2N\ge2, the relative undeformed SU(N)\mathrm{SU}(N) BFSS Hamiltonian (closure of 116∑α∥QαΨ∥2\frac1{16}\sum_\alpha\|Q_\alpha\Psi\|^2 on gauge-invariant states in L2(R9(N2−1))L^2(\mathbb R^{9(N^2-1)}) tensor a Clifford module) has a one-dimensional kernel, consisting of Spin(9)\mathrm{Spin}(9)-invariant even states. This gives both existence and uniqueness of the threshold bound state at each finite NN. Constants depend on NN; nothing is claimed about large-NN limits or the matrix-theory conjecture itself.

What the AI did

Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Witten's prediction as cited. Theorem 1.1 states dim⁡ker⁡HN=1\dim\ker H_N=1 for every N≥2N\ge2, with HNH_N the self-adjoint operator of the closed supercharge quadratic form on the gauge-invariant smooth compactly supported core; the kernel is Spin(9)\mathrm{Spin}(9)-invariant and even. The result is for that specific operator realization; the paper contrasts it with other realizations (Boulton et al.). The proof passes from a BMN mass-deformed Fredholm index to the undeformed kernel with cluster estimates. Not refereed here. No Lean formalization.

Sources

Changelog1 change

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