Threshold and positive-energy bound states of the BFSS matrix model

The explanation

Take two vectors x¹ and x² in a plane, and use the energy V = |x¹ ∧ x²|². This is the squared area of the parallelogram they span. Put x¹ = (r, 0) and x² = (0, s). Then V = r²s². At r = 10 and s = 0.1, V = 1. If the vectors point the same way, V = 0 however long they are. So the energy has flat valleys running off to infinity. A classical ball can roll out along such a valley. A quantum particle might not, because the valley gets narrower as r grows.

The BFSS model uses a potential of this kind. It has nine Hermitian matrices, plus fermion partners, linked by supersymmetry. Supersymmetric matrix quantum mechanics first appeared in the study of supermembranesSource 1. With SU(N) matrices it has 9(N² − 1) position coordinates. That is 27 for N = 2 and 72 for N = 3. Witten predicted that, with the centre of mass removed, every N has exactly one normalizable zero-energy stateSource 2. Normalizable means the wave stays concentrated, so the particles stay bound. In matrix theory, this state is the inner state of a single supergraviton carrying N units of momentumSource 3.

The state is hard to see because it sits at a threshold. The continuous spectrum, the energies of states that escape, starts at zeroSource 4. So the bound state has the same energy as states flying apart. Physicists counted zero-energy states with a supersymmetric index. In this noncompact problem, contributions from infinity are essentialSource 5. Sethi and Stern obtained the N = 2 index by including such a boundary termSource 6. A general-N argument by Konechny assumed that certain matrix differences all grow at the same rateSource 7. Controlling the regions where they do not is the main obstacle.

The first paper claims the full result. For every N ≥ 2, the model has exactly one normalizable zero-energy state after removing the centre of massSource 8. The paper calls this a proof of the threshold-bound-state conjecture.

The second paper looks at N = 2 above zero energy. The flat valleys appear when the three vectors in the model line upSource 9. The paper claims an unbounded sequence of positive energies with trapped wave functionsSource 10. In some symmetry sectors, oscillation across the valley costs enough energy to hold the state in. The paper says this refutes, at N = 2, the original BFSS paper's exclusion of such statesSource 11.

How sure is this? There is no Lean formalization for these papers. The claims rest on two papers, one of them long. No specialist review of them has been published.

What it doesn't show. This does not prove the BFSS matrix-theory conjecture itself, or anything about the large-N limit.