Bound states in BFSS matrix mechanics: one at zero, infinitely many above it

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Family 270 · Mathematical physics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

A quantum particle may remain bound even when classical motion offers a route to infinity. Transverse quantum vibrations can make escape expensive. Supersymmetry complicates this picture: bosonic and fermionic contributions can cancel that expense, allowing scattering at arbitrarily small energy.

The BFSS matrix model brings these effects together. Its variables are nine matrices rather than an ordinary particle position. When the matrices commute, the classical interaction vanishes, creating unbounded escape directions. Nevertheless, string-theoretic arguments predict exactly one normalizable zero-energy state after the freely moving center of mass is removed.

Family 270 contains two claims about this same spectral landscape. M1 reports the unique zero-energy state for every finite matrix size. M2 reports infinitely many positive-energy normalizable eigenstates when the matrices have size two. The latter challenges a separate exclusion stated in the original BFSS paper. These conclusions concern different energies and can coexist: uniqueness at zero does not mean uniqueness of all bound states.

2. Background and essential definitions

From positions to matrices

Fix $N\geq2$. The bosonic variables are nine traceless Hermitian $N\times N$ matrices $X_1,\ldots,X_9$. A Hermitian matrix equals its conjugate transpose. Tracelessness removes the scalar matrix coordinates that describe the free center of mass. Since the real dimension of traceless Hermitian matrices is $N^2-1$, the bosonic configuration space is

$$\mathbb R^{9(N^2-1)}.$$

There are also finitely many fermionic degrees of freedom, represented by a finite-dimensional complex space $F_N$. Wavefunctions are square-integrable $F_N$-valued functions on that Euclidean space.

The group $SU(N)$, consisting of unitary matrices with determinant one, acts by simultaneous conjugation on the bosonic variables and by a corresponding action on the fermions. Physical states are invariant under this action: conjugating every matrix describes a gauge change, not a new physical configuration. Thus

$$\mathcal H_N=\bigl[L^2(\mathbb R^{9(N^2-1)})\otimes F_N\bigr]^{SU(N)}.$$

The bosonic potential is built from squared commutators $[X_i,X_j]=X_iX_j-X_jX_i$. Commuting matrices therefore have zero potential even when their entries become large. Their simultaneous eigenvalues suggest separated clusters, but this geometric picture does not by itself determine the quantum spectrum.

Supersymmetry and normalizability

Sixteen differential operators $Q_\alpha$, called supercharges, combine momentum terms with matrix commutators and fermionic matrices. The manuscripts specify the Hamiltonian through the closed quadratic form

$$q_N(\Psi)=\frac1{16}\sum_{\alpha=1}^{16}\|Q_\alpha\Psi\|^2.$$

One begins with smooth compactly supported gauge-invariant functions and closes this form. The associated operator $H_N$ is nonnegative and self-adjoint. This specification matters: an operator restricted to a valley with an imposed boundary condition is a different spectral problem.

A normalizable eigenstate is a nonzero square-integrable vector satisfying $H_N\Psi=E\Psi$. After normalization, its squared norm integrates to one. A formal solution or a scattering wave need not satisfy this requirement.

The spectral set begins at zero and extends through nonnegative energies. A zero eigenstate is consequently at the threshold of the continuum, rather than below an isolated band of scattering energies. Positive normalizable eigenstates lie inside that spectral interval; such embedded eigenstates are logically compatible with the interval being the whole spectral set.

3. Development history and previous partial results

Supersymmetric matrix mechanics arose in supermembrane regularization. The BFSS conjecture of Banks, Fischler, Shenker and Susskind proposed a large-matrix-size description of M-theory. A prediction associated with Witten’s D0-brane analysis was that each finite size should have one relative zero-energy bound state. In the proposed interpretation, that internal state supports a single supergraviton carrying the corresponding longitudinal momentum.

De Wit, Lüscher and Nicolai established the nonnegative continuous spectral interval for the matrix Hamiltonians they studied. Their analysis explicitly left room for normalizable states inside it. Confusing a statement about the spectral set with an exclusion of eigenvectors would erase the distinction at issue here.

Supersymmetric indices supplied substantial evidence for the zero state. Such an index counts even zero modes minus odd zero modes. On a noncompact configuration space, contributions from infinity must be controlled. Yi analyzed the two-particle bulk contribution; Sethi and Stern included a boundary contribution to establish the two-particle bound state. Moore, Nekrasov and Shatashvili evaluated relevant matrix integrals. Green and Gutperle, and Konechny, developed general-rank cluster and asymptotic arguments.

M1 identifies a remaining analytic issue in passing from such calculations to a full-space kernel count: eigenvalue separations need not all grow at the same rate. Clusters can split into smaller clusters on nested scales, and normalizable mass must not disappear between those scales. Earlier rotation-invariance results of Sethi–Stern and Hasler–Hoppe are also important: in the grading used by M1, they make normalizable zero states even, allowing a signed count to become a dimension count.

Mass deformations offered another approach. Porrati–Rozenberg and Kac–Smilga emphasized normalizability when removing a deformation. The confining plane-wave deformation of Berenstein, Maldacena and Nastase has vacua indexed by partitions of $N$. Asymptotic wavefunction constructions by Halpern–Schwartz, Graf–Hoppe, Fröhlich and collaborators, and Lin–Yin clarified local cluster behavior. M1’s claimed advance is global control sufficient to remove the deformation without losing or inventing states.

Positive-energy binding has a separate history. Danielsson, Ferretti and Sundborg found excited normalizable wavefunctions in a Born–Oppenheimer approximation and discussed metastability beyond it. Symmetry-restricted confinement also appeared in related, simpler matrix models. M2 claims exact eigenvectors of the undeformed relative $SU(2)$ operator, beyond those approximate conclusions.

4. What the new manuscripts claim

Exactly one threshold state

M1 states

$$\dim\ker H_N=1\qquad\text{for every finite integer }N\geq2.$$

The kernel is the zero-energy eigenspace. Its states are invariant under $\operatorname{Spin}(9)$, the spin group implementing spatial rotations, and even in the specified fermion grading. Square integrability is over the entire configuration space. Constants may depend on the fixed value of $N$; this is not a uniform large-$N$ estimate.

The free $U(1)$ center of mass is absent. Adding it back introduces free motion, so the statement should not be rewritten as a normalizable bound state of all center-of-mass coordinates. Nor does uniqueness of a kernel supply a positive gap above it.

Infinitely many positive-energy states at size two

M2 uses the same full-space, gauge-invariant supercharge-form realization for the relative $SU(2)$ model. It states that there are orthonormal vectors $\Psi_j$ in the operator domain and numbers $E_j>0$ such that

$$H_2\Psi_j=E_j\Psi_j, \qquad E_j\longrightarrow\infty.$$

In particular, there are infinitely many distinct positive eigenvalues. M2 presents this as a counterexample, at $N=2$, to the original BFSS paper’s exclusion of other normalizable relative bound states beyond the threshold states. It does not present it as a disproof of the separate large-$N$ Matrix Theory conjecture.

The positive states occur in selected rotation sectors. They do not contradict the rotation-invariance theorem for zero-energy states, whose energy hypothesis is essential. M2 also does not assert that the entire operator has discrete spectrum.

5. Why the advance matters

Importance: 5/5, provisional. The threshold-state prediction is a central finite-size consistency question in matrix theory. A full-space existence-and-uniqueness theorem for every finite $N$, together with exact positive embedded states at $N=2$, would substantially sharpen the spectral foundations of this model.

The family is also instructive beyond its physical interpretation. It illustrates two distinct roles of symmetry: supersymmetry can permit escape channels, while a specified rotation type can prevent a state from using the low transverse modes needed for escape. The same operator can therefore support scattering and confined sectors.

This rating concerns the significance of the claims if correct. Their exposition does not independently establish the claimed proofs or settle the wider physical conjectures motivating the model.

6. The main idea

Count states without losing norm at infinity

M1 begins with a massive, confining deformation. A suitable restricted real supercharge has a Fredholm index equal to $p(N)$, the number of unordered integer partitions of $N$. A Fredholm index is a stable signed kernel count for an operator with controlled range and finite-dimensional defects.

As the mass is removed, the partitions acquire a second meaning. A proper partition describes separated blocks of smaller matrix sizes. Each block contributes its own internal zero state, while the block centers carry relative-motion variables. The intended accounting is one line for each of the $p(N)-1$ proper partitions, leaving one line for a bound state in which the whole system remains together.

That subtraction becomes meaningful only after proving that every limiting norm is accounted for. M1 separates fast transverse oscillators from slow cluster coordinates, corrects approximate test states so their charge outputs converge, and uses induction on block size. An exterior estimate based on the Hardy inequality controls intermediate scales; a separate massive estimate prevents escape to infinity.

The resulting decomposition preserves norms and inner products. Extraction forces existence of a whole-system zero state. Recovery of proposed profiles and a spectral-gap estimate in the massive comparison bound its multiplicity by one. This is the manuscript’s mechanism for turning a deformation count into the undeformed full-space statement.

Let rotation types forbid cheap escape

For $SU(2)$, write the bosonic coordinates as three vectors $x^1,x^2,x^3\in\mathbb R^9$. The potential in M2’s normalization is

$$V(x)=\sum_{a<b}|x^a\wedge x^b|^2.$$

It vanishes when the vectors are collinear. Fixing a nonzero vector $u$ exposes sixteen transverse coordinates for the other two vectors. They behave like an oscillator whose frequency grows with $|u|$.

The rotations fixing $u$ form $\operatorname{Spin}(8)$. Representation theory constrains which transverse oscillator levels can appear in a chosen $\operatorname{Spin}(9)$ sector. If its highest-weight label is $\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$, the relevant bound forces an oscillator level $n$ to satisfy $n\geq\lambda_2-M$, where $M$ depends only on the finite fermionic space.

Making the second weight coordinate large excludes many low oscillator levels. The remaining positive oscillator energy dominates the potentially negative fermionic term, which grows only linearly in the coordinates. The chosen sector then has compact resolvent: its spectrum consists of eigenvalues escaping to infinity. M2 constructs infinite-dimensional physical sectors with the required weights, so their eigenvectors are also eigenvectors of the full Hamiltonian.

7. Scope, limitations, and open questions

M1 is a theorem claimed for each fixed finite $N$, with the center of mass removed and a specified operator domain. It supplies neither an explicit closed-form wavefunction nor uniform control as $N\to\infty$. The existence of a threshold state does not separate it from the continuum by a gap.

M2 proves its positive-state statement only for relative $SU(2)$. It does not classify every eigenstate, determine exact energies, or extend the result to all matrix sizes. Its confined rotation sectors coexist with escape in other sectors. The stability of these embedded states under symmetry-breaking perturbations is a further question.

Scattering theory, detailed cluster asymptotics and the large-$N$ physical interpretation require additional work beyond the two headline statements. Keeping these boundaries explicit preserves the difference between resolving a precise spectral question and establishing an entire proposed physical duality.

8. Sources and further reading

The original BFSS and Sethi–Stern papers linked above provide the physical proposal and a major two-particle predecessor. For the spectral distinction, see de Wit, Lüscher and Nicolai, The supermembrane is unstable. For earlier approximate excited states, see Danielsson, Ferretti and Sundborg, D-particle dynamics and bound states. M1 and M2 below supply the claims and their stated operator conventions.

Source manuscripts

• M1. OpenAI, The unique threshold bound state of the SU(N) BFSS model (September 24, 2026).
• M2. OpenAI, Positive eigenvalues of the relative SU(2) BFSS Hamiltonian (October 5, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.