99.9...% of the SU(2) BFSS ground state

Alexander Michel; Jaroslav Scheinpflug; Xi Yin

AI summary · arXiv: 2611.09837v1 · hep-th

99.9...% of the SU(2) BFSS ground state

Alexander Michel, Jaroslav Scheinpflug, Xi Yin

Original paper Submitted to arXiv on 06 Nov. 2026 License: NONEXCLUSIVE-DISTRIB 1.0 Also in: fr

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The essentials

A symmetry-reduced variational calculation constructs an SU(2) BFSS state with a rigorous ground-subspace overlap bound above 0.999997 and reproduces the predicted asymptotic radial tail.

Question

Can the threshold ground state of the gapless SU(2) BFSS model be constructed globally with a variational method, and can its closeness to the ground subspace be established rigorously? The paper also tests whether the resulting state reproduces predicted ground-state moments and large-distance behavior.

Method

The authors use SU(2) × Spin(9) representation theory to reduce singlet wavefunctions to 2066 fermionic components, each depending on three singular values, and build a finite variational basis from 66 polynomial singlet seeds with anisotropic Gaussian profiles. They minimize an auxiliary Hamiltonian whose zero modes match those of the physical Hamiltonian and whose gap yields a rigorous fidelity bound; matrix elements are computed exactly using Wick contractions and elementary integrals. The reported state has energy at most 4.18 × 10^-8 in units of gYM^(2/3), and the calculation uses 256-bit ball arithmetic.

Findings

Result The variational state has a rigorously bounded ground-subspace overlap of at least 0.9999978.
See the passage in the paper
The expectation value of the auxiliary Hamiltonian bounds the ground-subspace overlap $F\equiv\|P_{\ker H}\psi\|^{2}$ via $F\geq[1-\kappa\langle H_{\mathrm{aux}}\rangle]_{+}$, with $\kappa=(36+8\sqrt{14})/25$ and the bracket denoting the positive part (Subsection III.2), so that $F\geq 0.9999978{}$.
Result The calculation obtains ground-state moments of ⟨R²⟩ = 8.07419(1) and ⟨R⁴⟩ = 68.7529(1), consistent with the paper’s comparatively loose bootstrap bounds.
See the passage in the paper
Beyond the tail, the state has $\langle R^{2}\rangle=8.07419(1){}$ and $\langle R^{4}\rangle=68.7529(1){}$, with $R^{2}=\sum_{a,i}X_{ai}^{2}$.
Result The variational state reproduces the predicted large-distance radial density exponent and coefficient.
See the passage in the paper
At $r$ large enough for the Coulomb-branch EFT to be reliable, the variational state agrees with it in Figs. 1–3, both in the radial density and in $\langle\rho^{2}\rangle_{r}$, $\langle\rho^{4}\rangle_{r}$.
Result The singlet-sector reduction leaves 2066 independent fermionic structures, each multiplying a function of three real variables.
See the passage in the paper
Using the Schur–Weyl duality [33, Thm. 6.3 and Ex. 6.11(b)], we construct an explicit basis for $\mathcal{F}_{*}$ and show that $$\dim\mathcal{F}_{*}=2066.$$

Limitations

The rigorous overlap bound does not rigorously control the reported expectation values of unbounded observables.
See the passage in the paper
The operators whose expectation values we evaluated above are unbounded, so we do not have rigorous control over their expectation values in the above sense.
The finite sum of Gaussian profiles eventually decays faster than the predicted radial tail, limiting the range over which the tail agreement holds.
See the passage in the paper
Since any truncated variational state, being a finite sum of Gaussian profiles, eventually decays faster than $p(r)$, the range of this agreement increases with the size of the basis.
The representation-theoretic reduction established here is specific to SU(2) and does not provide a comparable simplification for higher SU(N).
See the passage in the paper
For $\operatorname{SU}(N)$ with $N>2$, no comparable reduction occurs.

Why it matters

The authors argue that an auxiliary operator with a rigorously established gap makes Rayleigh–Ritz useful for a gapless BFSS Hamiltonian by converting a small variational objective into a rigorous ground-subspace overlap bound. For SU(2), representation theory makes a global variational construction tractable, enabling high-fidelity estimates of the state and tests of its predicted asymptotic behavior.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: BFSS matrix model SU(2) symmetry reduction variational method ground-state fidelity D0-brane asymptotics

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Comprehensive Summary
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Original abstract

The paper asks whether the ground state of the SU(2) BFSS matrix model can be computed globally, even though the model’s continuous spectrum extends down to zero. This threshold bound state is central to the BFSS description of D0-branes and M-theory. Conventional variational calculations face two challenges. First, the wavefunction appears to have an enormous number of fermionic components. Second, a low energy expectation value alone does not establish that a trial state is close to the ground state when the spectrum has no gap. The authors overcome these challenges by exploiting a symmetry reduction specific to SU(2). They express the bosonic matrices in singular-value form and impose gauge and Spin(9) invariance, showing that a singlet wavefunction is determined by functions of just three singular values and 2,066 allowed fermionic structures. This reduction makes a variational calculation feasible. The trial states combine polynomial singlets, generated in part by powers of the fermionic Yukawa interaction, with anisotropic Gaussian profiles designed to extend along the commuting valley, where the potential vanishes. The authors evaluate the required overlaps and matrix elements exactly using fermionic contractions, Gaussian integrals, and angular integrals. To establish fidelity rather than relying on energy alone, the authors introduce an auxiliary operator formed from the supercharges with the squared radial coordinate inserted between them. This operator shares the physical Hamiltonian’s zero-energy states and has a provable gap in the gauge-invariant Spin(9)-singlet sector. A Hardy-type inequality for spinor states then provides a quantitative bound: the auxiliary-operator expectation value limits the fraction of a normalized trial state that can lie outside the ground subspace. This approach requires neither a gap in the physical Hamiltonian nor a unique zero-energy state. For their optimized state, the authors report an energy upper bound of $4.18 \times 10^{-8}$, in units of the Yang–Mills coupling raised to the two-thirds power, and an auxiliary expectation value of about $8.21165 \times 10^{-7}$. These results yield a rigorous lower bound of 0.9999978 on the overlap with the zero-energy subspace. A smaller basis of 4,172 functions, built from 13 seeds, already gives an overlap bound above 0.999. The computation was performed on a laptop. The overlap result establishes closeness to the ground subspace, but does not provide a separate numerical error bound for every observable. The paper also tests the trial state against predictions for the large-separation region. Using a three-point amplitude for a bound state splitting into two D0-branes, the authors derive the asymptotic radial probability density. At large separation $r$, it scales as $1890\sqrt{2}\,r^{-10}$. Their variational state reproduces both the exponent and the coefficient over the range where the effective description applies. The authors report moments of the squared radius and its square of 8.07419 and 68.7529, respectively. These values are consistent with their matrix-bootstrap bounds, although those bounds are relatively loose. Because these are unbounded observables, the rigorous overlap bound does not certify their expectation values to the same degree. The authors therefore present the moment and tail comparisons as strong consistency checks rather than equivalent rigorous error bounds. The method’s most striking simplification is specific to SU(2). The authors argue that representation theory offers no comparable reduction at higher $N$, where other approaches may be needed. They identify higher-$N$ observables, scattering, and resonances as possible directions for future work.

How Much of the SU(2) BFSS Ground State Can We Capture?

The BFSS matrix model is a quantum-mechanical description of D0-branes that plays a central role in the BFSS proposal for M-theory. One of its most important features is a threshold bound state: a zero-energy ground state sitting at the bottom of a continuous spectrum that extends all the way down to zero.

That setting makes the ground state unusually difficult to study. A low energy from a trial calculation is not, by itself, proof that the trial state resembles the true ground state. With no gap separating the ground state from nearby states, even a very small energy does not directly measure how much of the state is correct. In their paper, “99.9...% of the SU(2) BFSS ground state,” Alexander Michel, Jaroslav Scheinpflug, and Xi Yin develop a way to address both the computational challenge and this question of reliability.

Turning a huge problem into a manageable one

A direct wavefunction calculation seems to require tracking an enormous number of fermionic components. The authors find a major simplification by focusing on SU(2) and restricting attention to states that are invariant under gauge transformations and Spin(9) rotations.

They express the bosonic matrices in singular-value form and show that a singlet wavefunction can be described using just three singular values, together with 2,066 allowed fermionic structures. This symmetry reduction turns a daunting problem into one that can be tackled with a variational calculation.

The trial states combine polynomial singlets with carefully chosen Gaussian profiles. Some of the polynomial structures are generated using powers of the fermionic Yukawa interaction. The Gaussian profiles are anisotropic, allowing the state to stretch along the commuting valley, a region where the potential vanishes. This matters because the ground state must account for the model’s long, flat directions, rather than being confined to a simple localized shape.

The authors evaluate the overlaps and matrix elements needed for their calculation using fermionic contractions, Gaussian integrals, and angular integrals. With the reduced basis, they can test and optimize a detailed trial state rather than relying on a crude approximation.

Measuring closeness without a physical energy gap

To establish that their trial state is close to the ground state, the authors introduce an auxiliary operator. It is built from the supercharges, with the squared radial coordinate inserted between them. Crucially, it has the same zero-energy states as the physical Hamiltonian, but it can be shown to have a gap within the gauge-invariant Spin(9)-singlet sector.

A Hardy-type inequality for spinor states then turns the auxiliary operator’s expectation value into a quantitative fidelity bound. In effect, it limits how much of a normalized trial state can lie outside the zero-energy subspace. The argument does not require the physical Hamiltonian to have a gap, or even require the zero-energy state to be unique.

For their optimized trial state, the authors report an energy upper bound of 4.18 × 10-8, in units of the Yang–Mills coupling raised to the two-thirds power. The auxiliary expectation value is about 8.21165 × 10-7. Together, these results give a rigorous lower bound of 0.9999978 on the state’s overlap with the zero-energy subspace. A smaller calculation, using 4,172 functions built from 13 seeds, already gives an overlap bound above 0.999. The authors report that the computation was performed on a laptop.

Checking the state in the long-distance region

The paper also tests whether the trial state behaves correctly when the D0-branes are far apart. Using a three-point amplitude for a bound state splitting into two D0-branes, the authors derive a prediction for the asymptotic radial probability density. At large separation r, it falls as 1890√2 r-10.

The variational state reproduces both the predicted power law and its coefficient across the range where the effective description applies. The authors also report expectation values of the squared radius and its square: 8.07419 and 68.7529. These values are consistent with their matrix-bootstrap bounds, though those bounds are comparatively loose.

There is an important distinction between these checks and the fidelity result. The rigorous overlap bound establishes closeness to the ground-state subspace, but it does not supply a separate numerical error bound for every observable. In particular, radius moments and tail probabilities are unbounded observables, so their expectation values are not certified to the same degree by the overlap estimate. The authors present these comparisons as strong consistency checks, not as equivalent rigorous error bounds.

What comes next

The striking efficiency of this method depends on the special representation theory of SU(2). The authors argue that higher values of N do not offer a comparable reduction, so different strategies may be necessary to study larger matrix models. They point to higher-N observables, scattering, and resonances as promising directions for future work.

For SU(2), however, the result is a notable advance: a calculation that is both practical and equipped with a rigorous measure of how much of the ground-state subspace it captures. Just as importantly, it shows why energy alone is not always enough—and how a carefully designed auxiliary operator can fill the gap when the physical spectrum has no gap of its own.

Created on 10 Nov. 2026

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99.9...% of the SU(2) BFSS ground state

The essentials

Question

Method

Findings

Limitations

Why it matters

Read further

Definitions

How Much of the SU(2) BFSS Ground State Can We Capture?

Turning a huge problem into a manageable one

Measuring closeness without a physical energy gap

Checking the state in the long-distance region

What comes next

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