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<p>We apply the quantum-mechanics bootstrap to supersymmetric quantum mechanics (SUSY QM) and to its matrix relative, the Marinari-Parisi model, which is conjectured to describe the worldvolume of unstable <i>D</i>0 branes. Using positivity of moment matrices together with Heisenberg, gauge, and (zero-temperature) thermal constraints, we obtain rigorous bounds on ground-state data. In the cases where SUSY is spontaneously broken, we find bounds that apply to the lowest-energy normalizable eigenstate.</p><p>For <i>N</i> = 1 SUSY QM with a cubic superpotential, we obtain tight bounds that agree well with available approximation methods. At weak coupling they match well with the semiclassical instanton contribution to SUSY-breaking ground-state energy, while at strong coupling they exhibit the expected scaling and match well with Hamiltonian truncation.</p><p>For the SUSY matrix QM, we construct a 44 × 44 bootstrap matrix and obtain bounds at large <i>N</i>. At strong coupling, we obtain the expected <i>E</i> ~ <i>κg</i><sup>2/3</sup> scaling of <i>E</i> with <i>g</i> and extract a lower bound on the coefficient <i>κ</i> &gt; .196. At small coupling, the theory has a critical point <i>g</i><sub><i>c</i></sub> where the two wells merge into one. We find a spurious kink at <i>g</i> = <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>g</mi> <mi>c</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">\( \sqrt{2}{g}_c \)</EquationSource> </InlineEquation>. We attribute this to truncation error and solver limitations, and discuss possible improvements.</p>

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Bootstrapping supersymmetric (matrix) quantum mechanics

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  • 发表日期：

    2026年4月22日

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  • Samuel Laliberte,
  • Brian McPeak

摘要

We apply the quantum-mechanics bootstrap to supersymmetric quantum mechanics (SUSY QM) and to its matrix relative, the Marinari-Parisi model, which is conjectured to describe the worldvolume of unstable D0 branes. Using positivity of moment matrices together with Heisenberg, gauge, and (zero-temperature) thermal constraints, we obtain rigorous bounds on ground-state data. In the cases where SUSY is spontaneously broken, we find bounds that apply to the lowest-energy normalizable eigenstate.

For N = 1 SUSY QM with a cubic superpotential, we obtain tight bounds that agree well with available approximation methods. At weak coupling they match well with the semiclassical instanton contribution to SUSY-breaking ground-state energy, while at strong coupling they exhibit the expected scaling and match well with Hamiltonian truncation.

For the SUSY matrix QM, we construct a 44 × 44 bootstrap matrix and obtain bounds at large N. At strong coupling, we obtain the expected E ~ κg2/3 scaling of E with g and extract a lower bound on the coefficient κ > .196. At small coupling, the theory has a critical point gc where the two wells merge into one. We find a spurious kink at g = 2 g c \( \sqrt{2}{g}_c \) . We attribute this to truncation error and solver limitations, and discuss possible improvements.