Nonperturbative completion and continuum interpretation of the Marinari-Parisi supersymmetric one-dimensional string

The supersymmetric one-dimensional string of Marinari and Parisi (Phys. Lett. B 240 (1990) 375) is a matrix quantum mechanics of an $N\times N$ Hermitian matrix superfield in one-dimensional superspace whose Feynman expansion discretizes random surfaces embedded in superspace; its equal-time observables coincide exactly with the ill-defined $d=0$ bosonic matrix integral, making it a candidate nonperturbative definition of pure two-dimensional quantum gravity and of a one-dimensional superstring. The source paper solved only the bosonic (zero-fermion-number) sector by WKB free fermions and found supersymmetry spontaneously broken by exponentially small effects, leaving five linked issues open: the spectrum in the fermionic sector under full supersymmetric constraints (including corrections to the Fermi energy neglected in WKB), the explicit exponentially small terms modifying the Painleve-analogue string equations and the principles fixing their acceptability, whether the supersymmetric $d=1$ theory nonperturbatively defines the $d=0$ bosonic theory, the vacuum-energy and genus-scaling status of the breaking, and the extension to target dimensions $d>1$. The surviving open core is threefold: (a) the exact nonperturbative supersymmetry-breaking data of the model at large $N$ (beyond the semiclassical one-instanton result; the super-eigenvalue dynamics, a supersymmetric Calogero-Moser system with cubic superpotential, has no known exact treatment); (b) the identification of the continuum string theory the model defines nonperturbatively (bosonic $c=0$ completion, exotic disordered theory, or two-dimensional type IIB/IIA superstring per the 2003-2004 D-particle and AdS$_2$ conjectures); (c) whether the Marinari-Parisi construction, or its direct supersymmetric matrix-model generalizations, extend to target dimensions $d>1$. Bootstrap work in 2026 still obtains only bounds on the ground-state data.

Background

Noncritical string theory - string theory formulated away from the critical spacetime dimension at which the worldsheet conformal anomaly cancels, so that the worldsheet gravity mode remains dynamical - and two-dimensional quantum gravity are rare regimes of quantum gravity in which the sum over worldsheets can be controlled beyond perturbation theory. The central tool is the matrix model: the Feynman-graph expansion of an $N\times N$ Hermitian matrix integral, or of a matrix quantum mechanics, generates discretized random surfaces, with powers of $1/N^{2}$ weighting the genus (the number of handles) of the surface. Sending $N\to\infty$ while tuning the matrix potential to a critical value at which arbitrarily large graphs dominate - the double-scaling limit - defines a continuum string theory whose string coupling $\varkappa$ is fixed by a combination of $N$ and the distance from criticality. Two benchmark cases are the $d=0$ model, a single matrix integral describing pure two-dimensional gravity whose double-scaled specific heat obeys a nonlinear Painleve-type ordinary differential equation (the string equation), and the $d=1$ model, a matrix quantum mechanics describing a string in one target dimension that is exactly solvable by mapping to noninteracting fermions. For target dimension $d>1$ matrix models cease to define sensible continuum strings (the $c=1$ barrier: the central charge $c$ counts the worldsheet matter degrees of freedom and equals the target dimension here, and above $c=1$ the dominant surfaces degenerate into branched polymers, i.e. tree-like random graphs).

A long-standing structural problem of the $d=0$ model is that its potential is unbounded below (bottomless): the defining integral diverges, so the theory exists only order by order in the genus expansion. A nonperturbative definition must stabilize the integral, and any stabilization modifies the theory by effects exponentially small in the string coupling - invisible at every finite genus order, but detectable as extra terms in the string equation (equivalently, in the Dyson-Schwinger equations, the exact identities expressing a field theory's correlation functions as functional integrals over its own action, which for these models reduce to the Painleve-type string equation). Which nonperturbative completion, if any, is physically correct became a classic issue of the field.

In this setting Marinari and Parisi (1990) proposed a supersymmetric discretized string: a sum over triangulations of a surface embedded in a one-dimensional superspace $(\tau,\theta,\bar\theta)$, realized as the quantum mechanics of a matrix superfield $\Phi=M+\theta\Psi+\bar\Psi\theta+\theta\bar\theta F$ (packaging a bosonic matrix $M$, a fermionic matrix $\Psi$, and an auxiliary field $F$ into one object with $\mathcal{N}=1$ supersymmetry along the time $\tau$), with cubic superpotential $W_0(\Phi)=\frac{1}{2}\Phi^{2}-\frac{\lambda}{3}\Phi^{3}$ (the superpotential is the single function of the superfields whose gradient generates all interaction terms, its square fixing the bosonic potential); the bosonic potential $V(M)=\frac{1}{2}M^{2}(1-\lambda M)^{2}$ is a symmetric double well with an unstable maximum in between. The model's equal-time (zero-length worldsheet) correlators coincide exactly with the ill-defined $d=0$ bosonic matrix integral, so the supersymmetric $d=1$ theory is a candidate nonperturbative definition of the $d=0$ theory. Equivalently, as the same authors showed in a companion paper, the construction is stochastic quantization: the extra dimension plays the role of the fictitious Langevin (fifth) time of Parisi-Wu stochastic quantization - a relaxation dynamics in an auxiliary time whose residual supersymmetry pairs the noise fields with the bosonic ones - and the exponentially small terms distinguishing different stochastic completions appear as corrections to the Dyson-Schwinger equations.

Marinari and Parisi solved the zero-fermion-number (singlet, bosonic) sector by diagonalizing $M$, which turns the Schrodinger problem into $N$ noninteracting fermions in the inverted potential, treated by WKB (the semiclassical Wentzel-Kramers-Brillouin approximation). They located the critical point $\lambda_c^2=1/(6\sqrt{3})$ with double scaling $\varkappa^{-1}=N(\lambda_c-\lambda)^{5/2}$, and concluded that supersymmetry is spontaneously broken - the ground state is not invariant under the supercharges, so the vacuum energy is positive - but only through effects exponentially small in the string coupling, invisible at every order in $1/N^{2}$. The paper and its companion explicitly left open the issues that define the present problem: the fermionic sector of the spectrum had not been computed under full supersymmetric constraints (the WKB treatment neglected corrections to the Fermi energy); the explicit form of the exponentially small terms in the Painleve-analogue equations, and the principles selecting an acceptable completion, were unknown; it was open whether the construction really defines the $d=0$ bosonic theory nonperturbatively, including its nonperturbative sectors; the vanishing of the planar vacuum energy and the genus-invisibility of the breaking were expectations rather than theorems, with the scaling of the breaking with genus undetermined; and the extension to $d>1$ was suggested but unworked.

Much of this has since been settled - most decisively by Dabholkar (1992), who constructed the manifestly supersymmetric eigenvalue formulation, computed the one-eigenvalue instanton (a tunneling event between the wells of the double-well potential, described in imaginary time) with its single fermion zero mode (a fermionic collective coordinate of the tunneling trajectory that costs no action and whose saturation fixes at which order in the fermions the breaking appears), and showed that the breaking parameter is nonzero, of order $\varkappa e^{-C/\varkappa}$ with $\varkappa^{2}$ the loop-counting parameter - the scaling that Shenker argued is generic for nonperturbative effects in string theory and much stronger than the $e^{-C/\varkappa^{2}}$ of ordinary field theory. What remains open is the subject of the problem statement.

Problem Statement

The problem is the nonperturbative completion and interpretation of the Marinari-Parisi supersymmetric one-dimensional string: to determine, with the full supersymmetric structure enforced, what the model defines nonperturbatively and what its vacuum structure is. The source paper left the following linked issues open (items (i) and (iv) have since been settled in substance), leaving the surviving open core (a)-(c):

Original issues from the source: (i) compute the spectrum of the model in the fermionic sector under full supersymmetric constraints, controlling the corrections to the Fermi energy neglected in the WKB analysis, and decide whether they change the scaling behavior at the critical point; (ii) determine the explicit form of the exponentially small terms that modify the Painleve-analogue string equations of the dimensionally reduced theory, and the general principles governing which nonperturbative completion is physically acceptable; (iii) decide whether the one-dimensional supersymmetric theory, whose equal-time observables coincide with the ill-defined zero-dimensional bosonic matrix integral, provides a nonperturbative definition of that bosonic theory beyond perturbation theory, including its nonperturbative sectors; (iv) establish the vacuum structure: whether the planar ($N\to\infty$ at fixed genus) vacuum energy vanishes identically, whether supersymmetry breaking is invisible at every order of the $1/N^{2}$ genus expansion, and how the nonperturbative, exponentially small breaking effects scale with the string coupling and genus; (v) extend the detailed one-dimensional analysis and its nonperturbative conclusions to target-space dimensions $d>1$, checking whether the same mechanisms and scalings persist.

Surviving open core (the target as of 2026), with the source issues still open subsumed by these branches - issue (ii) by (a) together with (b), issue (iii) by (b), and issue (v) by (c): (a) Exact nonperturbative vacuum data: determine, beyond the leading semiclassical one-instanton result $E_0\sim\varkappa e^{-C/\varkappa}$, the exact ground-state energy and supersymmetry-breaking parameter of the gauged model (gauged meaning that the model's $U(N)$ symmetry is treated as a local symmetry, projecting the theory onto its gauge-invariant singlet sector, as opposed to the ungauged model of distinguishable eigenvalues) in the double-scaling limit at large $N$ - equivalently, an exact treatment of its super-eigenvalue dynamics, the supersymmetric Calogero-Moser system (the standard quantum mechanics of $N$ particles on a line with pairwise inverse-square interactions, here extended by the fermionic partners of the eigenvalues) with cubic superpotential - including the exact instanton action at large $N$ and the resolution of the gauged-versus-ungauged and order-of-limits subtleties (supersymmetry is restored as $N\to\infty$ at fixed coupling, yet the breaking survives double scaling). (b) Continuum identification: identify which continuum string theory the model's double-scaling limit defines nonperturbatively - the bosonic $c=0$ string with an alternative nonperturbative completion, the exotic disordered theory of Chaudhuri-Polchinski, or the two-dimensional type IIB/IIA superstring of the D-particle and AdS$_2$ conjectures (D-particles are heavy, nonperturbative particle-like branes of string theory on which open strings can end; AdS$_2$ is two-dimensional anti-de Sitter space, the maximally symmetric spacetime of constant negative curvature) - as witnessed by the spacetime-field dictionary, the action of the matrix-model supersymmetry on target space, and nontrivial quantitative checks. (c) Higher dimensions: extend the detailed one-dimensional Marinari-Parisi analysis and its nonperturbative conclusions to target-space dimensions $d>1$ - determine whether the same critical behavior, nonperturbative completion, and supersymmetry-breaking mechanisms persist for the Marinari-Parisi construction or its direct supersymmetric matrix-model generalizations to higher target dimensions (in the sense of the source's proposal and the Hashimoto-Klebanov line), given the branched-polymer degeneration of supersymmetric matrix ensembles near criticality.

The branches (a)-(c) are facets of one and the same source program - the nonperturbative completion and continuum interpretation of the Marinari-Parisi model - not independent questions. A full resolution requires resolving all three branches (a)-(c).

References

1. E. Marinari and G. Parisi, 'The supersymmetric one-dimensional string', Phys. Lett. B 240 (1990) 375-380, DOI 10.1016/0370-2693(90)91115-R, https://doi.org/10.1016/0370-2693(90)91115-r
2. E. Marinari and G. Parisi, 'A non perturbative definition of 2D quantum gravity', Phys. Lett. B 247 (1990) 537, DOI 10.1016/0370-2693(90)91897-K, https://doi.org/10.1016/0370-2693(90)91897-K
3. M. Karliner and A. A. Migdal, 'Nonperturbative 2D quantum gravity via supersymmetric string', Mod. Phys. Lett. A 5 (1990) 2565, DOI 10.1142/S0217732390002985, https://doi.org/10.1142/S0217732390002985
4. S. Bellucci, T. R. Govindrajan, A. Kumar and R. N. Oerter, 'On the nonperturbative solution of the d = 1 superstring', Phys. Lett. B 249 (1990) 49, DOI 10.1016/0370-2693(90)90525-B, https://doi.org/10.1016/0370-2693(90)90525-B
5. D. Kutasov and N. Seiberg, 'Non-critical superstrings', Phys. Lett. B 251 (1990) 67, DOI 10.1016/0370-2693(90)90233-V, https://doi.org/10.1016/0370-2693(90)90233-V
6. S. H. Shenker, 'The Strength of nonperturbative effects in string theory', lectures at the Cargese Study Institute 'Random Surfaces, Quantum Gravity and Strings', Cargese, France, 27 May - 2 June 1990, pp. 191-200; reprinted in 'The Large N Expansion in Quantum Field Theory and Statistical Physics: From Spin Systems to Two-Dimensional Gravity' (World Scientific, Singapore, 1993), pp. 809-819, https://inspirehep.net/literature/298523
7. S. Chaudhuri and J. D. Lykken, 'Analyzing the solutions of Hermitian matrix models', Nucl. Phys. B 367 (1991) 614, DOI 10.1016/0550-3213(91)90010-U, https://doi.org/10.1016/0550-3213(91)90010-U
8. A. Jevicki and J. P. Rodrigues, 'Supersymmetric collective field theory', Phys. Lett. B 268 (1991) 53, DOI 10.1016/0370-2693(91)90921-C, https://doi.org/10.1016/0370-2693(91)90921-C
9. J. González, 'A new vacuum for the supersymmetric one-dimensional discretized string', Phys. Lett. B 255 (1991) 367, DOI 10.1016/0370-2693(91)90780-T, https://doi.org/10.1016/0370-2693(91)90780-T
10. A. Dabholkar, 'Fermions and nonperturbative supersymmetry breaking in the one-dimensional superstring', Nucl. Phys. B 368 (1992) 293, DOI 10.1016/0550-3213(92)90529-K, https://doi.org/10.1016/0550-3213(92)90529-K
11. J. D. Cohn and H. Dykstra, 'The Marinari-Parisi model and collective field theory', Mod. Phys. Lett. A 7 (1992) 1163, arXiv:hep-th/9202050, DOI 10.1142/S0217732392003621, https://arxiv.org/abs/hep-th/9202050
12. J. Feinberg, 'String field theory for d <= 0 matrix models via Marinari-Parisi', Phys. Lett. B 281 (1992) 225, arXiv:hep-th/9201037, DOI 10.1016/0370-2693(92)91133-T, https://arxiv.org/abs/hep-th/9201037
13. J. Greensite, 'Vanishing of the cosmological constant in stabilized quantum gravity', Phys. Lett. B 291 (1992) 405, arXiv:hep-th/9207028, DOI 10.1016/0370-2693(92)91395-P, https://arxiv.org/abs/hep-th/9207028
14. J. P. Rodrigues and A. J. van Tonder, 'Marinari-Parisi and supersymmetric collective field theory', Int. J. Mod. Phys. A 8 (1993) 2517, arXiv:hep-th/9204061, DOI 10.1142/S0217751X93001004, https://arxiv.org/abs/hep-th/9204061
15. S. Chaudhuri and J. Polchinski, 'Critical behavior of the Marinari-Parisi model', Phys. Lett. B 339 (1994) 309, arXiv:hep-th/9408005, DOI 10.1016/0370-2693(94)90624-6, https://arxiv.org/abs/hep-th/9408005
16. G. Ferretti, 'The untruncated Marinari-Parisi superstring', J. Math. Phys. 35 (1994) 4469, arXiv:hep-th/9310002, DOI 10.1063/1.530866, https://arxiv.org/abs/hep-th/9310002
17. A. Hashimoto and I. R. Klebanov, 'Matrix model approach to d > 2 noncritical superstrings', Mod. Phys. Lett. A 10 (1995) 2639, arXiv:hep-th/9507062, DOI 10.1142/S0217732395002775, https://arxiv.org/abs/hep-th/9507062
18. J. Ambjørn, Y. Makeenko and K. Zarembo, 'Supersymmetric matrix models and branched polymers', Nucl. Phys. B 482 (1996) 660, arXiv:cond-mat/9606041, DOI 10.1016/S0550-3213(96)00526-3, https://arxiv.org/abs/cond-mat/9606041
19. G. Veneziano and J. Wosiek, 'Planar quantum mechanics: an intriguing supersymmetric example', JHEP 01 (2006) 156, arXiv:hep-th/0512301, DOI 10.1088/1126-6708/2006/01/156, https://arxiv.org/abs/hep-th/0512301
20. J. McGreevy, S. Murthy and H. L. Verlinde, 'Two-dimensional superstrings and the supersymmetric matrix model', JHEP 04 (2004) 015, arXiv:hep-th/0308105, DOI 10.1088/1126-6708/2004/04/015, https://arxiv.org/abs/hep-th/0308105
21. H. L. Verlinde, 'Superstrings on AdS(2) and superconformal matrix quantum mechanics', arXiv:hep-th/0403024, https://arxiv.org/abs/hep-th/0403024
22. T. Kuroki and F. Sugino, 'Spontaneous supersymmetry breaking in matrix models from the viewpoints of localization and Nicolai mapping', Nucl. Phys. B 844 (2011) 409, arXiv:1009.6097, DOI 10.1016/j.nuclphysb.2010.11.015, https://arxiv.org/abs/1009.6097
23. D. Anninos, F. Denef and R. Monten, 'Grassmann matrix quantum mechanics', JHEP 04 (2016) 138, arXiv:1512.03803, DOI 10.1007/JHEP04(2016)138, https://arxiv.org/abs/1512.03803
24. S. Laliberte and B. McPeak, 'Bootstrapping supersymmetric (matrix) quantum mechanics', JHEP 04 (2026) 167, arXiv:2510.01356, DOI 10.1007/JHEP04(2026)167, https://arxiv.org/abs/2510.01356