Nonperturbative completion of the double-scaled one-matrix model for two-dimensional quantum gravity: selecting and constructing the physical solution of the string equation

Two-dimensional quantum gravity, formulated as a sum over random surfaces and over their genera, becomes exactly solvable in the double-scaling limit of a tuned hermitian one-matrix model: the specific heat $f(t)$, with $t$ the renormalized cosmological constant, obeys the string equation, which for pure gravity is, up to normalization, the first Painlevé equation (in minimal-string normalizations the string coupling $g_s$ multiplies the second-derivative term explicitly). The perturbative genus expansion of $f$ diverges with factorial coefficients and is not Borel-summable, and the string equation admits a family of solutions parametrized by nonperturbative data, so perturbation theory does not by itself define the theory. This problem, raised by the founding 1990 matrix-model papers and sharpened by three decades of subsequent work, asks for the nonperturbative completion of the theory: (1) a physical principle, derived from the underlying matrix integral or from the continuum loop equations, that selects the correct solution, equivalently fixes its transseries and Stokes data; (2) the global solution valid for all real $t$, resolving the double-pole structure of real solutions, the reality and positivity of the partition function as a sum over surfaces, and the physical meaning of the nonperturbative sectors; (3) consistent nonperturbative boundary conditions and spectrum for the associated Schrödinger operator $\hat H = -g_s^2 d^2/dt^2 + f(t)$, whose naive discrete spectrum conflicts with the continuum loop equations; (4) determination of which multicritical points admit a completion consistent with unitary matter coupled to gravity.

Background

Two-dimensional (2D) Euclidean quantum gravity is the path integral over equivalence classes of two-dimensional metrics. In two dimensions the Einstein term is topological, so the dynamical content reduces to the Liouville mode (the conformal factor of the metric, which becomes a dynamical field) coupled to conformal matter (two-dimensional matter field theories invariant under local rescalings of the metric, classified by a parameter $c$, the central charge), and the partition function is organized as a topological expansion: a formal sum over the genus $g$ of the underlying surface, weighted at each genus by powers of a coupling (the string coupling $g_s$) and by the exponential of the cosmological constant, the parameter controlling the area of the surface.

A remarkable representation of this theory uses random matrices. The integral over hermitian $N\times N$ matrices, $Z=\int dM\,\exp(-N\,\mathrm{Tr}\,V(M))$ with an even polynomial potential $V$, generates Feynman diagrams that are dual to triangulated surfaces, and the expansion in powers of $1/N$ is exactly an expansion in the genus of these discretized surfaces. Tuning $V$ to a critical point makes the lattice spacing negligible, and taking $N\to\infty$ simultaneously with the coupling approaching criticality — the double-scaling limit — yields continuum 2D quantum gravity at finite $g_s$. This was established exactly in 1990 by three independent groups: Gross and Migdal, Brézin and Kazakov, and Douglas and Shenker.

In the continuum limit the object that encodes the theory is the specific heat $f(t)$, the second derivative of the free energy with respect to the renormalized cosmological constant $t$. It obeys a nonlinear ordinary differential equation, the string equation, a member of the KdV (Korteweg–de Vries) hierarchy of integrable equations: for pure gravity ($k=2$) it is, up to normalization, exactly the first Painlevé equation — in the convention of the standard reviews, $u'' = 3u^2 - 3t$ (equivalently $t = u^2 - u''/3$), in the mathematics normalization $f'' = 6 f^2 + t$ — with the string coupling entering through the scaling $g_s \sim t^{-5/4}$ along the physical solution (in minimal-string normalizations $g_s$ multiplies the second-derivative term explicitly), while at the higher multicritical points $k \geq 3$ the string equation becomes a higher-order differential equation deforming the algebraic spherical equation. Its formal expansion in powers of $g_s^2 \sim t^{-5/2}$ at large $|t|$ reproduces the genus expansion of the sum over surfaces term by term. Tuning successive multicritical points of the potential yields the $(2,2k-1)$ minimal models coupled to gravity — the simplest family of two-dimensional conformal field theories, each with only finitely many fields, labeled by a pair of coprime integers: $k=2$ is pure gravity, while $k\geq 3$ corresponds to non-unitary matter ($k=3$ describes the Yang-Lee edge singularity, the critical point of an Ising model in an imaginary magnetic field); unitarity here means positive-norm (reflection-positive, probabilistic) matter content.

The nonperturbative problem appears at the next level. The genus series diverges with coefficients growing like $(2g)!$ and is not Borel summable — that is, its Borel transform (the series obtained by dividing each coefficient by a suitable factorial, whose convergence would allow the original series to be inverted by a real Laplace-type integral) has singularities on the integration path, so the series does not unambiguously reconstruct a function $f(t)$. Real solutions of the string equation develop infinitely many double poles on the real axis, on the side opposite the large-$|t|$ asymptotic region (in the reviews' convention above, on the negative real $t$ axis); at each pole the partition function, a sum of positive terms order by order in the genus expansion, has a zero. The continuum limit of the orthogonal-polynomial method (the standard technique for evaluating one-matrix integrals by recursion on polynomials orthogonal with respect to the measure $\mathrm{e}^{-N\mathrm{Tr}V}$) represents the matrix eigenvalue as a Schrödinger operator $\hat H = -g_s^2 d^2/dt^2 + f(t)$, whose spectral properties should encode the operators of the theory, but the discrete spectrum that follows from naive boundary conditions conflicts with the loop equations (the continuum Schwinger-Dyson equations, that is, the identities expressing the invariance of the matrix integral under infinitesimal changes of variables), which require a continuous spectrum. These obstructions were identified in the founding papers and catalogued in the authoritative reviews of the 1990s.

The modern language for the same problem is that of instantons (nonperturbative saddles with action $\sim 1/g_s$), Borel plane singularities, Stokes data (the constants governing how the asymptotic expansion jumps across sectors in the complex $t$ or coupling plane), and transseries (generalized expansions summing perturbative and instanton sectors with free parameters). In string-theoretic language the leading instantons of the $k=2$ model are identified with ZZ-branes (the pointlike D-branes, that is, extended objects on which open strings can end, appearing as pointlike states of Liouville theory and named after Zamolodchikov and Zamolodchikov). The present problem is the nonperturbative definition of the theory in these terms.

Problem Statement

Fix attention on the double-scaling limit of the $k$-th multicritical hermitian one-matrix model ($k=2$: pure 2D gravity), with renormalized cosmological constant $t$ and string coupling $g_s$. The specific heat $f(t)$ satisfies the string equation: for $k=2$ this is, up to normalization, the first Painlevé equation itself — in the convention of the standard reviews, $u'' = 3 u^2 - 3 t$ with the physical branch $u \sim t^{1/2}$ as $t \to +\infty$ (in the mathematics normalization, $f'' = 6 f^2 + t$ with $f \sim (-t)^{1/2}$ as $t \to -\infty$; the two conventions are related by $u(t) = 2 c^2 f(-c t)$ with $c = (3/2)^{1/5}$) — while for $k \geq 3$ it is a higher-order deformation of the algebraic spherical equation. Along the physical branch the formal expansion in powers of $g_s^2 \sim t^{-5/2}$ reproduces the genus expansion of the sum over surfaces, with $(2g)!$ non-Borel-summable coefficients.

The problem is the nonperturbative completion of this theory, with the following components, in which the solution selected by the principle in (1) is the solution whose global analytic structure is constructed in (2) and whose spectral data is analyzed in (3); component (4) also asks for the range of $k$ over which such a completion exists:

(1) Selection principle. Establish the physical principle — derived from the underlying regularized matrix integral (its instanton calculus, Stokes data inherited from finite $N$, or a stabilized/analytically continued ensemble) and/or from the continuum loop (Schwinger-Dyson) equations — that uniquely fixes the solution of the string equation, equivalently the physical meaning of the generalized transseries sectors and the resummation of the two-parameter transseries. The canonical and non-canonical Stokes data of the $k=2$ case are already computed (ref. 21).

(2) Global solution. Construct the selected solution as an actual function on the whole real $t$ axis (a resummation of the two-parameter transseries or an equivalent closed construction), resolving: the infinitely many double poles of real solutions on the real axis, which lie on the side opposite the large-$|t|$ asymptotic region (the negative real axis in the convention above) and produce zeros of the partition function; whether the physical solution is complex and in what sense the partition function is then real and positive as a sum over surfaces; and the monodromy/Painlevé property (the defining property of Painlevé equations, that the only movable singularities of solutions are poles) across the sectors traversed.

(3) Spectral and boundary-condition consistency. Determine the boundary conditions and spectrum of the Schrödinger operator $\hat H = -g_s^2 d^2/dt^2 + f(t)$ (with the normalization $g_s = 1$) that follow from the selected solution, and show consistency with the loop equations (which require a continuous spectrum) or else prove that no such consistency is possible. In the consistent case, extract at least one class of nonperturbative correlators (pointlike or macroscopic loop operators) from this data.

(4) Multicritical and unitary range. Determine which multicritical points ($k\geq 2$) admit such a completion, and, for those that do, whether the completion is unitary in the nonperturbative sense — that is, whether the full matter-plus-gravity theory is reflection-positive (defines a positive-definite measure on the loop operators) — as distinct from the perturbative matter-CFT unitarity. The perturbative identification of the multicritical points with the non-unitary $(2,2k-1)$ minimal models is already settled and does not by itself answer this, since for $k\geq 3$ the one-matrix multicritical points carry non-unitary matter and $k=2$ carries only the trivial $c=0$ matter.

Equivalently, the problem asks for such a completion or for a proof that none exists: a construction supplying (1)-(3) for the $k=2$ pure-gravity case with proofs of the claimed properties, together with, for each $k\geq 2$, the determination of whether such a completion exists and of its reflection-positivity status; or a rigorous impossibility result showing that no solution of the string equation — and no analytic continuation of the underlying matrix integral — satisfies all these conditions simultaneously. The conditions on a completion are the coherence condition above together with the following: the solution is selected by a principle derived from the regularized matrix integral or from the loop equations; the selected resummation is an actual function on the whole real $t$ axis whose partition function is real and positive in a stated sense (for example, that the resummation defines a real, positive-definite measure on the algebra of loop operators, or that its Borel resummation is real for real $t$ under a stated prescription); and the spectrum and boundary conditions of $\hat H$ are consistent with the loop equations.

References

1. David J. Gross and Alexander A. Migdal, "A nonperturbative treatment of two-dimensional quantum gravity", Nuclear Physics B 340 (1990) 333-365. DOI: 10.1016/0550-3213(90)90450-R. https://doi.org/10.1016/0550-3213(90)90450-R
2. Edward Brézin and Vladimir A. Kazakov, "Exactly solvable field theories of closed strings", Physics Letters B 236 (1990) 144-150. DOI: 10.1016/0370-2693(90)90818-Q. https://doi.org/10.1016/0370-2693(90)90818-Q
3. Michael R. Douglas and Stephen H. Shenker, "Strings in less than one dimension", Nuclear Physics B 335 (1990) 635-654. DOI: 10.1016/0550-3213(90)90522-F. https://doi.org/10.1016/0550-3213(90)90522-F
4. David J. Gross and Alexander A. Migdal, "Nonperturbative two-dimensional quantum gravity", Physical Review Letters 64 (1990) 127-130. DOI: 10.1103/PhysRevLett.64.127. https://doi.org/10.1103/PhysRevLett.64.127
5. François David, "Loop equations and non-perturbative effects in two-dimensional quantum gravity", Modern Physics Letters A 5 (1990) 1019-1030. DOI: 10.1142/S0217732390001141. https://doi.org/10.1142/S0217732390001141
6. François David, "Phases of the large-N matrix model and non-perturbative effects in 2D gravity", Nuclear Physics B 348 (1991) 507-524. DOI: 10.1016/0550-3213(91)90202-9. https://doi.org/10.1016/0550-3213(91)90202-9
7. Paul Ginsparg and Jean Zinn-Justin, "Large order behavior of non-perturbative gravity", Physics Letters B 255 (1991) 189-196. DOI: 10.1016/0370-2693(91)90234-H. https://doi.org/10.1016/0370-2693(91)90234-H
8. Bertrand Eynard and Jean Zinn-Justin, "Large order behaviour of 2D gravity coupled to d<1 matter", Physics Letters B 302 (1993) 396-402. DOI: 10.1016/0370-2693(93)90416-F. arXiv:hep-th/9301004. https://arxiv.org/abs/hep-th/9301004
9. Paul Ginsparg and Gregory Moore, "Lectures on 2D gravity and 2D string theory", lecture notes of the Theoretical Advanced Study Institute (TASI 1992), Boulder, Colorado; published in the TASI 1992 proceedings (World Scientific, Singapore, 1993), pp. 277-469. arXiv:hep-th/9304011. https://arxiv.org/abs/hep-th/9304011
10. P. Di Francesco, P. Ginsparg and J. Zinn-Justin, "2D Gravity and Random Matrices", Physics Reports 254 (1995) 1-133. DOI: 10.1016/0370-1573(94)00084-G. arXiv:hep-th/9306153. https://arxiv.org/abs/hep-th/9306153
11. René Lafrance and Robert C. Myers, "What unitary matrix models are not?", Physical Review D 47 (1993) 4785-4788. DOI: 10.1103/PhysRevD.47.4785. arXiv:hep-th/9206064. https://arxiv.org/abs/hep-th/9206064
12. Marek Karliner, Alexander Migdal and Boris Rusakov, "Ground state of 2D quantum gravity and spectral density of random matrices", Nuclear Physics B 399 (1993) 514-526. DOI: 10.1016/0550-3213(93)90507-L. arXiv:hep-th/9212114. https://arxiv.org/abs/hep-th/9212114
13. Simon Dalley, Clifford V. Johnson and Tim R. Morris, "Non-perturbative two-dimensional quantum gravity, again", Nuclear Physics B — Proceedings Supplements 25 (1992) 87-91. DOI: 10.1016/S0920-5632(05)80009-X. arXiv:hep-th/9108016. https://arxiv.org/abs/hep-th/9108016
14. Simon Dalley, Clifford V. Johnson and Tim R. Morris, "Non-perturbative two-dimensional quantum gravity", Nuclear Physics B 368 (1992) 655-670. DOI: 10.1016/0550-3213(92)90218-Z. https://doi.org/10.1016/0550-3213(92)90218-Z
15. Joshua Feinberg, "Stabilized matrix models for nonperturbative two-dimensional quantum gravity", International Journal of Modern Physics A 9 (1994) 3751-3772. DOI: 10.1142/S0217751X94001503. arXiv:hep-th/9212069. https://arxiv.org/abs/hep-th/9212069
16. J. Luis Miramontes and Joaquín Sánchez Guillen, "Instantons in the quantum framework of 2D gravity", Nuclear Physics B — Proceedings Supplements 25 (1992) 195-202. DOI: 10.1016/S0920-5632(05)80019-2. arXiv:hep-th/9110051. https://arxiv.org/abs/hep-th/9110051
17. Masafumi Fukuma and Shigeaki Yahikozawa, "Comments on D-instantons in c<1 strings", Physics Letters B 460 (1999) 71-78. DOI: 10.1016/S0370-2693(99)00744-3. arXiv:hep-th/9902169. https://arxiv.org/abs/hep-th/9902169
18. Masanori Hanada, Masashi Hayakawa, Nobuyuki Ishibashi, Hikaru Kawai, Tsunehide Kuroki, Yoshinori Matsuo and Tsukasa Tada, "Loops versus matrices: the nonperturbative aspects of noncritical string", Progress of Theoretical Physics 112 (2004) 131-181. DOI: 10.1143/PTP.112.131. arXiv:hep-th/0405076. https://arxiv.org/abs/hep-th/0405076
19. Robert de Mello Koch, Antal Jevicki and João P. Rodrigues, "Instantons in c=0 CSFT", Journal of High Energy Physics 0504 (2005) 011. DOI: 10.1088/1126-6708/2005/04/011. arXiv:hep-th/0412319. https://arxiv.org/abs/hep-th/0412319
20. Inês Aniceto, Ricardo Schiappa and Marcel Vonk, "The Resurgence of Instantons in String Theory", Communications in Number Theory and Physics 6 (2012) 339-496. DOI: 10.4310/CNTP.2012.v6.n2.a3. arXiv:1106.5922. https://arxiv.org/abs/1106.5922
21. Salvatore Baldino, Ricardo Schiappa, Maximilian Schwick and Roberto Vega, "Resurgent Stokes data for Painlevé equations and two-dimensional quantum (super)gravity", Communications in Number Theory and Physics 17 (2) (2023) 385-552. DOI: 10.4310/CNTP.2023.v17.n2.a5. arXiv:2203.13726. https://arxiv.org/abs/2203.13726
22. Carl M. Bender and Stefan Boettcher, "Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry", Physical Review Letters 80 (1998) 5243-5246. DOI: 10.1103/PhysRevLett.80.5243. arXiv:physics/9712001. https://arxiv.org/abs/physics/9712001
23. Carl M. Bender, Moshe Moshe and Sarben Sarkar, "PT-symmetric interpretation of double-scaling", Journal of Physics A: Mathematical and Theoretical 46 (2013) 102002. DOI: 10.1088/1751-8113/46/10/102002. arXiv:1206.4943. https://arxiv.org/abs/1206.4943
24. Carl M. Bender and Javad Komijani, "Painlevé Transcendents and PT-Symmetric Hamiltonians", arXiv:1502.04089 (2015). https://arxiv.org/abs/1502.04089