The real symmetric analog of the Kontsevich integral — a matrix model over real symmetric matrices coupled to an external field, whose Feynman expansion generates a sum over unoriented (non-orientable) surfaces — has resisted the treatment that its hermitian and unitary counterparts received. For the hermitian Kontsevich model, Gross and Newman (1992) rederived the Virasoro constraints of two-dimensional topological gravity directly from the eigenvalue Schwinger-Dyson equations; for the unitary model they found new constraints LnYˉ=−∂nYˉL_n\bar Y=-\partial_n\bar Y (the Brézin-Gross-Witten model, later identified with intersection theory of the Θ\Theta class on M‾g,n\overline{\mathcal{M}}_{g,n} and with super Weil-Petersson volumes). For the symmetric model, however, the pairwise term in the Schwinger-Dyson equations acquires a factor 12\frac12, the genus expansion acquires an order-1/N1/N contribution, the linearized 1/N1/N equation could not be solved and was conjectured to admit no closed form, and even the complete set of eigenvalue invariants (the analog of Kontsevich's times tkt_k) on which the partition function depends is unknown. The symmetric-matrix core — the unoriented analog of the Kontsevich/topological-gravity correspondence — remains open: identify the invariants and times, derive Virasoro-type constraints (or prove none exist), solve the 1/N1/N expansion in closed form, and find the moduli-space interpretation for unoriented surfaces.

Background

Large-NN matrix models are integrals over N×NN\times N matrices whose free energy admits a topological, or genus, expansion in powers of 1/N1/N: the perturbative Feynman diagrams are 'fat' ribbon graphs that thicken to a surface, and each power of 1/N21/N^2 counts surfaces of higher genus. In an external field, the weight of the integral contains a fixed matrix, so the partition function becomes a function of the eigenvalues of that external matrix. Brézin and Gross showed that the Schwinger-Dyson equations — the identities obtained by integrating total derivatives in the matrix integral, also called loop equations — can be rewritten as differential equations in the eigenvalues of the external field and used to compute the free energy to all orders in 1/N1/N. Gross and Newman applied this method to two integrals of central importance. The first is the Kontsevich integral Z=∫DM exp⁡Tr ⁣(−12AM2−g3M3)\mathcal{Z}=\int\mathcal{D}M\,\exp\mathrm{Tr}\!\left(-\tfrac12 AM^2-\tfrac{g}{3}M^3\right) over hermitian matrices MM with AA a fixed positive-definite hermitian matrix of eigenvalues μa\mu_a. Kontsevich proved that its ribbon-graph expansion gives a cell decomposition of the moduli space M‾g,n\overline{\mathcal{M}}_{g,n} of stable complex curves of genus gg with nn marked points, so that Z\mathcal{Z} generates the intersection numbers of the tautological ψ\psi classes (Chern classes of the cotangent line bundles at the marked points) — the correlators of two-dimensional topological gravity, whose generating function satisfies the Virasoro constraints conjectured by Witten. Gross and Newman gave a compact derivation of these constraints directly from the eigenvalue Schwinger-Dyson equations: after factoring out the Gaussian determinant, Z=∏a,b(μa+μb)−1/2Y\mathcal{Z}=\prod_{a,b}(\mu_a+\mu_b)^{-1/2}Y, and using Kontsevich's observation that the expansion depends on the eigenvalues only through the 'times' tk=−12k+1∑bμb−(2k+1),k≥0,t_k=-\frac{1}{2k+1}\sum_b\mu_b^{-(2k+1)},\qquad k\ge0, the Schwinger-Dyson equations imply LnY=12∂n+1YL_nY=\frac12\partial_{n+1}Y for n≥−1n\ge-1, where the LnL_n are differential operators in the tkt_k satisfying the Virasoro algebra — the Lie algebra of infinitesimal conformal transformations of the circle, with generators LnL_n and bracket [Lm,Ln]=(m−n)Lm+n[L_m,L_n]=(m-n)L_{m+n} up to a central term. The second integral runs over the unitary group, Zˉ=∫DU exp⁡Tr(A†U+U†A),\bar{\mathcal{Z}}=\int\mathcal{D}U\,\exp\mathrm{Tr}(A^\dagger U+U^\dagger A), and is the partition function of the Brézin-Gross-Witten (BGW) model of lattice gauge theory. The same method produced a new set of constraints LnYˉ=−∂nYˉL_n\bar Y=-\partial_n\bar Y (n≥0n\ge0) describing the zeroth multicritical point of the unitary series (the most degenerate of the nested critical points at which a unitary matrix model admits a continuum limit); the identical constraints were found independently from modified KdV flows by Hollowood, Miramontes, Pasquinucci and Nappi. Gross and Newman conjectured that, as in the hermitian case, this structure signals a Kontsevich-type representation of the unitary integral in terms of the moduli space of some unknown ensemble of surfaces, and left open what that ensemble is. This part of their question has since been answered: Mironov, Morozov and Semenoff identified the BGW partition function as a Generalized Kontsevich Model (the family obtained from the Kontsevich integral by replacing the cubic potential with an arbitrary function, here V(X)=1/X\mathcal{V}(X)=1/X); Norbury introduced the cohomology class Θ\Theta on M‾g,n\overline{\mathcal{M}}_{g,n} (the 'negative r-spin' class at r=2r=2, built from a square root of the canonical bundle) and conjectured that its descendant potential — the generating function of the intersection numbers with ψ\psi-class insertions — is the BGW tau function of the KdV (Korteweg-de Vries) hierarchy, an integrable hierarchy of evolution equations whose solutions are encoded in a single tau function; Chidambaram, Garcia-Failde and Giacchetto proved this conjecture; Alexandrov and Norbury proved that the generating function of super Weil-Petersson volumes of the moduli space of super Riemann surfaces (the supersymmetric analogs of Riemann surfaces, carrying fermionic as well as bosonic coordinates) with Neveu-Schwarz punctures (one of their two types of marked point) is the BGW KdV tau function, with the other type, Ramond punctures, governed by a generalized BGW tau function (citing Gross-Newman for the underlying constraints); and related work established a Hodge-BGW correspondence (relating Hodge-integral correlators on M‾g,n\overline{\mathcal{M}}_{g,n} to generalized BGW correlators) and identified the large-genus asymptotics of the BGW numbers with ψ\psi-Θ\Theta intersection numbers. The unsolved part of the source concerns real symmetric matrices. A matrix model over real symmetric matrices generates a sum over unoriented (non-orientable) surfaces, because the propagators cannot consistently orient the ribbon graphs. For Z~=∫∏i≤jdMij exp⁡N tr ⁣(XM−g3M3)\tilde{\mathcal{Z}}=\int\prod_{i\le j}dM_{ij}\,\exp N\,\mathrm{tr}\!\left(XM-\tfrac{g}{3}M^3\right) with XX a real symmetric external field of eigenvalues xax_a, the eigenvalue Schwinger-Dyson equations read ∂2Z~∂xa2+12∑b≠a1xa−xb(∂Z~∂xa−∂Z~∂xb)=N2g xa Z~,\frac{\partial^2\tilde{\mathcal{Z}}}{\partial x_a^2}+\frac12\sum_{b\ne a}\frac{1}{x_a-x_b}\left(\frac{\partial\tilde{\mathcal{Z}}}{\partial x_a}-\frac{\partial\tilde{\mathcal{Z}}}{\partial x_b}\right)=\frac{N^2}{g}\,x_a\,\tilde{\mathcal{Z}}, where the factor 12\frac12 in the pairwise term reflects that the elements of a symmetric matrix are not independent. This factor qualitatively changes the expansion: the cancellation that removes the order-1/N1/N terms in the hermitian and unitary cases fails, so the free energy has an order-1/N1/N contribution (the analog of non-orientable surfaces), and the equation linearized about the spherical (genus-zero) solution for this correction has a right-hand side of a type that had not appeared in the hermitian or unitary cases. Gross and Newman could not solve this equation, conjectured that it admits no closed-form solution, noted that it was then impossible even to write down the equations for higher 1/N1/N corrections, and computed the order-1/N1/N terms explicitly only through order g4g^4. Worse, for the symmetric Kontsevich-type integral Z=∫DM exp⁡N tr ⁣(−12AM2−g3M3),M=MT,\mathcal{Z}=\int\mathcal{D}M\,\exp N\,\mathrm{tr}\!\left(-\tfrac12 AM^2-\tfrac{g}{3}M^3\right),\qquad M=M^T, the perturbative expansion depends on the eigenvalues μa\mu_a of AA through invariants that proliferate: beyond the naive analogs of the Kontsevich times (the odd traces s2k+1=1N∑bμb−(2k+1)s_{2k+1}=\frac1N\sum_b\mu_b^{-(2k+1)}), new and increasingly complicated invariants appear at each order in g2g^2, so even the complete set of variables on which the partition function depends is unknown. No change of variables to a finite set of times, no Virasoro-type constraints, and no Kontsevich-type geometric interpretation is known for this ensemble; Gross and Newman concluded that further progress on the symmetric model by their method is impossible. This statement concerns the external-field cubic symmetric integral specifically: the b-deformed monotone-Hurwitz and non-oriented-map models of references 17-18 establish Virasoro/BKP structure for different ensembles — different matrices, potentials and counting problems — and do not supply the times, the Virasoro-type constraints, or the moduli-space interpretation sought here. (Their discussion's conjecture that the unitary model is a third example, after the Kontsevich model and the Penner model — the hermitian one-matrix integral whose expansion generates the virtual Euler characteristic of the moduli space of punctured surfaces — of an integral with a moduli-space representation referred to the unitary model; that specific question has been answered by the results quoted above at the level of the tau function (an identification of generating functions), rather than by a finite-NN ribbon-graph cell decomposition of a moduli space in the strict image of Kontsevich's theorem.)

Problem Statement

Determine whether the real symmetric external-field matrix integral — the unoriented-surface analog of the Kontsevich integral — possesses an analog of the Kontsevich-Witten structure. Concretely, for Z=∫DM exp⁡N tr ⁣(−12AM2−g3M3),M=MT∈RN×N,\mathcal{Z}=\int\mathcal{D}M\,\exp N\,\mathrm{tr}\!\left(-\tfrac12 AM^2-\tfrac{g}{3}M^3\right),\qquad M=M^T\in\mathbb{R}^{N\times N}, with AA a fixed real symmetric matrix of eigenvalues μa\mu_a — and likewise for its generating-function variant Z~=∫∏i≤jdMij exp⁡N tr ⁣(XM−g3M3)\tilde{\mathcal{Z}}=\int\prod_{i\le j}dM_{ij}\,\exp N\,\mathrm{tr}\!\left(XM-\tfrac{g}{3}M^3\right) over real symmetric MM with real symmetric external field XX of eigenvalues xax_a — the question is: (1) Times and variables: identify the complete set of invariants of the eigenvalues μa\mu_a on which the perturbative expansion of Z\mathcal{Z} depends — the analog of Kontsevich's times tk=−12k+1∑bμb−(2k+1)t_k=-\frac{1}{2k+1}\sum_b\mu_b^{-(2k+1)} — and determine whether a closed (ideally finite) system of universal times exists, performing the corresponding change of variables in the eigenvalue Schwinger-Dyson equations ∂2Z~∂xa2+12∑b≠a1xa−xb(∂Z~∂xa−∂Z~∂xb)=N2g xa Z~.\frac{\partial^2\tilde{\mathcal{Z}}}{\partial x_a^2}+\frac12\sum_{b\ne a}\frac{1}{x_a-x_b}\left(\frac{\partial\tilde{\mathcal{Z}}}{\partial x_a}-\frac{\partial\tilde{\mathcal{Z}}}{\partial x_b}\right)=\frac{N^2}{g}\,x_a\,\tilde{\mathcal{Z}}. (2) Constraints: derive the Virasoro-type constraints (or another closed algebraic system of constraints) satisfied by the partition function of this ensemble in these variables, or prove that no such finite constraint system exists. (3) Genus expansion: solve the equation obtained by writing F~=F~(0)+N−1F~(1)+⋯\tilde F=\tilde F^{(0)}+N^{-1}\tilde F^{(1)}+\cdots for the free energy and linearizing the displayed Schwinger-Dyson equation about its spherical (genus-zero) solution — the source's eq. (4.8) — for the order-1/N1/N correction to the free energy, and determine the closed-form structure of the full 1/N1/N expansion and of the higher-order corrections — or prove that no closed form exists, as Gross and Newman conjectured. (4) Geometric interpretation: identify the ensemble whose intersection-type invariants the symmetric Kontsevich-type integral generates — the analog of Kontsevich's cell decomposition of M‾g,n\overline{\mathcal{M}}_{g,n} for moduli spaces of unoriented (real or Klein) surfaces, the unoriented analogs of Riemann surfaces — or show that no such representation exists. These four questions are facets of one objective: the unoriented analog of the Kontsevich/topological-gravity correspondence for external-field matrix integrals. In the hermitian and unitary cases the analogs of (1), (2) and (4) were solved together by a single construction (times and constraints from the Schwinger-Dyson equations, geometry from the ribbon-graph expansion), and (3) follows from the constraint structure; the source's question is whether the same program can be carried out — or shown impossible — for symmetric matrices.

References

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