Skip to main content

Matrix Resolvent and the Discrete KdV Hierarchy


Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are obtained, and applications to enumeration of ribbon graphs with even valencies and to certain special cubic Hodge integrals are considered.

This is a preview of subscription content, access via your institution.


  1. The number \(a_g(2j_1,\ldots ,2j_k)\) has the alternative expression \(a_g(2j_1,\ldots ,2j_k) = \sum _G \frac{\prod _{\ell =1}^k (2 j_\ell )}{\#\, \mathrm{Sym} \, G}\), where \(\sum _G\) denotes summation over connected ribbon graphs G with unlabelled half-edges and unlabelled vertices of genus g with k vertices of valencies \(2j_1, \ldots , 2j_k\).

  2. We can say in a more accurate sense that the logarithmic derivatives are identified with the correlation functions, where the latter are defined as abstract differential polynomials; see for example [18] for the details.


  1. Bertola, M., Dubrovin, B., Yang, D.: Correlation functions of the KdV hierarchy and applications to intersection numbers over \(\overline{{\cal{M}}}_{g, n}\). Phys. D Nonlinear Phenom. 327, 30–57 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  2. Bertola, M., Dubrovin, B., Yang, D.: Simple Lie algebras and topological ODEs. IMRN 2016, 1368–1410 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Bertola, M., Dubrovin, B., Yang, D.: Simple Lie algebras, Drinfeld–Sokolov hierarchies, and multi-point correlation functions. Preprint arXiv:1610.07534v2

  4. Bessis, D., Itzykson, C., Zuber, J.-B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1, 109–157 (1980)

    MathSciNet  Article  Google Scholar 

  5. Borot, G., Garcia-Failde, E.: Simple maps, Hurwitz numbers, and Topological Recursion. Preprint arXiv:1710.07851v2

  6. Carlet, G., Dubrovin, B., Zhang, Y.: The extended Toda hierarchy. Mosc. Math. J. 4, 313–332 (2004)

    MathSciNet  Article  Google Scholar 

  7. Chekhov, L., Eynard, B.: Hermitean matrix model free energy: Feynman graph technique for all genera. JHEP 2006, 014 (2006)

    Article  Google Scholar 

  8. Claeys, T., Grava, T., McLaughlin, K.D.T.-R.: Asymptotics for the partition function in two-cut random matrix models. Commun. Math. Phys. 339, 513–587 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  9. Cunden, F. D., Dahlqvist A., O’Connell, N.: Integer moments of complex Wishart matrices and Hurwitz numbers. Preprint arXiv:1809.10033v2

  10. Deift, P.: Polynomials, Orthogonal, Random Matrices: A Riemann–Hilbert Approach. American Mathematical Society, Providence, RI (2000)

    MATH  Google Scholar 

  11. Di Francesco, P.: 2D quantum gravity, matrix models and graph combinatorics. In: Brézin, É., et al. (eds.) Applications of Random Matrices in Physics. Springer, Berlin (2006)

    Google Scholar 

  12. Dubrovin, B.: Geometry of 2D topological field theories. In: Francaviglia, M., Greco, S. (eds.) "Integrable Systems and Quantum Groups" (Montecatini Terme, 1993). Lecture Notes in Math., vol. 1620, pp. 120–348. Springer, Berlin (1996)

    Chapter  Google Scholar 

  13. Dubrovin, B.: Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions. In: Sidoravicius, V. (ed.) New trends in Mathematical Physics. Selected Contributions of the XVth International Congress on Mathematical Physics, pp. 231–276. Springer, Berlin (2009)

    Chapter  Google Scholar 

  14. Dubrovin, B., Yang, D.: Generating series for GUE correlators. Lett. Math. Phys. 107, 1971–2012 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  15. Dubrovin, B., Yang, D.: On cubic Hodge integrals and random matrices. Commun. Numb. Theory Phys. 11, 311–336 (2017)

    MathSciNet  Article  Google Scholar 

  16. Dubrovin, B., Yang, D.: On Gromov–Witten invariants of \({\mathbb{P}}^1\). Math. Res. Lett. 26, 729–748 (2019)

    MathSciNet  Article  Google Scholar 

  17. Dubrovin, B., Yang, D., Zagier, D.: Gromov–Witten invariants of the Riemann sphere. Pure Appl. Math. Q. 16, 153–190 (2020)

    Article  Google Scholar 

  18. Dubrovin, B., Yang, D., Zagier, D.: On tau-functions for the KdV hierarchy. arXiv:1812.08488

  19. Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs. Adv. Math. 293, 382–435 (2016)

    MathSciNet  Article  Google Scholar 

  20. Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge–GUE correspondence and the discrete KdV equation. Preprint arXiv:1612.02333v2

  21. Faddeev, L., Takhtajan, L.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1986)

    MATH  Google Scholar 

  22. Fernandes, R.L., Vanhaecke, P.: Hyperelliptic Prym varieties and integrable systems. Commun. Math. Phys. 221, 169–196 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  23. Flaschka, H.: On the Toda lattice. II. Inverse-scattering solution. Progr. Theoret. Phys. 51, 703–716 (1974)

    ADS  MathSciNet  Article  Google Scholar 

  24. Gerasimov, A., Marshakov, A., Mironov, A., Morozov, A., Orlov, A.: Matrix models of two-dimensional gravity and Toda theory. Nuclear Phys. B 357, 565–618 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  25. Gopakumar, R., Vafa, C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 5, 1415–1443 (1999)

    MathSciNet  Article  Google Scholar 

  26. Gisonni, M., Grava, T., Ruzza, G.: Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals. Preprint arXiv:1912.00525v3

  27. Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85, 457–485 (1986)

    ADS  MathSciNet  Article  Google Scholar 

  28. Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461–473 (1974)

    ADS  Article  Google Scholar 

  29. ’t Hooft, G.: A two-dimensional model for mesons. Nucl. Phys. B 75, 461–470 (1974)

    ADS  Article  Google Scholar 

  30. Kazakov, V., Kostov, I., Nekrasov, N.: D-particles, matrix integrals and KP hierarchy. Nuclear Phys. B 557, 413–442 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  31. Liu, C.-C.M., Liu, K., Zhou, J.: A proof of a conjecture of Mariño–Vafa on Hodge integrals. J. Differ. Geom. 65, 289–340 (2003)

    Article  Google Scholar 

  32. Manakov, S.V.: Complete integrability and stochastization of discrete dynamical systems. J. Exper. Theoret. Phys. 67: 543–555 (in Russian). English translation in Soviet Phys. JETP 40, 269–274 (1974)

  33. Mariño, M., Vafa, C.: Framed knots at large N. Contemp. Math. 310, 185–204 (2002)

    MathSciNet  Article  Google Scholar 

  34. Mehta, M.L.: Random Matrices, 2nd edn. Academic Press, New York (1991)

    MATH  Google Scholar 

  35. Morozov, A., Shakirov, S.: Exact 2-point function in Hermitian matrix model. JHEP 2009, 003 (2009)

    MathSciNet  Article  Google Scholar 

  36. Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and Geometry, pp. 271–328. Birkhäuser, Boston (1983)

  37. Okounkov, A., Pandharipande, R.: Hodge integrals and invariants of the unknot. Geomet. Topol. 8, 675–699 (2004)

    MathSciNet  Article  Google Scholar 

  38. Volterra, V.: Leçons sur la théorie mathématique de la lutte pour la vie. Gauthier-Villars, Paris (1936)

    MATH  Google Scholar 

  39. Witten, E.: Two-Dimensional Gravity and Intersection Theory on Moduli Space, Surveys in Differential Geometry (Cambridge, MA, 1990), vol. 1, pp. 243–310. Lehigh Univ, Bethlehem (1991)

    Google Scholar 

  40. Yang, D.: On tau-functions for the Toda lattice hierarchy. Lett. Math. Phys. 110, 555–583 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  41. Zhou, J.: On Absolute N-Point Function Associated with Gelfand–Dickey Polynomials (2015) (unpublished)

  42. Zhou, J.: Emergent Geometry of Matrix Models with Even Couplings. Preprint arXiv:1903.10767

Download references


We would like to thank the anonymous referee for valuable suggestions and constructive comments that improve a lot the presentation of the paper. One of the authors D.Y. is grateful to Youjin Zhang and Don Zagier for their advising, and to Giulio Ruzza for helpful discussions. Part of the work of D.Y. was done when he was a post-doc at MPIM, Bonn; he thanks MPIM for excellent working conditions and financial supports.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Di Yang.

Additional information

Communicated by P. Deift.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Boris Dubrovin: Deceased on March 19, 2019.

On Consequence of the Hodge–GUE Correspondence

On Consequence of the Hodge–GUE Correspondence

In this appendix, we derive a consequence of the Hodge–GUE correspondence that has a similar flavour to formula (121). Note that

$$\begin{aligned}&{\mathcal {H}}\bigl (\mathbf {t}(x,\mathbf {s}); \sqrt{2}\epsilon \bigr ) \\ {}&~ \; = \; \sum _{g\ge 0} 2^{g-1} \epsilon ^{2g-2} \sum _{k\ge 0} \frac{1}{k!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \prod _{m=1}^k \biggl (\sum _{i_m\ge 0} t_{i_m}(x,\mathbf {s}) \psi _m^{i_m}\biggr )\\ {}&~ \; = \; \sum _{g\ge 0} 2^{g-1} \epsilon ^{2g-2} \sum _{k\ge 0} \frac{1}{k!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \sum _{l=0}^k\left( {\begin{array}{c}k\\ l\end{array}}\right) \prod _{m=l+1}^k \biggl ((x -1) - \frac{\psi _m^2}{1-\psi _m} \biggr )\\ {}&\qquad \cdot \sum _{ p_1,\dots ,p_l} \prod _{m=1}^{l}\frac{p_m{\bar{s}}_{p_m}}{1- p_m\psi _m}. \end{aligned}$$

Then by comparing the coefficients of \(s_{p_1} \dots s_{p_l}\) of the both sides of (118) we get

$$\begin{aligned}&\langle \sigma _{p_1}\dots \sigma _{p_l}\rangle _g(x) \nonumber \\ {}&\quad = \sum _{k\ge l} \frac{1}{(k-l)!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \prod _{m={l+1}}^k \biggl ((x -1) - \frac{\psi _m^2}{1-\psi _m} \biggr ) \prod _{m=1}^{l}\frac{p_m \left( {\begin{array}{c}2p_m\\ p_m\end{array}}\right) }{1- p_m\psi _{m}} \nonumber \\ {}&\quad \; + \delta _{g,0} \delta _{l,2} \, \frac{p_1 \, p_2}{p_1+p_2} \left( {\begin{array}{c}2p_1\\ p_1\end{array}}\right) \left( {\begin{array}{c}2p_2\\ p_2\end{array}}\right) \, + \, \frac{1}{2}\delta _{g,0} \delta _{l,1} \left( {\begin{array}{c}2p_1\\ p_2\end{array}}\right) \biggl ( \frac{p_1}{1+p_1} - x \biggr ) . \end{aligned}$$

Here \(\langle \sigma _{p_1}\dots \sigma _{p_l}\rangle (x;\epsilon ) =: \sum _{g\ge 0} \epsilon ^{2g-2} \langle \sigma _{p_1}\dots \sigma _{p_l}\rangle _g(x)\), and \(\langle \sigma _{p_1}\dots \sigma _{p_l}\rangle (x;\epsilon ) \) are the modified GUE correlators with even couplings defined in (113). Taking \(x=1\) we find

$$\begin{aligned} \langle \sigma _{p_1}\dots \sigma _{p_l}\rangle _g|_{x=1} \; = \; \sum _{k\ge l} \frac{1}{(k-l)!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \prod _{m={l+1}}^k \biggl (- \frac{\psi _m^2}{1-\psi _m} \biggr ) \prod _{m=1}^{l}\frac{p_m \left( {\begin{array}{c}2p_m\\ p_m\end{array}}\right) }{1- p_m\psi _{m}} . \end{aligned}$$

A further consideration to (125) was given in [5].

Combining (124) with (115) we find for any fixed \(l\ge 1\), \(p_1,\dots ,p_l\ge 1\) the following identities:

$$\begin{aligned}&l! \, x^{2-2g -l+|j|} \sum _{\begin{array}{c} g_1,r\ge 0 \\ g_1+r=g \end{array}} \left( {\begin{array}{c}2-2g_1-l+|j|\\ 2r\end{array}}\right) \, \frac{E_{2r}}{2^{2r}} \,a_{g_1}(2p_1, \dots , 2p_l) \nonumber \\&\quad \; = \; \sum _{k\ge l} \frac{1}{(k-l)!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \prod _{m={l+1}}^k \biggl ((x -1) - \frac{\psi _m^2}{1-\psi _m} \biggr ) \prod _{m=1}^{l}\frac{p_m \left( {\begin{array}{c}2p_m\\ p_m\end{array}}\right) }{1- p_m\psi _{m}} \nonumber \\&\qquad \, + \, \delta _{g,0} \delta _{l,2} \, \frac{p_1 \, p_2}{p_1+p_2} \left( {\begin{array}{c}2p_1\\ p_1\end{array}}\right) \left( {\begin{array}{c}2p_2\\ p_2\end{array}}\right) \, + \, \frac{1}{2}\delta _{g,0} \delta _{l,1} \left( {\begin{array}{c}2p_1\\ p_2\end{array}}\right) \biggl ( \frac{p_1}{1+p_1} - x \biggr ), \qquad g\ge 0. \end{aligned}$$

Note that for any \(g\ge 0\), the RHS is a priori a power series of \(x-1\), but the LHS shows that it is actually a monomial of x and so is also a polynomial of \(x-1\). This subset of the identities deserve a further investigation. Moreover, the LHS vanishes when g is sufficiently large, an so is the RHS; this provides another subset of the identities for the cubic Hodge integrals.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dubrovin, B., Yang, D. Matrix Resolvent and the Discrete KdV Hierarchy. Commun. Math. Phys. 377, 1823–1852 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: