Skip to main content
SpringerLink
Log in

From wave functions to tau-functions for the Volterra lattice hierarchy

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

For an arbitrary solution to the Volterra lattice hierarchy, the logarithmic derivatives of the tau-function of the solution can be computed by the matrix-resolvent method. In this paper, we define a pair of wave functions of the solution and use them to give an expression of the matrix resolvent; based on this we obtain a new formula for the k-point functions for the Volterra lattice hierarchy in terms of wave functions. As an application, we give an explicit formula of k-point functions for the even GUE (Gaussian Unitary Ensemble) correlators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler M, van Moerbeke P. Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials. Duke Math J, 1995, 80: 863–911

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  3. Bertola M, Dubrovin B, Yang D. Simple Lie algebras and topological ODEs. Int Math Res Not, 2018, 2018: 1368–1410

    MathSciNet  Google Scholar 

  4. Bertola M, Dubrovin B, Yang D. Simple Lie algebras, Drinfeld–Sokolov hierarchies, and multi-point correlation functions. Mosc Math J, 2021, 21: 233–270

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Brézin E, Itzykson C, Parisi P, Zuber J B. Planar diagrams. Comm Math Phys, 1978, 59: 35–51

    Article  ADS  MathSciNet  Google Scholar 

  7. Cafasso M, Yang D. Tau-functions for the Ablowitz–Ladik hierarchy: the matrix-resolvent method. J Phys A: Math Theor, 2022, 55: 204001

    Article  ADS  MathSciNet  Google Scholar 

  8. Carlet G. The extended bigraded Toda hierarchy. J Phys A: Math Gen, 2006, 39: 9411–9435

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Deift P A. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Providence: American Mathematical Society, 1999

    Google Scholar 

  11. Dubrovin B. Algebraic spectral curves over Q and their tau-functions//Donagi R, Shaska T. Integrable Systems and Algebraic Geometry. Cambridge: Cambridge University Press, 2020: 41–91

    Chapter  Google Scholar 

  12. Dubrovin B. Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions//Sidoravicius V. New Trends in Mathematical Physics. Dordrecht: Springer, 2009: 231–276

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Dubrovin B, Liu S Q, Yang D, Zhang Y. Hodge-GUE correspondence and the discrete KdV equation. Comm Math Phys, 2020, 379: 461–490

    Article  ADS  MathSciNet  Google Scholar 

  15. Dubrovin B, Valeri D, Yang D. Affine Kac-Moody algebras and tau-functions for the Drinfeld-Sokolov hierarchies: the matrix-resolvent method. Symmetry Integrability Geom Methods Appl, 2022, 18: 077

    MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  17. Dubrovin B, Yang D. On cubic Hodge integrals and random matrices. Commun Number Theory Phys, 2017, 11: 311–336

    Article  MathSciNet  Google Scholar 

  18. Dubrovin B, Yang D. Matrix resolvent and the discrete KdV hierarchy. Comm Math Phys, 2020, 377: 1823–1852

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Dubrovin B, Yang D, Zagier D. On tau-functions for the KdV hierarchy. Sel Math, 2021, 27: Art 12

  21. Dubrovin B, Zhang Y. Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. arXiv:math/0108160

  22. Fu A, Yang D. The matrix-resolvent method to tau-functions for the nonlinear Schrödinger hierarchy. J Geom Phys, 2022, 179: 104592

    Article  Google Scholar 

  23. Gerasimov A, Marshakov A, Mironov A, et al. Matrix models of two dimensional gravity and Toda theory. Nuclear Physics B, 1991, 357: 565–618

    Article  ADS  MathSciNet  Google Scholar 

  24. Guo J, Yang D. On the large genus asymptotics of psi-class intersection numbers. Math Ann, 2022. DOI: https://doi.org/10.1007/s00208-022-02505-6

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

    Article  ADS  MathSciNet  Google Scholar 

  26. ’t Hooft G. A planar diagram theory for strong interactions. Nucl Phys B, 1974, 72: 461–473

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  29. Mehta M L. Random Matrices. New York: Academic Press, 1991

    Google Scholar 

  30. Morozov A, Shakirov S. Exact 2-point function in Hermitian matrix model. Journal of High Energy Physics, 2009, 12: Art 003

  31. Witten E. Two dimensional gravity and intersection theory on Moduli space. Surveys Diff Geom, 1991, 1: 243–310

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  33. Zhou J. Hermitian one-matrix model and KP hierarchy. arXiv:1809.07951

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    Ang Fu, Mingjin Li & Di Yang

Authors
  1. Ang Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingjin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Di Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Di Yang.

Ethics declarations

Conflict of Interest The authors declare no conflict of interest.

Additional information

The work was partially supported by the National Key R and D Program of China (2020YFA0713100).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, A., Li, M. & Yang, D. From wave functions to tau-functions for the Volterra lattice hierarchy. Acta Math Sci 44, 405–419 (2024). https://doi.org/10.1007/s10473-024-0201-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-024-0201-4

Key words

2020 MR Subject Classification

Search

Navigation