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On tau-functions for the KdV hierarchy

Selecta Mathematica volume 27, Article number: 12 (2021) Cite this article

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Abstract

For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in this paper two new formulae for the generating series by introducing a pair of wave functions of the solution. Applications to the Witten–Kontsevich tau-function, to the generalized Brézin–Gross–Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lamé tau-function are also given.

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Change history

  • 18 March 2021

    In PDF page 26, there are some blank spaces. Now, it has been removed.

Notes

  1. A dual wave function satisfies \(L^*\psi ^* = \lambda \, \psi ^*\), where \(L^*\) is the formal adjoint operator of L. For the KdV hierarchy, \(L^*=L\) and a dual wave function \(\psi ^*\) is also a wave function.

  2. We are grateful to Bumsig Kim for sharing with us his knowledge about this interesting point.

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Acknowledgements

We would like to express our thanks to Paul Norbury for helpful suggestions and to the anonymous referee for many constructive comments. D.Y. is grateful to Youjin Zhang for his advice and encouragement and to Chang-Shou Lin and Alexander Alexandrov for their interest. Part of the work of D.Y. was done in MPIM, Bonn while he was a postdoc; he acknowledges MPIM for excellent working conditions and financial supports.

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Author notes

  1. B. Dubrovin: Deceased on March 19, 2019.

Authors and Affiliations

  1. SISSA, via Bonomea 265, 34136, Trieste, Italy

    Boris Dubrovin

  2. School of Mathematical Sciences, University of Science and Technology of China, Jinzhai Road 96, Hefei, 230026, People’s Republic of China

    Di Yang

  3. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Don Zagier

  4. International Centre for Theoretical Physics, Strada Costiera 11, 34014, Trieste, Italy

    Don Zagier

Authors
  1. Boris Dubrovin
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  2. Di Yang
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  3. Don Zagier
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Correspondence to Di Yang.

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Dubrovin, B., Yang, D. & Zagier, D. On tau-functions for the KdV hierarchy. Sel. Math. New Ser. 27, 12 (2021). https://doi.org/10.1007/s00029-021-00620-x

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  • DOI: https://doi.org/10.1007/s00029-021-00620-x

Keywords

  • KdV hierarchy
  • Tau-function
  • Pair of wave functions
  • Matrix resolvent
  • Generating series

Mathematics Subject Classification

  • 37K10
  • 53D45
  • 14N35
  • 05A15
  • 33E15