On tau-functions for the KdV hierarchy

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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Notes

  1. 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. 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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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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Keywords

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

Mathematics Subject Classification

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