On tau-functions for the KdV hierarchy

• Boris Dubrovin ^[3] ; Di Yang ^[1] ; Don Zagier ^[2]
  1. [1] University of Science and Technology of China

     University of Science and Technology of China

     China

     
  2. [2] International Centre for Theoretical Physics

     International Centre for Theoretical Physics

     Trieste, Italia

     
  3. [3] SISSA, Italia

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 1, 2021
• Idioma: inglés
• DOI: 10.1007/s00029-021-00620-x
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• Resumen

  • 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.