On tau-functions for the KdV hierarchy
Mathematical Physics
2021-02-24 v2 math.MP
Exactly Solvable and Integrable Systems
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 Br\'ezin--Gross--Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lam\'e tau-function are also given.
Cite
@article{arxiv.1812.08488,
title = {On tau-functions for the KdV hierarchy},
author = {Boris Dubrovin and Di Yang and Don Zagier},
journal= {arXiv preprint arXiv:1812.08488},
year = {2021}
}
Comments
Minor changes: added remarks, added references, corrected typos; 32 pages