Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics

International Mathematics Research Notices. 2015. Vol. 2015. No. 8. P. 2082–2110.

Натанзон С. М., Забродин А. В.

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial constants and find recurrence relations for them. These results are used to obtain new formulas for the genus 0 double Hurwitz numbers. They can also serve as a starting point for a constructive approach to the Riemann mapping problem and the inverse potential problem in 2D.

Ключевые слова: числа Гурвица, Hurwitz numbers, конформное отображение, Toda hierarchy, conformal maps, Иерархия Тоды

ПУБЛИКАЦИЯ ПОДГОТОВЛЕНА ПО РЕЗУЛЬТАТАМ ПРОЕКТА: Числа Гурвица как реализация топологической теории струн (2013)