2D Toda τ functions, weighted Hurwitz numbers and the Cayley graph: Determinant representation and recursion formula

@article{Ding20222DT,
  title={2D Toda $\tau$ functions, weighted Hurwitz numbers and the Cayley graph: Determinant representation and recursion formula},
  author={Xiang-Mao Ding and Xiang Li},
  journal={Journal of Mathematical Physics},
  year={2022}
}
We generalize the determinant representation of the Kadomtsev–Petviashvili τ functions to the case of the 2D Toda τ functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda τ functions, for which we give a determinant representation of weighted Hurwitz numbers. Then, we can get a finite-dimensional equation system for the weighted Hurwitz numbers [Formula: see text] with the same dimension | σ| = | ω| = n. Using this equation system, we calculated… 

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