Geodesic
Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

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We derive an explicit formula for the connected $(n,m)$-point functions associated to an arbitrary diagonal tau-function $\tau_f(\mathbf{t}^+,\mathbf{t}^-)$ of the 2d Toda lattice hierarchy using fermionic computations and the boson-fermion correspondence. Then for fixed $\mathbf{t}^-$, we compute the KP-affine coordinates of $\tau_f(\mathbf{t}^+, \mathbf{t}^-)$. As applications, we present a unified approach to compute various types of connected double Hurwitz numbers, including the ordinary double Hurwitz numbers, the double Hurwitz numbers with completed $r$-cycles, and the mixed double Hurwitz numbers. We also apply this method to the computation of the stationary Gromov–Witten invariants of $\mathbb{P}^1$ relative to two points.
Keywords: 2d Toda lattice hierarchy, connected $(n,m)$-point functions, double Hurwitz numbers.
Mots-clés : boson-fermion correspondence 
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     author = {Zhiyuan Wang and Chenglang Yang},
     title = {Diagonal {Tau-Functions} of {2D} {Toda} {Lattice} {Hierarchy,} {Connected} $(n,m)${-Point} {Functions,} and {Double} {Hurwitz} {Numbers}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a84/}
}
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Zhiyuan Wang; Chenglang Yang. Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a84/

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