Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers

Abstract

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.

References in corpus (6)

• Hodge integrals, Hurwitz numbers, and Symmetric Groups
• Emergent Geometry and Mirror Symmetry of A Point
• BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions
• Kac-Schwarz Operators of Type $B$, Quantum Spectral Curves, and Spin Hurwitz Numbers
• BKP-Affine Coordinates and Emergent Geometry of Generalized Brézin-Gross-Witten Tau-Functions
• Connected $(n,m)$-Point Functions of Diagonal $2$-BKP Tau-Functions, and Spin Double Hurwitz Numbers

Cited by in corpus (1)

• Connected $(n,m)$-Point Functions of Diagonal $2$-BKP Tau-Functions, and Spin Double Hurwitz Numbers