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 τf(t+,t-) of the 2d Toda lattice hierarchy using fermionic computations and the boson-fermion correspondence. Then for fixed t-, we compute the KP-affine coordinates of τf(t+,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 P1 relative to two points.
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