For an arbitrary solution to the KdV hierarchy, we derive an explicit formula for the generating series of the logarithmic derivatives of the tau-function of the solution in terms of matrix resolvents. Then, using the wave functions, we derive two additional explicit formulae. We also extend these results to the Toda lattice hierarchy. Applications to computing certain enumerative invariants are given. The talk is based on a series of joint works with Marco Bertola, Boris Dubrovin and Don Zagier.
学术海报.pdf
