Gromov–Witten invariants of the Riemann sphere

Abstract

A conjectural formula for the k-point generating function of Gromov-Witten invariants of the Riemann sphere for all genera and all degrees was proposed in [11]. In this paper, we give a proof of this formula together with an explicit analytic (as opposed to formal) expression for the corresponding matrix resolvent. We also give a formula for the k-point function as a sum of (k − 1)! products of hypergeometric functions of one variable. We show that the k-point generating function coincides with the ǫ → 0 asymptotics of the analytic k-point function, and also compute three more asymptotics of the analytic function for ǫ → ∞, q → 0, q → ∞, thus defining new invariants for the Riemann sphere.

On tau-functions for the Toda lattice hierarchy

“…This is because, after the summation over the S k /C k and subtracting δ k,2 (λ 1 −λ 2 ) 2 , the poles in the diagonal cancel (cf. the Proposition 2 of [12] for a straightforward proof of this point). We note that, as formal power series, the coefficients of the both sides of (25) are in A.…”

“…Gromov-Witten invariants of P 1 in the stationary sector. The initial data for the Gromov-Witten solution to the Toda lattice hierarchy was for example derived in [10,12,11]. It has the following explicit expression:…”

“…Lemma 1 implies that the flows (12) all commute. So we can solve the whole Toda lattice hierarchy (12) together, which yields solutions of the form (v = v(n, t), w = w(n, t)). Here t := (t 0 , t 1 , .…”

“…Let V be a ring of functions of n closed under shifting n by ±1. For two given f (n), g(n) ∈ V , consider the initial value problem for (12) with the initial condition: v(n, 0) = f (n) , w(n, 0) = g(n) .…”

“…The corresponding unique solution governs the Gromov-Witten invariants of P 1 in the stationary sector in all genera and all degrees. 2 be an arbitrary solution to the Toda lattice hierarchy (12). Write Ω p,q (n, t) and S p (n, t) as the images of Ω p,q and S p under the substitutions…”