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The Toda equations and the Gromov-Witten theory of the Riemann sphere. (English) Zbl 0999.14020
This paper studies the (higher genus) Gromov-Witten theory of the Riemann sphere $$\mathbb P^1$$. The basic idea is to use the Toda equations to derive explicit expressions for (some) higher genus Gromov-Witten invariants on $$\mathbb P^1$$.
The Toda equations are certain differential equations for the Gromov-Witten potential, derived from so-called matrix models. They are very strong and allow to compute the Gromov-Witten potential from its degree zero part. They are thus stronger than the Virasoro constraints.
In this paper a proof of these equations in genus zero and one is explained, but in case of higher genus they remain conjectural. This means the explicit values for some Gromov-Witten invariants on $$\mathbb P^1$$ given in this paper, form now (strongly supported) conjectures. The derivation of the equations is very brief so that the non-specialist reader has to consult previous papers written in this context, but the basic ideas become clearly visible in this way.

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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