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The equivariant Gromov-Witten theory of \(\mathbb P^1\). (English) Zbl 1105.14077

The authors give an explicit description of the equivariant Gromov-Witten theory of \(\mathbb P^1\). Namely, they prove that the equivariant Gromov-Witten potential of \(\mathbb P^1\) satisfies the \(2\)nd Toda equation. In particular, all Gromov-Witten invariants of positive degree are determined by those of degree zero. This result also completes the proof of the GW/H-correspondence introduced in their previous article [Ann. Math. (2) 163, 517–560 (2006; Zbl 1105.14076)].
The proof is done in several steps. The key point is to prove that the generating function of the equivariant GW-invariants of \(\mathbb P^1\) equals a certain vacuum expectation value on the infinite wedge product (see Theorem 3). In order to prove Theorem 3, the authors proceed as follows:
First, using the equivariant localization formula of T. Graber and R. Pandharipande [Invent. Math. 135, 487–518 (1999; Zbl 0953.14035)], they express the generating function of the degree \(d\) equivariant GW-invariants of \(\mathbb P^1\) in terms of the Hodge integrals over the moduli space \({\overline M}_{g,n}\) (see Proposition 1).
Secondly, using the ‘ELSV-formula’ obtained by T. Ekedahl, S. Lando, M. Shapiro and A. Vainshtein [Invent. Math. 146, 297–327 (2001; Zbl 1073.14041)], and then dealing with complex analytical issues, the authors prove that the generating series for the \(n\)-point Hodge integrals equals the vacuum expectation of a certain operator on the infinite wedge space (see Theorem 2).
With the help of the formula given in Theorem 3, they establish that the Gromov-Witten potential satisfies the \(2\)nd Toda equation (see Theorem 5 and 7).
The results of this article are further used by the same authors in [Invent. Math. 163, 47–108 (2006; Zbl 1140.14047)] to establish the Virasoro constraints for the relative Gromov-Witten theory of smooth target curves.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H70 Relationships between algebraic curves and integrable systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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