TAILIEUCHUNG - Đề tài " The equivariant GromovWitten theory of P1 "

We present here the second in a sequence of three papers devoted to the Gromov-Witten theory of nonsingular target curves X. Let ω ∈ H2(X,Q) denote the Poincar´e dual of the point class. In the first paper [24], we considered the stationary sector of the Gromov-Witten theory of X formed by the descendents of ω. The stationary sector was identified in [24] with the Hurwitz theory of X with completed cycle insertions. The target P1 plays a distinguished role in the Gromov-Witten theory of target curves. Since P1 admits a C∗-action, equivariant localization may be used to study Gromov-Witten invariants [12]. The equivariant Poincar´e duals,. | Annals of Mathematics The equivariant Gromov-Witten theory of P1 By A. Okounkov and R. Pandharipande Annals of Mathematics 163 2006 561 605 The equivariant Gromov-Witten theory of P1 By A. Okounkov and R. Pandharipande Contents 0. Introduction . Overview . The equivariant Gromov-Witten theory of P1 . The equivariant Toda equation . Operator formalism . Plan of the paper . Acknowledgments 1. Localization for P1 . Hodge integrals . Equivariant n m-point functions . Localization vertex contributions . Localization global formulas 2. The operator formula for Hodge integrals . Review of the infinite wedge space . Hurwitz numbers and Hodge integrals . The opeartors A . Convergence of matrix elements . Series expansions of matrix elements . Commutation relations and rationality . Identification of H z u 3. The operator formula for Gromov-Witten invariants . Localization revisited . The T-function . The GW H correspondence 4. The 2-Toda hierarchy . Preliminaries of the 2-Toda hierarchy . String and divisor equations . The 2-Toda equation . The 2-Toda hierarchy 5. Commutation relations for operators A . Formula for the commutators . Some properties of the hypergeometric series . Conclusion of the proof of Theorem 1 562 A. OKOUNKOV AND R. PANDHARIPANDE 0. Introduction . Overview. . We present here the second in a sequence of three papers devoted to the Gromov-Witten theory of nonsingular target curves X. Let w e H2 X Q denote the Poincare dual of the point class. In the first paper 24 we considered the stationary sector of the Gromov-Witten theory of X formed by the descendents of w. The stationary sector was identified in 24 with the Hurwitz theory of X with completed cycle insertions. The target P1 plays a distinguished role in the Gromov-Witten theory of target curves. Since P1 admits a C -action equivariant localization may be used to study Gromov-Witten invariants 12 . The .

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