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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 0.1. Overview 0.2. The equivariant Gromov-Witten theory of P1 0.3. The equivariant Toda equation 0.4. Operator formalism 0.5. Plan of the paper 0.6. Acknowledgments 1. Localization for P1 1.1. Hodge integrals 1.2. Equivariant n m-point functions 1.3. Localization vertex contributions 1.4. Localization global formulas 2. The operator formula for Hodge integrals 2.0. Review of the infinite wedge space 2.1. Hurwitz numbers and Hodge integrals 2.2. The opeartors A 2.3. Convergence of matrix elements 2.4. Series expansions of matrix elements 2.5. Commutation relations and rationality 2.6. Identification of H z u 3. The operator formula for Gromov-Witten invariants 3.1. Localization revisited 3.2. The T-function 3.3. The GW H correspondence 4. The 2-Toda hierarchy 4.1. Preliminaries of the 2-Toda hierarchy 4.2. String and divisor equations 4.3. The 2-Toda equation 4.4. The 2-Toda hierarchy 5. Commutation relations for operators A 5.1. Formula for the commutators 5.2. Some properties of the hypergeometric series 5.3. Conclusion of the proof of Theorem 1 562 A. OKOUNKOV AND R. PANDHARIPANDE 0. Introduction 0.1. Overview. 0.1.1. 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 .
