## Elliptic Gromov-Witten invariants and the generalized mirror conjecture

Citations: | 49 - 5 self |

### BibTeX

@MISC{Givental_ellipticgromov-witten,

author = {Alexander Givental},

title = {Elliptic Gromov-Witten invariants and the generalized mirror conjecture},

year = {}

}

### OpenURL

### Abstract

A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov- Witten invariants of compact Kähler manifolds with isolated fixed points and for concave bundle spaces over such manifolds. Several results on genus 0 Gromov- Witten theory include: a non-linear Serre duality theorem, its application to the genus 0 mirror conjecture, a mirror theorem for concave bundle spaces over toric manifolds generalizing a recent result of B. Lian, K. Liu and S.-T. Yau. We also establish a correspondence (see the extensive footnote in section 4) between their new proof of the genus 0 mirror conjecture for quintic 3-folds and our proof of the same conjecture given two years ago. Research supported by NSF grants DMS-93-21915 and DMS-97-04774

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Citation Context ... q) −1/2 . 20 We reduce the group G here to the one-dimensional torus so that H ∗ G (CP 1 ) = Q[p, λ]/(p 2 − λ 2 ) where p is the equivariant Chern class of O(1). 21 The formula claimed by physicists =-=[4]-=- was first confirmed in [1] by toric methods and then rigorously justified by Yu. Manin, J. Bryan, R. Pandharipande by equivariant methods more elementary than the mirror theory. This example is also ... |

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Citation Context ... of this paper were first announced. 71 Gromov-Witten invariants and semi-simple Frobenius structures We review here some basic properties of Gromov-Witten invariants of compact symplectic manifolds =-=[17, 3, 2, 7, 18, 21, 6]-=-. Stable maps. Let (Σ, ε) denote a prestable marked curve, that is a compact connected complex curve Σ with at most double singular points and an ordered n-tuple (ε1, ..., εn) of distinct non-singular... |

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Citation Context ... of this paper were first announced. 71 Gromov-Witten invariants and semi-simple Frobenius structures We review here some basic properties of Gromov-Witten invariants of compact symplectic manifolds =-=[17, 3, 2, 7, 18, 21, 6]-=-. Stable maps. Let (Σ, ε) denote a prestable marked curve, that is a compact connected complex curve Σ with at most double singular points and an ordered n-tuple (ε1, ..., εn) of distinct non-singular... |

225 | The intrinsic normal cone
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Citation Context ... of this paper were first announced. 71 Gromov-Witten invariants and semi-simple Frobenius structures We review here some basic properties of Gromov-Witten invariants of compact symplectic manifolds =-=[17, 3, 2, 7, 18, 21, 6]-=-. Stable maps. Let (Σ, ε) denote a prestable marked curve, that is a compact connected complex curve Σ with at most double singular points and an ordered n-tuple (ε1, ..., εn) of distinct non-singular... |

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Citation Context ...structure. In this paper, we study the structure formed by rational and elliptic GW-invariants. The structure of rational GW-invariants alone is well-understood and has been formalized by B. Dubrovin =-=[6]-=- in the concept of Frobenius manifolds. The genus 0 GW-invariants define on the total cohomology space H := H ∗ (X) a Frobenius manifold structure; roughly speaking, it consists of the associative com... |

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Citation Context ...e fundamental classes [X0,n,d], being obvious in the orbifold case, can be easily extended to the general “virtual” case. A rigorous justification of these localization formulas was recently given in =-=[14]-=- on the basis of the algebraic-geometrical approach [2] to the virtual fundamental cycles. The idea of our proof of Theorem 2.1 can be now described as follows. Any fixed point of the torus G action o... |

95 | Stacks of stable maps and Gromov–Witten invariants - Manin - 1996 |

94 |
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Citation Context ...ar Serre duality theorem [9] (which relates genus 0 GW-invariants of a super-manifold and of the dual bundle space) in order to give a new proof of the mirror theorem for toric complete intersections =-=[11]-=-. The role of hyper-geometric functions is more transparent in this version of the proof. We believe that elliptic GW-invariants of toric complete intersections are expressible in terms of the rationa... |

92 | Equivariant Gromov-Witten invariants
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Citation Context ...-manifolds which are objects dual to the bundles of Section 4 and whose GW-invariants are to coincide with GW-invariants of toric complete intersections. We invoke the nonlinear Serre duality theorem =-=[9]-=- (which relates genus 0 GW-invariants of a super-manifold and of the dual bundle space) in order to give a new proof of the mirror theorem for toric complete intersections [11]. The role of hyper-geom... |

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Citation Context |

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Citation Context ... � −1 , and thus v1 ◦ ... ◦ vl = v1...vlϕ(±q). 38While the present paper was in preparation, a result equivalent to Theorem 4.2 in the case of concave bundles over projective spaces was published in =-=[19]-=-. 16 We outline below a proof of Theorem 4.2 which is completely parallel to the proof of the mirror theorem for projective and toric complete intersections given in [9] and [11] respectively and, as ... |

73 |
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Citation Context ...ation 14◦. Flatness of ∇� = �d − A1 is equivalent to A1 ∧ A1 = 0 and dA1 = 0. The first condition means commutativity [Aα, Aβ] = 0 while the second one implies that L is Lagrangian at generic points =-=[13]-=-. - The function u on L, may be multiple-valued, defined as a potential for the action 1-form ∑ pαdtα on T ∗H restricted to L. In our case of conformal ∑ frobenius structures u can be chosen as the re... |

63 | Topological field theory and rational curves
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(Show Context)
Citation Context ... group G here to the one-dimensional torus so that H ∗ G (CP 1 ) = Q[p, λ]/(p 2 − λ 2 ) where p is the equivariant Chern class of O(1). 21 The formula claimed by physicists [4] was first confirmed in =-=[1]-=- by toric methods and then rigorously justified by Yu. Manin, J. Bryan, R. Pandharipande by equivariant methods more elementary than the mirror theory. This example is also contained in [19] and [23].... |

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(Show Context)
Citation Context ... of the 1-form (∗). With this hypothesis in force, we arrive to the following Conjecture 0.1. The 1-form (*) satisfies all axioms for the genus 1 GW-invariant dG. In particular, E. Getzler’s relation =-=[8]-=- holods true for (∗). Remark on examples. We will see in Section 1 from the theory of Frobenius structures that differentials of the asymptotical coefficients Rα are expressible via the Hessians ∆β by... |

54 |
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(Show Context)
Citation Context ...olds) are coefficients of the linear differential equations satisfied by the gravitational GW-invariants in question. In fact such a relationship was the initial point of the whole project started by =-=[10, 12]-=- and completed in [9, 11]. Thus the two proofs of the same theorem appear to be variants of the same proof rather than two different ones, except that our reference to the general theory of equivarian... |

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(Show Context)
Citation Context ...th respect to the S 1 -action on CP 1 instead of GW-invariants on X — is borrowed in [19] from our paper [9], Sections 6 and 11. In fact this idea is profoundly rooted in the heuristic interpretation =-=[12]-=- of GW-invariants of X in terms of Floer cohomology theory on the loop space LX where the S 1 -action is given by rotation of loops. The generator in the cohomology algebra of BS 1 denoted � in our pa... |

17 |
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Citation Context ...ormal bundle to each fixed point component and reduces to integration over the Deligne-Mumford spaces. The idea to apply the localization technique to the moduli spaces Xg,n,d is due to M. Kontsevich =-=[16]-=- and was systematically exploited in [9, 11] and several other papers. The description used in [16, 9, 10] for localizations of the fundamental classes [X0,n,d], being obvious in the orbifold case, ca... |

9 | Mirror symmetry and elliptic curves
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(Show Context)
Citation Context .... n! and has a unique intersection with the hyperplane t0 = 0. We will use these facts in the following application of the string equation (notice that LT = 1 − (T(c) − T(0))/c). Proposition 3.1 (see =-=[6, 9, 5]-=-). s = e u/� , v = eu/x+u/y u log δ , µ = , ν = x + y 24 24 . Proof. The string equation implies Lu = 1, Ls = s v v , Lv = + � x y , Lδ = 0, Lµ = 1 , Lν = 0. 24 27At t0 = 0 the initial conditions u =... |

9 |
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Citation Context ...n theory. In singularity theory, the residue metric on Λ (which is the counterpart of the flat Poincare metric on H) has no reason to be flat. However, according to K. Saito theory of primitive forms =-=[22]-=- one can choose the holomorphic volume form v(z, λ)dz1 ∧ ... ∧ dzm (called primitive) in such a way that the corresponding residue metric is flat. Moreover, Saito’s theory can be reformulated as the t... |

8 | A rigorous definition of fiberwise quantum cohomology and equivariant Quantum cohomology - Lu |

5 | conjecture and Gromov-Witten invariants - Arnold - 1999 |