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## Gromov-Witten classes, quantum cohomology, and enumerative geometry (1994)

Venue: | Commun. Math. Phys |

Citations: | 475 - 3 self |

### Citations

539 | KodairaSpencer theory of gravity and exact results for quantum string amplitudes
- Bershadsky, Cecotti, et al.
- 1994
(Show Context)
Citation Context ...rm and remains invariant when one deforms the complex structure of V. In the infinite volume limit A-model should approximate the GW-model from 6.3, taking its existence for granted. One expects (see =-=[BCOV]-=-) that the difference between any correlators in A– and GW–models is a cohomology class of a moduli space Poincaré dual to a homology class supported on the boundary. B–model should depend only on the... |

521 | Homological algebra of mirror symmetry - Kontsevich - 1995 |

489 |
Two-dimensional gravity and intersection theory on moduli space
- Witten
- 1991
(Show Context)
Citation Context ... in [Y], [Ko], [Ma2]) which identifies Φ as a specific combination of hypergeometric functions. In this paper, we use a different tool, the so called “associativity” relations, or WDVV–equations (see =-=[W]-=-, [D]), in order to show that for Fano manifolds these equations tend to be so strong that they can define Φ uniquely up to a choice of a finite number of constants. (For Calabi–Yau varieties these eq... |

354 |
Koszul duality for operads
- Ginzburg, Kapranov
- 1994
(Show Context)
Citation Context ... their dependence on the geometry of V. In the remaining part of this section, we introduce an operadic firmalism for description of tree level CohFT’s. Our framework is similar to that of [V], [BG], =-=[GiK]-=-. 6.6. Trees. We will formally introduce trees describing combinatorial structure of a marked stable curve of arithmetical genus 0. Their vertices correspond to components, and edges to special points... |

285 |
A mathematical theory of quantum cohomology
- Ruan, Tian
- 1995
(Show Context)
Citation Context ... an elaboration of Witten’s treatment [W]. Unfortunately, the geometric construction of these classes to our knowledge has2 by the techniques of symplectic geometry going back to M. Gromov (see [R], =-=[RT]-=-), but they fall short of the complete picture. In 2.4 we sketch an algebro–geometric approach to this problem based upon a new notion of stable map due to one of us (M. K.) The axiomatic treatment of... |

202 |
Intersection theory of moduli space of stable n-pointed curves of genus zero
- Keel
(Show Context)
Citation Context ...ed upon the axiomatics of Gromov–Witten classes, another is based upon (a version of) operads. This formalism is used in §7 for a description of the cohomology of moduli spaces of genus zero. Keel in =-=[Ke]-=- described its ring structure in terms of generators, the classes of boundary divisors, and relations between them. We need more detailed understanding of linear relations between homology classes of ... |

120 |
Cubic forms: algebra, geometry, arithmetic Translated from Russian by M
- Manin
- 1974
(Show Context)
Citation Context ...enerated by its indecomposable elements Λ for r = 0, Λ − l1 and l1 for r = 1, and all exceptional classes for r ≥ 2. (Recall that l is exceptional iff (l2) = −1 and (−KV .β) = 1; for more details see =-=[Ma1]-=-). This allows us to rewrite (5.6) as an explicit sum over B. Writing γ = xe0 + yΛ + ze4 − ∑r check the following generalization of 5.2.1: i=1 yi li, Φ Vr (γ) = 1 6 (γ3 ) + ϕ(γ), we can easily29 5.2.... |

98 |
Integrable systems in topological field theory, Nuclear Phys
- Dubrovin
- 1992
(Show Context)
Citation Context ...Y], [Ko], [Ma2]) which identifies Φ as a specific combination of hypergeometric functions. In this paper, we use a different tool, the so called “associativity” relations, or WDVV–equations (see [W], =-=[D]-=-), in order to show that for Fano manifolds these equations tend to be so strong that they can define Φ uniquely up to a choice of a finite number of constants. (For Calabi–Yau varieties these equatio... |

80 | Topological field theory and rational curves
- Aspinwall, Morrison
- 1993
(Show Context)
Citation Context ...rational curves in the class dβ0, then the number of primitively parametrized curves with three marked points landing in β and incident to three fixed cycles dual to ∆1 must be d 3 N(d). According to =-=[AM]-=-, the parametrizations of degree m with three marked points must be counted with multiplicity m −3 . Hence we expect that Ñ(dβ0) = ∑ k/d N(k) k3 d 3 (5.12) which can also be taken as a formal definiti... |

55 |
Topological sigma model and Donaldson type invariants in Gromov theory
- Ruan
- 1996
(Show Context)
Citation Context ...is is an elaboration of Witten’s treatment [W]. Unfortunately, the geometric construction of these classes to our knowledge has2 by the techniques of symplectic geometry going back to M. Gromov (see =-=[R]-=-, [RT]), but they fall short of the complete picture. In 2.4 we sketch an algebro–geometric approach to this problem based upon a new notion of stable map due to one of us (M. K.) The axiomatic treatm... |

45 | Cyclic operads and cyclic homology - Getzler, Kapranov - 1995 |

30 |
Infinitesimal structure of moduli spaces of G–bundles, Int
- Beilinson, Ginzburg
- 1992
(Show Context)
Citation Context ...tizing their dependence on the geometry of V. In the remaining part of this section, we introduce an operadic firmalism for description of tree level CohFT’s. Our framework is similar to that of [V], =-=[BG]-=-, [GiK]. 6.6. Trees. We will formally introduce trees describing combinatorial structure of a marked stable curve of arithmetical genus 0. Their vertices correspond to components, and edges to special... |

13 | Topological field theories, string backgrounds and homotopy
- Voronov
- 1994
(Show Context)
Citation Context ...xiomatizing their dependence on the geometry of V. In the remaining part of this section, we introduce an operadic firmalism for description of tree level CohFT’s. Our framework is similar to that of =-=[V]-=-, [BG], [GiK]. 6.6. Trees. We will formally introduce trees describing combinatorial structure of a marked stable curve of arithmetical genus 0. Their vertices correspond to components, and edges to s... |

3 |
Problems on rational points and rational curves on algebraic varieties
- Manin
- 1995
(Show Context)
Citation Context ...y defined. In particular, when V is a Calabi–Yau manifold, Φ conjecturally describes a variation of Hodge structure of the mirror dual manifold in special coordinates (see contributions in [Y], [Ko], =-=[Ma2]-=-) which identifies Φ as a specific combination of hypergeometric functions. In this paper, we use a different tool, the so called “associativity” relations, or WDVV–equations (see [W], [D]), in order ... |

1 |
Counting rational curves on rational surfaces. Preprint Saclay T94/001
- Itzykson
(Show Context)
Citation Context ...at least for larger values of r? 5.2.4. Quadric. The quadric V = P 1 × P 1 is the last del Pezzo surface. Here Pic(V ) = Z 2 , −KV = (2, 2), and all associativity relations were written explicitly in =-=[I]-=-. In self-explanatory notation ϕ(γ) = ∑ a+b≥1 z N(a, b) 2a+2b−1 (2a + 2b − 1)! eay1 +by 2 , and the associativity relations together with initial conditions N(0, 1) = N(1, 0) = 1 imply the following r... |

1 |
E–mail addresses: maxim@ mpim–bonn.mpg.de, manin@ mpim–bonn.mpg.de
- Yau, ed
- 1992
(Show Context)
Citation Context ...mes uniquely defined. In particular, when V is a Calabi–Yau manifold, Φ conjecturally describes a variation of Hodge structure of the mirror dual manifold in special coordinates (see contributions in =-=[Y]-=-, [Ko], [Ma2]) which identifies Φ as a specific combination of hypergeometric functions. In this paper, we use a different tool, the so called “associativity” relations, or WDVV–equations (see [W], [D... |