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## Identification of the Givental formula with the spectral curve topological recursion procedure

Venue: | Comm. Math. Phys |

Citations: | 13 - 5 self |

### Citations

338 |
Geometry of 2-D topological field theories, in: Integrable Systems and Quantum Groups (Montecatini Terme
- Dubrovin
- 1993
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Citation Context ...eted as a formal Frobenius manifold with metric (2-7) ηαβ = ∂3F ∂t1∂tα∂tβ and Frobenius algebra structure cγαβ (2-8) cαβγ = ∂3F ∂tα∂tβ∂tγ . We can assume that ηαβ = δα+β,n+1 and e1 = e1. According to =-=[4]-=- it is always possible by an appropriate choice of these flat coordinates tµ. 2.2.1. Canonical coordinates. Another set of coordinates is given by the canonical coordinates {ui} which can be found as ... |

189 | Invariants of algebraic curves and topological expansion - Eynard, Orantin |

138 | Gromov-Witten invariants and quantization of quadratic Hamiltonians, Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary
- Givental
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Citation Context ...of the Gromov-Witten theory of CP1. 5.1. Gromov-Witten theory of CP1. The Gromov-Witten theory of CP1 is discussed from the geometric point of view in many sources, see e. g. [27]. Givental proved in =-=[18]-=- that his formula for the formal Gromov-Witten potential coincides with the geometric Gromov-Witten potential of CP1, so we discuss it here only from the Givental point of view, ignoring the geometric... |

93 | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, preprint 2001 http://arxiv.org/abs/math.DG/0108160 - Dubrovin, Zhang |

75 | Painlevé transcendents in two-dimensional topological field theory
- Dubrovin
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Citation Context ... manifold. Let Z({td,µ}) be the partition function of some N -dimensional semi-simple conformal cohomological field theory. We recall the construction (due to Givental [18, 17, 19], see also Dubrovin =-=[5]-=-) of an operator series R(z) as in the previous section whose quantization takes the product of N KdV τ -functions to Z. Let F be the restriction of log(Z) to the genus zero part without descendents. ... |

74 | Semisimple Frobenius structures at higher genus
- Givental
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Citation Context ...bury-Scott conjecture 28 5.3. Proof of the Norbury-Scott conjecture 29 References 34 1 2 P. DUNIN-BARKOWSKI, N. ORANTIN, S. SHADRIN, AND L. SPITZ 1. Introduction 1.1. Givental theory. Givental theory =-=[18, 17, 19]-=- is one of the most important tools in the study of Gromov-Witten invariants of target varieties and general cohomological field theories that allows, in particular, to obtain explicit relations betwe... |

52 | Symplectic Geometry of Frobenius Structures
- Givental
(Show Context)
Citation Context ...bury-Scott conjecture 28 5.3. Proof of the Norbury-Scott conjecture 29 References 34 1 2 P. DUNIN-BARKOWSKI, N. ORANTIN, S. SHADRIN, AND L. SPITZ 1. Introduction 1.1. Givental theory. Givental theory =-=[18, 17, 19]-=- is one of the most important tools in the study of Gromov-Witten invariants of target varieties and general cohomological field theories that allows, in particular, to obtain explicit relations betwe... |

43 | Tautological relations and the r-spin Witten conjecture, Annales scientifiques de l’ENS 43, fascicule 4(2010)621-658; arXiv:math/0612510 - Faber, Shadrin, et al. |

38 | Algebraic methods in random matrices and enumerative geometry,” arXiv:0811.3531 [math-ph
- Eynard, Orantin
(Show Context)
Citation Context ... is, the function parametrizing the intersection indices of ψ-classes on the moduli space of curves. 1.2. Topological recursion theory. The theory developed by Eynard and the second named author (see =-=[13, 15]-=-), is a procedure, called topological recursion, that takes the following objects as input. First, a particular Riemann surface, which is usually called the spectral curve. Second, two functions x and... |

29 | Recursion between Mumford volumes of moduli spaces, arXiv: 0706.4403
- Eynard
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Citation Context ... of intersection numbers: (3-10) ωg,n(z1, . . . , zn) =( − β 2α )2g+n−2 βg+n−1 ∑ α1,...,αn≥0 〈τa1 . . . τan〉g,n n∏ i=1 (2αi + 1)!! dzi z2αi+2i . This lemma was proved many times by direct computation =-=[10, 11, 15, 34]-=-, matching the topological recursion with the recursive definition of the intersection numbers. As a side note, the first few correlation functions are (3-11) ω0.3(z1, z2, z3) = −β 3 2α 3∏ i=1 dzi z2i... |

26 |
Weil-Petersson volume of moduli spaces, Mirzakhani’s recursion and matrix models, math-ph: arXiv:0705.3600v1
- Eynard, Orantin
(Show Context)
Citation Context ... β 2αh1 )2g+n−2 βg+n−1 ∞∑ m=0 (−1)m m!∑ ~α∈N∗m m∏ k=1 (2αk+1)!! h2αk+1 h1 n∏ i=1 (2di + 1)!! dzi z2di+2i 〈 n∏ j=1 τdj m∏ k=1 ταk+1 〉 g,n+m . Proof. Once again the proof can be found in the literature =-=[10, 14, 11]-=-. However, let us study a graphical interpretation of this result when considering an arbitrary convention for the topological recursion. For f(z) an analytic function around z → 0 and {Tk}k∈Z a set o... |

25 | Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture,” arXiv:1205.1103 [math-ph
- Eynard, Orantin
(Show Context)
Citation Context ...s often used to reproduce known partition functions, extracts from it some higher genus correlators which were up to now unreachable and gives new non-trivial relations for the correlators, see e. g. =-=[16]-=-. 1.3. Goals of the paper. As we see, there is a lot of similarity in both theories (which was first noted by Alexandrov, Mironov and Morozov in [1, 2, 3]). In both cases we have to start with a small... |

23 | Invariance of tautological equations. I. Conjectures and applications
- Lee
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Citation Context ...he topological recursion. 2.1. Givental group action. We remind the reader of the original formulation, due to Y.-P. Lee, of the infinitesimal Givental group action in terms of differential operators =-=[22, 23, 24]-=-. Consider the space of partition functions for N -dimensional cohomological field theories (2-1) Z = exp (∑ g≥0 ~ g−1Fg ) in variables vd,i, d ≥ 0, i = 1, . . . , N . There is a fixed scalar product ... |

21 | Witten’s conjecture and Virasoro conjecture for genus up to two
- Lee
- 2006
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Citation Context ...he topological recursion. 2.1. Givental group action. We remind the reader of the original formulation, due to Y.-P. Lee, of the infinitesimal Givental group action in terms of differential operators =-=[22, 23, 24]-=-. Consider the space of partition functions for N -dimensional cohomological field theories (2-1) Z = exp (∑ g≥0 ~ g−1Fg ) in variables vd,i, d ≥ 0, i = 1, . . . , N . There is a fixed scalar product ... |

20 | Invariance of tautological equations. II. Gromov-Witten theory
- Lee
(Show Context)
Citation Context ...he topological recursion. 2.1. Givental group action. We remind the reader of the original formulation, due to Y.-P. Lee, of the infinitesimal Givental group action in terms of differential operators =-=[22, 23, 24]-=-. Consider the space of partition functions for N -dimensional cohomological field theories (2-1) Z = exp (∑ g≥0 ~ g−1Fg ) in variables vd,i, d ≥ 0, i = 1, . . . , N . There is a fixed scalar product ... |

12 |
Intersection numbers of spectral curves
- Eynard
- 2011
(Show Context)
Citation Context ... of Givental’s formula for the formal Gromov-Witten potential, and in the case of topological recursion it is recovered locally in an expansion near a simple critical point of the spectral curve, see =-=[11]-=-). Moreover, in both cases we have an expansion of the correlators in terms of Feynman graphs, see [8] on the Givental side and [12, 16, 21] on the spectral curve side. So, the natural question is whe... |

11 | Invariants of spectral curves and intersection theory of moduli spaces of complex curves,
- Eynard
- 2014
(Show Context)
Citation Context ...n expansion near a simple critical point of the spectral curve, see [11]). Moreover, in both cases we have an expansion of the correlators in terms of Feynman graphs, see [8] on the Givental side and =-=[12, 16, 21]-=- on the spectral curve side. So, the natural question is whether we can precisely identify both theories in some setup. On the Givental side we restrict ourselves to a part of the Givental formula, na... |

10 |
Morozov ”M-Theory of Matrix Models
- Alexandrov, Mironov, et al.
(Show Context)
Citation Context ...as the original motivation for introducing the topological recursion; it is a natural generalization of the reconstruction procedure for the correlators of a certain class of matrix models, see, e.g. =-=[2]-=-), in some other cases they appear to be related to Gromov-Witten theory and to various intersection numbers on the moduli space of curves. 1The formula as appears here is missing one term correspondi... |

8 | The spectral curve of the EynardOrantin recursion via the Laplace transform,” arXiv:1202.1159v1 [math.AG
- Dumitrescu, Mulase, et al.
(Show Context)
Citation Context ...t of a partition function determines the two-point function, and the rest of the correlators can be reconstructed from these two via topological recursion, in terms of a proper expansion of ωg,n (see =-=[7]-=-). The topological recursion theory is often used to reproduce known partition functions, extracts from it some higher genus correlators which were up to now unreachable and gives new non-trivial rela... |

5 |
Deformations of cohomological field theories, preprint
- Kazarian
- 2007
(Show Context)
Citation Context ...l-defined. The main theorem of [9] states that this action preserves the property that Z is a generating function of the correlators of a cohomological field theory with target space (V, η) (see also =-=[20, 33]-=-). Remark 2.1. Note that this definition of R̂ differs from the one in [8] by the sign (−1)l. It is needed here to agree with Givental’s notation in Proposition 2.3, cf. [18, Proposition 7.3]. For the... |

4 | Givental graphs and inversion symmetry
- Dunin-Barkowski, Shadrin, et al.
(Show Context)
Citation Context ... it is recovered locally in an expansion near a simple critical point of the spectral curve, see [11]). Moreover, in both cases we have an expansion of the correlators in terms of Feynman graphs, see =-=[8]-=- on the Givental side and [12, 16, 21] on the spectral curve side. So, the natural question is whether we can precisely identify both theories in some setup. On the Givental side we restrict ourselves... |

3 |
A.Morozov, “Solving Virasoro Constraints
- Alexandrov
(Show Context)
Citation Context ...n-trivial relations for the correlators, see e. g. [16]. 1.3. Goals of the paper. As we see, there is a lot of similarity in both theories (which was first noted by Alexandrov, Mironov and Morozov in =-=[1, 2, 3]-=-). In both cases we have to start with a small amount of data fixed in genus zero, and in both cases the intersection indices of ψclasses on the moduli space of curves are a kind of structure constant... |