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...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 ...

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...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...

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... 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. ...

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...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...

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...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...

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... 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...

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... 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...

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... 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...

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...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...

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...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 ...

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...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 ...

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...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 ...

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... 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...

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...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...

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...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...

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...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...

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...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...

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... 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...

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...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...