Stable maps to rational curves and the relative Jacobian

Marcus, Steffen; Wise, Jonathan

doi:10.48550/arxiv.1310.5981

Cited by 16 publications

(22 citation statements)

References 7 publications

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“…The rubber moduli space carries a natural virtual fundamental class M There has been substantial work on double ramification cycles since questions motivated by Symplectic Field Theory were posed by Eliashberg in 2001. Examples of early results can be found in [7,9,13,17,23,24,33]. A complete formula was conjectured by Pixton in 2014 and proven in [28].…”

Section: Double Ramification Cyclesmentioning

confidence: 93%

Double ramification cycles with target varieties

Janda,

Pandharipande,

Pixton

et al. 2018

Preprint

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Let X be a nonsingular projective algebraic variety over C, and let M g,n,β (X) be the moduli space of stable maps f : (C, x 1 , . . . , x n ) → X from genus g, n-pointed curves C to X of degree β. Let S be a line bundle on X. Let A = (a 1 , . . . , a n ) be a vector of integers which satisfyConsider the following condition: the line bundle f * S has a meromorphic section with zeroes and poles exactly at the marked points x i with orders prescribed by the integers a i . In other words, we require f * S (− n i=1 a i x i ) to be the trivial line bundle on C. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over X and is denoted by M ∼ g,A,β (X, S). The moduli space carries a virtual fundamental classin Gromov-Witten theory. The main result of the paper is an explicit formula (in tautological classes) for the push-forward via the forgetful morphism of [M ∼ g,A,β (X, S)] vir to M g,n,β (X). In case X is a point, the result here specializes to Pixton's formula for the double ramification cycle proven in [28]. Several applications of the new formula are given.

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Section: Double Ramification Cyclesmentioning

confidence: 93%

Double ramification cycles with target varieties

Janda,

Pandharipande,

Pixton

et al. 2018

Preprint

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“…, a n homogeneous of degree 2g. It follows from Hain's formula [25], the result of [37] and the fact that λ g vanishes on M g,n \ M ct g,n , where M ct g,n is the moduli space of stable curves of compact type. Thus, the integral (2.4) can be written as a polynomial…”

Section: 2mentioning

confidence: 99%

“…, α n that satisfy equation (3.15). Hain's formula [25] together with the result of [37] implies that…”

Section: 2mentioning

confidence: 99%

Towards a description of the double ramification hierarchy for Witten's r-spin class

Buryak

Guéré

2016

Journal de Mathématiques Pures et Appliquées

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Abstract. The double ramification hierarchy is a new integrable hierarchy of hamiltonian PDEs introduced recently by the first author. It is associated to an arbitrary given cohomological field theory. In this paper we study the double ramification hierarchy associated to the cohomological field theory formed by Witten's r-spin classes. Using the formula for the product of the top Chern class of the Hodge bundle with Witten's class, found by the second author, we present an effective method for a computation of the double ramification hierarchy. We do explicit computations for r = 3, 4, 5 and prove that the double ramification hierarchy is Miura equivalent to the corresponding Dubrovin-Zhang hierarchy. As an application, this result together with a recent work of the first author with Paolo Rossi gives a quantization of the r-th Gelfand-Dickey hierarchy for r = 3, 4, 5.

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“…, a n of degree 2g with the coefficients in H 2g (M ct g,n ). This follows from Hain's formula [Hai13] for the version of the DR cycle defined using the universal Jacobian over M ct g,n and the result of the paper [MW13], where it is proved that the two versions of the DR cycle coincide on M ct g,n (the polynomiality of the DR cycle on M g,n is proved in [JPPZ17]). The polynomiality of the DR cycle on M ct g,n together with the fact that λ g vanishes on M g,n \ M ct g,n (see e.g.…”

Section: Construction Of the Double Ramification Hierarchy Denote Bymentioning

confidence: 88%

Towards a bihamiltonian structure for the double ramification hierarchy

Buryak,

Rossi,

Shadrin

2020

Preprint

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We propose a remarkably simple and explicit conjectural formula for a bihamiltonian structure of the double ramification hierarchy corresponding to an arbitrary homogeneous cohomological field theory. Various checks are presented to support the conjecture. ContentsIntroduction 1 Organization of the paper 3 Acknowledgements 3 1. Double ramification hierarchy and the main conjecture 3 1.1. Cohomological field theories 3 1.2. Double ramification hierarchy 5 1.3. Bihamiltonian structure for the double ramification hierarchy 9 2. Evidence I: checks for a general CohFT 10 2.1. Genus 0 10 2.2. The first equation of the bihamiltonian recursion 12 2.3. The Schouten-Nijenhuis bracket of two skew-symmetric operators 12 3. Evidence II: central invariants 15 3.1. Miura transformations 15 3.2. The central invariants of a pair of compatible Poisson operators 16 3.3. The central invariants of the pair (K 1 , K 2 ) 17 3.4. Relation with the Dubrovin-Zhang hierarchy 18 4. Evidence III: explicit examples 18 4.1. The trivial cohomological field theory 18 4.2. Higher r-spin theories 19 4.3. The Gromov-Witten theory of CP 1 24 References 26

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Stable maps to rational curves and the relative Jacobian

Cited by 16 publications

References 7 publications

Double ramification cycles with target varieties

Double ramification cycles with target varieties

Towards a description of the double ramification hierarchy for Witten's r-spin class

Towards a bihamiltonian structure for the double ramification hierarchy

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