Double ramification cycles on the moduli spaces of curves

Janda, Felix; Pandharipande, Rahul; Pixton, Aaron; Zvonkine, Dimitri

doi:10.48550/arxiv.1602.04705

Cited by 15 publications

(27 citation statements)

References 20 publications

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“…Proof. The proof here follows closely Pixton's proof of polynomiality in [17,Appendix]. We will use [17, Proposition A1], but with Pixton's r replaced with r − 1 (which we assume to be large enough).…”

Section: Proof Of Theoremmentioning

confidence: 91%

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Tautological relations via 𝑟-spin structures

Pandharipande¹,

Pixton²,

Zvonkine³

2019

J. Algebraic Geom.

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Relations among tautological classes on M g,n are obtained via the study of Witten's r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl 2 (C) is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten's r-spin class is obtained (along with tautological relations in higher degrees). As an application, the r = 4 relations are used to bound the Betti numbers of R * (M g ). At the second semisimple point, the form of the R-matrix implies a polynomiality property in r of Witten's r-spin class.In the Appendix (with F. Janda), a conjecture relating the r = 0 limit of Witten's r-spin class to the class of the moduli space of holomorphic differentials is presented. 0 Introduction 0.1 Overview Let M g,n be the moduli space of stable genus g curves with n markings. Letbe the subring of tautological classes in cohomology 1 . The subringsare defined together as the smallest system of Q-subalgebras closed under push-forward via all boundary and forgetful maps, see [6,7,12]. There has been substantial progress in the understanding of RH * (M g,n ) since the study began in the 1980s [21]. The subject took a new turn in 2012 with the family of relations conjectured in [25]. We refer the reader to [23] for a survey of recent developments.Witten's r-spin class defines a Cohomological Field Theory (CohFT) for each integer r ≥ 2. Witten's 2-spin theory concerns only the fundamental classes of the moduli spaces of curves (and leads to no new geometry). In our previous paper [24], we used Witten's 3-spin theory to construct a family of relations among tautological classes of M g,n equivalent (in cohomology) to the relations proposed in [25]. Our goal here is to extend our study of tautological relations to Witten's r-spin theory for all r ≥ 3.Taking [24] as a starting point, Janda has completed a formal study of tautological relations obtained from CohFTs. Two results of Janda are directly relevant here:(i) The relations for r = 3 are valid in Chow [14,16].(ii) The relations for r ≥ 4 are implied by the relations for r = 3 [14,15]. By (i) and (ii) together, all of the r-spin relations that we find will be valid in Chow. However, since our methods here are cohomological, we will use the language of cohomology throughout the paper.

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Section: Proof Of Theoremmentioning

confidence: 91%

“…Proof. Once again we follow Pixton's proof in [17,Appendix]. Let Q be a polynomial in N variables with (p-integral) Q-coefficients.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Tautological relations via 𝑟-spin structures

Pandharipande¹,

Pixton²,

Zvonkine³

2019

J. Algebraic Geom.

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“…. , a n ) in terms of stable graphs, obtained by expanding the exponential in the expression above, is given in [18], Corollary 4.…”

Section: Chiodo Classesmentioning

confidence: 99%

Chiodo formulas for the r-th roots and topological recursion

et al. 2016

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We analyze Chiodo's formulas for the Chern classes related to the r -th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of these classes with ψ-classes are reproduced via the Chekhov-Eynard-Orantin topological recursion. As an application, we prove that the Johnson-Pandharipande-Tseng formula for the orbifold Hurwitz numbers is equivalent to the topological recursion for the orbifold Hurwitz numbers. In particular, this gives a new proof of the topological recursion for the orbifold Hurwitz numbers.

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“…A stable curve is of compact type if and only if its Jacobian is an abelian variety, and the Abel-Jacobi map s A naturally extends to the moduli space of curves of compact type M ct g,n ⊂ M g,n (see formula (14) in [GZ14A] or Section 0.2.3 in [JPPZ16]). To define this extension, let (C, p 1 , .…”

Section: The θ-Relations and Pixton's Double Ramification Cycle Relat...mentioning

confidence: 99%

Powers of the Theta Divisor and Relations in the Tautological Ring

Clader

Grushevsky

Janda

et al. 2017

International Mathematics Research Notices

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Abstract. We show that the vanishing of the (g + 1)-st power of the theta divisor on the universal abelian variety X g implies, by pulling back along a collection of Abel-Jacobi maps, the vanishing results in the tautological ring of M g,n of Looijenga, Ionel, Graber-Vakil, and Faber-Pandharipande. We also show that Pixton's double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem ⋆ of Graber and Vakil, and we provide an explicit algorithm for expressing any tautological class on M g,n of sufficiently high codimension as a boundary class.

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Double ramification cycles on the moduli spaces of curves

Cited by 15 publications

References 20 publications

Tautological relations via 𝑟-spin structures

Tautological relations via 𝑟-spin structures

Chiodo formulas for the r-th roots and topological recursion

Powers of the Theta Divisor and Relations in the Tautological Ring

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