Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Tautological relations via $r$-spin structures

Authors: R. Pandharipande, A. Pixton and D. Zvonkine
Journal: J. Algebraic Geom. 28 (2019), 439-496
Published electronically: March 29, 2019
MathSciNet review: 3959068
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Abstract | References | Additional Information


Relations among tautological classes on $\overline {\mathcal {M}}_{g,n}$ 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 ${\mathsf {sl}}_2(\mathbb {C})$ is proven. The Givental-Teleman classification of CohFT (cohomological field theory) 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 $R^*(\mathcal {M}_g)$. At the second semisimple point, the form of the $R$-matrix implies a polynomiality property in $r$ of Witten’s $r$-spin class.

In Appendix A (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.

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Additional Information

R. Pandharipande
Affiliation: Departement Mathematik, ETH Zürich, Ramistrasse 101, 8092 Zurich, Switzerland
MR Author ID: 357813

A. Pixton
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
MR Author ID: 818035

D. Zvonkine
Affiliation: CNRS, Institut Mathématique de Jussieu, 75013 Paris, France
MR Author ID: 621483

Received by editor(s): October 11, 2016
Received by editor(s) in revised form: September 20, 2018
Published electronically: March 29, 2019
Additional Notes: The first author was partially supported by SNF-200021143274, SNF-200020162928, ERC-2012-AdG-320368-MCSK, ERC-2017-AdG-786580-MACI, SwissMAP, and the Einstein Stiftung. The second author was supported by a fellowship from the Clay Mathematics Institute. The third author was supported by the grants ANR-09-JCJC-0104-01 and ANR-18-CE40-0009 ENUMGEOM. This project received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 786580).
Article copyright: © Copyright 2019 University Press, Inc.