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.
DOI: https://doi.org/10.1090/jag/736
Published electronically: March 29, 2019
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Abstract | References | Additional Information

Abstract: 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
Email: rahul@math.ethz.ch

A. Pixton
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: apixton@mit.edu

D. Zvonkine
Affiliation: CNRS, Institut Mathématique de Jussieu, 75013 Paris, France
Email: dimitri.zvonkine@ump-prg.fr

DOI: https://doi.org/10.1090/jag/736
Received by editor(s): October 11, 2016
Received by editor(s) in revised form: September 1, 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.