Quantum Singularity Theory for A (r-1) and r-Spin Theory
Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2781-2802.

We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the r-spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity W of type A our construction of the stack of W-curves is canonically isomorphic to the stack of r-spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an r-spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the Witten Integrable Hierarchies Conjecture for r-spin curves applies to our theory for A-type singularities; that is, the total descendant potential function of our theory for A-type singularities satisfies the corresponding Gelfand-Dikii integrable hierarchy.

Nous passons en revue notre construction d’une théorie cohomologique des champs pour les singularités quasi-homogènes et la théorie des courbes r-spin de Jarvis-Kimura-Vaintrob. De plus, nous prouvons que pour une singularité W de type A notre construction du champ algébrique des W-courbes est canoniquement isomorphe au champ algébrique des courbes r-spin décrit par Abramovich et Jarvis. En outre, nous prouvons que notre théorie satisfait tous les axiomes de Jarvis-Kimura-Vaintrob pour une classe virtuelle r-spin. Par conséquent, la preuve de Faber-Shadrin-Zvonkine de la conjecture des hiérarchies intégrables de Witten pour les courbes r-spin s’applique à notre théorie des singularités de type A. C’est-à-dire, la fonction potentielle descendante totale de notre théorie des singularités de type A satisfait la hiérarchie intégrable de Gelfand-Dikii.

DOI: 10.5802/aif.2794
Classification: 14H70,  14H10,  14H81,  14B05,  32S25,  57R56,  14N35,  53D45
Keywords: FJRW, Mirror symmetry, r-spin curve, spin curve, Witten, Cohomological field theory, moduli, Gelfand-Dikii, integrable hierarchy
Fan, Huijun 1; Jarvis, Tyler 2; Ruan, Yongbin 3

1 School of Mathematical Sciences, Peking University, Beijing 100871, China
2 Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
3 Department of Mathematics, University of Michigan Ann Arbor, MI 48105 U.S.A
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     journal = {Annales de l'Institut Fourier},
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Fan, Huijun; Jarvis, Tyler; Ruan, Yongbin. Quantum Singularity Theory for $A_{(r - 1)}$ and $r$-Spin Theory. Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2781-2802. doi : 10.5802/aif.2794. http://archive.numdam.org/articles/10.5802/aif.2794/

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