Quantum Singularity Theory for $A_{(r-1)}$ and $r$-Spin Theory

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.

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.

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