Abstract
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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