We give a review of the quantum singularity theory of Fan-Jarvis-Ruan and the r-spin theory of Jarvis-Kimura-Vaintrob and describe the work of Abramovich-Jarvis showing that for the singularity A_{ r-1} = x^r the stack of A_{ r-1}-curves of is canonically isomorphic to the stack of r-spin curves. We prove that the A_{ r-1}-theory satisfies all the axioms of Jarvis-Kimura-Vaintrob for an r-spin virtual class. Therefore, the results of Lee, Faber-Shadrin-Zovonkine, and Givental all apply to the A_{ r-1}-theory. In particular, this shows that the Witten Integrable Hierarchies Conjecture is true for the A_{ r-1}-theory; that is, the total descendant potential function of the A_{ r-1}-theory satisfies the r-th Gelfand-Dikii hierarchy.
Categories: math.AG hep-th math.DG