  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.
