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

TL;DR

This paper reviews quantum singularity and r-spin theories, demonstrating their equivalence for the A_{r-1} singularity and confirming the Witten Integrable Hierarchies Conjecture for this case.

Contribution

It proves the equivalence of A_{r-1}-theory and r-spin theory, and verifies the Witten Hierarchies Conjecture for A_{r-1}.

Findings

A_{r-1}-theory is isomorphic to r-spin theory.

A_{r-1}-theory satisfies all axioms for an r-spin virtual class.

The total descendant potential satisfies the r-th Gelfand-Dikii hierarchy.

Abstract

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.