Quantum Singularity Theory for A_{r-1} and r-Spin Theory
Algebraic Geometry
2010-12-02 v1 High Energy Physics - Theory
Differential Geometry
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.
Cite
@article{arxiv.1012.0066,
title = {Quantum Singularity Theory for A_{r-1} and r-Spin Theory},
author = {Huijun Fan and Tyler J. Jarvis and Yongbin Ruan},
journal= {arXiv preprint arXiv:1012.0066},
year = {2010}
}