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arXiv · 1012.0066

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

Huijun Fan·Tyler J. Jarvis·Yongbin Ruan
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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.

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Huijun Fan, Tyler J. Jarvis, Yongbin Ruan. 2010-12-01. Quantum Singularity Theory for A_{r-1} and r-Spin Theory. https://arxiv.org/abs/1012.0066

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