Quantum Singularity Theory for A_r-1 and r-Spin Theory

2010/12/01 by Fan, Huijun, Jarvis, Tyler J., Ruan, Yongbin
#35Q53 #37K10 #37K20 #53D45 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Primary: 14N35 #Secondary: 32S05

paper · doi:10.48550/arxiv.1012.0066

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.

Related

• The Witten equation, mirror symmetry and quantum singularity theory
• Tautological relations via r-spin structures
• Algebraic virtual cycles for quantum singularity theories
• The Spin Gromov-Witten/Hurwitz correspondence for ℙ¹
• Singular vector structure of quantum curves