# Quantum Singularity Theory for ${A}_{(r-1)}$ and $r$-Spin Theory

Huijun Fan^{[1]}; Tyler Jarvis^{[2]}; Yongbin Ruan^{[3]}

- [1] School of Mathematical Sciences, Peking University, Beijing 100871, China
- [2] Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
- [3] Department of Mathematics, University of Michigan Ann Arbor, MI 48105 U.S.A

Annales de l’institut Fourier (2011)

- Volume: 61, Issue: 7, page 2781-2802
- ISSN: 0373-0956

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topFan, Huijun, Jarvis, Tyler, and Ruan, Yongbin. "Quantum Singularity Theory for $A_{(r - 1)}$ and $r$-Spin Theory." Annales de l’institut Fourier 61.7 (2011): 2781-2802. <http://eudml.org/doc/275544>.

@article{Fan2011,

abstract = {We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the $r$-spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity $W$ of type $A$ our construction of the stack of $W$-curves is canonically isomorphic to the stack of $r$-spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an $r$-spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the Witten Integrable Hierarchies Conjecture for $r$-spin curves applies to our theory for $A$-type singularities; that is, the total descendant potential function of our theory for $A$-type singularities satisfies the corresponding Gelfand-Dikii integrable hierarchy.},

affiliation = {School of Mathematical Sciences, Peking University, Beijing 100871, China; Department of Mathematics, Brigham Young University, Provo, UT 84602, USA; Department of Mathematics, University of Michigan Ann Arbor, MI 48105 U.S.A},

author = {Fan, Huijun, Jarvis, Tyler, Ruan, Yongbin},

journal = {Annales de l’institut Fourier},

keywords = {FJRW; Mirror symmetry; $r$-spin curve; spin curve; Witten; Cohomological field theory; moduli; Gelfand-Dikii; integrable hierarchy; mirror symmetry; -spin curve; cohomological field theory},

language = {eng},

number = {7},

pages = {2781-2802},

publisher = {Association des Annales de l’institut Fourier},

title = {Quantum Singularity Theory for $A_\{(r - 1)\}$ and $r$-Spin Theory},

url = {http://eudml.org/doc/275544},

volume = {61},

year = {2011},

}

TY - JOUR

AU - Fan, Huijun

AU - Jarvis, Tyler

AU - Ruan, Yongbin

TI - Quantum Singularity Theory for $A_{(r - 1)}$ and $r$-Spin Theory

JO - Annales de l’institut Fourier

PY - 2011

PB - Association des Annales de l’institut Fourier

VL - 61

IS - 7

SP - 2781

EP - 2802

AB - We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the $r$-spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity $W$ of type $A$ our construction of the stack of $W$-curves is canonically isomorphic to the stack of $r$-spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an $r$-spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the Witten Integrable Hierarchies Conjecture for $r$-spin curves applies to our theory for $A$-type singularities; that is, the total descendant potential function of our theory for $A$-type singularities satisfies the corresponding Gelfand-Dikii integrable hierarchy.

LA - eng

KW - FJRW; Mirror symmetry; $r$-spin curve; spin curve; Witten; Cohomological field theory; moduli; Gelfand-Dikii; integrable hierarchy; mirror symmetry; -spin curve; cohomological field theory

UR - http://eudml.org/doc/275544

ER -

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