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

Abstract

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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.

How to cite

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Fan, 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 -

References

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  4. C. Faber, S. Shadrin, D. Zvonkine, Tautological relations and the r -spin Witten conjecture, Annales Scientifiques de l’École Normal Supérieure. Quatrième Série 43 (2010), 621-658 Zbl1203.53090MR2722511
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  8. T. Jarvis, T. Kimura, A. Vaintrob, Moduli Spaces of Higher Spin Curves and Integrable Hierarchies, Compositio Mathematica 126 (2) (2001), 157-212 Zbl1015.14028MR1827643
  9. Y.-P. Lee, Witten’s conjecture and the Virasoro conjecture for genus up to two, Gromov-Witten theory of spin curves and orbifolds 403 (2006), 31-42, Amer. Math. Soc., Providence, RI Zbl1114.14034MR2234883
  10. A. Polishchuk, Witten’s top Chern class on the moduli space of higher spin curves, Frobenius manifolds (2004), 253-264, Vieweg, Wiesbaden Zbl1105.14010MR2115773
  11. A. Polishchuk, A. Vaintrob, Algebraic construction of Witten’s top Chern class, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) 276 (2001), 229-249, Amer. Math. Soc., Providence, RI Zbl1051.14007MR1837120
  12. C. T. C. Wall, A note on symmetry of singularities, Bull. London Math. Soc. 12 (1980), 169-175 Zbl0427.32010MR572095
  13. C. T. C. Wall, A second note on symmetry of singularities, Bull. London Math. Soc. 12 (1980), 347-354 Zbl0424.58006MR587705
  14. E. Witten, Two-dimensional gravity and intersection theory on the moduli space, Surveys in Diff. Geom. 1 (1991), 243-310 Zbl0757.53049MR1144529
  15. E. Witten, Algebraic geometry associated with matrix models of two-dimensional gravity, Topological models in modern mathematics (Stony Brook, NY, 1991) (1993), 235-249, Publish or Perish, Houston, TX Zbl0812.14017MR1215968

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