Quantum Singularity Theory for A (r-1) and r-Spin Theory
Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2781-2802.

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.

Nous passons en revue notre construction d’une théorie cohomologique des champs pour les singularités quasi-homogènes et la théorie des courbes r-spin de Jarvis-Kimura-Vaintrob. De plus, nous prouvons que pour une singularité W de type A notre construction du champ algébrique des W-courbes est canoniquement isomorphe au champ algébrique des courbes r-spin décrit par Abramovich et Jarvis. En outre, nous prouvons que notre théorie satisfait tous les axiomes de Jarvis-Kimura-Vaintrob pour une classe virtuelle r-spin. Par conséquent, la preuve de Faber-Shadrin-Zvonkine de la conjecture des hiérarchies intégrables de Witten pour les courbes r-spin s’applique à notre théorie des singularités de type A. C’est-à-dire, la fonction potentielle descendante totale de notre théorie des singularités de type A satisfait la hiérarchie intégrable de Gelfand-Dikii.

DOI: 10.5802/aif.2794
Classification: 14H70,  14H10,  14H81,  14B05,  32S25,  57R56,  14N35,  53D45
Keywords: FJRW, Mirror symmetry, r-spin curve, spin curve, Witten, Cohomological field theory, moduli, Gelfand-Dikii, integrable hierarchy
Fan, Huijun 1; Jarvis, Tyler 2; Ruan, Yongbin 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
@article{AIF_2011__61_7_2781_0,
     author = {Fan, Huijun and Jarvis, Tyler and Ruan, Yongbin},
     title = {Quantum {Singularity} {Theory} for $A_{(r - 1)}$ and $r${-Spin} {Theory}},
     journal = {Annales de l'Institut Fourier},
     pages = {2781--2802},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {7},
     year = {2011},
     doi = {10.5802/aif.2794},
     mrnumber = {3112508},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2794/}
}
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
DA  - 2011///
SP  - 2781
EP  - 2802
VL  - 61
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2794/
UR  - https://www.ams.org/mathscinet-getitem?mr=3112508
UR  - https://doi.org/10.5802/aif.2794
DO  - 10.5802/aif.2794
LA  - en
ID  - AIF_2011__61_7_2781_0
ER  - 
%0 Journal Article
%A Fan, Huijun
%A Jarvis, Tyler
%A Ruan, Yongbin
%T Quantum Singularity Theory for $A_{(r - 1)}$ and $r$-Spin Theory
%J Annales de l'Institut Fourier
%D 2011
%P 2781-2802
%V 61
%N 7
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2794
%R 10.5802/aif.2794
%G en
%F AIF_2011__61_7_2781_0
Fan, Huijun; Jarvis, Tyler; Ruan, Yongbin. Quantum Singularity Theory for $A_{(r - 1)}$ and $r$-Spin Theory. Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2781-2802. doi : 10.5802/aif.2794. http://archive.numdam.org/articles/10.5802/aif.2794/

[1] Abramovich, D.; Jarvis, T. Moduli of twisted spin curves, Proc. of the Amer. Math. Soc., Volume 131 (2003) no. 3, pp. 685-699 | MR | Zbl

[2] Chen, W.; Ruan, Y. A new cohomology theory for orbifold, Comm. Math. Phys., Volume 248 (2004) no. 1, pp. 1-31 | MR | Zbl

[3] Chiodo, A. The Witten top Chern class via K-theory, J. Alg. Geom., Volume 15 (2006) no. 4, pp. 691-707 | MR | Zbl

[4] Faber, C.; Shadrin, S.; Zvonkine, D. Tautological relations and the r-spin Witten conjecture, Annales Scientifiques de l’École Normal Supérieure. Quatrième Série, Volume 43 (2010) no. 4, pp. 621-658 | Numdam | MR | Zbl

[5] Fan, H.; Jarvis, T.; Ruan, Y. The Witten equation, mirror symmetry and quantum singularity theory (Preprint. http://arxiv.org/abs/0712.4021)

[6] Givental, A. A n-1 singularities and nKdV hierarchies, Mosc. Math. J., Volume 3 (2003) no. 2, p. 475-505, 743 | MR | Zbl

[7] Jarvis, T. Geometry of the moduli of higher spin curves, International Journal of Mathematics, Volume 11(5) (2001), pp. 637-663 | MR | Zbl

[8] Jarvis, T.; Kimura, T.; Vaintrob, A. Moduli Spaces of Higher Spin Curves and Integrable Hierarchies, Compositio Mathematica, Volume 126 (2) (2001), pp. 157-212 | MR | Zbl

[9] Lee, Y.-P. Witten’s conjecture and the Virasoro conjecture for genus up to two, Gromov-Witten theory of spin curves and orbifolds (Contemp. Math.), Volume 403, Amer. Math. Soc., Providence, RI, 2006, pp. 31-42 | MR | Zbl

[10] Polishchuk, A. Witten’s top Chern class on the moduli space of higher spin curves, Frobenius manifolds (Aspects Math., E36), Vieweg, Wiesbaden, 2004, pp. 253-264 | MR | Zbl

[11] Polishchuk, A.; Vaintrob, A. Algebraic construction of Witten’s top Chern class, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (Contemp. Math.), Volume 276, Amer. Math. Soc., Providence, RI, 2001, pp. 229-249 | MR | Zbl

[12] Wall, C. T. C. A note on symmetry of singularities, Bull. London Math. Soc., Volume 12 (1980) no. 3, pp. 169-175 | MR | Zbl

[13] Wall, C. T. C. A second note on symmetry of singularities, Bull. London Math. Soc., Volume 12 (1980) no. 5, pp. 347-354 | MR | Zbl

[14] Witten, E. Two-dimensional gravity and intersection theory on the moduli space, Surveys in Diff. Geom., Volume 1 (1991), pp. 243-310 | MR | Zbl

[15] Witten, E. Algebraic geometry associated with matrix models of two-dimensional gravity, Topological models in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 235-249 | MR | Zbl

Cited by Sources: