Quantum Singularity Theory for A (r-1) and r-Spin Theory
[Théorie Quantique des Singularités pour A (r-1) et Théorie des Courbes r-spin]
Fan, Huijun ; Jarvis, Tyler ; Ruan, Yongbin
Annales de l'Institut Fourier, Tome 61 (2011), p. 2781-2802 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

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.

Publié le : 2011-01-01
DOI : https://doi.org/10.5802/aif.2794
Classification:  14H70,  14H10,  14H81,  14B05,  32S25,  57R56,  14N35,  53D45
Mots clés: FJRW, symétrie miroir, courbe r-spin, courbe spin, Witten, théorie cohomologique des champs, module, Gelfand-Dikii, hiérarchie intégrable 
@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},
     volume = {61},
     year = {2011},
     pages = {2781-2802},
     doi = {10.5802/aif.2794},
     zbl = {pre06193027},
     mrnumber = {3112508},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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, Tome 61 (2011) pp. 2781-2802. doi : 10.5802/aif.2794. http://gdmltest.u-ga.fr/item/AIF_2011__61_7_2781_0/

[1] Abramovich, D.; Jarvis, T. Moduli of twisted spin curves, Proc. of the Amer. Math. Soc., Tome 131 (2003) no. 3, pp. 685-699 | MR 1937405 | Zbl 1037.14008

[2] Chen, W.; Ruan, Y. A new cohomology theory for orbifold, Comm. Math. Phys., Tome 248 (2004) no. 1, pp. 1-31 | MR 2104605 | Zbl 1063.53091

[3] Chiodo, A. The Witten top Chern class via K-theory, J. Alg. Geom., Tome 15 (2006) no. 4, pp. 691-707 | MR 2237266 | Zbl 1117.14008

[4] Faber, C.; Shadrin, S.; Zvonkine, D. Tautological relations and the r-spin Witten conjecture, Annales Scientifiques de l’École Normal Supérieure. Quatrième Série, Tome 43 (2010) no. 4, pp. 621-658 | Numdam | MR 2722511 | Zbl 1203.53090

[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., Tome 3 (2003) no. 2, p. 475-505, 743 | MR 2025270 | Zbl 1054.14067

[7] Jarvis, T. Geometry of the moduli of higher spin curves, International Journal of Mathematics, Tome 11(5) (2001), pp. 637-663 | MR 1780734 | Zbl 1094.14504

[8] Jarvis, T.; Kimura, T.; Vaintrob, A. Moduli Spaces of Higher Spin Curves and Integrable Hierarchies, Compositio Mathematica, Tome 126 (2) (2001), pp. 157-212 | MR 1827643 | Zbl 1015.14028

[9] Lee, Y.-P. Witten’s conjecture and the Virasoro conjecture for genus up to two, Gromov-Witten theory of spin curves and orbifolds, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 403 (2006), pp. 31-42 | MR 2234883 | Zbl 1114.14034

[10] Polishchuk, A. Witten’s top Chern class on the moduli space of higher spin curves, Frobenius manifolds, Vieweg, Wiesbaden (Aspects Math., E36) (2004), pp. 253-264 | MR 2115773 | Zbl 1105.14010

[11] Polishchuk, A.; Vaintrob, A. Algebraic construction of Witten’s top Chern class, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 276 (2001), pp. 229-249 | MR 1837120 | Zbl 1051.14007

[12] Wall, C. T. C. A note on symmetry of singularities, Bull. London Math. Soc., Tome 12 (1980) no. 3, pp. 169-175 | MR 572095 | Zbl 0427.32010

[13] Wall, C. T. C. A second note on symmetry of singularities, Bull. London Math. Soc., Tome 12 (1980) no. 5, pp. 347-354 | MR 587705 | Zbl 0424.58006

[14] Witten, E. Two-dimensional gravity and intersection theory on the moduli space, Surveys in Diff. Geom., Tome 1 (1991), pp. 243-310 | MR 1144529 | Zbl 0757.53049

[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 1215968 | Zbl 0812.14017