[Théorie Quantique des Singularités pour et Théorie des Courbes -spin]
Nous passons en revue notre construction d’une théorie cohomologique des champs pour les singularités quasi-homogènes et la théorie des courbes -spin de Jarvis-Kimura-Vaintrob. De plus, nous prouvons que pour une singularité de type notre construction du champ algébrique des -courbes est canoniquement isomorphe au champ algébrique des courbes -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 -spin. Par conséquent, la preuve de Faber-Shadrin-Zvonkine de la conjecture des hiérarchies intégrables de Witten pour les courbes -spin s’applique à notre théorie des singularités de type . C’est-à-dire, la fonction potentielle descendante totale de notre théorie des singularités de type 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 -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity of type our construction of the stack of -curves is canonically isomorphic to the stack of -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the Witten Integrable Hierarchies Conjecture for -spin curves applies to our theory for -type singularities; that is, the total descendant potential function of our theory for -type singularities satisfies the corresponding Gelfand-Dikii integrable hierarchy.
Keywords: FJRW, Mirror symmetry, $r$-spin curve, spin curve, Witten, Cohomological field theory, moduli, Gelfand-Dikii, integrable hierarchy
Mot 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},
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://www.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
SP - 2781
EP - 2802
VL - 61
IS - 7
PB - Association des Annales de l’institut Fourier
UR - http://www.numdam.org/articles/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 http://www.numdam.org/articles/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, Tome 61 (2011) no. 7, pp. 2781-2802. doi : 10.5802/aif.2794. http://www.numdam.org/articles/10.5802/aif.2794/
[1] Moduli of twisted spin curves, Proc. of the Amer. Math. Soc., Volume 131 (2003) no. 3, pp. 685-699 | MR | Zbl
[2] A new cohomology theory for orbifold, Comm. Math. Phys., Volume 248 (2004) no. 1, pp. 1-31 | MR | Zbl
[3] The Witten top Chern class via -theory, J. Alg. Geom., Volume 15 (2006) no. 4, pp. 691-707 | MR | Zbl
[4] Tautological relations and the -spin Witten conjecture, Annales Scientifiques de l’École Normal Supérieure. Quatrième Série, Volume 43 (2010) no. 4, pp. 621-658 | Numdam | MR | Zbl
[5] The Witten equation, mirror symmetry and quantum singularity theory (Preprint. http://arxiv.org/abs/0712.4021)
[6] singularities and KdV hierarchies, Mosc. Math. J., Volume 3 (2003) no. 2, p. 475-505, 743 | MR | Zbl
[7] Geometry of the moduli of higher spin curves, International Journal of Mathematics, Volume 11(5) (2001), pp. 637-663 | MR | Zbl
[8] Moduli Spaces of Higher Spin Curves and Integrable Hierarchies, Compositio Mathematica, Volume 126 (2) (2001), pp. 157-212 | MR | Zbl
[9] 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] 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] 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] A note on symmetry of singularities, Bull. London Math. Soc., Volume 12 (1980) no. 3, pp. 169-175 | MR | Zbl
[13] A second note on symmetry of singularities, Bull. London Math. Soc., Volume 12 (1980) no. 5, pp. 347-354 | MR | Zbl
[14] Two-dimensional gravity and intersection theory on the moduli space, Surveys in Diff. Geom., Volume 1 (1991), pp. 243-310 | MR | Zbl
[15] 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
Cité par Sources :
