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.
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.
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, Volume 61 (2011) no. 7, pp. 2781-2802. doi : 10.5802/aif.2794. http://www.numdam.org/articles/10.5802/aif.2794/
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