Geodesic
The Witten equation, mirror symmetry, and quantum singularity theory
Huijun Fan1 ; Tyler Jarvis2 ; Yongbin Ruan3
1 School of Mathematical Sciences, Peking University, Beijing 100871, China and Beijing International Center for Mathematical Research, Beijing 100871, China
2 Department of Mathematics, Brigham Young University, Provo, UT 84602
3 Yangtz Center of Mathematics at Sichuan University, Chengdu 610064, China and Department of Mathematics, University of Michigan, Ann Arbor, MI 48105
Annals of mathematics, Tome 178 (2013) no. 1, pp. 1-106.

Voir la notice de l'article provenant de la source Annals of Mathematics website

For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of $r$-spin curves, which corresponds to the simple singularity $A_{r-1}$.
We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual, and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.
DOI : 10.4007/annals.2013.178.1.1

Huijun Fan 1 ; Tyler Jarvis 2 ; Yongbin Ruan 3

1 School of Mathematical Sciences, Peking University, Beijing 100871, China and Beijing International Center for Mathematical Research, Beijing 100871, China
2 Department of Mathematics, Brigham Young University, Provo, UT 84602
3 Yangtz Center of Mathematics at Sichuan University, Chengdu 610064, China and Department of Mathematics, University of Michigan, Ann Arbor, MI 48105 
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Huijun Fan; Tyler Jarvis; Yongbin Ruan. The Witten equation, mirror symmetry,  and quantum singularity theory. Annals of mathematics, Tome 178 (2013) no. 1, pp. 1-106. doi : 10.4007/annals.2013.178.1.1. http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.1.1/

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