The Witten equation, mirror symmetry, and quantum singularity theory

• Autores: Hiujun Fan, Tyler J. Jarvis, Yongbin Ruan
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 178, Nº 1, 2013, págs. 1-106
• Idioma: inglés
• DOI: 10.4007/annals.2013.178.1.1
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• Resumen

  • 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 Ar-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.