The Witten equation and its virtual fundamental cycle
Abstract
We study a system of nonlinear elliptic PDEs associated with a quasi-homogeneous polynomial. These equations were proposed by Witten as the replacement for the Cauchy-Riemann equation in the singularity (Landau-Ginzburg) setting. We introduce a perturbation to the equation and construct a virtual cycle for the moduli space of its solutions. Then, we study the wall-crossing of the deformation of the virtual cycle under perturbation and match it to classical Picard-Lefschetz theory. An extended virtual cycle is obtained for the original equation. Finally, we prove that the extended virtual cycle satisfies a set of axioms similar to those of Gromov-Witten theory and r-spin theory.
Cite
@article{arxiv.0712.4025,
title = {The Witten equation and its virtual fundamental cycle},
author = {Huijun Fan and Tyler J. Jarvis and Yongbin Ruan},
journal= {arXiv preprint arXiv:0712.4025},
year = {2011}
}
Comments
Major revision. Additional axioms proved and additional details provided over previous version