The Witten equation and its virtual fundamental cycle

By Huijun Fan, Tyler Jarvis, Y. Ruan. 2007

134 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1f5732568ac57297da93e2374d1272dc3b45f6ba

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Kazushi Ueda
2. M. Futaki
3. Y. Ruan
4. A. Chiodo
5. P. Seidel
6. H. Iritani
7. Hsian-Hua Tseng
8. M. Yamazaki

In Statistical and Nonlinear Physics

1. A. Chiodo
2. M. Futaki
3. Y. Ruan
4. Kazushi Ueda

Citations in the corpus

• A new look at the path integral of quantum mechanics (2010, unknown): “For mathematical studies of this equation, with an elucidation of some important points, see [21,22].”
• Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations (2008, background): “nts W(Jm1,...,Jmn)g,n,G with Jm1,...,Jmn 6= 1 (as mentioned above in other cases the virtual cycle vanis hes by construction). In this situation the virtual fundamental cycle is constructed in [FJR1] [FJR3] for any quasihomogeneous nondegenerate polynomial W by defining the moduli stack W(Jm1,...,Jmn”
• Homological mirror symmetry for Brieskorn–Pham singularities (2009, background): “The corresponding theory on the Landau-Ginzburg side is recently developed by Fan, Jarvis and Ruan [7, 6] who call it Fan-Jarvis-Ruan-Witten theory.”