The Witten equation and its virtual fundamental cycle

TL;DR

This paper develops a virtual fundamental cycle for solutions to Witten's nonlinear elliptic PDEs associated with singularities, linking it to classical Picard-Lefschetz theory and Gromov-Witten axioms.

Contribution

It introduces a perturbation method, constructs a virtual cycle, and establishes its properties, extending the theory to the original Witten equations.

Findings

Constructed a virtual cycle for the moduli space of solutions.

Analyzed wall-crossing phenomena and matched with Picard-Lefschetz theory.

Proved the virtual cycle satisfies Gromov-Witten-like axioms.

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.