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The Witten equation and its virtual fundamental cycle

Huijun Fan, Tyler J. Jarvis, Yongbin Ruan

TL;DR

The paper develops a rigorous analytic framework for the Witten equation associated to a quasi-homogeneous potential $W$, incorporating perturbations $W+W_0$ to access Morse data and control moduli of solutions on orbicurves. It proves compactness and constructs a virtual fundamental cycle using a Kuranishi structure and multisection perturbations, while detailing wall-crossing behavior that mirrors Picard-Lefschetz theory. An extended virtual cycle for the unperturbed equation is obtained, and axioms analogous to Gromov-Witten and $r$-spin theories are established for the resulting invariants. The approach fuses delicate nonlinear analysis (interior estimates, exponential decay, Liouville-type results) with a robust algebro-geometric framework (rigidified $W$-structures, orbifold glueing) to produce a mathematically tractable theory with potential applications to mirror symmetry and singularity theory. Overall, the work lays analytic foundations for a computable, axiomatic theory of Witten-type invariants in the Landau-Ginzburg/Calabi-Yau context.

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.

The Witten equation and its virtual fundamental cycle

TL;DR

The paper develops a rigorous analytic framework for the Witten equation associated to a quasi-homogeneous potential , incorporating perturbations to access Morse data and control moduli of solutions on orbicurves. It proves compactness and constructs a virtual fundamental cycle using a Kuranishi structure and multisection perturbations, while detailing wall-crossing behavior that mirrors Picard-Lefschetz theory. An extended virtual cycle for the unperturbed equation is obtained, and axioms analogous to Gromov-Witten and -spin theories are established for the resulting invariants. The approach fuses delicate nonlinear analysis (interior estimates, exponential decay, Liouville-type results) with a robust algebro-geometric framework (rigidified -structures, orbifold glueing) to produce a mathematically tractable theory with potential applications to mirror symmetry and singularity theory. Overall, the work lays analytic foundations for a computable, axiomatic theory of Witten-type invariants in the Landau-Ginzburg/Calabi-Yau context.

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.

Paper Structure

This paper contains 33 sections, 85 theorems, 434 equations.

Table of Contents

  1. Introduction
  2. Subtleties of the Witten equation
  3. Outline of the construction and results
  4. Rigidified $W$-curves and their moduli
  5. The structure of the Deligne-Mumford space $\overline{{\mathscr M}}_{g,k}$
  6. The structure of the moduli space $\overline{{\mathscr W}}^{\mathrm{rig}}_{g,k}$
  7. The Witten equation
  8. The perturbed Witten equation
  9. Cylindrical metric and Witten-equation
  10. The perturbed Witten equation
  11. Witten equations over orbicurves
  12. Interior estimate of solutions
  13. Asymptotic behavior
  14. Exponential decay
  15. A Liouville type theorem for the $A_1$-equation
  16. ...and 18 more sections

Key Result

Theorem 1.2.1

If $W_0$ is strongly regular, then $\overline{{\mathscr W}}^{\mathrm{rig}}_{g,k}(\kappa_{j_1}, \dots, \kappa_{j_k})$ is compact and has a virtual fundamental cycle $[\overline{{\mathscr W}}^{\mathrm{rig}}_{g,k}(\kappa_{j_1}, \dots, \kappa_{j_k})]^{vir}$ of degree Here $\hat{c}_W$ is the central charge, $\iota_{\gamma_i}$ is the degree shifting number defined in FJR2, and $N_{\gamma_i}$ is the com

Theorems & Definitions (184)

  • Theorem 1.2.1
  • Theorem 1.2.2
  • Corollary 1.2.3
  • Definition 1.2.4
  • Theorem 1.2.5
  • Definition 2.0.1
  • Definition 2.0.2
  • Remark 2.0.3
  • Lemma 2.0.4
  • proof
  • ...and 174 more