[Paper Review] Virtual cycles of gauged Witten equation

Gang Tian, Guangbo Xu|arXiv (Cornell University)|Feb 24, 2016

Black Holes and Theoretical Physics30 references3 citations

TL;DR

This paper constructs virtual cycles on moduli spaces of perturbed gauged Witten equations over fixed smooth $r$-spin curves, establishing an oriented virtual orbifold atlas and defining a rational virtual cardinality. The construction, completed in conjunction with a wall-crossing formula in the companion paper, provides a rigorous mathematical foundation for the correlation functions in the gauged linear $\sigma$-model, proving their invariance under perturbations and completing the program initiated in [TX15, TX16].

ABSTRACT

We construct virtual cycles on moduli spaces of perturbed gauged Witten equation over a fixed smooth r -spin curve, under the framework of [TX15]. Together with the wall-crossing formula proved in the companion paper [TX19], it completes the construction of the correlation function for the gauged linear sigma model announced in [TX16] as well as the proof of its invariance.

Motivation & Objective

• To construct a virtual cycle on the moduli space of perturbed gauged Witten equations over a fixed smooth $r$-spin curve.
• To establish the existence of an oriented virtual orbifold atlas for the moduli space under strongly regular perturbations.
• To define a rational virtual cardinality for the moduli space, enabling the construction of correlation functions in the gauged linear $\sigma$-model.
• To complete the mathematical framework for the gauged linear $\sigma$-model by proving invariance of the correlation functions under perturbations.
• To provide a rigorous foundation for the Landau–Ginzburg/Calabi–Yau correspondence and geometric mirror symmetry via virtual cycle techniques.

Proposed method

• Utilizes the gauged Witten equation system combining the Witten equation and symplectic vortex equation on a $G$-bundle over a Riemann surface with a $G^\mathbb{C}$-action and moment map.
• Applies strongly regular perturbations $\underline{P}$ to the superpotential $W$, ensuring all critical values of the perturbed functions have distinct imaginary parts.
• Employs a good coordinate system with virtual dimension zero and constructs a transverse multi-valued perturbation to define the virtual cycle.
• Implements a virtual technique based on sequential compactness of the perturbed zero locus, ensuring finitely many contributions to the virtual cardinality.
• Uses topological transversality and boundary analysis in virtual dimension one to prove invariance of the virtual cardinality under choices of perturbations.
• Relies on results from [TX15] on compactness and Fredholm theory to ensure the moduli space admits a well-defined virtual structure.

Experimental results

Research questions

• RQ1 How can a virtual cycle be constructed on the moduli space of solutions to the perturbed gauged Witten equation over an $r$-spin curve?
• RQ2 What conditions on perturbations ensure the moduli space admits an oriented virtual orbifold atlas?
• RQ3 How is the virtual cardinality of the moduli space defined and why is it rational?
• RQ4 What is the role of the wall-crossing formula in ensuring invariance of the correlation function?
• RQ5 How can the virtual cardinality be shown to be independent of choices of perturbations and coordinate systems?

Key findings

• The moduli space $\mathcal{M}_{\underline{P}}(\mathcal{C},B,\underline{\kappa})$ of perturbed gauged Witten equations over a smooth $r$-spin curve $\mathcal{C}$ admits an oriented virtual orbifold atlas.
• The virtual cardinality $\#{\mathcal{M}}_{\underline{P}}(\mathcal{C},B,\underline{\kappa})$ is a well-defined rational number, finite and independent of choices under the given conditions.
• Strongly regular perturbations, which avoid a real analytic hypersurface in the perturbation space, ensure transversality and the existence of the virtual cycle.
• The virtual cardinality is invariant under perturbations due to the existence of a homotopy between different choices via a good coordinate system on $X \times [0,1]$.
• In the case where all charts are manifolds, the virtual cardinality is an integer, achieved through single-valued transverse perturbations.
• The boundary of a one-dimensional oriented good coordinate system has virtual cardinality zero, which implies invariance of the virtual cardinality under perturbation choices.