[Paper Review] Area dependence in gauged Gromov-Witten theory

Eduardo González, Chris Woodward|arXiv (Cornell University)|Nov 20, 2008

Algebraic Geometry and Number Theory63 references11 citations

TL;DR

This paper investigates how symplectic vortex invariants—defined on a fixed holomorphic curve—vary with changes in the area form, particularly for compact, convex varieties. It establishes a wall-crossing formula and proves a vortex version of the abelianization conjecture, linking invariants of a group and its maximal torus via symplectic vortices on non-trivial bundles.

ABSTRACT

Abstract. We study the variation of the moduli space of symplectic vortices on a fixed holomorphic curve with respect to the area form. For compact, convex varieties we define symplectic vortex invariants and prove a wall-crossing formula for them. As an application, we prove a vortex version of the abelianization conjecture of Bertram, Ciocan-Fontanine, and Kim [4], which related Gromov-Witten invariants of geometric invariant theory quotients by a group and its maximal torus, for vortices on non-trivial bundles.

Motivation & Objective

• To understand how the moduli space of symplectic vortices changes with respect to variations in the area form on a fixed holomorphic curve.
• To define symplectic vortex invariants for compact, convex symplectic varieties.
• To establish a wall-crossing formula governing the behavior of these invariants across area form chambers.
• To prove a vortex analogue of the abelianization conjecture by Bertram, Ciocan-Fontanine, and Kim, relating invariants of a group and its maximal torus.
• To extend this correspondence to the case of non-trivial bundles, generalizing previous results in Gromov-Witten theory.

Proposed method

• The authors analyze the moduli space of symplectic vortices as the area form varies, identifying chambers where the structure remains stable.
• They define symplectic vortex invariants using virtual fundamental classes associated to the moduli space of vortices.
• A wall-crossing formula is derived by studying the change in the virtual fundamental class when crossing area form walls.
• The approach leverages techniques from gauged Gromov-Witten theory and geometric invariant theory (GIT) quotients.
• The proof of the abelianization conjecture for vortices uses reduction to the maximal torus and compares invariants via wall-crossing behavior.
• The framework is extended to non-trivial G-bundles by analyzing the dependence of vortex invariants on the bundle structure and area form.

Experimental results

Research questions

• RQ1 How do symplectic vortex invariants change as the area form on a fixed holomorphic curve varies?
• RQ2 What is the precise structure of the wall-crossing formula for symplectic vortex invariants in compact, convex varieties?
• RQ3 To what extent does the abelianization conjecture for Gromov-Witten invariants extend to the setting of symplectic vortices?
• RQ4 How do vortex invariants behave on non-trivial G-bundles, and can they be related to invariants on the maximal torus?
• RQ5 What is the role of the area form in controlling the stability and structure of the moduli space of vortices?

Key findings

• A wall-crossing formula is established for symplectic vortex invariants, describing their discontinuous change across area form chambers.
• The abelianization conjecture is proven in the vortex setting, showing that invariants for a reductive group G and its maximal torus T are related via wall-crossing data.
• The invariants are well-defined and stable within each chamber of the area form moduli, enabling a chamber decomposition of the theory.
• The framework successfully extends the abelianization result to non-trivial G-bundles, generalizing previous results restricted to trivial bundles.
• The dependence of the moduli space on the area form is shown to be governed by a finite set of critical walls, with invariants changing predictably across them.
• The virtual fundamental class of the vortex moduli space transforms in a controlled way under area variation, enabling the wall-crossing computation.