$L^2$-moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends

TL;DR

This paper analyzes the structure and properties of $L^2$-moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends, revealing their critical points, flow behavior, and Fredholm properties.

Contribution

It establishes the Bott-Morse nature of the action functional, characterizes critical points as twisted sectors, and proves exponential convergence and Fredholm theory for the moduli spaces.

Findings

Critical points form twisted sectors of symplectic reduction

Gradient flow lines converge exponentially fast

Fredholm theory and compactness are established for the moduli spaces

Abstract

Let $(X,\omega)$ be a compact symplectic manifold with a Hamiltonian action of a compact Lie group $G$ and $\mu: X\to \mathfrak g$ be its moment map. In this paper, we study the $L^2$-moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends. We studied a circle-valued action functional whose gradient flow equation corresponds to the symplectic vortex equations on a cylinder $S^1\times \mathbb R$. Assume that $0$ is a regular value of the moment map $\mu$, we show that the functional is of Bott-Morse type and its critical points of the functional form twisted sectors of the symplectic reduction (the symplecitc orbifold $[\mu^{-1}(0)/G]$). We show that any gradient flow lines approaches its limit point exponentially fast. Fredholm theory and compactness property are then established for the $L^2$-Moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends.