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

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.

a few typo are corrected, 41 pages

References in corpus (2)

• Virtual neighborhood technique for pseudo-holomorphic spheres
• Vortices on the cylinder

Cited by in corpus (1)

• Equivariant commutative stringy cohomology rings on almost complex manifolds