Moduli space of twisted holomorphic maps with Lagrangian boundary condition: compactness

Abstract

Let $(X, \omega)$ be a compact symplectic manifold and $L$ be a Lagrangian submanifold. Suppose $(X, L)$ has a Hamiltonian $S^1$ action with moment map $\mu$. Take an invariant $\omega$-compatible almost complex structure, we consider tuples $(C, P, A, \varphi)$ where $C$ is a smooth bordered Riemann surface of fixed topological type, $P\to C$ is an $S^1$-principal bundle, $A$ is a connection on $P$ and $\varphi$ is a section of $P\times_{S^1} X$ satisfying $\ov\partial_A \varphi=0,\ \iota_\nu F_A+ \mu(\varphi)=c$ with boundary condition $\varphi(\partial C) \subset P \times_{S^1} L$. Here $F_A$ is the curvature of $A$ and $\nu$ is a volume form on $C$ and $c\in i{\mb R}$ is a constant. We compactify the moduli space of isomorphism classes of such objects with energy $\leq K$, where the energy is defined to be the Yang-Mills-Higgs functional $\| F_A\|_{L^2}^2+ \| d_A\varphi \|_{L^2}^2+ \| \mu(\varphi)-c \|_{L^2}^2.$ This generalizes the compactness theorem of Mundet-Tian \cite{Mundet_Tian_2009} in the case of closed Riemann surfaces.