  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.
