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 $(C,P,A,\phi )$ 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 $\phi$ is a section of $P{\times }_{S^1}X$ satisfying $\ov\partial_A φ=0,\ ι_νF_A+ μ(φ)=c$ with boundary condition $\phi (\partial C)\subset P{\times }_{S^1}L$. 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 $\le K$, where the energy is defined to be the Yang-Mills-Higgs functional $\Vert F_A{\Vert }_{L^2}^2+\Vert d_A\phi {\Vert }_{L^2}^2+\Vert \mu (\phi )-c{\Vert }_{L^2}^2.$ This generalizes the compactness theorem of Mundet-Tian \cite{Mundet_Tian_2009} in the case of closed Riemann surfaces.

56 pages, 10 figures. arXiv admin note: text overlap with arXiv:math/0404407 by other authors

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