The moduli space of twisted holomorphic maps with Lagrangian boundary condition: Compactness

Let (X,ω)(X,ω) be a compact symplectic manifold and LL be a compact Lagrangian submanifold. Suppose that (X,L)(X,L) has a Hamiltonian S1S1 action with moment map μμ. Take an S1S1-invariant, ωω-compatible almost complex structure, and we consider tuples (C,P,A,φ)(C,P,A,φ) where CC is a smooth bordered Riemann surface of fixed topological type, P→CP→C is an S1S1-principal bundle, AA is a connection on PP and φφ is a section of P×S1XP×S1X satisfying ∂¯Aφ=0,ινFA+μ(φ)=c with boundary condition φ(∂C)⊂P×S1Lφ(∂C)⊂P×S1L. Here FAFA is the curvature of AA and νν is a volume form on CC and c∈iRc∈iR is a constant. We compactify the moduli space of isomorphism classes of such objects with energy ≤K≤K, where the energy is defined to be the Yang–Mills–Higgs functional ‖FA‖L22+‖dAφ‖L22+‖μ(φ)−c‖L22. This generalizes the compactness theorem of Mundet–Tian (2009) [17] in the case of closed Riemann surfaces.