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

Abstract

Let (X,ω) be a compact symplectic manifold and L be a compact Lagrangian submanifold. Suppose that (X,L) has a Hamiltonian S1 action with moment map μ. Take an S1-invariant, ω-compatible almost complex structure, and we consider tuples (C,P,A,φ) where C is a smooth bordered Riemann surface of fixed topological type, P→C is an S1-principal bundle, A is a connection on P and φ is a section of P×S1X satisfying ∂¯Aφ=0,ινFA+μ(φ)=c with boundary condition φ(∂C)⊂P×S1L. Here FA is the curvature of A and ν is a volume form on C and c∈iR is a constant. We compactify the moduli space of isomorphism classes of such objects with energy ≤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.