A compactification of the moduli space of twisted holomorphic maps

Given compact symplectic manifold X   with a compatible almost complex structure and a Hamiltonian action of S1S1 with moment map μ:X→iR, and a real number K⩾0K⩾0, we compactify the moduli space of twisted holomorphic maps to X with energy ⩽K  . This moduli space parameterizes equivalence classes of tuples (C,P,A,ϕ)(C,P,A,ϕ), where C is a smooth compact complex curve of fixed genus g, P   is a principal S1S1 bundle over C, A is a connection on P and ϕ   is a section of P×S1XP×S1X satisfying∂¯Aϕ=0,ιvFA+μ(ϕ)=c,‖FA‖L22+‖dAϕ‖L22+‖μ(ϕ)−c‖L22⩽K. Here FAFA is the curvature of A, v is the restriction to C   of a volume form on the universal curve over M¯g and c   is a fixed constant. Two tuples (C,P,A,ϕ)(C,P,A,ϕ) and (C′,P′,A′,ϕ′)(C′,P′,A′,ϕ′) are equivalent if there is a morphism of bundles ρ:P→P′ lifting a biholomorphism C→C′C→C′ such that ρ∗v′=vρ∗v′=v, ρ∗A′=Aρ∗A′=A and ρ∗ϕ′=ϕρ∗ϕ′=ϕ. The topology of the moduli space is the quotient topology of the topology of C∞C∞ convergence on the set of tuples (C,P,A,ϕ)(C,P,A,ϕ). We also incorporate marked points in the picture.There are two sources of noncompactness. First, bubbling off phenomena, analogous to the one in Gromov–Witten theory. Second, degeneration of C to nodal curves. In this case, there appears a phenomenon which is not present in Gromov–Witten: near the nodes, the section ϕ   may degenerate to a chain of gradient flow lines of −iμ−iμ.