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 S¹ with moment map μ : X → i R, and a real number K ≥ 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{symbol}), where C is a smooth compact complex curve of fixed genus g, P is a principal S¹ bundle over C, A is a connection on P and φ{symbol} is a section of P ×_S1 X satisfyingover(∂, -)_A φ{symbol} = 0, ι_v F_A + μ (φ{symbol}) = c, {norm of matrix} F_A {norm of matrix}_L2² + {norm of matrix} d_A φ{symbol} {norm of matrix}_L2² + {norm of matrix} μ (φ{symbol}) - c {norm of matrix}_L2² ≤ K . Here F_A is the curvature of A, v is the restriction to C of a volume form on the universal curve over over(M, -)_g and c is a fixed constant. Two tuples (C, P, A, φ{symbol}) and (C^′, P^′, A^′, φ{symbol}^′) are equivalent if there is a morphism of bundles ρ : P → P^′ lifting a biholomorphism C → C^′ such that ρ^* v^′ = v, ρ^* A^′ = A and ρ^* φ{symbol}^′ = φ{symbol}. The topology of the moduli space is the quotient topology of the topology of C^∞ convergence on the set of tuples (C, P, A, φ{symbol}). 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{symbol} may degenerate to a chain of gradient flow lines of - i μ.