A compactification of the moduli space of twisted holomorphic maps

Abstract

Given compact symplectic manifold X with a compatible almost complex structure and a Hamiltonian action of S 1 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 ) , where C is a smooth compact complex curve of fixed genus g, P is a principal S 1 bundle over C, A is a connection on P and ϕ is a section of P × S 1 X satisfying ∂ ¯ A ϕ = 0 , ι v F A + μ ( ϕ ) = c , ‖ F A ‖ L 2 2 + ‖ d A ϕ ‖ L 2 2 + ‖ μ ( ϕ ) − c ‖ L 2 2 ⩽ K . Here F A 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 ) and ( C ′ , P ′ , A ′ , ϕ ′ ) 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 ′ = ϕ . 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 ) . 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 − i μ .