## 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{symbol}), 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{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}_{L2}^{2} + {norm of matrix} d_{A} φ{symbol} {norm of matrix}_{L2}^{2} + {norm of matrix} μ (φ{symbol}) - c {norm of matrix}_{L2}^{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 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.

Original language | English (US) |
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Pages (from-to) | 1117-1196 |

Number of pages | 80 |

Journal | Advances in Mathematics |

Volume | 222 |

Issue number | 4 |

DOIs | |

State | Published - Nov 10 2009 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

- Compactification of moduli spaces
- Hamiltonian actions
- Pseudoholomorphic curves
- Vortex equations