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\(J\)-holomorphic curves, moment maps, and invariants of Hamiltonian group actions. (English) Zbl 1083.53084
Summary: This paper outlines the construction of invariants of Hamiltonian group actions on symplectic manifolds. The invariants are derived from the solutions of a nonlinear first order elliptic partial differential equation involving the Cauchy-Riemann operator, the curvature, and the moment map. They are related to the Gromov invariants of the reduced spaces.
Instances of these equations include special cases of the vortex equations, the equations for Bradlow pairs, the anti-self-dual Yang-Mills equations, and the Seiberg-Witten equations.

MSC:
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53D20 Momentum maps; symplectic reduction
53D40 Symplectic aspects of Floer homology and cohomology
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